Group #13136067 — subgroups

Subgroups — H(3,𝔽₃)×#355 #13136067

Order 1728.

352 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 91 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	H(3,𝔽₃)×#124	C₂	1	1	yes	maximal, characteristic
2	H(3,𝔽₃)×#101	C₂	1	2	yes	maximal, complemented
3	C₃²×#355	C₃	1	4	yes	maximal, complemented
4	C₂×Q₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
5	C₂×D₄×H(3,𝔽₃)	C₄	1	1	yes	characteristic
6	M₁₆×H(3,𝔽₃)	—	2	2	no	—
7	C₂²⋊C₄×H(3,𝔽₃)	—	2	2	no	—
8	C₄²×H(3,𝔽₃)	C₄	1	1	yes	characteristic
9	C₃²×#124	C₂×C₃	1	4	yes	—
10	C₃²×#101	C₂×C₃	1	8	yes	complemented
11	C₂³×H(3,𝔽₃)	—	2	1	no	—
12	Q₈×H(3,𝔽₃)	—	4	1	no	—
13	D₄×H(3,𝔽₃)	—	4	1	no	—
14	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
15	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×H(3,𝔽₃)	D₄	1	2	yes	—
17	C₈×H(3,𝔽₃)	—	4	2	no	—
18	C₃×#355	—	3	4	no	—
19	C₃×#355	C₃²	1	1	yes	characteristic
20	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
21	C₂×C₃²×D₄	C₃×C₄	1	4	yes	—
22	C₃²×M₁₆	—	2	8	no	—
23	C₃²×C₂²⋊C₄	—	2	8	no	—
24	C₃²×C₄²	C₃×C₄	1	4	yes	—
25	C₂²×H(3,𝔽₃)	—	4	2	no	—
26	C₂²×H(3,𝔽₃)	—	4	1	no	—
27	C₂²×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
28	C₄×H(3,𝔽₃)	—	2	2	no	—
29	C₄×H(3,𝔽₃)	—	2	1	no	—
30	C₄×H(3,𝔽₃)	—	4	1	no	—
31	C₃×#124	—	3	4	no	—
32	C₃×#124	C₂×C₃²	1	1	yes	characteristic
33	C₃×#101	—	3	8	no	—
34	C₃×#101	C₂×C₃²	1	2	yes	—
35	C₂³×C₃²	—	2	4	no	—
36	C₃²×Q₈	—	4	4	no	—
37	C₃²×D₄	—	4	4	no	—
38	C₂×C₃²×C₄	—	2	4	no	—
39	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
40	C₂×C₃²×C₄	C₃×D₄	1	8	yes	—
41	C₃²×C₈	—	4	8	no	—
42	#355	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×H(3,𝔽₃)	—	2	1	no	—
44	C₂×H(3,𝔽₃)	—	8	1	no	—
45	C₂×H(3,𝔽₃)	#99	1	1	yes	characteristic
46	C₂×C₃×Q₈	—	3	4	no	—
47	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
48	C₂×C₃×D₄	—	3	4	no	—
49	C₂×C₃×D₄	C₃²×C₄	1	1	yes	characteristic
50	C₃×M₁₆	—	2	2	no	—
51	C₃×M₁₆	—	6	8	no	—
52	C₃×C₂²⋊C₄	—	2	2	no	—
53	C₃×C₂²⋊C₄	—	6	8	no	—
54	C₃×C₄²	—	3	4	no	—
55	C₃×C₄²	C₃²×C₄	1	1	yes	characteristic
56	C₂²×C₃²	—	4	8	no	—
57	C₂²×C₃²	—	4	4	no	—
58	C₂²×C₃²	C₃×C₂²⋊C₄	1	4	yes	—
59	C₃²×C₄	—	2	8	no	—
60	C₃²×C₄	—	2	4	no	—
61	C₃²×C₄	—	4	4	no	—
62	#124	C₂×H(3,𝔽₃)	1	1	yes	characteristic
63	#101	C₂×H(3,𝔽₃)	1	2	yes	complemented
64	H(3,𝔽₃)	#355	1	1	yes	characteristic, direct product factor, Sylow
65	C₂³×C₃	—	2	1	no	—
66	C₂³×C₃	—	6	4	no	—
67	C₃×Q₈	—	4	1	no	—
68	C₃×Q₈	—	12	4	no	—
69	C₃×D₄	—	4	1	no	—
70	C₃×D₄	—	12	4	no	—
71	C₂×C₃×C₄	—	2	1	no	—
72	C₂×C₃×C₄	—	3	4	no	—
73	C₂×C₃×C₄	—	3	8	no	—
74	C₂×C₃×C₄	—	6	4	no	—
75	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
76	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
77	C₃×C₈	—	4	2	no	—
78	C₃×C₈	—	12	8	no	—
79	C₂×C₃²	—	2	4	no	—
80	C₂×C₃²	—	8	4	no	—
81	C₂×C₃²	C₃×#99	1	4	yes	—
82	C₂×Q₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
83	C₂×D₄	C₄×H(3,𝔽₃)	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	C₂²⋊C₄	—	2	2	no	—
86	C₄²	C₄×H(3,𝔽₃)	1	1	yes	characteristic
87	C₂²×C₃	—	3	4	no	—
88	C₂²×C₃	—	4	2	no	—
89	C₂²×C₃	—	4	1	no	—
90	C₂²×C₃	—	12	4	no	—
91	C₂²×C₃	—	12	8	no	—
92	C₂²×C₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
93	C₃×C₄	—	2	2	no	—
94	C₃×C₄	—	2	1	no	—
95	C₃×C₄	—	4	1	no	—
96	C₃×C₄	—	6	8	no	—
97	C₃×C₄	—	6	4	no	—
98	C₃×C₄	—	12	4	no	—
99	C₃²	C₃×#355	1	4	yes	complemented
100	C₂³	—	2	1	no	—
101	Q₈	—	4	1	no	—
102	D₄	—	4	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	C₂×C₄×H(3,𝔽₃)	1	1	yes	characteristic
105	C₂×C₄	D₄×H(3,𝔽₃)	1	2	yes	—
106	C₈	—	4	2	no	—
107	C₂×C₃	—	2	1	no	—
108	C₂×C₃	—	3	4	no	—
109	C₂×C₃	—	6	4	no	—
110	C₂×C₃	—	8	1	no	—
111	C₂×C₃	—	24	4	no	—
112	C₂×C₃	C₃²×#99	1	1	yes	characteristic
113	C₂²	—	4	2	no	—
114	C₂²	—	4	1	no	—
115	C₂²	C₂²⋊C₄×H(3,𝔽₃)	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	1	no	—
118	C₄	—	4	1	no	—
119	C₃	—	3	4	no	—
120	C₃	C₃²×#355	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	8	1	no	—
123	C₂	H(3,𝔽₃)×#99	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

