Subgroups — C₃₁×#22824 #21479583

Order 7936.

636 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2295	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1985	C₂	1	2	yes	maximal
3	C₃₁×#1917	C₂	1	2	yes	maximal
4	C₃₁×#1916	C₂	1	2	yes	maximal
5	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1832	C₂	1	2	yes	maximal
7	C₃₁×#1771	C₂	1	2	yes	maximal
8	C₃₁×#1768	C₂	1	2	yes	maximal
9	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#97	C₂²	1	2	yes	—
18	C₂×C₃₁×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	C₃₁×#391	C₄	1	1	yes	characteristic
21	C₃₁×#391	C₄	1	1	yes	characteristic
22	C₃₁×#389	C₄	1	1	yes	characteristic
23	C₃₁×#389	C₄	1	1	yes	characteristic
24	C₃₁×#387	C₄	1	2	yes	—
25	C₃₁×#387	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#376	C₂²	1	2	yes	—
29	C₄³×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#336	C₂²	1	2	yes	—
31	C₃₁×#336	C₂²	1	2	yes	—
32	C₃₁×#336	C₂²	1	2	yes	—
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
42	C₄×Q₈×C₃₁	—	2	2	no	—
43	C₄×Q₈×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
55	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
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	2	no	—
61	C₃₁×C₄⋊C₈	—	2	4	no	—
62	C₃₁×C₄⋊C₈	—	4	1	no	—
63	C₃₁×C₄⋊C₈	—	4	1	no	—
64	C₃₁×C₄⋊C₈	Q₈	1	4	yes	complemented
65	C₃₁×#97	—	2	4	no	—
66	C₃₁×#97	—	4	2	no	—
67	C₄×C₈×C₃₁	—	4	2	no	—
68	C₃₁×#95	C₂×C₄	1	2	yes	—
69	C₃₁×#95	C₂×C₄	1	2	yes	—
70	C₃₁×#95	C₂×C₄	1	2	yes	—
71	C₃₁×#95	C₂×C₄	1	2	yes	—
72	C₃₁×#95	C₂×C₄	1	2	yes	—
73	C₃₁×#95	C₂×C₄	1	2	yes	—
74	C₃₁×#95	C₂×C₄	1	2	yes	—
75	C₂×Q₈×C₃₁	—	2	2	no	—
76	C₂×Q₈×C₃₁	—	4	1	no	—
77	C₂×Q₈×C₃₁	—	4	1	no	—
78	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
87	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	2	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
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	2	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
101	C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₄⋊C₄×C₃₁	—	2	4	no	—
103	C₄⋊C₄×C₃₁	—	2	4	no	—
104	C₄⋊C₄×C₃₁	—	4	1	no	—
105	C₄⋊C₄×C₃₁	—	4	1	no	—
106	C₄⋊C₄×C₃₁	—	4	2	no	—
107	C₄²×C₃₁	—	2	2	no	—
108	C₄²×C₃₁	—	2	2	no	—
109	C₄²×C₃₁	—	2	2	no	—
110	C₄²×C₃₁	—	2	2	no	—
111	C₄²×C₃₁	—	2	1	no	—
112	C₄²×C₃₁	—	2	1	no	—
113	C₄²×C₃₁	—	2	2	no	—
114	C₄²×C₃₁	—	4	1	no	—
115	C₄²×C₃₁	—	4	1	no	—
116	C₄²×C₃₁	—	4	1	no	—
117	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
118	C₄²×C₃₁	#41	1	2	yes	—
119	C₄²×C₃₁	#41	1	2	yes	—
120	C₄²×C₃₁	#41	1	2	yes	—
121	#22824	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
123	Q₈×C₃₁	—	4	4	no	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	2	1	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	2	1	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	1	no	—
132	C₂×C₄×C₃₁	—	2	1	no	—
133	C₂×C₄×C₃₁	—	2	1	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	2	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	2	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	2	2	no	—
147	C₂×C₄×C₃₁	—	2	2	no	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	#117	1	2	yes	—
150	C₂×C₄×C₃₁	#117	1	2	yes	—
151	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
152	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
153	C₂×C₄×C₃₁	#131	1	2	yes	—
154	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
155	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
160	C₈×C₃₁	—	2	4	no	—
161	C₈×C₃₁	—	4	2	no	—
162	C₈×C₃₁	—	4	2	no	—
163	C₈×C₃₁	—	4	2	no	—
164	#2295	C₂×C₃₁	1	1	yes	characteristic
165	#1985	C₂×C₃₁	1	2	yes	—
166	#1917	C₂×C₃₁	1	2	yes	—
167	#1916	C₂×C₃₁	1	2	yes	—
168	#1842	C₂×C₃₁	1	1	yes	characteristic
