Subgroups — C₃₁×#22758 #21479517

Order 7936.

978 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2354	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1936	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1921	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1879	C₂	1	2	yes	maximal
7	C₃₁×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1762	C₂	1	2	yes	maximal, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#117	—	2	1	no	—
17	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
18	C₄×D₈×C₃₁	C₄	1	8	yes	complemented
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#97	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#390	C₂²	1	2	yes	—
28	C₃₁×#389	C₂²	1	2	yes	—
29	C₃₁×#388	—	2	2	no	—
30	C₃₁×#385	—	2	2	no	—
31	C₃₁×#384	C₂²	1	1	yes	characteristic
32	C₃₁×#384	C₂²	1	1	yes	characteristic
33	C₃₁×#380	—	2	1	no	—
34	C₃₁×#380	—	2	1	no	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	—	2	1	no	—
45	C₂³×C₄×C₃₁	—	2	1	no	—
46	C₂³×C₄×C₃₁	—	2	2	no	—
47	C₂×D₈×C₃₁	—	2	2	no	—
48	C₂×D₈×C₃₁	—	2	2	no	—
49	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₄×D₄×C₃₁	—	2	4	no	—
54	C₄×D₄×C₃₁	—	2	2	no	—
55	C₄×D₄×C₃₁	—	2	2	no	—
56	C₄×D₄×C₃₁	—	4	2	no	—
57	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
58	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	complemented
59	C₃₁×#117	—	4	4	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
66	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₄²×C₃₁	—	2	2	no	—
78	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×#107	C₂×C₄	1	4	yes	complemented
81	C₃₁×C₄⋊C₈	—	2	2	no	—
82	C₃₁×#102	—	2	4	no	—
83	C₃₁×#102	—	4	2	no	—
84	C₃₁×#102	C₂×C₄	1	4	yes	—
85	C₃₁×#102	C₂×C₄	1	4	yes	complemented
86	C₃₁×#97	—	4	2	no	—
87	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
88	C₃₁×#95	—	2	2	no	—
89	C₃₁×#95	—	2	2	no	—
90	C₃₁×#95	D₄	1	2	yes	—
91	C₂⁴×C₃₁	—	2	1	no	—
92	C₂⁴×C₃₁	—	2	1	no	—
93	C₂×D₄×C₃₁	—	2	1	no	—
94	C₂×D₄×C₃₁	—	2	1	no	—
95	C₂×D₄×C₃₁	—	2	1	no	—
96	C₂×D₄×C₃₁	—	2	1	no	—
97	C₂×D₄×C₃₁	—	2	1	no	—
98	C₂×D₄×C₃₁	—	2	1	no	—
99	C₂×D₄×C₃₁	—	2	1	no	—
100	C₂×D₄×C₃₁	—	2	1	no	—
101	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
102	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
103	C₂×D₄×C₃₁	#41	1	2	yes	—
104	C₂×D₄×C₃₁	#41	1	2	yes	—
105	C₂²×C₄×C₃₁	—	2	1	no	—
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	2	4	no	—
108	C₂²×C₄×C₃₁	—	2	4	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	4	4	no	—
112	C₂²×C₄×C₃₁	—	4	4	no	—
113	C₂²×C₄×C₃₁	—	4	2	no	—
114	C₂²×C₄×C₃₁	—	4	2	no	—
115	C₂²×C₄×C₃₁	—	4	2	no	—
116	C₂²×C₄×C₃₁	—	4	2	no	—
117	C₂²×C₄×C₃₁	—	8	2	no	—
118	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
123	D₈×C₃₁	—	2	8	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
132	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	2	2	no	—
141	C₄⋊C₄×C₃₁	—	4	4	no	—
142	C₄⋊C₄×C₃₁	—	4	2	no	—
143	C₄⋊C₄×C₃₁	—	4	2	no	—
144	C₄⋊C₄×C₃₁	—	8	1	no	—
145	C₄⋊C₄×C₃₁	—	8	1	no	—
146	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
147	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
148	C₂²⋊C₄×C₃₁	—	2	4	no	—
149	C₂²⋊C₄×C₃₁	—	2	4	no	—
150	C₂²⋊C₄×C₃₁	—	4	4	no	—
151	C₂²⋊C₄×C₃₁	—	4	4	no	—
152	C₂²⋊C₄×C₃₁	—	8	2	no	—
153	C₄²×C₃₁	—	2	2	no	—
154	C₄²×C₃₁	—	2	2	no	—
155	C₄²×C₃₁	—	4	4	no	—
156	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
157	C₄²×C₃₁	#41	1	2	yes	—
158	#22758	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂³×C₃₁	—	2	2	no	—
160	C₂³×C₃₁	—	2	2	no	—
161	C₂³×C₃₁	—	2	2	no	—
162	C₂³×C₃₁	—	2	2	no	—
163	C₂³×C₃₁	—	4	1	no	—
164	C₂³×C₃₁	—	4	1	no	—
165	C₂³×C₃₁	—	4	1	no	—
166	C₂³×C₃₁	—	4	1	no	—
