Subgroups — C₃₁×#22835 #21479594

Order 7936.

924 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1954	C₂	1	2	yes	maximal
3	C₃₁×#1923	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1817	C₂	1	2	yes	maximal, complemented
6	C₈×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
7	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₃₁×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
10	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
11	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
16	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
19	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
23	C₄×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
24	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₄³×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#336	C₂²	1	1	yes	characteristic
27	C₃₁×#336	C₂²	1	2	yes	—
28	C₃₁×#336	C₂²	1	4	yes	—
29	C₃₁×#336	C₂²	1	2	yes	—
30	C₃₁×#336	C₂²	1	1	yes	characteristic
31	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₄×D₄×C₃₁	—	2	4	no	—
39	C₄×D₄×C₃₁	—	2	2	no	—
40	C₄×D₄×C₃₁	—	2	2	no	—
41	C₄×D₄×C₃₁	C₈	1	8	yes	complemented
42	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₃₁	—	4	4	no	—
49	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
56	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×C₄⋊C₈	—	2	4	no	—
60	C₃₁×C₄⋊C₈	—	2	4	no	—
61	C₃₁×C₂²⋊C₈	—	2	8	no	—
62	C₃₁×C₂²⋊C₈	—	4	4	no	—
63	C₄×C₈×C₃₁	—	2	4	no	—
64	C₄×C₈×C₃₁	—	4	4	no	—
65	C₃₁×#95	C₂×C₄	1	2	yes	—
66	C₃₁×#95	C₂×C₄	1	2	yes	—
67	C₃₁×#95	C₂×C₄	1	2	yes	—
68	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
69	C₂×D₄×C₃₁	—	2	2	no	—
70	C₂×D₄×C₃₁	—	2	2	no	—
71	C₂×D₄×C₃₁	C₂×C₈	1	4	yes	—
72	C₂²×C₄×C₃₁	—	2	4	no	—
73	C₂²×C₄×C₃₁	—	2	4	no	—
74	C₂²×C₄×C₃₁	—	2	4	no	—
75	C₂²×C₄×C₃₁	—	2	2	no	—
76	C₂²×C₄×C₃₁	—	2	2	no	—
77	C₂²×C₄×C₃₁	—	2	4	no	—
78	C₂²×C₄×C₃₁	—	4	2	no	—
79	C₂²×C₄×C₃₁	—	4	2	no	—
80	C₂²×C₄×C₃₁	—	4	2	no	—
81	C₂²×C₄×C₃₁	C₂×C₈	1	8	yes	—
82	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	—	2	2	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₂×C₈×C₃₁	—	2	4	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	—	2	4	no	—
98	C₂×C₈×C₃₁	—	2	4	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	#41	1	4	yes	—
101	C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
103	C₄⋊C₄×C₃₁	M₁₆	1	4	yes	—
104	C₂²⋊C₄×C₃₁	—	2	4	no	—
105	C₂²⋊C₄×C₃₁	—	2	4	no	—
106	C₂²⋊C₄×C₃₁	C₂×C₈	1	8	yes	—
107	C₄²×C₃₁	—	2	2	no	—
108	C₄²×C₃₁	—	2	2	no	—
109	C₄²×C₃₁	—	2	2	no	—
110	C₄²×C₃₁	—	2	4	no	—
111	C₄²×C₃₁	—	2	2	no	—
112	C₄²×C₃₁	—	4	2	no	—
113	C₄²×C₃₁	—	4	2	no	—
114	C₄²×C₃₁	—	4	1	no	—
115	C₄²×C₃₁	C₂×C₈	1	4	yes	—
116	C₄²×C₃₁	#41	1	2	yes	—
117	C₄²×C₃₁	#41	1	2	yes	—
118	#22835	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₃₁	—	2	2	no	—
120	C₂³×C₃₁	—	2	2	no	—
121	C₂³×C₃₁	—	2	2	no	—
122	C₂³×C₃₁	—	2	2	no	—
123	C₂³×C₃₁	—	2	2	no	—
124	C₂³×C₃₁	—	2	2	no	—
125	C₂³×C₃₁	C₂²×C₈	1	4	yes	—
126	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
127	D₄×C₃₁	—	2	8	no	—
128	C₂×C₄×C₃₁	—	2	1	no	—
129	C₂×C₄×C₃₁	—	2	1	no	—
130	C₂×C₄×C₃₁	—	2	1	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	1	no	—
134	C₂×C₄×C₃₁	—	2	1	no	—
135	C₂×C₄×C₃₁	—	2	1	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	2	1	no	—
142	C₂×C₄×C₃₁	—	2	2	no	—
143	C₂×C₄×C₃₁	—	2	2	no	—
144	C₂×C₄×C₃₁	—	2	1	no	—
145	C₂×C₄×C₃₁	—	2	1	no	—
146	C₂×C₄×C₃₁	—	4	8	no	—
147	C₂×C₄×C₃₁	—	4	8	no	—
148	C₂×C₄×C₃₁	—	4	4	no	—
149	C₂×C₄×C₃₁	—	4	4	no	—
150	C₂×C₄×C₃₁	#117	1	2	yes	—
151	C₂×C₄×C₃₁	#117	1	2	yes	—
152	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
153	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
154	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
155	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
156	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
157	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
158	C₂×C₄×C₃₁	#131	1	2	yes	—
159	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
160	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
161	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
163	C₈×C₃₁	—	2	4	no	—
164	C₈×C₃₁	—	4	2	no	—
165	C₈×C₃₁	—	4	4	no	—
166	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic
167	#1954	C₂×C₃₁	1	2	yes	—
168	#1923	C₂×C₃₁	1	4	yes	complemented
169	#1831	C₂×C₃₁	1	1	yes	characteristic, complemented
170	#1817	C₂×C₃₁	1	2	yes	complemented
