Subgroups — C₃₁×#22509 #21479268

Order 7936.

1064 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1985	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1984	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1798	C₂	1	2	yes	maximal
9	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1750	C₂	1	2	yes	maximal
11	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#533	—	2	4	no	—
13	C₃₁×#533	—	2	4	no	—
14	C₂×C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic
16	C₂×C₃₁×#125	C₄	1	2	yes	—
17	C₂×C₃₁×C₂²⋊Q₈	C₄	1	1	yes	characteristic
18	C₂×C₃₁×C₂²⋊Q₈	C₄	1	1	yes	characteristic
19	C₂×C₃₁×C₂²⋊Q₈	C₄	1	2	yes	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#117	C₂²	1	2	yes	—
24	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#446	—	2	4	no	—
26	C₈×Q₈×C₃₁	—	2	4	no	—
27	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
30	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
31	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#97	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₃₁×#336	C₂²	1	1	yes	characteristic
45	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
47	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×C₄⋊Q₈	—	2	4	no	—
58	C₃₁×C₄⋊Q₈	—	2	4	no	—
59	C₃₁×#125	—	2	8	no	—
60	C₃₁×C₂²⋊Q₈	—	2	4	no	—
61	C₃₁×C₂²⋊Q₈	—	2	4	no	—
62	C₃₁×C₂²⋊Q₈	—	2	8	no	—
63	C₄×Q₈×C₃₁	—	2	4	no	—
64	C₄×Q₈×C₃₁	—	2	4	no	—
65	C₄×Q₈×C₃₁	—	2	2	no	—
66	C₄×Q₈×C₃₁	—	2	2	no	—
67	C₄×Q₈×C₃₁	D₄	1	4	yes	—
68	C₄×Q₈×C₃₁	D₄	1	4	yes	—
69	C₃₁×#117	—	2	4	no	—
70	C₃₁×#117	—	2	4	no	—
71	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
75	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
78	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
79	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
81	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
82	C₂×C₄²×C₃₁	—	2	2	no	—
83	C₂×C₄²×C₃₁	—	2	2	no	—
84	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
88	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
90	C₃₁×C₄⋊C₈	—	2	2	no	—
91	C₃₁×C₄⋊C₈	—	2	4	no	—
92	C₃₁×C₄⋊C₈	—	2	2	no	—
93	C₃₁×C₄⋊C₈	—	2	4	no	—
94	C₃₁×C₂²⋊C₈	—	2	4	no	—
95	C₃₁×C₂²⋊C₈	—	2	2	no	—
96	C₃₁×C₂²⋊C₈	Q₈	1	4	yes	complemented
97	C₃₁×#97	—	2	4	no	—
98	C₄×C₈×C₃₁	—	2	2	no	—
99	C₄×C₈×C₃₁	—	2	4	no	—
100	C₄×C₈×C₃₁	—	2	2	no	—
101	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
102	C₂×Q₈×C₃₁	—	2	4	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	4	no	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	4	2	no	—
110	C₂×Q₈×C₃₁	C₂²⋊C₄	1	4	yes	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	1	no	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	4	1	no	—
121	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
126	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
127	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
131	C₂²×C₄×C₃₁	#41	1	2	yes	—
132	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	2	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
150	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
151	C₄⋊C₄×C₃₁	—	2	4	no	—
152	C₄⋊C₄×C₃₁	—	2	4	no	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	4	no	—
155	C₄⋊C₄×C₃₁	—	2	4	no	—
156	C₄⋊C₄×C₃₁	—	2	4	no	—
157	C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
159	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
160	C₂²⋊C₄×C₃₁	—	2	4	no	—
161	C₂²⋊C₄×C₃₁	—	2	4	no	—
162	C₄²×C₃₁	—	2	2	no	—
163	C₄²×C₃₁	—	2	2	no	—
164	C₄²×C₃₁	—	2	1	no	—
165	C₄²×C₃₁	—	2	1	no	—
166	C₄²×C₃₁	—	2	2	no	—
167	C₄²×C₃₁	—	2	2	no	—
168	C₄²×C₃₁	C₂×D₄	1	2	yes	—
169	C₄²×C₃₁	C₂×D₄	1	2	yes	—
170	C₄²×C₃₁	C₂×D₄	1	4	yes	—
171	C₄²×C₃₁	C₂×D₄	1	4	yes	—
172	#22509	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂³×C₃₁	—	2	1	no	—
