Group #28301484 — subgroups

Subgroups — #28301484

Order 13568.

294 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 96 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	#20290433	C₂	1	1	yes	maximal, characteristic
2	#20289929	C₂	1	1	yes	maximal, characteristic
3	#20289540	C₂	1	2	yes	maximal
4	#20289538	C₂	1	2	yes	maximal
5	#20272997	C₂	1	1	yes	maximal, characteristic
6	#16811641	C₂²	1	2	yes	—
7	#16811582	C₂²	1	1	yes	characteristic
8	#16811559	C₂²	1	1	yes	characteristic
9	(C₂×#6157)⋊C₈	C₄	1	4	yes	—
10	#88324⋊C₈	C₄	1	4	yes	complemented
11	#16810674	C₂²	1	2	yes	—
12	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
13	#16810345	C₄	1	2	yes	complemented
14	C₈×#88322	C₄	1	1	yes	characteristic, complemented
15	C₈²×C₅₃	C₄	1	1	yes	characteristic, complemented
16	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
18	#13133284	—	2	2	no	—
19	#13133274	—	2	2	no	—
20	#13133272	—	2	2	no	—
21	C₄×#88324	C₂×C₄	1	4	yes	—
22	(C₂×#6157)⋊C₄	—	2	2	no	—
23	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
24	C₄²×D₅₃	C₂³	1	1	yes	characteristic
25	#13133146	—	2	2	no	—
26	#13133129	—	2	2	no	—
27	C₄×#88322	C₂×C₄	1	2	yes	—
28	C₄×#88322	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×#6157	—	2	2	no	—
33	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×D₅₃	#41	1	2	yes	—
35	C₂×#88324	—	2	4	no	—
36	C₂×#88324	—	2	2	no	—
37	C₄×#6157	—	2	4	no	—
38	D₅₃⋊C₈	—	2	4	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	C₄²	1	2	yes	—
41	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₄×Dic₅₃	#41	1	2	yes	—
44	C₂×#88322	—	2	2	no	—
45	C₂×#88322	C₄²	1	2	yes	—
46	C₈×D₅₃	—	2	4	no	—
47	C₂²×D₅₃	#117	1	1	yes	characteristic
48	C₂×#6157	—	4	4	no	—
49	C₂×#6157	—	4	2	no	—
50	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#117	1	2	yes	—
52	C₂×Dic₅₃	#117	1	2	yes	—
53	C₂×Dic₅₃	#117	1	1	yes	characteristic
54	C₄×D₅₃	—	4	2	no	—
55	C₄×D₅₃	#131	1	4	yes	—
56	#88324	—	4	4	no	—
57	#88324	—	4	2	no	—
58	C₈×C₅₃	—	2	2	no	—
59	C₈×C₅₃	—	4	2	no	—
60	#88322	—	2	2	no	—
61	#88322	—	4	2	no	—
62	#8142	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	#376	1	1	yes	characteristic
64	C₂×D₅₃	—	2	2	no	—
65	C₂×D₅₃	#433	1	2	yes	—
66	#6157	—	8	2	no	—
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	#405	1	2	yes	—
69	Dic₅₃	—	2	2	no	—
70	Dic₅₃	#433	1	2	yes	—
71	#1746	—	53	1	no	—
72	#1452	—	53	1	no	—
73	#1450	—	53	1	no	—
74	#1275	—	53	2	no	—
75	#1274	—	53	2	no	—
76	C₂×C₅₃	#1746	1	1	yes	characteristic
77	C₂×C₅₃	#1747	1	2	yes	—
78	D₅₃	—	2	2	no	—
79	#431	—	53	2	no	—
80	#431	—	53	2	no	—
81	C₂×#97	—	53	1	no	—
82	C₂×C₄×C₈	—	53	1	no	—
83	#376	—	53	1	no	—
84	#336	—	53	4	no	—
85	#322	—	53	4	no	—
86	#322	—	53	2	no	—
87	C₈²	—	53	1	no	—
88	C₈²	#6157	1	1	yes	characteristic, complemented
89	C₅₃	#8142	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	53	2	no	—
91	C₂²×C₈	—	53	2	no	—
92	C₂×C₄²	—	53	1	no	—
93	C₂×C₄²	—	53	2	no	—
94	C₄⋊C₈	—	106	2	no	—
95	C₄⋊C₈	—	106	2	no	—
96	C₂²⋊C₈	—	106	2	no	—
97	C₂²⋊C₈	—	106	2	no	—
98	#97	—	106	2	no	—
99	C₄×C₈	—	53	2	no	—
100	C₄×C₈	—	53	4	no	—
101	C₄×C₈	—	53	1	no	—
102	C₄×C₈	—	53	2	no	—
103	C₄×C₈	C₂×#6157	1	2	yes	—
104	C₄×C₈	C₂×#6157	1	1	yes	characteristic
105	#95	—	106	2	no	—
106	C₂²×C₄	—	53	1	no	—
107	C₂²×C₄	—	53	2	no	—
108	C₂²×C₄	—	106	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	53	2	no	—
111	C₂×C₈	—	106	4	no	—
112	C₂×C₈	—	106	4	no	—
113	C₂×C₈	—	106	2	no	—
114	C₂×C₈	—	106	4	no	—
115	C₂×C₈	—	106	2	no	—
116	C₂×C₈	C₄×#6157	1	2	yes	—
117	C₄²	—	53	1	no	—
118	C₄²	—	53	2	no	—
119	C₄²	—	106	4	no	—
120	C₄²	C₂²×#6157	1	1	yes	characteristic
121	C₂³	—	53	1	no	—
122	C₂×C₄	—	53	1	no	—
123	C₂×C₄	—	53	4	no	—
124	C₂×C₄	—	53	2	no	—
125	C₂×C₄	—	212	2	no	—
126	C₂×C₄	—	212	2	no	—
127	C₂×C₄	—	212	4	no	—
128	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
129	C₂×C₄	#13133296	1	2	yes	—
130	C₈	—	2	2	no	—
131	C₈	—	4	2	no	—
132	C₈	—	106	2	no	—
133	C₈	—	212	2	no	—
134	C₈	—	212	2	no	—
135	C₈	—	212	4	no	—
136	C₂²	—	53	2	no	—
137	C₂²	—	106	2	no	—
138	C₂²	#16811559	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	53	2	no	—
141	C₄	—	106	2	no	—
142	C₄	—	424	2	no	—
143	C₄	#16811586	1	2	yes	—
144	C₂	—	106	2	no	—
145	C₂	#20290433	1	1	yes	characteristic
146	C₂	#20290436	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290433

