Group #11800059 — subgroups

Subgroups — #11800059

Order 1152.

198 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 40 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	#10593536	C₂	1	1	yes	maximal, characteristic, complemented
2	#10593293	C₂	1	1	yes	maximal, characteristic
3	#10593254	C₂	1	1	yes	maximal, characteristic, complemented
4	#10593160	C₂	1	4	yes	maximal, complemented
5	C₄⋊PSU(3,𝔽₂)	—	2	1	no	—
6	#64231	—	2	1	no	—
7	#64225	C₂²	1	2	yes	—
8	#64213	C₂²	1	2	yes	—
9	#63797	—	2	1	no	—
10	#63769	—	2	1	no	—
11	C₈×#157	C₂²	1	1	yes	characteristic
12	#63746	C₂²	1	2	yes	—
13	C₂×PSU(3,𝔽₂)	—	4	1	no	—
14	C₂×GO⁺(4,𝔽₂)	—	4	1	no	—
15	C₂×GO⁺(4,𝔽₂)	—	4	1	no	—
16	#3802	—	2	2	no	—
17	C₄⋊#157	—	2	2	no	—
18	C₄×#157	D₄	1	1	yes	characteristic
19	#3787	C₂³	1	1	yes	characteristic
20	#3777	—	4	1	no	—
21	#3774	—	2	4	no	—
22	#3771	—	2	2	no	—
23	C₈×C₃⋊D₃	D₄	1	1	yes	characteristic
24	C₃×D₂₄	—	4	1	no	—
25	#3713	—	4	1	no	—
26	#2240	—	9	1	no	maximal, Sylow
27	D₃×D₈	—	12	1	no	—
28	C₂×D₃²	—	4	2	no	—
29	C₂×#157	—	4	2	no	—
30	C₂×#157	D₈	1	1	yes	characteristic
31	C₂×#157	D₈	1	1	yes	characteristic
32	PSU(3,𝔽₂)	—	8	2	no	—
33	GO⁺(4,𝔽₂)	—	8	2	no	—
34	GO⁺(4,𝔽₂)	—	8	2	no	—
35	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
36	C₃×D₁₂	—	4	2	no	—
37	#625	—	4	2	no	—
38	#622	#41	1	2	yes	—
39	C₃²×C₈	SD₁₆	1	1	yes	characteristic
40	C₃⋊(C₃⋊C₈)	SD₁₆	1	1	yes	characteristic
41	#500	—	9	1	no	—
42	#493	—	9	1	no	—
43	#357	—	9	4	no	—
44	C₄×C₁₆	—	9	1	no	—
45	D₃×D₄	—	12	2	no	—
46	C₃×D₈	—	12	1	no	—
47	#213	—	12	2	no	—
48	D₂₄	—	4	1	no	—
49	D₃×C₈	—	12	1	no	—
50	C₂×C₃⋊D₃	C₂×D₈	1	1	yes	characteristic
51	C₂×C₃×D₃	—	8	2	no	—
52	D₃²	—	8	2	no	—
53	D₃²	—	8	2	no	—
54	#157	—	8	2	no	—
55	#157	D₁₆	1	2	yes	—
56	#157	SD₃₂	1	2	yes	—
57	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
58	C₃⋊Dic₃	#124	1	1	yes	characteristic
59	C₂×D₈	—	9	2	no	—
60	C₂×D₈	—	18	1	no	—
61	C₄⋊Q₈	—	18	1	no	—
62	#127	—	18	1	no	—
63	C₂×C₁₆	—	9	2	no	—
64	#107	—	9	2	no	—
65	#107	—	18	1	no	—
66	C₄×C₈	—	9	1	no	—
67	C₂²×D₃	—	24	2	no	—
68	C₃×D₄	—	12	2	no	—
69	#67	—	24	2	no	—
70	D₁₂	—	4	2	no	—
71	C₄×D₃	—	12	1	no	—
72	C₃×C₈	—	4	1	no	—
73	C₃⋊C₈	—	12	1	no	—
74	C₂×C₃²	#486	1	1	yes	characteristic
75	C₃⋊D₃	C₂×D₁₆	1	1	yes	characteristic
76	C₃⋊D₃	C₂×SD₃₂	1	1	yes	characteristic
77	C₃×D₃	—	16	2	no	—
78	C₂×Q₈	—	36	1	no	—
79	C₂×D₄	—	18	2	no	—
80	C₂×D₄	—	36	1	no	—
81	C₂×D₄	—	36	1	no	—
82	D₈	—	12	1	no	—
83	D₈	—	18	4	no	—
84	D₈	—	36	1	no	—
85	C₂×C₈	—	9	1	no	—
86	C₂×C₈	—	9	1	no	—
87	C₄⋊C₄	—	18	2	no	—
88	C₄⋊C₄	—	36	1	no	—
89	C₄²	—	9	1	no	—
90	C₁₆	—	18	2	no	—
91	C₂²×C₃	—	24	2	no	—
92	C₂×D₃	—	8	2	no	—
93	C₂×D₃	—	12	1	no	—
94	C₂×D₃	—	48	2	no	—
95	C₂×D₃	—	48	2	no	—
96	C₃×C₄	—	4	1	no	—
97	Dic₃	—	12	1	no	—
98	C₃²	#2240	1	1	yes	characteristic, complemented, Sylow
99	C₂³	—	36	2	no	—
100	Q₈	—	72	2	no	—
101	D₄	—	12	2	no	—
102	D₄	—	36	2	no	—
103	D₄	—	72	2	no	—
104	D₄	—	72	2	no	—
105	C₂×C₄	—	9	1	no	—
106	C₂×C₄	—	9	1	no	—
107	C₂×C₄	—	9	1	no	—
108	C₂×C₄	—	36	2	no	—
109	C₈	—	9	2	no	—
110	C₈	—	9	1	no	—
111	C₈	AΓL(1,𝔽₉)	1	1	yes	characteristic
112	C₂×C₃	—	4	1	no	—
113	C₂×C₃	—	48	2	no	—
114	D₃	—	12	1	no	—
115	D₃	—	12	1	no	—
116	D₃	—	16	2	no	—
117	C₂²	—	9	1	no	—
118	C₂²	—	24	2	no	—
119	C₂²	—	72	2	no	—
120	C₂²	—	72	2	no	—
121	C₄	—	9	2	no	—
122	C₄	—	9	2	no	—
123	C₄	—	9	1	no	—
124	C₄	—	72	2	no	—
125	C₄	C₂×AΓL(1,𝔽₉)	1	1	yes	characteristic
126	C₃	—	4	1	no	—
127	C₂	—	9	1	no	—
128	C₂	—	9	1	no	—
129	C₂	—	48	2	no	—
130	C₂	#10596402	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593536

