Subgroups — C₃₁×#25199 #21481958

Order 7936.

990 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3069	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3069	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3019	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#455	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×#2049	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2004	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1967	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×#1902	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×#137	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#137	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#137	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×#124	C₂²	1	1	yes	characteristic
27	C₃₁×#473	—	2	2	no	—
28	C₃₁×#473	—	2	2	no	—
29	C₃₁×#471	—	2	2	no	—
30	C₃₁×#471	—	2	2	no	—
31	C₃₁×#455	—	2	2	no	—
32	C₃₁×#455	—	2	2	no	—
33	C₃₁×#455	—	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×#419	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×#101	C₂²	1	1	yes	characteristic, complemented
52	C₂×C₃₁×#101	C₂²	1	1	yes	characteristic, complemented
53	C₂×C₃₁×#100	—	2	2	no	—
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
57	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
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×Q₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×D₄×C₃₁	—	2	2	no	—
74	C₂²×D₄×C₃₁	—	2	1	no	—
75	C₂²×D₄×C₃₁	—	2	1	no	—
76	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#137	—	2	1	no	—
78	C₃₁×#137	—	2	2	no	—
79	C₃₁×#137	—	2	2	no	—
80	C₃₁×#137	—	2	1	no	—
81	C₃₁×#137	—	2	1	no	—
82	C₃₁×#137	—	2	2	no	—
83	C₃₁×#137	—	2	2	no	—
84	C₃₁×#137	—	4	1	no	—
85	C₃₁×#137	—	4	1	no	—
86	C₃₁×#137	D₄	1	2	yes	—
87	C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
88	C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
89	C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
90	C₃₁×AGL(1,ℤ/8)	D₄	1	2	yes	—
91	C₃₁×#135	—	2	2	no	—
92	C₃₁×#135	—	4	1	no	—
93	C₂×Q₁₆×C₃₁	—	2	1	no	—
94	C₂×Q₁₆×C₃₁	—	2	1	no	—
95	C₂×Q₁₆×C₃₁	—	2	1	no	—
96	C₂×SD₁₆×C₃₁	—	2	1	no	—
97	C₂×SD₁₆×C₃₁	—	2	1	no	—
98	C₂×SD₁₆×C₃₁	—	2	1	no	—
99	C₃₁×#131	—	2	2	no	—
100	C₃₁×#131	—	2	2	no	—
101	C₃₁×#131	—	2	2	no	—
102	C₃₁×#131	—	2	2	no	—
103	C₃₁×#131	—	4	1	no	—
104	C₃₁×#131	—	4	1	no	—
105	C₂×M₁₆×C₃₁	—	2	2	no	—
106	C₂×M₁₆×C₃₁	—	2	2	no	—
107	C₂×M₁₆×C₃₁	—	4	1	no	—
108	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
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	C₃₁×#127	—	2	1	no	—
117	C₃₁×#124	—	2	1	no	—
118	C₃₁×#124	—	2	1	no	—
119	C₃₁×#124	—	2	1	no	—
