Subgroups — C₂×C₃₁×#2656 #21491330

Order 7936.

4886 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#121	C₂	1	6	yes	maximal, complemented
2	C₃₁×#2656	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃₁×#399	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×#392	C₂	1	4	yes	maximal, complemented
5	C₂²×C₃₁×#95	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄×C₃₁	C₂²	1	4	yes	—
7	C₂³×D₄×C₃₁	C₂²	1	6	yes	—
8	C₂⁴×C₄×C₃₁	C₂²	1	3	yes	—
9	C₂×C₃₁×#121	—	2	48	no	—
10	C₂×C₃₁×#121	C₂²	1	48	yes	complemented
11	C₂²×C₄⋊C₄×C₃₁	C₂²	1	6	yes	—
12	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	12	yes	—
13	C₃₁×#399	C₂²	1	32	yes	complemented
14	C₃₁×#392	C₂²	1	32	yes	complemented
15	C₂×C₃₁×#95	C₂²	1	4	yes	—
16	C₂×C₃₁×#95	C₂²	1	8	yes	complemented
17	C₂⁵×C₃₁	C₂³	1	4	yes	—
18	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
19	C₂²×D₄×C₃₁	—	2	48	no	—
20	C₂²×D₄×C₃₁	—	2	24	no	—
21	C₂²×D₄×C₃₁	—	4	24	no	—
22	C₂²×D₄×C₃₁	C₂³	1	16	yes	—
23	C₂²×D₄×C₃₁	C₂³	1	24	yes	—
24	C₂³×C₄×C₃₁	—	4	12	no	—
25	C₂³×C₄×C₃₁	D₄	1	24	yes	complemented
26	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
27	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
28	C₂³×C₄×C₃₁	C₂³	1	12	yes	—
29	C₃₁×#121	—	2	192	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	24	no	—
31	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
32	C₂×C₂²⋊C₄×C₃₁	—	2	48	no	—
33	C₂×C₂²⋊C₄×C₃₁	C₂³	1	48	yes	—
34	C₃₁×#95	C₂³	1	16	yes	—
35	C₂⁴×C₃₁	—	2	6	no	—
36	C₂⁴×C₃₁	—	2	2	no	—
37	C₂⁴×C₃₁	—	2	24	no	—
38	C₂⁴×C₃₁	—	2	24	no	—
39	C₂⁴×C₃₁	—	2	8	no	—
40	C₂⁴×C₃₁	C₂×D₄	1	12	yes	—
41	C₂⁴×C₃₁	C₂⁴	1	8	yes	—
42	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
43	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
44	C₂×D₄×C₃₁	—	2	96	no	—
45	C₂×D₄×C₃₁	—	2	48	no	—
46	C₂×D₄×C₃₁	—	4	48	no	—
47	C₂×D₄×C₃₁	—	4	96	no	—
48	C₂×D₄×C₃₁	—	4	64	no	—
49	C₂²×C₄×C₃₁	—	2	24	no	—
50	C₂²×C₄×C₃₁	—	2	24	no	—
51	C₂²×C₄×C₃₁	—	4	48	no	—
52	C₂²×C₄×C₃₁	—	4	24	no	—
53	C₂²×C₄×C₃₁	C₂×D₄	1	24	yes	—
54	C₂²×C₄×C₃₁	C₂×D₄	1	48	yes	complemented
55	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
56	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
57	C₄⋊C₄×C₃₁	—	2	48	no	—
58	C₂²⋊C₄×C₃₁	—	2	96	no	—
59	C₂×#2656	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₃₁	—	2	24	no	—
61	C₂³×C₃₁	—	2	24	no	—
62	C₂³×C₃₁	—	2	6	no	—
63	C₂³×C₃₁	—	2	2	no	—
64	C₂³×C₃₁	—	2	24	no	—
65	C₂³×C₃₁	—	2	24	no	—
66	C₂³×C₃₁	—	2	8	no	—
67	C₂³×C₃₁	—	4	16	no	—
68	C₂³×C₃₁	—	4	48	no	—
69	C₂³×C₃₁	—	4	48	no	—
70	C₂³×C₃₁	#127	1	16	yes	—
71	C₂³×C₃₁	C₂²×D₄	1	12	yes	—
72	C₂³×C₃₁	C₂²×D₄	1	6	yes	—
73	C₂³×C₃₁	H(5,𝔽₂)	1	6	yes	—
74	C₂³×C₃₁	2₋¹⁺⁴	1	2	yes	—
