Group #20273515 — subgroups

Subgroups — C₈×#11719924 #20273515

Order 6784.

180 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₄×#11719924	C₂	1	1	yes	maximal, characteristic
2	C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#16810373	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810350	C₂	1	1	yes	maximal, characteristic
6	C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719924	C₄	1	1	yes	characteristic
9	C₂×C₈×D₅₃	—	2	1	no	—
10	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
11	#13133148	C₄	1	1	yes	characteristic
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	#13133141	C₄	1	1	yes	characteristic
14	#13133137	C₄	1	1	yes	characteristic
15	#13133129	—	2	1	no	—
16	C₄×#88322	C₂²	1	1	yes	characteristic
17	C₄×#88322	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₅₃	—	2	1	no	—
20	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₅₃	—	2	1	no	—
22	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	—	2	1	no	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₅₃	—	2	1	no	—
32	C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	#11719924	C₈	1	4	yes	direct product factor
34	#11719921	—	2	1	no	—
35	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	C₂×C₄	1	1	yes	characteristic
39	C₂×#88322	C₂×C₄	1	1	yes	characteristic
40	C₈×D₅₃	—	4	2	no	—
41	C₂²×D₅₃	—	2	1	no	—
42	Q₈×C₅₃	—	2	1	no	—
43	Q₈×C₅₃	C₂×C₈	1	2	yes	complemented
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
49	C₂×Dic₅₃	—	2	1	no	—
50	D₂₁₂	—	2	1	no	—
51	D₂₁₂	C₂×C₈	1	2	yes	—
52	C₄×D₅₃	—	4	1	no	—
53	C₄×D₅₃	—	4	1	no	—
54	C₈×C₅₃	—	2	1	no	—
55	C₈×C₅₃	—	4	1	no	—
56	C₈×C₅₃	SD₁₆	1	2	yes	complemented
57	#88322	—	2	2	no	—
58	#88322	—	2	1	no	—
59	#88322	—	4	1	no	—
60	#88322	C₂×C₈	1	2	yes	complemented
61	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
62	C₂×D₅₃	—	2	2	no	—
63	C₂×D₅₃	—	4	1	no	—
64	C₂×D₅₃	—	4	1	no	—
65	C₄×C₅₃	—	2	2	no	—
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	—	4	1	no	—
68	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
69	C₄×C₅₃	#131	1	1	yes	characteristic
70	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₄×C₅₃	#135	1	1	yes	characteristic
72	Dic₅₃	—	4	1	no	—
73	C₈×SD₁₆	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
75	C₂×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
76	C₂×C₅₃	#443	1	1	yes	characteristic
77	D₅₃	—	4	2	no	—
78	C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
79	C₄×SD₁₆	—	53	1	no	—
80	C₈×D₄	—	53	1	no	—
81	#334	—	53	1	no	—
82	Q₈⋊C₈	—	53	1	no	—
83	D₄⋊C₈	—	53	1	no	—
84	C₈²	—	53	1	no	—
85	C₅₃	C₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	53	1	no	—
87	C₂²×C₈	—	106	1	no	—
88	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
89	C₄×D₄	—	53	1	no	—
90	#106	—	53	1	no	—
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	—	53	1	no	—
93	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
94	#103	—	53	1	no	—
95	#102	—	53	1	no	—
96	C₂²⋊C₈	—	106	1	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	53	1	no	—
99	C₄×C₈	—	53	1	no	—
100	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
101	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
102	C₂×D₄	—	53	1	no	—
103	C₂²×C₄	—	106	1	no	—
104	SD₁₆	—	53	4	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	53	1	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	—	106	1	no	—
110	C₂×C₈	—	212	2	no	—
111	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	53	1	no	—
115	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
116	C₂²⋊C₄	—	106	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	C₂²×D₅₃	1	1	yes	characteristic
119	C₂³	—	106	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	C₈×D₅₃	1	2	yes	complemented
122	D₄	—	53	2	no	—
123	D₄	—	106	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	106	1	no	—
127	C₂×C₄	—	212	1	no	—
128	C₂×C₄	—	212	1	no	—
129	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
130	C₂×C₄	#11719945	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	53	2	no	—
135	C₈	—	106	2	no	—
136	C₈	—	106	1	no	—
137	C₈	—	212	1	no	—
138	C₈	#11719924	1	2	yes	direct product factor
139	C₂²	—	106	2	no	—
140	C₂²	—	212	1	no	—
141	C₂²	—	212	1	no	—
142	C₂²	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	212	1	no	—
147	C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
148	C₄	#13133223	1	1	yes	characteristic
149	C₄	C₂×#11719924	1	1	yes	characteristic
150	C₄	#13133237	1	1	yes	characteristic
151	C₂	—	212	2	no	—
152	C₂	C₈×C₂²⋊D₅₃	1	1	yes	characteristic
153	C₂	C₄×#11719924	1	1	yes	characteristic
154	C₂	#16810738	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11719924

