Group #28298294 — subgroups

Subgroups — #28298294

Order 13568.

266 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 84 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	C₄×#13133127	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719910	C₄	1	4	yes	complemented
4	C₂²⋊#11719910	—	2	2	no	—
5	#16811511	—	2	2	no	—
6	C₂×#13133127	—	2	2	no	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂×#11719910	—	2	4	no	—
11	C₂²×#88322	D₄	1	2	yes	—
12	C₂×C₄×Dic₅₃	—	2	1	no	—
13	C₂×C₄×Dic₅₃	—	2	1	no	—
14	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
15	#13133127	—	2	8	no	—
16	C₄×#88322	C₂×C₄	1	2	yes	—
17	C₄×#88322	D₄	1	1	yes	characteristic
18	C₄×#88322	Q₈	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	—	2	1	no	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₅₃	—	2	1	no	—
22	C₄×C₈×C₅₃	C₈	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
26	C₂²×Dic₅₃	—	2	2	no	—
27	C₂×C₈×C₅₃	—	2	4	no	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₄²×C₅₃	C₄²	1	1	yes	characteristic
30	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
33	C₄×Dic₅₃	—	2	4	no	—
34	C₄×Dic₅₃	—	4	2	no	—
35	C₄×Dic₅₃	—	4	2	no	—
36	C₂×#88322	—	2	4	no	—
37	C₂×#88322	C₂²⋊C₄	1	2	yes	—
38	C₂×#88322	C₄⋊C₄	1	2	yes	—
39	#11719910	—	4	4	no	—
40	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
50	C₂×Dic₅₃	—	2	4	no	—
51	C₂×Dic₅₃	—	4	2	no	—
52	C₂×Dic₅₃	—	4	2	no	—
53	C₈×C₅₃	—	2	4	no	—
54	#88322	—	2	2	no	—
55	#88322	#104	1	4	yes	—
56	#7149	—	53	1	no	maximal, Sylow
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	—	2	1	no	—
59	C₂²×C₅₃	#336	1	1	yes	characteristic
60	C₂²×C₅₃	#349	1	2	yes	—
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	#339	1	2	yes	—
66	C₄×C₅₃	#347	1	2	yes	—
67	Dic₅₃	—	4	4	no	—
68	#2163	—	53	2	no	—
69	C₄×#97	—	53	1	no	—
70	C₂×C₅₃	—	2	2	no	—
71	C₂×C₅₃	#1276	1	1	yes	characteristic
72	C₂×C₅₃	#1375	1	1	yes	characteristic
73	C₂×C₅₃	#1377	1	1	yes	characteristic
74	C₂×#97	—	106	2	no	—
75	C₂×C₄×C₈	—	53	1	no	—
76	C₂×C₄×C₈	#6157	1	1	yes	characteristic
77	C₄³	—	53	1	no	—
78	C₄⋊C₁₆	—	53	4	no	—
79	C₂²⋊C₁₆	—	106	2	no	—
80	#346	—	106	2	no	—
81	C₅₃	#7149	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	—	53	2	no	—
84	C₂×C₄²	—	53	2	no	—
85	C₂×C₄²	—	106	1	no	—
86	C₂×C₄²	—	106	1	no	—
87	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
88	C₂×C₁₆	—	106	4	no	—
89	#97	—	106	8	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	—	53	2	no	—
92	C₄×C₈	—	53	1	no	—
93	C₄×C₈	—	53	1	no	—
94	C₄×C₈	#88324	1	2	yes	—
95	C₂²×C₄	—	106	2	no	—
96	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
97	C₂²×C₄	#11719936	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	53	2	no	—
102	C₂×C₈	—	53	2	no	—
103	C₂×C₈	—	106	4	no	—
104	C₄²	—	106	4	no	—
105	C₄²	—	212	2	no	—
106	C₄²	—	212	2	no	—
107	C₄²	C₄×#6157	1	1	yes	characteristic
108	C₄²	C₄⋊#6157	1	1	yes	characteristic
109	C₄²	C₂×#88324	1	1	yes	characteristic
110	C₄²	#11719940	1	1	yes	characteristic
111	C₁₆	—	212	4	no	—
112	C₂³	#13133274	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	106	4	no	—
117	C₂×C₄	—	212	2	no	—
118	C₂×C₄	—	212	2	no	—
119	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
120	C₂×C₄	C₄×#88324	1	1	yes	characteristic
121	C₂×C₄	#13133272	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊#88324	1	1	yes	characteristic
123	C₂×C₄	C₄⋊#88324	1	1	yes	characteristic
124	C₂×C₄	#13133284	1	1	yes	characteristic
125	C₈	—	2	4	no	—
126	C₈	—	53	4	no	—
127	C₈	—	106	2	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	#16811489	1	1	yes	characteristic
131	C₂²	#16811520	1	2	yes	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	212	4	no	—
137	C₄	#16811494	1	2	yes	—
138	C₄	#16811516	1	2	yes	—
139	C₂	—	2	2	no	—
140	C₂	#20289543	1	1	yes	characteristic
141	C₂	#20289757	1	1	yes	characteristic
142	C₂	#20289763	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291076

