Group #11798041 — subgroups

Subgroups — #11798041

Order 1152.

161 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 161 conjugacy classes of subgroups — too many to draw. Only its 61 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	#10593750	C₂	1	1	yes	maximal, characteristic
2	#10593263	C₂	1	2	yes	maximal
3	C₄⋊#3771	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#3771	C₂	1	2	yes	maximal
5	C₄×#3771	C₂	1	1	yes	maximal, characteristic
6	C₂×#3771	C₂²	1	1	yes	characteristic
7	C₂×#3771	C₂²	1	2	yes	—
8	C₂×#3771	C₂²	1	1	yes	characteristic
9	#64096	C₄	1	1	yes	characteristic
10	#63779	C₄	1	2	yes	—
11	C₄⋊#622	C₄	1	1	yes	characteristic
12	C₄×#622	C₂²	1	1	yes	characteristic
13	C₄×#622	C₂²	1	2	yes	—
14	C₂×#634	C₈	1	1	yes	characteristic
15	C₂×#622	C₂×C₄	1	2	yes	—
16	C₂×#622	C₂×C₄	1	1	yes	characteristic
17	C₂×#622	C₂³	1	1	yes	characteristic
18	#3771	—	2	1	no	—
19	#3771	—	2	2	no	—
20	#3771	Q₈	1	2	yes	complemented
21	C₃²×C₄⋊C₄	C₈	1	1	yes	characteristic
22	#3750	C₈	1	2	yes	—
23	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
25	#2183	—	9	1	no	maximal, Sylow
26	#887	—	12	1	no	—
27	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
28	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
29	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₃⋊Dic₃	C₂×C₈	1	2	yes	—
31	C₂×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
32	#634	—	2	2	no	—
33	#622	—	2	2	no	—
34	#622	—	2	1	no	—
35	#622	C₂×Q₈	1	1	yes	characteristic
36	#622	#41	1	1	yes	characteristic
37	C₈×Q₈	—	9	1	no	—
38	C₄⋊C₁₆	—	9	1	no	—
39	C₄⋊C₁₆	—	9	2	no	—
40	#346	—	9	2	no	—
41	C₄×C₁₆	—	9	1	no	—
42	C₂×Dic₆	—	12	1	no	—
43	C₃×C₄⋊C₄	—	4	1	no	—
44	Dic₃⋊C₄	—	12	2	no	—
45	C₄×Dic₃	—	12	2	no	—
46	C₄×Dic₃	—	12	1	no	—
47	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
48	C₃²×C₄	—	2	2	no	—
49	C₃²×C₄	M₃₂	1	2	yes	—
50	C₃⋊Dic₃	—	2	2	no	—
51	C₃⋊Dic₃	—	2	1	no	—
52	C₃⋊Dic₃	C₄×Q₈	1	1	yes	characteristic
53	C₃⋊Dic₃	#131	1	1	yes	characteristic
54	C₄×Q₈	—	9	1	no	—
55	C₂×C₁₆	—	9	1	no	—
56	C₂×C₁₆	—	9	1	no	—
57	C₂×C₁₆	—	9	2	no	—
58	C₄⋊C₈	—	9	1	no	—
59	C₄⋊C₈	—	9	2	no	—
60	C₄×C₈	—	9	1	no	—
61	C₄×C₈	—	9	2	no	—
62	C₂×C₃×C₄	—	4	1	no	—
63	C₂×C₃×C₄	—	4	2	no	—
64	C₂×Dic₃	—	12	2	no	—
65	C₂×Dic₃	—	12	1	no	—
66	C₂×Dic₃	—	12	1	no	—
67	Dic₆	—	12	4	no	—
68	C₂×C₃²	C₈×Q₈	1	1	yes	characteristic
69	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
70	C₂×C₃²	#504	1	1	yes	characteristic
71	C₂×Q₈	—	9	1	no	—
72	C₂×C₈	—	9	1	no	—
73	C₂×C₈	—	9	1	no	—
74	C₂×C₈	—	9	2	no	—
75	C₄⋊C₄	—	9	2	no	—
76	C₄⋊C₄	AGL(1,𝔽₉)	1	1	yes	characteristic
77	C₄²	—	9	1	no	—
78	C₄²	—	9	2	no	—
79	C₁₆	—	9	2	no	—
80	C₁₆	—	18	1	no	—
81	C₁₆	—	18	2	no	—
82	C₂²×C₃	—	4	1	no	—
83	C₃×C₄	—	4	2	no	—
84	C₃×C₄	—	8	2	no	—
85	Dic₃	—	12	1	no	—
86	Dic₃	—	12	1	no	—
87	Dic₃	—	12	4	no	—
88	Dic₃	—	24	1	no	—
89	C₃²	#2183	1	1	yes	characteristic, complemented, Sylow
90	Q₈	—	18	2	no	—
91	C₂×C₄	—	9	1	no	—
92	C₂×C₄	—	9	1	no	—
93	C₂×C₄	—	9	2	no	—
94	C₂×C₄	C₂×AGL(1,𝔽₉)	1	2	yes	—
95	C₂×C₄	C₂×AGL(1,𝔽₉)	1	1	yes	characteristic
96	C₈	—	9	1	no	—
97	C₈	—	9	1	no	—
98	C₈	—	18	1	no	—
99	C₈	—	18	2	no	—
100	C₂×C₃	—	4	1	no	—
101	C₂×C₃	—	4	1	no	—
102	C₂×C₃	—	4	1	no	—
103	C₂²	C₂²×AGL(1,𝔽₉)	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	9	1	no	—
106	C₄	—	9	1	no	—
107	C₄	—	18	2	no	—
108	C₄	—	18	1	no	—
109	C₄	#64214	1	2	yes	—
110	C₃	—	4	1	no	—
111	C₂	C₂×#64214	1	1	yes	characteristic
112	C₂	#10596485	1	1	yes	characteristic
113	C₂	Q₈×AGL(1,𝔽₉)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593750

