Group #11772226 — subgroups

Subgroups — #11772226

Order 1152.

169 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 169 conjugacy classes of subgroups — too many to draw. Only its 82 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	C₄×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₃×#4458	C₂	1	2	yes	maximal
3	#67469	—	3	2	no	maximal
4	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
5	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	1	yes	characteristic
6	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	1	yes	characteristic
7	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
8	C₃×#335	—	3	2	no	—
9	C₄×C₃⋊C₁₆	—	3	2	no	—
10	C₄×C₃⋊C₁₆	—	6	1	no	—
11	#4458	D₃	1	2	yes	—
12	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
14	C₃²×C₄²	D₄	1	1	yes	characteristic
15	C₂×C₃×C₃⋊C₈	—	4	2	no	—
16	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
17	C₃⋊(C₃⋊C₁₆)	C₈	1	2	yes	complemented
18	#1371	—	9	1	no	Sylow
19	C₃×C₄⋊C₈	—	6	2	no	—
20	C₃×C₄×C₈	—	2	1	no	—
21	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
22	C₂×C₃⋊C₁₆	—	3	2	no	—
23	C₂×C₃⋊C₁₆	—	3	2	no	—
24	C₂×C₃⋊C₁₆	—	6	1	no	—
25	C₂×C₃⋊C₁₆	—	6	1	no	—
26	C₄⋊(C₃⋊C₈)	—	2	2	no	—
27	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
29	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
30	C₃²×C₈	—	2	1	no	—
31	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
32	C₃²×C₈	M₁₆	1	1	yes	characteristic
33	C₃×C₃⋊C₈	—	8	2	no	—
34	C₄×C₁₆	—	9	1	no	—
35	#335	—	3	2	no	—
36	C₂×C₃×C₈	—	2	1	no	—
37	C₂×C₃×C₈	—	2	1	no	—
38	C₂×C₃×C₈	—	12	2	no	—
39	C₂×C₃×C₈	Dic₆	1	2	yes	—
40	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
41	C₃×C₄²	—	2	1	no	—
42	C₃×C₄²	#67	1	2	yes	—
43	C₂×C₃⋊C₈	—	4	2	no	—
44	C₃⋊C₁₆	—	3	4	no	—
45	C₃⋊C₁₆	—	6	2	no	—
46	C₃⋊C₁₆	—	6	2	no	—
47	C₃⋊C₁₆	—	6	2	no	—
48	C₂²×C₃²	#107	1	1	yes	characteristic
49	C₃²×C₄	—	2	1	no	—
50	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
51	C₃²×C₄	#108	1	1	yes	characteristic
52	C₃²×C₄	D₁₆	1	1	yes	characteristic
53	C₃²×C₄	Q₃₂	1	1	yes	characteristic
54	C₂×C₁₆	—	9	1	no	—
55	C₂×C₁₆	—	9	1	no	—
56	C₄⋊C₈	—	6	2	no	—
57	C₄×C₈	D₃²	1	1	yes	characteristic
58	C₂×C₃×C₄	—	2	1	no	—
59	C₂×C₃×C₄	—	2	1	no	—
60	C₂×C₃×C₄	—	2	1	no	—
61	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
62	C₂×C₃×C₄	#213	1	2	yes	—
63	C₂×C₃×C₄	#216	1	2	yes	—
64	C₃×C₈	—	2	2	no	—
65	C₃×C₈	—	2	2	no	—
66	C₃×C₈	—	2	1	no	—
67	C₃×C₈	—	2	1	no	—
68	C₃×C₈	—	24	2	no	—
69	C₃×C₈	D₃×C₈	1	2	yes	—
70	C₃×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
71	C₃⋊C₈	—	8	2	no	—
72	C₂×C₃²	#335	1	1	yes	characteristic
73	C₂×C₃²	#366	1	1	yes	characteristic
74	C₂×C₃²	#368	1	1	yes	characteristic
75	C₂×C₈	—	12	2	no	—
76	C₂×C₈	#624	1	1	yes	characteristic
77	C₂×C₈	#627	1	1	yes	characteristic
78	C₄²	#625	1	1	yes	characteristic
79	C₁₆	—	9	2	no	—
80	C₁₆	—	18	1	no	—
81	C₂²×C₃	—	2	1	no	—
82	C₂²×C₃	#807	1	2	yes	—
83	C₃×C₄	—	2	2	no	—
84	C₃×C₄	—	2	2	no	—
85	C₃×C₄	—	2	1	no	—
86	C₃×C₄	—	2	1	no	—
87	C₃×C₄	—	2	1	no	—
88	C₃×C₄	—	2	1	no	—
89	C₃×C₄	Dic₃⋊C₈	1	2	yes	—
90	C₃×C₄	#822	1	2	yes	—
91	C₃×C₄	#826	1	2	yes	—
92	C₃×C₄	#829	1	2	yes	—
93	C₃²	#1371	1	1	yes	characteristic, complemented, Sylow
94	C₂×C₄	#3713	1	1	yes	characteristic
95	C₂×C₄	#3718	1	1	yes	characteristic
96	C₂×C₄	#3724	1	1	yes	characteristic
97	C₈	—	2	1	no	—
98	C₈	—	24	2	no	—
99	C₈	#3710	1	1	yes	characteristic
100	C₈	#3712	1	1	yes	characteristic
101	C₂×C₃	—	2	1	no	—
102	C₂×C₃	—	2	1	no	—
103	C₂×C₃	—	2	1	no	—
104	C₂×C₃	#4479	1	2	yes	—
105	C₂×C₃	#4489	1	2	yes	—
106	C₂×C₃	#4493	1	2	yes	—
107	C₂²	#63576	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	#63545	1	1	yes	characteristic
110	C₄	#63550	1	1	yes	characteristic
111	C₄	#63572	1	1	yes	characteristic
112	C₄	#63578	1	1	yes	characteristic
113	C₃	—	2	1	no	—
114	C₃	#67469	1	2	yes	complemented
115	C₂	#10589100	1	1	yes	characteristic
116	C₂	#10589140	1	1	yes	characteristic
117	C₂	#10589144	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₃⋊(C₃⋊C₁₆)

