Group #11797782 — subgroups

Subgroups — #11797782

Order 1152.

444 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 145 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×#63628	C₂	1	1	yes	maximal, characteristic
2	C₄×#3746	C₂	1	2	yes	maximal
3	C₃²×#346	C₂	1	1	yes	maximal, characteristic
4	#10589371	C₂	1	1	yes	maximal, characteristic
5	#10589369	C₂	1	2	yes	maximal
6	#69082	—	3	4	no	maximal
7	C₂×#3746	C₂²	1	2	yes	—
8	C₄×#634	—	2	2	no	—
9	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	#63645	C₂²	1	2	yes	—
13	#63643	—	2	2	no	—
14	#63628	C₄	1	4	yes	complemented
15	C₂×Dic₂₄	—	3	4	no	—
16	C₄×Dic₁₂	—	3	8	no	—
17	C₃×#346	D₃	1	4	yes	—
18	#4505	—	3	4	no	—
19	#4503	—	3	8	no	—
20	C₂×#634	—	2	2	no	—
21	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
22	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
23	C₃²×C₄²	D₄	1	1	yes	characteristic
24	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
25	#3750	—	4	2	no	—
26	C₄×C₃⋊Dic₃	—	4	2	no	—
27	#3746	—	2	2	no	—
28	#3746	C₂×C₄	1	4	yes	—
29	C₃²×C₁₆	—	2	1	no	—
30	C₃²×C₁₆	C₂×C₄	1	2	yes	—
31	#2176	—	9	1	no	Sylow
32	C₂×Dic₁₂	—	3	8	no	—
33	C₄×Dic₆	—	6	8	no	—
34	C₂×C₃×C₁₆	C₂×D₃	1	4	yes	—
35	C₂×C₃×C₁₆	C₂×D₃	1	4	yes	—
36	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
37	#818	—	3	8	no	—
38	#816	—	6	8	no	—
39	Dic₂₄	—	3	16	no	—
40	C₂×C₃²×C₄	D₈	1	2	yes	—
41	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂×C₃⋊Dic₃	—	4	2	no	—
43	C₂×C₃⋊Dic₃	—	4	2	no	—
44	#634	—	2	4	no	—
45	#634	—	4	2	no	—
46	C₃²×C₈	—	2	1	no	—
47	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₃²×C₈	#41	1	1	yes	characteristic
49	C₂×Q₃₂	—	9	1	no	—
50	C₄×Q₁₆	—	9	2	no	—
51	#367	—	9	1	no	—
52	#358	—	9	2	no	—
53	#346	C₃⋊D₃	1	1	yes	characteristic
54	C₂×Dic₆	—	6	8	no	—
55	C₂×C₃×C₈	D₁₂	1	4	yes	—
56	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
57	C₃×C₄²	D₁₂	1	4	yes	—
58	C₄⋊Dic₃	—	6	8	no	—
59	Dic₃⋊C₄	—	12	8	no	—
60	C₄×Dic₃	—	12	8	no	—
61	Dic₁₂	—	3	16	no	—
62	Dic₁₂	—	6	8	no	—
63	C₃×C₁₆	—	2	4	no	—
64	C₃×C₁₆	C₄×D₃	1	8	yes	—
65	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
66	C₃²×C₄	—	2	2	no	—
67	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
68	C₃²×C₄	#135	1	1	yes	characteristic
69	C₃⋊Dic₃	—	4	4	no	—
70	C₃⋊Dic₃	—	8	2	no	—
71	C₂×Q₁₆	—	9	2	no	—
72	C₄×Q₈	—	18	2	no	—
73	Q₃₂	—	9	4	no	—
74	C₂×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
75	C₂×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
76	#107	—	9	2	no	—
77	#103	—	18	2	no	—
78	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
79	C₂×C₃×C₄	D₂₄	1	8	yes	—
80	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
81	C₂×Dic₃	—	12	8	no	—
82	C₂×Dic₃	—	12	8	no	—
83	Dic₆	—	6	16	no	—
84	Dic₆	—	12	8	no	—
85	C₃×C₈	—	2	4	no	—
86	C₃×C₈	C₂×C₄×D₃	1	4	yes	—
87	C₃×C₈	#235	1	4	yes	—
88	C₂×C₃²	C₄×D₈	1	1	yes	characteristic
89	C₂×C₃²	#510	1	2	yes	—
90	C₂×Q₈	—	18	2	no	—
91	Q₁₆	—	9	4	no	—
92	Q₁₆	—	18	2	no	—
93	C₂×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
94	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
95	C₄⋊C₄	—	18	2	no	—
96	C₄⋊C₄	—	36	2	no	—
97	C₄²	—	36	2	no	—
98	C₄²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
99	C₁₆	—	2	1	no	—
100	C₁₆	C₄×C₃⋊D₃	1	2	yes	—
101	C₂²×C₃	C₂×D₂₄	1	4	yes	—
102	C₃×C₄	—	2	8	no	—
103	C₃×C₄	C₄×D₁₂	1	4	yes	—
104	C₃×C₄	#904	1	4	yes	—
105	Dic₃	—	12	16	no	—
106	Dic₃	—	24	8	no	—
107	C₃²	#2176	1	1	yes	characteristic, complemented, Sylow
108	Q₈	—	18	4	no	—
109	Q₈	—	36	2	no	—
110	C₂×C₄	—	36	2	no	—
111	C₂×C₄	—	36	2	no	—
112	C₂×C₄	#3745	1	2	yes	—
113	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
116	C₈	#3828	1	1	yes	characteristic
117	C₂×C₃	C₄×D₂₄	1	4	yes	—
118	C₂×C₃	#4909	1	8	yes	—
119	C₂²	C₂×#3745	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	—	36	4	no	—
122	C₄	—	72	2	no	—
123	C₄	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
124	C₄	#64113	1	1	yes	characteristic
125	C₃	#69082	1	4	yes	complemented
126	C₂	C₄×#3745	1	1	yes	characteristic
127	C₂	#10592608	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#63628