C₂

yes

maximal, characteristic

#10593263

C₂

yes

maximal

C₄⋊#3771

C₂

yes

maximal, characteristic

C₄⋊#3771

C₂

yes

maximal

C₄×#3771

C₂

yes

maximal, characteristic

C₂×#3771

C₂²

yes

characteristic

C₂×#3771

C₂²

yes

—

C₂×#3771

C₂²

yes

characteristic

#64096

C₄

yes

characteristic

#63779

C₄

yes

—

C₄⋊#622

C₄

yes

characteristic

C₄×#622

C₂²

yes

characteristic

C₄×#622

C₂²

yes

—

C₂×#634

C₈

yes

characteristic

C₂×#622

C₂×C₄

yes

—

C₂×#622

C₂×C₄

yes

characteristic

C₂×#622

C₂³

yes

—

—

Q₈

yes

complemented

C₃²×C₄⋊C₄

C₈

yes

characteristic

#3750

C₈

yes

—

C₄×C₃⋊Dic₃

C₂×C₄

yes

characteristic

C₄×C₃⋊Dic₃

C₂×C₄

yes

—

#2183

—

maximal, Sylow

#887

—

C₂×C₃²×C₄

C₂×C₈

yes

—

C₂×C₃²×C₄

C₂×C₈

yes

characteristic

C₂×C₃⋊Dic₃

C₂×C₈

yes

characteristic

C₂×C₃⋊Dic₃

C₂×C₈

yes

—

C₂×C₃⋊Dic₃

C₂²×C₄

yes

—

—

—

C₂×Q₈

yes

yes

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

C₂²×C₈

yes

characteristic

C₃²×C₄

—

C₃²×C₄

M₃₂

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₃⋊Dic₃

C₄×Q₈

yes

characteristic

C₃⋊Dic₃

#131

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₂×C₃²

C₈×Q₈

yes

characteristic

C₂×C₃²

C₂×M₃₂

yes

characteristic

C₂×C₃²

#504

yes

characteristic

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

AGL(1,𝔽₉)

yes

characteristic

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

Dic₃

—

Dic₃

—

C₃²

#2183

yes

characteristic, complemented, Sylow

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×AGL(1,𝔽₉)

yes

—

C₂×C₄

C₂×AGL(1,𝔽₉)

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

100

C₂×C₃

—

101

C₂×C₃

—

102

C₂×C₃

—

103

C₂²

C₂²×AGL(1,𝔽₉)

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#64214

yes

—

110

C₃

—

111

C₂

C₂×#64214

yes

characteristic

112

C₂

#10596485

yes

characteristic

113

C₂

Q₈×AGL(1,𝔽₉)

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₉)
commutator, socle, prime radical	C₂²×C₃²	C₂²×C₈
Fitting	C₃²×C₄⋊C₄	C₈
cosocle	C₂×#622	C₂³
hypercenter	C₄⋊C₄	AGL(1,𝔽₉)
hypocenter	C₃²	#2183
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂²×AGL(1,𝔽₉)

commutator, socle, prime radical

C₂²×C₃²

C₂²×C₈

Fitting

C₃²×C₄⋊C₄

C₈

C₂³

C₄⋊C₄

C₃²

G ▷ C₂²×C₃² ▷ C₃²

C₁ ◁ C₂² ◁ C₄⋊C₄

N	H ≅ G/N	type
20. #3771	Q₈	semidirect
89. C₃²	#2183	semidirect

H ≅ G/N

type

20. #3771

Q₈

semidirect

89. C₃²

#2183

semidirect

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

subgroup	index
#10593750	2
#10593263	2
C₄⋊#3771	2
C₄×#3771	2
#2183	9

subgroup

index

Subgroups — #11798041

Conjugacy classes of subgroups

Complements

Maximal subgroups