Group #20291361 — subgroups

Subgroups — #20291361

Order 6784.

484 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 206 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×#13133305	C₂	1	2	yes	maximal
2	#16811630	C₂	1	6	yes	maximal, complemented
3	#16811627	C₂	1	3	yes	maximal, complemented
4	D₅₃⋊(C₄⋊C₈)	C₂	1	3	yes	maximal, complemented
5	C₄×#11719949	C₂	1	1	yes	maximal, characteristic
6	#13133305	—	2	8	no	—
7	C₂×#11719936	C₂²	1	6	yes	—
8	C₂×D₅₃⋊C₈	C₂²	1	6	yes	—
9	#13133284	C₂²	1	6	yes	—
10	C₄⋊#88324	C₂²	1	1	yes	characteristic
11	C₄⋊#88324	C₂²	1	3	yes	—
12	C₄⋊#88324	C₂²	1	6	yes	—
13	C₂×#11719949	C₂²	1	1	yes	characteristic
14	Q₈×Dic₅₃	C₄	1	1	yes	characteristic
15	C₄×D₂₁₂	C₄	1	3	yes	—
16	#13133179	C₄	1	3	yes	—
17	#13133175	C₂²	1	3	yes	—
18	C₄²×D₅₃	C₂²	1	3	yes	—
19	C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	#11719949	D₄	1	4	yes	—
22	#11719949	Q₈	1	4	yes	—
23	C₂×D₂₁₂	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
25	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
26	C₂×C₄×D₅₃	C₂³	1	3	yes	—
27	C₂×#88324	C₂³	1	6	yes	—
28	C₂×#88324	C₂³	1	2	yes	—
29	#11719936	—	2	12	no	—
30	D₅₃⋊C₈	—	2	12	no	—
31	C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
32	C₄²×C₅₃	C₂×C₄	1	3	yes	—
33	#11719921	C₂×C₄	1	6	yes	—
34	C₄⋊Dic₅₃	C₂×C₄	1	3	yes	—
35	C₄×Dic₅₃	C₂×C₄	1	3	yes	—
36	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
37	C₄×Dic₅₃	C₂³	1	3	yes	—
38	C₂²×D₅₃	C₂²×C₄	1	3	yes	—
39	Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
41	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₅₃	C₂²×C₄	1	3	yes	—
45	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
46	D₂₁₂	C₄⋊C₄	1	12	yes	—
47	C₄×D₅₃	—	2	6	no	—
48	C₄×D₅₃	—	4	3	no	—
49	C₄×D₅₃	C₂×D₄	1	6	yes	—
50	C₄×D₅₃	C₂×Q₈	1	6	yes	—
51	#88324	—	2	6	no	—
52	#88324	—	2	2	no	—
53	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
54	C₂×D₅₃	—	2	3	no	—
55	C₂×D₅₃	—	2	3	no	—
56	C₂×D₅₃	C₂×C₄⋊C₄	1	6	yes	—
57	C₄×C₅₃	—	2	3	no	—
58	C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
59	C₄×C₅₃	#131	1	2	yes	complemented
60	Dic₅₃	—	2	3	no	—
61	Dic₅₃	—	2	1	no	—
62	Dic₅₃	C₂²×D₄	1	1	yes	characteristic
63	Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
64	#2906	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
67	C₂×C₅₃	#568	1	1	yes	characteristic
68	D₅₃	—	2	6	no	—
69	C₂×#131	—	53	2	no	—
70	C₄×#41	—	53	1	no	—
71	#424	—	53	6	no	—
72	#423	—	53	3	no	—
73	C₂×C₄⋊C₈	—	53	3	no	—
74	C₅₃	#2906	1	1	yes	characteristic, complemented, Sylow
75	C₂×#41	—	53	1	no	—
76	#131	—	106	8	no	—
77	C₂×M₁₆	—	53	6	no	—
78	C₂²×C₈	—	53	6	no	—
79	C₄×Q₈	—	53	1	no	—
80	C₄×Q₈	#6157	1	1	yes	characteristic
81	C₄×D₄	—	53	3	no	—
82	C₄×D₄	—	53	3	no	—
83	#117	—	53	3	no	—
84	C₂×C₄²	—	53	3	no	—
85	C₄⋊C₈	—	53	1	no	—
86	C₄⋊C₈	—	53	3	no	—
87	C₄⋊C₈	—	53	6	no	—
88	C₄⋊C₈	—	53	6	no	—
89	#41	—	53	4	no	—
90	#41	—	53	4	no	—
91	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
92	C₂×D₄	—	53	3	no	—
93	C₂²×C₄	—	53	3	no	—
94	C₂²×C₄	—	53	3	no	—
95	C₂²×C₄	—	53	3	no	—
96	M₁₆	—	106	12	no	—
97	C₂×C₈	—	53	2	no	—
98	C₂×C₈	—	53	6	no	—
99	C₂×C₈	—	106	12	no	—
100	C₄⋊C₄	—	53	3	no	—
101	C₄⋊C₄	C₂×#6157	1	3	yes	—
102	C₂²⋊C₄	—	53	6	no	—
103	C₄²	—	53	1	no	—
104	C₄²	—	53	3	no	—
105	C₄²	—	53	3	no	—
106	C₄²	C₂×#6157	1	3	yes	—
107	C₂³	—	53	3	no	—
108	Q₈	C₄⋊#6157	1	4	yes	—
109	D₄	—	53	12	no	—
110	C₂×C₄	—	53	1	no	—
111	C₂×C₄	—	53	6	no	—
112	C₂×C₄	—	53	6	no	—
113	C₂×C₄	—	53	1	no	—
114	C₂×C₄	—	53	3	no	—
115	C₂×C₄	—	106	6	no	—
116	C₂×C₄	—	212	3	no	—
117	C₂×C₄	C₂²×#6157	1	3	yes	—
118	C₂×C₄	C₂²×#6157	1	3	yes	—
119	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
120	C₈	—	106	2	no	—
121	C₈	—	106	6	no	—
122	C₂²	—	53	6	no	—
123	C₂²	—	106	3	no	—
124	C₂²	—	106	3	no	—
125	C₂²	C₂³×#6157	1	1	yes	characteristic
126	C₄	—	2	3	no	—
127	C₄	—	53	1	no	—
128	C₄	—	53	1	no	—
129	C₄	—	106	3	no	—
130	C₄	—	106	1	no	—
131	C₄	C₂×C₄⋊#6157	1	6	yes	—
132	C₄	#13133305	1	2	yes	complemented
133	C₂	—	106	6	no	—
134	C₂	C₂²×C₄⋊#6157	1	1	yes	characteristic
135	C₂	C₂×#13133305	1	1	yes	characteristic
136	C₂	#16811702	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133305

