Subgroups — C₃₁×#62540 #21519299

Order 7936.

1754 conjugacy classes of subgroups.

The subgroup lattice has 690 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×#3503	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3372	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3329	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3320	C₂	1	8	yes	maximal, complemented
6	C₃₁×#3231	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3230	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3206	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3206	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3182	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3167	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3139	C₂	1	2	yes	maximal, complemented
13	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic, complemented
15	Q₈²×C₃₁	—	2	1	no	—
16	C₃₁×#557	C₂²	1	2	yes	—
17	C₃₁×#556	—	2	1	no	—
18	C₃₁×#555	C₂²	1	2	yes	—
19	C₃₁×#554	—	2	1	no	—
20	C₃₁×#554	—	2	2	no	—
21	C₃₁×#554	C₂²	1	2	yes	—
22	C₃₁×#553	C₂²	1	4	yes	—
23	D₄×Q₈×C₃₁	C₂²	1	4	yes	—
24	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#548	—	2	2	no	—
26	C₃₁×#547	—	2	1	no	—
27	C₃₁×#544	—	2	1	no	—
28	C₃₁×#544	—	2	2	no	—
29	C₃₁×#542	—	2	2	no	—
30	C₃₁×#537	—	2	1	no	—
31	C₃₁×#533	C₂²	1	1	yes	characteristic
32	C₃₁×#533	C₂²	1	2	yes	—
33	C₃₁×#533	C₂²	1	2	yes	—
34	C₃₁×#532	C₂²	1	1	yes	characteristic
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	—
38	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
39	C₃₁×#520	—	2	1	no	—
40	C₃₁×#520	—	2	1	no	—
41	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#500	C₂²	1	2	yes	—
43	C₃₁×#498	C₂²	1	1	yes	characteristic
44	C₃₁×#492	C₂²	1	1	yes	characteristic
45	C₃₁×#492	C₂²	1	2	yes	—
46	C₃₁×#491	C₂²	1	8	yes	—
47	C₃₁×#487	C₂²	1	4	yes	—
48	C₃₁×#487	C₂²	1	2	yes	—
49	C₃₁×#487	C₂²	1	2	yes	—
50	C₃₁×#487	C₂²	1	1	yes	characteristic
51	C₃₁×#487	C₂²	1	2	yes	—
52	C₃₁×#487	C₂²	1	2	yes	—
53	C₃₁×#486	C₂²	1	1	yes	characteristic
54	C₃₁×#485	C₂²	1	8	yes	—
55	C₃₁×#484	C₂²	1	4	yes	—
56	C₃₁×#479	C₂²	1	8	yes	—
57	C₃₁×#477	C₂²	1	4	yes	—
58	C₃₁×#468	C₂²	1	8	yes	—
59	C₃₁×#465	C₂²	1	4	yes	—
60	C₃₁×#461	C₂²	1	8	yes	—
61	C₃₁×#461	C₂²	1	4	yes	—
62	C₃₁×#448	C₂²	1	4	yes	—
63	C₃₁×#448	C₂²	1	8	yes	—
64	C₃₁×#446	C₂²	1	2	yes	—
65	C₃₁×#440	C₂²	1	8	yes	—
66	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
67	C₃₁×#436	C₂²	1	2	yes	—
68	C₃₁×#435	C₂²	1	4	yes	—
69	C₃₁×#431	C₂²	1	1	yes	characteristic
70	C₃₁×#431	C₂²	1	2	yes	—
71	C₃₁×#419	C₂²	1	8	yes	—
72	C₂×C₃₁×#103	C₂²	1	4	yes	—
73	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
74	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
75	C₂×#41×C₃₁	—	2	1	no	—
76	C₂×#41×C₃₁	—	2	2	no	—
77	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
78	C₂²×Q₈×C₃₁	—	2	1	no	—
