Group #27304721 — subgroups

Subgroups — #27304721

Order 13568.

166 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₁₆×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#20272910	C₂	1	2	yes	maximal
3	C₄×#13133123	C₂	1	1	yes	maximal, characteristic
4	C₃₂×Dic₅₃	C₂	1	2	yes	maximal
5	C₅₃×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic
7	C₂×#13133123	C₂²	1	1	yes	characteristic
8	C₂×#13133123	C₂²	1	1	yes	characteristic
9	C₄⋊#11719908	C₄	1	1	yes	characteristic
10	#16810441	C₄	1	2	yes	—
11	C₁₆×Dic₅₃	C₂²	1	2	yes	—
12	C₂×C₃₂×C₅₃	C₂²	1	2	yes	—
13	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₀₆	C₈	1	1	yes	characteristic
15	C₂×#11719908	C₂×C₄	1	2	yes	—
16	C₄⋊#88322	C₈	1	1	yes	characteristic
17	#13133146	C₈	1	2	yes	—
18	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
19	#13133123	—	2	1	no	—
20	#13133123	Q₈	1	2	yes	complemented
21	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₃₂×C₅₃	—	2	2	no	—
25	C₂×Dic₁₀₆	C₁₆	1	1	yes	characteristic
26	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
29	C₄⋊Dic₅₃	C₁₆	1	1	yes	characteristic
30	#11719919	C₁₆	1	2	yes	—
31	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
32	C₂×#88322	C₂×C₈	1	2	yes	—
33	C₁₆×C₅₃	—	2	1	no	—
34	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₁₆×C₅₃	#41	1	1	yes	characteristic
36	#11719908	—	2	2	no	—
37	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
40	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
41	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
42	Dic₁₀₆	C₃₂	1	4	yes	complemented
43	C₈×C₅₃	—	2	1	no	—
44	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
45	C₈×C₅₃	#131	1	1	yes	characteristic
46	#88322	—	2	2	no	—
47	Q₈×C₃₂	—	53	1	no	maximal, Sylow
48	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	C₂×C₃₂	1	2	yes	—
51	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
52	C₄×C₅₃	#504	1	1	yes	characteristic
53	Dic₅₃	—	2	2	no	—
54	Dic₅₃	C₂×C₃₂	1	4	yes	—
55	Q₈×C₁₆	—	53	1	no	—
56	C₄⋊C₃₂	—	53	2	no	—
57	C₄⋊C₃₂	D₅₃	1	1	yes	characteristic
58	C₄×C₃₂	—	53	2	no	—
59	C₄×C₃₂	—	53	1	no	—
60	C₂×C₅₃	Q₈×C₁₆	1	1	yes	characteristic
61	C₂×C₅₃	C₂²×C₃₂	1	1	yes	characteristic
62	C₂×C₅₃	#2258	1	1	yes	characteristic
63	C₈×Q₈	—	53	1	no	—
64	C₂×C₃₂	—	53	1	no	—
65	C₂×C₃₂	—	53	1	no	—
66	C₂×C₃₂	C₂×D₅₃	1	2	yes	—
67	C₄⋊C₁₆	—	53	2	no	—
68	C₄⋊C₁₆	—	53	1	no	—
69	C₄×C₁₆	—	53	2	no	—
70	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
71	C₅₃	Q₈×C₃₂	1	1	yes	characteristic, complemented, Sylow
72	C₄×Q₈	—	53	1	no	—
73	C₂×C₁₆	—	53	2	no	—
74	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
76	C₄⋊C₈	—	53	2	no	—
77	C₄⋊C₈	—	53	1	no	—
78	C₄×C₈	—	53	2	no	—
79	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
80	C₃₂	—	2	2	no	—
81	C₃₂	—	53	2	no	—
82	C₃₂	—	106	1	no	—
83	C₂×Q₈	—	53	1	no	—
84	C₂×C₈	—	53	2	no	—
85	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
86	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
87	C₄⋊C₄	—	53	2	no	—
88	C₄⋊C₄	—	53	1	no	—
89	C₄²	—	53	2	no	—
90	C₄²	C₈×D₅₃	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	106	2	no	—
93	C₁₆	#11719947	1	1	yes	characteristic
94	C₁₆	Q₈×D₅₃	1	1	yes	characteristic
95	Q₈	—	53	4	no	—
96	C₂×C₄	—	53	2	no	—
97	C₂×C₄	—	53	2	no	—
98	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
99	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
100	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	106	2	no	—
103	C₈	#13133182	1	1	yes	characteristic
104	C₈	#13133225	1	1	yes	characteristic
105	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	53	4	no	—
108	C₄	—	106	2	no	—
109	C₄	C₃₂×D₅₃	1	2	yes	—
110	C₄	#16810747	1	1	yes	characteristic
111	C₄	#16811041	1	1	yes	characteristic
112	C₂	#20276617	1	1	yes	characteristic
113	C₂	C₂×C₃₂×D₅₃	1	1	yes	characteristic
114	C₂	#20277010	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₀₆

