Group #20291498 — subgroups

Subgroups — Q₈×D₅₃⋊C₈ #20291498

Order 6784.

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

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 228 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	#16811672	C₂	1	6	yes	maximal, complemented
2	Q₈×#88324	C₂	1	2	yes	maximal, complemented
3	D₅₃⋊(C₄⋊C₈)	C₂	1	3	yes	maximal
4	C₄×D₅₃⋊C₈	C₂	1	3	yes	maximal
5	C₄×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×D₅₃⋊C₈	C₂²	1	3	yes	—
7	C₂×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
8	#13133284	C₂²	1	6	yes	—
9	C₄⋊#88324	C₂²	1	6	yes	—
10	C₄×#88324	C₂²	1	6	yes	—
11	C₄×#88324	C₂²	1	6	yes	—
12	C₂×Q₈×D₅₃	C₄	1	1	yes	characteristic
13	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₀₆	C₄	1	3	yes	—
15	#13133182	C₂²	1	3	yes	—
16	C₄⋊C₄×D₅₃	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	Q₈×D₅₃	C₈	1	8	yes	complemented
21	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
22	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
23	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₁₀₆	C₂×C₄	1	3	yes	—
25	C₂×#88324	C₂³	1	6	yes	—
26	C₂×#88324	C₂³	1	2	yes	—
27	D₅₃⋊C₈	—	2	6	no	—
28	D₅₃⋊C₈	Q₈	1	4	yes	direct product factor
29	C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
30	C₄²×C₅₃	C₂×C₄	1	3	yes	—
31	C₄⋊Dic₅₃	C₂×C₄	1	3	yes	—
32	#11719919	C₂×C₄	1	6	yes	—
33	C₄×Dic₅₃	C₂×C₄	1	3	yes	—
34	C₄×Dic₅₃	C₂³	1	3	yes	—
35	C₄×Dic₅₃	C₂³	1	3	yes	—
36	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₅₃	C₂×C₈	1	4	yes	complemented
38	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
40	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₅₃	C₂²×C₄	1	3	yes	—
42	C₂×Dic₅₃	C₂²×C₄	1	3	yes	—
43	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
44	C₄×D₅₃	—	2	6	no	—
45	C₄×D₅₃	C₂×C₈	1	12	yes	—
46	C₄×D₅₃	C₂×Q₈	1	2	yes	—
47	C₄×D₅₃	#41	1	2	yes	—
48	Dic₁₀₆	C₂×C₈	1	12	yes	complemented
49	#88324	—	2	6	no	—
50	#88324	C₂×Q₈	1	4	yes	complemented
51	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
52	C₂×D₅₃	C₄×Q₈	1	2	yes	—
53	C₂×D₅₃	C₂²×C₈	1	2	yes	—
54	C₂×D₅₃	#131	1	2	yes	—
55	C₄×C₅₃	—	2	3	no	—
56	C₄×C₅₃	C₄×Q₈	1	2	yes	—
