Subgroups — C₃₁×#34276 #21491035

Order 7936.

2930 conjugacy classes of subgroups.

The subgroup lattice has 608 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	C₂×C₃₁×#519	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2362	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2360	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2359	C₂	1	8	yes	maximal, complemented
7	C₃₁×#2327	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2326	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2317	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₃₁×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#124	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2297	C₂	1	1	yes	maximal, characteristic
13	C₂×C₃₁×#391	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#536	C₄	1	16	yes	complemented
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₃₁×#124	C₂²	1	2	yes	—
20	C₂×C₃₁×#123	C₂²	1	4	yes	—
21	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
22	C₃₁×#520	—	2	8	no	—
23	C₃₁×#519	—	2	8	no	—
24	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
31	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#117	C₂²	1	4	yes	—
33	C₂×C₃₁×#117	C₂²	1	2	yes	—
34	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
36	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
38	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#391	C₂²	1	1	yes	characteristic
40	C₃₁×#391	C₂²	1	2	yes	—
41	C₃₁×#391	C₂²	1	2	yes	—
42	C₃₁×#391	C₂²	1	1	yes	characteristic
43	C₃₁×#391	C₂²	1	4	yes	—
44	C₃₁×#390	C₂²	1	1	yes	characteristic
45	C₃₁×#390	C₂²	1	1	yes	characteristic
46	C₃₁×#389	C₂²	1	8	yes	—
47	C₃₁×#389	C₂²	1	4	yes	—
48	C₃₁×#388	C₂²	1	8	yes	—
49	C₃₁×#388	C₂²	1	8	yes	—
50	C₃₁×#387	C₂²	1	8	yes	—
51	C₃₁×#387	C₂²	1	8	yes	—
52	C₃₁×#386	C₂²	1	4	yes	—
53	C₃₁×#385	C₂²	1	4	yes	—
54	C₃₁×#385	C₂²	1	8	yes	—
55	C₃₁×#384	C₂²	1	2	yes	—
56	C₃₁×#382	C₂²	1	2	yes	—
57	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	—
58	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
59	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
60	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
61	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
62	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
63	C₃₁×#376	C₂²	1	2	yes	—
64	C₂×C₃₁×#95	C₂²	1	4	yes	—
65	C₄³×C₃₁	C₂²	1	2	yes	—
66	C₂×#41×C₃₁	—	2	4	no	—
67	C₂×#41×C₃₁	—	2	4	no	—
68	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
69	C₂²×Q₈×C₃₁	—	2	4	no	—
70	C₂²×Q₈×C₃₁	—	2	4	no	—
71	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
72	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
73	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
74	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
76	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
78	C₂³×C₄×C₃₁	C₂³	1	2	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	C₃₁×C₄⋊Q₈	C₂×C₄	1	16	yes	complemented
84	C₃₁×#124	C₂×C₄	1	16	yes	complemented
85	C₃₁×#123	C₂×C₄	1	32	yes	complemented
86	C₃₁×C₂²⋊Q₈	C₂×C₄	1	32	yes	complemented
87	C₄×Q₈×C₃₁	—	2	8	no	—
88	C₄×Q₈×C₃₁	—	2	8	no	—
89	C₄×Q₈×C₃₁	—	2	8	no	—
90	C₄×Q₈×C₃₁	—	2	8	no	—
91	C₄×Q₈×C₃₁	—	2	8	no	—
92	C₄×D₄×C₃₁	—	2	16	no	—
93	C₄×D₄×C₃₁	—	2	8	no	—
94	C₃₁×#117	—	2	4	no	—
95	C₃₁×#117	—	2	8	no	—
96	C₃₁×#117	—	2	16	no	—
97	C₃₁×#117	C₂×C₄	1	8	yes	complemented
98	C₃₁×#117	D₄	1	8	yes	—
99	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
100	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
113	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
117	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
119	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₂×C₄²×C₃₁	—	2	4	no	—
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
124	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
125	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
126	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
127	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
128	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
129	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
130	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
131	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
132	C₃₁×#95	C₂³	1	8	yes	—
133	C₃₁×#95	C₂³	1	8	yes	—
134	C₃₁×#95	C₂³	1	4	yes	—
135	C₃₁×#95	C₂³	1	4	yes	—
136	C₃₁×#95	C₂³	1	4	yes	—
137	C₃₁×#95	C₂³	1	8	yes	—
138	C₃₁×#95	C₂³	1	4	yes	—
