Subgroups — C₃₁×#34810 #21491569

Order 7936.

2738 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#525	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2504	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2503	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2502	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2501	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2500	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2499	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2498	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2412	C₂	1	4	yes	maximal, complemented
11	C₃₁×#2395	C₂	1	4	yes	maximal, complemented
12	C₄×C₃₁×#123	C₂	1	4	yes	maximal, complemented
13	C₃₁×#2297	C₂	1	2	yes	maximal, complemented
14	C₃₁×#2285	C₂	1	2	yes	maximal, complemented
15	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#525	—	2	8	no	—
19	C₂×C₃₁×#123	C₂²	1	2	yes	—
20	C₂×C₃₁×#123	C₂²	1	4	yes	—
21	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#120	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₃₁×#117	—	2	8	no	—
29	C₂×C₃₁×#117	C₂²	1	2	yes	—
30	C₂×C₃₁×#117	C₂²	1	2	yes	—
31	C₂×C₃₁×#117	C₂²	1	2	yes	—
32	C₂×C₃₁×#117	C₂²	1	2	yes	—
33	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
37	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
38	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
39	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
40	C₃₁×#400	C₂²	1	4	yes	complemented
41	C₃₁×#399	C₂²	1	4	yes	complemented
42	C₃₁×#398	C₂²	1	4	yes	complemented
43	C₃₁×#397	C₂²	1	8	yes	—
44	C₃₁×#397	C₂²	1	4	yes	complemented
45	C₃₁×#394	C₂²	1	8	yes	—
46	C₃₁×#391	C₂²	1	2	yes	complemented
47	C₃₁×#391	C₂²	1	2	yes	complemented
48	C₃₁×#390	C₂²	1	2	yes	complemented
49	C₃₁×#390	C₂²	1	2	yes	complemented
50	C₃₁×#388	C₂²	1	8	yes	complemented
51	C₃₁×#387	C₂²	1	8	yes	complemented
52	C₃₁×#386	C₂²	1	8	yes	—
53	C₃₁×#386	C₂²	1	4	yes	—
54	C₃₁×#385	C₂²	1	4	yes	—
55	C₃₁×#384	C₂²	1	2	yes	complemented
56	C₃₁×#384	C₂²	1	2	yes	complemented
57	C₃₁×#382	C₂²	1	2	yes	complemented
58	C₃₁×#382	C₂²	1	2	yes	complemented
59	C₃₁×#381	C₂²	1	4	yes	—
60	C₃₁×#380	C₂²	1	8	yes	—
61	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	complemented
62	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
64	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
65	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
66	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
67	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
68	C₂×C₃₁×#95	C₂²	1	2	yes	—
69	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×#41×C₃₁	—	2	4	no	—
71	C₂×#41×C₃₁	—	2	8	no	—
72	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
74	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
75	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
77	C₂³×C₄×C₃₁	—	2	8	no	—
78	C₂³×C₄×C₃₁	D₄	1	8	yes	—
79	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
80	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
81	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
82	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
88	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
89	C₃₁×#123	—	2	8	no	—
90	C₃₁×#123	—	2	16	no	—
91	C₃₁×C₂²⋊Q₈	—	2	4	no	—
92	C₃₁×C₂²⋊Q₈	—	2	4	no	—
93	C₃₁×#121	—	2	4	no	—
94	C₃₁×#121	—	2	4	no	—
95	C₃₁×#120	—	2	8	no	—
96	C₄×D₄×C₃₁	—	2	8	no	—
97	C₄×D₄×C₃₁	—	2	8	no	—
98	C₃₁×#117	—	2	8	no	—
99	C₃₁×#117	—	2	8	no	—
100	C₃₁×#117	—	2	8	no	—
101	C₃₁×#117	—	2	8	no	—
102	C₃₁×#117	—	2	4	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