167	C₂³×C₃₁	—	4	1	no	—
168	C₂³×C₃₁	—	4	1	no	—
169	C₂³×C₃₁	—	4	1	no	—
170	C₂³×C₃₁	—	4	1	no	—
171	C₂³×C₃₁	—	4	1	no	—
172	C₂³×C₃₁	—	4	1	no	—
173	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
174	D₄×C₃₁	—	2	2	no	—
175	D₄×C₃₁	—	2	2	no	—
176	D₄×C₃₁	—	2	2	no	—
177	D₄×C₃₁	—	2	2	no	—
178	D₄×C₃₁	—	2	2	no	—
179	D₄×C₃₁	—	4	1	no	—
180	D₄×C₃₁	—	4	1	no	—
181	D₄×C₃₁	#117	1	4	yes	complemented
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	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	2	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	8	no	—
197	C₂×C₄×C₃₁	—	4	8	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	8	8	no	—
205	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
206	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
207	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
208	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
209	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
210	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
213	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
214	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
215	C₈×C₃₁	—	2	2	no	—
216	C₈×C₃₁	—	2	2	no	—
217	C₈×C₃₁	—	4	2	no	—
218	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
219	#2354	C₂×C₃₁	1	1	yes	characteristic, complemented
220	D₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#1936	C₂×C₃₁	1	2	yes	complemented
222	#1921	C₂×C₃₁	1	2	yes	complemented
223	#1879	C₂×C₃₁	1	2	yes	—
224	#1834	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#1802	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#1802	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#1762	C₂×C₃₁	1	2	yes	complemented
229	C₂²×C₃₁	—	2	4	no	—
230	C₂²×C₃₁	—	4	2	no	—
231	C₂²×C₃₁	—	4	2	no	—
232	C₂²×C₃₁	—	4	2	no	—
233	C₂²×C₃₁	—	4	2	no	—
234	C₂²×C₃₁	—	4	2	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	8	1	no	—
240	C₂²×C₃₁	—	8	1	no	—
241	C₂²×C₃₁	—	8	1	no	—
242	C₂²×C₃₁	—	8	1	no	—
243	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
244	C₂²×C₃₁	#546	1	1	yes	characteristic
245	C₂²×C₃₁	#548	1	1	yes	characteristic
246	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₂²×C₃₁	#576	1	1	yes	characteristic
249	C₂²×C₃₁	#578	1	1	yes	characteristic
250	C₄×C₃₁	—	2	2	no	—
251	C₄×C₃₁	—	4	2	no	—
252	C₄×C₃₁	—	4	2	no	—
253	C₄×C₃₁	—	4	2	no	—
254	C₄×C₃₁	—	4	2	no	—
255	C₄×C₃₁	—	8	2	no	—
256	C₄×C₃₁	—	8	2	no	—
257	C₄×C₃₁	#442	1	4	yes	—
258	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
259	C₄×C₃₁	#520	1	1	yes	characteristic
260	C₄×C₃₁	#552	1	1	yes	characteristic
261	C₄×C₃₁	#553	1	1	yes	characteristic
262	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
264	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×#117	—	2	1	no	—
267	C₂²×C₄⋊C₄	—	2	1	no	—
268	C₄×D₈	C₄×C₃₁	1	8	yes	complemented
269	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×#102	C₂²×C₃₁	1	2	yes	—
272	C₂×#102	C₂²×C₃₁	1	2	yes	—
273	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×#97	C₂²×C₃₁	1	2	yes	—
276	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
277	#390	C₂²×C₃₁	1	2	yes	—
278	#389	C₂²×C₃₁	1	2	yes	—
279	#388	—	2	2	no	—
280	#385	—	2	2	no	—
281	#384	C₂²×C₃₁	1	1	yes	characteristic
282	#384	C₂²×C₃₁	1	1	yes	characteristic
283	#380	—	2	1	no	—
284	#380	—	2	1	no	—
285	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
286	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
287	#340	C₂²×C₃₁	1	2	yes	—
288	#340	C₂²×C₃₁	1	2	yes	—
289	#340	C₂²×C₃₁	1	2	yes	—
290	#340	C₂²×C₃₁	1	2	yes	—
291	#336	C₂²×C₃₁	1	2	yes	—
292	C₂×C₃₁	—	4	4	no	—
293	C₂×C₃₁	—	8	2	no	—
294	C₂×C₃₁	#2352	1	1	yes	characteristic
