Group #28298679 — subgroups

Subgroups — #28298679

Order 13568.

218 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 66 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×#16811515	C₂	1	2	yes	maximal
2	#20274850	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133291	—	2	2	no	—
4	#16811515	—	2	2	no	—
5	#16811515	C₄	1	4	yes	complemented
6	C₂×#13133128	C₂²	1	1	yes	characteristic
7	#16810489	C₄	1	1	yes	characteristic
8	#16810385	—	2	2	no	—
9	C₅₃×#384	C₄	1	1	yes	characteristic
10	#13133291	—	2	4	no	—
11	#13133291	—	2	4	no	—
12	C₂×#11719910	—	2	4	no	—
13	C₂²×#88322	D₄	1	2	yes	—
14	C₂×C₄⋊Dic₅₃	—	2	1	no	—
15	#13133128	C₂×C₄	1	2	yes	—
16	#13133128	D₄	1	1	yes	characteristic
17	#13133128	Q₈	1	1	yes	characteristic
18	#13133124	—	4	1	no	—
19	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	—	2	1	no	—
23	C₂²×C₄×C₅₃	—	4	1	no	—
24	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×Dic₅₃	—	4	1	no	—
26	C₄⋊C₄×C₅₃	—	4	2	no	—
27	C₄⋊C₄×C₅₃	—	8	2	no	—
28	C₄²×C₅₃	—	2	1	no	—
29	C₄²×C₅₃	C₄²	1	1	yes	characteristic
30	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₄⋊Dic₅₃	—	4	2	no	—
32	C₂×#88322	—	2	2	no	—
33	C₂×#88322	—	2	2	no	—
34	C₂×#88322	C₂²⋊C₄	1	2	yes	—
35	C₂×#88322	C₄⋊C₄	1	2	yes	—
36	#11719910	—	2	8	no	—
37	C₂³×C₅₃	#100	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	4	1	no	—
41	C₂×C₄×C₅₃	—	4	1	no	—
42	C₂×C₄×C₅₃	—	8	2	no	—
43	C₂×C₄×C₅₃	—	8	1	no	—
44	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#104	1	2	yes	—
47	C₂×C₄×C₅₃	#104	1	2	yes	—
48	C₂×Dic₅₃	—	8	1	no	—
49	C₂×Dic₅₃	—	8	1	no	—
50	C₂×Dic₅₃	—	8	1	no	—
51	#88322	—	2	2	no	—
52	#88322	—	2	2	no	—
53	#7266	—	53	1	no	maximal, Sylow
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	#329	1	1	yes	characteristic
57	C₂²×C₅₃	#329	1	1	yes	characteristic
58	C₂²×C₅₃	#343	1	1	yes	characteristic
59	C₄×C₅₃	—	4	1	no	—
60	C₄×C₅₃	—	4	1	no	—
61	C₄×C₅₃	—	16	1	no	—
62	C₄×C₅₃	#330	1	2	yes	—
63	C₄×C₅₃	#339	1	2	yes	—
64	Dic₅₃	—	16	1	no	—
65	C₂×#347	—	53	2	no	—
66	#1836	—	53	1	no	—
67	C₂×C₅₃	—	2	1	no	—
68	C₂×C₅₃	—	2	1	no	—
69	C₂×C₅₃	#1295	1	1	yes	characteristic
70	C₂×C₅₃	#1355	1	1	yes	characteristic
71	C₂×C₅₃	#1356	1	1	yes	characteristic
72	C₂×M₃₂	—	106	2	no	—
73	C₂×#97	—	53	1	no	—
74	#384	#6157	1	1	yes	characteristic
75	#382	—	53	1	no	—
76	#347	—	53	4	no	—
77	#347	—	106	2	no	—
78	#340	—	106	2	no	—
79	C₅₃	#7266	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	—	53	2	no	—
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	—	4	1	no	—
83	C₂×C₄⋊C₄	—	106	1	no	—
84	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
85	M₃₂	—	106	4	no	—
86	M₃₂	—	106	4	no	—
87	C₂×C₁₆	—	106	4	no	—
88	#97	—	53	2	no	—
89	#97	—	53	1	no	—
90	#97	—	53	1	no	—
91	#95	—	212	1	no	—
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	—	212	1	no	—
95	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
96	C₂×C₈	—	53	2	no	—
97	C₂×C₈	—	53	2	no	—
98	C₂×C₈	—	106	2	no	—
99	C₂×C₈	—	106	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	8	2	no	—
102	C₄⋊C₄	—	212	2	no	—
103	C₄²	—	2	1	no	—
104	C₄²	C₄×#6157	1	1	yes	characteristic
105	C₄²	C₄⋊#6157	1	1	yes	characteristic
106	C₁₆	—	106	8	no	—
107	C₂³	#13133277	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	8	2	no	—
113	C₂×C₄	—	8	1	no	—
114	C₂×C₄	—	424	1	no	—
115	C₂×C₄	—	424	1	no	—
116	C₂×C₄	—	424	1	no	—
117	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
118	C₂×C₄	#13133278	1	1	yes	characteristic
119	C₂×C₄	#13133281	1	2	yes	—
120	C₂×C₄	#13133282	1	2	yes	—
121	C₈	—	106	2	no	—
122	C₈	—	106	2	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	1	no	—
125	C₂²	#16811478	1	1	yes	characteristic
126	C₂²	#16811479	1	1	yes	characteristic
127	C₂²	#16811503	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	16	1	no	—
131	C₄	—	848	1	no	—
132	C₄	#16811480	1	2	yes	—
133	C₄	#16811493	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#20289591	1	1	yes	characteristic
137	C₂	#20289717	1	1	yes	characteristic
138	C₂	#20289720	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811515

