Subgroups — C₃₁×#24808 #21481567

Order 7936.

820 conjugacy classes of subgroups.

The subgroup lattice has 658 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×#3068	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#471	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#471	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2955	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2916	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2070	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2069	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2049	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2048	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1977	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1953	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1903	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1902	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1813	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#575	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#137	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#137	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	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#568	C₂²	1	1	yes	characteristic
23	C₃₁×#568	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
26	C₃₁×#473	—	2	2	no	—
27	C₃₁×#472	—	2	2	no	—
28	C₃₁×#471	—	2	2	no	—
29	C₃₁×#471	—	2	2	no	—
30	C₃₁×#471	—	2	2	no	—
31	C₃₁×#471	—	2	2	no	—
32	C₃₁×#471	—	2	2	no	—
33	C₃₁×#471	—	2	2	no	—
34	C₃₁×#444	—	2	2	no	—
35	C₃₁×#443	—	2	2	no	—
36	C₃₁×#430	—	2	2	no	—
37	C₃₁×#430	C₂²	1	1	yes	characteristic
38	C₃₁×#430	C₂²	1	1	yes	characteristic
39	C₃₁×#430	C₂²	1	1	yes	characteristic
40	C₃₁×#430	C₂²	1	1	yes	characteristic
41	C₃₁×#430	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#108	—	2	2	no	—
43	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
44	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
45	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
46	C₃₁×#421	C₂²	1	1	yes	characteristic
47	C₃₁×#421	C₂²	1	1	yes	characteristic
48	C₃₁×#419	C₂²	1	1	yes	characteristic
49	C₃₁×#418	C₂²	1	1	yes	characteristic
50	C₃₁×#413	C₂²	1	1	yes	characteristic
51	C₃₁×#413	C₂²	1	1	yes	characteristic
52	C₂×C₃₁×#101	C₂²	1	1	yes	characteristic
53	C₂×C₃₁×#101	C₂²	1	1	yes	characteristic
54	C₂×C₃₁×#101	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	C₃₁×#405	C₂²	1	1	yes	characteristic
59	C₃₁×#344	C₂²	1	1	yes	characteristic
60	C₃₁×#344	C₂²	1	1	yes	characteristic
61	C₃₁×#338	C₂²	1	1	yes	characteristic
62	C₃₁×#338	C₂²	1	1	yes	characteristic