C₂

yes

maximal, characteristic

C₈×D₂₁₂

C₂

yes

maximal, characteristic

#16810373

C₂

yes

maximal, characteristic, complemented

Q₈⋊#88322

C₂

yes

maximal, characteristic, complemented

#16810350

C₂

yes

maximal, characteristic

C₈×#88322

C₂

yes

maximal, characteristic, complemented

C₈×Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#11719924

C₄

yes

characteristic

C₂×C₈×D₅₃

—

C₄×D₂₁₂

C₂²

yes

characteristic

#13133148

C₄

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

#13133141

C₄

yes

characteristic

#13133137

C₄

yes

characteristic

#13133129

—

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₂₁₂

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719924

C₈

yes

direct product factor

#11719921

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

—

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₈×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

C₂×C₈

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

C₂×C₈

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

SD₁₆

yes

—

—

—

C₂×C₈

yes

complemented

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

#131

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

#135

yes

characteristic

Dic₅₃

—

C₈×SD₁₆

—

maximal, Sylow

C₂×C₅₃

C₈×D₄

yes

characteristic

C₂×C₅₃

C₄×SD₁₆

yes

characteristic

C₂×C₅₃

#443

yes

characteristic

D₅₃

—

C₈×Q₈

D₅₃

yes

characteristic, complemented

C₄×SD₁₆

—

C₈×D₄

—

#334

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₈²

—

C₅₃

C₈×SD₁₆

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂²×C₈

—

106

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#103

—

#102

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₅₃

yes

characteristic

101

C₂×Q₈

C₄×D₅₃

yes

characteristic

102

C₂×D₄

—

103

C₂²×C₄

—

106

—

104

SD₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

106

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

212

—

111

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

112

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₄×D₅₃

yes

characteristic

116

C₂²⋊C₄

—

106

—

117

C₄²

—

118

C₄²

C₂²×D₅₃

yes

characteristic

119

C₂³

—

106

—

120

Q₈

—

121

Q₈

C₈×D₅₃

yes

complemented

122

D₄

—

123

D₄

—

106

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

212

—

128

C₂×C₄

—

212

—

129

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

130

C₂×C₄

#11719945

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

106

—

136

C₈

—

106

—

137

C₈

—

212

—

138

C₈

#11719924

yes

direct product factor

139

C₂²

—

106

—

140

C₂²

—

212

—

141

C₂²

—

212

—

142

C₂²

C₄×C₂²⋊D₅₃

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

212

—

147

C₄

C₂×C₈×D₅₃

yes

characteristic

148

C₄

#13133223

yes

characteristic

149

C₄

C₂×#11719924

yes

characteristic

150

C₄

#13133237

yes

characteristic

151

C₂

—

212

—

152

C₂

C₈×C₂²⋊D₅₃

yes

characteristic

153

C₂

C₄×#11719924

yes

characteristic

154

C₂

#16810738

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₄²	C₂²×D₅₃
Fitting	C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	C₈×Q₈	D₅₃
hypocenter	C₅₃	C₈×SD₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂×C₈

C₂²⋊D₅₃

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₄²

C₂²×D₅₃

Fitting

C₈×Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₄×D₄

hypercenter

C₈×Q₈

D₅₃

hypocenter

C₅₃

C₈×SD₁₆

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₈

N	H ≅ G/N	type
3. #16810373	C₂	semidirect
4. Q₈⋊#88322	C₂	semidirect
6. C₈×#88322	C₂	semidirect
7. C₈×Q₈×C₅₃	C₂	semidirect
33. #11719924	C₈	direct (N × H)
43. Q₈×C₅₃	C₂×C₈	semidirect
56. C₈×C₅₃	SD₁₆	semidirect
60. #88322	C₂×C₈	semidirect
78. C₈×Q₈	D₅₃	semidirect
85. C₅₃	C₈×SD₁₆	semidirect
121. Q₈	C₈×D₅₃	semidirect
138. C₈	#11719924	direct (N × H)

H ≅ G/N

type

3. #16810373

C₂

semidirect

4. Q₈⋊#88322

C₂

semidirect

6. C₈×#88322

C₂

semidirect

7. C₈×Q₈×C₅₃

C₂

semidirect

33. #11719924

C₈

direct (N × H)

43. Q₈×C₅₃

C₂×C₈

semidirect

56. C₈×C₅₃

SD₁₆

semidirect

60. #88322

C₂×C₈

semidirect

78. C₈×Q₈

D₅₃

semidirect

85. C₅₃

C₈×SD₁₆

semidirect

121. Q₈

C₈×D₅₃

semidirect

138. C₈

#11719924

direct (N × H)

Subgroups — C₈×#11719924 #20273515

Conjugacy classes of subgroups

Complements

Subgroups — C8×#11719924 #20273515

Conjugacy classes of subgroups

Complements

Subgroups — C₈×#11719924 #20273515