Subgroups — C₃₁×#23676 #21480435

Order 7936.

1092 conjugacy classes of subgroups.

The subgroup lattice has 540 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	2	yes	maximal, complemented
2	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2083	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2071	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2067	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2054	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2041	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2011	C₂	1	1	yes	maximal, characteristic
10	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×#102	C₂	1	2	yes	maximal, complemented
12	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	—	2	1	no	—
15	C₂×C₄×D₄×C₃₁	—	2	1	no	—
16	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#117	—	2	1	no	—
19	C₄×D₈×C₃₁	—	2	8	no	—
20	C₂×C₃₁×#107	C₂²	1	2	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
27	C₃₁×#390	—	2	1	no	—
28	C₃₁×#390	—	2	1	no	—
29	C₃₁×#390	C₂²	1	1	yes	characteristic
30	C₃₁×#390	C₂²	1	2	yes	—
31	C₃₁×#390	C₂²	1	1	yes	characteristic
32	C₃₁×#389	C₂²	1	1	yes	characteristic
33	C₃₁×#389	C₂²	1	1	yes	characteristic
34	C₃₁×#388	—	2	1	no	—
35	C₃₁×#388	—	2	1	no	—
36	C₃₁×#384	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
39	C₄³×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	1	yes	characteristic
43	C₃₁×#336	C₂²	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
48	C₂³×C₄×C₃₁	—	2	1	no	—
49	C₂³×C₄×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	2	4	no	—
51	C₂×D₈×C₃₁	—	2	4	no	—
52	C₂×D₈×C₃₁	—	2	4	no	—
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₃₁×#127	—	2	4	no	—
56	C₄×D₄×C₃₁	—	2	4	no	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	—	2	4	no	—
59	C₄×D₄×C₃₁	—	4	4	no	—
60	C₄×D₄×C₃₁	—	4	4	no	—
61	C₄×D₄×C₃₁	D₄	1	4	yes	complemented
62	C₃₁×#117	—	4	4	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
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×C₂²⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
79	C₂×C₄²×C₃₁	—	2	1	no	—
80	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
83	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
87	C₃₁×#107	—	2	4	no	—
88	C₃₁×C₄⋊C₈	—	2	4	no	—
89	C₃₁×#102	—	2	4	no	—
90	C₃₁×#102	—	2	4	no	—
91	C₃₁×#102	—	2	4	no	—
92	C₃₁×#102	—	2	4	no	—
93	C₄×C₈×C₃₁	—	2	4	no	—
94	C₃₁×#95	—	2	1	no	—
95	C₃₁×#95	—	2	1	no	—
96	C₃₁×#95	—	2	1	no	—
97	C₂⁴×C₃₁	—	2	1	no	—
98	C₂⁴×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	4	no	—
100	C₂×D₄×C₃₁	—	2	4	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	4	2	no	—
110	C₂×D₄×C₃₁	—	4	2	no	—
111	C₂×D₄×C₃₁	—	4	2	no	—
112	C₂×D₄×C₃₁	—	4	2	no	—
113	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
114	C₂×D₄×C₃₁	#41	1	4	yes	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	4	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	4	2	no	—
121	C₂²×C₄×C₃₁	—	4	2	no	—
122	C₂²×C₄×C₃₁	—	4	4	no	—
123	C₂²×C₄×C₃₁	—	4	4	no	—
124	C₂²×C₄×C₃₁	—	4	4	no	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
