Group #22562035 — subgroups

Subgroups — #22562035

Order 8576.

164 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14523983	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863172	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14523983	—	2	2	no	—
9	C₂×C₆₇⋊C₁₆	—	2	2	no	—
10	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
17	C₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
18	C₂²⋊C₈×C₆₇	C₄	1	1	yes	characteristic
19	C₂²⋊C₈×C₆₇	C₄	1	1	yes	characteristic
20	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₆₇⋊C₁₆	—	2	1	no	—
22	C₆₇⋊C₁₆	—	2	1	no	—
23	C₆₇⋊C₁₆	—	2	1	no	—
24	C₆₇⋊C₁₆	D₄	1	2	yes	complemented
25	C₂×D₄×C₆₇	C₈	1	1	yes	characteristic
26	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₆₇	—	2	1	no	—
29	C₂×C₈×C₆₇	—	2	1	no	—
30	C₂×C₈×C₆₇	—	2	1	no	—
31	C₂×C₈×C₆₇	—	2	1	no	—
32	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₆₇	C₈	1	1	yes	characteristic
37	C₂²⋊C₄×C₆₇	C₈	1	1	yes	characteristic
38	C₂²⋊C₄×C₆₇	C₈	1	1	yes	characteristic
39	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₆₇	C₂×C₈	1	1	yes	characteristic
41	C₂³×C₆₇	C₂×C₈	1	1	yes	characteristic
42	D₄×C₆₇	—	2	2	no	—
43	C₂×C₄×C₆₇	—	2	1	no	—
44	C₂×C₄×C₆₇	—	2	1	no	—
45	C₂×C₄×C₆₇	—	2	1	no	—
46	C₂×C₄×C₆₇	—	2	1	no	—
47	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₈×C₆₇	—	2	1	no	—
53	C₈×C₆₇	—	2	1	no	—
54	C₈×C₆₇	—	2	1	no	—
55	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₈×C₆₇	#41	1	1	yes	characteristic
57	C₂²×C₆₇	—	2	1	no	—
58	C₂²×C₆₇	—	2	1	no	—
59	C₂²×C₆₇	—	2	1	no	—
60	C₂²×C₆₇	—	2	1	no	—
61	C₂²×C₆₇	—	2	1	no	—
62	C₂²×C₆₇	M₃₂	1	2	yes	complemented
63	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
64	C₄×C₆₇	—	2	1	no	—
65	C₄×C₆₇	—	2	1	no	—
66	C₄×C₆₇	—	2	1	no	—
67	C₄×C₆₇	—	2	1	no	—
68	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
69	C₄×C₆₇	#131	1	1	yes	characteristic
70	C₂×C₆₇	—	2	2	no	—
71	C₂×C₆₇	—	4	1	no	—
72	C₂×C₆₇	C₈×D₄	1	1	yes	characteristic
73	C₂×C₆₇	C₂×M₃₂	1	1	yes	characteristic
74	C₂×C₆₇	#504	1	1	yes	characteristic
75	#2168	—	67	1	no	maximal, Sylow
76	C₆₇	#2168	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₃₂	—	67	1	no	—
78	C₂²×C₁₆	—	67	1	no	—
79	C₈×D₄	D₆₇	1	1	yes	characteristic
80	C₄⋊C₁₆	—	67	1	no	—
81	C₂²⋊C₁₆	—	67	1	no	—
82	C₂²⋊C₁₆	—	67	1	no	—
83	#346	—	67	1	no	—
84	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
85	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
86	C₄×D₄	Dic₆₇	1	1	yes	characteristic
87	M₃₂	—	134	2	no	—
88	C₂×C₁₆	—	67	1	no	—
89	C₂×C₁₆	—	67	1	no	—
90	C₂×C₁₆	—	67	1	no	—
91	C₂×C₁₆	—	67	1	no	—
92	C₂×C₁₆	—	134	2	no	—
93	C₄⋊C₈	Dic₆₇	1	1	yes	characteristic
94	C₂²⋊C₈	Dic₆₇	1	1	yes	characteristic
95	C₂²⋊C₈	Dic₆₇	1	1	yes	characteristic
96	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
97	C₂×D₄	C₆₇⋊C₈	1	1	yes	characteristic
98	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
99	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
105	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
106	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
108	C₄⋊C₄	C₆₇⋊C₈	1	1	yes	characteristic
109	C₂²⋊C₄	C₆₇⋊C₈	1	1	yes	characteristic
110	C₂²⋊C₄	C₆₇⋊C₈	1	1	yes	characteristic
111	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
112	C₁₆	—	67	2	no	—
113	C₁₆	—	134	1	no	—
114	C₁₆	—	134	1	no	—
115	C₁₆	—	134	1	no	—
116	C₂³	C₂×C₆₇⋊C₈	1	1	yes	characteristic
117	C₂³	C₂×C₆₇⋊C₈	1	1	yes	characteristic
118	D₄	—	2	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
124	C₂×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
125	C₂×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
126	C₂×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
127	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	2	1	no	—
130	C₈	—	2	1	no	—
131	C₈	D₄×D₆₇	1	1	yes	characteristic
132	C₈	#11761627	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	#14523983	1	2	yes	complemented
139	C₂²	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	D₄×Dic₆₇	1	1	yes	characteristic
145	C₄	#14524050	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	—	4	1	no	—
148	C₂	D₄×C₆₇⋊C₈	1	1	yes	characteristic
149	C₂	C₂×#14523983	1	1	yes	characteristic
150	C₂	#16863813	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523983

