Group #20275554 — subgroups

Subgroups — #20275554

Order 6784.

396 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 97 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×#13133214	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133133	C₂	1	4	yes	maximal, complemented
3	C₄×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₂₁₂	C₄	1	4	yes	—
6	C₂²×D₂₁₂	C₂²	1	2	yes	—
7	C₂×#11719917	C₂²	1	2	yes	—
8	#13133214	—	2	1	no	—
9	#13133214	—	2	1	no	—
10	#13133214	—	2	2	no	—
11	#13133133	—	2	8	no	—
12	#13133128	—	2	1	no	—
13	C₄×#88322	—	2	1	no	—
14	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₅₃	—	2	2	no	—
17	C₂³×D₅₃	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂×D₂₁₂	—	2	4	no	—
21	C₂×D₂₁₂	—	2	4	no	—
22	C₂×D₂₁₂	—	2	4	no	—
23	C₂×D₂₁₂	—	2	2	no	—
24	C₂×D₂₁₂	—	2	2	no	—
25	C₂×D₂₁₂	—	2	4	no	—
26	C₂×D₂₁₂	—	4	8	no	—
27	M₁₆×C₅₃	—	2	4	no	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₄²×C₅₃	D₄	1	2	yes	—
32	#11719917	—	2	2	no	—
33	#11719917	D₄	1	4	yes	complemented
34	C₂×#88322	—	2	2	no	—
35	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×D₅₃	—	2	4	no	—
37	C₂²×D₅₃	—	2	8	no	—
38	C₂²×D₅₃	—	4	8	no	—
39	C₂×C₄×C₅₃	—	2	4	no	—
40	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	2	yes	—
46	D₂₁₂	—	4	8	no	—
47	D₂₁₂	—	4	8	no	—
48	D₂₁₂	—	4	4	no	—
49	D₂₁₂	—	4	2	no	—
50	D₂₁₂	—	4	2	no	—
51	C₈×C₅₃	—	4	2	no	—
52	#88322	—	2	4	no	—
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₅₃	C₄×D₄	1	2	yes	—
56	C₂×D₅₃	—	4	4	no	—
57	C₂×D₅₃	—	4	8	no	—
58	C₂×D₅₃	—	8	8	no	—
59	C₄×C₅₃	—	2	2	no	—
60	C₄×C₅₃	#100	1	4	yes	complemented
61	C₄×C₅₃	#121	1	2	yes	—
62	C₄×C₅₃	#124	1	1	yes	characteristic
63	C₄×C₅₃	#127	1	1	yes	characteristic
64	#1965	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	—	2	2	no	—
66	C₂×C₅₃	#390	1	1	yes	characteristic
67	C₂×C₅₃	C₂×#100	1	2	yes	—
68	D₅₃	—	8	4	no	—
69	C₂×#127	—	53	1	no	—
70	#423	D₅₃	1	1	yes	characteristic, complemented
71	C₂×#100	—	53	4	no	—
72	C₄×M₁₆	—	53	1	no	—
73	C₅₃	#1965	1	1	yes	characteristic, complemented, Sylow
74	C₂²×D₄	—	53	4	no	—
75	C₂²×D₄	—	53	2	no	—
76	C₂×M₁₆	—	53	2	no	—
77	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
78	#127	—	106	1	no	—
79	#127	—	106	1	no	—
80	#127	—	106	2	no	—
81	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
82	C₄⋊C₈	—	2	2	no	—
83	#100	—	106	8	no	—
84	#97	—	106	1	no	—
85	C₄×C₈	—	106	1	no	—
86	C₂⁴	—	53	4	no	—
87	C₂×D₄	—	106	2	no	—
88	C₂×D₄	—	106	4	no	—
89	C₂×D₄	—	106	4	no	—
90	C₂×D₄	—	106	2	no	—
91	C₂×D₄	—	106	4	no	—
92	C₂×D₄	—	106	4	no	—
93	C₂×D₄	—	212	8	no	—
94	C₂²×C₄	C₄×D₅₃	1	2	yes	—
95	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
96	M₁₆	—	2	4	no	—
97	M₁₆	—	53	4	no	—
98	M₁₆	—	106	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	106	2	no	—
101	C₄²	D₂₁₂	1	1	yes	characteristic
102	C₄²	D₂₁₂	1	1	yes	characteristic
103	C₄²	C₂²⋊D₅₃	1	2	yes	—
104	C₂³	—	106	8	no	—
105	C₂³	—	106	4	no	—
106	C₂³	—	212	8	no	—
107	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
108	D₄	—	212	8	no	—
109	D₄	—	212	8	no	—
110	D₄	—	212	4	no	—
111	D₄	—	212	2	no	—
112	D₄	—	212	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	#11719921	1	2	yes	—
115	C₂×C₄	#11719921	1	2	yes	—
116	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
117	C₂×C₄	D₄×D₅₃	1	2	yes	—
118	C₂×C₄	#11719949	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
120	C₈	—	4	2	no	—
121	C₈	—	106	4	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	424	8	no	—
124	C₂²	—	212	4	no	—
125	C₂²	—	212	8	no	—
126	C₂²	C₂×#11719921	1	1	yes	characteristic
127	C₂²	#13133179	1	2	yes	—
128	C₄	—	2	2	no	—
129	C₄	#13133133	1	4	yes	complemented
130	C₄	#13133192	1	2	yes	—
131	C₄	#13133206	1	1	yes	characteristic
132	C₄	#13133213	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	—	424	4	no	—
135	C₂	#16810524	1	1	yes	characteristic
136	C₂	C₂×#13133133	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133214

