Group #13144260 — subgroups

Subgroups — #13144260

Order 1728.

847 conjugacy classes of subgroups; only the 277 normal subgroups are shown.

normal subgroups only

This group has 847 conjugacy classes of subgroups — too many to draw. Only its 277 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×C₄²×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#1084	C₂	1	2	yes	maximal
3	#11720574	C₂	1	4	yes	maximal
4	C₃×#4936	—	3	4	no	maximal
5	C₂²×C₄×H(3,𝔽₃)	C₂²	1	2	yes	—
6	C₂²×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₂²×#1084	C₂²	1	4	yes	—
8	C₄²×H(3,𝔽₃)	C₄	1	4	yes	complemented
9	C₄⋊#1084	—	2	4	no	—
10	C₂×C₃²×C₄²	D₃	1	4	yes	—
11	C₂×C₃×C₄⋊Dic₃	—	3	8	no	—
12	C₃×C₃⋊#95	—	3	16	no	—
13	C₂³×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
14	C₂×C₄×H(3,𝔽₃)	—	2	4	no	—
15	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	4	yes	—
16	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	2	yes	—
17	C₂×C₄×H(3,𝔽₃)	D₄	1	2	yes	—
18	C₂×C₄×H(3,𝔽₃)	Q₈	1	2	yes	—
19	C₂×#1084	—	2	4	no	—
20	C₂×#1084	—	2	8	no	—
21	C₃×#382	—	9	1	no	—
22	#4936	—	3	4	no	—
23	C₂²×C₃²×C₄	C₂×D₃	1	8	yes	—
24	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
25	C₂²×C₃×Dic₃	—	3	16	no	—
26	C₃²×C₄²	Dic₃	1	16	yes	complemented
27	C₃×C₄⋊Dic₃	—	6	16	no	—
28	C₂²×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
29	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
30	C₂²×H(3,𝔽₃)	C₂×Q₈	1	1	yes	characteristic
31	C₂²×H(3,𝔽₃)	#41	1	4	yes	—
32	C₄×H(3,𝔽₃)	—	2	4	no	—
33	C₄×H(3,𝔽₃)	C₄⋊C₄	1	4	yes	—
34	#1084	—	4	4	no	—
35	C₂×C₃×C₄⋊C₄	—	9	2	no	—
36	C₂×C₃×C₄²	—	3	4	no	—
37	C₂×C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
38	C₂×C₄⋊Dic₃	—	3	8	no	—
39	C₃×#95	—	9	4	no	—
40	C₃⋊#95	—	3	16	no	—
41	C₂³×C₃²	C₂²×D₃	1	4	yes	—
42	C₂×C₃²×C₄	—	2	16	no	—
43	C₂×C₃²×C₄	Dic₆	1	8	yes	—
44	C₂×C₃²×C₄	D₁₂	1	8	yes	—
45	C₂×C₃²×C₄	C₂×Dic₃	1	16	yes	—
46	C₂×C₃²×C₄	C₂×Dic₃	1	8	yes	—
47	C₂×C₃×Dic₃	—	6	16	no	—
48	C₂×C₃×Dic₃	—	6	32	no	—
49	#382	—	9	1	no	Sylow
50	C₂×H(3,𝔽₃)	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×H(3,𝔽₃)	#117	1	2	yes	—
52	C₂×H(3,𝔽₃)	#124	1	2	yes	—
53	C₂×H(3,𝔽₃)	#125	1	2	yes	—
54	C₂²×C₃×C₄	—	3	4	no	—
55	C₂²×C₃×C₄	—	3	8	no	—
56	C₂²×C₃×C₄	—	9	4	no	—
57	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	2	yes	—
58	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
59	C₂²×Dic₃	—	3	16	no	—
60	C₃×C₄⋊C₄	—	18	4	no	—
61	C₃×C₄²	—	3	16	no	—
62	C₃×C₄²	C₃⋊Dic₃	1	4	yes	—
63	C₄⋊Dic₃	—	6	16	no	—
64	C₂²×C₃²	C₂×Dic₆	1	4	yes	—
65	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
66	C₂²×C₃²	#235	1	16	yes	—
67	C₂²×C₃²	C₂²×Dic₃	1	4	yes	—
68	C₃²×C₄	—	2	16	no	—
69	C₃²×C₄	C₄⋊Dic₃	1	16	yes	—
70	C₃×Dic₃	—	12	16	no	—
71	C₂×C₄⋊C₄	—	9	2	no	—
72	C₂×C₄²	#272	1	1	yes	characteristic
73	#95	—	9	4	no	—
74	H(3,𝔽₃)	#382	1	1	yes	characteristic, complemented, Sylow
75	C₂³×C₃	—	3	4	no	—
76	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
77	C₂×C₃×C₄	—	2	4	no	—
78	C₂×C₃×C₄	—	3	8	no	—
79	C₂×C₃×C₄	—	3	8	no	—
80	C₂×C₃×C₄	—	3	8	no	—
81	C₂×C₃×C₄	—	3	16	no	—
82	C₂×C₃×C₄	—	6	16	no	—
83	C₂×C₃×C₄	—	18	4	no	—
84	C₂×C₃×C₄	—	18	8	no	—
85	C₂×C₃×C₄	#634	1	2	yes	—
86	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
87	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
88	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
89	C₂×Dic₃	—	6	16	no	—
90	C₂×Dic₃	—	6	32	no	—
91	C₂×C₃²	#870	1	8	yes	—
92	C₂×C₃²	#875	1	8	yes	—
93	C₂×C₃²	C₂×C₄⋊Dic₃	1	4	yes	—
94	C₂×C₃²	#926	1	8	yes	—
95	C₂²×C₄	—	9	4	no	—
96	C₂²×C₄	C₂×#272	1	2	yes	—
97	C₂²×C₄	C₂×#272	1	1	yes	characteristic
98	C₄⋊C₄	—	18	4	no	—
99	C₄²	#1084	1	4	yes	complemented
100	C₂²×C₃	—	3	4	no	—
101	C₂²×C₃	—	3	4	no	—
102	C₂²×C₃	—	3	16	no	—
103	C₂²×C₃	—	3	4	no	—
104	C₂²×C₃	C₂×#634	1	1	yes	characteristic
105	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
106	C₂²×C₃	#3828	1	4	yes	—
107	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
108	C₃×C₄	—	2	4	no	—
109	C₃×C₄	—	3	16	no	—
110	C₃×C₄	—	6	16	no	—
111	C₃×C₄	—	36	4	no	—
112	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
113	Dic₃	—	12	16	no	—
114	C₃²	#4936	1	4	yes	complemented
115	C₂³	C₂²×#272	1	1	yes	characteristic
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	18	4	no	—
118	C₂×C₄	—	18	8	no	—
119	C₂×C₄	#6229	1	2	yes	—
120	C₂×C₄	C₄⋊#272	1	2	yes	—
121	C₂×C₄	C₂×#1084	1	4	yes	—
122	C₂×C₄	C₂×#1084	1	2	yes	—
123	C₂×C₃	—	3	4	no	—
124	C₂×C₃	—	3	8	no	—
125	C₂×C₃	—	3	8	no	—
126	C₂×C₃	—	3	8	no	—
127	C₂×C₃	#64079	1	2	yes	—
128	C₂×C₃	#64084	1	2	yes	—
129	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
130	C₂×C₃	#64135	1	2	yes	—
131	C₂²	C₂×#6229	1	1	yes	characteristic
132	C₂²	C₂×C₄⋊#272	1	1	yes	characteristic
133	C₂²	#88740	1	4	yes	—
134	C₂²	C₂²×#1084	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	36	4	no	—
137	C₄	C₄⋊#1084	1	4	yes	—
138	C₃	—	3	4	no	—
139	C₃	#10592635	1	1	yes	characteristic
140	C₂	#11721716	1	2	yes	—
141	C₂	#11721721	1	2	yes	—
142	C₂	C₂×C₄⋊#1084	1	1	yes	characteristic
143	C₂	#11721826	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×H(3,𝔽₃)

