Group #22559645 — subgroups

Subgroups — #22559645

Order 8576.

182 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16863742	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863602	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863588	C₂	1	1	yes	maximal, characteristic
4	#16863461	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863148	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863141	C₂	1	1	yes	maximal, characteristic, complemented
7	#331×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761617	—	2	1	no	—
9	C₂×#11761613	—	2	1	no	—
10	C₂×C₈×D₆₇	—	2	1	no	—
11	#14524043	C₂²	1	1	yes	characteristic
12	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
13	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	#14523968	—	2	1	no	—
15	#14523964	—	2	1	no	—
16	#14523964	—	2	1	no	—
17	#14523964	—	2	1	no	—
18	(C₂×D₆₇)⋊C₈	—	2	1	no	—
19	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	#127×C₆₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₄	—	2	1	no	—
24	C₂×D₂₆₈	D₄	1	1	yes	characteristic
25	C₂×C₄×D₆₇	—	2	1	no	—
26	#11761617	—	4	2	no	—
27	#11761613	—	4	2	no	—
28	C₂×C₆₇⋊C₈	—	2	1	no	—
29	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
30	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
31	C₈×D₆₇	—	4	2	no	—
32	C₄⋊Dic₆₇	—	2	1	no	—
33	C₄⋊Dic₆₇	—	2	1	no	—
34	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
35	(C₂×D₆₇)⋊C₄	—	2	1	no	—
36	C₂×D₄×C₆₇	—	2	1	no	—
37	C₂×D₄×C₆₇	—	2	1	no	—
38	C₂×D₄×C₆₇	—	2	1	no	—
39	C₂×C₈×C₆₇	—	2	1	no	—
40	C₂×C₈×C₆₇	—	2	1	no	—
41	C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₂³×C₆₇	—	4	1	no	—
43	C₂²×D₆₇	—	2	1	no	—
44	D₄×C₆₇	—	4	1	no	—
45	D₄×C₆₇	—	4	1	no	—
46	D₄×C₆₇	—	4	1	no	—
47	D₄×C₆₇	—	4	1	no	—
48	D₄×C₆₇	—	8	1	no	—
49	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
52	C₂×Dic₆₇	—	2	1	no	—
53	C₂×Dic₆₇	—	4	1	no	—
54	D₂₆₈	—	2	1	no	—
55	D₂₆₈	D₈	1	2	yes	—
56	C₄×D₆₇	—	4	1	no	—
57	C₄×D₆₇	—	4	1	no	—
58	Dic₁₃₄	—	4	1	no	—
59	Dic₁₃₄	—	4	1	no	—
60	C₈×C₆₇	—	4	1	no	—
61	C₈×C₆₇	—	4	1	no	—
62	C₆₇⋊C₈	—	2	1	no	—
63	C₆₇⋊C₈	—	2	1	no	—
64	C₆₇⋊C₈	—	4	1	no	—
65	C₂²×C₆₇	—	8	1	no	—
66	C₂²×C₆₇	—	8	1	no	—
67	C₂²×C₆₇	—	8	1	no	—
68	C₂²×C₆₇	#121	1	1	yes	characteristic
69	C₂×D₆₇	—	2	2	no	—
70	C₂×D₆₇	—	4	1	no	—
71	C₂×D₆₇	—	4	1	no	—
72	C₄×C₆₇	—	2	1	no	—
73	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
74	C₄×C₆₇	#135	1	1	yes	characteristic
75	C₄×C₆₇	#135	1	1	yes	characteristic
76	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	Dic₆₇	—	4	1	no	—
78	Dic₆₇	—	8	1	no	—
79	C₂×C₆₇	—	16	1	no	—
80	C₂×C₆₇	#463	1	1	yes	characteristic
81	C₂×C₆₇	#466	1	1	yes	characteristic
82	C₂×C₆₇	#471	1	1	yes	characteristic
83	D₆₇	—	4	2	no	—
84	#1681	—	67	1	no	maximal, Sylow
85	C₆₇	#1681	1	1	yes	characteristic, complemented, Sylow
86	#486	—	67	1	no	—
87	#461	—	67	1	no	—
88	#459	—	67	1	no	—
89	C₈×D₄	—	67	1	no	—
90	#335	—	67	1	no	—
91	#332	—	67	1	no	—
92	#331	D₆₇	1	1	yes	characteristic, complemented
93	C₂×SD₁₆	—	134	1	no	—
94	C₂×D₈	—	134	1	no	—
95	C₂²×C₈	—	134	1	no	—
96	C₄⋊Q₈	—	67	1	no	—
97	#127	C₂×D₆₇	1	1	yes	characteristic
98	C₄×D₄	—	67	1	no	—
99	C₄⋊C₈	—	67	1	no	—
100	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
101	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
102	#103	—	134	1	no	—
103	#102	—	134	1	no	—
104	#102	—	134	1	no	—
105	#102	—	134	1	no	—
106	C₂²⋊C₈	—	134	1	no	—
107	C₄×C₈	—	67	1	no	—
108	C₂×Q₈	—	134	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	67	1	no	—
113	C₂²×C₄	—	134	1	no	—
114	SD₁₆	—	268	2	no	—
115	D₈	—	268	2	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	268	2	no	—
119	C₂×C₈	—	67	1	no	—
120	C₂×C₈	—	67	1	no	—
121	C₂×C₈	—	134	1	no	—
122	C₄⋊C₄	—	67	1	no	—
123	C₄⋊C₄	—	134	1	no	—
124	C₄⋊C₄	—	134	1	no	—
125	C₂²⋊C₄	—	134	1	no	—
126	C₄²	C₂²×D₆₇	1	1	yes	characteristic
127	C₂³	—	4	1	no	—
128	C₂³	—	134	1	no	—
129	Q₈	—	268	1	no	—
130	Q₈	—	268	1	no	—
131	D₄	—	4	1	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	8	1	no	—
136	D₄	—	67	2	no	—
137	D₄	—	134	1	no	—
138	C₂×C₄	—	268	1	no	—
139	C₂×C₄	—	268	1	no	—
140	C₂×C₄	—	268	1	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	#11761628	1	1	yes	characteristic
145	C₈	—	4	1	no	—
146	C₈	—	4	1	no	—
147	C₈	—	268	1	no	—
148	C₈	—	134	1	no	—
149	C₈	—	134	1	no	—
150	C₂²	—	8	1	no	—
151	C₂²	—	8	1	no	—
152	C₂²	—	8	1	no	—
153	C₂²	—	268	1	no	—
154	C₂²	—	268	1	no	—
155	C₂²	—	134	2	no	—
156	C₂²	#14524016	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	268	1	no	—
159	C₄	—	536	1	no	—
160	C₄	D₈×D₆₇	1	1	yes	characteristic
161	C₄	#14524064	1	1	yes	characteristic
162	C₄	#14524064	1	1	yes	characteristic
163	C₄	#14524067	1	1	yes	characteristic
164	C₂	—	268	2	no	—
165	C₂	—	16	1	no	—
166	C₂	#16863610	1	1	yes	characteristic
167	C₂	#16863631	1	1	yes	characteristic
168	C₂	#16863658	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863742

