Group #22560985 — subgroups

Subgroups — #22560985

Order 8576.

200 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂×#14523979	C₂	1	2	yes	maximal
2	#16863420	C₂	1	1	yes	maximal, characteristic
3	C₄×#11761612	C₂	1	1	yes	maximal, characteristic
4	#16863161	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761612	C₂²	1	2	yes	—
7	C₂×#11761612	C₂²	1	1	yes	characteristic
8	C₂×#11761612	C₂²	1	1	yes	characteristic
9	#14523979	—	2	4	no	—
10	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
11	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
12	#14523954	—	2	1	no	—
13	C₄×C₆₇⋊C₈	—	2	1	no	—
14	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₆₇	C₄	1	2	yes	—
17	C₄⋊C₈×C₆₇	C₄	1	2	yes	—
18	#11761612	—	2	1	no	—
19	#11761612	—	2	1	no	—
20	#11761612	—	2	2	no	—
21	#11761612	—	2	2	no	—
22	#11761612	D₄	1	2	yes	—
23	#11761612	Q₈	1	2	yes	—
24	C₂×C₆₇⋊C₈	—	2	1	no	—
25	C₂×C₆₇⋊C₈	—	2	1	no	—
26	C₂×C₆₇⋊C₈	—	2	2	no	—
27	C₂²×C₄×C₆₇	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₆₇	—	2	2	no	—
31	C₂×C₈×C₆₇	—	2	2	no	—
32	C₂×C₈×C₆₇	C₂×C₄	1	4	yes	—
33	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₄²×C₆₇	D₄	1	1	yes	characteristic
37	C₂³×C₆₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₆₇	—	2	2	no	—
39	C₂×C₄×C₆₇	—	4	1	no	—
40	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
49	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
50	C₈×C₆₇	—	2	4	no	—
51	C₆₇⋊C₈	—	2	2	no	—
52	C₆₇⋊C₈	—	2	2	no	—
53	C₆₇⋊C₈	—	4	2	no	—
54	C₂²×C₆₇	—	2	1	no	—
55	C₂²×C₆₇	—	2	1	no	—
56	C₂²×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₆₇	#125	1	1	yes	characteristic
58	C₂²×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₄×C₆₇	—	2	1	no	—
60	C₄×C₆₇	—	2	1	no	—
61	C₄×C₆₇	—	2	1	no	—
62	C₄×C₆₇	#108	1	2	yes	complemented
63	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
64	C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
65	C₄×C₆₇	#121	1	1	yes	characteristic
66	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₆₇	—	2	1	no	—
68	C₂×C₆₇	—	2	1	no	—
69	C₂×C₆₇	#389	1	1	yes	characteristic
70	C₂×C₆₇	C₂×#108	1	1	yes	characteristic
71	C₂×C₆₇	#430	1	1	yes	characteristic
72	#1950	—	67	1	no	maximal, Sylow
73	C₆₇	#1950	1	1	yes	characteristic, complemented, Sylow
74	C₂×#108	—	67	2	no	—
75	#423	—	67	1	no	—
76	C₂×C₄⋊C₈	D₆₇	1	1	yes	characteristic
77	C₄×M₁₆	—	67	1	no	—
78	#341	—	67	2	no	—
79	C₂×M₁₆	—	67	1	no	—
80	C₂×M₁₆	—	67	1	no	—
81	C₂×M₁₆	—	67	2	no	—
82	C₂²×C₈	C₂×D₆₇	1	2	yes	—
83	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
84	#108	—	134	4	no	—
85	C₄⋊C₈	—	134	1	no	—
86	C₄⋊C₈	—	134	1	no	—
87	C₄⋊C₈	Dic₆₇	1	2	yes	—
88	C₄⋊C₈	Dic₆₇	1	2	yes	—
89	#97	—	134	1	no	—
90	C₄×C₈	—	134	1	no	—
91	C₂²×C₄	Dic₁₃₄	1	1	yes	characteristic
92	C₂²×C₄	Dic₁₃₄	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
94	M₁₆	—	67	2	no	—
95	M₁₆	—	67	2	no	—
96	M₁₆	—	134	2	no	—
97	M₁₆	—	134	2	no	—
98	M₁₆	—	134	1	no	—
99	M₁₆	—	134	1	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	134	2	no	—
103	C₂×C₈	—	134	1	no	—
104	C₂×C₈	—	134	1	no	—
105	C₂×C₈	C₂×Dic₆₇	1	4	yes	—
106	C₄²	D₂₆₈	1	1	yes	characteristic
107	C₄²	D₂₆₈	1	1	yes	characteristic
108	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
109	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
110	C₂³	C₂×Dic₁₃₄	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
114	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
115	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
116	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
118	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
119	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
121	C₂×C₄	#11761627	1	1	yes	characteristic
122	C₂×C₄	#11761629	1	1	yes	characteristic
123	C₈	—	2	4	no	—
124	C₈	—	268	2	no	—
125	C₈	—	134	2	no	—
126	C₈	—	134	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
130	C₂²	#14524034	1	1	yes	characteristic
131	C₂²	#14524042	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	#14523979	1	2	yes	complemented
136	C₄	D₄×Dic₆₇	1	1	yes	characteristic
137	C₄	Q₈×Dic₆₇	1	1	yes	characteristic
138	C₄	#14524018	1	1	yes	characteristic
139	C₄	#14524023	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	2	1	no	—
142	C₂	#16863295	1	1	yes	characteristic
143	C₂	C₂×#14523979	1	1	yes	characteristic
144	C₂	#16863445	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523979

