Group #11795372 — subgroups

Subgroups — #11795372

Order 1152.

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

normal subgroups only

This group has 340 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	#10593802	C₂	1	3	yes	maximal
2	#10593565	C₂	1	3	yes	maximal, complemented
3	#10590096	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	C₂×#3792	C₂²	1	3	yes	—
6	C₂×#3792	C₂²	1	3	yes	—
7	#63778	—	2	6	no	—
8	#63748	—	2	6	no	—
9	C₂³×C₃⋊D₃	Q₈	1	1	yes	characteristic
10	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
11	C₂²×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
12	C₂×C₄×C₃⋊D₃	—	2	3	no	—
13	C₂×C₄×C₃⋊D₃	—	2	3	no	—
14	#3792	—	2	6	no	—
15	#3792	—	2	6	no	—
16	C₂×#622	—	2	3	no	—
17	C₂×#622	D₄	1	6	yes	—
18	#2091	—	9	1	no	maximal, Sylow
19	C₂²×C₄×D₃	—	12	1	no	—
20	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
21	C₂²×C₃⋊D₃	—	2	3	no	—
22	C₂²×C₃⋊D₃	—	4	2	no	—
23	C₂×C₃²×C₄	—	2	3	no	—
24	C₂×C₃⋊Dic₃	C₂×D₄	1	3	yes	—
25	C₂×C₃⋊Dic₃	#41	1	3	yes	—
26	C₄×C₃⋊D₃	—	4	3	no	—
27	C₄×C₃⋊D₃	—	4	1	no	—
28	#622	—	2	6	no	—
29	#622	—	4	3	no	—
30	#430	—	9	3	no	—
31	#407	—	9	3	no	—
32	#338	—	9	1	no	—
33	C₂³×D₃	—	12	1	no	—
34	C₂²×C₃×C₄	—	4	1	no	—
35	C₂²×Dic₃	—	12	1	no	—
36	C₂×C₄×D₃	—	12	6	no	—
37	C₂×C₄×D₃	—	12	6	no	—
38	C₂²×C₃²	—	4	1	no	—
39	C₂²×C₃²	C₂²⋊Q₈	1	3	yes	—
40	C₂×C₃⋊D₃	—	2	6	no	—
41	C₂×C₃⋊D₃	—	4	3	no	—
42	C₂×C₃⋊D₃	—	4	1	no	—
43	C₃²×C₄	—	4	1	no	—
44	C₃⋊Dic₃	#123	1	3	yes	—
45	C₃⋊Dic₃	#127	1	1	yes	characteristic
46	C₂³×C₄	—	9	1	no	—
47	C₂×M₁₆	—	9	3	no	—
48	C₂×M₁₆	—	9	3	no	—
49	#108	—	18	6	no	—
50	C₂²⋊C₈	—	18	6	no	—
51	C₂³×C₃	—	4	1	no	—
52	C₂²×D₃	—	12	6	no	—
53	C₂²×D₃	—	12	8	no	—
54	C₂×C₃×C₄	—	4	6	no	—
55	C₂×Dic₃	—	12	3	no	—
56	C₂×Dic₃	—	12	3	no	—
57	C₄×D₃	—	12	12	no	—
58	C₄×D₃	—	12	4	no	—
59	C₂×C₃²	—	2	3	no	—
60	C₂×C₃²	#399	1	1	yes	characteristic
61	C₃⋊D₃	—	4	2	no	—
62	C₂⁴	—	9	1	no	—
63	C₂²×C₄	—	9	1	no	—
64	C₂²×C₄	—	18	3	no	—
65	C₂²×C₄	—	18	3	no	—
66	C₂²×C₄	PSU(3,𝔽₂)	1	1	yes	characteristic
67	M₁₆	—	18	6	no	—
68	M₁₆	—	18	6	no	—
69	C₂×C₈	—	9	6	no	—
70	C₂×C₈	—	18	3	no	—
71	C₂²×C₃	—	4	4	no	—
72	C₂²×C₃	—	4	3	no	—
73	C₂×D₃	—	12	12	no	—
74	C₂×D₃	—	12	4	no	—
75	C₂×D₃	—	12	12	no	—
76	C₃×C₄	—	4	4	no	—
77	Dic₃	—	12	3	no	—
78	Dic₃	—	12	1	no	—
79	C₃²	#2091	1	1	yes	characteristic, complemented, Sylow
80	C₂³	—	18	3	no	—
81	C₂³	—	36	2	no	—
82	C₂³	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
83	C₂×C₄	—	2	3	no	—
84	C₂×C₄	—	9	3	no	—
85	C₂×C₄	—	9	3	no	—
86	C₂×C₄	—	36	3	no	—
87	C₂×C₄	—	36	1	no	—
88	C₈	—	18	6	no	—
89	C₈	—	36	3	no	—
90	C₂×C₃	—	4	6	no	—
91	C₂×C₃	—	4	1	no	—
92	D₃	—	12	8	no	—
93	C₂²	—	4	1	no	—
94	C₂²	—	18	6	no	—
95	C₂²	—	36	3	no	—
96	C₂²	—	36	1	no	—
97	C₂²	C₂²⋊PSU(3,𝔽₂)	1	3	yes	—
98	C₄	—	4	1	no	—
99	C₄	—	9	3	no	—
100	C₄	—	9	1	no	—
101	C₃	—	4	1	no	—
102	C₂	—	2	3	no	—
103	C₂	—	36	2	no	—
104	C₂	#10593566	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593802