120	C₃₁×#124	—	4	1	no	—
121	C₃₁×#121	—	4	1	no	—
122	C₃₁×#120	—	2	2	no	—
123	C₃₁×#117	C₂³	1	1	yes	characteristic
124	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
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×#103	—	4	1	no	—
139	C₃₁×#101	—	2	1	no	—
140	C₃₁×#101	—	2	1	no	—
141	C₃₁×#101	—	2	2	no	—
142	C₃₁×#101	—	2	2	no	—
143	C₃₁×#101	D₄	1	2	yes	complemented
144	C₃₁×#100	—	2	1	no	—
145	C₃₁×#100	—	2	1	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	#41	1	2	yes	—
155	#41×C₃₁	—	2	1	no	—
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	—	4	2	no	—
166	#41×C₃₁	—	4	2	no	—
167	#41×C₃₁	—	4	2	no	—
168	#41×C₃₁	—	4	2	no	—
169	#41×C₃₁	C₂×D₄	1	2	yes	—
170	#41×C₃₁	C₂×D₄	1	2	yes	—
171	C₂×Q₈×C₃₁	—	2	1	no	—
172	C₂×Q₈×C₃₁	—	2	1	no	—
173	C₂×Q₈×C₃₁	—	2	1	no	—
174	C₂×Q₈×C₃₁	—	2	1	no	—
175	C₂×Q₈×C₃₁	—	2	1	no	—
176	C₂×Q₈×C₃₁	—	2	1	no	—
177	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₂×Q₈×C₃₁	#41	1	1	yes	characteristic
180	C₂×D₄×C₃₁	—	2	1	no	—
181	C₂×D₄×C₃₁	—	2	2	no	—
182	C₂×D₄×C₃₁	—	2	2	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	—	4	2	no	—
192	C₂×D₄×C₃₁	—	4	2	no	—
193	C₂×D₄×C₃₁	—	4	4	no	—
194	C₂×D₄×C₃₁	—	4	4	no	—
195	C₂×D₄×C₃₁	—	4	2	no	—
196	C₂×D₄×C₃₁	—	4	2	no	—
197	C₂×D₄×C₃₁	—	4	2	no	—
198	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
199	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
200	C₂×D₄×C₃₁	#41	1	1	yes	characteristic
201	C₂²×C₄×C₃₁	—	2	1	no	—
202	C₂²×C₄×C₃₁	—	2	2	no	—
203	C₂²×C₄×C₃₁	—	2	1	no	—
204	C₂²×C₄×C₃₁	—	2	1	no	—
205	C₂²×C₄×C₃₁	—	2	1	no	—
206	C₂²×C₄×C₃₁	—	2	1	no	—
207	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
208	Q₁₆×C₃₁	—	2	2	no	—
209	Q₁₆×C₃₁	—	4	2	no	—
210	Q₁₆×C₃₁	—	4	2	no	—
211	Q₁₆×C₃₁	—	4	1	no	—
212	Q₁₆×C₃₁	—	4	2	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	1	no	—
222	M₁₆×C₃₁	—	2	1	no	—
223	M₁₆×C₃₁	—	2	1	no	—
224	M₁₆×C₃₁	—	2	1	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	—	4	2	no	—
235	M₁₆×C₃₁	—	4	2	no	—
236	M₁₆×C₃₁	—	4	2	no	—
237	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
238	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
239	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
240	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
241	C₂×C₈×C₃₁	—	2	1	no	—
242	C₂×C₈×C₃₁	—	2	1	no	—
243	C₂×C₈×C₃₁	—	2	1	no	—
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	—	4	2	no	—
250	C₂×C₈×C₃₁	—	4	2	no	—
251	C₂×C₈×C₃₁	—	4	2	no	—
252	C₄⋊C₄×C₃₁	—	2	1	no	—
253	C₂²⋊C₄×C₃₁	—	2	1	no	—
254	C₂²⋊C₄×C₃₁	—	4	2	no	—
255	C₂²⋊C₄×C₃₁	—	4	2	no	—
256	C₂²⋊C₄×C₃₁	—	4	2	no	—
257	C₄²×C₃₁	—	2	1	no	—
258	C₄²×C₃₁	—	2	1	no	—
259	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
