Subgroups — C₃₁×#22548 #21479307

Order 7936.

1054 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#556	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2355	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1983	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1983	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1981	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1960	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1959	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1958	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1805	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1804	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1763	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1762	C₂	1	1	yes	maximal, characteristic
15	C₄×C₃₁×#103	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#556	—	2	4	no	—
17	C₃₁×#556	C₄	1	8	yes	complemented
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#126	—	2	1	no	—
20	C₂×C₃₁×#126	—	2	1	no	—
21	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
24	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#117	—	2	1	no	—
29	C₃₁×#446	—	2	4	no	—
30	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
41	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
43	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
44	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
45	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
46	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
47	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
48	C₃₁×#391	—	2	1	no	—
49	C₃₁×#387	—	2	1	no	—
50	C₃₁×#387	—	2	1	no	—
51	C₃₁×#384	C₂²	1	1	yes	characteristic
52	C₃₁×#382	C₂²	1	1	yes	characteristic
53	C₃₁×#382	C₂²	1	1	yes	characteristic
54	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
56	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
57	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
58	C₃₁×#340	C₂²	1	1	yes	characteristic
59	C₃₁×#340	C₂²	1	1	yes	characteristic
60	C₃₁×#340	C₂²	1	1	yes	characteristic
61	C₃₁×#340	C₂²	1	1	yes	characteristic
62	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
63	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	—	2	1	no	—
67	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×C₄⋊Q₈	—	2	2	no	—
72	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
73	C₃₁×#126	—	2	8	no	—
74	C₃₁×#126	—	2	8	no	—
75	C₃₁×#124	—	2	2	no	—
76	C₃₁×#124	—	2	2	no	—
77	C₃₁×#124	C₂×C₄	1	4	yes	—
78	C₃₁×#124	C₂×C₄	1	4	yes	—
79	C₃₁×C₂²⋊Q₈	—	2	8	no	—
80	C₄×Q₈×C₃₁	—	2	2	no	—
81	C₄×Q₈×C₃₁	—	2	2	no	—
82	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
83	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
84	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
85	C₄×Q₈×C₃₁	D₄	1	4	yes	—
86	C₄×D₄×C₃₁	—	2	2	no	—
87	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
88	C₃₁×#117	—	2	8	no	—
89	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
90	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
91	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