C₂

yes

maximal

#20274850

C₂

yes

maximal, characteristic

C₂×#13133291

—

#16811515

—

#16811515

C₄

yes

complemented

C₂×#13133128

C₂²

yes

characteristic

#16810489

C₄

yes

characteristic

#16810385

—

C₅₃×#384

C₄

yes

—

—

—

C₂²×#88322

D₄

yes

—

C₂×C₄⋊Dic₅₃

—

#13133128

C₂×C₄

yes

—

#13133128

D₄

yes

characteristic

#13133128

Q₈

yes

characteristic

#13133124

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂²⋊C₄

yes

—

C₂×#88322

C₄⋊C₄

yes

—

#11719910

—

C₂³×C₅₃

#100

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#104

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#330

yes

—

C₄×C₅₃

#339

yes

—

Dic₅₃

—

C₂×#347

—

#1836

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×M₃₂

—

106

—

C₂×#97

—

yes

—

—

—

106

—

#340

—

106

—

C₅₃

#7266

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂×#6157

yes

characteristic

M₃₂

—

106

—

M₃₂

—

106

—

C₂×C₁₆

—

106

—

—

—

—

—

212

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂²⋊#6157

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

212

—

103

C₄²

—

104

C₄²

C₄×#6157

yes

characteristic

105

C₄²

C₄⋊#6157

yes

characteristic

106

C₁₆

—

106

—

107

C₂³

#13133277

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

424

—

115

C₂×C₄

—

424

—

116

C₂×C₄

—

424

—

117

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

118

C₂×C₄

#13133278

yes

characteristic

119

C₂×C₄

#13133281

yes

—

120

C₂×C₄

#13133282

yes

—

121

C₈

—

106

—

122

C₈

—

106

—

123

C₂²

—

124

C₂²

—

125

C₂²

#16811478

yes

characteristic

126

C₂²

#16811479

yes

characteristic

127

C₂²

#16811503

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

848

—

132

C₄

#16811480

yes

—

133

C₄

#16811493

yes

—

134

C₂

—

135

C₂

—

136

C₂

#20289591

yes

characteristic

137

C₂

#20289717

yes

characteristic

138

C₂

#20289720

yes

characteristic

	subgroup	G/H
center	C₂²	#16811478
commutator	C₄²×C₅₃	C₄²
Frattini	C₂×C₄²	C₂×#6157
Fitting	C₅₃×#384	C₄
socle	C₂²×C₅₃	#329
prime radical	C₂³×C₅₃	#100
cosocle	C₂×#13133128	C₂²
hypercenter	#384	#6157
hypocenter	C₅₃	#7266
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂²

#16811478

commutator

C₄²×C₅₃

C₄²

Frattini

C₂×C₄²

C₂×#6157

Fitting

C₅₃×#384

C₄

C₂²×C₅₃

C₂³×C₅₃

C₂²

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
5. #16811515	C₄	semidirect
79. C₅₃	#7266	semidirect

H ≅ G/N

type

5. #16811515

C₄

semidirect

79. C₅₃

#7266

semidirect

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

subgroup	index
C₂×#16811515	2
#20274850	2
#7266	53

subgroup

index

C₂×#16811515

#20274850

#7266

Subgroups — #28298679

Conjugacy classes of subgroups

Complements

Maximal subgroups