Group #13135889 — subgroups

Subgroups — #13135889

Order 1728.

539 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 161 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×C₄⋊C₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×#88538	C₂	1	2	yes	maximal
3	C₃×#4530	—	3	4	no	maximal
4	C₂²×C₄×H(3,𝔽₃)	C₄	1	2	yes	—
5	C₂²×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₄⋊C₄×H(3,𝔽₃)	—	2	2	no	—
7	#88538	—	2	2	no	—
8	#88538	C₄	1	4	yes	complemented
9	C₂×#6180	—	2	2	no	—
10	C₂×C₃²×C₄⋊C₄	D₃	1	4	yes	—
11	C₂×C₃×#208	—	3	8	no	—
12	C₂³×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×H(3,𝔽₃)	—	2	2	no	—
14	C₂×C₄×H(3,𝔽₃)	—	4	2	no	—
15	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	2	yes	—
16	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
17	C₂×C₄×H(3,𝔽₃)	D₄	1	2	yes	—
18	C₂×C₄×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
19	#6180	—	2	4	no	—
20	C₃×#343	—	9	1	no	—
21	#4530	—	3	4	no	—
22	C₂²×C₃²×C₄	Dic₃	1	8	yes	—
23	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
24	C₃²×C₄⋊C₄	—	2	8	no	—
25	C₃×#208	—	3	16	no	—
26	C₃×#208	—	6	8	no	—
27	C₂×C₃×C₃⋊C₈	—	6	8	no	—
28	C₂²×H(3,𝔽₃)	—	2	1	no	—
29	C₂²×H(3,𝔽₃)	—	2	1	no	—
30	C₂²×H(3,𝔽₃)	C₄²	1	1	yes	characteristic
31	C₂²×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
33	C₄×H(3,𝔽₃)	—	4	2	no	—
34	C₄×H(3,𝔽₃)	C₂²⋊C₄	1	2	yes	—
35	C₄×H(3,𝔽₃)	C₄⋊C₄	1	2	yes	—
36	C₂×C₃×M₁₆	—	9	2	no	—
37	C₂×C₃×C₄⋊C₄	—	3	4	no	—
38	C₂×C₃×C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic
39	C₂×#208	—	3	8	no	—
40	C₂³×C₃²	C₂×Dic₃	1	4	yes	—
41	C₂×C₃²×C₄	—	2	8	no	—
42	C₂×C₃²×C₄	—	4	8	no	—
43	C₂×C₃²×C₄	Dic₆	1	4	yes	—
44	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
45	C₂×C₃²×C₄	D₁₂	1	4	yes	—
46	C₂×C₃²×C₄	#67	1	8	yes	—
47	C₃×C₃⋊C₈	—	6	16	no	—
48	#343	—	9	1	no	Sylow
49	C₂×H(3,𝔽₃)	—	2	1	no	—
50	C₂×H(3,𝔽₃)	—	2	1	no	—
51	C₂×H(3,𝔽₃)	#95	1	1	yes	characteristic
52	C₂×H(3,𝔽₃)	#100	1	1	yes	characteristic
53	C₂×H(3,𝔽₃)	#101	1	1	yes	characteristic
54	C₂²×C₃×C₄	—	3	8	no	—
55	C₂²×C₃×C₄	—	3	4	no	—
56	C₂²×C₃×C₄	C₃⋊Dic₃	1	2	yes	—
57	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
58	C₃×M₁₆	—	9	4	no	—
59	C₃×M₁₆	—	18	2	no	—
60	C₂×C₃×C₈	—	18	2	no	—
61	C₃×C₄⋊C₄	—	2	2	no	—
62	C₃×C₄⋊C₄	—	6	8	no	—
63	#208	—	3	16	no	—
64	#208	—	6	8	no	—
65	C₂×C₃⋊C₈	—	6	8	no	—
66	C₂²×C₃²	—	2	4	no	—
67	C₂²×C₃²	—	2	4	no	—
68	C₂²×C₃²	C₄×Dic₃	1	4	yes	—
69	C₂²×C₃²	C₄⋊Dic₃	1	4	yes	—
70	C₂²×C₃²	#217	1	4	yes	—
71	C₃²×C₄	—	4	8	no	—
