Subgroups — C₃₁×#62095 #21518854

Order 7936.

2238 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3499	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3358	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3334	C₂	1	6	yes	maximal, complemented
6	C₃₁×#3294	C₂	1	6	yes	maximal, complemented
7	C₃₁×#3245	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3226	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3100	C₂	1	3	yes	maximal, complemented
10	C₂×C₃₁×#474	C₂	1	3	yes	maximal, complemented
11	C₃₁×#2988	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×2₋¹⁺⁴	—	2	1	no	—
13	C₂²×#41×C₃₁	—	2	1	no	—
14	C₃₁×#550	C₂²	1	1	yes	characteristic
15	D₄×Q₈×C₃₁	—	2	1	no	—
16	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	D₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
18	C₃₁×#548	—	2	1	no	—
19	C₃₁×#547	—	2	3	no	—
20	C₃₁×#547	C₂²	1	3	yes	—
21	C₃₁×#547	C₂²	1	6	yes	complemented
22	C₃₁×#546	—	2	3	no	—
23	D₄²×C₃₁	C₂²	1	3	yes	—
24	C₃₁×#536	—	2	3	no	—
25	C₃₁×#532	C₂²	1	3	yes	—
26	C₃₁×#532	C₂²	1	3	yes	complemented
27	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	complemented
28	C₃₁×#525	—	2	3	no	—
29	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
31	C₃₁×#500	C₂²	1	6	yes	complemented
32	C₃₁×#486	C₂²	1	6	yes	complemented
33	C₃₁×#480	C₂²	1	6	yes	complemented
34	C₃₁×#480	C₂²	1	12	yes	complemented
35	C₃₁×#474	C₂²	1	3	yes	—
36	C₃₁×#474	C₂²	1	6	yes	—
37	C₃₁×#474	C₂²	1	6	yes	complemented
38	C₃₁×#474	C₂²	1	6	yes	complemented
39	C₃₁×#474	C₂²	1	3	yes	complemented
40	C₃₁×#474	C₂²	1	6	yes	complemented
41	C₃₁×#466	C₂²	1	12	yes	complemented
42	C₃₁×#459	C₂²	1	1	yes	characteristic
43	C₃₁×#459	C₂²	1	3	yes	complemented
44	C₃₁×#459	C₂²	1	6	yes	complemented
45	C₃₁×#447	C₂²	1	6	yes	—
46	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
47	C₄×D₈×C₃₁	C₂²	1	6	yes	—
48	C₄×D₈×C₃₁	C₂²	1	6	yes	complemented
49	C₈×D₄×C₃₁	C₂²	1	6	yes	complemented
50	C₃₁×#431	C₂²	1	3	yes	complemented
51	C₂×C₃₁×#107	C₂²	1	6	yes	complemented
52	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	complemented
53	C₂×C₃₁×#102	C₂²	1	6	yes	—
54	C₃₁×2₋¹⁺⁴	—	4	8	no	—
55	C₂×#41×C₃₁	—	2	1	no	—
56	C₂×#41×C₃₁	—	2	3	no	—
57	C₂×#41×C₃₁	—	2	6	no	—
58	C₂×#41×C₃₁	—	2	3	no	—
59	C₂×#41×C₃₁	—	2	3	no	—
60	C₂×#41×C₃₁	—	4	8	no	—
61	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
62	C₂²×Q₈×C₃₁	—	2	1	no	—
63	C₂²×Q₈×C₃₁	—	2	2	no	—
64	C₂²×Q₈×C₃₁	—	2	3	no	—
65	C₂²×D₄×C₃₁	—	2	3	no	—
66	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
67	C₂³×C₄×C₃₁	—	2	3	no	—
68	C₂×D₈×C₃₁	C₂³	1	2	yes	—
69	C₂×D₈×C₃₁	C₂³	1	6	yes	—
70	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
71	C₃₁×C₄⋊Q₈	—	2	3	no	—
72	C₃₁×C₄⋊Q₈	—	2	3	no	—
73	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
74	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
