Group #10596190 — subgroups

Subgroups — #10596190

Order 576.

720 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 183 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	C₃²×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	#64148	C₂	1	4	yes	maximal, complemented
3	#64147	C₂	1	2	yes	maximal, complemented
4	#64144	C₂	1	4	yes	maximal, complemented
5	D₄×C₃⋊Dic₃	C₂	1	2	yes	maximal, complemented
6	#64081	C₂	1	1	yes	maximal, characteristic, complemented
7	#64079	C₂	1	1	yes	maximal, characteristic, complemented
8	#5719	—	3	4	no	maximal
9	C₂×C₃²×D₄	C₂²	1	2	yes	—
10	C₂×C₃²×D₄	C₂²	1	4	yes	—
11	C₂×#638	C₂²	1	4	yes	—
12	C₂²×C₃⋊Dic₃	C₂²	1	4	yes	—
13	C₂×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
14	C₃²×C₄²	C₂²	1	1	yes	characteristic
15	#3757	C₂²	1	4	yes	—
16	#3757	C₂²	1	4	yes	—
17	#3752	C₂²	1	2	yes	—
18	C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
19	#3750	C₂²	1	4	yes	—
20	#3750	C₂²	1	2	yes	—
21	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
22	C₃×#127	D₃	1	4	yes	complemented
23	#939	—	3	16	no	—
24	#938	—	3	8	no	—
25	#935	—	3	16	no	—
26	D₄×Dic₃	—	3	8	no	—
27	#872	—	3	4	no	—
28	#870	—	3	4	no	—
29	C₂³×C₃²	C₂³	1	4	yes	—
30	C₂²×C₃⋊D₃	C₂³	1	1	yes	characteristic
31	C₃²×D₄	—	2	4	no	—
32	C₃²×D₄	—	4	4	no	—
33	C₂×C₃²×C₄	C₂³	1	2	yes	—
34	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
35	#638	—	4	4	no	—
36	C₂×C₃⋊Dic₃	—	4	4	no	—
37	C₂×C₃⋊Dic₃	C₂³	1	2	yes	—
38	C₂×C₃⋊Dic₃	C₂³	1	4	yes	—
39	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
40	C₄×C₃⋊D₃	—	2	2	no	—
41	#540	—	9	1	no	Sylow
42	C₂×C₃×D₄	C₂×D₃	1	8	yes	—
43	C₂×C₃×D₄	C₂×D₃	1	16	yes	—
44	C₂×#67	—	3	16	no	—
45	C₂²×Dic₃	—	3	16	no	—
46	C₂×C₄×D₃	—	3	4	no	—
47	C₃×C₄²	C₂×D₃	1	4	yes	—
48	#217	—	3	16	no	—
49	#217	—	3	16	no	—
50	(C₂×D₃)⋊C₄	—	3	8	no	—
51	C₄⋊Dic₃	—	3	8	no	—
52	Dic₃⋊C₄	—	3	16	no	—
53	Dic₃⋊C₄	—	3	8	no	—
54	C₄×Dic₃	—	3	4	no	—
55	C₂²×C₃²	—	2	4	no	—
56	C₂²×C₃²	—	2	4	no	—
57	C₂²×C₃²	—	2	4	no	—
58	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
59	C₂×C₃⋊D₃	—	2	1	no	—
60	C₂×C₃⋊D₃	—	2	2	no	—
61	C₃²×C₄	—	2	2	no	—
62	C₃²×C₄	#41	1	2	yes	—
63	C₃⋊Dic₃	—	2	1	no	—
