Subgroups — C₃₁×#24338 #21481097

Order 7936.

1296 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#442	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2580	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2058	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2023	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1879	C₂	1	2	yes	maximal
10	C₃₁×#1864	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1804	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₃₁	—	2	1	no	—
13	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
14	D₄×Q₈×C₃₁	—	2	4	no	—
15	D₄×Q₈×C₃₁	—	2	4	no	—
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
19	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#476	—	2	4	no	—
25	C₃₁×#442	—	2	4	no	—
26	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#106	C₂²	1	2	yes	—
28	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#102	C₂²	1	2	yes	—
32	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
34	C₃₁×#399	—	2	1	no	—
35	C₃₁×#393	—	2	1	no	—
36	C₃₁×#390	C₂²	1	2	yes	—
37	C₃₁×#389	C₂²	1	2	yes	—
38	C₃₁×#385	—	2	2	no	—
39	C₃₁×#382	C₂²	1	1	yes	characteristic
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₃₁×#336	C₂²	1	2	yes	—
45	C₂²×Q₈×C₃₁	—	2	8	no	—
46	C₂²×Q₈×C₃₁	—	2	1	no	—
47	C₂²×Q₈×C₃₁	—	2	2	no	—
48	C₂²×Q₈×C₃₁	—	4	8	no	—
49	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
50	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	—	2	2	no	—
53	C₂³×C₄×C₃₁	—	2	1	no	—
54	C₂³×C₄×C₃₁	—	2	1	no	—
55	C₂×D₈×C₃₁	—	2	2	no	—
56	C₂×D₈×C₃₁	—	2	2	no	—
57	C₂×D₈×C₃₁	Q₈	1	4	yes	complemented
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	—	2	4	no	—
62	C₃₁×C₄⋊Q₈	—	2	4	no	—
63	C₃₁×C₄⋊Q₈	—	2	2	no	—
64	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
65	C₃₁×C₂²⋊Q₈	—	2	8	no	—
66	C₃₁×C₂²⋊Q₈	—	2	16	no	—
67	C₄×Q₈×C₃₁	—	2	2	no	—
68	C₄×Q₈×C₃₁	—	2	2	no	—
69	C₄×D₄×C₃₁	—	2	4	no	—
70	C₄×D₄×C₃₁	—	2	4	no	—
71	C₄×D₄×C₃₁	—	2	2	no	—
72	C₄×D₄×C₃₁	—	2	2	no	—
73	C₄×D₄×C₃₁	—	2	2	no	—
74	C₄×D₄×C₃₁	D₄	1	4	yes	—
75	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
89	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
90	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
91	C₃₁×#106	—	2	2	no	—
92	C₃₁×#106	—	2	4	no	—
93	C₃₁×C₄⋊C₈	—	2	2	no	—
94	C₃₁×#102	—	2	2	no	—
95	C₃₁×#102	—	2	4	no	—
96	C₃₁×#102	—	2	2	no	—
97	C₃₁×#102	—	2	4	no	—
98	C₃₁×#97	—	2	2	no	—
99	C₃₁×#95	—	2	1	no	—
100	C₃₁×#95	—	2	1	no	—
101	C₃₁×#95	—	2	2	no	—
102	C₂⁴×C₃₁	—	2	1	no	—
103	C₂⁴×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	8	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	4	no	—
107	C₂×Q₈×C₃₁	—	2	4	no	—
108	C₂×Q₈×C₃₁	—	4	4	no	—
109	C₂×Q₈×C₃₁	—	4	4	no	—
110	C₂×Q₈×C₃₁	—	4	32	no	—
111	C₂×Q₈×C₃₁	—	4	4	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	1	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
121	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
122	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	complemented
123	C₂×D₄×C₃₁	#41	1	2	yes	—
124	C₂²×C₄×C₃₁	—	2	8	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	2	4	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	1	no	—
130	C₂²×C₄×C₃₁	—	4	4	no	—
131	C₂²×C₄×C₃₁	—	4	4	no	—
132	C₂²×C₄×C₃₁	—	4	4	no	—
133	C₂²×C₄×C₃₁	—	4	2	no	—
134	C₂²×C₄×C₃₁	—	8	2	no	—
135	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
140	D₈×C₃₁	—	2	8	no	—
141	C₂×C₈×C₃₁	—	2	1	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	2	no	—
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	complemented
150	C₄⋊C₄×C₃₁	—	2	8	no	—
151	C₄⋊C₄×C₃₁	—	2	4	no	—
152	C₄⋊C₄×C₃₁	—	2	4	no	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	2	1	no	—
156	C₄⋊C₄×C₃₁	—	2	1	no	—