79	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
80	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
81	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
82	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
83	C₂×M₁₆×C₃₁	C₂³	1	4	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	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	2	no	—
93	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
94	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
95	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
96	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
97	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
98	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
99	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
100	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
101	C₃₁×#127	C₂³	1	1	yes	characteristic
102	C₃₁×#126	—	2	4	no	—
103	C₃₁×#125	—	2	2	no	—
104	C₃₁×#125	—	2	4	no	—
105	C₃₁×#125	—	2	2	no	—
106	C₃₁×#125	—	2	1	no	—
107	C₃₁×#125	C₂³	1	4	yes	—
108	C₃₁×#124	—	2	4	no	—
109	C₃₁×#124	—	2	1	no	—
110	C₃₁×#124	—	2	2	no	—
111	C₃₁×#123	—	2	4	no	—
112	C₃₁×C₂²⋊Q₈	—	2	2	no	—
113	C₃₁×C₂²⋊Q₈	—	2	2	no	—
114	C₃₁×C₂²⋊Q₈	—	2	2	no	—
115	C₃₁×C₂²⋊Q₈	—	2	1	no	—
116	C₃₁×C₂²⋊Q₈	—	2	2	no	—
117	C₃₁×C₂²⋊Q₈	—	2	4	no	—
118	C₃₁×C₂²⋊Q₈	—	2	4	no	—
119	C₃₁×C₂²⋊Q₈	—	2	4	no	—
120	C₃₁×C₂²⋊Q₈	—	2	4	no	—
121	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
122	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
123	C₃₁×#121	—	2	2	no	—
124	C₃₁×#121	—	2	4	no	—
125	C₃₁×#121	—	2	2	no	—
126	C₃₁×#121	—	2	1	no	—
127	C₃₁×#121	C₂³	1	2	yes	—
128	C₃₁×#121	C₂³	1	4	yes	—
129	C₄×Q₈×C₃₁	—	2	2	no	—
130	C₄×Q₈×C₃₁	—	2	2	no	—
131	C₄×Q₈×C₃₁	—	2	2	no	—
132	C₄×Q₈×C₃₁	—	2	1	no	—
133	C₄×Q₈×C₃₁	—	2	2	no	—
134	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
135	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
136	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
137	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
138	C₄×D₄×C₃₁	—	2	4	no	—
139	C₄×D₄×C₃₁	—	2	2	no	—
140	C₄×D₄×C₃₁	—	2	4	no	—
141	C₄×D₄×C₃₁	—	2	2	no	—
142	C₄×D₄×C₃₁	—	2	1	no	—
143	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
144	C₄×D₄×C₃₁	D₄	1	2	yes	—
145	C₄×D₄×C₃₁	C₂³	1	2	yes	—
146	C₄×D₄×C₃₁	C₂³	1	4	yes	—
147	C₄×D₄×C₃₁	C₂³	1	2	yes	—
148	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
149	C₃₁×#117	—	2	2	no	—
150	C₃₁×#117	—	2	1	no	—
151	C₃₁×#117	—	2	2	no	—
152	C₃₁×#117	D₄	1	1	yes	characteristic
153	C₃₁×#117	D₄	1	2	yes	—
154	C₃₁×#117	C₂³	1	4	yes	—
155	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
156	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
162	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
163	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
164	C₃₁×#107	C₂³	1	8	yes	—
165	C₃₁×#106	C₂³	1	4	yes	—
