Group #20288116 — subgroups

Subgroups — C₂×#16811254 #20288116

Order 6784.

1080 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 565 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	#16811254	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13133216	C₂	1	4	yes	maximal, complemented
3	C₂×#13133208	C₂	1	4	yes	maximal, complemented
4	C₂×#13133196	C₂	1	4	yes	maximal
5	C₂×#13133176	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₀₆	C₂²	1	4	yes	—
8	#13133216	C₂²	1	32	yes	complemented
9	#13133208	C₂²	1	32	yes	complemented
10	#13133196	C₂²	1	32	yes	—
11	#13133176	C₂²	1	16	yes	—
12	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
15	C₂×#11719919	C₂²	1	8	yes	—
16	C₂×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
17	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
18	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
19	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	8	yes	—
22	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	—	4	4	no	—
24	C₂²×C₄×C₅₃	Q₈	1	8	yes	complemented
25	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₅₃	C₂³	1	8	yes	—
27	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
29	C₂²×Dic₅₃	C₂³	1	4	yes	—
30	C₂²×Dic₅₃	C₂³	1	4	yes	—
31	C₂×Dic₁₀₆	—	4	8	no	—
32	C₂×Dic₁₀₆	C₂³	1	16	yes	—
33	C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
34	C₂²⋊Dic₅₃	C₂³	1	16	yes	—
35	C₄⋊Dic₅₃	C₂³	1	8	yes	—
36	C₄⋊Dic₅₃	C₂³	1	8	yes	—
37	C₄⋊Dic₅₃	C₂³	1	16	yes	—
38	#11719919	C₂³	1	32	yes	—
39	C₄×Dic₅₃	C₂³	1	16	yes	—
40	C₂³×C₅₃	—	2	2	no	—
41	C₂³×C₅₃	—	2	2	no	—
42	C₂³×C₅₃	C₂×Q₈	1	4	yes	—
43	C₂³×C₅₃	C₂⁴	1	2	yes	—
44	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	8	no	—
46	C₂×C₄×C₅₃	—	4	8	no	—
47	C₂×C₄×C₅₃	C₂×Q₈	1	8	yes	—
48	C₂×C₄×C₅₃	C₂×Q₈	1	16	yes	complemented
49	C₂×C₄×C₅₃	C₂⁴	1	8	yes	—
50	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
51	C₂×Dic₅₃	—	2	8	no	—
52	C₂×Dic₅₃	—	2	8	no	—
53	C₂×Dic₅₃	C₂⁴	1	8	yes	—
54	C₂×Dic₅₃	C₂⁴	1	8	yes	—
55	Dic₁₀₆	—	4	16	no	—
56	C₂²×C₅₃	—	2	8	no	—
57	C₂²×C₅₃	—	2	2	no	—
58	C₂²×C₅₃	—	2	2	no	—
59	C₂²×C₅₃	C₂²×Q₈	1	4	yes	—
60	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
61	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
62	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
63	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
64	C₄×C₅₃	—	2	8	no	—
65	C₄×C₅₃	C₂²×Q₈	1	8	yes	—
66	Dic₅₃	—	2	8	no	—
67	Dic₅₃	—	2	8	no	—
68	C₂×#544	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	—	2	4	no	—
70	C₂×C₅₃	#544	1	4	yes	complemented
71	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
72	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
73	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
74	#544	—	53	16	no	—
75	C₂×C₄⋊Q₈	—	53	4	no	—
76	C₂×#125	—	53	4	no	—
77	C₂×C₂²⋊Q₈	—	53	4	no	—
78	C₂×#117	—	53	2	no	—
79	C₂²×C₄⋊C₄	D₅₃	1	1	yes	characteristic
80	C₅₃	C₂×#544	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	—	53	4	no	—
82	C₂³×C₄	C₂×D₅₃	1	2	yes	—
83	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
84	C₄⋊Q₈	—	53	32	no	—
85	#125	—	53	32	no	—
86	C₂²⋊Q₈	—	53	32	no	—
87	#117	—	53	16	no	—
88	C₂×C₄⋊C₄	—	53	2	no	—
89	C₂×C₄⋊C₄	—	53	2	no	—
90	C₂×C₄⋊C₄	—	53	4	no	—
91	C₂×C₄⋊C₄	—	53	8	no	—
92	C₂×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
93	C₂×C₄⋊C₄	C₂×D₅₃	1	8	yes	—
94	C₂×C₂²⋊C₄	—	53	4	no	—
95	C₂×C₄²	—	53	4	no	—
96	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
97	C₂×Q₈	—	53	16	no	—
98	C₂×Q₈	—	212	8	no	—
99	C₂²×C₄	—	4	4	no	—
100	C₂²×C₄	—	53	4	no	—
101	C₂²×C₄	—	53	4	no	—
102	C₂²×C₄	Dic₁₀₆	1	8	yes	complemented
103	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
104	C₂²×C₄	C₂²×D₅₃	1	8	yes	—
105	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
106	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
107	C₄⋊C₄	—	53	8	no	—
108	C₄⋊C₄	—	53	8	no	—
109	C₄⋊C₄	—	53	16	no	—
110	C₄⋊C₄	—	53	32	no	—
111	C₄⋊C₄	C₂²×D₅₃	1	16	yes	—
112	C₂²⋊C₄	—	53	16	no	—
113	C₄²	—	53	16	no	—
114	C₂³	—	2	2	no	—
115	C₂³	—	2	2	no	—
116	C₂³	C₂×Dic₁₀₆	1	4	yes	—
117	C₂³	C₂³×D₅₃	1	2	yes	—
118	C₂³	C₂³×D₅₃	1	1	yes	characteristic
119	Q₈	—	212	16	no	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	—	4	8	no	—
122	C₂×C₄	—	53	8	no	—
123	C₂×C₄	—	53	8	no	—
124	C₂×C₄	—	106	8	no	—
125	C₂×C₄	—	106	8	no	—
126	C₂×C₄	C₂×Dic₁₀₆	1	8	yes	—
127	C₂×C₄	C₂×Dic₁₀₆	1	16	yes	complemented
128	C₂×C₄	C₂³×D₅₃	1	8	yes	—
129	C₂×C₄	C₂³×D₅₃	1	4	yes	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	8	no	—
133	C₂²	C₂²×Dic₁₀₆	1	4	yes	—
134	C₂²	C₂²×Dic₁₀₆	1	2	yes	—
135	C₂²	#13133262	1	2	yes	—
136	C₂²	#13133265	1	2	yes	—
137	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
138	C₄	—	2	8	no	—
139	C₄	—	106	8	no	—
140	C₄	—	106	8	no	—
141	C₄	C₂²×Dic₁₀₆	1	8	yes	—
142	C₂	—	2	4	no	—
143	C₂	#16811254	1	4	yes	direct product factor
144	C₂	C₂³×Dic₁₀₆	1	1	yes	characteristic
145	C₂	C₂×#13133262	1	1	yes	characteristic
146	C₂	C₂×#13133265	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811254

