Group #20276422 — subgroups

Subgroups — #20276422

Order 6784.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂²×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810661	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810427	C₂	1	2	yes	maximal, complemented
4	#16810424	C₂	1	2	yes	maximal, complemented
5	C₂×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#13133176	—	2	1	no	—
7	C₂×#11719908	—	2	2	no	—
8	C₂×#11719908	C₂²	1	2	yes	—
9	#13133151	C₂²	1	2	yes	—
10	#13133149	—	2	1	no	—
11	C₂×#41×C₅₃	—	2	1	no	—
12	#135×C₅₃	—	2	2	no	—
13	#135×C₅₃	C₄	1	4	yes	complemented
14	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×SD₁₆×C₅₃	—	2	1	no	—
16	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	#41×C₅₃	—	2	4	no	—
19	#41×C₅₃	—	4	2	no	—
20	C₂×Q₈×C₅₃	—	2	1	no	—
21	C₂×D₄×C₅₃	—	2	1	no	—
22	C₂×D₄×C₅₃	—	4	1	no	—
23	C₂²×C₄×C₅₃	—	4	1	no	—
24	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
25	Q₁₆×C₅₃	—	2	1	no	—
26	Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
27	SD₁₆×C₅₃	—	2	4	no	—
28	D₈×C₅₃	—	2	1	no	—
29	D₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	D₄	1	2	yes	—
33	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²⋊Dic₅₃	—	4	1	no	—
35	C₄⋊Dic₅₃	—	2	2	no	—
36	C₄×Dic₅₃	—	4	1	no	—
37	#11719908	—	2	2	no	—
38	C₂³×C₅₃	—	4	1	no	—
39	C₂³×C₅₃	D₈	1	1	yes	characteristic
40	Q₈×C₅₃	—	2	2	no	—
41	Q₈×C₅₃	—	4	1	no	—
42	D₄×C₅₃	—	2	2	no	—
43	D₄×C₅₃	—	4	4	no	—
44	D₄×C₅₃	—	4	1	no	—
45	C₂×C₄×C₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	—	4	4	no	—
47	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₅₃	—	4	2	no	—
52	C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	—	4	2	no	—
57	C₂²×C₅₃	—	8	2	no	—
58	C₂²×C₅₃	#102	1	2	yes	—
59	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
60	C₄×C₅₃	—	4	2	no	—
61	C₄×C₅₃	#102	1	2	yes	—
62	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
64	Dic₅₃	—	8	2	no	—
65	#2138	—	53	1	no	maximal, Sylow
66	C₂×C₅₃	—	2	2	no	—
67	C₂×C₅₃	—	8	2	no	—
68	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
69	C₂×C₅₃	#508	1	2	yes	—
70	C₂×#135	D₅₃	1	1	yes	characteristic
71	C₂²×C₁₆	—	53	1	no	—
72	#427	—	53	1	no	—
73	#358	—	53	2	no	—
74	#357	—	53	2	no	—
75	C₅₃	#2138	1	1	yes	characteristic, complemented, Sylow
76	C₂×#41	—	2	1	no	—
77	#135	—	2	2	no	—
78	#135	Dic₅₃	1	4	yes	complemented
79	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
80	C₂×SD₁₆	—	2	1	no	—
81	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
83	#117	—	106	1	no	—
84	C₂×C₁₆	—	53	2	no	—
85	C₂×C₁₆	—	106	2	no	—
86	#107	—	53	2	no	—
87	#106	—	106	1	no	—
88	#41	—	2	4	no	—
89	#41	—	4	2	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	—	4	1	no	—
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
95	Q₁₆	—	2	1	no	—
96	Q₁₆	C₂×Dic₅₃	1	2	yes	—
97	SD₁₆	—	2	4	no	—
98	D₈	—	2	1	no	—
99	D₈	C₂×Dic₅₃	1	2	yes	—
100	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
101	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
102	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
103	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
104	C₄⋊C₄	—	106	2	no	—
105	C₂²⋊C₄	—	212	1	no	—
106	C₄²	—	212	1	no	—
107	C₁₆	—	106	2	no	—
108	C₂³	—	4	1	no	—
109	C₂³	#11719922	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	Q₈	—	4	1	no	—
112	D₄	—	2	2	no	—
113	D₄	—	4	4	no	—
114	D₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	212	2	no	—
118	C₂×C₄	#11719922	1	1	yes	characteristic
119	C₂×C₄	#11719923	1	2	yes	—
120	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
122	C₈	C₂²⋊Dic₅₃	1	2	yes	—
123	C₈	C₂²×Dic₅₃	1	1	yes	characteristic
124	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂²	—	2	2	no	—
126	C₂²	—	4	2	no	—
127	C₂²	—	8	2	no	—
128	C₂²	#13133138	1	2	yes	—
129	C₂²	C₂×#11719922	1	1	yes	characteristic
130	C₄	—	4	2	no	—
131	C₄	—	424	2	no	—
132	C₄	#13133138	1	2	yes	—
133	C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
134	C₄	C₂×#11719923	1	1	yes	characteristic
135	C₂	—	2	2	no	—
136	C₂	—	8	2	no	—
137	C₂	C₂×#13133138	1	1	yes	characteristic
138	C₂	#16811053	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719908