169	#1832	C₂×C₃₁	1	2	yes	—
170	#1771	C₂×C₃₁	1	2	yes	—
171	#1768	C₂×C₃₁	1	2	yes	—
172	#1767	C₂×C₃₁	1	1	yes	characteristic
173	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
174	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
175	C₂²×C₃₁	#520	1	1	yes	characteristic
176	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
177	C₂²×C₃₁	#568	1	1	yes	characteristic
178	C₂²×C₃₁	#568	1	1	yes	characteristic
179	C₂²×C₃₁	#568	1	1	yes	characteristic
180	C₄×C₃₁	—	2	2	no	—
181	C₄×C₃₁	—	2	2	no	—
182	C₄×C₃₁	—	2	2	no	—
183	C₄×C₃₁	—	2	2	no	—
184	C₄×C₃₁	—	2	2	no	—
185	C₄×C₃₁	—	4	2	no	—
186	C₄×C₃₁	—	4	2	no	—
187	C₄×C₃₁	—	4	2	no	—
188	C₄×C₃₁	—	4	1	no	—
189	C₄×C₃₁	—	4	2	no	—
190	C₄×C₃₁	#554	1	1	yes	characteristic
191	C₄×C₃₁	#556	1	1	yes	characteristic
192	C₄×C₃₁	#557	1	1	yes	characteristic
193	C₄×C₃₁	#559	1	1	yes	characteristic
194	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
195	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
196	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
197	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
198	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
199	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
200	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
201	C₂×#97	C₂²×C₃₁	1	2	yes	—
202	C₂×#97	C₂²×C₃₁	1	2	yes	—
203	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
204	#391	C₄×C₃₁	1	1	yes	characteristic
205	#391	C₄×C₃₁	1	1	yes	characteristic
206	#389	C₄×C₃₁	1	1	yes	characteristic
207	#389	C₄×C₃₁	1	1	yes	characteristic
208	#387	C₄×C₃₁	1	2	yes	—
209	#387	C₄×C₃₁	1	2	yes	—
210	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
211	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
212	#376	C₂²×C₃₁	1	2	yes	—
213	C₄³	C₂²×C₃₁	1	1	yes	characteristic
214	#336	C₂²×C₃₁	1	2	yes	—
215	#336	C₂²×C₃₁	1	2	yes	—
216	#336	C₂²×C₃₁	1	2	yes	—
217	#336	C₂²×C₃₁	1	2	yes	—
218	#336	C₂²×C₃₁	1	2	yes	—
219	#336	C₂²×C₃₁	1	2	yes	—
220	#336	C₂²×C₃₁	1	2	yes	—
221	C₂×C₃₁	#2355	1	1	yes	characteristic
222	C₂×C₃₁	#2921	1	1	yes	characteristic
223	C₂×C₃₁	#2925	1	1	yes	characteristic
224	C₂×C₃₁	#2961	1	1	yes	characteristic
225	C₂×C₃₁	#2962	1	1	yes	characteristic
226	C₂×C₃₁	#2992	1	1	yes	characteristic
227	C₂×C₃₁	#2995	1	1	yes	characteristic
228	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
229	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
230	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
231	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
232	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×Q₈	—	2	2	no	—
235	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
236	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
237	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
238	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
239	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
240	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
241	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
242	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
243	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
244	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
245	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
246	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
247	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
248	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
249	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
250	C₄⋊C₈	—	2	4	no	—
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	2	4	no	—
253	C₄⋊C₈	—	4	1	no	—
254	C₄⋊C₈	—	4	1	no	—
