Subgroups — C₃₁×#60723 #21517482

Order 7936.

1886 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3511	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3509	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3185	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3182	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3178	C₂	1	8	yes	maximal, complemented
6	C₃₁×#3170	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3167	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3165	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3152	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3063	C₂	1	4	yes	maximal, complemented
13	C₃₁×#2947	C₂	1	4	yes	maximal, complemented
14	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₃₁×#137	C₂²	1	4	yes	—
16	C₂×C₃₁×#131	C₂²	1	2	yes	—
17	C₃₁×#557	—	2	1	no	—
18	C₃₁×#557	—	2	1	no	—
19	C₃₁×#556	—	2	1	no	—
20	C₃₁×#556	—	2	1	no	—
21	C₃₁×#556	—	2	1	no	—
22	C₃₁×#555	—	2	1	no	—
23	C₃₁×#554	—	2	1	no	—
24	C₃₁×#553	—	2	2	no	—
25	C₃₁×#552	—	2	2	no	—
26	C₃₁×#552	—	2	2	no	—
27	C₃₁×#548	—	2	2	no	—
28	C₃₁×#547	—	2	1	no	—
29	C₃₁×#542	C₂²	1	4	yes	—
30	C₃₁×#541	C₂²	1	4	yes	—
31	C₃₁×#537	C₂²	1	2	yes	—
32	C₃₁×#536	C₂²	1	2	yes	—
33	C₃₁×#533	C₂²	1	2	yes	—
34	C₃₁×#533	C₂²	1	2	yes	—
35	C₃₁×#533	C₂²	1	2	yes	—
36	C₃₁×#533	C₂²	1	1	yes	characteristic
37	C₃₁×#532	C₂²	1	1	yes	characteristic
38	C₃₁×#532	C₂²	1	2	yes	—
39	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
41	C₃₁×#520	C₂²	1	2	yes	—
42	C₃₁×#519	C₂²	1	2	yes	—
43	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
44	C₃₁×#501	C₂²	1	4	yes	—
45	C₃₁×#500	C₂²	1	2	yes	—
46	C₃₁×#498	C₂²	1	2	yes	—
47	C₃₁×#497	C₂²	1	4	yes	—
48	C₃₁×#494	C₂²	1	2	yes	—
49	C₃₁×#492	C₂²	1	2	yes	—
50	C₃₁×#488	C₂²	1	4	yes	—
51	C₃₁×#487	C₂²	1	2	yes	—
52	C₃₁×#487	C₂²	1	4	yes	—
53	C₃₁×#487	C₂²	1	2	yes	—
54	C₃₁×#487	C₂²	1	1	yes	characteristic
55	C₃₁×#487	C₂²	1	1	yes	characteristic
56	C₃₁×#486	C₂²	1	1	yes	characteristic
57	C₃₁×#486	C₂²	1	1	yes	characteristic
58	C₃₁×#470	C₂²	1	8	yes	—
59	C₃₁×#468	C₂²	1	8	yes	—
60	C₃₁×#467	C₂²	1	8	yes	—
61	C₃₁×#465	C₂²	1	8	yes	—
62	C₃₁×#441	C₂²	1	8	yes	—
63	C₃₁×#440	C₂²	1	8	yes	—
64	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	—
65	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	—
66	C₃₁×#428	C₂²	1	4	yes	—
67	C₃₁×#416	C₂²	1	4	yes	—
68	C₂×C₃₁×#103	C₂²	1	4	yes	—
69	C₃₁×#405	C₂²	1	4	yes	—
70	C₃₁×#405	C₂²	1	2	yes	—
71	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
72	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
73	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
74	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
75	C₂×#41×C₃₁	—	2	2	no	—
76	C₂×#41×C₃₁	C₂³	1	2	yes	—
77	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
78	C₂²×Q₈×C₃₁	—	2	1	no	—
79	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
80	C₃₁×#137	—	2	16	no	—
81	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
82	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
83	C₃₁×#131	—	2	8	no	—
84	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
85	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
