Group #22620443 — subgroups

Subgroups — #22620443

Order 9344.

356 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 153 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	#17989140	C₂	1	2	yes	maximal
2	#17988991	C₂	1	2	yes	maximal, complemented
3	#17988988	C₂	1	2	yes	maximal, complemented
4	#17988987	C₂	1	2	yes	maximal, complemented
5	#17988984	C₂	1	2	yes	maximal, complemented
6	#17988625	C₂	1	1	yes	maximal, characteristic
7	#17988623	C₂	1	2	yes	maximal
8	C₂×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
9	#427×C₇₃	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10596831	C₂²	1	1	yes	characteristic
11	#14556590	C₂²	1	4	yes	—
12	#14556582	C₂²	1	4	yes	—
13	#14556568	—	2	4	no	—
14	C₄×Dic₁₄₆	—	2	4	no	—
15	#14556550	C₂²	1	2	yes	—
16	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
17	#14556525	C₂²	1	2	yes	—
18	#14556524	C₂²	1	4	yes	—
19	#14556523	C₂²	1	2	yes	—
20	#14556522	C₂²	1	4	yes	—
21	C₈×Dic₇₃	C₂²	1	1	yes	characteristic
22	C₈×Dic₇₃	C₂²	1	1	yes	characteristic
23	C₈×Dic₇₃	C₂²	1	1	yes	characteristic
24	C₈×Dic₇₃	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
26	#117×C₇₃	C₂²	1	2	yes	—
27	#107×C₇₃	C₂²	1	2	yes	—
28	#106×C₇₃	C₂²	1	2	yes	—
29	C₂×Dic₁₄₆	—	2	4	no	—
30	C₂²×Dic₇₃	D₄	1	2	yes	—
31	C₂×#10596831	Q₈	1	4	yes	complemented
32	C₂×#10596831	C₂³	1	1	yes	characteristic
33	C₂×#10596831	C₂³	1	1	yes	characteristic
34	C₄⋊Dic₇₃	—	2	4	no	—
35	C₄⋊Dic₇₃	C₂³	1	4	yes	—
36	#11924007	—	2	4	no	—
37	#11924007	—	2	4	no	—
38	C₂²⋊Dic₇₃	—	2	2	no	—
39	C₄×Dic₇₃	—	2	2	no	—
40	C₄×Dic₇₃	D₄	1	2	yes	—
41	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
42	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₇₃	Q₈	1	4	yes	complemented
45	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
48	C₂²⋊C₄×C₇₃	—	2	2	no	—
49	C₄²×C₇₃	—	2	2	no	—
50	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	—	2	4	no	—
52	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₇₃	—	2	4	no	—
57	C₂×Dic₇₃	—	2	4	no	—
58	C₂×Dic₇₃	C₂×D₄	1	2	yes	—
59	C₂×Dic₇₃	C₂×D₄	1	2	yes	—
60	Dic₁₄₆	—	4	8	no	—
61	C₈×C₇₃	C₂×Q₈	1	2	yes	—
62	C₈×C₇₃	C₂×Q₈	1	2	yes	—
63	#10596831	C₂×Q₈	1	2	yes	—
64	#10596831	C₂×Q₈	1	2	yes	—
65	C₂²×C₇₃	—	2	2	no	—
66	C₂²×C₇₃	C₄⋊Q₈	1	2	yes	—
67	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₇₃	—	4	4	no	—
69	C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
70	C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
71	C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
