Subgroups — C₃₁×#61335 #21518094

Order 7936.

1730 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3507	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3502	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3377	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3374	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3371	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3369	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3137	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3135	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#3084	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3078	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3044	C₂	1	2	yes	maximal, complemented
12	C₃₁×#3037	C₂	1	1	yes	maximal, characteristic
13	C₃₁×#2961	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2947	C₂	1	2	yes	maximal, complemented
15	C₂×C₃₁×#441	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#137	C₂²	1	2	yes	—
17	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#559	C₂²	1	2	yes	—
19	Q₈²×C₃₁	—	2	1	no	—
20	Q₈²×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#557	—	2	1	no	—
22	C₃₁×#557	—	2	2	no	—
23	C₃₁×#557	C₂²	1	2	yes	—
24	C₃₁×#557	C₂²	1	4	yes	—
25	C₃₁×#557	C₂²	1	2	yes	—
26	C₃₁×#556	—	2	1	no	—
27	C₃₁×#556	C₂²	1	4	yes	—
28	C₃₁×#554	—	2	1	no	—
29	C₃₁×#552	—	2	2	no	—
30	C₃₁×#544	—	2	2	no	—
31	C₃₁×#544	C₂²	1	2	yes	—
32	C₃₁×#539	—	2	2	no	—
33	C₃₁×#536	C₂²	1	2	yes	—
34	C₃₁×#533	—	2	2	no	—
35	C₃₁×#533	C₂²	1	1	yes	characteristic
36	C₃₁×#533	C₂²	1	1	yes	characteristic
37	C₃₁×#532	C₂²	1	1	yes	characteristic
38	C₃₁×#529	—	2	2	no	—
39	C₃₁×#520	C₂²	1	2	yes	—
40	C₃₁×#519	C₂²	1	2	yes	—
41	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
42	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×#491	C₂²	1	2	yes	—
44	C₃₁×#491	C₂²	1	4	yes	—
45	C₃₁×#490	C₂²	1	2	yes	—
46	C₃₁×#489	C₂²	1	2	yes	—
47	C₃₁×#489	C₂²	1	4	yes	—
48	C₃₁×#489	C₂²	1	2	yes	—
49	C₃₁×#488	C₂²	1	4	yes	—
50	C₃₁×#487	C₂²	1	4	yes	—
51	C₃₁×#479	C₂²	1	4	yes	—
52	C₃₁×#479	C₂²	1	4	yes	—
53	C₃₁×#478	C₂²	1	4	yes	—
54	C₃₁×#477	C₂²	1	2	yes	—
55	C₃₁×#476	C₂²	1	4	yes	—
56	C₃₁×#475	C₂²	1	2	yes	—
57	C₃₁×#475	C₂²	1	4	yes	—
58	C₃₁×#464	C₂²	1	4	yes	—
59	C₃₁×#464	C₂²	1	4	yes	—
60	C₃₁×#464	C₂²	1	4	yes	—
61	C₃₁×#462	C₂²	1	4	yes	—
62	C₃₁×#462	C₂²	1	2	yes	—
63	C₃₁×#462	C₂²	1	4	yes	—
64	C₃₁×#460	C₂²	1	2	yes	—
65	C₃₁×#446	C₂²	1	4	yes	—
66	C₃₁×#446	C₂²	1	4	yes	—
67	C₃₁×#441	C₂²	1	4	yes	—
68	C₃₁×#441	C₂²	1	4	yes	—
69	C₃₁×#441	C₂²	1	4	yes	—
70	C₃₁×#440	C₂²	1	4	yes	—
71	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
72	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
73	C₃₁×#428	C₂²	1	2	yes	—
74	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
75	C₃₁×#424	C₂²	1	2	yes	—
76	C₃₁×#423	C₂²	1	1	yes	characteristic
77	C₃₁×#419	C₂²	1	2	yes	—
78	C₃₁×#417	C₂²	1	2	yes	—
79	C₃₁×#416	C₂²	1	2	yes	—
80	C₃₁×#416	C₂²	1	2	yes	—
81	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
82	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
83	C₃₁×#405	C₂²	1	2	yes	—
84	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
85	C₂×#41×C₃₁	C₂³	1	2	yes	—
86	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₃₁×#137	—	2	8	no	—
