Group #14517538 — subgroups

Subgroups — C₄⋊#92305 #14517538

Order 1944.

364 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 60 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	C₂×#92305	C₂	1	2	yes	maximal, complemented
2	C₄×#6893	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₃⋊#272)	C₃	1	1	yes	maximal, characteristic
4	C₉×C₄⋊(C₃⋊D₃)	—	3	1	no	maximal
5	C₄⋊((C₃⋊D₃)⋊C₉)	—	3	3	no	maximal
6	C₂×#6893	C₂²	1	1	yes	characteristic
7	#92305	—	2	2	no	—
8	C₂×C₃⋊#272	C₂×C₃	1	2	yes	—
9	C₂×C₉×C₃⋊D₃	—	3	2	no	—
10	C₃×C₄×3₋¹⁺²	—	2	1	no	—
11	C₃×C₄×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
12	C₃²×C₄×C₉	D₃	1	1	yes	characteristic, complemented
13	C₂×(C₃⋊D₃)⋊C₉	—	3	6	no	—
14	C₄×C₃²⋊C₉	—	2	3	no	—
15	C₄×C₃²⋊C₉	D₃	1	3	yes	complemented
16	#6893	D₄	1	1	yes	characteristic, complemented, Sylow
17	C₃×C₄⋊(C₃⋊D₃)	—	3	1	no	—
18	C₃×C₄⋊(C₃⋊D₃)	—	3	3	no	—
19	C₄⋊#272	—	9	3	no	—
20	C₉×D₁₂	—	9	3	no	—
21	C₉×D₁₂	—	9	1	no	—
22	C₂×C₃×3₋¹⁺²	—	2	1	no	—
23	C₂×C₃×H(3,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
24	C₂×C₃²×C₉	C₂×D₃	1	1	yes	characteristic
25	C₃⋊#272	—	2	2	no	—
26	C₉×C₃⋊D₃	—	6	2	no	—
27	C₂×C₃²⋊C₉	—	2	3	no	—
28	C₂×C₃²⋊C₉	C₂×D₃	1	3	yes	—
29	(C₃⋊D₃)⋊C₉	—	6	6	no	—
30	C₂×C₃×C₃⋊D₃	—	3	2	no	—
31	C₂×C₃×C₃⋊D₃	—	3	6	no	—
32	C₃³×C₄	C₃×D₃	1	3	yes	—
33	C₃³×C₄	C₃×D₃	1	1	yes	characteristic
34	C₂×#272	—	9	6	no	—
35	C₂×D₃×C₉	—	9	6	no	—
36	C₂×D₃×C₉	—	9	2	no	—
37	C₄×3₋¹⁺²	—	6	3	no	—
38	C₄×H(3,𝔽₃)	—	3	3	no	—
39	C₃×C₄×C₉	—	2	1	no	—
40	C₃×C₄×C₉	—	2	3	no	—
41	C₃×C₄×C₉	—	3	3	no	—
42	C₃×C₄×C₉	C₃⋊D₃	1	1	yes	characteristic
43	C₃×3₋¹⁺²	—	2	1	no	—
44	C₃×H(3,𝔽₃)	C₃×D₄	1	1	yes	characteristic
45	C₃²×C₉	D₁₂	1	1	yes	characteristic, complemented
46	C₃²⋊C₉	—	2	3	no	—
47	C₃²⋊C₉	D₁₂	1	3	yes	complemented
48	C₄⋊(C₃⋊D₃)	—	3	1	no	—
49	C₄⋊(C₃⋊D₃)	—	3	3	no	—
50	C₃×D₁₂	—	9	3	no	—
51	C₃×D₁₂	—	9	1	no	—
52	C₃×D₁₂	—	27	3	no	—
53	D₄×C₉	—	27	1	no	—
54	C₂×C₃³	C₂×C₃×D₃	1	3	yes	—
55	C₂×C₃³	C₂×C₃×D₃	1	1	yes	characteristic
56	C₃×C₃⋊D₃	—	6	2	no	—
57	C₃×C₃⋊D₃	—	6	6	no	—
58	C₂×3₋¹⁺²	—	6	3	no	—
59	C₂×H(3,𝔽₃)	—	3	3	no	—
60	C₂×C₃×C₉	—	2	1	no	—
61	C₂×C₃×C₉	—	2	3	no	—
62	C₂×C₃×C₉	—	3	3	no	—
63	C₂×C₃×C₉	C₂×C₃⋊D₃	1	1	yes	characteristic
64	#272	—	18	6	no	—
65	D₃×C₉	—	18	6	no	—
66	D₃×C₉	—	18	2	no	—
