Group #13143816 — subgroups

Subgroups — #13143816

Order 1728.

605 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 109 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	#11721720	C₂	1	1	yes	maximal, characteristic
2	#11721716	C₂	1	1	yes	maximal, characteristic, complemented
3	H(3,𝔽₃)×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720554	C₂	1	4	yes	maximal, complemented
5	C₃×#4713	—	3	4	no	maximal
6	C₂×C₄⋊#272	—	2	2	no	—
7	C₂×C₄⋊#272	C₂²	1	2	yes	—
8	C₂×C₈×H(3,𝔽₃)	C₂²	1	2	yes	—
9	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
10	C₄⋊#1084	C₂²	1	2	yes	—
11	#88540	—	2	2	no	—
12	C₃×#874	—	3	4	no	—
13	C₃×#870	—	3	4	no	—
14	C₃²×#97	D₃	1	4	yes	complemented
15	C₃×#821	—	3	16	no	—
16	C₂²×#272	—	2	2	no	—
17	C₂×C₄×H(3,𝔽₃)	D₄	1	2	yes	—
18	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
19	C₂×#1084	—	2	2	no	—
20	C₄⋊#272	—	2	4	no	—
21	C₄⋊#272	—	4	4	no	—
22	C₈×H(3,𝔽₃)	—	2	2	no	—
23	C₃×#490	—	9	1	no	—
24	#4713	—	3	4	no	—
25	C₂×C₃×D₁₂	—	3	8	no	—
26	C₂×C₃×D₁₂	—	6	8	no	—
27	C₂×C₃²×C₈	C₂×D₃	1	8	yes	—
28	C₃²×C₄²	C₂×D₃	1	4	yes	—
29	C₃×C₄⋊Dic₃	—	3	8	no	—
30	C₃×Dic₃⋊C₄	—	6	8	no	—
31	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
32	C₂×#272	—	4	2	no	—
33	C₂×#272	—	4	4	no	—
34	C₄×H(3,𝔽₃)	—	2	2	no	—
35	C₄×H(3,𝔽₃)	#41	1	2	yes	—
36	#1084	—	4	2	no	—
37	C₃×#127	—	9	1	no	—
38	C₃×#125	—	9	1	no	—
39	#874	—	3	4	no	—
40	#870	—	3	4	no	—
41	C₃×#102	—	9	4	no	—
42	C₃×#97	—	3	4	no	—
43	C₃×#97	C₃⋊D₃	1	1	yes	characteristic
44	#821	—	3	16	no	—
45	C₂²×C₃×D₃	—	6	8	no	—
46	C₂×C₃²×C₄	D₁₂	1	8	yes	—
47	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
48	C₂×C₃×Dic₃	—	6	8	no	—
49	C₃×D₁₂	—	6	16	no	—
50	C₃×D₁₂	—	12	16	no	—
51	C₃²×C₈	—	2	8	no	—
52	#490	—	9	1	no	Sylow
53	C₂×H(3,𝔽₃)	#124	1	1	yes	characteristic
54	C₂×H(3,𝔽₃)	AGL(1,ℤ/8)	1	2	yes	—
55	#272	—	8	2	no	—
56	C₂×C₃×D₄	—	9	2	no	—
57	C₂×C₃×D₄	—	18	2	no	—
58	C₂×D₁₂	—	3	8	no	—
59	C₂×D₁₂	—	6	8	no	—
60	C₂×C₃×C₈	—	3	8	no	—
61	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	—
62	C₃×C₄⋊C₄	—	9	2	no	—
63	C₃×C₄⋊C₄	—	18	2	no	—
64	C₃×C₄²	—	3	4	no	—
