Subgroups — C₃₁×#60877 #21517636

Order 7936.

1786 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3508	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3503	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3241	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3228	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3228	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3227	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3226	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3165	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3096	C₂	1	4	yes	maximal, complemented
13	C₃₁×#3081	C₂	1	4	yes	maximal, complemented
14	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#559	—	2	1	no	—
16	C₃₁×#559	—	2	1	no	—
17	Q₈²×C₃₁	—	2	1	no	—
18	C₃₁×#557	—	2	1	no	—
19	C₃₁×#554	—	2	2	no	—
20	C₃₁×#554	—	2	1	no	—
21	C₃₁×#550	—	2	1	no	—
22	C₃₁×#550	—	2	1	no	—
23	D₄×Q₈×C₃₁	—	2	2	no	—
24	D₄×Q₈×C₃₁	—	2	1	no	—
25	C₃₁×#546	—	2	2	no	—
26	C₃₁×#546	—	2	2	no	—
27	C₃₁×#544	C₂²	1	2	yes	—
28	C₃₁×#544	C₂²	1	4	yes	complemented
29	C₃₁×#540	C₂²	1	4	yes	—
30	C₃₁×#535	C₂²	1	2	yes	—
31	C₃₁×#533	C₂²	1	1	yes	characteristic
32	C₃₁×#533	C₂²	1	1	yes	characteristic
33	C₃₁×#533	C₂²	1	2	yes	complemented
34	C₃₁×#532	C₂²	1	2	yes	—
35	C₃₁×#532	C₂²	1	2	yes	—
36	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	complemented
38	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
39	C₃₁×#520	C₂²	1	2	yes	—
40	C₃₁×#518	C₂²	1	2	yes	—
41	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#501	C₂²	1	4	yes	complemented
43	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
45	C₃₁×#500	C₂²	1	2	yes	complemented
46	C₃₁×#499	C₂²	1	4	yes	complemented
47	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
48	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
49	C₃₁×#498	C₂²	1	2	yes	complemented
50	C₃₁×#490	C₂²	1	4	yes	—
51	C₃₁×#486	C₂²	1	2	yes	—
52	C₃₁×#486	C₂²	1	2	yes	—
53	C₃₁×#486	C₂²	1	4	yes	—
54	C₃₁×#486	C₂²	1	2	yes	—
55	C₃₁×#486	C₂²	1	2	yes	—
56	C₃₁×#482	C₂²	1	8	yes	complemented
57	C₃₁×#482	C₂²	1	8	yes	complemented
58	C₃₁×#481	C₂²	1	8	yes	complemented
59	C₃₁×#480	C₂²	1	8	yes	complemented
60	C₃₁×#478	C₂²	1	8	yes	complemented
61	C₃₁×#478	C₂²	1	8	yes	complemented
62	C₃₁×#476	C₂²	1	8	yes	complemented
63	C₃₁×#474	C₂²	1	8	yes	complemented
64	C₃₁×#433	C₂²	1	4	yes	complemented
65	C₃₁×#431	C₂²	1	2	yes	complemented
66	C₃₁×#431	C₂²	1	2	yes	complemented
67	C₃₁×#428	C₂²	1	4	yes	—
68	C₃₁×#427	C₂²	1	4	yes	—
69	C₃₁×#424	C₂²	1	2	yes	—
70	C₃₁×#418	C₂²	1	4	yes	—
71	C₂×C₃₁×#102	C₂²	1	4	yes	—
72	C₃₁×#408	C₂²	1	2	yes	—
73	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
74	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
75	C₂×#41×C₃₁	—	2	2	no	—
76	C₂×#41×C₃₁	—	2	2	no	—
77	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
78	C₂²×Q₈×C₃₁	—	2	1	no	—
79	C₂²×Q₈×C₃₁	—	2	2	no	—
80	C₂²×D₄×C₃₁	—	2	1	no	—
81	C₂²×D₄×C₃₁	—	2	1	no	—
82	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
83	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
84	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
85	C₃₁×C₄⋊Q₈	—	2	2	no	—
86	C₃₁×C₄⋊Q₈	—	2	4	no	—
87	C₃₁×C₄⋊Q₈	—	2	1	no	—
88	C₃₁×C₄⋊Q₈	—	2	1	no	—
89	C₃₁×C₄⋊Q₈	—	2	2	no	—