260	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
261	#25199	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
262	C₂³×C₃₁	—	2	1	no	—
263	C₂³×C₃₁	—	2	2	no	—
264	C₂³×C₃₁	—	2	1	no	—
265	C₂³×C₃₁	—	2	2	no	—
266	C₂³×C₃₁	—	2	2	no	—
267	C₂³×C₃₁	—	2	1	no	—
268	C₂³×C₃₁	—	4	4	no	—
269	C₂³×C₃₁	—	8	2	no	—
270	C₂³×C₃₁	—	16	2	no	—
271	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
272	Q₈×C₃₁	—	2	1	no	—
273	Q₈×C₃₁	—	2	1	no	—
274	Q₈×C₃₁	—	2	1	no	—
275	Q₈×C₃₁	—	2	1	no	—
276	Q₈×C₃₁	—	2	1	no	—
277	Q₈×C₃₁	—	4	1	no	—
278	Q₈×C₃₁	—	4	1	no	—
279	Q₈×C₃₁	—	4	1	no	—
280	Q₈×C₃₁	—	4	1	no	—
281	Q₈×C₃₁	#121	1	2	yes	—
282	D₄×C₃₁	—	2	2	no	—
283	D₄×C₃₁	—	2	2	no	—
284	D₄×C₃₁	—	2	1	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	—	4	4	no	—
290	D₄×C₃₁	—	4	4	no	—
291	D₄×C₃₁	—	4	1	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	—	16	2	no	—
300	D₄×C₃₁	#121	1	2	yes	—
301	C₂×C₄×C₃₁	—	2	1	no	—
302	C₂×C₄×C₃₁	—	2	2	no	—
303	C₂×C₄×C₃₁	—	2	2	no	—
304	C₂×C₄×C₃₁	—	2	1	no	—
305	C₂×C₄×C₃₁	—	2	1	no	—
306	C₂×C₄×C₃₁	—	2	1	no	—
307	C₂×C₄×C₃₁	—	2	1	no	—
308	C₂×C₄×C₃₁	—	4	4	no	—
309	C₂×C₄×C₃₁	—	4	1	no	—
310	C₂×C₄×C₃₁	—	4	1	no	—
311	C₂×C₄×C₃₁	—	4	1	no	—
312	C₂×C₄×C₃₁	—	4	1	no	—
313	C₂×C₄×C₃₁	—	4	1	no	—
314	C₂×C₄×C₃₁	—	4	1	no	—
315	C₂×C₄×C₃₁	—	4	1	no	—
316	C₂×C₄×C₃₁	—	4	1	no	—
317	C₂×C₄×C₃₁	—	8	1	no	—
318	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
319	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
320	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
321	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
322	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
323	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
324	C₈×C₃₁	—	2	2	no	—
325	C₈×C₃₁	—	2	2	no	—
326	C₈×C₃₁	—	2	2	no	—
327	C₈×C₃₁	—	2	2	no	—
328	C₈×C₃₁	—	4	1	no	—
329	C₈×C₃₁	—	4	1	no	—
330	C₈×C₃₁	—	4	1	no	—
331	C₈×C₃₁	—	4	1	no	—
332	#3069	C₂×C₃₁	1	1	yes	characteristic, complemented
333	#3069	C₂×C₃₁	1	1	yes	characteristic, complemented
334	#3019	C₂×C₃₁	1	1	yes	characteristic
335	C₂×#455	C₂×C₃₁	1	1	yes	characteristic, complemented
336	#2956	C₂×C₃₁	1	1	yes	characteristic, complemented
337	#2888	C₂×C₃₁	1	1	yes	characteristic, complemented
338	#2069	C₂×C₃₁	1	1	yes	characteristic, complemented
339	#2050	C₂×C₃₁	1	1	yes	characteristic, complemented
340	#2049	C₂×C₃₁	1	1	yes	characteristic, complemented
341	#2004	C₂×C₃₁	1	1	yes	characteristic, complemented
342	#1967	C₂×C₃₁	1	1	yes	characteristic, complemented
343	#1932	C₂×C₃₁	1	1	yes	characteristic, complemented
344	#1914	C₂×C₃₁	1	1	yes	characteristic, complemented
345	#1902	C₂×C₃₁	1	1	yes	characteristic, complemented