75	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
76	C₃₁×#127	C₂³	1	1	yes	characteristic
77	C₃₁×#127	C₂³	1	3	yes	—
78	C₃₁×#124	—	2	3	no	—
79	C₃₁×#124	—	2	3	no	—
80	C₃₁×#123	—	2	6	no	—
81	C₃₁×#123	—	2	12	no	—
82	C₃₁×C₂²⋊Q₈	—	2	3	no	—
83	C₃₁×C₂²⋊Q₈	—	2	3	no	—
84	C₃₁×C₂²⋊Q₈	—	2	6	no	—
85	C₃₁×C₂²⋊Q₈	—	2	6	no	—
86	C₃₁×#121	—	2	3	no	—
87	C₃₁×#121	—	2	3	no	—
88	C₃₁×#121	—	2	3	no	—
89	C₃₁×#121	—	2	3	no	—
90	C₃₁×#121	C₂³	1	6	yes	—
91	C₃₁×#121	C₂³	1	6	yes	—
92	C₃₁×#120	—	2	6	no	—
93	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₄×D₄×C₃₁	—	2	3	no	—
96	C₄×D₄×C₃₁	—	2	3	no	—
97	C₄×D₄×C₃₁	—	2	3	no	—
98	C₄×D₄×C₃₁	—	2	3	no	—
99	C₄×D₄×C₃₁	D₄	1	3	yes	—
100	C₄×D₄×C₃₁	C₂³	1	3	yes	—
101	C₄×D₄×C₃₁	C₂³	1	6	yes	—
102	C₄×D₄×C₃₁	C₂³	1	3	yes	—
103	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
104	C₄×D₄×C₃₁	C₂³	1	3	yes	—
105	C₄×D₄×C₃₁	C₂³	1	3	yes	—
106	C₃₁×#117	D₄	1	3	yes	—
107	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
110	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
111	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
112	C₃₁×#107	C₂³	1	6	yes	—
113	C₃₁×#107	C₂³	1	12	yes	—
114	C₃₁×#107	C₂³	1	6	yes	—
115	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
116	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
117	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₃₁×#102	C₂³	1	6	yes	—
119	C₃₁×#102	C₂³	1	12	yes	—
120	C₃₁×#102	C₂³	1	6	yes	—
121	C₃₁×#102	C₂³	1	2	yes	—
122	C₃₁×#102	C₂³	1	6	yes	—
123	C₃₁×C₂²⋊C₈	C₂³	1	6	yes	—
124	C₄×C₈×C₃₁	C₂³	1	6	yes	—
125	C₂⁴×C₃₁	—	2	3	no	—
126	#41×C₃₁	—	4	4	no	—
127	#41×C₃₁	—	4	12	no	—
128	#41×C₃₁	—	4	24	no	—
129	#41×C₃₁	—	4	12	no	—
130	#41×C₃₁	—	4	12	no	—
131	#41×C₃₁	D₈	1	8	yes	complemented
132	C₂×Q₈×C₃₁	—	2	3	no	—
133	C₂×Q₈×C₃₁	—	2	6	no	—
134	C₂×Q₈×C₃₁	—	2	3	no	—
135	C₂×Q₈×C₃₁	—	4	4	no	—
136	C₂×Q₈×C₃₁	—	4	8	no	—
137	C₂×Q₈×C₃₁	—	4	12	no	—
138	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂×D₄×C₃₁	—	2	6	no	—
140	C₂×D₄×C₃₁	—	2	6	no	—
141	C₂×D₄×C₃₁	—	2	6	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	—	2	3	no	—
144	C₂×D₄×C₃₁	—	2	3	no	—
145	C₂×D₄×C₃₁	—	2	1	no	—
146	C₂×D₄×C₃₁	—	2	3	no	—
147	C₂×D₄×C₃₁	—	4	3	no	—
148	C₂×D₄×C₃₁	—	4	12	no	—
149	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
150	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
151	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
152	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	—	2	3	no	—
154	C₂²×C₄×C₃₁	—	2	6	no	—
155	C₂²×C₄×C₃₁	—	2	1	no	—
156	C₂²×C₄×C₃₁	—	2	3	no	—
157	C₂²×C₄×C₃₁	—	2	3	no	—
158	C₂²×C₄×C₃₁	—	2	3	no	—
159	C₂²×C₄×C₃₁	—	2	6	no	—
160	C₂²×C₄×C₃₁	—	4	12	no	—
161	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
162	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
163	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
164	D₈×C₃₁	—	2	4	no	—
165	D₈×C₃₁	—	4	6	no	—