115	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
117	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
119	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
121	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
122	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
123	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
125	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
126	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
127	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
128	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
129	C₂×C₄²×C₃₁	—	2	8	no	—
130	C₂×C₄²×C₃₁	—	2	8	no	—
131	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
132	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
133	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
134	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
135	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
136	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
137	C₃₁×#95	C₂³	1	8	yes	—
138	C₃₁×#95	C₂³	1	8	yes	—
139	C₃₁×#95	C₂³	1	8	yes	—
140	C₃₁×#95	C₂³	1	2	yes	—
141	C₃₁×#95	C₂³	1	2	yes	—
142	C₃₁×#95	C₂³	1	8	yes	—
143	C₃₁×#95	C₂³	1	4	yes	—
144	C₂⁴×C₃₁	—	2	4	no	—
145	C₂⁴×C₃₁	—	2	4	no	—
146	C₂⁴×C₃₁	—	4	2	no	—
147	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
148	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
149	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
150	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
151	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
152	#41×C₃₁	—	4	16	no	—
153	C₂×Q₈×C₃₁	—	2	2	no	—
154	C₂×Q₈×C₃₁	—	2	4	no	—
155	C₂×D₄×C₃₁	—	2	4	no	—
156	C₂×D₄×C₃₁	—	2	4	no	—
157	C₂×D₄×C₃₁	—	2	4	no	—
158	C₂×D₄×C₃₁	—	2	8	no	—
159	C₂×D₄×C₃₁	—	2	8	no	—
160	C₂×D₄×C₃₁	—	2	2	no	—
161	C₂×D₄×C₃₁	—	2	4	no	—
162	C₂×D₄×C₃₁	—	4	4	no	—
163	C₂²×C₄×C₃₁	—	2	4	no	—
164	C₂²×C₄×C₃₁	—	2	32	no	—
165	C₂²×C₄×C₃₁	—	2	4	no	—
166	C₂²×C₄×C₃₁	—	2	4	no	—
167	C₂²×C₄×C₃₁	—	2	4	no	—
168	C₂²×C₄×C₃₁	—	2	4	no	—
169	C₂²×C₄×C₃₁	—	2	4	no	—
170	C₂²×C₄×C₃₁	—	2	4	no	—
171	C₂²×C₄×C₃₁	—	2	8	no	—
172	C₂²×C₄×C₃₁	—	2	8	no	—
173	C₂²×C₄×C₃₁	—	2	4	no	—
174	C₂²×C₄×C₃₁	—	2	8	no	—
175	C₂²×C₄×C₃₁	—	2	8	no	—
176	C₂²×C₄×C₃₁	—	4	4	no	—
177	C₂²×C₄×C₃₁	—	4	4	no	—
178	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
179	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
180	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
181	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
182	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
183	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
184	C₂²×C₄×C₃₁	#41	1	8	yes	—
185	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
186	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
187	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
188	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
189	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
190	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
191	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
192	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
193	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
194	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
195	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
196	C₄⋊C₄×C₃₁	—	2	4	no	—
197	C₄⋊C₄×C₃₁	—	2	4	no	—
198	C₄⋊C₄×C₃₁	—	2	4	no	—
199	C₄⋊C₄×C₃₁	—	2	4	no	—
200	C₄⋊C₄×C₃₁	—	2	2	no	—
201	C₄⋊C₄×C₃₁	—	2	2	no	—
202	C₄⋊C₄×C₃₁	—	2	4	no	—
203	C₄⋊C₄×C₃₁	—	2	4	no	—
204	C₄⋊C₄×C₃₁	—	2	16	no	—
205	C₄⋊C₄×C₃₁	—	4	8	no	—