295	C₂×C₃₁	C₂×#442	1	1	yes	characteristic
296	C₂×C₃₁	#2951	1	1	yes	characteristic
297	C₂×C₃₁	#3325	1	1	yes	characteristic
298	C₂×C₃₁	#3330	1	1	yes	characteristic
299	C₂×C₃₁	#3366	1	1	yes	characteristic
300	C₂×C₃₁	#3367	1	1	yes	characteristic
301	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
302	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
303	C₂³×C₄	—	2	1	no	—
304	C₂³×C₄	—	2	1	no	—
305	C₂³×C₄	—	2	2	no	—
306	C₂×D₈	—	2	2	no	—
307	C₂×D₈	—	2	2	no	—
308	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
309	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
310	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
311	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
312	C₄×D₄	—	2	4	no	—
313	C₄×D₄	—	2	2	no	—
314	C₄×D₄	—	2	2	no	—
315	C₄×D₄	—	4	2	no	—
316	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
317	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	complemented
318	#117	—	4	4	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	2	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	—	4	4	no	—
325	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
326	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
327	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
328	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
329	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
330	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×C₂²⋊C₄	—	2	1	no	—
332	C₂×C₂²⋊C₄	—	2	1	no	—
333	C₂×C₂²⋊C₄	—	2	2	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₄²	—	2	2	no	—
337	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
338	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
339	#107	C₂×C₄×C₃₁	1	4	yes	complemented
340	C₄⋊C₈	—	2	2	no	—
341	#102	—	2	4	no	—
342	#102	—	4	2	no	—
343	#102	C₂×C₄×C₃₁	1	4	yes	—
344	#102	C₂×C₄×C₃₁	1	4	yes	complemented
345	#97	—	4	2	no	—
346	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
347	#95	—	2	2	no	—
348	#95	—	2	2	no	—
349	#95	D₄×C₃₁	1	2	yes	—
350	C₃₁	#22758	1	1	yes	characteristic, direct product factor, Sylow
351	C₂⁴	—	2	1	no	—
352	C₂⁴	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	—	2	1	no	—
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	—	2	1	no	—
361	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
362	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
363	C₂×D₄	#41×C₃₁	1	2	yes	—
364	C₂×D₄	#41×C₃₁	1	2	yes	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	4	no	—
368	C₂²×C₄	—	2	4	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	4	4	no	—
372	C₂²×C₄	—	4	4	no	—
373	C₂²×C₄	—	4	2	no	—
374	C₂²×C₄	—	4	2	no	—
375	C₂²×C₄	—	4	2	no	—
376	C₂²×C₄	—	4	2	no	—
377	C₂²×C₄	—	8	2	no	—
378	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
379	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
380	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
383	D₈	—	2	8	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
392	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	4	4	no	—
402	C₄⋊C₄	—	4	2	no	—
403	C₄⋊C₄	—	4	2	no	—
404	C₄⋊C₄	—	8	1	no	—
405	C₄⋊C₄	—	8	1	no	—
406	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
407	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
408	C₂²⋊C₄	—	2	4	no	—
409	C₂²⋊C₄	—	2	4	no	—
410	C₂²⋊C₄	—	4	4	no	—
411	C₂²⋊C₄	—	4	4	no	—
412	C₂²⋊C₄	—	8	2	no	—
413	C₄²	—	2	2	no	—
414	C₄²	—	2	2	no	—
415	C₄²	—	4	4	no	—
416	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
417	C₄²	#41×C₃₁	1	2	yes	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	2	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	4	1	no	—
423	C₂³	—	4	1	no	—
424	C₂³	—	4	1	no	—