67	C₂×C₃⋊D₃	—	3	2	no	—
68	C₂×C₃⋊D₃	—	3	6	no	—
69	C₂×C₃×D₃	—	9	6	no	—
70	C₂×C₃×D₃	—	9	2	no	—
71	C₂×C₃×D₃	—	27	6	no	—
72	C₃²×C₄	—	3	1	no	—
73	C₃²×C₄	—	3	3	no	—
74	C₃²×C₄	—	3	3	no	—
75	C₃²×C₄	—	6	1	no	—
76	C₃²×C₄	—	6	3	no	—
77	C₃²×C₄	(C₃⋊D₃)⋊C₃	1	3	yes	—
78	C₃²×C₄	C₃×C₃⋊D₃	1	1	yes	characteristic
79	C₂²×C₉	—	27	2	no	—
80	C₄×C₉	—	3	1	no	—
81	C₄×C₉	—	6	1	no	—
82	C₄×C₉	—	6	3	no	—
83	C₃³	C₃×D₁₂	1	3	yes	—
84	C₃³	C₃×D₁₂	1	1	yes	characteristic
85	3₋¹⁺²	—	6	3	no	—
86	H(3,𝔽₃)	—	3	3	no	—
87	C₃×C₉	—	2	1	no	—
88	C₃×C₉	—	2	3	no	—
89	C₃×C₉	—	3	3	no	—
90	C₃×C₉	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
91	C₃×D₄	—	27	1	no	—
92	D₁₂	—	9	3	no	—
93	D₁₂	—	9	1	no	—
94	D₁₂	—	27	3	no	—
95	C₂×C₃²	—	3	1	no	—
96	C₂×C₃²	—	3	3	no	—
97	C₂×C₃²	—	3	3	no	—
98	C₂×C₃²	—	6	1	no	—
99	C₂×C₃²	—	6	3	no	—
100	C₂×C₃²	C₂×(C₃⋊D₃)⋊C₃	1	3	yes	—
101	C₂×C₃²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
102	C₃⋊D₃	—	6	2	no	—
103	C₃⋊D₃	—	6	6	no	—
104	C₃×D₃	—	18	6	no	—
105	C₃×D₃	—	18	2	no	—
106	C₃×D₃	—	54	6	no	—
107	C₂×C₉	—	3	1	no	—
108	C₂×C₉	—	6	1	no	—
109	C₂×C₉	—	6	3	no	—
110	C₂×C₉	—	54	2	no	—
111	C₂²×C₃	—	27	2	no	—
112	C₂×D₃	—	9	6	no	—
113	C₂×D₃	—	9	2	no	—
114	C₂×D₃	—	27	6	no	—
115	C₃×C₄	—	2	1	no	—
116	C₃×C₄	—	3	3	no	—
117	C₃×C₄	—	9	3	no	—
118	C₃×C₄	C₃⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic
119	C₃×C₄	#4208	1	1	yes	characteristic
120	C₃²	—	3	1	no	—
121	C₃²	—	3	3	no	—
122	C₃²	—	3	3	no	—
123	C₃²	—	6	1	no	—
124	C₃²	—	6	3	no	—
125	C₃²	C₄⋊((C₃⋊D₃)⋊C₃)	1	3	yes	—
126	C₃²	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
127	C₉	—	3	1	no	—
128	C₉	—	6	1	no	—
129	C₉	—	6	3	no	—
130	D₄	—	27	1	no	Sylow
131	C₂×C₃	—	2	1	no	—
132	C₂×C₃	—	3	3	no	—
133	C₂×C₃	—	9	3	no	—
134	C₂×C₃	—	54	2	no	—
135	C₂×C₃	C₂×C₃⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic
136	C₂×C₃	C₂×#4208	1	1	yes	characteristic
137	D₃	—	18	6	no	—
138	D₃	—	18	2	no	—
139	D₃	—	54	6	no	—
140	C₂²	—	27	2	no	—
141	C₄	#92305	1	1	yes	characteristic, complemented
142	C₃	—	2	1	no	—
143	C₃	—	3	3	no	—
144	C₃	—	9	3	no	—
145	C₃	C₄⋊(C₃⋊((C₃⋊D₃)⋊C₃))	1	1	yes	characteristic
146	C₃	C₄⋊#4208	1	1	yes	characteristic
147	C₂	—	54	2	no	—
148	C₂	C₂×#92305	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#92305

