Subgroups — C₃₁×#34633 #21491392

Order 7936.

3166 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#525	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2621	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2620	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2619	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2618	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2617	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2392	C₂	1	8	yes	maximal, complemented
10	C₃₁×#2297	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#532	—	2	8	no	—
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃₁×#124	C₂²	1	2	yes	complemented
18	C₃₁×#525	—	2	16	no	—
19	C₂×C₃₁×#123	C₂²	1	4	yes	—
20	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
21	C₂×C₃₁×#121	C₂²	1	4	yes	—
22	C₂×C₃₁×#121	C₂²	1	4	yes	—
23	C₂×C₃₁×#120	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
26	C₂×C₃₁×#117	—	2	8	no	—
27	C₂×C₃₁×#117	C₂²	1	2	yes	—
28	C₂×C₃₁×#117	C₂²	1	2	yes	—
29	C₂×C₃₁×#117	C₂²	1	4	yes	—
30	C₂×C₃₁×#117	C₂²	1	2	yes	—
31	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
33	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
34	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#398	C₂²	1	8	yes	complemented
36	C₃₁×#397	C₂²	1	8	yes	—
37	C₃₁×#396	C₂²	1	16	yes	complemented
38	C₃₁×#394	C₂²	1	16	yes	—
39	C₃₁×#394	C₂²	1	8	yes	complemented
40	C₃₁×#392	C₂²	1	8	yes	—
41	C₃₁×#391	C₂²	1	4	yes	complemented
42	C₃₁×#390	C₂²	1	4	yes	complemented
43	C₃₁×#386	C₂²	1	8	yes	—
44	C₃₁×#384	C₂²	1	2	yes	complemented
45	C₃₁×#382	C₂²	1	2	yes	complemented
46	C₃₁×#380	C₂²	1	4	yes	—
47	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
48	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
49	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×#41×C₃₁	—	2	8	no	—
51	C₂×#41×C₃₁	—	2	8	no	—
52	C₂×#41×C₃₁	—	2	8	no	—
53	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
54	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
55	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
56	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
57	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
58	C₂³×C₄×C₃₁	—	2	4	no	—
59	C₂³×C₄×C₃₁	—	2	4	no	—
60	C₂³×C₄×C₃₁	D₄	1	8	yes	complemented
61	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
65	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
66	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
67	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×C₄⋊Q₈	—	2	4	no	—
69	C₃₁×#127	—	2	4	no	—
70	C₃₁×#124	—	2	8	no	—
71	C₃₁×#123	—	2	16	no	—
72	C₃₁×C₂²⋊Q₈	—	2	16	no	—
73	C₃₁×#121	—	2	16	no	—
74	C₃₁×#121	—	2	16	no	—
75	C₃₁×#120	—	2	16	no	—
76	C₄×D₄×C₃₁	—	2	32	no	—
77	C₄×D₄×C₃₁	—	2	16	no	—
78	C₃₁×#117	—	2	8	no	—
79	C₃₁×#117	—	2	16	no	—
80	C₃₁×#117	—	2	8	no	—
81	C₃₁×#117	—	2	8	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
89	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
90	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
92	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
93	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
94	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
97	C₂×C₄²×C₃₁	—	2	4	no	—
98	C₂×C₄²×C₃₁	D₄	1	8	yes	complemented
99	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
100	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
101	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
102	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
105	C₃₁×#95	C₂³	1	16	yes	—
106	C₃₁×#95	C₂³	1	8	yes	—
107	C₂⁴×C₃₁	—	2	2	no	—
108	C₂⁴×C₃₁	—	2	2	no	—
109	C₂⁴×C₃₁	—	4	4	no	—
110	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
111	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
112	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
113	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
