Subgroups — C₃₁×#60725 #21517484

Order 7936.

1962 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3512	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3508	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3180	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3176	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3174	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3171	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3166	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3135	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3062	C₂	1	4	yes	maximal, complemented
12	C₃₁×#2946	C₂	1	4	yes	maximal, complemented
13	C₃₁×#2875	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	—
15	C₂×C₃₁×#131	C₂²	1	2	yes	—
16	C₃₁×#559	—	2	2	no	—
17	C₃₁×#553	—	2	4	no	—
18	C₃₁×#553	—	2	2	no	—
19	C₃₁×#552	—	2	2	no	—
20	C₃₁×#550	—	2	2	no	—
21	C₃₁×#546	—	2	4	no	—
22	C₃₁×#541	C₂²	1	2	yes	complemented
23	C₃₁×#540	C₂²	1	4	yes	—
24	C₃₁×#540	C₂²	1	2	yes	complemented
25	C₃₁×#537	C₂²	1	2	yes	—
26	C₃₁×#535	C₂²	1	2	yes	—
27	C₃₁×#533	C₂²	1	1	yes	characteristic
28	C₃₁×#532	C₂²	1	1	yes	characteristic
29	C₃₁×#532	C₂²	1	4	yes	—
30	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
31	C₃₁×#529	C₂²	1	4	yes	complemented
32	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
33	C₃₁×#520	C₂²	1	2	yes	—
34	C₃₁×#518	C₂²	1	2	yes	—
35	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#501	C₂²	1	4	yes	complemented
37	C₃₁×#499	C₂²	1	4	yes	complemented
38	C₃₁×#496	C₂²	1	4	yes	—
39	C₃₁×#493	C₂²	1	2	yes	—
40	C₃₁×#492	C₂²	1	2	yes	—
41	C₃₁×#491	C₂²	1	2	yes	complemented
42	C₃₁×#490	C₂²	1	2	yes	—
43	C₃₁×#490	C₂²	1	4	yes	—
44	C₃₁×#490	C₂²	1	2	yes	—
45	C₃₁×#490	C₂²	1	2	yes	complemented
46	C₃₁×#489	C₂²	1	2	yes	complemented
47	C₃₁×#489	C₂²	1	2	yes	complemented
48	C₃₁×#469	C₂²	1	8	yes	complemented
49	C₃₁×#468	C₂²	1	8	yes	complemented
50	C₃₁×#466	C₂²	1	8	yes	complemented
51	C₃₁×#465	C₂²	1	8	yes	complemented
52	C₃₁×#442	C₂²	1	8	yes	complemented
53	C₃₁×#440	C₂²	1	8	yes	complemented
54	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	complemented
55	C₄×D₈×C₃₁	C₂²	1	8	yes	complemented
56	C₃₁×#428	C₂²	1	4	yes	—
57	C₃₁×#416	C₂²	1	4	yes	—
58	C₂×C₃₁×#102	C₂²	1	4	yes	—
59	C₃₁×#405	C₂²	1	2	yes	—
60	C₃₁×#405	C₂²	1	4	yes	complemented
61	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
62	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
63	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
64	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
65	C₂×#41×C₃₁	—	2	4	no	—
66	C₂×#41×C₃₁	C₂³	1	2	yes	—
67	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₃₁	—	2	2	no	—
69	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
70	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
71	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
72	C₂×D₈×C₃₁	C₂³	1	8	yes	—
73	C₃₁×#131	—	2	8	no	—
74	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