132	D₈×C₃₁	—	2	16	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	#41	1	4	yes	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	2	2	no	—
141	C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	4	2	no	—
145	C₄⋊C₄×C₃₁	—	4	2	no	—
146	C₄⋊C₄×C₃₁	—	4	2	no	—
147	C₄⋊C₄×C₃₁	—	4	2	no	—
148	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
149	C₂²⋊C₄×C₃₁	—	2	4	no	—
150	C₂²⋊C₄×C₃₁	—	4	2	no	—
151	C₂²⋊C₄×C₃₁	—	4	4	no	—
152	C₂²⋊C₄×C₃₁	—	4	4	no	—
153	C₂²⋊C₄×C₃₁	—	4	2	no	—
154	C₂²⋊C₄×C₃₁	—	4	2	no	—
155	C₂²⋊C₄×C₃₁	—	4	4	no	—
156	C₂²⋊C₄×C₃₁	—	4	2	no	—
157	C₄²×C₃₁	—	2	2	no	—
158	C₄²×C₃₁	—	2	2	no	—
159	C₄²×C₃₁	—	2	2	no	—
160	C₄²×C₃₁	—	2	2	no	—
161	C₄²×C₃₁	—	4	1	no	—
162	C₄²×C₃₁	—	4	2	no	—
163	C₄²×C₃₁	D₈	1	4	yes	complemented
164	C₄²×C₃₁	C₂×D₄	1	2	yes	—
165	C₄²×C₃₁	#41	1	2	yes	—
166	#23676	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₃₁	—	2	2	no	—
168	C₂³×C₃₁	—	2	4	no	—
169	C₂³×C₃₁	—	4	2	no	—
170	C₂³×C₃₁	—	4	1	no	—
171	C₂³×C₃₁	—	4	2	no	—
172	C₂³×C₃₁	—	4	1	no	—
173	C₂³×C₃₁	—	4	1	no	—
174	C₂³×C₃₁	—	4	2	no	—
175	C₂³×C₃₁	—	4	1	no	—
176	C₂³×C₃₁	—	4	2	no	—
177	C₂³×C₃₁	—	4	2	no	—
178	C₂³×C₃₁	—	4	1	no	—
179	C₂³×C₃₁	—	4	1	no	—
180	C₂³×C₃₁	—	4	2	no	—
181	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
182	D₄×C₃₁	—	2	4	no	—
183	D₄×C₃₁	—	2	4	no	—
184	D₄×C₃₁	—	2	4	no	—
185	D₄×C₃₁	—	4	8	no	—
186	D₄×C₃₁	—	4	8	no	—
187	D₄×C₃₁	—	4	2	no	—
188	D₄×C₃₁	—	4	2	no	—
189	D₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	—	2	1	no	—
195	C₂×C₄×C₃₁	—	2	1	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	1	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	8	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	1	no	—
207	C₂×C₄×C₃₁	—	4	1	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	8	8	no	—
215	C₂×C₄×C₃₁	#121	1	2	yes	—
216	C₂×C₄×C₃₁	#121	1	2	yes	—
217	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
218	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
219	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
220	C₂×C₄×C₃₁	#135	1	2	yes	—
221	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
222	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
224	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
225	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
226	C₈×C₃₁	—	2	4	no	—
227	C₈×C₃₁	—	4	2	no	—
228	C₂×C₄×D₈	C₂×C₃₁	1	2	yes	complemented
229	C₄×#127	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#2293	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#2083	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#2071	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#2067	C₂×C₃₁	1	2	yes	complemented
234	#2054	C₂×C₃₁	1	2	yes	complemented
235	#2041	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#2011	C₂×C₃₁	1	1	yes	characteristic
237	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₄×#102	C₂×C₃₁	1	2	yes	complemented
239	C₂²×C₃₁	—	4	4	no	—
240	C₂²×C₃₁	—	4	4	no	—
241	C₂²×C₃₁	—	4	2	no	—
242	C₂²×C₃₁	—	4	4	no	—
243	C₂²×C₃₁	—	4	2	no	—
244	C₂²×C₃₁	—	4	2	no	—
