Subgroups — C₃₁×#60726 #21517485

Order 7936.

1842 conjugacy classes of subgroups.

The subgroup lattice has 542 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
2	C₃₁×#3509	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3181	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3177	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3174	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3172	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3166	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3152	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3135	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3063	C₂	1	4	yes	maximal, complemented
12	C₃₁×#2947	C₂	1	4	yes	maximal, complemented
13	C₃₁×#2875	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₃₁×#137	C₂²	1	4	yes	—
15	C₂×C₃₁×#131	C₂²	1	2	yes	—
16	C₃₁×#557	—	2	2	no	—
17	C₃₁×#556	—	2	2	no	—
18	C₃₁×#553	—	2	4	no	—
19	C₃₁×#553	—	2	2	no	—
20	C₃₁×#552	—	2	2	no	—
21	C₃₁×#552	—	2	4	no	—
22	C₃₁×#541	C₂²	1	4	yes	—
23	C₃₁×#541	C₂²	1	2	yes	—
24	C₃₁×#540	C₂²	1	2	yes	—
25	C₃₁×#537	C₂²	1	2	yes	—
26	C₃₁×#536	C₂²	1	2	yes	—
27	C₃₁×#533	C₂²	1	4	yes	—
28	C₃₁×#533	C₂²	1	1	yes	characteristic
29	C₃₁×#532	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
31	C₃₁×#529	C₂²	1	4	yes	—
32	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
33	C₃₁×#520	C₂²	1	2	yes	—
34	C₃₁×#519	C₂²	1	2	yes	—
35	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#501	C₂²	1	4	yes	—
37	C₃₁×#499	C₂²	1	4	yes	—
38	C₃₁×#497	C₂²	1	4	yes	—
39	C₃₁×#494	C₂²	1	2	yes	—
40	C₃₁×#492	C₂²	1	2	yes	—
41	C₃₁×#491	C₂²	1	2	yes	—
42	C₃₁×#491	C₂²	1	4	yes	—
43	C₃₁×#491	C₂²	1	2	yes	—
44	C₃₁×#491	C₂²	1	2	yes	—
45	C₃₁×#490	C₂²	1	2	yes	—
46	C₃₁×#489	C₂²	1	2	yes	—
47	C₃₁×#489	C₂²	1	2	yes	—
48	C₃₁×#470	C₂²	1	8	yes	—
49	C₃₁×#468	C₂²	1	8	yes	—
50	C₃₁×#467	C₂²	1	8	yes	—
51	C₃₁×#465	C₂²	1	8	yes	—
52	C₃₁×#441	C₂²	1	8	yes	—
53	C₃₁×#440	C₂²	1	8	yes	—
54	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	—
55	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	—
56	C₃₁×#428	C₂²	1	4	yes	—
57	C₃₁×#416	C₂²	1	4	yes	—
58	C₂×C₃₁×#103	C₂²	1	4	yes	—
59	C₃₁×#405	C₂²	1	4	yes	—
60	C₃₁×#405	C₂²	1	2	yes	—
61	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
62	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
63	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
64	C₂×C₃₁×#97	C₂²	1	2	yes	—
65	C₂×#41×C₃₁	C₂³	1	2	yes	—
66	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
68	C₃₁×#137	—	2	16	no	—
69	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
70	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
71	C₃₁×#131	—	2	8	no	—
72	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
73	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
74	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
75	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
76	C₃₁×C₄⋊Q₈	—	2	2	no	—
77	C₃₁×C₄⋊Q₈	—	2	2	no	—
78	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