75	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
76	D₄×C₃₁	—	4	96	no	—
77	D₄×C₃₁	—	4	64	no	—
78	C₂×C₄×C₃₁	—	2	48	no	—
79	C₂×C₄×C₃₁	—	2	24	no	—
80	C₂×C₄×C₃₁	—	2	24	no	—
81	C₂×C₄×C₃₁	—	4	48	no	—
82	C₂×C₄×C₃₁	—	4	16	no	—
83	C₂×C₄×C₃₁	C₂²×D₄	1	24	yes	—
84	C₂²×#121	C₂×C₃₁	1	6	yes	complemented
85	#2656	C₂×C₃₁	1	16	yes	direct product factor
86	C₂×#399	C₂×C₃₁	1	4	yes	complemented
87	C₂×#392	C₂×C₃₁	1	4	yes	complemented
88	C₂²×#95	C₂×C₃₁	1	1	yes	characteristic, complemented
89	C₂²×C₃₁	—	2	32	no	—
90	C₂²×C₃₁	—	2	12	no	—
91	C₂²×C₃₁	—	2	12	no	—
92	C₂²×C₃₁	—	4	8	no	—
93	C₂²×C₃₁	—	4	24	no	—
94	C₂²×C₃₁	—	4	24	no	—
95	C₂²×C₃₁	—	8	32	no	—
96	C₂²×C₃₁	C₂×#127	1	8	yes	—
97	C₂²×C₃₁	C₂×#127	1	4	yes	—
98	C₂²×C₃₁	#534	1	12	yes	—
99	C₂²×C₃₁	#537	1	12	yes	—
100	C₂²×C₃₁	C₂³×D₄	1	3	yes	—
101	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	3	yes	—
102	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₃₁	—	2	24	no	—
104	C₄×C₃₁	—	4	8	no	—
105	C₂³×D₄	C₂²×C₃₁	1	4	yes	—
106	C₂³×D₄	C₂²×C₃₁	1	6	yes	—
107	C₂⁴×C₄	C₂²×C₃₁	1	3	yes	—
108	C₂×#121	—	2	48	no	—
109	C₂×#121	C₂²×C₃₁	1	48	yes	complemented
110	C₂²×C₄⋊C₄	C₂²×C₃₁	1	6	yes	—
111	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	12	yes	—
112	#399	C₂²×C₃₁	1	32	yes	complemented
113	#392	C₂²×C₃₁	1	32	yes	complemented
114	C₂×#95	C₂²×C₃₁	1	4	yes	—
115	C₂×#95	C₂²×C₃₁	1	8	yes	complemented
116	C₂×C₃₁	—	2	8	no	—
117	C₂×C₃₁	—	8	8	no	—
118	C₂×C₃₁	#2656	1	8	yes	direct product factor
119	C₂×C₃₁	C₂²×#127	1	1	yes	characteristic
120	C₂×C₃₁	C₂×#534	1	3	yes	—
121	C₂×C₃₁	C₂×#537	1	3	yes	—
122	C₂⁵	C₂³×C₃₁	1	4	yes	—
123	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
124	C₂²×D₄	—	2	48	no	—
125	C₂²×D₄	—	2	24	no	—
126	C₂²×D₄	—	4	24	no	—
127	C₂²×D₄	C₂³×C₃₁	1	16	yes	—
128	C₂²×D₄	C₂³×C₃₁	1	24	yes	—
129	C₂³×C₄	—	4	12	no	—
130	C₂³×C₄	D₄×C₃₁	1	24	yes	complemented
131	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
132	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
133	C₂³×C₄	C₂³×C₃₁	1	12	yes	—
134	#121	—	2	192	no	—
135	C₂×C₄⋊C₄	—	2	24	no	—
136	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
137	C₂×C₂²⋊C₄	—	2	48	no	—
138	C₂×C₂²⋊C₄	C₂³×C₃₁	1	48	yes	—
139	#95	C₂³×C₃₁	1	16	yes	—
140	C₃₁	C₂×#2656	1	1	yes	characteristic, direct product factor, Sylow
141	C₂⁴	—	2	6	no	—
142	C₂⁴	—	2	2	no	—
143	C₂⁴	—	2	24	no	—
144	C₂⁴	—	2	24	no	—
145	C₂⁴	—	2	8	no	—
146	C₂⁴	C₂×D₄×C₃₁	1	12	yes	—
147	C₂⁴	C₂⁴×C₃₁	1	8	yes	—
148	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
149	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
150	C₂×D₄	—	2	96	no	—
151	C₂×D₄	—	2	48	no	—
152	C₂×D₄	—	4	48	no	—
153	C₂×D₄	—	4	96	no	—
154	C₂×D₄	—	4	64	no	—