174	C₂³×C₃₁	—	2	1	no	—
175	C₂³×C₃₁	—	2	1	no	—
176	C₂³×C₃₁	—	2	1	no	—
177	C₂³×C₃₁	—	2	1	no	—
178	C₂³×C₃₁	—	2	1	no	—
179	C₂³×C₃₁	C₄×Q₈	1	2	yes	—
180	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
181	Q₈×C₃₁	—	2	8	no	—
182	Q₈×C₃₁	—	4	8	no	—
183	Q₈×C₃₁	—	4	4	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	4	4	no	—
208	C₂×C₄×C₃₁	—	4	4	no	—
209	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
210	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
211	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
212	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
213	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
214	C₂×C₄×C₃₁	#131	1	2	yes	—
215	C₂×C₄×C₃₁	#131	1	2	yes	—
216	C₂×C₄×C₃₁	#131	1	2	yes	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
222	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
223	C₈×C₃₁	—	2	2	no	—
224	C₈×C₃₁	—	2	2	no	—
225	C₈×C₃₁	—	4	1	no	—
226	C₈×C₃₁	—	4	2	no	—
227	C₈×C₃₁	—	4	1	no	—
228	C₈×C₃₁	—	4	2	no	—
229	C₂×#533	C₂×C₃₁	1	1	yes	characteristic
230	C₂×#446	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#1985	C₂×C₃₁	1	2	yes	complemented
233	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#1984	C₂×C₃₁	1	2	yes	complemented
236	#1798	C₂×C₃₁	1	2	yes	—
237	#1797	C₂×C₃₁	1	1	yes	characteristic
238	#1750	C₂×C₃₁	1	2	yes	—
239	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic
240	C₂²×C₃₁	—	2	2	no	—
241	C₂²×C₃₁	—	2	2	no	—
242	C₂²×C₃₁	—	2	2	no	—
243	C₂²×C₃₁	—	4	1	no	—
244	C₂²×C₃₁	—	4	1	no	—
245	C₂²×C₃₁	—	4	1	no	—
246	C₂²×C₃₁	—	4	1	no	—
247	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
248	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
249	C₂²×C₃₁	#519	1	1	yes	characteristic
250	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
251	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
252	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
253	C₂²×C₃₁	#568	1	1	yes	characteristic
254	C₄×C₃₁	—	2	2	no	—
255	C₄×C₃₁	—	2	4	no	—
256	C₄×C₃₁	—	2	2	no	—
257	C₄×C₃₁	—	2	2	no	—
258	C₄×C₃₁	—	2	4	no	—
259	C₄×C₃₁	—	4	1	no	—
260	C₄×C₃₁	—	4	2	no	—
261	C₄×C₃₁	—	4	2	no	—
262	C₄×C₃₁	#408	1	4	yes	—
263	C₄×C₃₁	#548	1	1	yes	characteristic
264	C₄×C₃₁	#548	1	1	yes	characteristic
265	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
266	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
267	#533	—	2	4	no	—
268	#533	—	2	4	no	—
269	C₂×C₄⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
270	C₂×C₄⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
271	C₂×#125	C₄×C₃₁	1	2	yes	—
272	C₂×C₂²⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
273	C₂×C₂²⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
274	C₂×C₂²⋊Q₈	C₄×C₃₁	1	2	yes	—
275	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
277	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×#117	C₂²×C₃₁	1	2	yes	—
279	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
280	#446	—	2	4	no	—
281	C₈×Q₈	—	2	4	no	—
282	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
284	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
285	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
286	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
287	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×#97	C₂²×C₃₁	1	2	yes	—
290	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
292	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
293	#336	C₂²×C₃₁	1	2	yes	—
294	#336	C₂²×C₃₁	1	2	yes	—
295	#336	C₂²×C₃₁	1	2	yes	—
296	#336	C₂²×C₃₁	1	1	yes	characteristic
297	#336	C₂²×C₃₁	1	2	yes	—
298	#336	C₂²×C₃₁	1	2	yes	—
299	#336	C₂²×C₃₁	1	1	yes	characteristic
300	C₂×C₃₁	—	4	2	no	—
301	C₂×C₃₁	Q₈×C₂²⋊C₄	1	1	yes	characteristic
302	C₂×C₃₁	C₂×#408	1	1	yes	characteristic