75	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
76	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
77	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
78	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
80	C₃₁×#127	—	2	2	no	—
81	C₃₁×#127	—	2	2	no	—
82	C₃₁×#127	C₂³	1	2	yes	—
83	C₃₁×#127	C₂³	1	2	yes	—
84	C₃₁×#127	C₂³	1	1	yes	characteristic
85	C₃₁×#126	—	2	4	no	—
86	C₃₁×#126	—	2	4	no	—
87	C₃₁×#125	—	2	4	no	—
88	C₃₁×#125	—	2	2	no	—
89	C₃₁×#125	C₂³	1	2	yes	—
90	C₃₁×#125	C₂³	1	2	yes	—
91	C₃₁×#125	C₂³	1	2	yes	—
92	C₃₁×#124	—	2	2	no	—
93	C₃₁×#124	—	2	4	no	—
94	C₃₁×#124	C₂³	1	2	yes	—
95	C₃₁×#123	—	2	4	no	—
96	C₃₁×#123	—	2	8	no	—
97	C₃₁×#123	—	2	4	no	—
98	C₃₁×C₂²⋊Q₈	—	2	2	no	—
99	C₃₁×C₂²⋊Q₈	—	2	2	no	—
100	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
101	C₃₁×#121	—	2	2	no	—
102	C₃₁×#121	—	2	4	no	—
103	C₃₁×#121	—	2	4	no	—
104	C₃₁×#121	—	2	2	no	—
105	C₃₁×#121	—	2	8	no	—
106	C₃₁×#121	—	2	4	no	—
107	C₃₁×#121	C₂³	1	8	yes	—
108	C₃₁×#121	C₂³	1	4	yes	—
109	C₃₁×#120	—	2	4	no	—
110	C₄×Q₈×C₃₁	—	2	2	no	—
111	C₄×Q₈×C₃₁	D₄	1	2	yes	—
112	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
113	C₄×D₄×C₃₁	—	2	4	no	—
114	C₄×D₄×C₃₁	—	2	4	no	—
115	C₄×D₄×C₃₁	—	2	4	no	—
116	C₄×D₄×C₃₁	—	2	4	no	—
117	C₄×D₄×C₃₁	—	2	2	no	—
118	C₄×D₄×C₃₁	D₄	1	4	yes	—
119	C₄×D₄×C₃₁	D₄	1	2	yes	—
120	C₄×D₄×C₃₁	C₂³	1	4	yes	—
121	C₄×D₄×C₃₁	C₂³	1	8	yes	—
122	C₃₁×#117	—	2	2	no	—
123	C₃₁×#117	—	2	4	no	—
124	C₃₁×#117	C₂³	1	2	yes	—
125	C₃₁×#117	C₂³	1	1	yes	characteristic
126	C₃₁×#117	C₂³	1	2	yes	—
127	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
130	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
131	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
133	C₃₁×#107	C₂³	1	8	yes	—
134	C₃₁×#106	C₂³	1	8	yes	—
135	C₃₁×#103	C₂³	1	4	yes	—
136	C₃₁×#103	C₂³	1	4	yes	—
137	C₃₁×#102	C₂³	1	8	yes	—
138	C₃₁×#102	C₂³	1	8	yes	—
139	C₃₁×#102	C₂³	1	4	yes	—
140	C₃₁×#102	C₂³	1	4	yes	—
141	C₃₁×#97	C₂³	1	1	yes	characteristic
142	C₃₁×#97	C₂³	1	1	yes	characteristic
143	C₃₁×#97	C₂³	1	2	yes	—
144	C₃₁×#97	C₂³	1	2	yes	—
145	C₃₁×#97	C₂³	1	4	yes	—
146	C₄×C₈×C₃₁	C₂³	1	2	yes	—
147	C₄×C₈×C₃₁	C₂³	1	4	yes	—
148	C₂⁴×C₃₁	—	2	2	no	—
149	#41×C₃₁	—	2	4	no	—
150	#41×C₃₁	—	2	8	no	—
151	#41×C₃₁	—	8	8	no	—
152	C₂×Q₈×C₃₁	—	2	2	no	—
153	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
155	C₂×D₄×C₃₁	—	2	2	no	—
156	C₂×D₄×C₃₁	—	2	4	no	—
157	C₂×D₄×C₃₁	—	2	8	no	—
158	C₂×D₄×C₃₁	—	2	2	no	—
159	C₂×D₄×C₃₁	—	2	4	no	—
160	C₂×D₄×C₃₁	—	2	2	no	—
161	C₂×D₄×C₃₁	—	2	4	no	—
162	C₂×D₄×C₃₁	—	2	8	no	—
163	C₂×D₄×C₃₁	—	8	4	no	—
164	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
165	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
167	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₃₁	—	2	2	no	—
169	C₂²×C₄×C₃₁	—	2	2	no	—
170	C₂²×C₄×C₃₁	—	2	8	no	—
171	C₂²×C₄×C₃₁	—	2	4	no	—
172	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
174	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