C₂

yes

maximal, characteristic

C₄×#3746

C₂

yes

maximal

C₃²×#346

C₂

yes

maximal, characteristic

#10589371

C₂

yes

maximal, characteristic

#10589369

C₂

yes

—

C₂²

yes

—

C₄×#634

—

C₂×C₃²×C₁₆

C₂²

yes

characteristic

C₂×C₃²×C₁₆

C₂²

yes

characteristic

C₃²×C₄×C₈

C₂²

yes

characteristic

#63645

C₂²

yes

—

#63643

—

#63628

C₄

yes

complemented

C₂×Dic₂₄

—

C₄×Dic₁₂

—

C₃×#346

D₃

yes

—

#4505

—

#4503

—

C₂×#634

—

C₂×C₃²×C₈

D₄

yes

characteristic

C₂×C₃²×C₈

C₂³

yes

characteristic

C₃²×C₄²

D₄

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

#3750

—

C₄×C₃⋊Dic₃

—

#3746

—

#3746

C₂×C₄

yes

—

C₃²×C₁₆

—

C₃²×C₁₆

C₂×C₄

yes

—

#2176

—

Sylow

C₂×Dic₁₂

—

C₄×Dic₆

—

C₂×C₃×C₁₆

C₂×D₃

yes

—

C₂×C₃×C₁₆

C₂×D₃

yes

—

C₃×C₄×C₈

C₂×D₃

yes

—

#818

—

#816

—

Dic₂₄

—

C₂×C₃²×C₄

D₈

yes

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

#634

—

C₃²×C₈

—

C₃²×C₈

C₂²×C₄

yes

characteristic

C₃²×C₈

#41

yes

characteristic

C₂×Q₃₂

—

C₄×Q₁₆

—

#367

—

#358

—

#346

C₃⋊D₃

yes

characteristic

C₂×Dic₆

—

C₂×C₃×C₈

D₁₂

yes

—

C₂×C₃×C₈

C₂²×D₃

yes

—

C₃×C₄²

D₁₂

yes

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

Dic₁₂

—

Dic₁₂

—

C₃×C₁₆

—

C₃×C₁₆

C₄×D₃

yes

—

C₂²×C₃²

C₂×D₈

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₄×D₄

yes

characteristic

C₃²×C₄

#135

yes

characteristic

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

C₂×C₃⋊D₃

yes

characteristic

C₂×C₁₆

C₂×C₃⋊D₃

yes

characteristic

#107

—

#103

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃×C₄

D₂₄

yes

—

C₂×C₃×C₄

C₂×D₁₂

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

C₂×C₄×D₃

yes

—

C₃×C₈

#235

yes

—

C₂×C₃²

C₄×D₈

yes

characteristic

C₂×C₃²

#510

yes

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

C₄⋊(C₃⋊D₃)

yes

characteristic

C₂×C₈

C₂²×C₃⋊D₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₄⋊(C₃⋊D₃)

yes

characteristic

C₁₆

—

100

C₁₆

C₄×C₃⋊D₃

yes

—

101

C₂²×C₃

C₂×D₂₄

yes

—

102

C₃×C₄

—

103

C₃×C₄

C₄×D₁₂

yes

—

104

C₃×C₄

#904

yes

—

105

Dic₃

—

106

Dic₃

—

107

C₃²

#2176

yes

characteristic, complemented, Sylow

108

Q₈

—

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#3745

yes

—

113

C₂×C₄

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

114

C₈

—

115

C₈

C₂×C₄×C₃⋊D₃

yes

characteristic

116

C₈

#3828

yes

characteristic

117

C₂×C₃

C₄×D₂₄

yes

—

118

C₂×C₃

#4909

yes

—

119

C₂²

C₂×#3745

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

C₄×C₄⋊(C₃⋊D₃)

yes

characteristic

124

C₄

#64113

yes

characteristic

125

C₃

#69082

yes

complemented

126

C₂

C₄×#3745

yes

characteristic

127

C₂

#10592608

yes

—

	subgroup	G/H
center	C₂²	C₂×#3745
commutator	C₃²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×C₃⋊D₃
Fitting	C₃²×#346	C₂
socle, prime radical	C₂²×C₃²	C₂×D₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	#346	C₃⋊D₃
hypocenter	C₃²	#2176
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

C₂×#3745

commutator

C₃²×C₈

C₂²×C₄

Frattini

C₂×C₈

C₂²×C₃⋊D₃

Fitting

C₃²×#346

C₂

socle, prime radical

C₂²×C₃²

C₂×D₈

cosocle

C₂×C₃²×C₈

C₂³

C₃⋊D₃

C₃²

G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
14. #63628	C₄	semidirect
107. C₃²	#2176	semidirect
125. C₃	#69082	semidirect

H ≅ G/N

type

14. #63628

C₄

semidirect

107. C₃²

#2176

semidirect

125. C₃

#69082

semidirect

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

subgroup	index
C₂×#63628	2
C₄×#3746	2
C₃²×#346	2
#10589371	2
#10589369	2
#69082	3

subgroup

index

C₃²×#346

Subgroups — #11797782

Conjugacy classes of subgroups

Complements

Maximal subgroups