Group #11772016 — subgroups

Subgroups — #11772016

Order 1152.

138 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 50 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	#10588365	C₂	1	1	yes	maximal, characteristic
2	C₉×#364	C₂	1	1	yes	maximal, characteristic
3	#10588140	C₂	1	1	yes	maximal, characteristic
4	#67572	—	3	1	no	maximal
5	C₂×Dic₃₆	—	2	1	no	—
6	C₂×Dic₃₆	—	2	1	no	—
7	#63431	C₄	1	2	yes	—
8	C₉×M₃₂	—	2	1	no	—
9	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
10	#63371	—	2	1	no	—
11	#4699	—	3	1	no	—
12	C₃×#364	D₃	1	1	yes	characteristic
13	#4463	—	3	1	no	—
14	C₂×Dic₁₈	—	2	2	no	—
15	C₂×C₈×C₉	D₄	1	1	yes	characteristic
16	C₂×C₈×C₉	D₄	1	1	yes	characteristic
17	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
18	C₄⋊Dic₉	—	4	2	no	—
19	Dic₃₆	—	4	1	no	—
20	Dic₃₆	—	4	1	no	—
21	Dic₃₆	—	4	1	no	—
22	Dic₃₆	—	4	1	no	—
23	C₉×C₁₆	—	4	1	no	—
24	C₉⋊C₁₆	—	4	1	no	—
25	#1358	—	9	1	no	Sylow
26	C₂×Dic₁₂	—	6	1	no	—
27	C₂×Dic₁₂	—	6	1	no	—
28	#869	—	3	2	no	—
29	C₃×M₃₂	—	2	1	no	—
30	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
31	#812	—	6	1	no	—
32	C₂×C₄×C₉	—	2	1	no	—
33	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×Dic₉	—	4	2	no	—
35	Dic₁₈	—	4	2	no	—
36	Dic₁₈	—	4	2	no	—
37	C₈×C₉	D₈	1	1	yes	characteristic
38	C₈×C₉	D₈	1	1	yes	characteristic
39	C₈×C₉	SD₁₆	1	1	yes	characteristic
40	C₈×C₉	SD₁₆	1	1	yes	characteristic
41	#494	—	9	1	no	—
42	#364	—	9	1	no	—
43	#364	D₉	1	1	yes	characteristic
44	C₂×Dic₆	—	6	2	no	—
45	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
46	C₂×C₃×C₈	#67	1	1	yes	characteristic
47	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
48	C₄⋊Dic₃	—	12	2	no	—
49	Dic₁₂	—	12	1	no	—
50	Dic₁₂	—	12	1	no	—
51	Dic₁₂	—	12	1	no	—
52	Dic₁₂	—	12	1	no	—
53	C₃×C₁₆	—	4	1	no	—
54	C₃⋊C₁₆	—	12	1	no	—
55	C₂²×C₉	#100	1	1	yes	characteristic
56	C₄×C₉	—	2	2	no	—
57	C₄×C₉	#102	1	1	yes	characteristic
58	C₄×C₉	#102	1	1	yes	characteristic
59	Dic₉	—	8	2	no	—
60	C₂×Q₁₆	—	18	1	no	—
61	C₂×Q₁₆	—	18	1	no	—
62	C₄⋊Q₈	—	9	2	no	—
63	M₃₂	—	2	1	no	—
64	M₃₂	—	18	1	no	—
65	C₄×C₈	C₂×D₉	1	1	yes	characteristic
66	C₂×C₃×C₄	—	2	1	no	—
67	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
68	C₂×Dic₃	—	12	2	no	—
69	Dic₆	—	12	2	no	—
70	Dic₆	—	12	2	no	—
71	C₃×C₈	#204	1	1	yes	characteristic
72	C₃×C₈	D₂₄	1	1	yes	characteristic
73	C₃×C₈	#213	1	1	yes	characteristic
74	C₃×C₈	#215	1	1	yes	characteristic
75	C₂×C₉	—	2	1	no	—
76	C₂×C₉	#331	1	1	yes	characteristic
77	C₂×Q₈	—	18	2	no	—
78	Q₁₆	—	36	1	no	—
79	Q₁₆	—	36	1	no	—
80	Q₁₆	—	36	1	no	—
81	Q₁₆	—	36	1	no	—
82	C₂×C₈	D₃₆	1	1	yes	characteristic
83	C₂×C₈	#611	1	1	yes	characteristic
84	C₄⋊C₄	—	36	2	no	—
85	C₄²	C₄×D₉	1	1	yes	characteristic
86	C₁₆	—	4	1	no	—
87	C₁₆	—	36	1	no	—
88	C₂²×C₃	#823	1	1	yes	characteristic
89	C₃×C₄	—	2	2	no	—
90	C₃×C₄	#809	1	1	yes	characteristic
91	C₃×C₄	#821	1	1	yes	characteristic
92	Dic₃	—	24	2	no	—
93	C₉	#1358	1	1	yes	characteristic, complemented, Sylow
94	Q₈	—	36	2	no	—
95	Q₈	—	36	2	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	36	2	no	—
98	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
99	C₈	#3664	1	1	yes	characteristic
100	C₈	D₇₂	1	1	yes	characteristic
101	C₈	#3673	1	1	yes	characteristic
102	C₈	#3675	1	1	yes	characteristic
103	C₂×C₃	—	2	1	no	—
104	C₂×C₃	#4485	1	1	yes	characteristic
105	C₂²	#63383	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	72	2	no	—
108	C₄	#63369	1	1	yes	characteristic
109	C₄	#63380	1	1	yes	characteristic
110	C₃	#67572	1	1	yes	characteristic
111	C₂	—	2	1	no	—
112	C₂	#10588163	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10588365

