Group #22562248 — subgroups

Subgroups — #22562248

Order 8576.

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

normal subgroups only

This group has 174 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×#14523987	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863771	C₂	1	1	yes	maximal, characteristic
4	#16863490	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863196	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×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×#11761613	—	2	1	no	—
11	#14524039	—	2	1	no	—
12	#14524008	—	2	1	no	—
13	#14523988	—	2	2	no	—
14	#14523987	—	2	2	no	—
15	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14523966	—	2	1	no	—
17	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×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×C₂²⋊D₆₇	—	4	1	no	—
24	C₂×C₂²⋊D₆₇	—	4	1	no	—
25	Dic₂₆₈	—	2	1	no	—
26	Dic₂₆₈	D₄	1	2	yes	complemented
27	D₅₃₆	—	2	1	no	—
28	D₅₃₆	—	2	1	no	—
29	#11761613	—	4	2	no	—
30	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
31	Dic₆₇⋊C₄	—	4	1	no	—
32	C₄×Dic₆₇	—	4	1	no	—
33	C₄×Dic₆₇	D₄	1	1	yes	characteristic
34	C₆₇⋊C₁₆	—	2	1	no	—
35	C₂×D₄×C₆₇	—	2	1	no	—
36	D₈×C₆₇	—	2	1	no	—
37	D₈×C₆₇	—	2	1	no	—
38	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₆₇	—	2	1	no	—
40	C₁₆×C₆₇	—	2	1	no	—
41	C₂³×C₆₇	—	4	1	no	—
42	C₂²×D₆₇	—	4	1	no	—
43	D₄×C₆₇	—	4	1	no	—
44	D₄×C₆₇	—	4	1	no	—
45	C₂×C₄×C₆₇	—	4	1	no	—
46	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂²⋊D₆₇	—	8	2	no	—
48	C₂²⋊D₆₇	—	8	2	no	—
49	C₂×Dic₆₇	—	4	1	no	—
50	C₂×Dic₆₇	D₈	1	1	yes	characteristic
51	C₂×Dic₆₇	D₈	1	1	yes	characteristic
52	D₂₆₈	—	4	1	no	—
53	D₂₆₈	—	4	1	no	—
54	Dic₁₃₄	—	2	2	no	—
55	Dic₁₃₄	—	4	1	no	—
56	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₈×C₆₇	#41	1	1	yes	characteristic
58	C₆₇⋊C₈	—	2	1	no	—
59	C₂²×C₆₇	—	8	1	no	—
60	C₂²×C₆₇	—	8	1	no	—
61	C₂²×C₆₇	—	8	1	no	—
62	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
63	C₂×D₆₇	—	8	1	no	—
64	C₂×D₆₇	—	8	1	no	—
65	C₂×D₆₇	—	8	1	no	—
66	C₄×C₆₇	—	8	1	no	—
67	C₄×C₆₇	#121	1	1	yes	characteristic
68	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
69	Dic₆₇	—	2	1	no	—
70	Dic₆₇	—	4	2	no	—
71	Dic₆₇	SD₃₂	1	2	yes	complemented
72	C₂×C₆₇	—	16	1	no	—
73	C₂×C₆₇	#459	1	1	yes	characteristic
74	C₂×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
75	C₂×C₆₇	#509	1	1	yes	characteristic
76	D₆₇	—	16	1	no	—
77	#2207	—	67	1	no	maximal, Sylow
78	C₆₇	#2207	1	1	yes	characteristic, complemented, Sylow
79	C₂×SD₃₂	—	67	1	no	—
80	C₂×SD₃₂	—	67	1	no	—
81	#493	—	67	1	no	—
82	C₄×Q₁₆	—	67	1	no	—
83	C₄⋊C₁₆	—	67	1	no	—
84	#357	—	67	1	no	—
85	#357	D₆₇	1	1	yes	characteristic, complemented
86	C₂×Q₁₆	—	67	1	no	—
87	C₂×D₈	—	67	1	no	—
88	C₂×D₈	—	134	1	no	—
89	C₂×D₈	C₂×D₆₇	1	1	yes	characteristic
90	#127	—	134	1	no	—
91	C₄×Q₈	—	134	1	no	—
92	SD₃₂	—	134	2	no	—
93	SD₃₂	—	134	2	no	—
94	C₂×C₁₆	—	67	1	no	—
95	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
96	#107	C₂×D₆₇	1	1	yes	characteristic
97	#103	—	134	1	no	—
98	C₄×C₈	—	67	1	no	—
99	C₂×Q₈	—	134	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	268	1	no	—
102	C₂×D₄	—	268	1	no	—
103	C₂×D₄	—	134	1	no	—
104	Q₁₆	—	67	2	no	—
105	Q₁₆	—	134	1	no	—
106	D₈	—	2	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	268	2	no	—
109	D₈	—	134	1	no	—
110	D₈	—	134	1	no	—
111	C₂×C₈	—	67	1	no	—
112	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	268	1	no	—
115	C₄²	—	268	1	no	—
116	C₄²	—	67	1	no	—
117	C₁₆	—	2	1	no	—
118	C₁₆	—	134	1	no	—
119	C₂³	—	4	1	no	—
120	C₂³	—	268	1	no	—
121	Q₈	—	268	1	no	—
122	Q₈	—	134	2	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	1	no	—
125	D₄	—	268	1	no	—
126	D₄	—	268	1	no	—
127	D₄	—	536	2	no	—
128	D₄	—	536	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	268	1	no	—
131	C₂×C₄	—	67	1	no	—
132	C₂×C₄	—	67	1	no	—
133	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
134	C₈	—	134	1	no	—
135	C₈	D₄×D₆₇	1	1	yes	characteristic
136	C₈	#11761628	1	1	yes	characteristic
137	C₂²	—	8	1	no	—
138	C₂²	—	8	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	—	536	1	no	—
141	C₂²	—	536	1	no	—
142	C₂²	—	536	1	no	—
143	C₂²	D₈×D₆₇	1	1	yes	characteristic
144	C₄	—	8	1	no	—
145	C₄	—	268	2	no	—
146	C₄	—	67	2	no	—
147	C₄	—	134	1	no	—
148	C₄	#14524016	1	1	yes	characteristic
149	C₄	#14524070	1	1	yes	characteristic
150	C₂	—	16	1	no	—
151	C₂	—	1072	1	no	—
152	C₂	#16863590	1	1	yes	characteristic
153	C₂	SD₃₂×D₆₇	1	1	yes	characteristic
154	C₂	#16863830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523988

