Group #22559366 — subgroups

Subgroups — #22559366

Order 8576.

188 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16864125	C₂	1	1	yes	maximal, characteristic
2	#16863746	C₂	1	1	yes	maximal, characteristic
3	#16863688	C₂	1	1	yes	maximal, characteristic
4	#16863141	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#16863126	C₂	1	1	yes	maximal, characteristic
7	#477×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14524043	—	2	1	no	—
9	#14524043	C₂²	1	1	yes	characteristic
10	#14524034	—	2	2	no	—
11	#14524034	—	2	1	no	—
12	Q₈×Dic₆₇	—	2	1	no	—
13	C₄×Dic₁₃₄	—	2	1	no	—
14	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
15	#14523976	—	2	1	no	—
16	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
17	#14523967	—	2	2	no	—
18	#14523966	—	2	1	no	—
19	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	#107×C₆₇	—	2	1	no	—
23	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#103×C₆₇	—	2	1	no	—
25	C₂×Dic₁₃₄	—	2	1	no	—
26	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
27	C₂×C₆₇⋊C₈	—	2	1	no	—
28	C₂×C₆₇⋊C₈	—	2	1	no	—
29	C₄⋊Dic₆₇	—	2	2	no	—
30	C₄⋊Dic₆₇	—	2	2	no	—
31	C₄⋊Dic₆₇	—	2	1	no	—
32	C₄⋊Dic₆₇	D₄	1	2	yes	—
33	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
34	Dic₆₇⋊C₄	—	2	2	no	—
35	Dic₆₇⋊C₄	—	2	1	no	—
36	C₄×Dic₆₇	—	2	2	no	—
37	C₄×Dic₆₇	—	2	1	no	—
38	C₂×Q₈×C₆₇	—	2	1	no	—
39	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₈×C₆₇	—	2	1	no	—
41	C₄⋊C₄×C₆₇	—	2	1	no	—
42	C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₄⋊C₄×C₆₇	—	2	1	no	—
44	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
45	C₄²×C₆₇	—	2	1	no	—
46	C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	Q₈×C₆₇	—	2	1	no	—
48	Q₈×C₆₇	—	4	2	no	—
49	Q₈×C₆₇	Q₁₆	1	2	yes	—
50	C₂×C₄×C₆₇	—	2	1	no	—
51	C₂×C₄×C₆₇	—	2	1	no	—
52	C₂×C₄×C₆₇	—	4	1	no	—
53	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₆₇	—	2	1	no	—
57	C₂×Dic₆₇	—	2	1	no	—
58	C₂×Dic₆₇	—	2	2	no	—
59	Dic₁₃₄	—	2	1	no	—
60	Dic₁₃₄	—	2	1	no	—
61	Dic₁₃₄	—	4	1	no	—
62	Dic₁₃₄	—	4	1	no	—
63	C₈×C₆₇	—	4	1	no	—
64	C₆₇⋊C₈	—	4	1	no	—
65	C₆₇⋊C₈	—	4	1	no	—
66	C₂²×C₆₇	#120	1	1	yes	characteristic
67	C₄×C₆₇	—	2	2	no	—
68	C₄×C₆₇	—	2	1	no	—
69	C₄×C₆₇	—	4	1	no	—
70	C₄×C₆₇	—	8	1	no	—
71	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
72	C₄×C₆₇	#135	1	1	yes	characteristic
73	C₄×C₆₇	#137	1	1	yes	characteristic
74	C₄×C₆₇	#137	1	1	yes	characteristic
75	Dic₆₇	—	4	1	no	—
76	Dic₆₇	—	4	2	no	—
77	Dic₆₇	—	4	1	no	—
78	C₂×C₆₇	#451	1	1	yes	characteristic
79	C₂×C₆₇	#452	1	1	yes	characteristic
80	C₂×C₆₇	#454	1	1	yes	characteristic
81	#1636	—	67	1	no	maximal, Sylow
82	C₆₇	#1636	1	1	yes	characteristic, complemented, Sylow
83	#557	—	67	1	no	—
84	#487	—	67	1	no	—
85	#477	—	67	1	no	—
86	#477	D₆₇	1	1	yes	characteristic
87	#332	—	67	1	no	—
88	Q₈⋊C₈	—	67	1	no	—
89	Q₈⋊C₈	—	67	1	no	—
90	C₄⋊Q₈	—	67	1	no	—
91	C₄⋊Q₈	—	134	1	no	—
92	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
93	#125	—	134	2	no	—
94	#125	—	134	1	no	—
95	C₄×Q₈	—	67	1	no	—
96	C₄×Q₈	—	134	1	no	—
97	C₄×Q₈	—	134	1	no	—
98	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
99	#107	—	2	1	no	—
100	#107	—	134	1	no	—
101	C₄⋊C₈	—	67	1	no	—
102	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
103	#103	—	2	1	no	—
104	#103	—	134	1	no	—
105	#103	—	134	2	no	—
106	C₄×C₈	—	67	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	67	1	no	—
109	C₂×Q₈	—	134	1	no	—
110	C₂×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	134	1	no	—
113	C₂×C₈	—	134	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	67	2	no	—
118	C₄⋊C₄	—	67	1	no	—
119	C₄⋊C₄	—	134	2	no	—
120	C₄⋊C₄	—	134	2	no	—
121	C₄⋊C₄	—	134	2	no	—
122	C₄⋊C₄	—	134	1	no	—
123	C₄⋊C₄	—	134	1	no	—
124	C₄⋊C₄	C₂²⋊D₆₇	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	—	134	2	no	—
127	C₄²	—	134	1	no	—
128	C₄²	C₂²×D₆₇	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	—	4	2	no	—
131	Q₈	—	268	1	no	—
132	Q₈	—	268	1	no	—
133	Q₈	—	134	1	no	—
134	Q₈	—	134	1	no	—
135	Q₈	#11761618	1	2	yes	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	134	1	no	—
140	C₂×C₄	—	134	2	no	—
141	C₂×C₄	—	134	1	no	—
142	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
143	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
145	C₈	—	4	1	no	—
146	C₈	—	268	1	no	—
147	C₈	—	268	1	no	—
148	C₂²	#14524011	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	8	1	no	—
153	C₄	—	268	2	no	—
154	C₄	—	268	1	no	—
155	C₄	—	268	1	no	—
156	C₄	C₂×#11761618	1	1	yes	characteristic
157	C₄	#14524066	1	1	yes	characteristic
158	C₄	#14524073	1	1	yes	characteristic
159	C₄	#14524075	1	1	yes	characteristic
160	C₂	#16863551	1	1	yes	characteristic
161	C₂	#16863557	1	1	yes	characteristic
162	C₂	#16863567	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864125

