Group #28298445 — subgroups

Subgroups — #28298445

Order 13568.

210 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 80 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	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719910	—	2	2	no	—
4	C₂²⋊#11719910	C₄	1	4	yes	—
5	#16811511	—	2	2	no	—
6	C₂×C₄×#88322	C₄	1	1	yes	characteristic
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	#16810345	—	2	2	no	—
9	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂×#11719910	—	2	4	no	—
11	C₂²×#88322	—	2	1	no	—
12	C₂²×#88322	C₂×C₄	1	2	yes	—
13	C₄×#88322	—	2	1	no	—
14	C₄×#88322	C₈	1	2	yes	—
15	C₄×#88322	D₄	1	1	yes	characteristic
16	C₄×#88322	D₄	1	2	yes	—
17	C₄×#88322	Q₈	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	—	2	1	no	—
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₅₃	—	2	1	no	—
21	C₄×C₈×C₅₃	C₈	1	2	yes	—
22	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	—	4	2	no	—
27	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂×#88322	—	2	4	no	—
32	C₂×#88322	—	2	2	no	—
33	C₂×#88322	—	4	2	no	—
34	C₂×#88322	C₂²⋊C₄	1	2	yes	—
35	C₂×#88322	C₄⋊C₄	1	2	yes	—
36	#11719910	—	4	4	no	—
37	C₂³×C₅₃	#97	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
47	C₈×C₅₃	—	4	2	no	—
48	#88322	—	2	4	no	—
49	#88322	—	2	2	no	—
50	#88322	—	2	2	no	—
51	#7196	—	53	1	no	maximal, Sylow
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	#336	1	1	yes	characteristic
55	C₂²×C₅₃	#347	1	2	yes	—
56	C₄×C₅₃	—	2	1	no	—
57	C₄×C₅₃	—	2	1	no	—
58	C₄×C₅₃	—	2	1	no	—
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	#344	1	2	yes	—
61	C₄×C₅₃	#364	1	2	yes	—
62	#2163	—	53	2	no	—
63	C₂×#322	—	53	1	no	—
64	C₂×C₅₃	—	2	2	no	—
65	C₂×C₅₃	#1281	1	1	yes	characteristic
66	C₂×C₅₃	#1375	1	1	yes	characteristic
67	C₂×C₅₃	#1376	1	1	yes	characteristic
68	C₂×C₄×C₈	—	53	1	no	—
69	C₂×C₄×C₈	—	53	1	no	—
70	C₂×C₄×C₈	#6157	1	1	yes	characteristic
71	C₄⋊C₁₆	—	106	2	no	—
72	C₂²⋊C₁₆	—	53	4	no	—
73	#346	—	106	2	no	—
74	#322	—	106	2	no	—
75	C₅₃	#7196	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	—	53	2	no	—
78	C₂²×C₈	—	106	1	no	—
79	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
80	C₂×C₁₆	—	106	4	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	—	53	1	no	—
83	C₄×C₈	—	53	1	no	—
84	C₄×C₈	—	53	2	no	—
85	C₄×C₈	—	53	2	no	—
86	C₄×C₈	—	106	1	no	—
87	C₄×C₈	#88324	1	2	yes	—
88	C₂²×C₄	#11719936	1	1	yes	characteristic
89	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
90	C₂²×C₄	#11719940	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	4	2	no	—
93	C₂×C₈	—	53	2	no	—
94	C₂×C₈	—	53	2	no	—
95	C₂×C₈	—	106	4	no	—
96	C₂×C₈	—	106	2	no	—
97	C₂×C₈	—	212	2	no	—
98	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
99	C₄²	C₄⋊#6157	1	1	yes	characteristic
100	C₄²	C₂×#88324	1	1	yes	characteristic
101	C₄²	C₂²⋊#6157	1	1	yes	characteristic
102	C₁₆	—	212	4	no	—
103	C₂³	#13133272	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
108	C₂×C₄	C₄×#88324	1	1	yes	characteristic
109	C₂×C₄	C₂²⋊#88324	1	1	yes	characteristic
110	C₂×C₄	#13133274	1	1	yes	characteristic
111	C₂×C₄	C₄⋊#88324	1	1	yes	characteristic
112	C₂×C₄	#13133284	1	1	yes	characteristic
113	C₈	—	4	2	no	—
114	C₈	—	106	4	no	—
115	C₈	—	106	2	no	—
116	C₈	—	106	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#16811489	1	1	yes	characteristic
120	C₂²	#16811515	1	2	yes	—
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	#16811505	1	2	yes	—
126	C₄	#16811545	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	#20289555	1	1	yes	characteristic
129	C₂	#20289756	1	1	yes	characteristic
130	C₂	#20289760	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291076