C₂

yes

maximal, direct product factor

C₂×#13133216

C₂

yes

maximal, complemented

C₂×#13133208

C₂

yes

maximal, complemented

C₂×#13133196

C₂

yes

maximal

C₂×#13133176

C₂

yes

maximal

C₂²×C₄⋊C₄×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

#13133216

C₂²

yes

complemented

#13133208

C₂²

yes

complemented

#13133196

C₂²

yes

—

#13133176

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

Q₈

yes

complemented

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

C₂⁴

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

complemented

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

Dic₁₀₆

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₂²×C₅₃

C₂⁵

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂²×Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

C₂×#544

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#544

yes

complemented

C₂×C₅₃

C₂³×Q₈

yes

characteristic

C₂×C₅₃

C₂×H(5,𝔽₂)

yes

characteristic

C₂×C₅₃

C₂×2₋¹⁺⁴

yes

characteristic

#544

—

C₂×C₄⋊Q₈

—

C₂×#125

—

C₂×C₂²⋊Q₈

—

C₂×#117

—

C₂²×C₄⋊C₄

D₅₃

yes

characteristic

C₅₃

C₂×#544

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂³×C₄

C₂×D₅₃

yes

—

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

212

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

Dic₁₀₆

yes

complemented

103

C₂²×C₄

C₂²×D₅₃

yes

—

104

C₂²×C₄

C₂²×D₅₃

yes

—

105

C₂²×C₄

C₂²×D₅₃

yes

—

106

C₂²×C₄

C₂²×D₅₃

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²×D₅₃

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₂³

—

115

C₂³

—

116

C₂³

C₂×Dic₁₀₆

yes

—

117

C₂³

C₂³×D₅₃

yes

—

118

C₂³

C₂³×D₅₃

yes

characteristic

119

Q₈

—

212

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

106

—

126

C₂×C₄

C₂×Dic₁₀₆

yes

—

127

C₂×C₄

C₂×Dic₁₀₆

yes

complemented

128

C₂×C₄

C₂³×D₅₃

yes

—

129

C₂×C₄

C₂³×D₅₃

yes

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

C₂²×Dic₁₀₆

yes

—

134

C₂²

C₂²×Dic₁₀₆

yes

—

135

C₂²

#13133262

yes

—

136

C₂²

#13133265

yes

—

137

C₂²

C₂⁴×D₅₃

yes

characteristic

138

C₄

—

139

C₄

—

106

—

140

C₄

—

106

—

141

C₄

C₂²×Dic₁₀₆

yes

—

142

C₂

—

143

C₂

#16811254

yes

direct product factor

144

C₂

C₂³×Dic₁₀₆

yes

characteristic

145

C₂

C₂×#13133262

yes

characteristic

146

C₂

C₂×#13133265

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂²×C₄⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂²×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂×#544
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂⁵

Frattini

C₂²

C₂⁴×D₅₃

Fitting

C₂²×C₄⋊C₄×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₂²×C₄⋊C₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
1. #16811254	C₂	direct (N × H)
2. C₂×#13133216	C₂	semidirect
3. C₂×#13133208	C₂	semidirect
8. #13133216	C₂²	semidirect
9. #13133208	C₂²	semidirect
24. C₂²×C₄×C₅₃	Q₈	semidirect
48. C₂×C₄×C₅₃	C₂×Q₈	semidirect
70. C₂×C₅₃	#544	semidirect
80. C₅₃	C₂×#544	semidirect
102. C₂²×C₄	Dic₁₀₆	semidirect
127. C₂×C₄	C₂×Dic₁₀₆	semidirect
143. C₂	#16811254	direct (N × H)

H ≅ G/N

type

1. #16811254

C₂

direct (N × H)

2. C₂×#13133216

C₂

semidirect

3. C₂×#13133208

C₂

semidirect

8. #13133216

C₂²

semidirect

9. #13133208

C₂²

semidirect

24. C₂²×C₄×C₅₃

Q₈

semidirect

48. C₂×C₄×C₅₃

C₂×Q₈

semidirect

70. C₂×C₅₃

#544

semidirect

80. C₅₃

C₂×#544

semidirect

102. C₂²×C₄

Dic₁₀₆

semidirect

127. C₂×C₄

C₂×Dic₁₀₆

semidirect

143. C₂

#16811254

direct (N × H)

Subgroups — C₂×#16811254 #20288116

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811254 #20288116

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811254 #20288116