Group #22561496 — subgroups

Subgroups — #22561496

Order 8576.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16864196	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523971	C₂	1	2	yes	maximal, complemented
3	#16863378	C₂	1	2	yes	maximal, complemented
4	C₆₇⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
5	#423×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11761628	C₂²	1	2	yes	—
7	#14524076	—	2	2	no	—
8	#14524071	—	2	2	no	—
9	#14524065	—	2	2	no	—
10	C₂×#11761612	C₂²	1	2	yes	—
11	#14523971	—	2	2	no	—
12	#14523971	—	2	2	no	—
13	#14523961	—	2	2	no	—
14	#14523959	—	2	2	no	—
15	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₆₇	—	2	2	no	—
19	#11761628	—	2	2	no	—
20	#11761628	—	2	2	no	—
21	#11761628	—	2	2	no	—
22	#11761628	—	2	2	no	—
23	C₂×Dic₁₃₄	D₄	1	2	yes	—
24	C₂×D₂₆₈	D₄	1	2	yes	—
25	C₂×C₂²⋊D₆₇	—	2	2	no	—
26	C₂×C₄×D₆₇	—	2	2	no	—
27	Dic₂₆₈	—	2	2	no	—
28	#11761617	—	2	4	no	—
29	D₅₃₆	—	2	2	no	—
30	#11761612	—	2	2	no	—
31	#11761612	—	2	2	no	—
32	C₂×C₆₇⋊C₈	—	2	2	no	—
33	C₂²×C₄×C₆₇	—	2	1	no	—
34	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₆₇	—	2	2	no	—
36	M₁₆×C₆₇	—	2	2	no	—
37	C₂×C₈×C₆₇	—	2	1	no	—
38	C₂×C₈×C₆₇	D₄	1	2	yes	—
39	C₄²×C₆₇	—	2	1	no	—
40	C₄²×C₆₇	D₄	1	1	yes	characteristic
41	C₄²×C₆₇	D₄	1	1	yes	characteristic
42	C₂³×C₆₇	#41	1	1	yes	characteristic
43	C₂²×D₆₇	—	2	2	no	—
44	C₂×C₄×C₆₇	—	2	1	no	—
45	C₂×C₄×C₆₇	—	2	1	no	—
46	C₂×C₄×C₆₇	—	4	2	no	—
47	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	#41	1	2	yes	—
51	C₂²⋊D₆₇	—	4	2	no	—
52	C₂²⋊D₆₇	—	4	2	no	—
53	C₂×Dic₆₇	—	2	2	no	—
54	D₂₆₈	—	2	2	no	—
55	D₂₆₈	—	2	2	no	—
56	C₄×D₆₇	—	4	2	no	—
57	C₄×D₆₇	—	4	2	no	—
58	Dic₁₃₄	—	2	2	no	—
59	Dic₁₃₄	—	2	2	no	—
60	C₈×C₆₇	—	2	2	no	—
61	C₈×C₆₇	—	4	1	no	—
62	C₆₇⋊C₈	—	4	2	no	—
63	C₂²×C₆₇	—	4	1	no	—
64	C₂²×C₆₇	#121	1	1	yes	characteristic
65	C₂²×C₆₇	#123	1	2	yes	—
66	C₂×D₆₇	—	4	2	no	—
67	C₂×D₆₇	—	8	2	no	—
68	C₄×C₆₇	—	2	2	no	—
69	C₄×C₆₇	—	4	1	no	—
70	C₄×C₆₇	#120	1	1	yes	characteristic
71	C₄×C₆₇	#121	1	2	yes	—
72	C₄×C₆₇	#124	1	1	yes	characteristic
73	Dic₆₇	—	4	2	no	—
74	C₂×C₆₇	—	2	1	no	—
75	C₂×C₆₇	—	2	2	no	—
76	C₂×C₆₇	#394	1	1	yes	characteristic
77	D₆₇	—	8	2	no	—
78	#2050	—	67	1	no	maximal, Sylow
79	C₆₇	#2050	1	1	yes	characteristic, complemented, Sylow
80	#575	—	67	1	no	—
81	#423	D₆₇	1	1	yes	characteristic, complemented
82	C₂×#104	—	67	2	no	—
83	#413	—	67	2	no	—
84	#338	—	67	1	no	—
85	C₂×#41	—	67	2	no	—
86	#137	—	134	2	no	—
87	AGL(1,ℤ/8)	—	134	2	no	—
88	#135	—	134	2	no	—
89	C₂×M₁₆	—	67	2	no	—
90	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
91	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
93	C₄⋊C₈	—	2	2	no	—
94	#104	—	134	2	no	—
95	#104	—	134	2	no	—
96	#101	—	134	2	no	—
97	#100	—	134	2	no	—
98	#41	—	134	2	no	—
99	#41	—	134	2	no	—
100	#41	—	134	2	no	—
101	#41	—	134	2	no	—
102	C₂×Q₈	—	67	2	no	—
103	C₂×D₄	—	67	2	no	—
104	C₂×D₄	—	134	2	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	134	2	no	—
107	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
108	Q₁₆	—	134	2	no	—
109	SD₁₆	—	134	4	no	—
110	D₈	—	134	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	2	2	no	—
113	M₁₆	—	134	2	no	—
114	M₁₆	—	134	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	134	2	no	—
117	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₄²	D₂₆₈	1	1	yes	characteristic
120	C₄²	D₂₆₈	1	1	yes	characteristic
121	C₂³	—	134	2	no	—
122	C₂³	#11761628	1	1	yes	characteristic
123	Q₈	—	134	2	no	—
124	Q₈	—	134	2	no	—
125	D₄	—	268	2	no	—
126	D₄	—	268	2	no	—
127	D₄	—	134	2	no	—
128	D₄	—	134	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	268	2	no	—
133	C₂×C₄	—	268	2	no	—
134	C₂×C₄	—	134	2	no	—
135	C₂×C₄	D₄×D₆₇	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
137	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
138	C₂×C₄	#11761629	1	2	yes	—
139	C₈	—	2	2	no	—
140	C₈	—	4	1	no	—
141	C₈	—	268	2	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	268	2	no	—
144	C₂²	—	536	2	no	—
145	C₂²	#14524017	1	1	yes	characteristic
146	C₂²	#14524028	1	2	yes	—
147	C₄	—	2	2	no	—
148	C₄	—	4	1	no	—
149	C₄	—	268	2	no	—
150	C₄	#14524011	1	1	yes	characteristic
151	C₄	#14524018	1	2	yes	—
152	C₄	#14524032	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	2	no	—
155	C₂	—	536	2	no	—
156	C₂	#16863313	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864196