90	C₃₁×C₄⋊Q₈	—	2	2	no	—
91	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
92	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
93	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
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×#127	—	2	2	no	—
99	C₃₁×#127	—	2	1	no	—
100	C₃₁×#127	—	2	1	no	—
101	C₃₁×#127	C₂³	1	2	yes	—
102	C₃₁×#127	C₂³	1	2	yes	—
103	C₃₁×#125	—	2	2	no	—
104	C₃₁×#125	—	2	4	no	—
105	C₃₁×#125	C₂³	1	4	yes	—
106	C₃₁×#125	C₂³	1	2	yes	—
107	C₃₁×#125	C₂³	1	2	yes	—
108	C₃₁×#124	—	2	2	no	—
109	C₃₁×#124	—	2	2	no	—
110	C₃₁×#124	—	2	2	no	—
111	C₃₁×#123	—	2	4	no	—
112	C₃₁×#123	—	2	4	no	—
113	C₃₁×C₂²⋊Q₈	—	2	1	no	—
114	C₃₁×C₂²⋊Q₈	—	2	1	no	—
115	C₃₁×C₂²⋊Q₈	—	2	1	no	—
116	C₃₁×C₂²⋊Q₈	—	2	1	no	—
117	C₃₁×C₂²⋊Q₈	—	2	2	no	—
118	C₃₁×C₂²⋊Q₈	—	2	2	no	—
119	C₃₁×C₂²⋊Q₈	—	2	4	no	—
120	C₃₁×C₂²⋊Q₈	—	2	4	no	—
121	C₃₁×C₂²⋊Q₈	—	2	4	no	—
122	C₃₁×#121	—	2	2	no	—
123	C₃₁×#121	—	2	2	no	—
124	C₃₁×#121	—	2	4	no	—
125	C₃₁×#121	—	2	4	no	—
126	C₃₁×#121	C₂³	1	4	yes	—
127	C₃₁×#121	C₂³	1	4	yes	—
128	C₃₁×#120	—	2	4	no	—
129	C₄×Q₈×C₃₁	—	2	2	no	—
130	C₄×Q₈×C₃₁	—	2	1	no	—
131	C₄×Q₈×C₃₁	—	2	1	no	—
132	C₄×Q₈×C₃₁	—	2	1	no	—
133	C₄×Q₈×C₃₁	—	2	1	no	—
134	C₄×Q₈×C₃₁	—	2	2	no	—
135	C₄×Q₈×C₃₁	—	2	2	no	—
136	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
137	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
138	C₄×D₄×C₃₁	—	2	4	no	—
139	C₄×D₄×C₃₁	—	2	2	no	—
140	C₄×D₄×C₃₁	—	2	2	no	—
141	C₄×D₄×C₃₁	—	2	2	no	—
142	C₄×D₄×C₃₁	—	2	2	no	—
143	C₄×D₄×C₃₁	—	2	2	no	—
144	C₄×D₄×C₃₁	D₄	1	2	yes	—
145	C₄×D₄×C₃₁	D₄	1	2	yes	—
146	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
147	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
148	C₄×D₄×C₃₁	C₂³	1	4	yes	—
149	C₄×D₄×C₃₁	C₂³	1	4	yes	—
150	C₃₁×#117	C₂³	1	2	yes	—
151	C₃₁×#117	C₂³	1	2	yes	—
152	C₃₁×#117	C₂³	1	4	yes	—
153	C₃₁×#117	C₂³	1	1	yes	characteristic
154	C₃₁×#117	C₂³	1	2	yes	—
155	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
156	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
158	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
159	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
160	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
161	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
162	C₃₁×#107	C₂³	1	4	yes	—
163	C₃₁×#107	C₂³	1	4	yes	—
164	C₃₁×#107	C₂³	1	8	yes	—
165	C₃₁×#106	C₂³	1	4	yes	—
166	C₃₁×#106	C₂³	1	4	yes	—
167	C₃₁×#106	C₂³	1	8	yes	—
168	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
169	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
170	C₃₁×#102	C₂³	1	8	yes	—
171	C₃₁×#102	C₂³	1	8	yes	—
172	C₃₁×#102	C₂³	1	8	yes	—
173	C₃₁×#102	C₂³	1	8	yes	—
174	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
175	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
176	C₃₁×#97	C₂³	1	2	yes	—
177	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
178	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
179	C₄×C₈×C₃₁	C₂³	1	2	yes	—
180	C₄×C₈×C₃₁	C₂³	1	2	yes	—
181	C₂⁴×C₃₁	—	2	2	no	—
182	#41×C₃₁	—	2	4	no	—
183	#41×C₃₁	—	8	4	no	—
184	#41×C₃₁	—	8	4	no	—
185	C₂×Q₈×C₃₁	—	2	1	no	—
186	C₂×Q₈×C₃₁	—	2	1	no	—
187	C₂×Q₈×C₃₁	—	2	1	no	—
188	C₂×Q₈×C₃₁	—	2	1	no	—
189	C₂×Q₈×C₃₁	—	2	2	no	—
190	C₂×Q₈×C₃₁	—	2	4	no	—
191	C₂×Q₈×C₃₁	—	2	2	no	—