139	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
140	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
141	#41×C₃₁	—	2	32	no	—
142	C₂×Q₈×C₃₁	—	2	32	no	—
143	C₂×Q₈×C₃₁	—	2	4	no	—
144	C₂×Q₈×C₃₁	—	2	4	no	—
145	C₂×Q₈×C₃₁	—	2	2	no	—
146	C₂×Q₈×C₃₁	—	2	4	no	—
147	C₂×Q₈×C₃₁	—	2	4	no	—
148	C₂×Q₈×C₃₁	—	2	2	no	—
149	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
150	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
151	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
152	C₂×D₄×C₃₁	—	2	4	no	—
153	C₂×D₄×C₃₁	—	2	4	no	—
154	C₂×D₄×C₃₁	—	2	2	no	—
155	C₂×D₄×C₃₁	—	2	2	no	—
156	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
157	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
158	C₂²×C₄×C₃₁	—	2	4	no	—
159	C₂²×C₄×C₃₁	—	2	4	no	—
160	C₂²×C₄×C₃₁	—	2	4	no	—
161	C₂²×C₄×C₃₁	—	2	4	no	—
162	C₂²×C₄×C₃₁	—	2	4	no	—
163	C₂²×C₄×C₃₁	—	2	4	no	—
164	C₂²×C₄×C₃₁	—	2	4	no	—
165	C₂²×C₄×C₃₁	—	2	8	no	—
166	C₂²×C₄×C₃₁	—	4	2	no	—
167	C₂²×C₄×C₃₁	—	4	8	no	—
168	C₂²×C₄×C₃₁	—	4	4	no	—
169	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
170	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
171	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
172	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
173	C₂²×C₄×C₃₁	#41	1	4	yes	—
174	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
179	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
186	C₄⋊C₄×C₃₁	—	2	4	no	—
187	C₄⋊C₄×C₃₁	—	2	8	no	—
188	C₄⋊C₄×C₃₁	—	2	4	no	—
189	C₄⋊C₄×C₃₁	—	2	8	no	—
190	C₄⋊C₄×C₃₁	—	2	8	no	—
191	C₄⋊C₄×C₃₁	—	2	2	no	—
192	C₄⋊C₄×C₃₁	—	2	2	no	—
193	C₄⋊C₄×C₃₁	—	4	8	no	—
194	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
195	C₄⋊C₄×C₃₁	C₂²×C₄	1	32	yes	—
196	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
197	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
198	C₂²⋊C₄×C₃₁	—	2	4	no	—
199	C₂²⋊C₄×C₃₁	—	2	8	no	—
200	C₂²⋊C₄×C₃₁	—	2	8	no	—
201	C₂²⋊C₄×C₃₁	—	4	4	no	—
202	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
203	C₂²⋊C₄×C₃₁	C₂²×C₄	1	32	yes	—
204	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
205	C₄²×C₃₁	—	2	8	no	—
206	C₄²×C₃₁	—	2	8	no	—
207	C₄²×C₃₁	—	2	8	no	—
208	C₄²×C₃₁	—	2	4	no	—
209	C₄²×C₃₁	—	2	2	no	—
210	C₄²×C₃₁	—	2	4	no	—
211	C₄²×C₃₁	—	2	2	no	—
212	C₄²×C₃₁	—	2	8	no	—
213	C₄²×C₃₁	—	4	4	no	—
214	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
215	C₄²×C₃₁	C₂×D₄	1	8	yes	—
216	#34276	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂³×C₃₁	—	2	2	no	—
218	C₂³×C₃₁	—	2	2	no	—
219	C₂³×C₃₁	—	2	2	no	—
220	C₂³×C₃₁	—	2	2	no	—
221	C₂³×C₃₁	—	2	4	no	—
222	C₂³×C₃₁	—	2	4	no	—
223	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
224	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
225	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
226	C₂³×C₃₁	C₂×#41	1	2	yes	—
227	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
228	Q₈×C₃₁	—	2	16	no	—
229	Q₈×C₃₁	—	2	16	no	—
230	Q₈×C₃₁	—	2	8	no	—
231	D₄×C₃₁	—	2	16	no	—
232	D₄×C₃₁	—	2	8	no	—
233	C₂×C₄×C₃₁	—	2	16	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	4	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	2	4	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	2	4	no	—
244	C₂×C₄×C₃₁	—	2	16	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	2	no	—
247	C₂×C₄×C₃₁	—	2	4	no	—
248	C₂×C₄×C₃₁	—	2	2	no	—
249	C₂×C₄×C₃₁	—	2	4	no	—
250	C₂×C₄×C₃₁	—	2	2	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	2	4	no	—
253	C₂×C₄×C₃₁	—	2	8	no	—
254	C₂×C₄×C₃₁	—	4	8	no	—
255	C₂×C₄×C₃₁	—	8	8	no	—
256	C₂×C₄×C₃₁	—	8	4	no	—
257	C₂×C₄×C₃₁	C₄×D₄	1	16	yes	—
258	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
259	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
260	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
261	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
262	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
263	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
264	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
265	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
266	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
267	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
268	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
269	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
270	C₂×#536	C₂×C₃₁	1	1	yes	characteristic
271	C₂×#520	C₂×C₃₁	1	1	yes	characteristic
272	C₂×#519	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#2362	C₂×C₃₁	1	4	yes	complemented
