Group #13048782 — subgroups

Subgroups — C₂×#10619905 #13048782

Order 1296.

275 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 68 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	C₂×#66410	C₂	1	1	yes	maximal, characteristic
2	C₂×#66409	C₂	1	2	yes	maximal
3	#10619905	C₂	1	4	yes	maximal, direct product factor
4	C₂×#6196	—	3	1	no	maximal
5	C₂×#6189	—	3	2	no	maximal
6	C₂²×#737	C₂²	1	1	yes	characteristic
7	#66410	C₂²	1	2	yes	—
8	#66409	C₂²	1	4	yes	—
9	C₂×#1084	D₃	1	1	yes	characteristic
10	C₂×C₃⋊Dic₉	D₃	1	2	yes	—
11	C₂×3₋¹⁺²⋊C₄	—	6	2	no	—
12	C₂×(C₃⋊Dic₃)⋊C₃	—	3	2	no	—
13	C₂×C₉×Dic₃	—	3	2	no	—
14	C₂×C₉×Dic₃	—	6	1	no	—
15	C₂×C₃×Dic₉	—	3	2	no	—
16	#6196	—	3	4	no	—
17	#6189	—	3	8	no	—
18	C₂×#737	Q₈	1	2	yes	complemented
19	C₂×#737	C₂³	1	1	yes	characteristic
20	C₂×#627	—	9	2	no	—
21	C₂×Dic₁₈	—	9	1	no	—
22	C₂²×3₋¹⁺²	—	2	2	no	—
23	C₂²×3₋¹⁺²	—	4	1	no	—
24	C₂²×H(3,𝔽₃)	C₂×D₃	1	1	yes	characteristic
25	C₂²×C₃×C₉	—	2	1	no	—
26	C₂²×C₃×C₉	C₂×D₃	1	2	yes	—
27	#1084	C₂×D₃	1	2	yes	—
28	C₃⋊Dic₉	C₂×D₃	1	4	yes	—
29	3₋¹⁺²⋊C₄	—	6	4	no	—
30	(C₃⋊Dic₃)⋊C₃	—	3	4	no	—
31	C₉×Dic₃	—	3	4	no	—
32	C₉×Dic₃	—	6	2	no	—
33	C₃×Dic₉	—	3	4	no	—
34	#737	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₃⋊Dic₃	—	3	2	no	—
36	C₂×C₃×Dic₃	—	3	2	no	—
37	C₂×C₃×Dic₃	—	6	1	no	—
38	C₂×C₃×Dic₃	—	9	2	no	—
39	#627	—	9	8	no	—
40	C₂×C₄×C₉	—	9	1	no	—
41	C₂×Dic₉	—	3	2	no	—
42	C₂×Dic₉	—	6	2	no	—
43	Dic₁₈	—	9	4	no	—
44	C₂×3₋¹⁺²	—	2	4	no	—
45	C₂×3₋¹⁺²	—	2	2	no	—
46	C₂×3₋¹⁺²	—	4	2	no	—
47	C₂×3₋¹⁺²	—	4	1	no	—
48	C₂×H(3,𝔽₃)	Dic₆	1	2	yes	—
49	C₂×H(3,𝔽₃)	C₂²×D₃	1	1	yes	characteristic
50	C₂×C₃×C₉	—	2	2	no	—
51	C₂×C₃×C₉	—	2	1	no	—
52	C₂×C₃×C₉	Dic₆	1	4	yes	complemented
53	C₂×C₃×C₉	C₂²×D₃	1	2	yes	—
54	C₂×Dic₆	—	27	2	no	—