89	C₂×Q₁₆×C₃₁	—	2	4	no	—
90	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
91	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
92	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
93	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
94	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₃₁×C₄⋊Q₈	—	2	2	no	—
97	C₃₁×C₄⋊Q₈	—	2	4	no	—
98	C₃₁×C₄⋊Q₈	—	2	2	no	—
99	C₃₁×C₄⋊Q₈	—	2	4	no	—
100	C₃₁×C₄⋊Q₈	—	2	2	no	—
101	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
102	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
103	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
104	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
105	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
106	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
107	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
108	C₃₁×#127	C₂³	1	1	yes	characteristic
109	C₃₁×#126	—	2	4	no	—
110	C₃₁×#126	—	2	4	no	—
111	C₃₁×#125	—	2	2	no	—
112	C₃₁×#125	—	2	4	no	—
113	C₃₁×#125	—	2	4	no	—
114	C₃₁×#125	—	2	4	no	—
115	C₃₁×#125	—	2	4	no	—
116	C₃₁×#125	—	2	2	no	—
117	C₃₁×#125	D₄	1	2	yes	—
118	C₃₁×#125	C₂³	1	4	yes	—
119	C₃₁×#125	C₂³	1	2	yes	—
120	C₃₁×#124	—	2	2	no	—
121	C₃₁×#124	C₂³	1	4	yes	—
122	C₃₁×#124	C₂³	1	2	yes	—
123	C₃₁×#123	—	2	4	no	—
124	C₃₁×#123	—	2	4	no	—
125	C₃₁×C₂²⋊Q₈	—	2	4	no	—
126	C₃₁×C₂²⋊Q₈	—	2	1	no	—
127	C₃₁×C₂²⋊Q₈	—	2	1	no	—
128	C₃₁×C₂²⋊Q₈	—	2	2	no	—
129	C₃₁×C₂²⋊Q₈	—	2	2	no	—
130	C₃₁×C₂²⋊Q₈	—	2	4	no	—
131	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
132	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
133	C₃₁×#121	—	2	1	no	—
134	C₃₁×#121	—	2	1	no	—
135	C₄×Q₈×C₃₁	—	2	2	no	—
136	C₄×Q₈×C₃₁	—	2	2	no	—
137	C₄×Q₈×C₃₁	—	2	2	no	—
138	C₄×Q₈×C₃₁	—	2	2	no	—
139	C₄×Q₈×C₃₁	—	2	4	no	—
140	C₄×Q₈×C₃₁	—	2	2	no	—
141	C₄×Q₈×C₃₁	—	2	1	no	—
142	C₄×Q₈×C₃₁	—	2	1	no	—
143	C₄×Q₈×C₃₁	—	2	1	no	—
144	C₄×Q₈×C₃₁	—	2	1	no	—
145	C₄×Q₈×C₃₁	—	2	1	no	—
146	C₄×Q₈×C₃₁	—	2	1	no	—
147	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
148	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
149	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
150	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
151	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
152	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
153	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
154	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
155	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
156	C₄×D₄×C₃₁	—	2	2	no	—
157	C₄×D₄×C₃₁	—	2	4	no	—
158	C₄×D₄×C₃₁	—	2	1	no	—
159	C₄×D₄×C₃₁	—	2	1	no	—
160	C₄×D₄×C₃₁	C₂³	1	4	yes	—
161	C₃₁×#117	—	2	2	no	—
162	C₃₁×#117	—	2	4	no	—
163	C₃₁×#117	—	2	4	no	—
164	C₃₁×#117	C₂³	1	2	yes	—
165	C₃₁×#117	C₂³	1	2	yes	—
166	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
168	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
170	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
171	C₂×C₄²×C₃₁	—	2	1	no	—
172	C₂×C₄²×C₃₁	—	2	1	no	—
173	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
174	C₃₁×#107	C₂³	1	4	yes	—
175	C₃₁×#106	C₂³	1	4	yes	—
176	C₃₁×#106	C₂³	1	4	yes	—
177	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