176	SD₁₆×C₃₁	—	2	16	no	—
177	D₈×C₃₁	—	2	16	no	—
178	M₁₆×C₃₁	—	2	8	no	—
179	M₁₆×C₃₁	#41	1	8	yes	complemented
180	C₂×C₈×C₃₁	—	2	4	no	—
181	C₂×C₈×C₃₁	—	2	8	no	—
182	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
183	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
184	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	—	2	4	no	—
187	C₄⋊C₄×C₃₁	—	2	4	no	—
188	C₄⋊C₄×C₃₁	—	2	4	no	—
189	C₄⋊C₄×C₃₁	—	2	4	no	—
190	C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₄⋊C₄×C₃₁	—	2	2	no	—
192	C₄⋊C₄×C₃₁	—	2	4	no	—
193	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
194	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
195	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
196	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
197	C₂²⋊C₄×C₃₁	—	2	2	no	—
198	C₂²⋊C₄×C₃₁	—	2	4	no	—
199	C₂²⋊C₄×C₃₁	—	2	8	no	—
200	C₂²⋊C₄×C₃₁	—	2	2	no	—
201	C₂²⋊C₄×C₃₁	—	2	4	no	—
202	C₂²⋊C₄×C₃₁	—	2	4	no	—
203	C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂²⋊C₄×C₃₁	—	2	8	no	—
205	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
206	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
207	C₄²×C₃₁	—	2	4	no	—
208	C₄²×C₃₁	—	2	2	no	—
209	C₄²×C₃₁	C₂×D₄	1	4	yes	—
210	C₄²×C₃₁	C₂×D₄	1	2	yes	—
211	C₄²×C₃₁	C₂⁴	1	2	yes	—
212	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
213	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
214	#60725	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂³×C₃₁	—	2	2	no	—
216	C₂³×C₃₁	—	2	4	no	—
217	C₂³×C₃₁	—	4	4	no	—
218	C₂³×C₃₁	—	8	4	no	—
219	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
220	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
221	Q₈×C₃₁	—	2	2	no	—
222	Q₈×C₃₁	—	2	4	no	—
223	Q₈×C₃₁	—	4	4	no	—
224	D₄×C₃₁	—	2	4	no	—
225	D₄×C₃₁	—	2	2	no	—
226	D₄×C₃₁	—	2	4	no	—
227	D₄×C₃₁	—	2	8	no	—
228	D₄×C₃₁	—	4	8	no	—
229	D₄×C₃₁	—	4	4	no	—
230	D₄×C₃₁	—	4	4	no	—
231	D₄×C₃₁	—	8	2	no	—
232	D₄×C₃₁	—	8	4	no	—
233	D₄×C₃₁	—	8	4	no	—
234	D₄×C₃₁	—	8	8	no	—
235	C₂×C₄×C₃₁	—	2	4	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	4	no	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	4	2	no	—
240	C₂×C₄×C₃₁	—	4	4	no	—
241	C₂×C₄×C₃₁	—	4	4	no	—
242	C₂×C₄×C₃₁	—	8	4	no	—
243	C₂×C₄×C₃₁	—	8	8	no	—
244	C₂×C₄×C₃₁	—	8	2	no	—
245	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
246	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
247	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
248	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
249	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
251	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
252	C₈×C₃₁	—	2	2	no	—
253	C₈×C₃₁	—	2	2	no	—
254	C₈×C₃₁	C₂×#41	1	8	yes	—
255	#3512	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#3508	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#3180	C₂×C₃₁	1	4	yes	complemented
258	#3176	C₂×C₃₁	1	4	yes	complemented
259	#3174	C₂×C₃₁	1	4	yes	complemented
