Group #13046826 — subgroups

Subgroups — #13046826

Order 1296.

200 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 35 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	Q₈×AΣL(1,𝔽₂₇)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊#4183	C₂	1	1	yes	maximal, characteristic
3	#10619509	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#3755	—	3	1	no	maximal
5	#3675⋊C₃	—	3	2	no	maximal
6	#3755⋊C₃	—	3	1	no	maximal
7	C₂×#4183	—	2	1	no	—
8	C₄×AΣL(1,𝔽₂₇)	—	2	1	no	—
9	C₄×AΣL(1,𝔽₂₇)	C₂²	1	1	yes	characteristic
10	C₃³×Q₈	D₃	1	1	yes	characteristic, complemented
11	C₃×C₄⋊(C₃⋊D₃)	—	3	1	no	—
12	C₃×C₃⋊(C₃⋊C₈)	—	3	1	no	—
13	Q₈×3₋¹⁺²	D₃	1	2	yes	complemented
14	Q₈×H(3,𝔽₃)	D₃	1	1	yes	characteristic, complemented
15	C₄⋊AGL(1,ℤ/9)	—	3	2	no	—
16	C₄⋊((C₃⋊D₃)⋊C₃)	—	3	1	no	—
17	3₋¹⁺²⋊C₈	—	3	2	no	—
18	(C₃⋊(C₃⋊C₈))⋊C₃	—	3	1	no	—
19	C₂×AΣL(1,𝔽₂₇)	D₄	1	1	yes	characteristic
20	#4183	—	4	1	no	—
21	#3755	—	3	1	no	—
22	#3755	—	3	1	no	—
23	C₃×#215	—	9	3	no	—
24	C₃×#215	—	9	1	no	—
25	#3675	—	3	2	no	—
26	C₂×C₃×C₃⋊D₃	—	6	1	no	—
27	C₃³×C₄	—	2	1	no	—
28	C₃³×C₄	C₂×D₃	1	1	yes	characteristic
29	C₂×AGL(1,ℤ/9)	—	6	2	no	—
30	C₂×(C₃⋊D₃)⋊C₃	—	6	1	no	—
31	C₄×3₋¹⁺²	—	2	2	no	—
32	C₄×3₋¹⁺²	C₂×D₃	1	2	yes	—
33	C₄×H(3,𝔽₃)	—	2	1	no	—
34	C₄×H(3,𝔽₃)	C₂×D₃	1	1	yes	characteristic
35	AΣL(1,𝔽₂₇)	SD₁₆	1	1	yes	characteristic, complemented, Sylow
36	C₃²×Q₈	—	3	1	no	—
37	C₃²×Q₈	—	3	3	no	—
38	C₃²×Q₈	—	3	1	no	—
39	C₃²×Q₈	C₃⋊D₃	1	1	yes	characteristic
40	C₄⋊(C₃⋊D₃)	—	3	1	no	—
41	C₄⋊(C₃⋊D₃)	—	3	1	no	—
42	C₃×D₁₂	—	9	3	no	—
43	C₃×D₁₂	—	9	1	no	—
44	C₃⋊(C₃⋊C₈)	—	3	1	no	—
45	C₃⋊(C₃⋊C₈)	—	3	1	no	—
46	C₃×C₃⋊C₈	—	9	3	no	—
47	C₃×C₃⋊C₈	—	9	1	no	—
48	Q₈×C₉	—	3	2	no	—
49	D₃₆	—	3	2	no	—
50	C₉⋊C₈	—	3	2	no	—
51	C₂×C₃³	#67	1	1	yes	characteristic
52	C₃×C₃⋊D₃	—	12	1	no	—
53	C₂×3₋¹⁺²	#67	1	2	yes	—
54	C₂×H(3,𝔽₃)	#67	1	1	yes	characteristic
55	AGL(1,ℤ/9)	—	12	2	no	—
56	(C₃⋊D₃)⋊C₃	—	12	1	no	—
57	C₃×SD₁₆	—	27	1	no	—
58	#215	—	9	3	no	—
59	#215	—	9	1	no	—
60	#215	—	27	1	no	—
61	C₂×C₃⋊D₃	—	6	1	no	—
62	C₂×C₃⋊D₃	—	6	1	no	—
63	C₂×C₃×D₃	—	18	3	no	—
64	C₂×C₃×D₃	—	18	1	no	—
65	C₃²×C₄	—	2	1	no	—
66	C₃²×C₄	—	3	1	no	—
67	C₃²×C₄	—	3	3	no	—
68	C₃²×C₄	—	3	1	no	—
69	C₃²×C₄	—	6	1	no	—
70	C₃²×C₄	—	6	1	no	—
71	C₃²×C₄	—	6	3	no	—
72	C₃²×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
73	C₂×D₉	—	6	2	no	—
74	C₄×C₉	—	3	2	no	—
75	C₄×C₉	—	6	2	no	—
76	C₃³	#215	1	1	yes	characteristic, complemented
77	3₋¹⁺²	#215	1	2	yes	complemented
78	H(3,𝔽₃)	#215	1	1	yes	characteristic, complemented
79	C₃×Q₈	—	3	1	no	—
80	C₃×Q₈	—	3	3	no	—
81	C₃×Q₈	—	9	1	no	—
82	C₃×Q₈	#272	1	1	yes	characteristic
83	C₃×D₄	—	27	1	no	—
84	D₁₂	—	9	3	no	—
85	D₁₂	—	9	1	no	—
86	D₁₂	—	27	1	no	—
87	C₃×C₈	—	27	1	no	—
88	C₃⋊C₈	—	9	3	no	—
89	C₃⋊C₈	—	9	1	no	—
90	C₃⋊C₈	—	27	1	no	—
91	C₂×C₃²	—	3	1	no	—
92	C₂×C₃²	—	3	3	no	—
93	C₂×C₃²	—	3	1	no	—
94	C₂×C₃²	#638	1	1	yes	characteristic
95	C₃⋊D₃	—	12	1	no	—
96	C₃⋊D₃	—	12	1	no	—
97	C₃×D₃	—	36	3	no	—
98	C₃×D₃	—	36	1	no	—
99	C₂×C₉	—	3	2	no	—
100	D₉	—	12	2	no	—
101	SD₁₆	—	27	1	no	Sylow
102	C₂²×C₃	—	54	1	no	—
103	C₂×D₃	—	18	3	no	—
104	C₂×D₃	—	18	1	no	—
105	C₂×D₃	—	54	1	no	—
106	C₃×C₄	—	2	1	no	—
107	C₃×C₄	—	3	3	no	—
108	C₃×C₄	—	3	1	no	—
109	C₃×C₄	—	6	1	no	—
110	C₃×C₄	—	6	3	no	—
111	C₃×C₄	—	9	1	no	—
112	C₃×C₄	—	18	1	no	—
113	C₃×C₄	C₂×#272	1	1	yes	characteristic
114	C₃²	—	3	3	no	—
115	C₃²	—	3	1	no	—
116	C₃²	—	3	1	no	—
117	C₃²	#3755	1	1	yes	characteristic
118	C₉	—	3	2	no	—
119	Q₈	#4183	1	1	yes	characteristic, complemented
120	D₄	—	27	1	no	—
121	C₈	—	27	1	no	—
122	C₂×C₃	—	3	3	no	—
123	C₂×C₃	—	3	1	no	—
124	C₂×C₃	—	9	1	no	—
125	C₂×C₃	—	108	1	no	—
126	C₂×C₃	#6235	1	1	yes	characteristic
127	D₃	—	36	3	no	—
128	D₃	—	36	1	no	—
129	D₃	—	108	1	no	—
130	C₂²	—	54	1	no	—
131	C₄	—	2	1	no	—
132	C₄	C₂×#4183	1	1	yes	characteristic
133	C₃	—	3	3	no	—
134	C₃	—	3	1	no	—
135	C₃	—	9	1	no	—
136	C₃	#88549	1	1	yes	characteristic
137	C₂	—	108	1	no	—
138	C₂	#10619679	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×AΣL(1,𝔽₂₇)

