Subgroups — C₃₁×#36708 #21493467

Order 7936.

2326 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2848	C₂	1	2	yes	maximal
4	C₃₁×#2690	C₂	1	8	yes	maximal, complemented
5	C₃₁×#2677	C₂	1	8	yes	maximal, complemented
6	C₃₁×#2627	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2561	C₂	1	2	yes	maximal
8	C₃₁×#2484	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2482	C₂	1	2	yes	maximal
10	C₃₁×#2434	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2313	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#544	—	2	8	no	—
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	4	yes	complemented
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	4	yes	complemented
17	C₂×C₃₁×#125	C₂²	1	4	yes	complemented
18	C₂×C₃₁×C₂²⋊Q₈	—	2	8	no	—
19	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	8	yes	—
21	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
23	C₂×C₃₁×#117	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#400	C₂²	1	4	yes	complemented
30	C₃₁×#400	C₂²	1	4	yes	complemented
31	C₃₁×#399	C₂²	1	4	yes	—
32	C₃₁×#398	C₂²	1	8	yes	complemented
33	C₃₁×#398	C₂²	1	4	yes	complemented
34	C₃₁×#397	C₂²	1	4	yes	—
35	C₃₁×#397	C₂²	1	4	yes	—
36	C₃₁×#395	C₂²	1	4	yes	complemented
37	C₃₁×#395	C₂²	1	8	yes	complemented
38	C₃₁×#393	C₂²	1	4	yes	—
39	C₃₁×#391	C₂²	1	4	yes	complemented
40	C₃₁×#388	C₂²	1	8	yes	—
41	C₃₁×#388	C₂²	1	8	yes	—
42	C₃₁×#387	C₂²	1	8	yes	complemented
43	C₃₁×#387	C₂²	1	8	yes	complemented
44	C₃₁×#385	C₂²	1	8	yes	—
45	C₃₁×#385	C₂²	1	4	yes	—
46	C₃₁×#385	C₂²	1	4	yes	—
47	C₃₁×#381	C₂²	1	2	yes	—
48	C₃₁×#381	C₂²	1	4	yes	—
49	C₃₁×#380	C₂²	1	2	yes	—
50	C₃₁×#380	C₂²	1	4	yes	—
51	C₃₁×#379	C₂²	1	4	yes	—
52	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
53	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×Q₈×C₃₁	—	2	4	no	—
55	C₂²×Q₈×C₃₁	—	4	4	no	—
56	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
57	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
58	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
59	C₂³×C₄×C₃₁	—	2	4	no	—
60	C₂³×C₄×C₃₁	—	4	4	no	—
61	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×C₄⋊Q₈	—	2	16	no	—
69	C₃₁×C₄⋊Q₈	—	4	8	no	—
70	C₃₁×#125	—	2	16	no	—
71	C₃₁×C₂²⋊Q₈	—	2	8	no	—
72	C₃₁×C₂²⋊Q₈	—	2	8	no	—
73	C₃₁×C₂²⋊Q₈	—	2	32	no	—
74	C₃₁×C₂²⋊Q₈	—	4	8	no	—
75	C₃₁×#117	—	2	8	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
78	C₂×C₄⋊C₄×C₃₁	D₄	1	8	yes	—
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
92	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
93	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
94	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
99	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
100	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
101	C₃₁×#95	C₂³	1	4	yes	—
102	C₃₁×#95	C₂³	1	8	yes	—
103	C₃₁×#95	C₂³	1	8	yes	—
104	C₃₁×#95	C₂³	1	4	yes	—
105	C₃₁×#95	C₂³	1	8	yes	—
106	C₃₁×#95	C₂³	1	8	yes	—
107	C₃₁×#95	C₂³	1	4	yes	—
108	C₃₁×#95	C₂³	1	4	yes	—
109	C₂⁴×C₃₁	—	2	2	no	—
110	C₂⁴×C₃₁	—	2	4	no	—
111	C₂⁴×C₃₁	—	2	4	no	—
112	C₂⁴×C₃₁	—	4	2	no	—
113	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
114	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
115	C₂×Q₈×C₃₁	—	2	8	no	—
116	C₂×Q₈×C₃₁	—	2	4	no	—
117	C₂×Q₈×C₃₁	—	2	4	no	—
118	C₂×Q₈×C₃₁	—	2	8	no	—
119	C₂×Q₈×C₃₁	—	4	4	no	—
120	C₂×Q₈×C₃₁	—	4	4	no	—
121	C₂×Q₈×C₃₁	—	4	4	no	—
122	C₂×Q₈×C₃₁	—	4	16	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	2	4	no	—