157	C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	—	8	2	no	—
161	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
162	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
163	C₂²⋊C₄×C₃₁	—	2	8	no	—
164	C₂²⋊C₄×C₃₁	—	2	4	no	—
165	C₂²⋊C₄×C₃₁	—	4	2	no	—
166	C₂²⋊C₄×C₃₁	—	4	4	no	—
167	C₂²⋊C₄×C₃₁	—	8	2	no	—
168	C₄²×C₃₁	—	2	2	no	—
169	C₄²×C₃₁	—	2	2	no	—
170	C₄²×C₃₁	C₂×D₄	1	2	yes	—
171	C₄²×C₃₁	C₂×D₄	1	2	yes	—
172	#24338	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂³×C₃₁	—	2	2	no	—
174	C₂³×C₃₁	—	2	2	no	—
175	C₂³×C₃₁	—	2	2	no	—
176	C₂³×C₃₁	—	2	2	no	—
177	C₂³×C₃₁	—	4	2	no	—
178	C₂³×C₃₁	—	4	2	no	—
179	C₂³×C₃₁	—	4	1	no	—
180	C₂³×C₃₁	—	4	1	no	—
181	C₂³×C₃₁	—	4	2	no	—
182	C₂³×C₃₁	—	4	2	no	—
183	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
184	Q₈×C₃₁	—	2	8	no	—
185	Q₈×C₃₁	—	4	8	no	—
186	Q₈×C₃₁	—	4	16	no	—
187	D₄×C₃₁	—	2	2	no	—
188	D₄×C₃₁	—	2	2	no	—
189	D₄×C₃₁	—	2	2	no	—
190	D₄×C₃₁	—	2	2	no	—
191	D₄×C₃₁	—	2	2	no	—
192	D₄×C₃₁	—	4	1	no	—
193	D₄×C₃₁	—	4	1	no	—
194	D₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
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	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	4	8	no	—
205	C₂×C₄×C₃₁	—	4	16	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	8	4	no	—
215	C₂×C₄×C₃₁	#120	1	2	yes	—
216	C₂×C₄×C₃₁	#120	1	2	yes	—
217	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
222	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
223	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
224	C₈×C₃₁	—	2	2	no	—
225	C₈×C₃₁	—	4	1	no	—
226	C₈×C₃₁	—	4	2	no	—
227	C₂×D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
228	C₂×#476	C₂×C₃₁	1	1	yes	characteristic, complemented
229	C₂×#442	C₂×C₃₁	1	1	yes	characteristic
230	#2580	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#2058	C₂×C₃₁	1	2	yes	complemented
232	#2023	C₂×C₃₁	1	2	yes	complemented
233	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#1985	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#1879	C₂×C₃₁	1	2	yes	—
236	#1864	C₂×C₃₁	1	2	yes	complemented
237	#1804	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₂²×C₃₁	—	2	4	no	—
239	C₂²×C₃₁	—	4	4	no	—
240	C₂²×C₃₁	—	4	4	no	—
241	C₂²×C₃₁	—	4	2	no	—
242	C₂²×C₃₁	—	4	2	no	—
243	C₂²×C₃₁	—	4	4	no	—
244	C₂²×C₃₁	—	4	2	no	—
245	C₂²×C₃₁	—	8	2	no	—
246	C₂²×C₃₁	—	8	1	no	—
247	C₂²×C₃₁	—	8	1	no	—
248	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
249	C₂²×C₃₁	#548	1	1	yes	characteristic
250	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
251	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
252	C₂²×C₃₁	#577	1	2	yes	—
253	C₄×C₃₁	—	2	2	no	—
254	C₄×C₃₁	—	2	4	no	—
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	4	4	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	8	2	no	—
260	C₄×C₃₁	—	8	1	no	—
261	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
262	C₄×C₃₁	#536	1	1	yes	characteristic
263	C₄×C₃₁	#548	1	1	yes	characteristic
264	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
265	C₂³×Q₈	—	2	1	no	—
266	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
267	D₄×Q₈	—	2	4	no	—
268	D₄×Q₈	—	2	4	no	—
269	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
270	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×C₂²⋊Q₈	—	2	2	no	—
272	C₂×C₂²⋊Q₈	—	2	1	no	—
273	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
276	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
277	#476	—	2	4	no	—
278	#442	—	2	4	no	—
279	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×#106	C₂²×C₃₁	1	2	yes	—
281	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×#102	C₂²×C₃₁	1	2	yes	—
283	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
284	C₂×#102	C₂²×C₃₁	1	2	yes	—