C₂

yes

maximal

#10593565

C₂

yes

maximal, complemented

#10590096

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×C₃⋊D₃

C₂²

yes

characteristic

C₂×#3792

C₂²

yes

—

C₂×#3792

C₂²

yes

—

#63778

—

#63748

—

C₂³×C₃⋊D₃

Q₈

yes

characteristic

C₂²×C₃²×C₄

Q₈

yes

characteristic

C₂²×C₃⋊Dic₃

C₂³

yes

characteristic

C₂×C₄×C₃⋊D₃

—

C₂×C₄×C₃⋊D₃

—

—

—

—

D₄

yes

—

#2091

—

maximal, Sylow

C₂²×C₄×D₃

—

C₂³×C₃²

C₂×Q₈

yes

characteristic

C₂²×C₃⋊D₃

—

C₂²×C₃⋊D₃

—

C₂×C₃²×C₄

—

C₂×C₃⋊Dic₃

C₂×D₄

yes

—

C₂×C₃⋊Dic₃

#41

yes

—

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

—

—

—

—

—

—

C₂³×D₃

—

C₂²×C₃×C₄

—

C₂²×Dic₃

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂²×C₃²

—

C₂²×C₃²

C₂²⋊Q₈

yes

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃⋊Dic₃

#123

yes

—

C₃⋊Dic₃

#127

yes

characteristic

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

#108

—

C₂²⋊C₈

—

C₂³×C₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₂×C₃²

—

C₂×C₃²

#399

yes

characteristic

C₃⋊D₃

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

PSU(3,𝔽₂)

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂²×C₃

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

Dic₃

—

Dic₃

—

C₃²

#2091

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

C₂×PSU(3,𝔽₂)

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

—

D₃

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²⋊PSU(3,𝔽₂)

yes

—

C₄

—

C₄

—

100

C₄

—

101

C₃

—

102

C₂

—

103

C₂

—

104

C₂

#10593566

yes

characteristic

	subgroup	G/H
center	C₂	#10593566
commutator, cosocle	C₂²×C₃⋊Dic₃	C₂³
Frattini	C₂³	C₂×PSU(3,𝔽₂)
Fitting	C₂²×C₃²×C₄	Q₈
socle	C₂×C₃²	#399
prime radical	C₂³×C₃⋊D₃	Q₈
hypercenter	C₂²×C₄	PSU(3,𝔽₂)
hypocenter	C₃²	#2091
LCS	G ▷ C₂²×C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄

subgroup

G/H

center

C₂

#10593566

commutator, cosocle

C₂²×C₃⋊Dic₃

C₂³

Frattini

C₂³

C₂×PSU(3,𝔽₂)

Fitting

C₂²×C₃²×C₄

Q₈

socle

C₂×C₃²

#399

prime radical

C₂³×C₃⋊D₃

Q₈

C₂²×C₄

C₃²

G ▷ C₂²×C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄

N	H ≅ G/N	type
2. #10593565	C₂	semidirect
3. #10590096	C₂	semidirect
79. C₃²	#2091	semidirect

H ≅ G/N

type

2. #10593565

C₂

semidirect

3. #10590096

C₂

semidirect

79. C₃²

#2091

semidirect

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

subgroup	index
#10593802	2
#10593565	2
#10590096	2
#2091	9

subgroup

index

Subgroups — #11795372

Conjugacy classes of subgroups

Complements

Maximal subgroups