Group #28298443 — subgroups

Subgroups — #28298443

Order 13568.

222 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 100 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	#20291072	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	C₄×#11719910	—	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	C₈	1	2	yes	—
12	C₂²×#88322	C₂×C₄	1	2	yes	—
13	C₄×#88322	—	2	1	no	—
14	C₄×#88322	—	2	1	no	—
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	C₈	1	2	yes	—
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₅₃	—	2	1	no	—
21	C₄×C₈×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×C₈	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	M₁₆	1	1	yes	characteristic
30	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
31	C₂×#88322	—	2	2	no	—
32	C₂×#88322	—	2	2	no	—
33	C₂×#88322	—	4	2	no	—
34	C₂×#88322	C₁₆	1	4	yes	complemented
35	C₂×#88322	C₂²⋊C₄	1	2	yes	—
36	C₂×#88322	C₄⋊C₄	1	2	yes	—
37	#11719910	—	4	4	no	—
38	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
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	#97	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	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
49	C₈×C₅₃	—	4	2	no	—
50	#88322	—	2	2	no	—
51	#88322	—	2	2	no	—
52	#88322	—	2	4	no	—
53	#7195	—	53	1	no	maximal, Sylow
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	#336	1	1	yes	characteristic
57	C₂²×C₅₃	C₄×C₁₆	1	1	yes	characteristic
58	C₂²×C₅₃	#346	1	1	yes	characteristic
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	#344	1	2	yes	—
62	C₄×C₅₃	C₂²⋊C₁₆	1	2	yes	—
63	C₄×C₅₃	#349	1	2	yes	—
64	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
65	#2162	—	53	2	no	—
66	C₂×#322	—	53	1	no	—
67	C₂×C₅₃	—	2	1	no	—
68	C₂×C₅₃	—	2	1	no	—
69	C₂×C₅₃	#1281	1	1	yes	characteristic
70	C₂×C₅₃	#1374	1	1	yes	characteristic
71	C₂×C₅₃	#1377	1	1	yes	characteristic
72	C₂×C₄×C₈	—	53	1	no	—
73	C₂×C₄×C₈	—	53	1	no	—
74	C₂×C₄×C₈	#6157	1	1	yes	characteristic
75	C₄⋊C₁₆	—	106	2	no	—
76	C₂²⋊C₁₆	—	53	4	no	—
77	C₄×C₁₆	—	106	2	no	—
78	#322	—	106	2	no	—
79	C₅₃	#7195	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	—	53	2	no	—
81	C₂²×C₈	—	53	2	no	—
82	C₂²×C₈	#88324	1	2	yes	—
83	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
84	C₂×C₁₆	—	106	4	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	2	1	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	—	106	1	no	—
92	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
93	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
94	C₂²×C₄	C₂×#88324	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	—	53	4	no	—
100	C₂×C₈	—	106	2	no	—
101	C₂×C₈	—	106	2	no	—
102	C₂×C₈	—	212	2	no	—
103	C₄²	#11719936	1	1	yes	characteristic
104	C₄²	C₄⋊#6157	1	1	yes	characteristic
105	C₄²	#11719940	1	1	yes	characteristic
106	C₄²	C₂²⋊#6157	1	1	yes	characteristic
107	C₁₆	—	212	4	no	—
108	C₂³	C₄×#88324	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
112	C₂×C₄	#13133272	1	1	yes	characteristic
113	C₂×C₄	C₂²⋊#88324	1	1	yes	characteristic
114	C₂×C₄	#13133274	1	1	yes	characteristic
115	C₂×C₄	C₄⋊#88324	1	1	yes	characteristic
116	C₂×C₄	#13133284	1	1	yes	characteristic
117	C₂×C₄	D₅₃⋊C₁₆	1	2	yes	—
118	C₂×C₄	#13133292	1	2	yes	—
119	C₈	—	4	2	no	—
120	C₈	—	106	2	no	—
121	C₈	—	106	2	no	—
122	C₈	—	106	4	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	1	no	—
125	C₂²	#16811489	1	1	yes	characteristic
126	C₂²	#16811509	1	1	yes	characteristic
127	C₂²	#16811512	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	#16811505	1	2	yes	—
131	C₄	#16811518	1	2	yes	—
132	C₄	#16811519	1	2	yes	—
133	C₄	#16811544	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#20289555	1	1	yes	characteristic
137	C₂	#20289754	1	1	yes	characteristic
138	C₂	#20289762	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291072