63	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#137	—	2	1	no	—
68	C₃₁×#137	—	2	2	no	—
69	C₃₁×#137	—	2	2	no	—
70	C₃₁×#137	—	2	2	no	—
71	C₃₁×#137	—	4	1	no	—
72	C₃₁×#137	D₄	1	2	yes	—
73	C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
74	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
75	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
76	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
77	C₃₁×AGL(1,ℤ/8)	—	4	1	no	—
78	C₃₁×AGL(1,ℤ/8)	D₄	1	2	yes	—
79	C₃₁×#135	—	2	2	no	—
80	C₂×Q₁₆×C₃₁	—	2	1	no	—
81	C₂×Q₁₆×C₃₁	—	2	1	no	—
82	C₂×SD₁₆×C₃₁	—	2	1	no	—
83	C₂×SD₁₆×C₃₁	—	2	1	no	—
84	C₂×SD₁₆×C₃₁	—	2	1	no	—
85	C₂×SD₁₆×C₃₁	—	2	1	no	—
86	C₂×SD₁₆×C₃₁	—	2	1	no	—
87	C₂×SD₁₆×C₃₁	—	2	1	no	—
88	C₂×SD₁₆×C₃₁	—	2	1	no	—
89	C₂×SD₁₆×C₃₁	—	2	1	no	—
90	C₂×SD₁₆×C₃₁	—	2	1	no	—
91	C₂×SD₁₆×C₃₁	—	2	1	no	—
92	C₂×D₈×C₃₁	—	2	1	no	—
93	C₂×D₈×C₃₁	—	2	1	no	—
94	C₃₁×#131	—	2	1	no	—
95	C₃₁×#131	—	2	1	no	—
96	C₃₁×#131	—	2	2	no	—
97	C₃₁×#131	—	2	2	no	—
98	C₃₁×#131	—	2	1	no	—
99	C₃₁×#131	—	2	1	no	—
100	C₃₁×#131	—	2	1	no	—
101	C₃₁×#131	—	2	1	no	—
102	C₃₁×#131	—	4	1	no	—
103	C₃₁×#131	—	4	1	no	—
104	C₃₁×#131	D₄	1	2	yes	—
105	C₃₁×#131	D₄	1	2	yes	—
106	C₂×M₁₆×C₃₁	—	2	1	no	—
107	C₂×M₁₆×C₃₁	—	2	1	no	—
108	C₂×M₁₆×C₃₁	—	2	2	no	—
109	C₂×M₁₆×C₃₁	—	2	2	no	—
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	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
118	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
119	C₂²×C₈×C₃₁	—	2	1	no	—
120	C₂²×C₈×C₃₁	—	2	1	no	—
121	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
122	C₃₁×#117	C₂³	1	1	yes	characteristic
123	C₃₁×#108	—	2	2	no	—
124	C₃₁×#108	—	2	2	no	—
125	C₃₁×#108	—	2	2	no	—
126	C₃₁×#108	—	2	2	no	—
127	C₃₁×#108	—	2	2	no	—
128	C₃₁×#108	—	2	1	no	—
129	C₃₁×#108	—	2	1	no	—
130	C₃₁×#108	—	2	1	no	—
131	C₃₁×#108	—	2	1	no	—
132	C₃₁×#108	D₄	1	2	yes	—
133	C₃₁×#108	D₄	1	2	yes	—
134	C₃₁×#104	—	2	2	no	—
135	C₃₁×#104	—	2	2	no	—
136	C₃₁×#103	—	4	1	no	—
137	C₃₁×#102	—	4	1	no	—
138	C₃₁×#101	—	2	1	no	—
139	C₃₁×#101	—	2	1	no	—
140	C₃₁×#101	—	2	2	no	—
141	C₃₁×#101	—	2	2	no	—
142	C₃₁×#101	—	2	2	no	—
143	C₃₁×#100	—	2	1	no	—
144	C₃₁×#100	—	2	1	no	—
145	C₃₁×#100	—	2	2	no	—
146	C₃₁×#100	—	2	2	no	—
147	C₃₁×#100	—	2	2	no	—
148	C₃₁×#97	—	2	1	no	—
149	C₄×C₈×C₃₁	—	2	1	no	—
150	C₂⁴×C₃₁	—	2	1	no	—
151	#41×C₃₁	—	2	1	no	—
152	#41×C₃₁	—	2	1	no	—
153	#41×C₃₁	—	2	1	no	—
154	#41×C₃₁	—	2	1	no	—
155	#41×C₃₁	—	2	1	no	—
156	#41×C₃₁	—	2	1	no	—
157	#41×C₃₁	C₂×D₄	1	2	yes	—
158	#41×C₃₁	C₂×D₄	1	2	yes	—
159	C₂×Q₈×C₃₁	—	2	1	no	—
160	C₂×Q₈×C₃₁	—	2	1	no	—
161	C₂×Q₈×C₃₁	—	2	1	no	—
162	C₂×Q₈×C₃₁	—	2	1	no	—
163	C₂×Q₈×C₃₁	—	2	1	no	—
164	C₂×Q₈×C₃₁	—	2	1	no	—