303	C₂×C₃₁	#2880	1	1	yes	characteristic
304	C₂×C₃₁	#2960	1	1	yes	characteristic
305	C₂×C₃₁	#2962	1	1	yes	characteristic
306	C₂×C₃₁	#2985	1	1	yes	characteristic
307	C₂×C₃₁	#2986	1	1	yes	characteristic
308	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
309	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
310	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
311	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
312	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
313	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
314	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
315	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
316	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
317	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
318	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
319	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
320	C₄⋊Q₈	—	2	4	no	—
321	C₄⋊Q₈	—	2	4	no	—
322	#125	—	2	8	no	—
323	C₂²⋊Q₈	—	2	4	no	—
324	C₂²⋊Q₈	—	2	4	no	—
325	C₂²⋊Q₈	—	2	8	no	—
326	C₄×Q₈	—	2	4	no	—
327	C₄×Q₈	—	2	4	no	—
328	C₄×Q₈	—	2	2	no	—
329	C₄×Q₈	—	2	2	no	—
330	C₄×Q₈	D₄×C₃₁	1	4	yes	—
331	C₄×Q₈	D₄×C₃₁	1	4	yes	—
332	#117	—	2	4	no	—
333	#117	—	2	4	no	—
334	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
335	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
336	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
337	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
338	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
339	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
340	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
341	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
342	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
343	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
344	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
345	C₂×C₄²	—	2	2	no	—
346	C₂×C₄²	—	2	2	no	—
347	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
348	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
349	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
350	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
351	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
352	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
353	C₄⋊C₈	—	2	2	no	—
354	C₄⋊C₈	—	2	4	no	—
355	C₄⋊C₈	—	2	2	no	—
356	C₄⋊C₈	—	2	4	no	—
357	C₂²⋊C₈	—	2	4	no	—
358	C₂²⋊C₈	—	2	2	no	—
359	C₂²⋊C₈	Q₈×C₃₁	1	4	yes	complemented
360	#97	—	2	4	no	—
361	C₄×C₈	—	2	2	no	—
362	C₄×C₈	—	2	4	no	—
363	C₄×C₈	—	2	2	no	—
364	C₃₁	#22509	1	1	yes	characteristic, direct product factor, Sylow
365	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
366	C₂×Q₈	—	2	4	no	—
367	C₂×Q₈	—	2	2	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	2	no	—
370	C₂×Q₈	—	2	4	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	—	4	2	no	—
374	C₂×Q₈	C₂²⋊C₄×C₃₁	1	4	yes	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	4	1	no	—
385	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
386	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
387	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
388	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
389	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
390	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
391	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
392	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
393	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
394	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
395	C₂²×C₄	#41×C₃₁	1	2	yes	—
396	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	1	no	—
407	C₂×C₈	—	2	1	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
