Subgroups — #30682666

Order 16832.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22554470	C₂	1	1	yes	maximal, characteristic
2	#22554468	C₂	1	2	yes	maximal
3	#22554462	C₂	1	2	yes	maximal
4	#22554460	C₂	1	4	yes	maximal
5	C₄×Dic₅₂₆	C₂	1	2	yes	maximal
6	C₄×Dic₅₂₆	C₂	1	1	yes	maximal, characteristic
7	#22554435	C₂	1	2	yes	maximal
8	C₄×Q₈×C₂₆₃	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₅₂₆	C₂²	1	2	yes	—
10	C₂×Dic₅₂₆	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₆₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₆₃	C₂²	1	2	yes	—
13	C₄⋊Dic₂₆₃	C₂²	1	2	yes	—
14	Dic₂₆₃⋊C₄	C₂²	1	4	yes	—
15	Dic₂₆₃⋊C₄	C₂²	1	2	yes	—
16	Dic₂₆₃⋊C₄	C₂²	1	4	yes	—
17	Dic₂₆₃⋊C₄	C₂²	1	4	yes	—
18	C₄×Dic₂₆₃	C₂²	1	4	yes	—
19	C₄×Dic₂₆₃	C₂²	1	2	yes	—
20	C₂×Q₈×C₂₆₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₂₆₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₂₆₃	C₂²	1	2	yes	—
23	C₄²×C₂₆₃	C₂²	1	2	yes	—
24	C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
25	Dic₅₂₆	—	2	4	no	—
26	Dic₅₂₆	Q₈	1	4	yes	complemented
27	C₂×Dic₂₆₃	C₂³	1	2	yes	—
28	C₂×Dic₂₆₃	C₂³	1	4	yes	—
29	C₂×Dic₂₆₃	C₂³	1	2	yes	—
30	Q₈×C₂₆₃	—	2	2	no	—
31	C₂×C₄×C₂₆₃	C₂³	1	2	yes	—
32	C₂×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₂₆₃	C₂³	1	2	yes	—
34	C₂×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
36	C₂²×C₂₆₃	C₂⁴	1	1	yes	characteristic
37	C₄×C₂₆₃	—	2	2	no	—
38	C₄×C₂₆₃	—	2	2	no	—
39	C₄×C₂₆₃	—	2	1	no	—
40	C₄×C₂₆₃	C₂×Q₈	1	2	yes	—
41	C₄×C₂₆₃	#41	1	2	yes	—
42	Dic₂₆₃	—	2	4	no	—
43	Dic₂₆₃	—	2	2	no	—
44	Dic₂₆₃	C₂×Q₈	1	4	yes	—
45	C₂×C₂₆₃	C₂²×Q₈	1	1	yes	characteristic
46	C₂×C₂₆₃	C₂×#41	1	1	yes	characteristic
47	C₂×C₂₆₃	2₋¹⁺⁴	1	1	yes	characteristic
48	C₂₆₃	#557	1	1	yes	characteristic, complemented, Sylow
49	#557	—	263	1	no	maximal, Sylow
50	C₄⋊Q₈	—	263	2	no	—
51	C₄⋊Q₈	—	263	1	no	—
52	#125	—	263	4	no	—
53	#125	—	263	2	no	—
54	C₄×Q₈	—	263	2	no	—
55	C₄×Q₈	—	263	2	no	—
56	C₄×Q₈	—	263	1	no	—
57	C₄×Q₈	D₂₆₃	1	1	yes	characteristic
58	C₂×Q₈	—	263	1	no	—
59	C₂×Q₈	—	263	2	no	—
60	C₂×Q₈	C₂×D₂₆₃	1	1	yes	characteristic
61	C₄⋊C₄	—	263	2	no	—
62	C₄⋊C₄	—	263	4	no	—
63	C₄⋊C₄	—	263	1	no	—
64	C₄⋊C₄	—	263	4	no	—
65	C₄⋊C₄	—	263	2	no	—
66	C₄⋊C₄	—	263	4	no	—
67	C₄⋊C₄	—	263	2	no	—
68	C₄⋊C₄	C₂×D₂₆₃	1	1	yes	characteristic
69	C₄⋊C₄	C₂×D₂₆₃	1	2	yes	—
70	C₄²	—	263	4	no	—
71	C₄²	—	263	2	no	—
72	C₄²	C₂×D₂₆₃	1	2	yes	—
73	C₄²	C₂×D₂₆₃	1	1	yes	characteristic
74	Q₈	—	2	2	no	—
75	Q₈	—	263	4	no	—
76	Q₈	—	526	4	no	—
77	C₂×C₄	—	263	2	no	—
78	C₂×C₄	—	263	2	no	—
79	C₂×C₄	—	263	4	no	—
80	C₂×C₄	C₂²×D₂₆₃	1	2	yes	—
81	C₂×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
82	C₂×C₄	C₂²×D₂₆₃	1	2	yes	—
83	C₂×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
84	C₂×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
85	C₂²	C₂³×D₂₆₃	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	2	no	—
88	C₄	—	2	1	no	—
89	C₄	—	263	4	no	—
90	C₄	—	526	4	no	—
91	C₄	—	526	2	no	—
92	C₄	Q₈×D₂₆₃	1	2	yes	—
93	C₄	#16862376	1	2	yes	—
94	C₂	C₂×Q₈×D₂₆₃	1	1	yes	characteristic
95	C₂	C₂×#16862376	1	1	yes	characteristic
96	C₂	#22554517	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₆₃
commutator, socle, prime radical, cosocle	C₂²×C₂₆₃	C₂⁴
Fitting	C₄×Q₈×C₂₆₃	C₂
hypercenter	C₄×Q₈	D₂₆₃
hypocenter	C₂₆₃	#557
LCS	G ▷ C₂²×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
26. Dic₅₂₆	Q₈	semidirect
48. C₂₆₃	#557	semidirect

Maximal subgroups

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

subgroup	index
#22554470	2
#22554468	2
#22554462	2
#22554460	2
C₄×Dic₅₂₆	2
#22554435	2
C₄×Q₈×C₂₆₃	2
#557	263