64	C₃⋊Dic₃	—	2	2	no	—
65	C₃⋊Dic₃	—	2	4	no	—
66	#127	C₃⋊D₃	1	1	yes	characteristic, complemented
67	#125	—	9	1	no	—
68	#123	—	9	4	no	—
69	#121	—	9	2	no	—
70	#121	—	9	4	no	—
71	C₄×D₄	—	9	2	no	—
72	#117	—	9	1	no	—
73	C₂³×C₃	C₂²×D₃	1	16	yes	—
74	C₂²×D₃	—	3	4	no	—
75	C₃×D₄	—	2	16	no	—
76	C₃×D₄	—	4	16	no	—
77	C₂×C₃×C₄	C₂²×D₃	1	8	yes	—
78	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
79	#67	—	12	16	no	—
80	C₂×Dic₃	—	3	8	no	—
81	C₂×Dic₃	—	3	16	no	—
82	C₂×Dic₃	—	3	4	no	—
83	C₂×Dic₃	—	12	16	no	—
84	C₄×D₃	—	6	8	no	—
85	C₂×C₃²	—	4	4	no	—
86	C₂×C₃²	C₂×#41	1	1	yes	characteristic
87	C₂×C₃²	H(5,𝔽₂)	1	2	yes	—
88	C₃⋊D₃	—	4	1	no	—
89	C₂×D₄	—	9	4	no	—
90	C₂×D₄	C₂×C₃⋊D₃	1	2	yes	—
91	C₂×D₄	C₂×C₃⋊D₃	1	4	yes	—
92	C₂²×C₄	—	9	4	no	—
93	C₂²×C₄	—	9	1	no	—
94	C₄⋊C₄	—	9	2	no	—
95	C₄⋊C₄	—	9	2	no	—
96	C₄⋊C₄	—	9	4	no	—
97	C₂²⋊C₄	—	9	4	no	—
98	C₂²⋊C₄	—	9	4	no	—
99	C₂²⋊C₄	—	9	2	no	—
100	C₄²	—	9	1	no	—
101	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
102	C₂²×C₃	—	2	16	no	—
103	C₂²×C₃	—	2	16	no	—
104	C₂²×C₃	—	2	16	no	—
105	C₂²×C₃	C₂³×D₃	1	4	yes	—
106	C₂×D₃	—	6	4	no	—
107	C₂×D₃	—	6	8	no	—
108	C₃×C₄	—	2	8	no	—
109	C₃×C₄	#237	1	8	yes	—
110	Dic₃	—	6	4	no	—
111	Dic₃	—	6	8	no	—
112	Dic₃	—	6	16	no	—
113	C₃²	#540	1	1	yes	characteristic, complemented, Sylow
114	C₂³	—	9	1	no	—
115	C₂³	C₂²×C₃⋊D₃	1	4	yes	—
116	D₄	—	2	4	no	—
117	D₄	—	4	4	no	—
118	D₄	—	36	4	no	—
119	C₂×C₄	—	9	2	no	—
120	C₂×C₄	—	9	1	no	—
121	C₂×C₄	—	9	4	no	—
122	C₂×C₄	—	18	2	no	—
123	C₂×C₄	—	36	4	no	—
124	C₂×C₄	C₂²×C₃⋊D₃	1	2	yes	—
125	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
126	C₂×C₃	—	4	16	no	—
127	C₂×C₃	C₂×#237	1	4	yes	—
128	C₂×C₃	#1004	1	8	yes	—
129	D₃	—	12	4	no	—
130	C₂²	—	2	4	no	—
131	C₂²	—	2	4	no	—
132	C₂²	—	2	4	no	—
133	C₂²	—	18	1	no	—
134	C₂²	—	18	2	no	—
135	C₂²	C₂³×C₃⋊D₃	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	18	1	no	—
138	C₄	—	18	2	no	—
139	C₄	—	18	4	no	—
140	C₄	#3830	1	2	yes	—
141	C₃	#5719	1	4	yes	complemented
142	C₂	—	4	4	no	—
143	C₂	—	36	1	no	—
144	C₂	C₂×#3830	1	1	yes	characteristic
145	C₂	#64361	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#127

