Group #13179695 — subgroups

Subgroups — #13179695

Order 1728.

2800 conjugacy classes of subgroups; only the 947 normal subgroups are shown.

normal subgroups only

This group has 2800 conjugacy classes of subgroups — too many to draw. Only its 947 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	C₃³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₃⋊(C₃⋊Dic₃)	C₂	1	2	yes	maximal
3	#11723931	C₂	1	6	yes	maximal
4	#11723910	C₂	1	3	yes	maximal
5	C₄×#6308	C₂	1	3	yes	maximal
6	#10596041	—	3	13	no	maximal
7	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
8	C₂×#6308	C₂²	1	3	yes	—
9	C₃³×C₄⋊C₄	C₂²	1	3	yes	—
10	C₃³×C₄²	C₂²	1	3	yes	—
11	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
12	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	6	yes	—
13	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	6	yes	—
14	#88857	C₂²	1	6	yes	—
15	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	6	yes	—
16	C₃²×C₄×Q₈	D₃	1	13	yes	—
17	Q₈×C₃⋊Dic₃	—	3	26	no	—
18	#64099	—	3	78	no	—
19	#64078	—	3	39	no	—
20	C₄×#634	—	3	39	no	—
21	C₃³×Q₈	Q₈	1	4	yes	complemented
22	C₂×C₃³×C₄	C₂³	1	3	yes	—
23	C₂×C₃³×C₄	C₂³	1	3	yes	—
24	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
25	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
26	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	6	yes	—
27	#6308	—	2	6	no	—
28	#5570	—	9	13	no	—
29	C₂×C₃²×Q₈	C₂×D₃	1	13	yes	—
30	C₂×#634	—	3	39	no	—
31	C₃²×C₄⋊C₄	C₂×D₃	1	39	yes	—
32	C₃²×C₄²	C₂×D₃	1	39	yes	—
33	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
34	C₄⋊(C₃⋊Dic₃)	—	3	78	no	—
35	C₄⋊(C₃⋊Dic₃)	—	3	78	no	—
36	#3750	—	3	78	no	—
37	C₄×C₃⋊Dic₃	—	3	78	no	—
38	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
39	C₃³×C₄	—	2	3	no	—
40	C₃³×C₄	C₂×Q₈	1	6	yes	—
41	C₃³×C₄	#41	1	2	yes	—
42	C₃⋊(C₃⋊Dic₃)	—	2	6	no	—
43	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
44	C₃×C₄×Q₈	C₃⋊D₃	1	13	yes	—
45	Q₈×Dic₃	—	9	26	no	—
46	#890	—	9	78	no	—
47	#869	—	9	39	no	—
48	C₄×Dic₆	—	9	39	no	—
49	C₃²×Q₈	Dic₆	1	52	yes	complemented
50	C₂×C₃²×C₄	C₂²×D₃	1	39	yes	—
51	C₂×C₃²×C₄	C₂²×D₃	1	39	yes	—
52	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
53	C₂×C₃⋊Dic₃	—	3	26	no	—
54	C₂×C₃⋊Dic₃	—	3	78	no	—
55	#634	—	6	78	no	—
56	#557	—	27	1	no	Sylow
57	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
58	C₂×C₃³	C₂×#41	1	1	yes	characteristic
59	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
60	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	13	yes	—
61	C₂×Dic₆	—	9	39	no	—
62	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	39	yes	—
63	C₃×C₄²	C₂×C₃⋊D₃	1	39	yes	—
64	C₄⋊Dic₃	—	9	13	no	—
65	C₄⋊Dic₃	—	9	78	no	—
66	C₄⋊Dic₃	—	9	78	no	—
67	Dic₃⋊C₄	—	9	78	no	—
68	C₄×Dic₃	—	9	78	no	—
69	C₂²×C₃²	C₂³×D₃	1	13	yes	—
70	C₃²×C₄	—	2	39	no	—
71	C₃²×C₄	C₂×Dic₆	1	78	yes	—
72	C₃²×C₄	#239	1	26	yes	—
73	C₃⋊Dic₃	—	6	78	no	—
74	C₃⋊Dic₃	—	6	26	no	—
75	C₄⋊Q₈	—	27	3	no	—
76	#125	—	27	6	no	—
77	C₄×Q₈	—	27	2	no	—
78	C₄×Q₈	—	27	3	no	—
79	C₄×Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
80	C₃³	#557	1	1	yes	characteristic, complemented, Sylow
81	C₃×Q₈	#634	1	52	yes	complemented
82	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	39	yes	—
83	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	39	yes	—
84	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
85	C₂×Dic₃	—	9	26	no	—
86	C₂×Dic₃	—	9	78	no	—
87	Dic₆	—	18	78	no	—
88	C₂×C₃²	C₂²×Dic₆	1	13	yes	—
89	C₂×C₃²	C₂×#239	1	13	yes	—
90	C₂×C₃²	#1010	1	13	yes	—
91	C₂×Q₈	—	27	3	no	—
92	C₂×Q₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
93	C₄⋊C₄	—	27	1	no	—
94	C₄⋊C₄	—	27	6	no	—
95	C₄⋊C₄	—	27	6	no	—
96	C₄⋊C₄	—	27	6	no	—
97	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	3	yes	—
98	C₄²	—	27	6	no	—
99	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	3	yes	—
100	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
101	C₃×C₄	—	2	39	no	—
102	C₃×C₄	C₂×#634	1	78	yes	—
103	C₃×C₄	#3832	1	26	yes	—
104	Dic₃	—	18	78	no	—
105	Dic₃	—	18	26	no	—
106	C₃²	#5570	1	13	yes	complemented
107	Q₈	—	54	6	no	—
108	Q₈	#6308	1	4	yes	complemented
109	C₂×C₄	—	27	2	no	—
110	C₂×C₄	—	27	6	no	—
111	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	3	yes	—
112	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	3	yes	—
113	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
114	C₂×C₃	C₂²×#634	1	13	yes	—
115	C₂×C₃	C₂×#3832	1	13	yes	—
116	C₂×C₃	#64367	1	13	yes	—
117	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
118	C₄	—	2	3	no	—
119	C₄	—	54	6	no	—
120	C₄	—	54	2	no	—
121	C₄	C₂×#6308	1	6	yes	—
122	C₄	#89080	1	2	yes	—
123	C₃	#10596041	1	13	yes	complemented
124	C₂	C₂²×#6308	1	1	yes	characteristic
125	C₂	C₂×#89080	1	1	yes	characteristic
126	C₂	#11724685	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×C₄×Q₈

