Group #22562323 — subgroups

Subgroups — #22562323

Order 8576.

160 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 49 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×#14523988	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14523982	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863638	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863633	C₂	1	1	yes	maximal, characteristic
5	#16863199	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863196	C₂	1	1	yes	maximal, characteristic, complemented
7	#366×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic
9	C₂×D₅₃₆	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
11	#14524023	—	2	1	no	—
12	#14524018	—	2	1	no	—
13	#14523988	—	2	2	no	—
14	#14523982	—	2	2	no	—
15	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14523966	—	2	1	no	—
17	#14523963	—	2	1	no	—
18	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	#107×C₆₇	C₂²	1	1	yes	characteristic
20	#107×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₃₄	—	2	1	no	—
22	C₂×D₂₆₈	—	2	1	no	—
23	C₂×D₂₆₈	—	4	1	no	—
24	C₂×C₄×D₆₇	D₄	1	1	yes	characteristic
25	Dic₂₆₈	—	2	1	no	—
26	Dic₂₆₈	—	2	1	no	—
27	D₅₃₆	—	2	1	no	—
28	D₅₃₆	—	2	1	no	—
29	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
30	C₈×D₆₇	—	2	1	no	—
31	C₈×D₆₇	—	2	1	no	—
32	C₄⋊Dic₆₇	—	4	1	no	—
33	(C₂×D₆₇)⋊C₄	—	4	1	no	—
34	(C₂×D₆₇)⋊C₄	—	4	1	no	—
35	C₆₇⋊C₁₆	—	2	1	no	—
36	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₁₆×C₆₇	D₄	1	2	yes	complemented
40	C₂²×D₆₇	—	4	1	no	—
41	C₂²×D₆₇	D₈	1	1	yes	characteristic
42	C₂×C₄×C₆₇	—	4	1	no	—
43	C₂×C₄×C₆₇	—	4	1	no	—
44	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₆₇	—	4	1	no	—
46	C₂×Dic₆₇	D₈	1	1	yes	characteristic
47	D₂₆₈	—	4	1	no	—
48	D₂₆₈	—	4	1	no	—
49	D₂₆₈	—	8	2	no	—
50	C₄×D₆₇	—	2	1	no	—
51	C₄×D₆₇	—	2	1	no	—
52	Dic₁₃₄	—	4	1	no	—
53	Dic₁₃₄	—	4	1	no	—
54	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₈×C₆₇	#41	1	1	yes	characteristic
56	C₆₇⋊C₈	—	2	1	no	—
57	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
58	C₂×D₆₇	—	2	1	no	—
59	C₂×D₆₇	—	2	1	no	—
60	C₂×D₆₇	—	2	1	no	—
61	C₂×D₆₇	—	8	1	no	—
62	C₂×D₆₇	—	8	1	no	—
63	C₂×D₆₇	—	8	1	no	—
64	C₄×C₆₇	—	8	1	no	—
65	C₄×C₆₇	—	8	1	no	—
66	C₄×C₆₇	#121	1	1	yes	characteristic
