Group #27298759 — subgroups

Subgroups — #27298759

Order 13568.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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	#20276997	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276938	C₂	1	2	yes	maximal, complemented
3	#20274304	C₂	1	2	yes	maximal, complemented
4	#20272631	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2246	C₂	1	1	yes	maximal, characteristic
6	#16811028	C₂²	1	2	yes	—
7	#16810995	—	2	2	no	—
8	#16810969	—	2	2	no	—
9	#16810451	—	2	2	no	—
10	#16810424	—	2	4	no	—
11	C₄×#11719908	C₂²	1	1	yes	characteristic
12	C₈×#88322	C₂²	1	1	yes	characteristic
13	C₂×D₁₆×C₅₃	—	2	2	no	—
14	C₅₃×#493	C₂²	1	2	yes	—
15	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#11719923	—	4	4	no	—
17	#13133217	—	2	2	no	—
18	C₂×#11719908	D₄	1	2	yes	—
19	#13133151	—	2	2	no	—
20	#13133151	—	2	2	no	—
21	#13133138	—	4	4	no	—
22	C₄×#88322	D₄	1	2	yes	—
23	C₂×D₈×C₅₃	—	2	2	no	—
24	C₂×D₈×C₅₃	—	2	2	no	—
25	#127×C₅₃	—	2	2	no	—
26	D₁₆×C₅₃	—	4	4	no	—
27	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×D₄×C₅₃	—	4	2	no	—
30	C₂×D₄×C₅₃	—	4	2	no	—
31	C₂×D₄×C₅₃	—	4	2	no	—
32	C₂×Dic₁₀₆	—	4	2	no	—
33	D₈×C₅₃	—	4	4	no	—
34	D₈×C₅₃	—	4	4	no	—
35	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
38	#11719923	—	8	8	no	—
39	C₄⋊Dic₅₃	—	4	2	no	—
40	C₄⋊Dic₅₃	—	4	2	no	—
41	C₂×#88322	D₈	1	4	yes	—
42	C₁₆×C₅₃	SD₁₆	1	4	yes	complemented
43	#11719908	—	2	2	no	—
44	C₂³×C₅₃	—	8	2	no	—
45	D₄×C₅₃	—	8	2	no	—
46	D₄×C₅₃	—	8	2	no	—
47	D₄×C₅₃	—	8	4	no	—
48	D₄×C₅₃	—	8	4	no	—
49	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
51	C₂×Dic₅₃	—	8	2	no	—
52	Dic₁₀₆	—	8	4	no	—
53	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
54	C₈×C₅₃	#135	1	2	yes	—
55	#88322	—	2	4	no	—
56	#12506	—	53	1	no	maximal, Sylow
57	C₂²×C₅₃	—	16	2	no	—
58	C₂²×C₅₃	—	16	2	no	—
59	C₂²×C₅₃	—	16	2	no	—
60	C₂²×C₅₃	#493	1	1	yes	characteristic
61	C₄×C₅₃	#492	1	1	yes	characteristic
62	C₄×C₅₃	#495	1	1	yes	characteristic
63	C₄×C₅₃	#509	1	4	yes	—
64	Dic₅₃	—	16	2	no	—
65	#2254	—	53	1	no	—
66	#2246	D₅₃	1	1	yes	characteristic
67	#2240	—	53	2	no	—
68	#1706	—	53	2	no	—
69	#1311	—	53	1	no	—
70	C₂×C₅₃	—	32	2	no	—
71	C₂×C₅₃	#1706	1	1	yes	characteristic
72	C₂×C₅₃	#2250	1	2	yes	—
73	C₂×D₁₆	—	2	2	no	—
74	#500	—	53	2	no	—
75	#493	C₂×D₅₃	1	2	yes	—
76	#492	—	106	2	no	—
77	#486	—	106	2	no	—
78	#367	—	106	2	no	—
79	#357	—	106	4	no	—
80	C₄×C₁₆	—	53	1	no	—
81	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
82	C₈²	—	53	1	no	—
83	C₅₃	#12506	1	1	yes	characteristic, complemented, Sylow
84	C₂×SD₁₆	—	212	4	no	—
85	C₂×D₈	—	2	2	no	—
86	C₂×D₈	—	2	2	no	—
87	C₄⋊Q₈	—	106	2	no	—
88	#127	—	2	2	no	—
89	D₁₆	—	4	4	no	—
90	C₂×C₁₆	—	53	2	no	—
91	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
92	#107	—	106	2	no	—
93	#107	—	106	2	no	—
94	#102	—	212	4	no	—
95	C₄×C₈	—	53	2	no	—
96	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
97	C₂×Q₈	—	212	2	no	—
98	C₂×D₄	—	4	2	no	—
99	C₂×D₄	—	4	2	no	—
100	C₂×D₄	—	4	2	no	—
101	SD₁₆	—	424	8	no	—
102	D₈	—	4	4	no	—
103	D₈	—	4	4	no	—
104	C₂×C₈	—	53	4	no	—
105	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
106	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
107	C₄⋊C₄	—	212	2	no	—
108	C₄⋊C₄	—	212	2	no	—
109	C₄²	D₄×D₅₃	1	1	yes	characteristic
110	C₁₆	—	106	2	no	—
111	C₁₆	#11719923	1	4	yes	complemented
112	C₂³	—	8	2	no	—
113	Q₈	—	424	4	no	—
114	D₄	—	8	4	no	—
115	D₄	—	8	4	no	—
116	D₄	—	8	2	no	—
117	D₄	—	8	2	no	—
118	C₂×C₄	—	424	2	no	—
119	C₂×C₄	#13133213	1	1	yes	characteristic
120	C₂×C₄	D₈×D₅₃	1	2	yes	—
121	C₈	—	106	4	no	—
122	C₈	C₂×#11719923	1	2	yes	—
123	C₈	#13133236	1	2	yes	—
124	C₂²	—	16	2	no	—
125	C₂²	—	16	2	no	—
126	C₂²	—	16	2	no	—
127	C₂²	#16810998	1	1	yes	characteristic
128	C₄	—	848	2	no	—
129	C₄	#16810995	1	1	yes	characteristic
130	C₄	#16811006	1	1	yes	characteristic
131	C₄	#16811057	1	4	yes	—
132	C₂	—	32	2	no	—
133	C₂	#20274304	1	1	yes	characteristic
134	C₂	#20276978	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276997

