Group #22560256 — subgroups

Subgroups — #22560256

Order 8576.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14524049	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863378	C₂	1	2	yes	maximal
4	C₆₇⋊#338	C₂	1	2	yes	maximal, complemented
5	C₂×#108×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11761628	C₂²	1	1	yes	characteristic
7	#14524049	—	2	2	no	—
8	#14524049	—	2	2	no	—
9	C₂×#11761612	C₂²	1	2	yes	—
10	C₂×#11761612	C₂²	1	1	yes	characteristic
11	C₂×#11761611	C₄	1	2	yes	—
12	C₂×C₈×D₆₇	C₄	1	2	yes	—
13	#14523979	—	2	2	no	—
14	#14523961	—	2	2	no	—
15	#14523959	—	2	2	no	—
16	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#108×C₆₇	—	2	2	no	—
19	#11761628	—	2	1	no	—
20	#11761628	—	2	1	no	—
21	#11761628	—	2	1	no	—
22	#11761628	—	2	1	no	—
23	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
24	C₂×D₂₆₈	D₄	1	1	yes	characteristic
25	C₂×C₂²⋊D₆₇	Q₈	1	2	yes	—
26	C₂×C₄×D₆₇	C₂×C₄	1	2	yes	—
27	#11761612	—	2	1	no	—
28	#11761612	—	2	1	no	—
29	#11761612	—	2	2	no	—
30	#11761612	—	2	2	no	—
31	#11761611	—	2	4	no	—
32	C₂×C₆₇⋊C₈	—	2	2	no	—
33	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
34	C₈×D₆₇	—	2	4	no	—
35	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	M₁₆×C₆₇	—	2	2	no	—
37	M₁₆×C₆₇	—	2	2	no	—
38	C₂×C₈×C₆₇	—	2	2	no	—
39	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	D₄	1	2	yes	—
42	C₂×C₈×C₆₇	D₄	1	2	yes	—
43	C₂³×C₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₂²×D₆₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
50	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
51	C₂²⋊D₆₇	—	2	2	no	—
52	C₂²⋊D₆₇	—	2	2	no	—
53	C₂×Dic₆₇	C₄⋊C₄	1	2	yes	—
54	D₂₆₈	—	2	1	no	—
55	D₂₆₈	—	2	1	no	—
56	C₄×D₆₇	—	2	2	no	—
57	C₄×D₆₇	—	2	2	no	—
58	Dic₁₃₄	—	2	1	no	—
59	Dic₁₃₄	—	2	1	no	—
60	C₈×C₆₇	—	4	2	no	—
61	C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
62	C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
63	C₆₇⋊C₈	—	2	2	no	—
64	C₆₇⋊C₈	—	4	2	no	—
65	C₂²×C₆₇	—	4	1	no	—
66	C₂²×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×D₆₇	—	2	2	no	—
70	C₂×D₆₇	—	4	2	no	—
71	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₆₇	#117	1	1	yes	characteristic
73	C₄×C₆₇	#121	1	1	yes	characteristic
74	C₄×C₆₇	#121	1	1	yes	characteristic
75	Dic₆₇	—	2	2	no	—
76	C₂×C₆₇	—	2	1	no	—
77	C₂×C₆₇	—	2	1	no	—
78	C₂×C₆₇	—	2	1	no	—
79	C₂×C₆₇	#380	1	1	yes	characteristic
80	D₆₇	—	4	2	no	—
81	#1813	—	67	1	no	maximal, Sylow
82	C₆₇	#1813	1	1	yes	characteristic, complemented, Sylow
83	C₂×#131	—	67	1	no	—
84	C₂×#108	—	67	1	no	—
85	C₂×#108	D₆₇	1	1	yes	characteristic, complemented
86	#413	—	67	2	no	—
87	#338	—	67	2	no	—
88	C₂×#41	—	67	1	no	—
89	#131	—	134	2	no	—
90	#131	—	134	2	no	—
91	C₂×M₁₆	—	67	1	no	—
92	C₂×M₁₆	—	67	2	no	—
93	C₂×M₁₆	—	67	2	no	—
94	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
95	C₂²×C₈	—	67	2	no	—
96	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
97	#108	—	2	2	no	—
98	#108	—	134	2	no	—
99	#101	—	134	2	no	—
100	#100	—	134	2	no	—
101	#41	—	134	1	no	—
102	#41	—	134	1	no	—
103	#41	—	134	1	no	—
104	#41	—	134	1	no	—
105	C₂×Q₈	—	67	1	no	—
106	C₂×D₄	—	67	2	no	—
107	C₂×D₄	—	67	1	no	—
108	C₂²×C₄	—	67	2	no	—
109	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
110	M₁₆	—	2	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	134	2	no	—
113	M₁₆	—	134	2	no	—
114	M₁₆	—	134	1	no	—
115	M₁₆	—	134	1	no	—
116	M₁₆	—	134	4	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	67	2	no	—
119	C₂×C₈	—	134	2	no	—
120	C₂×C₈	—	134	4	no	—
121	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
122	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
123	C₂×C₈	D₂₆₈	1	2	yes	—
124	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
125	C₂³	—	67	2	no	—
126	C₂³	Q₈×D₆₇	1	1	yes	characteristic
127	Q₈	—	134	1	no	—
128	Q₈	—	134	1	no	—
129	D₄	—	134	2	no	—
130	D₄	—	134	2	no	—
131	D₄	—	134	1	no	—
132	D₄	—	134	1	no	—
133	C₂×C₄	—	67	2	no	—
134	C₂×C₄	—	134	2	no	—
135	C₂×C₄	—	134	2	no	—
136	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
137	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
139	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
140	C₂×C₄	#11761627	1	1	yes	characteristic
141	C₂×C₄	#11761629	1	1	yes	characteristic
142	C₈	—	4	2	no	—
143	C₈	—	268	2	no	—
144	C₈	—	134	2	no	—
145	C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
146	C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
147	C₂²	—	4	1	no	—
148	C₂²	—	268	2	no	—
149	C₂²	—	134	2	no	—
150	C₂²	C₄⋊C₄×D₆₇	1	1	yes	characteristic
151	C₂²	#14524019	1	1	yes	characteristic
152	C₂²	#14524023	1	1	yes	characteristic
153	C₄	—	134	2	no	—
154	C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
155	C₄	#14524001	1	1	yes	characteristic
156	C₄	#14524014	1	1	yes	characteristic
157	C₄	#14524018	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	268	2	no	—
162	C₂	#16863253	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524049

