Group #27298821 — subgroups

Subgroups — #27298821

Order 13568.

360 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20276972	C₂	1	2	yes	maximal, complemented
2	#20276964	C₂	1	2	yes	maximal, complemented
3	#20274330	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272632	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1709	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133160	—	2	4	no	—
7	C₂×#13133158	—	2	4	no	—
8	#16811000	—	2	2	no	—
9	#16811000	C₂²	1	2	yes	—
10	C₄×#11719908	C₂²	1	2	yes	—
11	C₅₃×#494	C₂²	1	1	yes	characteristic
12	C₅₃×#493	C₂²	1	1	yes	characteristic
13	C₅₃×#486	—	2	2	no	—
14	C₈²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×D₄₂₄	—	2	2	no	—
16	C₂×D₄₂₄	—	2	2	no	—
17	C₂×D₄₂₄	—	4	4	no	—
18	#13133214	—	2	2	no	—
19	#13133160	—	4	8	no	—
20	#13133158	—	4	8	no	—
21	C₂×#11719908	D₄	1	4	yes	—
22	C₂×Q₁₆×C₅₃	—	2	2	no	—
23	C₂×D₈×C₅₃	—	2	2	no	—
24	C₄⋊Q₈×C₅₃	—	2	1	no	—
25	#127×C₅₃	—	2	1	no	—
26	#102×C₅₃	—	4	4	no	—
27	C₄×C₈×C₅₃	D₄	1	2	yes	—
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₅₃	—	4	1	no	—
30	C₂×D₄×C₅₃	—	4	1	no	—
31	C₂×D₄×C₅₃	—	4	2	no	—
32	C₂×D₂₁₂	—	4	2	no	—
33	C₂×D₂₁₂	—	4	4	no	—
34	Q₁₆×C₅₃	—	4	4	no	—
35	D₈×C₅₃	—	4	4	no	—
36	C₂×C₈×C₅₃	D₈	1	4	yes	—
37	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
38	C₄⋊C₄×C₅₃	—	4	2	no	—
39	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
40	D₄₂₄	—	4	4	no	—
41	D₄₂₄	—	4	4	no	—
42	D₄₂₄	—	8	8	no	—
43	#11719908	D₈	1	8	yes	complemented
44	C₂³×C₅₃	—	8	1	no	—
45	C₂²×D₅₃	—	8	2	no	—
46	Q₈×C₅₃	—	8	2	no	—
47	D₄×C₅₃	—	8	2	no	—
48	D₄×C₅₃	—	8	4	no	—
49	C₂×C₄×C₅₃	—	8	1	no	—
50	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
52	D₂₁₂	—	8	2	no	—
53	D₂₁₂	—	8	2	no	—
54	D₂₁₂	—	8	8	no	—
55	C₈×C₅₃	—	2	4	no	—
56	C₈×C₅₃	C₂×D₈	1	4	yes	—
57	#12519	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	—	16	1	no	—
59	C₂²×C₅₃	—	16	2	no	—
60	C₂²×C₅₃	#493	1	1	yes	characteristic
61	C₂×D₅₃	—	16	2	no	—
62	C₂×D₅₃	—	16	4	no	—
63	C₄×C₅₃	—	16	1	no	—
64	C₄×C₅₃	#493	1	2	yes	—
65	C₄×C₅₃	#509	1	4	yes	—
66	#2248	—	53	2	no	—
67	#2246	—	53	2	no	—
68	#1712	—	53	1	no	—
69	#1709	D₅₃	1	1	yes	characteristic, complemented
70	#1311	—	53	1	no	—
71	C₂×C₅₃	—	32	1	no	—
72	C₂×C₅₃	#1712	1	1	yes	characteristic
73	C₂×C₅₃	#2250	1	2	yes	—
74	D₅₃	—	32	2	no	—
75	C₂×SD₃₂	—	106	4	no	—
76	C₂×D₁₆	—	106	4	no	—
77	#494	C₂×D₅₃	1	1	yes	characteristic
78	#493	—	53	2	no	—
79	#493	—	106	2	no	—
80	#493	C₂×D₅₃	1	1	yes	characteristic
81	#486	—	2	2	no	—
82	C₄×C₁₆	—	53	2	no	—
83	C₈²	C₂×D₅₃	1	1	yes	characteristic
84	C₅₃	#12519	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	—	2	2	no	—
86	C₂×D₈	—	2	2	no	—
87	C₂×D₈	—	106	2	no	—
88	C₂×D₈	—	106	2	no	—
89	C₂×D₈	—	212	4	no	—
90	C₄⋊Q₈	—	2	1	no	—
91	#127	—	2	1	no	—
92	#127	—	106	2	no	—
93	SD₃₂	—	212	8	no	—
94	D₁₆	—	212	8	no	—
95	C₂×C₁₆	—	53	4	no	—
96	#102	—	4	4	no	—
97	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
98	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
99	C₂×Q₈	—	4	1	no	—
100	C₂×D₄	—	4	1	no	—
101	C₂×D₄	—	4	2	no	—
102	C₂×D₄	—	212	2	no	—
103	C₂×D₄	—	212	4	no	—
104	Q₁₆	—	4	4	no	—
105	D₈	—	4	4	no	—
106	D₈	—	424	8	no	—
107	D₈	—	212	4	no	—
108	D₈	—	212	4	no	—
109	C₂×C₈	#11719922	1	4	yes	—
110	C₂×C₈	D₄×D₅₃	1	2	yes	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
113	C₁₆	—	53	8	no	—
114	C₂³	—	8	1	no	—
115	C₂³	—	424	2	no	—
116	Q₈	—	8	2	no	—
117	D₄	—	8	4	no	—
118	D₄	—	8	2	no	—
119	D₄	—	424	8	no	—
120	D₄	—	424	2	no	—
121	D₄	—	424	2	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	#13133213	1	1	yes	characteristic
124	C₂×C₄	C₂×#11719922	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	D₈×D₅₃	1	4	yes	—
127	C₂²	—	16	1	no	—
128	C₂²	—	16	2	no	—
129	C₂²	—	848	2	no	—
130	C₂²	—	848	4	no	—
131	C₂²	#16810999	1	1	yes	characteristic
132	C₄	—	16	1	no	—
133	C₄	#16810998	1	2	yes	—
134	C₄	#16811058	1	4	yes	—
135	C₂	—	32	1	no	—
136	C₂	—	1696	2	no	—
137	C₂	#20274329	1	1	yes	characteristic
138	C₂	#20276982	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276972