C₂

yes

maximal, characteristic, complemented

C₂²×C₆₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₄⋊(C₆₇⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₆₇⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₆₇⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

#16863172

C₂

yes

maximal, characteristic, complemented

C₈×D₄×C₆₇

C₂

yes

maximal, characteristic

#14523983

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₄×D₄×C₆₇

C₄

yes

characteristic

C₄⋊C₈×C₆₇

C₄

yes

characteristic

C₂²⋊C₈×C₆₇

C₄

yes

characteristic

C₂²⋊C₈×C₆₇

C₄

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

D₄

yes

complemented

C₂×D₄×C₆₇

C₈

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₈

yes

characteristic

C₂²⋊C₄×C₆₇

C₈

yes

characteristic

C₂²⋊C₄×C₆₇

C₈

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂³×C₆₇

C₂×C₈

yes

characteristic

C₂³×C₆₇

C₂×C₈

yes

characteristic

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

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

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

M₃₂

yes

complemented

C₂²×C₆₇

C₂²×C₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×D₄

yes

characteristic

C₄×C₆₇

#131

yes

characteristic

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₈×D₄

yes

characteristic

C₂×C₆₇

C₂×M₃₂

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

C₈×D₄

D₆₇

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₄×D₄

Dic₆₇

yes

characteristic

M₃₂

—

134

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

134

—

C₄⋊C₈

Dic₆₇

yes

characteristic

C₂²⋊C₈

Dic₆₇

yes

characteristic

C₂²⋊C₈

Dic₆₇

yes

characteristic

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×D₄

C₆₇⋊C₈

yes

characteristic

C₂²×C₄

C₂×Dic₆₇

yes

characteristic

C₂²×C₄

C₂×Dic₆₇

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂×Dic₆₇

yes

characteristic

105

C₂×C₈

C₂×Dic₆₇

yes

characteristic

106

C₂×C₈

C₂×Dic₆₇

yes

characteristic

107

C₂×C₈

C₂²×D₆₇

yes

characteristic

108

C₄⋊C₄

C₆₇⋊C₈

yes

characteristic

109

C₂²⋊C₄

C₆₇⋊C₈

yes

characteristic

110

C₂²⋊C₄

C₆₇⋊C₈

yes

characteristic

111

C₄²

C₂×Dic₆₇

yes

characteristic

112

C₁₆

—

113

C₁₆

—

134

—

114

C₁₆

—

134

—

115

C₁₆

—

134

—

116

C₂³

C₂×C₆₇⋊C₈

yes

characteristic

117

C₂³

C₂×C₆₇⋊C₈

yes

characteristic

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂×C₆₇⋊C₈

yes

characteristic

124

C₂×C₄

C₂×C₆₇⋊C₈

yes

characteristic

125

C₂×C₄

C₂×C₆₇⋊C₈

yes

characteristic

126

C₂×C₄

C₂×C₆₇⋊C₈

yes

characteristic

127

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

D₄×D₆₇

yes

characteristic

132

C₈

#11761627

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#14523983

yes

complemented

139

C₂²

C₂²×C₆₇⋊C₈

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

D₄×Dic₆₇

yes

characteristic

145

C₄

#14524050

yes

characteristic

146

C₂

—

147

C₂

—

148

C₂

D₄×C₆₇⋊C₈

yes

characteristic

149

C₂

C₂×#14523983

yes

characteristic

150

C₂

#16863813

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂²×C₈
Fitting	C₈×D₄×C₆₇	C₂
hypercenter	C₈×D₄	D₆₇
hypocenter	C₆₇	#2168
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₆₇

commutator, socle

C₂²×C₆₇

C₂²×C₈

Fitting

C₈×D₄×C₆₇

C₂

hypercenter

C₈×D₄

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₈ ◁ C₈×D₄

N	H ≅ G/N	type
1. C₂×#14523983	C₂	semidirect
2. C₂²×C₆₇⋊C₁₆	C₂	semidirect
3. C₄⋊(C₆₇⋊C₁₆)	C₂	semidirect
4. C₂²⋊(C₆₇⋊C₁₆)	C₂	semidirect
5. C₂²⋊(C₆₇⋊C₁₆)	C₂	semidirect
6. #16863172	C₂	semidirect
24. C₆₇⋊C₁₆	D₄	semidirect
62. C₂²×C₆₇	M₃₂	semidirect
76. C₆₇	#2168	semidirect
138. C₂²	#14523983	semidirect

H ≅ G/N

type

1. C₂×#14523983

C₂

semidirect

2. C₂²×C₆₇⋊C₁₆

C₂

semidirect

3. C₄⋊(C₆₇⋊C₁₆)

C₂

semidirect

4. C₂²⋊(C₆₇⋊C₁₆)

C₂

semidirect

5. C₂²⋊(C₆₇⋊C₁₆)

C₂

semidirect

6. #16863172

C₂

semidirect

24. C₆₇⋊C₁₆

D₄

semidirect

62. C₂²×C₆₇

M₃₂

semidirect

76. C₆₇

#2168

semidirect

138. C₂²

#14523983

semidirect

Subgroups — #22562035

Conjugacy classes of subgroups

Complements