86	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
87	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
88	C₃₁×C₄⋊Q₈	—	2	2	no	—
89	C₃₁×C₄⋊Q₈	—	2	1	no	—
90	C₃₁×C₄⋊Q₈	—	2	1	no	—
91	C₃₁×C₄⋊Q₈	—	2	1	no	—
92	C₃₁×C₄⋊Q₈	—	2	1	no	—
93	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
94	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
95	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
96	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
97	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
98	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
99	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
100	C₃₁×#127	C₂³	1	1	yes	characteristic
101	C₃₁×#126	—	2	4	no	—
102	C₃₁×#126	—	2	4	no	—
103	C₃₁×#126	—	2	4	no	—
104	C₃₁×#125	—	2	4	no	—
105	C₃₁×#125	—	2	4	no	—
106	C₃₁×#125	—	2	2	no	—
107	C₃₁×#125	—	2	2	no	—
108	C₃₁×#125	—	2	2	no	—
109	C₃₁×#125	C₂³	1	2	yes	—
110	C₃₁×#124	—	2	2	no	—
111	C₃₁×#124	—	2	4	no	—
112	C₃₁×#124	—	2	2	no	—
113	C₃₁×#124	C₂³	1	2	yes	—
114	C₃₁×#123	—	2	4	no	—
115	C₃₁×#123	—	2	4	no	—
116	C₃₁×C₂²⋊Q₈	—	2	2	no	—
117	C₃₁×C₂²⋊Q₈	—	2	2	no	—
118	C₃₁×C₂²⋊Q₈	—	2	1	no	—
119	C₃₁×C₂²⋊Q₈	—	2	1	no	—
120	C₃₁×C₂²⋊Q₈	—	2	2	no	—
121	C₃₁×C₂²⋊Q₈	—	2	2	no	—
122	C₃₁×C₂²⋊Q₈	—	2	4	no	—
123	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
124	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
125	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
126	C₃₁×#121	—	2	2	no	—
127	C₃₁×#121	—	2	4	no	—
128	C₃₁×#121	—	2	1	no	—
129	C₃₁×#121	—	2	1	no	—
130	C₃₁×#121	—	2	2	no	—
131	C₃₁×#121	C₂³	1	4	yes	—
132	C₄×Q₈×C₃₁	—	2	2	no	—
133	C₄×Q₈×C₃₁	—	2	2	no	—
134	C₄×Q₈×C₃₁	—	2	2	no	—
135	C₄×Q₈×C₃₁	—	2	2	no	—
136	C₄×Q₈×C₃₁	—	2	2	no	—
137	C₄×Q₈×C₃₁	—	2	1	no	—
138	C₄×Q₈×C₃₁	—	2	1	no	—
139	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
140	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
141	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
142	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
143	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
144	C₄×D₄×C₃₁	—	2	4	no	—
145	C₄×D₄×C₃₁	—	2	1	no	—
146	C₄×D₄×C₃₁	—	2	2	no	—
147	C₄×D₄×C₃₁	—	2	1	no	—
148	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
149	C₄×D₄×C₃₁	D₄	1	2	yes	—
150	C₄×D₄×C₃₁	D₄	1	2	yes	—
151	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
152	C₄×D₄×C₃₁	C₂³	1	4	yes	—
153	C₃₁×#117	—	2	2	no	—
154	C₃₁×#117	—	2	2	no	—
155	C₃₁×#117	—	2	2	no	—
156	C₃₁×#117	—	2	2	no	—
157	C₃₁×#117	—	2	4	no	—
158	C₃₁×#117	C₂³	1	2	yes	—
159	C₃₁×#117	C₂³	1	1	yes	characteristic
160	C₃₁×#117	C₂³	1	2	yes	—
161	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
164	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
165	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
166	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
167	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
168	C₃₁×#107	C₂³	1	8	yes	—
169	C₃₁×#106	C₂³	1	8	yes	—
170	C₃₁×#103	C₂³	1	8	yes	—
171	C₃₁×#103	C₂³	1	8	yes	—
172	C₃₁×#103	C₂³	1	4	yes	—
173	C₃₁×#103	C₂³	1	4	yes	—
174	C₃₁×#102	C₂³	1	4	yes	—
