Subgroups — C₃₁×#60526 #21517285

Order 7936.

2166 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3489	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3098	C₂	1	6	yes	maximal, complemented
4	C₃₁×#3097	C₂	1	6	yes	maximal, complemented
5	C₃₁×#3095	C₂	1	3	yes	maximal, complemented
6	C₃₁×#3092	C₂	1	3	yes	maximal, complemented
7	C₃₁×#3091	C₂	1	3	yes	maximal, complemented
8	C₂×C₃₁×#484	C₂	1	3	yes	maximal, complemented
9	C₃₁×#2913	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2898	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2880	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#131	C₂²	1	2	yes	—
15	D₄×Q₈×C₃₁	—	2	1	no	—
16	C₃₁×#548	—	2	1	no	—
17	C₃₁×#547	—	2	3	no	—
18	C₃₁×#546	—	2	3	no	—
19	C₃₁×#537	C₂²	1	3	yes	—
20	C₃₁×#536	C₂²	1	3	yes	—
21	C₃₁×#525	C₂²	1	3	yes	—
22	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	3	yes	—
23	C₃₁×#520	—	2	3	no	—
24	C₃₁×#520	—	2	1	no	—
25	C₃₁×#520	C₂²	1	6	yes	—
26	C₃₁×#520	C₂²	1	2	yes	—
27	C₃₁×#520	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
29	C₂×C₄×D₄×C₃₁	—	2	3	no	—
30	C₂×C₃₁×#117	C₂²	1	3	yes	—
31	C₂²×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
32	C₃₁×#485	C₂²	1	6	yes	complemented
33	C₃₁×#485	C₂²	1	12	yes	complemented
34	C₃₁×#484	C₂²	1	3	yes	—
35	C₃₁×#484	C₂²	1	6	yes	—
36	C₃₁×#484	C₂²	1	12	yes	complemented
37	C₃₁×#483	C₂²	1	3	yes	—
38	C₃₁×#483	C₂²	1	6	yes	complemented
39	C₃₁×#482	C₂²	1	6	yes	complemented
40	C₃₁×#481	C₂²	1	3	yes	—
41	C₃₁×#480	C₂²	1	1	yes	characteristic
42	C₃₁×#480	C₂²	1	6	yes	complemented
43	C₃₁×#428	C₂²	1	6	yes	complemented
44	C₃₁×#428	C₂²	1	6	yes	complemented
45	C₃₁×#427	C₂²	1	6	yes	complemented
46	C₂×C₃₁×#107	C₂²	1	6	yes	complemented
47	C₃₁×#419	C₂²	1	6	yes	complemented
48	C₃₁×#417	C₂²	1	6	yes	complemented
49	C₃₁×#416	C₂²	1	6	yes	complemented
50	C₂×C₃₁×#103	C₂²	1	6	yes	complemented
51	C₃₁×#408	C₂²	1	6	yes	—
52	C₃₁×#407	C₂²	1	3	yes	—
53	C₂×C₃₁×C₂²⋊C₈	C₂²	1	3	yes	—
54	C₃₁×2₋¹⁺⁴	—	2	8	no	—
55	C₂×#41×C₃₁	—	2	8	no	—
56	C₂×#41×C₃₁	—	2	3	no	—
57	C₂×#41×C₃₁	D₄	1	3	yes	—
58	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
59	C₂×#41×C₃₁	C₂³	1	3	yes	—
60	C₂×#41×C₃₁	C₂³	1	3	yes	—
61	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₃₁	—	2	1	no	—
63	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
64	C₂²×Q₈×C₃₁	C₂³	1	3	yes	—
65	C₂²×D₄×C₃₁	D₄	1	3	yes	—
66	C₂³×C₄×C₃₁	—	2	3	no	—
67	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
68	C₃₁×#131	—	2	8	no	—
69	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
70	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
