Group #22562252 — subgroups

Subgroups — #22562252

Order 8576.

166 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 51 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×#14523987	C₂	1	1	yes	maximal, characteristic
2	#16863774	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863201	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863200	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic
9	C₂×#11761618	—	2	1	no	—
10	#14524041	—	2	1	no	—
11	D₄×Dic₆₇	—	2	1	no	—
12	#14523987	—	2	2	no	—
13	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	#14523977	C₂²	1	1	yes	characteristic
15	#14523964	—	2	1	no	—
16	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₆₇	C₂²	1	1	yes	characteristic
19	SD₃₂×C₆₇	—	2	2	no	—
20	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₃₄	—	2	1	no	—
22	C₂²×Dic₆₇	—	4	1	no	—
23	Dic₂₆₈	—	2	1	no	—
24	Dic₂₆₈	—	2	1	no	—
25	#11761618	—	4	2	no	—
26	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
27	C₄⋊Dic₆₇	—	2	1	no	—
28	Dic₆₇⋊C₄	—	4	1	no	—
29	Dic₆₇⋊C₄	—	4	1	no	—
30	C₂²⋊Dic₆₇	—	4	1	no	—
31	C₄×Dic₆₇	D₄	1	1	yes	characteristic
32	C₆₇⋊C₁₆	—	2	1	no	—
33	C₂×Q₈×C₆₇	—	2	1	no	—
34	C₂×D₄×C₆₇	—	2	1	no	—
35	Q₁₆×C₆₇	—	2	1	no	—
36	Q₁₆×C₆₇	—	2	1	no	—
37	D₈×C₆₇	—	2	1	no	—
38	D₈×C₆₇	D₄	1	2	yes	—
39	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₁₆×C₆₇	—	2	1	no	—
41	C₂³×C₆₇	—	4	1	no	—
42	Q₈×C₆₇	—	4	1	no	—
43	Q₈×C₆₇	—	4	1	no	—
44	D₄×C₆₇	—	2	2	no	—
45	D₄×C₆₇	—	4	1	no	—
46	C₂×C₄×C₆₇	—	4	1	no	—
47	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₆₇	—	4	1	no	—
49	C₂×Dic₆₇	—	4	1	no	—
50	C₂×Dic₆₇	—	8	2	no	—
51	C₂×Dic₆₇	D₈	1	1	yes	characteristic
52	C₂×Dic₆₇	D₈	1	1	yes	characteristic
53	Dic₁₃₄	—	4	1	no	—
54	Dic₁₃₄	—	4	1	no	—
55	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₈×C₆₇	#41	1	1	yes	characteristic
57	C₆₇⋊C₈	—	2	1	no	—
58	C₂²×C₆₇	—	4	2	no	—
59	C₂²×C₆₇	—	8	1	no	—
60	C₂²×C₆₇	—	8	1	no	—
61	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
62	C₄×C₆₇	—	8	1	no	—
63	C₄×C₆₇	#121	1	1	yes	characteristic
64	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	Dic₆₇	—	2	1	no	—
66	Dic₆₇	—	8	1	no	—
67	Dic₆₇	—	8	1	no	—
68	Dic₆₇	SD₃₂	1	2	yes	complemented
69	C₂×C₆₇	—	8	2	no	—
70	C₂×C₆₇	#459	1	1	yes	characteristic
71	C₂×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
72	C₂×C₆₇	#510	1	1	yes	characteristic
73	#2208	—	67	1	no	maximal, Sylow
74	C₆₇	#2208	1	1	yes	characteristic, complemented, Sylow
75	C₂×SD₃₂	—	67	1	no	—
76	C₂×SD₃₂	D₆₇	1	1	yes	characteristic
77	#494	—	67	1	no	—
78	C₄×D₈	—	67	1	no	—
79	C₄⋊C₁₆	—	67	1	no	—
80	#358	—	67	1	no	—
81	#358	—	67	1	no	—
82	C₂×Q₁₆	—	67	1	no	—
83	C₂×Q₁₆	—	134	1	no	—
84	C₂×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
85	C₂×D₈	C₂×D₆₇	1	1	yes	characteristic
86	C₄⋊Q₈	—	134	1	no	—
87	C₄×D₄	—	134	1	no	—
88	SD₃₂	—	2	2	no	—
89	SD₃₂	—	134	2	no	—
90	C₂×C₁₆	—	67	1	no	—
91	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
92	#107	—	67	1	no	—
93	#102	—	134	1	no	—
94	C₄×C₈	—	67	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	134	1	no	—
97	C₂×D₄	—	2	1	no	—
98	C₂²×C₄	—	268	1	no	—
99	Q₁₆	—	2	1	no	—
100	Q₁₆	—	2	1	no	—
101	Q₁₆	—	268	2	no	—
102	Q₁₆	—	134	1	no	—
103	Q₁₆	—	134	1	no	—
104	D₈	—	2	1	no	—
105	D₈	C₂²⋊D₆₇	1	2	yes	—
106	C₂×C₈	—	67	1	no	—
107	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
108	C₄⋊C₄	—	268	1	no	—
109	C₄⋊C₄	—	268	1	no	—
110	C₄⋊C₄	—	134	1	no	—
111	C₂²⋊C₄	—	268	1	no	—
112	C₄²	—	67	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	—	134	1	no	—
115	C₂³	—	4	1	no	—
116	Q₈	—	4	1	no	—
117	Q₈	—	4	1	no	—
118	Q₈	—	268	1	no	—
119	Q₈	—	268	1	no	—
120	D₄	—	2	2	no	—
121	D₄	—	4	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	268	1	no	—
124	C₂×C₄	—	268	1	no	—
125	C₂×C₄	—	536	2	no	—
126	C₂×C₄	—	67	1	no	—
127	C₂×C₄	—	67	1	no	—
128	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
129	C₈	—	134	1	no	—
130	C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
131	C₈	#11761627	1	1	yes	characteristic
132	C₂²	—	4	2	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	D₈×D₆₇	1	1	yes	characteristic
136	C₄	—	8	1	no	—
137	C₄	—	536	1	no	—
138	C₄	—	536	1	no	—
139	C₄	—	67	2	no	—
140	C₄	—	134	1	no	—
141	C₄	#14524015	1	1	yes	characteristic
142	C₄	#14524067	1	1	yes	characteristic
143	C₂	—	8	2	no	—
144	C₂	#16863589	1	1	yes	characteristic
145	C₂	SD₃₂×D₆₇	1	1	yes	characteristic
146	C₂	#16863835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523987

