Group #10592447 — subgroups

Subgroups — #10592447

Order 576.

312 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 75 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×#3754	C₂	1	1	yes	maximal, characteristic, complemented
2	#64141	C₂	1	1	yes	maximal, characteristic
3	#64118	C₂	1	1	yes	maximal, characteristic, complemented
4	#64116	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#100	C₂	1	1	yes	maximal, characteristic, complemented
6	#63651	C₂	1	1	yes	maximal, characteristic, complemented
7	#63649	C₂	1	1	yes	maximal, characteristic, complemented
8	#4748	—	3	4	no	maximal
9	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
10	#3828	C₂²	1	1	yes	characteristic
11	C₂×#634	C₂²	1	1	yes	characteristic
12	C₃²×M₁₆	C₂²	1	1	yes	characteristic
13	C₃²×M₁₆	C₂²	1	1	yes	characteristic
14	#3754	—	2	1	no	—
15	#3754	—	2	1	no	—
16	#3754	—	2	1	no	—
17	#3753	—	2	1	no	—
18	#3748	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
20	#3746	—	2	1	no	—
21	#3744	—	2	1	no	—
22	#3743	—	2	1	no	—
23	C₈×C₃⋊D₃	—	2	1	no	—
24	C₂×#214	—	3	4	no	—
25	#932	—	3	4	no	—
26	#909	—	3	4	no	—
27	#907	—	3	4	no	—
28	C₃×#100	D₃	1	4	yes	complemented
29	#824	—	3	4	no	—
30	#822	—	3	4	no	—
31	C₂³×C₃²	—	2	1	no	—
32	C₃²×D₄	—	2	1	no	—
33	C₃²×D₄	—	2	1	no	—
34	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
35	#638	—	2	1	no	—
36	C₂×C₃⋊Dic₃	—	2	1	no	—
37	C₄⋊(C₃⋊D₃)	D₄	1	1	yes	characteristic
38	C₄×C₃⋊D₃	—	2	1	no	—
39	#634	—	2	1	no	—
40	#634	—	2	1	no	—
41	#634	D₄	1	1	yes	characteristic
42	C₃²×C₈	—	2	1	no	—
43	C₃²×C₈	—	2	1	no	—
44	C₃⋊(C₃⋊C₈)	—	2	1	no	—
45	C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
46	C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
47	#471	—	9	1	no	Sylow
48	C₂×C₃×D₄	C₂×D₃	1	4	yes	—
49	#235	—	3	4	no	—
50	C₂×Dic₆	—	3	4	no	—
51	C₃×M₁₆	C₂×D₃	1	4	yes	—
52	C₃×M₁₆	C₂×D₃	1	4	yes	—
53	#214	—	6	4	no	—
54	#214	—	6	4	no	—
55	#214	—	6	4	no	—
56	#213	—	6	4	no	—
57	#208	—	3	4	no	—
58	C₂×C₃⋊C₈	—	3	4	no	—
59	Dic₁₂	—	6	4	no	—
60	#204	—	6	4	no	—
61	ΓL(1,𝔽₂₅)	—	6	4	no	—
62	D₃×C₈	—	6	4	no	—
63	C₂²×C₃²	—	4	1	no	—
64	C₂²×C₃²	—	8	1	no	—
65	C₂²×C₃²	#41	1	1	yes	characteristic
66	C₂×C₃⋊D₃	—	2	1	no	—
67	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