72	Dic₇₃	—	2	2	no	—
73	Dic₇₃	—	4	4	no	—
74	Dic₇₃	#135	1	4	yes	—
75	C₂×C₇₃	—	2	2	no	—
76	C₂×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₇₃	C₂×#135	1	2	yes	—
78	#3160	—	73	1	no	maximal, Sylow
79	C₇₃	#3160	1	1	yes	characteristic, complemented, Sylow
80	#533	—	73	2	no	—
81	#500	—	73	2	no	—
82	#499	—	73	2	no	—
83	#499	—	73	2	no	—
84	#498	—	73	2	no	—
85	#427	—	73	1	no	—
86	#427	—	73	2	no	—
87	#427	D₇₃	1	1	yes	characteristic
88	C₂×C₄×C₈	—	73	1	no	—
89	C₂²×C₈	—	73	1	no	—
90	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
91	C₄⋊Q₈	—	73	4	no	—
92	#125	—	73	4	no	—
93	C₂²⋊Q₈	—	146	4	no	—
94	C₄×Q₈	—	146	4	no	—
95	#117	—	73	2	no	—
96	#117	C₂×D₇₃	1	2	yes	—
97	C₂×C₄²	—	73	1	no	—
98	#107	—	73	2	no	—
99	#107	—	73	4	no	—
100	#107	C₂×D₇₃	1	2	yes	—
101	#106	—	73	2	no	—
102	#106	—	73	4	no	—
103	#106	C₂×D₇₃	1	2	yes	—
104	C₄×C₈	—	73	1	no	—
105	C₄×C₈	—	73	1	no	—
106	C₄×C₈	—	73	1	no	—
107	C₄×C₈	—	73	1	no	—
108	C₂×Q₈	—	146	4	no	—
109	C₂²×C₄	—	73	2	no	—
110	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
111	C₂×C₈	—	73	4	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	—	73	1	no	—
114	C₂×C₈	Dic₁₄₆	1	4	yes	complemented
115	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
117	C₄⋊C₄	—	73	4	no	—
118	C₄⋊C₄	—	146	4	no	—
119	C₄⋊C₄	—	146	4	no	—
120	C₄⋊C₄	—	146	4	no	—
121	C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
122	C₂²⋊C₄	—	2	2	no	—
123	C₂²⋊C₄	—	146	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	73	2	no	—
126	C₄²	—	73	1	no	—
127	C₄²	—	73	1	no	—
128	C₄²	—	146	2	no	—
129	C₂³	D₄×D₇₃	1	1	yes	characteristic
130	Q₈	—	292	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	73	2	no	—
133	C₂×C₄	—	73	2	no	—
134	C₂×C₄	—	146	4	no	—
135	C₂×C₄	—	146	4	no	—
136	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
137	C₂×C₄	Q₈×D₇₃	1	2	yes	—
138	C₂×C₄	C₂×Dic₁₄₆	1	2	yes	—
139	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
140	C₈	—	73	2	no	—
141	C₈	—	73	2	no	—
142	C₈	C₂×Dic₁₄₆	1	2	yes	—
143	C₈	C₂×Dic₁₄₆	1	2	yes	—
144	C₂²	—	2	2	no	—
145	C₂²	#14556590	1	2	yes	—
146	C₂²	C₂×D₄×D₇₃	1	1	yes	characteristic
147	C₄	—	4	4	no	—
148	C₄	—	292	4	no	—
149	C₄	—	73	4	no	—
150	C₄	—	146	2	no	—
151	C₄	#14556590	1	2	yes	—
152	C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
153	C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
154	C₂	—	2	2	no	—
155	C₂	C₂×#14556590	1	1	yes	characteristic
156	C₂	#135×D₇₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17989140

