Group #28308143 — subgroups

Subgroups — #28308143

Order 13568.

296 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 106 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	#20289653	C₂	1	4	yes	maximal
3	#20283650	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11719910	—	2	2	no	—
5	C₂²⋊#11719910	C₂²	1	4	yes	—
6	C₄×#11719910	—	2	2	no	—
7	#16810754	—	2	2	no	—
8	Q₈×#88322	—	2	2	no	—
9	#16810650	C₂²	1	2	yes	—
10	#16810647	C₄	1	1	yes	characteristic
11	#16810579	C₄	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₅₃×#533	C₄	1	1	yes	characteristic
14	C₂×#11719910	—	2	4	no	—
15	C₂×#11719917	C₂×C₄	1	2	yes	—
16	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
17	C₂²×#88322	C₂³	1	1	yes	characteristic
18	C₄⋊#88322	—	2	2	no	—
19	C₄⋊#88322	—	2	4	no	—
20	#13133128	—	2	2	no	—
21	C₄×#88322	—	2	2	no	—
22	C₄×#88322	D₄	1	2	yes	—
23	C₄×#88322	D₄	1	2	yes	—
24	C₄⋊Q₈×C₅₃	C₈	1	2	yes	—
25	#125×C₅₃	C₈	1	2	yes	—
26	C₂²⋊Q₈×C₅₃	—	2	2	no	—
27	C₄×Q₈×C₅₃	—	2	2	no	—
28	#117×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×Q₈×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₅₃	—	2	4	no	—
34	C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
36	C₂²⋊C₄×C₅₃	—	2	2	no	—
37	C₄²×C₅₃	—	2	2	no	—
38	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
41	#11719917	—	4	2	no	—
42	C₂×#88322	—	2	2	no	—
43	C₂×#88322	—	2	2	no	—
44	C₂×#88322	—	4	1	no	—
45	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×#88322	C₂×D₄	1	2	yes	—
48	#11719910	—	4	4	no	—
49	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
50	Q₈×C₅₃	—	4	4	no	—
51	C₂×C₄×C₅₃	—	2	4	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	2	4	no	—
55	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
59	#88322	—	2	2	no	—
60	#88322	—	2	1	no	—
61	#88322	—	4	2	no	—
62	#88322	#101	1	2	yes	—
63	#11496	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	—	2	2	no	—
65	C₂²×C₅₃	—	2	1	no	—
66	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	—	4	4	no	—
70	C₄×C₅₃	#350	1	4	yes	—
71	C₄×C₅₃	#410	1	1	yes	characteristic
72	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
73	#2960	—	53	1	no	—
74	#2162	—	53	2	no	—
75	#1323	—	53	4	no	—
76	C₂×C₅₃	—	4	1	no	—
77	C₂×C₅₃	#1461	1	1	yes	characteristic
78	C₂×C₅₃	C₂×#350	1	2	yes	—
79	#533	#6157	1	1	yes	characteristic
80	#446	—	106	2	no	—
81	C₈×Q₈	—	106	2	no	—
82	#424	—	53	2	no	—
83	#423	—	53	1	no	—
84	#405	—	53	2	no	—
85	C₂×C₄×C₈	—	53	1	no	—
86	C₄⋊C₁₆	—	106	2	no	—
87	C₂²⋊C₁₆	—	53	4	no	—
88	C₄×C₁₆	—	106	2	no	—
89	C₅₃	#11496	1	1	yes	characteristic, complemented, Sylow
90	C₂×M₁₆	—	53	2	no	—
91	C₂²×C₈	—	53	1	no	—
92	C₂²×C₈	—	53	1	no	—
93	C₄⋊Q₈	#88324	1	2	yes	—
94	#125	#88324	1	2	yes	—
95	C₂²⋊Q₈	—	2	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	#117	C₂×#6157	1	2	yes	—
98	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
99	C₂×C₁₆	—	106	4	no	—
100	C₄⋊C₈	—	106	4	no	—
101	C₄⋊C₈	—	106	2	no	—
102	#97	—	106	2	no	—
103	C₄×C₈	—	53	2	no	—
104	C₄×C₈	—	53	2	no	—
105	C₄×C₈	—	106	2	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂²×C₄	#11719940	1	2	yes	—
108	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
109	M₁₆	—	212	2	no	—
110	C₂×C₈	—	53	1	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	53	2	no	—
113	C₂×C₈	—	106	2	no	—
114	C₂×C₈	—	106	2	no	—
115	C₂×C₈	—	212	1	no	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	C₂×#88324	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	C₂×#88324	1	1	yes	characteristic
122	C₄²	C₂×#88324	1	1	yes	characteristic
123	C₄²	C₂²⋊#6157	1	2	yes	—
124	C₁₆	—	212	4	no	—
125	C₂³	C₂×#11719940	1	1	yes	characteristic
126	Q₈	—	4	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	C₂²⋊#88324	1	2	yes	—
132	C₂×C₄	C₂²⋊#88324	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
134	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
135	C₈	—	53	2	no	—
136	C₈	—	106	2	no	—
137	C₈	—	106	1	no	—
138	C₈	—	212	2	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×C₂²⋊#88324	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	4	no	—
145	C₄	#16811521	1	4	yes	—
146	C₄	#16811604	1	1	yes	characteristic
147	C₄	D₅₃⋊#101	1	1	yes	characteristic
148	C₂	—	4	1	no	—
149	C₂	#20289958	1	1	yes	characteristic
150	C₂	C₂×#16811521	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20291072