192	C₂×Q₈×C₃₁	—	8	2	no	—
193	C₂×Q₈×C₃₁	—	8	4	no	—
194	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₂×D₄×C₃₁	—	2	4	no	—
196	C₂×D₄×C₃₁	—	2	4	no	—
197	C₂×D₄×C₃₁	—	2	2	no	—
198	C₂×D₄×C₃₁	—	2	2	no	—
199	C₂×D₄×C₃₁	—	2	2	no	—
200	C₂×D₄×C₃₁	—	2	2	no	—
201	C₂×D₄×C₃₁	—	2	2	no	—
202	C₂×D₄×C₃₁	—	2	8	no	—
203	C₂×D₄×C₃₁	—	8	2	no	—
204	C₂×D₄×C₃₁	—	8	2	no	—
205	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
206	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
207	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
208	C₂²×C₄×C₃₁	—	2	4	no	—
209	C₂²×C₄×C₃₁	—	2	2	no	—
210	C₂²×C₄×C₃₁	—	2	4	no	—
211	C₂²×C₄×C₃₁	—	2	4	no	—
212	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
213	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
214	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
215	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
216	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
217	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
218	M₁₆×C₃₁	—	2	4	no	—
219	C₂×C₈×C₃₁	—	2	4	no	—
220	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
221	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
222	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
223	C₄⋊C₄×C₃₁	—	2	2	no	—
224	C₄⋊C₄×C₃₁	—	2	1	no	—
225	C₄⋊C₄×C₃₁	—	2	4	no	—
226	C₄⋊C₄×C₃₁	—	2	2	no	—
227	C₄⋊C₄×C₃₁	—	2	4	no	—
228	C₄⋊C₄×C₃₁	—	2	2	no	—
229	C₄⋊C₄×C₃₁	—	2	1	no	—
230	C₄⋊C₄×C₃₁	—	2	1	no	—
231	C₄⋊C₄×C₃₁	—	2	1	no	—
232	C₄⋊C₄×C₃₁	—	2	2	no	—
233	C₄⋊C₄×C₃₁	—	2	2	no	—
234	C₄⋊C₄×C₃₁	—	2	4	no	—
235	C₄⋊C₄×C₃₁	—	2	2	no	—
236	C₄⋊C₄×C₃₁	—	2	4	no	—
237	C₄⋊C₄×C₃₁	—	2	2	no	—
238	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
239	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
240	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
241	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
242	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
243	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
244	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
245	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
246	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
247	C₂²⋊C₄×C₃₁	—	2	2	no	—
248	C₂²⋊C₄×C₃₁	—	2	4	no	—
249	C₂²⋊C₄×C₃₁	—	2	2	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	C₂×D₄	1	2	yes	—
255	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
256	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
257	C₄²×C₃₁	—	2	2	no	—
258	C₄²×C₃₁	—	2	4	no	—
259	C₄²×C₃₁	—	2	2	no	—
260	C₄²×C₃₁	C₂×D₄	1	2	yes	—
261	C₄²×C₃₁	C₂×D₄	1	2	yes	—
262	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
263	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
264	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
265	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
266	C₄²×C₃₁	C₂⁴	1	2	yes	—
267	#60877	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
268	C₂³×C₃₁	—	2	4	no	—
269	C₂³×C₃₁	—	4	4	no	—
270	C₂³×C₃₁	—	8	2	no	—
271	C₂³×C₃₁	—	8	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	—	4	2	no	—
276	Q₈×C₃₁	—	4	4	no	—
277	Q₈×C₃₁	—	4	2	no	—
278	Q₈×C₃₁	—	4	2	no	—
279	Q₈×C₃₁	—	4	2	no	—
280	Q₈×C₃₁	—	8	4	no	—
281	Q₈×C₃₁	—	8	2	no	—
282	D₄×C₃₁	—	2	4	no	—
283	D₄×C₃₁	—	2	2	no	—
284	D₄×C₃₁	—	2	8	no	—
