Subgroups — C₃₁×#22474 #21479233

Order 7936.

1196 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#529	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1870	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1869	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1818	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1729	C₂	1	2	yes	maximal, complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#529	—	2	4	no	—
15	C₃₁×#529	—	2	4	no	—
16	C₂×C₃₁×#126	C₄	1	2	yes	—
17	C₂×C₃₁×#125	C₄	1	1	yes	characteristic
18	C₂×C₃₁×#124	C₄	1	1	yes	characteristic
19	C₂×C₃₁×#123	C₄	1	2	yes	—
20	C₂×C₃₁×C₂²⋊Q₈	C₄	1	1	yes	characteristic
21	C₂×C₃₁×#121	C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#117	C₂²	1	2	yes	—
26	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#436	—	2	4	no	—
28	C₈×D₄×C₃₁	—	2	4	no	—
29	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	1	yes	characteristic
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₃₁×#336	C₂²	1	1	yes	characteristic
45	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
47	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
52	C₂×M₁₆×C₃₁	—	2	4	no	—
53	C₂×M₁₆×C₃₁	—	2	4	no	—
54	C₂×M₁₆×C₃₁	—	4	2	no	—
55	C₂²×C₈×C₃₁	—	2	4	no	—
56	C₂²×C₈×C₃₁	—	2	4	no	—
57	C₂²×C₈×C₃₁	—	4	2	no	—
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
62	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₃₁×#126	—	2	8	no	—
65	C₃₁×#125	—	2	4	no	—
66	C₃₁×#124	—	2	4	no	—
67	C₃₁×#123	—	2	8	no	—
68	C₃₁×C₂²⋊Q₈	—	2	4	no	—
69	C₃₁×#121	—	2	4	no	—
70	C₄×Q₈×C₃₁	—	2	2	no	—
71	C₄×Q₈×C₃₁	—	2	2	no	—
72	C₄×D₄×C₃₁	—	2	4	no	—
73	C₄×D₄×C₃₁	—	2	4	no	—
74	C₄×D₄×C₃₁	D₄	1	4	yes	—
75	C₄×D₄×C₃₁	D₄	1	4	yes	—
76	C₃₁×#117	—	2	4	no	—
77	C₃₁×#117	—	2	4	no	—
78	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
79	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
85	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
86	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
87	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
88	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
89	C₂×C₄²×C₃₁	—	2	2	no	—
90	C₂×C₄²×C₃₁	—	2	2	no	—
91	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
95	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
96	C₃₁×C₄⋊C₈	—	2	2	no	—
97	C₃₁×C₄⋊C₈	—	2	2	no	—
98	C₃₁×C₂²⋊C₈	—	2	4	no	—
99	C₃₁×C₂²⋊C₈	—	2	4	no	—
100	C₃₁×C₂²⋊C₈	—	2	2	no	—
101	C₃₁×C₂²⋊C₈	—	2	2	no	—
102	C₃₁×C₂²⋊C₈	—	4	2	no	—
103	C₄×C₈×C₃₁	—	2	2	no	—
104	C₄×C₈×C₃₁	—	2	2	no	—
105	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
106	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	4	no	—
111	C₂×D₄×C₃₁	—	2	4	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	4	2	no	—
115	C₂×D₄×C₃₁	C₂²⋊C₄	1	4	yes	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	2	1	no	—
128	C₂²×C₄×C₃₁	—	2	1	no	—
129	C₂²×C₄×C₃₁	—	2	1	no	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	4	1	no	—
132	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
136	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
137	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
141	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
142	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
143	M₁₆×C₃₁	—	4	8	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	1	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	2	1	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	2	2	no	—
156	C₂×C₈×C₃₁	—	2	2	no	—
157	C₂×C₈×C₃₁	—	2	2	no	—
158	C₂×C₈×C₃₁	—	2	1	no	—
159	C₂×C₈×C₃₁	—	2	1	no	—
160	C₂×C₈×C₃₁	—	4	8	no	—
161	C₂×C₈×C₃₁	#41	1	2	yes	—
162	C₂×C₈×C₃₁	#41	1	2	yes	—
163	C₄⋊C₄×C₃₁	—	2	4	no	—
164	C₄⋊C₄×C₃₁	—	2	4	no	—
165	C₄⋊C₄×C₃₁	—	2	4	no	—
166	C₄⋊C₄×C₃₁	—	2	2	no	—
167	C₄⋊C₄×C₃₁	—	2	4	no	—
168	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
169	C₂²⋊C₄×C₃₁	—	2	4	no	—
170	C₂²⋊C₄×C₃₁	—	2	4	no	—
171	C₂²⋊C₄×C₃₁	—	2	4	no	—
172	C₂²⋊C₄×C₃₁	—	2	4	no	—
173	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
174	C₄²×C₃₁	—	2	2	no	—
175	C₄²×C₃₁	—	2	2	no	—
176	C₄²×C₃₁	—	2	1	no	—
177	C₄²×C₃₁	—	2	1	no	—
178	C₄²×C₃₁	—	2	2	no	—
179	C₄²×C₃₁	—	2	2	no	—
180	C₄²×C₃₁	C₂×D₄	1	2	yes	—
181	C₄²×C₃₁	C₂×D₄	1	2	yes	—
