Group #11772024 — subgroups

Subgroups — #11772024

Order 1152.

245 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 64 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	#10592398	C₂	1	1	yes	maximal, characteristic
2	C₃×#4463	C₂	1	2	yes	maximal
3	#67572	—	3	2	no	maximal
4	C₂×#3746	—	2	2	no	—
5	#64078	C₄	1	2	yes	—
6	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₃×#812	—	2	2	no	—
8	#4699	—	3	2	no	—
9	#4699	—	6	1	no	—
10	C₃×#364	—	3	2	no	—
11	#4463	D₃	1	2	yes	—
12	C₂×#634	—	2	2	no	—
13	C₂×C₃²×C₈	D₄	1	2	yes	—
14	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
15	C₄⋊(C₃⋊Dic₃)	—	4	2	no	—
16	#3746	—	4	2	no	—
17	#3746	—	4	2	no	—
18	C₃×C₃⋊C₁₆	—	4	2	no	—
19	#1358	—	9	1	no	Sylow
20	C₂×Dic₁₂	—	6	4	no	—
21	C₂×Dic₁₂	—	6	2	no	—
22	C₂×Dic₁₂	—	6	2	no	—
23	#869	—	3	4	no	—
24	#869	—	6	2	no	—
25	C₃×M₃₂	—	6	2	no	—
26	C₃×C₄×C₈	—	2	1	no	—
27	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
28	#812	—	2	2	no	—
29	C₂×C₃²×C₄	—	2	1	no	—
30	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₃⋊Dic₃	—	4	2	no	—
32	#634	—	4	4	no	—
33	C₃²×C₈	D₈	1	2	yes	—
34	C₃²×C₈	SD₁₆	1	2	yes	—
35	#494	—	9	1	no	—
36	#364	—	3	2	no	—
37	C₂×Dic₆	—	6	4	no	—
38	C₂×Dic₆	—	6	4	no	—
39	C₂×C₃×C₈	—	2	2	no	—
40	C₂×C₃×C₈	D₁₂	1	2	yes	—
41	C₂×C₃×C₈	#67	1	2	yes	—
42	C₃×C₄²	—	2	1	no	—
43	C₃×C₄²	C₄×D₃	1	2	yes	—
44	C₄⋊Dic₃	—	12	4	no	—
45	C₄⋊Dic₃	—	12	4	no	—
46	Dic₁₂	—	12	4	no	—
47	Dic₁₂	—	12	4	no	—
48	Dic₁₂	—	12	2	no	—
49	Dic₁₂	—	12	2	no	—
50	Dic₁₂	—	12	2	no	—
51	Dic₁₂	—	12	2	no	—
52	C₃×C₁₆	—	12	2	no	—
53	C₃⋊C₁₆	—	4	2	no	—
54	C₂²×C₃²	#100	1	1	yes	characteristic
55	C₃²×C₄	—	2	2	no	—
56	C₃²×C₄	#102	1	2	yes	—
57	C₃⋊Dic₃	—	8	2	no	—
58	C₂×Q₁₆	—	18	2	no	—
59	C₄⋊Q₈	—	9	2	no	—
60	M₃₂	—	6	2	no	—
61	C₄×C₈	D₃²	1	1	yes	characteristic
62	C₂×C₃×C₄	—	2	2	no	—
63	C₂×C₃×C₄	—	2	2	no	—
64	C₂×C₃×C₄	—	2	1	no	—
65	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
66	C₂×Dic₃	—	12	4	no	—
67	C₂×Dic₃	—	12	4	no	—
68	Dic₆	—	12	8	no	—
69	Dic₆	—	12	4	no	—
70	Dic₆	—	12	4	no	—
71	C₃×C₈	—	2	2	no	—
72	C₃×C₈	—	2	2	no	—
73	C₃×C₈	#204	1	2	yes	—
74	C₃×C₈	D₂₄	1	2	yes	—
75	C₃×C₈	#213	1	2	yes	—
76	C₃×C₈	#215	1	2	yes	—
77	C₂×C₃²	—	2	1	no	—
78	C₂×C₃²	#331	1	1	yes	characteristic
79	C₂×Q₈	—	18	2	no	—
80	Q₁₆	—	36	2	no	—
81	Q₁₆	—	36	2	no	—
82	C₂×C₈	#626	1	2	yes	—
83	C₄⋊C₄	—	36	2	no	—
84	C₄²	#624	1	1	yes	characteristic
85	C₁₆	—	12	2	no	—
86	C₂²×C₃	—	2	1	no	—
87	C₂²×C₃	#823	1	2	yes	—
88	C₃×C₄	—	2	4	no	—
89	C₃×C₄	—	2	4	no	—
90	C₃×C₄	—	2	2	no	—
91	C₃×C₄	#809	1	2	yes	—
92	C₃×C₄	#821	1	2	yes	—
93	Dic₃	—	24	4	no	—
94	Dic₃	—	24	4	no	—
95	C₃²	#1358	1	1	yes	characteristic, complemented, Sylow
96	Q₈	—	36	4	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	36	2	no	—
99	C₂×C₄	#3722	1	1	yes	characteristic
100	C₈	#3714	1	2	yes	—
101	C₈	#3717	1	2	yes	—
102	C₂×C₃	—	2	2	no	—
103	C₂×C₃	—	2	2	no	—
104	C₂×C₃	—	2	1	no	—
105	C₂×C₃	#4485	1	2	yes	—
106	C₂²	#63559	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	—	72	2	no	—
109	C₄	#63564	1	2	yes	—
110	C₃	—	2	1	no	—
111	C₃	#67572	1	2	yes	complemented
112	C₂	—	2	1	no	—
113	C₂	#10589091	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10592398

