Subgroups — C₃₁×#34056 #21490815

Order 7936.

2358 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2320	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2319	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2318	C₂	1	8	yes	maximal, complemented
5	C₃₁×#2298	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2297	C₂	1	2	yes	maximal
7	C₃₁×#2296	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2283	C₂	1	4	yes	maximal, complemented
9	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#376	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
12	C₄×#41×C₃₁	C₄	1	16	yes	complemented
13	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₃₁×#117	C₂²	1	2	yes	—
17	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#117	C₂²	1	2	yes	—
19	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
20	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#391	C₂²	1	4	yes	—
23	C₃₁×#390	C₂²	1	4	yes	—
24	C₃₁×#389	C₂²	1	8	yes	—
25	C₃₁×#388	C₂²	1	16	yes	—
26	C₃₁×#387	C₂²	1	16	yes	—
27	C₃₁×#386	C₂²	1	8	yes	—
28	C₃₁×#385	C₂²	1	8	yes	—
29	C₃₁×#383	C₂²	1	8	yes	—
30	C₃₁×#382	C₂²	1	8	yes	—
31	C₃₁×#381	C₂²	1	8	yes	—
32	C₃₁×#380	C₂²	1	8	yes	—
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	—
35	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
36	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
37	C₃₁×#376	C₂²	1	4	yes	—
38	C₃₁×#376	C₂²	1	2	yes	—
39	C₃₁×#376	C₂²	1	2	yes	—
40	C₃₁×#376	C₂²	1	2	yes	—
41	C₂×C₃₁×#95	C₂²	1	4	yes	—
42	C₄³×C₃₁	C₂²	1	2	yes	—
43	C₂×#41×C₃₁	—	2	8	no	—
44	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
45	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
47	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
49	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
50	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
51	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
52	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₄×Q₈×C₃₁	C₂×C₄	1	16	yes	complemented
54	C₄×D₄×C₃₁	C₂×C₄	1	32	yes	—
55	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
56	C₃₁×#117	—	2	8	no	—
57	C₃₁×#117	C₂×C₄	1	8	yes	—
58	C₃₁×#117	C₂×C₄	1	16	yes	complemented
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
64	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
65	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
67	C₂×C₄²×C₃₁	—	2	8	no	—
68	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	—
69	C₂×C₄²×C₃₁	C₂×C₄	1	16	yes	complemented
70	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
71	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
72	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
73	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
74	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
75	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
79	C₃₁×#95	C₂³	1	4	yes	—
80	C₃₁×#95	C₂³	1	4	yes	—
81	C₃₁×#95	C₂³	1	8	yes	—
82	C₃₁×#95	C₂³	1	8	yes	—
83	C₃₁×#95	C₂³	1	16	yes	—
84	C₃₁×#95	C₂³	1	16	yes	—
85	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
86	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
87	#41×C₃₁	—	2	32	no	—
88	C₂×Q₈×C₃₁	—	2	4	no	—
89	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
90	C₂×D₄×C₃₁	—	2	8	no	—
91	C₂×D₄×C₃₁	—	2	4	no	—
92	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
93	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
94	C₂²×C₄×C₃₁	—	2	8	no	—
95	C₂²×C₄×C₃₁	—	2	8	no	—
96	C₂²×C₄×C₃₁	—	4	4	no	—
97	C₂²×C₄×C₃₁	—	4	8	no	—
98	C₂²×C₄×C₃₁	—	4	8	no	—
99	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
100	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
101	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	—
102	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
103	C₂²×C₄×C₃₁	#41	1	8	yes	—
104	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
