Group #20291649 — subgroups

Subgroups — #20291649

Order 6784.

394 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 168 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	Q₈×#88324	C₂	1	2	yes	maximal, complemented
2	#16811657	C₂	1	6	yes	maximal, complemented
3	#16811642	C₂	1	3	yes	maximal, complemented
4	#16811627	C₂	1	3	yes	maximal, complemented
5	C₄×#11719949	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719936	C₂²	1	6	yes	—
7	C₄⋊#88324	C₂²	1	3	yes	—
8	C₄⋊#88324	C₂²	1	6	yes	—
9	C₄⋊#88324	C₂²	1	1	yes	characteristic
10	#13133274	C₂²	1	6	yes	—
11	C₄×#88324	C₂²	1	6	yes	—
12	C₂×#11719949	C₄	1	1	yes	characteristic
13	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₄×D₂₁₂	C₄	1	3	yes	—
15	#13133179	C₂²	1	3	yes	—
16	#13133175	C₄	1	3	yes	—
17	C₄²×D₅₃	C₂²	1	3	yes	—
18	C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
19	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	#11719949	—	2	4	no	—
21	C₂×D₂₁₂	C₂×C₄	1	3	yes	—
22	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
23	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₅₃	C₂³	1	3	yes	—
25	C₂×#88324	C₂³	1	2	yes	—
26	C₂×#88324	C₂³	1	6	yes	—
27	#11719936	—	4	6	no	—
28	C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
29	C₄²×C₅₃	C₂×C₄	1	3	yes	—
30	#11719921	C₂×C₄	1	6	yes	—
31	C₄⋊Dic₅₃	C₂×C₄	1	3	yes	—
32	C₄×Dic₅₃	C₂×C₄	1	3	yes	—
33	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
34	C₄×Dic₅₃	C₂³	1	3	yes	—
35	C₂²×D₅₃	C₂²×C₄	1	3	yes	—
36	Q₈×C₅₃	M₁₆	1	4	yes	complemented
37	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
38	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₅₃	C₂²×C₄	1	3	yes	—
42	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
43	D₂₁₂	—	2	6	no	—
44	C₄×D₅₃	—	2	6	no	—
45	C₄×D₅₃	—	2	6	no	—
46	C₄×D₅₃	—	2	3	no	—
47	C₄×D₅₃	—	2	3	no	—
48	#88324	—	2	6	no	—
49	#88324	—	2	2	no	—
50	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
51	C₂×D₅₃	—	2	3	no	—
52	C₂×D₅₃	—	2	3	no	—
53	C₂×D₅₃	—	2	3	no	—
54	C₄×C₅₃	—	2	3	no	—
55	C₄×C₅₃	C₂×M₁₆	1	6	yes	—
56	C₄×C₅₃	#131	1	2	yes	—
57	Dic₅₃	—	2	1	no	—
58	Dic₅₃	—	2	3	no	—
59	Dic₅₃	H(5,𝔽₂)	1	1	yes	characteristic
60	Dic₅₃	2₋¹⁺⁴	1	1	yes	characteristic
61	#2991	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	#520	1	1	yes	characteristic
63	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
64	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
65	D₅₃	—	4	3	no	—
66	C₄×#41	—	53	1	no	—
67	C₈×Q₈	—	53	2	no	—
68	#436	—	53	6	no	—
69	#431	—	53	3	no	—
70	#423	—	53	3	no	—
71	C₅₃	#2991	1	1	yes	characteristic, complemented, Sylow
72	C₂×#41	—	53	1	no	—
73	C₂×M₁₆	—	53	6	no	—
74	C₄×Q₈	—	53	1	no	—
75	C₄×Q₈	#6157	1	1	yes	characteristic
76	C₄×D₄	—	53	3	no	—
77	C₄×D₄	—	53	3	no	—
78	#117	—	53	3	no	—
79	C₂×C₄²	—	53	3	no	—
80	C₄⋊C₈	—	53	3	no	—
81	C₄⋊C₈	—	53	6	no	—
82	C₄⋊C₈	—	53	1	no	—
83	C₂²⋊C₈	—	53	6	no	—
84	C₄×C₈	—	53	6	no	—
85	#41	—	106	4	no	—
86	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
87	C₂×D₄	—	53	3	no	—
88	C₂²×C₄	—	53	3	no	—
89	C₂²×C₄	—	53	3	no	—
90	C₂²×C₄	—	53	3	no	—
91	M₁₆	—	212	6	no	—
92	C₂×C₈	—	53	2	no	—
93	C₂×C₈	—	53	6	no	—
94	C₄⋊C₄	—	53	3	no	—
95	C₄⋊C₄	C₂×#6157	1	3	yes	—
96	C₂²⋊C₄	—	53	6	no	—
97	C₄²	—	53	3	no	—
98	C₄²	—	53	1	no	—
99	C₄²	—	53	3	no	—
100	C₄²	C₂×#6157	1	3	yes	—
101	C₂³	—	53	3	no	—
102	Q₈	#11719936	1	4	yes	complemented
103	D₄	—	106	6	no	—
104	C₂×C₄	—	53	1	no	—
105	C₂×C₄	—	53	1	no	—
106	C₂×C₄	—	53	3	no	—
107	C₂×C₄	—	106	3	no	—
108	C₂×C₄	—	106	3	no	—
109	C₂×C₄	—	106	6	no	—
110	C₂×C₄	—	106	6	no	—
111	C₂×C₄	C₂²×#6157	1	3	yes	—
112	C₂×C₄	C₂²×#6157	1	3	yes	—
113	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
114	C₈	—	106	2	no	—
115	C₈	—	106	6	no	—
116	C₂²	—	106	3	no	—
117	C₂²	—	106	3	no	—
118	C₂²	—	106	3	no	—
119	C₂²	C₂³×#6157	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	53	1	no	—
122	C₄	—	53	1	no	—
123	C₄	—	106	1	no	—
124	C₄	—	106	3	no	—
125	C₄	C₂×#11719936	1	6	yes	—
126	C₄	#13133305	1	2	yes	—
127	C₂	—	212	3	no	—
128	C₂	#16811692	1	1	yes	characteristic
129	C₂	C₂²×#11719936	1	1	yes	characteristic
130	C₂	C₂×#13133305	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#88324