175	C₃₁×#102	C₂³	1	4	yes	—
176	C₃₁×#97	C₂³	1	4	yes	—
177	C₃₁×#97	C₂³	1	2	yes	—
178	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
179	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
180	C₄×C₈×C₃₁	C₂³	1	2	yes	—
181	C₄×C₈×C₃₁	C₂³	1	4	yes	—
182	C₄×C₈×C₃₁	C₂³	1	2	yes	—
183	#41×C₃₁	—	2	4	no	—
184	#41×C₃₁	—	2	8	no	—
185	#41×C₃₁	—	8	4	no	—
186	C₂×Q₈×C₃₁	—	2	1	no	—
187	C₂×Q₈×C₃₁	—	2	1	no	—
188	C₂×Q₈×C₃₁	—	2	2	no	—
189	C₂×Q₈×C₃₁	—	2	2	no	—
190	C₂×Q₈×C₃₁	—	2	2	no	—
191	C₂×Q₈×C₃₁	—	2	8	no	—
192	C₂×Q₈×C₃₁	—	8	2	no	—
193	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
195	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
196	C₂×D₄×C₃₁	—	2	2	no	—
197	C₂×D₄×C₃₁	—	2	4	no	—
198	C₂×D₄×C₃₁	—	2	1	no	—
199	C₂×D₄×C₃₁	—	2	1	no	—
200	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
201	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
202	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
203	C₂²×C₄×C₃₁	—	2	2	no	—
204	C₂²×C₄×C₃₁	—	2	2	no	—
205	C₂²×C₄×C₃₁	—	2	2	no	—
206	C₂²×C₄×C₃₁	—	2	4	no	—
207	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
208	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
209	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
210	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
211	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
212	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
213	Q₁₆×C₃₁	—	2	16	no	—
214	SD₁₆×C₃₁	—	2	16	no	—
215	M₁₆×C₃₁	—	2	8	no	—
216	M₁₆×C₃₁	#41	1	8	yes	complemented
217	C₂×C₈×C₃₁	—	2	4	no	—
218	C₂×C₈×C₃₁	—	2	8	no	—
219	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
220	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
221	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
222	C₄⋊C₄×C₃₁	—	2	2	no	—
223	C₄⋊C₄×C₃₁	—	2	2	no	—
224	C₄⋊C₄×C₃₁	—	2	2	no	—
225	C₄⋊C₄×C₃₁	—	2	2	no	—
226	C₄⋊C₄×C₃₁	—	2	4	no	—
227	C₄⋊C₄×C₃₁	—	2	2	no	—
228	C₄⋊C₄×C₃₁	—	2	2	no	—
229	C₄⋊C₄×C₃₁	—	2	2	no	—
230	C₄⋊C₄×C₃₁	—	2	1	no	—
231	C₄⋊C₄×C₃₁	—	2	1	no	—
232	C₄⋊C₄×C₃₁	—	2	1	no	—
233	C₄⋊C₄×C₃₁	—	2	4	no	—
234	C₄⋊C₄×C₃₁	—	2	4	no	—
235	C₄⋊C₄×C₃₁	—	2	1	no	—
236	C₄⋊C₄×C₃₁	—	2	4	no	—
237	C₄⋊C₄×C₃₁	—	2	2	no	—
238	C₄⋊C₄×C₃₁	—	2	2	no	—
239	C₄⋊C₄×C₃₁	—	2	4	no	—
240	C₄⋊C₄×C₃₁	—	2	4	no	—
241	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
242	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
243	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
244	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
245	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
246	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
247	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
248	C₂²⋊C₄×C₃₁	—	2	2	no	—
249	C₂²⋊C₄×C₃₁	—	2	4	no	—
250	C₂²⋊C₄×C₃₁	—	2	4	no	—
251	C₂²⋊C₄×C₃₁	—	2	4	no	—
252	C₂²⋊C₄×C₃₁	—	2	4	no	—
253	C₂²⋊C₄×C₃₁	—	2	4	no	—
254	C₂²⋊C₄×C₃₁	—	2	2	no	—
255	C₂²⋊C₄×C₃₁	—	2	2	no	—
256	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
257	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
258	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
259	C₄²×C₃₁	—	2	4	no	—
260	C₄²×C₃₁	—	2	4	no	—
261	C₄²×C₃₁	—	2	4	no	—