165	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂×Q₈×C₃₁	#41	1	1	yes	characteristic
167	C₂×D₄×C₃₁	—	2	1	no	—
168	C₂×D₄×C₃₁	—	2	1	no	—
169	C₂×D₄×C₃₁	—	2	1	no	—
170	C₂×D₄×C₃₁	—	2	1	no	—
171	C₂×D₄×C₃₁	—	2	1	no	—
172	C₂×D₄×C₃₁	—	2	1	no	—
173	C₂×D₄×C₃₁	—	2	1	no	—
174	C₂×D₄×C₃₁	—	2	1	no	—
175	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂×D₄×C₃₁	#41	1	1	yes	characteristic
177	C₂²×C₄×C₃₁	—	2	1	no	—
178	C₂²×C₄×C₃₁	—	2	1	no	—
179	C₂²×C₄×C₃₁	—	2	1	no	—
180	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
181	Q₁₆×C₃₁	—	2	2	no	—
182	Q₁₆×C₃₁	—	4	2	no	—
183	Q₁₆×C₃₁	—	4	2	no	—
184	SD₁₆×C₃₁	—	2	2	no	—
185	SD₁₆×C₃₁	—	2	2	no	—
186	SD₁₆×C₃₁	—	4	2	no	—
187	SD₁₆×C₃₁	—	4	2	no	—
188	SD₁₆×C₃₁	—	4	2	no	—
189	SD₁₆×C₃₁	—	4	2	no	—
190	SD₁₆×C₃₁	—	4	1	no	—
191	SD₁₆×C₃₁	—	4	1	no	—
192	SD₁₆×C₃₁	—	4	1	no	—
193	SD₁₆×C₃₁	—	4	1	no	—
194	D₈×C₃₁	—	2	2	no	—
195	D₈×C₃₁	—	4	2	no	—
196	D₈×C₃₁	—	4	2	no	—
197	M₁₆×C₃₁	—	2	1	no	—
198	M₁₆×C₃₁	—	2	1	no	—
199	M₁₆×C₃₁	—	2	1	no	—
200	M₁₆×C₃₁	—	2	1	no	—
201	M₁₆×C₃₁	—	2	2	no	—
202	M₁₆×C₃₁	—	2	2	no	—
203	M₁₆×C₃₁	—	2	1	no	—
204	M₁₆×C₃₁	—	2	1	no	—
205	M₁₆×C₃₁	—	2	1	no	—
206	M₁₆×C₃₁	—	2	1	no	—
207	M₁₆×C₃₁	—	2	1	no	—
208	M₁₆×C₃₁	—	2	1	no	—
209	M₁₆×C₃₁	—	2	1	no	—
210	M₁₆×C₃₁	—	2	1	no	—
211	M₁₆×C₃₁	—	2	1	no	—
212	M₁₆×C₃₁	—	2	1	no	—
213	M₁₆×C₃₁	—	4	2	no	—
214	M₁₆×C₃₁	—	4	2	no	—
215	M₁₆×C₃₁	—	4	1	no	—
216	M₁₆×C₃₁	—	4	1	no	—
217	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
218	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
219	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
220	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
221	C₂×C₈×C₃₁	—	2	1	no	—
222	C₂×C₈×C₃₁	—	2	1	no	—
223	C₂×C₈×C₃₁	—	2	1	no	—
224	C₂×C₈×C₃₁	—	2	1	no	—
225	C₂×C₈×C₃₁	—	2	2	no	—
226	C₂×C₈×C₃₁	—	2	2	no	—
227	C₂×C₈×C₃₁	—	2	1	no	—
228	C₂×C₈×C₃₁	—	2	1	no	—
229	C₂×C₈×C₃₁	—	2	1	no	—
230	C₂×C₈×C₃₁	—	2	1	no	—
231	C₂×C₈×C₃₁	—	2	1	no	—
232	C₂×C₈×C₃₁	—	4	2	no	—
233	C₂×C₈×C₃₁	—	4	2	no	—
234	C₂×C₈×C₃₁	—	4	1	no	—
235	C₂×C₈×C₃₁	—	4	1	no	—
236	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
237	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
238	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
239	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
240	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
241	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
242	C₄⋊C₄×C₃₁	—	2	1	no	—
243	C₂²⋊C₄×C₃₁	—	2	1	no	—
244	C₄²×C₃₁	—	2	1	no	—
245	#24808	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
246	C₂³×C₃₁	—	2	1	no	—
247	C₂³×C₃₁	—	2	1	no	—
248	C₂³×C₃₁	—	4	1	no	—
249	C₂³×C₃₁	—	4	1	no	—
250	C₂³×C₃₁	—	4	1	no	—
251	C₂³×C₃₁	—	16	1	no	—