92	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
93	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
94	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
95	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
105	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
106	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
108	C₂×C₄²×C₃₁	—	2	1	no	—
109	C₂×C₄²×C₃₁	—	2	1	no	—
110	C₂×C₄²×C₃₁	—	2	1	no	—
111	C₂×C₄²×C₃₁	—	2	1	no	—
112	C₂×C₄²×C₃₁	—	2	1	no	—
113	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
116	C₃₁×C₄⋊C₈	—	2	2	no	—
117	C₃₁×C₄⋊C₈	—	2	2	no	—
118	C₃₁×C₄⋊C₈	—	2	2	no	—
119	C₃₁×#103	—	2	2	no	—
120	C₃₁×#103	—	2	2	no	—
121	C₃₁×#103	—	2	2	no	—
122	C₃₁×#103	—	2	2	no	—
123	C₃₁×#103	—	4	1	no	—
124	C₃₁×#103	—	4	1	no	—
125	C₃₁×#103	—	4	1	no	—
126	C₃₁×#103	—	4	1	no	—
127	C₃₁×#102	—	2	2	no	—
128	C₃₁×#102	—	2	2	no	—
129	C₃₁×#102	—	4	1	no	—
130	C₃₁×#102	—	4	1	no	—
131	C₃₁×#97	—	2	2	no	—
132	C₃₁×#97	—	2	2	no	—
133	C₄×C₈×C₃₁	—	2	2	no	—
134	C₃₁×#95	—	2	1	no	—
135	C₃₁×#95	—	2	1	no	—
136	C₃₁×#95	—	2	1	no	—
137	C₃₁×#95	—	2	1	no	—
138	C₃₁×#95	—	2	1	no	—
139	C₃₁×#95	—	2	1	no	—
140	C₃₁×#95	—	2	1	no	—
141	C₂⁴×C₃₁	—	2	1	no	—
142	C₂×Q₈×C₃₁	—	2	1	no	—
143	C₂×Q₈×C₃₁	—	2	1	no	—
144	C₂×Q₈×C₃₁	—	2	1	no	—
145	C₂×Q₈×C₃₁	—	2	1	no	—
146	C₂×Q₈×C₃₁	—	2	1	no	—
147	C₂×Q₈×C₃₁	—	2	1	no	—
148	C₂×Q₈×C₃₁	—	2	1	no	—
149	C₂×Q₈×C₃₁	—	2	1	no	—
150	C₂×Q₈×C₃₁	—	2	1	no	—
151	C₂×Q₈×C₃₁	—	2	1	no	—
152	C₂×Q₈×C₃₁	—	4	1	no	—
153	C₂×Q₈×C₃₁	—	4	1	no	—
154	C₂×Q₈×C₃₁	C₂²⋊C₄	1	4	yes	—
155	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
156	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
157	C₂×D₄×C₃₁	—	2	1	no	—
158	C₂×D₄×C₃₁	—	2	1	no	—
159	C₂×D₄×C₃₁	—	2	1	no	—
160	C₂×D₄×C₃₁	—	2	1	no	—
161	C₂×D₄×C₃₁	—	2	1	no	—
162	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
163	C₂²×C₄×C₃₁	—	2	1	no	—
164	C₂²×C₄×C₃₁	—	2	1	no	—
165	C₂²×C₄×C₃₁	—	2	1	no	—
166	C₂²×C₄×C₃₁	—	2	1	no	—
167	C₂²×C₄×C₃₁	—	2	4	no	—
168	C₂²×C₄×C₃₁	—	2	1	no	—
169	C₂²×C₄×C₃₁	—	2	1	no	—
170	C₂²×C₄×C₃₁	—	2	1	no	—
171	C₂²×C₄×C₃₁	—	4	2	no	—
172	C₂²×C₄×C₃₁	—	4	2	no	—
173	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
180	C₂×C₈×C₃₁	—	2	1	no	—
181	C₂×C₈×C₃₁	—	2	1	no	—
182	C₂×C₈×C₃₁	—	2	1	no	—
183	C₂×C₈×C₃₁	—	2	1	no	—
184	C₂×C₈×C₃₁	—	2	1	no	—
185	C₂×C₈×C₃₁	—	2	1	no	—
186	C₂×C₈×C₃₁	—	2	1	no	—
187	C₂×C₈×C₃₁	—	2	1	no	—
188	C₂×C₈×C₃₁	—	2	1	no	—
189	C₂×C₈×C₃₁	—	2	1	no	—
190	C₂×C₈×C₃₁	—	2	1	no	—
191	C₂×C₈×C₃₁	#41	1	2	yes	—
192	C₄⋊C₄×C₃₁	—	2	4	no	—
193	C₄⋊C₄×C₃₁	—	2	4	no	—
194	C₄⋊C₄×C₃₁	—	2	4	no	—
195	C₄⋊C₄×C₃₁	—	2	4	no	—
196	C₄⋊C₄×C₃₁	—	2	1	no	—
197	C₄⋊C₄×C₃₁	—	2	1	no	—
198	C₄⋊C₄×C₃₁	—	2	1	no	—
199	C₄⋊C₄×C₃₁	—	2	1	no	—
200	C₄⋊C₄×C₃₁	—	2	1	no	—
201	C₄⋊C₄×C₃₁	—	2	1	no	—
202	C₄⋊C₄×C₃₁	—	2	1	no	—
203	C₄⋊C₄×C₃₁	—	2	1	no	—
204	C₄⋊C₄×C₃₁	—	4	2	no	—
205	C₄⋊C₄×C₃₁	—	8	1	no	—
206	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
207	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
208	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
209	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
210	C₄⋊C₄×C₃₁	#41	1	2	yes	—
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	4	no	—