262	C₄²×C₃₁	—	2	2	no	—
263	C₄²×C₃₁	C₂×D₄	1	2	yes	—
264	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
265	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
266	C₄²×C₃₁	C₂×D₄	1	2	yes	—
267	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
268	C₄²×C₃₁	C₂⁴	1	2	yes	—
269	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
270	#60723	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
271	C₂³×C₃₁	—	2	2	no	—
272	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
273	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
274	Q₈×C₃₁	—	2	2	no	—
275	Q₈×C₃₁	—	2	8	no	—
276	Q₈×C₃₁	—	2	4	no	—
277	Q₈×C₃₁	—	4	4	no	—
278	Q₈×C₃₁	—	4	2	no	—
279	Q₈×C₃₁	—	4	4	no	—
280	Q₈×C₃₁	—	4	2	no	—
281	Q₈×C₃₁	—	8	2	no	—
282	D₄×C₃₁	—	2	4	no	—
283	D₄×C₃₁	—	2	2	no	—
284	D₄×C₃₁	—	2	4	no	—
285	D₄×C₃₁	—	4	4	no	—
286	D₄×C₃₁	—	4	2	no	—
287	D₄×C₃₁	—	4	2	no	—
288	D₄×C₃₁	—	8	4	no	—
289	C₂×C₄×C₃₁	—	2	2	no	—
290	C₂×C₄×C₃₁	—	2	4	no	—
291	C₂×C₄×C₃₁	—	2	2	no	—
292	C₂×C₄×C₃₁	—	2	4	no	—
293	C₂×C₄×C₃₁	—	2	4	no	—
294	C₂×C₄×C₃₁	—	2	2	no	—
295	C₂×C₄×C₃₁	—	2	2	no	—
296	C₂×C₄×C₃₁	—	2	2	no	—
297	C₂×C₄×C₃₁	—	2	4	no	—
298	C₂×C₄×C₃₁	—	2	4	no	—
299	C₂×C₄×C₃₁	—	4	4	no	—
300	C₂×C₄×C₃₁	—	4	4	no	—
301	C₂×C₄×C₃₁	—	8	4	no	—
302	C₂×C₄×C₃₁	—	8	2	no	—
303	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
304	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
305	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
306	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
307	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
308	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
309	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
310	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
311	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
312	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
313	C₈×C₃₁	—	2	2	no	—
314	C₈×C₃₁	—	2	2	no	—
315	C₈×C₃₁	C₂×#41	1	8	yes	—
316	#3511	C₂×C₃₁	1	1	yes	characteristic
317	#3509	C₂×C₃₁	1	1	yes	characteristic, complemented
318	#3185	C₂×C₃₁	1	2	yes	complemented
319	#3182	C₂×C₃₁	1	2	yes	complemented
320	#3178	C₂×C₃₁	1	8	yes	complemented
321	#3170	C₂×C₃₁	1	2	yes	complemented
322	#3167	C₂×C₃₁	1	2	yes	complemented
323	#3165	C₂×C₃₁	1	1	yes	characteristic, complemented
324	#3152	C₂×C₃₁	1	1	yes	characteristic, complemented
325	#3131	C₂×C₃₁	1	1	yes	characteristic, complemented
326	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
327	#3063	C₂×C₃₁	1	4	yes	complemented
328	#2947	C₂×C₃₁	1	4	yes	complemented
329	C₄×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
330	C₂²×C₃₁	—	2	1	no	—
331	C₂²×C₃₁	—	2	1	no	—
332	C₂²×C₃₁	—	2	2	no	—
333	C₂²×C₃₁	—	2	2	no	—
334	C₂²×C₃₁	—	2	2	no	—
335	C₂²×C₃₁	—	4	2	no	—
336	C₂²×C₃₁	—	4	2	no	—
337	C₂²×C₃₁	—	4	2	no	—
338	C₂²×C₃₁	#532	1	2	yes	—
339	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
340	C₄×C₃₁	—	2	2	no	—
341	C₄×C₃₁	—	2	2	no	—
342	C₄×C₃₁	—	2	1	no	—
343	C₄×C₃₁	—	2	2	no	—
344	C₄×C₃₁	—	2	1	no	—
345	C₄×C₃₁	—	4	2	no	—
346	C₄×C₃₁	—	4	4	no	—
347	C₄×C₃₁	—	4	4	no	—
348	C₄×C₃₁	—	4	2	no	—
349	C₄×C₃₁	—	4	2	no	—
350	C₄×C₃₁	#532	1	2	yes	—
