Group #22622361 — subgroups

Subgroups — #22622361

Order 9344.

340 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 117 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	#17989311	C₂	1	2	yes	maximal
2	#17988930	C₂	1	4	yes	maximal, complemented
3	#17988869	C₂	1	2	yes	maximal, complemented
4	#17988829	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×D₂₉₂	C₂	1	1	yes	maximal, characteristic
7	#17988624	C₂	1	2	yes	maximal, complemented
8	#500×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₉₂	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₇₃	C₂²	1	2	yes	—
11	#14556586	—	2	4	no	—
12	#14556580	—	2	2	no	—
13	#14556571	C₂²	1	4	yes	—
14	Q₈×Dic₇₃	—	2	2	no	—
15	C₄×Dic₁₄₆	C₂²	1	2	yes	—
16	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
17	#14556549	C₂²	1	4	yes	—
18	#14556525	C₂²	1	1	yes	characteristic
19	#14556522	C₂²	1	4	yes	—
20	C₄⋊#10596831	C₂²	1	1	yes	characteristic
21	#14556516	C₂²	1	2	yes	—
22	#14556514	C₂²	1	4	yes	—
23	#14556503	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
25	#107×C₇₃	C₂²	1	2	yes	—
26	#107×C₇₃	C₂²	1	2	yes	—
27	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₄₆	C₂³	1	2	yes	—
29	C₂×D₂₉₂	D₄	1	1	yes	characteristic
30	C₂×C₄×D₇₃	C₂³	1	2	yes	—
31	Dic₂₉₂	—	2	2	no	—
32	C₂×#10596831	C₂³	1	2	yes	—
33	C₈×D₇₃	—	2	4	no	—
34	C₄⋊Dic₇₃	—	2	4	no	—
35	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
36	C₄⋊Dic₇₃	C₂³	1	2	yes	—
37	#11924007	—	2	2	no	—
38	#11924005	—	2	4	no	—
39	#11924005	—	2	4	no	—
40	#11924005	D₄	1	2	yes	—
41	C₄×Dic₇₃	—	2	2	no	—
42	C₄×Dic₇₃	—	2	4	no	—
43	C₂×Q₈×C₇₃	—	2	2	no	—
44	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₇₃	—	2	2	no	—
47	C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
48	C₄²×C₇₃	C₂³	1	1	yes	characteristic
49	C₂²×D₇₃	C₂×D₄	1	2	yes	—
50	Q₈×C₇₃	—	4	4	no	—
51	C₂×C₄×C₇₃	—	2	4	no	—
52	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₇₃	—	2	2	no	—
56	C₂×Dic₇₃	—	2	2	no	—
57	C₂×Dic₇₃	C₂×D₄	1	2	yes	—
58	D₂₉₂	—	2	2	no	—
59	C₄×D₇₃	—	2	2	no	—
60	C₄×D₇₃	—	2	2	no	—
61	Dic₁₄₆	—	2	2	no	—
62	Dic₁₄₆	—	2	2	no	—
63	C₈×C₇₃	—	2	1	no	—
64	C₈×C₇₃	#41	1	2	yes	—
65	#10596831	—	2	2	no	—
66	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₇₃	—	2	2	no	—
68	C₂×D₇₃	—	2	2	no	—
69	C₂×D₇₃	—	2	2	no	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	—	4	4	no	—
72	C₄×C₇₃	#135	1	2	yes	—
73	C₄×C₇₃	C₂×#41	1	1	yes	characteristic
74	C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
75	Dic₇₃	—	2	2	no	—
