Subgroups — C₃₁×#34266 #21491025

Order 7936.

2946 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#536	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#520	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2367	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2365	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2362	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2360	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2322	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2310	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₃₁×#123	C₂	1	4	yes	maximal, complemented
11	C₃₁×#2297	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#2296	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2274	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#536	C₄	1	16	yes	complemented
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₃₁×#124	C₂²	1	2	yes	—
19	C₂×C₃₁×#123	C₂²	1	4	yes	—
20	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
21	C₃₁×#520	—	2	8	no	—
22	C₂×C₄×Q₈×C₃₁	—	2	8	no	—
23	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	—
24	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
27	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#117	C₂²	1	2	yes	—
31	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
32	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
36	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
37	C₃₁×#391	C₂²	1	2	yes	—
38	C₃₁×#391	C₂²	1	2	yes	—
39	C₃₁×#391	C₂²	1	4	yes	—
40	C₃₁×#391	C₂²	1	2	yes	—
41	C₃₁×#390	C₂²	1	2	yes	—
42	C₃₁×#389	C₂²	1	8	yes	—
43	C₃₁×#389	C₂²	1	2	yes	—
44	C₃₁×#389	C₂²	1	2	yes	—
45	C₃₁×#388	C₂²	1	8	yes	—
46	C₃₁×#388	C₂²	1	8	yes	—
47	C₃₁×#387	C₂²	1	8	yes	—
48	C₃₁×#387	C₂²	1	8	yes	—
49	C₃₁×#386	C₂²	1	2	yes	—
50	C₃₁×#386	C₂²	1	2	yes	—
51	C₃₁×#385	C₂²	1	2	yes	—
52	C₃₁×#385	C₂²	1	8	yes	—
53	C₃₁×#385	C₂²	1	2	yes	—
54	C₃₁×#381	C₂²	1	4	yes	—
55	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
56	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	—
57	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
58	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
59	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
60	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
61	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
62	C₂×#41×C₃₁	—	2	4	no	—
63	C₂×#41×C₃₁	—	2	4	no	—
64	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
65	C₂²×Q₈×C₃₁	—	2	4	no	—
66	C₂²×Q₈×C₃₁	—	2	4	no	—
67	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
68	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
69	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
70	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
72	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
76	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
78	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
79	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊Q₈	C₂×C₄	1	16	yes	complemented
81	C₃₁×#124	C₂×C₄	1	16	yes	complemented
82	C₃₁×#123	C₂×C₄	1	32	yes	complemented
83	C₃₁×C₂²⋊Q₈	C₂×C₄	1	32	yes	complemented
84	C₄×Q₈×C₃₁	—	2	16	no	—
85	C₄×Q₈×C₃₁	—	2	16	no	—
86	C₄×Q₈×C₃₁	—	2	8	no	—
87	C₄×D₄×C₃₁	—	2	16	no	—
88	C₄×D₄×C₃₁	—	2	8	no	—
89	C₃₁×#117	—	2	4	no	—
90	C₃₁×#117	—	2	4	no	—
91	C₃₁×#117	—	2	8	no	—
92	C₃₁×#117	C₂×C₄	1	8	yes	complemented
93	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
94	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
95	C₂×C₄⋊C₄×C₃₁	D₄	1	8	yes	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
110	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
114	C₂×C₄²×C₃₁	—	2	8	no	—
115	C₂×C₄²×C₃₁	—	2	4	no	—
116	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
117	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
118	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
119	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
120	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
121	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
122	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
123	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
124	C₃₁×#95	C₂³	1	4	yes	—
125	C₃₁×#95	C₂³	1	8	yes	—
126	C₃₁×#95	C₂³	1	4	yes	—
127	C₃₁×#95	C₂³	1	8	yes	—
128	C₃₁×#95	C₂³	1	4	yes	—
