Group #14517206 — subgroups

Subgroups — #14517206

Order 1944.

209 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 49 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	C₃×#6204	C₃	1	1	yes	maximal, characteristic, complemented
2	#10619916	C₃	1	2	yes	maximal, complemented
3	#10619913	C₃	1	1	yes	maximal, characteristic, complemented
4	Q₈×AΣL(1,𝔽₂₇)	C₃	1	1	yes	maximal, characteristic, complemented
5	#10619585	C₃	1	2	yes	maximal, complemented
6	#10619582	C₃	1	2	yes	maximal, complemented
7	#10619581	C₃	1	2	yes	maximal, complemented
8	#10619579	C₃	1	1	yes	maximal, characteristic, complemented
9	#10619578	C₃	1	1	yes	maximal, characteristic, complemented
10	C₂×#6913	—	4	1	no	maximal
11	C₄×AΣL(1,𝔽₂₇)	—	3	1	no	—
12	#6913	—	4	1	no	Sylow
13	C₃³×Q₈	C₃²	1	1	yes	characteristic
14	Q₈×3₋¹⁺²	C₃²	1	2	yes	—
15	Q₈×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
16	Q₈⋊H(3,𝔽₃)	C₃²	1	2	yes	—
17	#6204	—	3	2	no	—
18	#6204	—	3	2	no	—
19	#6204	—	3	1	no	—
20	#6204	C₃²	1	2	yes	—
21	#6204	C₃²	1	2	yes	—
22	C₃×Q₈⋊C₉	—	3	1	no	—
23	C₃×Q₈⋊C₉	C₃²	1	1	yes	characteristic
24	C₃×Q₈⋊C₉	C₃²	1	1	yes	characteristic
25	C₃×Q₈⋊C₉	C₃²	1	1	yes	characteristic
26	C₉⋊SL(2,𝔽₃)	—	3	2	no	—
27	C₉⋊SL(2,𝔽₃)	—	3	2	no	—
28	C₉×SL(2,𝔽₃)	—	3	2	no	—
29	C₂×#737	—	4	2	no	—
30	C₂×#737	—	4	1	no	—
31	C₂×C₃×3₋¹⁺²	—	4	1	no	—
32	C₂×#733	—	4	2	no	—
33	C₂×#732	—	4	1	no	—
34	C₂×#731	—	4	2	no	—
35	C₂×#731	—	4	1	no	—
36	C₂×AΣL(1,𝔽₂₇)	—	4	2	no	—
37	C₂×AΣL(1,𝔽₂₇)	A₄	1	1	yes	characteristic
38	C₃³×C₄	—	3	1	no	—
39	C₄×3₋¹⁺²	—	3	2	no	—
40	C₄×H(3,𝔽₃)	—	3	1	no	—
41	#737	—	4	2	no	—
42	#737	—	4	1	no	—
43	C₃×3₋¹⁺²	—	4	1	no	—
44	#733	—	4	2	no	—
45	#732	—	4	1	no	—
46	#731	—	4	2	no	—
47	#731	—	4	1	no	—
48	AΣL(1,𝔽₂₇)	—	4	2	no	—
49	AΣL(1,𝔽₂₇)	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
50	C₃²×Q₈	—	3	1	no	—
51	C₃²×Q₈	—	3	1	no	—
52	C₃²×Q₈	—	9	1	no	—
53	C₃²×Q₈	C₃³	1	1	yes	characteristic
54	C₃×SL(2,𝔽₃)	—	3	2	no	—
55	Q₈×C₉	—	3	2	no	—
56	Q₈⋊C₉	—	3	2	no	—
57	Q₈⋊C₉	—	3	2	no	—
58	Q₈⋊C₉	—	3	1	no	—
59	Q₈⋊C₉	—	3	1	no	—
60	Q₈⋊C₉	H(3,𝔽₃)	1	2	yes	—
61	Q₈⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
62	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
63	C₂×3₋¹⁺²	—	4	2	no	—
64	C₂×3₋¹⁺²	—	4	2	no	—
65	C₂×3₋¹⁺²	—	12	2	no	—
66	C₂×3₋¹⁺²	—	12	2	no	—
67	C₂×3₋¹⁺²	—	12	2	no	—
68	C₂×3₋¹⁺²	—	12	1	no	—
69	C₂×3₋¹⁺²	—	12	2	no	—
70	C₂×3₋¹⁺²	C₃×A₄	1	2	yes	—
71	C₂×H(3,𝔽₃)	—	4	2	no	—
72	C₂×H(3,𝔽₃)	C₃×A₄	1	1	yes	characteristic
73	C₂×C₃×C₉	—	4	1	no	—
74	C₂×C₃×C₉	—	4	1	no	—
75	C₂×C₃×C₉	—	4	1	no	—
76	C₂×C₃×C₉	—	12	2	no	—
77	C₂×C₃×C₉	—	12	1	no	—
78	C₃²×C₄	—	3	1	no	—
79	C₃²×C₄	—	9	1	no	—
80	C₃²×C₄	—	9	1	no	—
81	C₃²×C₄	—	9	3	no	—
82	C₄×C₉	—	9	2	no	—
83	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
84	3₋¹⁺²	—	4	2	no	—
85	3₋¹⁺²	—	4	2	no	—
86	3₋¹⁺²	—	12	2	no	—
87	3₋¹⁺²	—	12	2	no	—
88	3₋¹⁺²	—	12	2	no	—
89	3₋¹⁺²	—	12	1	no	—
90	3₋¹⁺²	—	12	2	no	—
91	3₋¹⁺²	C₃×SL(2,𝔽₃)	1	2	yes	—
92	H(3,𝔽₃)	—	4	2	no	—
93	H(3,𝔽₃)	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
94	C₃×C₉	—	4	1	no	—
95	C₃×C₉	—	4	1	no	—
96	C₃×C₉	—	4	1	no	—
97	C₃×C₉	—	12	2	no	—
98	C₃×C₉	—	12	1	no	—
99	C₃×Q₈	—	3	1	no	—
100	C₃×Q₈	—	9	1	no	—
101	C₃×Q₈	—	9	1	no	—
102	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
103	SL(2,𝔽₃)	—	9	2	no	—
104	C₂×C₃²	—	3	1	no	—
105	C₂×C₃²	—	3	1	no	—
106	C₂×C₃²	—	9	1	no	—
107	C₂×C₃²	—	12	2	no	—
108	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
109	C₂×C₉	—	3	2	no	—
110	C₂×C₉	—	4	2	no	—
111	C₂×C₉	—	4	1	no	—
112	C₂×C₉	—	12	1	no	—
113	C₂×C₉	—	12	1	no	—
114	C₂×C₉	—	12	2	no	—
115	C₂×C₉	—	12	2	no	—
116	C₃×C₄	—	3	1	no	—
117	C₃×C₄	—	9	1	no	—
118	C₃×C₄	—	9	3	no	—
119	C₃×C₄	—	27	1	no	—
120	C₃²	—	3	1	no	—
121	C₃²	—	3	1	no	—
122	C₃²	—	9	1	no	—
123	C₃²	—	12	2	no	—
124	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
125	C₉	—	3	2	no	—
126	C₉	—	4	2	no	—
127	C₉	—	4	1	no	—
128	C₉	—	12	1	no	—
129	C₉	—	12	1	no	—
130	C₉	—	12	2	no	—
131	C₉	—	12	2	no	—
132	Q₈	#6913	1	1	yes	characteristic, complemented, Sylow
133	C₂×C₃	—	3	1	no	—
134	C₂×C₃	—	9	1	no	—
135	C₂×C₃	—	9	1	no	—
136	C₂×C₃	—	36	2	no	—
137	C₂×C₃	A₄×H(3,𝔽₃)	1	1	yes	characteristic
138	C₄	—	3	1	no	—
139	C₃	—	3	1	no	—
140	C₃	—	9	1	no	—
141	C₃	—	9	1	no	—
142	C₃	—	36	2	no	—
143	C₃	SL(2,𝔽₃)×H(3,𝔽₃)	1	1	yes	characteristic
144	C₂	#11757799	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6204

