Group #27304578 — subgroups

Subgroups — #27304578

Order 13568.

206 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 147 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×#16810457	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133123	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13133123	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic
5	#16810457	—	2	2	no	—
6	C₂×#13133123	—	2	2	no	—
7	C₂×#13133123	C₂²	1	2	yes	—
8	C₂×#13133123	C₂²	1	2	yes	—
9	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#405	C₄	1	1	yes	characteristic
12	C₅₃×#346	C₄	1	2	yes	—
13	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
14	#13133123	—	2	2	no	—
15	#13133123	—	2	2	no	—
16	C₂×M₁₆×C₅₃	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	#117×C₅₃	C₈	1	1	yes	characteristic
19	M₃₂×C₅₃	—	2	2	no	—
20	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
23	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
24	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
25	#97×C₅₃	C₈	1	2	yes	—
26	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
28	M₁₆×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
33	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₄⋊C₄×C₅₃	C₁₆	1	2	yes	—
35	C₂²⋊C₄×C₅₃	C₁₆	1	2	yes	—
36	C₄²×C₅₃	C₂×C₈	1	2	yes	—
37	C₁₆×C₅₃	—	2	2	no	—
38	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
44	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
46	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
47	C₈×C₅₃	—	2	2	no	—
48	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
49	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
51	#13628	—	53	1	no	maximal, Sylow
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
54	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
55	C₄×C₅₃	—	2	4	no	—
56	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
57	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
58	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
59	C₂×M₆₄	—	53	1	no	—
60	C₂²×C₃₂	—	53	1	no	—
61	#2108	D₅₃	1	1	yes	characteristic
62	C₄⋊C₃₂	—	53	4	no	—
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
65	C₂×C₅₃	#2258	1	2	yes	—
66	C₂×M₃₂	Dic₅₃	1	1	yes	characteristic
67	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
68	#405	Dic₅₃	1	1	yes	characteristic
69	M₆₄	—	106	2	no	—
70	C₂×C₃₂	—	53	2	no	—
71	C₂×C₃₂	—	53	2	no	—
72	C₂×C₃₂	—	106	2	no	—
73	#346	Dic₅₃	1	2	yes	—
74	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
75	C₅₃	#13628	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	#88322	1	1	yes	characteristic
77	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
78	#117	#88322	1	1	yes	characteristic
79	M₃₂	—	2	2	no	—
80	C₂×C₁₆	Dic₁₀₆	1	2	yes	—
81	C₂×C₁₆	D₂₁₂	1	2	yes	—
82	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
83	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
84	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
85	#97	#88322	1	2	yes	—
86	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
87	C₃₂	—	106	2	no	—
88	C₃₂	—	106	2	no	—
89	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	C₂×C₈	C₂×#88322	1	2	yes	—
92	C₂×C₈	C₂×#88322	1	1	yes	characteristic
93	C₂×C₈	#11719917	1	2	yes	—
94	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
95	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
96	C₄⋊C₄	#11719908	1	2	yes	—
97	C₂²⋊C₄	#11719908	1	2	yes	—
98	C₄²	C₂×#88322	1	2	yes	—
99	C₁₆	—	2	2	no	—
100	C₁₆	C₄⋊Dic₅₃	1	2	yes	—
101	C₁₆	C₂×Dic₁₀₆	1	1	yes	characteristic
102	C₁₆	C₂×D₂₁₂	1	1	yes	characteristic
103	C₂³	C₂×#11719908	1	1	yes	characteristic
104	C₂×C₄	C₄⋊#88322	1	2	yes	—
105	C₂×C₄	C₂×#11719908	1	4	yes	—
106	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
107	C₂×C₄	#13133157	1	2	yes	—
108	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	C₄⋊#88322	1	2	yes	—
111	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
112	C₈	C₂×#11719917	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	C₄⋊#11719908	1	2	yes	—
115	C₂²	C₂²×#11719908	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	C₄⋊#11719908	1	2	yes	—
118	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
119	C₄	C₂×#13133157	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₄⋊#11719908	1	1	yes	characteristic
122	C₂	#20277009	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810457

