Subgroups — C₃₁×#61020 #21517779

Order 7936.

1668 conjugacy classes of subgroups.

The subgroup lattice has 712 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×#3261	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3260	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3240	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3236	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3218	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3216	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3201	C₂	1	4	yes	maximal, complemented
10	C₃₁×#3181	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3172	C₂	1	2	yes	maximal, complemented
12	C₃₁×#3127	C₂	1	2	yes	maximal, complemented
13	C₃₁×#3126	C₂	1	2	yes	maximal, complemented
14	C₃₁×#3120	C₂	1	4	yes	maximal, complemented
15	C₃₁×#2923	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#557	—	2	2	no	—
17	C₃₁×#556	—	2	2	no	—
18	C₃₁×#553	—	2	2	no	—
19	C₃₁×#553	—	2	2	no	—
20	C₃₁×#553	—	2	2	no	—
21	C₃₁×#552	—	2	2	no	—
22	C₃₁×#552	—	2	2	no	—
23	C₃₁×#552	—	2	2	no	—
24	C₃₁×#541	C₂²	1	2	yes	—
25	C₃₁×#541	C₂²	1	1	yes	characteristic
26	C₃₁×#541	C₂²	1	2	yes	—
27	C₃₁×#541	C₂²	1	1	yes	characteristic
28	C₃₁×#540	C₂²	1	1	yes	characteristic
29	C₃₁×#540	C₂²	1	1	yes	characteristic
30	C₃₁×#537	C₂²	1	2	yes	—
31	C₃₁×#536	C₂²	1	2	yes	—
32	C₃₁×#533	C₂²	1	2	yes	—
33	C₃₁×#533	C₂²	1	2	yes	—
34	C₃₁×#533	C₂²	1	1	yes	characteristic
35	C₃₁×#532	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
37	C₃₁×#529	C₂²	1	2	yes	—
38	C₃₁×#529	C₂²	1	2	yes	—
39	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
40	C₃₁×#520	C₂²	1	2	yes	—
41	C₃₁×#519	C₂²	1	2	yes	—
42	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×#501	C₂²	1	2	yes	—
44	C₃₁×#501	C₂²	1	2	yes	—
45	C₃₁×#497	C₂²	1	2	yes	—
46	C₃₁×#497	C₂²	1	2	yes	—
47	C₃₁×#491	C₂²	1	2	yes	—
48	C₃₁×#491	C₂²	1	2	yes	—
49	C₃₁×#489	C₂²	1	2	yes	—
50	C₃₁×#489	C₂²	1	2	yes	—
51	C₃₁×#485	C₂²	1	4	yes	—
52	C₃₁×#484	C₂²	1	4	yes	—
53	C₃₁×#483	C₂²	1	4	yes	—
54	C₃₁×#482	C₂²	1	4	yes	—
55	C₃₁×#479	C₂²	1	4	yes	—
56	C₃₁×#479	C₂²	1	4	yes	—
57	C₃₁×#479	C₂²	1	4	yes	—
58	C₃₁×#478	C₂²	1	4	yes	—
59	C₃₁×#470	C₂²	1	4	yes	—
60	C₃₁×#468	C₂²	1	4	yes	—
61	C₃₁×#467	C₂²	1	4	yes	—
62	C₃₁×#465	C₂²	1	4	yes	—
63	C₃₁×#462	C₂²	1	4	yes	—
64	C₃₁×#462	C₂²	1	4	yes	—
65	C₃₁×#461	C₂²	1	4	yes	—
66	C₃₁×#460	C₂²	1	4	yes	—
67	C₃₁×#452	C₂²	1	4	yes	—
68	C₃₁×#452	C₂²	1	4	yes	—
69	C₃₁×#451	C₂²	1	4	yes	—
70	C₃₁×#448	C₂²	1	4	yes	—
71	C₃₁×#441	C₂²	1	4	yes	—
72	C₃₁×#440	C₂²	1	4	yes	—
73	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
74	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
75	C₃₁×#433	C₂²	1	2	yes	—
76	C₃₁×#433	C₂²	1	2	yes	—
77	C₃₁×#432	C₂²	1	2	yes	—
78	C₃₁×#432	C₂²	1	2	yes	—
79	C₃₁×#424	C₂²	1	2	yes	—
80	C₃₁×#408	C₂²	1	2	yes	—
81	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
82	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
83	C₂×#41×C₃₁	C₂³	1	2	yes	—
84	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
86	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
87	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