346	#1861	C₂×C₃₁	1	1	yes	characteristic, complemented
347	C₂²×C₃₁	—	2	2	no	—
348	C₂²×C₃₁	—	4	1	no	—
349	C₂²×C₃₁	—	4	1	no	—
350	C₂²×C₃₁	—	4	1	no	—
351	C₂²×C₃₁	—	4	2	no	—
352	C₂²×C₃₁	—	8	1	no	—
353	C₂²×C₃₁	—	8	1	no	—
354	C₂²×C₃₁	—	8	2	no	—
355	C₂²×C₃₁	—	8	2	no	—
356	C₂²×C₃₁	—	8	2	no	—
357	C₂²×C₃₁	—	8	1	no	—
358	C₂²×C₃₁	—	16	4	no	—
359	C₂²×C₃₁	#525	1	1	yes	characteristic
360	C₂²×C₃₁	#546	1	1	yes	characteristic
361	C₂²×C₃₁	#546	1	1	yes	characteristic
362	C₄×C₃₁	—	2	2	no	—
363	C₄×C₃₁	—	4	1	no	—
364	C₄×C₃₁	—	4	1	no	—
365	C₄×C₃₁	—	4	1	no	—
366	C₄×C₃₁	—	4	1	no	—
367	C₄×C₃₁	—	8	1	no	—
368	C₄×C₃₁	—	8	1	no	—
369	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
370	C₄×C₃₁	#538	1	1	yes	characteristic
371	C₄×C₃₁	D₄²	1	1	yes	characteristic
372	C₄×C₃₁	D₄²	1	1	yes	characteristic
373	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
374	#575	C₂²×C₃₁	1	1	yes	characteristic
375	#575	C₂²×C₃₁	1	1	yes	characteristic
376	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
377	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
378	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic, complemented
379	#568	C₂²×C₃₁	1	1	yes	characteristic
380	#568	C₂²×C₃₁	1	1	yes	characteristic
381	#568	C₂²×C₃₁	1	1	yes	characteristic, complemented
382	#535	C₂²×C₃₁	1	1	yes	characteristic
383	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
384	#473	—	2	2	no	—
385	#473	—	2	2	no	—
386	#471	—	2	2	no	—
387	#471	—	2	2	no	—
388	#455	—	2	2	no	—
389	#455	—	2	2	no	—
390	#455	—	2	2	no	—
391	#453	—	2	2	no	—
392	#444	—	2	2	no	—
393	#443	—	2	2	no	—
394	#430	C₂²×C₃₁	1	1	yes	characteristic, complemented
395	#430	C₂²×C₃₁	1	1	yes	characteristic, complemented
396	#430	C₂²×C₃₁	1	1	yes	characteristic, complemented
397	#423	C₂²×C₃₁	1	1	yes	characteristic, complemented
398	#421	C₂²×C₃₁	1	1	yes	characteristic
399	#421	C₂²×C₃₁	1	1	yes	characteristic
400	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
401	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
402	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic, complemented
403	#419	C₂²×C₃₁	1	1	yes	characteristic, complemented
404	#413	—	2	2	no	—
405	#413	C₂²×C₃₁	1	1	yes	characteristic
406	#413	C₂²×C₃₁	1	1	yes	characteristic
407	#413	C₂²×C₃₁	1	1	yes	characteristic, complemented
408	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic, complemented
409	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic, complemented
410	C₂×#100	—	2	2	no	—
411	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
412	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
413	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
