Subgroups — #30682550

Order 16832.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22554470	C₂	1	1	yes	maximal, characteristic, complemented
2	#22554463	C₂	1	4	yes	maximal
3	#22554462	C₂	1	1	yes	maximal, characteristic, complemented
4	#22554460	C₂	1	4	yes	maximal, complemented
5	#22554449	C₂	1	2	yes	maximal
6	C₄×Dic₅₂₆	C₂	1	2	yes	maximal, complemented
7	#117×C₂₆₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₂₆	C₂²	1	2	yes	—
9	C₄⋊Dic₂₆₃	C₂²	1	2	yes	—
10	C₄⋊Dic₂₆₃	C₂²	1	4	yes	—
11	Dic₂₆₃⋊C₄	C₂²	1	4	yes	—
12	Dic₂₆₃⋊C₄	C₂²	1	4	yes	—
13	Dic₂₆₃⋊C₄	C₂²	1	4	yes	—
14	C₂²⋊Dic₂₆₃	C₂²	1	4	yes	—
15	C₄×Dic₂₆₃	C₂²	1	4	yes	—
16	C₂²×C₄×C₂₆₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₆₃	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₂₆₃	C₂²	1	2	yes	—
19	C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
20	C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
21	Dic₅₂₆	—	2	4	no	—
22	C₂×Dic₂₆₃	C₂³	1	4	yes	—
23	C₂×Dic₂₆₃	C₂³	1	4	yes	—
24	C₂³×C₂₆₃	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₂₆₃	—	2	2	no	—
26	C₂×C₄×C₂₆₃	C₂³	1	2	yes	—
27	C₂×C₄×C₂₆₃	C₂³	1	2	yes	—
28	C₂×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
30	C₂²×C₂₆₃	—	2	1	no	—
31	C₂²×C₂₆₃	—	2	2	no	—
32	C₂²×C₂₆₃	C₂⁴	1	1	yes	characteristic
33	C₄×C₂₆₃	—	2	1	no	—
34	C₄×C₂₆₃	—	2	2	no	—
35	C₄×C₂₆₃	—	2	2	no	—
36	C₄×C₂₆₃	#41	1	2	yes	—
37	Dic₂₆₃	—	2	4	no	—
38	Dic₂₆₃	—	2	4	no	—
39	C₂×C₂₆₃	—	4	1	no	—
40	C₂×C₂₆₃	C₂×#41	1	1	yes	characteristic
41	C₂×C₂₆₃	2₋¹⁺⁴	1	2	yes	—
42	C₂₆₃	#541	1	1	yes	characteristic, complemented, Sylow
43	#541	—	263	1	no	maximal, Sylow
44	C₄⋊Q₈	—	263	1	no	—
45	#126	—	263	4	no	—
46	#125	—	263	4	no	—
47	#125	—	263	1	no	—
48	C₂²⋊Q₈	—	263	2	no	—
49	C₄×Q₈	—	263	2	no	—
50	#117	D₂₆₃	1	1	yes	characteristic
51	C₂×Q₈	—	263	2	no	—
52	C₂²×C₄	C₂×D₂₆₃	1	1	yes	characteristic
53	C₄⋊C₄	—	263	2	no	—
54	C₄⋊C₄	—	263	4	no	—
55	C₄⋊C₄	—	263	4	no	—
56	C₄⋊C₄	—	263	4	no	—
57	C₄⋊C₄	—	263	4	no	—
58	C₄⋊C₄	C₂×D₂₆₃	1	2	yes	—
59	C₂²⋊C₄	—	263	4	no	—
60	C₂²⋊C₄	C₂×D₂₆₃	1	2	yes	—
61	C₄²	—	263	4	no	—
62	C₄²	C₂×D₂₆₃	1	1	yes	characteristic
63	C₄²	C₂×D₂₆₃	1	1	yes	characteristic
64	C₂³	C₂²×D₂₆₃	1	1	yes	characteristic
65	Q₈	—	526	4	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	263	4	no	—
68	C₂×C₄	—	263	4	no	—
69	C₂×C₄	C₂²×D₂₆₃	1	2	yes	—
70	C₂×C₄	C₂²×D₂₆₃	1	2	yes	—
71	C₂×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
73	C₂²	—	2	1	no	—
74	C₂²	—	2	2	no	—
75	C₂²	C₂³×D₂₆₃	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	2	no	—
78	C₄	—	2	2	no	—
79	C₄	—	526	4	no	—
80	C₄	—	526	4	no	—
81	C₄	#16862376	1	2	yes	—
82	C₂	—	4	1	no	—
83	C₂	C₂×#16862376	1	1	yes	characteristic
84	C₂	#22554517	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₆₃
commutator, socle, cosocle	C₂²×C₂₆₃	C₂⁴
Fitting	#117×C₂₆₃	C₂
prime radical	C₂³×C₂₆₃	C₂³
hypercenter	#117	D₂₆₃
hypocenter	C₂₆₃	#541
LCS	G ▷ C₂²×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #22554470	C₂	semidirect
3. #22554462	C₂	semidirect
4. #22554460	C₂	semidirect
6. C₄×Dic₅₂₆	C₂	semidirect
42. C₂₆₃	#541	semidirect

Maximal subgroups

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

subgroup	index
#22554470	2
#22554463	2
#22554462	2
#22554460	2
#22554449	2
C₄×Dic₅₂₆	2
#117×C₂₆₃	2
#541	263