Group #22562007 — subgroups

Subgroups — #22562007

Order 8576.

160 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 81 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×C₈×D₆₇	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₆₇)⋊C₁₆	C₂	1	2	yes	maximal
4	C₁₆×Dic₆₇	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₆₇	C₄	1	1	yes	characteristic
7	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
8	C₄²×D₆₇	C₄	1	1	yes	characteristic
9	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
11	C₈×Dic₆₇	C₄	1	1	yes	characteristic
12	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₆₇	C₂²	1	2	yes	—
14	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₆₇	C₈	1	1	yes	characteristic
16	C₂×C₄×D₆₇	C₈	1	1	yes	characteristic
17	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₈×D₆₇	—	2	1	no	—
21	C₈×D₆₇	—	2	1	no	—
22	C₈×D₆₇	—	2	2	no	—
23	C₄×Dic₆₇	C₈	1	1	yes	characteristic
24	C₄×Dic₆₇	C₈	1	1	yes	characteristic
25	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
26	C₆₇⋊C₁₆	—	2	2	no	—
27	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₁₆×C₆₇	—	2	2	no	—
31	C₂²×D₆₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
36	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
37	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
38	C₄×D₆₇	—	2	1	no	—
39	C₄×D₆₇	—	2	1	no	—
40	C₄×D₆₇	—	2	2	no	—
41	C₄×D₆₇	C₁₆	1	4	yes	complemented
42	C₈×C₆₇	—	2	1	no	—
43	C₈×C₆₇	#41	1	1	yes	characteristic
44	C₈×C₆₇	#41	1	1	yes	characteristic
45	C₆₇⋊C₈	—	2	1	no	—
46	C₆₇⋊C₈	M₁₆	1	2	yes	—
47	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
48	C₂×D₆₇	—	2	1	no	—
49	C₂×D₆₇	—	2	1	no	—
50	C₂×D₆₇	C₂×C₁₆	1	2	yes	—
51	C₄×C₆₇	—	2	1	no	—
52	C₄×C₆₇	C₂×C₁₆	1	2	yes	—
53	C₄×C₆₇	#117	1	1	yes	characteristic
54	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
55	Dic₆₇	—	2	1	no	—
56	Dic₆₇	C₂×C₁₆	1	2	yes	—
57	Dic₆₇	M₃₂	1	2	yes	—
58	C₂×C₆₇	#431	1	1	yes	characteristic
59	C₂×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
60	C₂×C₆₇	C₂×M₃₂	1	1	yes	characteristic
61	D₆₇	—	2	2	no	—
62	#2162	—	67	1	no	maximal, Sylow
63	C₆₇	#2162	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄×C₈	—	67	1	no	—
65	C₄⋊C₁₆	—	67	1	no	—
66	C₄⋊C₁₆	D₆₇	1	1	yes	characteristic, complemented
67	C₂²⋊C₁₆	—	67	2	no	—
68	C₄×C₁₆	—	67	2	no	—
69	C₂²×C₈	—	67	1	no	—
70	C₂²×C₈	—	67	1	no	—
71	C₂×C₄²	—	67	1	no	—
72	C₂×C₁₆	—	67	2	no	—
73	C₂×C₁₆	C₂×D₆₇	1	2	yes	—
74	C₄×C₈	—	67	1	no	—
75	C₄×C₈	—	67	1	no	—
76	C₄×C₈	—	67	1	no	—
77	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
78	C₂²×C₄	—	67	1	no	—
79	C₂²×C₄	—	67	1	no	—
80	C₂²×C₄	—	67	1	no	—
81	C₂×C₈	—	67	1	no	—
82	C₂×C₈	—	67	1	no	—
83	C₂×C₈	—	134	2	no	—
84	C₂×C₈	—	134	1	no	—
85	C₂×C₈	—	134	1	no	—
86	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
87	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
88	C₄²	—	67	1	no	—
89	C₄²	—	67	1	no	—
90	C₄²	—	67	1	no	—
91	C₄²	C₄×D₆₇	1	1	yes	characteristic
92	C₁₆	—	2	2	no	—
93	C₁₆	—	134	2	no	—
94	C₂³	—	67	1	no	—
95	C₂×C₄	—	67	1	no	—
96	C₂×C₄	—	67	1	no	—
97	C₂×C₄	—	67	4	no	—
98	C₂×C₄	—	67	1	no	—
99	C₂×C₄	—	134	2	no	—
100	C₂×C₄	—	134	1	no	—
101	C₂×C₄	—	134	1	no	—
102	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
103	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	67	2	no	—
107	C₈	—	134	1	no	—
108	C₈	#11761627	1	1	yes	characteristic
109	C₈	#11761629	1	1	yes	characteristic
110	C₂²	—	67	2	no	—
111	C₂²	—	134	1	no	—
112	C₂²	—	134	1	no	—
113	C₂²	C₂×C₈×D₆₇	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	67	2	no	—
116	C₄	—	67	2	no	—
117	C₄	—	134	1	no	—
118	C₄	C₁₆×D₆₇	1	2	yes	—
119	C₄	#14524001	1	1	yes	characteristic
120	C₄	M₁₆×D₆₇	1	1	yes	characteristic
121	C₂	—	134	2	no	—
122	C₂	#16863448	1	1	yes	characteristic
123	C₂	C₂×C₁₆×D₆₇	1	1	yes	characteristic
124	C₂	M₃₂×D₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₈×D₆₇