C₂

yes

maximal, characteristic, complemented

#16810661

C₂

yes

maximal, characteristic, complemented

#16810427

C₂

yes

maximal, complemented

#16810424

C₂

yes

maximal, complemented

C₂×#135×C₅₃

C₂

yes

maximal, characteristic

#13133176

—

C₂×#11719908

—

C₂×#11719908

C₂²

yes

—

#13133151

C₂²

yes

—

#13133149

—

C₂×#41×C₅₃

—

#135×C₅₃

—

#135×C₅₃

C₄

yes

complemented

C₂×Q₁₆×C₅₃

C₂²

yes

characteristic

C₂×SD₁₆×C₅₃

—

C₂×D₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

C₂×C₄

yes

—

SD₁₆×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719908

—

C₂³×C₅₃

—

C₂³×C₅₃

D₈

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

C₂²⋊C₄

yes

—

C₈×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#102

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#102

yes

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

—

#2138

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#102

yes

characteristic

C₂×C₅₃

#508

yes

—

C₂×#135

D₅₃

yes

characteristic

C₂²×C₁₆

—

#427

—

#358

—

#357

—

C₅₃

#2138

yes

characteristic, complemented, Sylow

C₂×#41

—

#135

—

#135

Dic₅₃

yes

complemented

C₂×Q₁₆

C₂×D₅₃

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

#107

—

#106

—

106

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

Q₁₆

—

Q₁₆

C₂×Dic₅₃

yes

—

SD₁₆

—

D₈

—

D₈

C₂×Dic₅₃

yes

—

100

C₂×C₈

C₂×Dic₅₃

yes

—

101

C₂×C₈

C₂²⋊D₅₃

yes

—

102

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

103

C₂×C₈

C₂²×D₅₃

yes

characteristic

104

C₄⋊C₄

—

106

—

105

C₂²⋊C₄

—

212

—

106

C₄²

—

212

—

107

C₁₆

—

106

—

108

C₂³

—

109

C₂³

#11719922

yes

characteristic

110

Q₈

—

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

212

—

118

C₂×C₄

#11719922

yes

characteristic

119

C₂×C₄

#11719923

yes

—

120

C₂×C₄

C₂²⋊Dic₅₃

yes

—

121

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

122

C₈

C₂²⋊Dic₅₃

yes

—

123

C₈

C₂²×Dic₅₃

yes

characteristic

124

C₈

C₂×C₂²⋊D₅₃

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#13133138

yes

—

129

C₂²

C₂×#11719922

yes

characteristic

130

C₄

—

131

C₄

—

424

—

132

C₄

#13133138

yes

—

133

C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

134

C₄

C₂×#11719923

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

C₂×#13133138

yes

characteristic

138

C₂

#16811053

yes

—

	subgroup	G/H
center	C₂×C₄	#11719922
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂×#135×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₂×#135	D₅₃
hypocenter	C₅₃	#2138
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

subgroup

G/H

center

C₂×C₄

#11719922

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂×#135×C₅₃

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

N	H ≅ G/N	type
1. C₂²×#11719908	C₂	semidirect
2. #16810661	C₂	semidirect
3. #16810427	C₂	semidirect
4. #16810424	C₂	semidirect
13. #135×C₅₃	C₄	semidirect
75. C₅₃	#2138	semidirect
78. #135	Dic₅₃	semidirect

H ≅ G/N

type

1. C₂²×#11719908

C₂

semidirect

2. #16810661

C₂

semidirect

3. #16810427

C₂

semidirect

4. #16810424

C₂

semidirect

13. #135×C₅₃

C₄

semidirect

75. C₅₃

#2138

semidirect

78. #135

Dic₅₃

semidirect

Subgroups — #20276422

Conjugacy classes of subgroups

Complements