245	C₂²×C₃₁	—	8	1	no	—
246	C₂²×C₃₁	—	8	1	no	—
247	C₂²×C₃₁	—	8	1	no	—
248	C₂²×C₃₁	—	8	2	no	—
249	C₂²×C₃₁	—	8	2	no	—
250	C₂²×C₃₁	—	8	1	no	—
251	C₂²×C₃₁	—	8	2	no	—
252	C₂²×C₃₁	—	8	2	no	—
253	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
254	C₂²×C₃₁	#525	1	1	yes	characteristic
255	C₂²×C₃₁	#532	1	1	yes	characteristic
256	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
257	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
258	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
259	C₂²×C₃₁	#575	1	1	yes	characteristic
260	C₄×C₃₁	—	2	2	no	—
261	C₄×C₃₁	—	2	2	no	—
262	C₄×C₃₁	—	2	2	no	—
263	C₄×C₃₁	—	2	2	no	—
264	C₄×C₃₁	—	4	1	no	—
265	C₄×C₃₁	—	4	2	no	—
266	C₄×C₃₁	—	8	1	no	—
267	C₄×C₃₁	—	8	1	no	—
268	C₄×C₃₁	—	8	2	no	—
269	C₄×C₃₁	#459	1	4	yes	—
270	C₄×C₃₁	#525	1	2	yes	—
271	C₄×C₃₁	#529	1	2	yes	—
272	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
273	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×C₄×D₄	—	2	1	no	—
275	C₂×C₄×D₄	—	2	1	no	—
276	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
277	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×#117	—	2	1	no	—
279	C₄×D₈	—	2	8	no	—
280	C₂×#107	C₂²×C₃₁	1	2	yes	—
281	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
282	C₂×#102	C₂²×C₃₁	1	2	yes	—
283	C₂×#102	C₂²×C₃₁	1	2	yes	—
284	C₂×#102	C₂²×C₃₁	1	2	yes	—
285	C₂×#102	C₂²×C₃₁	1	2	yes	—
286	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
287	#390	—	2	1	no	—
288	#390	—	2	1	no	—
289	#390	C₂²×C₃₁	1	2	yes	—
290	#390	C₂²×C₃₁	1	1	yes	characteristic
291	#390	C₂²×C₃₁	1	1	yes	characteristic
292	#389	C₂²×C₃₁	1	1	yes	characteristic
293	#389	C₂²×C₃₁	1	1	yes	characteristic
294	#388	—	2	1	no	—
295	#388	—	2	1	no	—
296	#384	C₂²×C₃₁	1	1	yes	characteristic
297	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
298	C₄×C₂²⋊C₄	—	2	1	no	—
299	C₄³	C₂²×C₃₁	1	1	yes	characteristic
300	#340	C₂²×C₃₁	1	2	yes	—
301	#340	C₂²×C₃₁	1	2	yes	—
302	#336	C₂²×C₃₁	1	1	yes	characteristic
303	#336	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×C₃₁	—	8	4	no	—
305	C₂×C₃₁	—	8	2	no	—
306	C₂×C₃₁	#2392	1	1	yes	characteristic
307	C₂×C₃₁	C₂×#459	1	1	yes	characteristic
308	C₂×C₃₁	#3035	1	1	yes	characteristic
309	C₂×C₃₁	#3100	1	1	yes	characteristic
310	C₂×C₃₁	#3105	1	1	yes	characteristic
311	C₂×C₃₁	#3169	1	1	yes	characteristic
312	C₂×C₃₁	#3171	1	1	yes	characteristic
313	C₂²×D₄	—	2	1	no	—
314	C₂²×D₄	—	2	1	no	—
315	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
316	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
317	C₂³×C₄	—	2	1	no	—
318	C₂³×C₄	—	2	2	no	—
319	C₂×D₈	—	2	4	no	—
320	C₂×D₈	—	2	4	no	—
321	C₂×D₈	—	2	4	no	—
322	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
323	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
324	#127	—	2	4	no	—
325	C₄×D₄	—	2	4	no	—
326	C₄×D₄	—	2	2	no	—
327	C₄×D₄	—	2	4	no	—
328	C₄×D₄	—	4	4	no	—
329	C₄×D₄	—	4	4	no	—
330	C₄×D₄	D₄×C₃₁	1	4	yes	complemented
331	#117	—	4	4	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	1	no	—
336	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