C₃

yes

maximal, characteristic, complemented

#10619916

C₃

yes

maximal, complemented

#10619913

C₃

yes

maximal, characteristic, complemented

Q₈×AΣL(1,𝔽₂₇)

C₃

yes

maximal, characteristic, complemented

#10619585

C₃

yes

maximal, complemented

#10619582

C₃

yes

maximal, complemented

#10619581

C₃

yes

maximal, complemented

#10619579

C₃

yes

maximal, characteristic, complemented

#10619578

C₃

yes

maximal, characteristic, complemented

C₂×#6913

—

maximal

C₄×AΣL(1,𝔽₂₇)

—

#6913

—

Sylow

C₃³×Q₈

C₃²

yes

characteristic

Q₈×3₋¹⁺²

C₃²

yes

—

Q₈×H(3,𝔽₃)

C₃²

yes

characteristic

Q₈⋊H(3,𝔽₃)

C₃²

yes

—

—

—

—

C₃²

yes

—

#6204

C₃²

yes

—

C₃×Q₈⋊C₉

—

C₃×Q₈⋊C₉

C₃²

yes

characteristic

C₃×Q₈⋊C₉

C₃²

yes

characteristic

C₃×Q₈⋊C₉

C₃²

yes

characteristic

C₉⋊SL(2,𝔽₃)

—

C₉⋊SL(2,𝔽₃)

—

C₉×SL(2,𝔽₃)

—

C₂×#737

—

C₂×#737

—

C₂×C₃×3₋¹⁺²

—

—

—

—

—

C₂×AΣL(1,𝔽₂₇)

—

C₂×AΣL(1,𝔽₂₇)

