Group #14517388 — subgroups

Subgroups — C₄⋊#92283 #14517388

Order 1944.

284 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 63 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×#92283	C₂	1	2	yes	maximal, complemented
2	C₄×#6910	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄⋊#272	C₃	1	1	yes	maximal, characteristic
4	D₁₂×3₋¹⁺²	—	3	1	no	maximal
5	C₄⋊#4176	C₃	1	3	yes	maximal, complemented
6	C₂×#6910	C₂²	1	1	yes	characteristic
7	#92283	—	2	2	no	—
8	C₂×C₃×#272	C₂×C₃	1	2	yes	—
9	C₂×D₃×3₋¹⁺²	—	3	2	no	—
10	C₃×C₄×3₋¹⁺²	—	2	1	no	—
11	C₃×C₄×3₋¹⁺²	D₃	1	1	yes	characteristic, complemented
12	C₃×C₄×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
13	C₂×#4176	C₂×C₃	1	6	yes	complemented
14	C₄×#731	—	2	3	no	—
15	C₄×#731	C₂×C₃	1	3	yes	complemented
16	#6910	D₄	1	1	yes	characteristic, complemented, Sylow
17	C₃²×D₁₂	—	3	1	no	—
18	C₃²×D₁₂	—	9	1	no	—
19	D₄×3₋¹⁺²	—	9	1	no	—
20	C₄⋊#272	C₃²	1	1	yes	characteristic
21	C₉×D₁₂	—	3	3	no	—
22	C₂×C₃×3₋¹⁺²	—	2	1	no	—
23	C₂×C₃×3₋¹⁺²	C₂×D₃	1	1	yes	characteristic
24	C₂×C₃×H(3,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
25	C₃×#272	—	2	2	no	—
26	D₃×3₋¹⁺²	—	6	2	no	—
27	C₂×#731	—	2	3	no	—
28	C₂×#731	C₂²×C₃	1	3	yes	—
29	#4176	—	2	6	no	—
30	C₂×C₃²×D₃	—	3	2	no	—
31	C₂×C₃²×D₃	—	9	2	no	—
32	C₃³×C₄	—	3	1	no	—
33	C₃³×C₄	C₃×D₃	1	1	yes	characteristic
34	C₂²×3₋¹⁺²	—	9	2	no	—
35	C₂×#272	C₂×C₃²	1	2	yes	—
36	C₂×D₃×C₉	—	3	6	no	—
37	C₄×3₋¹⁺²	—	2	3	no	—
38	C₄×3₋¹⁺²	—	3	1	no	—
39	C₄×3₋¹⁺²	—	6	1	no	—
40	C₄×3₋¹⁺²	—	6	1	no	—
41	C₄×3₋¹⁺²	—	6	1	no	—
42	C₄×H(3,𝔽₃)	—	2	1	no	—
43	C₄×H(3,𝔽₃)	—	6	1	no	—
44	C₄×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
45	C₃×C₄×C₉	—	6	1	no	—
46	C₃×C₄×C₉	C₃×D₃	1	3	yes	complemented
47	C₃×3₋¹⁺²	—	2	1	no	—
48	C₃×3₋¹⁺²	D₁₂	1	1	yes	characteristic, complemented
49	C₃×H(3,𝔽₃)	C₃×D₄	1	1	yes	characteristic
50	#731	—	2	3	no	—
51	#731	C₃×D₄	1	3	yes	complemented
52	C₃²×D₄	—	9	1	no	—
53	C₃×D₁₂	—	3	1	no	—
54	C₃×D₁₂	—	9	1	no	—
55	C₃×D₁₂	—	9	1	no	—
56	C₃×D₁₂	—	9	3	no	—
57	D₄×C₉	—	9	3	no	—
58	C₂×C₃³	—	3	1	no	—
59	C₂×C₃³	C₂×C₃×D₃	1	1	yes	characteristic
60	C₃²×D₃	—	6	2	no	—
61	C₃²×D₃	—	18	2	no	—
62	C₂×3₋¹⁺²	—	2	3	no	—
63	C₂×3₋¹⁺²	—	3	1	no	—
64	C₂×3₋¹⁺²	—	6	1	no	—
65	C₂×3₋¹⁺²	—	6	1	no	—
66	C₂×3₋¹⁺²	—	6	1	no	—
67	C₂×3₋¹⁺²	—	18	2	no	—
68	C₂×H(3,𝔽₃)	—	2	1	no	—
69	C₂×H(3,𝔽₃)	—	6	1	no	—
70	C₂×H(3,𝔽₃)	C₂²×C₃²	1	1	yes	characteristic
71	C₂×C₃×C₉	—	6	1	no	—