285	D₄×C₃₁	—	4	4	no	—
286	D₄×C₃₁	—	4	4	no	—
287	D₄×C₃₁	—	4	4	no	—
288	D₄×C₃₁	—	8	2	no	—
289	D₄×C₃₁	—	8	2	no	—
290	D₄×C₃₁	—	8	4	no	—
291	D₄×C₃₁	—	8	4	no	—
292	C₂×C₄×C₃₁	—	2	2	no	—
293	C₂×C₄×C₃₁	—	2	4	no	—
294	C₂×C₄×C₃₁	—	2	2	no	—
295	C₂×C₄×C₃₁	—	2	4	no	—
296	C₂×C₄×C₃₁	—	2	4	no	—
297	C₂×C₄×C₃₁	—	2	2	no	—
298	C₂×C₄×C₃₁	—	2	2	no	—
299	C₂×C₄×C₃₁	—	2	2	no	—
300	C₂×C₄×C₃₁	—	2	4	no	—
301	C₂×C₄×C₃₁	—	2	2	no	—
302	C₂×C₄×C₃₁	—	8	4	no	—
303	C₂×C₄×C₃₁	—	8	4	no	—
304	C₂×C₄×C₃₁	—	8	2	no	—
305	C₂×C₄×C₃₁	—	8	4	no	—
306	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
307	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
308	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
309	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
310	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
311	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
312	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
313	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
314	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
315	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
316	C₈×C₃₁	—	2	2	no	—
317	C₈×C₃₁	—	2	2	no	—
318	C₈×C₃₁	—	2	4	no	—
319	#3508	C₂×C₃₁	1	1	yes	characteristic
320	#3503	C₂×C₃₁	1	1	yes	characteristic, complemented
321	#3241	C₂×C₃₁	1	8	yes	complemented
322	#3228	C₂×C₃₁	1	2	yes	complemented
323	#3228	C₂×C₃₁	1	2	yes	complemented
324	#3227	C₂×C₃₁	1	2	yes	complemented
325	#3226	C₂×C₃₁	1	2	yes	complemented
326	#3165	C₂×C₃₁	1	1	yes	characteristic, complemented
327	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
328	#3138	C₂×C₃₁	1	1	yes	characteristic, complemented
329	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
330	#3096	C₂×C₃₁	1	4	yes	complemented
331	#3081	C₂×C₃₁	1	4	yes	complemented
332	#2922	C₂×C₃₁	1	1	yes	characteristic, complemented
333	C₂²×C₃₁	—	2	1	no	—
334	C₂²×C₃₁	—	2	1	no	—
335	C₂²×C₃₁	—	2	2	no	—
336	C₂²×C₃₁	—	2	2	no	—
337	C₂²×C₃₁	—	2	2	no	—
338	C₂²×C₃₁	—	4	4	no	—
339	C₂²×C₃₁	—	4	4	no	—
340	C₂²×C₃₁	—	4	4	no	—
341	C₂²×C₃₁	—	16	4	no	—
342	C₂²×C₃₁	#536	1	2	yes	—
343	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
344	C₄×C₃₁	—	2	2	no	—
345	C₄×C₃₁	—	2	2	no	—
346	C₄×C₃₁	—	2	1	no	—
347	C₄×C₃₁	—	2	2	no	—
348	C₄×C₃₁	—	2	1	no	—
349	C₄×C₃₁	—	4	2	no	—
350	C₄×C₃₁	—	4	4	no	—
351	C₄×C₃₁	—	4	2	no	—
352	C₄×C₃₁	—	4	2	no	—
353	C₄×C₃₁	—	4	2	no	—
354	C₄×C₃₁	#536	1	2	yes	—
355	C₄×C₃₁	#576	1	2	yes	—
356	C₄×C₃₁	#577	1	2	yes	—
357	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
358	#559	—	2	1	no	—
359	#559	—	2	1	no	—
360	Q₈²	—	2	1	no	—
361	#557	—	2	1	no	—
362	#554	—	2	2	no	—
363	#554	—	2	1	no	—
364	#550	—	2	1	no	—
365	#550	—	2	1	no	—
366	D₄×Q₈	—	2	2	no	—
367	D₄×Q₈	—	2	1	no	—
368	#546	—	2	2	no	—
369	#546	—	2	2	no	—
370	#544	C₂²×C₃₁	1	2	yes	—
371	#544	C₂²×C₃₁	1	4	yes	complemented
372	#540	C₂²×C₃₁	1	4	yes	—
373	#535	C₂²×C₃₁	1	2	yes	—
374	#533	C₂²×C₃₁	1	1	yes	characteristic
375	#533	C₂²×C₃₁	1	1	yes	characteristic
376	#533	C₂²×C₃₁	1	2	yes	complemented
377	#532	C₂²×C₃₁	1	2	yes	—
378	#532	C₂²×C₃₁	1	2	yes	—
379	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
380	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
381	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