88	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
89	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
90	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
91	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
92	C₃₁×C₄⋊Q₈	—	2	2	no	—
93	C₃₁×C₄⋊Q₈	—	2	2	no	—
94	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
95	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
96	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
97	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
98	C₃₁×#127	C₂³	1	1	yes	characteristic
99	C₃₁×#126	—	2	4	no	—
100	C₃₁×#126	—	2	4	no	—
101	C₃₁×#126	—	2	2	no	—
102	C₃₁×#126	—	2	2	no	—
103	C₃₁×#126	—	2	2	no	—
104	C₃₁×#126	—	2	2	no	—
105	C₃₁×#125	—	2	2	no	—
106	C₃₁×#125	—	2	2	no	—
107	C₃₁×#125	—	2	4	no	—
108	C₃₁×#125	—	2	4	no	—
109	C₃₁×#125	—	2	2	no	—
110	C₃₁×#125	—	2	2	no	—
111	C₃₁×#125	—	2	2	no	—
112	C₃₁×#125	C₂³	1	2	yes	—
113	C₃₁×#125	C₂³	1	1	yes	characteristic
114	C₃₁×#125	C₂³	1	1	yes	characteristic
115	C₃₁×#125	C₂³	1	2	yes	—
116	C₃₁×#124	—	2	2	no	—
117	C₃₁×#124	C₂³	1	2	yes	—
118	C₃₁×#123	—	2	2	no	—
119	C₃₁×#123	—	2	2	no	—
120	C₃₁×#123	—	2	2	no	—
121	C₃₁×#123	—	2	2	no	—
122	C₃₁×#123	—	2	2	no	—
123	C₃₁×#123	—	2	2	no	—
124	C₃₁×C₂²⋊Q₈	—	2	2	no	—
125	C₃₁×C₂²⋊Q₈	—	2	2	no	—
126	C₃₁×C₂²⋊Q₈	—	2	2	no	—
127	C₃₁×C₂²⋊Q₈	—	2	2	no	—
128	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
129	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
130	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
131	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
132	C₃₁×#121	—	2	2	no	—
133	C₃₁×#121	—	2	2	no	—
134	C₃₁×#121	—	2	2	no	—
135	C₃₁×#121	—	2	2	no	—
136	C₃₁×#121	—	2	2	no	—
137	C₃₁×#121	—	2	2	no	—
138	C₃₁×#121	C₂³	1	2	yes	—
139	C₃₁×#121	C₂³	1	2	yes	—
140	C₄×Q₈×C₃₁	—	2	4	no	—
141	C₄×Q₈×C₃₁	—	2	2	no	—
142	C₄×Q₈×C₃₁	—	2	2	no	—
143	C₄×Q₈×C₃₁	—	2	2	no	—
144	C₄×Q₈×C₃₁	D₄	1	2	yes	—
145	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
146	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
147	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
148	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
149	C₄×D₄×C₃₁	—	2	2	no	—
150	C₄×D₄×C₃₁	—	2	2	no	—
151	C₄×D₄×C₃₁	—	2	2	no	—
152	C₄×D₄×C₃₁	—	2	2	no	—
153	C₄×D₄×C₃₁	—	2	2	no	—
154	C₄×D₄×C₃₁	D₄	1	2	yes	—
155	C₄×D₄×C₃₁	D₄	1	2	yes	—
156	C₄×D₄×C₃₁	D₄	1	2	yes	—
157	C₄×D₄×C₃₁	C₂³	1	2	yes	—
158	C₄×D₄×C₃₁	C₂³	1	2	yes	—
159	C₃₁×#117	—	2	2	no	—
160	C₃₁×#117	—	2	2	no	—
161	C₃₁×#117	—	2	2	no	—
162	C₃₁×#117	—	2	4	no	—
163	C₃₁×#117	—	2	2	no	—
164	C₃₁×#117	—	2	2	no	—
165	C₃₁×#117	C₂³	1	2	yes	—
166	C₃₁×#117	C₂³	1	1	yes	characteristic
167	C₃₁×#117	C₂³	1	1	yes	characteristic
168	C₃₁×#117	C₂³	1	1	yes	characteristic
169	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
170	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
171	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
173	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
174	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
175	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
176	C₃₁×#107	C₂³	1	4	yes	—
177	C₃₁×#107	C₂³	1	4	yes	—
