Group #28307859 — subgroups

Subgroups — #28307859

Order 13568.

246 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 108 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	#20291076	C₂	1	2	yes	maximal
2	#20289633	C₂	1	4	yes	maximal, complemented
3	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11719910	C₄	1	4	yes	—
5	C₂²⋊#11719910	C₄	1	4	yes	—
6	#16811511	C₂²	1	4	yes	—
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	#16810345	C₄	1	2	yes	—
10	#16810345	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#11719910	C₂×C₄	1	8	yes	—
13	C₂²×#88322	—	2	1	no	—
14	C₂²×#88322	C₂×C₄	1	2	yes	—
15	C₄×#88322	—	2	1	no	—
16	C₄×#88322	C₂×C₄	1	2	yes	—
17	C₄×#88322	C₂×C₄	1	1	yes	characteristic
18	C₄×#88322	C₂×C₄	1	2	yes	—
19	C₄×#88322	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	—	2	1	no	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	—	2	1	no	—
23	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	—	4	2	no	—
29	C₄²×C₅₃	C₄²	1	1	yes	characteristic
30	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
31	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
32	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×#88322	—	2	4	no	—
34	C₂×#88322	—	2	2	no	—
35	C₂×#88322	—	4	2	no	—
36	C₂×#88322	C₄²	1	2	yes	—
37	C₂×#88322	C₂²×C₄	1	2	yes	—
38	#11719910	—	2	8	no	—
39	C₂³×C₅₃	#97	1	1	yes	characteristic
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
49	C₈×C₅₃	—	4	2	no	—
50	#88322	—	2	2	no	—
51	#88322	—	4	2	no	—
52	#88322	#131	1	4	yes	—
53	#11415	—	53	1	no	maximal, Sylow
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	#347	1	2	yes	—
57	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
58	C₄×C₅₃	—	2	1	no	—
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	#347	1	2	yes	—
63	C₄×C₅₃	#405	1	2	yes	—
64	#2163	—	53	2	no	—
65	C₂×#322	—	53	1	no	—
66	#1313	—	53	4	no	—
67	C₂×C₅₃	—	2	2	no	—
68	C₂×C₅₃	#1452	1	1	yes	characteristic
69	C₂×C₅₃	C₂×#347	1	1	yes	characteristic
70	C₂×C₅₃	C₂×#347	1	1	yes	characteristic
71	C₂×C₄×C₈	—	53	1	no	—
72	C₂×C₄×C₈	—	53	1	no	—
73	C₂×C₄×C₈	#6157	1	1	yes	characteristic
74	C₄⋊C₁₆	—	53	4	no	—
75	C₂²⋊C₁₆	—	53	4	no	—
76	#346	—	53	4	no	—
77	#322	—	53	2	no	—
78	#322	—	53	2	no	—
79	C₅₃	#11415	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	—	2	1	no	—
81	C₂²×C₈	—	53	2	no	—
82	C₂²×C₈	—	106	1	no	—
83	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
84	C₂×C₁₆	—	53	8	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	53	2	no	—
87	C₄×C₈	—	53	1	no	—
88	C₄×C₈	—	53	1	no	—
89	C₄×C₈	—	53	2	no	—
90	C₄×C₈	—	106	1	no	—
91	C₄×C₈	C₂×#6157	1	2	yes	—
92	C₂²×C₄	#11719936	1	1	yes	characteristic
93	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
94	C₂²×C₄	#11719940	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	—	53	2	no	—
98	C₂×C₈	—	53	2	no	—
99	C₂×C₈	—	106	4	no	—
100	C₂×C₈	—	106	2	no	—
101	C₂×C₈	—	212	2	no	—
102	C₄²	#11719936	1	1	yes	characteristic
103	C₄²	C₄×#6157	1	1	yes	characteristic
104	C₄²	#11719940	1	1	yes	characteristic
105	C₄²	C₂²×#6157	1	1	yes	characteristic
106	C₁₆	—	106	8	no	—
107	C₂³	#13133272	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	#13133272	1	1	yes	characteristic
112	C₂×C₄	#13133272	1	1	yes	characteristic
113	C₂×C₄	#13133272	1	1	yes	characteristic
114	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
115	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
116	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
117	C₈	—	4	2	no	—
118	C₈	—	53	4	no	—
119	C₈	—	106	2	no	—
120	C₈	—	212	2	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	#16811515	1	2	yes	—
124	C₂²	C₂×#13133272	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	#16811516	1	2	yes	—
130	C₄	#16811586	1	2	yes	—
131	C₂	—	2	2	no	—
132	C₂	#20289928	1	1	yes	characteristic
133	C₂	C₂×#16811515	1	1	yes	characteristic
134	C₂	C₂×#16811516	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291076