125	C₂²×C₄×C₃₁	—	2	4	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	2	4	no	—
129	C₂²×C₄×C₃₁	—	2	8	no	—
130	C₂²×C₄×C₃₁	—	2	8	no	—
131	C₂²×C₄×C₃₁	—	4	2	no	—
132	C₂²×C₄×C₃₁	—	4	4	no	—
133	C₂²×C₄×C₃₁	—	4	2	no	—
134	C₂²×C₄×C₃₁	—	4	2	no	—
135	C₂²×C₄×C₃₁	—	4	16	no	—
136	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
137	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
138	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
140	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
142	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
145	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
146	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
148	C₄⋊C₄×C₃₁	—	2	8	no	—
149	C₄⋊C₄×C₃₁	—	2	8	no	—
150	C₄⋊C₄×C₃₁	—	2	4	no	—
151	C₄⋊C₄×C₃₁	—	2	4	no	—
152	C₄⋊C₄×C₃₁	—	2	4	no	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	8	no	—
155	C₄⋊C₄×C₃₁	—	2	8	no	—
156	C₄⋊C₄×C₃₁	—	2	16	no	—
157	C₄⋊C₄×C₃₁	—	4	4	no	—
158	C₄⋊C₄×C₃₁	—	4	8	no	—
159	C₄⋊C₄×C₃₁	C₂×D₄	1	16	yes	—
160	C₂²⋊C₄×C₃₁	—	2	8	no	—
161	C₂²⋊C₄×C₃₁	—	2	16	no	—
162	C₂²⋊C₄×C₃₁	—	2	8	no	—
163	C₂²⋊C₄×C₃₁	—	2	8	no	—
164	C₂²⋊C₄×C₃₁	—	2	8	no	—
165	C₂²⋊C₄×C₃₁	—	4	8	no	—
166	C₄²×C₃₁	—	2	8	no	—
167	C₄²×C₃₁	—	4	4	no	—
168	#36708	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂³×C₃₁	—	2	2	no	—
170	C₂³×C₃₁	—	2	2	no	—
171	C₂³×C₃₁	—	2	8	no	—
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	—	2	4	no	—
174	C₂³×C₃₁	—	2	2	no	—
175	C₂³×C₃₁	—	2	4	no	—
176	C₂³×C₃₁	—	4	16	no	—
177	C₂³×C₃₁	—	4	4	no	—
178	C₂³×C₃₁	—	4	4	no	—
179	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
180	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
181	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
182	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
183	Q₈×C₃₁	—	4	16	no	—
184	Q₈×C₃₁	—	8	4	no	—
185	Q₈×C₃₁	—	8	4	no	—
186	C₂×C₄×C₃₁	—	2	8	no	—
187	C₂×C₄×C₃₁	—	2	8	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	4	no	—
190	C₂×C₄×C₃₁	—	2	4	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	4	no	—
193	C₂×C₄×C₃₁	—	2	4	no	—
194	C₂×C₄×C₃₁	—	2	4	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	8	no	—
198	C₂×C₄×C₃₁	—	2	8	no	—
199	C₂×C₄×C₃₁	—	2	4	no	—
200	C₂×C₄×C₃₁	—	2	8	no	—
201	C₂×C₄×C₃₁	—	2	8	no	—
202	C₂×C₄×C₃₁	—	2	8	no	—
203	C₂×C₄×C₃₁	—	2	4	no	—
204	C₂×C₄×C₃₁	—	4	4	no	—
205	C₂×C₄×C₃₁	—	4	4	no	—
206	C₂×C₄×C₃₁	—	4	16	no	—
207	C₂×C₄×C₃₁	—	4	4	no	—
208	C₂×C₄×C₃₁	—	4	4	no	—
209	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
210	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
211	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
212	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
213	C₂×#544	C₂×C₃₁	1	1	yes	characteristic, complemented
214	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
215	#2848	C₂×C₃₁	1	2	yes	—
216	#2690	C₂×C₃₁	1	8	yes	complemented
217	#2677	C₂×C₃₁	1	8	yes	complemented
218	#2627	C₂×C₃₁	1	2	yes	complemented
219	#2561	C₂×C₃₁	1	2	yes	—
220	#2484	C₂×C₃₁	1	2	yes	complemented
221	#2482	C₂×C₃₁	1	2	yes	—
222	#2434	C₂×C₃₁	1	2	yes	complemented
223	#2313	C₂×C₃₁	1	1	yes	characteristic, complemented
224	C₂²×C₃₁	—	2	2	no	—
225	C₂²×C₃₁	—	2	2	no	—
226	C₂²×C₃₁	—	2	8	no	—
227	C₂²×C₃₁	—	2	4	no	—
228	C₂²×C₃₁	—	2	8	no	—
229	C₂²×C₃₁	—	4	16	no	—
230	C₂²×C₃₁	—	4	2	no	—
231	C₂²×C₃₁	—	4	2	no	—
232	C₂²×C₃₁	—	4	4	no	—
233	C₂²×C₃₁	#536	1	4	yes	—
234	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
