Group #11782606 — subgroups

Subgroups — #11782606

Order 1152.

421 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 120 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	C₃×#4699	C₂	1	2	yes	maximal
2	#10590312	C₂	1	4	yes	maximal
3	C₈×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#71298	—	3	2	no	maximal
5	C₂×C₃×Dic₁₂	—	2	4	no	—
6	C₃×#869	C₂²	1	4	yes	—
7	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
9	#63576	—	2	8	no	—
10	C₃×#494	—	3	2	no	—
11	#5078	—	3	8	no	—
12	#4699	D₃	1	2	yes	—
13	C₈×C₃⋊C₈	—	3	2	no	—
14	C₈×C₃⋊C₈	—	6	1	no	—
15	C₂×C₃×Dic₆	—	2	4	no	—
16	C₂×C₃²×C₈	D₄	1	2	yes	—
17	C₃²×C₄²	C₂³	1	1	yes	characteristic
18	C₂×C₃⋊(C₃⋊C₈)	D₄	1	4	yes	—
19	C₃×C₄⋊Dic₃	—	2	8	no	—
20	C₃×Dic₁₂	—	4	8	no	—
21	#1713	—	9	1	no	Sylow
22	C₂×C₃×Q₁₆	—	6	4	no	—
23	C₃×C₄⋊Q₈	—	3	4	no	—
24	C₂×Dic₁₂	—	2	4	no	—
25	#869	C₂×D₃	1	4	yes	—
26	C₃×C₄×C₈	—	2	1	no	—
27	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
28	#807	—	6	16	no	—
29	C₄×C₃⋊C₈	—	3	4	no	—
30	C₄×C₃⋊C₈	—	6	2	no	—
31	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
32	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
33	C₂×C₃×Dic₃	—	4	4	no	—
34	C₃×Dic₆	—	4	8	no	—
35	C₃²×C₈	D₈	1	4	yes	—
36	C₃⋊(C₃⋊C₈)	Q₁₆	1	8	yes	complemented
37	#500	—	9	4	no	—
38	#494	—	3	2	no	—
39	C₈²	—	9	1	no	—
40	C₂×C₃×Q₈	—	6	4	no	—
41	C₂×Dic₆	—	2	4	no	—
42	C₃×Q₁₆	—	12	8	no	—
43	C₂×C₃×C₈	—	2	2	no	—
44	C₂×C₃×C₈	#67	1	4	yes	—
45	C₃×C₄⋊C₄	—	6	8	no	—
46	C₃×C₄²	—	2	1	no	—
47	C₃×C₄²	C₂²×D₃	1	2	yes	—
48	C₄⋊Dic₃	—	2	8	no	—
49	C₂×C₃⋊C₈	—	3	8	no	—
50	C₂×C₃⋊C₈	—	6	4	no	—
51	Dic₁₂	—	4	8	no	—
52	C₂²×C₃²	#127	1	1	yes	characteristic
53	C₃²×C₄	C₂×D₈	1	2	yes	—
54	C₃²×C₄	C₂×Q₁₆	1	4	yes	—
55	C₃×Dic₃	—	8	4	no	—
56	C₂×Q₁₆	—	6	4	no	—
57	C₄⋊Q₈	—	3	4	no	—
58	#107	—	18	8	no	—
59	C₄×C₈	—	9	2	no	—
60	C₄×C₈	D₃²	1	1	yes	characteristic
61	C₃×Q₈	—	12	8	no	—
62	C₂×C₃×C₄	—	2	2	no	—
63	C₂×C₃×C₄	—	2	1	no	—
64	C₂×C₃×C₄	—	12	4	no	—
65	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
66	C₂×C₃×C₄	C₂×#67	1	2	yes	—
67	C₂×Dic₃	—	4	4	no	—
68	Dic₆	—	4	8	no	—
69	C₃×C₈	—	2	4	no	—
70	C₃×C₈	#213	1	8	yes	—
71	C₃⋊C₈	—	3	16	no	—
72	C₃⋊C₈	—	6	8	no	—
73	C₂×C₃²	#493	1	1	yes	characteristic
74	C₂×C₃²	#494	1	2	yes	—
75	C₂×Q₈	—	6	4	no	—
76	Q₁₆	—	12	8	no	—
77	C₂×C₈	—	9	4	no	—
78	C₂×C₈	#625	1	2	yes	—
79	C₄⋊C₄	—	6	8	no	—
80	C₄²	C₂×D₃²	1	1	yes	characteristic
81	C₂²×C₃	—	2	1	no	—
82	C₂²×C₃	#940	1	2	yes	—
83	C₃×C₄	—	2	4	no	—
84	C₃×C₄	—	2	2	no	—
85	C₃×C₄	—	24	4	no	—
86	C₃×C₄	D₃×Q₁₆	1	8	yes	—
87	C₃×C₄	C₂×#213	1	4	yes	—
88	Dic₃	—	8	4	no	—
89	C₃²	#1713	1	1	yes	characteristic, complemented, Sylow
90	Q₈	—	12	8	no	—
91	C₂×C₄	—	12	4	no	—
92	C₂×C₄	#3802	1	2	yes	—
93	C₂×C₄	C₂×#625	1	1	yes	characteristic
94	C₈	—	9	8	no	—
95	C₈	#3713	1	4	yes	—
96	C₂×C₃	—	2	1	no	—
97	C₂×C₃	—	2	2	no	—
98	C₂×C₃	#5073	1	2	yes	—
99	C₂×C₃	#5183	1	4	yes	—
100	C₂²	#63914	1	1	yes	characteristic
101	C₄	—	24	4	no	—
102	C₄	#63802	1	4	yes	—
103	C₄	C₂×#3713	1	2	yes	—
104	C₃	—	2	1	no	—
105	C₃	#71298	1	2	yes	complemented
106	C₂	#10590352	1	1	yes	characteristic
107	C₂	#10590905	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#4699