67	C₄×C₆₇	#135	1	1	yes	characteristic
68	Dic₆₇	—	2	1	no	—
69	Dic₆₇	—	8	1	no	—
70	C₂×C₆₇	#466	1	1	yes	characteristic
71	C₂×C₆₇	#509	1	1	yes	characteristic
72	C₂×C₆₇	#510	1	1	yes	characteristic
73	D₆₇	—	4	1	no	—
74	D₆₇	—	16	1	no	—
75	#2220	—	67	1	no	maximal, Sylow
76	C₆₇	#2220	1	1	yes	characteristic, complemented, Sylow
77	C₂×SD₃₂	—	67	1	no	—
78	C₂×M₃₂	—	67	1	no	—
79	#467	—	67	1	no	—
80	#466	—	67	1	no	—
81	#366	D₆₇	1	1	yes	characteristic, complemented
82	#358	—	67	1	no	—
83	#357	—	67	1	no	—
84	C₂×Q₁₆	—	67	1	no	—
85	C₂×D₈	—	67	1	no	—
86	C₂²×C₈	—	67	1	no	—
87	C₂²⋊Q₈	—	134	1	no	—
88	#121	—	134	1	no	—
89	SD₃₂	—	134	2	no	—
90	M₃₂	—	134	2	no	—
91	C₂×C₁₆	—	67	1	no	—
92	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
93	#107	C₂×D₆₇	1	1	yes	characteristic
94	#107	C₂×D₆₇	1	1	yes	characteristic
95	#103	—	134	1	no	—
96	#102	—	134	1	no	—
97	C₂×Q₈	—	134	1	no	—
98	C₂×D₄	—	268	1	no	—
99	C₂×D₄	—	134	1	no	—
100	C₂²×C₄	—	67	1	no	—
101	Q₁₆	—	134	1	no	—
102	Q₁₆	—	134	1	no	—
103	D₈	—	134	1	no	—
104	D₈	—	134	1	no	—
105	C₂×C₈	—	67	1	no	—
106	C₂×C₈	—	134	1	no	—
107	C₂×C₈	—	134	1	no	—
108	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	268	1	no	—
112	C₂²⋊C₄	—	268	1	no	—
113	C₂²⋊C₄	—	268	1	no	—
114	C₁₆	—	134	1	no	—
115	C₁₆	D₂₆₈	1	2	yes	complemented
116	C₂³	—	268	1	no	—
117	C₂³	—	67	1	no	—
118	Q₈	—	268	1	no	—
119	Q₈	—	268	1	no	—
120	D₄	—	268	1	no	—
121	D₄	—	268	1	no	—
122	D₄	—	536	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	268	1	no	—
126	C₂×C₄	—	67	1	no	—
127	C₂×C₄	—	134	1	no	—
128	C₂×C₄	—	134	1	no	—
129	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
130	C₈	—	134	1	no	—
131	C₈	C₂×D₂₆₈	1	1	yes	characteristic
132	C₈	#11761629	1	1	yes	characteristic
133	C₂²	—	536	1	no	—
134	C₂²	—	536	1	no	—
135	C₂²	—	536	1	no	—
136	C₂²	—	134	1	no	—
137	C₂²	—	134	1	no	—
138	C₂²	—	134	1	no	—
139	C₂²	D₈×D₆₇	1	1	yes	characteristic
140	C₄	—	8	1	no	—
141	C₄	—	8	1	no	—
142	C₄	—	536	1	no	—
143	C₄	—	134	1	no	—
144	C₄	#14524018	1	1	yes	characteristic
145	C₄	#14524066	1	1	yes	characteristic
146	C₂	—	268	1	no	—
147	C₂	—	1072	1	no	—
148	C₂	#16863633	1	1	yes	characteristic
149	C₂	#16863830	1	1	yes	characteristic
150	C₂	#16863838	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523988

