Group #11723991 — subgroups

Subgroups — #11723991

Order 864.

868 conjugacy classes of subgroups; only the 263 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 263 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×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	#89076	C₂	1	1	yes	maximal, characteristic, complemented
3	#88863	C₂	1	1	yes	maximal, characteristic, complemented
4	#88862	C₂	1	1	yes	maximal, characteristic, complemented
5	#88861	C₂	1	1	yes	maximal, characteristic, complemented
6	#88860	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
8	#64159	—	3	13	no	maximal
9	C₃³×Q₈	C₂²	1	1	yes	characteristic
10	C₃³×D₄	—	2	1	no	—
11	C₃³×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₃³×C₄	—	2	1	no	—
13	C₂×C₃³×C₄	C₂²	1	1	yes	characteristic
14	#6312	—	2	1	no	—
15	C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	1	yes	characteristic
16	C₄×C₃⋊(C₃⋊D₃)	—	2	1	no	—
17	#6308	C₂²	1	1	yes	characteristic
18	C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
19	C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
20	C₃²×#41	D₃	1	13	yes	complemented
21	#3828	—	3	13	no	—
22	#3756	—	3	13	no	—
23	#3755	—	3	13	no	—
24	#3754	—	3	13	no	—
25	#3753	—	3	13	no	—
26	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
27	C₂²×C₃³	—	2	1	no	—
28	C₂²×C₃³	D₄	1	1	yes	characteristic
29	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
30	C₃³×C₄	—	2	1	no	—
31	C₃³×C₄	D₄	1	1	yes	characteristic
32	C₃³×C₄	C₂³	1	1	yes	characteristic
33	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
34	#950	—	9	13	no	—
35	C₃²×Q₈	C₂×D₃	1	13	yes	—
36	C₃²×D₄	—	2	13	no	—
37	C₃²×D₄	C₂×D₃	1	13	yes	—
38	C₂×C₃²×C₄	—	2	13	no	—
39	C₂×C₃²×C₄	C₂×D₃	1	13	yes	—
40	#638	—	6	13	no	—
41	C₄⋊(C₃⋊D₃)	—	3	13	no	—
42	C₄×C₃⋊D₃	—	6	13	no	—
43	#634	—	3	13	no	—
44	C₃⋊(C₃⋊C₈)	—	3	13	no	—
45	C₃⋊(C₃⋊C₈)	—	3	13	no	—
46	C₂×C₃³	—	2	1	no	—
47	C₂×C₃³	—	4	1	no	—
48	C₂×C₃³	C₂×D₄	1	1	yes	characteristic
49	C₃⋊(C₃⋊D₃)	—	4	1	no	—
50	C₃×#41	C₃⋊D₃	1	13	yes	complemented
51	#235	—	9	13	no	—
52	#216	—	9	13	no	—
53	#215	—	9	13	no	—
54	#214	—	9	13	no	—
55	#213	—	9	13	no	—
56	C₂×C₃⋊C₈	—	9	13	no	—
57	C₂²×C₃²	—	2	13	no	—
58	C₂²×C₃²	#67	1	13	yes	—
59	C₂×C₃⋊D₃	—	6	13	no	—
60	C₃²×C₄	—	2	13	no	—
61	C₃²×C₄	#67	1	13	yes	—
62	C₃²×C₄	C₂²×D₃	1	13	yes	—
63	C₃⋊Dic₃	—	6	13	no	—
64	#135	—	27	1	no	Sylow
65	C₃³	#135	1	1	yes	characteristic, complemented, Sylow
66	C₃×Q₈	C₂×C₃⋊D₃	1	13	yes	—
67	C₃×D₄	—	2	13	no	—
68	C₃×D₄	C₂×C₃⋊D₃	1	13	yes	—
69	C₂×C₃×C₄	—	2	13	no	—
70	C₂×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
71	#67	—	18	13	no	—
72	D₁₂	—	9	13	no	—
73	C₄×D₃	—	18	13	no	—
74	Dic₆	—	9	13	no	—
75	C₃⋊C₈	—	9	13	no	—
76	C₃⋊C₈	—	9	13	no	—
77	C₂×C₃²	—	2	13	no	—
78	C₂×C₃²	—	4	13	no	—
79	C₂×C₃²	C₂×#67	1	13	yes	—
80	C₃⋊D₃	—	12	13	no	—
81	#41	—	27	1	no	—
82	#41	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
83	Q₁₆	—	27	1	no	—
84	SD₁₆	—	27	1	no	—
85	SD₁₆	—	27	1	no	—
86	D₈	—	27	1	no	—
87	C₂×C₈	—	27	1	no	—
88	C₂²×C₃	—	2	13	no	—
89	C₂²×C₃	#638	1	13	yes	—
90	C₂×D₃	—	18	13	no	—
91	C₃×C₄	—	2	13	no	—
92	C₃×C₄	#638	1	13	yes	—
93	C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
94	Dic₃	—	18	13	no	—
95	C₃²	#950	1	13	yes	complemented
96	Q₈	—	27	1	no	—
97	Q₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
98	D₄	—	2	1	no	—
99	D₄	—	27	1	no	—
100	D₄	—	54	1	no	—
101	D₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	54	1	no	—
104	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
105	C₈	—	27	1	no	—
106	C₈	—	27	1	no	—
107	C₂×C₃	—	2	13	no	—
108	C₂×C₃	—	4	13	no	—
109	C₂×C₃	C₂×#638	1	13	yes	—
110	D₃	—	36	13	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	54	1	no	—
113	C₂²	#6312	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	54	1	no	—
116	C₄	#6312	1	1	yes	characteristic
117	C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
118	C₃	#64159	1	13	yes	complemented
119	C₂	—	2	1	no	—
120	C₂	—	4	1	no	—
121	C₂	—	108	1	no	—
122	C₂	C₂×#6312	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×#41