C₂

yes

maximal, characteristic

C₂×#14523979

C₂

yes

maximal, characteristic, complemented

#16863378

C₂

yes

maximal

C₆₇⋊#338

C₂

yes

maximal, complemented

C₂×#108×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761628

C₂²

yes

—

—

C₂²

yes

—

C₂×#11761612

C₂²

yes

characteristic

C₂×#11761611

C₄

yes

—

C₂×C₈×D₆₇

C₄

yes

—

#14523979

—

#14523961

—

#14523959

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#108×C₆₇

—

—

—

—

—

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂×D₂₆₈

D₄

yes

characteristic

C₂×C₂²⋊D₆₇

Q₈

yes

—

C₂×C₄×D₆₇

C₂×C₄

yes

—

—

—

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₈×D₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂×Q₈

yes

characteristic

C₂²×D₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂²⋊D₆₇

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

C₄⋊C₄

yes

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂²⋊C₄

yes

—

C₈×C₆₇

C₂²⋊C₄

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#380

yes

characteristic

D₆₇

—

#1813

—

maximal, Sylow

C₆₇

#1813

yes

characteristic, complemented, Sylow

C₂×#131

—

C₂×#108

—

C₂×#108

D₆₇

yes

characteristic, complemented

#413

—

#338

—

C₂×#41

—

#131

—

134

—

#131

—

134

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

#108

—

#108

—

134

—

#101

—

134

—

100

#100

—

134

—

101

#41

—

134

—

102

#41

—

134

—

103

#41

—

134

—

104

#41

—

134

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂²×D₆₇

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

134

—

113

M₁₆

—

134

—

114

M₁₆

—

134

—

115

M₁₆

—

134

—

116

M₁₆

—

134

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

134

—

120

C₂×C₈

—

134

—

121

C₂×C₈

C₄×D₆₇

yes

characteristic

122

C₂×C₈

C₄×D₆₇

yes

characteristic

123

C₂×C₈

D₂₆₈

yes

—

124

C₂×C₈

C₂²⋊D₆₇

yes

—

125

C₂³

—

126

C₂³

Q₈×D₆₇

yes

characteristic

127

Q₈

—

134

—

128

Q₈

—

134

—

129

D₄

—

134

—

130

D₄

—

134

—

131

D₄

—

134

—

132

D₄

—

134

—

133

C₂×C₄

—

134

C₂×C₄

—

134

—

135

C₂×C₄

—

134

—

136

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

137

C₂×C₄

D₄×D₆₇

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

139

C₂×C₄

C₂×D₂₆₈

yes

characteristic

140

C₂×C₄

#11761627

yes

characteristic

141

C₂×C₄

#11761629

yes

characteristic

142

C₈

—

143

C₈

—

268

—

144

C₈

—

134

—

145

C₈

(C₂×D₆₇)⋊C₄

yes

—

146

C₈

(C₂×D₆₇)⋊C₄

yes

—

147

C₂²

—

148

C₂²

—

268

—

149

C₂²

—

134

—

150

C₂²

C₄⋊C₄×D₆₇

yes

characteristic

151

C₂²

#14524019

yes

characteristic

152

C₂²

#14524023

yes

characteristic

153

C₄

—

134

—

154

C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

155

C₄

#14524001

yes

characteristic

156

C₄

#14524014

yes

characteristic

157

C₄

#14524018

yes

characteristic

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

—

268

—

162

C₂

#16863253

yes

characteristic

	subgroup	G/H
center	C₄	#14524001
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#108×C₆₇	C₂
socle	C₂×C₆₇	#380
hypercenter	C₂×#108	D₆₇
hypocenter	C₆₇	#1813
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#14524001

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#108×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
2. C₂×#14523979	C₂	semidirect
4. C₆₇⋊#338	C₂	semidirect
5. C₂×#108×C₆₇	C₂	semidirect
82. C₆₇	#1813	semidirect
85. C₂×#108	D₆₇	semidirect

H ≅ G/N

type

2. C₂×#14523979

C₂

semidirect

4. C₆₇⋊#338

C₂

semidirect

5. C₂×#108×C₆₇

C₂

semidirect

82. C₆₇

#1813

semidirect

85. C₂×#108

D₆₇

semidirect

Subgroups — #22560256

Conjugacy classes of subgroups

Complements