79	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
81	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
82	C₃₁×#127	C₂³	1	1	yes	characteristic
83	C₃₁×#126	—	2	8	no	—
84	C₃₁×#126	—	2	4	no	—
85	C₃₁×#126	—	2	4	no	—
86	C₃₁×#125	—	2	4	no	—
87	C₃₁×#125	—	2	8	no	—
88	C₃₁×#125	—	2	2	no	—
89	C₃₁×#125	—	2	4	no	—
90	C₃₁×#125	C₂³	1	2	yes	—
91	C₃₁×#125	C₂³	1	2	yes	—
92	C₃₁×#125	C₂³	1	2	yes	—
93	C₃₁×#124	—	2	2	no	—
94	C₃₁×#124	C₂³	1	2	yes	—
95	C₃₁×#123	—	2	4	no	—
96	C₃₁×#123	—	2	4	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	—	2	4	no	—
101	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
102	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
103	C₃₁×#121	—	2	2	no	—
104	C₃₁×#121	—	2	4	no	—
105	C₃₁×#121	—	2	2	no	—
106	C₃₁×#121	—	2	4	no	—
107	C₃₁×#121	C₂³	1	4	yes	—
108	C₄×Q₈×C₃₁	—	2	4	no	—
109	C₄×Q₈×C₃₁	—	2	4	no	—
110	C₄×Q₈×C₃₁	—	2	2	no	—
111	C₄×Q₈×C₃₁	D₄	1	2	yes	—
112	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
113	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
114	C₄×D₄×C₃₁	—	2	4	no	—
115	C₄×D₄×C₃₁	—	2	4	no	—
116	C₄×D₄×C₃₁	—	2	2	no	—
117	C₄×D₄×C₃₁	D₄	1	4	yes	—
118	C₄×D₄×C₃₁	D₄	1	2	yes	—
119	C₄×D₄×C₃₁	C₂³	1	4	yes	—
120	C₃₁×#117	—	2	4	no	—
121	C₃₁×#117	—	2	2	no	—
122	C₃₁×#117	—	2	4	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	—	2	2	no	—
130	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
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	8	yes	—
136	C₃₁×#103	C₂³	1	8	yes	—
137	C₃₁×#103	C₂³	1	4	yes	—
138	C₃₁×#103	C₂³	1	4	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	4	yes	—
145	C₃₁×#97	C₂³	1	2	yes	—
146	C₄×C₈×C₃₁	C₂³	1	4	yes	—
147	C₄×C₈×C₃₁	C₂³	1	2	yes	—
148	#41×C₃₁	—	2	4	no	—
149	#41×C₃₁	—	2	8	no	—
150	C₂×Q₈×C₃₁	—	2	2	no	—
151	C₂×Q₈×C₃₁	—	2	2	no	—
152	C₂×Q₈×C₃₁	—	2	8	no	—
153	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
155	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
156	C₂×D₄×C₃₁	—	2	2	no	—
157	C₂×D₄×C₃₁	—	2	4	no	—
158	C₂×D₄×C₃₁	—	2	2	no	—
159	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
160	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₃₁	—	2	2	no	—
163	C₂²×C₄×C₃₁	—	2	2	no	—
164	C₂²×C₄×C₃₁	—	2	2	no	—
165	C₂²×C₄×C₃₁	—	2	4	no	—
166	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
170	Q₁₆×C₃₁	—	2	16	no	—
171	SD₁₆×C₃₁	—	2	16	no	—
172	M₁₆×C₃₁	—	2	8	no	—
173	M₁₆×C₃₁	#41	1	8	yes	—
174	C₂×C₈×C₃₁	—	2	4	no	—
175	C₂×C₈×C₃₁	—	2	8	no	—
176	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
177	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
178	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
179	C₄⋊C₄×C₃₁	—	2	2	no	—
180	C₄⋊C₄×C₃₁	—	2	4	no	—
181	C₄⋊C₄×C₃₁	—	2	4	no	—
182	C₄⋊C₄×C₃₁	—	2	4	no	—
183	C₄⋊C₄×C₃₁	—	2	4	no	—
184	C₄⋊C₄×C₃₁	—	2	4	no	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	—	2	8	no	—
187	C₄⋊C₄×C₃₁	—	2	4	no	—
188	C₄⋊C₄×C₃₁	—	2	2	no	—
189	C₄⋊C₄×C₃₁	—	2	4	no	—
190	C₄⋊C₄×C₃₁	—	2	8	no	—
191	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
192	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