235	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
236	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
237	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
238	C₄×C₃₁	—	2	4	no	—
239	C₄×C₃₁	—	2	4	no	—
240	C₄×C₃₁	—	2	4	no	—
241	C₄×C₃₁	—	2	4	no	—
242	C₄×C₃₁	—	4	2	no	—
243	C₄×C₃₁	—	4	2	no	—
244	C₄×C₃₁	—	4	4	no	—
245	C₄×C₃₁	—	4	4	no	—
246	C₄×C₃₁	—	4	8	no	—
247	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
248	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
249	#544	—	2	8	no	—
250	C₂×C₄⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
251	C₂×C₄⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
252	C₂×#125	C₂²×C₃₁	1	4	yes	complemented
253	C₂×C₂²⋊Q₈	—	2	8	no	—
254	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
255	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
256	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	8	yes	—
257	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
258	C₂×#117	C₂²×C₃₁	1	2	yes	—
259	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
260	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
261	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
262	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
263	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
264	#400	C₂²×C₃₁	1	4	yes	complemented
265	#400	C₂²×C₃₁	1	4	yes	complemented
266	#399	C₂²×C₃₁	1	4	yes	—
267	#398	C₂²×C₃₁	1	8	yes	complemented
268	#398	C₂²×C₃₁	1	4	yes	complemented
269	#397	C₂²×C₃₁	1	4	yes	—
270	#397	C₂²×C₃₁	1	4	yes	—
271	#395	C₂²×C₃₁	1	4	yes	complemented
272	#395	C₂²×C₃₁	1	8	yes	complemented
273	#393	C₂²×C₃₁	1	4	yes	—
274	#391	C₂²×C₃₁	1	4	yes	complemented
275	#388	C₂²×C₃₁	1	8	yes	—
276	#388	C₂²×C₃₁	1	8	yes	—
277	#387	C₂²×C₃₁	1	8	yes	complemented
278	#387	C₂²×C₃₁	1	8	yes	complemented
279	#385	C₂²×C₃₁	1	8	yes	—
280	#385	C₂²×C₃₁	1	4	yes	—
281	#385	C₂²×C₃₁	1	4	yes	—
282	#381	C₂²×C₃₁	1	4	yes	—
283	#381	C₂²×C₃₁	1	2	yes	—
284	#380	C₂²×C₃₁	1	2	yes	—
285	#380	C₂²×C₃₁	1	4	yes	—
286	#379	C₂²×C₃₁	1	4	yes	—
287	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
288	C₂×C₃₁	—	2	4	no	—
289	C₂×C₃₁	—	4	4	no	—
290	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
291	C₂×C₃₁	#3484	1	1	yes	characteristic
292	C₂×C₃₁	#3486	1	1	yes	characteristic
293	C₂×C₃₁	#3538	1	4	yes	—
294	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
295	C₂²×Q₈	—	2	4	no	—
296	C₂²×Q₈	—	4	4	no	—
297	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
298	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
299	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
300	C₂³×C₄	—	2	4	no	—
301	C₂³×C₄	—	4	4	no	—
302	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
303	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
304	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
305	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
306	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
307	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
308	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
309	C₄⋊Q₈	—	2	16	no	—
310	C₄⋊Q₈	—	4	8	no	—
311	#125	—	2	16	no	—
312	C₂²⋊Q₈	—	2	8	no	—
313	C₂²⋊Q₈	—	2	8	no	—
314	C₂²⋊Q₈	—	2	32	no	—
315	C₂²⋊Q₈	—	4	8	no	—
316	#117	—	2	8	no	—
317	C₂×C₄⋊C₄	—	2	4	no	—
318	C₂×C₄⋊C₄	—	2	4	no	—
319	C₂×C₄⋊C₄	D₄×C₃₁	1	8	yes	—
320	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
321	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
322	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