114	#41×C₃₁	—	4	32	no	—
115	C₂×Q₈×C₃₁	—	2	4	no	—
116	C₂×Q₈×C₃₁	—	2	4	no	—
117	C₂×Q₈×C₃₁	—	2	4	no	—
118	C₂×D₄×C₃₁	—	2	16	no	—
119	C₂×D₄×C₃₁	—	2	8	no	—
120	C₂×D₄×C₃₁	—	2	8	no	—
121	C₂×D₄×C₃₁	—	2	8	no	—
122	C₂×D₄×C₃₁	—	2	8	no	—
123	C₂×D₄×C₃₁	—	2	4	no	—
124	C₂×D₄×C₃₁	—	2	4	no	—
125	C₂×D₄×C₃₁	—	2	4	no	—
126	C₂×D₄×C₃₁	—	4	8	no	—
127	C₂×D₄×C₃₁	—	4	16	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	32	no	—
131	C₂²×C₄×C₃₁	—	2	4	no	—
132	C₂²×C₄×C₃₁	—	2	4	no	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	2	4	no	—
135	C₂²×C₄×C₃₁	—	2	4	no	—
136	C₂²×C₄×C₃₁	—	2	8	no	—
137	C₂²×C₄×C₃₁	—	2	8	no	—
138	C₂²×C₄×C₃₁	—	2	8	no	—
139	C₂²×C₄×C₃₁	—	4	4	no	—
140	C₂²×C₄×C₃₁	—	4	4	no	—
141	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
142	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
143	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
144	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
145	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
146	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
147	C₂²×C₄×C₃₁	#41	1	4	yes	—
148	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
150	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
152	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
156	C₄⋊C₄×C₃₁	—	2	8	no	—
157	C₄⋊C₄×C₃₁	—	2	8	no	—
158	C₄⋊C₄×C₃₁	—	2	8	no	—
159	C₄⋊C₄×C₃₁	—	2	8	no	—
160	C₄⋊C₄×C₃₁	—	2	16	no	—
161	C₄⋊C₄×C₃₁	—	4	8	no	—
162	C₂²⋊C₄×C₃₁	—	2	8	no	—
163	C₂²⋊C₄×C₃₁	—	2	16	no	—
164	C₂²⋊C₄×C₃₁	—	2	8	no	—
165	C₂²⋊C₄×C₃₁	—	2	8	no	—
166	C₂²⋊C₄×C₃₁	—	2	16	no	—
167	C₂²⋊C₄×C₃₁	—	2	16	no	—
168	C₂²⋊C₄×C₃₁	—	4	16	no	—
169	C₄²×C₃₁	—	2	4	no	—
170	C₄²×C₃₁	—	2	4	no	—
171	C₄²×C₃₁	—	2	8	no	—
172	C₄²×C₃₁	C₂×D₄	1	8	yes	—
173	C₄²×C₃₁	C₂×D₄	1	4	yes	—
174	C₄²×C₃₁	C₂×D₄	1	4	yes	—
175	#34633	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₃₁	—	2	4	no	—
177	C₂³×C₃₁	—	2	8	no	—
178	C₂³×C₃₁	—	2	2	no	—
179	C₂³×C₃₁	—	2	2	no	—
180	C₂³×C₃₁	—	2	8	no	—
181	C₂³×C₃₁	—	2	4	no	—
182	C₂³×C₃₁	—	2	4	no	—
183	C₂³×C₃₁	—	2	4	no	—
184	C₂³×C₃₁	—	2	16	no	—
185	C₂³×C₃₁	—	4	8	no	—
186	C₂³×C₃₁	—	4	8	no	—
187	C₂³×C₃₁	—	4	8	no	—
188	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
189	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
190	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
191	C₂³×C₃₁	C₂×#41	1	2	yes	—
192	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
193	Q₈×C₃₁	—	4	8	no	—
194	D₄×C₃₁	—	4	16	no	—
195	D₄×C₃₁	—	4	32	no	—
196	D₄×C₃₁	—	4	16	no	—
197	D₄×C₃₁	—	4	8	no	—
198	C₂×C₄×C₃₁	—	2	8	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	4	no	—
202	C₂×C₄×C₃₁	—	2	4	no	—
203	C₂×C₄×C₃₁	—	2	16	no	—
204	C₂×C₄×C₃₁	—	2	16	no	—
205	C₂×C₄×C₃₁	—	2	4	no	—
206	C₂×C₄×C₃₁	—	2	8	no	—
207	C₂×C₄×C₃₁	—	2	4	no	—
208	C₂×C₄×C₃₁	—	2	4	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	8	no	—
211	C₂×C₄×C₃₁	—	2	16	no	—
212	C₂×C₄×C₃₁	—	4	8	no	—
213	C₂×C₄×C₃₁	—	4	8	no	—
214	C₂×C₄×C₃₁	—	4	16	no	—
215	C₂×C₄×C₃₁	#121	1	16	yes	—
216	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
219	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
222	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
224	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
225	C₂×#532	C₂×C₃₁	1	1	yes	characteristic, complemented
226	C₂×#525	C₂×C₃₁	1	2	yes	complemented
227	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#2621	C₂×C₃₁	1	2	yes	complemented
229	#2620	C₂×C₃₁	1	4	yes	complemented
230	#2619	C₂×C₃₁	1	2	yes	complemented
231	#2618	C₂×C₃₁	1	4	yes	complemented
232	#2617	C₂×C₃₁	1	4	yes	complemented
233	#2392	C₂×C₃₁	1	8	yes	complemented
234	#2297	C₂×C₃₁	1	2	yes	complemented
235	#2285	C₂×C₃₁	1	1	yes	characteristic, complemented
236	C₂²×C₃₁	—	2	4	no	—
237	C₂²×C₃₁	—	2	4	no	—
238	C₂²×C₃₁	—	2	4	no	—
239	C₂²×C₃₁	—	2	4	no	—
240	C₂²×C₃₁	—	2	8	no	—
