Subgroups — C₃₁×#25196 #21481955

Order 7936.

1032 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3068	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3068	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3019	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#453	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2888	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2069	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2050	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2048	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2002	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1965	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1932	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1914	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1903	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1861	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
17	C₃₁×#575	C₂²	1	1	yes	characteristic
18	C₃₁×#575	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#568	C₂²	1	1	yes	characteristic
23	C₃₁×#568	C₂²	1	1	yes	characteristic
24	C₃₁×#568	C₂²	1	1	yes	characteristic, complemented
25	C₃₁×#535	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
27	C₃₁×#472	—	2	2	no	—
28	C₃₁×#472	—	2	2	no	—
29	C₃₁×#471	—	2	2	no	—
30	C₃₁×#471	—	2	2	no	—
31	C₃₁×#455	—	2	2	no	—
32	C₃₁×#453	—	2	2	no	—
33	C₃₁×#453	—	2	2	no	—
34	C₃₁×#453	—	2	2	no	—
35	C₃₁×#444	—	2	2	no	—
36	C₃₁×#443	—	2	2	no	—
37	C₃₁×#430	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#430	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#430	C₂²	1	1	yes	characteristic, complemented
40	C₃₁×#423	C₂²	1	1	yes	characteristic, complemented
41	C₃₁×#421	C₂²	1	1	yes	characteristic
42	C₃₁×#421	C₂²	1	1	yes	characteristic
43	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
44	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
45	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic, complemented
46	C₃₁×#418	C₂²	1	1	yes	characteristic, complemented
47	C₃₁×#413	—	2	2	no	—
48	C₃₁×#413	C₂²	1	1	yes	characteristic
49	C₃₁×#413	C₂²	1	1	yes	characteristic
50	C₃₁×#413	C₂²	1	1	yes	characteristic, complemented
51	C₂×C₃₁×#100	—	2	2	no	—
52	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
53	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
54	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
55	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
56	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic, complemented
57	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic, complemented
58	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
59	C₃₁×#344	C₂²	1	1	yes	characteristic, complemented
60	C₃₁×#344	C₂²	1	1	yes	characteristic, complemented
61	C₃₁×#339	C₂²	1	1	yes	characteristic, complemented
62	C₃₁×#338	C₂²	1	1	yes	characteristic, complemented
63	C₃₁×H(5,𝔽₂)	—	2	2	no	—
64	C₃₁×H(5,𝔽₂)	—	2	2	no	—
65	C₃₁×H(5,𝔽₂)	—	4	1	no	—
66	C₃₁×H(5,𝔽₂)	—	4	1	no	—
67	C₂×#41×C₃₁	—	2	1	no	—
68	C₂×#41×C₃₁	—	2	1	no	—
69	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×D₄×C₃₁	—	2	2	no	—
73	C₂²×D₄×C₃₁	—	2	1	no	—
74	C₂²×D₄×C₃₁	—	2	1	no	—
75	C₂²×D₄×C₃₁	—	2	2	no	—
