Group #22573074 — subgroups

Subgroups — #22573074

Order 8576.

352 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16864103	C₂	1	2	yes	maximal, complemented
2	#16863801	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863797	C₂	1	2	yes	maximal, complemented
4	#16863694	C₂	1	4	yes	maximal, complemented
5	#16863686	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×Dic₆₇	C₂	1	2	yes	maximal
8	C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#14524043	C₂²	1	2	yes	—
10	#14524034	C₂²	1	4	yes	—
11	#14524022	—	2	2	no	—
12	#14524020	—	2	4	no	—
13	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
14	D₄×Dic₆₇	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₆₇	—	2	2	no	—
16	#14523977	C₂²	1	2	yes	—
17	#14523977	C₂²	1	2	yes	—
18	#14523974	C₂²	1	4	yes	—
19	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₆₇⋊C₈	C₂²	1	2	yes	—
21	#14523964	C₂²	1	4	yes	—
22	C₈×Dic₆₇	C₂²	1	2	yes	—
23	C₂×D₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₆₇	C₂²	1	2	yes	—
25	#107×C₆₇	C₂²	1	1	yes	characteristic
26	#102×C₆₇	C₂²	1	2	yes	—
27	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₃₄	—	2	2	no	—
29	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
30	C₂²×Dic₆₇	—	2	4	no	—
31	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
32	C₄⋊Dic₆₇	—	2	2	no	—
33	C₄⋊Dic₆₇	—	2	4	no	—
34	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
35	C₄⋊Dic₆₇	C₂³	1	4	yes	—
36	Dic₆₇⋊C₄	—	2	4	no	—
37	Dic₆₇⋊C₄	D₄	1	2	yes	—
38	C₂²⋊Dic₆₇	—	2	4	no	—
39	C₄×Dic₆₇	C₂³	1	2	yes	—
40	C₂×D₄×C₆₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₆₇	—	2	2	no	—
42	D₈×C₆₇	Q₈	1	4	yes	complemented
43	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
46	C₂²⋊C₄×C₆₇	—	2	2	no	—
47	C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₂³×C₆₇	—	2	2	no	—
49	D₄×C₆₇	—	2	2	no	—
50	D₄×C₆₇	C₂×Q₈	1	4	yes	—
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	—	4	4	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	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₆₇	—	2	4	no	—
57	C₂×Dic₆₇	—	8	4	no	—
58	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
59	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
60	Dic₁₃₄	—	2	2	no	—
61	Dic₁₃₄	—	8	2	no	—
62	C₈×C₆₇	—	2	1	no	—
63	C₈×C₆₇	C₂×Q₈	1	2	yes	—
64	C₆₇⋊C₈	—	2	2	no	—
65	C₂²×C₆₇	—	2	4	no	—
66	C₂²×C₆₇	—	4	2	no	—
67	C₂²×C₆₇	—	4	2	no	—
68	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
69	C₄×C₆₇	—	2	1	no	—
70	C₄×C₆₇	—	4	2	no	—
71	C₄×C₆₇	#135	1	2	yes	—