71	C₃₁×C₄⋊Q₈	—	2	3	no	—
72	C₃₁×#124	—	2	3	no	—
73	C₃₁×#123	—	2	6	no	—
74	C₃₁×#123	—	2	3	no	—
75	C₃₁×C₂²⋊Q₈	—	2	3	no	—
76	C₃₁×C₂²⋊Q₈	—	2	3	no	—
77	C₃₁×C₂²⋊Q₈	—	2	3	no	—
78	C₃₁×C₂²⋊Q₈	C₂³	1	3	yes	—
79	C₃₁×C₂²⋊Q₈	C₂³	1	3	yes	—
80	C₃₁×C₂²⋊Q₈	C₂³	1	6	yes	—
81	C₃₁×#121	—	2	6	no	—
82	C₃₁×#121	—	2	3	no	—
83	C₃₁×#121	C₂³	1	1	yes	characteristic
84	C₃₁×#121	C₂³	1	3	yes	—
85	C₃₁×#120	—	2	3	no	—
86	C₄×Q₈×C₃₁	—	2	6	no	—
87	C₄×Q₈×C₃₁	—	2	2	no	—
88	C₄×Q₈×C₃₁	—	2	1	no	—
89	C₄×D₄×C₃₁	—	2	12	no	—
90	C₄×D₄×C₃₁	—	2	6	no	—
91	C₄×D₄×C₃₁	—	2	6	no	—
92	C₄×D₄×C₃₁	—	2	3	no	—
93	C₄×D₄×C₃₁	—	2	3	no	—
94	C₄×D₄×C₃₁	—	2	3	no	—
95	C₃₁×#117	—	2	3	no	—
96	C₃₁×#117	—	2	3	no	—
97	C₃₁×#117	C₂³	1	6	yes	—
98	C₃₁×#117	C₂³	1	3	yes	—
99	C₃₁×#117	C₂³	1	6	yes	—
100	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
101	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
102	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
109	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
110	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
111	C₂×C₄²×C₃₁	—	2	3	no	—
112	C₃₁×#107	C₂³	1	6	yes	—
113	C₃₁×#107	C₂³	1	12	yes	—
114	C₃₁×#107	C₂³	1	2	yes	—
115	C₃₁×#106	C₂³	1	6	yes	—
116	C₃₁×#106	C₂³	1	6	yes	—
117	C₃₁×#103	C₂³	1	6	yes	—
118	C₃₁×#103	C₂³	1	12	yes	—
119	C₃₁×#103	C₂³	1	6	yes	—
120	C₃₁×#102	C₂³	1	2	yes	—
121	C₃₁×#102	C₂³	1	6	yes	—
122	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
123	C₃₁×C₂²⋊C₈	C₂³	1	3	yes	—
124	C₃₁×C₂²⋊C₈	C₂³	1	6	yes	—
125	C₃₁×C₂²⋊C₈	C₂³	1	6	yes	—
126	C₂⁴×C₃₁	C₂×D₄	1	3	yes	—
127	#41×C₃₁	—	2	4	no	—
128	#41×C₃₁	—	2	12	no	—
129	#41×C₃₁	—	2	12	no	—
130	#41×C₃₁	—	2	12	no	—
131	#41×C₃₁	—	4	12	no	—
132	#41×C₃₁	#41	1	8	yes	—
133	C₂×Q₈×C₃₁	—	2	4	no	—
134	C₂×Q₈×C₃₁	—	2	3	no	—
135	C₂×Q₈×C₃₁	—	2	12	no	—
136	C₂×Q₈×C₃₁	—	4	4	no	—
137	C₂×Q₈×C₃₁	C₂×D₄	1	3	yes	—
138	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂×Q₈×C₃₁	C₂⁴	1	3	yes	—
140	C₂×D₄×C₃₁	—	2	12	no	—
141	C₂×D₄×C₃₁	—	2	1	no	—
142	C₂×D₄×C₃₁	—	2	3	no	—
143	C₂×D₄×C₃₁	—	2	3	no	—
144	C₂×D₄×C₃₁	C₂×D₄	1	6	yes	—
145	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
146	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
147	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	—	2	12	no	—
149	C₂²×C₄×C₃₁	—	2	1	no	—
150	C₂²×C₄×C₃₁	—	2	3	no	—
151	C₂²×C₄×C₃₁	—	2	3	no	—
152	C₂²×C₄×C₃₁	—	2	3	no	—
153	C₂²×C₄×C₃₁	—	2	6	no	—
154	C₂²×C₄×C₃₁	—	2	6	no	—
155	C₂²×C₄×C₃₁	—	2	3	no	—
156	C₂²×C₄×C₃₁	—	8	6	no	—
157	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
158	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
