Group #22574703 — subgroups

Subgroups — #22574703

Order 8576.

890 conjugacy classes of subgroups; only the 455 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 455 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇⋊#564	C₂	1	3	yes	maximal, complemented
2	#16864123	C₂	1	12	yes	maximal, complemented
3	#16864005	C₂	1	12	yes	maximal
4	#16863952	C₂	1	3	yes	maximal
5	#564×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14524042	C₂²	1	12	yes	—
7	#14524042	C₂²	1	6	yes	—
8	#14524038	C₂²	1	12	yes	—
9	#14524037	C₂²	1	4	yes	—
10	#14524035	C₂²	1	3	yes	—
11	#14524034	C₂²	1	12	yes	—
12	#14524033	C₂²	1	12	yes	—
13	#14524033	C₂²	1	12	yes	—
14	#14524033	C₂²	1	12	yes	—
15	#14524021	C₂²	1	12	yes	—
16	C₄×Dic₁₃₄	C₂²	1	12	yes	—
17	#14524008	C₂²	1	12	yes	—
18	#14524008	C₂²	1	12	yes	—
19	#14524001	C₂²	1	6	yes	—
20	C₄²×D₆₇	C₂²	1	1	yes	characteristic
21	#125×C₆₇	C₂²	1	12	yes	—
22	#125×C₆₇	C₂²	1	3	yes	—
23	C₂×Dic₁₃₄	C₂³	1	12	yes	—
24	C₂×C₄×D₆₇	C₂³	1	3	yes	—
25	C₄⋊Dic₆₇	C₂³	1	6	yes	—
26	C₄⋊Dic₆₇	C₂³	1	12	yes	—
27	Dic₆₇⋊C₄	C₂³	1	12	yes	—
28	Dic₆₇⋊C₄	C₂³	1	12	yes	—
29	Dic₆₇⋊C₄	C₂³	1	12	yes	—
30	Dic₆₇⋊C₄	C₂³	1	12	yes	—
31	(C₂×D₆₇)⋊C₄	C₂³	1	12	yes	—
32	C₄×Dic₆₇	C₂³	1	3	yes	—
33	C₄×Dic₆₇	C₂³	1	12	yes	—
34	C₄×Dic₆₇	C₂³	1	12	yes	—
35	C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
36	C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
37	C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
38	C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₄²×C₆₇	C₂³	1	4	yes	—
40	C₂²×D₆₇	C₂⁴	1	1	yes	characteristic
41	C₂×C₄×C₆₇	C₂⁴	1	12	yes	—
42	C₂×C₄×C₆₇	C₂⁴	1	3	yes	—
43	C₂×Dic₆₇	C₂⁴	1	12	yes	—
44	C₂×Dic₆₇	C₂⁴	1	3	yes	—
45	C₄×D₆₇	—	2	6	no	—
46	Dic₁₃₄	—	2	24	no	—
47	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
48	C₂×D₆₇	—	2	3	no	—
49	C₄×C₆₇	—	2	12	no	—
50	C₄×C₆₇	—	2	3	no	—
51	Dic₆₇	—	2	12	no	—
52	Dic₆₇	2₋¹⁺⁴	1	6	yes	—
53	C₂×C₆₇	C₂×2₋¹⁺⁴	1	3	yes	—
54	D₆₇	—	4	1	no	—
55	#3556	—	67	1	no	maximal, Sylow
56	C₆₇	#3556	1	1	yes	characteristic, complemented, Sylow
57	#564	—	67	3	no	—
58	#564	D₆₇	1	1	yes	characteristic, complemented
59	#557	—	67	12	no	—
60	#541	—	67	12	no	—
61	#533	—	67	3	no	—
62	C₄⋊Q₈	—	67	6	no	—
63	C₄⋊Q₈	—	67	12	no	—
64	#126	—	67	12	no	—
65	#126	—	67	4	no	—
66	#125	—	67	12	no	—
67	#125	—	67	12	no	—
68	#125	—	67	12	no	—
69	#125	—	67	12	no	—
70	#125	—	67	3	no	—
71	#125	C₂×D₆₇	1	3	yes	—
72	#125	C₂×D₆₇	1	12	yes	—
73	C₂²⋊Q₈	—	67	12	no	—
74	C₄×Q₈	—	67	12	no	—
75	C₄×Q₈	—	67	12	no	—
76	C₄×Q₈	—	67	12	no	—
77	#117	—	67	6	no	—
78	C₂×C₄²	—	67	1	no	—
79	C₂×Q₈	—	67	12	no	—
80	C₂²×C₄	—	67	3	no	—
81	C₄⋊C₄	—	67	6	no	—
82	C₄⋊C₄	—	67	12	no	—
83	C₄⋊C₄	—	67	12	no	—
84	C₄⋊C₄	—	67	12	no	—
85	C₄⋊C₄	—	67	12	no	—
86	C₄⋊C₄	—	67	12	no	—
87	C₄⋊C₄	C₂²×D₆₇	1	6	yes	—
88	C₄⋊C₄	C₂²×D₆₇	1	12	yes	—
89	C₄⋊C₄	C₂²×D₆₇	1	12	yes	—
90	C₂²⋊C₄	—	67	12	no	—
91	C₄²	—	67	3	no	—
92	C₄²	—	67	12	no	—
93	C₄²	—	67	12	no	—
94	C₄²	C₂²×D₆₇	1	1	yes	characteristic
95	C₄²	C₂²×D₆₇	1	4	yes	—
96	C₂³	—	67	1	no	—
97	Q₈	—	134	24	no	—
98	C₂×C₄	—	67	3	no	—
99	C₂×C₄	—	67	12	no	—
100	C₂×C₄	—	134	6	no	—
101	C₂×C₄	C₂³×D₆₇	1	12	yes	—
102	C₂×C₄	C₂³×D₆₇	1	3	yes	—
103	C₂²	—	134	3	no	—
104	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
105	C₄	—	2	3	no	—
106	C₄	—	2	12	no	—
107	C₄	—	67	6	no	—
108	C₄	—	134	12	no	—
109	C₂	—	268	1	no	—
110	C₂	2₋¹⁺⁴×D₆₇	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇⋊#564

