Group #20288525 — subgroups

Subgroups — #20288525

Order 6784.

964 conjugacy classes of subgroups; only the 553 normal subgroups are shown.

normal subgroups only

This group has 964 conjugacy classes of subgroups — too many to draw. Only its 553 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×Dic₁₀₆	C₂	1	4	yes	maximal, complemented
2	#16811352	C₂	1	4	yes	maximal, complemented
3	#16811349	C₂	1	8	yes	maximal, complemented
4	#16811254	C₂	1	6	yes	maximal
5	#16811180	C₂	1	6	yes	maximal
6	#16811101	C₂	1	2	yes	maximal
7	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	#13133217	C₂²	1	6	yes	—
9	#13133217	C₂²	1	6	yes	—
10	#13133216	C₂²	1	24	yes	—
11	#13133209	C₂²	1	12	yes	—
12	#13133208	C₂²	1	24	yes	—
13	#13133196	C₂²	1	4	yes	—
14	#13133196	C₂²	1	12	yes	—
15	Q₈×Dic₅₃	C₂²	1	8	yes	—
16	Q₈×Dic₅₃	C₂²	1	8	yes	—
17	C₄×Dic₁₀₆	C₂²	1	12	yes	—
18	C₄×Dic₁₀₆	C₂²	1	12	yes	—
19	#13133176	C₂²	1	12	yes	—
20	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₅₃	C₂²	1	4	yes	—
22	C₄×Q₈×C₅₃	C₂²	1	4	yes	—
23	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
24	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
25	C₂×Q₈×C₅₃	Q₈	1	8	yes	complemented
26	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
29	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
30	C₂×Dic₁₀₆	C₂³	1	4	yes	—
31	C₂×Dic₁₀₆	C₂³	1	12	yes	—
32	C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
33	C₄²×C₅₃	C₂³	1	6	yes	—
34	C₄²×C₅₃	C₂³	1	6	yes	—
35	C₂²⋊Dic₅₃	C₂³	1	8	yes	—
36	C₄⋊Dic₅₃	C₂³	1	12	yes	—
37	C₄⋊Dic₅₃	C₂³	1	12	yes	—
38	C₄⋊Dic₅₃	C₂³	1	4	yes	—
39	C₄⋊Dic₅₃	C₂³	1	12	yes	—
40	#11719919	C₂³	1	8	yes	—
41	#11719919	C₂³	1	24	yes	—
42	C₄×Dic₅₃	C₂³	1	24	yes	—
43	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
44	Q₈×C₅₃	C₂×Q₈	1	16	yes	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	4	3	no	—
47	C₂×C₄×C₅₃	C₂×Q₈	1	12	yes	—
48	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
49	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
51	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₅₃	C₂⁴	1	8	yes	—
53	C₂×Dic₅₃	C₂⁴	1	8	yes	—
54	Dic₁₀₆	—	2	8	no	—
55	Dic₁₀₆	—	4	12	no	—
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
59	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	—	2	6	no	—
62	C₄×C₅₃	C₂²×Q₈	1	12	yes	—
63	C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
64	Dic₅₃	—	2	8	no	—
65	Dic₅₃	—	2	8	no	—
66	#3502	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	—	2	2	no	—
68	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
69	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
70	C₂×C₅₃	#585	1	1	yes	characteristic
71	Q₈²	—	53	4	no	—
72	#557	—	53	8	no	—
73	#557	—	53	4	no	—
74	#544	—	53	6	no	—
75	#533	—	53	6	no	—
76	#519	—	53	2	no	—
77	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
78	C₅₃	#3502	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
80	C₄⋊Q₈	—	53	24	no	—
81	C₄⋊Q₈	—	53	6	no	—
82	C₄⋊Q₈	—	53	6	no	—
83	#125	—	53	24	no	—
84	#125	—	53	12	no	—
85	C₂²⋊Q₈	—	53	4	no	—
86	C₂²⋊Q₈	—	53	12	no	—
87	C₄×Q₈	—	53	8	no	—
88	C₄×Q₈	—	53	8	no	—
89	C₄×Q₈	—	53	12	no	—
90	C₄×Q₈	—	53	12	no	—
91	C₄×Q₈	C₂×D₅₃	1	4	yes	—
92	C₄×Q₈	C₂×D₅₃	1	4	yes	—
93	#117	—	53	12	no	—
94	C₂×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
95	C₂×C₄²	C₂×D₅₃	1	3	yes	—
96	C₂×Q₈	—	53	4	no	—
97	C₂×Q₈	—	53	12	no	—
98	C₂×Q₈	Dic₁₀₆	1	8	yes	complemented
99	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
100	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
101	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
102	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
103	C₄⋊C₄	—	53	12	no	—
104	C₄⋊C₄	—	53	12	no	—
105	C₄⋊C₄	—	53	8	no	—
106	C₄⋊C₄	—	53	4	no	—
107	C₄⋊C₄	—	53	12	no	—
108	C₄⋊C₄	—	53	24	no	—
109	C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
110	C₂²⋊C₄	—	53	8	no	—
111	C₄²	—	53	24	no	—
112	C₄²	C₂²×D₅₃	1	6	yes	—
113	C₄²	C₂²×D₅₃	1	6	yes	—
114	C₂³	C₂³×D₅₃	1	1	yes	characteristic
115	Q₈	—	106	8	no	—
116	Q₈	—	212	12	no	—
117	Q₈	C₂×Dic₁₀₆	1	16	yes	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	3	no	—
120	C₂×C₄	—	53	8	no	—
121	C₂×C₄	—	53	8	no	—
122	C₂×C₄	C₂×Dic₁₀₆	1	12	yes	—
123	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
124	C₂×C₄	C₂³×D₅₃	1	6	yes	—
125	C₂×C₄	C₂³×D₅₃	1	6	yes	—
126	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₂²×Dic₁₀₆	1	2	yes	—
130	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	6	no	—
133	C₄	—	106	8	no	—
134	C₄	—	106	8	no	—
135	C₄	C₂²×Dic₁₀₆	1	12	yes	—
136	C₄	#13133265	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂³×Dic₁₀₆	1	1	yes	characteristic
139	C₂	C₂×#13133265	1	1	yes	characteristic
140	C₂	#16811459	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₀₆