162	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
163	M₁₆×C₃₁	—	2	12	no	—
164	C₂×C₈×C₃₁	—	2	12	no	—
165	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
166	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
167	C₄⋊C₄×C₃₁	—	2	3	no	—
168	C₄⋊C₄×C₃₁	—	2	6	no	—
169	C₄⋊C₄×C₃₁	—	2	6	no	—
170	C₄⋊C₄×C₃₁	—	2	3	no	—
171	C₄⋊C₄×C₃₁	—	2	6	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	3	no	—
174	C₄⋊C₄×C₃₁	—	2	3	no	—
175	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
176	C₄⋊C₄×C₃₁	C₂⁴	1	6	yes	—
177	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
178	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
179	C₂²⋊C₄×C₃₁	—	2	3	no	—
180	C₂²⋊C₄×C₃₁	—	2	6	no	—
181	C₂²⋊C₄×C₃₁	—	2	6	no	—
182	C₂²⋊C₄×C₃₁	—	2	6	no	—
183	C₂²⋊C₄×C₃₁	—	2	3	no	—
184	C₂²⋊C₄×C₃₁	—	2	3	no	—
185	C₂²⋊C₄×C₃₁	—	2	1	no	—
186	C₂²⋊C₄×C₃₁	—	2	3	no	—
187	C₄²×C₃₁	—	2	6	no	—
188	C₄²×C₃₁	—	2	6	no	—
189	C₄²×C₃₁	—	2	3	no	—
190	#60526	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂³×C₃₁	—	2	6	no	—
192	C₂³×C₃₁	—	2	6	no	—
193	C₂³×C₃₁	—	2	1	no	—
194	C₂³×C₃₁	—	4	3	no	—
195	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
196	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
197	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
198	Q₈×C₃₁	—	2	6	no	—
199	Q₈×C₃₁	—	2	6	no	—
200	Q₈×C₃₁	—	4	6	no	—
201	Q₈×C₃₁	#123	1	4	yes	—
202	D₄×C₃₁	—	2	12	no	—
203	D₄×C₃₁	—	2	6	no	—
204	D₄×C₃₁	—	2	2	no	—
205	D₄×C₃₁	—	4	6	no	—
206	D₄×C₃₁	—	4	2	no	—
207	D₄×C₃₁	—	8	3	no	—
208	D₄×C₃₁	#123	1	12	yes	—
209	C₂×C₄×C₃₁	—	2	6	no	—
210	C₂×C₄×C₃₁	—	2	6	no	—
211	C₂×C₄×C₃₁	—	2	6	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	6	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	6	no	—
217	C₂×C₄×C₃₁	—	2	3	no	—
218	C₂×C₄×C₃₁	—	2	3	no	—
219	C₂×C₄×C₃₁	—	4	6	no	—
220	C₂×C₄×C₃₁	—	4	6	no	—
221	C₂×C₄×C₃₁	—	4	2	no	—
222	C₂×C₄×C₃₁	—	4	6	no	—
223	C₂×C₄×C₃₁	—	8	3	no	—
224	C₂×C₄×C₃₁	—	8	6	no	—
225	C₂×C₄×C₃₁	—	8	6	no	—
226	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
229	C₂×C₄×C₃₁	C₂×#41	1	12	yes	—
230	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
231	C₈×C₃₁	—	2	2	no	—
232	C₈×C₃₁	—	2	6	no	—
233	#3489	C₂×C₃₁	1	1	yes	characteristic
234	C₂×#520	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#3098	C₂×C₃₁	1	6	yes	complemented
236	#3097	C₂×C₃₁	1	6	yes	complemented
237	#3095	C₂×C₃₁	1	3	yes	complemented
238	#3092	C₂×C₃₁	1	3	yes	complemented
239	#3091	C₂×C₃₁	1	3	yes	complemented
240	C₂×#484	C₂×C₃₁	1	3	yes	complemented
241	#2913	C₂×C₃₁	1	2	yes	complemented
242	#2898	C₂×C₃₁	1	2	yes	complemented