C₂

yes

maximal

#10590312

C₂

yes

maximal

C₈×C₃⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic

#71298

—

maximal

C₂×C₃×Dic₁₂

—

C₃×#869

C₂²

yes

—

C₃²×C₄×C₈

C₂²

yes

characteristic

C₄×C₃⋊(C₃⋊C₈)

C₂²

yes

—

—

—

—

D₃

yes

—

C₈×C₃⋊C₈

—

C₈×C₃⋊C₈

—

C₂×C₃×Dic₆

—

C₂×C₃²×C₈

D₄

yes

—

C₃²×C₄²

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

—

C₃×C₄⋊Dic₃

—

C₃×Dic₁₂

—

#1713

—

Sylow

C₂×C₃×Q₁₆

—

C₃×C₄⋊Q₈

—

C₂×Dic₁₂

—

#869

C₂×D₃

yes

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×D₃

yes

—

#807

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂×C₃²×C₄

C₂×D₄

yes

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃²×C₈

D₈

yes

—

C₃⋊(C₃⋊C₈)

Q₁₆

yes

complemented

#500

—

#494

—

C₈²

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂²×D₃

yes

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₂²×C₃²

#127

yes

characteristic

C₃²×C₄

C₂×D₈

yes

—

C₃²×C₄

C₂×Q₁₆

yes

—

C₃×Dic₃

—

C₂×Q₁₆

—

C₄⋊Q₈

—

#107

—

C₄×C₈

—

C₄×C₈

D₃²

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₃×D₄

yes

—

C₂×C₃×C₄

C₂×#67

yes

—

C₂×Dic₃

—

Dic₆

—

C₃×C₈

—

C₃×C₈

#213

yes

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

#493

yes

characteristic

C₂×C₃²

#494

yes

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

#625

yes

—

C₄⋊C₄

—

C₄²

C₂×D₃²

yes

characteristic

C₂²×C₃

—

C₂²×C₃

#940

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

D₃×Q₁₆

yes

—

C₃×C₄

C₂×#213

yes

—

Dic₃

—

C₃²

#1713

yes

characteristic, complemented, Sylow

Q₈

—

C₂×C₄

—

C₂×C₄

#3802

yes

—

C₂×C₄

C₂×#625

yes

characteristic

C₈

—

C₈

#3713

yes

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

#5073

yes

—

C₂×C₃

#5183

yes

—

100

C₂²

#63914

yes

characteristic

101

C₄

—

102

C₄

#63802

yes

—

103

C₄

C₂×#3713

yes

—

104

C₃

—

105

C₃

#71298

yes

complemented

106

C₂

#10590352

yes

characteristic

107

C₂

#10590905

yes

—

	subgroup	G/H
center	C₂²	#63914
commutator, cosocle	C₃²×C₄²	C₂³
Frattini	C₄²	C₂×D₃²
Fitting	C₃²×C₄×C₈	C₂²
socle, prime radical	C₂²×C₃²	#127
hypercenter	C₄×C₈	D₃²
hypocenter	C₃²	#1713
LCS	G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

#63914

commutator, cosocle

C₃²×C₄²

C₂³

Frattini

C₄²

C₂×D₃²

Fitting

C₃²×C₄×C₈

C₂²

socle, prime radical

C₂²×C₃²

#127

hypercenter

C₄×C₈

D₃²

hypocenter

C₃²

#1713

LCS

G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
36. C₃⋊(C₃⋊C₈)	Q₁₆	semidirect
89. C₃²	#1713	semidirect
105. C₃	#71298	semidirect

H ≅ G/N

type

36. C₃⋊(C₃⋊C₈)

Q₁₆

semidirect

89. C₃²

#1713

semidirect

105. C₃

#71298

semidirect

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

subgroup	index
C₃×#4699	2
#10590312	2
C₈×C₃⋊(C₃⋊C₈)	2
#71298	3

subgroup

index

C₃×#4699

#10590312

C₈×C₃⋊(C₃⋊C₈)

#71298

Subgroups — #11782606

Conjugacy classes of subgroups

Complements

Maximal subgroups