382	#520	C₂²×C₃₁	1	2	yes	—
383	#518	C₂²×C₃₁	1	2	yes	—
384	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
385	#501	C₂²×C₃₁	1	4	yes	complemented
386	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
387	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
388	#500	C₂²×C₃₁	1	2	yes	complemented
389	#499	C₂²×C₃₁	1	4	yes	complemented
390	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
391	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
392	#498	C₂²×C₃₁	1	2	yes	complemented
393	#490	C₂²×C₃₁	1	4	yes	—
394	#486	C₂²×C₃₁	1	2	yes	—
395	#486	C₂²×C₃₁	1	2	yes	—
396	#486	C₂²×C₃₁	1	4	yes	—
397	#486	C₂²×C₃₁	1	2	yes	—
398	#486	C₂²×C₃₁	1	2	yes	—
399	#482	C₂²×C₃₁	1	8	yes	complemented
400	#482	C₂²×C₃₁	1	8	yes	complemented
401	#481	C₂²×C₃₁	1	8	yes	complemented
402	#480	C₂²×C₃₁	1	8	yes	complemented
403	#478	C₂²×C₃₁	1	8	yes	complemented
404	#478	C₂²×C₃₁	1	8	yes	complemented
405	#476	C₂²×C₃₁	1	8	yes	complemented
406	#474	C₂²×C₃₁	1	8	yes	complemented
407	#433	C₂²×C₃₁	1	4	yes	complemented
408	#431	C₂²×C₃₁	1	2	yes	complemented
409	#431	C₂²×C₃₁	1	2	yes	complemented
410	#428	C₂²×C₃₁	1	4	yes	—
411	#427	C₂²×C₃₁	1	4	yes	—
412	#424	C₂²×C₃₁	1	2	yes	—
413	#418	C₂²×C₃₁	1	4	yes	—
414	C₂×#102	C₂²×C₃₁	1	4	yes	—
415	#408	C₂²×C₃₁	1	2	yes	—
416	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
417	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
418	C₂×C₃₁	—	2	2	no	—
419	C₂×C₃₁	—	4	2	no	—
420	C₂×C₃₁	—	8	4	no	—
421	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
422	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
423	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
424	C₂×#41	—	2	2	no	—
425	C₂×#41	—	2	2	no	—
426	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
427	C₂²×Q₈	—	2	1	no	—
428	C₂²×Q₈	—	2	2	no	—
429	C₂²×D₄	—	2	1	no	—
430	C₂²×D₄	—	2	1	no	—
431	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
432	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
433	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
434	C₄⋊Q₈	—	2	2	no	—
435	C₄⋊Q₈	—	2	4	no	—
436	C₄⋊Q₈	—	2	1	no	—
437	C₄⋊Q₈	—	2	1	no	—
438	C₄⋊Q₈	—	2	2	no	—
439	C₄⋊Q₈	—	2	2	no	—
440	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
441	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
442	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
443	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
444	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
445	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
446	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
447	#127	—	2	2	no	—
448	#127	—	2	1	no	—
449	#127	—	2	1	no	—
450	#127	C₂³×C₃₁	1	2	yes	—
451	#127	C₂³×C₃₁	1	2	yes	—
452	#125	—	2	2	no	—
453	#125	—	2	4	no	—
454	#125	C₂³×C₃₁	1	2	yes	—
455	#125	C₂³×C₃₁	1	2	yes	—
456	#125	C₂³×C₃₁	1	4	yes	—
457	#124	—	2	2	no	—
458	#124	—	2	2	no	—
459	#124	—	2	2	no	—
460	#123	—	2	4	no	—
461	#123	—	2	4	no	—
462	C₂²⋊Q₈	—	2	1	no	—
463	C₂²⋊Q₈	—	2	1	no	—
464	C₂²⋊Q₈	—	2	1	no	—
465	C₂²⋊Q₈	—	2	1	no	—
466	C₂²⋊Q₈	—	2	2	no	—
467	C₂²⋊Q₈	—	2	2	no	—
468	C₂²⋊Q₈	—	2	4	no	—
469	C₂²⋊Q₈	—	2	4	no	—
470	C₂²⋊Q₈	—	2	4	no	—
471	#121	—	2	2	no	—
472	#121	—	2	2	no	—
473	#121	—	2	4	no	—
474	#121	—	2	4	no	—
475	#121	C₂³×C₃₁	1	4	yes	—
476	#121	C₂³×C₃₁	1	4	yes	—
