Group #11772447 — subgroups

Subgroups — #11772447

Order 1152.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#10593768	C₂	1	2	yes	maximal
2	C₂×#63765	C₂	1	1	yes	maximal, characteristic
3	#64286	—	2	2	no	—
4	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	C₂×#3708	C₄	1	2	yes	complemented
6	#63769	—	2	2	no	—
7	#63768	C₄	1	4	yes	complemented
8	#63765	—	2	2	no	—
9	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
10	C₂²⋊#157	—	4	2	no	—
11	C₄⋊#157	—	2	4	no	—
12	C₄×#157	—	4	2	no	—
13	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
14	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
15	C₈×C₃⋊D₃	C₂×C₄	1	2	yes	—
16	C₈×C₃⋊D₃	D₄	1	2	yes	—
17	#3708	—	2	2	no	—
18	#1384	—	9	1	no	maximal, Sylow
19	C₂×D₃×C₈	—	12	1	no	—
20	C₂²×C₃⋊D₃	D₈	1	1	yes	characteristic
21	C₂×#157	—	4	4	no	—
22	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
23	C₂×C₃⋊Dic₃	Q₁₆	1	1	yes	characteristic
24	C₄×C₃⋊D₃	C₂²⋊C₄	1	2	yes	—
25	C₄×C₃⋊D₃	SD₁₆	1	2	yes	—
26	C₃²×C₈	C₄⋊C₄	1	2	yes	—
27	C₃⋊(C₃⋊C₈)	C₄²	1	1	yes	characteristic
28	C₃⋊(C₃⋊C₈)	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×M₃₂	—	9	1	no	—
30	#427	—	9	2	no	—
31	C₂×C₄×D₃	—	12	1	no	—
32	C₂×C₃×C₈	—	4	1	no	—
33	C₂×C₃⋊C₈	—	12	1	no	—
34	D₃×C₈	—	12	2	no	—
35	D₃×C₈	—	12	2	no	—
36	C₂²×C₃²	#107	1	1	yes	characteristic
37	C₂×C₃⋊D₃	—	4	1	no	—
38	C₂×C₃⋊D₃	#102	1	2	yes	—
39	#157	—	8	4	no	—
40	C₃²×C₄	#95	1	1	yes	characteristic
41	C₃²×C₄	#106	1	1	yes	characteristic
42	C₃⋊Dic₃	#103	1	2	yes	—
43	C₂²×C₈	—	9	1	no	—
44	#117	—	18	2	no	—
45	M₃₂	—	18	2	no	—
46	C₂×C₁₆	—	9	2	no	—
47	#107	—	18	2	no	—
48	#106	—	9	4	no	—
49	C₂²×D₃	—	12	1	no	—
50	C₂×C₃×C₄	—	4	1	no	—
51	C₂×Dic₃	—	12	1	no	—
52	C₄×D₃	—	12	2	no	—
53	C₄×D₃	—	12	2	no	—
54	C₃×C₈	—	4	2	no	—
55	C₃⋊C₈	—	12	1	no	—
56	C₃⋊C₈	—	12	1	no	—
57	C₂×C₃²	—	2	1	no	—
58	C₂×C₃²	#340	1	1	yes	characteristic
59	C₃⋊D₃	—	2	2	no	—
60	C₂²×C₄	—	9	1	no	—
61	C₂×C₈	—	9	1	no	—
62	C₂×C₈	—	9	2	no	—
63	C₂×C₈	—	9	2	no	—
64	C₂×C₈	PSU(3,𝔽₂)	1	1	yes	characteristic
65	C₄⋊C₄	—	18	4	no	—
66	C₂²⋊C₄	—	36	2	no	—
67	C₄²	—	36	2	no	—
68	C₁₆	—	18	2	no	—
69	C₂²×C₃	—	4	1	no	—
70	C₂×D₃	—	12	2	no	—
71	C₂×D₃	—	12	4	no	—
72	C₃×C₄	—	4	1	no	—
73	C₃×C₄	—	4	1	no	—
74	Dic₃	—	12	2	no	—
75	C₃²	#1384	1	1	yes	characteristic, complemented, Sylow
76	C₂³	—	9	1	no	—
77	C₂×C₄	—	9	1	no	—
78	C₂×C₄	—	9	2	no	—
79	C₂×C₄	—	9	2	no	—
80	C₂×C₄	—	36	4	no	—
81	C₂×C₄	#3777	1	1	yes	characteristic
82	C₈	—	9	1	no	—
83	C₈	—	9	1	no	—
84	C₈	#3777	1	2	yes	—
85	C₂×C₃	—	4	2	no	—
86	C₂×C₃	—	4	1	no	—
87	D₃	—	12	4	no	—
88	C₂²	—	9	2	no	—
89	C₂²	—	36	1	no	—
90	C₂²	#63746	1	1	yes	characteristic
91	C₄	—	9	2	no	—
92	C₄	—	72	4	no	—
93	C₄	#63745	1	1	yes	characteristic
94	C₄	#63747	1	1	yes	characteristic
95	C₃	—	4	1	no	—
96	C₂	—	2	1	no	—
97	C₂	—	18	2	no	—
98	C₂	#10590093	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593768