C₂

yes

maximal, characteristic, complemented

C₄⋊#4183

C₂

yes

maximal, characteristic

#10619509

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₄×AΣL(1,𝔽₂₇)

—

C₄×AΣL(1,𝔽₂₇)

C₂²

yes

characteristic

C₃³×Q₈

D₃

yes

characteristic, complemented

C₃×C₄⋊(C₃⋊D₃)

—

C₃×C₃⋊(C₃⋊C₈)

—

Q₈×3₋¹⁺²

D₃

yes

complemented

Q₈×H(3,𝔽₃)

D₃

yes

characteristic, complemented

C₄⋊AGL(1,ℤ/9)

—

C₄⋊((C₃⋊D₃)⋊C₃)

—

3₋¹⁺²⋊C₈

—

(C₃⋊(C₃⋊C₈))⋊C₃

—

C₂×AΣL(1,𝔽₂₇)

D₄

yes

—

—

—

—

—

—

C₂×C₃×C₃⋊D₃

—

C₃³×C₄

—

C₃³×C₄

C₂×D₃

yes

characteristic

C₂×AGL(1,ℤ/9)

—

C₂×(C₃⋊D₃)⋊C₃

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₂×D₃

yes

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂×D₃

yes

characteristic

AΣL(1,𝔽₂₇)

SD₁₆

yes

characteristic, complemented, Sylow

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

C₃⋊D₃

yes

characteristic

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₃×D₁₂

—

C₃×D₁₂

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

C₃×C₃⋊C₈

—

C₃×C₃⋊C₈

—

Q₈×C₉

—

D₃₆

—

C₉⋊C₈

—

C₂×C₃³

#67

yes

characteristic

C₃×C₃⋊D₃

—

C₂×3₋¹⁺²

#67

yes

—

C₂×H(3,𝔽₃)