215	C₂²⋊C₄×C₃₁	—	2	4	no	—
216	C₄²×C₃₁	—	2	4	no	—
217	C₄²×C₃₁	—	2	4	no	—
218	C₄²×C₃₁	—	4	2	no	—
219	C₄²×C₃₁	—	8	1	no	—
220	C₄²×C₃₁	—	8	1	no	—
221	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
222	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
223	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
224	C₄²×C₃₁	C₂×D₄	1	2	yes	—
225	C₄²×C₃₁	C₂×D₄	1	2	yes	—
226	C₄²×C₃₁	C₂×D₄	1	2	yes	—
227	#22548	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₂³×C₃₁	—	2	2	no	—
229	C₂³×C₃₁	—	4	1	no	—
230	C₂³×C₃₁	—	4	1	no	—
231	C₂³×C₃₁	—	4	1	no	—
232	C₂³×C₃₁	—	4	1	no	—
233	C₂³×C₃₁	—	4	1	no	—
234	C₂³×C₃₁	—	4	1	no	—
235	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
236	Q₈×C₃₁	—	2	2	no	—
237	Q₈×C₃₁	—	2	2	no	—
238	Q₈×C₃₁	—	2	2	no	—
239	Q₈×C₃₁	—	2	2	no	—
240	Q₈×C₃₁	—	4	4	no	—
241	Q₈×C₃₁	—	4	1	no	—
242	Q₈×C₃₁	—	4	1	no	—
243	Q₈×C₃₁	—	4	1	no	—
244	Q₈×C₃₁	—	4	1	no	—
245	D₄×C₃₁	—	2	2	no	—
246	D₄×C₃₁	—	2	2	no	—
247	D₄×C₃₁	—	4	1	no	—
248	D₄×C₃₁	—	4	1	no	—
249	C₂×C₄×C₃₁	—	2	1	no	—
250	C₂×C₄×C₃₁	—	2	1	no	—
251	C₂×C₄×C₃₁	—	2	1	no	—
252	C₂×C₄×C₃₁	—	2	1	no	—
253	C₂×C₄×C₃₁	—	2	1	no	—
254	C₂×C₄×C₃₁	—	2	1	no	—
255	C₂×C₄×C₃₁	—	2	1	no	—
256	C₂×C₄×C₃₁	—	2	1	no	—
257	C₂×C₄×C₃₁	—	2	1	no	—
258	C₂×C₄×C₃₁	—	2	1	no	—
259	C₂×C₄×C₃₁	—	2	1	no	—
260	C₂×C₄×C₃₁	—	2	1	no	—
261	C₂×C₄×C₃₁	—	2	2	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	—	4	8	no	—
266	C₂×C₄×C₃₁	—	4	1	no	—
267	C₂×C₄×C₃₁	—	4	1	no	—
268	C₂×C₄×C₃₁	—	4	1	no	—
269	C₂×C₄×C₃₁	—	4	1	no	—
270	C₂×C₄×C₃₁	—	4	1	no	—
271	C₂×C₄×C₃₁	—	4	1	no	—
272	C₂×C₄×C₃₁	—	4	1	no	—
273	C₂×C₄×C₃₁	—	4	1	no	—
274	C₂×C₄×C₃₁	—	4	1	no	—
275	C₂×C₄×C₃₁	—	4	1	no	—
276	C₂×C₄×C₃₁	—	4	1	no	—
277	C₂×C₄×C₃₁	—	4	1	no	—
278	C₂×C₄×C₃₁	—	4	1	no	—
279	C₂×C₄×C₃₁	—	4	1	no	—
280	C₂×C₄×C₃₁	—	4	1	no	—
281	C₂×C₄×C₃₁	—	4	1	no	—
282	C₂×C₄×C₃₁	—	4	1	no	—
283	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
284	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
285	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
286	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
287	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
288	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
289	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
290	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
291	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
292	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
293	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
294	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
295	C₈×C₃₁	—	2	2	no	—
296	C₈×C₃₁	—	4	1	no	—
297	C₈×C₃₁	—	4	1	no	—
298	C₈×C₃₁	—	4	1	no	—
299	C₂×#556	C₂×C₃₁	1	1	yes	characteristic
300	C₂×#446	C₂×C₃₁	1	1	yes	characteristic, complemented
301	#2355	C₂×C₃₁	1	1	yes	characteristic, complemented
302	#1983	C₂×C₃₁	1	1	yes	characteristic
303	#1983	C₂×C₃₁	1	1	yes	characteristic
304	#1981	C₂×C₃₁	1	1	yes	characteristic, complemented
305	#1960	C₂×C₃₁	1	1	yes	characteristic, complemented
306	#1959	C₂×C₃₁	1	1	yes	characteristic
307	#1958	C₂×C₃₁	1	1	yes	characteristic
308	#1805	C₂×C₃₁	1	1	yes	characteristic, complemented
