Group #28310440 — subgroups

Subgroups — #28310440

Order 13568.

354 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 156 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	C₄×#13133291	C₂	1	2	yes	maximal
2	#20289757	C₂	1	4	yes	maximal
3	C₄×#13133127	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133291	C₂²	1	4	yes	—
5	#16811511	C₄	1	4	yes	complemented
6	C₄×#11719910	C₄	1	4	yes	complemented
7	C₂×#13133127	C₄	1	2	yes	—
8	C₂×#13133127	C₂²	1	2	yes	—
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	#13133291	—	2	8	no	—
13	C₂×#11719910	C₂×C₄	1	8	yes	—
14	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
15	C₂²×#88322	C₂³	1	1	yes	characteristic
16	C₂×C₄×Dic₅₃	—	2	2	no	—
17	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
18	#13133127	—	2	4	no	—
19	#13133127	C₈	1	8	yes	—
20	C₄×#88322	C₂×C₄	1	2	yes	—
21	C₄×#88322	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂²×Dic₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
31	C₄²×C₅₃	C₄²	1	1	yes	characteristic
32	C₄²×C₅₃	C₄²	1	1	yes	characteristic
33	C₄²×C₅₃	M₁₆	1	2	yes	—
34	C₄×Dic₅₃	—	2	2	no	—
35	C₄×Dic₅₃	—	4	4	no	—
36	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
37	C₂×#88322	C₄²	1	2	yes	—
38	C₂×#88322	C₂×C₈	1	4	yes	—
39	C₂×#88322	C₂²×C₄	1	2	yes	—
40	C₂×#88322	#41	1	4	yes	—
41	#11719910	—	2	8	no	—
42	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
43	C₂×C₄×C₅₃	—	2	4	no	—
44	C₂×C₄×C₅₃	#97	1	2	yes	—
45	C₂×C₄×C₅₃	#97	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#117	1	2	yes	—
48	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
50	C₂×Dic₅₃	—	2	4	no	—
51	C₂×Dic₅₃	—	4	4	no	—
52	C₈×C₅₃	—	2	4	no	—
53	#88322	C₄×C₈	1	4	yes	—
54	#88322	#117	1	4	yes	—
55	#12594	—	53	1	no	maximal, Sylow
56	C₂²×C₅₃	—	2	2	no	—
57	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
58	C₂²×C₅₃	#431	1	2	yes	—
59	C₄×C₅₃	—	2	2	no	—
60	C₄×C₅₃	#347	1	4	yes	—
61	C₄×C₅₃	#376	1	1	yes	characteristic
62	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄×C₅₃	#431	1	2	yes	—
64	Dic₅₃	—	4	4	no	—
65	C₄×M₃₂	—	53	2	no	—
66	C₄×#97	—	53	1	no	—
67	#1375	—	53	4	no	—
68	C₂×C₅₃	—	2	2	no	—
69	C₂×C₅₃	#1744	1	1	yes	characteristic
70	C₂×C₅₃	C₂×#347	1	2	yes	—
71	C₂×M₃₂	—	53	4	no	—
72	C₂×#97	—	53	2	no	—
73	C₂×#97	—	53	2	no	—
74	C₂×C₄×C₈	—	53	1	no	—
75	C₂×C₄×C₈	#6157	1	1	yes	characteristic
76	C₄³	—	53	1	no	—
77	#346	—	53	4	no	—
78	C₄×C₁₆	—	53	4	no	—
79	C₅₃	#12594	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	—	53	1	no	—
81	C₂²×C₈	—	53	1	no	—
82	C₂²×C₈	C₂×#6157	1	2	yes	—
83	C₂×C₄²	—	53	2	no	—
84	C₂×C₄²	—	106	2	no	—
85	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
86	M₃₂	—	106	8	no	—
87	C₂×C₁₆	—	53	8	no	—
88	#97	—	53	8	no	—
89	#97	—	106	4	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	—	53	2	no	—
92	C₄×C₈	—	53	2	no	—
93	C₂²×C₄	—	106	2	no	—
94	C₂²×C₄	#11719936	1	2	yes	—
95	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	53	2	no	—
99	C₂×C₈	—	53	2	no	—
100	C₂×C₈	—	53	4	no	—
101	C₂×C₈	—	53	4	no	—
102	C₂×C₈	D₅₃⋊C₈	1	4	yes	—
103	C₄²	—	53	4	no	—
104	C₄²	—	106	2	no	—
105	C₄²	—	212	4	no	—
106	C₄²	C₄×#6157	1	1	yes	characteristic
107	C₄²	C₄×#6157	1	1	yes	characteristic
108	C₄²	#11719940	1	2	yes	—
109	C₁₆	—	106	8	no	—
110	C₂³	C₂×#11719936	1	1	yes	characteristic
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	106	4	no	—
113	C₂×C₄	—	212	4	no	—
114	C₂×C₄	#13133272	1	2	yes	—
115	C₂×C₄	#13133272	1	2	yes	—
116	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
117	C₂×C₄	#13133296	1	2	yes	—
118	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
119	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
120	C₈	—	2	4	no	—
121	C₈	—	53	4	no	—
122	C₈	—	53	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	C₂×#13133272	1	1	yes	characteristic
125	C₂²	#16811642	1	2	yes	—
126	C₄	—	2	2	no	—
127	C₄	—	212	4	no	—
128	C₄	#16811516	1	4	yes	—
129	C₄	#16811559	1	1	yes	characteristic
130	C₄	C₄×D₅₃⋊C₈	1	1	yes	characteristic
131	C₄	#16811640	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	#20290430	1	1	yes	characteristic
134	C₂	C₂×#16811516	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133291