C₂

yes

maximal

#16811630

C₂

yes

maximal, complemented

#16811627

C₂

yes

maximal, complemented

D₅₃⋊(C₄⋊C₈)

C₂

yes

maximal, complemented

C₄×#11719949

C₂

yes

maximal, characteristic

#13133305

—

C₂×#11719936

C₂²

yes

—

C₂×D₅₃⋊C₈

C₂²

yes

—

#13133284

C₂²

yes

—

C₄⋊#88324

C₂²

yes

characteristic

C₄⋊#88324

C₂²

yes

—

C₄⋊#88324

C₂²

yes

—

C₂×#11719949

C₂²

yes

characteristic

Q₈×Dic₅₃

C₄

yes

characteristic

C₄×D₂₁₂

C₄

yes

—

#13133179

C₄

yes

—

#13133175

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

—

C₄×Q₈×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

#11719949

D₄

yes

—

#11719949

Q₈

yes

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

C₂×#88324

C₂³

yes

—

C₂×#88324

C₂³

yes

—

#11719936

—

D₅₃⋊C₈

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

#11719921

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₄×Dic₅₃

C₂³

yes

—

C₂²×D₅₃

C₂²×C₄

yes

—

Q₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂⁴

yes

characteristic

D₂₁₂

C₄⋊C₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×D₄

yes

—

C₄×D₅₃

C₂×Q₈

yes

—

#88324

—

#88324

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂×C₄⋊C₄

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₄×C₅₃

#131

yes

complemented

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂²×D₄

yes

characteristic

Dic₅₃

C₂²×Q₈

yes

characteristic

#2906

—

maximal, Sylow

C₂×C₅₃

C₂²×C₄⋊C₄

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂×#131

—

C₄×#41

—

#424

—

#423

—

C₂×C₄⋊C₈

—

C₅₃

#2906

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

106

—

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

#6157

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#41

—

#41

—

C₂×Q₈

C₂×#6157

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

100

C₄⋊C₄

—

101

C₄⋊C₄

C₂×#6157

yes

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₂×#6157

yes

—

107

C₂³

—

108

Q₈

C₄⋊#6157

yes

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

106

—

116

C₂×C₄

—

212

—

117

C₂×C₄

C₂²×#6157

yes

—

118

C₂×C₄

C₂²×#6157

yes

—

119

C₂×C₄

C₂²×#6157

yes

characteristic

120

C₈

—

106

—

121

C₈

—

106

—

122

C₂²

—

123

C₂²

—

106

—

124

C₂²

—

106

—

125

C₂²

C₂³×#6157

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

106

—

130

C₄

—

106

—

131

C₄

C₂×C₄⋊#6157

yes

—

132

C₄

#13133305

yes

complemented

133

C₂

—

106

—

134

C₂

C₂²×C₄⋊#6157

yes

characteristic

135

C₂

C₂×#13133305

yes

characteristic

136

C₂

#16811702

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₄×Q₈×C₅₃	C₄
hypercenter	C₄×Q₈	#6157
hypocenter	C₅₃	#2906
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#6157

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₄×Q₈×C₅₃

C₄

C₄×Q₈

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
2. #16811630	C₂	semidirect
3. #16811627	C₂	semidirect
4. D₅₃⋊(C₄⋊C₈)	C₂	semidirect
59. C₄×C₅₃	#131	semidirect
74. C₅₃	#2906	semidirect
132. C₄	#13133305	semidirect

H ≅ G/N

type

2. #16811630

C₂

semidirect

3. #16811627

C₂

semidirect

4. D₅₃⋊(C₄⋊C₈)

C₂

semidirect

59. C₄×C₅₃

#131

semidirect

74. C₅₃

#2906

semidirect

132. C₄

#13133305

semidirect

Subgroups — #20291361

Conjugacy classes of subgroups

Complements