Group #22571603 — subgroups

Subgroups — #22571603

Order 8576.

342 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16863953	C₂	1	2	yes	maximal
2	#16863682	C₂	1	4	yes	maximal, complemented
3	#16863654	C₂	1	4	yes	maximal
4	#16863607	C₂	1	4	yes	maximal, complemented
5	#423×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₆₈	C₂²	1	4	yes	—
7	#14524043	C₂²	1	2	yes	—
8	#14524043	C₂²	1	2	yes	—
9	#14524035	—	2	2	no	—
10	#14524022	—	2	2	no	—
11	#14524022	C₂²	1	4	yes	—
12	C₄×Dic₁₃₄	—	2	2	no	—
13	C₄×Dic₁₃₄	C₂²	1	4	yes	—
14	#14524002	—	2	2	no	—
15	#14523975	C₂²	1	4	yes	—
16	#14523968	C₂²	1	8	yes	—
17	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
20	C₂×Dic₁₃₄	—	2	2	no	—
21	C₂×Dic₁₃₄	C₂³	1	4	yes	—
22	Dic₂₆₈	—	4	4	no	—
23	C₄⋊Dic₆₇	—	2	2	no	—
24	C₄⋊Dic₆₇	—	2	2	no	—
25	C₄⋊Dic₆₇	—	2	2	no	—
26	C₄⋊Dic₆₇	C₂³	1	4	yes	—
27	Dic₆₇⋊C₄	—	2	4	no	—
28	Dic₆₇⋊C₄	—	2	4	no	—
29	C₂²⋊Dic₆₇	—	2	4	no	—
30	C₄×Dic₆₇	—	2	4	no	—
31	C₂²×C₄×C₆₇	D₄	1	2	yes	—
32	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	M₁₆×C₆₇	—	4	2	no	—
34	C₂×C₈×C₆₇	C₂³	1	4	yes	—
35	C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₄²×C₆₇	D₄	1	1	yes	characteristic
37	C₄²×C₆₇	C₂³	1	2	yes	—
38	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	—	2	4	no	—
40	C₂×C₄×C₆₇	—	2	1	no	—
41	C₂×C₄×C₆₇	—	2	1	no	—
42	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₆₇	—	2	4	no	—
47	C₂×Dic₆₇	—	2	4	no	—
48	Dic₁₃₄	—	2	4	no	—
49	Dic₁₃₄	—	2	4	no	—
50	Dic₁₃₄	—	4	4	no	—
51	C₈×C₆₇	—	2	4	no	—
52	C₂²×C₆₇	—	2	1	no	—
53	C₂²×C₆₇	—	2	2	no	—
54	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₆₇	—	2	1	no	—
56	C₄×C₆₇	—	2	2	no	—
57	C₄×C₆₇	#137	1	4	yes	—
58	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
59	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
60	Dic₆₇	—	4	4	no	—
61	Dic₆₇	—	4	4	no	—
62	C₂×C₆₇	—	4	1	no	—
63	C₂×C₆₇	#537	1	1	yes	characteristic
64	C₂×C₆₇	C₂×#137	1	2	yes	—
65	#3255	—	67	1	no	maximal, Sylow
66	C₆₇	#3255	1	1	yes	characteristic, complemented, Sylow
67	#533	—	67	2	no	—
68	#475	—	67	4	no	—
69	#470	—	67	4	no	—
70	#462	—	67	4	no	—
71	#423	D₆₇	1	1	yes	characteristic
72	C₂×Q₁₆	—	67	4	no	—
73	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
74	C₄⋊Q₈	—	67	2	no	—
75	C₄⋊Q₈	—	67	2	no	—
76	#125	—	134	2	no	—
77	C₂²⋊Q₈	—	67	4	no	—
78	C₂²⋊Q₈	—	134	2	no	—
79	C₄×Q₈	—	67	4	no	—
80	C₄×Q₈	—	134	2	no	—
81	#117	—	134	2	no	—
82	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
83	#106	—	67	4	no	—
84	C₄⋊C₈	C₂×D₆₇	1	4	yes	—
85	#103	—	67	8	no	—
86	C₂×Q₈	—	67	4	no	—
87	C₂×Q₈	—	134	2	no	—
88	C₂²×C₄	D₂₆₈	1	2	yes	—
89	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
90	Q₁₆	—	268	4	no	—
91	M₁₆	—	4	2	no	—
92	C₂×C₈	C₂²×D₆₇	1	4	yes	—
93	C₄⋊C₄	—	67	4	no	—
94	C₄⋊C₄	—	134	2	no	—
95	C₄⋊C₄	—	134	2	no	—
96	C₄⋊C₄	—	134	4	no	—
97	C₄⋊C₄	—	134	4	no	—
98	C₄⋊C₄	—	134	2	no	—
99	C₂²⋊C₄	—	134	4	no	—
100	C₄²	—	134	4	no	—
101	C₄²	D₂₆₈	1	1	yes	characteristic
102	C₄²	D₂₆₈	1	1	yes	characteristic
103	C₄²	C₂²×D₆₇	1	2	yes	—
104	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
105	Q₈	—	268	4	no	—
106	Q₈	—	134	4	no	—
107	Q₈	—	134	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	134	4	no	—
112	C₂×C₄	—	134	4	no	—
113	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
114	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
115	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
116	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	268	4	no	—
124	C₄	—	268	4	no	—
125	C₄	#14524076	1	4	yes	—
126	C₄	#14524088	1	1	yes	characteristic
127	C₄	#14524091	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#16863977	1	1	yes	characteristic
130	C₂	C₂×#14524076	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863953