129	C₃₁×#95	C₂³	1	8	yes	—
130	C₃₁×#95	C₂³	1	4	yes	—
131	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
132	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
133	#41×C₃₁	—	2	32	no	—
134	C₂×Q₈×C₃₁	—	2	32	no	—
135	C₂×Q₈×C₃₁	—	2	4	no	—
136	C₂×Q₈×C₃₁	—	2	4	no	—
137	C₂×Q₈×C₃₁	—	2	2	no	—
138	C₂×Q₈×C₃₁	—	2	4	no	—
139	C₂×Q₈×C₃₁	—	2	4	no	—
140	C₂×Q₈×C₃₁	—	2	2	no	—
141	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
142	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
143	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
144	C₂×D₄×C₃₁	—	2	4	no	—
145	C₂×D₄×C₃₁	—	2	4	no	—
146	C₂×D₄×C₃₁	—	2	2	no	—
147	C₂×D₄×C₃₁	—	2	2	no	—
148	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
149	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	4	no	—
152	C₂²×C₄×C₃₁	—	2	4	no	—
153	C₂²×C₄×C₃₁	—	2	4	no	—
154	C₂²×C₄×C₃₁	—	2	4	no	—
155	C₂²×C₄×C₃₁	—	2	4	no	—
156	C₂²×C₄×C₃₁	—	2	2	no	—
157	C₂²×C₄×C₃₁	—	2	8	no	—
158	C₂²×C₄×C₃₁	—	4	4	no	—
159	C₂²×C₄×C₃₁	—	4	2	no	—
160	C₂²×C₄×C₃₁	—	4	8	no	—
161	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
162	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
163	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
164	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
165	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
166	C₂²×C₄×C₃₁	#41	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
172	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
178	C₄⋊C₄×C₃₁	—	2	2	no	—
179	C₄⋊C₄×C₃₁	—	2	8	no	—
180	C₄⋊C₄×C₃₁	—	2	16	no	—
181	C₄⋊C₄×C₃₁	—	2	2	no	—
182	C₄⋊C₄×C₃₁	—	2	4	no	—
183	C₄⋊C₄×C₃₁	—	2	8	no	—
184	C₄⋊C₄×C₃₁	—	4	4	no	—
185	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
186	C₄⋊C₄×C₃₁	C₂²×C₄	1	32	yes	—
187	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
188	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
189	C₂²⋊C₄×C₃₁	—	2	4	no	—
190	C₂²⋊C₄×C₃₁	—	2	4	no	—
191	C₂²⋊C₄×C₃₁	—	2	8	no	—
192	C₂²⋊C₄×C₃₁	—	2	8	no	—
193	C₂²⋊C₄×C₃₁	—	4	4	no	—
194	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
195	C₂²⋊C₄×C₃₁	C₂²×C₄	1	32	yes	—
196	C₄²×C₃₁	—	2	16	no	—
197	C₄²×C₃₁	—	2	4	no	—
198	C₄²×C₃₁	—	2	4	no	—
199	C₄²×C₃₁	—	2	8	no	—
200	C₄²×C₃₁	—	2	4	no	—
201	C₄²×C₃₁	—	2	4	no	—
202	C₄²×C₃₁	—	4	8	no	—
203	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
204	#34266	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂³×C₃₁	—	2	2	no	—
206	C₂³×C₃₁	—	2	2	no	—
207	C₂³×C₃₁	—	2	2	no	—
208	C₂³×C₃₁	—	2	2	no	—
209	C₂³×C₃₁	—	2	4	no	—
210	C₂³×C₃₁	—	2	4	no	—
211	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
212	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
213	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
214	C₂³×C₃₁	C₂×#41	1	2	yes	—
215	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
216	Q₈×C₃₁	—	2	16	no	—
217	Q₈×C₃₁	—	2	16	no	—
218	Q₈×C₃₁	—	2	8	no	—
219	D₄×C₃₁	—	2	16	no	—
220	D₄×C₃₁	—	2	8	no	—
221	C₂×C₄×C₃₁	—	2	8	no	—
222	C₂×C₄×C₃₁	—	2	16	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	2	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	4	no	—
231	C₂×C₄×C₃₁	—	2	8	no	—
232	C₂×C₄×C₃₁	—	2	16	no	—
233	C₂×C₄×C₃₁	—	2	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	4	no	—
237	C₂×C₄×C₃₁	—	2	4	no	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	2	4	no	—
240	C₂×C₄×C₃₁	—	2	2	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	4	8	no	—
244	C₂×C₄×C₃₁	—	4	4	no	—
245	C₂×C₄×C₃₁	—	8	8	no	—
246	C₂×C₄×C₃₁	C₄×D₄	1	16	yes	—
247	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
248	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
249	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
250	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
251	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
252	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
253	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
256	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
257	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
258	C₂×#536	C₂×C₃₁	1	1	yes	characteristic
259	C₂×#520	C₂×C₃₁	1	1	yes	characteristic
260	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#2367	C₂×C₃₁	1	4	yes	complemented
262	#2365	C₂×C₃₁	1	4	yes	complemented
263	#2362	C₂×C₃₁	1	4	yes	complemented
264	#2360	C₂×C₃₁	1	4	yes	complemented