A₄

yes

characteristic

C₃³×C₄

—

C₄×3₋¹⁺²

—

C₄×H(3,𝔽₃)

—

#737

—

#737

—

C₃×3₋¹⁺²

—

—

—

—

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

SL(2,𝔽₃)

yes

characteristic, complemented

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

C₃³

yes

characteristic

C₃×SL(2,𝔽₃)

—

Q₈×C₉

—

Q₈⋊C₉

—

Q₈⋊C₉

—

Q₈⋊C₉

—

Q₈⋊C₉

—

Q₈⋊C₉

H(3,𝔽₃)

yes

—

Q₈⋊C₉

H(3,𝔽₃)

yes

characteristic

C₂×C₃³

C₃×A₄

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₃×A₄

yes

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₃×A₄

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₄×C₉

—

C₃³

C₃×SL(2,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×SL(2,𝔽₃)

yes

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₃×SL(2,𝔽₃)

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×Q₈

—

100

C₃×Q₈

—

101

C₃×Q₈

—

102

C₃×Q₈

C₃×H(3,𝔽₃)

yes

characteristic

103

SL(2,𝔽₃)

—

104

C₂×C₃²

—

105

C₂×C₃²

—

106

C₂×C₃²

—

107

C₂×C₃²

—

108

C₂×C₃²

C₃²×A₄

yes

characteristic

109

C₂×C₉

—

110

C₂×C₉

—

111

C₂×C₉

—

112

C₂×C₉

—

113

C₂×C₉

—

114

C₂×C₉

—

115

C₂×C₉

—

116

C₃×C₄

—

117

C₃×C₄

—

118

C₃×C₄

—

119

C₃×C₄

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

C₃²×SL(2,𝔽₃)

yes

characteristic

125

C₉

—

126

C₉

—

127

C₉

—

128

C₉

—

129

C₉

—

130

C₉

—

131

C₉

—

132

Q₈

#6913

yes

characteristic, complemented, Sylow

133

C₂×C₃

—

134

C₂×C₃

—

135

C₂×C₃

—

136

C₂×C₃

—

137

C₂×C₃

A₄×H(3,𝔽₃)

yes

characteristic

138

C₄

—

139

C₃

—

140

C₃

—

141

C₃

—

142

C₃

—

143

C₃

SL(2,𝔽₃)×H(3,𝔽₃)

yes

characteristic

144

C₂

#11757799

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃	A₄×H(3,𝔽₃)
commutator, cosocle	C₃²×Q₈	C₃³
Frattini	C₂×C₃²	C₃²×A₄
Fitting	Q₈×AΣL(1,𝔽₂₇)	C₃
hypercenter	C₂×AΣL(1,𝔽₂₇)	A₄
hypocenter	Q₈	#6913
LCS	G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃² ◁ C₂×AΣL(1,𝔽₂₇)

subgroup

G/H

center, socle

C₂×C₃

A₄×H(3,𝔽₃)

commutator, cosocle

C₃²×Q₈

C₃³

Frattini

C₂×C₃²

C₃²×A₄

Fitting

Q₈×AΣL(1,𝔽₂₇)

C₃

hypercenter

C₂×AΣL(1,𝔽₂₇)

A₄

hypocenter

Q₈

#6913

LCS

G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈

UCS

C₁ ◁ C₂×C₃ ◁ C₂×C₃² ◁ C₂×AΣL(1,𝔽₂₇)

N	H ≅ G/N	type
1. C₃×#6204	C₃	semidirect
2. #10619916	C₃	semidirect
3. #10619913	C₃	semidirect
4. Q₈×AΣL(1,𝔽₂₇)	C₃	semidirect
5. #10619585	C₃	semidirect
6. #10619582	C₃	semidirect
7. #10619581	C₃	semidirect
8. #10619579	C₃	semidirect
9. #10619578	C₃	semidirect
49. AΣL(1,𝔽₂₇)	SL(2,𝔽₃)	semidirect
132. Q₈	#6913	semidirect

H ≅ G/N

type

1. C₃×#6204

C₃

semidirect

2. #10619916

C₃

semidirect

3. #10619913

C₃

semidirect

4. Q₈×AΣL(1,𝔽₂₇)

C₃

semidirect

5. #10619585

C₃

semidirect

6. #10619582

C₃

semidirect

7. #10619581

C₃

semidirect

8. #10619579

C₃

semidirect

9. #10619578

C₃

semidirect

49. AΣL(1,𝔽₂₇)

SL(2,𝔽₃)

semidirect

132. Q₈

#6913

semidirect

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

subgroup	index
C₃×#6204	3
#10619916	3
#10619913	3
Q₈×AΣL(1,𝔽₂₇)	3
#10619585	3
#10619582	3
#10619581	3
#10619579	3
#10619578	3
C₂×#6913	4

subgroup

index

Q₈×AΣL(1,𝔽₂₇)

Subgroups — #14517206

Conjugacy classes of subgroups

Complements

Maximal subgroups