C₂

yes

maximal, characteristic, complemented

#16863602

C₂

yes

maximal, characteristic, complemented

#16863588

C₂

yes

maximal, characteristic

#16863461

C₂

yes

maximal, characteristic, complemented

#16863148

C₂

yes

maximal, characteristic, complemented

#16863141

C₂

yes

maximal, characteristic, complemented

#331×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761617

—

C₂×#11761613

—

C₂×C₈×D₆₇

—

#14524043

C₂²

yes

characteristic

C₄×D₂₆₈

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

—

—

—

(C₂×D₆₇)⋊C₈

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

#127×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×D₂₆₈

D₄

yes

characteristic

C₂×C₄×D₆₇

—

#11761617

—

#11761613

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₈×D₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

(C₂×D₆₇)⋊C₄

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

C₂²×D₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

D₈

yes

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×D₈

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#1681

—

maximal, Sylow

C₆₇

#1681

yes

characteristic, complemented, Sylow

#486

—

#461

—

#459

—

C₈×D₄

—

#335

—

#332

—

#331

D₆₇

yes

characteristic, complemented

C₂×SD₁₆

—

134

—

C₂×D₈

—

134

—

C₂²×C₈

—

134

—

C₄⋊Q₈

—

#127

C₂×D₆₇

yes

characteristic

C₄×D₄

—

C₄⋊C₈

—

100

C₄⋊C₈

C₂×D₆₇

yes

characteristic

101

C₄⋊C₈

C₂×D₆₇

yes

characteristic

102

#103

—

134

—

103

#102

—

134

—

104

#102

—

134

—

105

#102

—

134

—

106

C₂²⋊C₈

—

134

—

107

C₄×C₈

—

108

C₂×Q₈

—

134

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂²×C₄

—

134

—

114

SD₁₆

—

268

—

115

D₈

—

268

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

268

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

134

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

134

—

124

C₄⋊C₄

—

134

—

125

C₂²⋊C₄

—

134

—

126

C₄²

C₂²×D₆₇

yes

characteristic

127

C₂³

—

128

C₂³

—

134

—

129

Q₈

—

268

—

130

Q₈

—

268

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

134

—

138

C₂×C₄

—

268

—

139

C₂×C₄

—

268

—

140

C₂×C₄

—

268

—

141

C₂×C₄

—

134

—

142

C₂×C₄

D₄×D₆₇

yes

characteristic

143

C₂×C₄

D₄×D₆₇

yes

characteristic

144

C₂×C₄

#11761628

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

268

—

148

C₈

—

134

—

149

C₈

—

134

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

268

—

154

C₂²

—

268

—

155

C₂²

—

134

—

156

C₂²

#14524016

yes

characteristic

157

C₄

—

158

C₄

—

268

—

159

C₄

—

536

—

160

C₄

D₈×D₆₇

yes

characteristic

161

C₄

#14524064

yes

characteristic

162

C₄

#14524064

yes

characteristic

163

C₄

#14524067

yes

characteristic

164

C₂

—

268

—

165

C₂

—

166

C₂

#16863610

yes

characteristic

167

C₂

#16863631

yes

characteristic

168

C₂

#16863658

yes

characteristic

	subgroup	G/H
center	C₂²	#14524016
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#331×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#331	D₆₇
hypocenter	C₆₇	#1681
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#14524016

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#331×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
1. #16863742	C₂	semidirect
2. #16863602	C₂	semidirect
4. #16863461	C₂	semidirect
5. #16863148	C₂	semidirect
6. #16863141	C₂	semidirect
7. #331×C₆₇	C₂	semidirect
85. C₆₇	#1681	semidirect
92. #331	D₆₇	semidirect

H ≅ G/N

type

1. #16863742

C₂

semidirect

2. #16863602

C₂

semidirect

4. #16863461

C₂

semidirect

5. #16863148

C₂

semidirect

6. #16863141

C₂

semidirect

7. #331×C₆₇

C₂

semidirect

85. C₆₇

#1681

semidirect

92. #331

D₆₇

semidirect

Subgroups — #22559645

Conjugacy classes of subgroups

Complements