260	#3171	C₂×C₃₁	1	4	yes	complemented
261	#3166	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#3151	C₂×C₃₁	1	1	yes	characteristic, complemented
263	#3135	C₂×C₃₁	1	1	yes	characteristic, complemented
264	C₂×#490	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#3062	C₂×C₃₁	1	4	yes	complemented
266	#2946	C₂×C₃₁	1	4	yes	complemented
267	#2875	C₂×C₃₁	1	1	yes	characteristic, complemented
268	C₂²×C₃₁	—	2	2	no	—
269	C₂²×C₃₁	—	2	4	no	—
270	C₂²×C₃₁	—	2	2	no	—
271	C₂²×C₃₁	—	4	2	no	—
272	C₂²×C₃₁	—	4	8	no	—
273	C₂²×C₃₁	—	4	2	no	—
274	C₂²×C₃₁	—	4	2	no	—
275	C₂²×C₃₁	—	4	4	no	—
276	C₂²×C₃₁	—	16	4	no	—
277	C₂²×C₃₁	#532	1	2	yes	—
278	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
279	C₄×C₃₁	—	2	2	no	—
280	C₄×C₃₁	—	2	4	no	—
281	C₄×C₃₁	—	2	2	no	—
282	C₄×C₃₁	—	2	2	no	—
283	C₄×C₃₁	—	4	4	no	—
284	C₄×C₃₁	—	4	4	no	—
285	C₄×C₃₁	—	4	2	no	—
286	C₄×C₃₁	#532	1	2	yes	—
287	C₄×C₃₁	C₂²×#41	1	2	yes	—
288	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	—
289	C₂×#131	C₂²×C₃₁	1	2	yes	—
290	#559	—	2	2	no	—
291	#553	—	2	4	no	—
292	#553	—	2	2	no	—
293	#552	—	2	2	no	—
294	#550	—	2	2	no	—
295	#546	—	2	4	no	—
296	#541	C₂²×C₃₁	1	2	yes	complemented
297	#540	C₂²×C₃₁	1	4	yes	—
298	#540	C₂²×C₃₁	1	2	yes	complemented
299	#537	C₂²×C₃₁	1	2	yes	—
300	#535	C₂²×C₃₁	1	2	yes	—
301	#533	C₂²×C₃₁	1	1	yes	characteristic
302	#532	C₂²×C₃₁	1	1	yes	characteristic
303	#532	C₂²×C₃₁	1	4	yes	—
304	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
305	#529	C₂²×C₃₁	1	4	yes	complemented
306	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
307	#520	C₂²×C₃₁	1	2	yes	—
308	#518	C₂²×C₃₁	1	2	yes	—
309	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
310	#501	C₂²×C₃₁	1	4	yes	complemented
311	#499	C₂²×C₃₁	1	4	yes	complemented
312	#496	C₂²×C₃₁	1	4	yes	—
313	#493	C₂²×C₃₁	1	2	yes	—
314	#492	C₂²×C₃₁	1	2	yes	—
315	#491	C₂²×C₃₁	1	2	yes	complemented
316	#490	C₂²×C₃₁	1	2	yes	—
317	#490	C₂²×C₃₁	1	4	yes	—
318	#490	C₂²×C₃₁	1	2	yes	—
319	#490	C₂²×C₃₁	1	2	yes	complemented
320	#489	C₂²×C₃₁	1	2	yes	complemented
321	#489	C₂²×C₃₁	1	2	yes	complemented
322	#469	C₂²×C₃₁	1	8	yes	complemented
323	#468	C₂²×C₃₁	1	8	yes	complemented
324	#466	C₂²×C₃₁	1	8	yes	complemented
325	#465	C₂²×C₃₁	1	8	yes	complemented
326	#442	C₂²×C₃₁	1	8	yes	complemented
327	#440	C₂²×C₃₁	1	8	yes	complemented
328	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	complemented
329	C₄×D₈	C₂²×C₃₁	1	8	yes	complemented
330	#428	C₂²×C₃₁	1	4	yes	—
331	#416	C₂²×C₃₁	1	4	yes	—
332	C₂×#102	C₂²×C₃₁	1	4	yes	—
333	#405	C₂²×C₃₁	1	2	yes	—
334	#405	C₂²×C₃₁	1	4	yes	complemented
335	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
336	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
337	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
339	C₂×C₃₁	—	2	2	no	—
340	C₂×C₃₁	—	4	2	no	—
341	C₂×C₃₁	—	8	2	no	—
342	C₂×C₃₁	—	8	4	no	—
343	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
344	C₂×C₃₁	#3585	1	2	yes	—
