Group #22612016 — subgroups

Subgroups — #22612016

Order 9344.

174 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 49 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	C₂×#14556532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988824	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988820	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988412	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988389	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₈₄	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	#14556566	—	2	1	no	—
11	#14556562	—	2	1	no	—
12	#14556532	—	2	2	no	—
13	C₂×#11924003	C₂²	1	1	yes	characteristic
14	C₁₆×D₇₃	—	2	2	no	—
15	#14556525	C₂²	1	1	yes	characteristic
16	#14556512	—	2	1	no	—
17	#14556511	—	2	1	no	—
18	C₂×D₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
20	#107×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×D₂₉₂	—	2	1	no	—
22	C₂×D₂₉₂	—	4	1	no	—
23	C₂×C₂²⋊D₇₃	—	4	1	no	—
24	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
25	D₅₈₄	—	2	1	no	—
26	D₅₈₄	—	2	1	no	—
27	C₂×#10596831	D₄	1	1	yes	characteristic
28	C₈×D₇₃	—	2	1	no	—
29	C₈×D₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₂²⋊Dic₇₃	—	4	1	no	—
32	#11924005	—	4	1	no	—
33	#11924003	D₄	1	2	yes	complemented
34	C₂×D₄×C₇₃	—	2	1	no	—
35	D₈×C₇₃	—	2	1	no	—
36	D₈×C₇₃	—	2	1	no	—
37	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₁₆×C₇₃	—	2	1	no	—
40	C₂³×C₇₃	—	4	1	no	—
41	C₂²×D₇₃	—	4	1	no	—
42	C₂²×D₇₃	D₈	1	1	yes	characteristic
43	D₄×C₇₃	—	4	1	no	—
44	D₄×C₇₃	—	4	1	no	—
45	C₂×C₄×C₇₃	—	4	1	no	—
46	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂²⋊D₇₃	—	8	2	no	—
48	C₂×Dic₇₃	—	4	1	no	—
49	C₂×Dic₇₃	D₈	1	1	yes	characteristic
50	D₂₉₂	—	4	1	no	—
51	D₂₉₂	—	4	1	no	—
52	D₂₉₂	—	8	2	no	—
53	C₄×D₇₃	—	2	1	no	—
54	C₄×D₇₃	—	2	1	no	—
55	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₈×C₇₃	#41	1	1	yes	characteristic
57	#10596831	—	2	1	no	—
58	C₂²×C₇₃	—	8	1	no	—
59	C₂²×C₇₃	—	8	1	no	—
60	C₂²×C₇₃	—	8	1	no	—
61	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
62	C₂×D₇₃	—	2	1	no	—
63	C₂×D₇₃	—	2	1	no	—
64	C₂×D₇₃	—	8	1	no	—
65	C₂×D₇₃	—	8	1	no	—
66	C₂×D₇₃	—	8	1	no	—
67	C₂×D₇₃	D₁₆	1	2	yes	complemented
68	C₄×C₇₃	—	8	1	no	—
69	C₄×C₇₃	#121	1	1	yes	characteristic
70	C₄×C₇₃	#135	1	1	yes	characteristic
71	Dic₇₃	—	2	1	no	—
72	Dic₇₃	—	8	1	no	—