72	C₃²×C₄	Dic₃⋊C₄	1	8	yes	—
73	C₃²×C₄	(C₂×D₃)⋊C₄	1	8	yes	—
74	C₂×M₁₆	—	9	2	no	—
75	C₂×C₄⋊C₄	#272	1	1	yes	characteristic
76	H(3,𝔽₃)	#343	1	1	yes	characteristic, complemented, Sylow
77	C₂³×C₃	—	3	4	no	—
78	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
79	C₂×C₃×C₄	—	2	2	no	—
80	C₂×C₃×C₄	—	3	4	no	—
81	C₂×C₃×C₄	—	3	8	no	—
82	C₂×C₃×C₄	—	3	4	no	—
83	C₂×C₃×C₄	—	3	8	no	—
84	C₂×C₃×C₄	—	4	2	no	—
85	C₂×C₃×C₄	—	6	8	no	—
86	C₂×C₃×C₄	—	12	8	no	—
87	C₂×C₃×C₄	#634	1	1	yes	characteristic
88	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
89	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
90	C₂×C₃×C₄	#638	1	2	yes	—
91	C₃×C₈	—	18	4	no	—
92	C₃⋊C₈	—	6	16	no	—
93	C₂×C₃²	—	2	4	no	—
94	C₂×C₃²	—	2	4	no	—
95	C₂×C₃²	C₃⋊#95	1	4	yes	—
96	C₂×C₃²	#833	1	4	yes	—
97	C₂×C₃²	#836	1	4	yes	—
98	C₂²×C₄	#1084	1	2	yes	—
99	C₂²×C₄	C₂×#272	1	1	yes	characteristic
100	M₁₆	—	9	4	no	—
101	M₁₆	—	18	2	no	—
102	C₂×C₈	—	18	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₂²×C₃	—	2	1	no	—
105	C₂²×C₃	—	2	1	no	—
106	C₂²×C₃	—	3	4	no	—
107	C₂²×C₃	—	3	4	no	—
108	C₂²×C₃	—	3	4	no	—
109	C₂²×C₃	—	6	4	no	—
110	C₂²×C₃	—	6	4	no	—
111	C₂²×C₃	C₄×C₃⋊Dic₃	1	1	yes	characteristic
112	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
113	C₂²×C₃	#3757	1	1	yes	characteristic
114	C₃×C₄	—	3	8	no	—
115	C₃×C₄	—	3	8	no	—
116	C₃×C₄	—	4	2	no	—
117	C₃×C₄	—	12	8	no	—
118	C₃×C₄	#3750	1	2	yes	—
119	C₃×C₄	#3752	1	2	yes	—
120	C₃²	#4530	1	4	yes	complemented
121	C₂³	C₂×#1084	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	#6229	1	1	yes	characteristic
125	C₂×C₄	C₄×#272	1	2	yes	—
126	C₂×C₄	C₄⋊#272	1	1	yes	characteristic
127	C₂×C₄	#6235	1	2	yes	—
128	C₈	—	18	4	no	—
129	C₂×C₃	—	2	1	no	—
130	C₂×C₃	—	2	1	no	—
131	C₂×C₃	—	3	4	no	—
132	C₂×C₃	—	3	4	no	—
133	C₂×C₃	—	3	4	no	—
134	C₂×C₃	—	6	4	no	—
135	C₂×C₃	—	6	4	no	—
136	C₂×C₃	#63658	1	1	yes	characteristic
137	C₂×C₃	#63660	1	1	yes	characteristic
138	C₂×C₃	#63663	1	1	yes	characteristic
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₄×#1084	1	1	yes	characteristic
142	C₂²	C₄⋊#1084	1	1	yes	characteristic
143	C₂²	#88552	1	1	yes	characteristic
144	C₄	—	4	2	no	—
145	C₄	#88540	1	2	yes	—
146	C₄	#88542	1	2	yes	—
147	C₃	—	3	4	no	—
148	C₃	#10589396	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#11720574	1	1	yes	characteristic
152	C₂	#11720579	1	1	yes	characteristic
153	C₂	#11720584	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₄×H(3,𝔽₃)