C₂

yes

maximal, characteristic, complemented

#10593293

C₂

yes

maximal, characteristic

#10593254

C₂

yes

maximal, characteristic, complemented

#10593160

C₂

yes

maximal, complemented

C₄⋊PSU(3,𝔽₂)

—

#64231

—

#64225

C₂²

yes

—

#64213

C₂²

yes

—

#63797

—

#63769

—

C₈×#157

C₂²

yes

characteristic

#63746

C₂²

yes

—

C₂×PSU(3,𝔽₂)

—

C₂×GO⁺(4,𝔽₂)

—

C₂×GO⁺(4,𝔽₂)

—

#3802

—

C₄⋊#157

—

C₄×#157

D₄

yes

characteristic

#3787

C₂³

yes

—

—

—

C₈×C₃⋊D₃

D₄

yes

characteristic

C₃×D₂₄

—

#3713

—

#2240

—

maximal, Sylow

D₃×D₈

—

C₂×D₃²

—

C₂×#157

—

C₂×#157

D₈

yes

characteristic

C₂×#157

D₈

yes

characteristic

PSU(3,𝔽₂)

—

GO⁺(4,𝔽₂)

—

GO⁺(4,𝔽₂)

—

C₄×C₃⋊D₃

C₂×D₄

yes

characteristic

C₃×D₁₂

—

#625

—

#622

#41

yes

—

C₃²×C₈

SD₁₆

yes

characteristic

C₃⋊(C₃⋊C₈)

SD₁₆

yes

—

—

—

C₄×C₁₆

—

D₃×D₄

—

C₃×D₈

—

#213

—

D₂₄

—

D₃×C₈

—

C₂×C₃⋊D₃

C₂×D₈

yes

characteristic

C₂×C₃×D₃

—

D₃²

—

D₃²

—

#157

—

#157

D₁₆

yes

—

#157

SD₃₂

yes

—

C₃²×C₄

C₂×SD₁₆

yes

characteristic

C₃⋊Dic₃

#124

yes

characteristic

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

#107

—

#107

—

C₄×C₈

—

C₂²×D₃

—

C₃×D₄

—

#67

—

D₁₂

—

C₄×D₃

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

#486

yes

characteristic

C₃⋊D₃

C₂×D₁₆

yes

characteristic

C₃⋊D₃

C₂×SD₃₂

yes

characteristic

C₃×D₃

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₁₆

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

Dic₃

—

C₃²

#2240

yes

characteristic, complemented, Sylow

C₂³

—

100

Q₈

—

101

D₄

—

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₈

—

110

C₈

—

111

C₈

AΓL(1,𝔽₉)

yes

characteristic

112

C₂×C₃

—

113

C₂×C₃

—

114

D₃

—

115

D₃

—

116

D₃

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

C₂×AΓL(1,𝔽₉)

yes

characteristic

126

C₃

—

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#10596402

yes

characteristic

	subgroup	G/H
center	C₂	#10596402
commutator, cosocle	#3787	C₂³
Frattini	C₄	C₂×AΓL(1,𝔽₉)
Fitting	C₃²×C₈	SD₁₆
socle	C₂×C₃²	#486
prime radical	#10593293	C₂
hypercenter	C₈	AΓL(1,𝔽₉)
hypocenter	C₃²	#2240
LCS	G ▷ #3787 ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

subgroup

G/H

C₂

C₂³

C₄

C₃²×C₈

SD₁₆

C₂×C₃²

C₂

C₈

C₃²

G ▷ #3787 ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈

N	H ≅ G/N	type
1. #10593536	C₂	semidirect
3. #10593254	C₂	semidirect
4. #10593160	C₂	semidirect
98. C₃²	#2240	semidirect

H ≅ G/N

type

1. #10593536

C₂

semidirect

3. #10593254

C₂

semidirect

4. #10593160

C₂

semidirect

98. C₃²

#2240

semidirect

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

subgroup	index
#10593536	2
#10593293	2
#10593254	2
#10593160	2
#2240	9

subgroup

index

Subgroups — #11800059

Conjugacy classes of subgroups

Complements

Maximal subgroups