166	C₂×C₈×C₃₁	—	4	6	no	—
167	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
168	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
169	C₄⋊C₄×C₃₁	—	2	3	no	—
170	C₄⋊C₄×C₃₁	—	2	6	no	—
171	C₄⋊C₄×C₃₁	—	2	3	no	—
172	C₄⋊C₄×C₃₁	—	2	6	no	—
173	C₄⋊C₄×C₃₁	—	2	3	no	—
174	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
175	C₄⋊C₄×C₃₁	C₂×D₄	1	3	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	1	no	—
180	C₂²⋊C₄×C₃₁	—	2	3	no	—
181	C₂²⋊C₄×C₃₁	—	2	3	no	—
182	C₂²⋊C₄×C₃₁	—	2	3	no	—
183	C₂²⋊C₄×C₃₁	—	2	6	no	—
184	C₂²⋊C₄×C₃₁	—	2	6	no	—
185	C₂²⋊C₄×C₃₁	—	2	6	no	—
186	C₂²⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
187	C₄²×C₃₁	C₂×D₄	1	3	yes	—
188	C₄²×C₃₁	C₂×D₄	1	3	yes	—
189	C₄²×C₃₁	C₂⁴	1	3	yes	—
190	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
191	#62095	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂³×C₃₁	—	2	1	no	—
193	C₂³×C₃₁	—	2	2	no	—
194	C₂³×C₃₁	—	2	3	no	—
195	C₂³×C₃₁	—	4	6	no	—
196	C₂³×C₃₁	—	4	6	no	—
197	C₂³×C₃₁	—	8	3	no	—
198	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
199	Q₈×C₃₁	—	4	6	no	—
200	Q₈×C₃₁	—	4	12	no	—
201	Q₈×C₃₁	—	4	6	no	—
202	Q₈×C₃₁	C₂×D₈	1	4	yes	—
203	D₄×C₃₁	—	2	2	no	—
204	D₄×C₃₁	—	2	3	no	—
205	D₄×C₃₁	—	2	1	no	—
206	D₄×C₃₁	—	2	2	no	—
207	D₄×C₃₁	—	2	3	no	—
208	D₄×C₃₁	—	4	6	no	—
209	D₄×C₃₁	—	4	2	no	—
210	D₄×C₃₁	—	4	12	no	—
211	D₄×C₃₁	—	4	12	no	—
212	D₄×C₃₁	—	4	6	no	—
213	D₄×C₃₁	—	4	2	no	—
214	D₄×C₃₁	—	8	3	no	—
215	D₄×C₃₁	C₂×D₈	1	12	yes	—
216	D₄×C₃₁	2₋¹⁺⁴	1	2	yes	complemented
217	C₂×C₄×C₃₁	—	2	3	no	—
218	C₂×C₄×C₃₁	—	2	3	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	3	no	—
221	C₂×C₄×C₃₁	—	2	6	no	—
222	C₂×C₄×C₃₁	—	4	3	no	—
223	C₂×C₄×C₃₁	—	4	6	no	—
224	C₂×C₄×C₃₁	—	4	12	no	—
225	C₂×C₄×C₃₁	—	4	2	no	—
226	C₂×C₄×C₃₁	—	4	6	no	—
227	C₂×C₄×C₃₁	—	4	6	no	—
228	C₂×C₄×C₃₁	—	4	6	no	—
229	C₂×C₄×C₃₁	—	4	12	no	—
230	C₂×C₄×C₃₁	C₂×D₈	1	12	yes	—
231	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
233	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
234	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
236	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
237	C₈×C₃₁	—	2	2	no	—
238	C₈×C₃₁	—	2	6	no	—
239	#3499	C₂×C₃₁	1	1	yes	characteristic, complemented
240	D₄×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
241	Q₈×D₈	C₂×C₃₁	1	2	yes	complemented
242	#3358	C₂×C₃₁	1	2	yes	complemented
243	#3334	C₂×C₃₁	1	6	yes	complemented
244	#3294	C₂×C₃₁	1	6	yes	complemented
245	#3245	C₂×C₃₁	1	3	yes	complemented
246	#3226	C₂×C₃₁	1	3	yes	complemented
247	#3100	C₂×C₃₁	1	3	yes	complemented
248	C₂×#474	C₂×C₃₁	1	3	yes	complemented
249	#2988	C₂×C₃₁	1	1	yes	characteristic, complemented
250	C₂²×C₃₁	—	2	3	no	—
251	C₂²×C₃₁	—	2	3	no	—
252	C₂²×C₃₁	—	2	2	no	—
253	C₂²×C₃₁	—	4	1	no	—
254	C₂²×C₃₁	—	4	1	no	—
255	C₂²×C₃₁	—	4	2	no	—
256	C₂²×C₃₁	—	4	3	no	—
257	C₂²×C₃₁	—	4	3	no	—
258	C₂²×C₃₁	—	4	2	no	—