76	Dic₇₃	—	4	2	no	—
77	Dic₇₃	—	4	2	no	—
78	C₂×C₇₃	#547	1	1	yes	characteristic
79	C₂×C₇₃	C₂×#135	1	1	yes	characteristic
80	C₂×C₇₃	#578	1	1	yes	characteristic
81	D₇₃	—	4	2	no	—
82	#3342	—	73	1	no	maximal, Sylow
83	C₇₃	#3342	1	1	yes	characteristic, complemented, Sylow
84	#556	—	73	2	no	—
85	#500	D₇₃	1	1	yes	characteristic, complemented
86	#485	—	73	4	no	—
87	#475	—	73	2	no	—
88	#467	—	73	2	no	—
89	C₄×Q₁₆	—	73	1	no	—
90	C₈×D₄	—	73	1	no	—
91	#427	—	73	2	no	—
92	C₂×Q₁₆	—	73	1	no	—
93	C₂²×C₈	—	73	2	no	—
94	C₄⋊Q₈	C₂×D₇₃	1	2	yes	—
95	#126	—	146	4	no	—
96	#124	—	146	2	no	—
97	C₂²⋊Q₈	—	73	4	no	—
98	C₄×Q₈	—	73	2	no	—
99	C₄×Q₈	—	146	2	no	—
100	C₄×D₄	—	73	1	no	—
101	#117	—	73	4	no	—
102	#107	—	73	1	no	—
103	#107	C₂×D₇₃	1	2	yes	—
104	#107	C₂×D₇₃	1	2	yes	—
105	#106	—	73	4	no	—
106	C₄⋊C₈	—	73	1	no	—
107	#103	—	73	2	no	—
108	#103	—	73	4	no	—
109	C₂²⋊C₈	—	73	2	no	—
110	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	—	73	2	no	—
113	C₂×D₄	—	73	1	no	—
114	C₂²×C₄	—	73	2	no	—
115	Q₁₆	—	146	2	no	—
116	C₂×C₈	—	73	2	no	—
117	C₂×C₈	—	146	4	no	—
118	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
119	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	73	1	no	—
122	C₄⋊C₄	—	73	2	no	—
123	C₄⋊C₄	—	146	4	no	—
124	C₄⋊C₄	—	146	2	no	—
125	C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
126	C₂²⋊C₄	—	73	2	no	—
127	C₂²⋊C₄	—	146	4	no	—
128	C₂²⋊C₄	—	146	4	no	—
129	C₄²	—	146	4	no	—
130	C₄²	—	146	2	no	—
131	C₄²	C₂²×D₇₃	1	1	yes	characteristic
132	C₂³	—	73	2	no	—
133	Q₈	—	4	4	no	—
134	Q₈	—	146	2	no	—
135	Q₈	—	146	2	no	—
136	D₄	—	146	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	73	2	no	—
139	C₂×C₄	—	146	2	no	—
140	C₂×C₄	—	146	2	no	—
141	C₂×C₄	—	146	2	no	—
142	C₂×C₄	—	146	2	no	—
143	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
144	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
145	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	146	2	no	—
148	C₈	#11924029	1	2	yes	—
149	C₂²	—	146	2	no	—
150	C₂²	—	146	2	no	—
151	C₂²	—	146	2	no	—
152	C₂²	C₂×D₄×D₇₃	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	4	no	—
155	C₄	—	292	2	no	—
156	C₄	—	292	2	no	—
157	C₄	—	146	2	no	—
158	C₄	#14556610	1	2	yes	—
159	C₄	C₂×#11924029	1	1	yes	characteristic
160	C₄	#14556639	1	1	yes	characteristic
161	C₂	—	292	2	no	—
162	C₂	#17989240	1	1	yes	characteristic
163	C₂	C₂×#14556610	1	1	yes	characteristic
164	C₂	#17989400	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989311