252	C₂³×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
253	Q₈×C₃₁	—	2	1	no	—
254	Q₈×C₃₁	—	2	1	no	—
255	Q₈×C₃₁	—	2	1	no	—
256	Q₈×C₃₁	—	4	1	no	—
257	Q₈×C₃₁	—	4	1	no	—
258	Q₈×C₃₁	—	4	1	no	—
259	Q₈×C₃₁	—	4	1	no	—
260	Q₈×C₃₁	#121	1	2	yes	—
261	D₄×C₃₁	—	2	2	no	—
262	D₄×C₃₁	—	2	2	no	—
263	D₄×C₃₁	—	2	1	no	—
264	D₄×C₃₁	—	2	1	no	—
265	D₄×C₃₁	—	2	1	no	—
266	D₄×C₃₁	—	4	1	no	—
267	D₄×C₃₁	—	4	1	no	—
268	D₄×C₃₁	—	4	1	no	—
269	D₄×C₃₁	—	4	1	no	—
270	D₄×C₃₁	—	4	1	no	—
271	D₄×C₃₁	—	4	1	no	—
272	D₄×C₃₁	#121	1	2	yes	—
273	C₂×C₄×C₃₁	—	2	1	no	—
274	C₂×C₄×C₃₁	—	2	2	no	—
275	C₂×C₄×C₃₁	—	2	2	no	—
276	C₂×C₄×C₃₁	—	2	1	no	—
277	C₂×C₄×C₃₁	—	4	1	no	—
278	C₂×C₄×C₃₁	—	4	1	no	—
279	C₂×C₄×C₃₁	—	4	1	no	—
280	C₂×C₄×C₃₁	—	4	1	no	—
281	C₂×C₄×C₃₁	—	4	1	no	—
282	C₂×C₄×C₃₁	—	4	1	no	—
283	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
285	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
287	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
288	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
289	C₈×C₃₁	—	2	2	no	—
290	C₈×C₃₁	—	2	2	no	—
291	C₈×C₃₁	—	2	2	no	—
292	C₈×C₃₁	—	2	2	no	—
293	C₈×C₃₁	—	4	1	no	—
294	C₈×C₃₁	—	4	1	no	—
295	C₈×C₃₁	—	4	1	no	—
296	C₈×C₃₁	—	4	1	no	—
297	C₈×C₃₁	—	4	1	no	—
298	C₈×C₃₁	#121	1	1	yes	characteristic
299	C₈×C₃₁	#121	1	1	yes	characteristic
300	C₈×C₃₁	#121	1	1	yes	characteristic
301	C₈×C₃₁	#121	1	1	yes	characteristic
302	#3069	C₂×C₃₁	1	1	yes	characteristic, complemented
303	#3068	C₂×C₃₁	1	1	yes	characteristic
304	C₂×#471	C₂×C₃₁	1	1	yes	characteristic, complemented
305	C₂×#471	C₂×C₃₁	1	1	yes	characteristic, complemented
306	#2955	C₂×C₃₁	1	1	yes	characteristic, complemented
307	#2916	C₂×C₃₁	1	1	yes	characteristic, complemented
308	#2070	C₂×C₃₁	1	1	yes	characteristic, complemented
309	#2069	C₂×C₃₁	1	1	yes	characteristic, complemented
310	#2049	C₂×C₃₁	1	1	yes	characteristic, complemented
311	#2048	C₂×C₃₁	1	1	yes	characteristic, complemented
312	#1977	C₂×C₃₁	1	1	yes	characteristic, complemented
313	#1953	C₂×C₃₁	1	1	yes	characteristic, complemented
314	#1903	C₂×C₃₁	1	1	yes	characteristic, complemented
315	#1902	C₂×C₃₁	1	1	yes	characteristic, complemented
316	#1813	C₂×C₃₁	1	1	yes	characteristic, complemented
317	C₂²×C₃₁	—	2	2	no	—
318	C₂²×C₃₁	—	4	1	no	—
319	C₂²×C₃₁	—	4	1	no	—
320	C₂²×C₃₁	—	8	1	no	—
321	C₂²×C₃₁	—	8	1	no	—
322	C₂²×C₃₁	—	8	1	no	—
323	C₂²×C₃₁	—	16	1	no	—
324	C₂²×C₃₁	—	16	1	no	—
325	C₂²×C₃₁	—	16	1	no	—
326	C₂²×C₃₁	#537	1	1	yes	characteristic
327	C₂²×C₃₁	#547	1	1	yes	characteristic
328	C₂²×C₃₁	#548	1	1	yes	characteristic
329	C₄×C₃₁	—	2	2	no	—
330	C₄×C₃₁	—	4	1	no	—
331	C₄×C₃₁	—	8	1	no	—
332	C₄×C₃₁	—	8	1	no	—
333	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