255	C₄⋊C₈	Q₈×C₃₁	1	4	yes	complemented
256	#97	—	2	4	no	—
257	#97	—	4	2	no	—
258	C₄×C₈	—	4	2	no	—
259	#95	C₂×C₄×C₃₁	1	2	yes	—
260	#95	C₂×C₄×C₃₁	1	2	yes	—
261	#95	C₂×C₄×C₃₁	1	2	yes	—
262	#95	C₂×C₄×C₃₁	1	2	yes	—
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	#22824	1	1	yes	characteristic, direct product factor, Sylow
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	4	1	no	—
269	C₂×Q₈	—	4	1	no	—
270	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
271	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
272	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
273	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
274	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
275	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
276	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
277	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
278	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
279	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
280	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	—	2	4	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	4	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	2	1	no	—
304	C₄²	—	2	1	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	4	1	no	—
307	C₄²	—	4	1	no	—
308	C₄²	—	4	1	no	—
309	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
310	C₄²	#41×C₃₁	1	2	yes	—
311	C₄²	#41×C₃₁	1	2	yes	—
312	C₄²	#41×C₃₁	1	2	yes	—
313	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
314	Q₈	—	4	4	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	C₃₁×#117	1	2	yes	—
341	C₂×C₄	C₃₁×#117	1	2	yes	—
342	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
343	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
344	C₂×C₄	C₃₁×#131	1	2	yes	—
345	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
346	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
347	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
348	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
349	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
350	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
351	C₈	—	2	4	no	—
352	C₈	—	4	2	no	—
353	C₈	—	4	2	no	—
354	C₈	—	4	2	no	—
355	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
356	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
357	C₂²	C₃₁×#520	1	1	yes	characteristic
358	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
359	C₂²	C₃₁×#568	1	1	yes	characteristic
360	C₂²	C₃₁×#568	1	1	yes	characteristic
361	C₂²	C₃₁×#568	1	1	yes	characteristic
362	C₄	—	2	2	no	—
363	C₄	—	2	2	no	—
364	C₄	—	2	2	no	—
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	4	2	no	—
368	C₄	—	4	2	no	—
369	C₄	—	4	2	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	2	no	—
372	C₄	C₃₁×#554	1	1	yes	characteristic
373	C₄	C₃₁×#556	1	1	yes	characteristic
374	C₄	C₃₁×#557	1	1	yes	characteristic
375	C₄	C₃₁×#559	1	1	yes	characteristic
376	C₂	C₃₁×#2355	1	1	yes	characteristic
377	C₂	C₃₁×#2921	1	1	yes	characteristic
378	C₂	C₃₁×#2925	1	1	yes	characteristic
379	C₂	C₃₁×#2961	1	1	yes	characteristic
380	C₂	C₃₁×#2962	1	1	yes	characteristic
381	C₂	C₃₁×#2992	1	1	yes	characteristic
382	C₂	C₃₁×#2995	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
64. C₃₁×C₄⋊C₈	Q₈	semidirect
121. #22824	C₃₁	direct (N × H)
255. C₄⋊C₈	Q₈×C₃₁	semidirect
266. C₃₁	#22824	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2295	2
C₃₁×#1985	2
C₃₁×#1917	2
C₃₁×#1916	2
C₃₁×#1842	2
C₃₁×#1832	2
C₃₁×#1771	2
C₃₁×#1768	2
C₃₁×#1767	2
#22824	31

Subgroups — C31×#22824 #21479583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22824 #21479583