Group #11800169 — subgroups

Subgroups — #11800169

Order 1152.

190 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 36 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	#10593533	C₂	1	1	yes	maximal, characteristic, complemented
2	#10593293	C₂	1	1	yes	maximal, characteristic
3	#10593255	C₂	1	1	yes	maximal, characteristic, complemented
4	#10593160	C₂	1	4	yes	maximal, complemented
5	#64243	—	2	1	no	—
6	#64231	—	2	1	no	—
7	#64225	C₂²	1	2	yes	—
8	#64213	C₂²	1	2	yes	—
9	#63797	—	2	1	no	—
10	#63768	—	2	1	no	—
11	C₈×#157	C₂²	1	1	yes	characteristic
12	#63746	C₂²	1	2	yes	—
13	C₂×GO⁺(4,𝔽₂)	—	4	2	no	—
14	#3802	—	2	2	no	—
15	C₄⋊#157	—	2	2	no	—
16	C₄×#157	D₄	1	1	yes	characteristic
17	#3787	C₂³	1	1	yes	characteristic
18	#3777	—	4	2	no	—
19	#3774	—	2	4	no	—
20	#3771	—	2	2	no	—
21	C₈×C₃⋊D₃	D₄	1	1	yes	characteristic
22	C₃×D₂₄	—	4	1	no	—
23	#3713	—	4	1	no	—
24	#2244	—	9	1	no	maximal, Sylow
25	D₃×D₈	—	12	1	no	—
26	C₂×D₃²	—	4	2	no	—
27	C₂×#157	—	4	2	no	—
28	C₂×#157	D₈	1	2	yes	—
29	GO⁺(4,𝔽₂)	—	8	4	no	—
30	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
31	C₃×D₁₂	—	4	2	no	—
32	#625	—	4	2	no	—
33	#622	#41	1	2	yes	—
34	C₃²×C₈	SD₁₆	1	1	yes	characteristic
35	C₃⋊(C₃⋊C₈)	SD₁₆	1	1	yes	characteristic
36	#499	—	9	1	no	—
37	#493	—	9	1	no	—
38	#357	—	9	4	no	—
39	#346	—	9	1	no	—
40	D₃×D₄	—	12	2	no	—
41	C₃×D₈	—	12	1	no	—
42	#213	—	12	2	no	—
43	D₂₄	—	4	1	no	—
44	D₃×C₈	—	12	1	no	—
45	C₂×C₃⋊D₃	C₂×D₈	1	1	yes	characteristic
46	C₂×C₃×D₃	—	8	2	no	—
47	D₃²	—	8	4	no	—
48	#157	—	2	2	no	—
49	#157	—	8	2	no	—
50	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
51	C₃⋊Dic₃	#124	1	1	yes	characteristic
52	C₂×D₈	—	9	2	no	—
53	C₂×D₈	—	18	1	no	—
54	#127	—	18	1	no	—
55	#125	—	18	1	no	—
56	C₂×C₁₆	—	9	2	no	—
57	#107	—	9	2	no	—
58	#106	—	18	1	no	—
59	C₄×C₈	—	9	1	no	—
60	C₂²×D₃	—	24	2	no	—
61	C₃×D₄	—	12	2	no	—
62	#67	—	24	2	no	—
63	D₁₂	—	4	2	no	—
64	C₄×D₃	—	12	1	no	—
65	C₃×C₈	—	4	1	no	—
66	C₃⋊C₈	—	12	1	no	—
67	C₂×C₃²	#486	1	1	yes	characteristic
68	C₃⋊D₃	#509	1	2	yes	—
69	C₃×D₃	—	16	2	no	—
70	C₂×D₄	—	18	2	no	—
71	C₂×D₄	—	36	2	no	—
72	D₈	—	12	1	no	—
73	D₈	—	18	4	no	—
74	D₈	—	36	1	no	—
75	C₂×C₈	—	9	1	no	—
76	C₂×C₈	—	9	1	no	—
77	C₄⋊C₄	—	18	2	no	—
78	C₄⋊C₄	—	36	2	no	—
79	C₄²	—	9	1	no	—
80	C₁₆	—	18	2	no	—
81	C₂²×C₃	—	24	2	no	—
82	C₂×D₃	—	8	2	no	—
83	C₂×D₃	—	12	1	no	—
84	C₂×D₃	—	48	4	no	—
85	C₃×C₄	—	4	1	no	—
86	Dic₃	—	12	1	no	—
87	C₃²	#2244	1	1	yes	characteristic, complemented, Sylow
88	C₂³	—	36	2	no	—
89	D₄	—	12	2	no	—
90	D₄	—	36	2	no	—
91	D₄	—	72	4	no	—
92	C₂×C₄	—	9	1	no	—
93	C₂×C₄	—	9	2	no	—
94	C₂×C₄	—	36	2	no	—
95	C₈	—	9	2	no	—
96	C₈	—	9	1	no	—
97	C₈	AΓL(1,𝔽₉)	1	1	yes	characteristic
98	C₂×C₃	—	4	1	no	—
99	C₂×C₃	—	48	2	no	—
100	D₃	—	12	2	no	—
101	D₃	—	16	2	no	—
102	C₂²	—	9	1	no	—
103	C₂²	—	24	2	no	—
104	C₂²	—	72	4	no	—
105	C₄	—	9	1	no	—
106	C₄	—	18	2	no	—
107	C₄	—	72	2	no	—
108	C₄	C₂×AΓL(1,𝔽₉)	1	1	yes	characteristic
109	C₃	—	4	1	no	—
110	C₂	—	9	2	no	—
111	C₂	—	48	2	no	—
112	C₂	#10596402	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593533