C₂

yes

maximal, characteristic, complemented

C₂×#14523987

C₂

yes

maximal, characteristic, complemented

#16863771

C₂

yes

maximal, characteristic

#16863490

C₂

yes

maximal, characteristic, complemented

Dic₆₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#16863196

C₂

yes

maximal, characteristic, complemented

#357×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₆₈

C₂²

yes

characteristic

C₂×D₅₃₆

C₂²

yes

—

—

—

—

—

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

#14523966

—

C₈×Dic₆₇

C₂²

yes

characteristic

C₂×D₈×C₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

#107×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×D₂₆₈

—

C₂×C₂²⋊D₆₇

—

C₂×C₂²⋊D₆₇

—

Dic₂₆₈

—

Dic₂₆₈

D₄

yes

complemented

D₅₃₆

—

D₅₃₆

—

#11761613

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

D₄

yes

characteristic

C₆₇⋊C₁₆

—

C₂×D₄×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

—

C₂²×D₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²⋊D₆₇

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

D₈

yes

characteristic

C₂×Dic₆₇

D₈

yes

characteristic

D₂₆₈

—

D₂₆₈

—

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

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

Dic₆₇

SD₃₂

yes

complemented

C₂×C₆₇

—

C₂×C₆₇

#459

yes

characteristic

C₂×C₆₇

C₂×SD₃₂

yes

characteristic

C₂×C₆₇

#509

yes

characteristic

D₆₇

—

#2207

—

maximal, Sylow

C₆₇

#2207

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

C₂×SD₃₂

—

#493

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#357

—

#357

D₆₇

yes

characteristic, complemented

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

134

—

C₂×D₈

C₂×D₆₇

yes

characteristic

#127

—

134

—

C₄×Q₈

—

134

—

SD₃₂

—

134

—

SD₃₂

—

134

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

characteristic

#107

C₂×D₆₇

yes

characteristic

#103

—

134

—

C₄×C₈

—

C₂×Q₈

—

134

—

100

C₂×D₄

—

101

C₂×D₄

—

268

—

102

C₂×D₄

—

268

—

103

C₂×D₄

—

134

—

104

Q₁₆

—

105

Q₁₆

—

134

—

106

D₈

—

107

D₈

—

108

D₈

—

268

—

109

D₈

—

134

—

110

D₈

—

134

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₆₇

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

268

—

115

C₄²

—

268

—

116

C₄²

—

117

C₁₆

—

118

C₁₆

—

134

—

119

C₂³

—

120

C₂³

—

268

—

121

Q₈

—

268

—

122

Q₈

—

134

—

123

D₄

—

124

D₄

—

125

D₄

—

268

—

126

D₄

—

268

—

127

D₄

—

536

—

128

D₄

—

536

—

129

C₂×C₄

—

130

C₂×C₄

—

268

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

D₄×D₆₇

yes

characteristic

134

C₈

—

134

—

135

C₈

D₄×D₆₇

yes

characteristic

136

C₈

#11761628

yes

characteristic

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

536

—

141

C₂²

—

536

—

142

C₂²

—

536

—

143

C₂²

D₈×D₆₇

yes

characteristic

144

C₄

—

145

C₄

—

268

—

146

C₄

—

147

C₄

—

134

—

148

C₄

#14524016

yes

characteristic

149

C₄

#14524070

yes

characteristic

150

C₂

—

151

C₂

—

1072

—

152

C₂

#16863590

yes

characteristic

153

C₂

SD₃₂×D₆₇

yes

characteristic

154

C₂

#16863830

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#357×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#357	D₆₇
hypocenter	C₆₇	#2207
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₆₇

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#357×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 ◁ #357

N	H ≅ G/N	type
1. C₂×#14523988	C₂	semidirect
2. C₂×#14523987	C₂	semidirect
4. #16863490	C₂	semidirect
5. Dic₆₇⋊C₁₆	C₂	semidirect
6. #16863196	C₂	semidirect
7. #357×C₆₇	C₂	semidirect
26. Dic₂₆₈	D₄	semidirect
71. Dic₆₇	SD₃₂	semidirect
78. C₆₇	#2207	semidirect
85. #357	D₆₇	semidirect

H ≅ G/N

type

1. C₂×#14523988

C₂

semidirect

2. C₂×#14523987

C₂

semidirect

4. #16863490

C₂

semidirect

5. Dic₆₇⋊C₁₆

C₂

semidirect

6. #16863196

C₂

semidirect

7. #357×C₆₇

C₂

semidirect

26. Dic₂₆₈

D₄

semidirect

71. Dic₆₇

SD₃₂

semidirect

78. C₆₇

#2207

semidirect

85. #357

D₆₇

semidirect

Subgroups — #22562248

Conjugacy classes of subgroups

Complements