Subgroups — C₃₁×#12063 #21468822

Order 7936.

242 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2169	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1312	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×M₃₂	—	2	1	no	—
9	C₂²×C₁₆×C₃₁	—	2	1	no	—
10	C₄×D₈×C₃₁	C₄	1	1	yes	characteristic
11	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
16	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
17	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#335	C₄	1	1	yes	characteristic
20	C₃₁×D₄⋊C₈	C₄	1	1	yes	characteristic
21	C₃₁×D₄⋊C₈	C₄	1	1	yes	characteristic
22	C₈²×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×D₈×C₃₁	C₈	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	—	2	1	no	—
25	C₂²×C₈×C₃₁	—	2	1	no	—
26	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₃₁×M₃₂	—	4	2	no	—
29	C₂×C₁₆×C₃₁	—	2	1	no	—
30	C₂×C₁₆×C₃₁	—	2	1	no	—
31	C₂×C₁₆×C₃₁	—	2	1	no	—
32	C₂×C₁₆×C₃₁	—	4	2	no	—
33	C₂×C₁₆×C₃₁	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₃₁	D₄	1	1	yes	characteristic
35	C₃₁×#107	C₈	1	1	yes	characteristic
36	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₃₁×#102	C₈	1	1	yes	characteristic
39	C₃₁×#102	C₈	1	1	yes	characteristic
40	C₃₁×C₂²⋊C₈	—	2	1	no	—
41	C₃₁×C₂²⋊C₈	—	2	1	no	—
42	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₃₁	C₂×C₈	1	1	yes	characteristic
46	C₂×D₄×C₃₁	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	—	2	1	no	—
48	C₂²×C₄×C₃₁	—	2	1	no	—
49	D₈×C₃₁	—	2	2	no	—
50	C₂×C₈×C₃₁	—	2	1	no	—
51	C₂×C₈×C₃₁	—	2	1	no	—
52	C₂×C₈×C₃₁	—	2	1	no	—
53	C₂×C₈×C₃₁	—	4	1	no	—
54	C₂×C₈×C₃₁	—	4	1	no	—
55	C₂×C₈×C₃₁	—	4	1	no	—
56	C₂×C₈×C₃₁	—	4	1	no	—
57	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
58	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
61	C₄⋊C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
62	C₄⋊C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
63	C₂²⋊C₄×C₃₁	—	2	1	no	—
64	C₂²⋊C₄×C₃₁	—	2	1	no	—
65	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₁₆×C₃₁	—	2	1	no	—
67	C₁₆×C₃₁	—	2	1	no	—
68	C₁₆×C₃₁	—	4	1	no	—
69	C₁₆×C₃₁	—	4	1	no	—
70	C₁₆×C₃₁	—	4	1	no	—
71	#12063	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₃₁	—	2	1	no	—
73	C₂³×C₃₁	—	2	1	no	—
74	D₄×C₃₁	—	2	1	no	—
75	D₄×C₃₁	—	2	1	no	—
76	D₄×C₃₁	—	2	1	no	—
77	D₄×C₃₁	M₃₂	1	2	yes	complemented
78	C₂×C₄×C₃₁	—	2	1	no	—
79	C₂×C₄×C₃₁	—	2	1	no	—
80	C₂×C₄×C₃₁	—	4	1	no	—
81	C₂×C₄×C₃₁	—	4	1	no	—
82	C₂×C₄×C₃₁	—	4	1	no	—
83	C₂×C₄×C₃₁	—	4	1	no	—
84	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₃₁	#131	1	1	yes	characteristic
87	C₈×C₃₁	—	2	2	no	—
88	C₈×C₃₁	—	2	1	no	—
89	C₈×C₃₁	—	2	1	no	—
90	C₈×C₃₁	—	2	1	no	—
91	C₈×C₃₁	—	4	1	no	—
92	C₈×C₃₁	—	4	1	no	—
93	C₈×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₈×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
95	#2169	C₂×C₃₁	1	1	yes	characteristic, complemented
96	D₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
97	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic
98	#1367	C₂×C₃₁	1	1	yes	characteristic, complemented
99	D₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
100	D₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
101	#1312	C₂×C₃₁	1	1	yes	characteristic, complemented
102	C₂²×C₃₁	—	2	2	no	—
103	C₂²×C₃₁	—	4	1	no	—
104	C₂²×C₃₁	—	4	1	no	—
105	C₂²×C₃₁	—	4	1	no	—
106	C₂²×C₃₁	—	4	1	no	—
107	C₂²×C₃₁	—	4	1	no	—
108	C₂²×C₃₁	C₈×D₄	1	1	yes	characteristic
109	C₄×C₃₁	—	2	1	no	—
110	C₄×C₃₁	—	4	1	no	—
111	C₄×C₃₁	—	4	1	no	—
112	C₄×C₃₁	#442	1	1	yes	characteristic
113	C₄×C₃₁	#444	1	1	yes	characteristic
114	C₄×C₃₁	C₂×M₃₂	1	1	yes	characteristic
115	C₄×C₃₁	#504	1	1	yes	characteristic
116	C₂×M₃₂	—	2	1	no	—
117	C₂²×C₁₆	—	2	1	no	—
118	C₄×D₈	C₄×C₃₁	1	1	yes	characteristic
119	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
120	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
121	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
122	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
123	C₂²⋊C₁₆	—	2	1	no	—
124	C₂²⋊C₁₆	—	2	1	no	—