178	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
179	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
180	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
181	C₃₁×#103	C₂³	1	4	yes	—
182	C₃₁×#103	C₂³	1	4	yes	—
183	C₃₁×#103	C₂³	1	2	yes	—
184	C₃₁×#103	C₂³	1	2	yes	—
185	C₃₁×#103	C₂³	1	2	yes	—
186	C₃₁×#103	C₂³	1	2	yes	—
187	C₃₁×#103	C₂³	1	4	yes	—
188	C₃₁×#103	C₂³	1	4	yes	—
189	C₃₁×#103	C₂³	1	4	yes	—
190	C₃₁×#102	C₂³	1	2	yes	—
191	C₃₁×#102	C₂³	1	2	yes	—
192	C₃₁×#102	C₂³	1	2	yes	—
193	C₃₁×#102	C₂³	1	2	yes	—
194	C₃₁×#102	C₂³	1	4	yes	—
195	C₃₁×#97	C₂³	1	2	yes	—
196	C₃₁×#97	C₂³	1	2	yes	—
197	C₃₁×#97	C₂³	1	4	yes	—
198	C₄×C₈×C₃₁	C₂³	1	4	yes	—
199	#41×C₃₁	—	2	4	no	—
200	#41×C₃₁	—	2	4	no	—
201	C₂×Q₈×C₃₁	—	2	1	no	—
202	C₂×Q₈×C₃₁	—	2	1	no	—
203	C₂×Q₈×C₃₁	—	2	1	no	—
204	C₂×Q₈×C₃₁	—	2	1	no	—
205	C₂×Q₈×C₃₁	—	2	4	no	—
206	C₂×Q₈×C₃₁	—	2	2	no	—
207	C₂×Q₈×C₃₁	—	2	2	no	—
208	C₂×Q₈×C₃₁	—	2	2	no	—
209	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
210	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
211	C₂×Q₈×C₃₁	#41	1	4	yes	—
212	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
213	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
214	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
215	C₂×D₄×C₃₁	—	2	2	no	—
216	C₂×D₄×C₃₁	—	2	1	no	—
217	C₂×D₄×C₃₁	—	2	1	no	—
218	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
219	C₂²×C₄×C₃₁	—	2	1	no	—
220	C₂²×C₄×C₃₁	—	2	1	no	—
221	C₂²×C₄×C₃₁	—	2	1	no	—
222	C₂²×C₄×C₃₁	—	2	1	no	—
223	C₂²×C₄×C₃₁	—	2	2	no	—
224	C₂²×C₄×C₃₁	—	2	4	no	—
225	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
226	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
227	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
228	Q₁₆×C₃₁	—	2	8	no	—
229	Q₁₆×C₃₁	—	2	8	no	—
230	SD₁₆×C₃₁	—	2	8	no	—
231	M₁₆×C₃₁	—	2	4	no	—
232	C₂×C₈×C₃₁	—	2	2	no	—
233	C₂×C₈×C₃₁	—	2	2	no	—
234	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
235	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
236	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
237	C₄⋊C₄×C₃₁	—	2	1	no	—
238	C₄⋊C₄×C₃₁	—	2	4	no	—
239	C₄⋊C₄×C₃₁	—	2	2	no	—
240	C₄⋊C₄×C₃₁	—	2	2	no	—
241	C₄⋊C₄×C₃₁	—	2	2	no	—
242	C₄⋊C₄×C₃₁	—	2	2	no	—
243	C₄⋊C₄×C₃₁	—	2	1	no	—
244	C₄⋊C₄×C₃₁	—	2	4	no	—
245	C₄⋊C₄×C₃₁	—	2	4	no	—
246	C₄⋊C₄×C₃₁	—	2	4	no	—
247	C₄⋊C₄×C₃₁	—	2	1	no	—
248	C₄⋊C₄×C₃₁	—	2	1	no	—
249	C₄⋊C₄×C₃₁	—	2	2	no	—
250	C₄⋊C₄×C₃₁	—	2	1	no	—
251	C₄⋊C₄×C₃₁	—	2	4	no	—
252	C₄⋊C₄×C₃₁	—	2	1	no	—
253	C₄⋊C₄×C₃₁	—	2	4	no	—
254	C₄⋊C₄×C₃₁	—	2	4	no	—
255	C₄⋊C₄×C₃₁	—	2	2	no	—
256	C₄⋊C₄×C₃₁	—	2	4	no	—
257	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
258	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
259	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
260	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
261	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
262	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
263	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
264	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
265	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
266	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
267	C₂²⋊C₄×C₃₁	—	2	2	no	—
268	C₂²⋊C₄×C₃₁	—	2	2	no	—
269	C₂²⋊C₄×C₃₁	—	2	4	no	—
