Group #14517209 — subgroups

Subgroups — #14517209

Order 1944.

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

normal subgroups only

This group has 215 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	#10619916	C₃	1	2	yes	maximal, complemented
2	#10619913	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×SL(2,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	Q₈×#731	C₃	1	1	yes	maximal, characteristic, complemented
5	#10619583	C₃	1	2	yes	maximal, complemented
6	#10619581	C₃	1	2	yes	maximal, complemented
7	#10619580	C₃	1	1	yes	maximal, characteristic, complemented
8	#10619579	C₃	1	1	yes	maximal, characteristic, complemented
9	#10619574	C₃	1	2	yes	maximal, complemented
10	C₂×#6908	—	4	1	no	maximal
11	C₄×#731	—	3	1	no	—
12	#6908	—	4	1	no	Sylow
13	C₃²×SL(2,𝔽₃)	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	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
17	Q₈⋊H(3,𝔽₃)	C₃²	1	2	yes	—
18	#6204	—	3	2	no	—
19	#6204	C₃²	1	2	yes	—
20	#6204	C₃²	1	2	yes	—
21	C₃×Q₈⋊C₉	—	3	1	no	—
22	C₃×Q₈⋊C₉	C₃²	1	1	yes	characteristic
23	C₃×Q₈⋊C₉	C₃²	1	1	yes	characteristic
24	C₉⋊SL(2,𝔽₃)	—	3	4	no	—
25	C₉×SL(2,𝔽₃)	—	3	1	no	—
26	C₉×SL(2,𝔽₃)	—	3	2	no	—
27	C₉×SL(2,𝔽₃)	—	3	2	no	—
28	C₂×#737	—	4	2	no	—
29	C₂×#737	—	4	1	no	—
30	C₂×C₃²×C₉	—	4	1	no	—
31	C₂×#733	—	4	2	no	—
32	C₂×#732	—	4	1	no	—
33	C₂×#731	—	4	2	no	—
34	C₂×#731	A₄	1	1	yes	characteristic
35	C₂×AΣL(1,𝔽₂₇)	—	4	2	no	—
36	C₂×AΣL(1,𝔽₂₇)	—	4	1	no	—
37	C₄×3₋¹⁺²	—	3	2	no	—
38	C₄×H(3,𝔽₃)	—	3	1	no	—
39	C₃×C₄×C₉	—	3	1	no	—
40	#737	—	4	2	no	—
41	#737	—	4	1	no	—
42	C₃²×C₉	—	4	1	no	—
43	#733	—	4	2	no	—
44	#732	—	4	1	no	—
45	#731	—	4	2	no	—
46	#731	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
47	AΣL(1,𝔽₂₇)	—	4	2	no	—
48	AΣL(1,𝔽₂₇)	—	4	1	no	—
49	C₃²×Q₈	—	3	1	no	—
50	C₃²×Q₈	C₃³	1	1	yes	characteristic
51	C₃×SL(2,𝔽₃)	—	3	1	no	—
52	C₃×SL(2,𝔽₃)	—	3	3	no	—
53	C₃×SL(2,𝔽₃)	—	3	2	no	—
54	Q₈×C₉	—	3	1	no	—
55	Q₈×C₉	—	3	2	no	—
56	Q₈⋊C₉	—	3	1	no	—
57	Q₈⋊C₉	—	3	2	no	—
58	Q₈⋊C₉	—	3	2	no	—
59	Q₈⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
60	Q₈⋊C₉	H(3,𝔽₃)	1	2	yes	—
61	C₂×C₃³	—	4	1	no	—
62	C₂×3₋¹⁺²	—	4	2	no	—
63	C₂×3₋¹⁺²	—	4	2	no	—
64	C₂×3₋¹⁺²	—	12	4	no	—
65	C₂×3₋¹⁺²	—	12	2	no	—
66	C₂×3₋¹⁺²	C₃×A₄	1	2	yes	—
67	C₂×H(3,𝔽₃)	—	4	2	no	—
68	C₂×H(3,𝔽₃)	C₃×A₄	1	1	yes	characteristic
69	C₂×C₃×C₉	—	4	1	no	—
70	C₂×C₃×C₉	—	4	1	no	—
71	C₂×C₃×C₉	—	12	1	no	—
72	C₂×C₃×C₉	—	12	2	no	—
73	C₂×C₃×C₉	—	12	2	no	—
74	C₂×C₃×C₉	—	12	1	no	—
75	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
76	C₃²×C₄	—	3	1	no	—
77	C₃²×C₄	—	9	1	no	—
78	C₄×C₉	—	9	1	no	—
79	C₄×C₉	—	9	2	no	—
80	C₃³	—	4	1	no	—
81	3₋¹⁺²	—	4	2	no	—
82	3₋¹⁺²	—	4	2	no	—
83	3₋¹⁺²	—	12	4	no	—
84	3₋¹⁺²	—	12	2	no	—
85	3₋¹⁺²	C₃×SL(2,𝔽₃)	1	2	yes	—
86	H(3,𝔽₃)	—	4	2	no	—
87	H(3,𝔽₃)	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
88	C₃×C₉	—	4	1	no	—
89	C₃×C₉	—	4	1	no	—
90	C₃×C₉	—	12	1	no	—
91	C₃×C₉	—	12	2	no	—
92	C₃×C₉	—	12	2	no	—
93	C₃×C₉	—	12	1	no	—
94	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
95	C₃×Q₈	—	3	1	no	—
96	C₃×Q₈	—	9	1	no	—
97	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
98	SL(2,𝔽₃)	—	3	3	no	—
99	SL(2,𝔽₃)	—	9	2	no	—
100	C₂×C₃²	—	3	1	no	—
101	C₂×C₃²	—	12	3	no	—
102	C₂×C₃²	—	12	1	no	—
103	C₂×C₃²	—	12	2	no	—
104	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
105	C₂×C₉	—	3	1	no	—
106	C₂×C₉	—	3	2	no	—
107	C₂×C₉	—	4	1	no	—
108	C₂×C₉	—	4	2	no	—
109	C₂×C₉	—	12	1	no	—
110	C₂×C₉	—	12	2	no	—
111	C₂×C₉	—	12	2	no	—
112	C₃×C₄	—	3	1	no	—
113	C₃×C₄	—	9	1	no	—
114	C₃×C₄	—	27	1	no	—
115	C₃²	—	3	1	no	—
116	C₃²	—	12	3	no	—
117	C₃²	—	12	1	no	—
118	C₃²	—	12	2	no	—
119	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
120	C₉	—	3	1	no	—
121	C₉	—	3	2	no	—
122	C₉	—	4	1	no	—
123	C₉	—	4	2	no	—
124	C₉	—	12	1	no	—
125	C₉	—	12	2	no	—
126	C₉	—	12	2	no	—
127	Q₈	#6908	1	1	yes	characteristic, complemented, Sylow
128	C₂×C₃	—	3	1	no	—
129	C₂×C₃	—	9	1	no	—
130	C₂×C₃	—	12	3	no	—
131	C₂×C₃	—	36	2	no	—
132	C₂×C₃	A₄×H(3,𝔽₃)	1	1	yes	characteristic
133	C₄	—	3	1	no	—
134	C₃	—	3	1	no	—
135	C₃	—	9	1	no	—
136	C₃	—	12	3	no	—
137	C₃	—	36	2	no	—
138	C₃	SL(2,𝔽₃)×H(3,𝔽₃)	1	1	yes	characteristic
139	C₂	#11757802	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10619916

