Group #22570749 — subgroups

Subgroups — #22570749

Order 8576.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#16864024	C₂	1	2	yes	maximal
2	#16863803	C₂	1	4	yes	maximal, complemented
3	#16863800	C₂	1	2	yes	maximal, complemented
4	#16863797	C₂	1	2	yes	maximal, complemented
5	#16863439	C₂	1	2	yes	maximal
6	C₂×#14523977	C₂	1	1	yes	maximal, characteristic
7	#16863351	C₂	1	1	yes	maximal, characteristic
8	#427×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#11761612	C₂²	1	1	yes	characteristic
10	#14524043	—	2	2	no	—
11	#14524042	C₂²	1	4	yes	—
12	#14524035	—	2	2	no	—
13	#14524034	C₂²	1	4	yes	—
14	#14524020	—	2	4	no	—
15	#14524002	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
17	#14523977	C₂²	1	2	yes	—
18	#14523977	C₂²	1	2	yes	—
19	#14523976	C₂²	1	4	yes	—
20	#14523974	C₂²	1	4	yes	—
21	#14523953	C₂²	1	2	yes	—
22	C₈×Dic₆₇	C₂²	1	2	yes	—
23	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#117×C₆₇	C₂²	1	2	yes	—
25	#107×C₆₇	C₂²	1	1	yes	characteristic
26	#107×C₆₇	C₂²	1	1	yes	characteristic
27	#106×C₆₇	C₂²	1	2	yes	—
28	C₂×Dic₁₃₄	—	2	4	no	—
29	C₂²×Dic₆₇	—	2	2	no	—
30	#11761612	Q₈	1	4	yes	complemented
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	D₄	1	1	yes	characteristic
36	C₄⋊Dic₆₇	C₂³	1	4	yes	—
37	Dic₆₇⋊C₄	—	2	4	no	—
38	Dic₆₇⋊C₄	—	2	4	no	—
39	C₂²⋊Dic₆₇	D₄	1	2	yes	—
40	C₄×Dic₆₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×C₈×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	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
47	C₂²⋊C₄×C₆₇	—	2	2	no	—
48	C₄²×C₆₇	—	2	2	no	—
49	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	—	2	2	no	—
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₆₇	—	2	4	no	—
57	C₂×Dic₆₇	—	8	2	no	—
58	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
59	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
60	Dic₁₃₄	—	8	4	no	—
61	C₈×C₆₇	C₂×Q₈	1	2	yes	—
62	C₈×C₆₇	C₂×Q₈	1	2	yes	—
63	C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
64	C₂²×C₆₇	—	2	1	no	—
65	C₂²×C₆₇	—	2	1	no	—
66	C₂²×C₆₇	C₄⋊Q₈	1	2	yes	—
67	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₆₇	—	4	2	no	—
69	C₄×C₆₇	—	4	2	no	—
70	C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
71	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
72	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
73	Dic₆₇	—	2	2	no	—
74	Dic₆₇	—	2	2	no	—
75	Dic₆₇	—	4	4	no	—
76	C₂×C₆₇	—	2	2	no	—
77	C₂×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₆₇	#576	1	1	yes	characteristic
79	C₂×C₆₇	#578	1	1	yes	characteristic
80	#3164	—	67	1	no	maximal, Sylow
81	C₆₇	#3164	1	1	yes	characteristic, complemented, Sylow
82	#544	—	67	2	no	—
83	#501	—	67	4	no	—
84	#500	—	67	2	no	—
85	#499	—	67	2	no	—
86	#428	—	67	2	no	—
87	#427	D₆₇	1	1	yes	characteristic
88	C₂×#107	—	67	1	no	—
89	#405	—	67	1	no	—
90	C₂×M₁₆	—	67	1	no	—
91	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
92	C₄⋊Q₈	—	67	4	no	—
93	C₄⋊Q₈	—	134	2	no	—
94	#125	—	67	4	no	—
95	#125	—	134	2	no	—
96	C₂²⋊Q₈	—	134	4	no	—
97	#117	—	67	1	no	—
98	#117	C₂×D₆₇	1	2	yes	—
99	C₂×C₄⋊C₄	—	67	2	no	—
100	#107	—	67	2	no	—
101	#107	—	67	2	no	—
102	#107	—	67	4	no	—
103	#107	C₂×D₆₇	1	1	yes	characteristic
104	#107	C₂×D₆₇	1	1	yes	characteristic
105	#106	—	67	4	no	—
106	#106	C₂×D₆₇	1	2	yes	—
107	#97	—	67	2	no	—
108	C₄×C₈	—	67	2	no	—
109	C₂×Q₈	—	134	4	no	—
110	C₂²×C₄	—	134	2	no	—
111	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
112	M₁₆	—	67	4	no	—
113	C₂×C₈	—	67	2	no	—
114	C₂×C₈	Dic₁₃₄	1	4	yes	complemented
115	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
117	C₄⋊C₄	—	67	1	no	—
118	C₄⋊C₄	—	67	1	no	—
119	C₄⋊C₄	—	67	4	no	—
120	C₄⋊C₄	—	134	4	no	—
121	C₄⋊C₄	—	134	4	no	—
122	C₄⋊C₄	—	134	4	no	—
123	C₄⋊C₄	—	134	2	no	—
124	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
125	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	67	2	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	67	2	no	—
130	C₂³	D₄×D₆₇	1	1	yes	characteristic
131	Q₈	—	536	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	536	2	no	—
135	C₂×C₄	—	67	2	no	—
136	C₂×C₄	—	67	2	no	—
137	C₂×C₄	—	134	4	no	—
138	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
139	C₂×C₄	Q₈×D₆₇	1	2	yes	—
140	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
141	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
142	C₈	—	67	4	no	—
143	C₈	C₂×Dic₁₃₄	1	2	yes	—
144	C₈	C₂×Dic₁₃₄	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#14524042	1	2	yes	—
148	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
149	C₄	—	4	2	no	—
150	C₄	—	4	2	no	—
151	C₄	—	268	4	no	—
152	C₄	—	134	2	no	—
153	C₄	—	134	2	no	—
154	C₄	#14524042	1	2	yes	—
155	C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
156	C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	C₂×#14524042	1	1	yes	characteristic
159	C₂	#16864198	1	1	yes	characteristic
160	C₂	#16864210	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864024

