Group #11771699 — subgroups

Subgroups — #11771699

Order 1152.

258 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 109 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	#10592378	C₂	1	1	yes	maximal, characteristic
2	C₃²×#346	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₃⋊(C₃⋊C₁₆))	C₂	1	1	yes	maximal, characteristic
4	#67518	—	3	4	no	maximal
5	C₄×#634	C₄	1	1	yes	characteristic
6	C₂×C₃²×C₁₆	—	2	1	no	—
7	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
8	#63642	—	2	1	no	—
9	#63641	—	2	1	no	—
10	C₂×C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
11	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	1	yes	characteristic
12	#4679	—	3	4	no	—
13	C₃×#346	D₃	1	4	yes	—
14	C₄⋊(C₃⋊C₁₆)	—	3	4	no	—
15	C₂×#634	C₈	1	1	yes	characteristic
16	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
17	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
18	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
19	C₄⋊(C₃⋊Dic₃)	C₈	1	1	yes	characteristic
20	#3750	—	2	1	no	—
21	C₄×C₃⋊Dic₃	—	2	1	no	—
22	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
23	C₃²×C₁₆	—	2	2	no	—
24	C₃⋊(C₃⋊C₁₆)	—	4	1	no	—
25	#1338	—	9	1	no	Sylow
26	C₄×Dic₆	—	3	4	no	—
27	C₂×C₃×C₁₆	—	2	4	no	—
28	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
29	#815	—	6	4	no	—
30	Dic₃⋊C₈	—	6	4	no	—
31	C₂×C₃⋊C₁₆	—	6	4	no	—
32	C₄⋊(C₃⋊C₈)	—	3	4	no	—
33	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
35	C₂×C₃²×C₄	M₁₆	1	1	yes	characteristic
36	C₂×C₃⋊Dic₃	—	2	1	no	—
37	C₂×C₃⋊Dic₃	—	2	1	no	—
38	#634	—	4	1	no	—
39	C₃²×C₈	—	2	1	no	—
40	C₃²×C₈	SD₁₆	1	1	yes	characteristic
41	C₃²×C₈	Q₁₆	1	1	yes	characteristic
42	C₃⋊(C₃⋊C₈)	—	4	1	no	—
43	#446	—	9	1	no	—
44	C₄⋊C₁₆	—	9	1	no	—
45	#346	C₃⋊D₃	1	1	yes	characteristic
46	C₂×Dic₆	—	3	4	no	—
47	C₂×C₃×C₈	D₁₂	1	4	yes	—
48	C₂×C₃×C₈	#67	1	4	yes	—
49	C₃×C₄²	C₄×D₃	1	4	yes	—
50	C₄⋊Dic₃	—	3	4	no	—
51	Dic₃⋊C₄	—	6	4	no	—
52	C₄×Dic₃	—	6	4	no	—
53	C₂×C₃⋊C₈	—	6	4	no	—
54	C₃×C₁₆	—	2	8	no	—
55	C₃⋊C₁₆	—	12	4	no	—
56	C₂²×C₃²	C₂²⋊C₈	1	1	yes	characteristic
57	C₃²×C₄	—	2	1	no	—
58	C₃²×C₄	—	2	1	no	—
59	C₃²×C₄	#103	1	1	yes	characteristic
60	C₃²×C₄	#104	1	1	yes	characteristic
61	C₃⋊Dic₃	—	4	1	no	—
62	C₃⋊Dic₃	—	4	1	no	—
63	C₄×Q₈	—	9	1	no	—
64	C₂×C₁₆	—	2	1	no	—
65	C₂×C₁₆	—	18	1	no	—
66	C₄⋊C₈	—	9	1	no	—
67	C₄⋊C₈	—	18	1	no	—
68	#97	—	18	1	no	—
69	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
70	C₂×C₃×C₄	D₃×C₈	1	4	yes	—
71	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	4	yes	—
72	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
73	C₂×Dic₃	—	6	4	no	—
74	C₂×Dic₃	—	6	4	no	—
75	Dic₆	—	12	4	no	—
76	C₃×C₈	—	2	4	no	—
77	C₃×C₈	#204	1	4	yes	—
78	C₃×C₈	Dic₁₂	1	4	yes	—
79	C₃⋊C₈	—	12	4	no	—
80	C₂×C₃²	Q₈⋊C₈	1	1	yes	characteristic
81	C₂×C₃²	#349	1	1	yes	characteristic
82	C₂×C₃²	#350	1	1	yes	characteristic
83	C₂×Q₈	—	9	1	no	—
84	C₂×C₈	—	18	1	no	—
85	C₂×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
86	C₂×C₈	#638	1	1	yes	characteristic
87	C₄⋊C₄	—	9	1	no	—
88	C₄⋊C₄	—	18	1	no	—
89	C₄²	—	18	1	no	—
90	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
91	C₁₆	—	2	2	no	—
92	C₁₆	—	36	1	no	—
93	C₂²×C₃	(C₂×D₃)⋊C₈	1	4	yes	—
94	C₃×C₄	—	2	4	no	—
95	C₃×C₄	—	2	4	no	—
96	C₃×C₄	#805	1	4	yes	—
97	C₃×C₄	#816	1	4	yes	—
98	Dic₃	—	12	4	no	—
99	Dic₃	—	12	4	no	—
100	C₃²	#1338	1	1	yes	characteristic, complemented, Sylow
101	Q₈	—	36	1	no	—
102	C₂×C₄	—	18	1	no	—
103	C₂×C₄	—	18	1	no	—
104	C₂×C₄	C₈×C₃⋊D₃	1	1	yes	characteristic
105	C₂×C₄	#3743	1	1	yes	characteristic
106	C₂×C₄	#3752	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	36	1	no	—
109	C₈	#3744	1	1	yes	characteristic
110	C₈	#3746	1	1	yes	characteristic
111	C₂×C₃	#4456	1	4	yes	—
112	C₂×C₃	#4514	1	4	yes	—
113	C₂×C₃	#4515	1	4	yes	—
114	C₂²	#63647	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	1	no	—
117	C₄	—	36	1	no	—
118	C₄	—	36	1	no	—
119	C₄	#63632	1	1	yes	characteristic
120	C₄	#63643	1	1	yes	characteristic
121	C₃	#67518	1	4	yes	complemented
122	C₂	#10589322	1	1	yes	characteristic
123	C₂	#10589380	1	1	yes	characteristic
124	C₂	#10589381	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10592378