351	C₄×C₃₁	#577	1	2	yes	—
352	C₄×C₃₁	#578	1	2	yes	—
353	C₄×C₃₁	C₂²×#41	1	2	yes	—
354	C₂×#137	C₂²×C₃₁	1	4	yes	—
355	C₂×#131	C₂²×C₃₁	1	2	yes	—
356	#557	—	2	1	no	—
357	#557	—	2	1	no	—
358	#556	—	2	1	no	—
359	#556	—	2	1	no	—
360	#556	—	2	1	no	—
361	#555	—	2	1	no	—
362	#554	—	2	1	no	—
363	#553	—	2	2	no	—
364	#552	—	2	2	no	—
365	#552	—	2	2	no	—
366	#548	—	2	2	no	—
367	#547	—	2	1	no	—
368	#542	C₂²×C₃₁	1	4	yes	—
369	#541	C₂²×C₃₁	1	4	yes	—
370	#537	C₂²×C₃₁	1	2	yes	—
371	#536	C₂²×C₃₁	1	2	yes	—
372	#533	C₂²×C₃₁	1	2	yes	—
373	#533	C₂²×C₃₁	1	2	yes	—
374	#533	C₂²×C₃₁	1	2	yes	—
375	#533	C₂²×C₃₁	1	1	yes	characteristic
376	#532	C₂²×C₃₁	1	1	yes	characteristic
377	#532	C₂²×C₃₁	1	2	yes	—
378	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
379	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
380	#520	C₂²×C₃₁	1	2	yes	—
381	#519	C₂²×C₃₁	1	2	yes	—
382	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
383	#501	C₂²×C₃₁	1	4	yes	—
384	#500	C₂²×C₃₁	1	2	yes	—
385	#498	C₂²×C₃₁	1	2	yes	—
386	#497	C₂²×C₃₁	1	4	yes	—
387	#494	C₂²×C₃₁	1	2	yes	—
388	#492	C₂²×C₃₁	1	2	yes	—
389	#488	C₂²×C₃₁	1	4	yes	—
390	#487	C₂²×C₃₁	1	2	yes	—
391	#487	C₂²×C₃₁	1	4	yes	—
392	#487	C₂²×C₃₁	1	2	yes	—
393	#487	C₂²×C₃₁	1	1	yes	characteristic
394	#487	C₂²×C₃₁	1	1	yes	characteristic
395	#486	C₂²×C₃₁	1	1	yes	characteristic
396	#486	C₂²×C₃₁	1	1	yes	characteristic
397	#470	C₂²×C₃₁	1	8	yes	—
398	#468	C₂²×C₃₁	1	8	yes	—
399	#467	C₂²×C₃₁	1	8	yes	—
400	#465	C₂²×C₃₁	1	8	yes	—
401	#441	C₂²×C₃₁	1	8	yes	—
402	#440	C₂²×C₃₁	1	8	yes	—
403	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	—
404	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	—
405	#428	C₂²×C₃₁	1	4	yes	—
406	#416	C₂²×C₃₁	1	4	yes	—
407	C₂×#103	C₂²×C₃₁	1	4	yes	—
408	#405	C₂²×C₃₁	1	4	yes	—
409	#405	C₂²×C₃₁	1	2	yes	—
410	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
411	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
412	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
413	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
414	C₂×C₃₁	—	2	2	no	—
415	C₂×C₃₁	—	4	2	no	—
416	C₂×C₃₁	—	8	2	no	—
417	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
418	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
419	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
420	C₂×#41	—	2	2	no	—
421	C₂×#41	C₂³×C₃₁	1	2	yes	—
422	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
423	C₂²×Q₈	—	2	1	no	—
424	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
425	#137	—	2	16	no	—
426	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
427	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
428	#131	—	2	8	no	—
429	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
430	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
431	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
432	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
433	C₄⋊Q₈	—	2	2	no	—
434	C₄⋊Q₈	—	2	1	no	—
435	C₄⋊Q₈	—	2	1	no	—
436	C₄⋊Q₈	—	2	1	no	—
437	C₄⋊Q₈	—	2	1	no	—
438	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
439	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