334	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
335	C₄×C₃₁	D₄²	1	1	yes	characteristic
336	C₄×C₃₁	#553	1	1	yes	characteristic
337	#575	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
339	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
340	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
341	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
342	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
343	#568	C₂²×C₃₁	1	1	yes	characteristic
344	#568	C₂²×C₃₁	1	1	yes	characteristic
345	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
346	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
347	#473	—	2	2	no	—
348	#472	—	2	2	no	—
349	#471	—	2	2	no	—
350	#471	—	2	2	no	—
351	#471	—	2	2	no	—
352	#471	—	2	2	no	—
353	#471	—	2	2	no	—
354	#471	—	2	2	no	—
355	#444	—	2	2	no	—
356	#443	—	2	2	no	—
357	#430	—	2	2	no	—
358	#430	C₂²×C₃₁	1	1	yes	characteristic
359	#430	C₂²×C₃₁	1	1	yes	characteristic
360	#430	C₂²×C₃₁	1	1	yes	characteristic
361	#430	C₂²×C₃₁	1	1	yes	characteristic
362	#430	C₂²×C₃₁	1	1	yes	characteristic
363	C₂×#108	—	2	2	no	—
364	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
365	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
366	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
367	#421	C₂²×C₃₁	1	1	yes	characteristic
368	#421	C₂²×C₃₁	1	1	yes	characteristic
369	#419	C₂²×C₃₁	1	1	yes	characteristic
370	#418	C₂²×C₃₁	1	1	yes	characteristic
371	#413	C₂²×C₃₁	1	1	yes	characteristic
372	#413	C₂²×C₃₁	1	1	yes	characteristic
373	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
374	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
375	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
376	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
377	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
378	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
379	#405	C₂²×C₃₁	1	1	yes	characteristic
380	#344	C₂²×C₃₁	1	1	yes	characteristic
381	#344	C₂²×C₃₁	1	1	yes	characteristic
382	#338	C₂²×C₃₁	1	1	yes	characteristic
383	#338	C₂²×C₃₁	1	1	yes	characteristic
384	C₂×C₃₁	—	2	1	no	—
385	C₂×C₃₁	—	2	1	no	—
386	C₂×C₃₁	—	2	1	no	—
387	C₂×C₃₁	—	4	2	no	—
388	C₂×C₃₁	—	8	1	no	—
389	C₂×C₃₁	—	16	1	no	—
390	C₂×C₃₁	#2416	1	1	yes	characteristic
391	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
392	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
393	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
394	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
395	#137	—	2	1	no	—
396	#137	—	2	2	no	—
397	#137	—	2	2	no	—
398	#137	—	2	2	no	—
399	#137	—	4	1	no	—
400	#137	D₄×C₃₁	1	2	yes	—
