Group #28312636 — subgroups

Subgroups — #28312636

Order 13568.

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

normal subgroups only

This group has 202 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	C₂×#16811556	C₂	1	2	yes	maximal, complemented
2	C₄⋊#13133267	C₂	1	1	yes	maximal, characteristic, complemented
3	#20289829	C₂	1	2	yes	maximal, complemented
4	C₄×#13133267	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276534	C₂	1	1	yes	maximal, characteristic
6	#16811556	—	2	4	no	—
7	C₂×#13133267	C₂²	1	1	yes	characteristic
8	C₂×#13133267	C₂²	1	2	yes	—
9	C₂×#13133267	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
11	C₈×D₂₁₂	C₄	1	1	yes	characteristic
12	#16810402	C₄	1	2	yes	—
13	C₄×#11719908	C₂²	1	1	yes	characteristic
14	C₅₃×C₄⋊C₁₆	C₄	1	1	yes	characteristic
15	#13133267	—	2	2	no	—
16	#13133267	—	2	1	no	—
17	#13133267	D₄	1	2	yes	complemented
18	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
19	C₄×D₂₁₂	C₈	1	1	yes	characteristic
20	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
21	C₂×#11719908	C₂³	1	1	yes	characteristic
22	C₁₆×D₅₃	—	2	2	no	—
23	C₁₆×D₅₃	—	2	2	no	—
24	C₄⋊#88322	C₈	1	1	yes	characteristic
25	#13133129	C₈	1	2	yes	—
26	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×D₂₁₂	C₁₆	1	1	yes	characteristic
29	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
33	#11719921	C₁₆	1	2	yes	—
34	C₄⋊Dic₅₃	C₁₆	1	1	yes	characteristic
35	C₂×#88322	C₂×C₈	1	2	yes	—
36	C₈×D₅₃	—	2	2	no	—
37	C₈×D₅₃	—	2	2	no	—
38	C₁₆×C₅₃	—	2	2	no	—
39	#11719908	—	2	1	no	—
40	#11719908	C₂×D₄	1	1	yes	characteristic
41	#11719908	#41	1	1	yes	characteristic
42	C₂²×D₅₃	C₂×C₁₆	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
47	D₂₁₂	—	2	2	no	—
48	C₄×D₅₃	—	2	2	no	—
49	C₄×D₅₃	—	2	2	no	—
50	C₈×C₅₃	—	2	1	no	—
51	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
52	C₈×C₅₃	#131	1	1	yes	characteristic
53	#88322	—	2	2	no	—
54	#13646	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
56	C₂×D₅₃	—	2	2	no	—
57	C₂×D₅₃	—	2	2	no	—
58	C₂×D₅₃	—	2	2	no	—
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	M₆₄	1	2	yes	complemented
61	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
62	C₄×C₅₃	#504	1	1	yes	characteristic
63	Dic₅₃	—	2	2	no	—
64	C₂×M₆₄	—	53	2	no	—
65	D₄×C₁₆	—	53	1	no	—
66	C₄⋊C₃₂	—	53	1	no	—
67	C₂²⋊C₃₂	—	53	2	no	—
68	C₄×C₃₂	—	53	1	no	—
69	C₂×C₅₃	D₄×C₁₆	1	1	yes	characteristic
70	C₂×C₅₃	C₂×M₆₄	1	1	yes	characteristic
71	C₂×C₅₃	#2258	1	1	yes	characteristic
72	D₅₃	—	4	2	no	—
73	C₂²×C₁₆	—	53	2	no	—
74	C₈×D₄	—	53	1	no	—
75	M₆₄	—	106	4	no	—
76	C₂×C₃₂	—	53	1	no	—
77	C₂×C₃₂	—	53	1	no	—
78	C₂×C₃₂	—	53	2	no	—
79	C₄⋊C₁₆	#6157	1	1	yes	characteristic
80	C₂²⋊C₁₆	—	53	2	no	—
81	C₄×C₁₆	—	53	1	no	—
82	C₅₃	#13646	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₈	—	53	2	no	—
84	C₄×D₄	—	53	1	no	—
85	C₂×C₁₆	—	53	1	no	—
86	C₂×C₁₆	—	53	1	no	—
87	C₂×C₁₆	—	106	2	no	—
88	C₂×C₁₆	—	106	2	no	—
89	C₂×C₁₆	C₂×#6157	1	2	yes	—
90	C₄⋊C₈	—	53	1	no	—
91	C₂²⋊C₈	—	53	2	no	—
92	C₄×C₈	C₂×#6157	1	1	yes	characteristic
93	C₃₂	—	53	2	no	—
94	C₃₂	—	106	2	no	—
95	C₃₂	—	106	1	no	—
96	C₂×D₄	—	53	1	no	—
97	C₂²×C₄	—	53	2	no	—
98	C₂×C₈	—	53	2	no	—
99	C₂×C₈	—	106	2	no	—
100	C₂×C₈	—	106	2	no	—
101	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
102	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
103	C₄⋊C₄	—	53	1	no	—
104	C₂²⋊C₄	—	53	2	no	—
105	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
106	C₁₆	—	2	2	no	—
107	C₁₆	—	53	1	no	—
108	C₁₆	—	53	1	no	—
109	C₁₆	—	106	1	no	—
110	C₂³	—	53	2	no	—
111	D₄	—	106	2	no	—
112	C₂×C₄	—	53	2	no	—
113	C₂×C₄	—	106	2	no	—
114	C₂×C₄	—	106	2	no	—
115	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
116	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
117	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
118	C₈	—	2	1	no	—
119	C₈	—	106	2	no	—
120	C₈	D₄×#6157	1	1	yes	characteristic
121	C₈	#13133305	1	1	yes	characteristic
122	C₂²	—	106	2	no	—
123	C₂²	—	106	2	no	—
124	C₂²	—	106	2	no	—
125	C₂²	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	106	2	no	—
128	C₄	#16811556	1	2	yes	complemented
129	C₄	#16811651	1	1	yes	characteristic
130	C₄	#16811682	1	1	yes	characteristic
131	C₂	—	212	2	no	—
132	C₂	#20291094	1	1	yes	characteristic
133	C₂	C₂×#16811556	1	1	yes	characteristic
134	C₂	#20291139	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811556