477	#120	—	2	4	no	—
478	C₄×Q₈	—	2	2	no	—
479	C₄×Q₈	—	2	1	no	—
480	C₄×Q₈	—	2	1	no	—
481	C₄×Q₈	—	2	1	no	—
482	C₄×Q₈	—	2	1	no	—
483	C₄×Q₈	—	2	2	no	—
484	C₄×Q₈	—	2	2	no	—
485	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
486	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
487	C₄×D₄	—	2	4	no	—
488	C₄×D₄	—	2	2	no	—
489	C₄×D₄	—	2	2	no	—
490	C₄×D₄	—	2	2	no	—
491	C₄×D₄	—	2	2	no	—
492	C₄×D₄	—	2	2	no	—
493	C₄×D₄	D₄×C₃₁	1	2	yes	—
494	C₄×D₄	D₄×C₃₁	1	2	yes	—
495	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
496	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
497	C₄×D₄	C₂³×C₃₁	1	4	yes	—
498	C₄×D₄	C₂³×C₃₁	1	4	yes	—
499	#117	C₂³×C₃₁	1	2	yes	—
500	#117	C₂³×C₃₁	1	2	yes	—
501	#117	C₂³×C₃₁	1	4	yes	—
502	#117	C₂³×C₃₁	1	1	yes	characteristic
503	#117	C₂³×C₃₁	1	2	yes	—
504	C₂×C₄⋊C₄	—	2	1	no	—
505	C₂×C₄⋊C₄	—	2	2	no	—
506	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
507	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
508	C₂×C₂²⋊C₄	—	2	4	no	—
509	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
510	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
511	#107	C₂³×C₃₁	1	4	yes	—
512	#107	C₂³×C₃₁	1	4	yes	—
513	#107	C₂³×C₃₁	1	8	yes	—
514	#106	C₂³×C₃₁	1	4	yes	—
515	#106	C₂³×C₃₁	1	4	yes	—
516	#106	C₂³×C₃₁	1	8	yes	—
517	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
518	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
519	#102	C₂³×C₃₁	1	8	yes	—
520	#102	C₂³×C₃₁	1	8	yes	—
521	#102	C₂³×C₃₁	1	8	yes	—
522	#102	C₂³×C₃₁	1	8	yes	—
523	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
524	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
525	#97	C₂³×C₃₁	1	2	yes	—
526	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
527	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
528	C₄×C₈	C₂³×C₃₁	1	2	yes	—
529	C₄×C₈	C₂³×C₃₁	1	2	yes	—
530	C₃₁	#60877	1	1	yes	characteristic, direct product factor, Sylow
531	C₂⁴	—	2	2	no	—
532	#41	—	2	4	no	—
533	#41	—	8	4	no	—
534	#41	—	8	4	no	—
535	C₂×Q₈	—	2	1	no	—
536	C₂×Q₈	—	2	1	no	—
537	C₂×Q₈	—	2	1	no	—
538	C₂×Q₈	—	2	1	no	—
539	C₂×Q₈	—	2	2	no	—
540	C₂×Q₈	—	2	4	no	—
541	C₂×Q₈	—	2	2	no	—
542	C₂×Q₈	—	8	2	no	—
543	C₂×Q₈	—	8	4	no	—
544	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
545	C₂×D₄	—	2	4	no	—
546	C₂×D₄	—	2	4	no	—
547	C₂×D₄	—	2	2	no	—
548	C₂×D₄	—	2	2	no	—
549	C₂×D₄	—	2	2	no	—
550	C₂×D₄	—	2	2	no	—
551	C₂×D₄	—	2	2	no	—
552	C₂×D₄	—	2	8	no	—
553	C₂×D₄	—	8	2	no	—
554	C₂×D₄	—	8	2	no	—
555	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
556	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
557	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
558	C₂²×C₄	—	2	4	no	—
559	C₂²×C₄	—	2	2	no	—
560	C₂²×C₄	—	2	4	no	—
561	C₂²×C₄	—	2	4	no	—
562	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
563	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
564	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
565	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
566	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
567	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
568	M₁₆	—	2	4	no	—
569	C₂×C₈	—	2	4	no	—
570	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
571	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