76	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×#137	—	2	1	no	—
79	C₃₁×#137	—	2	1	no	—
80	C₃₁×#137	—	2	1	no	—
81	C₃₁×#137	D₄	1	2	yes	complemented
82	C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
83	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
84	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
85	C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
86	C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
87	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
88	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
89	C₃₁×AGL(1,ℤ/8)	—	4	1	no	—
90	C₃₁×AGL(1,ℤ/8)	—	4	1	no	—
91	C₃₁×AGL(1,ℤ/8)	D₄	1	2	yes	complemented
92	C₃₁×#135	—	2	2	no	—
93	C₃₁×#135	—	4	1	no	—
94	C₂×SD₁₆×C₃₁	—	2	1	no	—
95	C₂×SD₁₆×C₃₁	—	2	1	no	—
96	C₂×SD₁₆×C₃₁	—	2	1	no	—
97	C₂×D₈×C₃₁	—	2	1	no	—
98	C₂×D₈×C₃₁	—	2	1	no	—
99	C₂×D₈×C₃₁	—	2	1	no	—
100	C₃₁×#131	—	2	2	no	—
101	C₃₁×#131	—	2	2	no	—
102	C₃₁×#131	—	2	2	no	—
103	C₃₁×#131	—	2	2	no	—
104	C₃₁×#131	—	4	1	no	—
105	C₃₁×#131	—	4	1	no	—
106	C₂×M₁₆×C₃₁	—	2	2	no	—
107	C₂×M₁₆×C₃₁	—	2	2	no	—
108	C₂×M₁₆×C₃₁	—	4	1	no	—
109	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
110	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
111	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
112	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
113	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
114	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
115	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
116	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
117	C₃₁×#127	—	2	1	no	—
118	C₃₁×#127	—	2	1	no	—
119	C₃₁×#127	—	2	1	no	—
120	C₃₁×#127	—	4	1	no	—
121	C₃₁×#124	—	2	1	no	—
122	C₃₁×#121	—	4	1	no	—
123	C₃₁×#120	—	2	2	no	—
124	C₃₁×#117	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
126	C₃₁×#108	—	2	2	no	—
127	C₃₁×#108	—	2	2	no	—
128	C₃₁×#108	—	2	2	no	—
129	C₃₁×C₄⋊C₈	—	4	1	no	—
130	C₃₁×#104	—	2	2	no	—
131	C₃₁×#104	—	2	2	no	—
132	C₃₁×#104	—	2	1	no	—
133	C₃₁×#104	—	2	1	no	—
134	C₃₁×#104	—	2	1	no	—
135	C₃₁×#104	—	2	1	no	—
136	C₃₁×#104	D₄	1	2	yes	complemented
137	C₃₁×#104	D₄	1	2	yes	complemented
138	C₃₁×#102	—	4	1	no	—
139	C₃₁×#101	—	2	1	no	—
140	C₃₁×#101	—	2	1	no	—
141	C₃₁×#101	D₄	1	2	yes	complemented
142	C₃₁×#100	—	2	1	no	—
143	C₃₁×#100	—	2	1	no	—
144	C₃₁×#100	—	2	2	no	—
145	C₃₁×#100	—	2	2	no	—
146	C₃₁×#100	—	2	2	no	—
147	C₃₁×#100	—	2	2	no	—
148	C₃₁×#100	—	2	2	no	—
149	C₃₁×#100	—	2	2	no	—
150	C₃₁×#100	D₄	1	2	yes	complemented
151	C₃₁×#97	—	2	1	no	—
152	C₄×C₈×C₃₁	—	2	1	no	—
153	C₂⁴×C₃₁	—	2	2	no	—
154	C₂⁴×C₃₁	—	2	1	no	—
155	C₂⁴×C₃₁	#41	1	2	yes	—
156	#41×C₃₁	—	2	1	no	—
157	#41×C₃₁	—	2	1	no	—
158	#41×C₃₁	—	2	1	no	—
159	#41×C₃₁	—	2	1	no	—
160	#41×C₃₁	—	2	1	no	—
161	#41×C₃₁	—	2	1	no	—
162	#41×C₃₁	—	2	1	no	—
163	#41×C₃₁	—	2	1	no	—
164	#41×C₃₁	—	2	1	no	—
165	#41×C₃₁	—	2	1	no	—
166	#41×C₃₁	—	4	2	no	—
167	#41×C₃₁	—	4	2	no	—
168	#41×C₃₁	—	4	2	no	—
169	#41×C₃₁	—	4	2	no	—
170	#41×C₃₁	C₂×D₄	1	2	yes	—
171	#41×C₃₁	C₂×D₄	1	2	yes	—
172	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₂×Q₈×C₃₁	#41	1	1	yes	characteristic
175	C₂×D₄×C₃₁	—	2	1	no	—
176	C₂×D₄×C₃₁	—	2	2	no	—
177	C₂×D₄×C₃₁	—	2	2	no	—
178	C₂×D₄×C₃₁	—	2	1	no	—
179	C₂×D₄×C₃₁	—	2	1	no	—