72	C₂×C₃×C₉	C₂×C₃×D₃	1	3	yes	—
73	#272	—	2	2	no	—
74	D₃×C₉	—	6	6	no	—
75	C₂²×C₃²	—	9	2	no	—
76	C₂×C₃×D₃	—	3	2	no	—
77	C₂×C₃×D₃	—	9	2	no	—
78	C₂×C₃×D₃	—	9	2	no	—
79	C₂×C₃×D₃	—	9	6	no	—
80	C₃²×C₄	—	2	1	no	—
81	C₃²×C₄	—	3	1	no	—
82	C₃²×C₄	—	6	1	no	—
83	C₃²×C₄	—	9	1	no	—
84	C₃²×C₄	—	9	3	no	—
85	C₃²×C₄	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
86	C₃²×C₄	C₃²×D₃	1	1	yes	characteristic
87	C₂²×C₉	—	9	6	no	—
88	C₄×C₉	—	3	3	no	—
89	C₄×C₉	—	6	3	no	—
90	C₃³	—	3	1	no	—
91	C₃³	C₃×D₁₂	1	1	yes	characteristic
92	3₋¹⁺²	—	2	3	no	—
93	3₋¹⁺²	—	3	1	no	—
94	3₋¹⁺²	—	6	1	no	—
95	3₋¹⁺²	—	6	1	no	—
96	3₋¹⁺²	—	6	1	no	—
97	H(3,𝔽₃)	—	2	1	no	—
98	H(3,𝔽₃)	—	6	1	no	—
99	H(3,𝔽₃)	C₃²×D₄	1	1	yes	characteristic
100	C₃×C₉	—	6	1	no	—
101	C₃×C₉	C₃×D₁₂	1	3	yes	complemented
102	C₃×D₄	—	9	1	no	—
103	C₃×D₄	—	27	1	no	—
104	D₁₂	—	3	1	no	—
105	D₁₂	—	9	1	no	—
106	C₂×C₃²	—	2	1	no	—
107	C₂×C₃²	—	3	1	no	—
108	C₂×C₃²	—	6	1	no	—
109	C₂×C₃²	—	9	1	no	—
110	C₂×C₃²	—	9	3	no	—
111	C₂×C₃²	—	18	2	no	—
112	C₂×C₃²	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
113	C₂×C₃²	C₂×C₃²×D₃	1	1	yes	characteristic
114	C₃×D₃	—	6	2	no	—
115	C₃×D₃	—	18	2	no	—
116	C₃×D₃	—	18	2	no	—
117	C₃×D₃	—	18	6	no	—
118	C₂×C₉	—	3	3	no	—
119	C₂×C₉	—	6	3	no	—
120	C₂×C₉	—	18	6	no	—
121	C₂²×C₃	—	9	2	no	—
122	C₂²×C₃	—	27	2	no	—
123	C₂×D₃	—	3	2	no	—
124	C₂×D₃	—	9	2	no	—
125	C₃×C₄	—	3	1	no	—
126	C₃×C₄	—	3	1	no	—
127	C₃×C₄	—	6	1	no	—
128	C₃×C₄	—	9	1	no	—
129	C₃×C₄	—	18	1	no	—
130	C₃×C₄	C₃×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
131	C₃²	—	2	1	no	—
132	C₃²	—	3	1	no	—
133	C₃²	—	6	1	no	—
134	C₃²	—	9	1	no	—
135	C₃²	—	9	3	no	—
136	C₃²	C₄⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic
137	C₃²	C₃²×D₁₂	1	1	yes	characteristic
138	C₉	—	3	3	no	—
139	C₉	—	6	3	no	—
140	D₄	—	9	1	no	Sylow
141	C₂×C₃	—	3	1	no	—
142	C₂×C₃	—	3	1	no	—
143	C₂×C₃	—	6	1	no	—
144	C₂×C₃	—	9	1	no	—
145	C₂×C₃	—	18	2	no	—
146	C₂×C₃	—	18	1	no	—
147	C₂×C₃	—	54	2	no	—
148	C₂×C₃	C₂×C₃×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
149	D₃	—	6	2	no	—
150	D₃	—	18	2	no	—
151	C₂²	—	9	2	no	—
152	C₄	#92283	1	1	yes	characteristic, complemented
153	C₃	—	3	1	no	—
154	C₃	—	3	1	no	—
155	C₃	—	6	1	no	—
156	C₃	—	9	1	no	—
157	C₃	—	18	1	no	—
158	C₃	C₃×C₄⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic
159	C₂	—	18	2	no	—
160	C₂	C₂×#92283	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#92283

