Group #22571104 — subgroups

Subgroups — #22571104

Order 8576.

378 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16863967	C₂	1	2	yes	maximal
2	#16863738	C₂	1	4	yes	maximal, complemented
3	#16863654	C₂	1	2	yes	maximal, complemented
4	#16863637	C₂	1	2	yes	maximal, complemented
5	#16863554	C₂	1	4	yes	maximal, complemented
6	#408×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₃₄	C₂²	1	2	yes	—
8	C₂×Dic₂₆₈	C₂²	1	2	yes	—
9	C₂×Dic₂₆₈	C₂²	1	2	yes	—
10	#14524041	—	2	2	no	—
11	#14524030	—	2	2	no	—
12	#14524024	—	2	2	no	—
13	#14524024	—	2	2	no	—
14	#14524022	C₂²	1	4	yes	—
15	#14524022	C₂²	1	4	yes	—
16	#14524002	C₂²	1	2	yes	—
17	#14523977	C₂²	1	2	yes	—
18	#14523975	C₂²	1	2	yes	—
19	#14523968	C₂²	1	4	yes	—
20	#14523968	C₂²	1	4	yes	—
21	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
25	C₂×Dic₁₃₄	—	2	2	no	—
26	C₂×Dic₁₃₄	—	4	2	no	—
27	C₂×Dic₁₃₄	—	4	2	no	—
28	C₂×Dic₁₃₄	C₂³	1	4	yes	—
29	C₂²×Dic₆₇	—	2	2	no	—
30	Dic₂₆₈	—	4	2	no	—
31	Dic₂₆₈	—	4	2	no	—
32	C₄⋊Dic₆₇	C₂³	1	4	yes	—
33	Dic₆₇⋊C₄	—	2	4	no	—
34	Dic₆₇⋊C₄	—	2	4	no	—
35	C₂²⋊Dic₆₇	—	2	2	no	—
36	C₂²⋊Dic₆₇	—	2	4	no	—
37	C₂²⋊Dic₆₇	—	2	4	no	—
38	C₄×Dic₆₇	—	2	2	no	—
39	#41×C₆₇	—	4	1	no	—
40	#41×C₆₇	—	4	1	no	—
41	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂×D₄×C₆₇	D₄	1	1	yes	characteristic
43	C₂×D₄×C₆₇	D₄	1	2	yes	—
44	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
46	M₁₆×C₆₇	—	4	1	no	—
47	C₂×C₈×C₆₇	—	4	1	no	—
48	C₂×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂³×C₆₇	C₂×D₄	1	2	yes	—
52	Q₈×C₆₇	—	4	1	no	—
53	D₄×C₆₇	—	4	2	no	—
54	D₄×C₆₇	—	4	1	no	—
55	C₂×C₄×C₆₇	—	2	1	no	—
56	C₂×C₄×C₆₇	—	2	1	no	—
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₆₇	—	2	4	no	—
63	C₂×Dic₆₇	—	2	4	no	—
64	C₂×Dic₆₇	—	4	4	no	—
65	Dic₁₃₄	—	2	4	no	—
66	Dic₁₃₄	—	2	4	no	—
67	Dic₁₃₄	—	4	4	no	—
68	C₈×C₆₇	—	2	2	no	—
69	C₈×C₆₇	—	2	2	no	—
70	C₂²×C₆₇	—	2	2	no	—
71	C₂²×C₆₇	—	2	1	no	—
72	C₂²×C₆₇	—	2	2	no	—
73	C₂²×C₆₇	—	2	4	no	—
74	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
75	C₄×C₆₇	—	2	2	no	—
76	C₄×C₆₇	—	2	1	no	—
77	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
78	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
79	Dic₆₇	—	4	4	no	—
80	Dic₆₇	—	4	4	no	—
81	C₂×C₆₇	—	4	2	no	—
82	C₂×C₆₇	—	4	1	no	—
83	C₂×C₆₇	#534	1	1	yes	characteristic
84	C₂×C₆₇	#578	1	2	yes	—
85	#3202	—	67	1	no	maximal, Sylow
86	C₆₇	#3202	1	1	yes	characteristic, complemented, Sylow
87	#536	—	67	2	no	—
88	#485	—	67	4	no	—
89	#470	—	67	2	no	—
90	#467	—	67	2	no	—
91	#451	—	67	4	no	—
92	#408	D₆₇	1	1	yes	characteristic
93	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
94	C₂²×Q₈	—	67	2	no	—
95	C₂×Q₁₆	—	67	2	no	—
96	C₂×Q₁₆	—	67	2	no	—
97	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
98	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
99	C₄⋊Q₈	—	134	2	no	—
100	#124	—	134	2	no	—
101	#123	—	134	2	no	—
102	#123	—	134	2	no	—
103	C₂²⋊Q₈	—	67	4	no	—
104	C₂²⋊Q₈	—	67	4	no	—
105	#117	—	67	2	no	—
106	#107	—	67	2	no	—
107	#106	—	67	2	no	—
108	#103	—	67	4	no	—
109	#103	—	67	4	no	—
110	C₂²⋊C₈	C₂×D₆₇	1	4	yes	—
111	#41	—	4	1	no	—
112	#41	—	4	1	no	—
113	C₂×Q₈	—	268	2	no	—
114	C₂×Q₈	—	268	2	no	—
115	C₂×Q₈	—	67	4	no	—
116	C₂×Q₈	—	134	2	no	—
117	C₂×Q₈	D₂₆₈	1	1	yes	characteristic
118	C₂×D₄	D₂₆₈	1	1	yes	characteristic
119	C₂×D₄	D₂₆₈	1	2	yes	—
120	C₂²×C₄	—	134	2	no	—
121	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
122	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
123	Q₁₆	—	268	2	no	—
124	Q₁₆	—	268	2	no	—
125	M₁₆	—	4	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	C₂²×D₆₇	1	2	yes	—
128	C₂×C₈	C₂²×D₆₇	1	2	yes	—
129	C₄⋊C₄	—	67	4	no	—
130	C₄⋊C₄	—	134	4	no	—
131	C₄⋊C₄	—	134	4	no	—
132	C₂²⋊C₄	—	134	2	no	—
133	C₂²⋊C₄	—	134	4	no	—
134	C₂²⋊C₄	—	134	4	no	—
135	C₄²	—	134	2	no	—
136	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
137	C₂³	C₂×D₂₆₈	1	2	yes	—
138	Q₈	—	4	1	no	—
139	Q₈	—	268	4	no	—
140	Q₈	—	134	4	no	—
141	Q₈	—	134	4	no	—
142	D₄	—	4	2	no	—
143	D₄	—	4	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	268	4	no	—
149	C₂×C₄	—	134	4	no	—
150	C₂×C₄	—	134	4	no	—
151	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
152	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
153	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	4	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	1	no	—
163	C₄	—	268	4	no	—
164	C₄	—	268	4	no	—
165	C₄	#14524088	1	1	yes	characteristic
166	C₄	#14524088	1	1	yes	characteristic
167	C₂	—	4	2	no	—
168	C₂	—	4	1	no	—
169	C₂	#16863958	1	1	yes	characteristic
170	C₂	#16864213	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863967