265	#2322	C₂×C₃₁	1	4	yes	complemented
266	#2310	C₂×C₃₁	1	1	yes	characteristic, complemented
267	C₄×#123	C₂×C₃₁	1	4	yes	complemented
268	#2297	C₂×C₃₁	1	1	yes	characteristic
269	#2296	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#2274	C₂×C₃₁	1	1	yes	characteristic, complemented
271	C₂²×C₃₁	—	2	4	no	—
272	C₂²×C₃₁	—	2	2	no	—
273	C₂²×C₃₁	—	2	2	no	—
274	C₂²×C₃₁	—	2	4	no	—
275	C₂²×C₃₁	—	2	4	no	—
276	C₂²×C₃₁	—	2	8	no	—
277	C₂²×C₃₁	—	4	2	no	—
278	C₂²×C₃₁	—	4	2	no	—
279	C₂²×C₃₁	—	4	4	no	—
280	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
281	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
282	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
283	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
284	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
285	C₂²×C₃₁	#585	1	2	yes	—
286	C₄×C₃₁	—	2	4	no	—
287	C₄×C₃₁	—	2	8	no	—
288	C₄×C₃₁	—	2	8	no	—
289	C₄×C₃₁	—	2	8	no	—
290	C₄×C₃₁	—	2	4	no	—
291	C₄×C₃₁	—	4	2	no	—
292	C₄×C₃₁	—	4	2	no	—
293	C₄×C₃₁	—	4	2	no	—
294	C₄×C₃₁	—	4	4	no	—
295	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
296	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
297	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
298	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
299	#536	C₄×C₃₁	1	16	yes	complemented
300	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
301	C₂×#124	C₂²×C₃₁	1	2	yes	—
302	C₂×#123	C₂²×C₃₁	1	4	yes	—
303	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
304	#520	—	2	8	no	—
305	C₂×C₄×Q₈	—	2	8	no	—
306	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	—
307	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
308	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	—
309	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
310	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
311	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
312	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
313	C₂×#117	C₂²×C₃₁	1	2	yes	—
314	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
315	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
316	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
317	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
318	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
319	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
320	#391	C₂²×C₃₁	1	2	yes	—
321	#391	C₂²×C₃₁	1	2	yes	—
322	#391	C₂²×C₃₁	1	4	yes	—
323	#391	C₂²×C₃₁	1	2	yes	—
324	#390	C₂²×C₃₁	1	2	yes	—
325	#389	C₂²×C₃₁	1	8	yes	—
326	#389	C₂²×C₃₁	1	2	yes	—
327	#389	C₂²×C₃₁	1	2	yes	—
328	#388	C₂²×C₃₁	1	8	yes	—
329	#388	C₂²×C₃₁	1	8	yes	—
330	#387	C₂²×C₃₁	1	8	yes	—
331	#387	C₂²×C₃₁	1	8	yes	—
332	#386	C₂²×C₃₁	1	2	yes	—
333	#386	C₂²×C₃₁	1	2	yes	—
334	#385	C₂²×C₃₁	1	2	yes	—
335	#385	C₂²×C₃₁	1	8	yes	—
336	#385	C₂²×C₃₁	1	2	yes	—
337	#381	C₂²×C₃₁	1	4	yes	—
338	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
339	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	—
340	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
341	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
342	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
343	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
344	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
345	C₂×C₃₁	—	2	4	no	—
346	C₂×C₃₁	—	4	4	no	—
347	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
348	C₂×C₃₁	C₄×2₋¹⁺⁴	1	2	yes	—
349	C₂×C₃₁	#3485	1	1	yes	characteristic
350	C₂×C₃₁	#3486	1	1	yes	characteristic
351	C₂×C₃₁	#3507	1	2	yes	—
352	C₂×#41	—	2	4	no	—
353	C₂×#41	—	2	4	no	—
354	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
355	C₂²×Q₈	—	2	4	no	—
356	C₂²×Q₈	—	2	4	no	—
357	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
358	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
359	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
360	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
361	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
362	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
363	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
364	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
365	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
366	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
367	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