259	C₂²×C₃₁	—	4	2	no	—
260	C₂²×C₃₁	—	4	3	no	—
261	C₂²×C₃₁	—	8	12	no	—
262	C₂²×C₃₁	C₂²×D₈	1	6	yes	—
263	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
264	C₄×C₃₁	—	2	1	no	—
265	C₄×C₃₁	—	2	3	no	—
266	C₄×C₃₁	—	2	3	no	—
267	C₄×C₃₁	—	4	1	no	—
268	C₄×C₃₁	—	4	3	no	—
269	C₄×C₃₁	—	4	6	no	—
270	C₄×C₃₁	C₂²×D₈	1	2	yes	—
271	C₄×C₃₁	C₂²×D₈	1	6	yes	—
272	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
273	C₄×C₃₁	#585	1	1	yes	characteristic
274	C₂×2₋¹⁺⁴	—	2	1	no	—
275	C₂²×#41	—	2	1	no	—
276	#550	C₂²×C₃₁	1	1	yes	characteristic
277	D₄×Q₈	—	2	1	no	—
278	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
279	D₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
280	#548	—	2	1	no	—
281	#547	—	2	3	no	—
282	#547	C₂²×C₃₁	1	3	yes	—
283	#547	C₂²×C₃₁	1	6	yes	complemented
284	#546	—	2	3	no	—
285	D₄²	C₂²×C₃₁	1	3	yes	—
286	#536	—	2	3	no	—
287	#532	C₂²×C₃₁	1	3	yes	—
288	#532	C₂²×C₃₁	1	3	yes	complemented
289	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	complemented
290	#525	—	2	3	no	—
291	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
293	#500	C₂²×C₃₁	1	6	yes	complemented
294	#486	C₂²×C₃₁	1	6	yes	complemented
295	#480	C₂²×C₃₁	1	6	yes	complemented
296	#480	C₂²×C₃₁	1	12	yes	complemented
297	#474	C₂²×C₃₁	1	3	yes	—
298	#474	C₂²×C₃₁	1	6	yes	—
299	#474	C₂²×C₃₁	1	6	yes	complemented
300	#474	C₂²×C₃₁	1	6	yes	complemented
301	#474	C₂²×C₃₁	1	3	yes	complemented
302	#474	C₂²×C₃₁	1	6	yes	complemented
303	#466	C₂²×C₃₁	1	12	yes	complemented
304	#459	C₂²×C₃₁	1	1	yes	characteristic
305	#459	C₂²×C₃₁	1	3	yes	complemented
306	#459	C₂²×C₃₁	1	6	yes	complemented
307	#447	C₂²×C₃₁	1	6	yes	—
308	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
309	C₄×D₈	C₂²×C₃₁	1	6	yes	—
310	C₄×D₈	C₂²×C₃₁	1	6	yes	complemented
311	C₈×D₄	C₂²×C₃₁	1	6	yes	complemented
312	#431	C₂²×C₃₁	1	3	yes	complemented
313	C₂×#107	C₂²×C₃₁	1	6	yes	complemented
314	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	complemented
315	C₂×#102	C₂²×C₃₁	1	6	yes	—
316	C₂×C₃₁	—	2	6	no	—
317	C₂×C₃₁	—	4	2	no	—
318	C₂×C₃₁	—	8	2	no	—
319	C₂×C₃₁	—	8	3	no	—
320	C₂×C₃₁	#3489	1	1	yes	characteristic
321	C₂×C₃₁	C₂³×D₈	1	1	yes	characteristic
322	C₂×C₃₁	#3586	1	1	yes	characteristic
323	2₋¹⁺⁴	—	4	8	no	—
324	C₂×#41	—	2	1	no	—
325	C₂×#41	—	2	3	no	—
326	C₂×#41	—	2	6	no	—
327	C₂×#41	—	2	3	no	—
328	C₂×#41	—	2	3	no	—
329	C₂×#41	—	4	8	no	—
330	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
331	C₂²×Q₈	—	2	1	no	—
332	C₂²×Q₈	—	2	2	no	—
333	C₂²×Q₈	—	2	3	no	—
334	C₂²×D₄	—	2	3	no	—
335	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
336	C₂³×C₄	—	2	3	no	—
337	C₂×D₈	C₂³×C₃₁	1	2	yes	—
338	C₂×D₈	C₂³×C₃₁	1	6	yes	—
339	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
340	C₄⋊Q₈	—	2	3	no	—
341	C₄⋊Q₈	—	2	3	no	—
342	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
