Subgroups — #30682686

Order 16832.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22554458	C₂	1	2	yes	maximal, complemented
2	#22554455	C₂	1	4	yes	maximal, complemented
3	#22554448	C₂	1	4	yes	maximal, complemented
4	#22554445	C₂	1	4	yes	maximal, complemented
5	#125×C₂₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₂₆	C₂²	1	2	yes	—
7	C₂×D₁₀₅₂	C₂²	1	4	yes	—
8	C₂×D₁₀₅₂	C₂²	1	2	yes	—
9	C₂×C₄×D₂₆₃	C₂²	1	4	yes	—
10	Dic₂₆₃⋊C₄	C₂²	1	4	yes	—
11	(C₂×D₂₆₃)⋊C₄	C₂²	1	4	yes	—
12	(C₂×D₂₆₃)⋊C₄	C₂²	1	4	yes	—
13	(C₂×D₂₆₃)⋊C₄	C₂²	1	4	yes	—
14	C₄⋊C₄×C₂₆₃	C₂²	1	4	yes	—
15	C₄⋊C₄×C₂₆₃	C₂²	1	2	yes	—
16	C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
17	C₂²×D₂₆₃	C₂³	1	4	yes	—
18	Dic₅₂₆	—	4	2	no	—
19	D₁₀₅₂	—	4	4	no	—
20	D₁₀₅₂	—	4	2	no	—
21	C₂×Dic₂₆₃	C₂³	1	4	yes	—
22	C₄×D₂₆₃	—	4	4	no	—
23	C₂×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₂₆₃	C₂³	1	2	yes	—
25	C₂×C₄×C₂₆₃	C₂³	1	4	yes	—
26	C₂²×C₂₆₃	C₂⁴	1	1	yes	characteristic
27	C₂×D₂₆₃	—	2	4	no	—
28	C₂×D₂₆₃	—	2	4	no	—
29	C₂×D₂₆₃	—	2	4	no	—
30	C₄×C₂₆₃	—	2	4	no	—
31	C₄×C₂₆₃	—	2	2	no	—
32	C₄×C₂₆₃	—	2	1	no	—
33	Dic₂₆₃	—	2	4	no	—
34	C₂×C₂₆₃	H(5,𝔽₂)	1	2	yes	—
35	C₂×C₂₆₃	2₋¹⁺⁴	1	1	yes	characteristic
36	D₂₆₃	—	4	4	no	—
37	C₂₆₃	#562	1	1	yes	characteristic, complemented, Sylow
38	#562	—	263	1	no	maximal, Sylow
39	#125	D₂₆₃	1	1	yes	characteristic, complemented
40	#124	—	263	2	no	—
41	#123	—	263	4	no	—
42	C₂²⋊Q₈	—	263	4	no	—
43	#121	—	263	4	no	—
44	C₂×Q₈	—	263	2	no	—
45	C₂×D₄	—	263	4	no	—
46	C₂×D₄	—	263	2	no	—
47	C₂²×C₄	—	263	4	no	—
48	C₄⋊C₄	—	263	4	no	—
49	C₄⋊C₄	C₂×D₂₆₃	1	4	yes	—
50	C₄⋊C₄	C₂×D₂₆₃	1	2	yes	—
51	C₂²⋊C₄	—	263	4	no	—
52	C₂²⋊C₄	—	263	4	no	—
53	C₂²⋊C₄	—	263	4	no	—
54	C₄²	C₂×D₂₆₃	1	1	yes	characteristic
55	C₂³	—	263	4	no	—
56	Q₈	—	1052	2	no	—
57	D₄	—	1052	4	no	—
58	D₄	—	1052	2	no	—
59	C₂×C₄	—	263	4	no	—
60	C₂×C₄	—	1052	4	no	—
61	C₂×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₆₃	1	2	yes	—
63	C₂×C₄	C₂²×D₂₆₃	1	4	yes	—
64	C₂²	—	526	4	no	—
65	C₂²	—	526	4	no	—
66	C₂²	—	526	4	no	—
67	C₂²	C₂³×D₂₆₃	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	526	4	no	—
72	C₂	—	1052	4	no	—
73	C₂	#22554516	1	2	yes	—
74	C₂	#22554518	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₆₃
commutator, socle, cosocle	C₂²×C₂₆₃	C₂⁴
Fitting	#125×C₂₆₃	C₂
hypercenter	#125	D₂₆₃
hypocenter	C₂₆₃	#562
LCS	G ▷ C₂²×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #22554458	C₂	semidirect
2. #22554455	C₂	semidirect
3. #22554448	C₂	semidirect
4. #22554445	C₂	semidirect
5. #125×C₂₆₃	C₂	semidirect
37. C₂₆₃	#562	semidirect
39. #125	D₂₆₃	semidirect

Maximal subgroups

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

subgroup	index
#22554458	2
#22554455	2
#22554448	2
#22554445	2
#125×C₂₆₃	2
#562	263