Group #27299508 — subgroups

Subgroups — #27299508

Order 13568.

306 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 163 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×#16810453	C₂	1	2	yes	maximal
2	C₄×#13133153	C₂	1	2	yes	maximal
3	#20272809	C₂	1	2	yes	maximal
4	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133153	C₂²	1	4	yes	—
6	C₄×#11719917	—	2	2	no	—
7	#16810453	C₄	1	8	yes	complemented
8	#16810388	—	2	2	no	—
9	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#346	C₄	1	4	yes	—
13	C₂×#11719917	—	2	4	no	—
14	#13133153	—	2	4	no	—
15	#13133153	C₂×C₄	1	8	yes	—
16	#13133128	—	4	2	no	—
17	C₄×#88322	—	4	2	no	—
18	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₅₃	—	2	2	no	—
22	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
23	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₅₃	Q₈	1	2	yes	—
27	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
28	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
33	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₅₃	#41	1	2	yes	—
35	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₄²×C₅₃	D₈	1	2	yes	—
37	#11719917	—	2	8	no	—
38	#11719917	—	4	4	no	—
39	C₂×#88322	—	4	4	no	—
40	C₁₆×C₅₃	—	2	2	no	—
41	C₁₆×C₅₃	C₄²	1	4	yes	—
42	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
43	C₂×C₄×C₅₃	—	4	2	no	—
44	C₂×C₄×C₅₃	#107	1	4	yes	—
45	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#135	1	2	yes	—
49	C₈×C₅₃	—	2	2	no	—
50	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
51	C₈×C₅₃	#117	1	1	yes	characteristic
52	C₈×C₅₃	C₄×Q₈	1	2	yes	—
53	#88322	—	4	8	no	—
54	#12676	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
57	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
58	C₄×C₅₃	—	2	4	no	—
59	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₅₃	#427	1	1	yes	characteristic
61	C₄×C₅₃	C₄×D₈	1	2	yes	—
62	C₂×#368	—	53	2	no	—
63	C₂×#346	D₅₃	1	1	yes	characteristic
64	C₄×#108	—	53	2	no	—
65	#1381	—	53	2	no	—
66	C₂×C₅₃	—	2	2	no	—
67	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
68	C₂×C₅₃	#2161	1	2	yes	—
69	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
70	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
71	C₂×#108	—	53	4	no	—
72	C₄×M₁₆	—	106	2	no	—
73	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
74	#368	—	53	8	no	—
75	#346	Dic₅₃	1	4	yes	—
76	#341	—	106	2	no	—
77	C₅₃	#12676	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₁₆	—	106	4	no	—
79	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
80	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
81	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
84	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
85	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
86	#108	—	53	8	no	—
87	#108	—	106	4	no	—
88	#97	—	212	2	no	—
89	C₄×C₈	—	212	2	no	—
90	C₄×C₈	Dic₁₀₆	1	2	yes	—
91	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
92	C₂²×C₄	Dic₂₁₂	1	2	yes	—
93	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
94	M₁₆	—	106	8	no	—
95	M₁₆	—	212	4	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	212	4	no	—
98	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
99	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
100	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
101	C₂×C₈	#11719945	1	2	yes	—
102	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
103	C₄²	D₄₂₄	1	2	yes	—
104	C₄²	C₄⋊Dic₅₃	1	2	yes	—
105	C₁₆	—	2	2	no	—
106	C₁₆	C₄×Dic₅₃	1	4	yes	—
107	C₂³	C₂×Dic₂₁₂	1	1	yes	characteristic
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	#13133151	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
111	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
112	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
113	C₂×C₄	#13133239	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	212	8	no	—
116	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
117	C₈	#13133176	1	1	yes	characteristic
118	C₈	C₄×Dic₁₀₆	1	2	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	C₂×#13133151	1	1	yes	characteristic
121	C₂²	C₄×Dic₂₁₂	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
124	C₄	#16810661	1	1	yes	characteristic
125	C₄	C₄×D₄₂₄	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	C₄×#13133151	1	1	yes	characteristic
128	C₂	#20276510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810453