C₂

yes

maximal, characteristic

C₃×#4458

C₂

yes

maximal

#67469

—

maximal

C₃²×C₄×C₈

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₁₆)

C₄

yes

characteristic

C₂×C₃⋊(C₃⋊C₁₆)

C₄

yes

characteristic

C₃×C₄⋊(C₃⋊C₈)

—

C₃×#335

—

C₄×C₃⋊C₁₆

—

C₄×C₃⋊C₁₆

—

#4458

D₃

yes

—

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

C₂×C₃²×C₈

Q₈

yes

characteristic

C₃²×C₄²

D₄

yes

characteristic

C₂×C₃×C₃⋊C₈

—

C₃⋊(C₃⋊C₁₆)

—

C₃⋊(C₃⋊C₁₆)

C₈

yes

complemented

#1371

—

Sylow

C₃×C₄⋊C₈

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×D₃

yes

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₄⋊(C₃⋊C₈)

—

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃²×C₄

D₈

yes

characteristic

C₂×C₃²×C₄

Q₁₆

yes

characteristic

C₃²×C₈

—

C₃²×C₈

C₂×C₈

yes

characteristic

C₃²×C₈

M₁₆

yes

characteristic

C₃×C₃⋊C₈

—

C₄×C₁₆

—

#335

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

Dic₆

yes

—

C₂×C₃×C₈

C₄×D₃

yes

—

C₃×C₄²

—

C₃×C₄²

#67

yes

—

C₂×C₃⋊C₈

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

#107

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₄⋊C₈

yes

characteristic

C₃²×C₄

#108

yes

characteristic

C₃²×C₄

D₁₆

yes

characteristic

C₃²×C₄

Q₃₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄×C₈

D₃²

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

Dic₃⋊C₄

yes

—

C₂×C₃×C₄

#213

yes

—

C₂×C₃×C₄

#216

yes

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

D₃×C₈

yes

—

C₃×C₈

ΓL(1,𝔽₂₅)

yes

—

C₃⋊C₈

—

C₂×C₃²

yes

C₂×C₃²

yes

C₂×C₃²

yes

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₂²×C₃

—

C₂²×C₃

#807

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

Dic₃⋊C₈

yes

—

C₃×C₄

#822

yes

—

C₃×C₄

#826

yes

—

C₃×C₄

#829

yes

—

C₃²

#1371

yes

characteristic, complemented, Sylow

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

#3710

yes

characteristic

100

C₈

#3712

yes

characteristic

101

C₂×C₃

—

102

C₂×C₃

—

103

C₂×C₃

—

104

C₂×C₃

#4479

yes

—

105

C₂×C₃

#4489

yes

—

106

C₂×C₃

#4493

yes

—

107

C₂²

#63576

yes

characteristic

108

C₄

—

109

C₄

#63545

yes

characteristic

110

C₄

#63550

yes

characteristic

111

C₄

#63572

yes

characteristic

112

C₄

#63578

yes

characteristic

113

C₃

—

114

C₃

#67469

yes

complemented

115

C₂

#10589100

yes

characteristic

116

C₂

#10589140

yes

characteristic

117

C₂

#10589144

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#3713
commutator	C₃²×C₈	C₂×C₈
Frattini, hypercenter	C₄×C₈	D₃²
Fitting, cosocle	C₃²×C₄×C₈	C₂²
socle, prime radical	C₂²×C₃²	#107
hypocenter	C₃²	#1371
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂×C₄

#3713

commutator

C₃²×C₈

C₂×C₈

Frattini, hypercenter

C₄×C₈

D₃²

Fitting, cosocle

C₃²×C₄×C₈

C₂²

C₂²×C₃²

C₃²

G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
17. C₃⋊(C₃⋊C₁₆)	C₈	semidirect
93. C₃²	#1371	semidirect
114. C₃	#67469	semidirect

H ≅ G/N

type

17. C₃⋊(C₃⋊C₁₆)

C₈

semidirect

93. C₃²

#1371

semidirect

114. C₃

#67469

semidirect

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

subgroup	index
C₄×C₃⋊(C₃⋊C₁₆)	2
C₃×#4458	2
#67469	3

subgroup

index

C₄×C₃⋊(C₃⋊C₁₆)

C₃×#4458

#67469

Subgroups — #11772226

Conjugacy classes of subgroups

Complements

Maximal subgroups