243	#2880	C₂×C₃₁	1	1	yes	characteristic, complemented
244	C₂²×C₃₁	—	2	1	no	—
245	C₂²×C₃₁	—	2	3	no	—
246	C₂²×C₃₁	—	2	1	no	—
247	C₂²×C₃₁	—	2	3	no	—
248	C₂²×C₃₁	—	2	3	no	—
249	C₂²×C₃₁	—	2	3	no	—
250	C₂²×C₃₁	—	2	3	no	—
251	C₂²×C₃₁	—	4	12	no	—
252	C₂²×C₃₁	—	4	1	no	—
253	C₂²×C₃₁	—	4	1	no	—
254	C₂²×C₃₁	—	4	1	no	—
255	C₂²×C₃₁	C₂×#123	1	6	yes	—
256	C₂²×C₃₁	#578	1	2	yes	—
257	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
258	C₄×C₃₁	—	2	1	no	—
259	C₄×C₃₁	—	2	3	no	—
260	C₄×C₃₁	—	4	6	no	—
261	C₄×C₃₁	—	4	3	no	—
262	C₄×C₃₁	—	4	3	no	—
263	C₄×C₃₁	—	4	2	no	—
264	C₄×C₃₁	—	4	1	no	—
265	C₄×C₃₁	C₂×#123	1	6	yes	—
266	C₄×C₃₁	C₂²×#41	1	2	yes	—
267	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	1	yes	characteristic
268	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×#131	C₂²×C₃₁	1	2	yes	—
270	D₄×Q₈	—	2	1	no	—
271	#548	—	2	1	no	—
272	#547	—	2	3	no	—
273	#546	—	2	3	no	—
274	#537	C₂²×C₃₁	1	3	yes	—
275	#536	C₂²×C₃₁	1	3	yes	—
276	#525	C₂²×C₃₁	1	3	yes	—
277	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	3	yes	—
278	#520	—	2	3	no	—
279	#520	—	2	1	no	—
280	#520	C₂²×C₃₁	1	6	yes	—
281	#520	C₂²×C₃₁	1	2	yes	—
282	#520	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×C₄×Q₈	—	2	1	no	—
284	C₂×C₄×D₄	—	2	3	no	—
285	C₂×#117	C₂²×C₃₁	1	3	yes	—
286	C₂²×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
287	#485	C₂²×C₃₁	1	6	yes	complemented
288	#485	C₂²×C₃₁	1	12	yes	complemented
289	#484	C₂²×C₃₁	1	3	yes	—
290	#484	C₂²×C₃₁	1	6	yes	—
291	#484	C₂²×C₃₁	1	12	yes	complemented
292	#483	C₂²×C₃₁	1	3	yes	—
293	#483	C₂²×C₃₁	1	6	yes	complemented
294	#482	C₂²×C₃₁	1	6	yes	complemented
295	#481	C₂²×C₃₁	1	3	yes	—
296	#480	C₂²×C₃₁	1	1	yes	characteristic
297	#480	C₂²×C₃₁	1	6	yes	complemented
298	#428	C₂²×C₃₁	1	6	yes	complemented
299	#428	C₂²×C₃₁	1	6	yes	complemented
300	#427	C₂²×C₃₁	1	6	yes	complemented
301	C₂×#107	C₂²×C₃₁	1	6	yes	complemented
302	#419	C₂²×C₃₁	1	6	yes	complemented
303	#417	C₂²×C₃₁	1	6	yes	complemented
304	#416	C₂²×C₃₁	1	6	yes	complemented
305	C₂×#103	C₂²×C₃₁	1	6	yes	complemented
306	#408	C₂²×C₃₁	1	6	yes	—
307	#407	C₂²×C₃₁	1	3	yes	—
308	C₂×C₂²⋊C₈	C₂²×C₃₁	1	3	yes	—
309	C₂×C₃₁	—	2	2	no	—
310	C₂×C₃₁	—	2	6	no	—
311	C₂×C₃₁	—	4	3	no	—
312	C₂×C₃₁	—	8	1	no	—
313	C₂×C₃₁	C₂²×#123	1	1	yes	characteristic
314	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
315	C₂×C₃₁	#3585	1	1	yes	characteristic
316	2₋¹⁺⁴	—	2	8	no	—
317	C₂×#41	—	2	8	no	—
318	C₂×#41	—	2	3	no	—
319	C₂×#41	D₄×C₃₁	1	3	yes	—
320	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
321	C₂×#41	C₂³×C₃₁	1	3	yes	—
322	C₂×#41	C₂³×C₃₁	1	3	yes	—