166	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
167	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
168	C₃₁×#103	C₂³	1	8	yes	—
169	C₃₁×#103	C₂³	1	8	yes	—
170	C₃₁×#103	C₂³	1	4	yes	—
171	C₃₁×#103	C₂³	1	8	yes	—
172	C₃₁×#103	C₂³	1	8	yes	—
173	C₃₁×#102	C₂³	1	4	yes	—
174	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
175	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
176	C₃₁×#97	C₂³	1	4	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	#41×C₃₁	—	2	4	no	—
182	#41×C₃₁	—	4	4	no	—
183	#41×C₃₁	—	8	4	no	—
184	C₂×Q₈×C₃₁	—	2	1	no	—
185	C₂×Q₈×C₃₁	—	2	1	no	—
186	C₂×Q₈×C₃₁	—	2	2	no	—
187	C₂×Q₈×C₃₁	—	2	2	no	—
188	C₂×Q₈×C₃₁	—	2	4	no	—
189	C₂×Q₈×C₃₁	—	2	2	no	—
190	C₂×Q₈×C₃₁	—	4	4	no	—
191	C₂×Q₈×C₃₁	—	4	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	4	yes	—
195	C₂×Q₈×C₃₁	C₂⁴	1	2	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	4	no	—
204	C₂²×C₄×C₃₁	—	2	2	no	—
205	C₂²×C₄×C₃₁	—	2	2	no	—
206	C₂²×C₄×C₃₁	—	2	2	no	—
207	C₂²×C₄×C₃₁	—	2	4	no	—
208	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
209	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
210	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
211	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
212	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
213	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
214	SD₁₆×C₃₁	—	4	4	no	—
215	SD₁₆×C₃₁	—	4	8	no	—
216	M₁₆×C₃₁	—	4	4	no	—
217	C₂×C₈×C₃₁	—	4	2	no	—
218	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
219	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
220	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
221	C₄⋊C₄×C₃₁	—	2	2	no	—
222	C₄⋊C₄×C₃₁	—	2	1	no	—
223	C₄⋊C₄×C₃₁	—	2	2	no	—
224	C₄⋊C₄×C₃₁	—	2	2	no	—
225	C₄⋊C₄×C₃₁	—	2	1	no	—
226	C₄⋊C₄×C₃₁	—	2	2	no	—
227	C₄⋊C₄×C₃₁	—	2	4	no	—
228	C₄⋊C₄×C₃₁	—	2	4	no	—
229	C₄⋊C₄×C₃₁	—	2	2	no	—
230	C₄⋊C₄×C₃₁	—	2	2	no	—
231	C₄⋊C₄×C₃₁	—	2	4	no	—
232	C₄⋊C₄×C₃₁	—	2	2	no	—
233	C₄⋊C₄×C₃₁	—	2	1	no	—
234	C₄⋊C₄×C₃₁	—	2	1	no	—
235	C₄⋊C₄×C₃₁	—	2	4	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	C₂×D₄	1	1	yes	characteristic
241	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
242	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
243	C₄⋊C₄×C₃₁	C₂×D₄	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	4	yes	—
247	C₂²⋊C₄×C₃₁	—	2	4	no	—
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	2	no	—
252	C₂²⋊C₄×C₃₁	—	2	4	no	—
253	C₂²⋊C₄×C₃₁	—	2	2	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	4	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	1	yes	characteristic
262	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
263	C₄²×C₃₁	C₂×D₄	1	2	yes	—
264	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
265	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
266	C₄²×C₃₁	C₂⁴	1	2	yes	—
267	#62540	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