65	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
66	C₄⋊Dic₃	—	3	8	no	—
67	Dic₃⋊C₄	—	6	8	no	—
68	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
69	C₂×C₃×D₃	—	12	8	no	—
70	C₂×C₃×D₃	—	12	16	no	—
71	C₃²×C₄	—	2	8	no	—
72	C₃²×C₄	#235	1	8	yes	—
73	C₃×Dic₃	—	12	8	no	—
74	#127	—	9	1	no	—
75	#125	—	9	1	no	—
76	#102	—	9	4	no	—
77	#97	#272	1	1	yes	characteristic, complemented
78	H(3,𝔽₃)	#490	1	1	yes	characteristic, complemented, Sylow
79	C₂³×C₃	—	18	2	no	—
80	C₂²×D₃	—	6	8	no	—
81	C₃×D₄	—	18	4	no	—
82	C₃×D₄	—	36	4	no	—
83	C₂×C₃×C₄	—	3	8	no	—
84	C₂×C₃×C₄	—	3	4	no	—
85	C₂×C₃×C₄	—	18	2	no	—
86	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
87	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
88	C₂×Dic₃	—	6	8	no	—
89	D₁₂	—	6	16	no	—
90	D₁₂	—	12	16	no	—
91	C₃×C₈	—	2	2	no	—
92	C₃×C₈	—	6	8	no	—
93	C₂×C₃²	#875	1	4	yes	—
94	C₂×C₃²	#908	1	8	yes	—
95	C₃×D₃	—	24	8	no	—
96	C₂×D₄	—	9	2	no	—
97	C₂×D₄	—	18	2	no	—
98	C₂×C₈	C₂×#272	1	2	yes	—
99	C₄⋊C₄	—	9	2	no	—
100	C₄⋊C₄	—	18	2	no	—
101	C₄²	C₂×#272	1	1	yes	characteristic
102	C₂²×C₃	—	3	4	no	—
103	C₂²×C₃	—	36	2	no	—
104	C₂²×C₃	—	36	4	no	—
105	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
106	C₂×D₃	—	12	8	no	—
107	C₂×D₃	—	12	16	no	—
108	C₃×C₄	—	2	2	no	—
109	C₃×C₄	—	3	8	no	—
110	C₃×C₄	—	6	8	no	—
111	C₃×C₄	—	36	2	no	—
112	C₃×C₄	#3828	1	2	yes	—
113	Dic₃	—	12	8	no	—
114	C₃²	#4713	1	4	yes	complemented
115	C₂³	—	18	2	no	—
116	D₄	—	18	4	no	—
117	D₄	—	36	4	no	—
118	C₂×C₄	—	18	2	no	—
119	C₂×C₄	C₄⋊#272	1	2	yes	—
120	C₂×C₄	C₂²×#272	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₂×C₃	—	3	8	no	—
123	C₂×C₃	—	3	4	no	—
124	C₂×C₃	—	72	2	no	—
125	C₂×C₃	#64084	1	1	yes	characteristic
126	C₂×C₃	#64117	1	2	yes	—
127	D₃	—	24	8	no	—
128	C₂²	—	36	4	no	—
129	C₂²	—	36	2	no	—
130	C₂²	C₂×C₄⋊#272	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	36	2	no	—
133	C₄	#88740	1	2	yes	—
134	C₃	—	3	4	no	—
135	C₃	#10592412	1	1	yes	characteristic
136	C₂	—	72	2	no	—
137	C₂	#11721721	1	1	yes	characteristic
138	C₂	#11721787	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11721720