C₂

yes

maximal, characteristic

C₄⋊(C₆₇⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

(C₂×D₆₇)⋊C₁₆

C₂

yes

maximal

C₁₆×Dic₆₇

C₂

yes

maximal, complemented

C₄⋊C₁₆×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₆₇

C₄

yes

characteristic

C₂×C₈×D₆₇

C₂²

yes

characteristic

C₄²×D₆₇

C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂²

yes

—

C₄×C₆₇⋊C₈

C₄

yes

characteristic

C₈×Dic₆₇

C₄

yes

characteristic

C₈×Dic₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄×D₆₇

C₈

yes

characteristic

C₂×C₄×D₆₇

C₈

yes

characteristic

C₂×C₄×D₆₇

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₈×D₆₇

—

C₈×D₆₇

—

C₈×D₆₇

—

C₄×Dic₆₇

C₈

yes

characteristic

C₄×Dic₆₇

C₈

yes

characteristic

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₆₇⋊C₁₆

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₁₆×C₆₇

—

C₂²×D₆₇

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

C₂×Dic₆₇

C₂×C₈

yes

characteristic

C₂×Dic₆₇

C₂×C₈

yes

characteristic

C₂×Dic₆₇

C₂×C₈

yes

characteristic

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

C₁₆

yes

complemented

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₆₇⋊C₈

—

C₆₇⋊C₈

M₁₆

yes

—

C₂²×C₆₇

C₂²×C₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

C₂×C₁₆

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₁₆

yes

—

C₄×C₆₇

#117

yes

characteristic

C₄×C₆₇

C₂×M₁₆

yes

characteristic

Dic₆₇

—

Dic₆₇

C₂×C₁₆

yes

—

Dic₆₇

M₃₂

yes

—

C₂×C₆₇

#431

yes

characteristic

C₂×C₆₇

C₂²×C₁₆

yes

characteristic

C₂×C₆₇

C₂×M₃₂

yes

characteristic

D₆₇

—

#2162

—

maximal, Sylow

C₆₇

#2162

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

D₆₇

yes

characteristic, complemented

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₆₇

yes

characteristic

C₁₆

—

C₁₆

—

134

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

100

C₂×C₄

—

134

—

101

C₂×C₄

—

134

—

102

C₂×C₄

C₈×D₆₇

yes

characteristic

103

C₂×C₄

C₈×D₆₇

yes

characteristic

104

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

134

—

108

C₈

#11761627

yes

characteristic

109

C₈

#11761629

yes

characteristic

110

C₂²

—

111

C₂²

—

134

—

112

C₂²

—

134

—

113

C₂²

C₂×C₈×D₆₇

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

134

—

118

C₄

C₁₆×D₆₇

yes

—

119

C₄

#14524001

yes

characteristic

120

C₄

M₁₆×D₆₇

yes

characteristic

121

C₂

—

134

—

122

C₂

#16863448

yes

characteristic

123

C₂

C₂×C₁₆×D₆₇

yes

characteristic

124

C₂

M₃₂×D₆₇

yes

characteristic

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

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₆₇

commutator

C₂×C₆₇

C₂²×C₁₆

Fitting

C₄⋊C₁₆×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×C₈

hypercenter

C₄⋊C₁₆

D₆₇

C₆₇

G ▷ C₂×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
2. C₄⋊(C₆₇⋊C₁₆)	C₂	semidirect
4. C₁₆×Dic₆₇	C₂	semidirect
5. C₄⋊C₁₆×C₆₇	C₂	semidirect
41. C₄×D₆₇	C₁₆	semidirect
63. C₆₇	#2162	semidirect
66. C₄⋊C₁₆	D₆₇	semidirect

H ≅ G/N

type

2. C₄⋊(C₆₇⋊C₁₆)

C₂

semidirect

4. C₁₆×Dic₆₇

C₂

semidirect

5. C₄⋊C₁₆×C₆₇

C₂

semidirect

41. C₄×D₆₇

C₁₆

semidirect

63. C₆₇

#2162

semidirect

66. C₄⋊C₁₆

D₆₇

semidirect

Subgroups — #22562007

Conjugacy classes of subgroups

Complements