401	AGL(1,ℤ/8)	—	2	1	no	—
402	AGL(1,ℤ/8)	—	2	2	no	—
403	AGL(1,ℤ/8)	—	2	2	no	—
404	AGL(1,ℤ/8)	—	2	2	no	—
405	AGL(1,ℤ/8)	—	4	1	no	—
406	AGL(1,ℤ/8)	D₄×C₃₁	1	2	yes	—
407	#135	—	2	2	no	—
408	C₂×Q₁₆	—	2	1	no	—
409	C₂×Q₁₆	—	2	1	no	—
410	C₂×SD₁₆	—	2	1	no	—
411	C₂×SD₁₆	—	2	1	no	—
412	C₂×SD₁₆	—	2	1	no	—
413	C₂×SD₁₆	—	2	1	no	—
414	C₂×SD₁₆	—	2	1	no	—
415	C₂×SD₁₆	—	2	1	no	—
416	C₂×SD₁₆	—	2	1	no	—
417	C₂×SD₁₆	—	2	1	no	—
418	C₂×SD₁₆	—	2	1	no	—
419	C₂×SD₁₆	—	2	1	no	—
420	C₂×D₈	—	2	1	no	—
421	C₂×D₈	—	2	1	no	—
422	#131	—	2	1	no	—
423	#131	—	2	1	no	—
424	#131	—	2	2	no	—
425	#131	—	2	2	no	—
426	#131	—	2	1	no	—
427	#131	—	2	1	no	—
428	#131	—	2	1	no	—
429	#131	—	2	1	no	—
430	#131	—	4	1	no	—
431	#131	—	4	1	no	—
432	#131	D₄×C₃₁	1	2	yes	—
433	#131	D₄×C₃₁	1	2	yes	—
434	C₂×M₁₆	—	2	1	no	—
435	C₂×M₁₆	—	2	1	no	—
436	C₂×M₁₆	—	2	2	no	—
437	C₂×M₁₆	—	2	2	no	—
438	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
439	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
440	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
441	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
442	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
443	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
444	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
445	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
446	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
447	C₂²×C₈	—	2	1	no	—
448	C₂²×C₈	—	2	1	no	—
449	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
450	#117	C₂³×C₃₁	1	1	yes	characteristic
451	#108	—	2	2	no	—
452	#108	—	2	2	no	—
453	#108	—	2	2	no	—
454	#108	—	2	2	no	—
455	#108	—	2	2	no	—
456	#108	—	2	1	no	—
457	#108	—	2	1	no	—
458	#108	—	2	1	no	—
459	#108	—	2	1	no	—
460	#108	D₄×C₃₁	1	2	yes	—
461	#108	D₄×C₃₁	1	2	yes	—
462	#104	—	2	2	no	—
463	#104	—	2	2	no	—
464	#103	—	4	1	no	—
465	#102	—	4	1	no	—
466	#101	—	2	1	no	—
467	#101	—	2	1	no	—
468	#101	—	2	2	no	—
469	#101	—	2	2	no	—
470	#101	—	2	2	no	—
471	#100	—	2	1	no	—
472	#100	—	2	1	no	—
473	#100	—	2	2	no	—
474	#100	—	2	2	no	—
475	#100	—	2	2	no	—
476	#97	—	2	1	no	—
477	C₄×C₈	—	2	1	no	—
478	C₃₁	#24808	1	1	yes	characteristic, direct product factor, Sylow
479	C₂⁴	—	2	1	no	—
480	#41	—	2	1	no	—
481	#41	—	2	1	no	—
482	#41	—	2	1	no	—
483	#41	—	2	1	no	—
484	#41	—	2	1	no	—
485	#41	—	2	1	no	—
486	#41	C₂×D₄×C₃₁	1	2	yes	—
487	#41	C₂×D₄×C₃₁	1	2	yes	—
488	C₂×Q₈	—	2	1	no	—
489	C₂×Q₈	—	2	1	no	—
490	C₂×Q₈	—	2	1	no	—
491	C₂×Q₈	—	2	1	no	—
492	C₂×Q₈	—	2	1	no	—
493	C₂×Q₈	—	2	1	no	—
494	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
495	C₂×Q₈	#41×C₃₁	1	1	yes	characteristic
496	C₂×D₄	—	2	1	no	—
497	C₂×D₄	—	2	1	no	—