C₂

yes

maximal, characteristic

C₂×C₄⋊#1084

C₂

yes

maximal

#11720574

C₂

yes

maximal

C₃×#4936

—

maximal

C₂²×C₄×H(3,𝔽₃)

C₂²

yes

—

C₂²×C₄×H(3,𝔽₃)

C₂²

yes

characteristic

C₂²×#1084

C₂²

yes

—

C₄²×H(3,𝔽₃)

C₄

yes

complemented

C₄⋊#1084

—

C₂×C₃²×C₄²

D₃

yes

—

C₂×C₃×C₄⋊Dic₃

—

C₃×C₃⋊#95

—

C₂³×H(3,𝔽₃)

C₂³

yes

characteristic

C₂×C₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

—

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

—

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

C₂×C₄×H(3,𝔽₃)

Q₈

yes

—

—

—

—

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃×Dic₃

—

C₃²×C₄²

Dic₃

yes

complemented

C₃×C₄⋊Dic₃

—

C₂²×H(3,𝔽₃)

C₂²×C₄

yes

characteristic

C₂²×H(3,𝔽₃)

C₂×D₄

yes

characteristic

C₂²×H(3,𝔽₃)

C₂×Q₈

yes

characteristic

C₂²×H(3,𝔽₃)

#41

yes

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₄⋊C₄

yes

—

#1084

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₃⋊D₃

yes

characteristic

C₂×C₄⋊Dic₃

—

C₃×#95

—

C₃⋊#95

—

C₂³×C₃²

C₂²×D₃

yes

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

Dic₆

yes

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

#382

—

Sylow

C₂×H(3,𝔽₃)