72	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
73	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
74	Dic₆₇	—	2	2	no	—
75	Dic₆₇	—	2	2	no	—
76	Dic₆₇	—	4	4	no	—
77	C₂×C₆₇	—	4	4	no	—
78	C₂×C₆₇	D₄×Q₈	1	1	yes	characteristic
79	C₂×C₆₇	C₂×#135	1	1	yes	characteristic
80	C₂×C₆₇	#576	1	1	yes	characteristic
81	#3380	—	67	1	no	maximal, Sylow
82	C₆₇	#3380	1	1	yes	characteristic, complemented, Sylow
83	#554	—	67	2	no	—
84	#500	—	67	1	no	—
85	#499	—	67	2	no	—
86	#478	—	67	4	no	—
87	#476	—	67	2	no	—
88	C₈×Q₈	—	67	1	no	—
89	C₄×D₈	—	67	2	no	—
90	C₄×D₈	D₆₇	1	1	yes	characteristic
91	C₂×D₈	C₂×D₆₇	1	1	yes	characteristic
92	C₄⋊Q₈	—	67	2	no	—
93	#125	—	67	4	no	—
94	C₂²⋊Q₈	—	134	4	no	—
95	C₂²⋊Q₈	—	134	2	no	—
96	C₄×Q₈	—	67	1	no	—
97	C₄×D₄	—	67	4	no	—
98	C₄×D₄	C₂×D₆₇	1	2	yes	—
99	C₂×C₄⋊C₄	—	134	2	no	—
100	#107	—	67	2	no	—
101	#107	—	67	2	no	—
102	#107	C₂×D₆₇	1	1	yes	characteristic
103	#106	—	67	4	no	—
104	C₄⋊C₈	—	67	2	no	—
105	C₄⋊C₈	—	67	1	no	—
106	#102	—	67	4	no	—
107	#102	C₂×D₆₇	1	2	yes	—
108	C₄×C₈	—	67	2	no	—
109	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
110	C₂×Q₈	—	67	1	no	—
111	C₂×Q₈	—	134	2	no	—
112	C₂×D₄	C₂²×D₆₇	1	2	yes	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	—	134	4	no	—
115	D₈	Dic₁₃₄	1	4	yes	complemented
116	C₂×C₈	—	67	2	no	—
117	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
119	C₄⋊C₄	—	67	1	no	—
120	C₄⋊C₄	—	67	2	no	—
121	C₄⋊C₄	—	67	4	no	—
122	C₄⋊C₄	—	134	4	no	—
123	C₄⋊C₄	—	134	4	no	—
124	C₄⋊C₄	—	134	2	no	—
125	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	134	4	no	—
128	C₄²	—	67	2	no	—
129	C₄²	C₂²×D₆₇	1	1	yes	characteristic
130	C₂³	—	2	2	no	—
131	Q₈	—	536	2	no	—
132	Q₈	—	134	2	no	—
133	D₄	—	2	2	no	—
134	D₄	C₂×Dic₁₃₄	1	4	yes	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	536	4	no	—
138	C₂×C₄	—	67	2	no	—
139	C₂×C₄	—	67	2	no	—
140	C₂×C₄	—	134	4	no	—
141	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
142	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	134	2	no	—
146	C₈	C₂×Dic₁₃₄	1	2	yes	—
147	C₂²	—	2	4	no	—
148	C₂²	—	4	2	no	—
149	C₂²	—	4	2	no	—
150	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	4	2	no	—
153	C₄	—	268	4	no	—
154	C₄	—	134	2	no	—
155	C₄	—	134	2	no	—
156	C₄	#14524062	1	2	yes	—
157	C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
158	C₄	#14524090	1	1	yes	characteristic
159	C₂	—	4	4	no	—
160	C₂	D₄×Dic₁₃₄	1	1	yes	characteristic
161	C₂	C₂×#14524062	1	1	yes	characteristic
162	C₂	#16864198	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864103