C₂

yes

maximal

#20289653

C₂

yes

maximal

#20283650

C₂

yes

maximal, characteristic

C₄⋊#11719910

—

C₂²⋊#11719910

C₂²

yes

—

—

—

—

C₂²

yes

—

#16810647

C₄

yes

characteristic

#16810579

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₅₃×#533

C₄

yes

characteristic

C₂×#11719910

—

C₂×#11719917

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×#88322

D₄

yes

—

C₄⋊Q₈×C₅₃

C₈

yes

—

#125×C₅₃

C₈

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₈

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₂×D₄

yes

—

#11719910

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

—

—

—

yes

—

#11496

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#350

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#1461

yes

characteristic

C₂×C₅₃

C₂×#350

yes

—

yes

—

106

—

C₈×Q₈

—

106

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

106

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

106

—

C₅₃

#11496

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

#88324

yes

—

#125

#88324

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×#6157

yes

—

C₂×C₄²

C₂×#6157

yes

characteristic

C₂×C₁₆

—

106

—

100

C₄⋊C₈

—

106

—

101

C₄⋊C₈

—

106

—

102

#97

—

106

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

—

106

C₂×Q₈

—

107

C₂²×C₄

#11719940

yes

—

108

C₂²×C₄

C₂²×#6157

yes

characteristic

109

M₁₆

—

212

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

106

—

114

C₂×C₈

—

106

—

115

C₂×C₈

—

212

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂×#88324

yes

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

C₂×#88324

yes

characteristic

122

C₄²

C₂×#88324

yes

characteristic

123

C₄²

C₂²⋊#6157

yes

—

124

C₁₆

—

212

—

125

C₂³

C₂×#11719940

yes

characteristic

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²⋊#88324

yes

—

132

C₂×C₄

C₂²⋊#88324

yes

—

133

C₂×C₄

C₂×C₂²⋊#6157

yes

characteristic

134

C₂×C₄

C₂²×#88324

yes

characteristic

135

C₈

—

136

C₈

—

106

—

137

C₈

—

106

—

138

C₈

—

212

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₂²⋊#88324

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#16811521

yes

—

146

C₄

#16811604

yes

characteristic

147

C₄

D₅₃⋊#101

yes

characteristic

148

C₂

—

149

C₂

#20289958

yes

characteristic

150

C₂

C₂×#16811521

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×#533	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×#88322	C₂³
hypercenter	#533	#6157
hypocenter	C₅₃	#11496
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6157

Fitting

C₅₃×#533

C₄

socle

C₂²×C₅₃

C₂×C₂²⋊C₈

prime radical

C₂³×C₅₃

C₂×M₁₆

cosocle

C₂²×#88322

C₂³

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
89. C₅₃	#11496	semidirect

H ≅ G/N

type

89. C₅₃

#11496

semidirect

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

subgroup	index
#20291072	2
#20289653	2
#20283650	2
#11496	53

subgroup

index

Subgroups — #28308143

Conjugacy classes of subgroups

Complements

Maximal subgroups