180	C₂×D₄×C₃₁	—	2	1	no	—
181	C₂×D₄×C₃₁	—	2	1	no	—
182	C₂×D₄×C₃₁	—	2	1	no	—
183	C₂×D₄×C₃₁	—	2	1	no	—
184	C₂×D₄×C₃₁	—	2	1	no	—
185	C₂×D₄×C₃₁	—	2	1	no	—
186	C₂×D₄×C₃₁	—	2	1	no	—
187	C₂×D₄×C₃₁	—	2	1	no	—
188	C₂×D₄×C₃₁	—	2	1	no	—
189	C₂×D₄×C₃₁	—	2	1	no	—
190	C₂×D₄×C₃₁	—	2	1	no	—
191	C₂×D₄×C₃₁	—	2	1	no	—
192	C₂×D₄×C₃₁	—	4	2	no	—
193	C₂×D₄×C₃₁	—	4	2	no	—
194	C₂×D₄×C₃₁	—	4	2	no	—
195	C₂×D₄×C₃₁	—	4	2	no	—
196	C₂×D₄×C₃₁	—	4	4	no	—
197	C₂×D₄×C₃₁	—	4	4	no	—
198	C₂×D₄×C₃₁	—	4	2	no	—
199	C₂×D₄×C₃₁	—	4	2	no	—
200	C₂×D₄×C₃₁	—	4	2	no	—
201	C₂×D₄×C₃₁	—	8	4	no	—
202	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
203	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
204	C₂×D₄×C₃₁	#41	1	1	yes	characteristic
205	C₂²×C₄×C₃₁	—	2	1	no	—
206	C₂²×C₄×C₃₁	—	2	2	no	—
207	C₂²×C₄×C₃₁	—	2	1	no	—
208	C₂²×C₄×C₃₁	—	2	1	no	—
209	C₂²×C₄×C₃₁	—	2	1	no	—
210	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
211	Q₁₆×C₃₁	—	2	2	no	—
212	Q₁₆×C₃₁	—	4	1	no	—
213	SD₁₆×C₃₁	—	2	2	no	—
214	SD₁₆×C₃₁	—	2	2	no	—
215	SD₁₆×C₃₁	—	4	2	no	—
216	SD₁₆×C₃₁	—	4	2	no	—
217	SD₁₆×C₃₁	—	4	1	no	—
218	SD₁₆×C₃₁	—	4	1	no	—
219	SD₁₆×C₃₁	—	4	2	no	—
220	D₈×C₃₁	—	2	2	no	—
221	D₈×C₃₁	—	4	2	no	—
222	D₈×C₃₁	—	4	2	no	—
223	D₈×C₃₁	—	4	1	no	—
224	D₈×C₃₁	—	4	2	no	—
225	M₁₆×C₃₁	—	2	1	no	—
226	M₁₆×C₃₁	—	2	1	no	—
227	M₁₆×C₃₁	—	2	1	no	—
228	M₁₆×C₃₁	—	2	1	no	—
229	M₁₆×C₃₁	—	2	1	no	—
230	M₁₆×C₃₁	—	2	1	no	—
231	M₁₆×C₃₁	—	2	1	no	—
232	M₁₆×C₃₁	—	2	1	no	—
233	M₁₆×C₃₁	—	2	1	no	—
234	M₁₆×C₃₁	—	2	1	no	—
235	M₁₆×C₃₁	—	2	1	no	—
236	M₁₆×C₃₁	—	2	1	no	—
237	M₁₆×C₃₁	—	4	2	no	—
238	M₁₆×C₃₁	—	4	2	no	—
239	M₁₆×C₃₁	—	4	2	no	—
240	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
241	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
242	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
243	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
244	C₂×C₈×C₃₁	—	2	1	no	—
245	C₂×C₈×C₃₁	—	2	1	no	—
246	C₂×C₈×C₃₁	—	2	1	no	—
247	C₂×C₈×C₃₁	—	2	1	no	—
248	C₂×C₈×C₃₁	—	2	1	no	—
249	C₂×C₈×C₃₁	—	2	1	no	—
250	C₂×C₈×C₃₁	—	2	1	no	—
251	C₂×C₈×C₃₁	—	2	1	no	—
252	C₂×C₈×C₃₁	—	4	2	no	—
253	C₂×C₈×C₃₁	—	4	2	no	—
254	C₂×C₈×C₃₁	—	4	2	no	—
255	C₄⋊C₄×C₃₁	—	2	1	no	—
256	C₂²⋊C₄×C₃₁	—	2	1	no	—
257	C₂²⋊C₄×C₃₁	—	4	2	no	—
258	C₄²×C₃₁	—	2	1	no	—
259	C₄²×C₃₁	—	2	1	no	—
260	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
261	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
262	#25196	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
263	C₂³×C₃₁	—	2	1	no	—
264	C₂³×C₃₁	—	2	2	no	—
265	C₂³×C₃₁	—	2	1	no	—
266	C₂³×C₃₁	—	2	2	no	—
267	C₂³×C₃₁	—	2	1	no	—
268	C₂³×C₃₁	—	2	2	no	—
269	C₂³×C₃₁	—	4	4	no	—
270	C₂³×C₃₁	—	4	1	no	—
271	C₂³×C₃₁	—	4	1	no	—
272	C₂³×C₃₁	—	4	1	no	—
273	C₂³×C₃₁	—	8	2	no	—
274	C₂³×C₃₁	—	8	2	no	—
275	C₂³×C₃₁	—	16	2	no	—
276	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
277	Q₈×C₃₁	—	2	1	no	—
278	Q₈×C₃₁	—	2	1	no	—
279	Q₈×C₃₁	—	2	1	no	—
280	Q₈×C₃₁	—	2	1	no	—
281	Q₈×C₃₁	—	2	1	no	—
282	Q₈×C₃₁	#121	1	2	yes	—
283	D₄×C₃₁	—	2	2	no	—
284	D₄×C₃₁	—	2	2	no	—
285	D₄×C₃₁	—	2	1	no	—
286	D₄×C₃₁	—	2	1	no	—
287	D₄×C₃₁	—	2	1	no	—
288	D₄×C₃₁	—	2	1	no	—
289	D₄×C₃₁	—	2	1	no	—
290	D₄×C₃₁	—	4	4	no	—
291	D₄×C₃₁	—	4	4	no	—
292	D₄×C₃₁	—	4	1	no	—
293	D₄×C₃₁	—	4	1	no	—
294	D₄×C₃₁	—	4	1	no	—
