Group #22569298 — subgroups

Subgroups — #22569298

Order 8576.

364 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 221 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	#16863527	C₂	1	6	yes	maximal, complemented
2	Q₈×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
3	#16863453	C₂	1	3	yes	maximal
4	#16863449	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
7	C₄⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
8	C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂²⋊(C₆₇⋊C₈)	C₂²	1	3	yes	—
10	#14523954	C₂²	1	6	yes	—
11	C₄×C₆₇⋊C₈	C₂²	1	6	yes	—
12	C₂²×Q₈×C₆₇	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₆₇	C₄	1	3	yes	—
14	C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
15	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₆₇	C₂²	1	3	yes	—
17	C₂×C₄⋊C₄×C₆₇	C₄	1	3	yes	—
18	C₂×C₄²×C₆₇	C₂²	1	3	yes	—
19	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
20	C₂×C₆₇⋊C₈	C₂³	1	6	yes	—
21	C₂×Q₈×C₆₇	—	2	4	no	—
22	C₂×Q₈×C₆₇	C₂×C₄	1	3	yes	—
23	C₂×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	C₂×C₄	1	3	yes	—
25	C₂²×C₄×C₆₇	C₂×C₄	1	3	yes	—
26	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
28	C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
29	C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
30	C₄²×C₆₇	C₂×C₄	1	3	yes	—
31	C₄²×C₆₇	C₂×C₄	1	3	yes	—
32	C₄²×C₆₇	C₂³	1	3	yes	—
33	C₄²×C₆₇	C₂³	1	3	yes	—
34	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₆₇	—	2	6	no	—
36	Q₈×C₆₇	M₁₆	1	4	yes	—
37	C₂×C₄×C₆₇	—	2	6	no	—
38	C₂×C₄×C₆₇	—	2	6	no	—
39	C₂×C₄×C₆₇	—	2	2	no	—
40	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	C₂²×C₄	1	3	yes	—
42	C₂×C₄×C₆₇	C₂²×C₄	1	3	yes	—
43	C₂×C₄×C₆₇	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₆₇	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
46	C₆₇⋊C₈	—	2	2	no	—
47	C₆₇⋊C₈	—	2	6	no	—
48	C₂²×C₆₇	—	2	1	no	—
49	C₂²×C₆₇	—	2	1	no	—
50	C₂²×C₆₇	#131	1	2	yes	—
51	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
52	C₄×C₆₇	—	2	1	no	—
53	C₄×C₆₇	—	2	3	no	—
54	C₄×C₆₇	—	2	3	no	—
55	C₄×C₆₇	—	2	3	no	—
56	C₄×C₆₇	C₂×M₁₆	1	6	yes	—
57	C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
58	C₂×C₆₇	—	2	2	no	—
59	C₂×C₆₇	#519	1	1	yes	characteristic
60	C₂×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
61	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
62	#2984	—	67	1	no	maximal, Sylow
63	C₆₇	#2984	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄×Q₈	D₆₇	1	1	yes	characteristic
65	#446	—	67	6	no	—
66	C₈×Q₈	—	67	2	no	—
67	#432	—	67	3	no	—
68	#431	—	67	3	no	—
69	C₂²×Q₈	Dic₆₇	1	1	yes	characteristic
70	C₄×Q₈	Dic₆₇	1	3	yes	—
71	C₄×Q₈	Dic₆₇	1	1	yes	characteristic
72	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
73	C₄×Q₈	C₂×D₆₇	1	3	yes	—
74	C₂×C₄⋊C₄	Dic₆₇	1	3	yes	—
75	C₂×C₄²	C₂×D₆₇	1	3	yes	—
76	C₄⋊C₈	—	67	6	no	—
77	C₄⋊C₈	—	67	6	no	—
78	C₂²⋊C₈	—	67	1	no	—
79	C₂²⋊C₈	—	67	3	no	—
80	#97	—	67	6	no	—
81	C₄×C₈	—	67	6	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂×Q₈	C₂×Dic₆₇	1	3	yes	—
84	C₂×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₆₇	1	3	yes	—
86	C₂²×C₄	C₂×Dic₆₇	1	3	yes	—
87	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
88	C₂×C₈	—	67	2	no	—
89	C₂×C₈	—	67	6	no	—
90	C₄⋊C₄	C₂×Dic₆₇	1	6	yes	—
91	C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
92	C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
93	C₄²	C₂×Dic₆₇	1	3	yes	—
94	C₄²	C₂×Dic₆₇	1	3	yes	—
95	C₄²	C₂²×D₆₇	1	3	yes	—
96	C₄²	C₂²×D₆₇	1	3	yes	—
97	C₂³	C₂²×Dic₆₇	1	1	yes	characteristic
98	Q₈	—	2	6	no	—
99	Q₈	#11761612	1	4	yes	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	—	2	6	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
104	C₂×C₄	C₂²×Dic₆₇	1	3	yes	—
105	C₂×C₄	C₂²×Dic₆₇	1	3	yes	—
106	C₂×C₄	C₂²×Dic₆₇	1	3	yes	—
107	C₂×C₄	C₂²×Dic₆₇	1	3	yes	—
108	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
109	C₈	—	134	2	no	—
110	C₈	—	134	6	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#14524050	1	2	yes	—
114	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	2	3	no	—
119	C₄	C₂×#11761612	1	6	yes	—
120	C₄	#14524091	1	2	yes	—
121	C₂	—	2	2	no	—
122	C₂	#16863874	1	1	yes	characteristic
123	C₂	C₂²×#11761612	1	1	yes	characteristic
124	C₂	C₂×#14524050	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863527