C₂

yes

maximal, characteristic, complemented

#20276938

C₂

yes

maximal, complemented

#20274304

C₂

yes

maximal, complemented

#20272631

C₂

yes

maximal, characteristic, complemented

C₅₃×#2246

C₂

yes

maximal, characteristic

#16811028

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₈×#88322

C₂²

yes

characteristic

C₂×D₁₆×C₅₃

—

C₅₃×#493

C₂²

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

—

—

D₄

yes

—

—

—

—

D₄

yes

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

#127×C₅₃

—

D₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×Dic₁₀₆

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

#11719923

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

D₈

yes

—

C₁₆×C₅₃

SD₁₆

yes

complemented

#11719908

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

C₂×SD₁₆

yes

—

C₈×C₅₃

#135

yes

—

#88322

—

#12506

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

#2254

—

#2246

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#1706

yes

characteristic

C₂×C₅₃

#2250

yes

—

C₂×D₁₆

—

#500

—

#493

C₂×D₅₃

yes

—

#492

—

106

—

#486

—

106

—

#367

—

106

—

#357

—

106

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₈²

—

C₅₃

#12506

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

212

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

106

—

#127

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅₃

yes

—

#107

—

106

—

#107

—

106

—

#102

—

212

—

C₄×C₈

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

212

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

SD₁₆

—

424

—

102

D₈

—

103

D₈

—

104

C₂×C₈

—

105

C₂×C₈

D₄×D₅₃

yes

characteristic

106

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

107

C₄⋊C₄

—

212

—

108

C₄⋊C₄

—

212

—

109

C₄²

D₄×D₅₃

yes

characteristic

110

C₁₆

—

106

—

111

C₁₆

#11719923

yes

complemented

112

C₂³

—

113

Q₈

—

424

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

424

—

119

C₂×C₄

#13133213

yes

characteristic

120

C₂×C₄

D₈×D₅₃

yes

—

121

C₈

—

106

—

122

C₈

C₂×#11719923

yes

—

123

C₈

#13133236

yes

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

#16810998

yes

characteristic

128

C₄

—

848

—

129

C₄

#16810995

yes

characteristic

130

C₄

#16811006

yes

characteristic

131

C₄

#16811057

yes

—

132

C₂

—

133

C₂

#20274304

yes

characteristic

134

C₂

#20276978

yes

—

	subgroup	G/H
center	C₂²	#16810998
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting, prime radical	C₅₃×#2246	C₂
socle	C₂²×C₅₃	#493
hypercenter	#2246	D₅₃
hypocenter	C₅₃	#12506
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#16810998

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting, prime radical

C₅₃×#2246

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. #20276997	C₂	semidirect
2. #20276938	C₂	semidirect
3. #20274304	C₂	semidirect
4. #20272631	C₂	semidirect
42. C₁₆×C₅₃	SD₁₆	semidirect
83. C₅₃	#12506	semidirect
111. C₁₆	#11719923	semidirect

H ≅ G/N

type

1. #20276997

C₂

semidirect

2. #20276938

C₂

semidirect

3. #20274304

C₂

semidirect

4. #20272631

C₂

semidirect

42. C₁₆×C₅₃

SD₁₆

semidirect

83. C₅₃

#12506

semidirect

111. C₁₆

#11719923

semidirect

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

subgroup	index
#20276997	2
#20276938	2
#20274304	2
#20272631	2
C₅₃×#2246	2
#12506	53

subgroup

index

C₅₃×#2246

Subgroups — #27298759

Conjugacy classes of subgroups

Complements

Maximal subgroups