178	C₃₁×#106	C₂³	1	4	yes	—
179	C₃₁×#106	C₂³	1	4	yes	—
180	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
181	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
182	C₃₁×#103	C₂³	1	4	yes	—
183	C₃₁×#103	C₂³	1	4	yes	—
184	C₃₁×#103	C₂³	1	4	yes	—
185	C₃₁×#103	C₂³	1	4	yes	—
186	C₃₁×#103	C₂³	1	4	yes	—
187	C₃₁×#103	C₂³	1	4	yes	—
188	C₃₁×#102	C₂³	1	4	yes	—
189	C₃₁×#102	C₂³	1	4	yes	—
190	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
191	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
192	C₃₁×#97	C₂³	1	2	yes	—
193	C₃₁×#97	C₂³	1	2	yes	—
194	C₃₁×#97	C₂³	1	2	yes	—
195	C₄×C₈×C₃₁	C₂³	1	2	yes	—
196	#41×C₃₁	—	4	2	no	—
197	#41×C₃₁	—	4	4	no	—
198	C₂×Q₈×C₃₁	—	2	2	no	—
199	C₂×Q₈×C₃₁	—	2	2	no	—
200	C₂×Q₈×C₃₁	—	4	4	no	—
201	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
202	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
203	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
204	C₂×D₄×C₃₁	—	2	2	no	—
205	C₂×D₄×C₃₁	—	2	2	no	—
206	C₂×D₄×C₃₁	—	2	2	no	—
207	C₂×D₄×C₃₁	—	2	2	no	—
208	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
209	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
210	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
211	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
212	C₂²×C₄×C₃₁	—	2	2	no	—
213	C₂²×C₄×C₃₁	—	2	2	no	—
214	C₂²×C₄×C₃₁	—	2	2	no	—
215	C₂²×C₄×C₃₁	—	2	2	no	—
216	C₂²×C₄×C₃₁	—	2	2	no	—
217	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
218	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
219	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
220	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
221	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
222	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
223	Q₁₆×C₃₁	—	4	4	no	—
224	Q₁₆×C₃₁	—	4	4	no	—
225	SD₁₆×C₃₁	—	4	4	no	—
226	SD₁₆×C₃₁	—	4	4	no	—
227	M₁₆×C₃₁	—	4	2	no	—
228	C₂×C₈×C₃₁	—	4	2	no	—
229	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
230	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
231	C₄⋊C₄×C₃₁	—	2	2	no	—
232	C₄⋊C₄×C₃₁	—	2	2	no	—
233	C₄⋊C₄×C₃₁	—	2	2	no	—
234	C₄⋊C₄×C₃₁	—	2	2	no	—
235	C₄⋊C₄×C₃₁	—	2	2	no	—
236	C₄⋊C₄×C₃₁	—	2	2	no	—
237	C₄⋊C₄×C₃₁	—	2	2	no	—
238	C₄⋊C₄×C₃₁	—	2	2	no	—
239	C₄⋊C₄×C₃₁	—	2	2	no	—
240	C₄⋊C₄×C₃₁	—	2	4	no	—
241	C₄⋊C₄×C₃₁	—	2	2	no	—
242	C₄⋊C₄×C₃₁	—	2	4	no	—
243	C₄⋊C₄×C₃₁	—	2	4	no	—
244	C₄⋊C₄×C₃₁	—	2	2	no	—
245	C₄⋊C₄×C₃₁	—	2	2	no	—
246	C₄⋊C₄×C₃₁	—	2	2	no	—
247	C₄⋊C₄×C₃₁	—	2	4	no	—
248	C₄⋊C₄×C₃₁	—	2	4	no	—
249	C₄⋊C₄×C₃₁	—	2	4	no	—
250	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
251	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
252	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
253	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
254	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
255	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
256	C₂²⋊C₄×C₃₁	—	2	2	no	—
257	C₂²⋊C₄×C₃₁	—	2	4	no	—
258	C₂²⋊C₄×C₃₁	—	2	4	no	—
259	C₂²⋊C₄×C₃₁	—	2	2	no	—
260	C₂²⋊C₄×C₃₁	—	2	2	no	—
261	C₂²⋊C₄×C₃₁	—	2	2	no	—
262	C₂²⋊C₄×C₃₁	—	2	2	no	—
263	C₂²⋊C₄×C₃₁	—	2	2	no	—
264	C₂²⋊C₄×C₃₁	—	2	2	no	—
