Group #22569306 — subgroups

Subgroups — #22569306

Order 8576.

322 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 161 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	#16863953	C₂	1	1	yes	maximal, characteristic
2	#16863524	C₂	1	4	yes	maximal, complemented
3	#16863523	C₂	1	2	yes	maximal, complemented
4	#16863520	C₂	1	2	yes	maximal, complemented
5	#16863454	C₂	1	4	yes	maximal
6	#16863449	C₂	1	1	yes	maximal, characteristic
7	#432×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14524043	C₄	1	1	yes	characteristic
9	#14524043	C₄	1	1	yes	characteristic
10	#14524035	C₄	1	2	yes	—
11	#14524022	C₄	1	2	yes	—
12	#14524022	C₄	1	2	yes	—
13	C₄×Dic₁₃₄	C₂²	1	2	yes	—
14	C₄×Dic₁₃₄	C₂²	1	2	yes	—
15	#14524002	C₂²	1	2	yes	—
16	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
17	Dic₆₇⋊C₈	C₂²	1	4	yes	—
18	Dic₆₇⋊C₈	C₂²	1	4	yes	—
19	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
20	#14523953	C₂²	1	4	yes	—
21	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
23	C₈×Dic₆₇	C₂²	1	4	yes	—
24	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
27	#97×C₆₇	C₂²	1	2	yes	—
28	C₂×Dic₁₃₄	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₃₄	C₂×C₄	1	2	yes	—
30	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
31	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
32	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
33	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
34	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
35	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
36	Dic₆₇⋊C₄	C₂×C₄	1	4	yes	—
37	Dic₆₇⋊C₄	C₂×C₄	1	4	yes	—
38	C₂²⋊Dic₆₇	C₂×C₄	1	4	yes	—
39	C₄×Dic₆₇	C₂³	1	4	yes	—
40	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₂³	1	4	yes	—
44	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	—	2	2	no	—
50	C₂×C₄×C₆₇	—	2	2	no	—
51	C₂×C₄×C₆₇	—	4	1	no	—
52	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₆₇	C₂²×C₄	1	4	yes	—
59	C₂×Dic₆₇	C₂²×C₄	1	4	yes	—
60	Dic₁₃₄	—	2	4	no	—
61	Dic₁₃₄	—	4	2	no	—
62	C₈×C₆₇	—	2	4	no	—
63	C₆₇⋊C₈	—	2	2	no	—
64	C₆₇⋊C₈	—	2	2	no	—
65	C₂²×C₆₇	—	2	1	no	—
66	C₂²×C₆₇	—	2	1	no	—
67	C₂²×C₆₇	—	2	1	no	—
68	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₆₇	—	2	1	no	—
70	C₄×C₆₇	—	2	1	no	—
71	C₄×C₆₇	—	2	1	no	—
72	C₄×C₆₇	—	2	1	no	—
73	C₄×C₆₇	#131	1	2	yes	—
74	C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
75	Dic₆₇	—	2	4	no	—
76	Dic₆₇	—	2	4	no	—
77	C₂×C₆₇	—	4	1	no	—
78	C₂×C₆₇	#519	1	1	yes	characteristic
79	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
80	C₂×C₆₇	#568	1	1	yes	characteristic
81	#2985	—	67	1	no	maximal, Sylow
82	C₆₇	#2985	1	1	yes	characteristic, complemented, Sylow
83	#533	—	67	1	no	—
84	#446	—	67	2	no	—
85	#446	—	67	4	no	—
86	C₈×Q₈	—	67	2	no	—
87	#433	—	67	4	no	—
88	#432	D₆₇	1	1	yes	characteristic
89	#431	—	67	1	no	—
90	C₄⋊Q₈	—	67	1	no	—
91	C₄⋊Q₈	—	67	1	no	—
92	#125	—	67	2	no	—
93	C₂²⋊Q₈	—	67	2	no	—
94	C₂²⋊Q₈	—	67	2	no	—
95	C₄×Q₈	—	67	2	no	—
96	C₄×Q₈	—	67	2	no	—
97	#117	—	67	2	no	—
98	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
99	C₄⋊C₈	—	67	4	no	—
100	C₄⋊C₈	—	67	4	no	—
101	C₄⋊C₈	—	67	2	no	—
102	C₄⋊C₈	C₂×D₆₇	1	2	yes	—
103	C₂²⋊C₈	—	67	2	no	—
104	C₂²⋊C₈	C₂×D₆₇	1	2	yes	—
105	#97	—	67	4	no	—
106	#97	C₂×D₆₇	1	2	yes	—
107	C₄×C₈	—	67	4	no	—
108	C₄×C₈	—	67	1	no	—
109	C₄×C₈	—	67	1	no	—
110	C₂×Q₈	—	67	2	no	—
111	C₂×Q₈	—	67	2	no	—
112	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
113	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
115	C₂×C₈	—	67	2	no	—
116	C₂×C₈	—	67	2	no	—
117	C₂×C₈	C₂²×D₆₇	1	4	yes	—
118	C₄⋊C₄	—	67	2	no	—
119	C₄⋊C₄	—	67	2	no	—
120	C₄⋊C₄	—	67	4	no	—
121	C₄⋊C₄	—	67	4	no	—
122	C₄⋊C₄	—	67	2	no	—
123	C₄⋊C₄	—	67	2	no	—
124	C₂²⋊C₄	—	67	4	no	—
125	C₄²	—	67	4	no	—
126	C₄²	C₄×D₆₇	1	1	yes	characteristic
127	C₄²	C₄×D₆₇	1	1	yes	characteristic
128	C₄²	C₂²×D₆₇	1	1	yes	characteristic
129	C₄²	C₂²×D₆₇	1	1	yes	characteristic
130	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
131	Q₈	—	268	2	no	—
132	Q₈	—	134	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	67	4	no	—
137	C₂×C₄	—	67	4	no	—
138	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	134	2	no	—
146	C₈	—	134	2	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	134	4	no	—
156	C₄	—	134	4	no	—
157	C₄	#14524051	1	2	yes	—
158	C₄	#14524090	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#16863873	1	1	yes	characteristic
161	C₂	C₂×#14524051	1	1	yes	characteristic
162	C₂	#16864162	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863953