309	#1804	C₂×C₃₁	1	1	yes	characteristic
310	#1803	C₂×C₃₁	1	1	yes	characteristic, complemented
311	#1763	C₂×C₃₁	1	1	yes	characteristic, complemented
312	#1762	C₂×C₃₁	1	1	yes	characteristic
313	C₄×#103	C₂×C₃₁	1	1	yes	characteristic, complemented
314	C₂²×C₃₁	—	4	2	no	—
315	C₂²×C₃₁	—	4	2	no	—
316	C₂²×C₃₁	—	4	2	no	—
317	C₂²×C₃₁	—	8	1	no	—
318	C₂²×C₃₁	—	8	1	no	—
319	C₂²×C₃₁	—	8	1	no	—
320	C₂²×C₃₁	—	8	1	no	—
321	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
322	C₂²×C₃₁	#548	1	1	yes	characteristic
323	C₂²×C₃₁	#550	1	1	yes	characteristic
324	C₂²×C₃₁	#575	1	1	yes	characteristic
325	C₂²×C₃₁	#575	1	1	yes	characteristic
326	C₂²×C₃₁	#577	1	1	yes	characteristic
327	C₂²×C₃₁	#578	1	1	yes	characteristic
328	C₄×C₃₁	—	2	2	no	—
329	C₄×C₃₁	—	2	2	no	—
330	C₄×C₃₁	—	2	2	no	—
331	C₄×C₃₁	—	2	2	no	—
332	C₄×C₃₁	—	2	2	no	—
333	C₄×C₃₁	—	2	2	no	—
334	C₄×C₃₁	—	4	2	no	—
335	C₄×C₃₁	—	4	2	no	—
336	C₄×C₃₁	—	4	2	no	—
337	C₄×C₃₁	—	4	2	no	—
338	C₄×C₃₁	—	8	1	no	—
339	C₄×C₃₁	—	8	1	no	—
340	C₄×C₃₁	—	8	1	no	—
341	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
342	C₄×C₃₁	#520	1	1	yes	characteristic
343	C₄×C₃₁	#548	1	1	yes	characteristic
344	C₄×C₃₁	#550	1	1	yes	characteristic
345	#556	—	2	4	no	—
346	#556	C₄×C₃₁	1	8	yes	complemented
347	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
348	C₂×#126	—	2	1	no	—
349	C₂×#126	—	2	1	no	—
350	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
351	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
352	C₂×C₂²⋊Q₈	—	2	1	no	—
353	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
354	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
355	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
356	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
357	C₂×#117	—	2	1	no	—
358	#446	—	2	4	no	—
359	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
360	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
361	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
362	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
363	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
364	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
365	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
366	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
367	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
368	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
369	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
370	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
371	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
372	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
373	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
374	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
375	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
376	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
377	#391	—	2	1	no	—
378	#387	—	2	1	no	—
379	#387	—	2	1	no	—
380	#384	C₂²×C₃₁	1	1	yes	characteristic
381	#382	C₂²×C₃₁	1	1	yes	characteristic
382	#382	C₂²×C₃₁	1	1	yes	characteristic
383	C₄×C₄⋊C₄	—	2	1	no	—