C₂

yes

maximal, complemented

Q₈×C₆₇⋊C₈

C₂

yes

maximal, complemented

#16863453

C₂

yes

maximal

#16863449

C₂

yes

maximal

C₂×C₄×Q₈×C₆₇

C₂

yes

maximal, characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₂²⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523954

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

—

C₂²×Q₈×C₆₇

C₄

yes

characteristic

C₄×Q₈×C₆₇

C₄

yes

—

C₄×Q₈×C₆₇

C₄

yes

characteristic

C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₄×Q₈×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₄

yes

—

C₂×C₄²×C₆₇

C₂²

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂×Q₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂²×C₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

M₁₆

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#131

yes

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×M₁₆

yes

—

C₄×C₆₇

2₋¹⁺⁴

yes

—

C₂×C₆₇

—

C₂×C₆₇

#519

yes

characteristic

C₂×C₆₇

C₂²×M₁₆

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₆₇

yes

characteristic

#446

—

C₈×Q₈

—

#432

—

#431

—

C₂²×Q₈

Dic₆₇

yes

characteristic

C₄×Q₈

Dic₆₇

yes

—

C₄×Q₈

Dic₆₇

yes

characteristic

C₄×Q₈

C₂×D₆₇

yes

characteristic

C₄×Q₈

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

Dic₆₇

yes

—

C₂×C₄²

C₂×D₆₇

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₆₇

yes

—

C₂×Q₈

C₂×Dic₆₇

yes

characteristic

C₂²×C₄

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₆₇

yes

—

C₄⋊C₄

C₂×Dic₆₇

yes

—

C₄⋊C₄

C₂×Dic₆₇

yes

—

C₄²

C₂×Dic₆₇

yes

—

C₄²

C₂×Dic₆₇

yes

—

C₄²

C₂²×D₆₇

yes

—

C₄²

C₂²×D₆₇

yes

—

C₂³

C₂²×Dic₆₇

yes

characteristic

Q₈

—

Q₈

#11761612

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

104

C₂×C₄

C₂²×Dic₆₇

yes

—

105

C₂×C₄

C₂²×Dic₆₇

yes

—

106

C₂×C₄

C₂²×Dic₆₇

yes

—

107

C₂×C₄

C₂²×Dic₆₇

yes

—

108

C₂×C₄

C₂³×D₆₇

yes

characteristic

109

C₈

—

134

—

110

C₈

—

134

—

111

C₂²

—

112

C₂²

—

113

C₂²

#14524050

yes

—

114

C₂²

C₂³×Dic₆₇

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₂×#11761612

yes

—

120

C₄

#14524091

yes

—

121

C₂

—

122

C₂

#16863874

yes

characteristic

123

C₂

C₂²×#11761612

yes

characteristic

124

C₂

C₂×#14524050

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₆₇	C₂
hypercenter	C₂×C₄×Q₈	D₆₇
hypocenter	C₆₇	#2984
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₇

commutator, socle

C₂²×C₆₇

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₆₇

C₂

hypercenter

C₂×C₄×Q₈

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #16863527	C₂	semidirect
2. Q₈×C₆₇⋊C₈	C₂	semidirect
63. C₆₇	#2984	semidirect

H ≅ G/N

type

1. #16863527

C₂

semidirect

2. Q₈×C₆₇⋊C₈

C₂

semidirect

63. C₆₇

#2984

semidirect

Subgroups — #22569298

Conjugacy classes of subgroups

Complements