295	D₄×C₃₁	—	4	1	no	—
296	D₄×C₃₁	—	4	1	no	—
297	D₄×C₃₁	—	4	1	no	—
298	D₄×C₃₁	—	4	1	no	—
299	D₄×C₃₁	—	4	1	no	—
300	D₄×C₃₁	—	4	1	no	—
301	D₄×C₃₁	—	4	1	no	—
302	D₄×C₃₁	—	4	1	no	—
303	D₄×C₃₁	—	4	1	no	—
304	D₄×C₃₁	—	8	4	no	—
305	D₄×C₃₁	—	8	4	no	—
306	D₄×C₃₁	—	16	2	no	—
307	D₄×C₃₁	#121	1	2	yes	—
308	C₂×C₄×C₃₁	—	2	1	no	—
309	C₂×C₄×C₃₁	—	2	1	no	—
310	C₂×C₄×C₃₁	—	2	2	no	—
311	C₂×C₄×C₃₁	—	2	2	no	—
312	C₂×C₄×C₃₁	—	2	1	no	—
313	C₂×C₄×C₃₁	—	2	1	no	—
314	C₂×C₄×C₃₁	—	2	1	no	—
315	C₂×C₄×C₃₁	—	4	4	no	—
316	C₂×C₄×C₃₁	—	4	2	no	—
317	C₂×C₄×C₃₁	—	4	1	no	—
318	C₂×C₄×C₃₁	—	4	1	no	—
319	C₂×C₄×C₃₁	—	4	1	no	—
320	C₂×C₄×C₃₁	—	4	1	no	—
321	C₂×C₄×C₃₁	—	4	1	no	—
322	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
323	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
324	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
325	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
326	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
327	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
328	C₈×C₃₁	—	2	2	no	—
329	C₈×C₃₁	—	2	2	no	—
330	C₈×C₃₁	—	2	2	no	—
331	C₈×C₃₁	—	2	2	no	—
332	C₈×C₃₁	—	4	1	no	—
333	C₈×C₃₁	—	4	1	no	—
334	C₈×C₃₁	—	4	1	no	—
335	C₈×C₃₁	—	4	1	no	—
336	#3068	C₂×C₃₁	1	1	yes	characteristic, complemented
337	#3068	C₂×C₃₁	1	1	yes	characteristic, complemented
338	#3019	C₂×C₃₁	1	1	yes	characteristic, complemented
339	C₂×#453	C₂×C₃₁	1	1	yes	characteristic, complemented
340	#2956	C₂×C₃₁	1	1	yes	characteristic, complemented
341	#2888	C₂×C₃₁	1	1	yes	characteristic, complemented
342	#2069	C₂×C₃₁	1	1	yes	characteristic, complemented
343	#2050	C₂×C₃₁	1	1	yes	characteristic, complemented
344	#2048	C₂×C₃₁	1	1	yes	characteristic, complemented
345	#2002	C₂×C₃₁	1	1	yes	characteristic, complemented
346	#1965	C₂×C₃₁	1	1	yes	characteristic, complemented
347	#1932	C₂×C₃₁	1	1	yes	characteristic, complemented
348	#1914	C₂×C₃₁	1	1	yes	characteristic, complemented
349	#1903	C₂×C₃₁	1	1	yes	characteristic, complemented
350	#1861	C₂×C₃₁	1	1	yes	characteristic, complemented
351	C₂²×C₃₁	—	2	2	no	—
352	C₂²×C₃₁	—	4	1	no	—
353	C₂²×C₃₁	—	4	1	no	—
354	C₂²×C₃₁	—	4	1	no	—
355	C₂²×C₃₁	—	4	2	no	—
356	C₂²×C₃₁	—	8	1	no	—
357	C₂²×C₃₁	—	8	2	no	—
358	C₂²×C₃₁	—	8	1	no	—
359	C₂²×C₃₁	—	8	2	no	—
360	C₂²×C₃₁	—	8	2	no	—
361	C₂²×C₃₁	—	8	1	no	—
362	C₂²×C₃₁	—	8	2	no	—
363	C₂²×C₃₁	—	8	1	no	—
364	C₂²×C₃₁	—	16	1	no	—
365	C₂²×C₃₁	—	16	1	no	—
366	C₂²×C₃₁	—	16	4	no	—
367	C₂²×C₃₁	#525	1	1	yes	characteristic
368	C₂²×C₃₁	#546	1	1	yes	characteristic
369	C₂²×C₃₁	#546	1	1	yes	characteristic
370	C₄×C₃₁	—	2	2	no	—
371	C₄×C₃₁	—	2	2	no	—
372	C₄×C₃₁	—	4	1	no	—
373	C₄×C₃₁	—	4	1	no	—
374	C₄×C₃₁	—	4	1	no	—
375	C₄×C₃₁	—	8	1	no	—
376	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
377	C₄×C₃₁	#538	1	1	yes	characteristic
378	C₄×C₃₁	D₄²	1	1	yes	characteristic
379	C₄×C₃₁	D₄²	1	1	yes	characteristic
380	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
381	#575	C₂²×C₃₁	1	1	yes	characteristic
382	#575	C₂²×C₃₁	1	1	yes	characteristic
383	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
384	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
385	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic, complemented