384	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
385	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
386	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
387	#340	C₂²×C₃₁	1	1	yes	characteristic
388	#340	C₂²×C₃₁	1	1	yes	characteristic
389	#340	C₂²×C₃₁	1	1	yes	characteristic
390	#340	C₂²×C₃₁	1	1	yes	characteristic
391	C₂×C₃₁	—	8	2	no	—
392	C₂×C₃₁	#2341	1	1	yes	characteristic
393	C₂×C₃₁	#2896	1	1	yes	characteristic
394	C₂×C₃₁	#2898	1	1	yes	characteristic
395	C₂×C₃₁	#3376	1	1	yes	characteristic
396	C₂×C₃₁	#3377	1	1	yes	characteristic
397	C₂×C₃₁	#3401	1	1	yes	characteristic
398	C₂×C₃₁	#3402	1	1	yes	characteristic
399	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
400	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
401	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
402	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
403	C₂³×C₄	—	2	1	no	—
404	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
405	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
406	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
407	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
408	C₄⋊Q₈	—	2	2	no	—
409	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
410	#126	—	2	8	no	—
411	#126	—	2	8	no	—
412	#124	—	2	2	no	—
413	#124	—	2	2	no	—
414	#124	C₂×C₄×C₃₁	1	4	yes	—
415	#124	C₂×C₄×C₃₁	1	4	yes	—
416	C₂²⋊Q₈	—	2	8	no	—
417	C₄×Q₈	—	2	2	no	—
418	C₄×Q₈	—	2	2	no	—
419	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
420	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
421	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
422	C₄×Q₈	D₄×C₃₁	1	4	yes	—
423	C₄×D₄	—	2	2	no	—
424	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
425	#117	—	2	8	no	—
426	C₂×C₄⋊C₄	—	2	1	no	—
427	C₂×C₄⋊C₄	—	2	1	no	—
428	C₂×C₄⋊C₄	—	2	1	no	—
429	C₂×C₄⋊C₄	—	2	1	no	—
430	C₂×C₄⋊C₄	—	2	1	no	—
431	C₂×C₄⋊C₄	—	2	1	no	—
432	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
433	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
434	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
435	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
436	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
437	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
438	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
439	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
440	C₂×C₂²⋊C₄	—	2	1	no	—
441	C₂×C₂²⋊C₄	—	2	1	no	—
442	C₂×C₂²⋊C₄	—	2	1	no	—
443	C₂×C₂²⋊C₄	—	2	1	no	—
444	C₂×C₂²⋊C₄	—	2	1	no	—
445	C₂×C₄²	—	2	1	no	—
446	C₂×C₄²	—	2	1	no	—
447	C₂×C₄²	—	2	1	no	—
448	C₂×C₄²	—	2	1	no	—
449	C₂×C₄²	—	2	1	no	—
450	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
451	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
452	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
453	C₄⋊C₈	—	2	2	no	—
454	C₄⋊C₈	—	2	2	no	—
455	C₄⋊C₈	—	2	2	no	—
456	#103	—	2	2	no	—
457	#103	—	2	2	no	—
458	#103	—	2	2	no	—
459	#103	—	2	2	no	—
460	#103	—	4	1	no	—
461	#103	—	4	1	no	—
462	#103	—	4	1	no	—
463	#103	—	4	1	no	—
464	#102	—	2	2	no	—
465	#102	—	2	2	no	—
466	#102	—	4	1	no	—
467	#102	—	4	1	no	—
468	#97	—	2	2	no	—