C₂

yes

maximal, complemented

C₄×#6910

C₂

yes

maximal, characteristic, complemented

C₃×C₄⋊#272

C₃

yes

maximal, characteristic

D₁₂×3₋¹⁺²

—

maximal

C₄⋊#4176

C₃

yes

maximal, complemented

C₂×#6910

C₂²

yes

characteristic

#92283

—

C₂×C₃×#272

C₂×C₃

yes

—

C₂×D₃×3₋¹⁺²

—

C₃×C₄×3₋¹⁺²

—

C₃×C₄×3₋¹⁺²

D₃

yes

characteristic, complemented

C₃×C₄×H(3,𝔽₃)

C₂×C₃

yes

characteristic

C₂×#4176

C₂×C₃

yes

complemented

C₄×#731

—

C₄×#731

C₂×C₃

yes

complemented

#6910

D₄

yes

characteristic, complemented, Sylow

C₃²×D₁₂

—

C₃²×D₁₂

—

D₄×3₋¹⁺²

—

C₄⋊#272

C₃²

yes

characteristic

C₉×D₁₂

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

C₂×D₃

yes

characteristic

C₂×C₃×H(3,𝔽₃)

C₂²×C₃

yes

characteristic

C₃×#272

—

D₃×3₋¹⁺²

—

C₂×#731

—

C₂×#731

C₂²×C₃

yes

—

#4176

—

C₂×C₃²×D₃

—

C₂×C₃²×D₃

—

C₃³×C₄

—

C₃³×C₄

C₃×D₃

yes

characteristic

C₂²×3₋¹⁺²

—

C₂×#272

C₂×C₃²

yes

—

C₂×D₃×C₉

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂×C₃²

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

C₃×D₃

yes

complemented

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

D₁₂

yes

characteristic, complemented

C₃×H(3,𝔽₃)

C₃×D₄

yes

characteristic

#731

—

#731

C₃×D₄

yes

complemented

C₃²×D₄

—

C₃×D₁₂

—

C₃×D₁₂

—

C₃×D₁₂

—

C₃×D₁₂

—

D₄×C₉

—

C₂×C₃³

—

C₂×C₃³

C₂×C₃×D₃

yes

characteristic

C₃²×D₃

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₂²×C₃²

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂×C₃×D₃

yes

—

#272

—

D₃×C₉

—

C₂²×C₃²

—

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

characteristic

C₃²×C₄

C₃²×D₃

yes

characteristic

C₂²×C₉

—

C₄×C₉

—

C₄×C₉

—

C₃³

—

C₃³

C₃×D₁₂

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₃²×D₄

yes

characteristic

100

C₃×C₉

—

101

C₃×C₉

C₃×D₁₂

yes

complemented

102

C₃×D₄

—

103

C₃×D₄

—

104

D₁₂

—

105

D₁₂

—

106

C₂×C₃²

—

107

C₂×C₃²

—

108

C₂×C₃²

—

109

C₂×C₃²

—

110

C₂×C₃²

—

111

C₂×C₃²

—

112

C₂×C₃²

C₂×(C₃⋊D₃)⋊C₃

yes

characteristic

113

C₂×C₃²

C₂×C₃²×D₃

yes

characteristic

114

C₃×D₃

—

115

C₃×D₃

—

116

C₃×D₃

—

117

C₃×D₃

—

118

C₂×C₉

—

119

C₂×C₉

—

120

C₂×C₉

—

121

C₂²×C₃

—

122

C₂²×C₃

—

123

C₂×D₃

—

124

C₂×D₃

—

125

C₃×C₄

—

126

C₃×C₄

—

127

C₃×C₄

—

128

C₃×C₄

—

129

C₃×C₄

—

130

C₃×C₄

C₃×(C₃⋊D₃)⋊C₃

yes

characteristic

131

C₃²

—

132

C₃²

—

133

C₃²

—

134

C₃²

—

135

C₃²

—

136

C₃²

C₄⋊((C₃⋊D₃)⋊C₃)