268	C₂³×C₃₁	—	2	2	no	—
269	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
270	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
271	Q₈×C₃₁	—	2	2	no	—
272	Q₈×C₃₁	—	2	4	no	—
273	Q₈×C₃₁	—	2	8	no	—
274	Q₈×C₃₁	—	4	2	no	—
275	Q₈×C₃₁	—	4	2	no	—
276	Q₈×C₃₁	—	4	4	no	—
277	Q₈×C₃₁	—	8	2	no	—
278	Q₈×C₃₁	—	8	4	no	—
279	Q₈×C₃₁	—	8	2	no	—
280	D₄×C₃₁	—	2	4	no	—
281	D₄×C₃₁	—	2	2	no	—
282	D₄×C₃₁	—	2	4	no	—
283	D₄×C₃₁	—	4	4	no	—
284	D₄×C₃₁	—	4	2	no	—
285	D₄×C₃₁	—	4	2	no	—
286	D₄×C₃₁	—	8	4	no	—
287	C₂×C₄×C₃₁	—	2	4	no	—
288	C₂×C₄×C₃₁	—	2	2	no	—
289	C₂×C₄×C₃₁	—	2	4	no	—
290	C₂×C₄×C₃₁	—	2	2	no	—
291	C₂×C₄×C₃₁	—	2	4	no	—
292	C₂×C₄×C₃₁	—	2	2	no	—
293	C₂×C₄×C₃₁	—	2	2	no	—
294	C₂×C₄×C₃₁	—	2	4	no	—
295	C₂×C₄×C₃₁	—	2	4	no	—
296	C₂×C₄×C₃₁	—	2	2	no	—
297	C₂×C₄×C₃₁	—	4	4	no	—
298	C₂×C₄×C₃₁	—	8	4	no	—
299	C₂×C₄×C₃₁	—	8	2	no	—
300	C₂×C₄×C₃₁	—	8	4	no	—
301	C₂×C₄×C₃₁	—	8	2	no	—
302	C₂×C₄×C₃₁	#137	1	4	yes	—
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	1	yes	characteristic
306	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
307	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
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	C₂⁵	1	1	yes	characteristic
312	C₈×C₃₁	—	2	2	no	—
313	C₈×C₃₁	—	2	2	no	—
314	C₈×C₃₁	—	2	4	no	—
315	#3511	C₂×C₃₁	1	1	yes	characteristic
316	#3503	C₂×C₃₁	1	1	yes	characteristic, complemented
317	#3372	C₂×C₃₁	1	4	yes	complemented
318	#3329	C₂×C₃₁	1	4	yes	complemented
319	#3320	C₂×C₃₁	1	8	yes	complemented
320	#3231	C₂×C₃₁	1	2	yes	complemented
321	#3230	C₂×C₃₁	1	1	yes	characteristic, complemented
322	#3206	C₂×C₃₁	1	1	yes	characteristic, complemented
323	#3206	C₂×C₃₁	1	2	yes	complemented
324	#3182	C₂×C₃₁	1	2	yes	complemented
325	#3167	C₂×C₃₁	1	1	yes	characteristic, complemented
326	#3139	C₂×C₃₁	1	2	yes	complemented
327	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
328	#2965	C₂×C₃₁	1	1	yes	characteristic, complemented
329	C₂²×C₃₁	—	2	1	no	—
330	C₂²×C₃₁	—	2	1	no	—
331	C₂²×C₃₁	—	2	2	no	—
332	C₂²×C₃₁	—	2	2	no	—
333	C₂²×C₃₁	—	2	2	no	—
334	C₂²×C₃₁	—	4	2	no	—
335	C₂²×C₃₁	—	4	2	no	—
336	C₂²×C₃₁	—	4	2	no	—
337	C₂²×C₃₁	C₂×#137	1	2	yes	—
338	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
339	C₄×C₃₁	—	2	1	no	—
340	C₄×C₃₁	—	2	2	no	—
341	C₄×C₃₁	—	2	2	no	—
342	C₄×C₃₁	—	2	2	no	—
343	C₄×C₃₁	—	2	1	no	—
344	C₄×C₃₁	—	4	4	no	—
345	C₄×C₃₁	—	4	2	no	—
346	C₄×C₃₁	—	4	2	no	—
347	C₄×C₃₁	—	4	4	no	—
348	C₄×C₃₁	—	4	2	no	—
349	C₄×C₃₁	C₂×#137	1	2	yes	—
350	C₄×C₃₁	C₂×#137	1	2	yes	—
351	C₄×C₃₁	#577	1	2	yes	—
352	C₄×C₃₁	#585	1	2	yes	—
353	Q₈²	—	2	1	no	—
354	#557	C₂²×C₃₁	1	2	yes	—
355	#556	—	2	1	no	—
356	#555	C₂²×C₃₁	1	2	yes	—
357	#554	—	2	1	no	—
358	#554	—	2	2	no	—
359	#554	C₂²×C₃₁	1	2	yes	—
360	#553	C₂²×C₃₁	1	4	yes	—
361	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
362	D₄×Q₈	C₂²×C₃₁	1	4	yes	—
