Group #24923959 — subgroups

Subgroups — #24923959

Order 10688.

214 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 99 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	#19128825	C₂	1	2	yes	maximal, complemented
2	#19128824	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128822	C₂	1	2	yes	maximal, complemented
4	#19128813	C₂	1	2	yes	maximal
5	#19128811	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128811	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Dic₃₃₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128791	C₂	1	1	yes	maximal, characteristic, complemented
9	#19128788	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊Dic₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₄×C₁₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₃₃₄	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₆₇	C₂²	1	2	yes	—
14	C₄⋊Dic₁₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₆₇	C₂²	1	2	yes	—
17	C₄⋊Dic₁₆₇	C₂²	1	1	yes	characteristic
18	Dic₁₆₇⋊C₄	C₂²	1	2	yes	—
19	Dic₁₆₇⋊C₄	C₂²	1	2	yes	—
20	Dic₁₆₇⋊C₄	C₂²	1	2	yes	—
21	Dic₁₆₇⋊C₄	C₂²	1	2	yes	—
22	Dic₁₆₇⋊C₄	C₂²	1	2	yes	—
23	C₂²⋊Dic₁₆₇	C₂²	1	2	yes	—
24	C₂²⋊Dic₁₆₇	C₂²	1	2	yes	—
25	C₂²⋊Dic₁₆₇	C₂²	1	2	yes	—
26	C₄×Dic₁₆₇	C₂²	1	2	yes	—
27	C₄×Dic₁₆₇	C₂²	1	2	yes	—
28	C₂×D₄×C₁₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄×C₁₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄×C₁₆₇	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
32	C₂²⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
33	C₂²⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
34	C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
35	Dic₃₃₄	—	2	2	no	—
36	C₂×Dic₁₆₇	—	2	4	no	—
37	C₂×Dic₁₆₇	C₂³	1	2	yes	—
38	C₂×Dic₁₆₇	C₂³	1	2	yes	—
39	C₂×Dic₁₆₇	C₂³	1	2	yes	—
40	C₂×Dic₁₆₇	C₂³	1	2	yes	—
41	C₂³×C₁₆₇	C₂³	1	1	yes	characteristic
42	C₂³×C₁₆₇	C₂³	1	1	yes	characteristic
43	D₄×C₁₆₇	—	2	2	no	—
44	C₂×C₄×C₁₆₇	—	2	2	no	—
45	C₂×C₄×C₁₆₇	—	2	2	no	—
46	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₁₆₇	—	2	1	no	—
52	C₂²×C₁₆₇	—	2	1	no	—
53	C₂²×C₁₆₇	—	2	1	no	—
54	C₂²×C₁₆₇	—	2	1	no	—
55	C₂²×C₁₆₇	—	2	1	no	—
56	C₂²×C₁₆₇	#41	1	2	yes	—
57	C₂²×C₁₆₇	C₂⁴	1	1	yes	characteristic
58	C₄×C₁₆₇	—	2	1	no	—
59	C₄×C₁₆₇	—	2	1	no	—
60	C₄×C₁₆₇	—	2	1	no	—
61	C₄×C₁₆₇	—	2	1	no	—
62	C₄×C₁₆₇	#41	1	2	yes	—
63	Dic₁₆₇	—	2	2	no	—
64	Dic₁₆₇	—	2	2	no	—
65	Dic₁₆₇	—	2	2	no	—
66	Dic₁₆₇	—	2	2	no	—
67	C₂×C₁₆₇	—	2	2	no	—
68	C₂×C₁₆₇	—	4	1	no	—
69	C₂×C₁₆₇	C₂×#41	1	1	yes	characteristic
70	C₂×C₁₆₇	C₂×#41	1	1	yes	characteristic
71	C₂×C₁₆₇	2₋¹⁺⁴	1	1	yes	characteristic
72	C₁₆₇	#552	1	1	yes	characteristic, complemented, Sylow
73	#552	—	167	1	no	maximal, Sylow
74	#126	—	167	2	no	—
75	#125	—	167	2	no	—
76	#125	—	167	1	no	—
77	#123	—	167	2	no	—
78	C₂²⋊Q₈	—	167	1	no	—
79	C₂²⋊Q₈	—	167	1	no	—
80	C₄×Q₈	—	167	1	no	—
81	C₄×D₄	D₁₆₇	1	1	yes	characteristic
82	#117	—	167	1	no	—
83	#117	—	167	2	no	—
84	C₂×C₄⋊C₄	—	167	1	no	—
85	C₂×Q₈	—	167	1	no	—
86	C₂×D₄	C₂×D₁₆₇	1	1	yes	characteristic
87	C₂²×C₄	—	167	2	no	—
88	C₂²×C₄	C₂×D₁₆₇	1	1	yes	characteristic
89	C₂²×C₄	C₂×D₁₆₇	1	1	yes	characteristic
90	C₄⋊C₄	—	167	1	no	—
91	C₄⋊C₄	—	167	1	no	—
92	C₄⋊C₄	—	167	2	no	—
93	C₄⋊C₄	—	167	2	no	—
94	C₄⋊C₄	—	167	1	no	—
95	C₄⋊C₄	—	167	2	no	—
96	C₄⋊C₄	—	167	2	no	—
97	C₄⋊C₄	—	167	2	no	—
98	C₄⋊C₄	—	167	2	no	—
99	C₄⋊C₄	C₂×D₁₆₇	1	1	yes	characteristic
100	C₂²⋊C₄	—	167	2	no	—
101	C₂²⋊C₄	—	167	2	no	—
102	C₂²⋊C₄	—	167	2	no	—
103	C₂²⋊C₄	C₂×D₁₆₇	1	1	yes	characteristic
104	C₂²⋊C₄	C₂×D₁₆₇	1	1	yes	characteristic
105	C₄²	—	167	2	no	—
106	C₄²	—	167	2	no	—
107	C₄²	C₂×D₁₆₇	1	1	yes	characteristic
108	C₂³	C₂²×D₁₆₇	1	1	yes	characteristic
109	C₂³	C₂²×D₁₆₇	1	1	yes	characteristic
110	Q₈	—	334	2	no	—
111	D₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	334	4	no	—
115	C₂×C₄	—	167	2	no	—
116	C₂×C₄	—	167	2	no	—
117	C₂×C₄	—	167	2	no	—
118	C₂×C₄	—	167	2	no	—
119	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
122	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
123	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	#14581950	1	2	yes	—
130	C₂²	C₂³×D₁₆₇	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	334	2	no	—
136	C₄	—	334	2	no	—
137	C₄	—	334	2	no	—
138	C₄	—	334	2	no	—
139	C₄	#14581949	1	2	yes	—
140	C₂	—	2	2	no	—
141	C₂	—	4	1	no	—
142	C₂	C₂×#14581949	1	1	yes	characteristic
143	C₂	C₂×#14581950	1	1	yes	characteristic
144	C₂	#19128879	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128825