323	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
324	C₂²×Q₈	—	2	1	no	—
325	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
326	C₂²×Q₈	C₂³×C₃₁	1	3	yes	—
327	C₂²×D₄	D₄×C₃₁	1	3	yes	—
328	C₂³×C₄	—	2	3	no	—
329	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
330	#131	—	2	8	no	—
331	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
332	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
333	C₄⋊Q₈	—	2	3	no	—
334	#124	—	2	3	no	—
335	#123	—	2	6	no	—
336	#123	—	2	3	no	—
337	C₂²⋊Q₈	—	2	3	no	—
338	C₂²⋊Q₈	—	2	3	no	—
339	C₂²⋊Q₈	—	2	3	no	—
340	C₂²⋊Q₈	C₂³×C₃₁	1	3	yes	—
341	C₂²⋊Q₈	C₂³×C₃₁	1	6	yes	—
342	C₂²⋊Q₈	C₂³×C₃₁	1	3	yes	—
343	#121	—	2	6	no	—
344	#121	—	2	3	no	—
345	#121	C₂³×C₃₁	1	1	yes	characteristic
346	#121	C₂³×C₃₁	1	3	yes	—
347	#120	—	2	3	no	—
348	C₄×Q₈	—	2	6	no	—
349	C₄×Q₈	—	2	2	no	—
350	C₄×Q₈	—	2	1	no	—
351	C₄×D₄	—	2	12	no	—
352	C₄×D₄	—	2	6	no	—
353	C₄×D₄	—	2	6	no	—
354	C₄×D₄	—	2	3	no	—
355	C₄×D₄	—	2	3	no	—
356	C₄×D₄	—	2	3	no	—
357	#117	—	2	3	no	—
358	#117	—	2	3	no	—
359	#117	C₂³×C₃₁	1	6	yes	—
360	#117	C₂³×C₃₁	1	6	yes	—
361	#117	C₂³×C₃₁	1	3	yes	—
362	C₂×C₄⋊C₄	—	2	3	no	—
363	C₂×C₄⋊C₄	—	2	3	no	—
364	C₂×C₄⋊C₄	—	2	3	no	—
365	C₂×C₄⋊C₄	—	2	3	no	—
366	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
367	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
369	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
371	C₂×C₂²⋊C₄	—	2	3	no	—
372	C₂×C₂²⋊C₄	—	2	3	no	—
373	C₂×C₄²	—	2	3	no	—
374	#107	C₂³×C₃₁	1	6	yes	—
375	#107	C₂³×C₃₁	1	12	yes	—
376	#107	C₂³×C₃₁	1	2	yes	—
377	#106	C₂³×C₃₁	1	6	yes	—
378	#106	C₂³×C₃₁	1	6	yes	—
379	#103	C₂³×C₃₁	1	6	yes	—
380	#103	C₂³×C₃₁	1	12	yes	—
381	#103	C₂³×C₃₁	1	6	yes	—
382	#102	C₂³×C₃₁	1	2	yes	—
383	#102	C₂³×C₃₁	1	6	yes	—
384	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
385	C₂²⋊C₈	C₂³×C₃₁	1	3	yes	—
386	C₂²⋊C₈	C₂³×C₃₁	1	6	yes	—
387	C₂²⋊C₈	C₂³×C₃₁	1	6	yes	—
388	C₃₁	#60526	1	1	yes	characteristic, direct product factor, Sylow
389	C₂⁴	C₂×D₄×C₃₁	1	3	yes	—
390	#41	—	2	4	no	—
391	#41	—	2	12	no	—
392	#41	—	2	12	no	—
393	#41	—	2	12	no	—
394	#41	—	4	12	no	—
395	#41	#41×C₃₁	1	8	yes	—
396	C₂×Q₈	—	2	4	no	—
397	C₂×Q₈	—	2	3	no	—
398	C₂×Q₈	—	2	12	no	—
399	C₂×Q₈	—	4	4	no	—
400	C₂×Q₈	C₂×D₄×C₃₁	1	3	yes	—
401	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
402	C₂×Q₈	C₂⁴×C₃₁	1	3	yes	—
403	C₂×D₄	—	2	12	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	—	2	3	no	—
406	C₂×D₄	—	2	3	no	—
407	C₂×D₄	C₂×D₄×C₃₁	1	6	yes	—
408	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
409	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