C₂

yes

maximal

#16863738

C₂

yes

maximal, complemented

#16863654

C₂

yes

maximal, complemented

#16863637

C₂

yes

maximal, complemented

#16863554

C₂

yes

maximal, complemented

#408×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

C₂×Dic₂₆₈

C₂²

yes

—

C₂×Dic₂₆₈

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#14524022

C₂²

yes

—

#14524002

C₂²

yes

—

#14523977

C₂²

yes

—

#14523975

C₂²

yes

—

#14523968

C₂²

yes

—

#14523968

C₂²

yes

—

C₂×#41×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂²⋊C₈×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂³×C₆₇

C₂×D₄

yes

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×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

characteristic

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#534

yes

characteristic

C₂×C₆₇

#578

yes

—

#3202

—

maximal, Sylow

C₆₇

#3202

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₆₇

yes

characteristic

C₂×#41

C₂×D₆₇

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₄⋊Q₈

—

134

—

100

#124

—

134

—

101

#123

—

134

—

102

#123

—

134

—

103

C₂²⋊Q₈

—

104

C₂²⋊Q₈

—

105

#117

—

106

#107

—

107

#106

—

108

#103

—

109

#103

—

110

C₂²⋊C₈

C₂×D₆₇

yes

—

111

#41

—

112

#41

—

113

C₂×Q₈

—

268

—

114

C₂×Q₈

—

268

—

115

C₂×Q₈

—

116

C₂×Q₈

—

134

—

117

C₂×Q₈

D₂₆₈

yes

characteristic

118

C₂×D₄

D₂₆₈

yes

characteristic

119

C₂×D₄

D₂₆₈

yes

—

120

C₂²×C₄

—

134

—

121

C₂²×C₄

C₂²×D₆₇

yes

—

122

C₂²×C₄

C₂²×D₆₇

yes

characteristic

123

Q₁₆

—

268

—

124

Q₁₆

—

268

—

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

C₂²×D₆₇

yes

—

128

C₂×C₈

C₂²×D₆₇

yes

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

134

—

131

C₄⋊C₄

—

134

—

132

C₂²⋊C₄

—

134

—

133

C₂²⋊C₄

—

134

—

134

C₂²⋊C₄

—

134

—

135

C₄²

—

134

—

136

C₂³

C₂×D₂₆₈

yes

characteristic

137

C₂³

C₂×D₂₆₈

yes

—

138

Q₈

—

139

Q₈

—

268

—

140

Q₈

—

134

—

141

Q₈

—

134

—

142

D₄

—

143

D₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

268

—

149

C₂×C₄

—

134

—

150

C₂×C₄

—

134

—

151

C₂×C₄

C₂×D₂₆₈

yes

—

152

C₂×C₄

C₂×D₂₆₈

yes

characteristic

153

C₂×C₄

C₂³×D₆₇

yes

characteristic

154

C₈

—

155

C₈

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

C₂²×D₂₆₈

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

268

—

164

C₄

—

268

—

165

C₄

#14524088

yes

characteristic

166

C₄

#14524088

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#16863958

yes

characteristic

170

C₂

#16864213

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#408×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#408	D₆₇
hypocenter	C₆₇	#3202
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #408

subgroup

G/H

center

C₂²

C₂²×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#408×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ #408

N	H ≅ G/N	type
2. #16863738	C₂	semidirect
3. #16863654	C₂	semidirect
4. #16863637	C₂	semidirect
5. #16863554	C₂	semidirect
86. C₆₇	#3202	semidirect

H ≅ G/N

type

2. #16863738

C₂

semidirect

3. #16863654

C₂

semidirect

4. #16863637

C₂

semidirect

5. #16863554

C₂

semidirect

86. C₆₇

#3202

semidirect

Subgroups — #22571104

Conjugacy classes of subgroups

Complements