Group #11786111 — subgroups

Subgroups — #11786111

Order 1152.

540 conjugacy classes of subgroups; only the 257 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 257 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×#97	C₂	1	1	yes	maximal, characteristic
2	#10592623	C₂	1	2	yes	maximal
3	#10589418	C₂	1	4	yes	maximal
4	#69841	—	3	4	no	maximal
5	C₂²×C₃²×C₈	C₂²	1	2	yes	—
6	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
7	C₂×#3748	C₂²	1	4	yes	—
8	C₃²×#97	C₄	1	4	yes	—
9	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	4	no	—
10	C₂×C₃×#97	D₃	1	4	yes	—
11	#4924	—	3	8	no	—
12	#4552	—	3	16	no	—
13	C₂²×C₃²×C₄	Q₈	1	2	yes	—
14	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
15	C₂×C₃²×C₈	—	2	4	no	—
16	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
17	C₃²×C₄²	C₂×C₄	1	2	yes	—
18	C₃²×C₄²	D₄	1	2	yes	—
19	#3748	—	2	8	no	—
20	C₂×C₃⋊(C₃⋊C₈)	—	2	4	no	—
21	#1838	—	9	1	no	Sylow
22	C₂²×C₃×C₈	C₂×D₃	1	8	yes	—
23	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
24	C₂×#208	—	3	16	no	—
25	C₃×#97	Dic₃	1	16	yes	—
26	C₄⋊(C₃⋊C₈)	—	6	16	no	—
27	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₃²×C₄	—	4	2	no	—
29	C₂×C₃²×C₄	C₄⋊C₄	1	4	yes	—
30	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
32	C₂×C₃²×C₄	#41	1	4	yes	—
33	C₃²×C₈	—	2	4	no	—
34	C₃⋊(C₃⋊C₈)	—	4	4	no	—
35	#423	—	9	2	no	—
36	C₂×#97	C₃⋊D₃	1	1	yes	characteristic
37	#341	—	9	4	no	—
38	C₂²×C₃×C₄	Dic₆	1	8	yes	—
39	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
40	C₂×C₃×C₈	—	2	16	no	—
41	C₂×C₃×C₈	C₂×Dic₃	1	16	yes	—
42	C₃×C₄²	D₁₂	1	8	yes	—
43	C₃×C₄²	C₂×Dic₃	1	8	yes	—
44	#208	—	6	32	no	—
45	C₂×C₃⋊C₈	—	6	16	no	—
46	C₂²×C₃²	—	2	2	no	—
47	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₃²	#125	1	2	yes	—
49	C₃²×C₄	—	2	4	no	—
50	C₃²×C₄	#117	1	2	yes	—
51	C₃²×C₄	#124	1	2	yes	—
52	C₂×M₁₆	—	9	4	no	—
53	C₂²×C₈	C₂×C₃⋊D₃	1	2	yes	—
54	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
55	C₄⋊C₈	—	18	4	no	—
56	#97	C₃⋊Dic₃	1	4	yes	—
57	C₂³×C₃	C₂×Dic₆	1	4	yes	—
58	C₂×C₃×C₄	—	4	8	no	—
59	C₂×C₃×C₄	C₄⋊Dic₃	1	16	yes	—
60	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
61	C₂×C₃×C₄	#235	1	16	yes	—
62	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
63	C₃×C₈	—	2	16	no	—
64	C₃⋊C₈	—	12	16	no	—
65	C₂×C₃²	—	2	2	no	—
66	C₂×C₃²	#382	1	1	yes	characteristic
67	C₂×C₃²	#430	1	2	yes	—
68	C₂²×C₄	#634	1	2	yes	—
69	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
70	M₁₆	—	18	8	no	—
71	C₂×C₈	—	2	4	no	—
72	C₂×C₈	—	18	4	no	—
73	C₂×C₈	C₂×C₃⋊Dic₃	1	4	yes	—
74	C₄²	C₄⋊(C₃⋊D₃)	1	2	yes	—
75	C₄²	C₂×C₃⋊Dic₃	1	2	yes	—
76	C₂²×C₃	—	2	8	no	—
77	C₂²×C₃	#870	1	8	yes	—
78	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
79	C₃×C₄	—	2	16	no	—
80	C₃×C₄	#875	1	8	yes	—
81	C₃×C₄	#926	1	8	yes	—
82	C₃²	#1838	1	1	yes	characteristic, complemented, Sylow
83	C₂³	C₂×#634	1	1	yes	characteristic
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
86	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
87	C₂×C₄	#3828	1	4	yes	—
88	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
89	C₈	—	2	4	no	—
90	C₈	—	36	4	no	—
91	C₂×C₃	—	2	8	no	—
92	C₂×C₃	#4936	1	4	yes	—
93	C₂×C₃	#5125	1	8	yes	—
94	C₂²	—	2	2	no	—
95	C₂²	#64079	1	2	yes	—
96	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
97	C₄	—	2	4	no	—
98	C₄	#64084	1	2	yes	—
99	C₄	#64135	1	2	yes	—
100	C₃	#69841	1	4	yes	complemented
101	C₂	—	2	2	no	—
102	C₂	#10592635	1	1	yes	characteristic
103	C₂	#10592824	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃²×#97