265	C₂²⋊C₄×C₃₁	—	2	2	no	—
266	C₂²⋊C₄×C₃₁	—	2	2	no	—
267	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
268	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
269	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
270	C₄²×C₃₁	—	2	2	no	—
271	C₄²×C₃₁	—	2	2	no	—
272	C₄²×C₃₁	—	2	4	no	—
273	C₄²×C₃₁	—	2	4	no	—
274	C₄²×C₃₁	—	2	2	no	—
275	C₄²×C₃₁	C₂×D₄	1	2	yes	—
276	C₄²×C₃₁	C₂×D₄	1	2	yes	—
277	C₄²×C₃₁	C₂×D₄	1	2	yes	—
278	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
279	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
280	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
281	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
282	#61020	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
283	C₂³×C₃₁	—	2	2	no	—
284	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
285	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
286	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
287	Q₈×C₃₁	—	2	4	no	—
288	Q₈×C₃₁	—	2	4	no	—
289	Q₈×C₃₁	—	2	4	no	—
290	Q₈×C₃₁	—	4	1	no	—
291	Q₈×C₃₁	—	4	4	no	—
292	Q₈×C₃₁	—	4	4	no	—
293	D₄×C₃₁	—	2	4	no	—
294	D₄×C₃₁	—	4	1	no	—
295	D₄×C₃₁	—	4	1	no	—
296	D₄×C₃₁	—	4	1	no	—
297	D₄×C₃₁	—	4	4	no	—
298	D₄×C₃₁	—	8	2	no	—
299	D₄×C₃₁	—	8	2	no	—
300	D₄×C₃₁	—	8	2	no	—
301	C₂×C₄×C₃₁	—	2	2	no	—
302	C₂×C₄×C₃₁	—	2	2	no	—
303	C₂×C₄×C₃₁	—	2	2	no	—
304	C₂×C₄×C₃₁	—	2	4	no	—
305	C₂×C₄×C₃₁	—	2	2	no	—
306	C₂×C₄×C₃₁	—	2	4	no	—
307	C₂×C₄×C₃₁	—	2	2	no	—
308	C₂×C₄×C₃₁	—	2	2	no	—
309	C₂×C₄×C₃₁	—	4	2	no	—
310	C₂×C₄×C₃₁	—	4	2	no	—
311	C₂×C₄×C₃₁	—	4	2	no	—
312	C₂×C₄×C₃₁	—	4	4	no	—
313	C₂×C₄×C₃₁	—	4	4	no	—
314	C₂×C₄×C₃₁	—	8	2	no	—
315	C₂×C₄×C₃₁	—	8	2	no	—
316	C₂×C₄×C₃₁	—	8	2	no	—
317	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
318	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
319	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
320	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
321	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
322	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
323	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
324	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
325	C₈×C₃₁	—	2	4	no	—
326	C₈×C₃₁	—	2	4	no	—
327	#3512	C₂×C₃₁	1	1	yes	characteristic
328	#3509	C₂×C₃₁	1	1	yes	characteristic, complemented
329	#3261	C₂×C₃₁	1	2	yes	complemented
330	#3260	C₂×C₃₁	1	2	yes	complemented
331	#3240	C₂×C₃₁	1	2	yes	complemented
332	#3236	C₂×C₃₁	1	2	yes	complemented
333	#3218	C₂×C₃₁	1	2	yes	complemented
334	#3216	C₂×C₃₁	1	2	yes	complemented
335	#3201	C₂×C₃₁	1	4	yes	complemented
336	#3181	C₂×C₃₁	1	2	yes	complemented
337	#3172	C₂×C₃₁	1	2	yes	complemented
338	#3127	C₂×C₃₁	1	2	yes	complemented
339	#3126	C₂×C₃₁	1	2	yes	complemented
340	#3120	C₂×C₃₁	1	4	yes	complemented
341	#2923	C₂×C₃₁	1	1	yes	characteristic, complemented
342	C₂²×C₃₁	—	2	2	no	—
343	C₂²×C₃₁	—	2	2	no	—
344	C₂²×C₃₁	—	2	2	no	—
345	C₂²×C₃₁	—	2	1	no	—
346	C₂²×C₃₁	—	2	1	no	—
347	C₂²×C₃₁	—	2	1	no	—
348	C₂²×C₃₁	—	4	2	no	—
349	C₂²×C₃₁	—	4	2	no	—
350	C₂²×C₃₁	—	4	2	no	—
351	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
352	C₄×C₃₁	—	2	2	no	—
