Group #22608682 — subgroups

Subgroups — #22608682

Order 9344.

180 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 83 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	#17988639	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
3	C₄×#11924011	C₂	1	1	yes	maximal, characteristic
4	#17988339	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988338	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988337	C₂	1	1	yes	maximal, characteristic, complemented
7	#335×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924011	C₄	1	1	yes	characteristic
9	C₂×#11924011	C₄	1	1	yes	characteristic
10	C₂×C₈×D₇₃	—	2	1	no	—
11	C₄²×D₇₃	C₂²	1	1	yes	characteristic
12	C₄⋊#10596831	C₂²	1	1	yes	characteristic
13	C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	#14556520	—	2	1	no	—
15	#14556503	—	2	1	no	—
16	#14556501	C₄	1	1	yes	characteristic
17	#14556501	C₄	1	1	yes	characteristic
18	C₄×#10596831	C₂²	1	1	yes	characteristic
19	C₈×Dic₇₃	—	2	1	no	—
20	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
25	C₂×C₄×D₇₃	Q₈	1	1	yes	characteristic
26	#11924011	—	2	2	no	—
27	#11924011	C₈	1	4	yes	complemented
28	C₂×#10596831	—	2	1	no	—
29	C₂×#10596831	—	2	1	no	—
30	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
32	C₈×D₇₃	—	4	2	no	—
33	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₇₃	D₄	1	1	yes	characteristic
35	C₄×Dic₇₃	Q₈	1	1	yes	characteristic
36	C₂×C₈×C₇₃	—	2	1	no	—
37	C₂×C₈×C₇₃	—	2	1	no	—
38	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₇₃	C₂³	1	1	yes	characteristic
41	C₂²×D₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄×D₇₃	—	2	1	no	—
49	C₄×D₇₃	—	2	1	no	—
50	C₄×D₇₃	—	4	1	no	—
51	C₄×D₇₃	C₂×C₈	1	2	yes	—
52	C₄×D₇₃	M₁₆	1	2	yes	—
53	C₈×C₇₃	—	2	1	no	—
54	C₈×C₇₃	—	4	1	no	—
55	C₈×C₇₃	—	4	1	no	—
56	C₈×C₇₃	C₂×C₈	1	2	yes	complemented
57	#10596831	—	2	1	no	—
58	#10596831	—	4	1	no	—
59	#10596831	—	4	1	no	—
60	#10596831	C₂×C₈	1	2	yes	complemented
61	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×D₇₃	—	2	1	no	—
63	C₂×D₇₃	—	2	1	no	—
64	C₂×D₇₃	C₄⋊C₈	1	2	yes	—
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
67	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
68	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₇₃	#137	1	1	yes	characteristic
70	Dic₇₃	—	2	1	no	—
71	Dic₇₃	—	2	1	no	—
72	Dic₇₃	C₄⋊C₈	1	2	yes	—
73	C₂×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
74	C₂×C₇₃	#428	1	1	yes	characteristic
75	C₂×C₇₃	#430	1	1	yes	characteristic
76	D₇₃	—	2	2	no	—
77	#1565	—	73	1	no	maximal, Sylow
78	C₇₃	#1565	1	1	yes	characteristic, complemented, Sylow
79	#431	—	73	1	no	—
80	C₂×C₄⋊C₈	—	73	1	no	—
81	C₄×M₁₆	—	73	1	no	—
82	#335	—	73	1	no	—
83	#335	D₇₃	1	1	yes	characteristic, complemented
84	#334	—	73	1	no	—
85	#334	—	73	1	no	—
86	C₂×M₁₆	—	73	1	no	—
87	C₂×M₁₆	—	73	1	no	—
88	C₂²×C₈	—	146	1	no	—
89	C₂×C₄²	—	73	1	no	—
90	C₄⋊C₈	—	73	1	no	—
91	C₄⋊C₈	—	73	1	no	—
92	C₄⋊C₈	—	146	1	no	—
93	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
94	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
95	C₂²⋊C₈	—	146	1	no	—
96	#97	—	73	1	no	—
97	#97	—	73	1	no	—
98	C₄×C₈	—	73	1	no	—
99	C₄×C₈	—	146	1	no	—
100	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
101	C₂²×C₄	—	73	1	no	—
102	C₂²×C₄	—	73	1	no	—
103	C₂²×C₄	—	73	1	no	—
104	M₁₆	—	73	4	no	—
105	M₁₆	—	146	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	292	2	no	—
109	C₂×C₈	—	73	1	no	—
110	C₂×C₈	—	73	1	no	—
111	C₂×C₈	—	146	1	no	—
112	C₂×C₈	—	146	1	no	—
113	C₂×C₈	C₄×D₇₃	1	1	yes	characteristic
114	C₂×C₈	C₄×D₇₃	1	1	yes	characteristic
115	C₄²	—	73	1	no	—
116	C₄²	—	73	1	no	—
117	C₄²	—	73	1	no	—
118	C₄²	C₂²×D₇₃	1	1	yes	characteristic
119	C₂³	—	73	1	no	—
120	C₂×C₄	—	292	1	no	—
121	C₂×C₄	—	73	1	no	—
122	C₂×C₄	—	73	2	no	—
123	C₂×C₄	—	73	2	no	—
124	C₂×C₄	—	73	1	no	—
125	C₂×C₄	—	73	1	no	—
126	C₂×C₄	—	146	1	no	—
127	C₂×C₄	—	146	1	no	—
128	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
129	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
130	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	292	1	no	—
135	C₈	—	292	1	no	—
136	C₈	—	73	2	no	—
137	C₈	—	146	1	no	—
138	C₈	C₈×D₇₃	1	2	yes	complemented
139	C₂²	—	73	2	no	—
140	C₂²	—	146	1	no	—
141	C₂²	—	146	1	no	—
142	C₂²	C₄⋊C₄×D₇₃	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	73	2	no	—
145	C₄	—	146	1	no	—
146	C₄	—	146	1	no	—
147	C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
148	C₄	M₁₆×D₇₃	1	1	yes	characteristic
149	C₄	#14556615	1	1	yes	characteristic
150	C₄	#14556623	1	1	yes	characteristic
151	C₂	—	146	2	no	—
152	C₂	C₄⋊C₈×D₇₃	1	1	yes	characteristic
153	C₂	#17988628	1	1	yes	characteristic
154	C₂	#17988634	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988639