H(3,𝔽₃)×#124

C₂

yes

maximal, characteristic

H(3,𝔽₃)×#101

C₂

yes

maximal, complemented

C₃²×#355

C₃

yes

maximal, complemented

C₂×Q₈×H(3,𝔽₃)

C₂²

yes

characteristic

C₂×D₄×H(3,𝔽₃)

C₄

yes

characteristic

M₁₆×H(3,𝔽₃)

—

C₂²⋊C₄×H(3,𝔽₃)

—

C₄²×H(3,𝔽₃)

C₄

yes

characteristic

C₃²×#124

C₂×C₃

yes

—

C₃²×#101

C₂×C₃

yes

complemented

C₂³×H(3,𝔽₃)

—

Q₈×H(3,𝔽₃)

—

D₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

C₈×H(3,𝔽₃)

—

C₃×#355

—

C₃×#355

C₃²

yes

characteristic

C₂×C₃²×Q₈

C₂²×C₃

yes

—

C₂×C₃²×D₄

C₃×C₄

yes

—

C₃²×M₁₆

—

C₃²×C₂²⋊C₄

—

C₃²×C₄²

C₃×C₄

yes

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

C₂²⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₃×#124

—

C₃×#124

C₂×C₃²

yes

characteristic

C₃×#101

—

C₃×#101

C₂×C₃²

yes

—

C₂³×C₃²

—

C₃²×Q₈

—

C₃²×D₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×C₃×C₄

yes

—

C₂×C₃²×C₄

C₃×D₄

yes

—

C₃²×C₈

—

#355

H(3,𝔽₃)

yes

characteristic, direct product factor, Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

