Group #11772124 — subgroups

Subgroups — #11772124

Order 1152.

120 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 30 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	#10593395	C₂	1	1	yes	maximal, characteristic
2	#10593265	C₂	1	1	yes	maximal, characteristic
3	#10590089	C₂	1	1	yes	maximal, characteristic
4	C₂×#3771	—	2	1	no	—
5	#63919	—	2	1	no	—
6	#63779	—	2	1	no	—
7	C₄×#622	C₂²	1	1	yes	characteristic
8	#63743	—	2	1	no	—
9	#63743	C₄	1	2	yes	—
10	C₂×#627	—	4	1	no	—
11	C₂×#622	—	4	1	no	—
12	C₂×#622	C₂×C₄	1	1	yes	characteristic
13	C₂×#622	D₄	1	1	yes	characteristic
14	#3771	—	4	1	no	—
15	C₄×C₃⋊Dic₃	D₄	1	1	yes	characteristic
16	C₃×C₄⋊Dic₃	—	4	1	no	—
17	#3724	—	2	2	no	—
18	#1364	—	9	1	no	maximal, Sylow
19	#888	—	12	1	no	—
20	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
21	C₂×C₃⋊Dic₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₃⋊Dic₃	D₈	1	1	yes	characteristic
23	C₂×C₃×Dic₃	—	4	2	no	—
24	#627	—	8	1	no	—
25	#627	—	8	1	no	—
26	#622	—	2	1	no	—
27	#622	—	8	1	no	—
28	#622	SD₁₆	1	1	yes	characteristic
29	#622	Q₁₆	1	1	yes	characteristic
30	#500	—	9	1	no	—
31	C₄⋊C₁₆	—	9	1	no	—
32	#335	—	9	1	no	—
33	C₂×Dic₆	—	12	2	no	—
34	C₃×C₄⋊C₄	—	12	1	no	—
35	C₄⋊Dic₃	—	4	1	no	—
36	Dic₃⋊C₄	—	12	2	no	—
37	C₄×Dic₃	—	12	1	no	—
38	C₂²×C₃²	#102	1	1	yes	characteristic
39	C₃²×C₄	—	2	1	no	—
40	C₃⋊Dic₃	#101	1	1	yes	characteristic
41	C₃⋊Dic₃	#103	1	1	yes	characteristic
42	C₃⋊Dic₃	D₁₆	1	1	yes	characteristic
43	C₃⋊Dic₃	SD₃₂	1	1	yes	characteristic
44	C₃×Dic₃	—	8	2	no	—
45	C₄⋊Q₈	—	18	1	no	—
46	C₂×C₁₆	—	18	1	no	—
47	#107	—	9	2	no	—
48	#107	—	18	1	no	—
49	C₄⋊C₈	—	18	1	no	—
50	C₄×C₈	—	9	1	no	—
51	C₂×C₃×C₄	—	4	1	no	—
52	C₂×C₃×C₄	—	12	2	no	—
53	C₂×Dic₃	—	4	2	no	—
54	C₂×Dic₃	—	12	1	no	—
55	C₂×Dic₃	—	12	1	no	—
56	Dic₆	—	24	2	no	—
57	Dic₆	—	24	2	no	—
58	C₂×C₃²	#332	1	1	yes	characteristic
59	C₂×C₃²	#357	1	1	yes	characteristic
60	C₂×C₃²	#362	1	1	yes	characteristic
61	C₂×Q₈	—	36	1	no	—
62	C₂×C₈	—	9	1	no	—
63	C₂×C₈	—	9	1	no	—
64	C₂×C₈	—	36	1	no	—
65	C₄⋊C₄	—	12	1	no	—
66	C₄⋊C₄	—	18	2	no	—
67	C₄²	—	9	1	no	—
68	C₁₆	—	36	1	no	—
69	C₂²×C₃	—	4	1	no	—
70	C₃×C₄	—	4	2	no	—
71	C₃×C₄	—	24	2	no	—
72	Dic₃	—	8	2	no	—
73	Dic₃	—	12	1	no	—
74	Dic₃	—	12	1	no	—
75	Dic₃	—	12	1	no	—
76	Dic₃	—	12	1	no	—
77	C₃²	#1364	1	1	yes	characteristic, complemented, Sylow
78	Q₈	—	72	1	no	—
79	Q₈	—	72	1	no	—
80	C₂×C₄	—	9	1	no	—
81	C₂×C₄	—	9	1	no	—
82	C₂×C₄	—	12	2	no	—
83	C₂×C₄	AΓL(1,𝔽₉)	1	1	yes	characteristic
84	C₈	—	9	1	no	—
85	C₈	—	9	1	no	—
86	C₈	—	18	1	no	—
87	C₈	—	72	1	no	—
88	C₂×C₃	—	4	1	no	—
89	C₂×C₃	—	4	1	no	—
90	C₂×C₃	—	4	1	no	—
91	C₂²	GO⁺(4,𝔽₂)⋊C₄	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	9	1	no	—
94	C₄	—	9	1	no	—
95	C₄	—	9	1	no	—
96	C₄	—	9	1	no	—
97	C₄	—	24	2	no	—
98	C₃	—	4	1	no	—
99	C₂	#10593156	1	1	yes	characteristic
100	C₂	#10593160	1	1	yes	characteristic
101	C₂	#10593169	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593395