572	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	4	no	—
576	C₄⋊C₄	—	2	2	no	—
577	C₄⋊C₄	—	2	4	no	—
578	C₄⋊C₄	—	2	2	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	1	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	2	no	—
583	C₄⋊C₄	—	2	2	no	—
584	C₄⋊C₄	—	2	4	no	—
585	C₄⋊C₄	—	2	2	no	—
586	C₄⋊C₄	—	2	4	no	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
589	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
590	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
591	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
592	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
593	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
594	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
595	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
596	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
597	C₂²⋊C₄	—	2	2	no	—
598	C₂²⋊C₄	—	2	4	no	—
599	C₂²⋊C₄	—	2	2	no	—
600	C₂²⋊C₄	—	2	4	no	—
601	C₂²⋊C₄	—	2	4	no	—
602	C₂²⋊C₄	—	2	4	no	—
603	C₂²⋊C₄	—	2	4	no	—
604	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
605	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
606	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
607	C₄²	—	2	2	no	—
608	C₄²	—	2	4	no	—
609	C₄²	—	2	2	no	—
610	C₄²	C₂×D₄×C₃₁	1	2	yes	—
611	C₄²	C₂×D₄×C₃₁	1	2	yes	—
612	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
613	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
614	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
615	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
616	C₄²	C₂⁴×C₃₁	1	2	yes	—
617	C₂³	—	2	4	no	—
618	C₂³	—	4	4	no	—
619	C₂³	—	8	2	no	—
620	C₂³	—	8	2	no	—
621	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
622	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
623	Q₈	—	2	2	no	—
624	Q₈	—	4	2	no	—
625	Q₈	—	4	4	no	—
626	Q₈	—	4	2	no	—
627	Q₈	—	4	2	no	—
628	Q₈	—	4	2	no	—
629	Q₈	—	8	4	no	—
630	Q₈	—	8	2	no	—
631	D₄	—	2	4	no	—
632	D₄	—	2	2	no	—
633	D₄	—	2	8	no	—
634	D₄	—	4	4	no	—
635	D₄	—	4	4	no	—
636	D₄	—	4	4	no	—
637	D₄	—	8	2	no	—
638	D₄	—	8	2	no	—
639	D₄	—	8	4	no	—
640	D₄	—	8	4	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	4	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	4	no	—
645	C₂×C₄	—	2	4	no	—
646	C₂×C₄	—	2	2	no	—
647	C₂×C₄	—	2	2	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	4	no	—
650	C₂×C₄	—	2	2	no	—
651	C₂×C₄	—	8	4	no	—
652	C₂×C₄	—	8	4	no	—
653	C₂×C₄	—	8	2	no	—
654	C₂×C₄	—	8	4	no	—
655	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
656	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
657	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
658	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
659	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
660	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
661	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
662	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
663	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
664	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
665	C₈	—	2	2	no	—
666	C₈	—	2	2	no	—
667	C₈	—	2	4	no	—
668	C₂²	—	2	1	no	—
669	C₂²	—	2	1	no	—
670	C₂²	—	2	2	no	—
671	C₂²	—	2	2	no	—
672	C₂²	—	2	2	no	—
673	C₂²	—	4	4	no	—