414	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
415	C₄⋊C₄	—	2	4	no	—
416	C₄⋊C₄	—	2	4	no	—
417	C₄⋊C₄	—	2	4	no	—
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	4	no	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
423	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
424	C₂²⋊C₄	—	2	4	no	—
425	C₂²⋊C₄	—	2	4	no	—
426	C₄²	—	2	2	no	—
427	C₄²	—	2	2	no	—
428	C₄²	—	2	1	no	—
429	C₄²	—	2	1	no	—
430	C₄²	—	2	2	no	—
431	C₄²	—	2	2	no	—
432	C₄²	C₂×D₄×C₃₁	1	2	yes	—
433	C₄²	C₂×D₄×C₃₁	1	2	yes	—
434	C₄²	C₂×D₄×C₃₁	1	4	yes	—
435	C₄²	C₂×D₄×C₃₁	1	4	yes	—
436	C₂³	—	2	1	no	—
437	C₂³	—	2	1	no	—
438	C₂³	—	2	1	no	—
439	C₂³	—	2	1	no	—
440	C₂³	—	2	1	no	—
441	C₂³	—	2	1	no	—
442	C₂³	C₄×Q₈×C₃₁	1	2	yes	—
443	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
444	Q₈	—	2	8	no	—
445	Q₈	—	4	8	no	—
446	Q₈	—	4	4	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	4	4	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
473	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
474	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
475	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
476	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
477	C₂×C₄	C₃₁×#131	1	2	yes	—
478	C₂×C₄	C₃₁×#131	1	2	yes	—
479	C₂×C₄	C₃₁×#131	1	2	yes	—
480	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
481	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
482	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
485	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
486	C₈	—	2	2	no	—
487	C₈	—	2	2	no	—
488	C₈	—	4	1	no	—
489	C₈	—	4	2	no	—
490	C₈	—	4	1	no	—
491	C₈	—	4	2	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	2	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	4	1	no	—
496	C₂²	—	4	1	no	—
497	C₂²	—	4	1	no	—
498	C₂²	—	4	1	no	—
499	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
500	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
501	C₂²	C₃₁×#519	1	1	yes	characteristic
502	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
503	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
504	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
505	C₂²	C₃₁×#568	1	1	yes	characteristic
506	C₄	—	2	2	no	—
507	C₄	—	2	4	no	—
508	C₄	—	2	2	no	—
509	C₄	—	2	2	no	—
510	C₄	—	2	4	no	—
511	C₄	—	4	1	no	—
512	C₄	—	4	2	no	—
513	C₄	—	4	2	no	—
514	C₄	C₃₁×#408	1	4	yes	—
515	C₄	C₃₁×#548	1	1	yes	characteristic
516	C₄	C₃₁×#548	1	1	yes	characteristic
517	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
518	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
519	C₂	—	4	2	no	—
520	C₂	Q₈×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
521	C₂	C₂×C₃₁×#408	1	1	yes	characteristic
522	C₂	C₃₁×#2880	1	1	yes	characteristic
523	C₂	C₃₁×#2960	1	1	yes	characteristic
524	C₂	C₃₁×#2962	1	1	yes	characteristic
525	C₂	C₃₁×#2985	1	1	yes	characteristic
526	C₂	C₃₁×#2986	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂⁴×C₃₁	C₂²×C₄
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. C₂×C₃₁×#446	C₂	semidirect
3. C₂×C₈×Q₈×C₃₁	C₂	semidirect
4. C₃₁×#1985	C₂	semidirect
5. C₃₁×#1984	C₂	semidirect
6. C₃₁×#1984	C₂	semidirect
7. C₃₁×#1984	C₂	semidirect
96. C₃₁×C₂²⋊C₈	Q₈	semidirect
172. #22509	C₃₁	direct (N × H)
230. C₂×#446	C₂×C₃₁	semidirect
231. C₂×C₈×Q₈	C₂×C₃₁	semidirect
232. #1985	C₂×C₃₁	semidirect
233. #1984	C₂×C₃₁	semidirect
234. #1984	C₂×C₃₁	semidirect
235. #1984	C₂×C₃₁	semidirect
359. C₂²⋊C₈	Q₈×C₃₁	semidirect
364. C₃₁	#22509	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×#533	2
C₂×C₃₁×#446	2
C₂×C₈×Q₈×C₃₁	2
C₃₁×#1985	2
C₃₁×#1984	2
C₃₁×#1798	2
C₃₁×#1797	2
C₃₁×#1750	2
C₄×C₃₁×C₂²⋊C₈	2
#22509	31

Subgroups — C31×#22509 #21479268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22509 #21479268