353	C₄×C₃₁	—	2	2	no	—
354	C₄×C₃₁	—	2	2	no	—
355	C₄×C₃₁	—	2	2	no	—
356	C₄×C₃₁	—	2	1	no	—
357	C₄×C₃₁	—	2	1	no	—
358	C₄×C₃₁	—	2	1	no	—
359	C₄×C₃₁	—	4	2	no	—
360	C₄×C₃₁	—	4	2	no	—
361	C₄×C₃₁	—	4	4	no	—
362	C₄×C₃₁	—	4	2	no	—
363	C₄×C₃₁	—	4	4	no	—
364	C₄×C₃₁	#585	1	2	yes	—
365	#557	—	2	2	no	—
366	#556	—	2	2	no	—
367	#553	—	2	2	no	—
368	#553	—	2	2	no	—
369	#553	—	2	2	no	—
370	#552	—	2	2	no	—
371	#552	—	2	2	no	—
372	#552	—	2	2	no	—
373	#541	C₂²×C₃₁	1	2	yes	—
374	#541	C₂²×C₃₁	1	1	yes	characteristic
375	#541	C₂²×C₃₁	1	2	yes	—
376	#541	C₂²×C₃₁	1	1	yes	characteristic
377	#540	C₂²×C₃₁	1	1	yes	characteristic
378	#540	C₂²×C₃₁	1	1	yes	characteristic
379	#537	C₂²×C₃₁	1	2	yes	—
380	#536	C₂²×C₃₁	1	2	yes	—
381	#533	C₂²×C₃₁	1	2	yes	—
382	#533	C₂²×C₃₁	1	2	yes	—
383	#533	C₂²×C₃₁	1	1	yes	characteristic
384	#532	C₂²×C₃₁	1	1	yes	characteristic
385	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
386	#529	C₂²×C₃₁	1	2	yes	—
387	#529	C₂²×C₃₁	1	2	yes	—
388	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
389	#520	C₂²×C₃₁	1	2	yes	—
390	#519	C₂²×C₃₁	1	2	yes	—
391	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
392	#501	C₂²×C₃₁	1	2	yes	—
393	#501	C₂²×C₃₁	1	2	yes	—
394	#497	C₂²×C₃₁	1	2	yes	—
395	#497	C₂²×C₃₁	1	2	yes	—
396	#491	C₂²×C₃₁	1	2	yes	—
397	#491	C₂²×C₃₁	1	2	yes	—
398	#489	C₂²×C₃₁	1	2	yes	—
399	#489	C₂²×C₃₁	1	2	yes	—
400	#485	C₂²×C₃₁	1	4	yes	—
401	#484	C₂²×C₃₁	1	4	yes	—
402	#483	C₂²×C₃₁	1	4	yes	—
403	#482	C₂²×C₃₁	1	4	yes	—
404	#479	C₂²×C₃₁	1	4	yes	—
405	#479	C₂²×C₃₁	1	4	yes	—
406	#479	C₂²×C₃₁	1	4	yes	—
407	#478	C₂²×C₃₁	1	4	yes	—
408	#470	C₂²×C₃₁	1	4	yes	—
409	#468	C₂²×C₃₁	1	4	yes	—
410	#467	C₂²×C₃₁	1	4	yes	—
411	#465	C₂²×C₃₁	1	4	yes	—
412	#462	C₂²×C₃₁	1	4	yes	—
413	#462	C₂²×C₃₁	1	4	yes	—
414	#461	C₂²×C₃₁	1	4	yes	—
415	#460	C₂²×C₃₁	1	4	yes	—
416	#452	C₂²×C₃₁	1	4	yes	—
417	#452	C₂²×C₃₁	1	4	yes	—
418	#451	C₂²×C₃₁	1	4	yes	—
419	#448	C₂²×C₃₁	1	4	yes	—
420	#441	C₂²×C₃₁	1	4	yes	—
421	#440	C₂²×C₃₁	1	4	yes	—
422	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
423	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
424	#433	C₂²×C₃₁	1	2	yes	—
425	#433	C₂²×C₃₁	1	2	yes	—
426	#432	C₂²×C₃₁	1	2	yes	—
427	#432	C₂²×C₃₁	1	2	yes	—
428	#424	C₂²×C₃₁	1	2	yes	—
429	#408	C₂²×C₃₁	1	2	yes	—
430	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
431	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
432	C₂×C₃₁	—	4	1	no	—
433	C₂×C₃₁	—	4	1	no	—
434	C₂×C₃₁	—	4	1	no	—
435	C₂×C₃₁	—	8	2	no	—
436	C₂×C₃₁	#3449	1	1	yes	characteristic
437	C₂×C₃₁	#3586	1	2	yes	—
438	C₂×#41	C₂³×C₃₁	1	2	yes	—
439	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
440	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
441	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
442	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
443	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
444	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
445	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
446	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
447	C₄⋊Q₈	—	2	2	no	—
448	C₄⋊Q₈	—	2	2	no	—