C₂

yes

maximal, characteristic

#10592623

C₂

yes

maximal

#10589418

C₂

yes

maximal

#69841

—

maximal

C₂²×C₃²×C₈

C₂²

yes

—

C₂×C₃²×C₄²

C₂²

yes

characteristic

C₂×#3748

C₂²

yes

—

C₃²×#97

C₄

yes

—

C₄⋊(C₃⋊(C₃⋊C₈))

—

C₂×C₃×#97

D₃

yes

—

#4924

—

#4552

—

C₂²×C₃²×C₄

Q₈

yes

—

C₂²×C₃²×C₄

C₂³

yes

characteristic

C₂×C₃²×C₈

—

C₂×C₃²×C₈

C₂×C₄

yes

—

C₃²×C₄²

C₂×C₄

yes

—

C₃²×C₄²

D₄

yes

—

#3748

—

C₂×C₃⋊(C₃⋊C₈)

—

#1838

—

Sylow

C₂²×C₃×C₈

C₂×D₃

yes

—

C₂×C₃×C₄²

C₂×D₃

yes

—

C₂×#208

—

C₃×#97

Dic₃

yes

—

C₄⋊(C₃⋊C₈)

—

C₂³×C₃²

C₂×Q₈

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₄⋊C₄

yes

—

C₂×C₃²×C₄

C₂²×C₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

#41

yes

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

#423

—

C₂×#97

C₃⋊D₃

yes

characteristic

#341

—

C₂²×C₃×C₄

Dic₆

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×Dic₃

yes

—

C₃×C₄²

D₁₂

yes

—

C₃×C₄²

C₂×Dic₃

yes

—

#208

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃²

#125

yes

—

C₃²×C₄

—

C₃²×C₄

#117

yes

—

C₃²×C₄

#124

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×C₃⋊D₃

yes

—

C₂×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊C₈

—

#97

C₃⋊Dic₃

yes

—

C₂³×C₃

C₂×Dic₆

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊Dic₃

yes

—

C₂×C₃×C₄

C₂×D₁₂

yes

—

C₂×C₃×C₄

#235

yes

—

C₂×C₃×C₄

C₂²×Dic₃

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#382

yes

characteristic

C₂×C₃²

#430

yes

—

C₂²×C₄

#634

yes

—

C₂²×C₄

C₂²×C₃⋊D₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₃⋊Dic₃

yes

—

C₄²

C₄⋊(C₃⋊D₃)

yes

—

C₄²

C₂×C₃⋊Dic₃

yes

—

C₂²×C₃

—

C₂²×C₃

#870

yes

—

C₂²×C₃

C₂×C₄⋊Dic₃

yes

—

C₃×C₄

—

C₃×C₄

#875

yes

—

C₃×C₄

#926

yes

—

C₃²

#1838

yes

characteristic, complemented, Sylow

C₂³

C₂×#634

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄⋊(C₃⋊Dic₃)

yes

—

C₂×C₄

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

C₂×C₄

#3828

yes

—

C₂×C₄

C₂²×C₃⋊Dic₃

yes

characteristic

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

#4936

yes

—

C₂×C₃

#5125

yes

—

C₂²

—

C₂²

#64079

yes

—

C₂²

C₂×C₄⋊(C₃⋊Dic₃)

yes

characteristic

C₄

—

C₄

#64084

yes

—

C₄

#64135

yes

—

100

C₃

#69841

yes

complemented

101

C₂

—

102

C₂

#10592635

yes

characteristic

103

C₂

#10592824

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×#97	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₂×Q₈
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×#97	C₃⋊D₃
hypocenter	C₃²	#1838
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊(C₃⋊D₃)

commutator

C₂×C₃²×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×C₃⋊D₃

Fitting

C₂×C₃²×#97

C₂

socle

C₂²×C₃²

C₂×C₄⋊C₄

prime radical

C₂³×C₃²

C₂×Q₈

cosocle

C₂²×C₃²×C₄

C₂³

hypercenter

C₂×#97

C₃⋊D₃

hypocenter

C₃²

#1838

LCS

G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
82. C₃²	#1838	semidirect
100. C₃	#69841	semidirect

H ≅ G/N

type

82. C₃²

#1838

semidirect

100. C₃

#69841

semidirect

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

subgroup	index
C₂×C₃²×#97	2
#10592623	2
#10589418	2
#69841	3

subgroup

index

C₂×C₃²×#97

#10592623

#10589418

#69841

Subgroups — #11786111

Conjugacy classes of subgroups

Complements

Maximal subgroups