68	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
69	C₃⋊Dic₃	—	2	1	no	—
70	C₃⋊Dic₃	—	4	1	no	—
71	#137	—	9	1	no	—
72	AGL(1,ℤ/8)	—	9	1	no	—
73	C₂×SD₁₆	—	9	1	no	—
74	#131	—	9	1	no	—
75	#108	—	9	1	no	—
76	#101	—	9	1	no	—
77	#100	C₃⋊D₃	1	1	yes	characteristic, complemented
78	C₂³×C₃	—	2	4	no	—
79	C₃×D₄	—	2	4	no	—
80	C₃×D₄	—	2	4	no	—
81	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
82	#67	—	6	4	no	—
83	C₂×Dic₃	—	6	4	no	—
84	D₁₂	—	3	4	no	—
85	C₄×D₃	—	6	4	no	—
86	Dic₆	—	3	4	no	—
87	Dic₆	—	6	4	no	—
88	Dic₆	—	6	4	no	—
89	C₃×C₈	—	2	4	no	—
90	C₃×C₈	—	2	4	no	—
91	C₃⋊C₈	—	3	4	no	—
92	C₃⋊C₈	—	3	4	no	—
93	C₃⋊C₈	—	6	4	no	—
94	C₂×C₃²	—	2	1	no	—
95	C₂×C₃²	—	8	1	no	—
96	C₂×C₃²	#121	1	1	yes	characteristic
97	C₃⋊D₃	—	4	1	no	—
98	#41	—	9	1	no	—
99	C₂×Q₈	—	9	1	no	—
100	C₂×D₄	C₂×C₃⋊D₃	1	1	yes	characteristic
101	Q₁₆	—	18	1	no	—
102	SD₁₆	—	18	1	no	—
103	SD₁₆	—	18	1	no	—
104	SD₁₆	—	18	1	no	—
105	SD₁₆	—	18	1	no	—
106	D₈	—	18	1	no	—
107	M₁₆	—	9	1	no	—
108	M₁₆	—	18	1	no	—
109	M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
110	M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
111	C₂×C₈	—	9	1	no	—
112	C₂×C₈	—	18	1	no	—
113	C₂²×C₃	—	4	4	no	—
114	C₂²×C₃	—	8	4	no	—
115	C₂²×C₃	#235	1	4	yes	—
116	C₂×D₃	—	6	4	no	—
117	C₃×C₄	D₃×D₄	1	4	yes	—
118	C₃×C₄	D₃×D₄	1	4	yes	—
119	Dic₃	—	6	4	no	—
120	Dic₃	—	12	4	no	—
121	C₃²	#471	1	1	yes	characteristic, complemented, Sylow
122	C₂³	—	2	1	no	—
123	Q₈	—	9	1	no	—
124	Q₈	—	18	1	no	—
125	Q₈	—	18	1	no	—
126	D₄	—	2	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	9	1	no	—
129	D₄	—	18	1	no	—
130	C₂×C₄	—	18	1	no	—
131	C₂×C₄	—	18	1	no	—
132	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	9	1	no	—
136	C₈	—	9	1	no	—
137	C₈	—	18	1	no	—
138	C₂×C₃	—	2	4	no	—
139	C₂×C₃	—	8	4	no	—
140	C₂×C₃	#884	1	4	yes	—
141	D₃	—	12	4	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	18	1	no	—
145	C₂²	#3828	1	1	yes	characteristic
146	C₄	—	18	1	no	—
147	C₄	—	36	1	no	—
148	C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
149	C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
150	C₃	#4748	1	4	yes	complemented
151	C₂	—	2	1	no	—
152	C₂	—	8	1	no	—
153	C₂	—	36	1	no	—
154	C₂	#64093	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#3754