363	#548	—	2	2	no	—
364	#547	—	2	1	no	—
365	#544	—	2	1	no	—
366	#544	—	2	2	no	—
367	#542	—	2	2	no	—
368	#537	—	2	1	no	—
369	#533	C₂²×C₃₁	1	1	yes	characteristic
370	#533	C₂²×C₃₁	1	2	yes	—
371	#533	C₂²×C₃₁	1	2	yes	—
372	#532	C₂²×C₃₁	1	1	yes	characteristic
373	#532	C₂²×C₃₁	1	2	yes	—
374	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
375	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
376	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
377	#520	—	2	1	no	—
378	#520	—	2	1	no	—
379	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
380	#500	C₂²×C₃₁	1	2	yes	—
381	#498	C₂²×C₃₁	1	1	yes	characteristic
382	#492	C₂²×C₃₁	1	1	yes	characteristic
383	#492	C₂²×C₃₁	1	2	yes	—
384	#491	C₂²×C₃₁	1	8	yes	—
385	#487	C₂²×C₃₁	1	4	yes	—
386	#487	C₂²×C₃₁	1	2	yes	—
387	#487	C₂²×C₃₁	1	2	yes	—
388	#487	C₂²×C₃₁	1	1	yes	characteristic
389	#487	C₂²×C₃₁	1	2	yes	—
390	#487	C₂²×C₃₁	1	2	yes	—
391	#486	C₂²×C₃₁	1	1	yes	characteristic
392	#485	C₂²×C₃₁	1	8	yes	—
393	#484	C₂²×C₃₁	1	4	yes	—
394	#479	C₂²×C₃₁	1	8	yes	—
395	#477	C₂²×C₃₁	1	4	yes	—
396	#468	C₂²×C₃₁	1	8	yes	—
397	#465	C₂²×C₃₁	1	4	yes	—
398	#461	C₂²×C₃₁	1	4	yes	—
399	#461	C₂²×C₃₁	1	8	yes	—
400	#448	C₂²×C₃₁	1	8	yes	—
401	#448	C₂²×C₃₁	1	4	yes	—
402	#446	C₂²×C₃₁	1	2	yes	—
403	#440	C₂²×C₃₁	1	8	yes	—
404	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
405	#436	C₂²×C₃₁	1	2	yes	—
406	#435	C₂²×C₃₁	1	4	yes	—
407	#431	C₂²×C₃₁	1	1	yes	characteristic
408	#431	C₂²×C₃₁	1	2	yes	—
409	#419	C₂²×C₃₁	1	8	yes	—
410	C₂×#103	C₂²×C₃₁	1	4	yes	—
411	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
412	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
413	C₂×C₃₁	—	2	2	no	—
414	C₂×C₃₁	—	4	2	no	—
415	C₂×C₃₁	—	8	2	no	—
416	C₂×C₃₁	#3500	1	1	yes	characteristic
417	C₂×C₃₁	C₂²×#137	1	1	yes	characteristic
418	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
419	C₂×#41	—	2	1	no	—
420	C₂×#41	—	2	2	no	—
421	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
422	C₂²×Q₈	—	2	1	no	—
423	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
424	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
425	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
426	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
427	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
428	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
429	C₄⋊Q₈	—	2	2	no	—
430	C₄⋊Q₈	—	2	4	no	—
431	C₄⋊Q₈	—	2	1	no	—
432	C₄⋊Q₈	—	2	2	no	—
433	C₄⋊Q₈	—	2	1	no	—
434	C₄⋊Q₈	—	2	1	no	—
435	C₄⋊Q₈	—	2	1	no	—
436	C₄⋊Q₈	—	2	2	no	—
437	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
438	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
439	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
440	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
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	#125	—	2	2	no	—