241	C₂²×C₃₁	—	4	16	no	—
242	C₂²×C₃₁	—	4	4	no	—
243	C₂²×C₃₁	—	4	4	no	—
244	C₂²×C₃₁	—	4	4	no	—
245	C₂²×C₃₁	—	4	8	no	—
246	C₂²×C₃₁	—	4	8	no	—
247	C₂²×C₃₁	—	4	8	no	—
248	C₂²×C₃₁	C₂×#121	1	4	yes	—
249	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
250	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
251	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
252	C₂²×C₃₁	#585	1	4	yes	—
253	C₄×C₃₁	—	2	4	no	—
254	C₄×C₃₁	—	2	4	no	—
255	C₄×C₃₁	—	2	4	no	—
256	C₄×C₃₁	—	4	4	no	—
257	C₄×C₃₁	—	4	4	no	—
258	C₄×C₃₁	—	4	8	no	—
259	C₄×C₃₁	C₂×#121	1	4	yes	—
260	C₄×C₃₁	C₂×#121	1	4	yes	—
261	C₂²×#41	C₂²×C₃₁	1	2	yes	—
262	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
263	#532	—	2	8	no	—
264	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
265	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic, complemented
266	C₂×#124	C₂²×C₃₁	1	2	yes	complemented
267	#525	—	2	16	no	—
268	C₂×#123	C₂²×C₃₁	1	4	yes	—
269	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
270	C₂×#121	C₂²×C₃₁	1	4	yes	—
271	C₂×#121	C₂²×C₃₁	1	4	yes	—
272	C₂×#120	C₂²×C₃₁	1	4	yes	—
273	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	—
274	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
275	C₂×#117	—	2	8	no	—
276	C₂×#117	C₂²×C₃₁	1	2	yes	—
277	C₂×#117	C₂²×C₃₁	1	2	yes	—
278	C₂×#117	C₂²×C₃₁	1	4	yes	—
279	C₂×#117	C₂²×C₃₁	1	2	yes	—
280	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
281	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
282	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
283	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
284	#398	C₂²×C₃₁	1	8	yes	complemented
285	#397	C₂²×C₃₁	1	8	yes	—
286	#396	C₂²×C₃₁	1	16	yes	complemented
287	#394	C₂²×C₃₁	1	16	yes	—
288	#394	C₂²×C₃₁	1	8	yes	complemented
289	#392	C₂²×C₃₁	1	8	yes	—
290	#391	C₂²×C₃₁	1	4	yes	complemented
291	#390	C₂²×C₃₁	1	4	yes	complemented
292	#386	C₂²×C₃₁	1	8	yes	—
293	#384	C₂²×C₃₁	1	2	yes	complemented
294	#382	C₂²×C₃₁	1	2	yes	complemented
295	#380	C₂²×C₃₁	1	4	yes	—
296	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
297	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
298	C₂×C₃₁	—	2	4	no	—
299	C₂×C₃₁	—	4	4	no	—
300	C₂×C₃₁	—	8	4	no	—
301	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
302	C₂×C₃₁	#3449	1	2	yes	—
303	C₂×C₃₁	#3449	1	2	yes	—
304	C₂×C₃₁	#3455	1	2	yes	—
305	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
306	C₂×#41	—	2	8	no	—
307	C₂×#41	—	2	8	no	—
308	C₂×#41	—	2	8	no	—
309	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
310	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
311	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
312	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
313	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
314	C₂³×C₄	—	2	4	no	—
315	C₂³×C₄	—	2	4	no	—
316	C₂³×C₄	D₄×C₃₁	1	8	yes	complemented
317	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
318	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
319	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
320	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
321	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
322	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
323	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
324	C₄⋊Q₈	—	2	4	no	—
325	#127	—	2	4	no	—
326	#124	—	2	8	no	—
327	#123	—	2	16	no	—
328	C₂²⋊Q₈	—	2	16	no	—
329	#121	—	2	16	no	—
330	#121	—	2	16	no	—
331	#120	—	2	16	no	—
332	C₄×D₄	—	2	32	no	—
333	C₄×D₄	—	2	16	no	—
334	#117	—	2	8	no	—
335	#117	—	2	16	no	—
336	#117	—	2	8	no	—
337	#117	—	2	8	no	—
338	C₂×C₄⋊C₄	—	2	8	no	—
339	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
340	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
341	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
342	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