C₂

yes

maximal, characteristic, complemented

#10593293

C₂

yes

maximal, characteristic

#10593255

C₂

yes

maximal, characteristic, complemented

#10593160

C₂

yes

maximal, complemented

#64243

—

#64231

—

#64225

C₂²

yes

—

#64213

C₂²

yes

—

#63797

—

#63768

—

C₈×#157

C₂²

yes

characteristic

#63746

C₂²

yes

—

C₂×GO⁺(4,𝔽₂)

—

#3802

—

C₄⋊#157

—

C₄×#157

D₄

yes

characteristic

#3787

C₂³

yes

—

—

—

C₈×C₃⋊D₃

D₄

yes

characteristic

C₃×D₂₄

—

#3713

—

#2244

—

maximal, Sylow

D₃×D₈

—

C₂×D₃²

—

C₂×#157

—

C₂×#157

D₈

yes

—

GO⁺(4,𝔽₂)

—

C₄×C₃⋊D₃

C₂×D₄

yes

characteristic

C₃×D₁₂

—

#625

—

#622

#41

yes

—

C₃²×C₈

SD₁₆

yes

characteristic

C₃⋊(C₃⋊C₈)

SD₁₆

yes

—

—

—

—

D₃×D₄

—

C₃×D₈

—

#213

—

D₂₄

—

D₃×C₈

—

C₂×C₃⋊D₃

C₂×D₈

yes

characteristic

C₂×C₃×D₃

—

D₃²

—

#157

—

#157

—

C₃²×C₄

C₂×SD₁₆

yes

characteristic

C₃⋊Dic₃

#124

yes

characteristic

C₂×D₈

—

C₂×D₈

—

#127

—

#125

—

C₂×C₁₆

—

#107

—

#106

—

C₄×C₈

—

C₂²×D₃

—

C₃×D₄

—

#67

—

D₁₂

—

C₄×D₃

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

#486

yes

characteristic

C₃⋊D₃

#509

yes

—

C₃×D₃

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₁₆

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

Dic₃

—

C₃²

#2244

yes

characteristic, complemented, Sylow

C₂³

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₈

—

C₈

—

C₈

AΓL(1,𝔽₉)

yes

characteristic

C₂×C₃

—

C₂×C₃

—

100

D₃

—

101

D₃

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

C₂×AΓL(1,𝔽₉)

yes

characteristic

109

C₃

—

110

C₂

—

111

C₂

—

112

C₂

#10596402

yes

characteristic

	subgroup	G/H
center	C₂	#10596402
commutator, cosocle	#3787	C₂³
Frattini	C₄	C₂×AΓL(1,𝔽₉)
Fitting	C₃²×C₈	SD₁₆
socle	C₂×C₃²	#486
prime radical	#10593293	C₂
hypercenter	C₈	AΓL(1,𝔽₉)
hypocenter	C₃²	#2244
LCS	G ▷ #3787 ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

subgroup

G/H

C₂

C₂³

C₄

C₃²×C₈

SD₁₆

C₂×C₃²

C₂

C₈

C₃²

G ▷ #3787 ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈

N	H ≅ G/N	type
1. #10593533	C₂	semidirect
3. #10593255	C₂	semidirect
4. #10593160	C₂	semidirect
87. C₃²	#2244	semidirect

H ≅ G/N

type

1. #10593533

C₂

semidirect

3. #10593255

C₂

semidirect

4. #10593160

C₂

semidirect

87. C₃²

#2244

semidirect

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

subgroup	index
#10593533	2
#10593293	2
#10593255	2
#10593160	2
#2244	9

subgroup

index

Subgroups — #11800169

Conjugacy classes of subgroups

Complements

Maximal subgroups