Subgroups — C₃₁×#21675 #21478434

Order 7936.

674 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#125	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1917	C₂	1	2	yes	maximal
5	C₃₁×#1917	C₂	1	2	yes	maximal
6	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1771	C₂	1	2	yes	maximal
8	C₃₁×#1770	C₂	1	2	yes	maximal
9	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic
10	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
11	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×#125	C₄	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#389	C₄	1	1	yes	characteristic
26	C₃₁×#389	C₄	1	1	yes	characteristic
27	C₃₁×#387	C₄	1	2	yes	—
28	C₃₁×#387	C₄	1	2	yes	—
29	C₃₁×#382	C₄	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₄³×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#125	—	2	4	no	—
48	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×C₄⋊C₈	—	2	4	no	—
66	C₃₁×C₄⋊C₈	—	2	4	no	—
67	C₃₁×C₄⋊C₈	—	2	2	no	—
68	C₃₁×C₄⋊C₈	—	2	2	no	—
69	C₃₁×#97	—	2	2	no	—
70	C₃₁×#97	—	2	2	no	—
71	C₃₁×#97	—	2	2	no	—
72	C₃₁×#97	—	2	2	no	—
73	C₃₁×#97	—	2	4	no	—
74	C₄×C₈×C₃₁	—	2	4	no	—
75	C₄×C₈×C₃₁	—	2	2	no	—
76	C₄×C₈×C₃₁	Q₈	1	4	yes	—
77	C₃₁×#95	C₂×C₄	1	2	yes	—
78	C₃₁×#95	C₂×C₄	1	2	yes	—
79	C₃₁×#95	C₂×C₄	1	2	yes	—
80	C₃₁×#95	C₂×C₄	1	2	yes	—
81	C₃₁×#95	C₂×C₄	1	2	yes	—
82	C₃₁×#95	C₂×C₄	1	2	yes	—
83	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
94	C₂×C₈×C₃₁	—	2	1	no	—
95	C₂×C₈×C₃₁	—	2	1	no	—
96	C₂×C₈×C₃₁	—	2	1	no	—
97	C₂×C₈×C₃₁	—	2	1	no	—
98	C₂×C₈×C₃₁	—	2	1	no	—
99	C₂×C₈×C₃₁	—	2	1	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
108	C₂×C₈×C₃₁	#41	1	2	yes	—
109	C₂×C₈×C₃₁	#41	1	2	yes	—
110	C₂×C₈×C₃₁	#41	1	2	yes	—
111	C₂×C₈×C₃₁	#41	1	2	yes	—
112	C₄⋊C₄×C₃₁	—	2	4	no	—
113	C₄⋊C₄×C₃₁	—	2	4	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	4	2	no	—
118	C₄⋊C₄×C₃₁	—	4	2	no	—
119	C₄⋊C₄×C₃₁	M₁₆	1	4	yes	—
120	C₄²×C₃₁	—	2	2	no	—
121	C₄²×C₃₁	—	2	2	no	—
122	C₄²×C₃₁	—	2	2	no	—
123	C₄²×C₃₁	—	2	1	no	—
124	C₄²×C₃₁	—	2	1	no	—
125	C₄²×C₃₁	—	2	2	no	—
126	C₄²×C₃₁	—	2	1	no	—
127	C₄²×C₃₁	—	2	1	no	—
128	C₄²×C₃₁	—	2	2	no	—
129	C₄²×C₃₁	—	2	2	no	—
130	C₄²×C₃₁	—	4	1	no	—
131	C₄²×C₃₁	—	4	1	no	—
132	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
133	C₄²×C₃₁	#41	1	2	yes	—
134	#21675	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
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	1	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	—	2	2	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
162	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
163	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
164	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
165	C₂×C₄×C₃₁	#131	1	2	yes	—
166	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
167	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
172	C₈×C₃₁	—	2	2	no	—
173	C₈×C₃₁	—	2	2	no	—
174	C₈×C₃₁	—	2	2	no	—
175	C₈×C₃₁	—	4	2	no	—
176	C₈×C₃₁	—	4	2	no	—
177	C₈×C₃₁	#125	1	4	yes	complemented
178	C₄×#125	C₂×C₃₁	1	1	yes	characteristic
179	#1939	C₂×C₃₁	1	1	yes	characteristic
180	#1939	C₂×C₃₁	1	1	yes	characteristic
181	#1917	C₂×C₃₁	1	2	yes	—
182	#1917	C₂×C₃₁	1	2	yes	—
183	#1831	C₂×C₃₁	1	1	yes	characteristic
184	#1771	C₂×C₃₁	1	2	yes	—
185	#1770	C₂×C₃₁	1	2	yes	—
186	#1770	C₂×C₃₁	1	1	yes	characteristic
187	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
188	C₄×#97	C₂×C₃₁	1	1	yes	characteristic
189	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
190	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
191	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
192	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