#99

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₂²×C₃²

yes

characteristic

C₂×C₃×D₄

—

C₂×C₃×D₄

C₃²×C₄

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₃²×C₄

yes

characteristic

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₃×C₂²⋊C₄

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

#124

C₂×H(3,𝔽₃)

yes

characteristic

#101

C₂×H(3,𝔽₃)

yes

complemented

H(3,𝔽₃)

#355

yes

characteristic, direct product factor, Sylow

C₂³×C₃

—

C₂³×C₃

—

C₃×Q₈

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃²×C₄

yes

characteristic

C₂×C₃×C₄

C₃²×D₄

yes

—

C₃×C₈

—

C₃×C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₃×#99

yes

—

C₂×Q₈

C₂²×H(3,𝔽₃)

yes

characteristic

C₂×D₄

C₄×H(3,𝔽₃)

yes

characteristic

M₁₆

—

C₂²⋊C₄

—

C₄²

C₄×H(3,𝔽₃)

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₃²×C₂²⋊C₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃²

C₃×#355

yes

complemented

100

C₂³

—

101

Q₈

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₄×H(3,𝔽₃)

yes

characteristic

105

C₂×C₄

D₄×H(3,𝔽₃)

yes

—

106

C₈

—

107

C₂×C₃

—

108

C₂×C₃

—

109

C₂×C₃

—

110

C₂×C₃

—

111

C₂×C₃

—

112

C₂×C₃

C₃²×#99

yes

characteristic

113

C₂²

—

114

C₂²

—

115

C₂²

C₂²⋊C₄×H(3,𝔽₃)

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₃

—

120

C₃

C₃²×#355

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

H(3,𝔽₃)×#99

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₃²×#99
commutator	C₂×C₃×C₄	C₂×C₃²×C₄
Frattini, cosocle	C₂×C₃×Q₈	C₂²×C₃²
prime radical	C₂×D₄×H(3,𝔽₃)	C₄
LCS	G ▷ C₂×C₃×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×H(3,𝔽₃) ◁ C₂×Q₈×H(3,𝔽₃) ◁ G

subgroup

G/H

center, socle

C₂×C₃

C₃²×#99

commutator

C₂×C₃×C₄

C₂×C₃²×C₄

Frattini, cosocle

C₂×C₃×Q₈

C₂²×C₃²

prime radical

C₂×D₄×H(3,𝔽₃)

C₄

LCS

G ▷ C₂×C₃×C₄ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₃ ◁ C₂²×H(3,𝔽₃) ◁ C₂×Q₈×H(3,𝔽₃) ◁ G

N	H ≅ G/N	type
2. H(3,𝔽₃)×#101	C₂	semidirect
3. C₃²×#355	C₃	semidirect
10. C₃²×#101	C₂×C₃	semidirect
42. #355	H(3,𝔽₃)	direct (N × H)
63. #101	C₂×H(3,𝔽₃)	semidirect
64. H(3,𝔽₃)	#355	direct (N × H)
99. C₃²	C₃×#355	semidirect

H ≅ G/N

type

2. H(3,𝔽₃)×#101

C₂

semidirect

3. C₃²×#355

C₃

semidirect

10. C₃²×#101

C₂×C₃

semidirect

42. #355

H(3,𝔽₃)

direct (N × H)

63. #101

C₂×H(3,𝔽₃)

semidirect

64. H(3,𝔽₃)

#355

direct (N × H)

99. C₃²

C₃×#355

semidirect

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

subgroup	index
H(3,𝔽₃)×#124	2
H(3,𝔽₃)×#101	2
C₃²×#355	3

subgroup

index

H(3,𝔽₃)×#124

H(3,𝔽₃)×#101

C₃²×#355

Subgroups — H(3,𝔽₃)×#355 #13136067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽3)×#355 #13136067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#355 #13136067