C₂

yes

maximal

C₂×#16810345

C₂

yes

maximal, characteristic

C₄⋊#11719910

—

C₂²⋊#11719910

C₄

yes

—

#16811511

—

C₂×C₄×#88322

C₄

yes

characteristic

C₂×C₄×#88322

C₂²

yes

characteristic

#16810345

—

C₂×C₄×C₈×C₅₃

C₄

yes

characteristic

C₂×#11719910

—

C₂²×#88322

—

C₂²×#88322

C₂×C₄

yes

—

C₄×#88322

—

C₄×#88322

C₈

yes

—

C₄×#88322

D₄

yes

characteristic

C₄×#88322

D₄

yes

—

C₄×#88322

Q₈

yes

characteristic

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₈

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

C₂²⋊C₄

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

—

—

—

C₂²⋊C₄

yes

—

C₂×#88322

C₄⋊C₄

yes

—

#11719910

—

C₂³×C₅₃

#97

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#336

yes

characteristic

C₂²×C₅₃

#347

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#344

yes

—

C₄×C₅₃

#364

yes

—

#2163

—

C₂×#322

—

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

—

106

—

C₂²⋊C₁₆

—

#346

—

106

—

#322

—

106

—

C₅₃

#7196

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂×C₄²

C₂×#6157

yes

characteristic

C₂×C₁₆

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

106

—

C₄×C₈

#88324

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₄²

D₅₃⋊C₈

yes

characteristic

C₄²

C₄⋊#6157

yes

characteristic

100

C₄²

C₂×#88324

yes

characteristic

101

C₄²

C₂²⋊#6157

yes

characteristic

102

C₁₆

—

212

—

103

C₂³

#13133272

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

108

C₂×C₄

C₄×#88324

yes

characteristic

109

C₂×C₄

C₂²⋊#88324

yes

characteristic

110

C₂×C₄

#13133274

yes

characteristic

111

C₂×C₄

C₄⋊#88324

yes

characteristic

112

C₂×C₄

#13133284

yes

characteristic

113

C₈

—

114

C₈

—

106

—

115

C₈

—

106

—

116

C₈

—

106

—

117

C₂²

—

118

C₂²

—

119

C₂²

#16811489

yes

characteristic

120

C₂²

#16811515

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#16811505

yes

—

126

C₄

#16811545

yes

—

127

C₂

—

128

C₂

#20289555

yes

characteristic

129

C₂

#20289756

yes

characteristic

130

C₂

#20289760

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×#6157)⋊C₄
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₂×C₄²	C₂×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	#336
prime radical	C₂³×C₅₃	#97
cosocle	C₂×C₄×#88322	C₂²
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#7196
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

(C₂×#6157)⋊C₄

commutator

C₂×C₄×C₅₃

C₄×C₈

Frattini

C₂×C₄²

C₂×#6157

Fitting

C₂×C₄×C₈×C₅₃

C₄

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄×#88322

C₂²

C₂×C₄×C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
75. C₅₃	#7196	semidirect

H ≅ G/N

type

75. C₅₃

#7196

semidirect

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

subgroup	index
#20291076	2
C₂×#16810345	2
#7196	53

subgroup

index

#20291076

C₂×#16810345

#7196

Subgroups — #28298445

Conjugacy classes of subgroups

Complements

Maximal subgroups