498	C₂×D₄	—	2	1	no	—
499	C₂×D₄	—	2	1	no	—
500	C₂×D₄	—	2	1	no	—
501	C₂×D₄	—	2	1	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	2	1	no	—
504	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
505	C₂×D₄	#41×C₃₁	1	1	yes	characteristic
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
510	Q₁₆	—	2	2	no	—
511	Q₁₆	—	4	2	no	—
512	Q₁₆	—	4	2	no	—
513	SD₁₆	—	2	2	no	—
514	SD₁₆	—	2	2	no	—
515	SD₁₆	—	4	2	no	—
516	SD₁₆	—	4	2	no	—
517	SD₁₆	—	4	2	no	—
518	SD₁₆	—	4	2	no	—
519	SD₁₆	—	4	1	no	—
520	SD₁₆	—	4	1	no	—
521	SD₁₆	—	4	1	no	—
522	SD₁₆	—	4	1	no	—
523	D₈	—	2	2	no	—
524	D₈	—	4	2	no	—
525	D₈	—	4	2	no	—
526	M₁₆	—	2	1	no	—
527	M₁₆	—	2	1	no	—
528	M₁₆	—	2	1	no	—
529	M₁₆	—	2	1	no	—
530	M₁₆	—	2	2	no	—
531	M₁₆	—	2	2	no	—
532	M₁₆	—	2	1	no	—
533	M₁₆	—	2	1	no	—
534	M₁₆	—	2	1	no	—
535	M₁₆	—	2	1	no	—
536	M₁₆	—	2	1	no	—
537	M₁₆	—	2	1	no	—
538	M₁₆	—	2	1	no	—
539	M₁₆	—	2	1	no	—
540	M₁₆	—	2	1	no	—
541	M₁₆	—	2	1	no	—
542	M₁₆	—	4	2	no	—
543	M₁₆	—	4	2	no	—
544	M₁₆	—	4	1	no	—
545	M₁₆	—	4	1	no	—
546	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
547	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
548	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
549	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
550	C₂×C₈	—	2	1	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	1	no	—
553	C₂×C₈	—	2	1	no	—
554	C₂×C₈	—	2	2	no	—
555	C₂×C₈	—	2	2	no	—
556	C₂×C₈	—	2	1	no	—
557	C₂×C₈	—	2	1	no	—
558	C₂×C₈	—	2	1	no	—
559	C₂×C₈	—	2	1	no	—
560	C₂×C₈	—	2	1	no	—
561	C₂×C₈	—	4	2	no	—
562	C₂×C₈	—	4	2	no	—
563	C₂×C₈	—	4	1	no	—
564	C₂×C₈	—	4	1	no	—
565	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
566	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
567	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
568	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
569	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
570	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
571	C₄⋊C₄	—	2	1	no	—
572	C₂²⋊C₄	—	2	1	no	—
573	C₄²	—	2	1	no	—
574	C₂³	—	2	1	no	—
575	C₂³	—	2	1	no	—
576	C₂³	—	4	1	no	—
577	C₂³	—	4	1	no	—
578	C₂³	—	4	1	no	—
579	C₂³	—	16	1	no	—
580	C₂³	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
581	Q₈	—	2	1	no	—
582	Q₈	—	2	1	no	—
583	Q₈	—	2	1	no	—
584	Q₈	—	4	1	no	—
585	Q₈	—	4	1	no	—
586	Q₈	—	4	1	no	—
587	Q₈	—	4	1	no	—
588	Q₈	C₃₁×#121	1	2	yes	—
589	D₄	—	2	2	no	—
590	D₄	—	2	2	no	—
591	D₄	—	2	1	no	—
592	D₄	—	2	1	no	—
593	D₄	—	2	1	no	—
594	D₄	—	4	1	no	—
595	D₄	—	4	1	no	—
596	D₄	—	4	1	no	—
597	D₄	—	4	1	no	—
598	D₄	—	4	1	no	—
599	D₄	—	4	1	no	—
600	D₄	C₃₁×#121	1	2	yes	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