206	C₂²⋊C₄×C₃₁	—	2	8	no	—
207	C₂²⋊C₄×C₃₁	—	2	8	no	—
208	C₂²⋊C₄×C₃₁	—	2	4	no	—
209	C₂²⋊C₄×C₃₁	—	2	4	no	—
210	C₂²⋊C₄×C₃₁	—	2	4	no	—
211	C₂²⋊C₄×C₃₁	—	2	4	no	—
212	C₂²⋊C₄×C₃₁	—	2	4	no	—
213	C₂²⋊C₄×C₃₁	—	2	4	no	—
214	C₂²⋊C₄×C₃₁	—	2	8	no	—
215	C₂²⋊C₄×C₃₁	—	2	16	no	—
216	C₂²⋊C₄×C₃₁	—	4	8	no	—
217	C₂²⋊C₄×C₃₁	#41	1	8	yes	complemented
218	C₄²×C₃₁	—	2	8	no	—
219	C₄²×C₃₁	—	2	8	no	—
220	C₄²×C₃₁	—	2	4	no	—
221	C₄²×C₃₁	—	2	4	no	—
222	C₄²×C₃₁	—	2	8	no	—
223	C₄²×C₃₁	—	2	4	no	—
224	#34810	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂³×C₃₁	—	2	2	no	—
226	C₂³×C₃₁	—	2	2	no	—
227	C₂³×C₃₁	—	2	2	no	—
228	C₂³×C₃₁	—	2	2	no	—
229	C₂³×C₃₁	—	2	8	no	—
230	C₂³×C₃₁	—	2	8	no	—
231	C₂³×C₃₁	—	2	2	no	—
232	C₂³×C₃₁	—	2	2	no	—
233	C₂³×C₃₁	—	2	2	no	—
234	C₂³×C₃₁	—	2	2	no	—
235	C₂³×C₃₁	—	2	4	no	—
236	C₂³×C₃₁	—	2	4	no	—
237	C₂³×C₃₁	—	2	4	no	—
238	C₂³×C₃₁	—	4	4	no	—
239	C₂³×C₃₁	—	4	16	no	—
240	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
241	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
242	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
243	C₂³×C₃₁	C₂×#41	1	4	yes	—
244	C₂³×C₃₁	C₂×#41	1	4	yes	—
245	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
246	Q₈×C₃₁	—	4	4	no	—
247	D₄×C₃₁	—	4	8	no	—
248	D₄×C₃₁	—	4	8	no	—
249	D₄×C₃₁	—	4	8	no	—
250	D₄×C₃₁	—	4	4	no	—
251	C₂×C₄×C₃₁	—	2	8	no	—
252	C₂×C₄×C₃₁	—	2	4	no	—
253	C₂×C₄×C₃₁	—	2	2	no	—
254	C₂×C₄×C₃₁	—	2	2	no	—
255	C₂×C₄×C₃₁	—	2	4	no	—
256	C₂×C₄×C₃₁	—	2	8	no	—
257	C₂×C₄×C₃₁	—	2	8	no	—
258	C₂×C₄×C₃₁	—	2	8	no	—
259	C₂×C₄×C₃₁	—	2	8	no	—
260	C₂×C₄×C₃₁	—	2	4	no	—
261	C₂×C₄×C₃₁	—	2	4	no	—
262	C₂×C₄×C₃₁	—	2	2	no	—
263	C₂×C₄×C₃₁	—	2	2	no	—
264	C₂×C₄×C₃₁	—	2	2	no	—
265	C₂×C₄×C₃₁	—	2	4	no	—
266	C₂×C₄×C₃₁	—	2	4	no	—
267	C₂×C₄×C₃₁	—	2	8	no	—
268	C₂×C₄×C₃₁	—	2	4	no	—
269	C₂×C₄×C₃₁	—	2	4	no	—
270	C₂×C₄×C₃₁	—	2	4	no	—
271	C₂×C₄×C₃₁	—	2	8	no	—
272	C₂×C₄×C₃₁	—	2	4	no	—
273	C₂×C₄×C₃₁	—	4	16	no	—
274	C₂×C₄×C₃₁	—	4	8	no	—
275	C₂×C₄×C₃₁	—	4	8	no	—
276	C₂×C₄×C₃₁	—	4	8	no	—
277	C₂×C₄×C₃₁	#123	1	16	yes	complemented
278	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
279	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
280	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
281	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
282	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
283	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
284	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
285	C₂×#525	C₂×C₃₁	1	1	yes	characteristic
286	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
287	#2504	C₂×C₃₁	1	1	yes	characteristic, complemented
288	#2503	C₂×C₃₁	1	1	yes	characteristic, complemented
289	#2502	C₂×C₃₁	1	1	yes	characteristic, complemented
290	#2501	C₂×C₃₁	1	1	yes	characteristic, complemented
291	#2500	C₂×C₃₁	1	2	yes	complemented
292	#2499	C₂×C₃₁	1	2	yes	complemented
293	#2498	C₂×C₃₁	1	4	yes	complemented
294	#2412	C₂×C₃₁	1	4	yes	complemented
295	#2395	C₂×C₃₁	1	4	yes	complemented
296	C₄×#123	C₂×C₃₁	1	4	yes	complemented
297	#2297	C₂×C₃₁	1	2	yes	complemented
298	#2285	C₂×C₃₁	1	2	yes	complemented
299	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
300	C₂²×C₃₁	—	2	2	no	—
301	C₂²×C₃₁	—	2	2	no	—
302	C₂²×C₃₁	—	2	2	no	—
303	C₂²×C₃₁	—	2	2	no	—
304	C₂²×C₃₁	—	2	8	no	—
305	C₂²×C₃₁	—	2	8	no	—
306	C₂²×C₃₁	—	2	2	no	—
307	C₂²×C₃₁	—	2	2	no	—
308	C₂²×C₃₁	—	2	2	no	—
309	C₂²×C₃₁	—	4	16	no	—
310	C₂²×C₃₁	—	4	4	no	—
311	C₂²×C₃₁	—	4	2	no	—
312	C₂²×C₃₁	—	4	2	no	—
313	C₂²×C₃₁	—	4	2	no	—
314	C₂²×C₃₁	—	4	8	no	—
315	C₂²×C₃₁	C₂×#123	1	4	yes	—
316	C₂²×C₃₁	#532	1	4	yes	—