73	C₂×C₇₃	—	16	1	no	—
74	C₂×C₇₃	#466	1	1	yes	characteristic
75	C₂×C₇₃	C₂×D₁₆	1	1	yes	characteristic
76	C₂×C₇₃	#508	1	1	yes	characteristic
77	D₇₃	—	2	2	no	—
78	D₇₃	—	16	1	no	—
79	#2215	—	73	1	no	maximal, Sylow
80	C₇₃	#2215	1	1	yes	characteristic, complemented, Sylow
81	C₂×D₁₆	—	73	1	no	—
82	C₂²×C₁₆	—	73	1	no	—
83	#466	—	73	1	no	—
84	#466	—	73	1	no	—
85	#366	—	73	1	no	—
86	#357	—	73	1	no	—
87	#357	D₇₃	1	1	yes	characteristic, complemented
88	C₂×D₈	—	73	1	no	—
89	C₂×D₈	C₂×D₇₃	1	1	yes	characteristic
90	C₂²×C₈	—	73	1	no	—
91	#121	—	146	1	no	—
92	#121	—	146	1	no	—
93	D₁₆	—	146	2	no	—
94	C₂×C₁₆	—	73	1	no	—
95	C₂×C₁₆	—	146	2	no	—
96	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
97	#107	—	73	1	no	—
98	#107	C₂×D₇₃	1	1	yes	characteristic
99	#102	—	146	1	no	—
100	#102	—	146	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	292	1	no	—
103	C₂×D₄	—	292	1	no	—
104	C₂×D₄	—	146	1	no	—
105	C₂²×C₄	—	73	1	no	—
106	D₈	—	2	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	146	1	no	—
109	D₈	—	146	1	no	—
110	C₂×C₈	—	73	1	no	—
111	C₂×C₈	—	146	1	no	—
112	C₂×C₈	—	146	1	no	—
113	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	146	1	no	—
116	C₂²⋊C₄	—	292	1	no	—
117	C₂²⋊C₄	—	292	1	no	—
118	C₁₆	—	2	1	no	—
119	C₁₆	—	73	2	no	—
120	C₂³	—	4	1	no	—
121	C₂³	—	292	1	no	—
122	C₂³	—	73	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	1	no	—
125	D₄	—	292	1	no	—
126	D₄	—	292	1	no	—
127	D₄	—	584	2	no	—
128	D₄	—	584	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	292	1	no	—
131	C₂×C₄	—	73	1	no	—
132	C₂×C₄	—	146	1	no	—
133	C₂×C₄	—	146	1	no	—
134	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
135	C₈	—	146	1	no	—
136	C₈	D₄×D₇₃	1	1	yes	characteristic
137	C₈	#11924028	1	1	yes	characteristic
138	C₂²	—	8	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	—	8	1	no	—
141	C₂²	—	584	1	no	—
142	C₂²	—	584	1	no	—
143	C₂²	—	584	1	no	—
144	C₂²	—	73	2	no	—
145	C₂²	—	146	1	no	—
146	C₂²	—	146	1	no	—
147	C₂²	D₈×D₇₃	1	1	yes	characteristic
148	C₄	—	8	1	no	—
149	C₄	—	584	1	no	—
150	C₄	—	146	1	no	—
151	C₄	#14556564	1	1	yes	characteristic
152	C₄	#14556612	1	1	yes	characteristic
153	C₂	—	16	1	no	—
154	C₂	—	1168	1	no	—
155	C₂	—	146	2	no	—
156	C₂	#17988822	1	1	yes	characteristic
157	C₂	D₁₆×D₇₃	1	1	yes	characteristic
158	C₂	#17989018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556532