469	#97	—	2	2	no	—
470	C₄×C₈	—	2	2	no	—
471	#95	—	2	1	no	—
472	#95	—	2	1	no	—
473	#95	—	2	1	no	—
474	#95	—	2	1	no	—
475	#95	—	2	1	no	—
476	#95	—	2	1	no	—
477	#95	—	2	1	no	—
478	C₃₁	#22548	1	1	yes	characteristic, direct product factor, Sylow
479	C₂⁴	—	2	1	no	—
480	C₂×Q₈	—	2	1	no	—
481	C₂×Q₈	—	2	1	no	—
482	C₂×Q₈	—	2	1	no	—
483	C₂×Q₈	—	2	1	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	—	2	1	no	—
487	C₂×Q₈	—	2	1	no	—
488	C₂×Q₈	—	2	1	no	—
489	C₂×Q₈	—	2	1	no	—
490	C₂×Q₈	—	4	1	no	—
491	C₂×Q₈	—	4	1	no	—
492	C₂×Q₈	C₂²⋊C₄×C₃₁	1	4	yes	—
493	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
494	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
495	C₂×D₄	—	2	1	no	—
496	C₂×D₄	—	2	1	no	—
497	C₂×D₄	—	2	1	no	—
498	C₂×D₄	—	2	1	no	—
499	C₂×D₄	—	2	1	no	—
500	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
501	C₂²×C₄	—	2	1	no	—
502	C₂²×C₄	—	2	1	no	—
503	C₂²×C₄	—	2	1	no	—
504	C₂²×C₄	—	2	1	no	—
505	C₂²×C₄	—	2	4	no	—
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	—	4	2	no	—
510	C₂²×C₄	—	4	2	no	—
511	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
512	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
513	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
514	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
515	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
516	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
517	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
518	C₂×C₈	—	2	1	no	—
519	C₂×C₈	—	2	1	no	—
520	C₂×C₈	—	2	1	no	—
521	C₂×C₈	—	2	1	no	—
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	2	1	no	—
524	C₂×C₈	—	2	1	no	—
525	C₂×C₈	—	2	1	no	—
526	C₂×C₈	—	2	1	no	—
527	C₂×C₈	—	2	1	no	—
528	C₂×C₈	—	2	1	no	—
529	C₂×C₈	#41×C₃₁	1	2	yes	—
530	C₄⋊C₄	—	2	4	no	—
531	C₄⋊C₄	—	2	4	no	—
532	C₄⋊C₄	—	2	4	no	—
533	C₄⋊C₄	—	2	4	no	—
534	C₄⋊C₄	—	2	1	no	—
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	2	1	no	—
537	C₄⋊C₄	—	2	1	no	—
538	C₄⋊C₄	—	2	1	no	—
539	C₄⋊C₄	—	2	1	no	—
540	C₄⋊C₄	—	2	1	no	—
541	C₄⋊C₄	—	2	1	no	—
542	C₄⋊C₄	—	4	2	no	—
543	C₄⋊C₄	—	8	1	no	—
544	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
545	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
546	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
547	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
548	C₄⋊C₄	#41×C₃₁	1	2	yes	—
549	C₂²⋊C₄	—	2	4	no	—
550	C₂²⋊C₄	—	2	4	no	—
551	C₂²⋊C₄	—	2	4	no	—
552	C₂²⋊C₄	—	2	4	no	—
553	C₂²⋊C₄	—	2	4	no	—
554	C₄²	—	2	4	no	—
555	C₄²	—	2	4	no	—
556	C₄²	—	4	2	no	—
557	C₄²	—	8	1	no	—
558	C₄²	—	8	1	no	—
559	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
560	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
561	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
562	C₄²	C₂×D₄×C₃₁	1	2	yes	—
563	C₄²	C₂×D₄×C₃₁	1	2	yes	—
564	C₄²	C₂×D₄×C₃₁	1	2	yes	—
565	C₂³	—	2	2	no	—
566	C₂³	—	4	1	no	—
567	C₂³	—	4	1	no	—
568	C₂³	—	4	1	no	—
569	C₂³	—	4	1	no	—
570	C₂³	—	4	1	no	—
571	C₂³	—	4	1	no	—
572	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
573	Q₈	—	2	2	no	—
574	Q₈	—	2	2	no	—
575	Q₈	—	2	2	no	—
576	Q₈	—	2	2	no	—
577	Q₈	—	4	4	no	—
578	Q₈	—	4	1	no	—