674	C₂²	—	4	4	no	—
675	C₂²	—	4	4	no	—
676	C₂²	—	16	4	no	—
677	C₂²	C₃₁×#536	1	2	yes	—
678	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
679	C₄	—	2	2	no	—
680	C₄	—	2	2	no	—
681	C₄	—	2	1	no	—
682	C₄	—	2	2	no	—
683	C₄	—	2	1	no	—
684	C₄	—	4	2	no	—
685	C₄	—	4	4	no	—
686	C₄	—	4	2	no	—
687	C₄	—	4	2	no	—
688	C₄	—	4	2	no	—
689	C₄	C₃₁×#536	1	2	yes	—
690	C₄	C₃₁×#576	1	2	yes	—
691	C₄	C₃₁×#577	1	2	yes	—
692	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
693	C₂	—	2	2	no	—
694	C₂	—	4	2	no	—
695	C₂	—	8	4	no	—
696	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
697	C₂	C₂×C₃₁×#576	1	1	yes	characteristic
698	C₂	C₂×C₃₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3508	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×#3503	C₂	semidirect
3. C₃₁×#3241	C₂	semidirect
4. C₃₁×#3228	C₂	semidirect
5. C₃₁×#3228	C₂	semidirect
6. C₃₁×#3227	C₂	semidirect
7. C₃₁×#3226	C₂	semidirect
8. C₃₁×#3165	C₂	semidirect
9. C₃₁×#3160	C₂	semidirect
10. C₃₁×#3138	C₂	semidirect
11. C₂×C₃₁×#486	C₂	semidirect
12. C₃₁×#3096	C₂	semidirect
13. C₃₁×#3081	C₂	semidirect
14. C₃₁×#2922	C₂	semidirect
28. C₃₁×#544	C₂²	semidirect
33. C₃₁×#533	C₂²	semidirect
37. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
42. C₃₁×#501	C₂²	semidirect
43. C₃₁×#500	C₂²	semidirect
44. C₃₁×#500	C₂²	semidirect
45. C₃₁×#500	C₂²	semidirect
46. C₃₁×#499	C₂²	semidirect
47. C₃₁×#498	C₂²	semidirect
48. C₃₁×#498	C₂²	semidirect
49. C₃₁×#498	C₂²	semidirect
56. C₃₁×#482	C₂²	semidirect
57. C₃₁×#482	C₂²	semidirect
58. C₃₁×#481	C₂²	semidirect
59. C₃₁×#480	C₂²	semidirect
60. C₃₁×#478	C₂²	semidirect
61. C₃₁×#478	C₂²	semidirect
62. C₃₁×#476	C₂²	semidirect
63. C₃₁×#474	C₂²	semidirect
64. C₃₁×#433	C₂²	semidirect
65. C₃₁×#431	C₂²	semidirect
66. C₃₁×#431	C₂²	semidirect
267. #60877	C₃₁	direct (N × H)
320. #3503	C₂×C₃₁	semidirect
321. #3241	C₂×C₃₁	semidirect
322. #3228	C₂×C₃₁	semidirect
323. #3228	C₂×C₃₁	semidirect
324. #3227	C₂×C₃₁	semidirect
325. #3226	C₂×C₃₁	semidirect
326. #3165	C₂×C₃₁	semidirect
327. #3160	C₂×C₃₁	semidirect
328. #3138	C₂×C₃₁	semidirect
329. C₂×#486	C₂×C₃₁	semidirect
330. #3096	C₂×C₃₁	semidirect
331. #3081	C₂×C₃₁	semidirect
332. #2922	C₂×C₃₁	semidirect
371. #544	C₂²×C₃₁	semidirect
376. #533	C₂²×C₃₁	semidirect
380. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
385. #501	C₂²×C₃₁	semidirect
386. #500	C₂²×C₃₁	semidirect
387. #500	C₂²×C₃₁	semidirect
388. #500	C₂²×C₃₁	semidirect
389. #499	C₂²×C₃₁	semidirect
390. #498	C₂²×C₃₁	semidirect
391. #498	C₂²×C₃₁	semidirect
392. #498	C₂²×C₃₁	semidirect
399. #482	C₂²×C₃₁	semidirect
400. #482	C₂²×C₃₁	semidirect
401. #481	C₂²×C₃₁	semidirect
402. #480	C₂²×C₃₁	semidirect
403. #478	C₂²×C₃₁	semidirect
404. #478	C₂²×C₃₁	semidirect
405. #476	C₂²×C₃₁	semidirect
406. #474	C₂²×C₃₁	semidirect
407. #433	C₂²×C₃₁	semidirect
408. #431	C₂²×C₃₁	semidirect
409. #431	C₂²×C₃₁	semidirect
530. C₃₁	#60877	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×#3508	2
C₃₁×#3503	2
C₃₁×#3241	2
C₃₁×#3228	2
C₃₁×#3227	2
C₃₁×#3226	2
C₃₁×#3165	2
C₃₁×#3160	2
C₃₁×#3138	2
C₂×C₃₁×#486	2
C₃₁×#3096	2
C₃₁×#3081	2
C₃₁×#2922	2
#60877	31

Subgroups — C31×#60877 #21517636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60877 #21517636