Subgroups — C₃₁×#22821 #21479580

Order 7936.

784 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1984	C₂	1	2	yes	maximal
3	C₃₁×#1916	C₂	1	4	yes	maximal
4	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1831	C₂	1	2	yes	maximal
6	C₈×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal
7	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₄×C₃₁×C₄⋊C₈	C₂	1	2	yes	maximal
9	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
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	2	yes	—
21	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
22	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₄³×C₃₁	C₂²	1	2	yes	—
26	C₄³×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#336	C₂²	1	4	yes	—
28	C₃₁×#336	C₂²	1	4	yes	—
29	C₃₁×#336	C₂²	1	2	yes	—
30	C₃₁×#336	C₂²	1	4	yes	—
31	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
35	C₄×Q₈×C₃₁	—	2	4	no	—
36	C₄×Q₈×C₃₁	—	2	2	no	—
37	C₄×Q₈×C₃₁	—	2	2	no	—
38	C₄×Q₈×C₃₁	C₈	1	8	yes	complemented
39	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
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	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
54	C₂×C₄²×C₃₁	C₂³	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	C₃₁×C₄⋊C₈	—	2	4	no	—
59	C₃₁×C₄⋊C₈	—	2	2	no	—
60	C₃₁×C₄⋊C₈	—	2	4	no	—
61	C₃₁×C₄⋊C₈	—	4	2	no	—
62	C₃₁×C₄⋊C₈	Q₈	1	4	yes	complemented
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	C₂×Q₈×C₃₁	—	2	2	no	—
69	C₂×Q₈×C₃₁	—	2	2	no	—
70	C₂×Q₈×C₃₁	C₂×C₈	1	4	yes	—
71	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	2	4	no	—
87	C₂×C₈×C₃₁	—	2	4	no	—
88	C₂×C₈×C₃₁	—	2	4	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
91	C₄⋊C₄×C₃₁	—	2	4	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	C₂×C₈	1	8	yes	—
95	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
96	C₄⋊C₄×C₃₁	M₁₆	1	4	yes	—
97	C₄²×C₃₁	—	2	4	no	—
98	C₄²×C₃₁	—	2	2	no	—
99	C₄²×C₃₁	—	2	2	no	—
100	C₄²×C₃₁	—	2	4	no	—
101	C₄²×C₃₁	—	2	2	no	—
102	C₄²×C₃₁	—	2	2	no	—
103	C₄²×C₃₁	—	2	2	no	—
104	C₄²×C₃₁	—	2	1	no	—
105	C₄²×C₃₁	—	2	2	no	—
106	C₄²×C₃₁	—	2	1	no	—
107	C₄²×C₃₁	—	4	1	no	—
108	C₄²×C₃₁	—	4	2	no	—
109	C₄²×C₃₁	C₂×C₈	1	8	yes	—
110	C₄²×C₃₁	C₂×C₈	1	4	yes	—
111	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
112	C₄²×C₃₁	#41	1	2	yes	—
113	C₄²×C₃₁	#41	1	2	yes	—
114	C₄²×C₃₁	#41	1	2	yes	—
115	#22821	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
117	Q₈×C₃₁	—	2	8	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	1	no	—
121	C₂×C₄×C₃₁	—	2	2	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	2	2	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	2	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	—	2	1	no	—
132	C₂×C₄×C₃₁	—	2	1	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	1	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	#117	1	2	yes	—
137	C₂×C₄×C₃₁	#117	1	2	yes	—
138	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
139	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
140	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
141	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
142	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
143	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
144	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
145	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
146	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
147	C₂×C₄×C₃₁	#131	1	2	yes	—
148	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
154	C₈×C₃₁	—	2	4	no	—
155	C₈×C₃₁	—	4	2	no	—
156	C₈×C₃₁	—	4	4	no	—
157	C₄²×Q₈	C₂×C₃₁	1	1	yes	characteristic
158	#1984	C₂×C₃₁	1	2	yes	—
159	#1916	C₂×C₃₁	1	4	yes	—
160	#1842	C₂×C₃₁	1	1	yes	characteristic
161	#1831	C₂×C₃₁	1	2	yes	—
162	C₈×C₄⋊C₄	C₂×C₃₁	1	2	yes	—
163	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
164	C₄×C₄⋊C₈	C₂×C₃₁	1	2	yes	—
165	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
166	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₂²×C₃₁	#520	1	1	yes	characteristic
168	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
169	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
170	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
171	C₂²×C₃₁	#568	1	1	yes	characteristic
172	C₄×C₃₁	—	2	4	no	—
173	C₄×C₃₁	—	2	2	no	—
174	C₄×C₃₁	—	2	2	no	—
175	C₄×C₃₁	—	2	4	no	—
176	C₄×C₃₁	—	2	2	no	—
177	C₄×C₃₁	—	2	4	no	—
178	C₄×C₃₁	—	4	2	no	—
179	C₄×C₃₁	—	4	1	no	—
180	C₄×C₃₁	#431	1	4	yes	—
181	C₄×C₃₁	C₈×Q₈	1	4	yes	—
182	C₄×C₃₁	#554	1	1	yes	characteristic
183	C₄×C₃₁	#556	1	1	yes	characteristic
184	C₄×C₃₁	#557	1	1	yes	characteristic
185	C₄×C₃₁	#559	1	1	yes	characteristic
186	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
187	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
188	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
189	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
190	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
191	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
192	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
193	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
194	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
195	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
196	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
197	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
198	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
199	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
200	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
201	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
202	C₄³	C₂²×C₃₁	1	2	yes	—
203	C₄³	C₂²×C₃₁	1	1	yes	characteristic
204	#336	C₂²×C₃₁	1	4	yes	—
205	#336	C₂²×C₃₁	1	4	yes	—
206	#336	C₂²×C₃₁	1	2	yes	—
207	#336	C₂²×C₃₁	1	4	yes	—
208	C₂×C₃₁	#2355	1	1	yes	characteristic
209	C₂×C₃₁	C₂×#431	1	1	yes	characteristic
210	C₂×C₃₁	#2925	1	1	yes	characteristic