C₂

yes

maximal, characteristic

#16863524

C₂

yes

maximal, complemented

#16863523

C₂

yes

maximal, complemented

#16863520

C₂

yes

maximal, complemented

#16863454

C₂

yes

maximal

#16863449

C₂

yes

maximal, characteristic

#432×C₆₇

C₂

yes

maximal, characteristic

#14524043

C₄

yes

characteristic

#14524043

C₄

yes

characteristic

#14524035

C₄

yes

—

#14524022

C₄

yes

—

#14524022

C₄

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

#14524002

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

Dic₆₇⋊C₈

C₂²

yes

—

Dic₆₇⋊C₈

C₂²

yes

—

C₂²⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523953

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

—

C₂²⋊C₈×C₆₇

C₂²

yes

—

#97×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

C₂×C₄

yes

—

C₂×Dic₁₃₄

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

Dic₆₇⋊C₄

C₂×C₄

yes

—

Dic₆₇⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

—

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

C₂²×C₄

yes

—

C₂×Dic₆₇

C₂²×C₄

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#131

yes

—

C₄×C₆₇

2₋¹⁺⁴

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#533

—

#446

—

#446

—

C₈×Q₈

—

#433

—

#432

D₆₇

yes

characteristic

#431

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

C₂×D₆₇

yes

—

103

C₂²⋊C₈

—

104

C₂²⋊C₈

C₂×D₆₇

yes

—

105

#97

—

106

#97

C₂×D₆₇

yes

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂²×C₄

C₄×D₆₇

yes

characteristic

113

C₂²×C₄

C₄×D₆₇

yes

characteristic

114

C₂²×C₄

C₂²×D₆₇

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₆₇

yes

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

C₄×D₆₇

yes

characteristic

127

C₄²

C₄×D₆₇

yes

characteristic

128

C₄²

C₂²×D₆₇

yes

characteristic

129

C₄²

C₂²×D₆₇

yes

characteristic

130

C₂³

C₂×C₄×D₆₇

yes

characteristic

131

Q₈

—

268

—

132

Q₈

—

134

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

139

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

140

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

141

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

142

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

143

C₂×C₄

C₂³×D₆₇

yes

characteristic

144

C₈

—

145

C₈

—

134

—

146

C₈

—

134

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂²×C₄×D₆₇

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

134

—

156

C₄

—

134

—

157

C₄

#14524051

yes

—

158

C₄

#14524090

yes

—

159

C₂

—

160

C₂

#16863873

yes

characteristic

161

C₂

C₂×#14524051

yes

characteristic

162

C₂

#16864162

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	#432×C₆₇	C₂
hypercenter	#432	D₆₇
hypocenter	C₆₇	#2985
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #432

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₇

commutator, socle

C₂²×C₆₇

C₂³×C₄

Fitting

#432×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄ ◁ #432

N	H ≅ G/N	type
2. #16863524	C₂	semidirect
3. #16863523	C₂	semidirect
4. #16863520	C₂	semidirect
82. C₆₇	#2985	semidirect

H ≅ G/N

type

2. #16863524

C₂

semidirect

3. #16863523

C₂

semidirect

4. #16863520

C₂

semidirect

82. C₆₇

#2985

semidirect

Subgroups — #22569306

Conjugacy classes of subgroups

Complements