270	C₂²⋊C₄×C₃₁	—	2	4	no	—
271	C₂²⋊C₄×C₃₁	—	2	2	no	—
272	C₂²⋊C₄×C₃₁	—	2	4	no	—
273	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
274	C₄²×C₃₁	—	2	2	no	—
275	C₄²×C₃₁	—	2	2	no	—
276	C₄²×C₃₁	—	2	2	no	—
277	C₄²×C₃₁	—	2	2	no	—
278	C₄²×C₃₁	—	2	4	no	—
279	C₄²×C₃₁	—	2	4	no	—
280	C₄²×C₃₁	—	2	4	no	—
281	C₄²×C₃₁	—	2	2	no	—
282	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
283	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
284	C₄²×C₃₁	C₂⁴	1	4	yes	—
285	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
286	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
287	#61335	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
288	C₂³×C₃₁	—	2	2	no	—
289	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
290	Q₈×C₃₁	—	2	4	no	—
291	Q₈×C₃₁	—	2	2	no	—
292	Q₈×C₃₁	—	2	2	no	—
293	Q₈×C₃₁	—	2	2	no	—
294	Q₈×C₃₁	—	2	2	no	—
295	Q₈×C₃₁	—	2	2	no	—
296	Q₈×C₃₁	—	4	2	no	—
297	Q₈×C₃₁	—	4	2	no	—
298	Q₈×C₃₁	—	4	2	no	—
299	Q₈×C₃₁	—	4	2	no	—
300	Q₈×C₃₁	—	8	1	no	—
301	Q₈×C₃₁	—	8	4	no	—
302	Q₈×C₃₁	C₂×#41	1	4	yes	—
303	D₄×C₃₁	—	2	2	no	—
304	D₄×C₃₁	—	2	2	no	—
305	D₄×C₃₁	—	4	2	no	—
306	D₄×C₃₁	—	4	4	no	—
307	D₄×C₃₁	—	8	1	no	—
308	C₂×C₄×C₃₁	—	2	4	no	—
309	C₂×C₄×C₃₁	—	2	2	no	—
310	C₂×C₄×C₃₁	—	2	2	no	—
311	C₂×C₄×C₃₁	—	2	2	no	—
312	C₂×C₄×C₃₁	—	2	2	no	—
313	C₂×C₄×C₃₁	—	2	2	no	—
314	C₂×C₄×C₃₁	—	2	2	no	—
315	C₂×C₄×C₃₁	—	2	2	no	—
316	C₂×C₄×C₃₁	—	4	1	no	—
317	C₂×C₄×C₃₁	—	4	1	no	—
318	C₂×C₄×C₃₁	—	4	2	no	—
319	C₂×C₄×C₃₁	—	4	4	no	—
320	C₂×C₄×C₃₁	—	8	4	no	—
321	C₂×C₄×C₃₁	—	8	1	no	—
322	C₂×C₄×C₃₁	—	8	1	no	—
323	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
324	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
325	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
326	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
327	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
328	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
329	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
330	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
331	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
332	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
333	C₈×C₃₁	—	2	2	no	—
334	C₈×C₃₁	—	2	4	no	—
335	C₈×C₃₁	—	2	2	no	—
336	#3507	C₂×C₃₁	1	1	yes	characteristic
337	#3502	C₂×C₃₁	1	1	yes	characteristic, complemented
338	#3377	C₂×C₃₁	1	4	yes	complemented
339	#3374	C₂×C₃₁	1	4	yes	complemented
340	#3371	C₂×C₃₁	1	4	yes	complemented
341	#3369	C₂×C₃₁	1	4	yes	complemented
342	#3137	C₂×C₃₁	1	2	yes	complemented
343	#3135	C₂×C₃₁	1	1	yes	characteristic
344	#3084	C₂×C₃₁	1	2	yes	complemented
345	#3078	C₂×C₃₁	1	1	yes	characteristic, complemented
346	#3044	C₂×C₃₁	1	2	yes	complemented
347	#3037	C₂×C₃₁	1	1	yes	characteristic
348	#2961	C₂×C₃₁	1	1	yes	characteristic, complemented
349	#2947	C₂×C₃₁	1	2	yes	complemented
350	C₂×#441	C₂×C₃₁	1	1	yes	characteristic, complemented
351	C₂²×C₃₁	—	2	1	no	—
352	C₂²×C₃₁	—	2	1	no	—
353	C₂²×C₃₁	—	4	2	no	—
354	C₂²×C₃₁	—	4	2	no	—
355	C₂²×C₃₁	—	4	2	no	—
356	C₂²×C₃₁	#548	1	2	yes	—
357	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
358	C₄×C₃₁	—	2	1	no	—
359	C₄×C₃₁	—	2	1	no	—
360	C₄×C₃₁	—	2	1	no	—
361	C₄×C₃₁	—	2	1	no	—
362	C₄×C₃₁	—	2	4	no	—
363	C₄×C₃₁	—	2	4	no	—