105	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
106	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
107	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
108	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
109	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
110	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₃₁	—	2	8	no	—
113	C₄⋊C₄×C₃₁	—	4	16	no	—
114	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
115	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
116	C₂²⋊C₄×C₃₁	—	2	8	no	—
117	C₂²⋊C₄×C₃₁	—	4	16	no	—
118	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
119	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
120	C₄²×C₃₁	—	2	8	no	—
121	C₄²×C₃₁	—	2	8	no	—
122	C₄²×C₃₁	—	2	8	no	—
123	C₄²×C₃₁	—	4	8	no	—
124	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
125	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
126	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
127	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
128	C₄²×C₃₁	C₂²×C₄	1	16	yes	—
129	#34056	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂³×C₃₁	—	2	4	no	—
131	C₂³×C₃₁	—	2	4	no	—
132	C₂³×C₃₁	—	2	8	no	—
133	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
134	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
135	C₂³×C₃₁	C₂×#41	1	4	yes	—
136	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
137	Q₈×C₃₁	—	2	8	no	—
138	D₄×C₃₁	—	2	16	no	—
139	D₄×C₃₁	—	2	8	no	—
140	C₂×C₄×C₃₁	—	2	4	no	—
141	C₂×C₄×C₃₁	—	2	4	no	—
142	C₂×C₄×C₃₁	—	2	8	no	—
143	C₂×C₄×C₃₁	—	2	4	no	—
144	C₂×C₄×C₃₁	—	2	16	no	—
145	C₂×C₄×C₃₁	—	2	8	no	—
146	C₂×C₄×C₃₁	—	2	8	no	—
147	C₂×C₄×C₃₁	—	2	16	no	—
148	C₂×C₄×C₃₁	—	2	8	no	—
149	C₂×C₄×C₃₁	—	2	8	no	—
150	C₂×C₄×C₃₁	—	4	8	no	—
151	C₂×C₄×C₃₁	—	4	16	no	—
152	C₂×C₄×C₃₁	—	4	16	no	—
153	C₂×C₄×C₃₁	#117	1	16	yes	—
154	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
155	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
156	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
157	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
158	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
159	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
160	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
161	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
162	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic
163	#2320	C₂×C₃₁	1	4	yes	complemented
164	#2319	C₂×C₃₁	1	4	yes	complemented
165	#2318	C₂×C₃₁	1	8	yes	complemented
166	#2298	C₂×C₃₁	1	4	yes	complemented
167	#2297	C₂×C₃₁	1	2	yes	—
168	#2296	C₂×C₃₁	1	2	yes	complemented
169	#2283	C₂×C₃₁	1	4	yes	complemented
170	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
171	C₂×#376	C₂×C₃₁	1	1	yes	characteristic, complemented
172	C₂²×C₃₁	—	2	4	no	—
173	C₂²×C₃₁	—	2	8	no	—
174	C₂²×C₃₁	—	2	4	no	—
175	C₂²×C₃₁	—	2	8	no	—
176	C₂²×C₃₁	—	4	8	no	—
177	C₂²×C₃₁	C₂×#117	1	4	yes	—
178	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
179	C₂²×C₃₁	C₂²×#41	1	2	yes	—
180	C₂²×C₃₁	#585	1	4	yes	—
181	C₄×C₃₁	—	2	4	no	—
182	C₄×C₃₁	—	2	4	no	—
183	C₄×C₃₁	—	2	4	no	—
184	C₄×C₃₁	—	2	8	no	—
185	C₄×C₃₁	—	4	8	no	—
186	C₄×C₃₁	—	4	8	no	—
187	C₄×C₃₁	C₂×#117	1	4	yes	—
188	C₄×C₃₁	C₂×#117	1	4	yes	—
189	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
190	C₄×#41	C₄×C₃₁	1	16	yes	complemented
191	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
192	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
193	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
194	C₂×#117	C₂²×C₃₁	1	2	yes	—
195	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
196	C₂×#117	C₂²×C₃₁	1	2	yes	—
197	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
198	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
199	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
200	#391	C₂²×C₃₁	1	4	yes	—
201	#390	C₂²×C₃₁	1	4	yes	—
202	#389	C₂²×C₃₁	1	8	yes	—