C₂

yes

maximal, complemented

#16863801

C₂

yes

maximal, characteristic, complemented

#16863797

C₂

yes

maximal, complemented

#16863694

C₂

yes

maximal, complemented

#16863686

C₂

yes

maximal, complemented

C₈×Dic₁₃₄

C₂

yes

maximal, characteristic, complemented

D₈×Dic₆₇

C₂

yes

maximal

C₄×D₈×C₆₇

C₂

yes

maximal, characteristic

#14524043

C₂²

yes

—

#14524034

C₂²

yes

—

#14524022

—

#14524020

—

C₄×Dic₁₃₄

C₂²

yes

characteristic

D₄×Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

—

#14523977

C₂²

yes

—

#14523977

C₂²

yes

—

#14523974

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

C₂²

yes

—

#14523964

C₂²

yes

—

C₈×Dic₆₇

C₂²

yes

—

C₂×D₈×C₆₇

C₂²

yes

characteristic

C₄×D₄×C₆₇

C₂²

yes

—

#107×C₆₇

C₂²

yes

characteristic

#102×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

D₄

yes

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

C₂³

yes

—

C₂×D₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

—

D₈×C₆₇

Q₈

yes

complemented

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂³

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

C₂×D₄

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂×Q₈

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#135

yes

—

C₄×C₆₇

C₂²×Q₈

yes

characteristic

C₄×C₆₇

2₋¹⁺⁴

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

D₄×Q₈

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₈×Q₈

—

C₄×D₈

—

C₄×D₈

D₆₇

yes

characteristic

C₂×D₈

C₂×D₆₇

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

134

—

C₂²⋊Q₈

—

134

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

—

134

—

100

#107

—

101

#107

—

102

#107

C₂×D₆₇

yes

characteristic

103

#106

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

#102

—

107

#102

C₂×D₆₇

yes

—

108

C₄×C₈

—

109

C₄×C₈

C₂×D₆₇

yes

characteristic

110

C₂×Q₈

—

111

C₂×Q₈

—

134

—

112

C₂×D₄

C₂²×D₆₇

yes

—

113

C₂²×C₄

—

114

C₂²×C₄

—

134

—

115

D₈

Dic₁₃₄

yes

complemented

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₆₇

yes

characteristic

118

C₂×C₈

C₂²×D₆₇

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

134

—

123

C₄⋊C₄

—

134

—

124

C₄⋊C₄

—

134

—

125

C₄⋊C₄

C₂²×D₆₇

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

134

—

128

C₄²

—

129

C₄²

C₂²×D₆₇

yes

characteristic

130

C₂³

—

131

Q₈

—

536

—

132

Q₈

—

134

—

133

D₄

—

134

D₄

C₂×Dic₁₃₄

yes

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

536

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

134

—

141

C₂×C₄

D₄×D₆₇

yes

characteristic

142

C₂×C₄

D₄×D₆₇

yes

characteristic

143

C₂×C₄

C₂³×D₆₇

yes

characteristic

144

C₈

—

145

C₈

—

134

—

146

C₈

C₂×Dic₁₃₄

yes

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂×D₄×D₆₇

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

268

—

154

C₄

—

134

—

155

C₄

—

134

—

156

C₄

#14524062

yes

—

157

C₄

C₂²×Dic₁₃₄

yes

characteristic

158

C₄

#14524090

yes

characteristic

159

C₂

—

160

C₂

D₄×Dic₁₃₄

yes

characteristic

161

C₂

C₂×#14524062

yes

characteristic

162

C₂

#16864198

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₄×D₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₄×D₈	D₆₇
hypocenter	C₆₇	#3380
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₄×D₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

hypercenter

C₄×D₈

D₆₇

hypocenter

C₆₇

#3380

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. #16864103	C₂	semidirect
2. #16863801	C₂	semidirect
3. #16863797	C₂	semidirect
4. #16863694	C₂	semidirect
5. #16863686	C₂	semidirect
6. C₈×Dic₁₃₄	C₂	semidirect
42. D₈×C₆₇	Q₈	semidirect
82. C₆₇	#3380	semidirect
115. D₈	Dic₁₃₄	semidirect

H ≅ G/N

type

1. #16864103

C₂

semidirect

2. #16863801

C₂

semidirect

3. #16863797

C₂

semidirect

4. #16863694

C₂

semidirect

5. #16863686

C₂

semidirect

6. C₈×Dic₁₃₄

C₂

semidirect

42. D₈×C₆₇

Q₈

semidirect

82. C₆₇

#3380

semidirect

115. D₈

Dic₁₃₄

semidirect

Subgroups — #22573074

Conjugacy classes of subgroups

Complements