182	#22474	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂³×C₃₁	—	2	1	no	—
184	C₂³×C₃₁	—	2	2	no	—
185	C₂³×C₃₁	—	2	1	no	—
186	C₂³×C₃₁	—	2	2	no	—
187	C₂³×C₃₁	—	2	1	no	—
188	C₂³×C₃₁	—	2	1	no	—
189	C₂³×C₃₁	—	2	1	no	—
190	C₂³×C₃₁	—	2	2	no	—
191	C₂³×C₃₁	—	2	2	no	—
192	C₂³×C₃₁	—	2	2	no	—
193	C₂³×C₃₁	—	2	1	no	—
194	C₂³×C₃₁	—	2	2	no	—
195	C₂³×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
196	C₂³×C₃₁	#131	1	2	yes	—
197	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
198	Q₈×C₃₁	—	4	4	no	—
199	D₄×C₃₁	—	2	8	no	—
200	D₄×C₃₁	—	4	8	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	4	no	—
206	C₂×C₄×C₃₁	—	2	4	no	—
207	C₂×C₄×C₃₁	—	2	4	no	—
208	C₂×C₄×C₃₁	—	2	4	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	1	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	1	no	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	1	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	4	4	no	—
225	C₂×C₄×C₃₁	—	4	1	no	—
226	C₂×C₄×C₃₁	—	4	1	no	—
227	C₂×C₄×C₃₁	—	4	1	no	—
228	C₂×C₄×C₃₁	—	4	1	no	—
229	C₂×C₄×C₃₁	—	4	4	no	—
230	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
231	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
232	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
233	C₂×C₄×C₃₁	#131	1	2	yes	—
234	C₂×C₄×C₃₁	#131	1	2	yes	—
235	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
240	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
241	C₈×C₃₁	—	2	2	no	—
242	C₈×C₃₁	—	2	2	no	—
243	C₈×C₃₁	—	4	2	no	—
244	C₈×C₃₁	—	4	2	no	—
245	C₈×C₃₁	—	4	1	no	—
246	C₈×C₃₁	—	4	1	no	—
247	C₂×#529	C₂×C₃₁	1	1	yes	characteristic
248	C₂×#436	C₂×C₃₁	1	1	yes	characteristic, complemented
249	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1870	C₂×C₃₁	1	2	yes	complemented
253	#1869	C₂×C₃₁	1	2	yes	complemented
254	#1818	C₂×C₃₁	1	2	yes	complemented
255	#1797	C₂×C₃₁	1	1	yes	characteristic, complemented
256	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#1729	C₂×C₃₁	1	2	yes	complemented
258	C₂²×C₃₁	—	2	4	no	—
259	C₂²×C₃₁	—	2	4	no	—
260	C₂²×C₃₁	—	2	2	no	—
261	C₂²×C₃₁	—	2	4	no	—
262	C₂²×C₃₁	—	2	2	no	—
263	C₂²×C₃₁	—	2	2	no	—
264	C₂²×C₃₁	—	4	1	no	—
265	C₂²×C₃₁	—	4	1	no	—
266	C₂²×C₃₁	—	4	2	no	—
267	C₂²×C₃₁	—	4	1	no	—
268	C₂²×C₃₁	—	4	2	no	—
269	C₂²×C₃₁	—	4	1	no	—
270	C₂²×C₃₁	—	4	2	no	—
271	C₂²×C₃₁	—	4	2	no	—
272	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
273	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
274	C₂²×C₃₁	#520	1	1	yes	characteristic
275	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
276	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
277	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
278	C₂²×C₃₁	#568	1	1	yes	characteristic
279	C₄×C₃₁	—	2	4	no	—
280	C₄×C₃₁	—	2	2	no	—
281	C₄×C₃₁	—	4	1	no	—
282	C₄×C₃₁	—	4	1	no	—
283	C₄×C₃₁	—	4	2	no	—
284	C₄×C₃₁	—	4	1	no	—
285	C₄×C₃₁	—	4	2	no	—
286	C₄×C₃₁	#408	1	4	yes	—
287	C₄×C₃₁	#546	1	1	yes	characteristic
288	C₄×C₃₁	#546	1	1	yes	characteristic
289	C₄×C₃₁	#547	1	1	yes	characteristic
290	C₄×C₃₁	#547	1	1	yes	characteristic
291	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
292	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
293	#529	—	2	4	no	—
294	#529	—	2	4	no	—
295	C₂×#126	C₄×C₃₁	1	2	yes	—
296	C₂×#125	C₄×C₃₁	1	1	yes	characteristic
297	C₂×#124	C₄×C₃₁	1	1	yes	characteristic
298	C₂×#123	C₄×C₃₁	1	2	yes	—
299	C₂×C₂²⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
300	C₂×#121	C₄×C₃₁	1	1	yes	characteristic
301	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
302	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
303	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×#117	C₂²×C₃₁	1	2	yes	—
305	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
306	#436	—	2	4	no	—
307	C₈×D₄	—	2	4	no	—
308	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
310	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
311	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
312	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
313	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
314	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
315	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