155	C₂²×C₄	—	2	24	no	—
156	C₂²×C₄	—	2	24	no	—
157	C₂²×C₄	—	4	48	no	—
158	C₂²×C₄	—	4	24	no	—
159	C₂²×C₄	C₂×D₄×C₃₁	1	24	yes	—
160	C₂²×C₄	C₂×D₄×C₃₁	1	48	yes	complemented
161	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
162	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
163	C₄⋊C₄	—	2	48	no	—
164	C₂²⋊C₄	—	2	96	no	—
165	C₂³	—	2	24	no	—
166	C₂³	—	2	24	no	—
167	C₂³	—	2	6	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	24	no	—
170	C₂³	—	2	24	no	—
171	C₂³	—	2	8	no	—
172	C₂³	—	4	16	no	—
173	C₂³	—	4	48	no	—
174	C₂³	—	4	48	no	—
175	C₂³	C₃₁×#127	1	16	yes	—
176	C₂³	C₂²×D₄×C₃₁	1	12	yes	—
177	C₂³	C₂²×D₄×C₃₁	1	6	yes	—
178	C₂³	C₃₁×H(5,𝔽₂)	1	6	yes	—
179	C₂³	C₃₁×2₋¹⁺⁴	1	2	yes	—
180	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
181	D₄	—	4	96	no	—
182	D₄	—	4	64	no	—
183	C₂×C₄	—	2	48	no	—
184	C₂×C₄	—	2	24	no	—
185	C₂×C₄	—	2	24	no	—
186	C₂×C₄	—	4	48	no	—
187	C₂×C₄	—	4	16	no	—
188	C₂×C₄	C₂²×D₄×C₃₁	1	24	yes	—
189	C₂²	—	2	32	no	—
190	C₂²	—	2	12	no	—
191	C₂²	—	2	12	no	—
192	C₂²	—	4	8	no	—
193	C₂²	—	4	24	no	—
194	C₂²	—	4	24	no	—
195	C₂²	—	8	32	no	—
196	C₂²	C₂×C₃₁×#127	1	8	yes	—
197	C₂²	C₂×C₃₁×#127	1	4	yes	—
198	C₂²	C₃₁×#534	1	12	yes	—
199	C₂²	C₃₁×#537	1	12	yes	—
200	C₂²	C₂³×D₄×C₃₁	1	3	yes	—
201	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	3	yes	—
202	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
203	C₄	—	2	24	no	—
204	C₄	—	4	8	no	—
205	C₂	—	2	8	no	—
206	C₂	—	8	8	no	—
207	C₂	C₃₁×#2656	1	8	yes	direct product factor
208	C₂	C₂²×C₃₁×#127	1	1	yes	characteristic
209	C₂	C₂×C₃₁×#534	1	3	yes	—
210	C₂	C₂×C₃₁×#537	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×#121	C₂	semidirect
2. C₃₁×#2656	C₂	direct (N × H)
3. C₂×C₃₁×#399	C₂	semidirect
4. C₂×C₃₁×#392	C₂	semidirect
5. C₂²×C₃₁×#95	C₂	semidirect
10. C₂×C₃₁×#121	C₂²	semidirect
13. C₃₁×#399	C₂²	semidirect
14. C₃₁×#392	C₂²	semidirect
16. C₂×C₃₁×#95	C₂²	semidirect
25. C₂³×C₄×C₃₁	D₄	semidirect
54. C₂²×C₄×C₃₁	C₂×D₄	semidirect
59. C₂×#2656	C₃₁	direct (N × H)
84. C₂²×#121	C₂×C₃₁	semidirect
85. #2656	C₂×C₃₁	direct (N × H)
86. C₂×#399	C₂×C₃₁	semidirect
87. C₂×#392	C₂×C₃₁	semidirect
88. C₂²×#95	C₂×C₃₁	semidirect
109. C₂×#121	C₂²×C₃₁	semidirect
112. #399	C₂²×C₃₁	semidirect
113. #392	C₂²×C₃₁	semidirect
115. C₂×#95	C₂²×C₃₁	semidirect
118. C₂×C₃₁	#2656	direct (N × H)
130. C₂³×C₄	D₄×C₃₁	semidirect
140. C₃₁	C₂×#2656	direct (N × H)
160. C₂²×C₄	C₂×D₄×C₃₁	semidirect
207. C₂	C₃₁×#2656	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#121	2
C₃₁×#2656	2
C₂×C₃₁×#399	2
C₂×C₃₁×#392	2
C₂²×C₃₁×#95	2
C₂×#2656	31

Subgroups — C2×C31×#2656 #21491330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2656 #21491330