C₂

yes

maximal, characteristic

C₉×#364

C₂

yes

maximal, characteristic

#10588140

C₂

yes

maximal, characteristic

#67572

—

maximal

C₂×Dic₃₆

—

C₂×Dic₃₆

—

#63431

C₄

yes

—

C₉×M₃₂

—

C₄×C₈×C₉

C₂²

yes

—

—

D₃

yes

characteristic

#4463

—

C₂×Dic₁₈

—

C₂×C₈×C₉

D₄

yes

characteristic

C₂×C₈×C₉

D₄

yes

characteristic

C₄²×C₉

C₂×C₄

yes

characteristic

C₄⋊Dic₉

—

Dic₃₆

—

Dic₃₆

—

Dic₃₆

—

Dic₃₆

—

C₉×C₁₆

—

C₉⋊C₁₆

—

#1358

—

Sylow

C₂×Dic₁₂

—

C₂×Dic₁₂

—

#869

—

C₃×M₃₂

—

C₃×C₄×C₈

C₂×D₃

yes

characteristic

#812

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

C₈×C₉

D₈

yes

characteristic

C₈×C₉

D₈

yes

characteristic

C₈×C₉

SD₁₆

yes

characteristic

C₈×C₉

SD₁₆

yes

—

—

D₉

yes

characteristic

C₂×Dic₆

—

C₂×C₃×C₈

D₁₂

yes

characteristic

C₂×C₃×C₈

#67

yes

characteristic

C₃×C₄²

C₄×D₃

yes

characteristic

C₄⋊Dic₃

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₉

#100

yes

characteristic

C₄×C₉

—

C₄×C₉

yes

C₄×C₉

yes

Dic₉

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

M₃₂

—

M₃₂

—

C₄×C₈

C₂×D₉

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

characteristic

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

#204

yes

characteristic

C₃×C₈

D₂₄

yes

C₃×C₈

yes

C₃×C₈

yes

C₂×C₉

—

C₂×C₉

#331

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

D₃₆

yes

characteristic

C₂×C₈

#611

yes

characteristic

C₄⋊C₄

—

C₄²

C₄×D₉

yes

characteristic

C₁₆

—

C₁₆

—

C₂²×C₃

#823

yes

characteristic

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

Dic₃

—

C₉

#1358

yes

characteristic, complemented, Sylow

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₉)⋊C₄

yes

characteristic

C₈

#3664

yes

characteristic

100

C₈

D₇₂

yes

characteristic

101

C₈

#3673

yes

characteristic

102

C₈

#3675

yes

characteristic

103

C₂×C₃

—

104

C₂×C₃

#4485

yes

characteristic

105

C₂²

#63383

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

#63369

yes

characteristic

109

C₄

#63380

yes

characteristic

110

C₃

#67572

yes

characteristic

111

C₂

—

112

C₂

#10588163

yes

characteristic

	subgroup	G/H
center	C₂	#10588163
commutator	C₄²×C₉	C₂×C₄
Frattini	C₃×C₄×C₈	C₂×D₃
Fitting	C₉×#364	C₂
socle	C₂×C₃	#4485
prime radical	C₂²×C₃	#823
cosocle	C₄×C₈×C₉	C₂²
hypercenter	#364	D₉
hypocenter	C₉	#1358
LCS	G ▷ C₄²×C₉ ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#10588163

commutator

C₄²×C₉

C₂×C₄

Frattini

C₃×C₄×C₈

C₂×D₃

C₂

C₂×C₃

C₂²×C₃

C₄×C₈×C₉

C₂²

D₉

C₉

G ▷ C₄²×C₉ ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
93. C₉	#1358	semidirect

H ≅ G/N

type

93. C₉

#1358

semidirect

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

subgroup	index
#10588365	2
C₉×#364	2
#10588140	2
#67572	3

subgroup

index

Subgroups — #11772016

Conjugacy classes of subgroups

Complements

Maximal subgroups