57	C₄×C₅₃	C₂²×C₈	1	6	yes	—
58	Dic₅₃	—	2	3	no	—
59	Dic₅₃	C₂²×C₈	1	6	yes	—
60	Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
61	Dic₅₃	C₂×#41	1	1	yes	characteristic
62	C₂×C₈×Q₈	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
64	C₂×C₅₃	C₂³×C₈	1	1	yes	characteristic
65	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
66	D₅₃	C₈×Q₈	1	4	yes	complemented
67	C₂×C₄×Q₈	—	53	1	no	—
68	C₈×Q₈	—	53	2	no	—
69	C₈×Q₈	—	53	6	no	—
70	C₂×C₄⋊C₈	—	53	3	no	—
71	C₂×C₄×C₈	—	53	3	no	—
72	C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	53	1	no	—
74	C₂²×C₈	—	53	1	no	—
75	C₂²×C₈	—	53	3	no	—
76	C₄×Q₈	—	53	1	no	—
77	C₄×Q₈	—	53	3	no	—
78	C₄×Q₈	—	53	3	no	—
79	C₄×Q₈	#6157	1	1	yes	characteristic
80	C₂×C₄⋊C₄	—	53	3	no	—
81	C₂×C₄²	—	53	3	no	—
82	C₄⋊C₈	—	53	6	no	—
83	C₄⋊C₈	—	53	6	no	—
84	C₄×C₈	—	53	6	no	—
85	C₄×C₈	—	53	6	no	—
86	C₂×Q₈	—	53	8	no	—
87	C₂×Q₈	—	53	3	no	—
88	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
89	C₂²×C₄	—	53	3	no	—
90	C₂²×C₄	—	53	1	no	—
91	C₂²×C₄	—	53	3	no	—
92	C₂×C₈	—	53	2	no	—
93	C₂×C₈	—	53	4	no	—
94	C₂×C₈	—	53	6	no	—
95	C₂×C₈	—	106	6	no	—
96	C₄⋊C₄	—	53	6	no	—
97	C₄⋊C₄	—	53	3	no	—
98	C₄⋊C₄	C₂×#6157	1	3	yes	—
99	C₄²	—	53	3	no	—
100	C₄²	—	53	3	no	—
101	C₄²	—	53	3	no	—
102	C₄²	C₂×#6157	1	3	yes	—
103	C₂³	—	53	1	no	—
104	Q₈	—	53	12	no	—
105	Q₈	D₅₃⋊C₈	1	4	yes	direct product factor
106	C₂×C₄	—	53	3	no	—
107	C₂×C₄	—	53	1	no	—
108	C₂×C₄	—	53	2	no	—
109	C₂×C₄	—	53	2	no	—
110	C₂×C₄	—	53	12	no	—
111	C₂×C₄	—	53	3	no	—
112	C₂×C₄	—	106	6	no	—
113	C₂×C₄	C₂²×#6157	1	3	yes	—
114	C₂×C₄	C₂²×#6157	1	3	yes	—
115	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
116	C₈	—	53	4	no	—
117	C₈	—	106	6	no	—
118	C₂²	—	53	2	no	—
119	C₂²	—	53	2	no	—
120	C₂²	—	53	2	no	—
121	C₂²	C₂³×#6157	1	1	yes	characteristic
122	C₄	—	2	3	no	—
123	C₄	—	53	1	no	—
124	C₄	—	53	1	no	—
125	C₄	—	53	6	no	—
126	C₄	—	106	3	no	—
127	C₄	Q₈×#6157	1	2	yes	—
128	C₄	C₂×D₅₃⋊C₈	1	6	yes	—
129	C₂	—	53	4	no	—
130	C₂	C₂×Q₈×#6157	1	1	yes	characteristic
131	C₂	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
132	C₂	D₅₃⋊#131	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811672