425	C₂³	—	4	1	no	—
426	C₂³	—	4	1	no	—
427	C₂³	—	4	1	no	—
428	C₂³	—	4	1	no	—
429	C₂³	—	4	1	no	—
430	C₂³	—	4	1	no	—
431	C₂³	—	4	1	no	—
432	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
433	D₄	—	2	2	no	—
434	D₄	—	2	2	no	—
435	D₄	—	2	2	no	—
436	D₄	—	2	2	no	—
437	D₄	—	2	2	no	—
438	D₄	—	4	1	no	—
439	D₄	—	4	1	no	—
440	D₄	C₃₁×#117	1	4	yes	complemented
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	8	no	—
456	C₂×C₄	—	4	8	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	8	8	no	—
464	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
465	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
466	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
467	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
468	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
469	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
470	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
471	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
472	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
473	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
474	C₈	—	2	2	no	—
475	C₈	—	2	2	no	—
476	C₈	—	4	2	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	8	1	no	—
488	C₂²	—	8	1	no	—
489	C₂²	—	8	1	no	—
490	C₂²	—	8	1	no	—
491	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
492	C₂²	C₃₁×#546	1	1	yes	characteristic
493	C₂²	C₃₁×#548	1	1	yes	characteristic
494	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
495	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
496	C₂²	C₃₁×#576	1	1	yes	characteristic
497	C₂²	C₃₁×#578	1	1	yes	characteristic
498	C₄	—	2	2	no	—
499	C₄	—	4	2	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	2	no	—
503	C₄	—	8	2	no	—
504	C₄	—	8	2	no	—
505	C₄	C₃₁×#442	1	4	yes	—
506	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
507	C₄	C₃₁×#520	1	1	yes	characteristic
508	C₄	C₃₁×#552	1	1	yes	characteristic
509	C₄	C₃₁×#553	1	1	yes	characteristic
510	C₂	—	4	4	no	—
511	C₂	—	8	2	no	—
512	C₂	C₃₁×#2352	1	1	yes	characteristic
513	C₂	C₂×C₃₁×#442	1	1	yes	characteristic
514	C₂	C₃₁×#2951	1	1	yes	characteristic
515	C₂	C₃₁×#3325	1	1	yes	characteristic
516	C₂	C₃₁×#3330	1	1	yes	characteristic
517	C₂	C₃₁×#3366	1	1	yes	characteristic
518	C₂	C₃₁×#3367	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×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2354	C₂	semidirect
3. D₄×C₄⋊C₄×C₃₁	C₂	semidirect
4. C₃₁×#1936	C₂	semidirect
5. C₃₁×#1921	C₂	semidirect
7. C₃₁×#1834	C₂	semidirect
8. C₃₁×#1802	C₂	semidirect
9. C₃₁×#1802	C₂	semidirect
10. C₃₁×#1770	C₂	semidirect
11. C₃₁×#1762	C₂	semidirect
18. C₄×D₈×C₃₁	C₄	semidirect
58. C₄×D₄×C₃₁	C₂×C₄	semidirect
80. C₃₁×#107	C₂×C₄	semidirect
85. C₃₁×#102	C₂×C₄	semidirect
87. C₄×C₈×C₃₁	C₂×C₄	semidirect
158. #22758	C₃₁	direct (N × H)
181. D₄×C₃₁	#117	semidirect
219. #2354	C₂×C₃₁	semidirect
220. D₄×C₄⋊C₄	C₂×C₃₁	semidirect
221. #1936	C₂×C₃₁	semidirect
222. #1921	C₂×C₃₁	semidirect
224. #1834	C₂×C₃₁	semidirect
225. #1802	C₂×C₃₁	semidirect
226. #1802	C₂×C₃₁	semidirect
227. #1770	C₂×C₃₁	semidirect
228. #1762	C₂×C₃₁	semidirect
268. C₄×D₈	C₄×C₃₁	semidirect
317. C₄×D₄	C₂×C₄×C₃₁	semidirect
339. #107	C₂×C₄×C₃₁	semidirect
344. #102	C₂×C₄×C₃₁	semidirect
346. C₄×C₈	C₂×C₄×C₃₁	semidirect
350. C₃₁	#22758	direct (N × H)
440. D₄	C₃₁×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₃₁	2
C₃₁×#2354	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#1936	2
C₃₁×#1921	2
C₃₁×#1879	2
C₃₁×#1834	2
C₃₁×#1802	2
C₃₁×#1770	2
C₃₁×#1762	2
#22758	31

Subgroups — C31×#22758 #21479517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22758 #21479517