193	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
194	C₂²×C₃₁	#568	1	1	yes	characteristic
195	C₂²×C₃₁	#568	1	1	yes	characteristic
196	C₄×C₃₁	—	2	2	no	—
197	C₄×C₃₁	—	2	4	no	—
198	C₄×C₃₁	—	4	1	no	—
199	C₄×C₃₁	—	4	1	no	—
200	C₄×C₃₁	—	4	2	no	—
201	C₄×C₃₁	—	4	2	no	—
202	C₄×C₃₁	—	4	1	no	—
203	C₄×C₃₁	—	4	1	no	—
204	C₄×C₃₁	—	4	2	no	—
205	C₄×C₃₁	#446	1	4	yes	—
206	C₄×C₃₁	C₂×#125	1	1	yes	characteristic
207	C₄×C₃₁	#529	1	1	yes	characteristic
208	C₄×C₃₁	#529	1	1	yes	characteristic
209	C₄×C₃₁	#533	1	1	yes	characteristic
210	C₂×#125	C₄×C₃₁	1	1	yes	characteristic
211	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
212	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
213	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
214	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
215	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
216	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
217	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
219	C₂×#97	C₂²×C₃₁	1	2	yes	—
220	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
221	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
222	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
223	#389	C₄×C₃₁	1	1	yes	characteristic
224	#389	C₄×C₃₁	1	1	yes	characteristic
225	#387	C₄×C₃₁	1	2	yes	—
226	#387	C₄×C₃₁	1	2	yes	—
227	#382	C₄×C₃₁	1	1	yes	characteristic
228	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
229	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
230	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
231	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
232	C₄³	C₂²×C₃₁	1	1	yes	characteristic
233	#336	C₂²×C₃₁	1	2	yes	—
234	#336	C₂²×C₃₁	1	2	yes	—
235	#336	C₂²×C₃₁	1	2	yes	—
236	#336	C₂²×C₃₁	1	2	yes	—
237	#336	C₂²×C₃₁	1	2	yes	—
238	#336	C₂²×C₃₁	1	2	yes	—
239	C₂×C₃₁	C₄×#125	1	1	yes	characteristic
240	C₂×C₃₁	C₂×#446	1	1	yes	characteristic
241	C₂×C₃₁	#2961	1	1	yes	characteristic
242	C₂×C₃₁	#2969	1	1	yes	characteristic
243	C₂×C₃₁	#2969	1	1	yes	characteristic
244	C₂×C₃₁	#2970	1	1	yes	characteristic
245	C₂×C₃₁	#2971	1	1	yes	characteristic
246	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
247	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
248	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
249	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
250	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
251	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
252	#125	—	2	4	no	—
253	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
254	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
255	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
256	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
259	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
260	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
261	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
262	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
263	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
264	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
266	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
268	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
269	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
270	C₄⋊C₈	—	2	4	no	—
271	C₄⋊C₈	—	2	4	no	—
272	C₄⋊C₈	—	2	2	no	—
273	C₄⋊C₈	—	2	2	no	—
274	#97	—	2	2	no	—
275	#97	—	2	2	no	—
276	#97	—	2	2	no	—
277	#97	—	2	2	no	—
278	#97	—	2	4	no	—
279	C₄×C₈	—	2	4	no	—
280	C₄×C₈	—	2	2	no	—
281	C₄×C₈	Q₈×C₃₁	1	4	yes	—
282	#95	C₂×C₄×C₃₁	1	2	yes	—
283	#95	C₂×C₄×C₃₁	1	2	yes	—
284	#95	C₂×C₄×C₃₁	1	2	yes	—
