Subgroups — #16848888

Order 3968.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14520929	C₂	1	12	yes	maximal, complemented
2	#14520917	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
4	C₂²×D₄×C₃₁	C₂²	1	6	yes	—
5	C₂³×Dic₃₁	C₂²	1	4	yes	—
6	C₃₁×#127	C₄	1	8	yes	complemented
7	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊Dic₃₁	C₂²	1	12	yes	—
9	C₃₁⋊#95	C₂²	1	12	yes	—
10	C₂⁴×C₃₁	C₂³	1	4	yes	—
11	C₂×D₄×C₃₁	—	4	6	no	—
12	C₂×D₄×C₃₁	—	4	6	no	—
13	C₂×D₄×C₃₁	C₂×C₄	1	24	yes	complemented
14	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
15	C₂²×Dic₃₁	—	4	12	no	—
16	C₂²×Dic₃₁	C₂³	1	8	yes	—
17	C₄²×C₃₁	C₂×C₄	1	4	yes	complemented
18	C₂²⋊Dic₃₁	—	4	12	no	—
19	C₂³×C₃₁	—	2	12	no	—
20	C₂³×C₃₁	—	2	12	no	—
21	C₂³×C₃₁	C₂²×C₄	1	8	yes	—
22	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
23	D₄×C₃₁	—	4	24	no	—
24	C₂×C₄×C₃₁	—	2	6	no	—
25	C₂×C₄×C₃₁	C₂²×C₄	1	6	yes	—
26	C₂×Dic₃₁	—	2	24	no	—
27	C₂×Dic₃₁	—	4	16	no	—
28	#2380	—	31	1	no	maximal, Sylow
29	C₂²×C₃₁	—	2	24	no	—
30	C₂²×C₃₁	—	4	4	no	—
31	C₂²×C₃₁	—	4	12	no	—
32	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
33	C₂²×C₃₁	H(5,𝔽₂)	1	6	yes	—
34	C₄×C₃₁	—	2	6	no	—
35	Dic₃₁	—	4	8	no	—
36	C₂×#127	D₃₁	1	1	yes	characteristic
37	#386	—	31	12	no	—
38	#383	—	31	2	no	—
39	C₂×C₃₁	—	4	8	no	—
40	C₂×C₃₁	#518	1	3	yes	—
41	C₂×C₃₁	#560	1	4	yes	—
42	C₂²×D₄	C₂×D₃₁	1	6	yes	—
43	C₂³×C₄	—	31	4	no	—
44	#127	Dic₃₁	1	8	yes	complemented
45	C₂×C₂²⋊C₄	—	31	12	no	—
46	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
47	#95	—	31	12	no	—
48	C₃₁	#2380	1	1	yes	characteristic, complemented, Sylow
49	C₂⁴	C₂²×D₃₁	1	4	yes	—
50	C₂×D₄	—	4	6	no	—
51	C₂×D₄	—	4	6	no	—
52	C₂×D₄	C₂×Dic₃₁	1	24	yes	complemented
53	C₂²×C₄	—	31	8	no	—
54	C₂²×C₄	—	124	12	no	—
55	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
56	C₂²⋊C₄	—	124	12	no	—
57	C₄²	C₂×Dic₃₁	1	4	yes	complemented
58	C₂³	—	2	12	no	—
59	C₂³	—	2	12	no	—
60	C₂³	C₂²×Dic₃₁	1	8	yes	—
61	C₂³	C₂³×D₃₁	1	1	yes	characteristic
62	D₄	—	4	24	no	—
63	C₂×C₄	—	2	6	no	—
64	C₂×C₄	—	62	24	no	—
65	C₂×C₄	—	124	16	no	—
66	C₂×C₄	C₂²×Dic₃₁	1	6	yes	—
67	C₂²	—	2	24	no	—
68	C₂²	—	4	4	no	—
69	C₂²	—	4	12	no	—
70	C₂²	#11758561	1	6	yes	—
71	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
72	C₄	—	2	6	no	—
73	C₄	—	124	8	no	—
74	C₂	—	4	8	no	—
75	C₂	#14521524	1	3	yes	—
76	C₂	#14521786	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting, prime radical	C₂×C₃₁×#127	C₂
socle, cosocle	C₂³×C₃₁	C₂⁴
hypercenter	C₂×#127	D₃₁
hypocenter	C₃₁	#2380
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. #14520929	C₂	semidirect
2. #14520917	C₂	semidirect
6. C₃₁×#127	C₄	semidirect
13. C₂×D₄×C₃₁	C₂×C₄	semidirect
17. C₄²×C₃₁	C₂×C₄	semidirect
44. #127	Dic₃₁	semidirect
48. C₃₁	#2380	semidirect
52. C₂×D₄	C₂×Dic₃₁	semidirect
57. C₄²	C₂×Dic₃₁	semidirect

Maximal subgroups

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

subgroup	index
#14520929	2
#14520917	2
C₂×C₃₁×#127	2
#2380	31