579	Q₈	—	4	1	no	—
580	Q₈	—	4	1	no	—
581	Q₈	—	4	1	no	—
582	D₄	—	2	2	no	—
583	D₄	—	2	2	no	—
584	D₄	—	4	1	no	—
585	D₄	—	4	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	4	8	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
621	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
622	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
623	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
624	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
625	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
626	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
627	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
628	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
629	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
630	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
631	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
632	C₈	—	2	2	no	—
633	C₈	—	4	1	no	—
634	C₈	—	4	1	no	—
635	C₈	—	4	1	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	2	no	—
638	C₂²	—	4	2	no	—
639	C₂²	—	8	1	no	—
640	C₂²	—	8	1	no	—
641	C₂²	—	8	1	no	—
642	C₂²	—	8	1	no	—
643	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
644	C₂²	C₃₁×#548	1	1	yes	characteristic
645	C₂²	C₃₁×#550	1	1	yes	characteristic
646	C₂²	C₃₁×#575	1	1	yes	characteristic
647	C₂²	C₃₁×#575	1	1	yes	characteristic
648	C₂²	C₃₁×#577	1	1	yes	characteristic
649	C₂²	C₃₁×#578	1	1	yes	characteristic
650	C₄	—	2	2	no	—
651	C₄	—	2	2	no	—
652	C₄	—	2	2	no	—
653	C₄	—	2	2	no	—
654	C₄	—	2	2	no	—
655	C₄	—	2	2	no	—
656	C₄	—	4	2	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	2	no	—
659	C₄	—	4	2	no	—
660	C₄	—	8	1	no	—
661	C₄	—	8	1	no	—
662	C₄	—	8	1	no	—
663	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
664	C₄	C₃₁×#520	1	1	yes	characteristic
665	C₄	C₃₁×#548	1	1	yes	characteristic
666	C₄	C₃₁×#550	1	1	yes	characteristic
667	C₂	—	8	2	no	—
668	C₂	C₃₁×#2341	1	1	yes	characteristic
669	C₂	C₃₁×#2896	1	1	yes	characteristic
670	C₂	C₃₁×#2898	1	1	yes	characteristic
671	C₂	C₃₁×#3376	1	1	yes	characteristic
672	C₂	C₃₁×#3377	1	1	yes	characteristic
673	C₂	C₃₁×#3401	1	1	yes	characteristic
674	C₂	C₃₁×#3402	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#446	C₂	semidirect
3. C₃₁×#2355	C₂	semidirect
6. C₃₁×#1981	C₂	semidirect
7. C₃₁×#1960	C₂	semidirect
10. C₃₁×#1805	C₂	semidirect
12. C₃₁×#1803	C₂	semidirect
13. C₃₁×#1763	C₂	semidirect
15. C₄×C₃₁×#103	C₂	semidirect
17. C₃₁×#556	C₄	semidirect
227. #22548	C₃₁	direct (N × H)
300. C₂×#446	C₂×C₃₁	semidirect
301. #2355	C₂×C₃₁	semidirect
304. #1981	C₂×C₃₁	semidirect
305. #1960	C₂×C₃₁	semidirect
308. #1805	C₂×C₃₁	semidirect
310. #1803	C₂×C₃₁	semidirect
311. #1763	C₂×C₃₁	semidirect
313. C₄×#103	C₂×C₃₁	semidirect
346. #556	C₄×C₃₁	semidirect
478. C₃₁	#22548	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×#556	2
C₂×C₃₁×#446	2
C₃₁×#2355	2
C₃₁×#1983	2
C₃₁×#1981	2
C₃₁×#1960	2
C₃₁×#1959	2
C₃₁×#1958	2
C₃₁×#1805	2
C₃₁×#1804	2
C₃₁×#1803	2
C₃₁×#1763	2
C₃₁×#1762	2
C₄×C₃₁×#103	2
#22548	31

Subgroups — C31×#22548 #21479307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22548 #21479307