345	C₂×#41	—	2	4	no	—
346	C₂×#41	C₂³×C₃₁	1	2	yes	—
347	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
348	C₂²×D₄	—	2	2	no	—
349	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
350	AGL(1,ℤ/8)	—	2	16	no	—
351	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
352	C₂×D₈	C₂³×C₃₁	1	8	yes	—
353	#131	—	2	8	no	—
354	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
355	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
356	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
357	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
358	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
359	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
360	#127	—	2	2	no	—
361	#127	—	2	2	no	—
362	#127	C₂³×C₃₁	1	2	yes	—
363	#127	C₂³×C₃₁	1	2	yes	—
364	#127	C₂³×C₃₁	1	1	yes	characteristic
365	#126	—	2	4	no	—
366	#126	—	2	4	no	—
367	#125	—	2	4	no	—
368	#125	—	2	2	no	—
369	#125	C₂³×C₃₁	1	2	yes	—
370	#125	C₂³×C₃₁	1	2	yes	—
371	#125	C₂³×C₃₁	1	2	yes	—
372	#124	—	2	2	no	—
373	#124	—	2	4	no	—
374	#124	C₂³×C₃₁	1	2	yes	—
375	#123	—	2	4	no	—
376	#123	—	2	8	no	—
377	#123	—	2	4	no	—
378	C₂²⋊Q₈	—	2	2	no	—
379	C₂²⋊Q₈	—	2	2	no	—
380	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
381	#121	—	2	2	no	—
382	#121	—	2	4	no	—
383	#121	—	2	4	no	—
384	#121	—	2	2	no	—
385	#121	—	2	8	no	—
386	#121	—	2	4	no	—
387	#121	C₂³×C₃₁	1	4	yes	—
388	#121	C₂³×C₃₁	1	8	yes	—
389	#120	—	2	4	no	—
390	C₄×Q₈	—	2	2	no	—
391	C₄×Q₈	D₄×C₃₁	1	2	yes	—
392	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
393	C₄×D₄	—	2	4	no	—
394	C₄×D₄	—	2	4	no	—
395	C₄×D₄	—	2	4	no	—
396	C₄×D₄	—	2	4	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	D₄×C₃₁	1	4	yes	—
399	C₄×D₄	D₄×C₃₁	1	2	yes	—
400	C₄×D₄	C₂³×C₃₁	1	8	yes	—
401	C₄×D₄	C₂³×C₃₁	1	4	yes	—
402	#117	—	2	2	no	—
403	#117	—	2	4	no	—
404	#117	C₂³×C₃₁	1	2	yes	—
405	#117	C₂³×C₃₁	1	1	yes	characteristic
406	#117	C₂³×C₃₁	1	2	yes	—
407	C₂×C₄⋊C₄	—	2	2	no	—
408	C₂×C₄⋊C₄	—	2	2	no	—
409	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
410	C₂×C₂²⋊C₄	—	2	4	no	—
411	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
412	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
413	#107	C₂³×C₃₁	1	8	yes	—
414	#106	C₂³×C₃₁	1	8	yes	—
415	#103	C₂³×C₃₁	1	4	yes	—
416	#103	C₂³×C₃₁	1	4	yes	—
417	#102	C₂³×C₃₁	1	8	yes	—
418	#102	C₂³×C₃₁	1	8	yes	—
419	#102	C₂³×C₃₁	1	4	yes	—
420	#102	C₂³×C₃₁	1	4	yes	—
421	#97	C₂³×C₃₁	1	1	yes	characteristic
422	#97	C₂³×C₃₁	1	1	yes	characteristic
423	#97	C₂³×C₃₁	1	2	yes	—
424	#97	C₂³×C₃₁	1	2	yes	—
425	#97	C₂³×C₃₁	1	4	yes	—
426	C₄×C₈	C₂³×C₃₁	1	2	yes	—
427	C₄×C₈	C₂³×C₃₁	1	4	yes	—
428	C₃₁	#60725	1	1	yes	characteristic, direct product factor, Sylow
429	C₂⁴	—	2	2	no	—
430	#41	—	2	4	no	—
431	#41	—	2	8	no	—
432	#41	—	8	8	no	—
433	C₂×Q₈	—	2	2	no	—
434	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
435	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	2	4	no	—