274	#2360	C₂×C₃₁	1	4	yes	complemented
275	#2359	C₂×C₃₁	1	8	yes	complemented
276	#2327	C₂×C₃₁	1	2	yes	complemented
277	#2326	C₂×C₃₁	1	2	yes	complemented
278	#2317	C₂×C₃₁	1	1	yes	characteristic, complemented
279	C₄×C₄⋊Q₈	C₂×C₃₁	1	2	yes	complemented
280	C₄×#124	C₂×C₃₁	1	2	yes	complemented
281	#2297	C₂×C₃₁	1	1	yes	characteristic
282	C₂×#391	C₂×C₃₁	1	1	yes	characteristic, complemented
283	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
284	C₂²×C₃₁	—	2	4	no	—
285	C₂²×C₃₁	—	2	4	no	—
286	C₂²×C₃₁	—	2	2	no	—
287	C₂²×C₃₁	—	2	2	no	—
288	C₂²×C₃₁	—	2	4	no	—
289	C₂²×C₃₁	—	2	8	no	—
290	C₂²×C₃₁	—	4	4	no	—
291	C₂²×C₃₁	—	4	2	no	—
292	C₂²×C₃₁	—	4	2	no	—
293	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
294	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
295	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
296	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
297	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
298	C₂²×C₃₁	#585	1	2	yes	—
299	C₄×C₃₁	—	2	4	no	—
300	C₄×C₃₁	—	2	8	no	—
301	C₄×C₃₁	—	2	8	no	—
302	C₄×C₃₁	—	2	4	no	—
303	C₄×C₃₁	—	2	4	no	—
304	C₄×C₃₁	—	2	4	no	—
305	C₄×C₃₁	—	4	2	no	—
306	C₄×C₃₁	—	4	2	no	—
307	C₄×C₃₁	—	4	2	no	—
308	C₄×C₃₁	—	4	4	no	—
309	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
310	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
311	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
312	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
313	#536	C₄×C₃₁	1	16	yes	complemented
314	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
315	C₂×#124	C₂²×C₃₁	1	2	yes	—
316	C₂×#123	C₂²×C₃₁	1	4	yes	—
317	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
318	#520	—	2	8	no	—
319	#519	—	2	8	no	—
320	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
321	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
322	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
323	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
324	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
325	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
326	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
327	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
328	C₂×#117	C₂²×C₃₁	1	4	yes	—
329	C₂×#117	C₂²×C₃₁	1	2	yes	—
330	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
332	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
333	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
334	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
335	#391	C₂²×C₃₁	1	1	yes	characteristic
336	#391	C₂²×C₃₁	1	2	yes	—
337	#391	C₂²×C₃₁	1	2	yes	—
338	#391	C₂²×C₃₁	1	1	yes	characteristic
339	#391	C₂²×C₃₁	1	4	yes	—
340	#390	C₂²×C₃₁	1	1	yes	characteristic
341	#390	C₂²×C₃₁	1	1	yes	characteristic
342	#389	C₂²×C₃₁	1	8	yes	—
343	#389	C₂²×C₃₁	1	4	yes	—
344	#388	C₂²×C₃₁	1	8	yes	—
345	#388	C₂²×C₃₁	1	8	yes	—
346	#387	C₂²×C₃₁	1	8	yes	—
347	#387	C₂²×C₃₁	1	8	yes	—
348	#386	C₂²×C₃₁	1	4	yes	—
349	#385	C₂²×C₃₁	1	4	yes	—
350	#385	C₂²×C₃₁	1	8	yes	—
351	#384	C₂²×C₃₁	1	2	yes	—
352	#382	C₂²×C₃₁	1	2	yes	—
353	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	—
354	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
355	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
356	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
357	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
358	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
359	#376	C₂²×C₃₁	1	2	yes	—
360	C₂×#95	C₂²×C₃₁	1	4	yes	—
361	C₄³	C₂²×C₃₁	1	2	yes	—
362	C₂×C₃₁	—	2	4	no	—
363	C₂×C₃₁	—	4	4	no	—
364	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
365	C₂×C₃₁	C₄×2₋¹⁺⁴	1	2	yes	—
366	C₂×C₃₁	#3495	1	2	yes	—
367	C₂×C₃₁	#3499	1	1	yes	characteristic
368	C₂×C₃₁	#3500	1	1	yes	characteristic
369	C₂×#41	—	2	4	no	—
370	C₂×#41	—	2	4	no	—
371	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
372	C₂²×Q₈	—	2	4	no	—
373	C₂²×Q₈	—	2	4	no	—
374	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
375	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
376	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
377	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
378	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
379	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
380	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
381	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
382	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