C₂

yes

maximal

C₂×#63765

C₂

yes

maximal, characteristic

#64286

—

C₂×C₈×C₃⋊D₃

C₂²

yes

characteristic

C₂×#3708

C₄

yes

complemented

#63769

—

#63768

C₄

yes

complemented

#63765

—

C₂×C₄×C₃⋊D₃

D₄

yes

characteristic

C₂²⋊#157

—

C₄⋊#157

—

C₄×#157

—

C₂×C₃²×C₈

Q₈

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

characteristic

C₈×C₃⋊D₃

C₂×C₄

yes

—

C₈×C₃⋊D₃

D₄

yes

—

#3708

—

#1384

—

maximal, Sylow

C₂×D₃×C₈

—

C₂²×C₃⋊D₃

D₈

yes

characteristic

C₂×#157

—

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

Q₁₆

yes

characteristic

C₄×C₃⋊D₃

C₂²⋊C₄

yes

—

C₄×C₃⋊D₃

SD₁₆

yes

—

C₃²×C₈

C₄⋊C₄

yes

—

C₃⋊(C₃⋊C₈)

C₄²

yes

characteristic

C₃⋊(C₃⋊C₈)

C₂²⋊C₄

yes

characteristic

C₂×M₃₂

—

#427

—

C₂×C₄×D₃

—

C₂×C₃×C₈

—

C₂×C₃⋊C₈

—

D₃×C₈

—

D₃×C₈

—

C₂²×C₃²

#107

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

#102

yes

—

#157

—

C₃²×C₄

yes

C₃²×C₄

yes

C₃⋊Dic₃

yes

—

C₂²×C₈

—

#117

—

M₃₂

—

C₂×C₁₆

—

#107

—

#106

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#340

yes

characteristic

C₃⋊D₃

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

PSU(3,𝔽₂)

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

C₃²

#1384

yes

characteristic, complemented, Sylow

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#3777

yes

characteristic

C₈

—

C₈

—

C₈

#3777

yes

—

C₂×C₃

—

C₂×C₃

—

D₃

—

C₂²

—

C₂²

—

C₂²

#63746

yes

characteristic

C₄

—

C₄

—

C₄

yes

C₄

yes

C₃

—

C₂

—

C₂

—

C₂

#10590093

yes

characteristic

	subgroup	G/H
center	C₄	#63745
commutator	C₃⋊(C₃⋊C₈)	C₄²
Frattini, hypercenter	C₂×C₈	PSU(3,𝔽₂)
Fitting	C₂×C₃²×C₈	Q₈
socle	C₂×C₃²	#340
prime radical	C₂²×C₃⋊D₃	D₈
cosocle	C₂×C₈×C₃⋊D₃	C₂²
hypocenter	C₃²	#1384
LCS	G ▷ C₃⋊(C₃⋊C₈) ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center

C₄

#63745

commutator

C₃⋊(C₃⋊C₈)

C₄²

Frattini, hypercenter

C₂×C₈

PSU(3,𝔽₂)

Fitting

C₂×C₃²×C₈

Q₈

socle

C₂×C₃²

#340

prime radical

C₂²×C₃⋊D₃

D₈

cosocle

C₂×C₈×C₃⋊D₃

C₂²

hypocenter

C₃²

#1384

LCS

G ▷ C₃⋊(C₃⋊C₈) ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
5. C₂×#3708	C₄	semidirect
7. #63768	C₄	semidirect
75. C₃²	#1384	semidirect

H ≅ G/N

type

5. C₂×#3708

C₄

semidirect

7. #63768

C₄

semidirect

75. C₃²

#1384

semidirect

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

subgroup	index
#10593768	2
C₂×#63765	2
#1384	9

subgroup

index

#10593768

C₂×#63765

#1384

Subgroups — #11772447

Conjugacy classes of subgroups

Complements

Maximal subgroups