Group #22562116 — subgroups

Subgroups — #22562116

Order 8576.

126 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#16863172	C₂	1	1	yes	maximal, characteristic
5	#16863171	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
9	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
12	Dic₆₇⋊C₈	C₄	1	1	yes	characteristic
13	Dic₆₇⋊C₈	C₄	1	1	yes	characteristic
14	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
15	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₃₄	C₈	1	1	yes	characteristic
20	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
23	Dic₆₇⋊C₄	C₈	1	1	yes	characteristic
24	Dic₆₇⋊C₄	C₈	1	1	yes	characteristic
25	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
27	C₆₇⋊C₁₆	—	2	1	no	—
28	C₆₇⋊C₁₆	Q₈	1	2	yes	complemented
29	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₁₆×C₆₇	—	2	1	no	—
33	C₁₆×C₆₇	—	2	1	no	—
34	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
39	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
40	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
41	Dic₁₃₄	—	2	2	no	—
42	C₈×C₆₇	—	2	1	no	—
43	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₈×C₆₇	#41	1	1	yes	characteristic
45	C₆₇⋊C₈	—	2	1	no	—
46	C₆₇⋊C₈	—	2	1	no	—
47	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₆₇	—	2	1	no	—
49	C₄×C₆₇	—	2	1	no	—
50	C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
51	C₄×C₆₇	#131	1	1	yes	characteristic
52	Dic₆₇	—	2	1	no	—
53	Dic₆₇	—	2	1	no	—
54	Dic₆₇	—	2	1	no	—
55	Dic₆₇	M₃₂	1	2	yes	—
56	C₂×C₆₇	C₈×Q₈	1	1	yes	characteristic
57	C₂×C₆₇	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₆₇	#504	1	1	yes	characteristic
59	#2183	—	67	1	no	maximal, Sylow
60	C₆₇	#2183	1	1	yes	characteristic, complemented, Sylow
61	C₈×Q₈	—	67	1	no	—
62	C₄⋊C₁₆	—	67	1	no	—
63	C₄⋊C₁₆	—	67	1	no	—
64	C₄⋊C₁₆	D₆₇	1	1	yes	characteristic
65	#346	—	67	1	no	—
66	#346	—	67	1	no	—
67	C₄×C₁₆	—	67	1	no	—
68	C₄×Q₈	—	67	1	no	—
69	C₂×C₁₆	—	67	1	no	—
70	C₂×C₁₆	—	67	1	no	—
71	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
72	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
73	C₄⋊C₈	—	67	1	no	—
74	C₄⋊C₈	—	67	1	no	—
75	C₄⋊C₈	—	67	1	no	—
76	C₄×C₈	—	67	1	no	—
77	C₄×C₈	—	67	1	no	—
78	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
79	C₂×Q₈	—	67	1	no	—
80	C₂×C₈	—	67	1	no	—
81	C₂×C₈	—	67	1	no	—
82	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
83	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
84	C₄⋊C₄	—	67	1	no	—
85	C₄⋊C₄	—	67	1	no	—
86	C₄⋊C₄	—	67	1	no	—
87	C₄²	—	67	1	no	—
88	C₄²	—	67	1	no	—
89	C₄²	C₄×D₆₇	1	1	yes	characteristic
90	C₁₆	—	2	1	no	—
91	C₁₆	—	2	1	no	—
92	C₁₆	—	67	2	no	—
93	C₁₆	—	134	1	no	—
94	Q₈	—	134	2	no	—
95	C₂×C₄	—	67	1	no	—
96	C₂×C₄	—	67	1	no	—
97	C₂×C₄	—	67	1	no	—
98	C₂×C₄	—	67	1	no	—
99	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
100	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	134	1	no	—
104	C₈	—	134	1	no	—
105	C₈	Q₈×D₆₇	1	1	yes	characteristic
106	C₈	#11761627	1	1	yes	characteristic
107	C₂²	C₂×C₈×D₆₇	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	67	2	no	—
111	C₄	—	134	1	no	—
112	C₄	—	134	1	no	—
113	C₄	—	134	1	no	—
114	C₄	#14524008	1	1	yes	characteristic
115	C₄	#14524051	1	1	yes	characteristic
116	C₂	#16863520	1	1	yes	characteristic
117	C₂	M₃₂×D₆₇	1	1	yes	characteristic
118	C₂	#16863812	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₁₃₄

C₂

yes

maximal, characteristic

Dic₆₇⋊C₁₆

C₂

yes

maximal, characteristic

Dic₆₇⋊C₁₆

C₂

yes

maximal, characteristic

#16863172

C₂

yes

maximal, characteristic

#16863171

C₂

yes

maximal, characteristic

C₁₆×Dic₆₇

C₂

yes

maximal, characteristic

C₄⋊C₁₆×C₆₇

C₂

yes

maximal, characteristic

C₄×Dic₁₃₄

C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₄

yes

characteristic

Dic₆₇⋊C₈

C₄

yes

characteristic

Dic₆₇⋊C₈

C₄

yes

characteristic

C₈×Dic₆₇

C₂²

yes

characteristic

C₈×Dic₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

C₈

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄⋊Dic₆₇

C₈

yes

characteristic

Dic₆₇⋊C₄

C₈

yes

characteristic

Dic₆₇⋊C₄

C₈

yes

characteristic

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

Q₈

yes

complemented

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

—

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

C₂×Dic₆₇

C₂×C₈

yes

characteristic

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂×Q₈

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂²×C₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×Q₈

yes

characteristic

C₄×C₆₇

#131

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

M₃₂

yes

—

C₂×C₆₇

C₈×Q₈

yes

characteristic

C₂×C₆₇

C₂×M₃₂

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

D₆₇

yes

characteristic

#346

—

#346

—

C₄×C₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

characteristic

C₂×C₁₆

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₄×D₆₇

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

134

—

Q₈

—

134

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₆₇

yes

characteristic

100

C₂×C₄

C₈×D₆₇

yes

characteristic

101

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

102

C₈

—

103

C₈

—

134

—

104

C₈

—

134

—

105

C₈

Q₈×D₆₇

yes

characteristic

106

C₈

#11761627

yes

characteristic

107

C₂²

C₂×C₈×D₆₇

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

134

—

112

C₄

—

134

—

113

C₄

—

134

—

114

C₄

#14524008

yes

characteristic

115

C₄

#14524051

yes

characteristic

116

C₂

#16863520

yes

characteristic

117

C₂

M₃₂×D₆₇

yes

characteristic

118

C₂

#16863812

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂²×C₈
Fitting	C₄⋊C₁₆×C₆₇	C₂
hypercenter	C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	#2183
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₆₇

commutator, socle

C₂²×C₆₇

C₂²×C₈

Fitting

C₄⋊C₁₆×C₆₇

C₂

hypercenter

C₄⋊C₁₆

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
28. C₆₇⋊C₁₆	Q₈	semidirect
60. C₆₇	#2183	semidirect

H ≅ G/N

type

28. C₆₇⋊C₁₆

Q₈

semidirect

60. C₆₇

#2183

semidirect

Subgroups — #22562116

Conjugacy classes of subgroups

Complements