Group #11772129 — subgroups

Subgroups — #11772129

Order 1152.

129 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 129 conjugacy classes of subgroups — too many to draw. Only its 66 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	C₉×#500	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#10588135	C₂	1	1	yes	maximal, characteristic
4	#67802	—	3	1	no	maximal
5	C₉×C₄⋊Q₈	—	2	1	no	—
6	C₉×#107	—	2	1	no	—
7	C₉×#107	C₄	1	2	yes	—
8	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉⋊C₁₆	—	2	1	no	—
10	C₄⋊(C₉⋊C₈)	—	2	1	no	—
11	C₃×#500	D₃	1	1	yes	characteristic
12	C₄⋊(C₃⋊C₁₆)	—	3	1	no	—
13	#4458	—	3	1	no	—
14	C₂×Q₈×C₉	—	4	1	no	—
15	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₉	D₄	1	1	yes	characteristic
17	C₉×C₄⋊C₄	—	2	2	no	—
18	C₉×C₄⋊C₄	—	4	1	no	—
19	C₄²×C₉	D₄	1	1	yes	characteristic
20	C₂×C₉⋊C₈	—	4	1	no	—
21	C₉⋊C₁₆	—	4	1	no	—
22	#1365	—	9	1	no	Sylow
23	C₃×C₄⋊Q₈	—	2	1	no	—
24	C₃×#107	—	2	1	no	—
25	C₃×#107	Dic₃	1	2	yes	—
26	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
27	C₂×C₃⋊C₁₆	—	6	1	no	—
28	C₄⋊(C₃⋊C₈)	—	6	1	no	—
29	Q₈×C₉	—	8	2	no	—
30	C₂×C₄×C₉	—	4	2	no	—
31	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₉	D₈	1	1	yes	characteristic
33	C₂×C₄×C₉	SD₁₆	1	1	yes	characteristic
34	C₈×C₉	—	2	1	no	—
35	C₈×C₉	SD₁₆	1	1	yes	characteristic
36	C₈×C₉	Q₁₆	1	1	yes	characteristic
37	C₉⋊C₈	—	8	1	no	—
38	#500	D₉	1	1	yes	characteristic
39	C₄⋊C₁₆	—	9	1	no	—
40	#335	—	9	1	no	—
41	C₂×C₃×Q₈	—	4	1	no	—
42	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
43	C₂×C₃×C₈	#67	1	1	yes	characteristic
44	C₃×C₄⋊C₄	—	2	2	no	—
45	C₃×C₄⋊C₄	—	4	1	no	—
46	C₃×C₄²	#67	1	1	yes	characteristic
47	C₂×C₃⋊C₈	—	12	1	no	—
48	C₃⋊C₁₆	—	12	1	no	—
49	C₂²×C₉	#102	1	1	yes	characteristic
50	C₄×C₉	—	2	1	no	—
51	C₄×C₉	—	8	2	no	—
52	C₄×C₉	#101	1	1	yes	characteristic
53	C₄×C₉	#103	1	1	yes	characteristic
54	C₄×C₉	SD₃₂	1	1	yes	characteristic
55	C₄×C₉	Q₃₂	1	1	yes	characteristic
56	C₄⋊Q₈	—	2	1	no	—
57	C₂×C₁₆	—	18	1	no	—
58	#107	—	2	1	no	—
59	#107	Dic₉	1	2	yes	—
60	C₄⋊C₈	—	18	1	no	—
61	C₄×C₈	C₂×D₉	1	1	yes	characteristic
62	C₃×Q₈	—	8	2	no	—
63	C₂×C₃×C₄	—	4	2	no	—
64	C₂×C₃×C₄	#213	1	1	yes	characteristic
65	C₂×C₃×C₄	#214	1	1	yes	characteristic
66	C₂×C₃×C₄	#217	1	1	yes	characteristic
67	C₃×C₈	—	2	1	no	—
68	C₃×C₈	#215	1	1	yes	characteristic
69	C₃×C₈	#216	1	1	yes	characteristic
70	C₃⋊C₈	—	24	1	no	—
71	C₂×C₉	#332	1	1	yes	characteristic
72	C₂×C₉	#358	1	1	yes	characteristic
73	C₂×C₉	#361	1	1	yes	characteristic
74	C₂×Q₈	—	4	1	no	—
75	C₂×C₈	—	36	1	no	—
76	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
77	C₂×C₈	#611	1	1	yes	characteristic
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	4	1	no	—
80	C₄²	#611	1	1	yes	characteristic
81	C₁₆	—	36	1	no	—
82	C₂²×C₃	#832	1	1	yes	characteristic
83	C₃×C₄	—	2	1	no	—
84	C₃×C₄	—	8	2	no	—
85	C₃×C₄	#828	1	1	yes	characteristic
86	C₃×C₄	#829	1	1	yes	characteristic
87	C₃×C₄	#835	1	1	yes	characteristic
88	C₃×C₄	#836	1	1	yes	characteristic
89	C₉	#1365	1	1	yes	characteristic, complemented, Sylow
90	Q₈	—	8	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	#3672	1	1	yes	characteristic
93	C₂×C₄	#3673	1	1	yes	characteristic
94	C₂×C₄	#3676	1	1	yes	characteristic
95	C₈	—	2	1	no	—
96	C₈	—	72	1	no	—
97	C₈	#3674	1	1	yes	characteristic
98	C₈	#3675	1	1	yes	characteristic
99	C₂×C₃	#4547	1	1	yes	characteristic
100	C₂×C₃	#4563	1	1	yes	characteristic
101	C₂×C₃	#4564	1	1	yes	characteristic
102	C₂²	#63392	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	8	2	no	—
105	C₄	#63387	1	1	yes	characteristic
106	C₄	#63388	1	1	yes	characteristic
107	C₄	#63394	1	1	yes	characteristic
108	C₄	#63395	1	1	yes	characteristic
109	C₃	#67802	1	1	yes	characteristic
110	C₂	#10588224	1	1	yes	characteristic
111	C₂	#10588240	1	1	yes	characteristic
112	C₂	#10588241	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#500