285	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
287	#399	—	2	1	no	—
288	#393	—	2	1	no	—
289	#390	C₂²×C₃₁	1	2	yes	—
290	#389	C₂²×C₃₁	1	2	yes	—
291	#385	—	2	2	no	—
292	#382	C₂²×C₃₁	1	1	yes	characteristic
293	#340	C₂²×C₃₁	1	2	yes	—
294	#340	C₂²×C₃₁	1	2	yes	—
295	#340	C₂²×C₃₁	1	2	yes	—
296	#336	C₂²×C₃₁	1	2	yes	—
297	#336	C₂²×C₃₁	1	2	yes	—
298	C₂×C₃₁	—	4	4	no	—
299	C₂×C₃₁	—	8	2	no	—
300	C₂×C₃₁	#2409	1	1	yes	characteristic
301	C₂×C₃₁	#3003	1	1	yes	characteristic
302	C₂×C₃₁	#3010	1	1	yes	characteristic
303	C₂×C₃₁	#3373	1	2	yes	—
304	C₂×C₃₁	#3384	1	2	yes	—
305	C₂²×Q₈	—	2	8	no	—
306	C₂²×Q₈	—	2	1	no	—
307	C₂²×Q₈	—	2	2	no	—
308	C₂²×Q₈	—	4	8	no	—
309	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
310	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
311	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂³×C₄	—	2	2	no	—
313	C₂³×C₄	—	2	1	no	—
314	C₂³×C₄	—	2	1	no	—
315	C₂×D₈	—	2	2	no	—
316	C₂×D₈	—	2	2	no	—
317	C₂×D₈	Q₈×C₃₁	1	4	yes	complemented
318	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
319	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
320	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
321	C₄⋊Q₈	—	2	4	no	—
322	C₄⋊Q₈	—	2	4	no	—
323	C₄⋊Q₈	—	2	2	no	—
324	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
325	C₂²⋊Q₈	—	2	8	no	—
326	C₂²⋊Q₈	—	2	16	no	—
327	C₄×Q₈	—	2	2	no	—
328	C₄×Q₈	—	2	2	no	—
329	C₄×D₄	—	2	4	no	—
330	C₄×D₄	—	2	4	no	—
331	C₄×D₄	—	2	2	no	—
332	C₄×D₄	—	2	2	no	—
333	C₄×D₄	—	2	2	no	—
334	C₄×D₄	D₄×C₃₁	1	4	yes	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	2	no	—
338	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
339	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
340	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
341	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
342	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
343	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
344	C₂×C₂²⋊C₄	—	2	2	no	—
345	C₂×C₂²⋊C₄	—	2	1	no	—
346	C₂×C₂²⋊C₄	—	2	1	no	—
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
350	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
351	#106	—	2	2	no	—
352	#106	—	2	4	no	—
353	C₄⋊C₈	—	2	2	no	—
354	#102	—	2	2	no	—
355	#102	—	2	4	no	—
356	#102	—	2	2	no	—
357	#102	—	2	4	no	—
358	#97	—	2	2	no	—
359	#95	—	2	1	no	—
360	#95	—	2	1	no	—
361	#95	—	2	2	no	—
362	C₃₁	#24338	1	1	yes	characteristic, direct product factor, Sylow
363	C₂⁴	—	2	1	no	—
364	C₂⁴	—	2	1	no	—
365	C₂×Q₈	—	2	8	no	—
366	C₂×Q₈	—	2	2	no	—
367	C₂×Q₈	—	2	4	no	—
368	C₂×Q₈	—	2	4	no	—
369	C₂×Q₈	—	4	4	no	—
370	C₂×Q₈	—	4	4	no	—
371	C₂×Q₈	—	4	32	no	—
372	C₂×Q₈	—	4	4	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	—	2	1	no	—
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	—	2	1	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
382	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
383	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	complemented
384	C₂×D₄	#41×C₃₁	1	2	yes	—
385	C₂²×C₄	—	2	8	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	4	4	no	—
392	C₂²×C₄	—	4	4	no	—
393	C₂²×C₄	—	4	4	no	—
394	C₂²×C₄	—	4	2	no	—
395	C₂²×C₄	—	8	2	no	—
396	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
397	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
400	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
401	D₈	—	2	8	no	—
402	C₂×C₈	—	2	1	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	1	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	complemented
411	C₄⋊C₄	—	2	8	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	—	8	2	no	—
422	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
423	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
424	C₂²⋊C₄	—	2	8	no	—