317	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
318	C₂²×C₃₁	C₂²×#41	1	2	yes	—
319	C₂²×C₃₁	C₂²×#41	1	2	yes	—
320	C₂²×C₃₁	#585	1	2	yes	—
321	C₄×C₃₁	—	2	2	no	—
322	C₄×C₃₁	—	2	2	no	—
323	C₄×C₃₁	—	2	8	no	—
324	C₄×C₃₁	—	4	2	no	—
325	C₄×C₃₁	—	4	4	no	—
326	C₄×C₃₁	—	4	2	no	—
327	C₄×C₃₁	—	4	4	no	—
328	C₄×C₃₁	—	4	2	no	—
329	C₄×C₃₁	—	4	4	no	—
330	C₄×C₃₁	C₂×#123	1	4	yes	—
331	C₄×C₃₁	C₂×#123	1	4	yes	—
332	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
333	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
334	#525	—	2	8	no	—
335	C₂×#123	C₂²×C₃₁	1	2	yes	—
336	C₂×#123	C₂²×C₃₁	1	4	yes	—
337	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
339	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
340	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
341	C₂×#120	C₂²×C₃₁	1	2	yes	—
342	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
343	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
344	C₂×#117	—	2	8	no	—
345	C₂×#117	C₂²×C₃₁	1	2	yes	—
346	C₂×#117	C₂²×C₃₁	1	2	yes	—
347	C₂×#117	C₂²×C₃₁	1	2	yes	—
348	C₂×#117	C₂²×C₃₁	1	2	yes	—
349	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
350	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
351	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
352	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
353	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
354	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
355	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
356	#400	C₂²×C₃₁	1	4	yes	complemented
357	#399	C₂²×C₃₁	1	4	yes	complemented
358	#398	C₂²×C₃₁	1	4	yes	complemented
359	#397	C₂²×C₃₁	1	8	yes	—
360	#397	C₂²×C₃₁	1	4	yes	complemented
361	#394	C₂²×C₃₁	1	8	yes	—
362	#391	C₂²×C₃₁	1	2	yes	complemented
363	#391	C₂²×C₃₁	1	2	yes	complemented
364	#390	C₂²×C₃₁	1	2	yes	complemented
365	#390	C₂²×C₃₁	1	2	yes	complemented
366	#388	C₂²×C₃₁	1	8	yes	complemented
367	#387	C₂²×C₃₁	1	8	yes	complemented
368	#386	C₂²×C₃₁	1	8	yes	—
369	#386	C₂²×C₃₁	1	4	yes	—
370	#385	C₂²×C₃₁	1	4	yes	—
371	#384	C₂²×C₃₁	1	2	yes	complemented
372	#384	C₂²×C₃₁	1	2	yes	complemented
373	#382	C₂²×C₃₁	1	2	yes	complemented
374	#382	C₂²×C₃₁	1	2	yes	complemented
375	#381	C₂²×C₃₁	1	4	yes	—
376	#380	C₂²×C₃₁	1	8	yes	—
377	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	complemented
378	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
379	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
380	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
381	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
382	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
383	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
384	C₂×#95	C₂²×C₃₁	1	2	yes	—
385	C₂×C₃₁	—	2	4	no	—
386	C₂×C₃₁	—	2	4	no	—
387	C₂×C₃₁	—	4	2	no	—
388	C₂×C₃₁	—	8	2	no	—
389	C₂×C₃₁	C₂²×#123	1	1	yes	characteristic
390	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
391	C₂×C₃₁	#3449	1	1	yes	characteristic
392	C₂×C₃₁	#3475	1	4	yes	—
393	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
394	C₂×#41	—	2	4	no	—
395	C₂×#41	—	2	8	no	—
396	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
397	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
398	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
399	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
400	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
401	C₂³×C₄	—	2	8	no	—
402	C₂³×C₄	D₄×C₃₁	1	8	yes	—
403	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
404	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
405	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