410	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
411	C₂²×C₄	—	2	12	no	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	3	no	—
414	C₂²×C₄	—	2	3	no	—
415	C₂²×C₄	—	2	3	no	—
416	C₂²×C₄	—	2	6	no	—
417	C₂²×C₄	—	2	6	no	—
418	C₂²×C₄	—	2	3	no	—
419	C₂²×C₄	—	8	6	no	—
420	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
421	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
422	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
423	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
425	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
426	M₁₆	—	2	12	no	—
427	C₂×C₈	—	2	12	no	—
428	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
429	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
430	C₄⋊C₄	—	2	3	no	—
431	C₄⋊C₄	—	2	6	no	—
432	C₄⋊C₄	—	2	6	no	—
433	C₄⋊C₄	—	2	3	no	—
434	C₄⋊C₄	—	2	6	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	3	no	—
437	C₄⋊C₄	—	2	3	no	—
438	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
439	C₄⋊C₄	C₂⁴×C₃₁	1	6	yes	—
440	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
441	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
442	C₂²⋊C₄	—	2	3	no	—
443	C₂²⋊C₄	—	2	6	no	—
444	C₂²⋊C₄	—	2	6	no	—
445	C₂²⋊C₄	—	2	6	no	—
446	C₂²⋊C₄	—	2	3	no	—
447	C₂²⋊C₄	—	2	3	no	—
448	C₂²⋊C₄	—	2	1	no	—
449	C₂²⋊C₄	—	2	3	no	—
450	C₄²	—	2	6	no	—
451	C₄²	—	2	6	no	—
452	C₄²	—	2	3	no	—
453	C₂³	—	2	6	no	—
454	C₂³	—	2	6	no	—
455	C₂³	—	2	1	no	—
456	C₂³	—	4	3	no	—
457	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
458	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
459	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
460	Q₈	—	2	6	no	—
461	Q₈	—	2	6	no	—
462	Q₈	—	4	6	no	—
463	Q₈	C₃₁×#123	1	4	yes	—
464	D₄	—	2	12	no	—
465	D₄	—	2	6	no	—
466	D₄	—	2	2	no	—
467	D₄	—	4	6	no	—
468	D₄	—	4	2	no	—
469	D₄	—	8	3	no	—
470	D₄	C₃₁×#123	1	12	yes	—
471	C₂×C₄	—	2	6	no	—
472	C₂×C₄	—	2	6	no	—
473	C₂×C₄	—	2	6	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	6	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	6	no	—
479	C₂×C₄	—	2	3	no	—
480	C₂×C₄	—	2	3	no	—
481	C₂×C₄	—	4	6	no	—
482	C₂×C₄	—	4	6	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	6	no	—
485	C₂×C₄	—	8	3	no	—
486	C₂×C₄	—	8	6	no	—
487	C₂×C₄	—	8	6	no	—
488	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
489	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
490	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
491	C₂×C₄	C₂×#41×C₃₁	1	12	yes	—
492	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
493	C₈	—	2	2	no	—
494	C₈	—	2	6	no	—
495	C₂²	—	2	1	no	—
