Subgroups — C₃₁×#9665 #21466424

Order 7936.

586 conjugacy classes of subgroups.

The subgroup lattice has 268 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×#1954	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
8	C₄×D₈×C₃₁	—	2	2	no	—
9	C₄×D₈×C₃₁	—	2	2	no	—
10	C₂×C₃₁×#107	C₄	1	1	yes	characteristic
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₃₁×#102	C₄	1	2	yes	—
13	C₂×C₃₁×C₂²⋊C₈	—	2	4	no	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#334	—	2	2	no	—
18	C₃₁×D₄⋊C₈	—	2	4	no	—
19	C₃₁×#322	—	2	2	no	—
20	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₃₁	—	2	2	no	—
22	C₂×D₈×C₃₁	—	2	4	no	—
23	C₂×D₈×C₃₁	—	2	2	no	—
24	C₂²×C₈×C₃₁	—	2	1	no	—
25	C₂²×C₈×C₃₁	—	2	2	no	—
26	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	D₄	1	2	yes	—
29	C₄×D₄×C₃₁	—	2	2	no	—
30	C₄×D₄×C₃₁	—	2	2	no	—
31	C₄×D₄×C₃₁	—	2	2	no	—
32	C₄×D₄×C₃₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
36	C₃₁×#107	—	2	2	no	—
37	C₃₁×C₄⋊C₈	—	2	2	no	—
38	C₃₁×C₄⋊C₈	—	2	2	no	—
39	C₃₁×#102	—	2	4	no	—
40	C₃₁×C₂²⋊C₈	—	4	8	no	—
41	C₄×C₈×C₃₁	—	2	1	no	—
42	C₄×C₈×C₃₁	—	2	1	no	—
43	C₄×C₈×C₃₁	—	2	1	no	—
44	C₄×C₈×C₃₁	D₄	1	2	yes	—
45	C₄×C₈×C₃₁	D₄	1	2	yes	—
46	C₄×C₈×C₃₁	D₄	1	2	yes	—
47	C₂⁴×C₃₁	—	2	2	no	—
48	C₂×D₄×C₃₁	—	2	2	no	—
49	C₂×D₄×C₃₁	—	2	2	no	—
50	C₂×D₄×C₃₁	—	2	2	no	—
51	C₂×D₄×C₃₁	—	2	2	no	—
52	C₂×D₄×C₃₁	—	2	2	no	—
53	C₂×D₄×C₃₁	M₁₆	1	4	yes	—
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	—	4	2	no	—
56	C₂²×C₄×C₃₁	—	4	2	no	—
57	C₂²×C₄×C₃₁	—	4	4	no	—
58	C₂²×C₄×C₃₁	—	4	4	no	—
59	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
62	D₈×C₃₁	—	2	8	no	—
63	C₂×C₈×C₃₁	—	2	4	no	—
64	C₂×C₈×C₃₁	—	2	2	no	—
65	C₂×C₈×C₃₁	—	2	2	no	—
66	C₂×C₈×C₃₁	—	2	2	no	—
67	C₂×C₈×C₃₁	—	2	2	no	—
68	C₂×C₈×C₃₁	—	2	4	no	—
69	C₂×C₈×C₃₁	—	4	2	no	—
70	C₂×C₈×C₃₁	—	4	4	no	—
71	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
73	C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₂²⋊C₄×C₃₁	—	4	2	no	—
76	C₂²⋊C₄×C₃₁	—	4	2	no	—
77	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₃₁	#41	1	1	yes	characteristic
81	#9665	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₃₁	—	2	4	no	—
83	C₂³×C₃₁	—	4	2	no	—
84	C₂³×C₃₁	—	4	2	no	—
85	C₂³×C₃₁	—	4	4	no	—
86	C₂³×C₃₁	—	4	4	no	—
87	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
88	D₄×C₃₁	—	2	4	no	—
89	D₄×C₃₁	—	2	4	no	—
90	D₄×C₃₁	—	4	2	no	—
91	D₄×C₃₁	—	4	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	4	2	no	—
94	C₂×C₄×C₃₁	—	4	4	no	—
95	C₂×C₄×C₃₁	—	8	4	no	—
96	C₂×C₄×C₃₁	—	8	2	no	—
97	C₂×C₄×C₃₁	—	8	2	no	—
98	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
101	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
102	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
103	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
105	C₂×C₄×C₃₁	#131	1	2	yes	—
106	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
107	C₈×C₃₁	—	2	4	no	—
108	C₈×C₃₁	—	2	2	no	—
109	C₈×C₃₁	—	2	2	no	—
110	C₈×C₃₁	—	4	2	no	—
111	C₈×C₃₁	—	8	2	no	—
112	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
113	#1954	C₂×C₃₁	1	2	yes	complemented
114	C₂×#334	C₂×C₃₁	1	1	yes	characteristic, complemented
115	C₂×D₄⋊C₈	C₂×C₃₁	1	2	yes	complemented
116	C₂×#322	C₂×C₃₁	1	1	yes	characteristic, complemented
117	C₂²×C₃₁	—	4	8	no	—
118	C₂²×C₃₁	—	4	4	no	—
119	C₂²×C₃₁	—	8	4	no	—
120	C₂²×C₃₁	—	8	2	no	—
121	C₂²×C₃₁	—	8	2	no	—
122	C₂²×C₃₁	#390	1	1	yes	characteristic
123	C₂²×C₃₁	#435	1	2	yes	—
124	C₂²×C₃₁	#442	1	2	yes	—
125	C₂²×C₃₁	#443	1	2	yes	—
126	C₄×C₃₁	—	2	1	no	—
127	C₄×C₃₁	—	2	1	no	—
128	C₄×C₃₁	—	8	2	no	—
129	C₄×C₃₁	#407	1	1	yes	characteristic
130	C₄×C₃₁	#408	1	1	yes	characteristic
131	C₄×C₃₁	#435	1	2	yes	—
132	C₄×C₃₁	#469	1	2	yes	—
133	C₄×C₃₁	#496	1	2	yes	—
134	C₂²×D₈	C₄×C₃₁	1	1	yes	characteristic
135	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
136	C₄×D₈	—	2	2	no	—
137	C₄×D₈	—	2	2	no	—