414	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
415	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
416	#344	C₂²×C₃₁	1	1	yes	characteristic, complemented
417	#344	C₂²×C₃₁	1	1	yes	characteristic, complemented
418	#339	C₂²×C₃₁	1	1	yes	characteristic, complemented
419	#338	C₂²×C₃₁	1	1	yes	characteristic, complemented
420	C₂×C₃₁	—	2	1	no	—
421	C₂×C₃₁	—	2	1	no	—
422	C₂×C₃₁	—	2	1	no	—
423	C₂×C₃₁	—	4	2	no	—
424	C₂×C₃₁	—	8	1	no	—
425	C₂×C₃₁	—	8	2	no	—
426	C₂×C₃₁	—	8	1	no	—
427	C₂×C₃₁	#2424	1	1	yes	characteristic
428	H(5,𝔽₂)	—	2	2	no	—
429	H(5,𝔽₂)	—	2	2	no	—
430	H(5,𝔽₂)	—	4	1	no	—
431	H(5,𝔽₂)	—	4	1	no	—
432	C₂×#41	—	2	1	no	—
433	C₂×#41	—	2	1	no	—
434	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
435	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
436	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
437	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
438	C₂²×D₄	—	2	2	no	—
439	C₂²×D₄	—	2	1	no	—
440	C₂²×D₄	—	2	1	no	—
441	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
442	#137	—	2	1	no	—
443	#137	—	2	2	no	—
444	#137	—	2	2	no	—
445	#137	—	2	1	no	—
446	#137	—	2	1	no	—
447	#137	—	2	2	no	—
448	#137	—	2	2	no	—
449	#137	—	4	1	no	—
450	#137	—	4	1	no	—
451	#137	D₄×C₃₁	1	2	yes	—
452	AGL(1,ℤ/8)	—	2	1	no	—
453	AGL(1,ℤ/8)	—	2	1	no	—
454	AGL(1,ℤ/8)	—	2	1	no	—
455	AGL(1,ℤ/8)	D₄×C₃₁	1	2	yes	—
456	#135	—	2	2	no	—
457	#135	—	4	1	no	—
458	C₂×Q₁₆	—	2	1	no	—
459	C₂×Q₁₆	—	2	1	no	—
460	C₂×Q₁₆	—	2	1	no	—
461	C₂×SD₁₆	—	2	1	no	—
462	C₂×SD₁₆	—	2	1	no	—
463	C₂×SD₁₆	—	2	1	no	—
464	#131	—	2	2	no	—
465	#131	—	2	2	no	—
466	#131	—	2	2	no	—
467	#131	—	2	2	no	—
468	#131	—	4	1	no	—
469	#131	—	4	1	no	—
470	C₂×M₁₆	—	2	2	no	—
471	C₂×M₁₆	—	2	2	no	—
472	C₂×M₁₆	—	4	1	no	—
473	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
474	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
475	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
476	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
477	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
478	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
479	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
480	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
481	#127	—	2	1	no	—
482	#124	—	2	1	no	—
483	#124	—	2	1	no	—
484	#124	—	2	1	no	—
485	#124	—	4	1	no	—
486	#121	—	4	1	no	—
487	#120	—	2	2	no	—
488	#117	C₂³×C₃₁	1	1	yes	characteristic
489	C₂×C₂²⋊C₄	—	2	2	no	—
490	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
491	#108	—	2	2	no	—
492	#108	—	2	2	no	—
493	#108	—	2	2	no	—
494	C₄⋊C₈	—	4	1	no	—
495	#104	—	2	2	no	—
496	#104	—	2	2	no	—
497	#104	—	2	1	no	—
498	#104	—	2	1	no	—
499	#104	—	2	1	no	—
