Group #22561438 — subgroups

Subgroups — #22561438

Order 8576.

290 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 69 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	#16863977	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523971	C₂	1	2	yes	maximal, complemented
3	#16863395	C₂	1	2	yes	maximal, complemented
4	#16863165	C₂	1	1	yes	maximal, characteristic, complemented
5	#382×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11761628	C₂²	1	2	yes	—
7	C₂×#11761612	C₂²	1	2	yes	—
8	#14524023	—	2	2	no	—
9	#14524018	—	2	2	no	—
10	#14524017	—	2	2	no	—
11	#14523971	—	2	4	no	—
12	#14523966	—	2	2	no	—
13	#14523963	—	2	2	no	—
14	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#95×C₆₇	—	2	2	no	—
18	#11761628	—	2	2	no	—
19	#11761628	—	2	2	no	—
20	#11761628	—	2	2	no	—
21	#11761628	—	2	2	no	—
22	C₂×Dic₁₃₄	D₄	1	2	yes	—
23	C₂×D₂₆₈	—	2	2	no	—
24	C₂×D₂₆₈	D₄	1	2	yes	—
25	C₂×C₂²⋊D₆₇	—	2	2	no	—
26	C₂×C₄×D₆₇	—	2	2	no	—
27	#11761612	—	2	2	no	—
28	#11761612	—	2	2	no	—
29	C₂×C₆₇⋊C₈	—	2	2	no	—
30	C₄⋊Dic₆₇	—	2	2	no	—
31	(C₂×D₆₇)⋊C₄	—	2	4	no	—
32	C₂²×C₄×C₆₇	—	2	1	no	—
33	C₂²×C₄×C₆₇	—	2	1	no	—
34	C₂²×C₄×C₆₇	D₄	1	2	yes	—
35	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₄²×C₆₇	—	2	1	no	—
39	C₄²×C₆₇	D₄	1	2	yes	complemented
40	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₆₇	—	2	2	no	—
42	C₂×C₄×C₆₇	—	2	2	no	—
43	C₂×C₄×C₆₇	—	2	2	no	—
44	C₂×C₄×C₆₇	—	4	1	no	—
45	C₂×C₄×C₆₇	—	4	1	no	—
46	C₂×C₄×C₆₇	—	4	2	no	—
47	C₂×C₄×C₆₇	—	4	2	no	—
48	C₂×C₄×C₆₇	—	4	1	no	—
49	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
52	C₂×C₄×C₆₇	#41	1	2	yes	—
53	C₂²⋊D₆₇	—	4	2	no	—
54	C₂²⋊D₆₇	—	4	2	no	—
55	C₂×Dic₆₇	—	2	2	no	—
56	D₂₆₈	—	2	2	no	—
57	D₂₆₈	—	2	2	no	—
58	D₂₆₈	—	8	2	no	—
59	C₄×D₆₇	—	4	2	no	—
60	C₄×D₆₇	—	4	2	no	—
61	Dic₁₃₄	—	2	2	no	—
62	Dic₁₃₄	—	2	2	no	—
63	C₆₇⋊C₈	—	4	2	no	—
64	C₂²×C₆₇	—	2	1	no	—
65	C₂²×C₆₇	—	2	1	no	—
66	C₂²×C₆₇	#120	1	1	yes	characteristic
67	C₂²×C₆₇	#121	1	1	yes	characteristic
68	C₂²×C₆₇	#124	1	1	yes	characteristic
69	C₂×D₆₇	—	4	2	no	—
70	C₂×D₆₇	—	4	2	no	—
71	C₂×D₆₇	—	4	2	no	—
72	C₄×C₆₇	—	4	2	no	—
73	C₄×C₆₇	—	4	2	no	—
74	C₄×C₆₇	—	8	1	no	—
75	C₄×C₆₇	#121	1	2	yes	—
76	C₄×C₆₇	#123	1	2	yes	—
77	Dic₆₇	—	4	2	no	—
78	C₂×C₆₇	—	2	1	no	—
79	C₂×C₆₇	—	2	1	no	—
80	C₂×C₆₇	#394	1	1	yes	characteristic
81	C₂×C₆₇	#454	1	1	yes	characteristic
82	C₂×C₆₇	#455	1	1	yes	characteristic
83	D₆₇	—	8	2	no	—
84	#2040	—	67	1	no	maximal, Sylow
85	C₆₇	#2040	1	1	yes	characteristic, complemented, Sylow
86	#537	—	67	1	no	—
87	C₂×#104	—	67	2	no	—
88	#417	—	67	2	no	—
89	#382	D₆₇	1	1	yes	characteristic, complemented
90	#343	—	67	1	no	—
91	C₂×#41	—	67	2	no	—
92	C₂×M₁₆	—	67	2	no	—
93	C₂²⋊Q₈	—	134	2	no	—
94	#121	—	134	2	no	—
95	#121	—	134	2	no	—
96	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
98	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
99	#104	—	134	4	no	—
100	#103	—	134	2	no	—
101	#102	—	134	2	no	—
102	#95	—	2	2	no	—
103	#41	—	134	2	no	—
104	#41	—	134	2	no	—
105	#41	—	134	2	no	—
106	#41	—	134	2	no	—
107	C₂×Q₈	—	67	2	no	—
108	C₂×D₄	—	67	2	no	—
109	C₂×D₄	—	134	2	no	—
110	C₂×D₄	—	134	2	no	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	134	2	no	—
114	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
115	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
116	M₁₆	—	134	2	no	—
117	M₁₆	—	134	2	no	—
118	C₂×C₈	—	134	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	134	2	no	—
122	C₂²⋊C₄	—	134	4	no	—
123	C₄²	—	2	1	no	—
124	C₄²	D₂₆₈	1	2	yes	complemented
125	C₂³	—	134	2	no	—
126	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
127	Q₈	—	134	2	no	—
128	Q₈	—	134	2	no	—
129	D₄	—	268	2	no	—
130	D₄	—	268	2	no	—
131	D₄	—	536	2	no	—
132	D₄	—	134	2	no	—
133	D₄	—	134	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	268	2	no	—
142	C₂×C₄	—	268	2	no	—
143	C₂×C₄	—	134	2	no	—
144	C₂×C₄	D₄×D₆₇	1	2	yes	—
145	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
146	C₂×C₄	#11761628	1	1	yes	characteristic
147	C₂×C₄	#11761629	1	2	yes	—
148	C₈	—	268	2	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	268	2	no	—
152	C₂²	—	268	2	no	—
153	C₂²	—	268	2	no	—
154	C₂²	#14524011	1	1	yes	characteristic
155	C₂²	#14524017	1	1	yes	characteristic
156	C₂²	#14524032	1	1	yes	characteristic
157	C₄	—	4	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	8	1	no	—
160	C₄	—	268	2	no	—
161	C₄	#14524018	1	2	yes	—
162	C₄	#14524028	1	2	yes	—
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	536	2	no	—
166	C₂	#16863313	1	1	yes	characteristic
167	C₂	#16863568	1	1	yes	characteristic
168	C₂	#16863573	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863977

