Group #22562008 — subgroups

Subgroups — #22562008

Order 8576.

160 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₄⋊(C₆₇⋊C₁₆)	C₂	1	2	yes	maximal, complemented
2	C₂²⋊(C₆₇⋊C₁₆)	C₂	1	2	yes	maximal
3	C₄×C₆₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₇⋊C₁₆	C₂²	1	4	yes	—
6	C₂²×C₈×C₆₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₆₇	C₄	1	1	yes	characteristic
9	C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
10	C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
11	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₆₇⋊C₁₆	—	2	4	no	—
14	C₂²×C₄×C₆₇	C₈	1	1	yes	characteristic
15	C₂²×C₄×C₆₇	C₈	1	1	yes	characteristic
16	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₆₇	—	2	2	no	—
18	C₂×C₈×C₆₇	—	2	2	no	—
19	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
23	C₄²×C₆₇	C₈	1	1	yes	characteristic
24	C₄²×C₆₇	C₈	1	1	yes	characteristic
25	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₆₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₆₇	—	2	2	no	—
29	C₂×C₄×C₆₇	—	2	1	no	—
30	C₂×C₄×C₆₇	—	2	1	no	—
31	C₂×C₄×C₆₇	C₁₆	1	4	yes	complemented
32	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₆₇	—	2	1	no	—
39	C₈×C₆₇	—	2	1	no	—
40	C₈×C₆₇	M₁₆	1	2	yes	—
41	C₈×C₆₇	#41	1	2	yes	—
42	C₂²×C₆₇	—	2	1	no	—
43	C₂²×C₆₇	—	2	1	no	—
44	C₂²×C₆₇	C₂×C₁₆	1	2	yes	—
45	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
46	C₄×C₆₇	—	2	1	no	—
47	C₄×C₆₇	—	2	1	no	—
48	C₄×C₆₇	C₂×C₁₆	1	2	yes	—
49	C₄×C₆₇	C₂×C₁₆	1	2	yes	—
50	C₄×C₆₇	M₃₂	1	2	yes	—
51	C₄×C₆₇	#117	1	1	yes	characteristic
52	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₆₇	—	2	2	no	—
54	C₂×C₆₇	#431	1	1	yes	characteristic
55	C₂×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
56	C₂×C₆₇	C₂×M₃₂	1	1	yes	characteristic
57	#2162	—	67	1	no	maximal, Sylow
58	C₆₇	#2162	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₄×C₈	D₆₇	1	1	yes	characteristic
60	C₄⋊C₁₆	—	67	2	no	—
61	C₂²⋊C₁₆	—	67	2	no	—
62	C₄×C₁₆	—	67	2	no	—
63	C₂²×C₈	Dic₆₇	1	1	yes	characteristic
64	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
65	C₂×C₄²	Dic₆₇	1	1	yes	characteristic
66	C₂×C₁₆	—	67	4	no	—
67	C₄×C₈	Dic₆₇	1	1	yes	characteristic
68	C₄×C₈	Dic₆₇	1	1	yes	characteristic
69	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
70	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
71	C₂²×C₄	C₆₇⋊C₈	1	1	yes	characteristic
72	C₂²×C₄	C₆₇⋊C₈	1	1	yes	characteristic
73	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
77	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
78	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
79	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
80	C₄²	C₆₇⋊C₈	1	1	yes	characteristic
81	C₄²	C₆₇⋊C₈	1	1	yes	characteristic
82	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
83	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
84	C₁₆	—	134	4	no	—
85	C₂³	C₂×C₆₇⋊C₈	1	1	yes	characteristic
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	C₆₇⋊C₁₆	1	4	yes	complemented
90	C₂×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
91	C₂×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
92	C₂×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
93	C₂×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
94	C₂×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
95	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	2	1	no	—
98	C₈	#11761612	1	2	yes	—
99	C₈	#11761628	1	2	yes	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	C₂×C₆₇⋊C₁₆	1	2	yes	—
103	C₂²	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	C₂×C₆₇⋊C₁₆	1	2	yes	—
107	C₄	C₂×C₆₇⋊C₁₆	1	2	yes	—
108	C₄	#14523983	1	2	yes	—
109	C₄	#14524002	1	1	yes	characteristic
110	C₄	C₂×#11761612	1	1	yes	characteristic
111	C₂	—	2	2	no	—
112	C₂	#16863449	1	1	yes	characteristic
113	C₂	C₂²×C₆₇⋊C₁₆	1	1	yes	characteristic
114	C₂	C₂×#14523983	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊(C₆₇⋊C₁₆)