C₂

yes

maximal

#17988930

C₂

yes

maximal, complemented

#17988869

C₂

yes

maximal, complemented

#17988829

C₂

yes

maximal, complemented

C₄×Dic₂₉₂

C₂

yes

maximal, characteristic, complemented

C₈×D₂₉₂

C₂

yes

maximal, characteristic

#17988624

C₂

yes

maximal, complemented

#500×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₉₂

C₂²

yes

characteristic

C₂×C₈×D₇₃

C₂²

yes

—

#14556586

—

#14556580

—

#14556571

C₂²

yes

—

Q₈×Dic₇₃

—

C₄×Dic₁₄₆

C₂²

yes

—

C₄×D₂₉₂

C₂²

yes

characteristic

#14556549

C₂²

yes

—

#14556525

C₂²

yes

characteristic

#14556522

C₂²

yes

—

C₄⋊#10596831

C₂²

yes

characteristic

#14556516

C₂²

yes

—

#14556514

C₂²

yes

—

#14556503

C₂²

yes

—

C₄⋊Q₈×C₇₃

C₂²

yes

—

#107×C₇₃

C₂²

yes

—

#107×C₇₃

C₂²

yes

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

C₂³

yes

—

C₂×D₂₉₂

D₄

yes

characteristic

C₂×C₄×D₇₃

C₂³

yes

—

Dic₂₉₂

—

C₂×#10596831

C₂³

yes

—

C₈×D₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

C₂×D₄

yes

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₄

yes

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

#41

yes

—

#10596831

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#135

yes

—

C₄×C₇₃

C₂×#41

yes

characteristic

C₄×C₇₃

2₋¹⁺⁴

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#3342

—

maximal, Sylow

C₇₃

#3342

yes

characteristic, complemented, Sylow

#556

—

#500

D₇₃

yes

characteristic, complemented

#485

—

#475

—

#467

—

C₄×Q₁₆

—

C₈×D₄

—

#427

—

C₂×Q₁₆

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₇₃

yes

—

#126

—

146

—

#124

—

146

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

146

—

100

C₄×D₄

—

101

#117

—

102

#107

—

103

#107

C₂×D₇₃

yes

—

104

#107

C₂×D₇₃

yes

—

105

#106

—

106

C₄⋊C₈

—

107

#103

—

108

#103

—

109

C₂²⋊C₈

—

110

C₄×C₈

C₂×D₇₃

yes

characteristic

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

Q₁₆

—

146

—

116

C₂×C₈

—

117

C₂×C₈

—

146

—

118

C₂×C₈

C₂²×D₇₃

yes

characteristic

119

C₂×C₈

C₂²×D₇₃

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

146

—

124

C₄⋊C₄

—

146

—

125

C₄⋊C₄

C₂²×D₇₃

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

146

—

128

C₂²⋊C₄

—

146

—

129

C₄²

—

146

—

130

C₄²

—

146

—

131

C₄²

C₂²×D₇₃

yes

characteristic

132

C₂³

—

133

Q₈

—

134

Q₈

—

146

—

135

Q₈

—

146

—

136

D₄

—

146

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

146

—

140

C₂×C₄

—

146

—

141

C₂×C₄

—

146

—

142

C₂×C₄

—

146

—

143

C₂×C₄

D₄×D₇₃

yes

characteristic

144

C₂×C₄

D₄×D₇₃

yes

characteristic

145

C₂×C₄

C₂³×D₇₃

yes

characteristic

146

C₈

—

147

C₈

—

146

—

148

C₈

#11924029

yes

—

149

C₂²

—

146

—

150

C₂²

—

146

—

151

C₂²

—

146

—

152

C₂²

C₂×D₄×D₇₃

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

292

—

156

C₄

—

292

—

157

C₄

—

146

—

158

C₄

#14556610

yes

—

159

C₄

C₂×#11924029

yes

characteristic

160

C₄

#14556639

yes

characteristic

161

C₂

—

292

—

162

C₂

#17989240

yes

characteristic

163

C₂

C₂×#14556610

yes

characteristic

164

C₂

#17989400

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#500×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#500	D₇₃
hypocenter	C₇₃	#3342
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#500×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
2. #17988930	C₂	semidirect
3. #17988869	C₂	semidirect
4. #17988829	C₂	semidirect
5. C₄×Dic₂₉₂	C₂	semidirect
7. #17988624	C₂	semidirect
8. #500×C₇₃	C₂	semidirect
83. C₇₃	#3342	semidirect
85. #500	D₇₃	semidirect

H ≅ G/N

type

2. #17988930

C₂

semidirect

3. #17988869

C₂

semidirect

4. #17988829

C₂

semidirect

5. C₄×Dic₂₉₂

C₂

semidirect

7. #17988624

C₂

semidirect

8. #500×C₇₃

C₂

semidirect

83. C₇₃

#3342

semidirect

85. #500

D₇₃

semidirect

Subgroups — #22622361

Conjugacy classes of subgroups

Complements