211	C₂×C₃₁	C₂×C₈×Q₈	1	1	yes	characteristic
212	C₂×C₃₁	Q₈×M₁₆	1	1	yes	characteristic
213	C₂×C₃₁	#2988	1	1	yes	characteristic
214	C₂×C₃₁	#2990	1	1	yes	characteristic
215	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
217	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
218	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
219	C₄×Q₈	—	2	4	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	C₈×C₃₁	1	8	yes	complemented
223	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
224	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
225	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
226	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
227	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
228	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
229	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
230	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
231	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
232	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
233	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
234	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
235	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
236	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
237	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
238	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
239	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
240	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
241	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
242	C₄⋊C₈	—	2	4	no	—
243	C₄⋊C₈	—	2	2	no	—
244	C₄⋊C₈	—	2	4	no	—
245	C₄⋊C₈	—	4	2	no	—
246	C₄⋊C₈	Q₈×C₃₁	1	4	yes	complemented
247	C₄×C₈	—	2	4	no	—
248	C₄×C₈	—	4	4	no	—
249	#95	C₂×C₄×C₃₁	1	2	yes	—
250	#95	C₂×C₄×C₃₁	1	2	yes	—
251	#95	C₂×C₄×C₃₁	1	2	yes	—
252	C₃₁	#22821	1	1	yes	characteristic, direct product factor, Sylow
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	C₂×C₈×C₃₁	1	4	yes	—
256	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
257	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
258	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
259	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
260	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
261	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
262	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
263	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
264	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
265	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
266	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	C₂×C₈×C₃₁	1	8	yes	—
280	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
281	C₄⋊C₄	M₁₆×C₃₁	1	4	yes	—
282	C₄²	—	2	4	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	4	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	4	2	no	—
294	C₄²	C₂×C₈×C₃₁	1	8	yes	—
295	C₄²	C₂×C₈×C₃₁	1	4	yes	—
296	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
297	C₄²	#41×C₃₁	1	2	yes	—
298	C₄²	#41×C₃₁	1	2	yes	—
299	C₄²	#41×C₃₁	1	2	yes	—
300	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
301	Q₈	—	2	8	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	C₃₁×#117	1	2	yes	—
321	C₂×C₄	C₃₁×#117	1	2	yes	—
322	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
323	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
324	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
325	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
326	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
327	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
328	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
329	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
330	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
331	C₂×C₄	C₃₁×#131	1	2	yes	—
332	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
333	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
334	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
335	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
336	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
337	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
338	C₈	—	2	4	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	4	no	—
341	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
342	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
343	C₂²	C₃₁×#520	1	1	yes	characteristic
344	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
345	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
346	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
347	C₂²	C₃₁×#568	1	1	yes	characteristic
348	C₄	—	2	4	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	4	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	4	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	1	no	—
356	C₄	C₃₁×#431	1	4	yes	—
357	C₄	C₈×Q₈×C₃₁	1	4	yes	—
358	C₄	C₃₁×#554	1	1	yes	characteristic
359	C₄	C₃₁×#556	1	1	yes	characteristic
360	C₄	C₃₁×#557	1	1	yes	characteristic
361	C₄	C₃₁×#559	1	1	yes	characteristic
362	C₂	C₃₁×#2355	1	1	yes	characteristic
363	C₂	C₂×C₃₁×#431	1	1	yes	characteristic
364	C₂	C₃₁×#2925	1	1	yes	characteristic
365	C₂	C₂×C₈×Q₈×C₃₁	1	1	yes	characteristic
366	C₂	Q₈×M₁₆×C₃₁	1	1	yes	characteristic
367	C₂	C₃₁×#2988	1	1	yes	characteristic
368	C₂	C₃₁×#2990	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle, prime radical	C₂³×C₃₁	C₂³×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
38. C₄×Q₈×C₃₁	C₈	semidirect
62. C₃₁×C₄⋊C₈	Q₈	semidirect
115. #22821	C₃₁	direct (N × H)
222. C₄×Q₈	C₈×C₃₁	semidirect
246. C₄⋊C₈	Q₈×C₃₁	semidirect
252. C₃₁	#22821	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×Q₈×C₃₁	2
C₃₁×#1984	2
C₃₁×#1916	2
C₃₁×#1842	2
C₃₁×#1831	2
C₈×C₄⋊C₄×C₃₁	2
C₄×C₃₁×C₄⋊C₈	2
#22821	31

Subgroups — C31×#22821 #21479580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22821 #21479580