386	#568	C₂²×C₃₁	1	1	yes	characteristic
387	#568	C₂²×C₃₁	1	1	yes	characteristic
388	#568	C₂²×C₃₁	1	1	yes	characteristic, complemented
389	#535	C₂²×C₃₁	1	1	yes	characteristic
390	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
391	#472	—	2	2	no	—
392	#472	—	2	2	no	—
393	#471	—	2	2	no	—
394	#471	—	2	2	no	—
395	#455	—	2	2	no	—
396	#453	—	2	2	no	—
397	#453	—	2	2	no	—
398	#453	—	2	2	no	—
399	#444	—	2	2	no	—
400	#443	—	2	2	no	—
401	#430	C₂²×C₃₁	1	1	yes	characteristic, complemented
402	#430	C₂²×C₃₁	1	1	yes	characteristic, complemented
403	#430	C₂²×C₃₁	1	1	yes	characteristic, complemented
404	#423	C₂²×C₃₁	1	1	yes	characteristic, complemented
405	#421	C₂²×C₃₁	1	1	yes	characteristic
406	#421	C₂²×C₃₁	1	1	yes	characteristic
407	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
408	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
409	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic, complemented
410	#418	C₂²×C₃₁	1	1	yes	characteristic, complemented
411	#413	—	2	2	no	—
412	#413	C₂²×C₃₁	1	1	yes	characteristic
413	#413	C₂²×C₃₁	1	1	yes	characteristic
414	#413	C₂²×C₃₁	1	1	yes	characteristic, complemented
415	C₂×#100	—	2	2	no	—
416	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
417	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
418	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
419	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
420	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic, complemented
421	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic, complemented
422	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
423	#344	C₂²×C₃₁	1	1	yes	characteristic, complemented
424	#344	C₂²×C₃₁	1	1	yes	characteristic, complemented
425	#339	C₂²×C₃₁	1	1	yes	characteristic, complemented
426	#338	C₂²×C₃₁	1	1	yes	characteristic, complemented
427	C₂×C₃₁	—	2	1	no	—
428	C₂×C₃₁	—	2	1	no	—
429	C₂×C₃₁	—	2	1	no	—
430	C₂×C₃₁	—	4	2	no	—
431	C₂×C₃₁	—	8	1	no	—
432	C₂×C₃₁	—	8	2	no	—
433	C₂×C₃₁	—	8	1	no	—
434	C₂×C₃₁	—	16	1	no	—
435	C₂×C₃₁	#2424	1	1	yes	characteristic
436	H(5,𝔽₂)	—	2	2	no	—
437	H(5,𝔽₂)	—	2	2	no	—
438	H(5,𝔽₂)	—	4	1	no	—
439	H(5,𝔽₂)	—	4	1	no	—
440	C₂×#41	—	2	1	no	—
441	C₂×#41	—	2	1	no	—
442	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
443	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
444	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
445	C₂²×D₄	—	2	2	no	—
446	C₂²×D₄	—	2	1	no	—
447	C₂²×D₄	—	2	1	no	—
448	C₂²×D₄	—	2	2	no	—
449	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
450	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
451	#137	—	2	1	no	—
452	#137	—	2	1	no	—
453	#137	—	2	1	no	—
454	#137	D₄×C₃₁	1	2	yes	complemented
455	AGL(1,ℤ/8)	—	2	1	no	—
456	AGL(1,ℤ/8)	—	2	2	no	—
457	AGL(1,ℤ/8)	—	2	2	no	—
458	AGL(1,ℤ/8)	—	2	1	no	—
459	AGL(1,ℤ/8)	—	2	1	no	—
460	AGL(1,ℤ/8)	—	2	2	no	—
461	AGL(1,ℤ/8)	—	2	2	no	—
462	AGL(1,ℤ/8)	—	4	1	no	—
463	AGL(1,ℤ/8)	—	4	1	no	—
464	AGL(1,ℤ/8)	D₄×C₃₁	1	2	yes	complemented
465	#135	—	2	2	no	—
466	#135	—	4	1	no	—
467	C₂×SD₁₆	—	2	1	no	—
468	C₂×SD₁₆	—	2	1	no	—
469	C₂×SD₁₆	—	2	1	no	—
470	C₂×D₈	—	2	1	no	—
471	C₂×D₈	—	2	1	no	—
472	C₂×D₈	—	2	1	no	—
473	#131	—	2	2	no	—
474	#131	—	2	2	no	—
475	#131	—	2	2	no	—