438	C₂×D₄	—	2	8	no	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	8	no	—
444	C₂×D₄	—	8	4	no	—
445	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
446	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
447	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
448	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	2	no	—
451	C₂²×C₄	—	2	8	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
454	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
455	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
456	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
457	SD₁₆	—	2	16	no	—
458	D₈	—	2	16	no	—
459	M₁₆	—	2	8	no	—
460	M₁₆	#41×C₃₁	1	8	yes	complemented
461	C₂×C₈	—	2	4	no	—
462	C₂×C₈	—	2	8	no	—
463	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
464	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
465	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	4	no	—
468	C₄⋊C₄	—	2	4	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	4	no	—
474	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
475	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
476	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
477	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
478	C₂²⋊C₄	—	2	2	no	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	—	2	8	no	—
481	C₂²⋊C₄	—	2	2	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	2	no	—
485	C₂²⋊C₄	—	2	8	no	—
486	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
487	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
488	C₄²	—	2	4	no	—
489	C₄²	—	2	2	no	—
490	C₄²	C₂×D₄×C₃₁	1	4	yes	—
491	C₄²	C₂×D₄×C₃₁	1	2	yes	—
492	C₄²	C₂⁴×C₃₁	1	2	yes	—
493	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
494	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
495	C₂³	—	2	2	no	—
496	C₂³	—	2	4	no	—
497	C₂³	—	4	4	no	—
498	C₂³	—	8	4	no	—
499	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
500	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
501	Q₈	—	2	2	no	—
502	Q₈	—	2	4	no	—
503	Q₈	—	4	4	no	—
504	D₄	—	2	4	no	—
505	D₄	—	2	2	no	—
506	D₄	—	2	4	no	—
507	D₄	—	2	8	no	—
508	D₄	—	4	8	no	—
509	D₄	—	4	4	no	—
510	D₄	—	4	4	no	—
511	D₄	—	8	2	no	—
512	D₄	—	8	4	no	—
513	D₄	—	8	4	no	—
514	D₄	—	8	8	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	4	no	—
521	C₂×C₄	—	4	4	no	—
522	C₂×C₄	—	8	4	no	—
523	C₂×C₄	—	8	8	no	—
524	C₂×C₄	—	8	2	no	—
525	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
526	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
527	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
528	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
529	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
530	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
531	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
532	C₈	—	2	2	no	—
533	C₈	—	2	2	no	—
534	C₈	C₂×#41×C₃₁	1	8	yes	—
535	C₂²	—	2	2	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	4	no	—
538	C₂²	—	4	2	no	—