406	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
407	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
408	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
409	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
410	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
411	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
412	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
413	#123	—	2	8	no	—
414	#123	—	2	16	no	—
415	C₂²⋊Q₈	—	2	4	no	—
416	C₂²⋊Q₈	—	2	4	no	—
417	#121	—	2	4	no	—
418	#121	—	2	4	no	—
419	#120	—	2	8	no	—
420	C₄×D₄	—	2	8	no	—
421	C₄×D₄	—	2	8	no	—
422	#117	—	2	8	no	—
423	#117	—	2	8	no	—
424	#117	—	2	8	no	—
425	#117	—	2	8	no	—
426	#117	—	2	4	no	—
427	C₂×C₄⋊C₄	—	2	4	no	—
428	C₂×C₄⋊C₄	—	2	4	no	—
429	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
430	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
431	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
432	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
433	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
434	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
435	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
436	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
437	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
438	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
439	C₂×C₂²⋊C₄	—	2	8	no	—
440	C₂×C₂²⋊C₄	—	2	8	no	—
441	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
442	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
443	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
444	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
445	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
446	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
447	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
448	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
449	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
450	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
451	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
452	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
453	C₂×C₄²	—	2	8	no	—
454	C₂×C₄²	—	2	8	no	—
455	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
456	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
457	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
458	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
459	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
460	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
461	#95	C₂³×C₃₁	1	4	yes	—
462	#95	C₂³×C₃₁	1	2	yes	—
463	#95	C₂³×C₃₁	1	2	yes	—
464	#95	C₂³×C₃₁	1	8	yes	—
465	#95	C₂³×C₃₁	1	8	yes	—
466	#95	C₂³×C₃₁	1	8	yes	—
467	#95	C₂³×C₃₁	1	8	yes	—
468	C₃₁	#34810	1	1	yes	characteristic, direct product factor, Sylow
469	C₂⁴	—	2	4	no	—
470	C₂⁴	—	2	4	no	—
471	C₂⁴	—	4	2	no	—
472	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
473	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
474	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
475	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
476	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
477	#41	—	4	16	no	—
478	C₂×Q₈	—	2	2	no	—
479	C₂×Q₈	—	2	4	no	—
480	C₂×D₄	—	2	4	no	—
481	C₂×D₄	—	2	4	no	—
482	C₂×D₄	—	2	4	no	—
483	C₂×D₄	—	2	8	no	—
484	C₂×D₄	—	2	8	no	—
485	C₂×D₄	—	2	2	no	—
486	C₂×D₄	—	2	4	no	—
487	C₂×D₄	—	4	4	no	—
488	C₂²×C₄	—	2	4	no	—
489	C₂²×C₄	—	2	32	no	—
490	C₂²×C₄	—	2	4	no	—
491	C₂²×C₄	—	2	4	no	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	—	2	4	no	—
494	C₂²×C₄	—	2	4	no	—