C₃

yes

maximal, complemented

#10619913

C₃

yes

maximal, characteristic, complemented

C₃×C₉×SL(2,𝔽₃)

C₃

yes

maximal, characteristic, complemented

Q₈×#731

C₃

yes

maximal, characteristic, complemented

#10619583

C₃

yes

maximal, complemented

#10619581

C₃

yes

maximal, complemented

#10619580

C₃

yes

maximal, characteristic, complemented

#10619579

C₃

yes

maximal, characteristic, complemented

#10619574

C₃

yes

maximal, complemented

—

—

—

C₃²×SL(2,𝔽₃)

C₃²

yes

characteristic

Q₈×3₋¹⁺²

C₃²

yes

—

Q₈×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×Q₈×C₉

C₃²

yes

characteristic

Q₈⋊H(3,𝔽₃)

C₃²

yes

—

#6204

—

#6204

C₃²

yes

—

#6204

C₃²

yes

—

C₃×Q₈⋊C₉

—

C₃×Q₈⋊C₉

C₃²

yes

characteristic

C₃×Q₈⋊C₉

C₃²

yes

characteristic

C₉⋊SL(2,𝔽₃)

—

C₉×SL(2,𝔽₃)

—

C₉×SL(2,𝔽₃)

—

C₉×SL(2,𝔽₃)

