Group #11782610 — subgroups

Subgroups — #11782610

Order 1152.

290 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 59 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	#10593450	C₂	1	2	yes	maximal
2	#10593395	C₂	1	2	yes	maximal
3	#10590902	C₂	1	2	yes	maximal
4	C₈×#622	C₂	1	1	yes	maximal, characteristic
5	C₂×#3776	—	2	4	no	—
6	#63919	C₂²	1	4	yes	—
7	C₄×#622	C₂²	1	1	yes	characteristic
8	C₄×#622	C₂²	1	1	yes	characteristic
9	#63743	—	2	4	no	—
10	C₈×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
11	C₃×#818	—	2	2	no	—
12	#63576	—	2	2	no	—
13	#4874	—	6	2	no	—
14	C₂×#627	—	2	4	no	—
15	C₂×#622	D₄	1	2	yes	—
16	C₂×#622	D₄	1	2	yes	—
17	#3776	—	4	8	no	—
18	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
19	C₄×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
20	C₂×C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
21	C₃×C₄⋊Dic₃	—	2	4	no	—
22	#3724	—	2	4	no	—
23	#1713	—	9	1	no	maximal, Sylow
24	#888	—	6	4	no	—
25	C₃×#107	—	6	2	no	—
26	#818	—	2	2	no	—
27	C₈×Dic₃	—	6	2	no	—
28	#807	—	6	4	no	—
29	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
30	C₂×C₃⋊Dic₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₃⋊Dic₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₃×Dic₃	—	4	4	no	—
33	#627	—	4	8	no	—
34	#622	D₈	1	4	yes	—
35	#622	Q₁₆	1	4	yes	complemented
36	C₃²×C₈	Q₁₆	1	2	yes	complemented
37	C₃⋊(C₃⋊C₈)	Q₁₆	1	2	yes	complemented
38	#500	—	9	2	no	—
39	#500	—	9	2	no	—
40	#494	—	9	2	no	—
41	C₈²	—	9	1	no	—
42	C₂×Dic₆	—	12	4	no	—
43	C₂×C₃×C₈	—	2	2	no	—
44	C₃×C₄⋊C₄	—	6	4	no	—
45	C₄⋊Dic₃	—	2	4	no	—
46	Dic₃⋊C₄	—	12	4	no	—
47	C₄×Dic₃	—	6	2	no	—
48	C₂×C₃⋊C₈	—	6	2	no	—
49	C₂²×C₃²	#127	1	1	yes	characteristic
50	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
51	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
52	C₃⋊Dic₃	C₂×D₈	1	2	yes	—
53	C₃⋊Dic₃	C₂×Q₁₆	1	2	yes	—
54	C₃×Dic₃	—	8	4	no	—
55	C₂×Q₁₆	—	18	4	no	—
56	C₄⋊Q₈	—	9	4	no	—
57	#107	—	6	2	no	—
58	#107	—	18	2	no	—
59	#107	—	18	4	no	—
60	C₄×C₈	—	9	1	no	—
61	C₄×C₈	—	9	1	no	—
62	C₄×C₈	—	9	1	no	—
63	C₂×C₃×C₄	—	2	2	no	—
64	C₂×C₃×C₄	—	12	4	no	—
65	C₂×Dic₃	—	4	4	no	—
66	C₂×Dic₃	—	6	2	no	—
67	C₂×Dic₃	—	6	2	no	—
68	Dic₆	—	24	8	no	—
69	C₃×C₈	—	2	4	no	—
70	C₃⋊C₈	—	6	4	no	—
71	C₂×C₃²	#493	1	1	yes	characteristic
72	C₂×C₃²	#494	1	1	yes	characteristic
73	C₂×C₃²	#494	1	1	yes	characteristic
74	C₂×Q₈	—	18	4	no	—
75	Q₁₆	—	36	8	no	—
76	C₂×C₈	—	9	2	no	—
77	C₂×C₈	—	9	2	no	—
78	C₂×C₈	—	9	1	no	—
79	C₂×C₈	GO⁺(4,𝔽₂)	1	1	yes	characteristic
80	C₄⋊C₄	—	6	4	no	—
81	C₄⋊C₄	—	18	4	no	—
82	C₄²	—	9	1	no	—
83	C₂²×C₃	—	2	2	no	—
84	C₃×C₄	—	2	2	no	—
85	C₃×C₄	—	2	2	no	—
86	C₃×C₄	—	24	4	no	—
87	Dic₃	—	6	4	no	—
88	Dic₃	—	6	4	no	—
89	Dic₃	—	8	4	no	—
90	C₃²	#1713	1	1	yes	characteristic, complemented, Sylow
91	Q₈	—	36	8	no	—
92	C₂×C₄	—	9	1	no	—
93	C₂×C₄	—	9	1	no	—
94	C₂×C₄	—	12	4	no	—
95	C₂×C₄	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic
96	C₈	—	9	4	no	—
97	C₈	—	9	4	no	—
98	C₈	—	9	2	no	—
99	C₈	#3776	1	2	yes	complemented
100	C₂×C₃	—	2	2	no	—
101	C₂×C₃	—	2	2	no	—
102	C₂×C₃	—	2	2	no	—
103	C₂²	#64231	1	1	yes	characteristic
104	C₄	—	9	2	no	—
105	C₄	—	9	2	no	—
106	C₄	—	24	4	no	—
107	C₄	#64228	1	1	yes	characteristic
108	C₄	C₂×#3776	1	1	yes	characteristic
109	C₃	—	2	2	no	—
110	C₂	#10593293	1	1	yes	characteristic
111	C₂	#10593295	1	1	yes	characteristic
112	C₂	#10593450	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593450