440	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
441	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
442	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
443	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
444	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
445	#127	C₂³×C₃₁	1	1	yes	characteristic
446	#126	—	2	4	no	—
447	#126	—	2	4	no	—
448	#126	—	2	4	no	—
449	#125	—	2	4	no	—
450	#125	—	2	4	no	—
451	#125	—	2	2	no	—
452	#125	—	2	2	no	—
453	#125	—	2	2	no	—
454	#125	C₂³×C₃₁	1	2	yes	—
455	#124	—	2	2	no	—
456	#124	—	2	4	no	—
457	#124	—	2	2	no	—
458	#124	C₂³×C₃₁	1	2	yes	—
459	#123	—	2	4	no	—
460	#123	—	2	4	no	—
461	C₂²⋊Q₈	—	2	2	no	—
462	C₂²⋊Q₈	—	2	2	no	—
463	C₂²⋊Q₈	—	2	1	no	—
464	C₂²⋊Q₈	—	2	1	no	—
465	C₂²⋊Q₈	—	2	2	no	—
466	C₂²⋊Q₈	—	2	2	no	—
467	C₂²⋊Q₈	—	2	4	no	—
468	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
469	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
470	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
471	#121	—	2	2	no	—
472	#121	—	2	4	no	—
473	#121	—	2	1	no	—
474	#121	—	2	1	no	—
475	#121	—	2	2	no	—
476	#121	C₂³×C₃₁	1	4	yes	—
477	C₄×Q₈	—	2	2	no	—
478	C₄×Q₈	—	2	2	no	—
479	C₄×Q₈	—	2	2	no	—
480	C₄×Q₈	—	2	2	no	—
481	C₄×Q₈	—	2	2	no	—
482	C₄×Q₈	—	2	1	no	—
483	C₄×Q₈	—	2	1	no	—
484	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
485	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
486	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
487	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
488	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
489	C₄×D₄	—	2	4	no	—
490	C₄×D₄	—	2	1	no	—
491	C₄×D₄	—	2	2	no	—
492	C₄×D₄	—	2	1	no	—
493	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
494	C₄×D₄	D₄×C₃₁	1	2	yes	—
495	C₄×D₄	D₄×C₃₁	1	2	yes	—
496	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
497	C₄×D₄	C₂³×C₃₁	1	4	yes	—
498	#117	—	2	2	no	—
499	#117	—	2	2	no	—
500	#117	—	2	2	no	—
501	#117	—	2	2	no	—
502	#117	—	2	4	no	—
503	#117	C₂³×C₃₁	1	2	yes	—
504	#117	C₂³×C₃₁	1	1	yes	characteristic
505	#117	C₂³×C₃₁	1	2	yes	—
506	C₂×C₄⋊C₄	—	2	1	no	—
507	C₂×C₄⋊C₄	—	2	1	no	—
508	C₂×C₄⋊C₄	—	2	1	no	—
509	C₂×C₄⋊C₄	—	2	2	no	—
510	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
511	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
512	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
513	#107	C₂³×C₃₁	1	8	yes	—
514	#106	C₂³×C₃₁	1	8	yes	—
515	#103	C₂³×C₃₁	1	8	yes	—
516	#103	C₂³×C₃₁	1	8	yes	—
517	#103	C₂³×C₃₁	1	4	yes	—
518	#103	C₂³×C₃₁	1	4	yes	—
519	#102	C₂³×C₃₁	1	4	yes	—
520	#102	C₂³×C₃₁	1	4	yes	—
521	#97	C₂³×C₃₁	1	4	yes	—
522	#97	C₂³×C₃₁	1	2	yes	—
523	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
524	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
525	C₄×C₈	C₂³×C₃₁	1	2	yes	—
526	C₄×C₈	C₂³×C₃₁	1	4	yes	—
527	C₄×C₈	C₂³×C₃₁	1	2	yes	—
528	C₃₁	#60723	1	1	yes	characteristic, direct product factor, Sylow
529	#41	—	2	4	no	—
530	#41	—	2	8	no	—
531	#41	—	8	4	no	—
532	C₂×Q₈	—	2	1	no	—
533	C₂×Q₈	—	2	1	no	—
534	C₂×Q₈	—	2	2	no	—
535	C₂×Q₈	—	2	2	no	—
536	C₂×Q₈	—	2	2	no	—
537	C₂×Q₈	—	2	8	no	—
538	C₂×Q₈	—	8	2	no	—