C₂

yes

maximal, complemented

#19128824

C₂

yes

maximal, characteristic, complemented

#19128822

C₂

yes

maximal, complemented

#19128813

C₂

yes

maximal

#19128811

C₂

yes

maximal, characteristic, complemented

#19128811

C₂

yes

maximal, characteristic, complemented

C₄×Dic₃₃₄

C₂

yes

maximal, characteristic, complemented

#19128791

C₂

yes

maximal, characteristic, complemented

#19128788

C₂

yes

maximal, complemented

C₂×C₄⋊Dic₁₆₇

C₂

yes

maximal, characteristic, complemented

C₄×D₄×C₁₆₇

C₂

yes

maximal, characteristic

C₂×Dic₃₃₄

C₂²

yes

characteristic

C₂²×Dic₁₆₇

C₂²

yes

—

C₄⋊Dic₁₆₇

C₂²

yes

characteristic

C₄⋊Dic₁₆₇

C₂²

yes

characteristic

C₄⋊Dic₁₆₇

C₂²

yes

—

C₄⋊Dic₁₆₇

C₂²

yes

characteristic

Dic₁₆₇⋊C₄

C₂²

yes

—

Dic₁₆₇⋊C₄

C₂²

yes

—

Dic₁₆₇⋊C₄

C₂²

yes

—

Dic₁₆₇⋊C₄

C₂²

yes

—

Dic₁₆₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₆₇

C₂²

yes

—

C₂²⋊Dic₁₆₇

C₂²

yes

—

C₂²⋊Dic₁₆₇

C₂²

yes

—

C₄×Dic₁₆₇

C₂²

yes

—

C₄×Dic₁₆₇

C₂²

yes

—

C₂×D₄×C₁₆₇

C₂²

yes

characteristic

C₂²×C₄×C₁₆₇

C₂²

yes

characteristic

C₂²×C₄×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₆₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₆₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₆₇