C₂

yes

maximal, characteristic

C₂×#14523971

C₂

yes

maximal, complemented

#16863395

C₂

yes

maximal, complemented

#16863165

C₂

yes

maximal, characteristic, complemented

#382×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761628

C₂²

yes

—

C₂×#11761612

C₂²

yes

—

—

—

—

—

—

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#95×C₆₇

—

—

—

—

—

C₂×Dic₁₃₄

D₄

yes

—

C₂×D₂₆₈

—

C₂×D₂₆₈

D₄

yes

—

C₂×C₂²⋊D₆₇

—

C₂×C₄×D₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

(C₂×D₆₇)⋊C₄

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

D₄

yes

complemented

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂²⋊D₆₇

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#121

yes

—

C₄×C₆₇

#123

yes

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#2040

—

maximal, Sylow

C₆₇

#2040

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

characteristic, complemented

#343

—

C₂×#41

—

C₂×M₁₆

—

C₂²⋊Q₈

—

134

—

#121

—

134

—

#121

—

134

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

C₂×C₄²

C₂×D₆₇

yes

characteristic

#104

—

134

—

100

#103

—

134

—

101

#102

—

134

—

102

#95

—

103

#41

—

134

—

104

#41

—

134

—

105

#41

—

134

—

106

#41

—

134

—

107

C₂×Q₈

—

108

C₂×D₄

—

109

C₂×D₄

—

134

—

110

C₂×D₄

—

134

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

134

—

114

C₂²×C₄

C₂²⋊D₆₇

yes

—

115

C₂²×C₄

C₂²×D₆₇

yes

characteristic

116

M₁₆

—

134

—

117

M₁₆

—

134

—

118

C₂×C₈

—

134

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

134

—

122

C₂²⋊C₄

—

134

—

123

C₄²

—

124

C₄²

D₂₆₈

yes

complemented

125

C₂³

—

134

—

126

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

127

Q₈

—

134

—

128

Q₈

—

134

—

129

D₄

—

268

—

130

D₄

—

268

—

131

D₄

—

536

—

132

D₄

—

134

—

133

D₄

—

134

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

268

—

142

C₂×C₄

—

268

—

143

C₂×C₄

—

134

—

144

C₂×C₄

D₄×D₆₇

yes

—

145

C₂×C₄

C₂×D₂₆₈

yes

characteristic

146

C₂×C₄

#11761628

yes

characteristic

147

C₂×C₄

#11761629

yes

—

148

C₈

—

268

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

268

—

152

C₂²

—

268

—

153

C₂²

—

268

—

154

C₂²

#14524011

yes

characteristic

155

C₂²

#14524017

yes

characteristic

156

C₂²

#14524032

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

268

—

161

C₄

#14524018

yes

—

162

C₄

#14524028

yes

—

163

C₂

—

164

C₂

—

165

C₂

—

536

—

166

C₂

#16863313

yes

characteristic

167

C₂

#16863568

yes

characteristic

168

C₂

#16863573

yes

characteristic

	subgroup	G/H
center	C₂²	#14524011
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#382×C₆₇	C₂
socle	C₂²×C₆₇	#120
hypercenter	#382	D₆₇
hypocenter	C₆₇	#2040
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #382

subgroup

G/H

center

C₂²

#14524011

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#382×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #382

N	H ≅ G/N	type
2. C₂×#14523971	C₂	semidirect
3. #16863395	C₂	semidirect
4. #16863165	C₂	semidirect
5. #382×C₆₇	C₂	semidirect
39. C₄²×C₆₇	D₄	semidirect
85. C₆₇	#2040	semidirect
89. #382	D₆₇	semidirect
124. C₄²	D₂₆₈	semidirect

H ≅ G/N

type

2. C₂×#14523971

C₂

semidirect

3. #16863395

C₂

semidirect

4. #16863165

C₂

semidirect

5. #382×C₆₇

C₂

semidirect

39. C₄²×C₆₇

D₄

semidirect

85. C₆₇

#2040

semidirect

89. #382

D₆₇

semidirect

124. C₄²

D₂₆₈

semidirect

Subgroups — #22561438

Conjugacy classes of subgroups

Complements