343	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
345	C₂×C₂²⋊C₄	—	2	8	no	—
346	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
347	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
348	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
349	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
350	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
351	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
353	C₂×C₄²	—	2	4	no	—
354	C₂×C₄²	D₄×C₃₁	1	8	yes	complemented
355	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
356	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
357	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
358	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
359	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
360	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
361	#95	C₂³×C₃₁	1	16	yes	—
362	#95	C₂³×C₃₁	1	8	yes	—
363	C₃₁	#34633	1	1	yes	characteristic, direct product factor, Sylow
364	C₂⁴	—	2	2	no	—
365	C₂⁴	—	2	2	no	—
366	C₂⁴	—	4	4	no	—
367	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
368	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
369	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
370	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
371	#41	—	4	32	no	—
372	C₂×Q₈	—	2	4	no	—
373	C₂×Q₈	—	2	4	no	—
374	C₂×Q₈	—	2	4	no	—
375	C₂×D₄	—	2	16	no	—
376	C₂×D₄	—	2	8	no	—
377	C₂×D₄	—	2	8	no	—
378	C₂×D₄	—	2	8	no	—
379	C₂×D₄	—	2	8	no	—
380	C₂×D₄	—	2	4	no	—
381	C₂×D₄	—	2	4	no	—
382	C₂×D₄	—	2	4	no	—
383	C₂×D₄	—	4	8	no	—
384	C₂×D₄	—	4	16	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	32	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	2	4	no	—
393	C₂²×C₄	—	2	8	no	—
394	C₂²×C₄	—	2	8	no	—
395	C₂²×C₄	—	2	8	no	—
396	C₂²×C₄	—	4	4	no	—
397	C₂²×C₄	—	4	4	no	—
398	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
399	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
400	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
401	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
402	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
403	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
404	C₂²×C₄	#41×C₃₁	1	4	yes	—
405	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
406	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
407	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
408	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
409	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
410	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
411	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
412	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
413	C₄⋊C₄	—	2	8	no	—
414	C₄⋊C₄	—	2	8	no	—
415	C₄⋊C₄	—	2	8	no	—
416	C₄⋊C₄	—	2	8	no	—
417	C₄⋊C₄	—	2	16	no	—
418	C₄⋊C₄	—	4	8	no	—
419	C₂²⋊C₄	—	2	8	no	—
420	C₂²⋊C₄	—	2	16	no	—
421	C₂²⋊C₄	—	2	8	no	—
422	C₂²⋊C₄	—	2	8	no	—
423	C₂²⋊C₄	—	2	16	no	—
424	C₂²⋊C₄	—	2	16	no	—
425	C₂²⋊C₄	—	4	16	no	—
426	C₄²	—	2	4	no	—
427	C₄²	—	2	4	no	—
428	C₄²	—	2	8	no	—
429	C₄²	C₂×D₄×C₃₁	1	8	yes	—
430	C₄²	C₂×D₄×C₃₁	1	4	yes	—
431	C₄²	C₂×D₄×C₃₁	1	4	yes	—
432	C₂³	—	2	4	no	—
433	C₂³	—	2	8	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	2	2	no	—
436	C₂³	—	2	8	no	—
437	C₂³	—	2	4	no	—
438	C₂³	—	2	4	no	—
439	C₂³	—	2	4	no	—
440	C₂³	—	2	16	no	—
441	C₂³	—	4	8	no	—
442	C₂³	—	4	8	no	—
443	C₂³	—	4	8	no	—
444	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
445	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
446	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
447	C₂³	C₂×#41×C₃₁	1	2	yes	—
448	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
449	Q₈	—	4	8	no	—
450	D₄	—	4	16	no	—
451	D₄	—	4	32	no	—
452	D₄	—	4	16	no	—
453	D₄	—	4	8	no	—
454	C₂×C₄	—	2	8	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	16	no	—