343	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
344	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
345	#127	C₂³×C₃₁	1	3	yes	—
346	#127	C₂³×C₃₁	1	1	yes	characteristic
347	#124	—	2	3	no	—
348	#124	—	2	3	no	—
349	#123	—	2	6	no	—
350	#123	—	2	12	no	—
351	C₂²⋊Q₈	—	2	3	no	—
352	C₂²⋊Q₈	—	2	3	no	—
353	C₂²⋊Q₈	—	2	6	no	—
354	C₂²⋊Q₈	—	2	6	no	—
355	#121	—	2	3	no	—
356	#121	—	2	3	no	—
357	#121	—	2	3	no	—
358	#121	—	2	3	no	—
359	#121	C₂³×C₃₁	1	6	yes	—
360	#121	C₂³×C₃₁	1	6	yes	—
361	#120	—	2	6	no	—
362	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
363	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
364	C₄×D₄	—	2	3	no	—
365	C₄×D₄	—	2	3	no	—
366	C₄×D₄	—	2	3	no	—
367	C₄×D₄	—	2	3	no	—
368	C₄×D₄	D₄×C₃₁	1	3	yes	—
369	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
370	C₄×D₄	C₂³×C₃₁	1	3	yes	—
371	C₄×D₄	C₂³×C₃₁	1	3	yes	—
372	C₄×D₄	C₂³×C₃₁	1	3	yes	—
373	C₄×D₄	C₂³×C₃₁	1	6	yes	—
374	C₄×D₄	C₂³×C₃₁	1	3	yes	—
375	#117	D₄×C₃₁	1	3	yes	—
376	C₂×C₄⋊C₄	—	2	3	no	—
377	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
378	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
379	C₂×C₂²⋊C₄	—	2	3	no	—
380	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
381	#107	C₂³×C₃₁	1	6	yes	—
382	#107	C₂³×C₃₁	1	12	yes	—
383	#107	C₂³×C₃₁	1	6	yes	—
384	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
385	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
386	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
387	#102	C₂³×C₃₁	1	6	yes	—
388	#102	C₂³×C₃₁	1	12	yes	—
389	#102	C₂³×C₃₁	1	6	yes	—
390	#102	C₂³×C₃₁	1	2	yes	—
391	#102	C₂³×C₃₁	1	6	yes	—
392	C₂²⋊C₈	C₂³×C₃₁	1	6	yes	—
393	C₄×C₈	C₂³×C₃₁	1	6	yes	—
394	C₃₁	#62095	1	1	yes	characteristic, direct product factor, Sylow
395	C₂⁴	—	2	3	no	—
396	#41	—	4	4	no	—
397	#41	—	4	12	no	—
398	#41	—	4	24	no	—
399	#41	—	4	12	no	—
400	#41	—	4	12	no	—
401	#41	D₈×C₃₁	1	8	yes	complemented
402	C₂×Q₈	—	2	3	no	—
403	C₂×Q₈	—	2	6	no	—
404	C₂×Q₈	—	2	3	no	—
405	C₂×Q₈	—	4	4	no	—
406	C₂×Q₈	—	4	8	no	—
407	C₂×Q₈	—	4	12	no	—
408	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
409	C₂×D₄	—	2	6	no	—
410	C₂×D₄	—	2	6	no	—
411	C₂×D₄	—	2	6	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	3	no	—
414	C₂×D₄	—	2	3	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	3	no	—
417	C₂×D₄	—	4	3	no	—
418	C₂×D₄	—	4	12	no	—
419	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
420	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
421	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
422	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
423	C₂²×C₄	—	2	3	no	—
424	C₂²×C₄	—	2	6	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	3	no	—
427	C₂²×C₄	—	2	3	no	—
428	C₂²×C₄	—	2	3	no	—
429	C₂²×C₄	—	2	6	no	—
430	C₂²×C₄	—	4	12	no	—
431	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
432	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