C₂

yes

maximal, complemented

C₄×#6893

C₂

yes

maximal, characteristic, complemented

C₄⋊(C₃⋊#272)

C₃

yes

maximal, characteristic

C₉×C₄⋊(C₃⋊D₃)

—

maximal

C₄⋊((C₃⋊D₃)⋊C₉)

—

maximal

C₂×#6893

C₂²

yes

characteristic

#92305

—

C₂×C₃⋊#272

C₂×C₃

yes

—

C₂×C₉×C₃⋊D₃

—

C₃×C₄×3₋¹⁺²

—

C₃×C₄×H(3,𝔽₃)

C₂×C₃

yes

characteristic

C₃²×C₄×C₉

D₃

yes

characteristic, complemented

C₂×(C₃⋊D₃)⋊C₉

—

C₄×C₃²⋊C₉

—

C₄×C₃²⋊C₉

D₃

yes

complemented

#6893

D₄

yes

characteristic, complemented, Sylow

C₃×C₄⋊(C₃⋊D₃)

—

C₃×C₄⋊(C₃⋊D₃)

—

C₄⋊#272

—

C₉×D₁₂

—

C₉×D₁₂

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×H(3,𝔽₃)

C₂²×C₃

yes

characteristic

C₂×C₃²×C₉

C₂×D₃

yes

characteristic

C₃⋊#272

—

C₉×C₃⋊D₃

—

C₂×C₃²⋊C₉

—

C₂×C₃²⋊C₉

C₂×D₃

yes

—

(C₃⋊D₃)⋊C₉

—

C₂×C₃×C₃⋊D₃

—

C₂×C₃×C₃⋊D₃

—

C₃³×C₄

C₃×D₃

yes

—

C₃³×C₄

C₃×D₃

yes

characteristic

C₂×#272

—

C₂×D₃×C₉

—

C₂×D₃×C₉

—

C₄×3₋¹⁺²

—

C₄×H(3,𝔽₃)

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₃⋊D₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

C₃×D₄

yes

characteristic

C₃²×C₉

D₁₂

yes

characteristic, complemented

C₃²⋊C₉

—

C₃²⋊C₉

D₁₂

yes

complemented

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₃×D₁₂

—

C₃×D₁₂

—

C₃×D₁₂

—

D₄×C₉

—

C₂×C₃³

C₂×C₃×D₃

yes

—

C₂×C₃³

C₂×C₃×D₃

yes

characteristic

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₂×3₋¹⁺²

—

C₂×H(3,𝔽₃)

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂×C₃⋊D₃

yes

characteristic

#272

—

D₃×C₉

—

D₃×C₉

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

(C₃⋊D₃)⋊C₃

yes

—

C₃²×C₄

C₃×C₃⋊D₃

yes

characteristic

C₂²×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₃³

C₃×D₁₂

yes

—

C₃³

C₃×D₁₂

yes

characteristic

3₋¹⁺²

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₄⋊(C₃⋊D₃)

yes

characteristic

C₃×D₄

—

D₁₂

—

D₁₂

—

D₁₂

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

100

C₂×C₃²

C₂×(C₃⋊D₃)⋊C₃

yes

—

101

C₂×C₃²

C₂×C₃×C₃⋊D₃

yes

characteristic

102

C₃⋊D₃

—

103

C₃⋊D₃

—

104

C₃×D₃

—

105

C₃×D₃

—

106

C₃×D₃

—

107

C₂×C₉

—

108

C₂×C₉

—

109

C₂×C₉

—

110

C₂×C₉

—

111

C₂²×C₃

—

112

C₂×D₃

—

113

C₂×D₃

—

114

C₂×D₃

—

115

C₃×C₄

—

116

C₃×C₄

—

117

C₃×C₄

—

118

C₃×C₄

C₃⋊((C₃⋊D₃)⋊C₃)