C₂

yes

maximal

#20289633

C₂

yes

maximal, complemented

C₂×#16810345

C₂

yes

maximal, characteristic

C₄⋊#11719910

C₄

yes

—

C₂²⋊#11719910

C₄

yes

—

#16811511

C₂²

yes

—

C₂×C₄×#88322

C₄

yes

characteristic

C₂×C₄×#88322

C₂²

yes

characteristic

#16810345

C₄

yes

—

#16810345

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₄

yes

characteristic

C₂×#11719910

C₂×C₄

yes

—

C₂²×#88322

—

C₂²×#88322

C₂×C₄

yes

—

C₄×#88322

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

characteristic

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

C₂³

yes

characteristic

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

M₁₆

yes

characteristic

C₂²×C₄×C₅₃

M₁₆

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

M₁₆

yes

characteristic

C₄²×C₅₃

M₁₆

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

—

—

—

C₄²

yes

—

C₂×#88322

C₂²×C₄

yes

—

#11719910

—

C₂³×C₅₃

#97

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₈×C₅₃

—

—

—

yes

—

#11415

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#347

yes

—

C₂²×C₅₃

C₂×#97

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#347

yes

—

C₄×C₅₃

#405

yes

—

#2163

—

C₂×#322

—

#1313

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#6157

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

—

#322

—

C₅₃

#11415

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂×C₄²

C₂×#6157

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

106

—

C₄×C₈

C₂×#6157

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

100

C₂×C₈

—

106

—

101

C₂×C₈

—

212

—

102

C₄²

#11719936

yes

characteristic

103

C₄²

C₄×#6157

yes

characteristic

104

C₄²

#11719940

yes

characteristic

105

C₄²

C₂²×#6157

yes

characteristic

106

C₁₆

—

106

—

107

C₂³

#13133272

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#13133272

yes

characteristic

112

C₂×C₄

#13133272

yes

characteristic

113

C₂×C₄

#13133272

yes

characteristic

114

C₂×C₄

C₂×C₄×#6157

yes

characteristic

115

C₂×C₄

C₂×#11719940

yes

characteristic

116

C₂×C₄

C₂×#11719936

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

106

—

120

C₈

—

212

—

121

C₂²

—

122

C₂²

—

123

C₂²

#16811515

yes

—

124

C₂²

C₂×#13133272

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#16811516

yes

—

130

C₄

#16811586

yes

—

131

C₂

—

132

C₂

#20289928

yes

characteristic

133

C₂

C₂×#16811515

yes

characteristic

134

C₂

C₂×#16811516

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6157
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₄²	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	#97
cosocle	C₄×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#11415
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×C₄×#6157

commutator

C₂×C₄×C₅₃

C₂×C₄²

Frattini

C₄²

C₂²×#6157

Fitting

C₂×C₄×C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×#97

C₂³×C₅₃

C₂³

C₂×C₄×C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #20289633	C₂	semidirect
79. C₅₃	#11415	semidirect

H ≅ G/N

type

2. #20289633

C₂

semidirect

79. C₅₃

#11415

semidirect

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

subgroup	index
#20291076	2
#20289633	2
C₂×#16810345	2
#11415	53

subgroup

index

Subgroups — #28307859

Conjugacy classes of subgroups

Complements

Maximal subgroups