#67

yes

characteristic

AGL(1,ℤ/9)

—

(C₃⋊D₃)⋊C₃

—

C₃×SD₁₆

—

#215

—

#215

—

#215

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×C₃⋊D₃

yes

characteristic

C₂×D₉

—

C₄×C₉

—

C₄×C₉

—

C₃³

#215

yes

characteristic, complemented

3₋¹⁺²

yes

yes

characteristic, complemented

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

#272

yes

characteristic

C₃×D₄

—

D₁₂

—

D₁₂

—

D₁₂

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#638

yes

characteristic

C₃⋊D₃

—

C₃⋊D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

100

D₉

—

101

SD₁₆

—

Sylow

102

C₂²×C₃

—

103

C₂×D₃

—

104

C₂×D₃

—

105

C₂×D₃

—

106

C₃×C₄

—

107

C₃×C₄

—

108

C₃×C₄

—

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

—

112

C₃×C₄

—

113

C₃×C₄

C₂×#272

yes

characteristic

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

#3755

yes

characteristic

118

C₉

—

119

Q₈

#4183

yes

characteristic, complemented

120

D₄

—

121

C₈

—

122

C₂×C₃

—

123

C₂×C₃

—

124

C₂×C₃

—

125

C₂×C₃

—

108

—

126

C₂×C₃

#6235

yes

characteristic

127

D₃

—

128

D₃

—

129

D₃

—

108

—

130

C₂²

—

131

C₄

—

132

C₄

C₂×#4183

yes

characteristic

133

C₃

—

134

C₃

—

135

C₃

—

136

C₃

#88549

yes

characteristic

137

C₂

—

108

—

138

C₂

#10619679

yes

characteristic

	subgroup	G/H
center	C₂	#10619679
commutator, cosocle	C₄×AΣL(1,𝔽₂₇)	C₂²
Frattini	C₃²×C₄	C₂×C₃⋊D₃
Fitting	Q₈×AΣL(1,𝔽₂₇)	C₂
socle	C₂×C₃	#6235
prime radical	C₄⋊#4183	C₂
hypercenter	Q₈	#4183
hypocenter	AΣL(1,𝔽₂₇)	SD₁₆
LCS	G ▷ C₄×AΣL(1,𝔽₂₇) ▷ C₂×AΣL(1,𝔽₂₇) ▷ AΣL(1,𝔽₂₇)
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

subgroup

G/H

center

C₂

#10619679

commutator, cosocle

C₄×AΣL(1,𝔽₂₇)

C₂²

Frattini

C₃²×C₄

C₂×C₃⋊D₃

Fitting

Q₈×AΣL(1,𝔽₂₇)

C₂

C₂×C₃

C₂

Q₈

AΣL(1,𝔽₂₇)

SD₁₆

LCS

G ▷ C₄×AΣL(1,𝔽₂₇) ▷ C₂×AΣL(1,𝔽₂₇) ▷ AΣL(1,𝔽₂₇)

UCS

C₁ ◁ C₂ ◁ C₄ ◁ Q₈

N	H ≅ G/N	type
1. Q₈×AΣL(1,𝔽₂₇)	C₂	semidirect
3. #10619509	C₂	semidirect
10. C₃³×Q₈	D₃	semidirect
13. Q₈×3₋¹⁺²	D₃	semidirect
14. Q₈×H(3,𝔽₃)	D₃	semidirect
35. AΣL(1,𝔽₂₇)	SD₁₆	semidirect
76. C₃³	#215	semidirect
77. 3₋¹⁺²	#215	semidirect
78. H(3,𝔽₃)	#215	semidirect
119. Q₈	#4183	semidirect

H ≅ G/N

type

1. Q₈×AΣL(1,𝔽₂₇)

C₂

semidirect

3. #10619509

C₂

semidirect

10. C₃³×Q₈

D₃

semidirect

13. Q₈×3₋¹⁺²

D₃

semidirect

14. Q₈×H(3,𝔽₃)

D₃

semidirect

35. AΣL(1,𝔽₂₇)

SD₁₆

semidirect

76. C₃³

#215

semidirect

77. 3₋¹⁺²

#215

semidirect

78. H(3,𝔽₃)

#215

semidirect

119. Q₈

#4183

semidirect

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

subgroup	index
Q₈×AΣL(1,𝔽₂₇)	2
C₄⋊#4183	2
#10619509	2
C₃×#3755	3
#3675⋊C₃	3
#3755⋊C₃	3

subgroup

index

Q₈×AΣL(1,𝔽₂₇)

Subgroups — #13046826

Conjugacy classes of subgroups

Complements

Maximal subgroups