C₂

yes

maximal, characteristic

#20289929

C₂

yes

maximal, characteristic

#20289540

C₂

yes

maximal

#20289538

C₂

yes

maximal

#20272997

C₂

yes

maximal, characteristic

#16811641

C₂²

yes

—

#16811582

C₂²

yes

characteristic

#16811559

C₂²

yes

characteristic

(C₂×#6157)⋊C₈

C₄

yes

—

#88324⋊C₈

C₄

yes

complemented

#16810674

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810345

C₄

yes

complemented

C₈×#88322

C₄

yes

characteristic, complemented

C₈²×C₅₃

C₄

yes

characteristic, complemented

C₂×D₅₃⋊C₈

C₂×C₄

yes

—

C₂×C₄×#6157

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

(C₂×#6157)⋊C₄

—

C₂×C₈×D₅₃

C₂×C₄

yes

—

C₄²×D₅₃

C₂³

yes

—

—

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×#6157

—

C₂×C₄×D₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×D₅₃

#41

yes

—

C₂×#88324

—

C₂×#88324

—

C₄×#6157

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

#41

yes

—

C₂×#88322

—

C₂×#88322

C₄²

yes

—

C₈×D₅₃

—

C₂²×D₅₃

yes

—

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×Dic₅₃

#117

yes

—

C₂×Dic₅₃

#117

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

#131

yes

—

#88324

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

#433

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

#405

yes

—

Dic₅₃

—

Dic₅₃

#433

yes

—

—

—

—

—

—

C₂×C₅₃

#1746

yes

characteristic

C₂×C₅₃

#1747

yes

—

D₅₃

—

#431

—

#431

—

C₂×#97

—

C₂×C₄×C₈

—

—

—

—

—

C₈²

—

C₈²

#6157

yes

characteristic, complemented

C₅₃

#8142

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

#97

—

106

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂×#6157

yes

—

104

C₄×C₈

C₂×#6157

yes

characteristic

105

#95

—

106

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

106

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

106

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

106

—

114

C₂×C₈

—

106

—

115

C₂×C₈

—

106

—

116

C₂×C₈

C₄×#6157

yes

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

106

—

120

C₄²

C₂²×#6157

yes

characteristic

121

C₂³

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

212

—

126

C₂×C₄

—

212

—

127

C₂×C₄

—

212

—

128

C₂×C₄

C₂×C₄×#6157

yes

characteristic

129

C₂×C₄

#13133296

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

106

—

133

C₈

—

212

—

134

C₈

—

212

—

135

C₈

—

212

—

136

C₂²

—

137

C₂²

—

106

—

138

C₂²

#16811559

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

106

—

142

C₄

—

424

—

143

C₄

#16811586

yes

—

144

C₂

—

106

—

145

C₂

#20290433

yes

characteristic

146

C₂

#20290436

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6157
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₄²	C₂²×#6157
Fitting	C₈²×C₅₃	C₄
socle	C₂²×C₅₃	#376
prime radical	C₂²×D₅₃	#117
cosocle	C₄²×D₅₃	C₂³
hypercenter	C₈²	#6157
hypocenter	C₅₃	#8142
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈²

subgroup

G/H

center

C₂×C₄

C₂×C₄×#6157

commutator

C₂×C₄×C₅₃

C₂×C₄²

Frattini

C₄²

C₂²×#6157

Fitting

C₈²×C₅₃

C₄

C₂²×C₅₃

C₂²×D₅₃

C₄²×D₅₃

C₂³

C₈²

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₈²

N	H ≅ G/N	type
10. #88324⋊C₈	C₄	semidirect
13. #16810345	C₄	semidirect
14. C₈×#88322	C₄	semidirect
15. C₈²×C₅₃	C₄	semidirect
88. C₈²	#6157	semidirect
89. C₅₃	#8142	semidirect

H ≅ G/N

type

10. #88324⋊C₈

C₄

semidirect

13. #16810345

C₄

semidirect

14. C₈×#88322

C₄

semidirect

15. C₈²×C₅₃

C₄

semidirect

88. C₈²

#6157

semidirect

89. C₅₃

#8142

semidirect

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

subgroup	index
#20290433	2
#20289929	2
#20289540	2
#20289538	2
#20272997	2
#8142	53

subgroup

index

Subgroups — #28301484

Conjugacy classes of subgroups

Complements

Maximal subgroups