C₂

yes

maximal

#17988991

C₂

yes

maximal, complemented

#17988988

C₂

yes

maximal, complemented

#17988987

C₂

yes

maximal, complemented

#17988984

C₂

yes

maximal, complemented

#17988625

C₂

yes

maximal, characteristic

#17988623

C₂

yes

maximal

C₂×C₈×Dic₇₃

C₂

yes

maximal, characteristic

#427×C₇₃

C₂

yes

maximal, characteristic

C₂²×#10596831

C₂²

yes

characteristic

#14556590

C₂²

yes

—

#14556582

C₂²

yes

—

#14556568

—

C₄×Dic₁₄₆

—

#14556550

C₂²

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

#14556525

C₂²

yes

—

#14556524

C₂²

yes

—

#14556523

C₂²

yes

—

#14556522

C₂²

yes

—

C₈×Dic₇₃

C₂²

yes

characteristic

C₈×Dic₇₃

C₂²

yes

characteristic

C₈×Dic₇₃

C₂²

yes

characteristic

C₈×Dic₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

#117×C₇₃

C₂²

yes

—

#107×C₇₃

C₂²

yes

—

#106×C₇₃

C₂²

yes

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

D₄

yes

—

C₂×#10596831

Q₈

yes

complemented

C₂×#10596831

C₂³

yes

characteristic

C₂×#10596831

C₂³

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂³

yes

—

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

—

C₄×Dic₇₃

C₂³

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

Q₈

yes

complemented

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

C₂³

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₄

yes

—

C₂×Dic₇₃

C₂×D₄

yes

—

Dic₁₄₆

—

C₈×C₇₃

C₂×Q₈

yes

—

C₈×C₇₃

C₂×Q₈

yes

—

#10596831

C₂×Q₈

yes

—

#10596831

C₂×Q₈

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄⋊Q₈

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₄⋊Q₈

yes

—

C₄×C₇₃

C₂²×Q₈

yes

characteristic

C₄×C₇₃

C₂²×Q₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#135

yes

—

C₂×C₇₃

—

C₂×C₇₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇₃

C₂×#135

yes

—

#3160

—

maximal, Sylow

C₇₃

#3160

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

—

D₇₃

yes

characteristic

C₂×C₄×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₃

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

146

—

C₄×Q₈

—

146

—

#117

—

#117

C₂×D₇₃

yes

—

C₂×C₄²

—

#107

—

#107

—

100

#107

C₂×D₇₃

yes

—

101

#106

—

102

#106

—

103

#106

C₂×D₇₃

yes

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₂×Q₈

—

146

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²×D₇₃

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

Dic₁₄₆

yes

complemented

115

C₂×C₈

C₂²×D₇₃

yes

characteristic

116

C₂×C₈

C₂²×D₇₃

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

146

—

119

C₄⋊C₄

—

146

—

120

C₄⋊C₄

—

146

—

121

C₄⋊C₄

C₂²×D₇₃

yes

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

146

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

146

—

129

C₂³

D₄×D₇₃

yes

characteristic

130

Q₈

—

292

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

146

—

135

C₂×C₄

—

146

—

136

C₂×C₄

D₄×D₇₃

yes

characteristic

137

C₂×C₄

Q₈×D₇₃

yes

—

138

C₂×C₄

C₂×Dic₁₄₆

yes

—

139

C₂×C₄

C₂³×D₇₃

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

C₂×Dic₁₄₆

yes

—

143

C₈

C₂×Dic₁₄₆

yes

—

144

C₂²

—

145

C₂²

#14556590

yes

—

146

C₂²

C₂×D₄×D₇₃

yes

characteristic

147

C₄

—

148

C₄

—

292

—

149

C₄

—

150

C₄

—

146

—

151

C₄

#14556590

yes

—

152

C₄

C₂×Q₈×D₇₃

yes

characteristic

153

C₄

C₂²×Dic₁₄₆

yes

characteristic

154

C₂

—

155

C₂

C₂×#14556590

yes

characteristic

156

C₂

#135×D₇₃

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#427×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#427	D₇₃
hypocenter	C₇₃	#3160
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#427×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #17988991	C₂	semidirect
3. #17988988	C₂	semidirect
4. #17988987	C₂	semidirect
5. #17988984	C₂	semidirect
31. C₂×#10596831	Q₈	semidirect
44. C₂×C₈×C₇₃	Q₈	semidirect
79. C₇₃	#3160	semidirect
114. C₂×C₈	Dic₁₄₆	semidirect

H ≅ G/N

type

2. #17988991

C₂

semidirect

3. #17988988

C₂

semidirect

4. #17988987

C₂

semidirect

5. #17988984

C₂

semidirect

31. C₂×#10596831

Q₈

semidirect

44. C₂×C₈×C₇₃

Q₈

semidirect

79. C₇₃

#3160

semidirect

114. C₂×C₈

Dic₁₄₆

semidirect

Subgroups — #22620443

Conjugacy classes of subgroups

Complements