316	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
317	#336	C₂²×C₃₁	1	2	yes	—
318	#336	C₂²×C₃₁	1	1	yes	characteristic
319	#336	C₂²×C₃₁	1	2	yes	—
320	#336	C₂²×C₃₁	1	2	yes	—
321	#336	C₂²×C₃₁	1	2	yes	—
322	#336	C₂²×C₃₁	1	2	yes	—
323	#336	C₂²×C₃₁	1	1	yes	characteristic
324	C₂×C₃₁	—	4	4	no	—
325	C₂×C₃₁	—	4	2	no	—
326	C₂×C₃₁	#2339	1	1	yes	characteristic
327	C₂×C₃₁	C₂×#408	1	1	yes	characteristic
328	C₂×C₃₁	#2880	1	1	yes	characteristic
329	C₂×C₃₁	#2966	1	1	yes	characteristic
330	C₂×C₃₁	#2969	1	1	yes	characteristic
331	C₂×C₃₁	#2993	1	1	yes	characteristic
332	C₂×C₃₁	#2994	1	1	yes	characteristic
333	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
334	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
335	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
336	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
337	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
338	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
339	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
340	C₂×M₁₆	—	2	4	no	—
341	C₂×M₁₆	—	2	4	no	—
342	C₂×M₁₆	—	4	2	no	—
343	C₂²×C₈	—	2	4	no	—
344	C₂²×C₈	—	2	4	no	—
345	C₂²×C₈	—	4	2	no	—
346	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
347	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
348	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
349	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
350	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
351	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
352	#126	—	2	8	no	—
353	#125	—	2	4	no	—
354	#124	—	2	4	no	—
355	#123	—	2	8	no	—
356	C₂²⋊Q₈	—	2	4	no	—
357	#121	—	2	4	no	—
358	C₄×Q₈	—	2	2	no	—
359	C₄×Q₈	—	2	2	no	—
360	C₄×D₄	—	2	4	no	—
361	C₄×D₄	—	2	4	no	—
362	C₄×D₄	D₄×C₃₁	1	4	yes	—
363	C₄×D₄	D₄×C₃₁	1	4	yes	—
364	#117	—	2	4	no	—
365	#117	—	2	4	no	—
366	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
367	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
368	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
369	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
371	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
372	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
373	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
374	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
375	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
376	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
377	C₂×C₄²	—	2	2	no	—
378	C₂×C₄²	—	2	2	no	—
379	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
380	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
381	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
382	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
383	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
384	C₄⋊C₈	—	2	2	no	—
385	C₄⋊C₈	—	2	2	no	—
386	C₂²⋊C₈	—	2	4	no	—
387	C₂²⋊C₈	—	2	4	no	—
388	C₂²⋊C₈	—	2	2	no	—
389	C₂²⋊C₈	—	2	2	no	—
390	C₂²⋊C₈	—	4	2	no	—
391	C₄×C₈	—	2	2	no	—
392	C₄×C₈	—	2	2	no	—
393	C₃₁	#22474	1	1	yes	characteristic, direct product factor, Sylow
394	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
395	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
396	C₂×Q₈	—	2	2	no	—
397	C₂×Q₈	—	2	2	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×D₄	—	2	4	no	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	4	2	no	—
404	C₂×D₄	C₂²⋊C₄×C₃₁	1	4	yes	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	4	1	no	—
421	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
422	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
423	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
425	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
426	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
427	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
428	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
429	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
430	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
431	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
432	M₁₆	—	4	8	no	—
433	C₂×C₈	—	2	1	no	—
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	2	no	—
