Group #27299528 — subgroups

Subgroups — #27299528

Order 13568.

958 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 219 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×C₂²⋊#11719908	C₂	1	6	yes	maximal, complemented
2	C₂²×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11719908	C₂²	1	4	yes	—
4	C₂²⋊#11719908	—	2	12	no	—
5	C₂²×#41×C₅₃	C₄	1	1	yes	characteristic
6	C₂×#131×C₅₃	—	2	12	no	—
7	C₂²×M₁₆×C₅₃	C₄	1	3	yes	—
8	C₂³×C₈×C₅₃	C₂²	1	3	yes	—
9	C₂×#11719908	—	2	12	no	—
10	C₂×#41×C₅₃	—	2	6	no	—
11	C₂×#41×C₅₃	—	2	6	no	—
12	C₂²×Q₈×C₅₃	C₈	1	1	yes	characteristic
13	C₂²×D₄×C₅₃	C₈	1	3	yes	—
14	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
15	#131×C₅₃	—	2	32	no	—
16	C₂×M₁₆×C₅₃	—	2	12	no	—
17	C₂×M₁₆×C₅₃	—	2	6	no	—
18	C₂²×C₈×C₅₃	—	2	12	no	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
20	C₂²×C₈×C₅₃	D₄	1	12	yes	—
21	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	C₂⁴×C₅₃	C₂×C₈	1	3	yes	—
23	#41×C₅₃	—	2	8	no	—
24	#41×C₅₃	—	2	24	no	—
25	C₂×Q₈×C₅₃	—	2	6	no	—
26	C₂×D₄×C₅₃	—	2	12	no	—
27	C₂×D₄×C₅₃	—	2	6	no	—
28	C₂²×C₄×C₅₃	—	2	6	no	—
29	C₂²×C₄×C₅₃	—	2	6	no	—
30	C₂²×C₄×C₅₃	C₂²⋊C₄	1	6	yes	—
31	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
32	C₂²×C₄×C₅₃	M₁₆	1	6	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	M₁₆×C₅₃	—	2	24	no	—
35	C₂×C₈×C₅₃	—	2	6	no	—
36	C₂×C₈×C₅₃	—	2	24	no	—
37	C₂×C₈×C₅₃	C₂²⋊C₄	1	6	yes	—
38	C₂×C₈×C₅₃	C₂×D₄	1	6	yes	—
39	#11719908	—	4	4	no	—
40	C₂³×C₅₃	—	2	6	no	—
41	C₂³×C₅₃	—	2	12	no	—
42	C₂³×C₅₃	C₂²⋊C₈	1	6	yes	—
43	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
44	Q₈×C₅₃	—	2	8	no	—
45	D₄×C₅₃	—	2	24	no	—
46	C₂×C₄×C₅₃	—	2	6	no	—
47	C₂×C₄×C₅₃	—	2	6	no	—
48	C₂×C₄×C₅₃	—	2	6	no	—
49	C₂×C₄×C₅₃	—	2	12	no	—
50	C₂×C₄×C₅₃	C₂²⋊C₈	1	6	yes	—
51	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
52	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
53	C₈×C₅₃	—	2	6	no	—
54	C₈×C₅₃	#120	1	4	yes	—
55	#12680	—	53	1	no	maximal, Sylow
56	C₂²×C₅₃	—	2	12	no	—
57	C₂²×C₅₃	—	2	6	no	—
58	C₂²×C₅₃	—	4	12	no	—
59	C₂²×C₅₃	C₂×C₂²⋊C₈	1	3	yes	—
60	C₂²×C₅₃	#504	1	4	yes	—
61	C₄×C₅₃	—	2	6	no	—
62	C₄×C₅₃	#379	1	1	yes	characteristic
63	C₄×C₅₃	#407	1	3	yes	—
64	C₂²×#131	D₅₃	1	1	yes	characteristic
65	C₂×C₂²⋊C₁₆	—	53	6	no	—
66	C₂×C₅₃	—	4	6	no	—
67	C₂×C₅₃	#1779	1	1	yes	characteristic
68	C₂×C₅₃	#2113	1	6	yes	—
69	C₂²×#41	Dic₅₃	1	1	yes	characteristic
70	C₂×#131	—	2	12	no	—
71	C₂²×M₁₆	Dic₅₃	1	3	yes	—
72	C₂³×C₈	C₂×D₅₃	1	3	yes	—
73	C₂²×C₁₆	—	53	4	no	—
74	C₂²⋊C₁₆	—	106	12	no	—
75	C₅₃	#12680	1	1	yes	characteristic, complemented, Sylow
76	C₂×#41	—	2	6	no	—
77	C₂×#41	—	2	6	no	—
78	C₂²×Q₈	#88322	1	1	yes	characteristic
79	C₂²×D₄	#88322	1	3	yes	—
80	C₂³×C₄	C₂×Dic₅₃	1	3	yes	—
81	#131	—	2	32	no	—
82	C₂×M₁₆	—	2	6	no	—
83	C₂×M₁₆	—	2	12	no	—
84	C₂²×C₈	—	2	12	no	—
85	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
86	C₂²×C₈	C₂²⋊D₅₃	1	12	yes	—
87	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
88	C₂×C₁₆	—	106	12	no	—
89	C₂⁴	C₂×#88322	1	3	yes	—
90	#41	—	2	8	no	—
91	#41	—	2	24	no	—
92	C₂×Q₈	—	2	6	no	—
93	C₂×D₄	—	2	12	no	—
94	C₂×D₄	—	2	6	no	—
95	C₂²×C₄	—	2	6	no	—
96	C₂²×C₄	—	2	6	no	—
97	C₂²×C₄	C₂×#88322	1	3	yes	—
98	C₂²×C₄	#11719917	1	6	yes	—
99	C₂²×C₄	C₂²⋊Dic₅₃	1	6	yes	—
100	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
101	M₁₆	—	2	24	no	—
102	C₂×C₈	—	2	6	no	—
103	C₂×C₈	—	2	24	no	—
104	C₂×C₈	C₂²⋊Dic₅₃	1	6	yes	—
105	C₂×C₈	C₂×C₂²⋊D₅₃	1	6	yes	—
106	C₁₆	—	212	4	no	—
107	C₂³	—	2	12	no	—
108	C₂³	—	2	6	no	—
109	C₂³	C₂²⋊#88322	1	6	yes	—
110	C₂³	C₂²×#88322	1	1	yes	characteristic
111	Q₈	—	2	8	no	—
112	D₄	—	2	24	no	—
113	C₂×C₄	—	2	6	no	—
114	C₂×C₄	—	2	12	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	2	6	no	—
117	C₂×C₄	C₂²⋊#88322	1	6	yes	—
118	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	3	yes	—
119	C₂×C₄	C₂×#11719917	1	3	yes	—
120	C₈	—	2	6	no	—
121	C₈	#13133186	1	4	yes	—
122	C₂²	—	2	12	no	—
123	C₂²	—	2	6	no	—
124	C₂²	—	4	12	no	—
125	C₂²	C₂×C₂²⋊#88322	1	3	yes	—
126	C₂²	#16811040	1	4	yes	—
127	C₄	—	2	6	no	—
128	C₄	#16810479	1	1	yes	characteristic
129	C₄	#16810587	1	3	yes	—
130	C₂	—	4	6	no	—
131	C₂	#20274625	1	1	yes	characteristic
132	C₂	#20276329	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊#11719908