337	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
338	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
339	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
340	C₂×C₂²⋊C₄	—	2	1	no	—
341	C₂×C₂²⋊C₄	—	2	1	no	—
342	C₂×C₂²⋊C₄	—	2	1	no	—
343	C₂×C₂²⋊C₄	—	2	2	no	—
344	C₂×C₂²⋊C₄	—	2	2	no	—
345	C₂×C₂²⋊C₄	—	2	2	no	—
346	C₂×C₂²⋊C₄	—	2	1	no	—
347	C₂×C₂²⋊C₄	—	2	1	no	—
348	C₂×C₄²	—	2	1	no	—
349	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
350	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
351	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
352	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
353	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
354	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
355	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
356	#107	—	2	4	no	—
357	C₄⋊C₈	—	2	4	no	—
358	#102	—	2	4	no	—
359	#102	—	2	4	no	—
360	#102	—	2	4	no	—
361	#102	—	2	4	no	—
362	C₄×C₈	—	2	4	no	—
363	#95	—	2	1	no	—
364	#95	—	2	1	no	—
365	#95	—	2	1	no	—
366	C₃₁	#23676	1	1	yes	characteristic, direct product factor, Sylow
367	C₂⁴	—	2	1	no	—
368	C₂⁴	—	2	2	no	—
369	C₂×D₄	—	2	4	no	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	4	2	no	—
380	C₂×D₄	—	4	2	no	—
381	C₂×D₄	—	4	2	no	—
382	C₂×D₄	—	4	2	no	—
383	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
384	C₂×D₄	#41×C₃₁	1	4	yes	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	4	2	no	—
391	C₂²×C₄	—	4	2	no	—
392	C₂²×C₄	—	4	4	no	—
393	C₂²×C₄	—	4	4	no	—
394	C₂²×C₄	—	4	4	no	—
395	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
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	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
401	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
402	D₈	—	2	16	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	#41×C₃₁	1	4	yes	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	4	2	no	—
415	C₄⋊C₄	—	4	2	no	—
416	C₄⋊C₄	—	4	2	no	—
417	C₄⋊C₄	—	4	2	no	—
418	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
419	C₂²⋊C₄	—	2	4	no	—
420	C₂²⋊C₄	—	4	2	no	—
421	C₂²⋊C₄	—	4	4	no	—
422	C₂²⋊C₄	—	4	4	no	—
423	C₂²⋊C₄	—	4	2	no	—
424	C₂²⋊C₄	—	4	2	no	—
425	C₂²⋊C₄	—	4	4	no	—
426	C₂²⋊C₄	—	4	2	no	—
427	C₄²	—	2	2	no	—
428	C₄²	—	2	2	no	—
429	C₄²	—	2	2	no	—
430	C₄²	—	2	2	no	—
431	C₄²	—	4	1	no	—
432	C₄²	—	4	2	no	—
433	C₄²	D₈×C₃₁	1	4	yes	complemented
434	C₄²	C₂×D₄×C₃₁	1	2	yes	—
435	C₄²	#41×C₃₁	1	2	yes	—
436	C₂³	—	2	2	no	—
437	C₂³	—	2	4	no	—
438	C₂³	—	4	2	no	—
439	C₂³	—	4	1	no	—
440	C₂³	—	4	2	no	—
441	C₂³	—	4	1	no	—
442	C₂³	—	4	1	no	—
443	C₂³	—	4	2	no	—
444	C₂³	—	4	1	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	4	1	no	—
448	C₂³	—	4	1	no	—
449	C₂³	—	4	2	no	—
450	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
451	D₄	—	2	4	no	—
452	D₄	—	2	4	no	—
453	D₄	—	2	4	no	—
454	D₄	—	4	8	no	—
455	D₄	—	4	8	no	—
456	D₄	—	4	2	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	2	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	8	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	8	8	no	—