C₂

yes

maximal

C₄×#13133153

C₂

yes

maximal

#20272809

C₂

yes

maximal

C₂×C₅₃×#346

C₂

yes

maximal, characteristic

C₂×#13133153

C₂²

yes

—

C₄×#11719917

—

#16810453

C₄

yes

complemented

#16810388

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

C₄

yes

—

C₂×#11719917

—

#13133153

—

#13133153

C₂×C₄

yes

—

#13133128

—

C₄×#88322

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

Q₁₆

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

D₈

yes

—

#11719917

—

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₄²

yes

—

C₂³×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₄²

yes

characteristic

C₈×C₅₃

#117

yes

characteristic

C₈×C₅₃

C₄×Q₈

yes

—

#88322

—

#12676

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#107

yes

characteristic

C₂²×C₅₃

C₄×Q₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₄×C₅₃

#427

yes

characteristic

C₄×C₅₃

C₄×D₈

yes

—

C₂×#368

—

C₂×#346

D₅₃

yes

characteristic

C₄×#108

—

#1381

—

C₂×C₅₃

—

C₂×C₅₃

C₄×#107

yes

characteristic

C₂×C₅₃

#2161

yes

—

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

C₂×#108

—

C₄×M₁₆

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#368

—

#346

Dic₅₃

yes

—

#341

—

106

—

C₅₃

#12676

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

D₂₁₂

yes

characteristic

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅₃

yes

—

C₂×C₁₆

C₂×Dic₅₃

yes

—

C₂×C₁₆

C₂×Dic₅₃

yes

—

#108

—

#108

—

106

—

#97

—

212

—

C₄×C₈

—

212

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₂²×C₄

Dic₂₁₂

yes

—

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

100

C₂×C₈

C₂×C₄×D₅₃

yes

—

101

C₂×C₈

#11719945

yes

—

102

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

103

C₄²

D₄₂₄

yes

—

104

C₄²

C₄⋊Dic₅₃

yes

—

105

C₁₆

—

106

C₁₆

C₄×Dic₅₃

yes

—

107

C₂³

C₂×Dic₂₁₂

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

#13133151

yes

—

110

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

111

C₂×C₄

C₄×D₂₁₂

yes

—

112

C₂×C₄

C₂×D₄₂₄

yes

characteristic

113

C₂×C₄

#13133239

yes

—

114

C₈

—

115

C₈

—

212

—

116

C₈

C₂×C₄×Dic₅₃

yes

characteristic

117

C₈

#13133176

yes

characteristic

118

C₈

C₄×Dic₁₀₆

yes

—

119

C₂²

—

120

C₂²

C₂×#13133151

yes

characteristic

121

C₂²

C₄×Dic₂₁₂

yes

—

122

C₄

—

123

C₄

C₄×C₄⋊Dic₅₃

yes

characteristic

124

C₄

#16810661

yes

characteristic

125

C₄

C₄×D₄₂₄

yes

—

126

C₂

—

127

C₂

C₄×#13133151

yes

characteristic

128

C₂

#20276510

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#12676
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₄₂₄

commutator

C₈×C₅₃

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#346

C₂

socle

C₂²×C₅₃

C₂×#107

prime radical

C₂³×C₅₃

C₂×Q₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
7. #16810453	C₄	semidirect
77. C₅₃	#12676	semidirect

H ≅ G/N

type

7. #16810453

C₄

semidirect

77. C₅₃

#12676

semidirect

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

subgroup	index
C₂×#16810453	2
C₄×#13133153	2
#20272809	2
C₂×C₅₃×#346	2
#12676	53

subgroup

index

C₂×C₅₃×#346

Subgroups — #27299508

Conjugacy classes of subgroups

Complements

Maximal subgroups