500	#104	—	2	1	no	—
501	#104	D₄×C₃₁	1	2	yes	complemented
502	#104	D₄×C₃₁	1	2	yes	complemented
503	#103	—	4	1	no	—
504	#101	—	2	1	no	—
505	#101	—	2	1	no	—
506	#101	—	2	2	no	—
507	#101	—	2	2	no	—
508	#101	D₄×C₃₁	1	2	yes	complemented
509	#100	—	2	1	no	—
510	#100	—	2	1	no	—
511	#100	—	2	2	no	—
512	#100	—	2	2	no	—
513	#100	—	2	2	no	—
514	#100	—	2	2	no	—
515	#100	D₄×C₃₁	1	2	yes	complemented
516	#97	—	2	1	no	—
517	C₄×C₈	—	2	1	no	—
518	C₃₁	#25199	1	1	yes	characteristic, direct product factor, Sylow
519	C₂⁴	—	2	2	no	—
520	C₂⁴	#41×C₃₁	1	2	yes	—
521	#41	—	2	1	no	—
522	#41	—	2	1	no	—
523	#41	—	2	1	no	—
524	#41	—	2	1	no	—
525	#41	—	2	1	no	—
526	#41	—	2	1	no	—
527	#41	—	2	1	no	—
528	#41	—	2	1	no	—
529	#41	—	2	1	no	—
530	#41	—	2	1	no	—
531	#41	—	4	2	no	—
532	#41	—	4	2	no	—
533	#41	—	4	2	no	—
534	#41	—	4	2	no	—
535	#41	C₂×D₄×C₃₁	1	2	yes	—
536	#41	C₂×D₄×C₃₁	1	2	yes	—
537	C₂×Q₈	—	2	1	no	—
538	C₂×Q₈	—	2	1	no	—
539	C₂×Q₈	—	2	1	no	—
540	C₂×Q₈	—	2	1	no	—
541	C₂×Q₈	—	2	1	no	—
542	C₂×Q₈	—	2	1	no	—
543	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
544	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
545	C₂×Q₈	#41×C₃₁	1	1	yes	characteristic
546	C₂×D₄	—	2	1	no	—
547	C₂×D₄	—	2	2	no	—
548	C₂×D₄	—	2	2	no	—
549	C₂×D₄	—	2	1	no	—
550	C₂×D₄	—	2	1	no	—
551	C₂×D₄	—	2	1	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	—	4	2	no	—
558	C₂×D₄	—	4	2	no	—
559	C₂×D₄	—	4	4	no	—
560	C₂×D₄	—	4	4	no	—
561	C₂×D₄	—	4	2	no	—
562	C₂×D₄	—	4	2	no	—
563	C₂×D₄	—	4	2	no	—
564	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
565	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
566	C₂×D₄	#41×C₃₁	1	1	yes	characteristic
567	C₂²×C₄	—	2	1	no	—
568	C₂²×C₄	—	2	2	no	—
569	C₂²×C₄	—	2	1	no	—
570	C₂²×C₄	—	2	1	no	—
571	C₂²×C₄	—	2	1	no	—
572	C₂²×C₄	—	2	1	no	—
573	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
574	Q₁₆	—	2	2	no	—
575	Q₁₆	—	4	2	no	—
576	Q₁₆	—	4	2	no	—
577	Q₁₆	—	4	1	no	—
578	Q₁₆	—	4	2	no	—
579	SD₁₆	—	2	2	no	—
580	SD₁₆	—	2	2	no	—
581	SD₁₆	—	4	2	no	—
582	SD₁₆	—	4	2	no	—
583	SD₁₆	—	4	1	no	—
584	SD₁₆	—	4	1	no	—
585	SD₁₆	—	4	2	no	—
586	D₈	—	2	2	no	—
587	D₈	—	4	1	no	—
588	M₁₆	—	2	1	no	—
589	M₁₆	—	2	1	no	—
590	M₁₆	—	2	1	no	—
591	M₁₆	—	2	1	no	—
592	M₁₆	—	2	1	no	—
593	M₁₆	—	2	1	no	—
594	M₁₆	—	2	1	no	—
595	M₁₆	—	2	1	no	—
596	M₁₆	—	2	1	no	—
597	M₁₆	—	2	1	no	—
598	M₁₆	—	2	1	no	—
599	M₁₆	—	2	1	no	—
600	M₁₆	—	4	2	no	—
601	M₁₆	—	4	2	no	—
602	M₁₆	—	4	2	no	—
603	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
604	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
605	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
606	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