C₂²

yes

characteristic

C₄²×C₁₆₇

C₂²

yes

characteristic

Dic₃₃₄

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

C₂³

yes

—

C₂×Dic₁₆₇

C₂³

yes

—

C₂×Dic₁₆₇

C₂³

yes

—

C₂×Dic₁₆₇

C₂³

yes

—

C₂³×C₁₆₇

C₂³

yes

characteristic

C₂³×C₁₆₇

C₂³

yes

characteristic

D₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

#41

yes

—

C₂²×C₁₆₇

C₂⁴

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

#41

yes

—

Dic₁₆₇

—

Dic₁₆₇

—

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

C₂×#41

yes

characteristic

C₂×C₁₆₇

C₂×#41

yes

characteristic

C₂×C₁₆₇

2₋¹⁺⁴

yes

characteristic

C₁₆₇

#552

yes

characteristic, complemented, Sylow

#552

—

167

maximal, Sylow

#126

—

167

—

#125

—

167

—

#125

—

167

—

#123

—

167

—

C₂²⋊Q₈

—

167

—

C₂²⋊Q₈

—

167

—

C₄×Q₈

—

167

—

C₄×D₄

D₁₆₇

yes

characteristic

#117

—

167

—

#117

—

167

—

C₂×C₄⋊C₄

—

167

—

C₂×Q₈

—

167

—

C₂×D₄

C₂×D₁₆₇

yes

characteristic

C₂²×C₄

—

167

—

C₂²×C₄

C₂×D₁₆₇

yes

characteristic

C₂²×C₄

C₂×D₁₆₇

yes

characteristic

C₄⋊C₄

—

167

—

C₄⋊C₄

—

167

—

C₄⋊C₄

—

167

—

C₄⋊C₄

—

167

—

C₄⋊C₄

—

167

—

C₄⋊C₄

—

167

—

C₄⋊C₄

—

167

—

C₄⋊C₄

—

167

—

C₄⋊C₄

—

167

—

C₄⋊C₄

C₂×D₁₆₇

yes

characteristic

100

C₂²⋊C₄

—

167

—

101

C₂²⋊C₄

—

167

—

102

C₂²⋊C₄

—

167

—

103

C₂²⋊C₄

C₂×D₁₆₇

yes

characteristic

104

C₂²⋊C₄

C₂×D₁₆₇

yes

characteristic

105

C₄²

—

167

—

106

C₄²

—

167

—

107

C₄²

C₂×D₁₆₇

yes

characteristic

108

C₂³

C₂²×D₁₆₇

yes

characteristic

109

C₂³

C₂²×D₁₆₇

yes

characteristic

110

Q₈

—

334

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

334

—

115

C₂×C₄

—

167

—

116

C₂×C₄

—

167

—

117

C₂×C₄

—

167

—

118

C₂×C₄

—

167

—

119

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

120

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

121

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

122

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

123

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#14581950

yes

—

130

C₂²

C₂³×D₁₆₇

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

334

—

136

C₄

—

334

—

137

C₄

—

334

—

138

C₄

—

334

—

139

C₄

#14581949

yes

—

140

C₂

—

141

C₂

—

142

C₂

C₂×#14581949

yes

characteristic

143

C₂

C₂×#14581950

yes

characteristic

144

C₂

#19128879

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₆₇
commutator, socle, cosocle	C₂²×C₁₆₇	C₂⁴
Fitting	C₄×D₄×C₁₆₇	C₂
prime radical	C₂×D₄×C₁₆₇	C₂²
hypercenter	C₄×D₄	D₁₆₇
hypocenter	C₁₆₇	#552
LCS	G ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₆₇

commutator, socle, cosocle

C₂²×C₁₆₇

C₂⁴

Fitting

C₄×D₄×C₁₆₇

C₂

prime radical

C₂×D₄×C₁₆₇

C₂²

hypercenter

C₄×D₄

D₁₆₇

C₁₆₇

G ▷ C₂²×C₁₆₇ ▷ C₁₆₇

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #19128825	C₂	semidirect
2. #19128824	C₂	semidirect
3. #19128822	C₂	semidirect
5. #19128811	C₂	semidirect
6. #19128811	C₂	semidirect
7. C₄×Dic₃₃₄	C₂	semidirect
8. #19128791	C₂	semidirect
9. #19128788	C₂	semidirect
10. C₂×C₄⋊Dic₁₆₇	C₂	semidirect
72. C₁₆₇	#552	semidirect

H ≅ G/N

type

1. #19128825

C₂

semidirect

2. #19128824

C₂

semidirect

3. #19128822

C₂

semidirect

5. #19128811

C₂

semidirect

6. #19128811

C₂

semidirect

7. C₄×Dic₃₃₄

C₂

semidirect

8. #19128791

C₂

semidirect

9. #19128788

C₂

semidirect

10. C₂×C₄⋊Dic₁₆₇

C₂

semidirect

72. C₁₆₇

#552

semidirect

Subgroups — #24923959

Conjugacy classes of subgroups

Complements