460	C₂×C₄	—	2	16	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	8	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	16	no	—
468	C₂×C₄	—	4	8	no	—
469	C₂×C₄	—	4	8	no	—
470	C₂×C₄	—	4	16	no	—
471	C₂×C₄	C₃₁×#121	1	16	yes	—
472	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
473	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
474	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
475	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
476	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
477	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
478	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
479	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
480	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
481	C₂²	—	2	4	no	—
482	C₂²	—	2	4	no	—
483	C₂²	—	2	4	no	—
484	C₂²	—	2	4	no	—
485	C₂²	—	2	8	no	—
486	C₂²	—	4	16	no	—
487	C₂²	—	4	4	no	—
488	C₂²	—	4	4	no	—
489	C₂²	—	4	4	no	—
490	C₂²	—	4	8	no	—
491	C₂²	—	4	8	no	—
492	C₂²	—	4	8	no	—
493	C₂²	C₂×C₃₁×#121	1	4	yes	—
494	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
495	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
496	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
497	C₂²	C₃₁×#585	1	4	yes	—
498	C₄	—	2	4	no	—
499	C₄	—	2	4	no	—
500	C₄	—	2	4	no	—
501	C₄	—	4	4	no	—
502	C₄	—	4	4	no	—
503	C₄	—	4	8	no	—
504	C₄	C₂×C₃₁×#121	1	4	yes	—
505	C₄	C₂×C₃₁×#121	1	4	yes	—
506	C₂	—	2	4	no	—
507	C₂	—	4	4	no	—
508	C₂	—	8	4	no	—
509	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic
510	C₂	C₃₁×#3449	1	2	yes	—
511	C₂	C₃₁×#3449	1	2	yes	—
512	C₂	C₃₁×#3455	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
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
1. C₂×C₃₁×#532	C₂	semidirect
2. C₂×C₃₁×#525	C₂	semidirect
3. C₂²×C₃₁×#117	C₂	semidirect
4. C₃₁×#2621	C₂	semidirect
5. C₃₁×#2620	C₂	semidirect
6. C₃₁×#2619	C₂	semidirect
7. C₃₁×#2618	C₂	semidirect
8. C₃₁×#2617	C₂	semidirect
9. C₃₁×#2392	C₂	semidirect
10. C₃₁×#2297	C₂	semidirect
11. C₃₁×#2285	C₂	semidirect
15. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
16. C₂×C₃₁×#127	C₂²	semidirect
17. C₂×C₃₁×#124	C₂²	semidirect
25. C₂×C₄×D₄×C₃₁	C₂²	semidirect
35. C₃₁×#398	C₂²	semidirect
37. C₃₁×#396	C₂²	semidirect
39. C₃₁×#394	C₂²	semidirect
41. C₃₁×#391	C₂²	semidirect
42. C₃₁×#390	C₂²	semidirect
44. C₃₁×#384	C₂²	semidirect
45. C₃₁×#382	C₂²	semidirect
47. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
48. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
60. C₂³×C₄×C₃₁	D₄	semidirect
98. C₂×C₄²×C₃₁	D₄	semidirect
175. #34633	C₃₁	direct (N × H)
225. C₂×#532	C₂×C₃₁	semidirect
226. C₂×#525	C₂×C₃₁	semidirect
227. C₂²×#117	C₂×C₃₁	semidirect
228. #2621	C₂×C₃₁	semidirect
229. #2620	C₂×C₃₁	semidirect
230. #2619	C₂×C₃₁	semidirect
231. #2618	C₂×C₃₁	semidirect
232. #2617	C₂×C₃₁	semidirect
233. #2392	C₂×C₃₁	semidirect
234. #2297	C₂×C₃₁	semidirect
235. #2285	C₂×C₃₁	semidirect
264. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
265. C₂×#127	C₂²×C₃₁	semidirect
266. C₂×#124	C₂²×C₃₁	semidirect
274. C₂×C₄×D₄	C₂²×C₃₁	semidirect
284. #398	C₂²×C₃₁	semidirect
286. #396	C₂²×C₃₁	semidirect
288. #394	C₂²×C₃₁	semidirect
290. #391	C₂²×C₃₁	semidirect
291. #390	C₂²×C₃₁	semidirect
293. #384	C₂²×C₃₁	semidirect
294. #382	C₂²×C₃₁	semidirect
296. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
297. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
316. C₂³×C₄	D₄×C₃₁	semidirect
354. C₂×C₄²	D₄×C₃₁	semidirect
363. C₃₁	#34633	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×#532	2
C₂×C₃₁×#525	2
C₂²×C₃₁×#117	2
C₃₁×#2621	2
C₃₁×#2620	2
C₃₁×#2619	2
C₃₁×#2618	2
C₃₁×#2617	2
C₃₁×#2392	2
C₃₁×#2297	2
C₃₁×#2285	2
#34633	31

Subgroups — C31×#34633 #21491392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34633 #21491392