C₂

yes

maximal, complemented

#20276964

C₂

yes

maximal, complemented

#20274330

C₂

yes

maximal, characteristic, complemented

#20272632

C₂

yes

maximal, characteristic, complemented

C₅₃×#1709

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

—

C₄×#11719908

C₂²

yes

—

C₅₃×#494

C₂²

yes

characteristic

C₅₃×#493

C₂²

yes

characteristic

C₅₃×#486

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

—

—

—

D₄

yes

—

C₂×Q₁₆×C₅₃

—

C₂×D₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

#127×C₅₃

—

#102×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

Q₁₆×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

D₄₂₄

—

D₄₂₄

—

D₄₂₄

—

#11719908

D₈

yes

complemented

C₂³×C₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₈

yes

—

#12519

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#493

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#493

yes

—

C₄×C₅₃

#509

yes

—

—

—

—

D₅₃

yes

characteristic, complemented

#1311

—

C₂×C₅₃

—

C₂×C₅₃

#1712

yes

characteristic

C₂×C₅₃

#2250

yes

—

D₅₃

—

C₂×SD₃₂

—

106

—

C₂×D₁₆

—

106

—

#494

C₂×D₅₃

yes

characteristic

#493

—

#493

—

106

—

#493

C₂×D₅₃

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12519

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

106

—

C₂×D₈

—

106

—

C₂×D₈

—

212

—

C₄⋊Q₈

—

#127

—

#127

—

106

—

SD₃₂

—

212

—

D₁₆

—

212

—

C₂×C₁₆

—

#102

—

C₄×C₈

C₂²⋊D₅₃

yes

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

212

—

103

C₂×D₄

—

212

—

104

Q₁₆

—

105

D₈

—

106

D₈

—

424

—

107

D₈

—

212

—

108

D₈

—

212

—

109

C₂×C₈

#11719922

yes

—

110

C₂×C₈

D₄×D₅₃

yes

—

111

C₄⋊C₄

—

112

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

113

C₁₆

—

114

C₂³

—

115

C₂³

—

424

—

116

Q₈

—

117

D₄

—

118

D₄

—

119

D₄

—

424

—

120

D₄

—

424

—

121

D₄

—

424

—

122

C₂×C₄

—

123

C₂×C₄

#13133213

yes

characteristic

124

C₂×C₄

C₂×#11719922

yes

—

125

C₈

—

126

C₈

D₈×D₅₃

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

848

—

130

C₂²

—

848

—

131

C₂²

#16810999

yes

characteristic

132

C₄

—

133

C₄

#16810998

yes

—

134

C₄

#16811058

yes

—

135

C₂

—

136

C₂

—

1696

—

137

C₂

#20274329

yes

characteristic

138

C₂

#20276982

yes

—

	subgroup	G/H
center	C₂²	#16810999
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1709	C₂
socle	C₂²×C₅₃	#493
hypercenter	#1709	D₅₃
hypocenter	C₅₃	#12519
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#16810999

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1709

C₂

C₂²×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #20276972	C₂	semidirect
2. #20276964	C₂	semidirect
3. #20274330	C₂	semidirect
4. #20272632	C₂	semidirect
5. C₅₃×#1709	C₂	semidirect
43. #11719908	D₈	semidirect
69. #1709	D₅₃	semidirect
84. C₅₃	#12519	semidirect

H ≅ G/N

type

1. #20276972

C₂

semidirect

2. #20276964

C₂

semidirect

3. #20274330

C₂

semidirect

4. #20272632

C₂

semidirect

5. C₅₃×#1709

C₂

semidirect

43. #11719908

D₈

semidirect

69. #1709

D₅₃

semidirect

84. C₅₃

#12519

semidirect

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

subgroup	index
#20276972	2
#20276964	2
#20274330	2
#20272632	2
C₅₃×#1709	2
#12519	53

subgroup

index

C₅₃×#1709

Subgroups — #27298821

Conjugacy classes of subgroups

Complements

Maximal subgroups