171	C₈×C₄⋊C₄	C₂×C₃₁	1	2	yes	complemented
172	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
173	C₄×C₂²⋊C₈	C₂×C₃₁	1	2	yes	complemented
174	C₂²×C₃₁	—	2	4	no	—
175	C₂²×C₃₁	—	2	4	no	—
176	C₂²×C₃₁	—	2	4	no	—
177	C₂²×C₃₁	—	4	2	no	—
178	C₂²×C₃₁	—	4	2	no	—
179	C₂²×C₃₁	—	4	2	no	—
180	C₂²×C₃₁	—	4	2	no	—
181	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
182	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
183	C₂²×C₃₁	#518	1	1	yes	characteristic
184	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
185	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
186	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
187	C₂²×C₃₁	#568	1	1	yes	characteristic
188	C₄×C₃₁	—	2	2	no	—
189	C₄×C₃₁	—	2	2	no	—
190	C₄×C₃₁	—	2	2	no	—
191	C₄×C₃₁	—	2	4	no	—
192	C₄×C₃₁	—	2	2	no	—
193	C₄×C₃₁	—	2	4	no	—
194	C₄×C₃₁	—	4	2	no	—
195	C₄×C₃₁	—	4	1	no	—
196	C₄×C₃₁	#431	1	4	yes	—
197	C₄×C₃₁	#553	1	2	yes	—
198	C₄×C₃₁	#559	1	2	yes	—
199	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
200	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
201	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
202	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
203	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
204	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
205	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
206	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
207	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
208	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
209	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
210	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
211	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
212	C₄×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
213	C₄×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
214	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
215	C₄³	C₂²×C₃₁	1	1	yes	characteristic
216	#336	C₂²×C₃₁	1	1	yes	characteristic
217	#336	C₂²×C₃₁	1	2	yes	—
218	#336	C₂²×C₃₁	1	4	yes	—
219	#336	C₂²×C₃₁	1	2	yes	—
220	#336	C₂²×C₃₁	1	1	yes	characteristic
221	C₂×C₃₁	—	4	4	no	—
222	C₂×C₃₁	#2357	1	1	yes	characteristic
223	C₂×C₃₁	C₂×#431	1	1	yes	characteristic
224	C₂×C₃₁	#2925	1	1	yes	characteristic
225	C₂×C₃₁	C₈×#41	1	1	yes	characteristic
226	C₂×C₃₁	#2971	1	1	yes	characteristic
227	C₂×C₃₁	#2972	1	1	yes	characteristic
228	C₂×C₃₁	#2974	1	1	yes	characteristic
229	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
230	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
231	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
232	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
233	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
234	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
235	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
236	C₄×D₄	—	2	4	no	—
237	C₄×D₄	—	2	2	no	—
238	C₄×D₄	—	2	2	no	—
239	C₄×D₄	C₈×C₃₁	1	8	yes	complemented
240	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
241	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
242	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
243	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
244	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
245	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
246	C₂×C₄²	—	4	4	no	—
247	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
248	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
249	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
250	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
251	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
252	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
253	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
254	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
255	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
256	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
257	C₄⋊C₈	—	2	4	no	—
258	C₄⋊C₈	—	2	4	no	—
259	C₂²⋊C₈	—	2	8	no	—
260	C₂²⋊C₈	—	4	4	no	—
261	C₄×C₈	—	2	4	no	—
262	C₄×C₈	—	4	4	no	—
263	#95	C₂×C₄×C₃₁	1	2	yes	—
264	#95	C₂×C₄×C₃₁	1	2	yes	—
265	#95	C₂×C₄×C₃₁	1	2	yes	—
266	C₃₁	#22835	1	1	yes	characteristic, direct product factor, Sylow
267	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	C₂×C₈×C₃₁	1	4	yes	—
271	C₂²×C₄	—	2	4	no	—
272	C₂²×C₄	—	2	4	no	—
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	4	no	—