449	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
450	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
451	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
452	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
453	#127	C₂³×C₃₁	1	1	yes	characteristic
454	#126	—	2	4	no	—
455	#126	—	2	4	no	—
456	#126	—	2	2	no	—
457	#126	—	2	2	no	—
458	#126	—	2	2	no	—
459	#126	—	2	2	no	—
460	#125	—	2	2	no	—
461	#125	—	2	2	no	—
462	#125	—	2	4	no	—
463	#125	—	2	4	no	—
464	#125	—	2	2	no	—
465	#125	—	2	2	no	—
466	#125	—	2	2	no	—
467	#125	C₂³×C₃₁	1	2	yes	—
468	#125	C₂³×C₃₁	1	1	yes	characteristic
469	#125	C₂³×C₃₁	1	1	yes	characteristic
470	#125	C₂³×C₃₁	1	2	yes	—
471	#124	—	2	2	no	—
472	#124	C₂³×C₃₁	1	2	yes	—
473	#123	—	2	2	no	—
474	#123	—	2	2	no	—
475	#123	—	2	2	no	—
476	#123	—	2	2	no	—
477	#123	—	2	2	no	—
478	#123	—	2	2	no	—
479	C₂²⋊Q₈	—	2	2	no	—
480	C₂²⋊Q₈	—	2	2	no	—
481	C₂²⋊Q₈	—	2	2	no	—
482	C₂²⋊Q₈	—	2	2	no	—
483	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
484	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
485	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
486	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
487	#121	—	2	2	no	—
488	#121	—	2	2	no	—
489	#121	—	2	2	no	—
490	#121	—	2	2	no	—
491	#121	—	2	2	no	—
492	#121	—	2	2	no	—
493	#121	C₂³×C₃₁	1	2	yes	—
494	#121	C₂³×C₃₁	1	2	yes	—
495	C₄×Q₈	—	2	4	no	—
496	C₄×Q₈	—	2	2	no	—
497	C₄×Q₈	—	2	2	no	—
498	C₄×Q₈	—	2	2	no	—
499	C₄×Q₈	D₄×C₃₁	1	2	yes	—
500	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
501	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
502	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
503	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
504	C₄×D₄	—	2	2	no	—
505	C₄×D₄	—	2	2	no	—
506	C₄×D₄	—	2	2	no	—
507	C₄×D₄	—	2	2	no	—
508	C₄×D₄	—	2	2	no	—
509	C₄×D₄	D₄×C₃₁	1	2	yes	—
510	C₄×D₄	D₄×C₃₁	1	2	yes	—
511	C₄×D₄	D₄×C₃₁	1	2	yes	—
512	C₄×D₄	C₂³×C₃₁	1	2	yes	—
513	C₄×D₄	C₂³×C₃₁	1	2	yes	—
514	#117	—	2	2	no	—
515	#117	—	2	2	no	—
516	#117	—	2	2	no	—
517	#117	—	2	4	no	—
518	#117	—	2	2	no	—
519	#117	—	2	2	no	—
520	#117	C₂³×C₃₁	1	2	yes	—
521	#117	C₂³×C₃₁	1	1	yes	characteristic
522	#117	C₂³×C₃₁	1	1	yes	characteristic
523	#117	C₂³×C₃₁	1	1	yes	characteristic
524	C₂×C₄⋊C₄	—	2	2	no	—
525	C₂×C₄⋊C₄	—	2	2	no	—
526	C₂×C₄⋊C₄	—	2	2	no	—
527	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
528	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
529	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
530	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
531	#107	C₂³×C₃₁	1	4	yes	—
532	#107	C₂³×C₃₁	1	4	yes	—
533	#106	C₂³×C₃₁	1	4	yes	—
534	#106	C₂³×C₃₁	1	4	yes	—
535	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
536	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
537	#103	C₂³×C₃₁	1	4	yes	—
538	#103	C₂³×C₃₁	1	4	yes	—
539	#103	C₂³×C₃₁	1	4	yes	—
540	#103	C₂³×C₃₁	1	4	yes	—
541	#103	C₂³×C₃₁	1	4	yes	—
542	#103	C₂³×C₃₁	1	4	yes	—
543	#102	C₂³×C₃₁	1	4	yes	—
544	#102	C₂³×C₃₁	1	4	yes	—
545	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
546	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