C₂

yes

maximal

#16863682

C₂

yes

maximal, complemented

#16863654

C₂

yes

maximal

#16863607

C₂

yes

maximal, complemented

#423×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₂²

yes

—

#14524043

C₂²

yes

—

#14524043

C₂²

yes

—

#14524035

—

#14524022

—

#14524022

C₂²

yes

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

C₂²

yes

—

#14524002

—

#14523975

C₂²

yes

—

#14523968

C₂²

yes

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂³

yes

—

Dic₂₆₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#137

yes

—

C₄×C₆₇

H(5,𝔽₂)

yes

characteristic

C₄×C₆₇

2₋¹⁺⁴

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#537

yes

characteristic

C₂×C₆₇

C₂×#137

yes

—

#3255

—

maximal, Sylow

C₆₇

#3255

yes

characteristic, complemented, Sylow

—

—

—

—

D₆₇

yes

characteristic

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

134

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

134

—

C₄×Q₈

—

C₄×Q₈

—

134

—

#117

—

134

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#106

—

C₄⋊C₈

C₂×D₆₇

yes

—

#103

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂²×C₄

D₂₆₈

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

Q₁₆

—

268

—

M₁₆

—

C₂×C₈

C₂²×D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

134

—

100

C₄²

—

134

—

101

C₄²

D₂₆₈

yes

characteristic

102

C₄²

D₂₆₈

yes

characteristic

103

C₄²

C₂²×D₆₇

yes

—

104

C₂³

C₂×D₂₆₈

yes

characteristic

105

Q₈

—

268

—

106

Q₈

—

134

—

107

Q₈

—

134

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

134

—

112

C₂×C₄

—

134

—

113

C₂×C₄

C₂×D₂₆₈

yes

characteristic

114

C₂×C₄

C₂×D₂₆₈

yes

—

115

C₂×C₄

C₂×D₂₆₈

yes

—

116

C₂×C₄

C₂³×D₆₇

yes

characteristic

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂²×D₂₆₈

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

268

—

124

C₄

—

268

—

125

C₄

#14524076

yes

—

126

C₄

#14524088

yes

characteristic

127

C₄

#14524091

yes

characteristic

128

C₂

—

129

C₂

#16863977

yes

characteristic

130

C₂

C₂×#14524076

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#423×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#423	D₆₇
hypocenter	C₆₇	#3255
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#423×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. #16863682	C₂	semidirect
4. #16863607	C₂	semidirect
66. C₆₇	#3255	semidirect

H ≅ G/N

type

2. #16863682

C₂

semidirect

4. #16863607

C₂

semidirect

66. C₆₇

#3255

semidirect

Subgroups — #22571603

Conjugacy classes of subgroups

Complements