364	C₄×C₃₁	—	2	4	no	—
365	C₄×C₃₁	—	4	2	no	—
366	C₄×C₃₁	—	4	2	no	—
367	C₄×C₃₁	—	4	2	no	—
368	C₄×C₃₁	—	4	2	no	—
369	C₄×C₃₁	—	4	2	no	—
370	C₄×C₃₁	—	4	2	no	—
371	C₄×C₃₁	#548	1	2	yes	—
372	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
373	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
374	C₂×#137	C₂²×C₃₁	1	2	yes	—
375	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
376	#559	C₂²×C₃₁	1	2	yes	—
377	Q₈²	—	2	1	no	—
378	Q₈²	C₂²×C₃₁	1	2	yes	—
379	#557	—	2	1	no	—
380	#557	—	2	2	no	—
381	#557	C₂²×C₃₁	1	2	yes	—
382	#557	C₂²×C₃₁	1	4	yes	—
383	#557	C₂²×C₃₁	1	2	yes	—
384	#556	—	2	1	no	—
385	#556	C₂²×C₃₁	1	4	yes	—
386	#554	—	2	1	no	—
387	#552	—	2	2	no	—
388	#544	—	2	2	no	—
389	#544	C₂²×C₃₁	1	2	yes	—
390	#539	—	2	2	no	—
391	#536	C₂²×C₃₁	1	2	yes	—
392	#533	—	2	2	no	—
393	#533	C₂²×C₃₁	1	1	yes	characteristic
394	#533	C₂²×C₃₁	1	1	yes	characteristic
395	#532	C₂²×C₃₁	1	1	yes	characteristic
396	#529	—	2	2	no	—
397	#520	C₂²×C₃₁	1	2	yes	—
398	#519	C₂²×C₃₁	1	2	yes	—
399	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
400	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
401	#491	C₂²×C₃₁	1	2	yes	—
402	#491	C₂²×C₃₁	1	4	yes	—
403	#490	C₂²×C₃₁	1	2	yes	—
404	#489	C₂²×C₃₁	1	2	yes	—
405	#489	C₂²×C₃₁	1	4	yes	—
406	#489	C₂²×C₃₁	1	2	yes	—
407	#488	C₂²×C₃₁	1	4	yes	—
408	#487	C₂²×C₃₁	1	4	yes	—
409	#479	C₂²×C₃₁	1	4	yes	—
410	#479	C₂²×C₃₁	1	4	yes	—
411	#478	C₂²×C₃₁	1	4	yes	—
412	#477	C₂²×C₃₁	1	2	yes	—
413	#476	C₂²×C₃₁	1	4	yes	—
414	#475	C₂²×C₃₁	1	2	yes	—
415	#475	C₂²×C₃₁	1	4	yes	—
416	#464	C₂²×C₃₁	1	4	yes	—
417	#464	C₂²×C₃₁	1	4	yes	—
418	#464	C₂²×C₃₁	1	4	yes	—
419	#462	C₂²×C₃₁	1	4	yes	—
420	#462	C₂²×C₃₁	1	2	yes	—
421	#462	C₂²×C₃₁	1	4	yes	—
422	#460	C₂²×C₃₁	1	2	yes	—
423	#446	C₂²×C₃₁	1	4	yes	—
424	#446	C₂²×C₃₁	1	4	yes	—
425	#441	C₂²×C₃₁	1	4	yes	—
426	#441	C₂²×C₃₁	1	4	yes	—
427	#441	C₂²×C₃₁	1	4	yes	—
428	#440	C₂²×C₃₁	1	4	yes	—
429	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
430	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
431	#428	C₂²×C₃₁	1	2	yes	—
432	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
433	#424	C₂²×C₃₁	1	2	yes	—
434	#423	C₂²×C₃₁	1	1	yes	characteristic
435	#419	C₂²×C₃₁	1	2	yes	—
436	#417	C₂²×C₃₁	1	2	yes	—
437	#416	C₂²×C₃₁	1	2	yes	—
438	#416	C₂²×C₃₁	1	2	yes	—
439	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
440	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
441	#405	C₂²×C₃₁	1	2	yes	—
442	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
443	C₂×C₃₁	—	2	2	no	—
444	C₂×C₃₁	—	8	2	no	—
445	C₂×C₃₁	C₂×#548	1	1	yes	characteristic
446	C₂×C₃₁	#3586	1	1	yes	characteristic
447	C₂×C₃₁	#3586	1	1	yes	characteristic
448	C₂×#41	C₂³×C₃₁	1	2	yes	—
449	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
450	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
451	#137	—	2	8	no	—
452	C₂×Q₁₆	—	2	4	no	—
453	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
454	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
455	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
456	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
457	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
458	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
459	C₄⋊Q₈	—	2	2	no	—
