Group #28309973 — subgroups

Subgroups — #28309973

Order 13568.

226 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 76 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	#20290303	C₂	1	2	yes	maximal
2	#20289746	C₂	1	1	yes	maximal, characteristic
3	#20289672	C₂	1	2	yes	maximal
4	C₁₆×#88324	C₂	1	1	yes	maximal, characteristic
5	#20276573	C₂	1	1	yes	maximal, characteristic
6	#16811672	—	2	2	no	—
7	#16811509	C₂²	1	1	yes	characteristic
8	#16811486	C₂²	1	2	yes	—
9	#16811473	—	2	2	no	—
10	C₈×#88324	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₂₄	C₄	1	1	yes	characteristic
12	#16810717	C₂²	1	2	yes	—
13	#16810426	C₄	1	2	yes	—
14	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#366	C₄	1	1	yes	characteristic
16	C₂×#11719910	—	2	1	no	—
17	#13133284	—	4	2	no	—
18	C₄⋊#88324	—	2	2	no	—
19	C₄×#88324	—	4	2	no	—
20	C₄×#88324	D₄	1	1	yes	characteristic
21	C₄×#88324	D₄	1	1	yes	characteristic
22	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
23	#13133182	—	2	2	no	—
24	Dic₄₂₄	C₈	1	4	yes	complemented
25	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
26	#13133140	—	2	2	no	—
27	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
29	#107×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×Dic₁₀₆	—	2	2	no	—
31	C₂×#88324	—	2	1	no	—
32	C₂×#88324	—	4	2	no	—
33	C₂×#88324	D₈	1	1	yes	characteristic
34	C₂×#88324	D₈	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₄⋊C₄×C₅₃	—	2	2	no	—
37	#11719919	—	4	2	no	—
38	C₄×Dic₅₃	—	4	2	no	—
39	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×#88322	#41	1	1	yes	characteristic
41	Dic₂₁₂	—	2	2	no	—
42	Dic₂₁₂	C₂×C₈	1	4	yes	—
43	#11719910	—	2	2	no	—
44	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
45	#11719908	—	2	1	no	—
46	C₂×C₄×C₅₃	—	4	2	no	—
47	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
48	C₂×Dic₅₃	—	4	2	no	—
49	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₅₃	#135	1	1	yes	characteristic
51	Dic₁₀₆	—	2	4	no	—
52	Dic₁₀₆	—	4	2	no	—
53	#88324	—	2	2	no	—
54	#88324	—	2	1	no	—
55	#88324	—	8	2	no	—
56	#88324	Q₃₂	1	2	yes	complemented
57	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
58	C₈×C₅₃	#131	1	1	yes	characteristic
59	#88322	—	2	1	no	—
60	C₈×Q₃₂	—	53	1	no	maximal, Sylow
61	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
62	C₄×C₅₃	—	8	2	no	—
63	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
64	C₄×C₅₃	#443	1	1	yes	characteristic
65	Dic₅₃	—	2	1	no	—
66	Dic₅₃	—	4	4	no	—
67	Dic₅₃	C₂×Q₃₂	1	1	yes	characteristic
68	Dic₅₃	#508	1	1	yes	characteristic
69	C₄×Q₃₂	—	53	1	no	—
70	C₈×Q₁₆	—	53	2	no	—
71	#1371	—	53	1	no	—
72	#1332	—	53	2	no	—
73	C₈×C₁₆	—	53	1	no	—
74	C₂×C₅₃	C₈×D₈	1	1	yes	characteristic
75	C₂×C₅₃	C₄×Q₃₂	1	1	yes	characteristic
76	C₂×C₅₃	#2178	1	1	yes	characteristic
77	C₂×Q₃₂	—	53	1	no	—
78	C₈×Q₈	—	106	2	no	—
79	C₄×Q₁₆	—	53	2	no	—
80	#366	#6157	1	1	yes	characteristic
81	#358	—	53	2	no	—
82	C₄×C₁₆	—	53	1	no	—
83	C₄×C₁₆	—	53	1	no	—
84	#335	—	53	2	no	—
85	Q₈⋊C₈	—	106	2	no	—
86	C₈²	—	53	1	no	—
87	C₅₃	C₈×Q₃₂	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	—	53	2	no	—
89	C₄×Q₈	—	106	2	no	—
90	Q₃₂	—	53	4	no	—
91	C₂×C₁₆	—	53	1	no	—
92	C₂×C₁₆	—	106	1	no	—
93	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
94	#107	C₂×#6157	1	2	yes	—
95	C₄⋊C₈	—	106	2	no	—
96	C₄⋊C₈	—	212	2	no	—
97	#103	—	106	2	no	—
98	C₄×C₈	—	53	1	no	—
99	C₄×C₈	—	53	1	no	—
100	C₄×C₈	—	53	1	no	—
101	C₄×C₈	—	212	2	no	—
102	C₂×Q₈	—	106	2	no	—
103	Q₁₆	—	53	4	no	—
104	Q₁₆	—	106	2	no	—
105	C₂×C₈	—	53	1	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	53	1	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	—	212	2	no	—
110	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	212	2	no	—
113	C₄²	—	53	1	no	—
114	C₄²	—	212	2	no	—
115	C₁₆	—	106	2	no	—
116	C₁₆	—	106	1	no	—
117	C₁₆	D₅₃⋊C₈	1	2	yes	—
118	Q₈	—	106	4	no	—
119	Q₈	—	212	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	53	1	no	—
122	C₂×C₄	—	53	1	no	—
123	C₂×C₄	—	212	2	no	—
124	C₂×C₄	D₄×#6157	1	1	yes	characteristic
125	C₈	—	53	2	no	—
126	C₈	—	106	1	no	—
127	C₈	—	106	2	no	—
128	C₈	—	106	1	no	—
129	C₈	—	424	2	no	—
130	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
131	C₈	#13133305	1	1	yes	characteristic
132	C₂²	D₈×#6157	1	1	yes	characteristic
133	C₄	—	8	2	no	—
134	C₄	—	53	1	no	—
135	C₄	—	53	1	no	—
136	C₄	—	106	1	no	—
137	C₄	—	212	4	no	—
138	C₄	#16811651	1	1	yes	characteristic
139	C₄	#16811667	1	1	yes	characteristic
140	C₂	#20290296	1	1	yes	characteristic
141	C₂	Q₃₂×#6157	1	1	yes	characteristic
142	C₂	#20291114	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290303