496	C₂²	—	2	3	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	2	3	no	—
499	C₂²	—	2	3	no	—
500	C₂²	—	2	3	no	—
501	C₂²	—	2	3	no	—
502	C₂²	—	4	12	no	—
503	C₂²	—	4	1	no	—
504	C₂²	—	4	1	no	—
505	C₂²	—	4	1	no	—
506	C₂²	C₂×C₃₁×#123	1	6	yes	—
507	C₂²	C₃₁×#578	1	2	yes	—
508	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
509	C₄	—	2	1	no	—
510	C₄	—	2	3	no	—
511	C₄	—	4	3	no	—
512	C₄	—	4	2	no	—
513	C₄	—	4	1	no	—
514	C₄	—	4	6	no	—
515	C₄	—	4	3	no	—
516	C₄	C₂×C₃₁×#123	1	6	yes	—
517	C₄	C₂²×#41×C₃₁	1	2	yes	—
518	C₂	—	2	2	no	—
519	C₂	—	2	6	no	—
520	C₂	—	4	3	no	—
521	C₂	—	8	1	no	—
522	C₂	C₂²×C₃₁×#123	1	1	yes	characteristic
523	C₂	C₂×C₃₁×#578	1	1	yes	characteristic
524	C₂	C₃₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3489	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#520	C₂	semidirect
3. C₃₁×#3098	C₂	semidirect
4. C₃₁×#3097	C₂	semidirect
5. C₃₁×#3095	C₂	semidirect
6. C₃₁×#3092	C₂	semidirect
7. C₃₁×#3091	C₂	semidirect
8. C₂×C₃₁×#484	C₂	semidirect
9. C₃₁×#2913	C₂	semidirect
10. C₃₁×#2898	C₂	semidirect
11. C₃₁×#2880	C₂	semidirect
32. C₃₁×#485	C₂²	semidirect
33. C₃₁×#485	C₂²	semidirect
36. C₃₁×#484	C₂²	semidirect
38. C₃₁×#483	C₂²	semidirect
39. C₃₁×#482	C₂²	semidirect
42. C₃₁×#480	C₂²	semidirect
43. C₃₁×#428	C₂²	semidirect
44. C₃₁×#428	C₂²	semidirect
45. C₃₁×#427	C₂²	semidirect
46. C₂×C₃₁×#107	C₂²	semidirect
47. C₃₁×#419	C₂²	semidirect
48. C₃₁×#417	C₂²	semidirect
49. C₃₁×#416	C₂²	semidirect
50. C₂×C₃₁×#103	C₂²	semidirect
190. #60526	C₃₁	direct (N × H)
234. C₂×#520	C₂×C₃₁	semidirect
235. #3098	C₂×C₃₁	semidirect
236. #3097	C₂×C₃₁	semidirect
237. #3095	C₂×C₃₁	semidirect
238. #3092	C₂×C₃₁	semidirect
239. #3091	C₂×C₃₁	semidirect
240. C₂×#484	C₂×C₃₁	semidirect
241. #2913	C₂×C₃₁	semidirect
242. #2898	C₂×C₃₁	semidirect
243. #2880	C₂×C₃₁	semidirect
287. #485	C₂²×C₃₁	semidirect
288. #485	C₂²×C₃₁	semidirect
291. #484	C₂²×C₃₁	semidirect
293. #483	C₂²×C₃₁	semidirect
294. #482	C₂²×C₃₁	semidirect
297. #480	C₂²×C₃₁	semidirect
298. #428	C₂²×C₃₁	semidirect
299. #428	C₂²×C₃₁	semidirect
300. #427	C₂²×C₃₁	semidirect
301. C₂×#107	C₂²×C₃₁	semidirect
302. #419	C₂²×C₃₁	semidirect
303. #417	C₂²×C₃₁	semidirect
304. #416	C₂²×C₃₁	semidirect
305. C₂×#103	C₂²×C₃₁	semidirect
388. C₃₁	#60526	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×#3489	2
C₂×C₃₁×#520	2
C₃₁×#3098	2
C₃₁×#3097	2
C₃₁×#3095	2
C₃₁×#3092	2
C₃₁×#3091	2
C₂×C₃₁×#484	2
C₃₁×#2913	2
C₃₁×#2898	2
C₃₁×#2880	2
#60526	31

Subgroups — C31×#60526 #21517285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60526 #21517285