Group #11772453 — subgroups

Subgroups — #11772453

Order 1152.

372 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 193 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×C₃²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#63646	C₂	1	2	yes	maximal
3	#67820	—	3	4	no	maximal
4	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
5	C₂×#3748	—	2	2	no	—
6	C₃²×M₃₂	—	2	2	no	—
7	C₂×C₃²×C₁₆	C₄	1	2	yes	—
8	#63646	—	2	2	no	—
9	#63646	C₄	1	4	yes	—
10	C₂×C₃×M₃₂	D₃	1	4	yes	—
11	C₂×#819	—	3	8	no	—
12	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
13	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
15	C₂×C₃²×C₈	D₄	1	2	yes	—
16	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
17	#3748	—	2	4	no	—
18	#3748	—	4	2	no	—
19	C₂×C₃⋊(C₃⋊C₈)	—	4	2	no	—
20	C₃²×C₁₆	—	2	2	no	—
21	#1385	—	9	1	no	Sylow
22	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
23	C₂×#208	—	6	8	no	—
24	C₃×M₃₂	—	2	8	no	—
25	C₂×C₃×C₁₆	Dic₃	1	8	yes	—
26	#819	—	3	16	no	—
27	#819	—	6	8	no	—
28	C₂³×C₃²	Q₁₆	1	1	yes	characteristic
29	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
30	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
32	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
33	C₃²×C₈	C₄²	1	1	yes	characteristic
34	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
35	C₃²×C₈	C₄⋊C₄	1	2	yes	—
36	C₃⋊(C₃⋊C₈)	—	4	4	no	—
37	C₂×M₃₂	C₃⋊D₃	1	1	yes	characteristic
38	C₂×#108	—	9	2	no	—
39	C₂²×C₃×C₄	D₁₂	1	4	yes	—
40	C₂×C₃×C₈	Dic₆	1	4	yes	—
41	C₂×C₃×C₈	C₄×D₃	1	8	yes	—
42	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
43	C₂×C₃×C₈	#67	1	8	yes	—
44	#208	—	6	16	no	—
45	#208	—	12	8	no	—
46	C₂×C₃⋊C₈	—	12	8	no	—
47	C₃×C₁₆	—	2	8	no	—
48	C₂²×C₃²	—	4	1	no	—
49	C₂²×C₃²	#103	1	2	yes	—
50	C₂²×C₃²	#107	1	1	yes	characteristic
51	C₃²×C₄	#95	1	1	yes	characteristic
52	C₃²×C₄	#102	1	2	yes	—
53	C₃²×C₄	#106	1	1	yes	characteristic
54	C₂×M₁₆	—	18	2	no	—
55	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
56	M₃₂	—	2	2	no	—
57	C₂×C₁₆	C₃⋊Dic₃	1	2	yes	—
58	#108	—	9	4	no	—
59	#108	—	18	2	no	—
60	C₂³×C₃	Dic₁₂	1	4	yes	—
61	C₂×C₃×C₄	#204	1	8	yes	—
62	C₂×C₃×C₄	D₂₄	1	4	yes	—
63	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
64	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	8	yes	—
65	C₃×C₈	C₄×Dic₃	1	4	yes	—
66	C₃×C₈	Dic₃⋊C₄	1	8	yes	—
67	C₃×C₈	#217	1	4	yes	—
68	C₃⋊C₈	—	12	16	no	—
69	C₂×C₃²	—	2	2	no	—
70	C₂×C₃²	—	2	1	no	—
71	C₂×C₃²	#340	1	1	yes	characteristic
72	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
73	M₁₆	—	18	4	no	—
74	M₁₆	—	36	2	no	—
75	C₂×C₈	—	36	2	no	—
76	C₂×C₈	#634	1	1	yes	characteristic
77	C₂×C₈	C₄×C₃⋊D₃	1	2	yes	—
78	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
79	C₂×C₈	#638	1	2	yes	—
80	C₁₆	—	2	2	no	—
81	C₂²×C₃	—	4	4	no	—
82	C₂²×C₃	#816	1	8	yes	—
83	C₂²×C₃	#818	1	4	yes	—
84	C₃×C₄	#817	1	4	yes	—
85	C₃×C₄	#821	1	8	yes	—
86	C₃×C₄	C₃⋊#95	1	4	yes	—
87	C₃²	#1385	1	1	yes	characteristic, complemented, Sylow
88	C₂³	#3746	1	1	yes	characteristic
89	C₂×C₄	#3744	1	2	yes	—
90	C₂×C₄	#3745	1	1	yes	characteristic
91	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
92	C₂×C₄	#3752	1	2	yes	—
93	C₈	—	36	4	no	—
94	C₈	C₄×C₃⋊Dic₃	1	1	yes	characteristic
95	C₈	#3750	1	2	yes	—
96	C₈	#3757	1	1	yes	characteristic
97	C₂×C₃	—	2	8	no	—
98	C₂×C₃	—	2	4	no	—
99	C₂×C₃	#4551	1	4	yes	—
100	C₂²	—	4	1	no	—
101	C₂²	#63643	1	2	yes	—
102	C₂²	#63645	1	1	yes	characteristic
103	C₄	#63644	1	1	yes	characteristic
104	C₄	#63648	1	2	yes	—
105	C₄	#63658	1	1	yes	characteristic
106	C₃	#67820	1	4	yes	complemented
107	C₂	—	2	2	no	—
108	C₂	—	2	1	no	—
109	C₂	#10589417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃²×M₃₂