55	C₂×Dic₆	—	27	1	no	—
56	C₂²×C₃²	—	2	1	no	—
57	C₂²×C₃²	D₃²	1	2	yes	—
58	C₃⋊Dic₃	—	3	4	no	—
59	C₃×Dic₃	—	3	4	no	—
60	C₃×Dic₃	—	6	2	no	—
61	C₃×Dic₃	—	9	4	no	—
62	C₂²×C₉	—	2	2	no	—
63	C₂²×C₉	—	4	1	no	—
64	C₂²×C₉	D₃²	1	1	yes	characteristic
65	C₄×C₉	—	9	2	no	—
66	Dic₉	—	3	4	no	—
67	Dic₉	—	6	4	no	—
68	3₋¹⁺²	—	2	2	no	—
69	3₋¹⁺²	—	4	1	no	—
70	H(3,𝔽₃)	C₂×Dic₆	1	1	yes	characteristic
71	C₃×C₉	—	2	1	no	—
72	C₃×C₉	C₂×Dic₆	1	2	yes	complemented
73	C₂×C₃×C₄	—	9	1	no	—
74	C₂×C₃×C₄	—	27	2	no	—
75	C₂×Dic₃	—	3	2	no	—
76	C₂×Dic₃	—	6	1	no	—
77	C₂×Dic₃	—	9	2	no	—
78	C₂×Dic₃	—	27	2	no	—
79	Dic₆	—	27	8	no	—
80	Dic₆	—	27	4	no	—
81	C₂×C₃²	—	2	2	no	—
82	C₂×C₃²	—	2	1	no	—
83	C₂×C₃²	#627	1	4	yes	—
84	C₂×C₃²	C₂×D₃²	1	2	yes	—
85	C₂×C₉	—	2	4	no	—
86	C₂×C₉	—	2	2	no	—
87	C₂×C₉	—	4	2	no	—
88	C₂×C₉	—	4	1	no	—
89	C₂×C₉	#627	1	2	yes	—
90	C₂×C₉	C₂×D₃²	1	1	yes	characteristic
91	C₂×Q₈	—	81	1	no	Sylow
92	C₂²×C₃	—	3	2	no	—
93	C₂²×C₃	—	6	1	no	—
94	C₂²×C₃	#1113	1	1	yes	characteristic
95	C₃×C₄	—	9	2	no	—
96	C₃×C₄	—	27	4	no	—
97	Dic₃	—	3	4	no	—
98	Dic₃	—	6	2	no	—
99	Dic₃	—	9	4	no	—
100	Dic₃	—	27	4	no	—
101	C₃²	—	2	1	no	—
102	C₃²	C₂×#627	1	2	yes	—
103	C₉	—	2	2	no	—
104	C₉	—	4	1	no	—
105	C₉	C₂×#627	1	1	yes	characteristic
106	Q₈	—	81	4	no	—
107	C₂×C₄	—	9	1	no	—
108	C₂×C₄	—	27	2	no	—
109	C₂×C₃	—	3	4	no	—
110	C₂×C₃	—	3	2	no	—
111	C₂×C₃	—	6	2	no	—
112	C₂×C₃	—	6	1	no	—
113	C₂×C₃	#6296	1	2	yes	—
114	C₂×C₃	C₂×#1113	1	1	yes	characteristic
115	C₂²	#4207⋊C₂	1	1	yes	characteristic
116	C₄	—	9	2	no	—
117	C₄	—	27	4	no	—
118	C₃	—	3	2	no	—
119	C₃	—	6	1	no	—
120	C₃	C₂×#6296	1	1	yes	characteristic
121	C₂	#10619905	1	2	yes	direct product factor
122	C₂	C₂×#4207⋊C₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#66410