476	#131	—	2	2	no	—
477	#131	—	4	1	no	—
478	#131	—	4	1	no	—
479	C₂×M₁₆	—	2	2	no	—
480	C₂×M₁₆	—	2	2	no	—
481	C₂×M₁₆	—	4	1	no	—
482	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
483	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
484	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
485	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
486	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
487	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
488	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
489	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
490	#127	—	2	1	no	—
491	#127	—	2	1	no	—
492	#127	—	2	1	no	—
493	#127	—	4	1	no	—
494	#124	—	2	1	no	—
495	#121	—	4	1	no	—
496	#120	—	2	2	no	—
497	#117	C₂³×C₃₁	1	1	yes	characteristic
498	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
499	#108	—	2	2	no	—
500	#108	—	2	2	no	—
501	#108	—	2	2	no	—
502	C₄⋊C₈	—	4	1	no	—
503	#104	—	2	2	no	—
504	#104	—	2	2	no	—
505	#104	—	2	1	no	—
506	#104	—	2	1	no	—
507	#104	—	2	1	no	—
508	#104	—	2	1	no	—
509	#104	D₄×C₃₁	1	2	yes	complemented
510	#104	D₄×C₃₁	1	2	yes	complemented
511	#102	—	4	1	no	—
512	#101	—	2	1	no	—
513	#101	—	2	1	no	—
514	#101	D₄×C₃₁	1	2	yes	complemented
515	#100	—	2	1	no	—
516	#100	—	2	1	no	—
517	#100	—	2	2	no	—
518	#100	—	2	2	no	—
519	#100	—	2	2	no	—
520	#100	—	2	2	no	—
521	#100	—	2	2	no	—
522	#100	—	2	2	no	—
523	#100	D₄×C₃₁	1	2	yes	complemented
524	#97	—	2	1	no	—
525	C₄×C₈	—	2	1	no	—
526	C₃₁	#25196	1	1	yes	characteristic, direct product factor, Sylow
527	C₂⁴	—	2	2	no	—
528	C₂⁴	—	2	1	no	—
529	C₂⁴	#41×C₃₁	1	2	yes	—
530	#41	—	2	1	no	—
531	#41	—	2	1	no	—
532	#41	—	2	1	no	—
533	#41	—	2	1	no	—
534	#41	—	2	1	no	—
535	#41	—	2	1	no	—
536	#41	—	2	1	no	—
537	#41	—	2	1	no	—
538	#41	—	2	1	no	—
539	#41	—	2	1	no	—
540	#41	—	4	2	no	—
541	#41	—	4	2	no	—
542	#41	—	4	2	no	—
543	#41	—	4	2	no	—
544	#41	C₂×D₄×C₃₁	1	2	yes	—
545	#41	C₂×D₄×C₃₁	1	2	yes	—
546	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
547	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
548	C₂×Q₈	#41×C₃₁	1	1	yes	characteristic
549	C₂×D₄	—	2	1	no	—
550	C₂×D₄	—	2	2	no	—
551	C₂×D₄	—	2	2	no	—
552	C₂×D₄	—	2	1	no	—
553	C₂×D₄	—	2	1	no	—
554	C₂×D₄	—	2	1	no	—
555	C₂×D₄	—	2	1	no	—
556	C₂×D₄	—	2	1	no	—
557	C₂×D₄	—	2	1	no	—
558	C₂×D₄	—	2	1	no	—
559	C₂×D₄	—	2	1	no	—
560	C₂×D₄	—	2	1	no	—
561	C₂×D₄	—	2	1	no	—
562	C₂×D₄	—	2	1	no	—
563	C₂×D₄	—	2	1	no	—
564	C₂×D₄	—	2	1	no	—
565	C₂×D₄	—	2	1	no	—
566	C₂×D₄	—	4	2	no	—
567	C₂×D₄	—	4	2	no	—
568	C₂×D₄	—	4	2	no	—
569	C₂×D₄	—	4	2	no	—
570	C₂×D₄	—	4	4	no	—
571	C₂×D₄	—	4	4	no	—
572	C₂×D₄	—	4	2	no	—
573	C₂×D₄	—	4	2	no	—
574	C₂×D₄	—	4	2	no	—
575	C₂×D₄	—	8	4	no	—
576	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
577	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
578	C₂×D₄	#41×C₃₁	1	1	yes	characteristic
579	C₂²×C₄	—	2	1	no	—
580	C₂²×C₄	—	2	2	no	—
581	C₂²×C₄	—	2	1	no	—
582	C₂²×C₄	—	2	1	no	—
583	C₂²×C₄	—	2	1	no	—
584	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
585	Q₁₆	—	2	2	no	—
586	Q₁₆	—	4	1	no	—
587	SD₁₆	—	2	2	no	—
588	SD₁₆	—	2	2	no	—
589	SD₁₆	—	4	2	no	—
590	SD₁₆	—	4	2	no	—
591	SD₁₆	—	4	1	no	—
592	SD₁₆	—	4	1	no	—
593	SD₁₆	—	4	2	no	—
594	D₈	—	2	2	no	—