193	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
194	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
195	C₂²⋊C₄×C₃₁	—	2	2	no	—
196	C₂²⋊C₄×C₃₁	—	2	8	no	—
197	C₂²⋊C₄×C₃₁	—	2	4	no	—
198	C₂²⋊C₄×C₃₁	—	2	2	no	—
199	C₂²⋊C₄×C₃₁	—	2	4	no	—
200	C₂²⋊C₄×C₃₁	—	2	4	no	—
201	C₂²⋊C₄×C₃₁	—	2	2	no	—
202	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
203	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
204	C₄²×C₃₁	—	2	4	no	—
205	C₄²×C₃₁	—	2	8	no	—
206	C₄²×C₃₁	—	2	2	no	—
207	C₄²×C₃₁	C₂×D₄	1	4	yes	—
208	C₄²×C₃₁	C₂×D₄	1	2	yes	—
209	C₄²×C₃₁	C₂⁴	1	2	yes	—
210	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
211	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
212	#60726	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂³×C₃₁	—	2	2	no	—
214	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
215	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
216	Q₈×C₃₁	—	2	2	no	—
217	Q₈×C₃₁	—	2	8	no	—
218	Q₈×C₃₁	—	2	4	no	—
219	Q₈×C₃₁	—	4	4	no	—
220	Q₈×C₃₁	—	4	4	no	—
221	D₄×C₃₁	—	2	4	no	—
222	D₄×C₃₁	—	2	2	no	—
223	D₄×C₃₁	—	2	4	no	—
224	D₄×C₃₁	—	4	4	no	—
225	D₄×C₃₁	—	8	2	no	—
226	D₄×C₃₁	—	8	4	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	4	no	—
231	C₂×C₄×C₃₁	—	2	4	no	—
232	C₂×C₄×C₃₁	—	4	4	no	—
233	C₂×C₄×C₃₁	—	4	2	no	—
234	C₂×C₄×C₃₁	—	4	4	no	—
235	C₂×C₄×C₃₁	—	4	4	no	—
236	C₂×C₄×C₃₁	—	8	4	no	—
237	C₂×C₄×C₃₁	—	8	2	no	—
238	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
239	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
240	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
244	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
245	C₈×C₃₁	—	2	2	no	—
246	C₈×C₃₁	—	2	2	no	—
247	C₈×C₃₁	C₂×#41	1	8	yes	—
248	#3512	C₂×C₃₁	1	1	yes	characteristic
249	#3509	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#3181	C₂×C₃₁	1	4	yes	complemented
251	#3177	C₂×C₃₁	1	4	yes	complemented
252	#3174	C₂×C₃₁	1	4	yes	complemented
253	#3172	C₂×C₃₁	1	4	yes	complemented
254	#3166	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#3152	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#3135	C₂×C₃₁	1	1	yes	characteristic, complemented
257	C₂×#491	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#3063	C₂×C₃₁	1	4	yes	complemented
259	#2947	C₂×C₃₁	1	4	yes	complemented
260	#2875	C₂×C₃₁	1	1	yes	characteristic, complemented
261	C₂²×C₃₁	—	2	2	no	—
262	C₂²×C₃₁	—	2	4	no	—
263	C₂²×C₃₁	—	2	2	no	—
264	C₂²×C₃₁	—	4	2	no	—
265	C₂²×C₃₁	—	4	2	no	—
266	C₂²×C₃₁	—	4	2	no	—
267	C₂²×C₃₁	#532	1	2	yes	—
268	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
269	C₄×C₃₁	—	2	2	no	—
270	C₄×C₃₁	—	2	4	no	—
271	C₄×C₃₁	—	2	2	no	—
272	C₄×C₃₁	—	2	2	no	—
273	C₄×C₃₁	—	4	4	no	—
274	C₄×C₃₁	—	4	4	no	—
275	C₄×C₃₁	—	4	2	no	—
276	C₄×C₃₁	—	4	4	no	—
277	C₄×C₃₁	#532	1	2	yes	—
278	C₄×C₃₁	C₂²×#41	1	2	yes	—
279	C₂×#137	C₂²×C₃₁	1	4	yes	—
280	C₂×#131	C₂²×C₃₁	1	2	yes	—
281	#557	—	2	2	no	—
282	#556	—	2	2	no	—
283	#553	—	2	4	no	—
284	#553	—	2	2	no	—
285	#552	—	2	2	no	—
286	#552	—	2	4	no	—
287	#541	C₂²×C₃₁	1	4	yes	—