C₂

yes

maximal, characteristic, complemented

#89076

C₂

yes

maximal, characteristic, complemented

#88863

C₂

yes

maximal, characteristic, complemented

#88862

C₂

yes

maximal, characteristic, complemented

#88861

C₂

yes

maximal, characteristic, complemented

#88860

C₂

yes

maximal, characteristic, complemented

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

C₂

yes

maximal, characteristic, complemented

#64159

—

maximal

C₃³×Q₈

C₂²

yes

characteristic

C₃³×D₄

—

C₃³×D₄

C₂²

yes

characteristic

C₂×C₃³×C₄

—

C₂×C₃³×C₄

C₂²

yes

characteristic

#6312

—

C₄⋊(C₃⋊(C₃⋊D₃))

C₂²

yes

characteristic

C₄×C₃⋊(C₃⋊D₃)

—

#6308

C₂²

yes

characteristic

C₃⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

characteristic

C₃⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

characteristic

C₃²×#41

D₃

yes

—

—

—

—

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂²×C₃³

—

C₂²×C₃³

D₄

yes

characteristic

C₂×C₃⋊(C₃⋊D₃)

—

C₃³×C₄

—

C₃³×C₄

D₄

yes

characteristic

C₃³×C₄

C₂³

yes

characteristic

C₃⋊(C₃⋊Dic₃)

—

#950

—

C₃²×Q₈

C₂×D₃

yes

—

C₃²×D₄

—

C₃²×D₄

C₂×D₃

yes

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×D₃

yes

—

#638

—

C₄⋊(C₃⋊D₃)

—

C₄×C₃⋊D₃

—

#634

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

C₂×D₄

yes

characteristic

C₃⋊(C₃⋊D₃)

—

C₃×#41

C₃⋊D₃

yes

—

—

—

—

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

#67

yes

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃²×C₄

#67

yes

—

C₃²×C₄

C₂²×D₃

yes

—

C₃⋊Dic₃

—

#135

—

Sylow

C₃³

#135

yes

characteristic, complemented, Sylow

C₃×Q₈

C₂×C₃⋊D₃

yes

—

C₃×D₄

—

C₃×D₄

C₂×C₃⋊D₃

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃⋊D₃

yes

—

#67

—

D₁₂

—

C₄×D₃

—

Dic₆

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂×#67

yes

—

C₃⋊D₃

—

#41

—

#41

C₃⋊(C₃⋊D₃)

yes

characteristic, complemented

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂²×C₃

—

C₂²×C₃

#638

yes

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

#638

yes

—

C₃×C₄

C₂²×C₃⋊D₃

yes

—

Dic₃

—

C₃²

#950

yes

complemented

Q₈

—

Q₈

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

D₄

—

D₄

—

100

D₄

—

101

D₄

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

105

C₈

—

106

C₈

—

107

C₂×C₃

—

108

C₂×C₃

—

109

C₂×C₃

C₂×#638

yes

—

110

D₃

—

111

C₂²

—

112

C₂²

—

113

C₂²

#6312

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

#6312

yes

characteristic

117

C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

118

C₃

#64159

yes

complemented

119

C₂

—

120

C₂

—

121

C₂

—

108

—

122

C₂

C₂×#6312

yes

characteristic

	subgroup	G/H
center	C₄	#6312
commutator, cosocle	C₃³×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₃³×#41	C₂
socle	C₂×C₃³	C₂×D₄
hypercenter	#41	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#135
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

subgroup

G/H

center

C₄

#6312

commutator, cosocle

C₃³×C₄

C₂³

Frattini

C₄

C₂²×C₃⋊(C₃⋊D₃)

Fitting

C₃³×#41

C₂

socle

C₂×C₃³

C₂×D₄

C₃⋊(C₃⋊D₃)

C₃³

G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

N	H ≅ G/N	type
1. C₃³×#41	C₂	semidirect
2. #89076	C₂	semidirect
3. #88863	C₂	semidirect
4. #88862	C₂	semidirect
5. #88861	C₂	semidirect
6. #88860	C₂	semidirect
7. C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
20. C₃²×#41	D₃	semidirect
50. C₃×#41	C₃⋊D₃	semidirect
65. C₃³	#135	semidirect
82. #41	C₃⋊(C₃⋊D₃)	semidirect
95. C₃²	#950	semidirect
118. C₃	#64159	semidirect

H ≅ G/N

type

1. C₃³×#41

C₂

semidirect

2. #89076

C₂

semidirect

3. #88863

C₂

semidirect

4. #88862

C₂

semidirect

5. #88861

C₂

semidirect

6. #88860

C₂

semidirect

7. C₂×C₃⋊(C₃⋊(C₃⋊C₈))

C₂

semidirect

20. C₃²×#41

D₃

semidirect

50. C₃×#41

C₃⋊D₃

semidirect

65. C₃³

#135

semidirect

82. #41

C₃⋊(C₃⋊D₃)

semidirect

95. C₃²

#950

semidirect

118. C₃

#64159

semidirect

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

subgroup	index
C₃³×#41	2
#89076	2
#88863	2
#88862	2
#88861	2
#88860	2
C₂×C₃⋊(C₃⋊(C₃⋊C₈))	2
#64159	3

subgroup

index

C₃³×#41

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

Subgroups — #11723991

Conjugacy classes of subgroups

Complements

Maximal subgroups