368	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
369	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
370	C₄⋊Q₈	C₂×C₄×C₃₁	1	16	yes	complemented
371	#124	C₂×C₄×C₃₁	1	16	yes	complemented
372	#123	C₂×C₄×C₃₁	1	32	yes	complemented
373	C₂²⋊Q₈	C₂×C₄×C₃₁	1	32	yes	complemented
374	C₄×Q₈	—	2	16	no	—
375	C₄×Q₈	—	2	16	no	—
376	C₄×Q₈	—	2	8	no	—
377	C₄×D₄	—	2	16	no	—
378	C₄×D₄	—	2	8	no	—
379	#117	—	2	4	no	—
380	#117	—	2	4	no	—
381	#117	—	2	8	no	—
382	#117	C₂×C₄×C₃₁	1	8	yes	complemented
383	C₂×C₄⋊C₄	—	2	8	no	—
384	C₂×C₄⋊C₄	—	2	8	no	—
385	C₂×C₄⋊C₄	D₄×C₃₁	1	8	yes	—
386	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
388	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
389	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
391	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
392	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
393	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
394	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
395	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
396	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
404	C₂×C₄²	—	2	8	no	—
405	C₂×C₄²	—	2	4	no	—
406	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
407	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
408	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
409	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
410	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
411	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
412	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
413	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
414	#95	C₂³×C₃₁	1	4	yes	—
415	#95	C₂³×C₃₁	1	8	yes	—
416	#95	C₂³×C₃₁	1	4	yes	—
417	#95	C₂³×C₃₁	1	8	yes	—
418	#95	C₂³×C₃₁	1	4	yes	—
419	#95	C₂³×C₃₁	1	8	yes	—
420	#95	C₂³×C₃₁	1	4	yes	—
421	C₃₁	#34266	1	1	yes	characteristic, direct product factor, Sylow
422	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
423	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
424	#41	—	2	32	no	—
425	C₂×Q₈	—	2	32	no	—
426	C₂×Q₈	—	2	4	no	—
427	C₂×Q₈	—	2	4	no	—
428	C₂×Q₈	—	2	2	no	—
429	C₂×Q₈	—	2	4	no	—
430	C₂×Q₈	—	2	4	no	—
431	C₂×Q₈	—	2	2	no	—
432	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
433	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
434	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
435	C₂×D₄	—	2	4	no	—
436	C₂×D₄	—	2	4	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	2	no	—
439	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
440	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	4	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	4	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	8	no	—
449	C₂²×C₄	—	4	4	no	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	—	4	8	no	—
452	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
453	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
454	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
455	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
456	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
457	C₂²×C₄	#41×C₃₁	1	4	yes	—
458	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
459	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
460	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
461	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
462	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
463	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
464	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
465	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
466	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
467	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
468	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	—	2	8	no	—
471	C₄⋊C₄	—	2	16	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	4	no	—
474	C₄⋊C₄	—	2	8	no	—
475	C₄⋊C₄	—	4	4	no	—
476	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
477	C₄⋊C₄	C₂²×C₄×C₃₁	1	32	yes	—
478	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
479	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
480	C₂²⋊C₄	—	2	4	no	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	8	no	—
483	C₂²⋊C₄	—	2	8	no	—
484	C₂²⋊C₄	—	4	4	no	—
