Group #13135921 — subgroups

Subgroups — #13135921

Order 1728.

320 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₃×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	2	yes	maximal
3	#10589412	—	3	1	no	maximal
4	#10588223	—	3	3	no	maximal
5	C₂×C₃×Q₈×C₉	C₄	1	2	yes	—
6	C₃×C₉×C₄⋊C₄	—	2	2	no	—
7	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₃⋊(C₉⋊C₈)	—	2	2	no	—
9	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic
10	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	2	no	—
11	C₉×C₄⋊Q₈	D₃	1	3	yes	—
12	C₄⋊(C₉⋊C₈)	—	3	6	no	—
13	C₃×Q₈×C₉	—	4	2	no	—
14	C₂×C₃×C₄×C₉	—	2	2	no	—
15	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
17	C₃⋊(C₉⋊C₈)	—	4	2	no	—
18	#4546	—	9	3	no	—
19	#4546	—	9	1	no	—
20	C₂×C₃²×Q₈	Dic₃	1	2	yes	—
21	C₃²×C₄⋊C₄	—	2	2	no	—
22	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
23	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
24	C₂×Q₈×C₉	Dic₃	1	6	yes	—
25	C₉×C₄⋊C₄	—	2	6	no	—
26	C₄²×C₉	C₂×D₃	1	3	yes	—
27	C₂×C₉⋊C₈	—	6	6	no	—
28	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₃×C₄×C₉	—	2	1	no	—
30	C₃×C₄×C₉	—	4	2	no	—
31	C₃×C₄×C₉	SD₁₆	1	2	yes	—
32	C₃×C₄×C₉	Q₁₆	1	2	yes	—
33	C₃×C₄⋊Q₈	D₉	1	3	yes	—
34	C₃×C₄⋊Q₈	C₃⋊D₃	1	1	yes	characteristic
35	C₄⋊(C₃⋊C₈)	—	9	6	no	—
36	C₄⋊(C₃⋊C₈)	—	9	2	no	—
37	C₃²×Q₈	—	4	2	no	—
38	C₂×C₃²×C₄	—	2	2	no	—
39	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
40	C₂×C₃²×C₄	#67	1	2	yes	—
41	C₃⋊(C₃⋊C₈)	—	12	2	no	—
42	Q₈×C₉	—	4	6	no	—
43	C₂×C₄×C₉	—	2	6	no	—
44	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
45	C₂×C₄×C₉	#67	1	6	yes	—
46	C₉⋊C₈	—	12	6	no	—
47	#333	—	27	1	no	Sylow
48	C₂×C₃×C₉	#100	1	1	yes	characteristic
49	C₂×C₃×C₉	#103	1	2	yes	—
50	C₂×C₃×Q₈	Dic₉	1	6	yes	—
51	C₂×C₃×Q₈	C₃⋊Dic₃	1	2	yes	—
52	C₃×C₄⋊C₄	—	2	6	no	—
53	C₃×C₄⋊C₄	—	2	2	no	—
54	C₃×C₄²	C₂×D₉	1	3	yes	—
55	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
56	C₂×C₃⋊C₈	—	18	6	no	—
57	C₂×C₃⋊C₈	—	18	2	no	—
58	C₂²×C₃²	#217	1	1	yes	characteristic
59	C₃²×C₄	—	2	1	no	—
60	C₃²×C₄	—	4	2	no	—
61	C₃²×C₄	#215	1	2	yes	—
62	C₃²×C₄	#216	1	2	yes	—
63	C₂²×C₉	#217	1	3	yes	—
64	C₄×C₉	—	2	3	no	—
65	C₄×C₉	—	4	6	no	—
66	C₄×C₉	#215	1	6	yes	—
67	C₄×C₉	#216	1	6	yes	—
68	C₄⋊Q₈	C₃⋊D₉	1	1	yes	characteristic
69	C₄⋊C₈	—	27	2	no	—
70	C₃×C₉	#333	1	1	yes	characteristic, complemented, Sylow
71	C₃×Q₈	—	4	2	no	—
72	C₃×Q₈	—	4	6	no	—
73	C₂×C₃×C₄	—	2	6	no	—
74	C₂×C₃×C₄	—	2	2	no	—
75	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
76	C₂×C₃×C₄	#611	1	6	yes	—
77	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
78	C₂×C₃×C₄	#638	1	2	yes	—
79	C₃⋊C₈	—	36	6	no	—
80	C₃⋊C₈	—	36	2	no	—
81	C₂×C₃²	#833	1	1	yes	characteristic
82	C₂×C₃²	#835	1	2	yes	—
83	C₂×C₉	#833	1	3	yes	—
84	C₂×C₉	#835	1	6	yes	—
85	C₂×Q₈	C₃⋊Dic₉	1	2	yes	—
86	C₂×C₈	—	54	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
89	C₂²×C₃	#3676	1	3	yes	—
90	C₂²×C₃	#3757	1	1	yes	characteristic
91	C₃×C₄	—	2	3	no	—
92	C₃×C₄	—	2	1	no	—
93	C₃×C₄	—	4	6	no	—
94	C₃×C₄	—	4	2	no	—
95	C₃×C₄	#3674	1	6	yes	—
96	C₃×C₄	#3675	1	6	yes	—
97	C₃×C₄	#3755	1	2	yes	—
98	C₃×C₄	#3756	1	2	yes	—
99	C₃²	#4546	1	1	yes	characteristic
100	C₉	#4546	1	3	yes	complemented
101	Q₈	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
104	C₂×C₄	#6233	1	2	yes	—
105	C₈	—	108	2	no	—
106	C₂×C₃	#63391	1	3	yes	—
107	C₂×C₃	#63395	1	6	yes	—
108	C₂×C₃	#63660	1	1	yes	characteristic
109	C₂×C₃	#63662	1	2	yes	—
110	C₂²	#88551	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	4	2	no	—
113	C₄	#88547	1	2	yes	—
114	C₄	#88548	1	2	yes	—
115	C₃	#10588223	1	3	yes	complemented
116	C₃	#10589412	1	1	yes	characteristic
117	C₂	#11720575	1	1	yes	characteristic
118	C₂	#11720582	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₄⋊(C₃⋊(C₉⋊C₈))