425	C₂²⋊C₄	—	2	4	no	—
426	C₂²⋊C₄	—	4	2	no	—
427	C₂²⋊C₄	—	4	4	no	—
428	C₂²⋊C₄	—	8	2	no	—
429	C₄²	—	2	2	no	—
430	C₄²	—	2	2	no	—
431	C₄²	C₂×D₄×C₃₁	1	2	yes	—
432	C₄²	C₂×D₄×C₃₁	1	2	yes	—
433	C₂³	—	2	2	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	2	2	no	—
436	C₂³	—	2	2	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	2	no	—
439	C₂³	—	4	1	no	—
440	C₂³	—	4	1	no	—
441	C₂³	—	4	2	no	—
442	C₂³	—	4	2	no	—
443	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
444	Q₈	—	2	8	no	—
445	Q₈	—	4	8	no	—
446	Q₈	—	4	16	no	—
447	D₄	—	2	2	no	—
448	D₄	—	2	2	no	—
449	D₄	—	2	2	no	—
450	D₄	—	2	2	no	—
451	D₄	—	2	2	no	—
452	D₄	—	4	1	no	—
453	D₄	—	4	1	no	—
454	D₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
455	C₂×C₄	—	2	2	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	4	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	4	8	no	—
465	C₂×C₄	—	4	16	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	8	4	no	—
475	C₂×C₄	C₃₁×#120	1	2	yes	—
476	C₂×C₄	C₃₁×#120	1	2	yes	—
477	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
478	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
479	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
480	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
481	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
482	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
483	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
484	C₈	—	2	2	no	—
485	C₈	—	4	1	no	—
486	C₈	—	4	2	no	—
487	C₂²	—	2	4	no	—
488	C₂²	—	4	4	no	—
489	C₂²	—	4	4	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	4	4	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	8	2	no	—
495	C₂²	—	8	1	no	—
496	C₂²	—	8	1	no	—
497	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
498	C₂²	C₃₁×#548	1	1	yes	characteristic
499	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
500	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
501	C₂²	C₃₁×#577	1	2	yes	—
502	C₄	—	2	2	no	—
503	C₄	—	2	4	no	—
504	C₄	—	2	2	no	—
505	C₄	—	4	4	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	2	no	—
508	C₄	—	8	2	no	—
509	C₄	—	8	1	no	—
510	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
511	C₄	C₃₁×#536	1	1	yes	characteristic
512	C₄	C₃₁×#548	1	1	yes	characteristic
513	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
514	C₂	—	4	4	no	—
515	C₂	—	8	2	no	—
516	C₂	C₃₁×#2409	1	1	yes	characteristic
517	C₂	C₃₁×#3003	1	1	yes	characteristic
518	C₂	C₃₁×#3010	1	1	yes	characteristic
519	C₂	C₃₁×#3373	1	2	yes	—
520	C₂	C₃₁×#3384	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂²×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
1. C₂×D₄×Q₈×C₃₁	C₂	semidirect
2. C₂×C₃₁×#476	C₂	semidirect
4. C₃₁×#2580	C₂	semidirect
5. C₃₁×#2058	C₂	semidirect
6. C₃₁×#2023	C₂	semidirect
7. C₃₁×#1993	C₂	semidirect
8. C₃₁×#1985	C₂	semidirect
10. C₃₁×#1864	C₂	semidirect
11. C₃₁×#1804	C₂	semidirect
57. C₂×D₈×C₃₁	Q₈	semidirect
122. C₂×D₄×C₃₁	C₂×Q₈	semidirect
149. C₂×C₈×C₃₁	C₂×Q₈	semidirect
172. #24338	C₃₁	direct (N × H)
227. C₂×D₄×Q₈	C₂×C₃₁	semidirect
228. C₂×#476	C₂×C₃₁	semidirect
230. #2580	C₂×C₃₁	semidirect
231. #2058	C₂×C₃₁	semidirect
232. #2023	C₂×C₃₁	semidirect
233. #1993	C₂×C₃₁	semidirect
234. #1985	C₂×C₃₁	semidirect
236. #1864	C₂×C₃₁	semidirect
237. #1804	C₂×C₃₁	semidirect
317. C₂×D₈	Q₈×C₃₁	semidirect
362. C₃₁	#24338	direct (N × H)
383. C₂×D₄	C₂×Q₈×C₃₁	semidirect
410. C₂×C₈	C₂×Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈×C₃₁	2
C₂×C₃₁×#476	2
C₂×C₃₁×#442	2
C₃₁×#2580	2
C₃₁×#2058	2
C₃₁×#2023	2
C₃₁×#1993	2
C₃₁×#1985	2
C₃₁×#1879	2
C₃₁×#1864	2
C₃₁×#1804	2
#24338	31

Subgroups — C31×#24338 #21481097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24338 #21481097