Group #28309914 — subgroups

Subgroups — #28309914

Order 13568.

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

normal subgroups only

This group has 242 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	#20290273	C₂	1	2	yes	maximal
2	#20289749	C₂	1	4	yes	maximal, complemented
3	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#16811627	—	2	2	no	—
5	#16811486	—	2	2	no	—
6	#16811486	C₂²	1	4	yes	—
7	C₂×C₁₆×D₅₃	C₄	1	2	yes	—
8	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₄×#11719908	C₄	1	1	yes	characteristic
10	C₁₆×Dic₅₃	C₂²	1	2	yes	—
11	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#11719936	—	4	2	no	—
13	C₄⋊#88324	—	2	4	no	—
14	C₄⋊#88324	—	4	2	no	—
15	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
17	C₄²×D₅₃	D₄	1	1	yes	characteristic
18	C₂×#11719908	C₂×C₄	1	2	yes	—
19	C₁₆×D₅₃	—	2	4	no	—
20	C₄×#88322	Q₈	1	1	yes	characteristic
21	C₈×Dic₅₃	D₄	1	1	yes	characteristic
22	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
27	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
28	C₂×#88324	—	4	4	no	—
29	#11719936	—	8	4	no	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₄×Dic₅₃	D₈	1	1	yes	characteristic
34	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
35	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
37	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
38	C₈×D₅₃	—	2	2	no	—
39	C₈×D₅₃	—	2	2	no	—
40	C₁₆×C₅₃	—	2	2	no	—
41	#11719908	M₁₆	1	4	yes	complemented
42	C₂²×D₅₃	#107	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×Dic₅₃	#107	1	1	yes	characteristic
47	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
49	C₄×D₅₃	—	2	2	no	—
50	C₄×D₅₃	—	2	1	no	—
51	C₄×D₅₃	—	2	1	no	—
52	C₄×D₅₃	SD₃₂	1	4	yes	—
53	#88324	—	8	4	no	—
54	C₈×C₅₃	#108	1	2	yes	—
55	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
56	#88322	—	2	1	no	—
57	#88322	—	2	1	no	—
58	#12023	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
60	C₂×D₅₃	—	2	1	no	—
61	C₂×D₅₃	—	2	1	no	—
62	C₂×D₅₃	#367	1	2	yes	—
63	C₄×C₅₃	#367	1	2	yes	—
64	C₄×C₅₃	#368	1	2	yes	—
65	C₄×C₅₃	#423	1	1	yes	characteristic
66	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
67	Dic₅₃	—	2	1	no	—
68	Dic₅₃	—	2	1	no	—
69	Dic₅₃	C₂×SD₃₂	1	2	yes	—
70	C₂×C₄×C₁₆	—	53	1	no	—
71	#1567	—	53	2	no	—
72	#1372	—	53	4	no	—
73	C₂×C₅₃	#1567	1	1	yes	characteristic
74	C₂×C₅₃	C₂×#367	1	1	yes	characteristic
75	C₂×C₅₃	C₂×#368	1	1	yes	characteristic
76	D₅₃	—	2	2	no	—
77	C₂²×C₁₆	—	53	2	no	—
78	#423	—	106	2	no	—
79	C₂×C₄×C₈	—	53	1	no	—
80	C₄×C₁₆	—	53	2	no	—
81	C₄×C₁₆	—	53	1	no	—
82	C₄×C₁₆	#6157	1	1	yes	characteristic
83	#335	—	53	4	no	—
84	#335	—	106	2	no	—
85	C₅₃	#12023	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₁₆	—	212	2	no	—
87	C₂²×C₈	—	53	1	no	—
88	C₂²×C₈	—	53	1	no	—
89	C₂×C₄²	—	53	1	no	—
90	C₂×C₁₆	—	53	2	no	—
91	C₂×C₁₆	—	106	4	no	—
92	C₂×C₁₆	C₂×#6157	1	2	yes	—
93	C₄⋊C₈	—	106	4	no	—
94	C₄⋊C₈	—	212	2	no	—
95	C₄×C₈	—	53	1	no	—
96	C₄×C₈	—	53	1	no	—
97	C₄×C₈	—	53	1	no	—
98	C₄×C₈	C₂×#6157	1	1	yes	characteristic
99	C₂²×C₄	—	53	1	no	—
100	C₂²×C₄	—	53	1	no	—
101	C₂²×C₄	—	53	1	no	—
102	M₁₆	—	424	4	no	—
103	C₂×C₈	—	53	1	no	—
104	C₂×C₈	—	53	1	no	—
105	C₂×C₈	—	106	2	no	—
106	C₂×C₈	—	106	2	no	—
107	C₂×C₈	—	212	4	no	—
108	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
109	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
110	C₄²	—	53	1	no	—
111	C₄²	—	53	1	no	—
112	C₄²	—	53	1	no	—
113	C₄²	C₄⋊#6157	1	1	yes	characteristic
114	C₁₆	—	2	2	no	—
115	C₁₆	—	53	4	no	—
116	C₂³	—	53	1	no	—
117	C₂×C₄	—	53	1	no	—
118	C₂×C₄	—	53	4	no	—
119	C₂×C₄	—	53	1	no	—
120	C₂×C₄	—	53	1	no	—
121	C₂×C₄	—	106	2	no	—
122	C₂×C₄	—	106	1	no	—
123	C₂×C₄	—	106	1	no	—
124	C₂×C₄	#13133286	1	1	yes	characteristic
125	C₂×C₄	#13133286	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
127	C₈	—	106	1	no	—
128	C₈	—	106	1	no	—
129	C₈	—	424	4	no	—
130	C₈	#13133287	1	2	yes	—
131	C₈	D₅₃⋊M₁₆	1	2	yes	—
132	C₂²	—	53	2	no	—
133	C₂²	—	106	1	no	—
134	C₂²	—	106	1	no	—
135	C₂²	C₂×#13133286	1	1	yes	characteristic
136	C₄	—	53	2	no	—
137	C₄	—	106	1	no	—
138	C₄	—	106	1	no	—
139	C₄	#16811550	1	2	yes	—
140	C₄	#16811551	1	2	yes	—
141	C₄	#16811628	1	1	yes	characteristic
142	C₄	C₂×#13133287	1	1	yes	characteristic
143	C₂	—	106	2	no	—
144	C₂	#20290274	1	1	yes	characteristic
145	C₂	C₂×#16811550	1	1	yes	characteristic
146	C₂	C₂×#16811551	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290273