C₂×C₄⋊C₄

yes

characteristic

C₂×H(3,𝔽₃)

#117

yes

—

C₂×H(3,𝔽₃)

#124

yes

—

C₂×H(3,𝔽₃)

#125

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

—

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

characteristic

C₂²×Dic₃

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₃⋊Dic₃

yes

—

C₄⋊Dic₃

—

C₂²×C₃²

C₂×Dic₆

yes

—

C₂²×C₃²

C₂×D₁₂

yes

—

C₂²×C₃²

#235

yes

—

C₂²×C₃²

C₂²×Dic₃

yes

—

C₃²×C₄

—

C₃²×C₄

C₄⋊Dic₃

yes

—

C₃×Dic₃

—

C₂×C₄⋊C₄

—

C₂×C₄²

yes

—

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

C₂²×C₃⋊D₃

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#634

yes

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×C₃²

#870

yes

—

C₂×C₃²

#875

yes

—

C₂×C₃²

C₂×C₄⋊Dic₃

yes

—

C₂×C₃²

#926

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×#272

yes

—

C₂²×C₄

C₂×#272

yes

characteristic

C₄⋊C₄

—

C₄²

#1084

yes

complemented

100

C₂²×C₃

—

101

C₂²×C₃

—

102

C₂²×C₃

—

103

C₂²×C₃

—

104

C₂²×C₃

C₂×#634

yes

characteristic

105

C₂²×C₃

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

106

C₂²×C₃

#3828

yes

—

107

C₂²×C₃

C₂²×C₃⋊Dic₃

yes

characteristic

108

C₃×C₄

—

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

—

112

C₃×C₄

C₄⋊(C₃⋊Dic₃)

yes

—

113

Dic₃

—

114

C₃²

#4936

yes

complemented

115

C₂³

C₂²×#272

yes

characteristic

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#6229

yes

—

120

C₂×C₄

C₄⋊#272

yes

—

121

C₂×C₄

C₂×#1084

yes

—

122

C₂×C₄

C₂×#1084

yes

—

123

C₂×C₃

—

124

C₂×C₃

—

125

C₂×C₃

—

126

C₂×C₃

—

127

C₂×C₃

#64079

yes

—

128

C₂×C₃

#64084

yes

—

129

C₂×C₃

C₂×C₄⋊(C₃⋊Dic₃)

yes

characteristic

130

C₂×C₃

#64135

yes

—

131

C₂²

C₂×#6229

yes

characteristic

132

C₂²

C₂×C₄⋊#272

yes

characteristic

133

C₂²

#88740

yes

—

134

C₂²

C₂²×#1084

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

C₄⋊#1084

yes

—

138

C₃

—

139

C₃

#10592635

yes

characteristic

140

C₂

#11721716

yes

—

141

C₂

#11721721

yes

—

142

C₂

C₂×C₄⋊#1084

yes

characteristic

143

C₂

#11721826

yes

—

	subgroup	G/H
center, Frattini, socle	C₂³×C₃	C₂²×C₃⋊D₃
commutator	C₂²×H(3,𝔽₃)	C₂²×C₄
Fitting	C₂×C₄²×H(3,𝔽₃)	C₂
prime radical, cosocle	C₂³×H(3,𝔽₃)	C₂³
hypercenter	C₂×C₃×C₄²	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#382
LCS	G ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄²

subgroup

G/H

center, Frattini, socle

C₂³×C₃

C₂²×C₃⋊D₃

commutator

C₂²×H(3,𝔽₃)

C₂²×C₄

Fitting

C₂×C₄²×H(3,𝔽₃)

C₂

prime radical, cosocle

C₂³×H(3,𝔽₃)

C₂³

hypercenter

C₂×C₃×C₄²

C₃⋊D₃

G ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄²

N	H ≅ G/N	type
8. C₄²×H(3,𝔽₃)	C₄	semidirect
26. C₃²×C₄²	Dic₃	semidirect
74. H(3,𝔽₃)	#382	semidirect
99. C₄²	#1084	semidirect
114. C₃²	#4936	semidirect

H ≅ G/N

type

8. C₄²×H(3,𝔽₃)

C₄

semidirect

26. C₃²×C₄²

Dic₃

semidirect

74. H(3,𝔽₃)

#382

semidirect

99. C₄²

#1084

semidirect

114. C₃²

#4936

semidirect

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

subgroup	index
C₂×C₄²×H(3,𝔽₃)	2
C₂×C₄⋊#1084	2
#11720574	2
C₃×#4936	3

subgroup

index

C₂×C₄²×H(3,𝔽₃)

C₂×C₄⋊#1084

#11720574

C₃×#4936

Subgroups — #13144260

Conjugacy classes of subgroups

Complements

Maximal subgroups