444	C₂×C₈	—	2	2	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	—	2	2	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	4	8	no	—
450	C₂×C₈	#41×C₃₁	1	2	yes	—
451	C₂×C₈	#41×C₃₁	1	2	yes	—
452	C₄⋊C₄	—	2	4	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
458	C₂²⋊C₄	—	2	4	no	—
459	C₂²⋊C₄	—	2	4	no	—
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
463	C₄²	—	2	2	no	—
464	C₄²	—	2	2	no	—
465	C₄²	—	2	1	no	—
466	C₄²	—	2	1	no	—
467	C₄²	—	2	2	no	—
468	C₄²	—	2	2	no	—
469	C₄²	C₂×D₄×C₃₁	1	2	yes	—
470	C₄²	C₂×D₄×C₃₁	1	2	yes	—
471	C₂³	—	2	1	no	—
472	C₂³	—	2	2	no	—
473	C₂³	—	2	1	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	2	1	no	—
476	C₂³	—	2	1	no	—
477	C₂³	—	2	1	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	1	no	—
482	C₂³	—	2	2	no	—
483	C₂³	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
484	C₂³	C₃₁×#131	1	2	yes	—
485	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
486	Q₈	—	4	4	no	—
487	D₄	—	2	8	no	—
488	D₄	—	4	8	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	4	4	no	—
513	C₂×C₄	—	4	4	no	—
514	C₂×C₄	—	4	1	no	—
515	C₂×C₄	—	4	1	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
519	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
520	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
521	C₂×C₄	C₃₁×#131	1	2	yes	—
522	C₂×C₄	C₃₁×#131	1	2	yes	—
523	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
524	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
525	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
526	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
527	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
528	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
529	C₈	—	2	2	no	—
530	C₈	—	2	2	no	—
531	C₈	—	4	2	no	—
532	C₈	—	4	2	no	—
533	C₈	—	4	1	no	—
534	C₈	—	4	1	no	—
535	C₂²	—	2	4	no	—
536	C₂²	—	2	4	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	4	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	4	1	no	—
542	C₂²	—	4	1	no	—
543	C₂²	—	4	2	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	4	2	no	—
546	C₂²	—	4	1	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
550	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
551	C₂²	C₃₁×#520	1	1	yes	characteristic
552	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
553	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
554	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
555	C₂²	C₃₁×#568	1	1	yes	characteristic
556	C₄	—	2	4	no	—
557	C₄	—	2	2	no	—
558	C₄	—	4	1	no	—
559	C₄	—	4	1	no	—
560	C₄	—	4	2	no	—
561	C₄	—	4	1	no	—
562	C₄	—	4	2	no	—
563	C₄	C₃₁×#408	1	4	yes	—
564	C₄	C₃₁×#546	1	1	yes	characteristic
565	C₄	C₃₁×#546	1	1	yes	characteristic
566	C₄	C₃₁×#547	1	1	yes	characteristic
567	C₄	C₃₁×#547	1	1	yes	characteristic
568	C₂	—	4	4	no	—
569	C₂	—	4	2	no	—
570	C₂	C₃₁×#2339	1	1	yes	characteristic
571	C₂	C₂×C₃₁×#408	1	1	yes	characteristic
572	C₂	C₃₁×#2880	1	1	yes	characteristic
573	C₂	C₃₁×#2966	1	1	yes	characteristic
574	C₂	C₃₁×#2969	1	1	yes	characteristic
575	C₂	C₃₁×#2993	1	1	yes	characteristic
576	C₂	C₃₁×#2994	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂×C₃₁×#121	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
2. C₂×C₃₁×#436	C₂	semidirect
3. C₂×C₈×D₄×C₃₁	C₂	semidirect
4. C₃₁×#1984	C₂	semidirect
5. C₃₁×#1984	C₂	semidirect
6. C₃₁×#1870	C₂	semidirect
7. C₃₁×#1869	C₂	semidirect
8. C₃₁×#1818	C₂	semidirect
9. C₃₁×#1797	C₂	semidirect
10. C₄×C₃₁×C₂²⋊C₈	C₂	semidirect
11. C₃₁×#1729	C₂	semidirect
182. #22474	C₃₁	direct (N × H)
248. C₂×#436	C₂×C₃₁	semidirect
249. C₂×C₈×D₄	C₂×C₃₁	semidirect
250. #1984	C₂×C₃₁	semidirect
251. #1984	C₂×C₃₁	semidirect
252. #1870	C₂×C₃₁	semidirect
253. #1869	C₂×C₃₁	semidirect
254. #1818	C₂×C₃₁	semidirect
255. #1797	C₂×C₃₁	semidirect
256. C₄×C₂²⋊C₈	C₂×C₃₁	semidirect
257. #1729	C₂×C₃₁	semidirect
393. C₃₁	#22474	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#529	2
C₂×C₃₁×#436	2
C₂×C₈×D₄×C₃₁	2
C₃₁×#1984	2
C₃₁×#1870	2
C₃₁×#1869	2
C₃₁×#1818	2
C₃₁×#1797	2
C₄×C₃₁×C₂²⋊C₈	2
C₃₁×#1729	2
#22474	31

Subgroups — C31×#22474 #21479233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22474 #21479233