C₂

yes

maximal

C₄×#13133127

C₂

yes

maximal, characteristic

C₄⋊#11719910

C₄

yes

—

—

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₄²×Dic₅₃

C₄

yes

characteristic

C₂×C₄×C₈×C₅₃

C₄

yes

characteristic

C₂×#11719910

—

C₂²×#88322

D₄

yes

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₈

yes

—

#13133127

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

D₄

yes

characteristic

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

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

M₁₆

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

M₁₆

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

C₂²⋊C₄

yes

—

C₂×#88322

C₄⋊C₄

yes

—

#11719910

—

C₂³×C₅₃

C₂²⋊C₈

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

#97

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

#88322

#104

yes

—

#7149

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#336

yes

characteristic

C₂²×C₅₃

#349

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#339

yes

—

C₄×C₅₃

#347

yes

—

Dic₅₃

—

#2163

—

C₄×#97

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×#97

—

106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#6157

yes

characteristic

C₄³

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

106

—

#346

—

106

—

C₅₃

#7149

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

106

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×#6157

yes

characteristic

C₂×C₁₆

—

106

—

#97

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#88324

yes

—

C₂²×C₄

—

106

—

C₂²×C₄

D₅₃⋊C₈

yes

characteristic

C₂²×C₄

#11719936

yes

characteristic

C₂²×C₄

C₂²⋊#6157

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

106

—

104

C₄²

—

106

—

105

C₄²

—

212

—

106

C₄²

—

212

—

107

C₄²

C₄×#6157

yes

characteristic

108

C₄²

C₄⋊#6157

yes

characteristic

109

C₄²

C₂×#88324

yes

characteristic

110

C₄²

#11719940

yes

characteristic

111

C₁₆

—

212

—

112

C₂³

#13133274

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

106

—

117

C₂×C₄

—

212

—

118

C₂×C₄

—

212

—

119

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

120

C₂×C₄

C₄×#88324

yes

characteristic

121

C₂×C₄

#13133272

yes

characteristic

122

C₂×C₄

C₂²⋊#88324

yes

characteristic

123

C₂×C₄

C₄⋊#88324

yes

characteristic

124

C₂×C₄

#13133284

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

106

—

128

C₂²

—

129

C₂²

—

130

C₂²

#16811489

yes

characteristic

131

C₂²

#16811520

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

212

—

137

C₄

#16811494

yes

—

138

C₄

#16811516

yes

—

139

C₂

—

140

C₂

#20289543

yes

characteristic

141

C₂

#20289757

yes

characteristic

142

C₂

#20289763

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	C₂²⋊C₈
cosocle	C₂×C₄×#88322	C₂²
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#7149
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

socle

C₂²×C₅₃

#336

prime radical

C₂³×C₅₃

C₂²⋊C₈

cosocle

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
3. C₄⋊#11719910	C₄	semidirect
81. C₅₃	#7149	semidirect

H ≅ G/N

type

3. C₄⋊#11719910

C₄

semidirect

81. C₅₃

#7149

semidirect

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

subgroup	index
#20291076	2
C₄×#13133127	2
#7149	53

subgroup

index

#20291076

C₄×#13133127

#7149

Subgroups — #28298294

Conjugacy classes of subgroups

Complements

Maximal subgroups