Subgroups — #14580725

Order 2624.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10620273	C₂	1	2	yes	maximal, complemented
2	C₂×D₃₂₈	C₂	1	2	yes	maximal, complemented
3	#13050161	C₂	1	1	yes	maximal, characteristic, complemented
4	#13050152	C₂	1	1	yes	maximal, characteristic, complemented
5	#97×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆₄	—	2	2	no	—
7	C₂×D₁₆₄	C₂²	1	1	yes	characteristic
8	C₂×D₁₆₄	C₂²	1	2	yes	—
9	C₂×Dic₈₂	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₄₁	C₂²	1	2	yes	—
11	C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	#10620281	—	2	2	no	—
13	D₃₂₈	—	2	4	no	—
14	#10620273	—	2	4	no	—
15	C₂²×D₄₁	—	2	1	no	—
16	C₂²×D₄₁	—	2	2	no	—
17	C₂×C₄×C₄₁	D₄	1	2	yes	—
18	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
19	C₂×Dic₄₁	—	2	1	no	—
20	D₁₆₄	—	2	2	no	—
21	D₁₆₄	—	2	2	no	—
22	D₁₆₄	—	2	2	no	—
23	D₁₆₄	—	4	4	no	—
24	Dic₈₂	—	2	2	no	—
25	C₈×C₄₁	D₄	1	4	yes	complemented
26	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
27	C₂×D₄₁	—	4	2	no	—
28	C₂×D₄₁	—	4	1	no	—
29	C₂×D₄₁	—	4	4	no	—
30	C₂×D₄₁	—	4	2	no	—
31	C₄×C₄₁	—	2	2	no	—
32	C₄×C₄₁	C₂×D₄	1	2	yes	—
33	Dic₄₁	—	4	1	no	—
34	C₂×C₄₁	#127	1	1	yes	characteristic
35	C₂×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
36	D₄₁	—	8	2	no	—
37	D₄₁	—	8	1	no	—
38	#496	—	41	1	no	maximal, Sylow
39	C₄₁	#496	1	1	yes	characteristic, complemented, Sylow
40	C₂×SD₁₆	—	41	2	no	—
41	C₂×D₈	—	41	2	no	—
42	#127	—	41	1	no	—
43	#124	—	41	1	no	—
44	#97	D₄₁	1	1	yes	characteristic, complemented
45	C₂×Q₈	—	41	1	no	—
46	C₂×D₄	—	41	1	no	—
47	C₂×D₄	—	41	2	no	—
48	C₂×D₄	—	82	2	no	—
49	SD₁₆	—	82	4	no	—
50	D₈	—	82	4	no	—
51	C₂×C₈	C₂×D₄₁	1	2	yes	—
52	C₂²⋊C₄	—	82	2	no	—
53	C₄²	C₂×D₄₁	1	1	yes	characteristic
54	C₂³	—	82	1	no	—
55	C₂³	—	82	2	no	—
56	Q₈	—	82	2	no	—
57	D₄	—	82	2	no	—
58	D₄	—	82	2	no	—
59	D₄	—	82	2	no	—
60	D₄	—	164	4	no	—
61	C₂×C₄	—	82	1	no	—
62	C₂×C₄	D₁₆₄	1	2	yes	—
63	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
64	C₈	D₁₆₄	1	4	yes	complemented
65	C₂²	—	164	1	no	—
66	C₂²	—	164	2	no	—
67	C₂²	—	164	4	no	—
68	C₂²	—	164	2	no	—
69	C₂²	C₂×D₁₆₄	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	164	1	no	—
72	C₄	C₂×D₁₆₄	1	2	yes	—
73	C₂	—	328	2	no	—
74	C₂	—	328	1	no	—
75	C₂	#13050161	1	1	yes	characteristic
76	C₂	#13050192	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₆₄
commutator, cosocle	C₂×C₄×C₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	#97×C₄₁	C₂
socle	C₂²×C₄₁	C₂×D₄
hypercenter	#97	D₄₁
hypocenter	C₄₁	#496
LCS	G ▷ C₂×C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
1. C₂×#10620273	C₂	semidirect
2. C₂×D₃₂₈	C₂	semidirect
3. #13050161	C₂	semidirect
4. #13050152	C₂	semidirect
5. #97×C₄₁	C₂	semidirect
25. C₈×C₄₁	D₄	semidirect
39. C₄₁	#496	semidirect
44. #97	D₄₁	semidirect
64. C₈	D₁₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10620273	2
C₂×D₃₂₈	2
#13050161	2
#13050152	2
#97×C₄₁	2
#496	41