323	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
324	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
325	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
326	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
327	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
328	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
329	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
330	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
331	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
332	C₂×C₂²⋊C₄	—	2	4	no	—
333	C₂×C₂²⋊C₄	—	2	4	no	—
334	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
335	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
336	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
337	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
338	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
339	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
340	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
341	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
342	#95	C₂³×C₃₁	1	4	yes	—
343	#95	C₂³×C₃₁	1	4	yes	—
344	#95	C₂³×C₃₁	1	8	yes	—
345	#95	C₂³×C₃₁	1	4	yes	—
346	#95	C₂³×C₃₁	1	4	yes	—
347	#95	C₂³×C₃₁	1	8	yes	—
348	#95	C₂³×C₃₁	1	8	yes	—
349	#95	C₂³×C₃₁	1	8	yes	—
350	C₃₁	#36708	1	1	yes	characteristic, direct product factor, Sylow
351	C₂⁴	—	2	2	no	—
352	C₂⁴	—	2	4	no	—
353	C₂⁴	—	2	4	no	—
354	C₂⁴	—	4	2	no	—
355	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
356	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
357	C₂×Q₈	—	2	8	no	—
358	C₂×Q₈	—	2	4	no	—
359	C₂×Q₈	—	2	4	no	—
360	C₂×Q₈	—	2	8	no	—
361	C₂×Q₈	—	4	4	no	—
362	C₂×Q₈	—	4	4	no	—
363	C₂×Q₈	—	4	4	no	—
364	C₂×Q₈	—	4	16	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	4	no	—
367	C₂²×C₄	—	2	4	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	4	no	—
371	C₂²×C₄	—	2	8	no	—
372	C₂²×C₄	—	2	8	no	—
373	C₂²×C₄	—	4	2	no	—
374	C₂²×C₄	—	4	4	no	—
375	C₂²×C₄	—	4	2	no	—
376	C₂²×C₄	—	4	2	no	—
377	C₂²×C₄	—	4	16	no	—
378	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
379	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
380	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
381	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
382	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
383	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
384	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
385	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
386	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
387	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
388	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
389	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
390	C₄⋊C₄	—	2	8	no	—
391	C₄⋊C₄	—	2	8	no	—
392	C₄⋊C₄	—	2	4	no	—
393	C₄⋊C₄	—	2	4	no	—
394	C₄⋊C₄	—	2	4	no	—
395	C₄⋊C₄	—	2	4	no	—
396	C₄⋊C₄	—	2	8	no	—
397	C₄⋊C₄	—	2	8	no	—
398	C₄⋊C₄	—	2	16	no	—
399	C₄⋊C₄	—	4	4	no	—
400	C₄⋊C₄	—	4	8	no	—
401	C₄⋊C₄	C₂×D₄×C₃₁	1	16	yes	—
402	C₂²⋊C₄	—	2	8	no	—
403	C₂²⋊C₄	—	2	16	no	—
404	C₂²⋊C₄	—	2	8	no	—
405	C₂²⋊C₄	—	2	8	no	—
406	C₂²⋊C₄	—	2	8	no	—
407	C₂²⋊C₄	—	4	8	no	—
408	C₄²	—	2	8	no	—
409	C₄²	—	4	4	no	—
410	C₂³	—	2	2	no	—
411	C₂³	—	2	2	no	—
412	C₂³	—	2	8	no	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	4	no	—
415	C₂³	—	2	2	no	—
416	C₂³	—	2	4	no	—
417	C₂³	—	4	16	no	—
418	C₂³	—	4	4	no	—
419	C₂³	—	4	4	no	—
420	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