C₂

yes

maximal, characteristic

C₃×#4463

C₂

yes

—

—

C₄

yes

—

C₃²×C₄×C₈

C₂²

yes

—

—

—

—

D₃

yes

—

C₂×#634

—

C₂×C₃²×C₈

D₄

yes

—

C₃²×C₄²

C₂×C₄

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

#3746

—

#3746

—

C₃×C₃⋊C₁₆

—

#1358

—

Sylow

C₂×Dic₁₂

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

#869

—

#869

—

C₃×M₃₂

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×D₃

yes

—

#812

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

—

#634

—

C₃²×C₈

D₈

yes

—

C₃²×C₈

SD₁₆

yes

—

#494

—

#364

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₁₂

yes

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄²

—

C₃×C₄²

C₄×D₃

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

#100

yes

characteristic

C₃²×C₄

—

C₃²×C₄

#102

yes

—

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₄⋊Q₈

—

M₃₂

—

C₄×C₈

D₃²

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

#204

yes

—

C₃×C₈

D₂₄

yes

—

C₃×C₈

#213

yes

—

C₃×C₈

#215

yes

—

C₂×C₃²

—

C₂×C₃²

#331

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

#626

yes

—

C₄⋊C₄

—

C₄²

#624

yes

characteristic

C₁₆

—

C₂²×C₃

—

C₂²×C₃

#823

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#809

yes

—

C₃×C₄

#821

yes

—

Dic₃

—

Dic₃

—

C₃²

#1358

yes

characteristic, complemented, Sylow

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#3722

yes

characteristic

100

C₈

#3714

yes

—

101

C₈

#3717

yes

—

102

C₂×C₃

—

103

C₂×C₃

—

104

C₂×C₃

—

105

C₂×C₃

#4485

yes

—

106

C₂²

#63559

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

#63564

yes

—

110

C₃

—

111

C₃

#67572

yes

complemented

112

C₂

—

113

C₂

#10589091

yes

characteristic

	subgroup	G/H
center	C₂	#10589091
commutator	C₃²×C₄²	C₂×C₄
Frattini, hypercenter	C₄×C₈	D₃²
Fitting, cosocle	C₃²×C₄×C₈	C₂²
socle	C₂×C₃²	#331
prime radical	C₂²×C₃²	#100
hypocenter	C₃²	#1358
LCS	G ▷ C₃²×C₄² ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈

subgroup

G/H

center

C₂

#10589091

commutator

C₃²×C₄²

C₂×C₄

Frattini, hypercenter

C₄×C₈

D₃²

Fitting, cosocle

C₃²×C₄×C₈

C₂²

C₂×C₃²

C₂²×C₃²

C₃²

G ▷ C₃²×C₄² ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈

N	H ≅ G/N	type
95. C₃²	#1358	semidirect
111. C₃	#67572	semidirect

H ≅ G/N

type

95. C₃²

#1358

semidirect

111. C₃

#67572

semidirect

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

subgroup	index
#10592398	2
C₃×#4463	2
#67572	3

subgroup

index

#10592398

C₃×#4463

#67572

Subgroups — #11772024

Conjugacy classes of subgroups

Complements

Maximal subgroups