Group #22569294 — subgroups

Subgroups — #22569294

Order 8576.

356 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 217 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	8	yes	maximal, complemented
2	#16863453	C₂	1	4	yes	maximal
3	#16863449	C₂	1	2	yes	maximal
4	C₂×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
6	C₄⋊(C₆₇⋊C₈)	C₂²	1	8	yes	—
7	C₂²⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
8	#14523954	C₂²	1	8	yes	—
9	C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
10	C₂²×Q₈×C₆₇	C₄	1	1	yes	characteristic
11	C₄×Q₈×C₆₇	C₄	1	4	yes	—
12	C₄×Q₈×C₆₇	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
14	C₂×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
17	C₂×C₆₇⋊C₈	C₂³	1	8	yes	—
18	C₂×Q₈×C₆₇	—	2	4	no	—
19	C₂×Q₈×C₆₇	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
26	C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
27	C₄²×C₆₇	C₂×C₄	1	4	yes	—
28	C₄²×C₆₇	C₂×C₄	1	2	yes	—
29	C₄²×C₆₇	C₂³	1	2	yes	—
30	C₄²×C₆₇	C₂³	1	4	yes	—
31	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
32	Q₈×C₆₇	—	2	8	no	—
33	C₂×C₄×C₆₇	—	2	4	no	—
34	C₂×C₄×C₆₇	—	2	2	no	—
35	C₂×C₄×C₆₇	—	4	1	no	—
36	C₂×C₄×C₆₇	—	4	2	no	—
37	C₂×C₄×C₆₇	M₁₆	1	4	yes	—
38	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
41	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
42	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
44	C₆₇⋊C₈	—	2	8	no	—
45	C₂²×C₆₇	—	2	1	no	—
46	C₂²×C₆₇	—	2	1	no	—
47	C₂²×C₆₇	C₂×M₁₆	1	2	yes	—
48	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
49	C₄×C₆₇	—	2	1	no	—
50	C₄×C₆₇	—	2	4	no	—
51	C₄×C₆₇	—	2	2	no	—
52	C₄×C₆₇	—	2	4	no	—
53	C₄×C₆₇	C₂×M₁₆	1	4	yes	—
54	C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
55	C₂×C₆₇	—	2	2	no	—
56	C₂×C₆₇	#519	1	1	yes	characteristic
57	C₂×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
58	C₂×C₆₇	#568	1	1	yes	characteristic
59	#2983	—	67	1	no	maximal, Sylow
60	C₆₇	#2983	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₄×Q₈	D₆₇	1	1	yes	characteristic
62	#446	—	67	8	no	—
63	#432	—	67	4	no	—
64	#431	—	67	2	no	—
65	C₂²×Q₈	Dic₆₇	1	1	yes	characteristic
66	C₄×Q₈	Dic₆₇	1	4	yes	—
67	C₄×Q₈	C₂×D₆₇	1	4	yes	—
68	C₂×C₄⋊C₄	Dic₆₇	1	2	yes	—
69	C₂×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
70	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
71	C₂×C₄²	C₂×D₆₇	1	2	yes	—
72	C₄⋊C₈	—	67	4	no	—
73	C₄⋊C₈	—	67	8	no	—
74	C₂²⋊C₈	—	67	4	no	—
75	#97	—	67	8	no	—
76	C₄×C₈	—	67	4	no	—
77	C₂×Q₈	—	2	4	no	—
78	C₂×Q₈	C₂×Dic₆₇	1	4	yes	—
79	C₂²×C₄	C₂×Dic₆₇	1	2	yes	—
80	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
81	C₂²×C₄	C₂×Dic₆₇	1	2	yes	—
82	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
83	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
84	C₂×C₈	—	67	8	no	—
85	C₄⋊C₄	C₂×Dic₆₇	1	8	yes	—
86	C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
87	C₄²	C₂×Dic₆₇	1	4	yes	—
88	C₄²	C₂×Dic₆₇	1	2	yes	—
89	C₄²	C₂²×D₆₇	1	2	yes	—
90	C₄²	C₂²×D₆₇	1	4	yes	—
91	C₂³	C₂²×Dic₆₇	1	1	yes	characteristic
92	Q₈	—	2	8	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	#11761612	1	4	yes	—
98	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
99	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
100	C₂×C₄	C₂²×Dic₆₇	1	4	yes	—
101	C₂×C₄	C₂²×Dic₆₇	1	4	yes	—
102	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
103	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
104	C₈	—	134	8	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	C₂×#11761612	1	2	yes	—
108	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	2	4	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	4	no	—
113	C₄	C₂×#11761612	1	4	yes	—
114	C₄	#14524091	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	#16863874	1	1	yes	characteristic
117	C₂	C₂²×#11761612	1	1	yes	characteristic
118	C₂	#16864163	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863527

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

—

#14523954

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

—

C₂²×Q₈×C₆₇

C₄

yes

characteristic

C₄×Q₈×C₆₇

C₄

yes

—

C₄×Q₈×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₄

yes

—

C₂×C₄⋊C₄×C₆₇

C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

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

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

M₁₆

yes

—

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

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×M₁₆

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

—

—

—

C₂²×Q₈

Dic₆₇

yes

characteristic

C₄×Q₈

Dic₆₇

yes

—

C₄×Q₈

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

Dic₆₇

yes

—

C₂×C₄⋊C₄

Dic₆₇

yes

characteristic

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₂×C₄²

C₂×D₆₇

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂×Dic₆₇

yes

characteristic

C₂²×C₄

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂×Dic₆₇

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

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

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#11761612

yes

—

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

C₂×C₄

C₂²×Dic₆₇

yes

—

100

C₂×C₄

C₂²×Dic₆₇

yes

—

101

C₂×C₄

C₂²×Dic₆₇

yes

—

102

C₂×C₄

C₂²×Dic₆₇

yes

—

103

C₂×C₄

C₂³×D₆₇

yes

characteristic

104

C₈

—

134

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×#11761612

yes

—

108

C₂²

C₂³×Dic₆₇

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

C₂×#11761612

yes

—

114

C₄

#14524091

yes

—

115

C₂

—

116

C₂

#16863874

yes

characteristic

117

C₂

C₂²×#11761612

yes

characteristic

118

C₂

#16864163

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	#2983
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
60. C₆₇	#2983	semidirect

H ≅ G/N

type

1. #16863527

C₂

semidirect

60. C₆₇

#2983

semidirect

Subgroups — #22569294

Conjugacy classes of subgroups

Complements