288	#541	C₂²×C₃₁	1	2	yes	—
289	#540	C₂²×C₃₁	1	2	yes	—
290	#537	C₂²×C₃₁	1	2	yes	—
291	#536	C₂²×C₃₁	1	2	yes	—
292	#533	C₂²×C₃₁	1	4	yes	—
293	#533	C₂²×C₃₁	1	1	yes	characteristic
294	#532	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
296	#529	C₂²×C₃₁	1	4	yes	—
297	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
298	#520	C₂²×C₃₁	1	2	yes	—
299	#519	C₂²×C₃₁	1	2	yes	—
300	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
301	#501	C₂²×C₃₁	1	4	yes	—
302	#499	C₂²×C₃₁	1	4	yes	—
303	#497	C₂²×C₃₁	1	4	yes	—
304	#494	C₂²×C₃₁	1	2	yes	—
305	#492	C₂²×C₃₁	1	2	yes	—
306	#491	C₂²×C₃₁	1	2	yes	—
307	#491	C₂²×C₃₁	1	4	yes	—
308	#491	C₂²×C₃₁	1	2	yes	—
309	#491	C₂²×C₃₁	1	2	yes	—
310	#490	C₂²×C₃₁	1	2	yes	—
311	#489	C₂²×C₃₁	1	2	yes	—
312	#489	C₂²×C₃₁	1	2	yes	—
313	#470	C₂²×C₃₁	1	8	yes	—
314	#468	C₂²×C₃₁	1	8	yes	—
315	#467	C₂²×C₃₁	1	8	yes	—
316	#465	C₂²×C₃₁	1	8	yes	—
317	#441	C₂²×C₃₁	1	8	yes	—
318	#440	C₂²×C₃₁	1	8	yes	—
319	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	—
320	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	—
321	#428	C₂²×C₃₁	1	4	yes	—
322	#416	C₂²×C₃₁	1	4	yes	—
323	C₂×#103	C₂²×C₃₁	1	4	yes	—
324	#405	C₂²×C₃₁	1	4	yes	—
325	#405	C₂²×C₃₁	1	2	yes	—
326	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
327	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
328	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
329	C₂×#97	C₂²×C₃₁	1	2	yes	—
330	C₂×C₃₁	—	2	2	no	—
331	C₂×C₃₁	—	4	2	no	—
332	C₂×C₃₁	—	8	2	no	—
333	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
334	C₂×C₃₁	#3586	1	2	yes	—
335	C₂×#41	C₂³×C₃₁	1	2	yes	—
336	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
337	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
338	#137	—	2	16	no	—
339	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
340	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
341	#131	—	2	8	no	—
342	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
343	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
344	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
345	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
346	C₄⋊Q₈	—	2	2	no	—
347	C₄⋊Q₈	—	2	2	no	—
348	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
349	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
350	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
351	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
352	#127	C₂³×C₃₁	1	1	yes	characteristic
353	#126	—	2	8	no	—
354	#126	—	2	4	no	—
355	#126	—	2	4	no	—
356	#125	—	2	4	no	—
357	#125	—	2	8	no	—
358	#125	—	2	2	no	—
359	#125	—	2	4	no	—
360	#125	C₂³×C₃₁	1	2	yes	—
361	#125	C₂³×C₃₁	1	2	yes	—
362	#125	C₂³×C₃₁	1	2	yes	—
363	#124	—	2	2	no	—
364	#124	C₂³×C₃₁	1	2	yes	—
365	#123	—	2	4	no	—
366	#123	—	2	4	no	—
367	#123	—	2	4	no	—
368	C₂²⋊Q₈	—	2	2	no	—
369	C₂²⋊Q₈	—	2	2	no	—
370	C₂²⋊Q₈	—	2	4	no	—
371	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
372	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
373	#121	—	2	2	no	—
374	#121	—	2	4	no	—