C₂

yes

—

—

C₂×C₃×Q₈×C₉

C₄

yes

—

C₃×C₉×C₄⋊C₄

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₂×C₃⋊(C₉⋊C₈)

—

C₃²×C₄⋊Q₈

D₃

yes

characteristic

C₄⋊(C₃⋊(C₃⋊C₈))

—

C₉×C₄⋊Q₈

D₃

yes

—

C₄⋊(C₉⋊C₈)

—

C₃×Q₈×C₉

—

C₂×C₃×C₄×C₉

—

C₂×C₃×C₄×C₉

C₂×C₄

yes

characteristic

C₂×C₃×C₄×C₉

D₄

yes

—

C₃⋊(C₉⋊C₈)

—

#4546

—

#4546

—

C₂×C₃²×Q₈

Dic₃

yes

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

—

C₂×Q₈×C₉

Dic₃

yes

—

C₉×C₄⋊C₄

—

C₄²×C₉

C₂×D₃

yes

—

C₂×C₉⋊C₈

—

C₂²×C₃×C₉

C₂²⋊C₄

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

SD₁₆

yes

—

C₃×C₄×C₉

Q₁₆

yes

—

C₃×C₄⋊Q₈

D₉

yes

—

C₃×C₄⋊Q₈

C₃⋊D₃

yes

characteristic

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×Dic₃

yes

characteristic

C₂×C₃²×C₄

#67

yes

—

C₃⋊(C₃⋊C₈)

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×Dic₃

yes

—

C₂×C₄×C₉

#67

yes

—

C₉⋊C₈

—

—

C₂×C₃×C₉

yes

C₂×C₃×C₉

yes

—

C₂×C₃×Q₈

Dic₉

yes

—

C₂×C₃×Q₈

C₃⋊Dic₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×D₉

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

#217

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#215

yes

—

C₃²×C₄

#216

yes

—

C₂²×C₉

#217

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

#215

yes

—

C₄×C₉

#216

yes

—

C₄⋊Q₈

C₃⋊D₉

yes

characteristic

C₄⋊C₈

—

C₃×C₉

#333

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×Dic₉

yes

—

C₂×C₃×C₄

#611

yes

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

characteristic

C₂×C₃×C₄

#638

yes

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

#833

yes

characteristic

C₂×C₃²

#835

yes

—

C₂×C₉

#833

yes

—

C₂×C₉

#835

yes

—

C₂×Q₈

C₃⋊Dic₉

yes

—

C₂×C₈

—

C₄⋊C₄

—

C₄²

C₂×C₃⋊D₉

yes

characteristic

C₂²×C₃

#3676

yes

—

C₂²×C₃

#3757

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#3674

yes

—

C₃×C₄

#3675

yes

—

C₃×C₄

#3755

yes

—

C₃×C₄

#3756

yes

—

C₃²

#4546

yes

characteristic

100

C₉

#4546

yes

complemented

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

C₂×C₃⋊Dic₉

yes

characteristic

104

C₂×C₄

#6233

yes

—

105

C₈

—

108

—

106

C₂×C₃

#63391

yes

—

107

C₂×C₃

#63395

yes

—

108

C₂×C₃

#63660

yes

characteristic

109

C₂×C₃

#63662

yes

—

110

C₂²

#88551

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

#88547

yes

—

114

C₄

#88548

yes

—

115

C₃

#10588223

yes

complemented

116

C₃

#10589412

yes

characteristic

117

C₂

#11720575

yes

characteristic

118

C₂

#11720582

yes

—

	subgroup	G/H
center	C₂²	#88551
commutator	C₂×C₃×C₄×C₉	C₂×C₄
Frattini	C₃×C₄²	C₂×C₃⋊D₃
Fitting	C₃×C₉×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃²	#217
cosocle	C₃×C₄²×C₉	C₂²
hypercenter	C₄⋊Q₈	C₃⋊D₉
hypocenter	C₃×C₉	#333
LCS	G ▷ C₂×C₃×C₄×C₉ ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

#88551

commutator

C₂×C₃×C₄×C₉

C₂×C₄

Frattini

C₃×C₄²

C₂×C₃⋊D₃

Fitting

C₃×C₉×C₄⋊Q₈

C₂

socle, prime radical

C₂²×C₃²

#217

cosocle

C₃×C₄²×C₉

C₂²

hypercenter

C₄⋊Q₈

C₃⋊D₉

hypocenter

C₃×C₉

#333

LCS

G ▷ C₂×C₃×C₄×C₉ ▷ C₂²×C₃×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
70. C₃×C₉	#333	semidirect
100. C₉	#4546	semidirect
115. C₃	#10588223	semidirect

H ≅ G/N

type

70. C₃×C₉

#333

semidirect

100. C₉

#4546

semidirect

115. C₃

#10588223

semidirect

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

subgroup	index
C₃×C₉×C₄⋊Q₈	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
#10589412	3
#10588223	3

subgroup

index

C₃×C₉×C₄⋊Q₈

C₄⋊(C₃⋊(C₉⋊C₈))

#10589412

#10588223

Subgroups — #13135921

Conjugacy classes of subgroups

Complements

Maximal subgroups