C₂

yes

maximal, characteristic, complemented

C₂×#14523982

C₂

yes

maximal, characteristic, complemented

#16863638

C₂

yes

maximal, characteristic, complemented

#16863633

C₂

yes

maximal, characteristic

#16863199

C₂

yes

maximal, characteristic, complemented

#16863196

C₂

yes

maximal, characteristic, complemented

#366×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₆₈

C₂²

yes

characteristic

C₂×D₅₃₆

C₂²

yes

characteristic

C₂×C₈×D₆₇

C₂²

yes

—

—

—

—

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

#14523966

—

#14523963

—

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

#107×C₆₇

C₂²

yes

characteristic

#107×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×C₄×D₆₇

D₄

yes

characteristic

Dic₂₆₈

—

Dic₂₆₈

—

D₅₃₆

—

D₅₃₆

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₈×D₆₇

—

C₈×D₆₇

—

C₄⋊Dic₆₇

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₆₇⋊C₁₆

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₁₆×C₆₇

D₄

yes

complemented

C₂²×D₆₇

—

C₂²×D₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

D₈

yes

characteristic

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

D₆₇

—

#2220

—

maximal, Sylow

C₆₇

#2220

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

C₂×M₃₂

—

#467

—

#466

—

#366

D₆₇

yes

characteristic, complemented

#358

—

#357

—

C₂×Q₁₆

—

C₂×D₈

—

C₂²×C₈

—

C₂²⋊Q₈

—

134

—

#121

—

134

—

SD₃₂

—

134

—

M₃₂

—

134

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

characteristic

#107

C₂×D₆₇

yes

characteristic

#107

C₂×D₆₇

yes

characteristic

#103

—

134

—

#102

—

134

—

C₂×Q₈

—

134

—

C₂×D₄

—

268

—

C₂×D₄

—

134

—

100

C₂²×C₄

—

101

Q₁₆

—

134

—

102

Q₁₆

—

134

—

103

D₈

—

134

—

104

D₈

—

134

—

105

C₂×C₈

—

106

C₂×C₈

—

134

—

107

C₂×C₈

—

134

—

108

C₂×C₈

C₂²×D₆₇

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

268

—

112

C₂²⋊C₄

—

268

—

113

C₂²⋊C₄

—

268

—

114

C₁₆

—

134

—

115

C₁₆

D₂₆₈

yes

complemented

116

C₂³

—

268

—

117

C₂³

—

118

Q₈

—

268

—

119

Q₈

—

268

—

120

D₄

—

268

—

121

D₄

—

268

—

122

D₄

—

536

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

268

—

126

C₂×C₄

—

127

C₂×C₄

—

134

—

128

C₂×C₄

—

134

—

129

C₂×C₄

D₄×D₆₇

yes

characteristic

130

C₈

—

134

—

131

C₈

C₂×D₂₆₈

yes

characteristic

132

C₈

#11761629

yes

characteristic

133

C₂²

—

536

—

134

C₂²

—

536

—

135

C₂²

—

536

—

136

C₂²

—

134

—

137

C₂²

—

134

—

138

C₂²

—

134

—

139

C₂²

D₈×D₆₇

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

536

—

143

C₄

—

134

—

144

C₄

#14524018

yes

characteristic

145

C₄

#14524066

yes

characteristic

146

C₂

—

268

—

147

C₂

—

1072

—

148

C₂

#16863633

yes

characteristic

149

C₂

#16863830

yes

characteristic

150

C₂

#16863838

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#366×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#366	D₆₇
hypocenter	C₆₇	#2220
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

subgroup

G/H

center

C₂²

D₈×D₆₇

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#366×C₆₇

C₂

socle

C₂²×C₆₇

C₂×D₈

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

N	H ≅ G/N	type
1. C₂×#14523988	C₂	semidirect
2. C₂×#14523982	C₂	semidirect
3. #16863638	C₂	semidirect
5. #16863199	C₂	semidirect
6. #16863196	C₂	semidirect
7. #366×C₆₇	C₂	semidirect
39. C₁₆×C₆₇	D₄	semidirect
76. C₆₇	#2220	semidirect
81. #366	D₆₇	semidirect
115. C₁₆	D₂₆₈	semidirect

H ≅ G/N

type

1. C₂×#14523988

C₂

semidirect

2. C₂×#14523982

C₂

semidirect

3. #16863638

C₂

semidirect

5. #16863199

C₂

semidirect

6. #16863196

C₂

semidirect

7. #366×C₆₇

C₂

semidirect

39. C₁₆×C₆₇

D₄

semidirect

76. C₆₇

#2220

semidirect

81. #366

D₆₇

semidirect

115. C₁₆

D₂₆₈

semidirect

Subgroups — #22562323

Conjugacy classes of subgroups

Complements