375	#121	—	2	2	no	—
376	#121	—	2	4	no	—
377	#121	C₂³×C₃₁	1	4	yes	—
378	C₄×Q₈	—	2	4	no	—
379	C₄×Q₈	—	2	4	no	—
380	C₄×Q₈	—	2	2	no	—
381	C₄×Q₈	D₄×C₃₁	1	2	yes	—
382	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
383	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
384	C₄×D₄	—	2	4	no	—
385	C₄×D₄	—	2	4	no	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	D₄×C₃₁	1	4	yes	—
388	C₄×D₄	D₄×C₃₁	1	2	yes	—
389	C₄×D₄	C₂³×C₃₁	1	4	yes	—
390	#117	—	2	4	no	—
391	#117	—	2	2	no	—
392	#117	—	2	4	no	—
393	#117	—	2	4	no	—
394	#117	C₂³×C₃₁	1	2	yes	—
395	#117	C₂³×C₃₁	1	1	yes	characteristic
396	#117	C₂³×C₃₁	1	2	yes	—
397	C₂×C₄⋊C₄	—	2	2	no	—
398	C₂×C₄⋊C₄	—	2	2	no	—
399	C₂×C₄⋊C₄	—	2	2	no	—
400	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
401	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
402	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
403	#107	C₂³×C₃₁	1	8	yes	—
404	#106	C₂³×C₃₁	1	8	yes	—
405	#103	C₂³×C₃₁	1	8	yes	—
406	#103	C₂³×C₃₁	1	8	yes	—
407	#103	C₂³×C₃₁	1	4	yes	—
408	#103	C₂³×C₃₁	1	4	yes	—
409	#102	C₂³×C₃₁	1	4	yes	—
410	#102	C₂³×C₃₁	1	4	yes	—
411	#97	C₂³×C₃₁	1	1	yes	characteristic
412	#97	C₂³×C₃₁	1	1	yes	characteristic
413	#97	C₂³×C₃₁	1	2	yes	—
414	#97	C₂³×C₃₁	1	4	yes	—
415	#97	C₂³×C₃₁	1	2	yes	—
416	C₄×C₈	C₂³×C₃₁	1	4	yes	—
417	C₄×C₈	C₂³×C₃₁	1	2	yes	—
418	C₃₁	#60726	1	1	yes	characteristic, direct product factor, Sylow
419	#41	—	2	4	no	—
420	#41	—	2	8	no	—
421	C₂×Q₈	—	2	2	no	—
422	C₂×Q₈	—	2	2	no	—
423	C₂×Q₈	—	2	8	no	—
424	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
425	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
426	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	2	4	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
431	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
432	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
438	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
439	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
440	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
441	Q₁₆	—	2	16	no	—
442	SD₁₆	—	2	16	no	—
443	M₁₆	—	2	8	no	—
444	M₁₆	#41×C₃₁	1	8	yes	—
445	C₂×C₈	—	2	4	no	—
446	C₂×C₈	—	2	8	no	—
447	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
448	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
449	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	4	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	2	8	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	—	2	4	no	—
461	C₄⋊C₄	—	2	8	no	—
462	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
463	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
464	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
465	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
466	C₂²⋊C₄	—	2	2	no	—
467	C₂²⋊C₄	—	2	8	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₂²⋊C₄	—	2	2	no	—
470	C₂²⋊C₄	—	2	4	no	—
471	C₂²⋊C₄	—	2	4	no	—
472	C₂²⋊C₄	—	2	2	no	—
473	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
474	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
