Group #13144223 — subgroups

Subgroups — #13144223

Order 1728.

462 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 71 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	#11721808	C₂	1	2	yes	maximal, complemented
2	H(3,𝔽₃)×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#11720569	C₂	1	2	yes	maximal, complemented
4	#11720512	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₆×#272	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4919	—	3	4	no	maximal
7	#88748	—	2	2	no	—
8	Q₁₆×H(3,𝔽₃)	C₂²	1	2	yes	—
9	#88549	—	2	2	no	—
10	#88529	C₂²	1	2	yes	—
11	C₈×#272	C₂²	1	1	yes	characteristic
12	C₁₆×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
13	#88386	C₂²	1	1	yes	characteristic
14	C₃×#919	—	3	8	no	—
15	C₃²×Q₃₂	D₃	1	4	yes	complemented
16	C₃×#828	—	3	8	no	—
17	C₃×D₄₈	—	3	4	no	—
18	C₃×D₃×C₁₆	—	3	4	no	—
19	Q₈×H(3,𝔽₃)	—	2	2	no	—
20	C₄⋊#272	—	2	2	no	—
21	C₄⋊#272	—	4	2	no	—
22	C₄×#272	—	4	2	no	—
23	C₄×#272	D₄	1	1	yes	characteristic
24	C₈×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
25	#6180	D₄	1	1	yes	characteristic
26	C₃×#508	—	9	1	no	—
27	#4919	—	3	4	no	—
28	C₃×#239	—	6	8	no	—
29	C₃²×Q₁₆	C₂×D₃	1	8	yes	—
30	C₃×#215	—	6	8	no	—
31	C₃×D₂₄	—	3	8	no	—
32	C₃×D₃×C₈	—	3	4	no	—
33	C₃²×C₁₆	C₂×D₃	1	4	yes	—
34	C₃×C₃⋊C₁₆	—	3	4	no	—
35	C₂×#272	—	4	2	no	—
36	C₂×#272	D₈	1	1	yes	characteristic
37	C₄×H(3,𝔽₃)	—	4	2	no	—
38	C₄×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
39	#1084	D₈	1	1	yes	characteristic
40	C₃×#135	—	9	2	no	—
41	#919	—	3	8	no	—
42	C₃×Q₃₂	—	3	4	no	—
43	C₃×Q₃₂	C₃⋊D₃	1	1	yes	characteristic
44	C₃×SD₃₂	—	9	2	no	—
45	C₃×D₁₆	—	9	1	no	—
46	C₂×C₃×C₁₆	—	9	1	no	—
47	#828	—	3	8	no	—
48	D₄₈	—	3	4	no	—
49	D₃×C₁₆	—	3	4	no	—
50	C₃²×Q₈	—	2	8	no	—
51	C₃×D₁₂	—	6	8	no	—
52	C₃×D₁₂	—	12	8	no	—
53	C₃×C₄×D₃	—	3	4	no	—
54	C₃×C₄×D₃	—	12	8	no	—
55	C₃²×C₈	C₂²×D₃	1	4	yes	—
56	C₃×C₃⋊C₈	—	3	4	no	—
57	#508	—	9	1	no	Sylow
58	C₂×H(3,𝔽₃)	C₂×D₈	1	1	yes	characteristic
59	#272	—	2	1	no	—
60	#272	—	8	2	no	—
61	C₃×#41	—	18	2	no	—
62	#239	—	6	8	no	—
63	C₃×Q₁₆	—	3	8	no	—
64	C₃×Q₁₆	C₂×C₃⋊D₃	1	2	yes	—
65	C₃×SD₁₆	—	18	2	no	—
66	C₃×D₈	—	9	2	no	—
67	C₂×C₃×C₈	—	9	1	no	—
68	#215	—	6	8	no	—
69	D₂₄	—	3	8	no	—
70	D₃×C₈	—	3	4	no	—
71	C₃×C₁₆	—	3	4	no	—
72	C₃×C₁₆	—	9	1	no	—
73	C₃×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
74	C₃⋊C₁₆	—	3	4	no	—
75	C₂×C₃×D₃	—	3	4	no	—
76	C₂×C₃×D₃	—	12	8	no	—
77	C₃²×C₄	—	4	8	no	—
78	C₃²×C₄	D₃×D₄	1	4	yes	—
79	C₃×Dic₃	—	3	4	no	—
80	#135	—	9	2	no	—
81	Q₃₂	#272	1	1	yes	characteristic, complemented
82	SD₃₂	—	9	2	no	—
83	D₁₆	—	9	1	no	—
84	C₂×C₁₆	—	9	1	no	—
85	H(3,𝔽₃)	#508	1	1	yes	characteristic, complemented, Sylow
86	C₃×Q₈	—	2	2	no	—
87	C₃×Q₈	—	6	8	no	—
88	C₃×D₄	—	18	2	no	—
89	C₃×D₄	—	36	2	no	—
90	C₂×C₃×C₄	—	9	1	no	—
91	C₂×C₃×C₄	—	36	2	no	—
92	D₁₂	—	6	8	no	—
93	D₁₂	—	12	8	no	—
94	C₄×D₃	—	3	4	no	—
95	C₄×D₃	—	12	8	no	—
96	C₃×C₈	—	3	4	no	—
97	C₃×C₈	—	9	1	no	—
98	C₃×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
99	C₃⋊C₈	—	3	4	no	—
100	C₂×C₃²	D₃×D₈	1	4	yes	—
101	C₃×D₃	—	6	4	no	—
102	C₃×D₃	—	24	8	no	—
103	#41	—	18	2	no	—
104	Q₁₆	C₂×#272	1	2	yes	—
105	SD₁₆	—	18	2	no	—
106	D₈	—	9	2	no	—
107	C₂×C₈	—	9	1	no	—
108	C₁₆	—	9	1	no	—
109	C₁₆	C₂×#272	1	1	yes	characteristic
110	C₂²×C₃	—	9	1	no	—
111	C₂²×C₃	—	36	2	no	—
112	C₂×D₃	—	3	4	no	—
113	C₂×D₃	—	12	8	no	—
114	C₃×C₄	—	3	4	no	—
115	C₃×C₄	—	4	2	no	—
116	C₃×C₄	—	9	1	no	—
117	C₃×C₄	—	12	8	no	—
118	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
119	Dic₃	—	3	4	no	—
120	C₃²	#4919	1	4	yes	complemented
121	Q₈	—	2	2	no	—
122	D₄	—	18	2	no	—
123	D₄	—	36	2	no	—
124	C₂×C₄	—	9	1	no	—
125	C₂×C₄	—	36	2	no	—
126	C₈	—	9	1	no	—
127	C₈	C₂²×#272	1	1	yes	characteristic
128	C₂×C₃	—	3	4	no	—
129	C₂×C₃	—	18	1	no	—
130	C₂×C₃	—	72	2	no	—
131	C₂×C₃	D₈×C₃⋊D₃	1	1	yes	characteristic
132	D₃	—	6	4	no	—
133	D₃	—	24	8	no	—
134	C₂²	—	9	1	no	—
135	C₂²	—	36	2	no	—
136	C₄	—	4	2	no	—
137	C₄	—	9	1	no	—
138	C₄	D₄×#272	1	1	yes	characteristic
139	C₃	—	3	4	no	—
140	C₃	#10592618	1	1	yes	characteristic
141	C₂	—	18	1	no	—
142	C₂	—	72	2	no	—
143	C₂	D₈×#272	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11721808