C₂

yes

maximal, complemented

Q₈×#88324

C₂

yes

maximal, complemented

D₅₃⋊(C₄⋊C₈)

C₂

yes

maximal

C₄×D₅₃⋊C₈

C₂

yes

maximal

C₄×Q₈×D₅₃

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₈

C₂²

yes

—

C₂×D₅₃⋊C₈

C₂²

yes

characteristic

#13133284

C₂²

yes

—

C₄⋊#88324

C₂²

yes

—

C₄×#88324

C₂²

yes

—

C₄×#88324

C₂²

yes

—

C₂×Q₈×D₅₃

C₄

yes

characteristic

Q₈×Dic₅₃

C₂²

yes

characteristic

C₄×Dic₁₀₆

C₄

yes

—

#13133182

C₂²

yes

—

C₄⋊C₄×D₅₃

C₄

yes

—

C₄²×D₅₃

C₂²

yes

—

C₄×Q₈×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

Q₈×D₅₃

C₈

yes

complemented

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×#88324

C₂³

yes

—

C₂×#88324

C₂³

yes

—

D₅₃⋊C₈

—

D₅₃⋊C₈

Q₈

yes

direct product factor

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂²×D₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

C₂×C₈

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

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂⁴

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₂×C₈

yes

—

C₄×D₅₃

C₂×Q₈

yes

—

C₄×D₅₃

#41

yes

—

Dic₁₀₆

C₂×C₈

yes

complemented

#88324

—

#88324

C₂×Q₈

yes

complemented

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

C₄×Q₈

yes

—

C₂×D₅₃

C₂²×C₈

yes

—

C₂×D₅₃

#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

C₂²×C₈

yes

—

Dic₅₃

—

Dic₅₃

C₂²×C₈

yes

—

Dic₅₃

C₂²×Q₈

yes

characteristic

Dic₅₃

C₂×#41

yes

characteristic

C₂×C₈×Q₈

—

maximal, Sylow

C₂×C₅₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₅₃

C₂³×C₈

yes

characteristic

C₂×C₅₃

C₂×#131

yes

characteristic

D₅₃

C₈×Q₈

yes

complemented

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₅₃

C₂×C₈×Q₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

#6157

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×#6157

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#6157

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂×#6157

yes

—

103

C₂³

—

104

Q₈

—

105

Q₈

D₅₃⋊C₈

yes

direct product factor

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

106

—

113

C₂×C₄

C₂²×#6157

yes

—

114

C₂×C₄

C₂²×#6157

yes

—

115

C₂×C₄

C₂²×#6157

yes

characteristic

116

C₈

—

117

C₈

—

106

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂³×#6157

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

106

—

127

C₄

Q₈×#6157

yes

—

128

C₄

C₂×D₅₃⋊C₈

yes

—

129

C₂

—

130

C₂

C₂×Q₈×#6157

yes

characteristic

131

C₂

C₂²×D₅₃⋊C₈

yes

characteristic

132

C₂

D₅₃⋊#131

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂³×C₈
Frattini	C₂²	C₂³×#6157
Fitting	C₄×Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂³×C₄
hypercenter	C₄×Q₈	#6157
hypocenter	C₅₃	C₂×C₈×Q₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

subgroup

G/H

center

C₂×C₄

C₂²×#6157

commutator

C₂×C₅₃

C₂³×C₈

Frattini

C₂²

C₂³×#6157

Fitting

C₄×Q₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂³×C₄

hypercenter

C₄×Q₈

#6157

hypocenter

C₅₃

C₂×C₈×Q₈

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×Q₈

N	H ≅ G/N	type
1. #16811672	C₂	semidirect
2. Q₈×#88324	C₂	semidirect
20. Q₈×D₅₃	C₈	semidirect
28. D₅₃⋊C₈	Q₈	direct (N × H)
37. Q₈×C₅₃	C₂×C₈	semidirect
48. Dic₁₀₆	C₂×C₈	semidirect
50. #88324	C₂×Q₈	semidirect
66. D₅₃	C₈×Q₈	semidirect
72. C₅₃	C₂×C₈×Q₈	semidirect
105. Q₈	D₅₃⋊C₈	direct (N × H)

H ≅ G/N

type

1. #16811672

C₂

semidirect

2. Q₈×#88324

C₂

semidirect

20. Q₈×D₅₃

C₈

semidirect

28. D₅₃⋊C₈

Q₈

direct (N × H)

37. Q₈×C₅₃

C₂×C₈

semidirect

48. Dic₁₀₆

C₂×C₈

semidirect

50. #88324

C₂×Q₈

semidirect

66. D₅₃

C₈×Q₈

semidirect

72. C₅₃

C₂×C₈×Q₈

semidirect

105. Q₈

D₅₃⋊C₈

direct (N × H)

Subgroups — Q₈×D₅₃⋊C₈ #20291498

Conjugacy classes of subgroups

Complements

Subgroups — Q8×D53⋊C8 #20291498

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×D₅₃⋊C₈ #20291498