277	C₂²×C₄	—	4	2	no	—
278	C₂²×C₄	—	4	2	no	—
279	C₂²×C₄	—	4	2	no	—
280	C₂²×C₄	C₂×C₈×C₃₁	1	8	yes	—
281	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
282	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
283	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
284	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
285	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
286	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
287	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
288	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
289	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
290	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	4	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	4	no	—
297	C₂×C₈	—	2	4	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	#41×C₃₁	1	4	yes	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
302	C₄⋊C₄	M₁₆×C₃₁	1	4	yes	—
303	C₂²⋊C₄	—	2	4	no	—
304	C₂²⋊C₄	—	2	4	no	—
305	C₂²⋊C₄	C₂×C₈×C₃₁	1	8	yes	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	2	4	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	4	2	no	—
312	C₄²	—	4	2	no	—
313	C₄²	—	4	1	no	—
314	C₄²	C₂×C₈×C₃₁	1	4	yes	—
315	C₄²	#41×C₃₁	1	2	yes	—
316	C₄²	#41×C₃₁	1	2	yes	—
317	C₂³	—	2	2	no	—
318	C₂³	—	2	2	no	—
319	C₂³	—	2	2	no	—
320	C₂³	—	2	2	no	—
321	C₂³	—	2	2	no	—
322	C₂³	—	2	2	no	—
323	C₂³	C₂²×C₈×C₃₁	1	4	yes	—
324	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
325	D₄	—	2	8	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	4	8	no	—
345	C₂×C₄	—	4	8	no	—
346	C₂×C₄	—	4	4	no	—
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	C₃₁×#117	1	2	yes	—
349	C₂×C₄	C₃₁×#117	1	2	yes	—
350	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
351	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
352	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
353	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
354	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
355	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
356	C₂×C₄	C₃₁×#131	1	2	yes	—
357	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
358	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
359	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
360	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
361	C₈	—	2	4	no	—
362	C₈	—	4	2	no	—
363	C₈	—	4	4	no	—
364	C₂²	—	2	4	no	—
365	C₂²	—	2	4	no	—
366	C₂²	—	2	4	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	2	no	—
370	C₂²	—	4	2	no	—
371	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
372	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
373	C₂²	C₃₁×#518	1	1	yes	characteristic
374	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
375	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
376	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
377	C₂²	C₃₁×#568	1	1	yes	characteristic
378	C₄	—	2	2	no	—
379	C₄	—	2	2	no	—
380	C₄	—	2	2	no	—
381	C₄	—	2	4	no	—
382	C₄	—	2	2	no	—
383	C₄	—	2	4	no	—
384	C₄	—	4	2	no	—
385	C₄	—	4	1	no	—
386	C₄	C₃₁×#431	1	4	yes	—
387	C₄	C₃₁×#553	1	2	yes	—
388	C₄	C₃₁×#559	1	2	yes	—
389	C₂	—	4	4	no	—
390	C₂	C₃₁×#2357	1	1	yes	characteristic
391	C₂	C₂×C₃₁×#431	1	1	yes	characteristic
392	C₂	C₃₁×#2925	1	1	yes	characteristic
393	C₂	C₈×#41×C₃₁	1	1	yes	characteristic
394	C₂	C₃₁×#2971	1	1	yes	characteristic
395	C₂	C₃₁×#2972	1	1	yes	characteristic
396	C₂	C₃₁×#2974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂²×D₄×C₃₁	C₂×C₄
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
3. C₃₁×#1923	C₂	semidirect
4. C₃₁×#1831	C₂	semidirect
5. C₃₁×#1817	C₂	semidirect
6. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
7. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
8. C₄×C₃₁×C₂²⋊C₈	C₂	semidirect
41. C₄×D₄×C₃₁	C₈	semidirect
118. #22835	C₃₁	direct (N × H)
168. #1923	C₂×C₃₁	semidirect
169. #1831	C₂×C₃₁	semidirect
170. #1817	C₂×C₃₁	semidirect
171. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
172. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
173. C₄×C₂²⋊C₈	C₂×C₃₁	semidirect
239. C₄×D₄	C₈×C₃₁	semidirect
266. C₃₁	#22835	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×D₄×C₃₁	2
C₃₁×#1954	2
C₃₁×#1923	2
C₃₁×#1831	2
C₃₁×#1817	2
C₈×C₄⋊C₄×C₃₁	2
C₄×C₃₁×C₄⋊C₈	2
C₄×C₃₁×C₂²⋊C₈	2
#22835	31

Subgroups — C31×#22835 #21479594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22835 #21479594