C₂

yes

maximal

#20289749

C₂

yes

maximal, complemented

C₄×C₁₆×D₅₃

C₂

yes

maximal, characteristic

#16811627

—

#16811486

—

#16811486

C₂²

yes

—

C₂×C₁₆×D₅₃

C₄

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

C₄×#11719908

C₄

yes

characteristic

C₁₆×Dic₅₃

C₂²

yes

—

C₄×C₁₆×C₅₃

C₄

yes

—

—

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

Q₈

yes

characteristic

C₄²×D₅₃

D₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

—

C₁₆×D₅₃

—

C₄×#88322

Q₈

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₈×Dic₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₅₃

D₈

yes

characteristic

C₂×C₄×D₅₃

Q₁₆

yes

characteristic

C₂×#88324

—

#11719936

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

D₈

yes

characteristic

C₄×Dic₅₃

Q₁₆

yes

characteristic

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₂×Q₈

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

M₁₆

yes

C₂²×D₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

#107

yes

characteristic

C₂×Dic₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

C₂×Q₁₆

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

SD₃₂

yes

—

#88324

—

C₈×C₅₃

#108

yes

—

C₈×C₅₃

C₂×M₁₆

yes

—

—

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#367

yes

—

C₄×C₅₃

#367

yes

—

C₄×C₅₃

#368

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×SD₃₂

yes

—

C₂×C₄×C₁₆

—

#1567

—

#1372

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂²×C₁₆

—

#423

—

106

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

yes

—

—

106

—

C₅₃

#12023

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

212

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂×#6157

yes

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

212

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#6157

yes

characteristic

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

M₁₆

—

424

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

—

106

C₂×C₈

—

106

—

107

C₂×C₈

—

212

—

108

C₂×C₈

C₄⋊#6157

yes

characteristic

109

C₂×C₈

C₂²×#6157

yes

characteristic

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

C₄⋊#6157

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₂³

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

106

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

106

—

124

C₂×C₄

#13133286

yes

characteristic

125

C₂×C₄

#13133286

yes

characteristic

126

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

127

C₈

—

106

—

128

C₈

—

106

—

129

C₈

—

424

—

130

C₈

#13133287

yes

—

131

C₈

D₅₃⋊M₁₆

yes

—

132

C₂²

—

133

C₂²

—

106

—

134

C₂²

—

106

—

135

C₂²

C₂×#13133286

yes

characteristic

136

C₄

—

137

C₄

—

106

—

138

C₄

—

106

—

139

C₄

#16811550

yes

—

140

C₄

#16811551

yes

—

141

C₄

#16811628

yes

characteristic

142

C₄

C₂×#13133287

yes

characteristic

143

C₂

—

106

—

144

C₂

#20290274

yes

characteristic

145

C₂

C₂×#16811550

yes

characteristic

146

C₂

C₂×#16811551

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13133286
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₄×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂²×D₅₃	#107
cosocle	C₈×Dic₅₃	C₂³
hypercenter	C₄×C₁₆	#6157
hypocenter	C₅₃	#12023
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#13133286

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₄×C₁₆×C₅₃

C₄

C₂²×C₅₃

C₂²×D₅₃

C₈×Dic₅₃

C₂³

C₄×C₁₆

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. #20289749	C₂	semidirect
41. #11719908	M₁₆	semidirect
85. C₅₃	#12023	semidirect

H ≅ G/N

type

2. #20289749

C₂

semidirect

41. #11719908

M₁₆

semidirect

85. C₅₃

#12023

semidirect

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

subgroup	index
#20290273	2
#20289749	2
C₄×C₁₆×D₅₃	2
#12023	53

subgroup

index

#20290273

#20289749

C₄×C₁₆×D₅₃

#12023

Subgroups — #28309914

Conjugacy classes of subgroups

Complements

Maximal subgroups