—

C₂×#737

—

C₂×#737

—

C₂×C₃²×C₉

—

—

—

—

A₄

yes

characteristic

C₂×AΣL(1,𝔽₂₇)

—

C₂×AΣL(1,𝔽₂₇)

—

C₄×3₋¹⁺²

—

C₄×H(3,𝔽₃)

—

C₃×C₄×C₉

—

#737

—

#737

—

C₃²×C₉

—

—

—

—

yes

characteristic, complemented

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₃²×Q₈

—

C₃²×Q₈

C₃³

yes

characteristic

C₃×SL(2,𝔽₃)

—

C₃×SL(2,𝔽₃)

—

C₃×SL(2,𝔽₃)

—

Q₈×C₉

—

Q₈×C₉

—

Q₈⋊C₉

—

Q₈⋊C₉

—

Q₈⋊C₉

—

Q₈⋊C₉

H(3,𝔽₃)

yes

characteristic

Q₈⋊C₉

H(3,𝔽₃)

yes

—

C₂×C₃³

—

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

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₃×A₄

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₄×C₉

—

C₄×C₉

—

C₃³

—

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×C₉

—

C₃×C₉

C₃×SL(2,𝔽₃)

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

C₃×H(3,𝔽₃)

yes

characteristic

SL(2,𝔽₃)

—

SL(2,𝔽₃)

—

100

C₂×C₃²

—

101

C₂×C₃²

—

102

C₂×C₃²

—

103

C₂×C₃²

—

104

C₂×C₃²

C₃²×A₄

yes

characteristic

105

C₂×C₉

—

106

C₂×C₉

—

107

C₂×C₉

—

108

C₂×C₉

—

109

C₂×C₉

—

110

C₂×C₉

—

111

C₂×C₉

—

112

C₃×C₄

—

113

C₃×C₄

—

114

C₃×C₄

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

C₃²×SL(2,𝔽₃)

yes

characteristic

120

C₉

—

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

—

126

C₉

—

127

Q₈

#6908

yes

characteristic, complemented, Sylow

128

C₂×C₃

—

129

C₂×C₃

—

130

C₂×C₃

—

131

C₂×C₃

—

132

C₂×C₃

A₄×H(3,𝔽₃)

yes

characteristic

133

C₄

—

134

C₃

—

135

C₃

—

136

C₃

—

137

C₃

—

138

C₃

SL(2,𝔽₃)×H(3,𝔽₃)

yes

characteristic

139

C₂

#11757802

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×#731	C₃
hypercenter	C₂×#731	A₄
hypocenter	Q₈	#6908
LCS	G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃² ◁ C₂×#731

subgroup

G/H

center, socle

C₂×C₃

A₄×H(3,𝔽₃)

commutator, cosocle

C₃²×Q₈

C₃³

Frattini

C₂×C₃²

C₃²×A₄

C₃

A₄

Q₈

G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈

UCS

C₁ ◁ C₂×C₃ ◁ C₂×C₃² ◁ C₂×#731

N	H ≅ G/N	type
1. #10619916	C₃	semidirect
2. #10619913	C₃	semidirect
3. C₃×C₉×SL(2,𝔽₃)	C₃	semidirect
4. Q₈×#731	C₃	semidirect
5. #10619583	C₃	semidirect
6. #10619581	C₃	semidirect
7. #10619580	C₃	semidirect
8. #10619579	C₃	semidirect
9. #10619574	C₃	semidirect
46. #731	SL(2,𝔽₃)	semidirect
127. Q₈	#6908	semidirect

H ≅ G/N

type

1. #10619916

C₃

semidirect

2. #10619913

C₃

semidirect

3. C₃×C₉×SL(2,𝔽₃)

C₃

semidirect

4. Q₈×#731

C₃

semidirect

5. #10619583

C₃

semidirect

6. #10619581

C₃

semidirect

7. #10619580

C₃

semidirect

8. #10619579

C₃

semidirect

9. #10619574

C₃

semidirect

46. #731

SL(2,𝔽₃)

semidirect

127. Q₈

#6908

semidirect

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

subgroup	index
#10619916	3
#10619913	3
C₃×C₉×SL(2,𝔽₃)	3
Q₈×#731	3
#10619583	3
#10619581	3
#10619580	3
#10619579	3
#10619574	3
C₂×#6908	4

subgroup

index

C₃×C₉×SL(2,𝔽₃)

Subgroups — #14517209

Conjugacy classes of subgroups

Complements

Maximal subgroups