C₂

yes

maximal

C₂×#16810345

C₂

yes

maximal, characteristic

C₄⋊#11719910

—

C₂²⋊#11719910

C₄

yes

—

C₄×#11719910

—

C₂×C₄×#88322

C₄

yes

characteristic

C₂×C₄×#88322

C₂²

yes

characteristic

#16810345

—

C₂×C₄×C₈×C₅₃

C₄

yes

characteristic

C₂×#11719910

—

C₂²×#88322

C₈

yes

—

C₂²×#88322

C₂×C₄

yes

—

C₄×#88322

—

C₄×#88322

—

C₄×#88322

D₄

yes

characteristic

C₄×#88322

D₄

yes

—

C₄×#88322

Q₈

yes

characteristic

C₂²×C₈×C₅₃

C₈

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

M₁₆

yes

characteristic

C₄²×C₅₃

M₁₆

yes

—

—

—

C₁₆

yes

complemented

C₂×#88322

C₂²⋊C₄

yes

—

C₂×#88322

C₄⋊C₄

yes

—

#11719910

—

C₂³×C₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

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

C₂×C₄×C₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

M₃₂

yes

—

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#336

yes

characteristic

C₂²×C₅₃

C₄×C₁₆

yes

characteristic

C₂²×C₅₃

#346

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#344

yes

—

C₄×C₅₃

C₂²⋊C₁₆

yes

—

C₄×C₅₃

#349

yes

—

C₄×C₅₃

C₄⋊C₁₆

yes

—

#2162

—

C₂×#322

—

C₂×C₅₃

—

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

—

C₄×C₁₆

—

106

—

#322

—

106

—

C₅₃

#7195

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

#88324

yes

—

C₂×C₄²

C₂×#6157

yes

characteristic

C₂×C₁₆

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

D₅₃⋊C₈

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

106

—

101

C₂×C₈

—

106

—

102

C₂×C₈

—

212

—

103

C₄²

#11719936

yes

characteristic

104

C₄²

C₄⋊#6157

yes

characteristic

105

C₄²

#11719940

yes

characteristic

106

C₄²

C₂²⋊#6157

yes

characteristic

107

C₁₆

—

212

—

108

C₂³

C₄×#88324

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

112

C₂×C₄

#13133272

yes

characteristic

113

C₂×C₄

C₂²⋊#88324

yes

characteristic

114

C₂×C₄

#13133274

yes

characteristic

115

C₂×C₄

C₄⋊#88324

yes

characteristic

116

C₂×C₄

#13133284

yes

characteristic

117

C₂×C₄

D₅₃⋊C₁₆

yes

—

118

C₂×C₄

#13133292

yes

—

119

C₈

—

120

C₈

—

106

—

121

C₈

—

106

—

122

C₈

—

106

—

123

C₂²

—

124

C₂²

—

125

C₂²

#16811489

yes

characteristic

126

C₂²

#16811509

yes

characteristic

127

C₂²

#16811512

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

#16811505

yes

—

131

C₄

#16811518

yes

—

132

C₄

#16811519

yes

—

133

C₄

#16811544

yes

—

134

C₂

—

135

C₂

—

136

C₂

#20289555

yes

characteristic

137

C₂

#20289754

yes

characteristic

138

C₂

#20289762

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊#6157
commutator	C₂²×C₅₃	C₄×C₁₆
Frattini	C₂×C₄²	C₂×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	#336
prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₂×C₄×#88322	C₂²
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#7195
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄²

C₂²⋊#6157

commutator

C₂²×C₅₃

C₄×C₁₆

Frattini

C₂×C₄²

C₂×#6157

Fitting

C₂×C₄×C₈×C₅₃

C₄

socle

C₂²×C₅₃

#336

prime radical

C₂³×C₅₃

C₄×C₈

cosocle

C₂×C₄×#88322

C₂²

C₂×C₄×C₈

C₅₃

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
34. C₂×#88322	C₁₆	semidirect
79. C₅₃	#7195	semidirect

H ≅ G/N

type

34. C₂×#88322

C₁₆

semidirect

79. C₅₃

#7195

semidirect

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

subgroup	index
#20291072	2
C₂×#16810345	2
#7195	53

subgroup

index

#20291072

C₂×#16810345

#7195

Subgroups — #28298443

Conjugacy classes of subgroups

Complements

Maximal subgroups