484	C₂×C₄	C₃₁×#121	1	2	yes	—
485	C₂×C₄	C₃₁×#121	1	2	yes	—
486	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
487	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
488	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
489	C₂×C₄	C₃₁×#135	1	2	yes	—
490	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
491	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
492	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
493	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
494	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
495	C₈	—	2	4	no	—
496	C₈	—	4	2	no	—
497	C₂²	—	4	4	no	—
498	C₂²	—	4	4	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	4	no	—
501	C₂²	—	4	2	no	—
502	C₂²	—	4	2	no	—
503	C₂²	—	8	1	no	—
504	C₂²	—	8	1	no	—
505	C₂²	—	8	1	no	—
506	C₂²	—	8	2	no	—
507	C₂²	—	8	2	no	—
508	C₂²	—	8	1	no	—
509	C₂²	—	8	2	no	—
510	C₂²	—	8	2	no	—
511	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
512	C₂²	C₃₁×#525	1	1	yes	characteristic
513	C₂²	C₃₁×#532	1	1	yes	characteristic
514	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
515	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
516	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
517	C₂²	C₃₁×#575	1	1	yes	characteristic
518	C₄	—	2	2	no	—
519	C₄	—	2	2	no	—
520	C₄	—	2	2	no	—
521	C₄	—	2	2	no	—
522	C₄	—	4	1	no	—
523	C₄	—	4	2	no	—
524	C₄	—	8	1	no	—
525	C₄	—	8	1	no	—
526	C₄	—	8	2	no	—
527	C₄	C₃₁×#459	1	4	yes	—
528	C₄	C₃₁×#525	1	2	yes	—
529	C₄	C₃₁×#529	1	2	yes	—
530	C₂	—	8	4	no	—
531	C₂	—	8	2	no	—
532	C₂	C₃₁×#2392	1	1	yes	characteristic
533	C₂	C₂×C₃₁×#459	1	1	yes	characteristic
534	C₂	C₃₁×#3035	1	1	yes	characteristic
535	C₂	C₃₁×#3100	1	1	yes	characteristic
536	C₂	C₃₁×#3105	1	1	yes	characteristic
537	C₂	C₃₁×#3169	1	1	yes	characteristic
538	C₂	C₃₁×#3171	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂²×D₄
prime radical	C₃₁×#2011	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×D₈×C₃₁	C₂	semidirect
2. C₄×C₃₁×#127	C₂	semidirect
3. C₃₁×#2293	C₂	semidirect
4. C₃₁×#2083	C₂	semidirect
5. C₃₁×#2071	C₂	semidirect
6. C₃₁×#2067	C₂	semidirect
7. C₃₁×#2054	C₂	semidirect
8. C₃₁×#2041	C₂	semidirect
10. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
11. C₄×C₃₁×#102	C₂	semidirect
61. C₄×D₄×C₃₁	D₄	semidirect
163. C₄²×C₃₁	D₈	semidirect
166. #23676	C₃₁	direct (N × H)
228. C₂×C₄×D₈	C₂×C₃₁	semidirect
229. C₄×#127	C₂×C₃₁	semidirect
230. #2293	C₂×C₃₁	semidirect
231. #2083	C₂×C₃₁	semidirect
232. #2071	C₂×C₃₁	semidirect
233. #2067	C₂×C₃₁	semidirect
234. #2054	C₂×C₃₁	semidirect
235. #2041	C₂×C₃₁	semidirect
237. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
238. C₄×#102	C₂×C₃₁	semidirect
330. C₄×D₄	D₄×C₃₁	semidirect
366. C₃₁	#23676	direct (N × H)
433. C₄²	D₈×C₃₁	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
C₄×C₃₁×#127	2
C₃₁×#2293	2
C₃₁×#2083	2
C₃₁×#2071	2
C₃₁×#2067	2
C₃₁×#2054	2
C₃₁×#2041	2
C₃₁×#2011	2
C₄×C₃₁×C₄⋊C₈	2
C₄×C₃₁×#102	2
#23676	31

Subgroups — C31×#23676 #21480435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23676 #21480435