C₂

yes

maximal, characteristic

Q₈×C₃⋊(C₃⋊Dic₃)

C₂

yes

maximal

#11723931

C₂

yes

maximal

#11723910

C₂

yes

maximal

C₄×#6308

C₂

yes

maximal

#10596041

—

maximal

C₂×C₃³×Q₈

C₂²

yes

characteristic

C₂×#6308

C₂²

yes

—

C₃³×C₄⋊C₄

C₂²

yes

—

C₃³×C₄²

C₂²

yes

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

characteristic

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

—

#88857

C₂²

yes

—

C₄×C₃⋊(C₃⋊Dic₃)

C₂²

yes

—

C₃²×C₄×Q₈

D₃

yes

—

Q₈×C₃⋊Dic₃

—

#64099

—

#64078

—

C₄×#634

—

C₃³×Q₈

Q₈

yes

complemented

C₂×C₃³×C₄

C₂³

yes

—

C₂×C₃³×C₄

C₂³

yes

—

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

C₂³

yes

—

C₂×C₃⋊(C₃⋊Dic₃)

C₂³

yes

—

#6308

—

#5570

—

C₂×C₃²×Q₈

C₂×D₃

yes

—

C₂×#634

—

C₃²×C₄⋊C₄

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

#3750

—

C₄×C₃⋊Dic₃

—

C₂²×C₃³

C₂⁴

yes

characteristic

C₃³×C₄

—

C₃³×C₄

C₂×Q₈

yes

—

C₃³×C₄

#41

yes

—

C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊Dic₃)

—

C₃×C₄×Q₈

C₃⋊D₃

yes

—

Q₈×Dic₃

—

#890

—

#869

—

C₄×Dic₆

—

C₃²×Q₈

Dic₆

yes

complemented

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

#557

—

Sylow

C₂×C₃³

C₂²×Q₈

yes

characteristic

C₂×C₃³

C₂×#41

yes

characteristic

C₂×C₃³

2₋¹⁺⁴

yes

characteristic

C₂×C₃×Q₈

C₂×C₃⋊D₃

yes

—

C₂×Dic₆

—

C₃×C₄⋊C₄

C₂×C₃⋊D₃

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂²×C₃²

C₂³×D₃

yes

—

C₃²×C₄

—

C₃²×C₄

C₂×Dic₆

yes

—

C₃²×C₄

#239

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₄⋊Q₈

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₃⋊(C₃⋊D₃)

yes

characteristic

C₃³

#557

yes

characteristic, complemented, Sylow

C₃×Q₈

#634

yes

complemented

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₂×C₃²

C₂²×Dic₆

yes

—

C₂×C₃²

C₂×#239

yes

—

C₂×C₃²

#1010

yes

—

C₂×Q₈

—

C₂×Q₈

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₃⋊(C₃⋊D₃)

yes

—

C₄²

—

C₄²

C₂×C₃⋊(C₃⋊D₃)

yes

—

100

C₂²×C₃

C₂³×C₃⋊D₃

yes

—

101

C₃×C₄

—

102

C₃×C₄

C₂×#634

yes

—

103

C₃×C₄

#3832

yes

—

104

Dic₃

—

105

Dic₃

—

106

C₃²

#5570

yes

complemented

107

Q₈

—

108

Q₈

#6308

yes

complemented

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

—

112

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

—

113

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

114

C₂×C₃

C₂²×#634

yes

—

115

C₂×C₃

C₂×#3832

yes

—

116

C₂×C₃

#64367

yes

—

117

C₂²

C₂³×C₃⋊(C₃⋊D₃)

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂×#6308

yes

—

122

C₄

#89080

yes

—

123

C₃

#10596041

yes

complemented

124

C₂

C₂²×#6308

yes

characteristic

125

C₂

C₂×#89080

yes

characteristic

126

C₂

#11724685

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊(C₃⋊D₃)
commutator, socle, prime radical, cosocle	C₂²×C₃³	C₂⁴
Fitting	C₃³×C₄×Q₈	C₂
hypercenter	C₄×Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#557
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×C₃⋊(C₃⋊D₃)

commutator, socle, prime radical, cosocle

C₂²×C₃³

C₂⁴

Fitting

C₃³×C₄×Q₈

C₂

hypercenter

C₄×Q₈

C₃⋊(C₃⋊D₃)

C₃³

G ▷ C₂²×C₃³ ▷ C₃³

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
21. C₃³×Q₈	Q₈	semidirect
49. C₃²×Q₈	Dic₆	semidirect
80. C₃³	#557	semidirect
81. C₃×Q₈	#634	semidirect
106. C₃²	#5570	semidirect
108. Q₈	#6308	semidirect
123. C₃	#10596041	semidirect

H ≅ G/N

type

21. C₃³×Q₈

Q₈

semidirect

49. C₃²×Q₈

Dic₆

semidirect

80. C₃³

#557

semidirect

81. C₃×Q₈

#634

semidirect

106. C₃²

#5570

semidirect

108. Q₈

#6308

semidirect

123. C₃

#10596041

semidirect

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

subgroup	index
C₃³×C₄×Q₈	2
Q₈×C₃⋊(C₃⋊Dic₃)	2
#11723931	2
#11723910	2
C₄×#6308	2
#10596041	3

subgroup

index

C₃³×C₄×Q₈

Q₈×C₃⋊(C₃⋊Dic₃)

Subgroups — #13179695

Conjugacy classes of subgroups

Complements

Maximal subgroups