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₇₃

C₂

yes

maximal, characteristic, complemented

#17988824

C₂

yes

maximal, characteristic, complemented

#17988820

C₂

yes

maximal, characteristic, complemented

#17988412

C₂

yes

maximal, characteristic, complemented

#17988389

C₂

yes

maximal, characteristic, complemented

#357×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×D₅₈₄

C₂²

yes

characteristic

C₂×C₈×D₇₃

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₁₆×D₇₃

—

#14556525

C₂²

yes

characteristic

#14556512

—

#14556511

—

C₂×D₈×C₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

#107×C₇₃

C₂²

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×C₂²⋊D₇₃

—

C₂×C₄×D₇₃

D₄

yes

characteristic

D₅₈₄

—

D₅₈₄

—

C₂×#10596831

D₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

#11924005

—

#11924003

D₄

yes

complemented

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₁₆×C₇₃

—

C₂³×C₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

D₈

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

D₈

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

C₂×D₄

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

D₁₆

yes

complemented

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#466

yes

characteristic

C₂×C₇₃

C₂×D₁₆

yes

characteristic

C₂×C₇₃

#508

yes

characteristic

D₇₃

—

D₇₃

—

#2215

—

maximal, Sylow

C₇₃

#2215

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₂²×C₁₆

—

—

—

—

—

D₇₃

yes

characteristic, complemented

C₂×D₈

—

C₂×D₈

C₂×D₇₃

yes

characteristic

C₂²×C₈

—

#121

—

146

—

#121

—

146

—

D₁₆

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×D₇₃

yes

characteristic

#107

—

#107

C₂×D₇₃

yes

characteristic

#102

—

146

—

100

#102

—

146

—

101

C₂×D₄

—

102

C₂×D₄

—

292

—

103

C₂×D₄

—

292

—

104

C₂×D₄

—

146

—

105

C₂²×C₄

—

106

D₈

—

107

D₈

—

108

D₈

—

146

—

109

D₈

—

146

—

110

C₂×C₈

—

111

C₂×C₈

—

146

—

112

C₂×C₈

—

146

—

113

C₂×C₈

C₂²×D₇₃

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

146

—

116

C₂²⋊C₄

—

292

—

117

C₂²⋊C₄

—

292

—

118

C₁₆

—

119

C₁₆

—

120

C₂³

—

121

C₂³

—

292

—

122

C₂³

—

123

D₄

—

124

D₄

—

125

D₄

—

292

—

126

D₄

—

292

—

127

D₄

—

584

—

128

D₄

—

584

—

129

C₂×C₄

—

130

C₂×C₄

—

292

—

131

C₂×C₄

—

132

C₂×C₄

—

146

—

133

C₂×C₄

—

146

—

134

C₂×C₄

D₄×D₇₃

yes

characteristic

135

C₈

—

146

—

136

C₈

D₄×D₇₃

yes

characteristic

137

C₈

#11924028

yes

characteristic

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

584

—

142

C₂²

—

584

—

143

C₂²

—

584

—

144

C₂²

—

145

C₂²

—

146

—

146

C₂²

—

146

—

147

C₂²

D₈×D₇₃

yes

characteristic

148

C₄

—

149

C₄

—

584

—

150

C₄

—

146

—

151

C₄

#14556564

yes

characteristic

152

C₄

#14556612

yes

characteristic

153

C₂

—

154

C₂

—

1168

—

155

C₂

—

146

—

156

C₂

#17988822

yes

characteristic

157

C₂

D₁₆×D₇₃

yes

characteristic

158

C₂

#17989018

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₃
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#357×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	#357	D₇₃
hypocenter	C₇₃	#2215
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₇₃

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#357×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. C₂×#14556532	C₂	semidirect
2. C₂×C₁₆×D₇₃	C₂	semidirect
3. #17988824	C₂	semidirect
4. #17988820	C₂	semidirect
5. #17988412	C₂	semidirect
6. #17988389	C₂	semidirect
7. #357×C₇₃	C₂	semidirect
33. #11924003	D₄	semidirect
67. C₂×D₇₃	D₁₆	semidirect
80. C₇₃	#2215	semidirect
87. #357	D₇₃	semidirect

H ≅ G/N

type

1. C₂×#14556532

C₂

semidirect

2. C₂×C₁₆×D₇₃

C₂

semidirect

3. #17988824

C₂

semidirect

4. #17988820

C₂

semidirect

5. #17988412

C₂

semidirect

6. #17988389

C₂

semidirect

7. #357×C₇₃

C₂

semidirect

33. #11924003

D₄

semidirect

67. C₂×D₇₃

D₁₆

semidirect

80. C₇₃

#2215

semidirect

87. #357

D₇₃

semidirect

Subgroups — #22612016

Conjugacy classes of subgroups

Complements