Group #27536606 — subgroups

Subgroups — #27536606

Order 13568.

1734 conjugacy classes of subgroups; only the 395 normal subgroups are shown.

normal subgroups only

This group has 1734 conjugacy classes of subgroups — too many to draw. Only its 395 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	C₂×#16810995	C₂	1	12	yes	maximal, complemented
2	#20283113	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2867	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719923	C₂²	1	16	yes	—
6	C₂×#13133217	C₂²	1	6	yes	—
7	#16810995	—	2	48	no	—
8	C₂×C₄×#88322	C₂²	1	6	yes	—
9	#16810493	—	2	4	no	—
10	C₂×#127×C₅₃	—	2	4	no	—
11	C₂×#127×C₅₃	C₂²	1	6	yes	—
12	C₄³×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₀₆	C₂³	1	4	yes	—
14	C₂×#11719923	—	2	96	no	—
15	C₂²×#88322	C₂³	1	4	yes	—
16	#13133217	—	2	24	no	—
17	C₂×C₄⋊Dic₅₃	—	2	12	no	—
18	C₄×#88322	D₄	1	24	yes	complemented
19	C₂²×D₄×C₅₃	—	2	12	no	—
20	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
21	#127×C₅₃	—	2	24	no	—
22	#127×C₅₃	—	4	16	no	—
23	C₂×C₄²×C₅₃	D₄	1	4	yes	—
24	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
25	C₂⁴×C₅₃	—	2	4	no	—
26	C₂×D₄×C₅₃	—	2	24	no	—
27	C₂×D₄×C₅₃	—	4	24	no	—
28	C₂×D₄×C₅₃	—	4	48	no	—
29	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
30	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
31	C₂²×Dic₅₃	—	2	4	no	—
32	C₂×Dic₁₀₆	—	2	24	no	—
33	C₄²×C₅₃	SD₁₆	1	16	yes	complemented
34	C₄²×C₅₃	C₂×D₄	1	12	yes	—
35	#11719923	—	4	64	no	—
36	C₄⋊Dic₅₃	—	4	24	no	—
37	C₂×#88322	C₂×D₄	1	24	yes	—
38	C₂³×C₅₃	—	4	12	no	—
39	C₂³×C₅₃	—	4	16	no	—
40	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
41	D₄×C₅₃	—	4	16	no	—
42	D₄×C₅₃	—	8	48	no	—
43	C₂×C₄×C₅₃	#127	1	12	yes	—
44	C₂×C₄×C₅₃	C₂×SD₁₆	1	24	yes	—
45	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
46	C₂×Dic₅₃	—	4	12	no	—
47	Dic₁₀₆	—	4	16	no	—
48	#88322	#127	1	16	yes	complemented
49	#33300	—	53	1	no	maximal, Sylow
50	C₂²×C₅₃	—	8	4	no	—
51	C₂²×C₅₃	—	8	24	no	—
52	C₂²×C₅₃	C₂×#127	1	3	yes	—
53	C₂²×C₅₃	C₂²×SD₁₆	1	4	yes	—
54	C₄×C₅₃	#492	1	24	yes	complemented
55	C₄×C₅₃	C₂×#127	1	4	yes	—
56	Dic₅₃	—	8	4	no	—
57	C₂×#492	—	53	12	no	—
58	#2868	—	53	1	no	—
59	#2867	D₅₃	1	1	yes	characteristic
60	C₄²×C₈	—	53	1	no	—
61	C₂×C₅₃	—	16	4	no	—
62	C₂×C₅₃	#2867	1	1	yes	characteristic
63	C₂×C₅₃	C₂×#492	1	6	yes	—
64	C₂²×SD₁₆	—	53	16	no	—
65	C₂×C₄⋊Q₈	—	53	6	no	—
66	C₂×#127	—	2	4	no	—
67	C₂×#127	C₂×D₅₃	1	6	yes	—
68	#492	—	106	48	no	—
69	C₂×C₄×C₈	—	53	6	no	—
70	#384	—	106	4	no	—
71	C₄³	C₂×D₅₃	1	1	yes	characteristic
72	C₅₃	#33300	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	53	4	no	—
74	C₂²×D₄	—	2	12	no	—
75	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
76	C₂×SD₁₆	—	106	96	no	—
77	C₂²×C₈	—	53	4	no	—
78	C₄⋊Q₈	—	106	24	no	—
79	#127	—	2	24	no	—
80	#127	—	4	16	no	—
81	C₂×C₄⋊C₄	—	106	12	no	—
82	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
83	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
84	C₄×C₈	—	53	24	no	—
85	C₂⁴	—	2	4	no	—
86	C₂×Q₈	—	106	24	no	—
87	C₂×D₄	—	2	24	no	—
88	C₂×D₄	—	4	24	no	—
89	C₂×D₄	—	4	48	no	—
90	C₂²×C₄	—	106	4	no	—
91	C₂²×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
92	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
93	SD₁₆	—	212	64	no	—
94	C₂×C₈	—	53	24	no	—
95	C₄⋊C₄	—	212	24	no	—
96	C₄²	#11719923	1	16	yes	complemented
97	C₄²	D₄×D₅₃	1	12	yes	—
98	C₂³	—	4	16	no	—
99	C₂³	—	4	12	no	—
100	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
101	Q₈	—	212	16	no	—
102	D₄	—	4	16	no	—
103	D₄	—	8	48	no	—
104	C₂×C₄	—	212	12	no	—
105	C₂×C₄	#13133213	1	12	yes	—
106	C₂×C₄	C₂×#11719923	1	24	yes	—
107	C₂×C₄	C₂×D₄×D₅₃	1	6	yes	—
108	C₈	—	53	16	no	—
109	C₂²	—	8	24	no	—
110	C₂²	—	8	4	no	—
111	C₂²	C₂×#13133213	1	3	yes	—
112	C₂²	C₂²×#11719923	1	4	yes	—
113	C₄	—	424	4	no	—
114	C₄	#16810995	1	24	yes	complemented
115	C₄	#127×D₅₃	1	4	yes	—
116	C₂	—	16	4	no	—
117	C₂	#20283109	1	1	yes	characteristic
118	C₂	C₂×#16810995	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810995