C₂

yes

maximal, characteristic

#16863746

C₂

yes

maximal, characteristic

#16863688

C₂

yes

maximal, characteristic

#16863141

C₂

yes

maximal, characteristic

Q₈⋊(C₆₇⋊C₈)

C₂

yes

maximal, characteristic

#16863126

C₂

yes

maximal, characteristic

#477×C₆₇

C₂

yes

maximal, characteristic

#14524043

—

#14524043

C₂²

yes

characteristic

#14524034

—

#14524034

—

Q₈×Dic₆₇

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

C₂²

yes

characteristic

#14523976

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

#14523967

—

#14523966

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₄×Q₈×C₆₇

C₂²

yes

characteristic

#107×C₆₇

—

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

#103×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

—

C₄⋊Dic₆₇

D₄

yes

characteristic

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

Q₁₆

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#120

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×Q₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

characteristic

#332

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

#125

—

134

—

#125

—

134

—

C₄×Q₈

—

C₄×Q₈

—

134

—

C₄×Q₈

—

134

—

C₄×Q₈

C₂×D₆₇

yes

characteristic

#107

—

100

#107

—

134

—

101

C₄⋊C₈

—

102

C₄⋊C₈

C₂×D₆₇

yes

characteristic

103

#103

—

104

#103

—

134

—

105

#103

—

134

—

106

C₄×C₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

134

—

110

C₂×Q₈

C₂²⋊D₆₇

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

134

—

113

C₂×C₈

—

134

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

134

—

120

C₄⋊C₄

—

134

—

121

C₄⋊C₄

—

134

—

122

C₄⋊C₄

—

134

—

123

C₄⋊C₄

—

134

—

124

C₄⋊C₄

C₂²⋊D₆₇

yes

characteristic

125

C₄²

—

126

C₄²

—

134

—

127

C₄²

—

134

—

128

C₄²

C₂²×D₆₇

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

268

—

132

Q₈

—

268

—

133

Q₈

—

134

—

134

Q₈

—

134

—

135

Q₈

#11761618

yes

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

134

—

140

C₂×C₄

—

134

—

141

C₂×C₄

—

134

—

142

C₂×C₄

D₄×D₆₇

yes

characteristic

143

C₂×C₄

D₄×D₆₇

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

145

C₈

—

146

C₈

—

268

—

147

C₈

—

268

—

148

C₂²

#14524011

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

268

—

154

C₄

—

268

—

155

C₄

—

268

—

156

C₄

C₂×#11761618

yes

characteristic

157

C₄

#14524066

yes

characteristic

158

C₄

#14524073

yes

characteristic

159

C₄

#14524075

yes

characteristic

160

C₂

#16863551

yes

characteristic

161

C₂

#16863557

yes

characteristic

162

C₂

#16863567

yes

characteristic

	subgroup	G/H
center	C₂²	#14524011
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#477×C₆₇	C₂
socle	C₂²×C₆₇	#120
hypercenter	#477	D₆₇
hypocenter	C₆₇	#1636
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#14524011

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#477×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
82. C₆₇	#1636	semidirect

H ≅ G/N

type

82. C₆₇

#1636

semidirect

Subgroups — #22559366

Conjugacy classes of subgroups

Complements