C₂

yes

maximal, characteristic

C₃²×#346

C₂

yes

maximal, characteristic

C₄⋊(C₃⋊(C₃⋊C₁₆))

C₂

yes

maximal, characteristic

#67518

—

maximal

C₄×#634

C₄

yes

characteristic

C₂×C₃²×C₁₆

—

C₃²×C₄×C₈

C₂²

yes

characteristic

#63642

—

#63641

—

C₂×C₃⋊(C₃⋊C₁₆)

—

C₄⋊(C₃⋊(C₃⋊C₈))

C₄

yes

characteristic

#4679

—

C₃×#346

D₃

yes

—

C₄⋊(C₃⋊C₁₆)

—

C₂×#634

C₈

yes

characteristic

C₂×C₃²×C₈

D₄

yes

characteristic

C₂×C₃²×C₈

D₄

yes

characteristic

C₃²×C₄²

C₂×C₄

yes

characteristic

C₄⋊(C₃⋊Dic₃)

C₈

yes

characteristic

#3750

—

C₄×C₃⋊Dic₃

—

C₂×C₃⋊(C₃⋊C₈)

—

C₃²×C₁₆

—

C₃⋊(C₃⋊C₁₆)

—

#1338

—

Sylow

C₄×Dic₆

—

C₂×C₃×C₁₆

—

C₃×C₄×C₈

C₂×D₃

yes

—

#815

—

Dic₃⋊C₈

—

C₂×C₃⋊C₁₆

—

C₄⋊(C₃⋊C₈)

—

C₂×C₃²×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃²×C₄

C₂×C₈

yes

characteristic

C₂×C₃²×C₄

M₁₆

yes

characteristic

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

C₃²×C₈

—

C₃²×C₈

SD₁₆

yes

characteristic

C₃²×C₈

Q₁₆

yes

characteristic

C₃⋊(C₃⋊C₈)

—

#446

—

C₄⋊C₁₆

—

#346

C₃⋊D₃

yes

characteristic

C₂×Dic₆

—

C₂×C₃×C₈

D₁₂

yes

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄²

C₄×D₃

yes

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

C₂²⋊C₈

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

yes

C₃²×C₄

yes

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃×C₄

D₃×C₈

yes

—

C₂×C₃×C₄

ΓL(1,𝔽₂₅)

yes

—

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₃×C₈

—

C₃×C₈

#204

yes

—

C₃×C₈

Dic₁₂

yes

—

C₃⋊C₈

—

C₂×C₃²

Q₈⋊C₈

yes

C₂×C₃²

yes

C₂×C₃²

yes

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

C₄⋊(C₃⋊D₃)

yes

characteristic

C₂×C₈

#638

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₄×C₃⋊D₃

yes

characteristic

C₁₆

—

C₁₆

—

C₂²×C₃

(C₂×D₃)⋊C₈

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#805

yes

—

C₃×C₄

#816

yes

—

Dic₃

—

Dic₃

—

100

C₃²

#1338

yes

characteristic, complemented, Sylow

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₈×C₃⋊D₃

yes

characteristic

105

C₂×C₄

#3743

yes

characteristic

106

C₂×C₄

#3752

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

#3744

yes

characteristic

110

C₈

#3746

yes

characteristic

111

C₂×C₃

#4456

yes

—

112

C₂×C₃

#4514

yes

—

113

C₂×C₃

#4515

yes

—

114

C₂²

#63647

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#63632

yes

characteristic

120

C₄

#63643

yes

characteristic

121

C₃

#67518

yes

complemented

122

C₂

#10589322

yes

characteristic

123

C₂

#10589380

yes

characteristic

124

C₂

#10589381

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#3752
commutator	C₂×C₃²×C₄	C₂×C₈
Frattini	C₄×C₈	C₂×C₃⋊D₃
Fitting	C₃²×#346	C₂
socle, prime radical	C₂²×C₃²	C₂²⋊C₈
cosocle	C₃²×C₄×C₈	C₂²
hypercenter	#346	C₃⋊D₃
hypocenter	C₃²	#1338
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂×C₄

#3752

commutator

C₂×C₃²×C₄

C₂×C₈

Frattini

C₄×C₈

C₂×C₃⋊D₃

Fitting

C₃²×#346

C₂

socle, prime radical

C₂²×C₃²

C₂²⋊C₈

cosocle

C₃²×C₄×C₈

C₂²

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
100. C₃²	#1338	semidirect
121. C₃	#67518	semidirect

H ≅ G/N

type

100. C₃²

#1338

semidirect

121. C₃

#67518

semidirect

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

subgroup	index
#10592378	2
C₃²×#346	2
C₄⋊(C₃⋊(C₃⋊C₁₆))	2
#67518	3

subgroup

index

#10592378

C₃²×#346

C₄⋊(C₃⋊(C₃⋊C₁₆))

#67518

Subgroups — #11771699

Conjugacy classes of subgroups

Complements

Maximal subgroups