C₂

yes

maximal

#20289746

C₂

yes

maximal, characteristic

#20289672

C₂

yes

maximal

C₁₆×#88324

C₂

yes

maximal, characteristic

#20276573

C₂

yes

maximal, characteristic

#16811672

—

#16811509

C₂²

yes

characteristic

#16811486

C₂²

yes

—

#16811473

—

C₈×#88324

C₂²

yes

characteristic

C₂×Dic₄₂₄

C₄

yes

characteristic

#16810717

C₂²

yes

—

#16810426

C₄

yes

—

C₁₆×Dic₅₃

C₂²

yes

characteristic

C₅₃×#366

C₄

yes

—

—

—

—

D₄

yes

characteristic

C₄×#88324

D₄

yes

characteristic

C₂×Dic₂₁₂

C₂×C₄

yes

—

#13133182

—

Dic₄₂₄

C₈

yes

complemented

C₂×#11719908

C₂×C₄

yes

characteristic

#13133140

—

C₈×Dic₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

#107×C₅₃

C₂×C₄

yes

—

C₂×Dic₁₀₆

—

C₂×#88324

—

C₂×#88324

—

C₂×#88324

D₈

yes

characteristic

C₂×#88324

D₈

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×D₄

yes

yes

Dic₂₁₂

—

Dic₂₁₂

C₂×C₈

yes

—

#11719910

—

C₁₆×C₅₃

C₂×C₈

yes

—

#11719908

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

#135

yes

characteristic

Dic₁₀₆

—

Dic₁₀₆

—

—

—

—

Q₃₂

yes

complemented

C₈×C₅₃

C₂²×C₈

yes

C₈×C₅₃

yes

—

C₈×Q₃₂

—

maximal, Sylow

C₂²×C₅₃

C₄×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

#443

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×Q₃₂

yes

characteristic

Dic₅₃

#508

yes

characteristic

C₄×Q₃₂

—

C₈×Q₁₆

—

#1371

—

#1332

—

C₈×C₁₆

—

C₂×C₅₃

C₈×D₈

yes

characteristic

C₂×C₅₃

C₄×Q₃₂

yes

characteristic

C₂×C₅₃

#2178

yes

characteristic

C₂×Q₃₂

—

C₈×Q₈

—

106

—

C₄×Q₁₆

—

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

106

—

C₈²

—

C₅₃

C₈×Q₃₂

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

106

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

yes

yes

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

212

—

#103

—

106

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

212

—

102

C₂×Q₈

—

106

—

103

Q₁₆

—

104

Q₁₆

—

106

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

106

—

109

C₂×C₈

—

212

—

110

C₂×C₈

C₂²×#6157

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

212

—

113

C₄²

—

114

C₄²

—

212

—

115

C₁₆

—

106

—

116

C₁₆

—

106

—

117

C₁₆

D₅₃⋊C₈

yes

—

118

Q₈

—

106

—

119

Q₈

—

212

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

212

—

124

C₂×C₄

D₄×#6157

yes

characteristic

125

C₈

—

126

C₈

—

106

—

127

C₈

—

106

—

128

C₈

—

106

—

129

C₈

—

424

—

130

C₈

C₂×D₅₃⋊C₈

yes

characteristic

131

C₈

#13133305

yes

characteristic

132

C₂²

D₈×#6157

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

106

—

137

C₄

—

212

—

138

C₄

#16811651

yes

characteristic

139

C₄

#16811667

yes

characteristic

140

C₂

#20290296

yes

characteristic

141

C₂

Q₃₂×#6157

yes

characteristic

142

C₂

#20291114

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×#6157
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#366	C₄
socle, prime radical	C₂²×C₅₃	C₄×D₈
cosocle	C₈×Dic₅₃	C₂³
hypercenter	#366	#6157
hypocenter	C₅₃	C₈×Q₃₂
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

subgroup

G/H

center

C₂²

D₈×#6157

commutator

C₈×C₅₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₅₃×#366

C₄

socle, prime radical

C₂²×C₅₃

C₄×D₈

cosocle

C₈×Dic₅₃

C₂³

C₅₃

C₈×Q₃₂

LCS

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

N	H ≅ G/N	type
24. Dic₄₂₄	C₈	semidirect
56. #88324	Q₃₂	semidirect
87. C₅₃	C₈×Q₃₂	semidirect

H ≅ G/N

type

24. Dic₄₂₄

C₈

semidirect

56. #88324

Q₃₂

semidirect

87. C₅₃

C₈×Q₃₂

semidirect

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

subgroup	index
#20290303	2
#20289746	2
#20289672	2
C₁₆×#88324	2
#20276573	2
C₈×Q₃₂	53

subgroup

index

C₁₆×#88324

C₈×Q₃₂

Subgroups — #28309973

Conjugacy classes of subgroups

Complements

Maximal subgroups