C₂

yes

maximal, complemented

#16811657

C₂

yes

maximal, complemented

#16811642

C₂

yes

maximal, complemented

#16811627

C₂

yes

maximal, complemented

C₄×#11719949

C₂

yes

maximal, characteristic

C₂×#11719936

C₂²

yes

—

C₄⋊#88324

C₂²

yes

—

C₄⋊#88324

C₂²

yes

—

C₄⋊#88324

C₂²

yes

characteristic

#13133274

C₂²

yes

—

C₄×#88324

C₂²

yes

—

C₂×#11719949

C₄

yes

characteristic

Q₈×Dic₅₃

C₂²

yes

characteristic

C₄×D₂₁₂

C₄

yes

—

#13133179

C₂²

yes

—

#13133175

C₄

yes

—

C₄²×D₅₃

C₂²

yes

—

C₄×Q₈×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

#11719949

—

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

—

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

M₁₆

yes

complemented

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×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

#88324

—

#88324

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

#131

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

H(5,𝔽₂)

yes

characteristic

Dic₅₃

2₋¹⁺⁴

yes

—

C₂×C₅₃

yes

C₂×C₅₃

C₂²×M₁₆

yes

characteristic

C₂×C₅₃

C₂×#131

yes

characteristic

D₅₃

—

C₄×#41

—

C₈×Q₈

—

#436

—

#431

—

#423

—

C₅₃

#2991

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

#6157

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#41

—

106

—

C₂×Q₈

C₂×#6157

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#6157

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

C₂×#6157

yes

—

101

C₂³

—

102

Q₈

#11719936

yes

complemented

103

D₄

—

106

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

106

—

108

C₂×C₄

—

106

—

109

C₂×C₄

—

106

—

110

C₂×C₄

—

106

—

111

C₂×C₄

C₂²×#6157

yes

—

112

C₂×C₄

C₂²×#6157

yes

—

113

C₂×C₄

C₂²×#6157

yes

characteristic

114

C₈

—

106

—

115

C₈

—

106

—

116

C₂²

—

106

—

117

C₂²

—

106

—

118

C₂²

—

106

—

119

C₂²

C₂³×#6157

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

106

—

124

C₄

—

106

—

125

C₄

C₂×#11719936

yes

—

126

C₄

#13133305

yes

—

127

C₂

—

212

—

128

C₂

#16811692

yes

characteristic

129

C₂

C₂²×#11719936

yes

characteristic

130

C₂

C₂×#13133305

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	#2991
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
1. Q₈×#88324	C₂	semidirect
2. #16811657	C₂	semidirect
3. #16811642	C₂	semidirect
4. #16811627	C₂	semidirect
36. Q₈×C₅₃	M₁₆	semidirect
71. C₅₃	#2991	semidirect
102. Q₈	#11719936	semidirect

H ≅ G/N

type

1. Q₈×#88324

C₂

semidirect

2. #16811657

C₂

semidirect

3. #16811642

C₂

semidirect

4. #16811627

C₂

semidirect

36. Q₈×C₅₃

M₁₆

semidirect

71. C₅₃

#2991

semidirect

102. Q₈

#11719936

semidirect

Subgroups — #20291649

Conjugacy classes of subgroups

Complements