460	C₄⋊Q₈	—	2	4	no	—
461	C₄⋊Q₈	—	2	2	no	—
462	C₄⋊Q₈	—	2	4	no	—
463	C₄⋊Q₈	—	2	2	no	—
464	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
465	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
466	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
467	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
468	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
469	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
470	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
471	#127	C₂³×C₃₁	1	1	yes	characteristic
472	#126	—	2	4	no	—
473	#126	—	2	4	no	—
474	#125	—	2	2	no	—
475	#125	—	2	4	no	—
476	#125	—	2	4	no	—
477	#125	—	2	4	no	—
478	#125	—	2	4	no	—
479	#125	—	2	2	no	—
480	#125	D₄×C₃₁	1	2	yes	—
481	#125	C₂³×C₃₁	1	2	yes	—
482	#125	C₂³×C₃₁	1	4	yes	—
483	#124	—	2	2	no	—
484	#124	C₂³×C₃₁	1	2	yes	—
485	#124	C₂³×C₃₁	1	4	yes	—
486	#123	—	2	4	no	—
487	#123	—	2	4	no	—
488	C₂²⋊Q₈	—	2	4	no	—
489	C₂²⋊Q₈	—	2	1	no	—
490	C₂²⋊Q₈	—	2	1	no	—
491	C₂²⋊Q₈	—	2	2	no	—
492	C₂²⋊Q₈	—	2	2	no	—
493	C₂²⋊Q₈	—	2	4	no	—
494	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
495	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
496	#121	—	2	1	no	—
497	#121	—	2	1	no	—
498	C₄×Q₈	—	2	2	no	—
499	C₄×Q₈	—	2	2	no	—
500	C₄×Q₈	—	2	2	no	—
501	C₄×Q₈	—	2	2	no	—
502	C₄×Q₈	—	2	4	no	—
503	C₄×Q₈	—	2	2	no	—
504	C₄×Q₈	—	2	1	no	—
505	C₄×Q₈	—	2	1	no	—
506	C₄×Q₈	—	2	1	no	—
507	C₄×Q₈	—	2	1	no	—
508	C₄×Q₈	—	2	1	no	—
509	C₄×Q₈	—	2	1	no	—
510	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
511	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
512	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
513	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
514	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
515	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
516	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
517	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
518	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
519	C₄×D₄	—	2	2	no	—
520	C₄×D₄	—	2	4	no	—
521	C₄×D₄	—	2	1	no	—
522	C₄×D₄	—	2	1	no	—
523	C₄×D₄	C₂³×C₃₁	1	4	yes	—
524	#117	—	2	2	no	—
525	#117	—	2	4	no	—
526	#117	—	2	4	no	—
527	#117	C₂³×C₃₁	1	2	yes	—
528	#117	C₂³×C₃₁	1	2	yes	—
529	C₂×C₄⋊C₄	—	2	1	no	—
530	C₂×C₄⋊C₄	—	2	1	no	—
531	C₂×C₄⋊C₄	—	2	2	no	—
532	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
533	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
534	C₂×C₄²	—	2	1	no	—
535	C₂×C₄²	—	2	1	no	—
536	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
537	#107	C₂³×C₃₁	1	4	yes	—
538	#106	C₂³×C₃₁	1	4	yes	—
539	#106	C₂³×C₃₁	1	4	yes	—
540	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
541	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
542	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
543	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
544	#103	C₂³×C₃₁	1	4	yes	—
545	#103	C₂³×C₃₁	1	4	yes	—
546	#103	C₂³×C₃₁	1	2	yes	—
547	#103	C₂³×C₃₁	1	2	yes	—
548	#103	C₂³×C₃₁	1	2	yes	—
549	#103	C₂³×C₃₁	1	2	yes	—
550	#103	C₂³×C₃₁	1	4	yes	—
551	#103	C₂³×C₃₁	1	4	yes	—
552	#103	C₂³×C₃₁	1	4	yes	—
553	#102	C₂³×C₃₁	1	2	yes	—
554	#102	C₂³×C₃₁	1	2	yes	—
555	#102	C₂³×C₃₁	1	2	yes	—
556	#102	C₂³×C₃₁	1	2	yes	—