C₂

yes

maximal, characteristic, complemented

#64141

C₂

yes

maximal, characteristic

#64118

C₂

yes

maximal, characteristic, complemented

#64116

C₂

yes

maximal, characteristic, complemented

C₃²×#100

C₂

yes

maximal, characteristic, complemented

#63651

C₂

yes

maximal, characteristic, complemented

#63649

C₂

yes

maximal, characteristic, complemented

#4748

—

maximal

C₂×C₃²×D₄

C₂²

yes

characteristic

#3828

C₂²

yes

characteristic

C₂×#634

C₂²

yes

characteristic

C₃²×M₁₆

C₂²

yes

characteristic

C₃²×M₁₆

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₂²

yes

—

—

—

C₈×C₃⋊D₃

—

—

—

—

—

D₃

yes

complemented

#824

—

#822

—

C₂³×C₃²

—

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

C₂³

yes

characteristic

#638

—

C₂×C₃⋊Dic₃

—

C₄⋊(C₃⋊D₃)

D₄

yes

characteristic

C₄×C₃⋊D₃

—

#634

—

#634

—

#634

D₄

yes

characteristic

C₃²×C₈

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

D₄

yes

characteristic

C₃⋊(C₃⋊C₈)

D₄

yes

characteristic

#471

—

Sylow

C₂×C₃×D₄

C₂×D₃

yes

—

#235

—

C₂×Dic₆

—

C₃×M₁₆

C₂×D₃

yes

—

C₃×M₁₆

C₂×D₃

yes

—

—

—

—

—

—

C₂×C₃⋊C₈

—

Dic₁₂

—

#204

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

#41

yes

characteristic

C₂×C₃⋊D₃

—

C₃²×C₄

C₂×D₄

yes

characteristic

C₃²×C₄

C₂×D₄

yes

characteristic

C₃⋊Dic₃

—

C₃⋊Dic₃

—

#137

—

AGL(1,ℤ/8)

—

C₂×SD₁₆

—

—

—

—

C₃⋊D₃

yes

characteristic, complemented

C₂³×C₃

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

C₂²×D₃

yes

—

#67

—

C₂×Dic₃

—

D₁₂

—

C₄×D₃

—

Dic₆

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#121

yes

characteristic

C₃⋊D₃

—

#41

—

C₂×Q₈

—

100

C₂×D₄

C₂×C₃⋊D₃

yes

characteristic

101

Q₁₆

—

102

SD₁₆

—

103

SD₁₆

—

104

SD₁₆

—

105

SD₁₆

—

106

D₈

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

C₂×C₃⋊D₃

yes

characteristic

110

M₁₆

C₂×C₃⋊D₃

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂²×C₃

—

114

C₂²×C₃

—

115

C₂²×C₃

#235

yes

—

116

C₂×D₃

—

117

C₃×C₄

D₃×D₄

yes

—

118

C₃×C₄

D₃×D₄

yes

—

119

Dic₃

—

120

Dic₃

—

121

C₃²

#471

yes

characteristic, complemented, Sylow

122

C₂³

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

C₂²×C₃⋊D₃

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₂×C₃

—

139

C₂×C₃

—

140

C₂×C₃

#884

yes

—

141

D₃

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#3828

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

D₄×C₃⋊D₃

yes

characteristic

149

C₄

D₄×C₃⋊D₃

yes

characteristic

150

C₃

#4748

yes

complemented

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

#64093

yes

characteristic

	subgroup	G/H
center	C₂	#64093
commutator, cosocle	C₂×C₃²×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#100	C₂
socle	C₂×C₃²	#121
prime radical	#64141	C₂
hypercenter	#100	C₃⋊D₃
hypocenter	C₃²	#471
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

subgroup

G/H

center

C₂

#64093

commutator, cosocle

C₂×C₃²×C₄

C₂³

Frattini

C₂×C₄

C₂²×C₃⋊D₃

Fitting

C₃²×#100

C₂

C₂×C₃²

C₂

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

N	H ≅ G/N	type
1. C₂×#3754	C₂	semidirect
3. #64118	C₂	semidirect
4. #64116	C₂	semidirect
5. C₃²×#100	C₂	semidirect
6. #63651	C₂	semidirect
7. #63649	C₂	semidirect
28. C₃×#100	D₃	semidirect
77. #100	C₃⋊D₃	semidirect
121. C₃²	#471	semidirect
150. C₃	#4748	semidirect

H ≅ G/N

type

1. C₂×#3754

C₂

semidirect

3. #64118

C₂

semidirect

4. #64116

C₂

semidirect

5. C₃²×#100

C₂

semidirect

6. #63651

C₂

semidirect

7. #63649

C₂

semidirect

28. C₃×#100

D₃

semidirect

77. #100

C₃⋊D₃

semidirect

121. C₃²

#471

semidirect

150. C₃

#4748

semidirect

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

subgroup	index
C₂×#3754	2
#64141	2
#64118	2
#64116	2
C₃²×#100	2
#63651	2
#63649	2
#4748	3

subgroup

index

C₃²×#100

Subgroups — #10592447

Conjugacy classes of subgroups

Complements

Maximal subgroups