C₂

yes

maximal

#16863420

C₂

yes

maximal, characteristic

C₄×#11761612

C₂

yes

maximal, characteristic

#16863161

C₂

yes

maximal

C₂×C₄⋊C₈×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

—

C₂×#11761612

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

characteristic

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₄

yes

—

C₄⋊C₈×C₆₇

C₄

yes

—

—

—

—

—

D₄

yes

—

#11761612

Q₈

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

Q₈

yes

characteristic

C₂²×C₄×C₆₇

Q₈

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₂³×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₇

#125

yes

characteristic

C₂²×C₆₇

C₄⋊Q₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#108

yes

complemented

C₄×C₆₇

C₄×D₄

yes

characteristic

C₄×C₆₇

C₄×Q₈

yes

characteristic

C₄×C₆₇

#121

yes

characteristic

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×#108

—

#423

—

C₂×C₄⋊C₈

D₆₇

yes

characteristic

C₄×M₁₆

—

#341

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₆₇

yes

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#108

—

134

—

C₄⋊C₈

—

134

—

C₄⋊C₈

—

134

—

C₄⋊C₈

Dic₆₇

yes

—

C₄⋊C₈

Dic₆₇

yes

—

#97

—

134

—

C₄×C₈

—

134

—

C₂²×C₄

Dic₁₃₄

yes

characteristic

C₂²×C₄

Dic₁₃₄

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

134

—

M₁₆

—

134

—

M₁₆

—

134

—

M₁₆

—

134

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

134

—

103

C₂×C₈

—

134

—

104

C₂×C₈

—

134

—

105

C₂×C₈

C₂×Dic₆₇

yes

—

106

C₄²

D₂₆₈

yes

characteristic

107

C₄²

D₂₆₈

yes

characteristic

108

C₄²

C₂×Dic₆₇

yes

characteristic

109

C₄²

C₂×Dic₆₇

yes

characteristic

110

C₂³

C₂×Dic₁₃₄

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

114

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

115

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

116

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

117

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

118

C₂×C₄

D₄×D₆₇

yes

characteristic

119

C₂×C₄

C₂×D₂₆₈

yes

characteristic

120

C₂×C₄

Q₈×D₆₇

yes

characteristic

121

C₂×C₄

#11761627

yes

characteristic

122

C₂×C₄

#11761629

yes

characteristic

123

C₈

—

124

C₈

—

268

—

125

C₈

—

134

—

126

C₈

—

134

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×C₄⋊Dic₆₇

yes

characteristic

130

C₂²

#14524034

yes

characteristic

131

C₂²

#14524042

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#14523979

yes

complemented

136

C₄

D₄×Dic₆₇

yes

characteristic

137

C₄

Q₈×Dic₆₇

yes

characteristic

138

C₄

#14524018

yes

characteristic

139

C₄

#14524023

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

#16863295

yes

characteristic

143

C₂

C₂×#14523979

yes

characteristic

144

C₂

#16863445

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×C₄⋊C₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₄
hypercenter	C₂×C₄⋊C₈	D₆₇
hypocenter	C₆₇	#1950
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×C₄⋊C₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂×C₄⋊C₄

hypercenter

C₂×C₄⋊C₈

D₆₇

hypocenter

C₆₇

#1950

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
62. C₄×C₆₇	#108	semidirect
73. C₆₇	#1950	semidirect
135. C₄	#14523979	semidirect

H ≅ G/N

type

62. C₄×C₆₇

#108

semidirect

73. C₆₇

#1950

semidirect

135. C₄

#14523979

semidirect

Subgroups — #22560985

Conjugacy classes of subgroups

Complements