383	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
384	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
385	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
386	C₄⋊Q₈	C₂×C₄×C₃₁	1	16	yes	complemented
387	#124	C₂×C₄×C₃₁	1	16	yes	complemented
388	#123	C₂×C₄×C₃₁	1	32	yes	complemented
389	C₂²⋊Q₈	C₂×C₄×C₃₁	1	32	yes	complemented
390	C₄×Q₈	—	2	8	no	—
391	C₄×Q₈	—	2	8	no	—
392	C₄×Q₈	—	2	8	no	—
393	C₄×Q₈	—	2	8	no	—
394	C₄×Q₈	—	2	8	no	—
395	C₄×D₄	—	2	16	no	—
396	C₄×D₄	—	2	8	no	—
397	#117	—	2	4	no	—
398	#117	—	2	8	no	—
399	#117	—	2	16	no	—
400	#117	C₂×C₄×C₃₁	1	8	yes	complemented
401	#117	D₄×C₃₁	1	8	yes	—
402	C₂×C₄⋊C₄	—	2	4	no	—
403	C₂×C₄⋊C₄	—	2	8	no	—
404	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
405	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
407	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
410	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
411	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
412	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
413	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
414	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
415	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
416	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
423	C₂×C₄²	—	2	4	no	—
424	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
425	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
426	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
427	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
428	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
429	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
430	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
432	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
433	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
434	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
435	#95	C₂³×C₃₁	1	8	yes	—
436	#95	C₂³×C₃₁	1	8	yes	—
437	#95	C₂³×C₃₁	1	4	yes	—
438	#95	C₂³×C₃₁	1	4	yes	—
439	#95	C₂³×C₃₁	1	4	yes	—
440	#95	C₂³×C₃₁	1	8	yes	—
441	#95	C₂³×C₃₁	1	4	yes	—
442	C₃₁	#34276	1	1	yes	characteristic, direct product factor, Sylow
443	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
444	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
445	#41	—	2	32	no	—
446	C₂×Q₈	—	2	32	no	—
447	C₂×Q₈	—	2	4	no	—
448	C₂×Q₈	—	2	4	no	—
449	C₂×Q₈	—	2	2	no	—
450	C₂×Q₈	—	2	4	no	—
451	C₂×Q₈	—	2	4	no	—
452	C₂×Q₈	—	2	2	no	—
453	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
454	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
455	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
461	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	2	8	no	—
470	C₂²×C₄	—	4	2	no	—
471	C₂²×C₄	—	4	8	no	—
472	C₂²×C₄	—	4	4	no	—
473	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
474	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
475	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
476	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
477	C₂²×C₄	#41×C₃₁	1	4	yes	—
478	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
479	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
480	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
481	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
482	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
483	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
484	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
485	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
486	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
487	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
488	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
489	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
490	C₄⋊C₄	—	2	4	no	—
491	C₄⋊C₄	—	2	8	no	—
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	8	no	—
494	C₄⋊C₄	—	2	8	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	4	8	no	—
498	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
499	C₄⋊C₄	C₂²×C₄×C₃₁	1	32	yes	—
500	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
501	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	2	8	no	—
504	C₂²⋊C₄	—	2	8	no	—
505	C₂²⋊C₄	—	4	4	no	—
506	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
507	C₂²⋊C₄	C₂²×C₄×C₃₁	1	32	yes	—
508	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
509	C₄²	—	2	8	no	—
510	C₄²	—	2	8	no	—
511	C₄²	—	2	8	no	—