C₂

yes

maximal, characteristic

C₄⋊C₈×D₇₃

C₂

yes

maximal, characteristic

C₄×#11924011

C₂

yes

maximal, characteristic

#17988339

C₂

yes

maximal, characteristic, complemented

#17988338

C₂

yes

maximal, characteristic, complemented

#17988337

C₂

yes

maximal, characteristic, complemented

#335×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924011

C₄

yes

characteristic

C₂×#11924011

C₄

yes

characteristic

C₂×C₈×D₇₃

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

C₄

yes

characteristic

#14556501

C₄

yes

characteristic

C₄×#10596831

C₂²

yes

characteristic

C₈×Dic₇₃

—

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

Q₈

yes

characteristic

#11924011

—

#11924011

C₈

yes

—

—

C₂×C₄

yes

characteristic

C₂×#10596831

C₂×C₄

yes

characteristic

C₈×D₇₃

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

Q₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×Dic₇₃

C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×C₈

yes

—

C₄×D₇₃

M₁₆

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×C₈

yes

—

—

—

C₂×C₈

yes

complemented

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₄⋊C₈

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂²×C₈

yes

characteristic

C₄×C₇₃

C₂×M₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₄⋊C₈

yes

—

C₂×C₇₃

C₂×C₄⋊C₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1565

—

maximal, Sylow

C₇₃

#1565

yes

characteristic, complemented, Sylow

#431

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

#335

—

#335

D₇₃

yes

characteristic, complemented

#334

—

#334

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

146

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

C₄⋊C₈

C₂×D₇₃

yes

characteristic

C₄⋊C₈

C₂×D₇₃

yes

characteristic

C₂²⋊C₈

—

146

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

146

—

100

C₄×C₈

C₂×D₇₃

yes

characteristic

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

M₁₆

—

105

M₁₆

—

146

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

292

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

146

—

112

C₂×C₈

—

146

—

113

C₂×C₈

C₄×D₇₃

yes

characteristic

114

C₂×C₈

C₄×D₇₃

yes

characteristic

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₂²×D₇₃

yes

characteristic

119

C₂³

—

120

C₂×C₄

—

292

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

146

—

127

C₂×C₄

—

146

—

128

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

129

C₂×C₄

D₄×D₇₃

yes

characteristic

130

C₂×C₄

Q₈×D₇₃

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

292

—

135

C₈

—

292

—

136

C₈

—

137

C₈

—

146

—

138

C₈

C₈×D₇₃

yes

complemented

139

C₂²

—

140

C₂²

—

146

—

141

C₂²

—

146

—

142

C₂²

C₄⋊C₄×D₇₃

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

—

146

C₄

—

146

—

147

C₄

C₂×C₈×D₇₃

yes

characteristic

148

C₄

M₁₆×D₇₃

yes

characteristic

149

C₄

#14556615

yes

characteristic

150

C₄

#14556623

yes

characteristic

151

C₂

—

146

—

152

C₂

C₄⋊C₈×D₇₃

yes

characteristic

153

C₂

#17988628

yes

characteristic

154

C₂

#17988634

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₇₃
commutator	C₄×C₇₃	C₂²×C₈
Frattini	C₄²	C₂²×D₇₃
Fitting	#335×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₄⋊C₄
hypercenter	#335	D₇₃
hypocenter	C₇₃	#1565
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×D₇₃

commutator

C₄×C₇₃

C₂²×C₈

Frattini

C₄²

C₂²×D₇₃

Fitting

#335×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₄⋊C₄

D₇₃

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
4. #17988339	C₂	semidirect
5. #17988338	C₂	semidirect
6. #17988337	C₂	semidirect
7. #335×C₇₃	C₂	semidirect
27. #11924011	C₈	semidirect
56. C₈×C₇₃	C₂×C₈	semidirect
60. #10596831	C₂×C₈	semidirect
78. C₇₃	#1565	semidirect
83. #335	D₇₃	semidirect
138. C₈	C₈×D₇₃	semidirect

H ≅ G/N

type

4. #17988339

C₂

semidirect

5. #17988338

C₂

semidirect

6. #17988337

C₂

semidirect

7. #335×C₇₃

C₂

semidirect

27. #11924011

C₈

semidirect

56. C₈×C₇₃

C₂×C₈

semidirect

60. #10596831

C₂×C₈

semidirect

78. C₇₃

#1565

semidirect

83. #335

D₇₃

semidirect

138. C₈

C₈×D₇₃

semidirect

Subgroups — #22608682

Conjugacy classes of subgroups

Complements