138	C₂×#107	C₄×C₃₁	1	1	yes	characteristic
139	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
140	C₂×#102	C₄×C₃₁	1	2	yes	—
141	C₂×C₂²⋊C₈	—	2	4	no	—
142	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
145	#334	—	2	2	no	—
146	D₄⋊C₈	—	2	4	no	—
147	#322	—	2	2	no	—
148	C₂×C₃₁	—	8	4	no	—
149	C₂×C₃₁	#1586	1	4	yes	—
150	C₂×C₃₁	#1955	1	1	yes	characteristic
151	C₂×C₃₁	#1972	1	1	yes	characteristic
152	C₂×C₃₁	#1974	1	1	yes	characteristic
153	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
154	C₂³×C₄	—	2	2	no	—
155	C₂×D₈	—	2	4	no	—
156	C₂×D₈	—	2	2	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
160	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
161	C₂²×C₈	D₄×C₃₁	1	2	yes	—
162	C₄×D₄	—	2	2	no	—
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	—	2	2	no	—
165	C₄×D₄	—	2	2	no	—
166	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
167	C₂×C₂²⋊C₄	—	2	2	no	—
168	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
169	#107	—	2	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	#102	—	2	4	no	—
173	C₂²⋊C₈	—	4	8	no	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	—	2	1	no	—
177	C₄×C₈	D₄×C₃₁	1	2	yes	—
178	C₄×C₈	D₄×C₃₁	1	2	yes	—
179	C₄×C₈	D₄×C₃₁	1	2	yes	—
180	C₃₁	#9665	1	1	yes	characteristic, direct product factor, Sylow
181	C₂⁴	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	M₁₆×C₃₁	1	4	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
195	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
196	D₈	—	2	8	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	4	no	—
205	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
206	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
212	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
213	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
214	C₄²	#41×C₃₁	1	1	yes	characteristic
215	C₂³	—	2	4	no	—
216	C₂³	—	4	2	no	—
217	C₂³	—	4	4	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	4	2	no	—
220	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
221	D₄	—	2	4	no	—
222	D₄	—	2	4	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
232	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
233	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
234	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
235	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
236	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
237	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
238	C₂×C₄	C₃₁×#131	1	2	yes	—
239	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
240	C₈	—	2	4	no	—
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	8	2	no	—
245	C₂²	—	4	8	no	—
246	C₂²	—	4	4	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	C₃₁×#390	1	1	yes	characteristic
251	C₂²	C₃₁×#435	1	2	yes	—
252	C₂²	C₃₁×#442	1	2	yes	—
253	C₂²	C₃₁×#443	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	8	2	no	—
257	C₄	C₃₁×#407	1	1	yes	characteristic
258	C₄	C₃₁×#408	1	1	yes	characteristic
259	C₄	C₃₁×#435	1	2	yes	—
260	C₄	C₃₁×#469	1	2	yes	—
261	C₄	C₃₁×#496	1	2	yes	—
262	C₂	—	8	4	no	—
263	C₂	C₃₁×#1586	1	4	yes	—
264	C₂	C₃₁×#1955	1	1	yes	characteristic
265	C₂	C₃₁×#1972	1	1	yes	characteristic
266	C₂	C₃₁×#1974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂³×C₃₁	C₄×D₄
prime radical	C₂²×D₈×C₃₁	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1954	C₂	semidirect
3. C₂×C₃₁×#334	C₂	semidirect
4. C₂×C₃₁×D₄⋊C₈	C₂	semidirect
5. C₂×C₃₁×#322	C₂	semidirect
81. #9665	C₃₁	direct (N × H)
113. #1954	C₂×C₃₁	semidirect
114. C₂×#334	C₂×C₃₁	semidirect
115. C₂×D₄⋊C₈	C₂×C₃₁	semidirect
116. C₂×#322	C₂×C₃₁	semidirect
180. C₃₁	#9665	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×C₄×D₈×C₃₁	2
C₃₁×#1954	2
C₂×C₃₁×#334	2
C₂×C₃₁×D₄⋊C₈	2
C₂×C₃₁×#322	2
#9665	31

Subgroups — C31×#9665 #21466424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9665 #21466424