C₂

yes

maximal, characteristic

C₂×#66409

C₂

yes

maximal

#10619905

C₂

yes

maximal, direct product factor

—

—

C₂²×#737

C₂²

yes

characteristic

#66410

C₂²

yes

—

#66409

C₂²

yes

—

C₂×#1084

D₃

yes

characteristic

C₂×C₃⋊Dic₉

D₃

yes

—

C₂×3₋¹⁺²⋊C₄

—

C₂×(C₃⋊Dic₃)⋊C₃

—

C₂×C₉×Dic₃

—

C₂×C₉×Dic₃

—

C₂×C₃×Dic₉

—

#6196

—

#6189

—

C₂×#737

Q₈

yes

complemented

C₂×#737

C₂³

yes

characteristic

C₂×#627

—

C₂×Dic₁₈

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

—

C₂²×H(3,𝔽₃)

C₂×D₃

yes

characteristic

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₂×D₃

yes

—

#1084

C₂×D₃

yes

—

C₃⋊Dic₉

C₂×D₃

yes

—

3₋¹⁺²⋊C₄

—

(C₃⋊Dic₃)⋊C₃

—

C₉×Dic₃

—

C₉×Dic₃

—

C₃×Dic₉

—

#737

C₂×Q₈

yes

characteristic, complemented, Sylow

C₂×C₃⋊Dic₃

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

#627

—

C₂×C₄×C₉

—

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×H(3,𝔽₃)

Dic₆

yes

—

C₂×H(3,𝔽₃)

C₂²×D₃

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

Dic₆

yes

complemented

C₂×C₃×C₉

C₂²×D₃

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂²×C₃²

—

C₂²×C₃²

D₃²

yes

—

C₃⋊Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

D₃²

yes

characteristic

C₄×C₉

—

Dic₉

—

Dic₉

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

C₂×Dic₆

yes

characteristic

C₃×C₉

—

C₃×C₉

C₂×Dic₆

yes

complemented

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#627

yes

—

C₂×C₃²

C₂×D₃²

yes

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

#627

yes

—

C₂×C₉

C₂×D₃²

yes

characteristic

C₂×Q₈

—

Sylow

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#1113

yes

characteristic

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

Dic₃

—

100

Dic₃

—

101

C₃²

—

102

C₃²

C₂×#627

yes

—

103

C₉

—

104

C₉

—

105

C₉

C₂×#627

yes

characteristic

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₃

—

110

C₂×C₃

—

111

C₂×C₃

—

112

C₂×C₃

—

113

C₂×C₃

#6296

yes

—

114

C₂×C₃

C₂×#1113

yes

characteristic

115

C₂²

#4207⋊C₂

yes

characteristic

116

C₄

—

117

C₄

—

118

C₃

—

119

C₃

—

120

C₃

C₂×#6296

yes

characteristic

121

C₂

#10619905

yes

direct product factor

122

C₂

C₂×#4207⋊C₂

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂²	#4207⋊C₂
commutator, cosocle	C₂×#737	C₂³
Frattini	C₂×C₃	C₂×#1113
Fitting	C₂²×#737	C₂²
socle	C₂²×C₃	#1113
prime radical	C₂²×H(3,𝔽₃)	C₂×D₃
hypocenter	#737	C₂×Q₈
LCS	G ▷ C₂×#737 ▷ #737
UCS	C₁ ◁ C₂²

subgroup

G/H

C₂²

C₂³

C₂×C₃

C₂²×#737

C₂²

socle

C₂²×C₃

#1113

prime radical

C₂²×H(3,𝔽₃)

C₂×D₃

C₂×Q₈

G ▷ C₂×#737 ▷ #737

C₁ ◁ C₂²

N	H ≅ G/N	type
3. #10619905	C₂	direct (N × H)
18. C₂×#737	Q₈	semidirect
34. #737	C₂×Q₈	semidirect
52. C₂×C₃×C₉	Dic₆	semidirect
72. C₃×C₉	C₂×Dic₆	semidirect
121. C₂	#10619905	direct (N × H)

H ≅ G/N

type

3. #10619905

C₂

direct (N × H)

18. C₂×#737

Q₈

semidirect

34. #737

C₂×Q₈

semidirect

52. C₂×C₃×C₉

Dic₆

semidirect

72. C₃×C₉

C₂×Dic₆

semidirect

121. C₂

#10619905

direct (N × H)

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

subgroup	index
C₂×#66410	2
C₂×#66409	2
#10619905	2
C₂×#6196	3
C₂×#6189	3

subgroup

index

Subgroups — C₂×#10619905 #13048782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#10619905 #13048782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10619905 #13048782