Group #11771266 — subgroups

Subgroups — #11771266

Order 1152.

184 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 41 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	#10593731	C₂	1	2	yes	maximal
2	#10593728	C₂	1	1	yes	maximal, characteristic
3	C₂×#3788	C₄	1	2	yes	—
4	C₄²×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	#63779	—	2	2	no	—
6	#63778	C₄	1	4	yes	—
7	#63777	—	2	2	no	—
8	C₄⋊#622	—	2	2	no	—
9	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄×C₃⋊D₃	C₂×C₄	1	2	yes	—
11	C₂×#622	—	2	2	no	—
12	C₂×#622	—	2	4	no	—
13	#3788	—	4	2	no	—
14	C₃²×C₄²	Q₈	1	1	yes	characteristic
15	C₄×C₃⋊Dic₃	D₄	1	2	yes	—
16	C₄×C₃⋊Dic₃	D₄	1	1	yes	characteristic
17	#1307	—	9	1	no	maximal, Sylow
18	C₄²×D₃	—	12	1	no	—
19	C₂²×C₃⋊D₃	C₄²	1	1	yes	characteristic
20	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
22	C₂×C₃⋊Dic₃	C₂²⋊C₄	1	2	yes	—
23	C₂×C₃⋊Dic₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₄×C₃⋊D₃	—	2	1	no	—
25	C₄×C₃⋊D₃	—	2	1	no	—
26	C₄×C₃⋊D₃	—	4	2	no	—
27	#622	—	4	4	no	—
28	#622	—	4	2	no	—
29	#432	—	9	2	no	—
30	#423	—	9	1	no	—
31	C₂×C₄×D₃	—	12	1	no	—
32	C₂×C₄×D₃	—	12	2	no	—
33	C₃×C₄²	—	4	1	no	—
34	C₄×Dic₃	—	12	1	no	—
35	C₄×Dic₃	—	12	2	no	—
36	C₂²×C₃²	#95	1	1	yes	characteristic
37	C₂×C₃⋊D₃	—	2	1	no	—
38	C₂×C₃⋊D₃	—	2	2	no	—
39	C₃²×C₄	—	2	1	no	—
40	C₃²×C₄	—	2	2	no	—
41	C₃⋊Dic₃	—	2	2	no	—
42	C₃⋊Dic₃	#100	1	1	yes	characteristic
43	C₃⋊Dic₃	#101	1	1	yes	characteristic
44	C₂×M₁₆	—	9	2	no	—
45	C₂×C₄²	—	9	1	no	—
46	C₄⋊C₈	—	18	2	no	—
47	C₄⋊C₈	—	18	2	no	—
48	C₂²⋊C₈	—	9	4	no	—
49	#97	—	18	2	no	—
50	C₂²×D₃	—	12	1	no	—
51	C₂×C₃×C₄	—	4	1	no	—
52	C₂×C₃×C₄	—	4	2	no	—
53	C₂×Dic₃	—	12	1	no	—
54	C₂×Dic₃	—	12	2	no	—
55	C₄×D₃	—	12	2	no	—
56	C₄×D₃	—	12	2	no	—
57	C₄×D₃	—	12	8	no	—
58	C₂×C₃²	#343	1	1	yes	characteristic
59	C₂×C₃²	#344	1	2	yes	—
60	C₃⋊D₃	—	4	1	no	—
61	C₂²×C₄	—	9	1	no	—
62	C₂²×C₄	—	9	2	no	—
63	M₁₆	—	36	2	no	—
64	C₂×C₈	—	18	4	no	—
65	C₂×C₈	—	18	2	no	—
66	C₄²	—	9	1	no	—
67	C₄²	—	9	2	no	—
68	C₄²	PSU(3,𝔽₂)	1	1	yes	characteristic
69	C₂²×C₃	—	4	1	no	—
70	C₂×D₃	—	12	2	no	—
71	C₂×D₃	—	12	4	no	—
72	C₃×C₄	—	4	2	no	—
73	C₃×C₄	—	4	4	no	—
74	Dic₃	—	12	1	no	—
75	Dic₃	—	12	1	no	—
76	Dic₃	—	12	4	no	—
77	C₃²	#1307	1	1	yes	characteristic, complemented, Sylow
78	C₂³	—	9	1	no	—
79	C₂×C₄	—	9	1	no	—
80	C₂×C₄	—	9	2	no	—
81	C₂×C₄	—	18	1	no	—
82	C₂×C₄	—	18	1	no	—
83	C₂×C₄	—	36	2	no	—
84	C₂×C₄	#3777	1	1	yes	characteristic
85	C₂×C₄	#3777	1	2	yes	—
86	C₈	—	36	2	no	—
87	C₈	—	36	4	no	—
88	C₂×C₃	—	4	1	no	—
89	C₂×C₃	—	4	2	no	—
90	D₃	—	12	4	no	—
91	C₂²	—	18	1	no	—
92	C₂²	—	18	2	no	—
93	C₂²	#63747	1	1	yes	characteristic
94	C₄	—	2	1	no	—
95	C₄	—	2	2	no	—
96	C₄	—	9	1	no	—
97	C₄	—	9	1	no	—
98	C₄	—	18	2	no	—
99	C₃	—	4	1	no	—
100	C₂	—	36	1	no	—
101	C₂	#10590094	1	1	yes	characteristic
102	C₂	#10590095	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10593731