512	C₄²	—	2	4	no	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	4	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	8	no	—
517	C₄²	—	4	4	no	—
518	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
519	C₄²	C₂×D₄×C₃₁	1	8	yes	—
520	C₂³	—	2	2	no	—
521	C₂³	—	2	2	no	—
522	C₂³	—	2	2	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	2	4	no	—
526	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
527	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
528	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
529	C₂³	C₂×#41×C₃₁	1	2	yes	—
530	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
531	Q₈	—	2	16	no	—
532	Q₈	—	2	16	no	—
533	Q₈	—	2	8	no	—
534	D₄	—	2	16	no	—
535	D₄	—	2	8	no	—
536	C₂×C₄	—	2	16	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	16	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	8	no	—
557	C₂×C₄	—	4	8	no	—
558	C₂×C₄	—	8	8	no	—
559	C₂×C₄	—	8	4	no	—
560	C₂×C₄	C₄×D₄×C₃₁	1	16	yes	—
561	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
562	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
563	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
564	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
565	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
566	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
567	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
568	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
569	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
570	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
571	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
572	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
573	C₂²	—	2	4	no	—
574	C₂²	—	2	4	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	2	4	no	—
578	C₂²	—	2	8	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	2	no	—
582	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
583	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
584	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
585	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
586	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
587	C₂²	C₃₁×#585	1	2	yes	—
588	C₄	—	2	4	no	—
589	C₄	—	2	8	no	—
590	C₄	—	2	8	no	—
591	C₄	—	2	4	no	—
592	C₄	—	2	4	no	—
593	C₄	—	2	4	no	—
594	C₄	—	4	2	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	4	no	—
597	C₄	—	4	2	no	—
598	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
599	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
600	C₂	—	2	4	no	—
601	C₂	—	4	4	no	—
602	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
603	C₂	C₄×C₃₁×2₋¹⁺⁴	1	2	yes	—
604	C₂	C₃₁×#3495	1	2	yes	—
605	C₂	C₃₁×#3499	1	1	yes	characteristic
606	C₂	C₃₁×#3500	1	1	yes	characteristic

	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. C₂×C₃₁×#519	C₂	semidirect
4. C₃₁×#2362	C₂	semidirect
5. C₃₁×#2360	C₂	semidirect
6. C₃₁×#2359	C₂	semidirect
7. C₃₁×#2327	C₂	semidirect
8. C₃₁×#2326	C₂	semidirect
9. C₃₁×#2317	C₂	semidirect
10. C₄×C₃₁×C₄⋊Q₈	C₂	semidirect
11. C₄×C₃₁×#124	C₂	semidirect
13. C₂×C₃₁×#391	C₂	semidirect
14. C₄×C₃₁×#117	C₂	semidirect
17. C₃₁×#536	C₄	semidirect
71. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
83. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
84. C₃₁×#124	C₂×C₄	semidirect
85. C₃₁×#123	C₂×C₄	semidirect
86. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
97. C₃₁×#117	C₂×C₄	semidirect
216. #34276	C₃₁	direct (N × H)
272. C₂×#519	C₂×C₃₁	semidirect
273. #2362	C₂×C₃₁	semidirect
274. #2360	C₂×C₃₁	semidirect
275. #2359	C₂×C₃₁	semidirect
276. #2327	C₂×C₃₁	semidirect
277. #2326	C₂×C₃₁	semidirect
278. #2317	C₂×C₃₁	semidirect
279. C₄×C₄⋊Q₈	C₂×C₃₁	semidirect
280. C₄×#124	C₂×C₃₁	semidirect
282. C₂×#391	C₂×C₃₁	semidirect
283. C₄×#117	C₂×C₃₁	semidirect
313. #536	C₄×C₃₁	semidirect
374. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
386. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
387. #124	C₂×C₄×C₃₁	semidirect
388. #123	C₂×C₄×C₃₁	semidirect
389. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
400. #117	C₂×C₄×C₃₁	semidirect
442. C₃₁	#34276	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
C₂×C₃₁×#519	2
C₃₁×#2362	2
C₃₁×#2360	2
C₃₁×#2359	2
C₃₁×#2327	2
C₃₁×#2326	2
C₃₁×#2317	2
C₄×C₃₁×C₄⋊Q₈	2
C₄×C₃₁×#124	2
C₃₁×#2297	2
C₂×C₃₁×#391	2
C₄×C₃₁×#117	2
#34276	31

Subgroups — C31×#34276 #21491035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34276 #21491035