C₂

yes

maximal, characteristic

#11721716

C₂

yes

maximal, characteristic, complemented

H(3,𝔽₃)×#97

C₂

yes

maximal, characteristic, complemented

#11720554

C₂

yes

maximal, complemented

C₃×#4713

—

maximal

C₂×C₄⋊#272

—

C₂×C₄⋊#272

C₂²

yes

—

C₂×C₈×H(3,𝔽₃)

C₂²

yes

—

C₄²×H(3,𝔽₃)

C₂²

yes

characteristic

C₄⋊#1084

C₂²

yes

—

#88540

—

C₃×#874

—

C₃×#870

—

C₃²×#97

D₃

yes

complemented

C₃×#821

—

C₂²×#272

—

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

C₂×C₄×H(3,𝔽₃)

C₂³

yes

—

—

—

C₈×H(3,𝔽₃)

—

C₃×#490

—

#4713

—

C₂×C₃×D₁₂

—

C₂×C₃×D₁₂

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

C₃×C₄⋊Dic₃

—

C₃×Dic₃⋊C₄

—

C₂²×H(3,𝔽₃)

C₂×D₄

yes

characteristic

C₂×#272

—

C₂×#272

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

#41

yes

—

—

—

—

—

—

—

C₃×#97

—

C₃×#97

C₃⋊D₃

yes

characteristic

#821

—

C₂²×C₃×D₃

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃×Dic₃

—

C₃×D₁₂

—

C₃×D₁₂

—

C₃²×C₈

—

—

C₂×H(3,𝔽₃)

yes

C₂×H(3,𝔽₃)

yes

—

#272

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₂²×C₃²

C₂×D₁₂

yes

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃²×C₄

#235

yes

—

C₃×Dic₃

—

—

—

—

yes

characteristic, complemented

H(3,𝔽₃)

#490

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂²×D₃

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

C₂×Dic₃

—

D₁₂

—

D₁₂

—

C₃×C₈

—

C₃×C₈

—

C₂×C₃²

#875

yes

—

C₂×C₃²

#908

yes

—

C₃×D₃

—

C₂×D₄

—

C₂×D₄

—

C₂×C₈

C₂×#272

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

C₂×#272

yes

characteristic

102

C₂²×C₃

—

103

C₂²×C₃

—

104

C₂²×C₃

—

105

C₂²×C₃

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

106

C₂×D₃

—

107

C₂×D₃

—

108

C₃×C₄

—

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

—

112

C₃×C₄

#3828

yes

—

113

Dic₃

—

114

C₃²

#4713

yes

complemented

115

C₂³

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

C₄⋊#272

yes

—

120

C₂×C₄

C₂²×#272

yes

characteristic

121

C₈

—

122

C₂×C₃

—

123

C₂×C₃

—

124

C₂×C₃

—

125

C₂×C₃

#64084

yes

characteristic

126

C₂×C₃

#64117

yes

—

127

D₃

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×C₄⋊#272

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

#88740

yes

—

134

C₃

—

135

C₃

#10592412

yes

characteristic

136

C₂

—

137

C₂

#11721721

yes

characteristic

138

C₂

#11721787

yes

—

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₄×H(3,𝔽₃)	C₂³
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	H(3,𝔽₃)×#97	C₂
prime radical	#11721720	C₂
hypercenter	C₃×#97	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#490
LCS	G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#97

subgroup

G/H

center, socle

C₂²×C₃

C₂×C₄⋊(C₃⋊D₃)

commutator, cosocle

C₂×C₄×H(3,𝔽₃)

C₂³

Frattini

C₂×C₃×C₄

C₂²×C₃⋊D₃

C₂

C₂

C₃×#97

C₃⋊D₃

G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#97

N	H ≅ G/N	type
2. #11721716	C₂	semidirect
3. H(3,𝔽₃)×#97	C₂	semidirect
4. #11720554	C₂	semidirect
14. C₃²×#97	D₃	semidirect
77. #97	#272	semidirect
78. H(3,𝔽₃)	#490	semidirect
114. C₃²	#4713	semidirect

H ≅ G/N

type

2. #11721716

C₂

semidirect

3. H(3,𝔽₃)×#97

C₂

semidirect

4. #11720554

C₂

semidirect

14. C₃²×#97

D₃

semidirect

77. #97

#272

semidirect

78. H(3,𝔽₃)

#490

semidirect

114. C₃²

#4713

semidirect

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

subgroup	index
#11721720	2
#11721716	2
H(3,𝔽₃)×#97	2
#11720554	2
C₃×#4713	3

subgroup

index

Subgroups — #13143816

Conjugacy classes of subgroups

Complements

Maximal subgroups