Group #22562039 — subgroups

Subgroups — #22562039

Order 8576.

162 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 71 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×#14523982	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×C₂²⋊D₆₇	C₂	1	1	yes	maximal, characteristic
3	Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₆₇)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
9	C₄×C₂²⋊D₆₇	C₄	1	1	yes	characteristic
10	#14523982	—	2	2	no	—
11	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	Dic₆₇⋊C₈	C₄	1	1	yes	characteristic
14	C₂²⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
15	(C₂×D₆₇)⋊C₈	C₄	1	1	yes	characteristic
16	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	M₃₂×C₆₇	—	2	2	no	—
19	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊D₆₇	C₈	1	1	yes	characteristic
22	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₈×D₆₇	—	2	1	no	—
26	C₈×D₆₇	—	2	1	no	—
27	Dic₆₇⋊C₄	C₈	1	1	yes	characteristic
28	C₂²⋊Dic₆₇	C₈	1	1	yes	characteristic
29	(C₂×D₆₇)⋊C₄	C₈	1	1	yes	characteristic
30	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
31	C₆₇⋊C₁₆	—	2	1	no	—
32	C₆₇⋊C₁₆	—	2	1	no	—
33	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₇	—	2	1	no	—
35	C₂×C₈×C₆₇	—	2	1	no	—
36	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
38	C₁₆×C₆₇	—	2	1	no	—
39	C₁₆×C₆₇	D₄	1	2	yes	complemented
40	C₂³×C₆₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₆₇	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₆₇	—	2	1	no	—
43	C₂×C₄×C₆₇	—	2	1	no	—
44	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²⋊D₆₇	—	2	2	no	—
47	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
48	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
49	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
50	C₄×D₆₇	—	2	1	no	—
51	C₄×D₆₇	—	2	1	no	—
52	C₈×C₆₇	—	2	1	no	—
53	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₈×C₆₇	#41	1	1	yes	characteristic
55	C₆₇⋊C₈	—	2	1	no	—
56	C₆₇⋊C₈	—	2	1	no	—
57	C₂²×C₆₇	—	2	1	no	—
58	C₂²×C₆₇	—	2	1	no	—
59	C₂²×C₆₇	—	2	1	no	—
60	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
61	C₂×D₆₇	—	2	1	no	—
62	C₂×D₆₇	—	2	1	no	—
63	C₂×D₆₇	—	2	1	no	—
64	C₄×C₆₇	—	2	1	no	—
65	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
66	C₄×C₆₇	#131	1	1	yes	characteristic
67	Dic₆₇	—	2	1	no	—
68	Dic₆₇	—	2	1	no	—
69	Dic₆₇	M₃₂	1	2	yes	complemented
70	C₂×C₆₇	—	4	1	no	—
71	C₂×C₆₇	C₈×D₄	1	1	yes	characteristic
72	C₂×C₆₇	C₂×M₃₂	1	1	yes	characteristic
73	C₂×C₆₇	#504	1	1	yes	characteristic
74	D₆₇	—	4	1	no	—
75	#2169	—	67	1	no	maximal, Sylow
76	C₆₇	#2169	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₃₂	—	67	1	no	—
78	C₂×M₃₂	D₆₇	1	1	yes	characteristic, complemented
79	C₈×D₄	—	67	1	no	—
80	C₄⋊C₁₆	—	67	1	no	—
81	C₂²⋊C₁₆	—	67	1	no	—
82	C₂²⋊C₁₆	—	67	1	no	—
83	C₄×C₁₆	—	67	1	no	—
84	C₂²×C₈	—	67	1	no	—
85	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
86	C₄×D₄	—	67	1	no	—
87	M₃₂	—	2	2	no	—
88	M₃₂	—	134	2	no	—
89	C₂×C₁₆	—	67	1	no	—
90	C₂×C₁₆	—	67	1	no	—
91	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
92	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
93	C₄⋊C₈	—	67	1	no	—
94	C₂²⋊C₈	—	67	1	no	—
95	C₂²⋊C₈	—	67	1	no	—
96	C₄×C₈	—	67	1	no	—
97	C₂×D₄	—	67	1	no	—
98	C₂²×C₄	—	67	1	no	—
99	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	67	1	no	—
103	C₂×C₈	—	67	1	no	—
104	C₂×C₈	—	134	1	no	—
105	C₂×C₈	—	134	1	no	—
106	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
108	C₄⋊C₄	—	67	1	no	—
109	C₂²⋊C₄	—	67	1	no	—
110	C₂²⋊C₄	—	67	1	no	—
111	C₄²	—	67	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	134	1	no	—
114	C₁₆	—	134	1	no	—
115	C₁₆	C₂²⋊D₆₇	1	2	yes	complemented
116	C₂³	—	67	1	no	—
117	C₂³	C₈×D₆₇	1	1	yes	characteristic
118	D₄	—	134	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	67	1	no	—
122	C₂×C₄	—	67	1	no	—
123	C₂×C₄	—	67	1	no	—
124	C₂×C₄	—	134	1	no	—
125	C₂×C₄	—	134	1	no	—
126	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	134	1	no	—
130	C₈	—	134	1	no	—
131	C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
132	C₈	#11761628	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	134	1	no	—
137	C₂²	—	134	1	no	—
138	C₂²	—	134	1	no	—
139	C₂²	C₂×C₈×D₆₇	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	67	2	no	—
142	C₄	—	134	1	no	—
143	C₄	—	134	1	no	—
144	C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
145	C₄	#14524049	1	1	yes	characteristic
146	C₂	—	4	1	no	—
147	C₂	—	268	1	no	—
148	C₂	C₈×C₂²⋊D₆₇	1	1	yes	characteristic
149	C₂	M₃₂×D₆₇	1	1	yes	characteristic
150	C₂	#16863814	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523982