547	#97	C₂³×C₃₁	1	2	yes	—
548	#97	C₂³×C₃₁	1	2	yes	—
549	#97	C₂³×C₃₁	1	2	yes	—
550	C₄×C₈	C₂³×C₃₁	1	2	yes	—
551	C₃₁	#61020	1	1	yes	characteristic, direct product factor, Sylow
552	#41	—	4	2	no	—
553	#41	—	4	4	no	—
554	C₂×Q₈	—	2	2	no	—
555	C₂×Q₈	—	2	2	no	—
556	C₂×Q₈	—	4	4	no	—
557	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
558	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
559	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
560	C₂×D₄	—	2	2	no	—
561	C₂×D₄	—	2	2	no	—
562	C₂×D₄	—	2	2	no	—
563	C₂×D₄	—	2	2	no	—
564	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
565	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
566	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
567	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
568	C₂²×C₄	—	2	2	no	—
569	C₂²×C₄	—	2	2	no	—
570	C₂²×C₄	—	2	2	no	—
571	C₂²×C₄	—	2	2	no	—
572	C₂²×C₄	—	2	2	no	—
573	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
574	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
575	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
576	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
577	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
578	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
579	Q₁₆	—	4	4	no	—
580	Q₁₆	—	4	4	no	—
581	SD₁₆	—	4	4	no	—
582	SD₁₆	—	4	4	no	—
583	M₁₆	—	4	2	no	—
584	C₂×C₈	—	4	2	no	—
585	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
586	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	—	2	2	no	—
589	C₄⋊C₄	—	2	2	no	—
590	C₄⋊C₄	—	2	2	no	—
591	C₄⋊C₄	—	2	2	no	—
592	C₄⋊C₄	—	2	2	no	—
593	C₄⋊C₄	—	2	2	no	—
594	C₄⋊C₄	—	2	2	no	—
595	C₄⋊C₄	—	2	2	no	—
596	C₄⋊C₄	—	2	4	no	—
597	C₄⋊C₄	—	2	2	no	—
598	C₄⋊C₄	—	2	4	no	—
599	C₄⋊C₄	—	2	4	no	—
600	C₄⋊C₄	—	2	2	no	—
601	C₄⋊C₄	—	2	2	no	—
602	C₄⋊C₄	—	2	2	no	—
603	C₄⋊C₄	—	2	4	no	—
604	C₄⋊C₄	—	2	4	no	—
605	C₄⋊C₄	—	2	4	no	—
606	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
607	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
608	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
609	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
610	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
611	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
612	C₂²⋊C₄	—	2	2	no	—
613	C₂²⋊C₄	—	2	4	no	—
614	C₂²⋊C₄	—	2	4	no	—
615	C₂²⋊C₄	—	2	2	no	—
616	C₂²⋊C₄	—	2	2	no	—
617	C₂²⋊C₄	—	2	2	no	—
618	C₂²⋊C₄	—	2	2	no	—
619	C₂²⋊C₄	—	2	2	no	—
620	C₂²⋊C₄	—	2	2	no	—
621	C₂²⋊C₄	—	2	2	no	—
622	C₂²⋊C₄	—	2	2	no	—
623	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
624	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
625	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
626	C₄²	—	2	2	no	—
627	C₄²	—	2	2	no	—
628	C₄²	—	2	4	no	—
629	C₄²	—	2	4	no	—
630	C₄²	—	2	2	no	—
631	C₄²	C₂×D₄×C₃₁	1	2	yes	—
632	C₄²	C₂×D₄×C₃₁	1	2	yes	—
633	C₄²	C₂×D₄×C₃₁	1	2	yes	—
634	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
635	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
636	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
637	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
638	C₂³	—	2	2	no	—