607	C₂×C₈	—	2	1	no	—
608	C₂×C₈	—	2	1	no	—
609	C₂×C₈	—	2	1	no	—
610	C₂×C₈	—	2	1	no	—
611	C₂×C₈	—	2	1	no	—
612	C₂×C₈	—	2	1	no	—
613	C₂×C₈	—	2	1	no	—
614	C₂×C₈	—	2	1	no	—
615	C₂×C₈	—	4	2	no	—
616	C₂×C₈	—	4	2	no	—
617	C₂×C₈	—	4	2	no	—
618	C₄⋊C₄	—	2	1	no	—
619	C₂²⋊C₄	—	2	1	no	—
620	C₂²⋊C₄	—	4	2	no	—
621	C₂²⋊C₄	—	4	2	no	—
622	C₂²⋊C₄	—	4	2	no	—
623	C₄²	—	2	1	no	—
624	C₄²	—	2	1	no	—
625	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
626	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
627	C₂³	—	2	1	no	—
628	C₂³	—	2	2	no	—
629	C₂³	—	2	1	no	—
630	C₂³	—	2	2	no	—
631	C₂³	—	2	2	no	—
632	C₂³	—	2	1	no	—
633	C₂³	—	4	4	no	—
634	C₂³	—	8	2	no	—
635	C₂³	—	16	2	no	—
636	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
637	Q₈	—	2	1	no	—
638	Q₈	—	2	1	no	—
639	Q₈	—	2	1	no	—
640	Q₈	—	2	1	no	—
641	Q₈	—	2	1	no	—
642	Q₈	—	4	1	no	—
643	Q₈	—	4	1	no	—
644	Q₈	—	4	1	no	—
645	Q₈	—	4	1	no	—
646	Q₈	C₃₁×#121	1	2	yes	—
647	D₄	—	2	2	no	—
648	D₄	—	2	2	no	—
649	D₄	—	2	1	no	—
650	D₄	—	2	1	no	—
651	D₄	—	2	1	no	—
652	D₄	—	2	1	no	—
653	D₄	—	2	1	no	—
654	D₄	—	4	4	no	—
655	D₄	—	4	4	no	—
656	D₄	—	4	1	no	—
657	D₄	—	4	1	no	—
658	D₄	—	4	1	no	—
659	D₄	—	4	1	no	—
660	D₄	—	4	1	no	—
661	D₄	—	4	1	no	—
662	D₄	—	4	1	no	—
663	D₄	—	4	1	no	—
664	D₄	—	16	2	no	—
665	D₄	C₃₁×#121	1	2	yes	—
666	C₂×C₄	—	2	1	no	—
667	C₂×C₄	—	2	2	no	—
668	C₂×C₄	—	2	2	no	—
669	C₂×C₄	—	2	1	no	—
670	C₂×C₄	—	2	1	no	—
671	C₂×C₄	—	2	1	no	—
672	C₂×C₄	—	2	1	no	—
673	C₂×C₄	—	4	4	no	—
674	C₂×C₄	—	4	1	no	—
675	C₂×C₄	—	4	1	no	—
676	C₂×C₄	—	4	1	no	—
677	C₂×C₄	—	4	1	no	—
678	C₂×C₄	—	4	1	no	—
679	C₂×C₄	—	4	1	no	—
680	C₂×C₄	—	4	1	no	—
681	C₂×C₄	—	4	1	no	—
682	C₂×C₄	—	8	1	no	—
683	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
684	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
685	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
686	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
687	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
688	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
689	C₈	—	2	2	no	—
690	C₈	—	2	2	no	—
691	C₈	—	2	2	no	—
692	C₈	—	2	2	no	—
693	C₈	—	4	1	no	—
694	C₈	—	4	1	no	—
695	C₈	—	4	1	no	—
696	C₈	—	4	1	no	—
697	C₂²	—	2	2	no	—
698	C₂²	—	4	1	no	—
699	C₂²	—	4	1	no	—
700	C₂²	—	4	1	no	—
701	C₂²	—	4	2	no	—
702	C₂²	—	8	1	no	—
703	C₂²	—	8	1	no	—
704	C₂²	—	8	2	no	—
705	C₂²	—	8	2	no	—
706	C₂²	—	8	2	no	—
707	C₂²	—	8	1	no	—
708	C₂²	—	16	4	no	—
709	C₂²	C₃₁×#525	1	1	yes	characteristic
710	C₂²	C₃₁×#546	1	1	yes	characteristic
711	C₂²	C₃₁×#546	1	1	yes	characteristic
712	C₄	—	2	2	no	—
713	C₄	—	4	1	no	—
714	C₄	—	4	1	no	—
715	C₄	—	4	1	no	—
716	C₄	—	4	1	no	—
717	C₄	—	8	1	no	—
718	C₄	—	8	1	no	—