C₂

yes

maximal

#20289757

C₂

yes

maximal

C₄×#13133127

C₂

yes

maximal, characteristic

C₂×#13133291

C₂²

yes

—

#16811511

C₄

yes

complemented

C₄×#11719910

C₄

yes

complemented

C₂×#13133127

C₄

yes

—

C₂×#13133127

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₄²×Dic₅₃

C₄

yes

characteristic

C₂×C₄×C₈×C₅₃

C₄

yes

characteristic

#13133291

—

C₂×#11719910

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂³

yes

characteristic

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

#13133127

—

#13133127

C₈

yes

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

M₁₆

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₈

yes

—

C₂×#88322

C₄²

yes

—

C₂×#88322

C₂×C₈

yes

—

C₂×#88322

C₂²×C₄

yes

—

C₂×#88322

#41

yes

—

#11719910

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#97

yes

—

C₂×C₄×C₅₃

#97

yes

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

C₄×C₈

yes

—

#88322

#117

yes

—

#12594

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#97

yes

characteristic

C₂²×C₅₃

#431

yes

—

C₄×C₅₃

—

C₄×C₅₃

#347

yes

—

C₄×C₅₃

#376

yes

characteristic

C₄×C₅₃

C₂×C₄×C₈

yes

characteristic

C₄×C₅₃

#431

yes

—

Dic₅₃

—

C₄×M₃₂

—

C₄×#97

—

#1375

—

C₂×C₅₃

—

C₂×C₅₃

#1744

yes

characteristic

C₂×C₅₃

C₂×#347

yes

—

C₂×M₃₂

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#6157

yes

characteristic

C₄³

—

#346

—

C₄×C₁₆

—

C₅₃

#12594

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#6157

yes

—

C₂×C₄²

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×#6157

yes

characteristic

M₃₂

—

106

—

C₂×C₁₆

—

#97

—

#97

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

106

—

C₂²×C₄

#11719936

yes

—

C₂²×C₄

C₂²×#6157

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

D₅₃⋊C₈

yes

—

103

C₄²

—

104

C₄²

—

106

—

105

C₄²

—

212

—

106

C₄²

C₄×#6157

yes

characteristic

107

C₄²

C₄×#6157

yes

characteristic

108

C₄²

#11719940

yes

—

109

C₁₆

—

106

—

110

C₂³

C₂×#11719936

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

106

—

113

C₂×C₄

—

212

—

114

C₂×C₄

#13133272

yes

—

115

C₂×C₄

#13133272

yes

—

116

C₂×C₄

C₂×C₄×#6157

yes

characteristic

117

C₂×C₄

#13133296

yes

—

118

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

119

C₂×C₄

C₂×#11719940

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

C₂×#13133272

yes

characteristic

125

C₂²

#16811642

yes

—

126

C₄

—

127

C₄

—

212

—

128

C₄

#16811516

yes

—

129

C₄

#16811559

yes

characteristic

130

C₄

C₄×D₅₃⋊C₈

yes

characteristic

131

C₄

#16811640

yes

—

132

C₂

—

133

C₂

#20290430

yes

characteristic

134

C₂

C₂×#16811516

yes

—

	subgroup	G/H
center	C₄²	C₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#12594
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄²

C₄×#6157

commutator

C₄×C₅₃

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×#6157

Fitting

C₂×C₄×C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×#97

prime radical

C₂³×C₅₃

C₂×M₁₆

cosocle

C₂²×#88322

C₂³

C₂×C₄×C₈

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
5. #16811511	C₄	semidirect
6. C₄×#11719910	C₄	semidirect
79. C₅₃	#12594	semidirect

H ≅ G/N

type

5. #16811511

C₄

semidirect

6. C₄×#11719910

C₄

semidirect

79. C₅₃

#12594

semidirect

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

subgroup	index
C₄×#13133291	2
#20289757	2
C₄×#13133127	2
#12594	53

subgroup

index

Subgroups — #28310440

Conjugacy classes of subgroups

Complements

Maximal subgroups