C₂

yes

maximal

#10593395

C₂

yes

maximal

#10590902

C₂

yes

maximal

C₈×#622

C₂

yes

maximal, characteristic

C₂×#3776

—

#63919

C₂²

yes

—

C₄×#622

C₂²

yes

characteristic

C₄×#622

C₂²

yes

characteristic

#63743

—

C₈×C₃⋊Dic₃

C₂²

yes

—

—

—

—

D₄

yes

—

C₂×#622

D₄

yes

—

#3776

—

C₂×C₃²×C₈

D₄

yes

characteristic

C₄×C₃⋊Dic₃

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

characteristic

C₃×C₄⋊Dic₃

—

—

—

—

—

—

C₈×Dic₃

—

#807

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃⋊Dic₃

C₂×D₄

yes

characteristic

C₂×C₃⋊Dic₃

C₂×D₄

yes

characteristic

C₂×C₃×Dic₃

—

#627

—

#622

D₈

yes

—

#622

Q₁₆

yes

complemented

C₃²×C₈

Q₁₆

yes

complemented

C₃⋊(C₃⋊C₈)

Q₁₆

yes

—

—

—

C₈²

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂²×C₃²

#127

yes

characteristic

C₃²×C₄

C₂×Q₁₆

yes

characteristic

C₃²×C₄

C₂×Q₁₆

yes

characteristic

C₃⋊Dic₃

C₂×D₈

yes

—

C₃⋊Dic₃

C₂×Q₁₆

yes

—

C₃×Dic₃

—

C₂×Q₁₆

—

C₄⋊Q₈

—

#107

—

#107

—

#107

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

yes

C₂×C₃²

yes

C₂×C₃²

yes

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

GO⁺(4,𝔽₂)

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

Dic₃

—

C₃²

#1713

yes

characteristic, complemented, Sylow

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×GO⁺(4,𝔽₂)

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

#3776

yes

complemented

100

C₂×C₃

—

101

C₂×C₃

—

102

C₂×C₃

—

103

C₂²

#64231

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

#64228

yes

characteristic

108

C₄

C₂×#3776

yes

characteristic

109

C₃

—

110

C₂

#10593293

yes

characteristic

111

C₂

#10593295

yes

characteristic

112

C₂

#10593450

yes

characteristic

	subgroup	G/H
center	C₂²	#64231
commutator, cosocle	C₄×C₃⋊Dic₃	C₂³
Frattini	C₂×C₄	C₂×GO⁺(4,𝔽₂)
Fitting	C₂×C₃²×C₈	D₄
socle, prime radical	C₂²×C₃²	#127
hypercenter	C₂×C₈	GO⁺(4,𝔽₂)
hypocenter	C₃²	#1713
LCS	G ▷ C₄×C₃⋊Dic₃ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center

C₂²

#64231

commutator, cosocle

C₄×C₃⋊Dic₃

C₂³

Frattini

C₂×C₄

C₂×GO⁺(4,𝔽₂)

Fitting

C₂×C₃²×C₈

D₄

socle, prime radical

C₂²×C₃²

#127

hypercenter

C₂×C₈

GO⁺(4,𝔽₂)

hypocenter

C₃²

#1713

LCS

G ▷ C₄×C₃⋊Dic₃ ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
35. #622	Q₁₆	semidirect
36. C₃²×C₈	Q₁₆	semidirect
37. C₃⋊(C₃⋊C₈)	Q₁₆	semidirect
90. C₃²	#1713	semidirect
99. C₈	#3776	semidirect

H ≅ G/N

type

35. #622

Q₁₆

semidirect

36. C₃²×C₈

Q₁₆

semidirect

37. C₃⋊(C₃⋊C₈)

Q₁₆

semidirect

90. C₃²

#1713

semidirect

99. C₈

#3776

semidirect

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

subgroup	index
#10593450	2
#10593395	2
#10590902	2
C₈×#622	2
#1713	9

subgroup

index

Subgroups — #11782610

Conjugacy classes of subgroups

Complements

Maximal subgroups