Subgroups — C₃₁×#22823 #21479582

Order 7936.

736 conjugacy classes of subgroups.

The subgroup lattice has 494 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×#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×#1831	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1771	C₂	1	2	yes	maximal
9	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic
11	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	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×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#97	C₂²	1	2	yes	—
22	C₂×C₃₁×#97	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₄³×C₃₁	C₂²	1	1	yes	characteristic
34	C₄³×C₃₁	C₂²	1	1	yes	characteristic
35	C₄³×C₃₁	C₂²	1	1	yes	characteristic
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	C₃₁×#336	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₄×Q₈×C₃₁	—	2	4	no	—
49	C₄×Q₈×C₃₁	—	2	4	no	—
50	C₄×Q₈×C₃₁	—	2	2	no	—
51	C₄×Q₈×C₃₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊C₈	—	2	4	no	—
81	C₃₁×C₄⋊C₈	—	2	2	no	—
82	C₃₁×C₄⋊C₈	—	2	2	no	—
83	C₃₁×C₄⋊C₈	—	2	2	no	—
84	C₃₁×C₄⋊C₈	—	4	2	no	—
85	C₃₁×C₄⋊C₈	Q₈	1	4	yes	complemented
86	C₃₁×#97	—	2	4	no	—
87	C₃₁×#97	—	4	2	no	—
88	C₄×C₈×C₃₁	—	4	2	no	—
89	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
90	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
91	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
92	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
93	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
94	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
95	C₂×Q₈×C₃₁	—	2	2	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	M₁₆	1	4	yes	—
134	C₄²×C₃₁	—	2	2	no	—
135	C₄²×C₃₁	—	2	2	no	—
136	C₄²×C₃₁	—	2	2	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	2	2	no	—
139	C₄²×C₃₁	—	2	2	no	—
140	C₄²×C₃₁	—	2	2	no	—
141	C₄²×C₃₁	—	2	2	no	—
142	C₄²×C₃₁	—	2	1	no	—
143	C₄²×C₃₁	—	2	1	no	—
144	C₄²×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	—	2	1	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	C₄²×C₃₁	—	2	1	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	—	2	1	no	—
152	C₄²×C₃₁	—	4	1	no	—
153	C₄²×C₃₁	—	4	1	no	—
154	C₄²×C₃₁	—	4	1	no	—
155	C₄²×C₃₁	M₁₆	1	4	yes	—
156	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
157	C₄²×C₃₁	#41	1	2	yes	—
158	C₄²×C₃₁	#41	1	2	yes	—
159	C₄²×C₃₁	#41	1	2	yes	—
160	#22823	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
162	Q₈×C₃₁	—	2	8	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	#117	1	2	yes	—
195	C₂×C₄×C₃₁	#117	1	2	yes	—
196	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
197	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
198	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
199	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
200	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
201	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
202	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
203	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
204	C₂×C₄×C₃₁	#131	1	2	yes	—
205	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
206	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
208	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
209	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
210	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
211	C₈×C₃₁	—	2	4	no	—
212	C₈×C₃₁	—	4	2	no	—
213	C₈×C₃₁	—	4	2	no	—
214	C₈×C₃₁	—	4	2	no	—
215	C₄²×Q₈	C₂×C₃₁	1	1	yes	characteristic
216	#1985	C₂×C₃₁	1	2	yes	—
217	#1917	C₂×C₃₁	1	2	yes	—
218	#1916	C₂×C₃₁	1	2	yes	—
219	#1842	C₂×C₃₁	1	1	yes	characteristic
220	#1831	C₂×C₃₁	1	1	yes	characteristic
221	#1831	C₂×C₃₁	1	1	yes	characteristic
222	#1771	C₂×C₃₁	1	2	yes	—
223	#1767	C₂×C₃₁	1	1	yes	characteristic
224	#1767	C₂×C₃₁	1	1	yes	characteristic
225	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
226	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
227	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
228	C₂²×C₃₁	#520	1	1	yes	characteristic
229	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
230	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
231	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
232	C₂²×C₃₁	#568	1	1	yes	characteristic
233	C₄×C₃₁	—	2	2	no	—
234	C₄×C₃₁	—	2	2	no	—
235	C₄×C₃₁	—	2	2	no	—
236	C₄×C₃₁	—	2	2	no	—
237	C₄×C₃₁	—	2	2	no	—
238	C₄×C₃₁	—	2	2	no	—
239	C₄×C₃₁	—	2	2	no	—
240	C₄×C₃₁	—	2	2	no	—
241	C₄×C₃₁	—	4	1	no	—
242	C₄×C₃₁	—	4	1	no	—
243	C₄×C₃₁	—	4	1	no	—
244	C₄×C₃₁	—	4	1	no	—
245	C₄×C₃₁	—	4	1	no	—
246	C₄×C₃₁	#446	1	4	yes	—
247	C₄×C₃₁	#554	1	1	yes	characteristic
248	C₄×C₃₁	#556	1	1	yes	characteristic
249	C₄×C₃₁	#557	1	1	yes	characteristic
250	C₄×C₃₁	#559	1	1	yes	characteristic
251	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
252	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
253	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
258	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
259	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×#97	C₂²×C₃₁	1	2	yes	—
261	C₂×#97	C₂²×C₃₁	1	2	yes	—
262	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
263	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
264	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
265	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
266	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
267	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
268	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
269	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
270	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
271	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
272	C₄³	C₂²×C₃₁	1	1	yes	characteristic
273	C₄³	C₂²×C₃₁	1	1	yes	characteristic
274	C₄³	C₂²×C₃₁	1	1	yes	characteristic
275	#336	C₂²×C₃₁	1	2	yes	—
276	#336	C₂²×C₃₁	1	2	yes	—
277	#336	C₂²×C₃₁	1	2	yes	—
278	#336	C₂²×C₃₁	1	2	yes	—
279	#336	C₂²×C₃₁	1	2	yes	—
280	#336	C₂²×C₃₁	1	2	yes	—
281	#336	C₂²×C₃₁	1	2	yes	—
282	C₂×C₃₁	#2355	1	1	yes	characteristic
283	C₂×C₃₁	#2920	1	1	yes	characteristic
284	C₂×C₃₁	#2925	1	1	yes	characteristic
285	C₂×C₃₁	C₂×#446	1	1	yes	characteristic
286	C₂×C₃₁	Q₈×M₁₆	1	1	yes	characteristic
287	C₂×C₃₁	#2989	1	1	yes	characteristic
288	C₂×C₃₁	#2990	1	1	yes	characteristic