125	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
126	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
127	#335	C₄×C₃₁	1	1	yes	characteristic
128	D₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
129	D₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
130	C₈²	C₂²×C₃₁	1	1	yes	characteristic
131	C₂×C₃₁	—	4	2	no	—
132	C₂×C₃₁	—	8	1	no	—
133	C₂×C₃₁	#1580	1	1	yes	characteristic
134	C₂×C₃₁	#2168	1	1	yes	characteristic
135	C₂×C₃₁	#2170	1	1	yes	characteristic
136	C₂×D₈	C₈×C₃₁	1	1	yes	characteristic
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	—	2	1	no	—
139	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
140	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
141	M₃₂	—	4	2	no	—
142	C₂×C₁₆	—	2	1	no	—
143	C₂×C₁₆	—	2	1	no	—
144	C₂×C₁₆	—	2	1	no	—
145	C₂×C₁₆	—	4	2	no	—
146	C₂×C₁₆	D₄×C₃₁	1	1	yes	characteristic
147	C₂×C₁₆	D₄×C₃₁	1	1	yes	characteristic
148	#107	C₈×C₃₁	1	1	yes	characteristic
149	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
150	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
151	#102	C₈×C₃₁	1	1	yes	characteristic
152	#102	C₈×C₃₁	1	1	yes	characteristic
153	C₂²⋊C₈	—	2	1	no	—
154	C₂²⋊C₈	—	2	1	no	—
155	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
156	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
157	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
158	C₃₁	#12063	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×D₄	C₂×C₈×C₃₁	1	1	yes	characteristic
160	C₂×D₄	C₂×C₈×C₃₁	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	D₈	—	2	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
172	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
174	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
175	C₄⋊C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
176	C₄⋊C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
177	C₂²⋊C₄	—	2	1	no	—
178	C₂²⋊C₄	—	2	1	no	—
179	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
180	C₁₆	—	2	1	no	—
181	C₁₆	—	2	1	no	—
182	C₁₆	—	4	1	no	—
183	C₁₆	—	4	1	no	—
184	C₁₆	—	4	1	no	—
185	C₂³	—	2	1	no	—
186	C₂³	—	2	1	no	—
187	D₄	—	2	1	no	—
188	D₄	—	2	1	no	—
189	D₄	—	2	1	no	—
190	D₄	C₃₁×M₃₂	1	2	yes	complemented
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
198	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
199	C₂×C₄	C₃₁×#131	1	1	yes	characteristic
200	C₈	—	2	2	no	—
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	4	1	no	—
205	C₈	—	4	1	no	—
206	C₈	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₈	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
208	C₂²	—	2	2	no	—
209	C₂²	—	4	1	no	—
210	C₂²	—	4	1	no	—
211	C₂²	—	4	1	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	C₈×D₄×C₃₁	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	4	1	no	—
217	C₄	—	4	1	no	—
218	C₄	C₃₁×#442	1	1	yes	characteristic
219	C₄	C₃₁×#444	1	1	yes	characteristic
220	C₄	C₂×C₃₁×M₃₂	1	1	yes	characteristic
221	C₄	C₃₁×#504	1	1	yes	characteristic
222	C₂	—	4	2	no	—
223	C₂	—	8	1	no	—
224	C₂	C₃₁×#1580	1	1	yes	characteristic
225	C₂	C₃₁×#2168	1	1	yes	characteristic
226	C₂	C₃₁×#2170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₈×D₄
prime radical	C₂×D₈×C₃₁	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₄×C₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2169	C₂	semidirect
2. D₄×C₁₆×C₃₁	C₂	semidirect
4. C₃₁×#1367	C₂	semidirect
5. C₃₁×D₄⋊C₁₆	C₂	semidirect
6. C₃₁×D₄⋊C₁₆	C₂	semidirect
7. C₃₁×#1312	C₂	semidirect
71. #12063	C₃₁	direct (N × H)
77. D₄×C₃₁	M₃₂	semidirect
95. #2169	C₂×C₃₁	semidirect
96. D₄×C₁₆	C₂×C₃₁	semidirect
98. #1367	C₂×C₃₁	semidirect
99. D₄⋊C₁₆	C₂×C₃₁	semidirect
100. D₄⋊C₁₆	C₂×C₃₁	semidirect
101. #1312	C₂×C₃₁	semidirect
158. C₃₁	#12063	direct (N × H)
190. D₄	C₃₁×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2169	2
D₄×C₁₆×C₃₁	2
C₈×D₈×C₃₁	2
C₃₁×#1367	2
C₃₁×D₄⋊C₁₆	2
C₃₁×#1312	2
#12063	31

Subgroups — C31×#12063 #21468822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12063 #21468822