539	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
540	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
541	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
542	C₂×D₄	—	2	2	no	—
543	C₂×D₄	—	2	4	no	—
544	C₂×D₄	—	2	1	no	—
545	C₂×D₄	—	2	1	no	—
546	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
547	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
548	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
549	C₂²×C₄	—	2	2	no	—
550	C₂²×C₄	—	2	2	no	—
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	4	no	—
553	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
554	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
555	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
556	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
557	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
558	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
559	Q₁₆	—	2	16	no	—
560	SD₁₆	—	2	16	no	—
561	M₁₆	—	2	8	no	—
562	M₁₆	#41×C₃₁	1	8	yes	complemented
563	C₂×C₈	—	2	4	no	—
564	C₂×C₈	—	2	8	no	—
565	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
566	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
567	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
568	C₄⋊C₄	—	2	2	no	—
569	C₄⋊C₄	—	2	2	no	—
570	C₄⋊C₄	—	2	2	no	—
571	C₄⋊C₄	—	2	2	no	—
572	C₄⋊C₄	—	2	4	no	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	2	2	no	—
575	C₄⋊C₄	—	2	2	no	—
576	C₄⋊C₄	—	2	1	no	—
577	C₄⋊C₄	—	2	1	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	2	4	no	—
580	C₄⋊C₄	—	2	4	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	4	no	—
583	C₄⋊C₄	—	2	2	no	—
584	C₄⋊C₄	—	2	2	no	—
585	C₄⋊C₄	—	2	4	no	—
586	C₄⋊C₄	—	2	4	no	—
587	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
588	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
589	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
590	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
591	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
592	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
593	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
594	C₂²⋊C₄	—	2	2	no	—
595	C₂²⋊C₄	—	2	4	no	—
596	C₂²⋊C₄	—	2	4	no	—
597	C₂²⋊C₄	—	2	4	no	—
598	C₂²⋊C₄	—	2	4	no	—
599	C₂²⋊C₄	—	2	4	no	—
600	C₂²⋊C₄	—	2	2	no	—
601	C₂²⋊C₄	—	2	2	no	—
602	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
603	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
604	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
605	C₄²	—	2	4	no	—
606	C₄²	—	2	4	no	—
607	C₄²	—	2	4	no	—
608	C₄²	—	2	2	no	—
609	C₄²	C₂×D₄×C₃₁	1	2	yes	—
610	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
611	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
612	C₄²	C₂×D₄×C₃₁	1	2	yes	—
613	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
614	C₄²	C₂⁴×C₃₁	1	2	yes	—
615	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
616	C₂³	—	2	2	no	—
617	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
618	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
619	Q₈	—	2	2	no	—
620	Q₈	—	2	8	no	—
621	Q₈	—	2	4	no	—
622	Q₈	—	4	4	no	—
623	Q₈	—	4	2	no	—
624	Q₈	—	4	4	no	—