yes

characteristic

119

C₃×C₄

#4208

yes

characteristic

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

C₄⋊((C₃⋊D₃)⋊C₃)

yes

—

126

C₃²

C₃×C₄⋊(C₃⋊D₃)

yes

characteristic

127

C₉

—

128

C₉

—

129

C₉

—

130

D₄

—

Sylow

131

C₂×C₃

—

132

C₂×C₃

—

133

C₂×C₃

—

134

C₂×C₃

—

135

C₂×C₃

C₂×C₃⋊((C₃⋊D₃)⋊C₃)

yes

characteristic

136

C₂×C₃

C₂×#4208

yes

characteristic

137

D₃

—

138

D₃

—

139

D₃

—

140

C₂²

—

141

C₄

#92305

yes

characteristic, complemented

142

C₃

—

143

C₃

—

144

C₃

—

145

C₃

C₄⋊(C₃⋊((C₃⋊D₃)⋊C₃))

yes

characteristic

146

C₃

C₄⋊#4208

yes

characteristic

147

C₂

—

148

C₂

C₂×#92305

yes

characteristic

	subgroup	G/H
center	C₂×C₃	C₂×C₃⋊((C₃⋊D₃)⋊C₃)
commutator, cosocle	C₂×C₃×H(3,𝔽₃)	C₂²×C₃
Frattini, socle	C₂×C₃²	C₂×C₃×C₃⋊D₃
Fitting	C₄×#6893	C₂
prime radical	C₄⋊(C₃⋊#272)	C₃
hypercenter	C₃×C₄	C₃⋊((C₃⋊D₃)⋊C₃)
hypocenter	C₃×H(3,𝔽₃)	C₃×D₄
LCS	G ▷ C₂×C₃×H(3,𝔽₃) ▷ C₃×H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

subgroup

G/H

center

C₂×C₃

C₂×C₃⋊((C₃⋊D₃)⋊C₃)

commutator, cosocle

C₂×C₃×H(3,𝔽₃)

C₂²×C₃

Frattini, socle

C₂×C₃²

C₂×C₃×C₃⋊D₃

C₂

C₃

C₃×C₄

C₃⋊((C₃⋊D₃)⋊C₃)

hypocenter

C₃×H(3,𝔽₃)

C₃×D₄

LCS

G ▷ C₂×C₃×H(3,𝔽₃) ▷ C₃×H(3,𝔽₃)

UCS

C₁ ◁ C₂×C₃ ◁ C₃×C₄

N	H ≅ G/N	type
1. C₂×#92305	C₂	semidirect
2. C₄×#6893	C₂	semidirect
12. C₃²×C₄×C₉	D₃	semidirect
15. C₄×C₃²⋊C₉	D₃	semidirect
16. #6893	D₄	semidirect
45. C₃²×C₉	D₁₂	semidirect
47. C₃²⋊C₉	D₁₂	semidirect
141. C₄	#92305	semidirect

H ≅ G/N

type

1. C₂×#92305

C₂

semidirect

2. C₄×#6893

C₂

semidirect

12. C₃²×C₄×C₉

D₃

semidirect

15. C₄×C₃²⋊C₉

D₃

semidirect

16. #6893

D₄

semidirect

45. C₃²×C₉

D₁₂

semidirect

47. C₃²⋊C₉

D₁₂

semidirect

141. C₄

#92305

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#92305	2
C₄×#6893	2
C₄⋊(C₃⋊#272)	3
C₉×C₄⋊(C₃⋊D₃)	3
C₄⋊((C₃⋊D₃)⋊C₉)	3

subgroup

index

C₂×#92305

C₄×#6893

C₄⋊(C₃⋊#272)

C₉×C₄⋊(C₃⋊D₃)

C₄⋊((C₃⋊D₃)⋊C₉)

Subgroups — C₄⋊#92305 #14517538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4⋊#92305 #14517538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊#92305 #14517538