C₂

yes

maximal, characteristic

#10593265

C₂

yes

maximal, characteristic

#10590089

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

#63743

—

#63743

C₄

yes

—

C₂×#627

—

C₂×#622

—

C₂×#622

C₂×C₄

yes

characteristic

C₂×#622

D₄

yes

characteristic

#3771

—

C₄×C₃⋊Dic₃

D₄

yes

characteristic

C₃×C₄⋊Dic₃

—

—

—

—

C₂×C₃²×C₄

SD₁₆

yes

characteristic

C₂×C₃⋊Dic₃

C₂²⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

D₈

yes

characteristic

C₂×C₃×Dic₃

—

—

—

—

—

SD₁₆

yes

characteristic

#622

Q₁₆

yes

characteristic

#500

—

C₄⋊C₁₆

—

#335

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂²×C₃²

#102

yes

characteristic

C₃²×C₄

—

C₃⋊Dic₃

yes

C₃⋊Dic₃

yes

C₃⋊Dic₃

D₁₆

yes

characteristic

C₃⋊Dic₃

SD₃₂

yes

characteristic

C₃×Dic₃

—

C₄⋊Q₈

—

C₂×C₁₆

—

#107

—

#107

—

C₄⋊C₈

—

C₄×C₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₂×C₃²

yes

C₂×C₃²

yes

C₂×C₃²

yes

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₁₆

—

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

Dic₃

—

Dic₃

—

Dic₃

—

C₃²

#1364

yes

characteristic, complemented, Sylow

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

AΓL(1,𝔽₉)

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂²

GO⁺(4,𝔽₂)⋊C₄

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₃

—

C₂

#10593156

yes

characteristic

100

C₂

#10593160

yes

characteristic

101

C₂

#10593169

yes

characteristic

	subgroup	G/H
center	C₂²	GO⁺(4,𝔽₂)⋊C₄
commutator	C₂×#622	C₂×C₄
Frattini, hypercenter	C₂×C₄	AΓL(1,𝔽₉)
Fitting	C₂×C₃²×C₄	SD₁₆
socle, prime radical	C₂²×C₃²	#102
cosocle	C₄×#622	C₂²
hypocenter	C₃²	#1364
LCS	G ▷ C₂×#622 ▷ C₂×C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

subgroup

G/H

center

C₂²

GO⁺(4,𝔽₂)⋊C₄

commutator

C₂×#622

C₂×C₄

Frattini, hypercenter

C₂×C₄

AΓL(1,𝔽₉)

Fitting

C₂×C₃²×C₄

SD₁₆

C₂²×C₃²

C₂²

C₃²

G ▷ C₂×#622 ▷ C₂×C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₂×C₄

N	H ≅ G/N	type
77. C₃²	#1364	semidirect

H ≅ G/N

type

77. C₃²

#1364

semidirect

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

subgroup	index
#10593395	2
#10593265	2
#10590089	2
#1364	9

subgroup

index

Subgroups — #11772124

Conjugacy classes of subgroups

Complements

Maximal subgroups