C₂

yes

maximal

#16863803

C₂

yes

maximal, complemented

#16863800

C₂

yes

maximal, complemented

#16863797

C₂

yes

maximal, complemented

#16863439

C₂

yes

maximal

C₂×#14523977

C₂

yes

maximal, characteristic

#16863351

C₂

yes

maximal, characteristic

#427×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

characteristic

#14524043

—

#14524042

C₂²

yes

—

#14524035

—

#14524034

C₂²

yes

—

#14524020

—

#14524002

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

#14523977

C₂²

yes

—

#14523977

C₂²

yes

—

#14523976

C₂²

yes

—

#14523974

C₂²

yes

—

#14523953

C₂²

yes

—

C₈×Dic₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

—

#107×C₆₇

C₂²

yes

characteristic

#107×C₆₇

C₂²

yes

characteristic

#106×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

#11761612

Q₈

yes

complemented

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

C₄⋊Dic₆₇

D₄

yes

characteristic

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

D₄

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

complemented

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

C₂×D₄

yes

—

Dic₁₃₄

—

C₈×C₆₇

C₂×Q₈

yes

—

C₈×C₆₇

C₂×Q₈

yes

—

C₆₇⋊C₈

C₂×Q₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄⋊Q₈

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄⋊Q₈

yes

—

C₄×C₆₇

C₂²×Q₈

yes

characteristic

C₄×C₆₇

C₂²×Q₈

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×C₄⋊Q₈

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₆₇

yes

characteristic

C₂×#107

—

#405

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

#125

—

#125

—

134

—

C₂²⋊Q₈

—

134

—

#117

—

#117

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

—

100

#107

—

101

#107

—

102

#107

—

103

#107

C₂×D₆₇

yes

characteristic

104

#107

C₂×D₆₇

yes

characteristic

105

#106

—

106

#106

C₂×D₆₇

yes

—

107

#97

—

108

C₄×C₈

—

109

C₂×Q₈

—

134

—

110

C₂²×C₄

—

134

—

111

C₂²×C₄

C₂²×D₆₇

yes

characteristic

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

Dic₁₃₄

yes

complemented

115

C₂×C₈

C₂²×D₆₇

yes

characteristic

116

C₂×C₈

C₂²×D₆₇

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

134

—

121

C₄⋊C₄

—

134

—

122

C₄⋊C₄

—

134

—

123

C₄⋊C₄

—

134

—

124

C₄⋊C₄

C₂²×D₆₇

yes

—

125

C₄⋊C₄

C₂²×D₆₇

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

—

130

C₂³

D₄×D₆₇

yes

characteristic

131

Q₈

—

536

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

536

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

134

—

138

C₂×C₄

D₄×D₆₇

yes

characteristic

139

C₂×C₄

Q₈×D₆₇

yes

—

140

C₂×C₄

C₂×Dic₁₃₄

yes

—

141

C₂×C₄

C₂³×D₆₇

yes

characteristic

142

C₈

—

143

C₈

C₂×Dic₁₃₄

yes

—

144

C₈

C₂×Dic₁₃₄

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

#14524042

yes

—

148

C₂²

C₂×D₄×D₆₇

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

268

—

152

C₄

—

134

—

153

C₄

—

134

—

154

C₄

#14524042

yes

—

155

C₄

C₂×Q₈×D₆₇

yes

characteristic

156

C₄

C₂²×Dic₁₃₄

yes

characteristic

157

C₂

—

158

C₂

C₂×#14524042

yes

characteristic

159

C₂

#16864198

yes

characteristic

160

C₂

#16864210

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#427×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#427	D₆₇
hypocenter	C₆₇	#3164
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#427×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #16863803	C₂	semidirect
3. #16863800	C₂	semidirect
4. #16863797	C₂	semidirect
30. #11761612	Q₈	semidirect
42. C₂×C₈×C₆₇	Q₈	semidirect
81. C₆₇	#3164	semidirect
114. C₂×C₈	Dic₁₃₄	semidirect

H ≅ G/N

type

2. #16863803

C₂

semidirect

3. #16863800

C₂

semidirect

4. #16863797

C₂

semidirect

30. #11761612

Q₈

semidirect

42. C₂×C₈×C₆₇

Q₈

semidirect

81. C₆₇

#3164

semidirect

114. C₂×C₈

Dic₁₃₄

semidirect

Subgroups — #22570749

Conjugacy classes of subgroups

Complements