203	#388	C₂²×C₃₁	1	16	yes	—
204	#387	C₂²×C₃₁	1	16	yes	—
205	#386	C₂²×C₃₁	1	8	yes	—
206	#385	C₂²×C₃₁	1	8	yes	—
207	#383	C₂²×C₃₁	1	8	yes	—
208	#382	C₂²×C₃₁	1	8	yes	—
209	#381	C₂²×C₃₁	1	8	yes	—
210	#380	C₂²×C₃₁	1	8	yes	—
211	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
212	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	—
213	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
214	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
215	#376	C₂²×C₃₁	1	4	yes	—
216	#376	C₂²×C₃₁	1	2	yes	—
217	#376	C₂²×C₃₁	1	2	yes	—
218	#376	C₂²×C₃₁	1	2	yes	—
219	C₂×#95	C₂²×C₃₁	1	4	yes	—
220	C₄³	C₂²×C₃₁	1	2	yes	—
221	C₂×C₃₁	—	2	4	no	—
222	C₂×C₃₁	—	4	4	no	—
223	C₂×C₃₁	C₂²×#117	1	1	yes	characteristic
224	C₂×C₃₁	#3428	1	2	yes	—
225	C₂×C₃₁	#3455	1	4	yes	—
226	C₂×#41	—	2	8	no	—
227	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
228	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
229	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
230	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
231	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
232	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
233	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
234	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
235	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
236	C₄×Q₈	C₂×C₄×C₃₁	1	16	yes	complemented
237	C₄×D₄	C₂×C₄×C₃₁	1	32	yes	—
238	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
239	#117	—	2	8	no	—
240	#117	C₂×C₄×C₃₁	1	8	yes	—
241	#117	C₂×C₄×C₃₁	1	16	yes	complemented
242	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
243	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	—
244	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
245	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
246	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
247	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
248	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
249	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
250	C₂×C₄²	—	2	8	no	—
251	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	—
252	C₂×C₄²	C₂×C₄×C₃₁	1	16	yes	complemented
253	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
254	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
255	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
256	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
257	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
258	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
259	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
260	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
261	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
262	#95	C₂³×C₃₁	1	4	yes	—
263	#95	C₂³×C₃₁	1	4	yes	—
264	#95	C₂³×C₃₁	1	8	yes	—
265	#95	C₂³×C₃₁	1	8	yes	—
266	#95	C₂³×C₃₁	1	16	yes	—
267	#95	C₂³×C₃₁	1	16	yes	—
268	C₃₁	#34056	1	1	yes	characteristic, direct product factor, Sylow
269	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
270	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
271	#41	—	2	32	no	—
272	C₂×Q₈	—	2	4	no	—
273	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
274	C₂×D₄	—	2	8	no	—
275	C₂×D₄	—	2	4	no	—
276	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
277	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
278	C₂²×C₄	—	2	8	no	—
279	C₂²×C₄	—	2	8	no	—
280	C₂²×C₄	—	4	4	no	—
281	C₂²×C₄	—	4	8	no	—
282	C₂²×C₄	—	4	8	no	—
283	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
284	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
285	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	—
286	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
287	C₂²×C₄	#41×C₃₁	1	8	yes	—
288	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
289	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