539	C₂²	—	4	4	no	—
540	C₂²	—	4	2	no	—
541	C₂²	—	4	8	no	—
542	C₂²	—	4	2	no	—
543	C₂²	—	16	4	no	—
544	C₂²	C₃₁×#532	1	2	yes	—
545	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
546	C₄	—	2	2	no	—
547	C₄	—	2	2	no	—
548	C₄	—	2	4	no	—
549	C₄	—	2	2	no	—
550	C₄	—	4	2	no	—
551	C₄	—	4	4	no	—
552	C₄	—	4	4	no	—
553	C₄	C₃₁×#532	1	2	yes	—
554	C₄	C₂²×#41×C₃₁	1	2	yes	—
555	C₂	—	2	2	no	—
556	C₂	—	4	2	no	—
557	C₂	—	8	4	no	—
558	C₂	—	8	2	no	—
559	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
560	C₂	C₃₁×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3512	C₂	semidirect
2. C₃₁×#3508	C₂	semidirect
3. C₃₁×#3180	C₂	semidirect
4. C₃₁×#3176	C₂	semidirect
5. C₃₁×#3174	C₂	semidirect
6. C₃₁×#3171	C₂	semidirect
7. C₃₁×#3166	C₂	semidirect
8. C₃₁×#3151	C₂	semidirect
9. C₃₁×#3135	C₂	semidirect
10. C₂×C₃₁×#490	C₂	semidirect
11. C₃₁×#3062	C₂	semidirect
12. C₃₁×#2946	C₂	semidirect
13. C₃₁×#2875	C₂	semidirect
22. C₃₁×#541	C₂²	semidirect
24. C₃₁×#540	C₂²	semidirect
31. C₃₁×#529	C₂²	semidirect
36. C₃₁×#501	C₂²	semidirect
37. C₃₁×#499	C₂²	semidirect
41. C₃₁×#491	C₂²	semidirect
45. C₃₁×#490	C₂²	semidirect
46. C₃₁×#489	C₂²	semidirect
47. C₃₁×#489	C₂²	semidirect
48. C₃₁×#469	C₂²	semidirect
49. C₃₁×#468	C₂²	semidirect
50. C₃₁×#466	C₂²	semidirect
51. C₃₁×#465	C₂²	semidirect
52. C₃₁×#442	C₂²	semidirect
53. C₃₁×#440	C₂²	semidirect
54. C₄×SD₁₆×C₃₁	C₂²	semidirect
55. C₄×D₈×C₃₁	C₂²	semidirect
60. C₃₁×#405	C₂²	semidirect
62. C₄×M₁₆×C₃₁	C₂²	semidirect
64. C₂×C₃₁×#97	C₂²	semidirect
179. M₁₆×C₃₁	#41	semidirect
214. #60725	C₃₁	direct (N × H)
255. #3512	C₂×C₃₁	semidirect
256. #3508	C₂×C₃₁	semidirect
257. #3180	C₂×C₃₁	semidirect
258. #3176	C₂×C₃₁	semidirect
259. #3174	C₂×C₃₁	semidirect
260. #3171	C₂×C₃₁	semidirect
261. #3166	C₂×C₃₁	semidirect
262. #3151	C₂×C₃₁	semidirect
263. #3135	C₂×C₃₁	semidirect
264. C₂×#490	C₂×C₃₁	semidirect
265. #3062	C₂×C₃₁	semidirect
266. #2946	C₂×C₃₁	semidirect
267. #2875	C₂×C₃₁	semidirect
296. #541	C₂²×C₃₁	semidirect
298. #540	C₂²×C₃₁	semidirect
305. #529	C₂²×C₃₁	semidirect
310. #501	C₂²×C₃₁	semidirect
311. #499	C₂²×C₃₁	semidirect
315. #491	C₂²×C₃₁	semidirect
319. #490	C₂²×C₃₁	semidirect
320. #489	C₂²×C₃₁	semidirect
321. #489	C₂²×C₃₁	semidirect
322. #469	C₂²×C₃₁	semidirect
323. #468	C₂²×C₃₁	semidirect
324. #466	C₂²×C₃₁	semidirect
325. #465	C₂²×C₃₁	semidirect
326. #442	C₂²×C₃₁	semidirect
327. #440	C₂²×C₃₁	semidirect
328. C₄×SD₁₆	C₂²×C₃₁	semidirect
329. C₄×D₈	C₂²×C₃₁	semidirect
334. #405	C₂²×C₃₁	semidirect
336. C₄×M₁₆	C₂²×C₃₁	semidirect
338. C₂×#97	C₂²×C₃₁	semidirect
428. C₃₁	#60725	direct (N × H)
460. M₁₆	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3512	2
C₃₁×#3508	2
C₃₁×#3180	2
C₃₁×#3176	2
C₃₁×#3174	2
C₃₁×#3171	2
C₃₁×#3166	2
C₃₁×#3151	2
C₃₁×#3135	2
C₂×C₃₁×#490	2
C₃₁×#3062	2
C₃₁×#2946	2
C₃₁×#2875	2
#60725	31

Subgroups — C31×#60725 #21517484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60725 #21517484