C₂

yes

maximal, complemented

C₄⋊#13133267

C₂

yes

maximal, characteristic, complemented

#20289829

C₂

yes

maximal, complemented

C₄×#13133267

C₂

yes

maximal, characteristic, complemented

#20276534

C₂

yes

maximal, characteristic

#16811556

—

C₂×#13133267

C₂²

yes

characteristic

C₂×#13133267

C₂²

yes

—

C₂×#13133267

C₂²

yes

characteristic

C₂×C₁₆×D₅₃

C₂²

yes

—

C₈×D₂₁₂

C₄

yes

characteristic

#16810402

C₄

yes

—

C₄×#11719908

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

C₄

yes

—

—

D₄

yes

complemented

C₂×C₈×D₅₃

C₂×C₄

yes

—

C₄×D₂₁₂

C₈

yes

characteristic

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

C₂³

yes

characteristic

C₁₆×D₅₃

—

C₁₆×D₅₃

—

C₄⋊#88322

C₈

yes

characteristic

#13133129

C₈

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₂₁₂

C₁₆

yes

characteristic

C₂×C₄×D₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

#11719921

C₁₆

yes

—

C₄⋊Dic₅₃

C₁₆

yes

characteristic

C₂×#88322

C₂×C₈

yes

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

—

#11719908

C₂×D₄

yes

yes

C₂²×D₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₁₆

yes

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄×D₄

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

C₂²×C₁₆

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

M₆₄

yes

complemented

C₄×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

#504

yes

characteristic

Dic₅₃

—

C₂×M₆₄

—

D₄×C₁₆

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

C₄×C₃₂

—

C₂×C₅₃

D₄×C₁₆

yes

characteristic

C₂×C₅₃

C₂×M₆₄

yes

characteristic

C₂×C₅₃

#2258

yes

characteristic

D₅₃

—

C₂²×C₁₆

—

C₈×D₄

—

M₆₄

—

106

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

#6157

yes

characteristic

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₅₃

#13646

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂×#6157

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂×#6157

yes

characteristic

C₃₂

—

C₃₂

—

106

—

C₃₂

—

106

—

C₂×D₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

106

—

100

C₂×C₈

—

106

—

101

C₂×C₈

D₅₃⋊C₈

yes

characteristic

102

C₂×C₈

C₂²×#6157

yes

characteristic

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

D₅₃⋊C₈

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₁₆

—

109

C₁₆

—

106

—

110

C₂³

—

111

D₄

—

106

—

112

C₂×C₄

—

113

C₂×C₄

—

106

—

114

C₂×C₄

—

106

—

115

C₂×C₄

D₅₃⋊C₁₆

yes

characteristic

116

C₂×C₄

D₅₃⋊C₁₆

yes

characteristic

117

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

118

C₈

—

119

C₈

—

106

—

120

C₈

D₄×#6157

yes

characteristic

121

C₈

#13133305

yes

characteristic

122

C₂²

—

106

—

123

C₂²

—

106

—

124

C₂²

—

106

—

125

C₂²

C₂×D₅₃⋊C₁₆

yes

characteristic

126

C₄

—

127

C₄

—

106

—

128

C₄

#16811556

yes

complemented

129

C₄

#16811651

yes

characteristic

130

C₄

#16811682

yes

characteristic

131

C₂

—

212

—

132

C₂

#20291094

yes

characteristic

133

C₂

C₂×#16811556

yes

characteristic

134

C₂

#20291139

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₅₃×C₄⋊C₁₆	C₄
prime radical	C₂×D₂₁₂	C₁₆
cosocle	C₂×#11719908	C₂³
hypercenter	C₄⋊C₁₆	#6157
hypocenter	C₅₃	#13646
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#6157

commutator, socle

C₂²×C₅₃

C₂²×C₁₆

Fitting

C₅₃×C₄⋊C₁₆

C₄

prime radical

C₂×D₂₁₂

C₁₆

C₂³

C₄⋊C₁₆

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
1. C₂×#16811556	C₂	semidirect
2. C₄⋊#13133267	C₂	semidirect
3. #20289829	C₂	semidirect
4. C₄×#13133267	C₂	semidirect
17. #13133267	D₄	semidirect
60. C₄×C₅₃	M₆₄	semidirect
82. C₅₃	#13646	semidirect
128. C₄	#16811556	semidirect

H ≅ G/N

type

1. C₂×#16811556

C₂

semidirect

2. C₄⋊#13133267

C₂

semidirect

3. #20289829

C₂

semidirect

4. C₄×#13133267

C₂

semidirect

17. #13133267

D₄

semidirect

60. C₄×C₅₃

M₆₄

semidirect

82. C₅₃

#13646

semidirect

128. C₄

#16811556

semidirect

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

subgroup	index
C₂×#16811556	2
C₄⋊#13133267	2
#20289829	2
C₄×#13133267	2
#20276534	2
#13646	53

subgroup

index

Subgroups — #28312636

Conjugacy classes of subgroups

Complements

Maximal subgroups