C₂

yes

maximal, characteristic

C₂×#13133133

C₂

yes

maximal, complemented

C₄×#11719917

C₂

yes

maximal, characteristic, complemented

C₅₃×#423

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₄

yes

—

C₂²×D₂₁₂

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

—

—

—

—

—

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂³×D₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

#11719917

—

#11719917

D₄

yes

complemented

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#100

yes

complemented

C₄×C₅₃

#121

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#390

yes

characteristic

C₂×C₅₃

C₂×#100

yes

—

D₅₃

—

C₂×#127

—

#423

D₅₃

yes

characteristic, complemented

C₂×#100

—

C₄×M₁₆

—

C₅₃

#1965

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

#127

—

106

—

#127

—

106

—

#127

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

#100

—

106

—

#97

—

106

—

C₄×C₈

—

106

—

C₂⁴

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

C₂×C₈

—

100

C₂×C₈

—

106

—

101

C₄²

D₂₁₂

yes

characteristic

102

C₄²

D₂₁₂

yes

characteristic

103

C₄²

C₂²⋊D₅₃

yes

—

104

C₂³

—

106

—

105

C₂³

—

106

—

106

C₂³

—

212

—

107

C₂³

C₂×C₄×D₅₃

yes

characteristic

108

D₄

—

212

—

109

D₄

—

212

—

110

D₄

—

212

—

111

D₄

—

212

—

112

D₄

—

212

—

113

C₂×C₄

—

114

C₂×C₄

#11719921

yes

—

115

C₂×C₄

#11719921

yes

—

116

C₂×C₄

C₂×D₂₁₂

yes

characteristic

117

C₂×C₄

D₄×D₅₃

yes

—

118

C₂×C₄

#11719949

yes

—

119

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

120

C₈

—

121

C₈

—

106

—

122

C₂²

—

123

C₂²

—

424

—

124

C₂²

—

212

—

125

C₂²

—

212

—

126

C₂²

C₂×#11719921

yes

characteristic

127

C₂²

#13133179

yes

—

128

C₄

—

129

C₄

#13133133

yes

complemented

130

C₄

#13133192

yes

—

131

C₄

#13133206

yes

characteristic

132

C₄

#13133213

yes

characteristic

133

C₂

—

134

C₂

—

424

—

135

C₂

#16810524

yes

characteristic

136

C₂

C₂×#13133133

yes

—

	subgroup	G/H
center	C₂²	C₂×#11719921
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#423	D₅₃
hypocenter	C₅₃	#1965
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#11719921

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#423

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#13133133	C₂	semidirect
3. C₄×#11719917	C₂	semidirect
4. C₅₃×#423	C₂	semidirect
33. #11719917	D₄	semidirect
60. C₄×C₅₃	#100	semidirect
70. #423	D₅₃	semidirect
73. C₅₃	#1965	semidirect
129. C₄	#13133133	semidirect

H ≅ G/N

type

2. C₂×#13133133

C₂

semidirect

3. C₄×#11719917

C₂

semidirect

4. C₅₃×#423

C₂

semidirect

33. #11719917

D₄

semidirect

60. C₄×C₅₃

#100

semidirect

70. #423

D₅₃

semidirect

73. C₅₃

#1965

semidirect

129. C₄

#13133133

semidirect

Subgroups — #20275554

Conjugacy classes of subgroups

Complements