475	C₄²	—	2	4	no	—
476	C₄²	—	2	8	no	—
477	C₄²	—	2	2	no	—
478	C₄²	C₂×D₄×C₃₁	1	4	yes	—
479	C₄²	C₂×D₄×C₃₁	1	2	yes	—
480	C₄²	C₂⁴×C₃₁	1	2	yes	—
481	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
482	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
483	C₂³	—	2	2	no	—
484	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
485	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
486	Q₈	—	2	2	no	—
487	Q₈	—	2	8	no	—
488	Q₈	—	2	4	no	—
489	Q₈	—	4	4	no	—
490	Q₈	—	4	4	no	—
491	D₄	—	2	4	no	—
492	D₄	—	2	2	no	—
493	D₄	—	2	4	no	—
494	D₄	—	4	4	no	—
495	D₄	—	8	2	no	—
496	D₄	—	8	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	4	4	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	4	no	—
505	C₂×C₄	—	4	4	no	—
506	C₂×C₄	—	8	4	no	—
507	C₂×C₄	—	8	2	no	—
508	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
509	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
510	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
511	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
512	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
513	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
514	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
515	C₈	—	2	2	no	—
516	C₈	—	2	2	no	—
517	C₈	C₂×#41×C₃₁	1	8	yes	—
518	C₂²	—	2	2	no	—
519	C₂²	—	2	2	no	—
520	C₂²	—	2	4	no	—
521	C₂²	—	4	2	no	—
522	C₂²	—	4	2	no	—
523	C₂²	—	4	2	no	—
524	C₂²	C₃₁×#532	1	2	yes	—
525	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
526	C₄	—	2	2	no	—
527	C₄	—	2	2	no	—
528	C₄	—	2	4	no	—
529	C₄	—	2	2	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	4	no	—
532	C₄	—	4	4	no	—
533	C₄	—	4	4	no	—
534	C₄	C₃₁×#532	1	2	yes	—
535	C₄	C₂²×#41×C₃₁	1	2	yes	—
536	C₂	—	2	2	no	—
537	C₂	—	4	2	no	—
538	C₂	—	8	2	no	—
539	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
540	C₂	C₃₁×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3512	C₂
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
2. C₃₁×#3509	C₂	semidirect
3. C₃₁×#3181	C₂	semidirect
4. C₃₁×#3177	C₂	semidirect
5. C₃₁×#3174	C₂	semidirect
6. C₃₁×#3172	C₂	semidirect
7. C₃₁×#3166	C₂	semidirect
8. C₃₁×#3152	C₂	semidirect
9. C₃₁×#3135	C₂	semidirect
10. C₂×C₃₁×#491	C₂	semidirect
11. C₃₁×#3063	C₂	semidirect
12. C₃₁×#2947	C₂	semidirect
13. C₃₁×#2875	C₂	semidirect
212. #60726	C₃₁	direct (N × H)
249. #3509	C₂×C₃₁	semidirect
250. #3181	C₂×C₃₁	semidirect
251. #3177	C₂×C₃₁	semidirect
252. #3174	C₂×C₃₁	semidirect
253. #3172	C₂×C₃₁	semidirect
254. #3166	C₂×C₃₁	semidirect
255. #3152	C₂×C₃₁	semidirect
256. #3135	C₂×C₃₁	semidirect
257. C₂×#491	C₂×C₃₁	semidirect
258. #3063	C₂×C₃₁	semidirect
259. #2947	C₂×C₃₁	semidirect
260. #2875	C₂×C₃₁	semidirect
418. C₃₁	#60726	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×#3512	2
C₃₁×#3509	2
C₃₁×#3181	2
C₃₁×#3177	2
C₃₁×#3174	2
C₃₁×#3172	2
C₃₁×#3166	2
C₃₁×#3152	2
C₃₁×#3135	2
C₂×C₃₁×#491	2
C₃₁×#3063	2
C₃₁×#2947	2
C₃₁×#2875	2
#60726	31

Subgroups — C31×#60726 #21517485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60726 #21517485