290	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
291	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
292	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
293	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
294	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
295	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
296	C₄⋊C₄	—	2	8	no	—
297	C₄⋊C₄	—	4	16	no	—
298	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
299	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
300	C₂²⋊C₄	—	2	8	no	—
301	C₂²⋊C₄	—	4	16	no	—
302	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
303	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
304	C₄²	—	2	8	no	—
305	C₄²	—	2	8	no	—
306	C₄²	—	2	8	no	—
307	C₄²	—	4	8	no	—
308	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
309	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
310	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
311	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
312	C₄²	C₂²×C₄×C₃₁	1	16	yes	—
313	C₂³	—	2	4	no	—
314	C₂³	—	2	4	no	—
315	C₂³	—	2	8	no	—
316	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
317	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
318	C₂³	C₂×#41×C₃₁	1	4	yes	—
319	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
320	Q₈	—	2	8	no	—
321	D₄	—	2	16	no	—
322	D₄	—	2	8	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	8	no	—
326	C₂×C₄	—	2	4	no	—
327	C₂×C₄	—	2	16	no	—
328	C₂×C₄	—	2	8	no	—
329	C₂×C₄	—	2	8	no	—
330	C₂×C₄	—	2	16	no	—
331	C₂×C₄	—	2	8	no	—
332	C₂×C₄	—	2	8	no	—
333	C₂×C₄	—	4	8	no	—
334	C₂×C₄	—	4	16	no	—
335	C₂×C₄	—	4	16	no	—
336	C₂×C₄	C₃₁×#117	1	16	yes	—
337	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
338	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
339	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
340	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
341	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
342	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
343	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
344	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
345	C₂²	—	2	4	no	—
346	C₂²	—	2	8	no	—
347	C₂²	—	2	4	no	—
348	C₂²	—	2	8	no	—
349	C₂²	—	4	8	no	—
350	C₂²	C₂×C₃₁×#117	1	4	yes	—
351	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
352	C₂²	C₂²×#41×C₃₁	1	2	yes	—
353	C₂²	C₃₁×#585	1	4	yes	—
354	C₄	—	2	4	no	—
355	C₄	—	2	4	no	—
356	C₄	—	2	4	no	—
357	C₄	—	2	8	no	—
358	C₄	—	4	8	no	—
359	C₄	—	4	8	no	—
360	C₄	C₂×C₃₁×#117	1	4	yes	—
361	C₄	C₂×C₃₁×#117	1	4	yes	—
362	C₂	—	2	4	no	—
363	C₂	—	4	4	no	—
364	C₂	C₂²×C₃₁×#117	1	1	yes	characteristic
365	C₂	C₃₁×#3428	1	2	yes	—
366	C₂	C₃₁×#3455	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2320	C₂	semidirect
3. C₃₁×#2319	C₂	semidirect
4. C₃₁×#2318	C₂	semidirect
5. C₃₁×#2298	C₂	semidirect
7. C₃₁×#2296	C₂	semidirect
8. C₃₁×#2283	C₂	semidirect
9. C₄×C₃₁×#117	C₂	semidirect
10. C₂×C₃₁×#376	C₂	semidirect
12. C₄×#41×C₃₁	C₄	semidirect
53. C₄×Q₈×C₃₁	C₂×C₄	semidirect
55. C₄×D₄×C₃₁	C₂×C₄	semidirect
58. C₃₁×#117	C₂×C₄	semidirect
69. C₂×C₄²×C₃₁	C₂×C₄	semidirect
129. #34056	C₃₁	direct (N × H)
163. #2320	C₂×C₃₁	semidirect
164. #2319	C₂×C₃₁	semidirect
165. #2318	C₂×C₃₁	semidirect
166. #2298	C₂×C₃₁	semidirect
168. #2296	C₂×C₃₁	semidirect
169. #2283	C₂×C₃₁	semidirect
170. C₄×#117	C₂×C₃₁	semidirect
171. C₂×#376	C₂×C₃₁	semidirect
190. C₄×#41	C₄×C₃₁	semidirect
236. C₄×Q₈	C₂×C₄×C₃₁	semidirect
238. C₄×D₄	C₂×C₄×C₃₁	semidirect
241. #117	C₂×C₄×C₃₁	semidirect
252. C₂×C₄²	C₂×C₄×C₃₁	semidirect
268. C₃₁	#34056	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×C₄×#41×C₃₁	2
C₃₁×#2320	2
C₃₁×#2319	2
C₃₁×#2318	2
C₃₁×#2298	2
C₃₁×#2297	2
C₃₁×#2296	2
C₃₁×#2283	2
C₄×C₃₁×#117	2
C₂×C₃₁×#376	2
#34056	31

Subgroups — C31×#34056 #21490815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34056 #21490815