557	#102	C₂³×C₃₁	1	4	yes	—
558	#97	C₂³×C₃₁	1	2	yes	—
559	#97	C₂³×C₃₁	1	2	yes	—
560	#97	C₂³×C₃₁	1	4	yes	—
561	C₄×C₈	C₂³×C₃₁	1	4	yes	—
562	C₃₁	#61335	1	1	yes	characteristic, direct product factor, Sylow
563	#41	—	2	4	no	—
564	#41	—	2	4	no	—
565	C₂×Q₈	—	2	1	no	—
566	C₂×Q₈	—	2	1	no	—
567	C₂×Q₈	—	2	1	no	—
568	C₂×Q₈	—	2	1	no	—
569	C₂×Q₈	—	2	4	no	—
570	C₂×Q₈	—	2	2	no	—
571	C₂×Q₈	—	2	2	no	—
572	C₂×Q₈	—	2	2	no	—
573	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
574	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
575	C₂×Q₈	#41×C₃₁	1	4	yes	—
576	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
577	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
578	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
579	C₂×D₄	—	2	2	no	—
580	C₂×D₄	—	2	1	no	—
581	C₂×D₄	—	2	1	no	—
582	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
583	C₂²×C₄	—	2	1	no	—
584	C₂²×C₄	—	2	1	no	—
585	C₂²×C₄	—	2	1	no	—
586	C₂²×C₄	—	2	1	no	—
587	C₂²×C₄	—	2	2	no	—
588	C₂²×C₄	—	2	4	no	—
589	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
590	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
591	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
592	Q₁₆	—	2	8	no	—
593	Q₁₆	—	2	8	no	—
594	SD₁₆	—	2	8	no	—
595	M₁₆	—	2	4	no	—
596	C₂×C₈	—	2	2	no	—
597	C₂×C₈	—	2	2	no	—
598	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
599	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
600	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
601	C₄⋊C₄	—	2	1	no	—
602	C₄⋊C₄	—	2	4	no	—
603	C₄⋊C₄	—	2	2	no	—
604	C₄⋊C₄	—	2	2	no	—
605	C₄⋊C₄	—	2	2	no	—
606	C₄⋊C₄	—	2	2	no	—
607	C₄⋊C₄	—	2	1	no	—
608	C₄⋊C₄	—	2	4	no	—
609	C₄⋊C₄	—	2	4	no	—
610	C₄⋊C₄	—	2	4	no	—
611	C₄⋊C₄	—	2	1	no	—
612	C₄⋊C₄	—	2	1	no	—
613	C₄⋊C₄	—	2	2	no	—
614	C₄⋊C₄	—	2	1	no	—
615	C₄⋊C₄	—	2	4	no	—
616	C₄⋊C₄	—	2	1	no	—
617	C₄⋊C₄	—	2	4	no	—
618	C₄⋊C₄	—	2	4	no	—
619	C₄⋊C₄	—	2	2	no	—
620	C₄⋊C₄	—	2	4	no	—
621	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
622	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
623	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
624	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
625	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
626	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
627	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
628	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
629	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
630	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
631	C₂²⋊C₄	—	2	2	no	—
632	C₂²⋊C₄	—	2	2	no	—
633	C₂²⋊C₄	—	2	4	no	—
634	C₂²⋊C₄	—	2	4	no	—
635	C₂²⋊C₄	—	2	2	no	—
636	C₂²⋊C₄	—	2	4	no	—
637	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
638	C₄²	—	2	2	no	—
639	C₄²	—	2	2	no	—
640	C₄²	—	2	2	no	—
641	C₄²	—	2	2	no	—
642	C₄²	—	2	4	no	—
643	C₄²	—	2	4	no	—
644	C₄²	—	2	4	no	—
645	C₄²	—	2	2	no	—
646	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
647	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
648	C₄²	C₂⁴×C₃₁	1	4	yes	—
649	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
650	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
651	C₂³	—	2	2	no	—
652	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
653	Q₈	—	2	4	no	—