C₂

yes

maximal, complemented

C₂²×#131×C₅₃

C₂

yes

maximal, characteristic

C₂²×#11719908

C₂²

yes

—

C₂²⋊#11719908

—

C₂²×#41×C₅₃

C₄

yes

characteristic

C₂×#131×C₅₃

—

C₂²×M₁₆×C₅₃

C₄

yes

—

C₂³×C₈×C₅₃

C₂²

yes

—

C₂×#11719908

—

C₂×#41×C₅₃

—

C₂×#41×C₅₃

—

C₂²×Q₈×C₅₃

C₈

yes

characteristic

C₂²×D₄×C₅₃

C₈

yes

—

C₂³×C₄×C₅₃

C₂×C₄

yes

—

#131×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂⁴×C₅₃

C₂×C₈

yes

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

—

C₂²×C₄×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

#11719908

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₈

yes

—

C₂³×C₅₃

C₂²×C₈

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

#120

yes

—

#12680

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

—

C₂²×C₅₃

#504

yes

—

C₄×C₅₃

—

C₄×C₅₃

#379

yes

characteristic

C₄×C₅₃

#407

yes

—

C₂²×#131

D₅₃

yes

characteristic

C₂×C₂²⋊C₁₆

—

C₂×C₅₃

—

C₂×C₅₃

#1779

yes

characteristic

C₂×C₅₃

#2113

yes

—

C₂²×#41

Dic₅₃

yes

characteristic

C₂×#131

—

C₂²×M₁₆

Dic₅₃

yes

—

C₂³×C₈

C₂×D₅₃

yes

—

C₂²×C₁₆

—

C₂²⋊C₁₆

—

106

—

C₅₃

#12680

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×#41

—

C₂²×Q₈

#88322

yes

characteristic

C₂²×D₄

#88322

yes

—

C₂³×C₄

C₂×Dic₅₃

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₅₃

yes

—

C₂²×C₈

C₂²⋊D₅₃

yes

—

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₂×C₁₆

—

106

—

C₂⁴

C₂×#88322

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#88322

yes

—

C₂²×C₄

#11719917

yes

—

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

100

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂²⋊Dic₅₃

yes

—

105

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

106

C₁₆

—

212

—

107

C₂³

—

108

C₂³

—

109

C₂³

C₂²⋊#88322

yes

—

110

C₂³

C₂²×#88322

yes

characteristic

111

Q₈

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²⋊#88322

yes

—

118

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

—

119

C₂×C₄

C₂×#11719917

yes

—

120

C₈

—

121

C₈

#13133186

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×C₂²⋊#88322

yes

—

126

C₂²

#16811040

yes

—

127

C₄

—

128

C₄

#16810479

yes

characteristic

129

C₄

#16810587

yes

—

130

C₂

—

131

C₂

#20274625

yes

characteristic

132

C₂

#20276329

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂²×C₈
Fitting	C₂²×#131×C₅₃	C₂
prime radical	C₂²×#41×C₅₃	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×#131	D₅₃
hypocenter	C₅₃	#12680
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×#131

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₅₃

commutator, socle

C₂³×C₅₃

C₂²×C₈

Fitting

C₂²×#131×C₅₃

C₂

prime radical

C₂²×#41×C₅₃

C₄

cosocle

C₂²×C₈×C₅₃

C₂³

hypercenter

C₂²×#131

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₈ ◁ C₂²×#131

N	H ≅ G/N	type
1. C₂×C₂²⋊#11719908	C₂	semidirect
75. C₅₃	#12680	semidirect

H ≅ G/N

type

1. C₂×C₂²⋊#11719908

C₂

semidirect

75. C₅₃

#12680

semidirect

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

subgroup	index
C₂×C₂²⋊#11719908	2
C₂²×#131×C₅₃	2
#12680	53

subgroup

index

C₂×C₂²⋊#11719908

C₂²×#131×C₅₃

#12680

Subgroups — #27299528

Conjugacy classes of subgroups

Complements

Maximal subgroups