C₂

yes

maximal, characteristic, complemented

#64148

C₂

yes

maximal, complemented

#64147

C₂

yes

maximal, complemented

#64144

C₂

yes

maximal, complemented

D₄×C₃⋊Dic₃

C₂

yes

maximal, complemented

#64081

C₂

yes

maximal, characteristic, complemented

#64079

C₂

yes

maximal, characteristic, complemented

#5719

—

maximal

C₂×C₃²×D₄

C₂²

yes

—

C₂×C₃²×D₄

C₂²

yes

—

C₂×#638

C₂²

yes

—

C₂²×C₃⋊Dic₃

C₂²

yes

—

C₂×C₄×C₃⋊D₃

C₂²

yes

characteristic

C₃²×C₄²

C₂²

yes

characteristic

#3757

C₂²

yes

—

#3757

C₂²

yes

—

#3752

C₂²

yes

—

C₄⋊(C₃⋊Dic₃)

C₂²

yes

—

#3750

C₂²

yes

—

#3750

C₂²

yes

—

C₄×C₃⋊Dic₃

C₂²

yes

characteristic

C₃×#127

D₃

yes

—

—

—

D₄×Dic₃

—

#872

—

#870

—

C₂³×C₃²

C₂³

yes

—

C₂²×C₃⋊D₃

C₂³

yes

characteristic

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

C₂³

yes

—

C₂×C₃²×C₄

C₂³

yes

characteristic

#638

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

C₂³

yes

—

C₂×C₃⋊Dic₃

C₂³

yes

—

C₂×C₃⋊Dic₃

C₂³

yes

characteristic

C₄×C₃⋊D₃

—

#540

—

Sylow

C₂×C₃×D₄

C₂×D₃

yes

—

C₂×C₃×D₄

C₂×D₃

yes

—

C₂×#67

—

C₂²×Dic₃

—

C₂×C₄×D₃

—

C₃×C₄²

C₂×D₃

yes

—

#217

—

#217

—

(C₂×D₃)⋊C₄

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂⁴

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃²×C₄

#41

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

#127

C₃⋊D₃

yes

characteristic, complemented

—

—

—

—

C₄×D₄

—

#117

—

C₂³×C₃

C₂²×D₃

yes

—

C₂²×D₃

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

C₂²×D₃

yes

—

C₂×C₃×C₄

C₂²×D₃

yes

—

#67

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

C₂×C₃²

—

C₂×C₃²

C₂×#41

yes

characteristic

C₂×C₃²

H(5,𝔽₂)

yes

—

C₃⋊D₃

—

C₂×D₄

—

C₂×D₄

C₂×C₃⋊D₃

yes

—

C₂×D₄

C₂×C₃⋊D₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₄²

—

101

C₄²

C₂×C₃⋊D₃

yes

characteristic

102

C₂²×C₃

—

103

C₂²×C₃

—

104

C₂²×C₃

—

105

C₂²×C₃

C₂³×D₃

yes

—

106

C₂×D₃

—

107

C₂×D₃

—

108

C₃×C₄

—

109

C₃×C₄

#237

yes

—

110

Dic₃

—

111

Dic₃

—

112

Dic₃

—

113

C₃²

#540

yes

characteristic, complemented, Sylow

114

C₂³

—

115

C₂³

C₂²×C₃⋊D₃

yes

—

116

D₄

—

117

D₄

—

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂²×C₃⋊D₃

yes

—

125

C₂×C₄

C₂²×C₃⋊D₃

yes

characteristic

126

C₂×C₃

—

127

C₂×C₃

C₂×#237

yes

—

128

C₂×C₃

#1004

yes

—

129

D₃

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

C₂³×C₃⋊D₃

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#3830

yes

—

141

C₃

#5719

yes

complemented

142

C₂

—

143

C₂

—

144

C₂

C₂×#3830

yes

characteristic

145

C₂

#64361

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊D₃
commutator, socle, cosocle	C₂²×C₃²	C₂⁴
Fitting	C₃²×#127	C₂
hypercenter	#127	C₃⋊D₃
hypocenter	C₃²	#540
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ #127

subgroup

G/H

center, Frattini

C₂²

C₂³×C₃⋊D₃

commutator, socle, cosocle

C₂²×C₃²

C₂⁴

Fitting

C₃²×#127

C₂

C₃⋊D₃

C₃²

G ▷ C₂²×C₃² ▷ C₃²

C₁ ◁ C₂² ◁ #127

N	H ≅ G/N	type
1. C₃²×#127	C₂	semidirect
2. #64148	C₂	semidirect
3. #64147	C₂	semidirect
4. #64144	C₂	semidirect
5. D₄×C₃⋊Dic₃	C₂	semidirect
6. #64081	C₂	semidirect
7. #64079	C₂	semidirect
22. C₃×#127	D₃	semidirect
66. #127	C₃⋊D₃	semidirect
113. C₃²	#540	semidirect
141. C₃	#5719	semidirect

H ≅ G/N

type

1. C₃²×#127

C₂

semidirect

2. #64148

C₂

semidirect

3. #64147

C₂

semidirect

4. #64144

C₂

semidirect

5. D₄×C₃⋊Dic₃

C₂

semidirect

6. #64081

C₂

semidirect

7. #64079

C₂

semidirect

22. C₃×#127

D₃

semidirect

66. #127

C₃⋊D₃

semidirect

113. C₃²

#540

semidirect

141. C₃

#5719

semidirect

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

subgroup	index
C₃²×#127	2
#64148	2
#64147	2
#64144	2
D₄×C₃⋊Dic₃	2
#64081	2
#64079	2
#5719	3

subgroup

index

C₃²×#127

D₄×C₃⋊Dic₃

Subgroups — #10596190

Conjugacy classes of subgroups

Complements

Maximal subgroups