C₂

yes

maximal, complemented

#16864123

C₂

yes

maximal, complemented

#16864005

C₂

yes

maximal

#16863952

C₂

yes

maximal

#564×C₆₇

C₂

yes

maximal, characteristic, complemented

#14524042

C₂²

yes

—

#14524042

C₂²

yes

—

#14524038

C₂²

yes

—

#14524037

C₂²

yes

—

#14524035

C₂²

yes

—

#14524034

C₂²

yes

—

#14524033

C₂²

yes

—

#14524033

C₂²

yes

—

#14524033

C₂²

yes

—

#14524021

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

#14524008

C₂²

yes

—

#14524008

C₂²

yes

—

#14524001

C₂²

yes

—

C₄²×D₆₇

C₂²

yes

characteristic

#125×C₆₇

C₂²

yes

—

#125×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂×C₄×D₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

C₂³

yes

—

Dic₆₇⋊C₄

C₂³

yes

—

Dic₆₇⋊C₄

C₂³

yes

—

Dic₆₇⋊C₄

C₂³

yes

—

(C₂×D₆₇)⋊C₄

C₂³

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄²×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

C₂³

yes

—

C₂²×D₆₇

C₂⁴

yes

characteristic

C₂×C₄×C₆₇

C₂⁴

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

—

C₂×Dic₆₇

C₂⁴

yes

—

C₂×Dic₆₇

C₂⁴

yes

—

C₄×D₆₇

—

Dic₁₃₄

—

C₂²×C₆₇

C₂⁵

yes

characteristic

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

Dic₆₇

—

Dic₆₇

2₋¹⁺⁴

yes

—

C₂×C₆₇

C₂×2₋¹⁺⁴

yes

—

D₆₇

—

#3556

—

maximal, Sylow

C₆₇

#3556

yes

characteristic, complemented, Sylow

#564

—

#564

D₆₇

yes

characteristic, complemented

#557

—

#541

—

#533

—

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

—

—

—

—

C₂×D₆₇

yes

—

#125

C₂×D₆₇

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×Q₈

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₆₇

yes

—

C₄⋊C₄

C₂²×D₆₇

yes

—

C₄⋊C₄

C₂²×D₆₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×D₆₇

yes

characteristic

C₄²

C₂²×D₆₇

yes

—

C₂³

—

Q₈

—

134

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

134

—

101

C₂×C₄

C₂³×D₆₇

yes

—

102

C₂×C₄

C₂³×D₆₇

yes

—

103

C₂²

—

134

—

104

C₂²

C₂⁴×D₆₇

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

134

—

109

C₂

—

268

—

110

C₂

2₋¹⁺⁴×D₆₇

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	#564×C₆₇	C₂
hypercenter	#564	D₆₇
hypocenter	C₆₇	#3556
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₆₇

commutator, socle

C₂²×C₆₇

C₂⁵

Fitting

#564×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
1. C₆₇⋊#564	C₂	semidirect
2. #16864123	C₂	semidirect
5. #564×C₆₇	C₂	semidirect
56. C₆₇	#3556	semidirect
58. #564	D₆₇	semidirect

H ≅ G/N

type

1. C₆₇⋊#564

C₂

semidirect

2. #16864123

C₂

semidirect

5. #564×C₆₇

C₂

semidirect

56. C₆₇

#3556

semidirect

58. #564

D₆₇

semidirect

Subgroups — #22574703

Conjugacy classes of subgroups

Complements