495	C₂²×C₄	—	2	4	no	—
496	C₂²×C₄	—	2	8	no	—
497	C₂²×C₄	—	2	8	no	—
498	C₂²×C₄	—	2	4	no	—
499	C₂²×C₄	—	2	8	no	—
500	C₂²×C₄	—	2	8	no	—
501	C₂²×C₄	—	4	4	no	—
502	C₂²×C₄	—	4	4	no	—
503	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
504	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
505	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
506	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
507	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
508	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
509	C₂²×C₄	#41×C₃₁	1	8	yes	—
510	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
511	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
512	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
513	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
514	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
515	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
516	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
517	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
518	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
519	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
520	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
521	C₄⋊C₄	—	2	4	no	—
522	C₄⋊C₄	—	2	4	no	—
523	C₄⋊C₄	—	2	4	no	—
524	C₄⋊C₄	—	2	4	no	—
525	C₄⋊C₄	—	2	2	no	—
526	C₄⋊C₄	—	2	2	no	—
527	C₄⋊C₄	—	2	4	no	—
528	C₄⋊C₄	—	2	4	no	—
529	C₄⋊C₄	—	2	16	no	—
530	C₄⋊C₄	—	4	8	no	—
531	C₂²⋊C₄	—	2	8	no	—
532	C₂²⋊C₄	—	2	8	no	—
533	C₂²⋊C₄	—	2	4	no	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₂²⋊C₄	—	2	4	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	2	8	no	—
540	C₂²⋊C₄	—	2	16	no	—
541	C₂²⋊C₄	—	4	8	no	—
542	C₂²⋊C₄	#41×C₃₁	1	8	yes	complemented
543	C₄²	—	2	8	no	—
544	C₄²	—	2	8	no	—
545	C₄²	—	2	4	no	—
546	C₄²	—	2	4	no	—
547	C₄²	—	2	8	no	—
548	C₄²	—	2	4	no	—
549	C₂³	—	2	2	no	—
550	C₂³	—	2	2	no	—
551	C₂³	—	2	2	no	—
552	C₂³	—	2	2	no	—
553	C₂³	—	2	8	no	—
554	C₂³	—	2	8	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	2	2	no	—
557	C₂³	—	2	2	no	—
558	C₂³	—	2	2	no	—
559	C₂³	—	2	4	no	—
560	C₂³	—	2	4	no	—
561	C₂³	—	2	4	no	—
562	C₂³	—	4	4	no	—
563	C₂³	—	4	16	no	—
564	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
565	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
566	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
567	C₂³	C₂×#41×C₃₁	1	4	yes	—
568	C₂³	C₂×#41×C₃₁	1	4	yes	—
569	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
570	Q₈	—	4	4	no	—
571	D₄	—	4	8	no	—
572	D₄	—	4	8	no	—
573	D₄	—	4	8	no	—
574	D₄	—	4	4	no	—
575	C₂×C₄	—	2	8	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	8	no	—
581	C₂×C₄	—	2	8	no	—
582	C₂×C₄	—	2	8	no	—
583	C₂×C₄	—	2	8	no	—
584	C₂×C₄	—	2	4	no	—
585	C₂×C₄	—	2	4	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	4	no	—
590	C₂×C₄	—	2	4	no	—
591	C₂×C₄	—	2	8	no	—
592	C₂×C₄	—	2	4	no	—
593	C₂×C₄	—	2	4	no	—
594	C₂×C₄	—	2	4	no	—
595	C₂×C₄	—	2	8	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	4	16	no	—
598	C₂×C₄	—	4	8	no	—
599	C₂×C₄	—	4	8	no	—
600	C₂×C₄	—	4	8	no	—
601	C₂×C₄	C₃₁×#123	1	16	yes	complemented
602	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
603	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
604	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
605	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
606	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
607	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
608	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
609	C₂²	—	2	2	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	2	2	no	—