485	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
486	C₂²⋊C₄	C₂²×C₄×C₃₁	1	32	yes	—
487	C₄²	—	2	16	no	—
488	C₄²	—	2	4	no	—
489	C₄²	—	2	4	no	—
490	C₄²	—	2	8	no	—
491	C₄²	—	2	4	no	—
492	C₄²	—	2	4	no	—
493	C₄²	—	4	8	no	—
494	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	2	no	—
497	C₂³	—	2	2	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	4	no	—
500	C₂³	—	2	4	no	—
501	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
502	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
503	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
504	C₂³	C₂×#41×C₃₁	1	2	yes	—
505	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
506	Q₈	—	2	16	no	—
507	Q₈	—	2	16	no	—
508	Q₈	—	2	8	no	—
509	D₄	—	2	16	no	—
510	D₄	—	2	8	no	—
511	C₂×C₄	—	2	8	no	—
512	C₂×C₄	—	2	16	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	8	no	—
522	C₂×C₄	—	2	16	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	4	8	no	—
534	C₂×C₄	—	4	4	no	—
535	C₂×C₄	—	8	8	no	—
536	C₂×C₄	C₄×D₄×C₃₁	1	16	yes	—
537	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
538	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
539	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
540	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
541	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
542	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
543	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
544	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
545	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
546	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
547	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
548	C₂²	—	2	4	no	—
549	C₂²	—	2	2	no	—
550	C₂²	—	2	2	no	—
551	C₂²	—	2	4	no	—
552	C₂²	—	2	4	no	—
553	C₂²	—	2	8	no	—
554	C₂²	—	4	2	no	—
555	C₂²	—	4	2	no	—
556	C₂²	—	4	4	no	—
557	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
558	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
559	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
560	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
561	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
562	C₂²	C₃₁×#585	1	2	yes	—
563	C₄	—	2	4	no	—
564	C₄	—	2	8	no	—
565	C₄	—	2	8	no	—
566	C₄	—	2	8	no	—
567	C₄	—	2	4	no	—
568	C₄	—	4	2	no	—
569	C₄	—	4	2	no	—
570	C₄	—	4	2	no	—
571	C₄	—	4	4	no	—
572	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
573	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
574	C₂	—	2	4	no	—
575	C₂	—	4	4	no	—
576	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
577	C₂	C₄×C₃₁×2₋¹⁺⁴	1	2	yes	—
578	C₂	C₃₁×#3485	1	1	yes	characteristic
579	C₂	C₃₁×#3486	1	1	yes	characteristic
580	C₂	C₃₁×#3507	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄×Q₈×C₃₁	C₂	semidirect
4. C₃₁×#2367	C₂	semidirect
5. C₃₁×#2365	C₂	semidirect
6. C₃₁×#2362	C₂	semidirect
7. C₃₁×#2360	C₂	semidirect
8. C₃₁×#2322	C₂	semidirect
9. C₃₁×#2310	C₂	semidirect
10. C₄×C₃₁×#123	C₂	semidirect
12. C₃₁×#2296	C₂	semidirect
13. C₃₁×#2274	C₂	semidirect
16. C₃₁×#536	C₄	semidirect
67. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
80. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
81. C₃₁×#124	C₂×C₄	semidirect
82. C₃₁×#123	C₂×C₄	semidirect
83. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
92. C₃₁×#117	C₂×C₄	semidirect
204. #34266	C₃₁	direct (N × H)
260. C₂²×C₄×Q₈	C₂×C₃₁	semidirect
261. #2367	C₂×C₃₁	semidirect
262. #2365	C₂×C₃₁	semidirect
263. #2362	C₂×C₃₁	semidirect
264. #2360	C₂×C₃₁	semidirect
265. #2322	C₂×C₃₁	semidirect
266. #2310	C₂×C₃₁	semidirect
267. C₄×#123	C₂×C₃₁	semidirect
269. #2296	C₂×C₃₁	semidirect
270. #2274	C₂×C₃₁	semidirect
299. #536	C₄×C₃₁	semidirect
357. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
370. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
371. #124	C₂×C₄×C₃₁	semidirect
372. #123	C₂×C₄×C₃₁	semidirect
373. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
382. #117	C₂×C₄×C₃₁	semidirect
421. C₃₁	#34266	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×#536	2
C₂×C₃₁×#520	2
C₂²×C₄×Q₈×C₃₁	2
C₃₁×#2367	2
C₃₁×#2365	2
C₃₁×#2362	2
C₃₁×#2360	2
C₃₁×#2322	2
C₃₁×#2310	2
C₄×C₃₁×#123	2
C₃₁×#2297	2
C₃₁×#2296	2
C₃₁×#2274	2
#34266	31

Subgroups — C31×#34266 #21491025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34266 #21491025