719	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
720	C₄	C₃₁×#538	1	1	yes	characteristic
721	C₄	D₄²×C₃₁	1	1	yes	characteristic
722	C₄	D₄²×C₃₁	1	1	yes	characteristic
723	C₂	—	2	1	no	—
724	C₂	—	2	1	no	—
725	C₂	—	2	1	no	—
726	C₂	—	4	2	no	—
727	C₂	—	8	2	no	—
728	C₂	—	8	1	no	—
729	C₂	—	8	1	no	—
730	C₂	C₃₁×#2424	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#2424
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3019	C₂
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×#3069	C₂	semidirect
2. C₃₁×#3069	C₂	semidirect
4. C₂×C₃₁×#455	C₂	semidirect
5. C₃₁×#2956	C₂	semidirect
6. C₃₁×#2888	C₂	semidirect
7. C₃₁×#2069	C₂	semidirect
8. C₃₁×#2050	C₂	semidirect
9. C₃₁×#2049	C₂	semidirect
10. C₃₁×#2004	C₂	semidirect
11. C₃₁×#1967	C₂	semidirect
12. C₃₁×#1932	C₂	semidirect
13. C₃₁×#1914	C₂	semidirect
14. C₃₁×#1902	C₂	semidirect
15. C₃₁×#1861	C₂	semidirect
21. C₂×C₃₁×#137	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×#419	C₂²	semidirect
50. C₃₁×#413	C₂²	semidirect
51. C₂×C₃₁×#101	C₂²	semidirect
52. C₂×C₃₁×#101	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
136. C₃₁×#104	D₄	semidirect
137. C₃₁×#104	D₄	semidirect
143. C₃₁×#101	D₄	semidirect
150. C₃₁×#100	D₄	semidirect
261. #25199	C₃₁	direct (N × H)
332. #3069	C₂×C₃₁	semidirect
333. #3069	C₂×C₃₁	semidirect
335. C₂×#455	C₂×C₃₁	semidirect
336. #2956	C₂×C₃₁	semidirect
337. #2888	C₂×C₃₁	semidirect
338. #2069	C₂×C₃₁	semidirect
339. #2050	C₂×C₃₁	semidirect
340. #2049	C₂×C₃₁	semidirect
341. #2004	C₂×C₃₁	semidirect
342. #1967	C₂×C₃₁	semidirect
343. #1932	C₂×C₃₁	semidirect
344. #1914	C₂×C₃₁	semidirect
345. #1902	C₂×C₃₁	semidirect
346. #1861	C₂×C₃₁	semidirect
378. C₂×#137	C₂²×C₃₁	semidirect
381. #568	C₂²×C₃₁	semidirect
394. #430	C₂²×C₃₁	semidirect
395. #430	C₂²×C₃₁	semidirect
396. #430	C₂²×C₃₁	semidirect
397. #423	C₂²×C₃₁	semidirect
402. C₂×#104	C₂²×C₃₁	semidirect
403. #419	C₂²×C₃₁	semidirect
407. #413	C₂²×C₃₁	semidirect
408. C₂×#101	C₂²×C₃₁	semidirect
409. C₂×#101	C₂²×C₃₁	semidirect
415. C₄×M₁₆	C₂²×C₃₁	semidirect
416. #344	C₂²×C₃₁	semidirect
417. #344	C₂²×C₃₁	semidirect
418. #339	C₂²×C₃₁	semidirect
419. #338	C₂²×C₃₁	semidirect
501. #104	D₄×C₃₁	semidirect
502. #104	D₄×C₃₁	semidirect
508. #101	D₄×C₃₁	semidirect
515. #100	D₄×C₃₁	semidirect
518. C₃₁	#25199	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×#3069	2
C₃₁×#3019	2
C₂×C₃₁×#455	2
C₃₁×#2956	2
C₃₁×#2888	2
C₃₁×#2069	2
C₃₁×#2050	2
C₃₁×#2049	2
C₃₁×#2004	2
C₃₁×#1967	2
C₃₁×#1932	2
C₃₁×#1914	2
C₃₁×#1902	2
C₃₁×#1861	2
#25199	31

Subgroups — C31×#25199 #21481958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25199 #21481958