289	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
291	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
292	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
293	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
294	C₄×Q₈	—	2	4	no	—
295	C₄×Q₈	—	2	4	no	—
296	C₄×Q₈	—	2	2	no	—
297	C₄×Q₈	—	2	2	no	—
298	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
299	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
300	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
301	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
306	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
307	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
309	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
310	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
311	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
312	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
313	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
314	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
315	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
316	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
317	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
318	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
319	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
320	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
321	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
322	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
323	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
324	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
325	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
326	C₄⋊C₈	—	2	4	no	—
327	C₄⋊C₈	—	2	2	no	—
328	C₄⋊C₈	—	2	2	no	—
329	C₄⋊C₈	—	2	2	no	—
330	C₄⋊C₈	—	4	2	no	—
331	C₄⋊C₈	Q₈×C₃₁	1	4	yes	complemented
332	#97	—	2	4	no	—
333	#97	—	4	2	no	—
334	C₄×C₈	—	4	2	no	—
335	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
336	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
337	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
338	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
339	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
340	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
341	C₃₁	#22823	1	1	yes	characteristic, direct product factor, Sylow
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
346	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
347	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
348	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
349	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
350	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
351	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
353	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
354	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
356	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
357	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
358	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
359	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	M₁₆×C₃₁	1	4	yes	—
381	C₄²	—	2	2	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	2	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	2	2	no	—
389	C₄²	—	2	1	no	—
390	C₄²	—	2	1	no	—
391	C₄²	—	2	1	no	—
392	C₄²	—	2	1	no	—
393	C₄²	—	2	1	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	2	1	no	—
397	C₄²	—	2	1	no	—
398	C₄²	—	2	1	no	—
399	C₄²	—	4	1	no	—
400	C₄²	—	4	1	no	—
401	C₄²	—	4	1	no	—
402	C₄²	M₁₆×C₃₁	1	4	yes	—
403	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
404	C₄²	#41×C₃₁	1	2	yes	—
405	C₄²	#41×C₃₁	1	2	yes	—
406	C₄²	#41×C₃₁	1	2	yes	—
407	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
408	Q₈	—	2	8	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	C₃₁×#117	1	2	yes	—
441	C₂×C₄	C₃₁×#117	1	2	yes	—
442	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
443	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
444	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
445	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
446	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
447	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
448	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
449	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
450	C₂×C₄	C₃₁×#131	1	2	yes	—
451	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
452	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
453	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
454	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
455	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
456	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
457	C₈	—	2	4	no	—
458	C₈	—	4	2	no	—
459	C₈	—	4	2	no	—
460	C₈	—	4	2	no	—
461	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
462	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
463	C₂²	C₃₁×#520	1	1	yes	characteristic
464	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
465	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
466	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
467	C₂²	C₃₁×#568	1	1	yes	characteristic
468	C₄	—	2	2	no	—
469	C₄	—	2	2	no	—
470	C₄	—	2	2	no	—
471	C₄	—	2	2	no	—
472	C₄	—	2	2	no	—
473	C₄	—	2	2	no	—
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	4	1	no	—
477	C₄	—	4	1	no	—
478	C₄	—	4	1	no	—
479	C₄	—	4	1	no	—
480	C₄	—	4	1	no	—
481	C₄	C₃₁×#446	1	4	yes	—
482	C₄	C₃₁×#554	1	1	yes	characteristic
483	C₄	C₃₁×#556	1	1	yes	characteristic
484	C₄	C₃₁×#557	1	1	yes	characteristic
485	C₄	C₃₁×#559	1	1	yes	characteristic
486	C₂	C₃₁×#2355	1	1	yes	characteristic
487	C₂	C₃₁×#2920	1	1	yes	characteristic
488	C₂	C₃₁×#2925	1	1	yes	characteristic
489	C₂	C₂×C₃₁×#446	1	1	yes	characteristic
490	C₂	Q₈×M₁₆×C₃₁	1	1	yes	characteristic
491	C₂	C₃₁×#2989	1	1	yes	characteristic
492	C₂	C₃₁×#2990	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
85. C₃₁×C₄⋊C₈	Q₈	semidirect
160. #22823	C₃₁	direct (N × H)
331. C₄⋊C₈	Q₈×C₃₁	semidirect
341. C₃₁	#22823	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×#1985	2
C₃₁×#1917	2
C₃₁×#1916	2
C₃₁×#1842	2
C₃₁×#1831	2
C₃₁×#1771	2
C₃₁×#1767	2
C₄×C₃₁×C₄⋊C₈	2
#22823	31

Subgroups — C31×#22823 #21479582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22823 #21479582