C₂

yes

maximal, complemented

C₂²⋊(C₆₇⋊C₁₆)

C₂

yes

maximal

C₄×C₆₇⋊C₁₆

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₆₇

C₂

yes

maximal, characteristic

C₂×C₆₇⋊C₁₆

C₂²

yes

—

C₂²×C₈×C₆₇

C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₄

yes

characteristic

C₄×C₈×C₆₇

C₄

yes

characteristic

C₄×C₈×C₆₇

C₄

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₈

yes

characteristic

C₂²×C₄×C₆₇

C₈

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

C₈

yes

characteristic

C₄²×C₆₇

C₈

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂³×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₁₆

yes

complemented

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

M₁₆

yes

—

C₈×C₆₇

#41

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₁₆

yes

—

C₂²×C₆₇

C₂²×C₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₁₆

yes

—

C₄×C₆₇

C₂×C₁₆

yes

—

C₄×C₆₇

M₃₂

yes

—

C₄×C₆₇

#117

yes

characteristic

C₄×C₆₇

C₂×M₁₆

yes

characteristic

C₂×C₆₇

—

C₂×C₆₇

#431

yes

characteristic

C₂×C₆₇

C₂²×C₁₆

yes

characteristic

C₂×C₆₇

C₂×M₃₂

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

D₆₇

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂²×C₈

Dic₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄²

Dic₆₇

yes

characteristic

C₂×C₁₆

—

C₄×C₈

Dic₆₇

yes

characteristic

C₄×C₈

Dic₆₇

yes

characteristic

C₄×C₈

C₂×D₆₇

yes

characteristic

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂²×C₄

C₆₇⋊C₈

yes

characteristic

C₂²×C₄

C₆₇⋊C₈

yes

characteristic

C₂²×C₄

C₂×Dic₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄²

C₆₇⋊C₈

yes

characteristic

C₄²

C₆₇⋊C₈

yes

characteristic

C₄²

C₂×Dic₆₇

yes

characteristic

C₄²

C₂×Dic₆₇

yes

characteristic

C₁₆

—

134

—

C₂³

C₂×C₆₇⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₆₇⋊C₁₆

yes

complemented

C₂×C₄

C₂×C₆₇⋊C₈

yes

characteristic

C₂×C₄

C₂×C₆₇⋊C₈

yes

characteristic

C₂×C₄

C₂×C₆₇⋊C₈

yes

characteristic

C₂×C₄

C₂×C₆₇⋊C₈

yes

characteristic

C₂×C₄

C₂×C₆₇⋊C₈

yes

characteristic

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

C₈

—

C₈

—

C₈

#11761612

yes

—

C₈

#11761628

yes

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂×C₆₇⋊C₁₆

yes

—

103

C₂²

C₂²×C₆₇⋊C₈

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

C₂×C₆₇⋊C₁₆

yes

—

107

C₄

C₂×C₆₇⋊C₁₆

yes

—

108

C₄

#14523983

yes

—

109

C₄

#14524002

yes

characteristic

110

C₄

C₂×#11761612

yes

characteristic

111

C₂

—

112

C₂

#16863449

yes

characteristic

113

C₂

C₂²×C₆₇⋊C₁₆

yes

characteristic

114

C₂

C₂×#14523983

yes

characteristic

	subgroup	G/H
center	C₄×C₈	C₂×D₆₇
commutator	C₂×C₆₇	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	C₂×C₄×C₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₈
hypercenter	C₂×C₄×C₈	D₆₇
hypocenter	C₆₇	#2162
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄×C₈

C₂×D₆₇

commutator

C₂×C₆₇

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

C₂×C₄×C₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×C₈

hypercenter

C₂×C₄×C₈

D₆₇

C₆₇

G ▷ C₂×C₆₇ ▷ C₆₇

C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
1. C₄⋊(C₆₇⋊C₁₆)	C₂	semidirect
3. C₄×C₆₇⋊C₁₆	C₂	semidirect
31. C₂×C₄×C₆₇	C₁₆	semidirect
58. C₆₇	#2162	semidirect
89. C₂×C₄	C₆₇⋊C₁₆	semidirect

H ≅ G/N

type

1. C₄⋊(C₆₇⋊C₁₆)

C₂

semidirect

3. C₄×C₆₇⋊C₁₆

C₂

semidirect

31. C₂×C₄×C₆₇

C₁₆

semidirect

58. C₆₇

#2162

semidirect

89. C₂×C₄

C₆₇⋊C₁₆

semidirect

Subgroups — #22562008

Conjugacy classes of subgroups

Complements