285	#95	C₂×C₄×C₃₁	1	2	yes	—
286	#95	C₂×C₄×C₃₁	1	2	yes	—
287	#95	C₂×C₄×C₃₁	1	2	yes	—
288	C₃₁	#21675	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
290	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
291	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
292	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
293	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
294	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
295	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
296	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
297	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
298	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
299	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
313	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
314	C₂×C₈	#41×C₃₁	1	2	yes	—
315	C₂×C₈	#41×C₃₁	1	2	yes	—
316	C₂×C₈	#41×C₃₁	1	2	yes	—
317	C₂×C₈	#41×C₃₁	1	2	yes	—
318	C₄⋊C₄	—	2	4	no	—
319	C₄⋊C₄	—	2	4	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	4	2	no	—
324	C₄⋊C₄	—	4	2	no	—
325	C₄⋊C₄	M₁₆×C₃₁	1	4	yes	—
326	C₄²	—	2	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	2	2	no	—
329	C₄²	—	2	1	no	—
330	C₄²	—	2	1	no	—
331	C₄²	—	2	2	no	—
332	C₄²	—	2	1	no	—
333	C₄²	—	2	1	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	4	1	no	—
337	C₄²	—	4	1	no	—
338	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
339	C₄²	#41×C₃₁	1	2	yes	—
340	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
367	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
368	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
369	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
370	C₂×C₄	C₃₁×#131	1	2	yes	—
371	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
372	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
373	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
374	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
375	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
376	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
377	C₈	—	2	2	no	—
378	C₈	—	2	2	no	—
379	C₈	—	2	2	no	—
380	C₈	—	4	2	no	—
381	C₈	—	4	2	no	—
382	C₈	C₃₁×#125	1	4	yes	complemented
383	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
384	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
385	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
386	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
387	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
388	C₂²	C₃₁×#568	1	1	yes	characteristic
389	C₂²	C₃₁×#568	1	1	yes	characteristic
390	C₄	—	2	2	no	—
391	C₄	—	2	4	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	1	no	—
394	C₄	—	4	2	no	—
395	C₄	—	4	2	no	—
396	C₄	—	4	1	no	—
397	C₄	—	4	1	no	—
398	C₄	—	4	2	no	—
399	C₄	C₃₁×#446	1	4	yes	—
400	C₄	C₂×C₃₁×#125	1	1	yes	characteristic
401	C₄	C₃₁×#529	1	1	yes	characteristic
402	C₄	C₃₁×#529	1	1	yes	characteristic
403	C₄	C₃₁×#533	1	1	yes	characteristic
404	C₂	C₄×C₃₁×#125	1	1	yes	characteristic
405	C₂	C₂×C₃₁×#446	1	1	yes	characteristic
406	C₂	C₃₁×#2961	1	1	yes	characteristic
407	C₂	C₃₁×#2969	1	1	yes	characteristic
408	C₂	C₃₁×#2969	1	1	yes	characteristic
409	C₂	C₃₁×#2970	1	1	yes	characteristic
410	C₂	C₃₁×#2971	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
134. #21675	C₃₁	direct (N × H)
177. C₈×C₃₁	#125	semidirect
288. C₃₁	#21675	direct (N × H)
382. C₈	C₃₁×#125	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#125	2
C₃₁×#1939	2
C₃₁×#1917	2
C₃₁×#1831	2
C₃₁×#1771	2
C₃₁×#1770	2
C₈×C₄⋊C₄×C₃₁	2
C₄×C₃₁×#97	2
#21675	31

Subgroups — C31×#21675 #21478434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21675 #21478434