448	#125	—	2	4	no	—
449	#125	—	2	2	no	—
450	#125	—	2	1	no	—
451	#125	C₂³×C₃₁	1	4	yes	—
452	#124	—	2	4	no	—
453	#124	—	2	1	no	—
454	#124	—	2	2	no	—
455	#123	—	2	4	no	—
456	C₂²⋊Q₈	—	2	2	no	—
457	C₂²⋊Q₈	—	2	2	no	—
458	C₂²⋊Q₈	—	2	2	no	—
459	C₂²⋊Q₈	—	2	1	no	—
460	C₂²⋊Q₈	—	2	2	no	—
461	C₂²⋊Q₈	—	2	4	no	—
462	C₂²⋊Q₈	—	2	4	no	—
463	C₂²⋊Q₈	—	2	4	no	—
464	C₂²⋊Q₈	—	2	4	no	—
465	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
466	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
467	#121	—	2	2	no	—
468	#121	—	2	4	no	—
469	#121	—	2	2	no	—
470	#121	—	2	1	no	—
471	#121	C₂³×C₃₁	1	2	yes	—
472	#121	C₂³×C₃₁	1	4	yes	—
473	C₄×Q₈	—	2	2	no	—
474	C₄×Q₈	—	2	2	no	—
475	C₄×Q₈	—	2	2	no	—
476	C₄×Q₈	—	2	1	no	—
477	C₄×Q₈	—	2	2	no	—
478	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
479	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
480	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
481	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
482	C₄×D₄	—	2	2	no	—
483	C₄×D₄	—	2	4	no	—
484	C₄×D₄	—	2	4	no	—
485	C₄×D₄	—	2	2	no	—
486	C₄×D₄	—	2	1	no	—
487	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
488	C₄×D₄	D₄×C₃₁	1	2	yes	—
489	C₄×D₄	C₂³×C₃₁	1	4	yes	—
490	C₄×D₄	C₂³×C₃₁	1	2	yes	—
491	C₄×D₄	C₂³×C₃₁	1	2	yes	—
492	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
493	#117	—	2	2	no	—
494	#117	—	2	1	no	—
495	#117	—	2	2	no	—
496	#117	D₄×C₃₁	1	1	yes	characteristic
497	#117	D₄×C₃₁	1	2	yes	—
498	#117	C₂³×C₃₁	1	4	yes	—
499	C₂×C₄⋊C₄	—	2	1	no	—
500	C₂×C₄⋊C₄	—	2	2	no	—
501	C₂×C₄⋊C₄	—	2	1	no	—
502	C₂×C₄⋊C₄	—	2	1	no	—
503	C₂×C₄⋊C₄	—	2	2	no	—
504	C₂×C₄⋊C₄	—	2	2	no	—
505	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
506	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
507	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
508	#107	C₂³×C₃₁	1	8	yes	—
509	#106	C₂³×C₃₁	1	4	yes	—
510	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
511	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
512	#103	C₂³×C₃₁	1	8	yes	—
513	#103	C₂³×C₃₁	1	8	yes	—
514	#103	C₂³×C₃₁	1	4	yes	—
515	#103	C₂³×C₃₁	1	8	yes	—
516	#103	C₂³×C₃₁	1	8	yes	—
517	#102	C₂³×C₃₁	1	4	yes	—
518	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
519	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
520	#97	C₂³×C₃₁	1	4	yes	—
521	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
522	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
523	C₄×C₈	C₂³×C₃₁	1	2	yes	—
524	C₄×C₈	C₂³×C₃₁	1	2	yes	—
525	C₃₁	#62540	1	1	yes	characteristic, direct product factor, Sylow
526	#41	—	2	4	no	—
527	#41	—	4	4	no	—
528	#41	—	8	4	no	—
529	C₂×Q₈	—	2	1	no	—
530	C₂×Q₈	—	2	1	no	—
531	C₂×Q₈	—	2	2	no	—
532	C₂×Q₈	—	2	2	no	—
533	C₂×Q₈	—	2	4	no	—
534	C₂×Q₈	—	2	2	no	—
535	C₂×Q₈	—	4	4	no	—
536	C₂×Q₈	—	4	8	no	—
537	C₂×Q₈	—	8	2	no	—
538	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
