Subgroups — #46780574

Order 33664.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂₆₃⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₂₆₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₆₃⋊C₈	C₄	1	4	yes	—
4	C₄×C₂₆₃⋊C₈	—	2	2	no	—
5	C₄×C₂₆₃⋊C₈	—	2	2	no	—
6	C₂²×C₈×C₂₆₃	C₄	1	2	yes	—
7	C₂×C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₂₆₃	—	2	2	no	—
9	C₂×C₂₆₃⋊C₈	—	2	4	no	—
10	C₂×C₂₆₃⋊C₈	—	2	4	no	—
11	C₂×C₂₆₃⋊C₈	C₈	1	8	yes	complemented
12	C₂²×C₄×C₂₆₃	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₂₆₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₂₆₃	—	2	2	no	—
15	C₂×C₈×C₂₆₃	—	2	2	no	—
16	C₂×C₈×C₂₆₃	C₈	1	4	yes	complemented
17	C₄²×C₂₆₃	D₄	1	2	yes	—
18	C₄²×C₂₆₃	D₄	1	1	yes	characteristic
19	C₄²×C₂₆₃	Q₈	1	1	yes	characteristic
20	C₂₆₃⋊C₈	—	2	8	no	—
21	C₂³×C₂₆₃	C₄²	1	1	yes	characteristic
22	C₂×C₄×C₂₆₃	C₂²⋊C₄	1	2	yes	—
23	C₂×C₄×C₂₆₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₆₃	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₂₆₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₆₃	C₂×C₈	1	4	yes	—
27	C₂×C₄×C₂₆₃	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₂₆₃	M₁₆	1	4	yes	—
29	C₂×C₄×C₂₆₃	M₁₆	1	2	yes	—
30	C₈×C₂₆₃	—	2	4	no	—
31	C₂²×C₂₆₃	#95	1	1	yes	characteristic
32	C₂²×C₂₆₃	C₄×C₈	1	2	yes	—
33	C₂²×C₂₆₃	C₄×C₈	1	1	yes	characteristic
34	C₂²×C₂₆₃	#97	1	2	yes	—
35	C₂²×C₂₆₃	#97	1	1	yes	characteristic
36	C₄×C₂₆₃	C₂²⋊C₈	1	4	yes	—
37	C₄×C₂₆₃	C₂²⋊C₈	1	2	yes	—
38	C₄×C₂₆₃	C₄⋊C₈	1	4	yes	—
39	C₄×C₂₆₃	C₄⋊C₈	1	2	yes	—
40	C₂×C₂₆₃	C₈²	1	1	yes	characteristic
41	C₂×C₂₆₃	#322	1	1	yes	characteristic
42	C₂×C₂₆₃	#322	1	2	yes	—
43	C₂×C₂₆₃	#336	1	2	yes	—
44	C₂×C₂₆₃	#336	1	1	yes	characteristic
45	C₂₆₃	#1273	1	1	yes	characteristic, complemented, Sylow
46	#1273	—	263	1	no	maximal, Sylow
47	C₂×C₄×C₈	—	263	2	no	—
48	C₂×C₄×C₈	D₂₆₃	1	1	yes	characteristic
49	C₂²×C₈	—	263	4	no	—
50	C₂²×C₈	Dic₂₆₃	1	2	yes	—
51	C₂×C₄²	C₂×D₂₆₃	1	1	yes	characteristic
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	—	526	2	no	—
54	C₄×C₈	—	526	2	no	—
55	C₂²×C₄	C₄×D₂₆₃	1	2	yes	—
56	C₂²×C₄	C₂×Dic₂₆₃	1	1	yes	characteristic
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	263	8	no	—
60	C₂×C₈	—	526	4	no	—
61	C₂×C₈	—	526	4	no	—
62	C₂×C₈	C₂₆₃⋊C₈	1	4	yes	complemented
63	C₄²	C₂²⋊D₂₆₃	1	2	yes	—
64	C₄²	D₁₀₅₂	1	1	yes	characteristic
65	C₄²	Dic₅₂₆	1	1	yes	characteristic
66	C₂³	C₄×Dic₂₆₃	1	1	yes	characteristic
67	C₂×C₄	(C₂×D₂₆₃)⋊C₄	1	2	yes	—
68	C₂×C₄	C₂²⋊Dic₂₆₃	1	1	yes	characteristic
69	C₂×C₄	Dic₂₆₃⋊C₄	1	2	yes	—
70	C₂×C₄	C₄⋊Dic₂₆₃	1	1	yes	characteristic
71	C₂×C₄	C₈×D₂₆₃	1	4	yes	—
72	C₂×C₄	C₂×C₂₆₃⋊C₈	1	2	yes	—
73	C₂×C₄	#16862359	1	4	yes	—
74	C₂×C₄	#16862360	1	2	yes	—
75	C₈	—	2	4	no	—
76	C₈	—	526	8	no	—
77	C₂²	C₂₆₃⋊#95	1	1	yes	characteristic
78	C₂²	C₈×Dic₂₆₃	1	2	yes	—
79	C₂²	C₄×C₂₆₃⋊C₈	1	1	yes	characteristic
80	C₂²	#22554380	1	2	yes	—
81	C₂²	#22554381	1	1	yes	characteristic
82	C₄	(C₂×D₂₆₃)⋊C₈	1	4	yes	—
83	C₄	C₂²⋊(C₂₆₃⋊C₈)	1	2	yes	—
84	C₄	Dic₂₆₃⋊C₈	1	4	yes	—
85	C₄	C₄⋊(C₂₆₃⋊C₈)	1	2	yes	—
86	C₂	C₈×C₂₆₃⋊C₈	1	1	yes	characteristic
87	C₂	#30681659	1	1	yes	characteristic
88	C₂	#30681660	1	2	yes	—
89	C₂	#30681692	1	2	yes	—
90	C₂	#30681693	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂×D₂₆₃
commutator	C₂×C₂₆₃	C₈²
Fitting	C₂×C₄×C₈×C₂₆₃	C₂
socle, prime radical	C₂³×C₂₆₃	C₄²
cosocle	C₂×C₄²×C₂₆₃	C₂²
hypercenter	C₂×C₄×C₈	D₂₆₃
hypocenter	C₂₆₃	#1273
LCS	G ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
11. C₂×C₂₆₃⋊C₈	C₈	semidirect
16. C₂×C₈×C₂₆₃	C₈	semidirect
45. C₂₆₃	#1273	semidirect
62. C₂×C₈	C₂₆₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂₆₃⋊C₈	2
C₂×C₄×C₈×C₂₆₃	2
#1273	263