yes

characteristic

137

C₃²

C₃²×D₁₂

yes

characteristic

138

C₉

—

139

C₉

—

140

D₄

—

Sylow

141

C₂×C₃

—

142

C₂×C₃

—

143

C₂×C₃

—

144

C₂×C₃

—

145

C₂×C₃

—

146

C₂×C₃

—

147

C₂×C₃

—

148

C₂×C₃

C₂×C₃×(C₃⋊D₃)⋊C₃

yes

characteristic

149

D₃

—

150

D₃

—

151

C₂²

—

152

C₄

#92283

yes

characteristic, complemented

153

C₃

—

154

C₃

—

155

C₃

—

156

C₃

—

157

C₃

—

158

C₃

C₃×C₄⋊((C₃⋊D₃)⋊C₃)

yes

characteristic

159

C₂

—

160

C₂

C₂×#92283

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂×C₃×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₂×H(3,𝔽₃)	C₂²×C₃²
Frattini	C₂×C₃²	C₂×C₃²×D₃
Fitting	C₄×#6910	C₂
prime radical	C₃×C₄⋊#272	C₃
hypercenter	C₃²×C₄	(C₃⋊D₃)⋊C₃
hypocenter	H(3,𝔽₃)	C₃²×D₄
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃²×C₄

subgroup

G/H

center, socle

C₂×C₃

C₂×C₃×(C₃⋊D₃)⋊C₃

commutator, cosocle

C₂×H(3,𝔽₃)

C₂²×C₃²

Frattini

C₂×C₃²

C₂×C₃²×D₃

Fitting

C₄×#6910

C₂

prime radical

C₃×C₄⋊#272

C₃

hypercenter

C₃²×C₄

(C₃⋊D₃)⋊C₃

hypocenter

H(3,𝔽₃)

C₃²×D₄

LCS

G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂×C₃ ◁ C₃²×C₄

N	H ≅ G/N	type
1. C₂×#92283	C₂	semidirect
2. C₄×#6910	C₂	semidirect
5. C₄⋊#4176	C₃	semidirect
11. C₃×C₄×3₋¹⁺²	D₃	semidirect
13. C₂×#4176	C₂×C₃	semidirect
15. C₄×#731	C₂×C₃	semidirect
16. #6910	D₄	semidirect
46. C₃×C₄×C₉	C₃×D₃	semidirect
48. C₃×3₋¹⁺²	D₁₂	semidirect
51. #731	C₃×D₄	semidirect
101. C₃×C₉	C₃×D₁₂	semidirect
152. C₄	#92283	semidirect

H ≅ G/N

type

1. C₂×#92283

C₂

semidirect

2. C₄×#6910

C₂

semidirect

5. C₄⋊#4176

C₃

semidirect

11. C₃×C₄×3₋¹⁺²

D₃

semidirect

13. C₂×#4176

C₂×C₃

semidirect

15. C₄×#731

C₂×C₃

semidirect

16. #6910

D₄

semidirect

46. C₃×C₄×C₉

C₃×D₃

semidirect

48. C₃×3₋¹⁺²

D₁₂

semidirect

51. #731

C₃×D₄

semidirect

101. C₃×C₉

C₃×D₁₂

semidirect

152. C₄

#92283

semidirect

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

subgroup	index
C₂×#92283	2
C₄×#6910	2
C₃×C₄⋊#272	3
D₁₂×3₋¹⁺²	3
C₄⋊#4176	3

subgroup

index

C₂×#92283

C₄×#6910

C₃×C₄⋊#272

D₁₂×3₋¹⁺²

C₄⋊#4176

Subgroups — C₄⋊#92283 #14517388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4⋊#92283 #14517388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊#92283 #14517388