C₂

yes

maximal, characteristic, complemented

C₈×C₂²⋊D₆₇

C₂

yes

maximal, characteristic

Dic₆₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₆₇⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

(C₂×D₆₇)⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₆₇

C₂

yes

maximal, characteristic, complemented

C₂×M₃₂×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₆₇

C₂²

yes

characteristic

C₄×C₂²⋊D₆₇

C₄

yes

characteristic

#14523982

—

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

Dic₆₇⋊C₈

C₄

yes

characteristic

C₂²⋊(C₆₇⋊C₈)

C₄

yes

characteristic

(C₂×D₆₇)⋊C₈

C₄

yes

characteristic

C₈×Dic₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×C₂²⋊D₆₇

C₈

yes

characteristic

C₂×C₄×D₆₇

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₈×D₆₇

—

C₈×D₆₇

—

Dic₆₇⋊C₄

C₈

yes

characteristic

C₂²⋊Dic₆₇

C₈

yes

characteristic

(C₂×D₆₇)⋊C₄

C₈

yes

characteristic

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

D₄

yes

complemented

C₂³×C₆₇

C₂×C₈

yes

characteristic

C₂²×D₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²⋊D₆₇

—

C₂×Dic₆₇

C₂×C₈

yes

characteristic

C₂×Dic₆₇

C₂×C₈

yes

characteristic

C₂×Dic₆₇

C₂×C₈

yes

characteristic

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×C₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×D₄

yes

characteristic

C₄×C₆₇

#131

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

M₃₂

yes

complemented

C₂×C₆₇

—

C₂×C₆₇

C₈×D₄

yes

characteristic

C₂×C₆₇

C₂×M₃₂

yes

characteristic

C₂×C₆₇

#504

yes

characteristic

D₆₇

—

#2169

—

maximal, Sylow

C₆₇

#2169

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

D₆₇

yes

characteristic, complemented

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₄×D₄

—

M₃₂

—

M₃₂

—

134

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

characteristic

C₂×C₁₆

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

134

—

105

C₂×C₈

—

134

—

106

C₂×C₈

C₄×D₆₇

yes

characteristic

107

C₂×C₈

C₂²×D₆₇

yes

characteristic

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₁₆

—

113

C₁₆

—

134

—

114

C₁₆

—

134

—

115

C₁₆

C₂²⋊D₆₇

yes

complemented

116

C₂³

—

117

C₂³

C₈×D₆₇

yes

characteristic

118

D₄

—

134

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

134

—

125

C₂×C₄

—

134

—

126

C₂×C₄

C₈×D₆₇

yes

characteristic

127

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

128

C₈

—

129

C₈

—

134

—

130

C₈

—

134

—

131

C₈

C₂×C₂²⋊D₆₇

yes

characteristic

132

C₈

#11761628

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

134

—

137

C₂²

—

134

—

138

C₂²

—

134

—

139

C₂²

C₂×C₈×D₆₇

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

134

—

143

C₄

—

134

—

144

C₄

C₄×C₂²⋊D₆₇

yes

characteristic

145

C₄

#14524049

yes

characteristic

146

C₂

—

147

C₂

—

268

—

148

C₂

C₈×C₂²⋊D₆₇

yes

characteristic

149

C₂

M₃₂×D₆₇

yes

characteristic

150

C₂

#16863814

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂²×C₈
Fitting	C₂×M₃₂×C₆₇	C₂
hypercenter	C₂×M₃₂	D₆₇
hypocenter	C₆₇	#2169
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₆₇

commutator, socle

C₂²×C₆₇

C₂²×C₈

Fitting

C₂×M₃₂×C₆₇

C₂

hypercenter

C₂×M₃₂

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
1. C₂×#14523982	C₂	semidirect
3. Dic₆₇⋊C₁₆	C₂	semidirect
4. C₂²⋊(C₆₇⋊C₁₆)	C₂	semidirect
5. (C₂×D₆₇)⋊C₁₆	C₂	semidirect
6. C₁₆×Dic₆₇	C₂	semidirect
7. C₂×M₃₂×C₆₇	C₂	semidirect
39. C₁₆×C₆₇	D₄	semidirect
69. Dic₆₇	M₃₂	semidirect
76. C₆₇	#2169	semidirect
78. C₂×M₃₂	D₆₇	semidirect
115. C₁₆	C₂²⋊D₆₇	semidirect

H ≅ G/N

type

1. C₂×#14523982

C₂

semidirect

3. Dic₆₇⋊C₁₆

C₂

semidirect

4. C₂²⋊(C₆₇⋊C₁₆)

C₂

semidirect

5. (C₂×D₆₇)⋊C₁₆

C₂

semidirect

6. C₁₆×Dic₆₇

C₂

semidirect

7. C₂×M₃₂×C₆₇

C₂

semidirect

39. C₁₆×C₆₇

D₄

semidirect

69. Dic₆₇

M₃₂

semidirect

76. C₆₇

#2169

semidirect

78. C₂×M₃₂

D₆₇

semidirect

115. C₁₆

C₂²⋊D₆₇

semidirect

Subgroups — #22562039

Conjugacy classes of subgroups

Complements