433	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
434	D₈	—	2	4	no	—
435	D₈	—	4	6	no	—
436	C₂×C₈	—	4	6	no	—
437	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
438	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
439	C₄⋊C₄	—	2	3	no	—
440	C₄⋊C₄	—	2	6	no	—
441	C₄⋊C₄	—	2	3	no	—
442	C₄⋊C₄	—	2	6	no	—
443	C₄⋊C₄	—	2	3	no	—
444	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
445	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
446	C₄⋊C₄	C₂⁴×C₃₁	1	6	yes	—
447	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
448	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
449	C₂²⋊C₄	—	2	1	no	—
450	C₂²⋊C₄	—	2	3	no	—
451	C₂²⋊C₄	—	2	3	no	—
452	C₂²⋊C₄	—	2	3	no	—
453	C₂²⋊C₄	—	2	6	no	—
454	C₂²⋊C₄	—	2	6	no	—
455	C₂²⋊C₄	—	2	6	no	—
456	C₂²⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
457	C₄²	C₂×D₄×C₃₁	1	3	yes	—
458	C₄²	C₂×D₄×C₃₁	1	3	yes	—
459	C₄²	C₂⁴×C₃₁	1	3	yes	—
460	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
461	C₂³	—	2	1	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	2	3	no	—
464	C₂³	—	4	6	no	—
465	C₂³	—	4	6	no	—
466	C₂³	—	8	3	no	—
467	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
468	Q₈	—	4	6	no	—
469	Q₈	—	4	12	no	—
470	Q₈	—	4	6	no	—
471	Q₈	C₂×D₈×C₃₁	1	4	yes	—
472	D₄	—	2	2	no	—
473	D₄	—	2	3	no	—
474	D₄	—	2	1	no	—
475	D₄	—	2	2	no	—
476	D₄	—	2	3	no	—
477	D₄	—	4	6	no	—
478	D₄	—	4	2	no	—
479	D₄	—	4	12	no	—
480	D₄	—	4	12	no	—
481	D₄	—	4	6	no	—
482	D₄	—	4	2	no	—
483	D₄	—	8	3	no	—
484	D₄	C₂×D₈×C₃₁	1	12	yes	—
485	D₄	C₃₁×2₋¹⁺⁴	1	2	yes	complemented
486	C₂×C₄	—	2	3	no	—
487	C₂×C₄	—	2	3	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	3	no	—
490	C₂×C₄	—	2	6	no	—
491	C₂×C₄	—	4	3	no	—
492	C₂×C₄	—	4	6	no	—
493	C₂×C₄	—	4	12	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	4	6	no	—
496	C₂×C₄	—	4	6	no	—
497	C₂×C₄	—	4	6	no	—
498	C₂×C₄	—	4	12	no	—
499	C₂×C₄	C₂×D₈×C₃₁	1	12	yes	—
500	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
501	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
502	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
503	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
504	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
505	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
506	C₈	—	2	2	no	—
507	C₈	—	2	6	no	—
508	C₂²	—	2	3	no	—
509	C₂²	—	2	3	no	—
510	C₂²	—	2	2	no	—
511	C₂²	—	4	1	no	—
512	C₂²	—	4	1	no	—
513	C₂²	—	4	3	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	4	3	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	3	no	—
519	C₂²	—	8	12	no	—
520	C₂²	C₂²×D₈×C₃₁	1	6	yes	—
521	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
522	C₄	—	2	3	no	—
523	C₄	—	2	1	no	—
524	C₄	—	2	3	no	—
525	C₄	—	4	1	no	—
526	C₄	—	4	3	no	—
527	C₄	—	4	6	no	—
528	C₄	C₂²×D₈×C₃₁	1	2	yes	—
529	C₄	C₂²×D₈×C₃₁	1	6	yes	—