C₂

yes

maximal, characteristic

#20272910

C₂

yes

maximal

C₄×#13133123

C₂

yes

maximal, characteristic

C₃₂×Dic₅₃

C₂

yes

maximal

C₅₃×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₈×Dic₁₀₆

C₄

yes

characteristic

C₂×#13133123

C₂²

yes

characteristic

C₂×#13133123

C₂²

yes

characteristic

C₄⋊#11719908

C₄

yes

characteristic

#16810441

C₄

yes

—

C₁₆×Dic₅₃

C₂²

yes

—

C₂×C₃₂×C₅₃

C₂²

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₄×Dic₁₀₆

C₈

yes

characteristic

C₂×#11719908

C₂×C₄

yes

—

C₄⋊#88322

C₈

yes

characteristic

#13133146

C₈

yes

—

C₈×Dic₅₃

C₂×C₄

yes

—

#13133123

—

#13133123

Q₈

yes

complemented

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₃₂×C₅₃

—

C₂×Dic₁₀₆

C₁₆

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄⋊Dic₅₃

C₁₆

yes

characteristic

#11719919

C₁₆

yes

—

C₄×Dic₅₃

C₂×C₈

yes

—

C₂×#88322

C₂×C₈

yes

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₂×Q₈

yes

C₁₆×C₅₃

yes

—

C₂×C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₁₆

yes

—

C₂×Dic₅₃

C₂×C₁₆

yes

—

Dic₁₀₆

C₃₂

yes

complemented

C₈×C₅₃

—

C₈×C₅₃

C₄×Q₈

yes

C₈×C₅₃

yes

—

Q₈×C₃₂

—

maximal, Sylow

C₂²×C₅₃

C₂²×C₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×C₃₂

yes

—

C₄×C₅₃

C₈×Q₈

yes

characteristic

C₄×C₅₃

#504

yes

characteristic

Dic₅₃

—

Dic₅₃

C₂×C₃₂

yes

—

Q₈×C₁₆

—

C₄⋊C₃₂

—

C₄⋊C₃₂

D₅₃

yes

characteristic

C₄×C₃₂

—

C₄×C₃₂

—

C₂×C₅₃

Q₈×C₁₆

yes

characteristic

C₂×C₅₃

C₂²×C₃₂

yes

characteristic

C₂×C₅₃

#2258

yes

characteristic

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₃

yes

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₅₃

Q₈×C₃₂

yes

characteristic, complemented, Sylow

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅₃

yes

characteristic

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₅₃

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

106

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

C₈×D₅₃

yes

characteristic

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₈×D₅₃

yes

characteristic

C₁₆

—

C₁₆

—

106

—

C₁₆

#11719947

yes

characteristic

C₁₆

Q₈×D₅₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₁₆×D₅₃

yes

characteristic

C₂×C₄

C₁₆×D₅₃

yes

characteristic

100

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

101

C₈

—

102

C₈

—

106

—

103

C₈

#13133182

yes

characteristic

104

C₈

#13133225

yes

characteristic

105

C₂²

C₂×C₁₆×D₅₃

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

106

—

109

C₄

C₃₂×D₅₃

yes

—

110

C₄

#16810747

yes

characteristic

111

C₄

#16811041

yes

characteristic

112

C₂

#20276617

yes

characteristic

113

C₂

C₂×C₃₂×D₅₃

yes

characteristic

114

C₂

#20277010

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₃₂
Fitting	C₅₃×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₅₃	C₂²×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₄⋊C₃₂	D₅₃
hypocenter	C₅₃	Q₈×C₃₂
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₅₃

commutator

C₂×C₅₃

C₂²×C₃₂

Fitting

C₅₃×C₄⋊C₃₂

C₂

socle, prime radical

C₂²×C₅₃

C₂²×C₁₆

cosocle

C₂×C₁₆×C₅₃

C₂³

hypercenter

C₄⋊C₃₂

D₅₃

hypocenter

C₅₃

Q₈×C₃₂

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

N	H ≅ G/N	type
20. #13133123	Q₈	semidirect
42. Dic₁₀₆	C₃₂	semidirect
71. C₅₃	Q₈×C₃₂	semidirect

H ≅ G/N

type

20. #13133123

Q₈

semidirect

42. Dic₁₀₆

C₃₂

semidirect

71. C₅₃

Q₈×C₃₂

semidirect

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

subgroup	index
C₁₆×Dic₁₀₆	2
#20272910	2
C₄×#13133123	2
C₃₂×Dic₅₃	2
C₅₃×C₄⋊C₃₂	2
Q₈×C₃₂	53

subgroup

index

C₁₆×Dic₁₀₆

#20272910

C₄×#13133123

C₃₂×Dic₅₃

C₅₃×C₄⋊C₃₂

Q₈×C₃₂

Subgroups — #27304721

Conjugacy classes of subgroups

Complements

Maximal subgroups