625	Q₈	—	4	2	no	—
626	Q₈	—	8	2	no	—
627	D₄	—	2	4	no	—
628	D₄	—	2	2	no	—
629	D₄	—	2	4	no	—
630	D₄	—	4	4	no	—
631	D₄	—	4	2	no	—
632	D₄	—	4	2	no	—
633	D₄	—	8	4	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	4	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	4	no	—
638	C₂×C₄	—	2	4	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	4	no	—
643	C₂×C₄	—	2	4	no	—
644	C₂×C₄	—	4	4	no	—
645	C₂×C₄	—	4	4	no	—
646	C₂×C₄	—	8	4	no	—
647	C₂×C₄	—	8	2	no	—
648	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
649	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
650	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
651	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
652	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
653	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
654	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
655	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
656	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
657	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
658	C₈	—	2	2	no	—
659	C₈	—	2	2	no	—
660	C₈	C₂×#41×C₃₁	1	8	yes	—
661	C₂²	—	2	1	no	—
662	C₂²	—	2	1	no	—
663	C₂²	—	2	2	no	—
664	C₂²	—	2	2	no	—
665	C₂²	—	2	2	no	—
666	C₂²	—	4	2	no	—
667	C₂²	—	4	2	no	—
668	C₂²	—	4	2	no	—
669	C₂²	C₃₁×#532	1	2	yes	—
670	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
671	C₄	—	2	2	no	—
672	C₄	—	2	2	no	—
673	C₄	—	2	1	no	—
674	C₄	—	2	2	no	—
675	C₄	—	2	1	no	—
676	C₄	—	4	2	no	—
677	C₄	—	4	4	no	—
678	C₄	—	4	4	no	—
679	C₄	—	4	2	no	—
680	C₄	—	4	2	no	—
681	C₄	C₃₁×#532	1	2	yes	—
682	C₄	C₃₁×#577	1	2	yes	—
683	C₄	C₃₁×#578	1	2	yes	—
684	C₄	C₂²×#41×C₃₁	1	2	yes	—
685	C₂	—	2	2	no	—
686	C₂	—	4	2	no	—
687	C₂	—	8	2	no	—
688	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
689	C₂	C₂×C₃₁×#577	1	1	yes	characteristic
690	C₂	C₂×C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3511	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×#3185	C₂	semidirect
4. C₃₁×#3182	C₂	semidirect
5. C₃₁×#3178	C₂	semidirect
6. C₃₁×#3170	C₂	semidirect
7. C₃₁×#3167	C₂	semidirect
8. C₃₁×#3165	C₂	semidirect
9. C₃₁×#3152	C₂	semidirect
10. C₃₁×#3131	C₂	semidirect
11. C₂×C₃₁×#487	C₂	semidirect
12. C₃₁×#3063	C₂	semidirect
13. C₃₁×#2947	C₂	semidirect
14. C₄×C₃₁×#131	C₂	semidirect
216. M₁₆×C₃₁	#41	semidirect
270. #60723	C₃₁	direct (N × H)
317. #3509	C₂×C₃₁	semidirect
318. #3185	C₂×C₃₁	semidirect
319. #3182	C₂×C₃₁	semidirect
320. #3178	C₂×C₃₁	semidirect
321. #3170	C₂×C₃₁	semidirect
322. #3167	C₂×C₃₁	semidirect
323. #3165	C₂×C₃₁	semidirect
324. #3152	C₂×C₃₁	semidirect
325. #3131	C₂×C₃₁	semidirect
326. C₂×#487	C₂×C₃₁	semidirect
327. #3063	C₂×C₃₁	semidirect
328. #2947	C₂×C₃₁	semidirect
329. C₄×#131	C₂×C₃₁	semidirect
528. C₃₁	#60723	direct (N × H)
562. 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×#3511	2
C₃₁×#3509	2
C₃₁×#3185	2
C₃₁×#3182	2
C₃₁×#3178	2
C₃₁×#3170	2
C₃₁×#3167	2
C₃₁×#3165	2
C₃₁×#3152	2
C₃₁×#3131	2
C₂×C₃₁×#487	2
C₃₁×#3063	2
C₃₁×#2947	2
C₄×C₃₁×#131	2
#60723	31

Subgroups — C31×#60723 #21517482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60723 #21517482