530	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
531	C₄	C₃₁×#585	1	1	yes	characteristic
532	C₂	—	2	6	no	—
533	C₂	—	4	2	no	—
534	C₂	—	8	3	no	—
535	C₂	—	8	2	no	—
536	C₂	C₃₁×#3489	1	1	yes	characteristic
537	C₂	C₂³×D₈×C₃₁	1	1	yes	characteristic
538	C₂	C₃₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
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
1. C₃₁×#3499	C₂	semidirect
2. D₄×#41×C₃₁	C₂	semidirect
3. Q₈×D₈×C₃₁	C₂	semidirect
4. C₃₁×#3358	C₂	semidirect
5. C₃₁×#3334	C₂	semidirect
6. C₃₁×#3294	C₂	semidirect
7. C₃₁×#3245	C₂	semidirect
8. C₃₁×#3226	C₂	semidirect
9. C₃₁×#3100	C₂	semidirect
10. C₂×C₃₁×#474	C₂	semidirect
11. C₃₁×#2988	C₂	semidirect
17. D₄×Q₈×C₃₁	C₂²	semidirect
21. C₃₁×#547	C₂²	semidirect
26. C₃₁×#532	C₂²	semidirect
27. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
31. C₃₁×#500	C₂²	semidirect
32. C₃₁×#486	C₂²	semidirect
33. C₃₁×#480	C₂²	semidirect
34. C₃₁×#480	C₂²	semidirect
37. C₃₁×#474	C₂²	semidirect
38. C₃₁×#474	C₂²	semidirect
39. C₃₁×#474	C₂²	semidirect
40. C₃₁×#474	C₂²	semidirect
41. C₃₁×#466	C₂²	semidirect
43. C₃₁×#459	C₂²	semidirect
44. C₃₁×#459	C₂²	semidirect
46. C₈×Q₈×C₃₁	C₂²	semidirect
48. C₄×D₈×C₃₁	C₂²	semidirect
49. C₈×D₄×C₃₁	C₂²	semidirect
50. C₃₁×#431	C₂²	semidirect
51. C₂×C₃₁×#107	C₂²	semidirect
52. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
131. #41×C₃₁	D₈	semidirect
191. #62095	C₃₁	direct (N × H)
216. D₄×C₃₁	2₋¹⁺⁴	semidirect
239. #3499	C₂×C₃₁	semidirect
240. D₄×#41	C₂×C₃₁	semidirect
241. Q₈×D₈	C₂×C₃₁	semidirect
242. #3358	C₂×C₃₁	semidirect
243. #3334	C₂×C₃₁	semidirect
244. #3294	C₂×C₃₁	semidirect
245. #3245	C₂×C₃₁	semidirect
246. #3226	C₂×C₃₁	semidirect
247. #3100	C₂×C₃₁	semidirect
248. C₂×#474	C₂×C₃₁	semidirect
249. #2988	C₂×C₃₁	semidirect
279. D₄×Q₈	C₂²×C₃₁	semidirect
283. #547	C₂²×C₃₁	semidirect
288. #532	C₂²×C₃₁	semidirect
289. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
293. #500	C₂²×C₃₁	semidirect
294. #486	C₂²×C₃₁	semidirect
295. #480	C₂²×C₃₁	semidirect
296. #480	C₂²×C₃₁	semidirect
299. #474	C₂²×C₃₁	semidirect
300. #474	C₂²×C₃₁	semidirect
301. #474	C₂²×C₃₁	semidirect
302. #474	C₂²×C₃₁	semidirect
303. #466	C₂²×C₃₁	semidirect
305. #459	C₂²×C₃₁	semidirect
306. #459	C₂²×C₃₁	semidirect
308. C₈×Q₈	C₂²×C₃₁	semidirect
310. C₄×D₈	C₂²×C₃₁	semidirect
311. C₈×D₄	C₂²×C₃₁	semidirect
312. #431	C₂²×C₃₁	semidirect
313. C₂×#107	C₂²×C₃₁	semidirect
314. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
394. C₃₁	#62095	direct (N × H)
401. #41	D₈×C₃₁	semidirect
485. D₄	C₃₁×2₋¹⁺⁴	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3499	2
D₄×#41×C₃₁	2
Q₈×D₈×C₃₁	2
C₃₁×#3358	2
C₃₁×#3334	2
C₃₁×#3294	2
C₃₁×#3245	2
C₃₁×#3226	2
C₃₁×#3100	2
C₂×C₃₁×#474	2
C₃₁×#2988	2
#62095	31

Subgroups — C31×#62095 #21518854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62095 #21518854