421	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
422	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
423	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
424	Q₈	—	4	16	no	—
425	Q₈	—	8	4	no	—
426	Q₈	—	8	4	no	—
427	C₂×C₄	—	2	8	no	—
428	C₂×C₄	—	2	8	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	4	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	8	no	—
439	C₂×C₄	—	2	8	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	2	8	no	—
443	C₂×C₄	—	2	8	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	4	4	no	—
446	C₂×C₄	—	4	4	no	—
447	C₂×C₄	—	4	16	no	—
448	C₂×C₄	—	4	4	no	—
449	C₂×C₄	—	4	4	no	—
450	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
451	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
452	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
453	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
454	C₂²	—	2	2	no	—
455	C₂²	—	2	2	no	—
456	C₂²	—	2	8	no	—
457	C₂²	—	2	4	no	—
458	C₂²	—	2	8	no	—
459	C₂²	—	4	16	no	—
460	C₂²	—	4	2	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	4	4	no	—
463	C₂²	C₃₁×#536	1	4	yes	—
464	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
465	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
466	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
467	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
468	C₄	—	2	4	no	—
469	C₄	—	2	4	no	—
470	C₄	—	2	4	no	—
471	C₄	—	2	4	no	—
472	C₄	—	4	2	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	4	no	—
475	C₄	—	4	4	no	—
476	C₄	—	4	8	no	—
477	C₂	—	2	4	no	—
478	C₂	—	4	4	no	—
479	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
480	C₂	C₃₁×#3484	1	1	yes	characteristic
481	C₂	C₃₁×#3486	1	1	yes	characteristic
482	C₂	C₃₁×#3538	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂⁵×C₃₁	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#544	C₂	semidirect
4. C₃₁×#2690	C₂	semidirect
5. C₃₁×#2677	C₂	semidirect
6. C₃₁×#2627	C₂	semidirect
8. C₃₁×#2484	C₂	semidirect
10. C₃₁×#2434	C₂	semidirect
11. C₃₁×#2313	C₂	semidirect
15. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
16. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
17. C₂×C₃₁×#125	C₂²	semidirect
29. C₃₁×#400	C₂²	semidirect
30. C₃₁×#400	C₂²	semidirect
32. C₃₁×#398	C₂²	semidirect
33. C₃₁×#398	C₂²	semidirect
36. C₃₁×#395	C₂²	semidirect
37. C₃₁×#395	C₂²	semidirect
39. C₃₁×#391	C₂²	semidirect
42. C₃₁×#387	C₂²	semidirect
43. C₃₁×#387	C₂²	semidirect
168. #36708	C₃₁	direct (N × H)
213. C₂×#544	C₂×C₃₁	semidirect
216. #2690	C₂×C₃₁	semidirect
217. #2677	C₂×C₃₁	semidirect
218. #2627	C₂×C₃₁	semidirect
220. #2484	C₂×C₃₁	semidirect
222. #2434	C₂×C₃₁	semidirect
223. #2313	C₂×C₃₁	semidirect
250. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
251. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
252. C₂×#125	C₂²×C₃₁	semidirect
264. #400	C₂²×C₃₁	semidirect
265. #400	C₂²×C₃₁	semidirect
267. #398	C₂²×C₃₁	semidirect
268. #398	C₂²×C₃₁	semidirect
271. #395	C₂²×C₃₁	semidirect
272. #395	C₂²×C₃₁	semidirect
274. #391	C₂²×C₃₁	semidirect
277. #387	C₂²×C₃₁	semidirect
278. #387	C₂²×C₃₁	semidirect
350. C₃₁	#36708	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#544	2
C₂²×C₃₁×C₂²⋊Q₈	2
C₃₁×#2848	2
C₃₁×#2690	2
C₃₁×#2677	2
C₃₁×#2627	2
C₃₁×#2561	2
C₃₁×#2484	2
C₃₁×#2482	2
C₃₁×#2434	2
C₃₁×#2313	2
#36708	31

Subgroups — C31×#36708 #21493467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#36708 #21493467