C₂

yes

maximal

#10593728

C₂

yes

maximal, characteristic

C₂×#3788

C₄

yes

—

C₄²×C₃⋊D₃

C₂²

yes

characteristic

#63779

—

#63778

C₄

yes

—

#63777

—

C₄⋊#622

—

C₂×C₄×C₃⋊D₃

C₂×C₄

yes

characteristic

C₂×C₄×C₃⋊D₃

C₂×C₄

yes

—

C₂×#622

—

C₂×#622

—

#3788

—

C₃²×C₄²

Q₈

yes

characteristic

C₄×C₃⋊Dic₃

D₄

yes

—

C₄×C₃⋊Dic₃

D₄

yes

characteristic

#1307

—

maximal, Sylow

C₄²×D₃

—

C₂²×C₃⋊D₃

C₄²

yes

characteristic

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃²×C₄

C₄⋊C₄

yes

—

C₂×C₃⋊Dic₃

C₂²⋊C₄

yes

—

C₂×C₃⋊Dic₃

C₂²⋊C₄

yes

characteristic

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

—

—

—

—

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₃×C₄²

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

#95

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃²×C₄

—

C₃⋊Dic₃

—

C₃⋊Dic₃

yes

C₃⋊Dic₃

yes

C₂×M₁₆

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₂×C₃²

#343

yes

characteristic

C₂×C₃²

#344

yes

—

C₃⋊D₃

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

PSU(3,𝔽₂)

yes

characteristic

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

Dic₃

—

C₃²

#1307

yes

characteristic, complemented, Sylow

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#3777

yes

characteristic

C₂×C₄

#3777

yes

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

—

D₃

—

C₂²

—

C₂²

—

C₂²

#63747

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₃

—

100

C₂

—

101

C₂

#10590094

yes

characteristic

102

C₂

#10590095

yes

—

	subgroup	G/H
center	C₂²	#63747
commutator, prime radical	C₂²×C₃⋊D₃	C₄²
Frattini, hypercenter	C₄²	PSU(3,𝔽₂)
Fitting	C₃²×C₄²	Q₈
socle	C₂²×C₃²	#95
cosocle	C₄²×C₃⋊D₃	C₂²
hypocenter	C₃²	#1307
LCS	G ▷ C₂²×C₃⋊D₃ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

#63747

commutator, prime radical

C₂²×C₃⋊D₃

C₄²

Frattini, hypercenter

C₄²

PSU(3,𝔽₂)

Fitting

C₃²×C₄²

Q₈

socle

C₂²×C₃²

#95

cosocle

C₄²×C₃⋊D₃

C₂²

hypocenter

C₃²

#1307

LCS

G ▷ C₂²×C₃⋊D₃ ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
77. C₃²	#1307	semidirect

H ≅ G/N

type

77. C₃²

#1307

semidirect

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

subgroup	index
#10593731	2
#10593728	2
#1307	9

subgroup

index

#10593731

#10593728

#1307

Subgroups — #11771266

Conjugacy classes of subgroups

Complements

Maximal subgroups