C₂

yes

maximal, characteristic

C₂×#88538

C₂

yes

maximal

C₃×#4530

—

maximal

C₂²×C₄×H(3,𝔽₃)

C₄

yes

—

C₂²×C₄×H(3,𝔽₃)

C₂²

yes

characteristic

C₄⋊C₄×H(3,𝔽₃)

—

#88538

—

#88538

C₄

yes

complemented

C₂×#6180

—

C₂×C₃²×C₄⋊C₄

D₃

yes

—

C₂×C₃×#208

—

C₂³×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

—

C₂×C₄×H(3,𝔽₃)

D₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

C₂×C₄×H(3,𝔽₃)

Q₈

yes

—

—

—

C₂²×C₃²×C₄

Dic₃

yes

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₃²×C₄⋊C₄

—

C₃×#208

—

C₃×#208

—

C₂×C₃×C₃⋊C₈

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

C₄²

yes

characteristic

C₂²×H(3,𝔽₃)

C₂²⋊C₄

yes

characteristic

C₂²×H(3,𝔽₃)

C₄⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂²⋊C₄

yes

—

C₄×H(3,𝔽₃)

C₄⋊C₄

yes

—

C₂×C₃×M₁₆

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₃⋊D₃

yes

characteristic

C₂×#208

—

C₂³×C₃²

C₂×Dic₃

yes

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

Dic₆

yes

—

C₂×C₃²×C₄

C₄×D₃

yes

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

#67

yes

—

C₃×C₃⋊C₈

—

#343

—

Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

yes

C₂×H(3,𝔽₃)

yes

C₂×H(3,𝔽₃)

yes

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₃⋊Dic₃

yes

—

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₄×Dic₃

yes

—

C₂²×C₃²

C₄⋊Dic₃

yes

—

C₂²×C₃²

#217

yes

—

C₃²×C₄

—

C₃²×C₄

Dic₃⋊C₄

yes

—

C₃²×C₄

(C₂×D₃)⋊C₄

yes

—

C₂×M₁₆

—

C₂×C₄⋊C₄

yes

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

C₂×C₃⋊Dic₃

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

characteristic

C₂×C₃×C₄

C₄×C₃⋊D₃

yes

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

characteristic

C₂×C₃×C₄

#638

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₃⋊#95

yes

—

C₂×C₃²

#833

yes

—

C₂×C₃²

#836

yes

—

C₂²×C₄

#1084

yes

—

C₂²×C₄

C₂×#272

yes

characteristic

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₂²×C₃

—

105

C₂²×C₃

—

106

C₂²×C₃

—

107

C₂²×C₃

—

108

C₂²×C₃

—

109

C₂²×C₃

—

110

C₂²×C₃

—

111

C₂²×C₃

C₄×C₃⋊Dic₃

yes

characteristic

112

C₂²×C₃

C₄⋊(C₃⋊Dic₃)

yes

characteristic

113

C₂²×C₃

#3757

yes

characteristic

114

C₃×C₄

—

115

C₃×C₄

—

116

C₃×C₄

—

117

C₃×C₄

—

118

C₃×C₄

#3750

yes

—

119

C₃×C₄

#3752

yes

—

120

C₃²

#4530

yes

complemented

121

C₂³

C₂×#1084

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#6229

yes

characteristic

125

C₂×C₄

C₄×#272

yes

—

126

C₂×C₄

C₄⋊#272

yes

characteristic

127

C₂×C₄

#6235

yes

—

128

C₈

—

129

C₂×C₃

—

130

C₂×C₃

—

131

C₂×C₃

—

132

C₂×C₃

—

133

C₂×C₃

—

134

C₂×C₃

—

135

C₂×C₃

—

136

C₂×C₃

#63658

yes

characteristic

137

C₂×C₃

#63660

yes

characteristic

138

C₂×C₃

#63663

yes

characteristic

139

C₂²

—

140

C₂²

—

141

C₂²

C₄×#1084

yes

characteristic

142

C₂²

C₄⋊#1084

yes

characteristic

143

C₂²

#88552

yes

characteristic

144

C₄

—

145

C₄

#88540

yes

—

146

C₄

#88542

yes

—

147

C₃

—

148

C₃

#10589396

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#11720574

yes

characteristic

152

C₂

#11720579

yes

characteristic

153

C₂

#11720584

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃	#3757
commutator	C₂²×H(3,𝔽₃)	C₄²
Frattini	C₂²×C₃×C₄	C₂×C₃⋊D₃
Fitting	C₂×C₄⋊C₄×H(3,𝔽₃)	C₂
prime radical	C₂³×H(3,𝔽₃)	C₂×C₄
cosocle	C₂²×C₄×H(3,𝔽₃)	C₂²
hypercenter	C₂×C₃×C₄⋊C₄	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#343
LCS	G ▷ C₂²×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄⋊C₄

subgroup

G/H

center, socle

C₂²×C₃

#3757

commutator

C₂²×H(3,𝔽₃)

C₄²

Frattini

C₂²×C₃×C₄

C₂×C₃⋊D₃

Fitting

C₂×C₄⋊C₄×H(3,𝔽₃)

C₂

prime radical

C₂³×H(3,𝔽₃)

C₂×C₄

cosocle

C₂²×C₄×H(3,𝔽₃)

C₂²

hypercenter

C₂×C₃×C₄⋊C₄

C₃⋊D₃

G ▷ C₂²×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄⋊C₄

N	H ≅ G/N	type
8. #88538	C₄	semidirect
76. H(3,𝔽₃)	#343	semidirect
120. C₃²	#4530	semidirect

H ≅ G/N

type

8. #88538

C₄

semidirect

76. H(3,𝔽₃)

#343

semidirect

120. C₃²

#4530

semidirect

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

subgroup	index
C₂×C₄⋊C₄×H(3,𝔽₃)	2
C₂×#88538	2
C₃×#4530	3

subgroup

index

C₂×C₄⋊C₄×H(3,𝔽₃)

C₂×#88538

C₃×#4530

Subgroups — #13135889

Conjugacy classes of subgroups

Complements

Maximal subgroups