C₂

yes

maximal, complemented

H(3,𝔽₃)×Q₃₂

C₂

yes

maximal, characteristic, complemented

#11720569

C₂

yes

maximal, complemented

#11720512

C₂

yes

maximal, characteristic, complemented

C₁₆×#272

C₂

yes

maximal, characteristic, complemented

C₃×#4919

—

maximal

#88748

—

Q₁₆×H(3,𝔽₃)

C₂²

yes

—

#88549

—

#88529

C₂²

yes

—

C₈×#272

C₂²

yes

characteristic

C₁₆×H(3,𝔽₃)

C₂²

yes

characteristic

#88386

C₂²

yes

characteristic

C₃×#919

—

C₃²×Q₃₂

D₃

yes

complemented

C₃×#828

—

C₃×D₄₈

—

C₃×D₃×C₁₆

—

Q₈×H(3,𝔽₃)

—

—

—

—

D₄

yes

characteristic

C₈×H(3,𝔽₃)

C₂³

yes

characteristic

#6180

D₄

yes

—

—

—

C₃²×Q₁₆

C₂×D₃

yes

—

C₃×#215

—

C₃×D₂₄

—

C₃×D₃×C₈

—

C₃²×C₁₆

C₂×D₃

yes

—

C₃×C₃⋊C₁₆

—

C₂×#272

—

C₂×#272

D₈

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂×D₄

yes

characteristic

#1084

D₈

yes

characteristic

C₃×#135

—

#919

—

C₃×Q₃₂

—

C₃×Q₃₂

C₃⋊D₃

yes

characteristic

C₃×SD₃₂

—

C₃×D₁₆

—

C₂×C₃×C₁₆

—

#828

—

D₄₈

—

D₃×C₁₆

—

C₃²×Q₈

—

C₃×D₁₂

—

C₃×D₁₂

—

C₃×C₄×D₃

—

C₃×C₄×D₃

—

C₃²×C₈

C₂²×D₃

yes

—

C₃×C₃⋊C₈

—

#508

—

Sylow

C₂×H(3,𝔽₃)