612	C₂²	—	2	2	no	—
613	C₂²	—	2	8	no	—
614	C₂²	—	2	8	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	2	no	—
617	C₂²	—	2	2	no	—
618	C₂²	—	4	16	no	—
619	C₂²	—	4	4	no	—
620	C₂²	—	4	2	no	—
621	C₂²	—	4	2	no	—
622	C₂²	—	4	2	no	—
623	C₂²	—	4	8	no	—
624	C₂²	C₂×C₃₁×#123	1	4	yes	—
625	C₂²	C₃₁×#532	1	4	yes	—
626	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
627	C₂²	C₂²×#41×C₃₁	1	2	yes	—
628	C₂²	C₂²×#41×C₃₁	1	2	yes	—
629	C₂²	C₃₁×#585	1	2	yes	—
630	C₄	—	2	2	no	—
631	C₄	—	2	2	no	—
632	C₄	—	2	8	no	—
633	C₄	—	4	2	no	—
634	C₄	—	4	4	no	—
635	C₄	—	4	2	no	—
636	C₄	—	4	4	no	—
637	C₄	—	4	2	no	—
638	C₄	—	4	4	no	—
639	C₄	C₂×C₃₁×#123	1	4	yes	—
640	C₄	C₂×C₃₁×#123	1	4	yes	—
641	C₂	—	2	4	no	—
642	C₂	—	2	4	no	—
643	C₂	—	4	2	no	—
644	C₂	—	8	2	no	—
645	C₂	C₂²×C₃₁×#123	1	1	yes	characteristic
646	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
647	C₂	C₃₁×#3449	1	1	yes	characteristic
648	C₂	C₃₁×#3475	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₂×C₃₁×#525	C₂
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. C₂²×C₃₁×#117	C₂	semidirect
3. C₃₁×#2504	C₂	semidirect
4. C₃₁×#2503	C₂	semidirect
5. C₃₁×#2502	C₂	semidirect
6. C₃₁×#2501	C₂	semidirect
7. C₃₁×#2500	C₂	semidirect
8. C₃₁×#2499	C₂	semidirect
9. C₃₁×#2498	C₂	semidirect
10. C₃₁×#2412	C₂	semidirect
11. C₃₁×#2395	C₂	semidirect
12. C₄×C₃₁×#123	C₂	semidirect
13. C₃₁×#2297	C₂	semidirect
14. C₃₁×#2285	C₂	semidirect
15. C₂×C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
40. C₃₁×#400	C₂²	semidirect
41. C₃₁×#399	C₂²	semidirect
42. C₃₁×#398	C₂²	semidirect
44. C₃₁×#397	C₂²	semidirect
46. C₃₁×#391	C₂²	semidirect
47. C₃₁×#391	C₂²	semidirect
48. C₃₁×#390	C₂²	semidirect
49. C₃₁×#390	C₂²	semidirect
50. C₃₁×#388	C₂²	semidirect
51. C₃₁×#387	C₂²	semidirect
55. C₃₁×#384	C₂²	semidirect
56. C₃₁×#384	C₂²	semidirect
57. C₃₁×#382	C₂²	semidirect
58. C₃₁×#382	C₂²	semidirect
61. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
67. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
217. C₂²⋊C₄×C₃₁	#41	semidirect
224. #34810	C₃₁	direct (N × H)
277. C₂×C₄×C₃₁	#123	semidirect
286. C₂²×#117	C₂×C₃₁	semidirect
287. #2504	C₂×C₃₁	semidirect
288. #2503	C₂×C₃₁	semidirect
289. #2502	C₂×C₃₁	semidirect
290. #2501	C₂×C₃₁	semidirect
291. #2500	C₂×C₃₁	semidirect
292. #2499	C₂×C₃₁	semidirect
293. #2498	C₂×C₃₁	semidirect
294. #2412	C₂×C₃₁	semidirect
295. #2395	C₂×C₃₁	semidirect
296. C₄×#123	C₂×C₃₁	semidirect
297. #2297	C₂×C₃₁	semidirect
298. #2285	C₂×C₃₁	semidirect
299. C₂×C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
356. #400	C₂²×C₃₁	semidirect
357. #399	C₂²×C₃₁	semidirect
358. #398	C₂²×C₃₁	semidirect
360. #397	C₂²×C₃₁	semidirect
362. #391	C₂²×C₃₁	semidirect
363. #391	C₂²×C₃₁	semidirect
364. #390	C₂²×C₃₁	semidirect
365. #390	C₂²×C₃₁	semidirect
366. #388	C₂²×C₃₁	semidirect
367. #387	C₂²×C₃₁	semidirect
371. #384	C₂²×C₃₁	semidirect
372. #384	C₂²×C₃₁	semidirect
373. #382	C₂²×C₃₁	semidirect
374. #382	C₂²×C₃₁	semidirect
377. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
383. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
468. C₃₁	#34810	direct (N × H)
542. C₂²⋊C₄	#41×C₃₁	semidirect
601. C₂×C₄	C₃₁×#123	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#525	2
C₂²×C₃₁×#117	2
C₃₁×#2504	2
C₃₁×#2503	2
C₃₁×#2502	2
C₃₁×#2501	2
C₃₁×#2500	2
C₃₁×#2499	2
C₃₁×#2498	2
C₃₁×#2412	2
C₃₁×#2395	2
C₄×C₃₁×#123	2
C₃₁×#2297	2
C₃₁×#2285	2
C₂×C₄×C₂²⋊C₄×C₃₁	2
#34810	31

Subgroups — C31×#34810 #21491569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34810 #21491569