539	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
540	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
541	C₂×D₄	—	2	2	no	—
542	C₂×D₄	—	2	4	no	—
543	C₂×D₄	—	2	1	no	—
544	C₂×D₄	—	2	1	no	—
545	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
546	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
547	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
548	C₂²×C₄	—	2	4	no	—
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	2	yes	—
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	1	yes	characteristic
557	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
558	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
559	SD₁₆	—	4	4	no	—
560	SD₁₆	—	4	8	no	—
561	M₁₆	—	4	4	no	—
562	C₂×C₈	—	4	2	no	—
563	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
564	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
565	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
566	C₄⋊C₄	—	2	2	no	—
567	C₄⋊C₄	—	2	1	no	—
568	C₄⋊C₄	—	2	2	no	—
569	C₄⋊C₄	—	2	2	no	—
570	C₄⋊C₄	—	2	1	no	—
571	C₄⋊C₄	—	2	2	no	—
572	C₄⋊C₄	—	2	4	no	—
573	C₄⋊C₄	—	2	4	no	—
574	C₄⋊C₄	—	2	2	no	—
575	C₄⋊C₄	—	2	2	no	—
576	C₄⋊C₄	—	2	4	no	—
577	C₄⋊C₄	—	2	2	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	4	no	—
581	C₄⋊C₄	—	2	4	no	—
582	C₄⋊C₄	—	2	2	no	—
583	C₄⋊C₄	—	2	2	no	—
584	C₄⋊C₄	—	2	4	no	—
585	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
586	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
587	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
588	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
589	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
590	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
591	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
592	C₂²⋊C₄	—	2	4	no	—
593	C₂²⋊C₄	—	2	2	no	—
594	C₂²⋊C₄	—	2	4	no	—
595	C₂²⋊C₄	—	2	4	no	—
596	C₂²⋊C₄	—	2	2	no	—
597	C₂²⋊C₄	—	2	4	no	—
598	C₂²⋊C₄	—	2	2	no	—
599	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
600	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
601	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
602	C₄²	—	2	4	no	—
603	C₄²	—	2	4	no	—
604	C₄²	—	2	2	no	—
605	C₄²	C₂×D₄×C₃₁	1	2	yes	—
606	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
607	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
608	C₄²	C₂×D₄×C₃₁	1	2	yes	—
609	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
610	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
611	C₄²	C₂⁴×C₃₁	1	2	yes	—
612	C₂³	—	2	2	no	—
613	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
614	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
615	Q₈	—	2	2	no	—
616	Q₈	—	2	4	no	—
617	Q₈	—	2	8	no	—
618	Q₈	—	4	2	no	—
619	Q₈	—	4	2	no	—
620	Q₈	—	4	4	no	—
621	Q₈	—	8	2	no	—
622	Q₈	—	8	4	no	—
623	Q₈	—	8	2	no	—
624	D₄	—	2	4	no	—
625	D₄	—	2	2	no	—
626	D₄	—	2	4	no	—