C₂

yes

maximal, characteristic

C₂×#14523971

C₂

yes

maximal, complemented

#16863378

C₂

yes

maximal, complemented

C₆₇⋊#338

C₂

yes

maximal, characteristic, complemented

#423×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761628

C₂²

yes

—

—

—

—

C₂²

yes

—

—

—

—

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

—

—

—

—

—

C₂×Dic₁₃₄

D₄

yes

—

C₂×D₂₆₈

D₄

yes

—

C₂×C₂²⋊D₆₇

—

C₂×C₄×D₆₇

—

Dic₂₆₈

—

#11761617

—

D₅₃₆

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₂³×C₆₇

#41

yes

characteristic

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂²⋊D₆₇

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₂²×C₆₇

#123

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#120

yes

characteristic

C₄×C₆₇

#121

yes

—

C₄×C₆₇

#124

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#394

yes

characteristic

D₆₇

—

#2050

—

maximal, Sylow

C₆₇

#2050

yes

characteristic, complemented, Sylow

#575

—

#423

D₆₇

yes

characteristic, complemented

C₂×#104

—

#413

—

#338

—

C₂×#41

—

#137

—

134

—

AGL(1,ℤ/8)

—

134

—

#135

—

134

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

#104

—

134

—

#104

—

134

—

#101

—

134

—

#100

—

134

—

#41

—

134

—

#41

—

134

—

100

#41

—

134

—

101

#41

—

134

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

134

—

105

C₂²×C₄

—

106

C₂²×C₄

—

134

—

107

C₂²×C₄

C₂²×D₆₇

yes

characteristic

108

Q₁₆

—

134

—

109

SD₁₆

—

134

—

110

D₈

—

134

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

134

—

114

M₁₆

—

134

—

115

C₂×C₈

—

116

C₂×C₈

—

134

—

117

C₂×C₈

C₂²⋊D₆₇

yes

—

118

C₄²

—

119

C₄²

D₂₆₈

yes

characteristic

120

C₄²

D₂₆₈

yes

characteristic

121

C₂³

—

134

—

122

C₂³

#11761628

yes

characteristic

123

Q₈

—

134

—

124

Q₈

—

134

—

125

D₄

—

268

—

126

D₄

—

268

—

127

D₄

—

134

—

128

D₄

—

134

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

268

—

133

C₂×C₄

—

268

—

134

C₂×C₄

—

134

—

135

C₂×C₄

D₄×D₆₇

yes

—

136

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

137

C₂×C₄

C₂×D₂₆₈

yes

characteristic

138

C₂×C₄

#11761629

yes

—

139

C₈

—

140

C₈

—

141

C₈

—

268

—

142

C₂²

—

143

C₂²

—

268

—

144

C₂²

—

536

—

145

C₂²

#14524017

yes

characteristic

146

C₂²

#14524028

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

268

—

150

C₄

#14524011

yes

characteristic

151

C₄

#14524018

yes

—

152

C₄

#14524032

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

536

—

156

C₂

#16863313

yes

characteristic

	subgroup	G/H
center	C₄	#14524032
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#423×C₆₇	C₂
socle	C₂×C₆₇	#394
hypercenter	#423	D₆₇
hypocenter	C₆₇	#2050
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

subgroup

G/H

center

C₄

#14524032

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#423×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

N	H ≅ G/N	type
2. C₂×#14523971	C₂	semidirect
3. #16863378	C₂	semidirect
4. C₆₇⋊#338	C₂	semidirect
5. #423×C₆₇	C₂	semidirect
79. C₆₇	#2050	semidirect
81. #423	D₆₇	semidirect

H ≅ G/N

type

2. C₂×#14523971

C₂

semidirect

3. #16863378

C₂

semidirect

4. C₆₇⋊#338

C₂

semidirect

5. #423×C₆₇

C₂

semidirect

79. C₆₇

#2050

semidirect

81. #423

D₆₇

semidirect

Subgroups — #22561496

Conjugacy classes of subgroups

Complements