C₂

yes

maximal, characteristic

C₂²×#13133123

C₂

yes

maximal, characteristic

C₄⋊#13133123

C₂

yes

maximal, complemented

C₅₃×#2108

C₂

yes

maximal, characteristic

#16810457

—

C₂×#13133123

—

C₂×#13133123

C₂²

yes

—

C₂×#13133123

C₂²

yes

—

C₂×M₃₂×C₅₃

C₄

yes

characteristic

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₅₃×#405

C₄

yes

characteristic

C₅₃×#346

C₄

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

—

#13133123

—

#13133123

—

C₂×M₁₆×C₅₃

C₈

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

#117×C₅₃

C₈

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

D₄

yes

—

C₂×C₁₆×C₅₃

Q₈

yes

—

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

#97×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₁₆

yes

—

C₂²⋊C₄×C₅₃

C₁₆

yes

—

C₄²×C₅₃

C₂×C₈

yes

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₄⋊C₄

yes

—

C₁₆×C₅₃

C₂×D₄

yes

characteristic

C₁₆×C₅₃

C₂×Q₈

yes

characteristic

C₂³×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

M₃₂

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

characteristic

#13628

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊C₁₆

yes

—

C₂²×C₅₃

C₂²×C₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄⋊C₁₆

yes

—

C₄×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅₃

C₂×M₃₂

yes

characteristic

C₂×M₆₄

—

C₂²×C₃₂

—

#2108

D₅₃

yes

characteristic

C₄⋊C₃₂

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₅₃

#2258

yes

—

C₂×M₃₂

Dic₅₃

yes

characteristic

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

#405

Dic₅₃

yes

characteristic

M₆₄

—

106

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

106

—

#346

Dic₅₃

yes

—

C₄×C₁₆

C₂×D₅₃

yes

—

C₅₃

#13628

yes

characteristic, complemented, Sylow

C₂×M₁₆

#88322

yes

characteristic

C₂²×C₈

C₂×Dic₅₃

yes

yes

M₃₂

—

C₂×C₁₆

Dic₁₀₆

yes

—

C₂×C₁₆

D₂₁₂

yes

—

C₂×C₁₆

C₂×Dic₅₃

yes

—

C₂×C₁₆

C₂×Dic₅₃

yes

characteristic

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

#97

#88322

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₃₂

—

106

—

C₃₂

—

106

—

C₂²×C₄

C₂×#88322

yes

characteristic

M₁₆

—

C₂×C₈

C₂×#88322

yes

—

C₂×C₈

C₂×#88322

yes

characteristic

C₂×C₈

#11719917

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

C₄⋊C₄

#11719908

yes

—

C₂²⋊C₄

#11719908

yes

—

C₄²

C₂×#88322

yes

—

C₁₆

—

100

C₁₆

C₄⋊Dic₅₃

yes

—

101

C₁₆

C₂×Dic₁₀₆

yes

characteristic

102

C₁₆

C₂×D₂₁₂

yes

characteristic

103

C₂³

C₂×#11719908

yes

characteristic

104

C₂×C₄

C₄⋊#88322

yes

—

105

C₂×C₄

C₂×#11719908

yes

—

106

C₂×C₄

C₂×#11719908

yes

characteristic

107

C₂×C₄

#13133157

yes

—

108

C₂×C₄

C₂²×#88322

yes

characteristic

109

C₈

—

110

C₈

C₄⋊#88322

yes

—

111

C₈

C₂×C₄⋊Dic₅₃

yes

characteristic

112

C₈

C₂×#11719917

yes

characteristic

113

C₂²

—

114

C₂²

C₄⋊#11719908

yes

—

115

C₂²

C₂²×#11719908

yes

characteristic

116

C₄

—

117

C₄

C₄⋊#11719908

yes

—

118

C₄

C₂×C₄⋊#88322

yes

characteristic

119

C₄

C₂×#13133157

yes

characteristic

120

C₂

—

121

C₂

C₂×C₄⋊#11719908

yes

characteristic

122

C₂

#20277009

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₅₃×#2108	C₂
prime radical	C₂³×C₅₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	#2108	D₅₃
hypocenter	C₅₃	#13628
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ #2108

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂²×C₁₆

Fitting

C₅₃×#2108

C₂

prime radical

C₂³×C₅₃

C₂×C₁₆

cosocle

C₂×C₁₆×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₁₆ ◁ #2108

N	H ≅ G/N	type
3. C₄⋊#13133123	C₂	semidirect
75. C₅₃	#13628	semidirect

H ≅ G/N

type

3. C₄⋊#13133123

C₂

semidirect

75. C₅₃

#13628

semidirect

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

subgroup	index
C₂×#16810457	2
C₂²×#13133123	2
C₄⋊#13133123	2
C₅₃×#2108	2
#13628	53

subgroup

index

C₅₃×#2108

Subgroups — #27304578

Conjugacy classes of subgroups

Complements

Maximal subgroups