Subgroups — #14581073

Order 2624.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13050158	C₂	1	2	yes	maximal, complemented
2	#13050145	C₂	1	3	yes	maximal, complemented
3	#13050136	C₂	1	6	yes	maximal, complemented
4	#13050132	C₂	1	3	yes	maximal, complemented
5	#127×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄₁	C₂²	1	1	yes	characteristic
7	C₂×D₄×C₄₁	C₂²	1	3	yes	—
8	C₂×D₄×C₄₁	C₂²	1	3	yes	complemented
9	C₂×C₂²⋊D₄₁	C₂²	1	6	yes	—
10	C₂²×Dic₄₁	C₂²	1	3	yes	complemented
11	C₄²×C₄₁	C₂²	1	1	yes	characteristic, complemented
12	C₂²⋊Dic₄₁	C₂²	1	3	yes	—
13	C₂²⋊Dic₄₁	C₂²	1	3	yes	complemented
14	#10620281	C₂²	1	6	yes	—
15	#10620279	C₂²	1	6	yes	complemented
16	C₂³×C₄₁	C₂³	1	1	yes	characteristic
17	C₂³×C₄₁	C₂³	1	3	yes	—
18	C₂²×D₄₁	—	4	3	no	—
19	C₂²×D₄₁	C₂³	1	2	yes	—
20	D₄×C₄₁	—	4	3	no	—
21	D₄×C₄₁	—	4	3	no	—
22	C₂×C₄×C₄₁	C₂³	1	3	yes	—
23	C₂²⋊D₄₁	—	4	6	no	—
24	C₂×Dic₄₁	—	4	3	no	—
25	C₂×Dic₄₁	C₂³	1	6	yes	—
26	C₂²×C₄₁	—	2	3	no	—
27	C₂²×C₄₁	—	2	6	no	—
28	C₂²×C₄₁	—	2	3	no	—
29	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
30	C₂×D₄₁	—	2	6	no	—
31	C₂×D₄₁	—	4	4	no	—
32	C₄×C₄₁	—	2	3	no	—
33	Dic₄₁	—	2	6	no	—
34	C₂×C₄₁	—	4	1	no	—
35	C₂×C₄₁	—	4	3	no	—
36	C₂×C₄₁	H(5,𝔽₂)	1	3	yes	—
37	D₄₁	—	4	2	no	—
38	#560	—	41	1	no	maximal, Sylow
39	C₄₁	#560	1	1	yes	characteristic, complemented, Sylow
40	#127	D₄₁	1	1	yes	characteristic, complemented
41	#126	—	41	2	no	—
42	#123	—	41	3	no	—
43	#121	—	41	6	no	—
44	#120	—	41	3	no	—
45	C₂⁴	—	41	1	no	—
46	C₂×D₄	—	41	6	no	—
47	C₂×D₄	C₂×D₄₁	1	3	yes	—
48	C₂×D₄	C₂×D₄₁	1	3	yes	complemented
49	C₂²×C₄	—	41	3	no	—
50	C₄⋊C₄	—	41	6	no	—
51	C₂²⋊C₄	—	41	3	no	—
52	C₂²⋊C₄	—	41	3	no	—
53	C₂²⋊C₄	—	41	6	no	—
54	C₄²	C₂×D₄₁	1	1	yes	characteristic, complemented
55	C₂³	—	41	2	no	—
56	C₂³	—	164	3	no	—
57	C₂³	C₂²×D₄₁	1	1	yes	characteristic
58	C₂³	C₂²×D₄₁	1	3	yes	—
59	D₄	—	4	3	no	—
60	D₄	—	4	3	no	—
61	D₄	—	164	6	no	—
62	C₂×C₄	—	41	6	no	—
63	C₂×C₄	—	164	3	no	—
64	C₂×C₄	C₂²×D₄₁	1	3	yes	—
65	C₂²	—	2	3	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	82	6	no	—
69	C₂²	—	164	4	no	—
70	C₂²	C₂³×D₄₁	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	82	6	no	—
73	C₂	—	4	1	no	—
74	C₂	—	4	3	no	—
75	C₂	—	164	2	no	—
76	C₂	#13050209	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator, socle, cosocle	C₂²×C₄₁	C₂⁴
Fitting	#127×C₄₁	C₂
hypercenter	#127	D₄₁
hypocenter	C₄₁	#560
LCS	G ▷ C₂²×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #13050158	C₂	semidirect
2. #13050145	C₂	semidirect
3. #13050136	C₂	semidirect
4. #13050132	C₂	semidirect
5. #127×C₄₁	C₂	semidirect
8. C₂×D₄×C₄₁	C₂²	semidirect
10. C₂²×Dic₄₁	C₂²	semidirect
11. C₄²×C₄₁	C₂²	semidirect
13. C₂²⋊Dic₄₁	C₂²	semidirect
15. #10620279	C₂²	semidirect
39. C₄₁	#560	semidirect
40. #127	D₄₁	semidirect
48. C₂×D₄	C₂×D₄₁	semidirect
54. C₄²	C₂×D₄₁	semidirect

Maximal subgroups

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

subgroup	index
#13050158	2
#13050145	2
#13050136	2
#13050132	2
#127×C₄₁	2
#560	41