654	Q₈	—	2	2	no	—
655	Q₈	—	2	2	no	—
656	Q₈	—	2	2	no	—
657	Q₈	—	2	2	no	—
658	Q₈	—	2	2	no	—
659	Q₈	—	4	2	no	—
660	Q₈	—	4	2	no	—
661	Q₈	—	4	2	no	—
662	Q₈	—	4	2	no	—
663	Q₈	—	8	1	no	—
664	Q₈	—	8	4	no	—
665	Q₈	C₂×#41×C₃₁	1	4	yes	—
666	D₄	—	2	2	no	—
667	D₄	—	2	2	no	—
668	D₄	—	4	2	no	—
669	D₄	—	4	4	no	—
670	D₄	—	8	1	no	—
671	C₂×C₄	—	2	4	no	—
672	C₂×C₄	—	2	2	no	—
673	C₂×C₄	—	2	2	no	—
674	C₂×C₄	—	2	2	no	—
675	C₂×C₄	—	2	2	no	—
676	C₂×C₄	—	2	2	no	—
677	C₂×C₄	—	2	2	no	—
678	C₂×C₄	—	2	2	no	—
679	C₂×C₄	—	4	1	no	—
680	C₂×C₄	—	4	1	no	—
681	C₂×C₄	—	4	2	no	—
682	C₂×C₄	—	4	4	no	—
683	C₂×C₄	—	8	4	no	—
684	C₂×C₄	—	8	1	no	—
685	C₂×C₄	—	8	1	no	—
686	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
687	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
688	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
689	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
690	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
691	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
692	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
693	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
694	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
695	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
696	C₈	—	2	2	no	—
697	C₈	—	2	4	no	—
698	C₈	—	2	2	no	—
699	C₂²	—	2	1	no	—
700	C₂²	—	2	1	no	—
701	C₂²	—	4	2	no	—
702	C₂²	—	4	2	no	—
703	C₂²	—	4	2	no	—
704	C₂²	C₃₁×#548	1	2	yes	—
705	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
706	C₄	—	2	4	no	—
707	C₄	—	2	1	no	—
708	C₄	—	2	1	no	—
709	C₄	—	2	1	no	—
710	C₄	—	2	1	no	—
711	C₄	—	2	4	no	—
712	C₄	—	2	4	no	—
713	C₄	—	4	2	no	—
714	C₄	—	4	2	no	—
715	C₄	—	4	2	no	—
716	C₄	—	4	2	no	—
717	C₄	—	4	2	no	—
718	C₄	—	4	2	no	—
719	C₄	C₃₁×#548	1	2	yes	—
720	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
721	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
722	C₂	—	2	2	no	—
723	C₂	—	8	2	no	—
724	C₂	C₂×C₃₁×#548	1	1	yes	characteristic
725	C₂	C₃₁×#3586	1	1	yes	characteristic
726	C₂	C₃₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#533 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3502	C₂	semidirect
3. C₃₁×#3377	C₂	semidirect
4. C₃₁×#3374	C₂	semidirect
5. C₃₁×#3371	C₂	semidirect
6. C₃₁×#3369	C₂	semidirect
7. C₃₁×#3137	C₂	semidirect
9. C₃₁×#3084	C₂	semidirect
10. C₃₁×#3078	C₂	semidirect
11. C₃₁×#3044	C₂	semidirect
13. C₃₁×#2961	C₂	semidirect
14. C₃₁×#2947	C₂	semidirect
15. C₂×C₃₁×#441	C₂	semidirect
287. #61335	C₃₁	direct (N × H)
337. #3502	C₂×C₃₁	semidirect
338. #3377	C₂×C₃₁	semidirect
339. #3374	C₂×C₃₁	semidirect
340. #3371	C₂×C₃₁	semidirect
341. #3369	C₂×C₃₁	semidirect
342. #3137	C₂×C₃₁	semidirect
344. #3084	C₂×C₃₁	semidirect
345. #3078	C₂×C₃₁	semidirect
346. #3044	C₂×C₃₁	semidirect
348. #2961	C₂×C₃₁	semidirect
349. #2947	C₂×C₃₁	semidirect
350. C₂×#441	C₂×C₃₁	semidirect
562. C₃₁	#61335	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×#3507	2
C₃₁×#3502	2
C₃₁×#3377	2
C₃₁×#3374	2
C₃₁×#3371	2
C₃₁×#3369	2
C₃₁×#3137	2
C₃₁×#3135	2
C₃₁×#3084	2
C₃₁×#3078	2
C₃₁×#3044	2
C₃₁×#3037	2
C₃₁×#2961	2
C₃₁×#2947	2
C₂×C₃₁×#441	2
#61335	31

Subgroups — C31×#61335 #21518094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61335 #21518094