612	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
613	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
614	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
615	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
616	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
617	C₈	—	2	2	no	—
618	C₈	—	2	2	no	—
619	C₈	—	2	2	no	—
620	C₈	—	2	2	no	—
621	C₈	—	4	1	no	—
622	C₈	—	4	1	no	—
623	C₈	—	4	1	no	—
624	C₈	—	4	1	no	—
625	C₈	—	4	1	no	—
626	C₈	C₃₁×#121	1	1	yes	characteristic
627	C₈	C₃₁×#121	1	1	yes	characteristic
628	C₈	C₃₁×#121	1	1	yes	characteristic
629	C₈	C₃₁×#121	1	1	yes	characteristic
630	C₂²	—	2	2	no	—
631	C₂²	—	4	1	no	—
632	C₂²	—	4	1	no	—
633	C₂²	—	8	1	no	—
634	C₂²	—	8	1	no	—
635	C₂²	—	8	1	no	—
636	C₂²	—	16	1	no	—
637	C₂²	—	16	1	no	—
638	C₂²	—	16	1	no	—
639	C₂²	C₃₁×#537	1	1	yes	characteristic
640	C₂²	C₃₁×#547	1	1	yes	characteristic
641	C₂²	C₃₁×#548	1	1	yes	characteristic
642	C₄	—	2	2	no	—
643	C₄	—	4	1	no	—
644	C₄	—	8	1	no	—
645	C₄	—	8	1	no	—
646	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
647	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
648	C₄	D₄²×C₃₁	1	1	yes	characteristic
649	C₄	C₃₁×#553	1	1	yes	characteristic
650	C₂	—	2	1	no	—
651	C₂	—	2	1	no	—
652	C₂	—	2	1	no	—
653	C₂	—	4	2	no	—
654	C₂	—	8	1	no	—
655	C₂	—	16	1	no	—
656	C₂	C₃₁×#2416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#2416
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3068	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
3. C₂×C₃₁×#471	C₂	semidirect
4. C₂×C₃₁×#471	C₂	semidirect
5. C₃₁×#2955	C₂	semidirect
6. C₃₁×#2916	C₂	semidirect
7. C₃₁×#2070	C₂	semidirect
8. C₃₁×#2069	C₂	semidirect
9. C₃₁×#2049	C₂	semidirect
10. C₃₁×#2048	C₂	semidirect
11. C₃₁×#1977	C₂	semidirect
12. C₃₁×#1953	C₂	semidirect
13. C₃₁×#1903	C₂	semidirect
14. C₃₁×#1902	C₂	semidirect
15. C₃₁×#1813	C₂	semidirect
245. #24808	C₃₁	direct (N × H)
302. #3069	C₂×C₃₁	semidirect
304. C₂×#471	C₂×C₃₁	semidirect
305. C₂×#471	C₂×C₃₁	semidirect
306. #2955	C₂×C₃₁	semidirect
307. #2916	C₂×C₃₁	semidirect
308. #2070	C₂×C₃₁	semidirect
309. #2069	C₂×C₃₁	semidirect
310. #2049	C₂×C₃₁	semidirect
311. #2048	C₂×C₃₁	semidirect
312. #1977	C₂×C₃₁	semidirect
313. #1953	C₂×C₃₁	semidirect
314. #1903	C₂×C₃₁	semidirect
315. #1902	C₂×C₃₁	semidirect
316. #1813	C₂×C₃₁	semidirect
478. C₃₁	#24808	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×#3068	2
C₂×C₃₁×#471	2
C₃₁×#2955	2
C₃₁×#2916	2
C₃₁×#2070	2
C₃₁×#2069	2
C₃₁×#2049	2
C₃₁×#2048	2
C₃₁×#1977	2
C₃₁×#1953	2
C₃₁×#1903	2
C₃₁×#1902	2
C₃₁×#1813	2
#24808	31

Subgroups — C31×#24808 #21481567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24808 #21481567