C₂

yes

maximal, complemented

#16811352

C₂

yes

maximal, complemented

#16811349

C₂

yes

maximal, complemented

#16811254

C₂

yes

maximal

#16811180

C₂

yes

maximal

#16811101

C₂

yes

maximal

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

#13133217

C₂²

yes

—

#13133217

C₂²

yes

—

#13133216

C₂²

yes

—

#13133209

C₂²

yes

—

#13133208

C₂²

yes

—

#13133196

C₂²

yes

—

#13133196

C₂²

yes

—

Q₈×Dic₅₃

C₂²

yes

—

Q₈×Dic₅₃

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133176

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

—

C₄×Q₈×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

Q₈

yes

complemented

C₂×Q₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂×Dic₁₀₆

C₂³

yes

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

C₂²⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

C₂³

yes

—

#11719919

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

C₂⁵

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×Q₈

yes

—

C₄×C₅₃

2₋¹⁺⁴

yes

—

Dic₅₃

—

Dic₅₃

—

#3502

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂³×Q₈

yes

characteristic

C₂×C₅₃

C₂×2₋¹⁺⁴

yes

characteristic

C₂×C₅₃

#585

yes

characteristic

Q₈²

—

—

—

—

—

—

C₂×C₄×Q₈

D₅₃

yes

characteristic

C₅₃

#3502

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₅₃

yes

—

C₄×Q₈

C₂×D₅₃

yes

—

#117

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₀₆

yes

complemented

C₂×Q₈

C₂²×D₅₃

yes

—

100

C₂²×C₄

C₂²×D₅₃

yes

characteristic

101

C₂²×C₄

C₂²×D₅₃

yes

—

102

C₂²×C₄

C₂²×D₅₃

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

C₂²×D₅₃

yes

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

C₂²×D₅₃

yes

—

113

C₄²

C₂²×D₅₃

yes

—

114

C₂³

C₂³×D₅₃

yes

characteristic

115

Q₈

—

106

—

116

Q₈

—

212

—

117

Q₈

C₂×Dic₁₀₆

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×Dic₁₀₆

yes

—

123

C₂×C₄

C₂³×D₅₃

yes

characteristic

124

C₂×C₄

C₂³×D₅₃

yes

—

125

C₂×C₄

C₂³×D₅₃

yes

—

126

C₂×C₄

C₂³×D₅₃

yes

characteristic

127

C₂²

—

128

C₂²

—

129

C₂²

C₂²×Dic₁₀₆

yes

—

130

C₂²

C₂⁴×D₅₃

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

106

—

134

C₄

—

106

—

135

C₄

C₂²×Dic₁₀₆

yes

—

136

C₄

#13133265

yes

—

137

C₂

—

138

C₂

C₂³×Dic₁₀₆

yes

characteristic

139

C₂

C₂×#13133265

yes

characteristic

140

C₂

#16811459

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₂×C₄×Q₈×C₅₃	C₂
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	#3502
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₅₃

commutator, socle

C₂²×C₅₃

C₂⁵

Fitting

C₂×C₄×Q₈×C₅₃

C₂

hypercenter

C₂×C₄×Q₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. Q₈×Dic₁₀₆	C₂	semidirect
2. #16811352	C₂	semidirect
3. #16811349	C₂	semidirect
25. C₂×Q₈×C₅₃	Q₈	semidirect
78. C₅₃	#3502	semidirect
98. C₂×Q₈	Dic₁₀₆	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₀₆

C₂

semidirect

2. #16811352

C₂

semidirect

3. #16811349

C₂

semidirect

25. C₂×Q₈×C₅₃

Q₈

semidirect

78. C₅₃

#3502

semidirect

98. C₂×Q₈

Dic₁₀₆

semidirect

Subgroups — #20288525

Conjugacy classes of subgroups

Complements