Subgroups — #14580504

Order 2624.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13050189	C₂	1	2	yes	maximal, complemented
2	#13050160	C₂	1	1	yes	maximal, characteristic, complemented
3	#13050091	C₂	1	2	yes	maximal, complemented
4	#13050081	C₂	1	1	yes	maximal, characteristic, complemented
5	H(5,𝔽₂)×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₄₁	—	2	2	no	—
7	#41×C₄₁	C₂²	1	2	yes	—
8	C₂×D₄×C₄₁	—	2	2	no	—
9	C₂×D₄×C₄₁	—	2	2	no	—
10	C₂×D₄×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₄₁	—	2	2	no	—
12	C₂×D₁₆₄	C₂²	1	1	yes	characteristic, complemented
13	#10620284	—	2	2	no	—
14	#10620282	—	2	2	no	—
15	C₄×Dic₄₁	C₂²	1	1	yes	characteristic, complemented
16	#10620277	C₂²	1	2	yes	complemented
17	C₂³×C₄₁	—	2	2	no	—
18	C₂³×C₄₁	D₄	1	2	yes	—
19	C₂²×D₄₁	—	2	1	no	—
20	Q₈×C₄₁	D₄	1	2	yes	complemented
21	D₄×C₄₁	—	2	4	no	—
22	D₄×C₄₁	—	2	2	no	—
23	D₄×C₄₁	—	2	2	no	—
24	D₄×C₄₁	D₄	1	2	yes	complemented
25	C₂×C₄×C₄₁	—	2	2	no	—
26	C₂×C₄×C₄₁	—	2	2	no	—
27	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
28	C₂²⋊D₄₁	—	4	4	no	—
29	C₂×Dic₄₁	—	2	1	no	—
30	D₁₆₄	—	2	2	no	—
31	#66491	—	2	2	no	—
32	C₂²×C₄₁	—	2	2	no	—
33	C₂²×C₄₁	—	2	2	no	—
34	C₂²×C₄₁	—	4	2	no	—
35	C₂²×C₄₁	—	8	2	no	—
36	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
37	C₂×D₄₁	—	4	2	no	—
38	C₂×D₄₁	—	4	1	no	—
39	C₄×C₄₁	—	2	2	no	—
40	C₄×C₄₁	C₂×D₄	1	2	yes	—
41	Dic₄₁	—	2	2	no	—
42	C₂×C₄₁	—	2	1	no	—
43	C₂×C₄₁	—	4	2	no	—
44	C₂×C₄₁	—	4	2	no	—
45	C₂×C₄₁	#120	1	1	yes	characteristic
46	D₄₁	—	8	1	no	—
47	#453	—	41	1	no	maximal, Sylow
48	C₄₁	#453	1	1	yes	characteristic, complemented, Sylow
49	H(5,𝔽₂)	D₄₁	1	1	yes	characteristic, complemented
50	AGL(1,ℤ/8)	—	41	2	no	—
51	#127	—	41	1	no	—
52	#104	—	41	2	no	—
53	#100	—	41	1	no	—
54	#41	—	2	2	no	—
55	#41	C₂×D₄₁	1	2	yes	—
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	—	41	1	no	—
59	C₂×D₄	—	82	2	no	—
60	C₂×D₄	C₂×D₄₁	1	1	yes	characteristic
61	SD₁₆	—	82	2	no	—
62	D₈	—	82	2	no	—
63	M₁₆	—	41	2	no	—
64	C₄²	—	41	1	no	—
65	C₂³	—	2	2	no	—
66	C₂³	—	82	1	no	—
67	C₂³	C₂²⋊D₄₁	1	2	yes	—
68	Q₈	C₂²⋊D₄₁	1	2	yes	complemented
69	D₄	—	2	4	no	—
70	D₄	—	2	2	no	—
71	D₄	—	2	2	no	—
72	D₄	—	82	2	no	—
73	D₄	—	164	4	no	—
74	D₄	C₂²⋊D₄₁	1	2	yes	complemented
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	82	1	no	—
78	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
79	C₈	—	82	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	8	2	no	—
84	C₂²	—	164	2	no	—
85	C₂²	—	164	1	no	—
86	C₂²	C₂×C₂²⋊D₄₁	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	82	2	no	—
89	C₄	C₂×C₂²⋊D₄₁	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	4	2	no	—
93	C₂	—	328	1	no	—
94	C₂	#13050133	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13050133
commutator, cosocle	C₂×C₄×C₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	H(5,𝔽₂)×C₄₁	C₂
socle	C₂×C₄₁	#120
hypercenter	H(5,𝔽₂)	D₄₁
hypocenter	C₄₁	#453
LCS	G ▷ C₂×C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #13050189	C₂	semidirect
2. #13050160	C₂	semidirect
3. #13050091	C₂	semidirect
4. #13050081	C₂	semidirect
5. H(5,𝔽₂)×C₄₁	C₂	semidirect
12. C₂×D₁₆₄	C₂²	semidirect
15. C₄×Dic₄₁	C₂²	semidirect
16. #10620277	C₂²	semidirect
20. Q₈×C₄₁	D₄	semidirect
24. D₄×C₄₁	D₄	semidirect
48. C₄₁	#453	semidirect
49. H(5,𝔽₂)	D₄₁	semidirect
68. Q₈	C₂²⋊D₄₁	semidirect
74. D₄	C₂²⋊D₄₁	semidirect

Maximal subgroups

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

subgroup	index
#13050189	2
#13050160	2
#13050091	2
#13050081	2
H(5,𝔽₂)×C₄₁	2
#453	41