C₂

yes

maximal, characteristic

C₄⋊(C₉⋊C₁₆)

C₂

yes

maximal, characteristic

#10588135

C₂

yes

maximal, characteristic

#67802

—

maximal

C₉×C₄⋊Q₈

—

C₉×#107

—

C₉×#107

C₄

yes

—

C₄×C₈×C₉

C₂²

yes

characteristic

C₂×C₉⋊C₁₆

—

C₄⋊(C₉⋊C₈)

—

C₃×#500

D₃

yes

characteristic

C₄⋊(C₃⋊C₁₆)

—

#4458

—

C₂×Q₈×C₉

—

C₂×C₈×C₉

C₂×C₄

yes

characteristic

C₂×C₈×C₉

D₄

yes

characteristic

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₄²×C₉

D₄

yes

characteristic

C₂×C₉⋊C₈

—

C₉⋊C₁₆

—

#1365

—

Sylow

C₃×C₄⋊Q₈

—

C₃×#107

—

C₃×#107

Dic₃

yes

—

C₃×C₄×C₈

C₂×D₃

yes

characteristic

C₂×C₃⋊C₁₆

—

C₄⋊(C₃⋊C₈)

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉

D₈

yes

characteristic

C₂×C₄×C₉

SD₁₆

yes

characteristic

C₈×C₉

—

C₈×C₉

SD₁₆

yes

characteristic

C₈×C₉

Q₁₆

yes

characteristic

C₉⋊C₈

—

#500

D₉

yes

characteristic

C₄⋊C₁₆

—

#335

—

C₂×C₃×Q₈

—

C₂×C₃×C₈

C₂×Dic₃

yes

characteristic

C₂×C₃×C₈

#67

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

#67

yes

characteristic

C₂×C₃⋊C₈

—

C₃⋊C₁₆

—

C₂²×C₉

#102

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

yes

C₄×C₉

yes

C₄×C₉

SD₃₂

yes

characteristic

C₄×C₉

Q₃₂

yes

characteristic

C₄⋊Q₈

—

C₂×C₁₆

—

#107

—

#107

Dic₉

yes

—

C₄⋊C₈

—

C₄×C₈

C₂×D₉

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

yes

C₂×C₃×C₄

yes

C₂×C₃×C₄

yes

C₃×C₈

—

C₃×C₈

yes

C₃×C₈

yes

C₃⋊C₈

—

C₂×C₉

yes

C₂×C₉

yes

C₂×C₉

yes

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₉

yes

characteristic

C₂×C₈

#611

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

#611

yes

characteristic

C₁₆

—

C₂²×C₃

#832

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

C₃×C₄

yes

C₃×C₄

yes

C₉

yes

characteristic, complemented, Sylow

Q₈

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂×C₃

yes

100

C₂×C₃

#4563

yes

characteristic

101

C₂×C₃

#4564

yes

characteristic

102

C₂²

#63392

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

#63387

yes

characteristic

106

C₄

#63388

yes

characteristic

107

C₄

#63394

yes

characteristic

108

C₄

#63395

yes

characteristic

109

C₃

#67802

yes

characteristic

110

C₂

#10588224

yes

characteristic

111

C₂

#10588240

yes

characteristic

112

C₂

#10588241

yes

characteristic

	subgroup	G/H
center	C₂²	#63392
commutator	C₂×C₈×C₉	C₂×C₄
Frattini	C₃×C₄×C₈	C₂×D₃
Fitting	C₉×#500	C₂
socle, prime radical	C₂²×C₃	#832
cosocle	C₄×C₈×C₉	C₂²
hypercenter	#500	D₉
hypocenter	C₉	#1365
LCS	G ▷ C₂×C₈×C₉ ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

subgroup

G/H

center

C₂²

#63392

commutator

C₂×C₈×C₉

C₂×C₄

Frattini

C₃×C₄×C₈

C₂×D₃

C₂

C₂²×C₃

C₄×C₈×C₉

C₂²

D₉

C₉

G ▷ C₂×C₈×C₉ ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

N	H ≅ G/N	type
89. C₉	#1365	semidirect

H ≅ G/N

type

89. C₉

#1365

semidirect

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

subgroup	index
C₉×#500	2
C₄⋊(C₉⋊C₁₆)	2
#10588135	2
#67802	3

subgroup

index

C₉×#500

C₄⋊(C₉⋊C₁₆)

#10588135

#67802

Subgroups — #11772129

Conjugacy classes of subgroups

Complements

Maximal subgroups