C₂

yes

maximal, characteristic

C₂×#63646

C₂

yes

maximal

#67820

—

maximal

C₂²×C₃²×C₈

C₂²

yes

characteristic

C₂×#3748

—

C₃²×M₃₂

—

C₂×C₃²×C₁₆

C₄

yes

—

#63646

—

#63646

C₄

yes

—

C₂×C₃×M₃₂

D₃

yes

—

C₂×#819

—

C₂²×C₃²×C₄

D₄

yes

characteristic

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

C₂×C₃²×C₈

C₂×C₄

yes

—

C₂×C₃²×C₈

D₄

yes

—

C₂×C₃²×C₈

Q₈

yes

characteristic

#3748

—

#3748

—

C₂×C₃⋊(C₃⋊C₈)

—

C₃²×C₁₆

—

#1385

—

Sylow

C₂²×C₃×C₈

C₂×D₃

yes

—

C₂×#208

—

C₃×M₃₂

—

C₂×C₃×C₁₆

Dic₃

yes

—

#819

—

#819

—

C₂³×C₃²

Q₁₆

yes

characteristic

C₂×C₃²×C₄

C₂²⋊C₄

yes

—

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃²×C₄

D₈

yes

characteristic

C₂×C₃²×C₄

SD₁₆

yes

—

C₃²×C₈

C₄²

yes

characteristic

C₃²×C₈

C₂²⋊C₄

yes

characteristic

C₃²×C₈

C₄⋊C₄

yes

—

C₃⋊(C₃⋊C₈)

—

C₂×M₃₂

C₃⋊D₃

yes

characteristic

C₂×#108

—

C₂²×C₃×C₄

D₁₂

yes

—

C₂×C₃×C₈

Dic₆

yes

—

C₂×C₃×C₈

C₄×D₃

yes

—

C₂×C₃×C₈

C₂×Dic₃

yes

—

C₂×C₃×C₈

#67

yes

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

#103

yes

—

C₂²×C₃²

yes

C₃²×C₄

yes

C₃²×C₄

yes

—

C₃²×C₄

#106

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₂×C₃⋊D₃

yes

characteristic

M₃₂

—

C₂×C₁₆

C₃⋊Dic₃

yes

—

#108

—

#108

—

C₂³×C₃

Dic₁₂

yes

—

C₂×C₃×C₄

#204

yes

—

C₂×C₃×C₄

D₂₄

yes

—

C₂×C₃×C₄

C₄⋊Dic₃

yes

—

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

—

C₃×C₈

C₄×Dic₃

yes

—

C₃×C₈

Dic₃⋊C₄

yes

—

C₃×C₈

#217

yes

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#340

yes

characteristic

C₂²×C₄

C₄⋊(C₃⋊D₃)

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

#634

yes

characteristic

C₂×C₈

C₄×C₃⋊D₃

yes

—

C₂×C₈

C₂×C₃⋊Dic₃

yes

characteristic

C₂×C₈

#638

yes

—

C₁₆

—

C₂²×C₃

—

C₂²×C₃

#816

yes

—

C₂²×C₃

#818

yes

—

C₃×C₄

#817

yes

—

C₃×C₄

#821

yes

—

C₃×C₄

C₃⋊#95

yes

—

C₃²

#1385

yes

characteristic, complemented, Sylow

C₂³

#3746

yes

characteristic

C₂×C₄

#3744

yes

—

C₂×C₄

#3745

yes

characteristic

C₂×C₄

C₄⋊(C₃⋊Dic₃)

yes

characteristic

C₂×C₄

#3752

yes

—

C₈

—

C₈

C₄×C₃⋊Dic₃

yes

characteristic

C₈

#3750

yes

—

C₈

#3757

yes

characteristic

C₂×C₃

—

C₂×C₃

—

C₂×C₃

#4551

yes

—

100

C₂²

—

101

C₂²

#63643

yes

—

102

C₂²

#63645

yes

characteristic

103

C₄

#63644

yes

characteristic

104

C₄

#63648

yes

—

105

C₄

#63658

yes

characteristic

106

C₃

#67820

yes

complemented

107

C₂

—

108

C₂

—

109

C₂

#10589417

yes

characteristic

	subgroup	G/H
center	C₄	#63644
commutator	C₃²×C₈	C₄²
Frattini	C₂²×C₈	C₂×C₃⋊D₃
Fitting	C₂×C₃²×M₃₂	C₂
socle	C₂×C₃²	#340
prime radical	C₂³×C₃²	Q₁₆
cosocle	C₂²×C₃²×C₈	C₂²
hypercenter	C₂×M₃₂	C₃⋊D₃
hypocenter	C₃²	#1385
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₄

#63644

commutator

C₃²×C₈

C₄²

Frattini

C₂²×C₈

C₂×C₃⋊D₃

Fitting

C₂×C₃²×M₃₂

C₂

socle

C₂×C₃²

#340

prime radical

C₂³×C₃²

Q₁₆

cosocle

C₂²×C₃²×C₈

C₂²

hypercenter

C₂×M₃₂

C₃⋊D₃

hypocenter

C₃²

#1385

LCS

G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
87. C₃²	#1385	semidirect
106. C₃	#67820	semidirect

H ≅ G/N

type

87. C₃²

#1385

semidirect

106. C₃

#67820

semidirect

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

subgroup	index
C₂×C₃²×M₃₂	2
C₂×#63646	2
#67820	3

subgroup

index

C₂×C₃²×M₃₂

C₂×#63646

#67820

Subgroups — #11772453

Conjugacy classes of subgroups

Complements

Maximal subgroups