Group #22571554 — subgroups

Subgroups — #22571554

Order 8576.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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	#16863692	C₂	1	4	yes	maximal, complemented
3	#16863637	C₂	1	4	yes	maximal
4	#16863607	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊C₈×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	#14523977	C₂²	1	4	yes	—
16	#14523968	C₂²	1	8	yes	—
17	C₂²×C₈×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	1	yes	characteristic
32	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₆₇	—	4	2	no	—
35	C₂×C₈×C₆₇	C₂³	1	4	yes	—
36	C₄²×C₆₇	D₄	1	2	yes	—
37	C₄²×C₆₇	C₂³	1	2	yes	—
38	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	—	2	1	no	—
40	C₂×C₄×C₆₇	—	2	1	no	—
41	C₂×C₄×C₆₇	—	4	1	no	—
42	C₂×C₄×C₆₇	Q₁₆	1	4	yes	complemented
43	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₆₇	—	2	4	no	—
48	C₂×Dic₆₇	—	2	4	no	—
49	Dic₁₃₄	—	2	4	no	—
50	Dic₁₃₄	—	2	4	no	—
51	Dic₁₃₄	—	4	4	no	—
52	C₈×C₆₇	—	2	4	no	—
53	C₂²×C₆₇	—	2	1	no	—
54	C₂²×C₆₇	—	2	1	no	—
55	C₂²×C₆₇	C₂×Q₁₆	1	2	yes	—
56	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₆₇	—	2	1	no	—
58	C₄×C₆₇	—	2	2	no	—
59	C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
60	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
61	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
62	Dic₆₇	—	4	4	no	—
63	Dic₆₇	—	4	4	no	—
64	C₂×C₆₇	—	2	2	no	—
65	C₂×C₆₇	#537	1	1	yes	characteristic
66	C₂×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
67	C₂×C₆₇	#575	1	1	yes	characteristic
68	#3249	—	67	1	no	maximal, Sylow
69	C₆₇	#3249	1	1	yes	characteristic, complemented, Sylow
70	#533	—	67	2	no	—
71	#477	—	67	4	no	—
72	#467	—	67	4	no	—
73	#462	—	67	4	no	—
74	C₂×C₄⋊C₈	D₆₇	1	1	yes	characteristic
75	C₂×Q₁₆	—	67	4	no	—
76	C₂²×C₈	C₂×D₆₇	1	2	yes	—
77	C₄⋊Q₈	—	67	2	no	—
78	C₄⋊Q₈	—	67	2	no	—
79	#125	—	134	2	no	—
80	C₂²⋊Q₈	—	67	4	no	—
81	C₂²⋊Q₈	—	134	2	no	—
82	C₄×Q₈	—	67	4	no	—
83	C₄×Q₈	—	134	2	no	—
84	#117	—	134	2	no	—
85	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
86	#107	—	67	4	no	—
87	C₄⋊C₈	C₂×D₆₇	1	4	yes	—
88	#103	—	67	8	no	—
89	C₂×Q₈	—	67	4	no	—
90	C₂×Q₈	—	134	2	no	—
91	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
92	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
94	Q₁₆	—	268	4	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	C₂²×D₆₇	1	4	yes	—
97	C₄⋊C₄	—	67	4	no	—
98	C₄⋊C₄	—	134	2	no	—
99	C₄⋊C₄	—	134	2	no	—
100	C₄⋊C₄	—	134	4	no	—
101	C₄⋊C₄	—	134	4	no	—
102	C₄⋊C₄	—	134	2	no	—
103	C₂²⋊C₄	—	134	4	no	—
104	C₄²	—	134	4	no	—
105	C₄²	D₂₆₈	1	2	yes	—
106	C₄²	C₂²×D₆₇	1	2	yes	—
107	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
108	Q₈	—	268	4	no	—
109	Q₈	—	134	4	no	—
110	Q₈	—	134	4	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	134	4	no	—
115	C₂×C₄	—	134	4	no	—
116	C₂×C₄	Dic₂₆₈	1	4	yes	complemented
117	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
118	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
119	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
120	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×Dic₂₆₈	1	2	yes	—
125	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	1	no	—
128	C₄	—	268	4	no	—
129	C₄	—	268	4	no	—
130	C₄	C₂×Dic₂₆₈	1	4	yes	—
131	C₄	#14524088	1	1	yes	characteristic
132	C₄	#14524091	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	#16863977	1	1	yes	characteristic
135	C₂	C₂²×Dic₂₆₈	1	1	yes	characteristic
136	C₂	#16864196	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863953

C₂

yes

maximal

#16863692

C₂

yes

maximal, complemented

#16863637

C₂

yes

maximal

#16863607

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×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

—

#14523977

C₂²

yes

—

#14523968

C₂²

yes

—

C₂²×C₈×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

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄²×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

Q₁₆

yes

complemented

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

C₂×Q₁₆

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×Q₁₆

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

C₂²×Q₁₆

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×C₄⋊C₈

D₆₇

yes

characteristic

C₂×Q₁₆

—

C₂²×C₈

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

#107

—

C₄⋊C₈

C₂×D₆₇

yes

—

#103

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂²×C₄

D₂₆₈

yes

characteristic

C₂²×C₄

D₂₆₈

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

characteristic

Q₁₆

—

268

—

C₂×C₈

—

C₂×C₈

C₂²×D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

100

C₄⋊C₄

—

134

—

101

C₄⋊C₄

—

134

—

102

C₄⋊C₄

—

134

—

103

C₂²⋊C₄

—

134

—

104

C₄²

—

134

—

105

C₄²

D₂₆₈

yes

—

106

C₄²

C₂²×D₆₇

yes

—

107

C₂³

C₂×D₂₆₈

yes

characteristic

108

Q₈

—

268

—

109

Q₈

—

134

—

110

Q₈

—

134

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

134

—

115

C₂×C₄

—

134

—

116

C₂×C₄

Dic₂₆₈

yes

complemented

117

C₂×C₄

C₂×D₂₆₈

yes

characteristic

118

C₂×C₄

C₂×D₂₆₈

yes

—

119

C₂×C₄

C₂×D₂₆₈

yes

—

120

C₂×C₄

C₂³×D₆₇

yes

characteristic

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×Dic₂₆₈

yes

—

125

C₂²

C₂²×D₂₆₈

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

268

—

129

C₄

—

268

—

130

C₄

C₂×Dic₂₆₈

yes

—

131

C₄

#14524088

yes

characteristic

132

C₄

#14524091

yes

characteristic

133

C₂

—

134

C₂

#16863977

yes

characteristic

135

C₂

C₂²×Dic₂₆₈

yes

characteristic

136

C₂

#16864196

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×C₄⋊C₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₂×C₄⋊C₈	D₆₇
hypocenter	C₆₇	#3249
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²

C₂²×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×C₄⋊C₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

hypercenter

C₂×C₄⋊C₈

D₆₇

hypocenter

C₆₇

#3249

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. #16863692	C₂	semidirect
4. #16863607	C₂	semidirect
42. C₂×C₄×C₆₇	Q₁₆	semidirect
69. C₆₇	#3249	semidirect
116. C₂×C₄	Dic₂₆₈	semidirect

H ≅ G/N

type

2. #16863692

C₂

semidirect

4. #16863607

C₂

semidirect

42. C₂×C₄×C₆₇

Q₁₆

semidirect

69. C₆₇

#3249

semidirect

116. C₂×C₄

Dic₂₆₈

semidirect

Subgroups — #22571554

Conjugacy classes of subgroups

Complements