627	D₄	—	4	4	no	—
628	D₄	—	4	2	no	—
629	D₄	—	4	2	no	—
630	D₄	—	8	4	no	—
631	C₂×C₄	—	2	4	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	4	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	4	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	4	no	—
639	C₂×C₄	—	2	4	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	4	4	no	—
642	C₂×C₄	—	8	4	no	—
643	C₂×C₄	—	8	2	no	—
644	C₂×C₄	—	8	4	no	—
645	C₂×C₄	—	8	2	no	—
646	C₂×C₄	C₃₁×#137	1	4	yes	—
647	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
648	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
649	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
650	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
651	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
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	C₂⁵×C₃₁	1	1	yes	characteristic
656	C₈	—	2	2	no	—
657	C₈	—	2	2	no	—
658	C₈	—	2	4	no	—
659	C₂²	—	2	1	no	—
660	C₂²	—	2	1	no	—
661	C₂²	—	2	2	no	—
662	C₂²	—	2	2	no	—
663	C₂²	—	2	2	no	—
664	C₂²	—	4	2	no	—
665	C₂²	—	4	2	no	—
666	C₂²	—	4	2	no	—
667	C₂²	C₂×C₃₁×#137	1	2	yes	—
668	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
669	C₄	—	2	1	no	—
670	C₄	—	2	2	no	—
671	C₄	—	2	2	no	—
672	C₄	—	2	2	no	—
673	C₄	—	2	1	no	—
674	C₄	—	4	4	no	—
675	C₄	—	4	2	no	—
676	C₄	—	4	2	no	—
677	C₄	—	4	4	no	—
678	C₄	—	4	2	no	—
679	C₄	C₂×C₃₁×#137	1	2	yes	—
680	C₄	C₂×C₃₁×#137	1	2	yes	—
681	C₄	C₃₁×#577	1	2	yes	—
682	C₄	C₃₁×#585	1	2	yes	—
683	C₂	—	2	2	no	—
684	C₂	—	4	2	no	—
685	C₂	—	8	2	no	—
686	C₂	C₃₁×#3500	1	1	yes	characteristic
687	C₂	C₂²×C₃₁×#137	1	1	yes	characteristic
688	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×#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×#3503	C₂	semidirect
3. C₃₁×#3372	C₂	semidirect
4. C₃₁×#3329	C₂	semidirect
5. C₃₁×#3320	C₂	semidirect
6. C₃₁×#3231	C₂	semidirect
7. C₃₁×#3230	C₂	semidirect
8. C₃₁×#3206	C₂	semidirect
9. C₃₁×#3206	C₂	semidirect
10. C₃₁×#3182	C₂	semidirect
11. C₃₁×#3167	C₂	semidirect
12. C₃₁×#3139	C₂	semidirect
13. C₂×C₃₁×#487	C₂	semidirect
14. C₃₁×#2965	C₂	semidirect
267. #62540	C₃₁	direct (N × H)
316. #3503	C₂×C₃₁	semidirect
317. #3372	C₂×C₃₁	semidirect
318. #3329	C₂×C₃₁	semidirect
319. #3320	C₂×C₃₁	semidirect
320. #3231	C₂×C₃₁	semidirect
321. #3230	C₂×C₃₁	semidirect
322. #3206	C₂×C₃₁	semidirect
323. #3206	C₂×C₃₁	semidirect
324. #3182	C₂×C₃₁	semidirect
325. #3167	C₂×C₃₁	semidirect
326. #3139	C₂×C₃₁	semidirect
327. C₂×#487	C₂×C₃₁	semidirect
328. #2965	C₂×C₃₁	semidirect
525. C₃₁	#62540	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×#3511	2
C₃₁×#3503	2
C₃₁×#3372	2
C₃₁×#3329	2
C₃₁×#3320	2
C₃₁×#3231	2
C₃₁×#3230	2
C₃₁×#3206	2
C₃₁×#3182	2
C₃₁×#3167	2
C₃₁×#3139	2
C₂×C₃₁×#487	2
C₃₁×#2965	2
#62540	31

Subgroups — C31×#62540 #21519299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62540 #21519299