595	D₈	—	4	2	no	—
596	D₈	—	4	2	no	—
597	D₈	—	4	1	no	—
598	D₈	—	4	2	no	—
599	M₁₆	—	2	1	no	—
600	M₁₆	—	2	1	no	—
601	M₁₆	—	2	1	no	—
602	M₁₆	—	2	1	no	—
603	M₁₆	—	2	1	no	—
604	M₁₆	—	2	1	no	—
605	M₁₆	—	2	1	no	—
606	M₁₆	—	2	1	no	—
607	M₁₆	—	2	1	no	—
608	M₁₆	—	2	1	no	—
609	M₁₆	—	2	1	no	—
610	M₁₆	—	2	1	no	—
611	M₁₆	—	4	2	no	—
612	M₁₆	—	4	2	no	—
613	M₁₆	—	4	2	no	—
614	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
615	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
616	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
617	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
618	C₂×C₈	—	2	1	no	—
619	C₂×C₈	—	2	1	no	—
620	C₂×C₈	—	2	1	no	—
621	C₂×C₈	—	2	1	no	—
622	C₂×C₈	—	2	1	no	—
623	C₂×C₈	—	2	1	no	—
624	C₂×C₈	—	2	1	no	—
625	C₂×C₈	—	2	1	no	—
626	C₂×C₈	—	4	2	no	—
627	C₂×C₈	—	4	2	no	—
628	C₂×C₈	—	4	2	no	—
629	C₄⋊C₄	—	2	1	no	—
630	C₂²⋊C₄	—	2	1	no	—
631	C₂²⋊C₄	—	4	2	no	—
632	C₄²	—	2	1	no	—
633	C₄²	—	2	1	no	—
634	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
635	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
636	C₂³	—	2	1	no	—
637	C₂³	—	2	2	no	—
638	C₂³	—	2	1	no	—
639	C₂³	—	2	2	no	—
640	C₂³	—	2	1	no	—
641	C₂³	—	2	2	no	—
642	C₂³	—	4	4	no	—
643	C₂³	—	4	1	no	—
644	C₂³	—	4	1	no	—
645	C₂³	—	4	1	no	—
646	C₂³	—	8	2	no	—
647	C₂³	—	8	2	no	—
648	C₂³	—	16	2	no	—
649	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
650	Q₈	—	2	1	no	—
651	Q₈	—	2	1	no	—
652	Q₈	—	2	1	no	—
653	Q₈	—	2	1	no	—
654	Q₈	—	2	1	no	—
655	Q₈	C₃₁×#121	1	2	yes	—
656	D₄	—	2	2	no	—
657	D₄	—	2	2	no	—
658	D₄	—	2	1	no	—
659	D₄	—	2	1	no	—
660	D₄	—	2	1	no	—
661	D₄	—	2	1	no	—
662	D₄	—	2	1	no	—
663	D₄	—	4	4	no	—
664	D₄	—	4	4	no	—
665	D₄	—	4	1	no	—
666	D₄	—	4	1	no	—
667	D₄	—	4	1	no	—
668	D₄	—	4	1	no	—
669	D₄	—	4	1	no	—
670	D₄	—	4	1	no	—
671	D₄	—	4	1	no	—
672	D₄	—	4	1	no	—
673	D₄	—	4	1	no	—
674	D₄	—	4	1	no	—
675	D₄	—	4	1	no	—
676	D₄	—	4	1	no	—
677	D₄	—	8	4	no	—
678	D₄	—	8	4	no	—
679	D₄	—	16	2	no	—
680	D₄	C₃₁×#121	1	2	yes	—
681	C₂×C₄	—	2	1	no	—
682	C₂×C₄	—	2	1	no	—
683	C₂×C₄	—	2	2	no	—
684	C₂×C₄	—	2	2	no	—
685	C₂×C₄	—	2	1	no	—
686	C₂×C₄	—	2	1	no	—
687	C₂×C₄	—	2	1	no	—
688	C₂×C₄	—	4	4	no	—
689	C₂×C₄	—	4	2	no	—
690	C₂×C₄	—	4	1	no	—
691	C₂×C₄	—	4	1	no	—
692	C₂×C₄	—	4	1	no	—
693	C₂×C₄	—	4	1	no	—
694	C₂×C₄	—	4	1	no	—
695	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
696	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
697	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
698	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
699	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
700	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
701	C₈	—	2	2	no	—
702	C₈	—	2	2	no	—
703	C₈	—	2	2	no	—
704	C₈	—	2	2	no	—
705	C₈	—	4	1	no	—
706	C₈	—	4	1	no	—
707	C₈	—	4	1	no	—
708	C₈	—	4	1	no	—
709	C₂²	—	2	2	no	—
710	C₂²	—	4	1	no	—
711	C₂²	—	4	1	no	—
712	C₂²	—	4	1	no	—
713	C₂²	—	4	2	no	—
714	C₂²	—	8	1	no	—
715	C₂²	—	8	2	no	—
716	C₂²	—	8	1	no	—
717	C₂²	—	8	2	no	—
718	C₂²	—	8	2	no	—
719	C₂²	—	8	1	no	—
720	C₂²	—	8	2	no	—
721	C₂²	—	8	1	no	—
722	C₂²	—	16	1	no	—
723	C₂²	—	16	1	no	—
724	C₂²	—	16	4	no	—
725	C₂²	C₃₁×#525	1	1	yes	characteristic
726	C₂²	C₃₁×#546	1	1	yes	characteristic
727	C₂²	C₃₁×#546	1	1	yes	characteristic