C₂

yes

maximal, complemented

#20283113

C₂

yes

maximal, characteristic, complemented

C₄²×#88322

C₂

yes

maximal, characteristic, complemented

C₅₃×#2867

C₂

yes

maximal, characteristic

C₂²×#11719923

C₂²

yes

—

C₂×#13133217

C₂²

yes

—

#16810995

—

C₂×C₄×#88322

C₂²

yes

—

#16810493

—

C₂×#127×C₅₃

—

C₂×#127×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×Dic₁₀₆

C₂³

yes

—

C₂×#11719923

—

C₂²×#88322

C₂³

yes

—

#13133217

—

C₂×C₄⋊Dic₅₃

—

C₄×#88322

D₄

yes

complemented

C₂²×D₄×C₅₃

—

C₂²×D₄×C₅₃

C₂³

yes

—

#127×C₅₃

—

#127×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

—

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₄²×C₅₃

SD₁₆

yes

complemented

C₄²×C₅₃

C₂×D₄

yes

—

#11719923

—

C₄⋊Dic₅₃

—

C₂×#88322

C₂×D₄

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²×D₄

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#127

yes

—

C₂²×C₅₃

C₂²×SD₁₆

yes

—

C₄×C₅₃

#492

yes

complemented

C₄×C₅₃

C₂×#127

yes

—

Dic₅₃

—

C₂×#492

—

#2868

—

#2867

D₅₃

yes

characteristic

C₄²×C₈

—

C₂×C₅₃

—

C₂×C₅₃

#2867

yes

characteristic

C₂×C₅₃

C₂×#492

yes

—

C₂²×SD₁₆

—

C₂×C₄⋊Q₈

—

C₂×#127

—

C₂×#127

C₂×D₅₃

yes

—

#492

—

106

—

C₂×C₄×C₈

—

#384

—

106

—

C₄³

C₂×D₅₃

yes

characteristic

C₅₃

#33300

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

C₂²×D₅₃

yes

—

C₂×SD₁₆

—

106

—

C₂²×C₈

—

C₄⋊Q₈

—

106

—

#127

—

#127

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂²⋊D₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

—

C₄×C₈

—

C₂⁴

—

C₂×Q₈

—

106

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂³×D₅₃

yes

characteristic

SD₁₆

—

212

—

C₂×C₈

—

C₄⋊C₄

—

212

—

C₄²

#11719923

yes

complemented

C₄²

D₄×D₅₃

yes

—

C₂³

—

C₂³

—

100

C₂³

C₂²×C₂²⋊D₅₃

yes

characteristic

101

Q₈

—

212

—

102

D₄

—

103

D₄

—

104

C₂×C₄

—

212

—

105

C₂×C₄

#13133213

yes

—

106

C₂×C₄

C₂×#11719923

yes

—

107

C₂×C₄

C₂×D₄×D₅₃

yes

—

108

C₈

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂×#13133213

yes

—

112

C₂²

C₂²×#11719923

yes

—

113

C₄

—

424

—

114

C₄

#16810995

yes

complemented

115

C₄

#127×D₅₃

yes

—

116

C₂

—

117

C₂

#20283109

yes

characteristic

118

C₂

C₂×#16810995

yes

—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₅₃×#2867	C₂
socle	C₂³×C₅₃	C₂²×D₄
hypercenter	#2867	D₅₃
hypocenter	C₅₃	#33300
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2867

subgroup

G/H

center

C₂³

C₂²×C₂²⋊D₅₃

commutator, cosocle

C₂²×C₄×C₅₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₅₃

Fitting, prime radical

C₅₃×#2867

C₂

socle

C₂³×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2867

N	H ≅ G/N	type
1. C₂×#16810995	C₂	semidirect
2. #20283113	C₂	semidirect
3. C₄²×#88322	C₂	semidirect
18. C₄×#88322	D₄	semidirect
33. C₄²×C₅₃	SD₁₆	semidirect
48. #88322	#127	semidirect
54. C₄×C₅₃	#492	semidirect
72. C₅₃	#33300	semidirect
96. C₄²	#11719923	semidirect
114. C₄	#16810995	semidirect

H ≅ G/N

type

1. C₂×#16810995

C₂

semidirect

2. #20283113

C₂

semidirect

3. C₄²×#88322

C₂

semidirect

18. C₄×#88322

D₄

semidirect

33. C₄²×C₅₃

SD₁₆

semidirect

48. #88322

#127

semidirect

54. C₄×C₅₃

#492

semidirect

72. C₅₃

#33300

semidirect

96. C₄²

#11719923

semidirect

114. C₄

#16810995

semidirect

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

subgroup	index
C₂×#16810995	2
#20283113	2
C₄²×#88322	2
C₅₃×#2867	2
#33300	53

subgroup

index

C₂×#16810995

#20283113

C₄²×#88322

C₅₃×#2867

#33300

Subgroups — #27536606

Conjugacy classes of subgroups

Complements

Maximal subgroups