C₂×D₈

yes

characteristic

#272

—

#272

—

C₃×#41

—

#239

—

C₃×Q₁₆

—

C₃×Q₁₆

C₂×C₃⋊D₃

yes

—

C₃×SD₁₆

—

C₃×D₈

—

C₂×C₃×C₈

—

#215

—

D₂₄

—

D₃×C₈

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃×C₁₆

C₂×C₃⋊D₃

yes

characteristic

C₃⋊C₁₆

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃²×C₄

D₃×D₄

yes

—

C₃×Dic₃

—

#135

—

Q₃₂

#272

yes

characteristic, complemented

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

H(3,𝔽₃)

#508

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

D₁₂

—

D₁₂

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₂²×C₃⋊D₃

yes

characteristic

C₃⋊C₈

—

100

C₂×C₃²

D₃×D₈

yes

—

101

C₃×D₃

—

102

C₃×D₃

—

103

#41

—

104

Q₁₆

C₂×#272

yes

—

105

SD₁₆

—

106

D₈

—

107

C₂×C₈

—

108

C₁₆

—

109

C₁₆

C₂×#272

yes

characteristic

110

C₂²×C₃

—

111

C₂²×C₃

—

112

C₂×D₃

—

113

C₂×D₃

—

114

C₃×C₄

—

115

C₃×C₄

—

116

C₃×C₄

—

117

C₃×C₄

—

118

C₃×C₄

D₄×C₃⋊D₃

yes

characteristic

119

Dic₃

—

120

C₃²

#4919

yes

complemented

121

Q₈

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₈

—

127

C₈

C₂²×#272

yes

characteristic

128

C₂×C₃

—

129

C₂×C₃

—

130

C₂×C₃

—

131

C₂×C₃

D₈×C₃⋊D₃

yes

characteristic

132

D₃

—

133

D₃

—

134

C₂²

—

135

C₂²

—

136

C₄

—

137

C₄

—

138

C₄

D₄×#272

yes

characteristic

139

C₃

—

140

C₃

#10592618

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

D₈×#272

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃	D₈×C₃⋊D₃
commutator, cosocle	C₈×H(3,𝔽₃)	C₂³
Frattini	C₃×C₈	C₂²×C₃⋊D₃
Fitting	H(3,𝔽₃)×Q₃₂	C₂
hypercenter	C₃×Q₃₂	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#508
LCS	G ▷ C₈×H(3,𝔽₃) ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈ ◁ C₃×Q₃₂

subgroup

G/H

center, socle

C₂×C₃

D₈×C₃⋊D₃

commutator, cosocle

C₈×H(3,𝔽₃)

C₂³

Frattini

C₃×C₈

C₂²×C₃⋊D₃

Fitting

H(3,𝔽₃)×Q₃₂

C₂

hypercenter

C₃×Q₃₂

C₃⋊D₃

G ▷ C₈×H(3,𝔽₃) ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈ ◁ C₃×Q₃₂

N	H ≅ G/N	type
1. #11721808	C₂	semidirect
2. H(3,𝔽₃)×Q₃₂	C₂	semidirect
3. #11720569	C₂	semidirect
4. #11720512	C₂	semidirect
5. C₁₆×#272	C₂	semidirect
15. C₃²×Q₃₂	D₃	semidirect
81. Q₃₂	#272	semidirect
85. H(3,𝔽₃)	#508	semidirect
120. C₃²	#4919	semidirect

H ≅ G/N

type

1. #11721808

C₂

semidirect

2. H(3,𝔽₃)×Q₃₂

C₂

semidirect

3. #11720569

C₂

semidirect

4. #11720512

C₂

semidirect

5. C₁₆×#272

C₂

semidirect

15. C₃²×Q₃₂

D₃

semidirect

81. Q₃₂

#272

semidirect

85. H(3,𝔽₃)

#508

semidirect

120. C₃²

#4919

semidirect

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

subgroup	index
#11721808	2
H(3,𝔽₃)×Q₃₂	2
#11720569	2
#11720512	2
C₁₆×#272	2
C₃×#4919	3

subgroup

index

H(3,𝔽₃)×Q₃₂

C₁₆×#272

Subgroups — #13144223

Conjugacy classes of subgroups

Complements

Maximal subgroups