728	C₄	—	2	2	no	—
729	C₄	—	2	2	no	—
730	C₄	—	4	1	no	—
731	C₄	—	4	1	no	—
732	C₄	—	4	1	no	—
733	C₄	—	8	1	no	—
734	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
735	C₄	C₃₁×#538	1	1	yes	characteristic
736	C₄	D₄²×C₃₁	1	1	yes	characteristic
737	C₄	D₄²×C₃₁	1	1	yes	characteristic
738	C₂	—	2	1	no	—
739	C₂	—	2	1	no	—
740	C₂	—	2	1	no	—
741	C₂	—	4	2	no	—
742	C₂	—	8	2	no	—
743	C₂	—	8	1	no	—
744	C₂	—	8	1	no	—
745	C₂	—	16	1	no	—
746	C₂	C₃₁×#2424	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#2424
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃₁×#3068	C₂	semidirect
2. C₃₁×#3068	C₂	semidirect
3. C₃₁×#3019	C₂	semidirect
4. C₂×C₃₁×#453	C₂	semidirect
5. C₃₁×#2956	C₂	semidirect
6. C₃₁×#2888	C₂	semidirect
7. C₃₁×#2069	C₂	semidirect
8. C₃₁×#2050	C₂	semidirect
9. C₃₁×#2048	C₂	semidirect
10. C₃₁×#2002	C₂	semidirect
11. C₃₁×#1965	C₂	semidirect
12. C₃₁×#1932	C₂	semidirect
13. C₃₁×#1914	C₂	semidirect
14. C₃₁×#1903	C₂	semidirect
15. C₃₁×#1861	C₂	semidirect
21. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
24. C₃₁×#568	C₂²	semidirect
37. C₃₁×#430	C₂²	semidirect
38. C₃₁×#430	C₂²	semidirect
39. C₃₁×#430	C₂²	semidirect
40. C₃₁×#423	C₂²	semidirect
45. C₂×C₃₁×#104	C₂²	semidirect
46. C₃₁×#418	C₂²	semidirect
50. C₃₁×#413	C₂²	semidirect
56. C₂×C₃₁×#100	C₂²	semidirect
57. C₂×C₃₁×#100	C₂²	semidirect
58. C₄×M₁₆×C₃₁	C₂²	semidirect
59. C₃₁×#344	C₂²	semidirect
60. C₃₁×#344	C₂²	semidirect
61. C₃₁×#339	C₂²	semidirect
62. C₃₁×#338	C₂²	semidirect
81. C₃₁×#137	D₄	semidirect
91. C₃₁×AGL(1,ℤ/8)	D₄	semidirect
136. C₃₁×#104	D₄	semidirect
137. C₃₁×#104	D₄	semidirect
141. C₃₁×#101	D₄	semidirect
150. C₃₁×#100	D₄	semidirect
262. #25196	C₃₁	direct (N × H)
336. #3068	C₂×C₃₁	semidirect
337. #3068	C₂×C₃₁	semidirect
338. #3019	C₂×C₃₁	semidirect
339. C₂×#453	C₂×C₃₁	semidirect
340. #2956	C₂×C₃₁	semidirect
341. #2888	C₂×C₃₁	semidirect
342. #2069	C₂×C₃₁	semidirect
343. #2050	C₂×C₃₁	semidirect
344. #2048	C₂×C₃₁	semidirect
345. #2002	C₂×C₃₁	semidirect
346. #1965	C₂×C₃₁	semidirect
347. #1932	C₂×C₃₁	semidirect
348. #1914	C₂×C₃₁	semidirect
349. #1903	C₂×C₃₁	semidirect
350. #1861	C₂×C₃₁	semidirect
385. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
388. #568	C₂²×C₃₁	semidirect
401. #430	C₂²×C₃₁	semidirect
402. #430	C₂²×C₃₁	semidirect
403. #430	C₂²×C₃₁	semidirect
404. #423	C₂²×C₃₁	semidirect
409. C₂×#104	C₂²×C₃₁	semidirect
410. #418	C₂²×C₃₁	semidirect
414. #413	C₂²×C₃₁	semidirect
420. C₂×#100	C₂²×C₃₁	semidirect
421. C₂×#100	C₂²×C₃₁	semidirect
422. C₄×M₁₆	C₂²×C₃₁	semidirect
423. #344	C₂²×C₃₁	semidirect
424. #344	C₂²×C₃₁	semidirect
425. #339	C₂²×C₃₁	semidirect
426. #338	C₂²×C₃₁	semidirect
454. #137	D₄×C₃₁	semidirect
464. AGL(1,ℤ/8)	D₄×C₃₁	semidirect
509. #104	D₄×C₃₁	semidirect
510. #104	D₄×C₃₁	semidirect
514. #101	D₄×C₃₁	semidirect
523. #100	D₄×C₃₁	semidirect
526. C₃₁	#25196	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3068	2
C₃₁×#3019	2
C₂×C₃₁×#453	2
C₃₁×#2956	2
C₃₁×#2888	2
C₃₁×#2069	2
C₃₁×#2050	2
C₃₁×#2048	2
C₃₁×#2002	2
C₃₁×#1965	2
C₃₁×#1932	2
C₃₁×#1914	2
C₃₁×#1903	2
C₃₁×#1861	2
#25196	31

Subgroups — C31×#25196 #21481955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25196 #21481955