C₂

yes

maximal, characteristic

#16863774

C₂

yes

maximal, characteristic, complemented

D₈×Dic₆₇

C₂

yes

maximal, characteristic

Dic₆₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#16863201

C₂

yes

maximal, characteristic, complemented

#16863200

C₂

yes

maximal, characteristic, complemented

C₂×SD₃₂×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₂²

yes

characteristic

C₂×#11761618

—

#14524041

—

D₄×Dic₆₇

—

#14523987

—

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

#14523977

C₂²

yes

characteristic

#14523964

—

C₈×Dic₆₇

C₂²

yes

characteristic

C₂×Q₁₆×C₆₇

C₂²

yes

characteristic

C₂×D₈×C₆₇

C₂²

yes

characteristic

SD₃₂×C₆₇

—

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

#11761618

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

D₄

yes

characteristic

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

C₂×D₄×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

D₈

yes

characteristic

C₂×Dic₆₇

D₈

yes

characteristic

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

SD₃₂

yes

complemented

C₂×C₆₇

—

C₂×C₆₇

#459

yes

characteristic

C₂×C₆₇

C₂×SD₃₂

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

C₂×SD₃₂

D₆₇

yes

characteristic

#494

—

C₄×D₈

—

C₄⋊C₁₆

—

#358

—

#358

—

C₂×Q₁₆

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

C₂×D₆₇

yes

characteristic

C₂×D₈

C₂×D₆₇

yes

characteristic

C₄⋊Q₈

—

134

—

C₄×D₄

—

134

—

SD₃₂

—

SD₃₂

—

134

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

characteristic

#107

—

#102

—

134

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×D₄

—

C₂²×C₄

—

268

—

Q₁₆

—

100

Q₁₆

—

101

Q₁₆

—

268

—

102

Q₁₆

—

134

—

103

Q₁₆

—

134

—

104

D₈

—

105

D₈

C₂²⋊D₆₇

yes

—

106

C₂×C₈

—

107

C₂×C₈

C₂²×D₆₇

yes

characteristic

108

C₄⋊C₄

—

268

—

109

C₄⋊C₄

—

268

—

110

C₄⋊C₄

—

134

—

111

C₂²⋊C₄

—

268

—

112

C₄²

—

113

C₁₆

—

114

C₁₆

—

134

—

115

C₂³

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

268

—

119

Q₈

—

268

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

268

—

124

C₂×C₄

—

268

—

125

C₂×C₄

—

536

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

D₄×D₆₇

yes

characteristic

129

C₈

—

134

—

130

C₈

C₂×C₂²⋊D₆₇

yes

characteristic

131

C₈

#11761627

yes

characteristic

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

D₈×D₆₇

yes

characteristic

136

C₄

—

137

C₄

—

536

—

138

C₄

—

536

—

139

C₄

—

140

C₄

—

134

—

141

C₄

#14524015

yes

characteristic

142

C₄

#14524067

yes

characteristic

143

C₂

—

144

C₂

#16863589

yes

characteristic

145

C₂

SD₃₂×D₆₇

yes

characteristic

146

C₂

#16863835

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	C₂×SD₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	C₂×SD₃₂	D₆₇
hypocenter	C₆₇	#2208
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×SD₃₂

subgroup

G/H

center

C₂²

D₈×D₆₇

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

C₂×SD₃₂×C₆₇

C₂

socle

C₂²×C₆₇

C₂×D₈

hypercenter

C₂×SD₃₂

D₆₇

hypocenter

C₆₇

#2208

LCS

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×SD₃₂

N	H ≅ G/N	type
2. #16863774	C₂	semidirect
4. Dic₆₇⋊C₁₆	C₂	semidirect
5. #16863201	C₂	semidirect
6. #16863200	C₂	semidirect
68. Dic₆₇	SD₃₂	semidirect
74. C₆₇	#2208	semidirect

H ≅ G/N

type

2. #16863774

C₂

semidirect

4. Dic₆₇⋊C₁₆

C₂

semidirect

5. #16863201

C₂

semidirect

6. #16863200

C₂

semidirect

68. Dic₆₇

SD₃₂

semidirect

74. C₆₇

#2208

semidirect

Subgroups — #22562252

Conjugacy classes of subgroups

Complements