639	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
640	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
641	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
642	Q₈	—	2	4	no	—
643	Q₈	—	2	4	no	—
644	Q₈	—	2	4	no	—
645	Q₈	—	4	1	no	—
646	Q₈	—	4	4	no	—
647	Q₈	—	4	4	no	—
648	D₄	—	2	4	no	—
649	D₄	—	4	1	no	—
650	D₄	—	4	1	no	—
651	D₄	—	4	1	no	—
652	D₄	—	4	4	no	—
653	D₄	—	8	2	no	—
654	D₄	—	8	2	no	—
655	D₄	—	8	2	no	—
656	C₂×C₄	—	2	2	no	—
657	C₂×C₄	—	2	2	no	—
658	C₂×C₄	—	2	2	no	—
659	C₂×C₄	—	2	4	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	2	4	no	—
662	C₂×C₄	—	2	2	no	—
663	C₂×C₄	—	2	2	no	—
664	C₂×C₄	—	4	2	no	—
665	C₂×C₄	—	4	2	no	—
666	C₂×C₄	—	4	2	no	—
667	C₂×C₄	—	4	4	no	—
668	C₂×C₄	—	4	4	no	—
669	C₂×C₄	—	8	2	no	—
670	C₂×C₄	—	8	2	no	—
671	C₂×C₄	—	8	2	no	—
672	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
673	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
674	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
675	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
676	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
677	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
678	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
679	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
680	C₈	—	2	4	no	—
681	C₈	—	2	4	no	—
682	C₂²	—	2	1	no	—
683	C₂²	—	2	1	no	—
684	C₂²	—	2	1	no	—
685	C₂²	—	2	2	no	—
686	C₂²	—	2	2	no	—
687	C₂²	—	2	2	no	—
688	C₂²	—	4	2	no	—
689	C₂²	—	4	2	no	—
690	C₂²	—	4	2	no	—
691	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
692	C₄	—	2	1	no	—
693	C₄	—	2	1	no	—
694	C₄	—	2	1	no	—
695	C₄	—	2	2	no	—
696	C₄	—	2	2	no	—
697	C₄	—	2	2	no	—
698	C₄	—	2	2	no	—
699	C₄	—	4	2	no	—
700	C₄	—	4	4	no	—
701	C₄	—	4	2	no	—
702	C₄	—	4	2	no	—
703	C₄	—	4	4	no	—
704	C₄	C₃₁×#585	1	2	yes	—
705	C₂	—	4	1	no	—
706	C₂	—	4	1	no	—
707	C₂	—	4	1	no	—
708	C₂	—	8	2	no	—
709	C₂	C₃₁×#3449	1	1	yes	characteristic
710	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×#3261	C₂	semidirect
4. C₃₁×#3260	C₂	semidirect
5. C₃₁×#3240	C₂	semidirect
6. C₃₁×#3236	C₂	semidirect
7. C₃₁×#3218	C₂	semidirect
8. C₃₁×#3216	C₂	semidirect
9. C₃₁×#3201	C₂	semidirect
10. C₃₁×#3181	C₂	semidirect
11. C₃₁×#3172	C₂	semidirect
12. C₃₁×#3127	C₂	semidirect
13. C₃₁×#3126	C₂	semidirect
14. C₃₁×#3120	C₂	semidirect
15. C₃₁×#2923	C₂	semidirect
282. #61020	C₃₁	direct (N × H)
328. #3509	C₂×C₃₁	semidirect
329. #3261	C₂×C₃₁	semidirect
330. #3260	C₂×C₃₁	semidirect
331. #3240	C₂×C₃₁	semidirect
332. #3236	C₂×C₃₁	semidirect
333. #3218	C₂×C₃₁	semidirect
334. #3216	C₂×C₃₁	semidirect
335. #3201	C₂×C₃₁	semidirect
336. #3181	C₂×C₃₁	semidirect
337. #3172	C₂×C₃₁	semidirect
338. #3127	C₂×C₃₁	semidirect
339. #3126	C₂×C₃₁	semidirect
340. #3120	C₂×C₃₁	semidirect
341. #2923	C₂×C₃₁	semidirect
551. C₃₁	#61020	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×#3261	2
C₃₁×#3260	2
C₃₁×#3240	2
C₃₁×#3236	2
C₃₁×#3218	2
C₃₁×#3216	2
C₃₁×#3201	2
C₃₁×#3181	2
C₃₁×#3172	2
C₃₁×#3127	2
C₃₁×#3126	2
C₃₁×#3120	2
C₃₁×#2923	2
#61020	31

Subgroups — C31×#61020 #21517779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61020 #21517779