Group #22610013 — subgroups

Subgroups — #22610013

Order 9344.

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

normal subgroups only

This group has 272 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	C₂×#14556549	C₂	1	1	yes	maximal, characteristic
2	#17988550	C₂	1	2	yes	maximal
3	#17988356	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988355	C₂	1	2	yes	maximal, complemented
5	#343×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₇₃	C₂²	1	1	yes	characteristic
7	C₂×#11924012	C₂²	1	2	yes	—
8	#14556549	—	2	4	no	—
9	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
10	C₂×#11924005	C₄	1	2	yes	—
11	C₂×C₄×Dic₇₃	C₄	1	2	yes	—
12	#14556503	—	2	4	no	—
13	C₂×M₁₆×C₇₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
15	C₂³×D₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
18	C₂×C₄×D₇₃	—	2	1	no	—
19	C₂×C₄×D₇₃	—	2	2	no	—
20	C₂×C₄×D₇₃	—	2	1	no	—
21	C₂×C₄×D₇₃	D₄	1	4	yes	—
22	#11924012	—	2	2	no	—
23	#11924012	—	2	2	no	—
24	C₂×#10596831	—	2	2	no	—
25	C₄⋊Dic₇₃	—	2	2	no	—
26	#11924005	—	2	4	no	—
27	C₄×Dic₇₃	—	2	4	no	—
28	C₂²×C₄×C₇₃	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
30	M₁₆×C₇₃	—	2	2	no	—
31	M₁₆×C₇₃	—	2	2	no	—
32	C₂×C₈×C₇₃	—	2	2	no	—
33	C₄⋊C₄×C₇₃	—	2	2	no	—
34	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×D₇₃	—	2	1	no	—
36	C₂²×D₇₃	—	2	1	no	—
37	C₂²×D₇₃	—	4	1	no	—
38	C₂²×D₇₃	—	4	1	no	—
39	C₂²×D₇₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₇₃	—	2	2	no	—
41	C₂×C₄×C₇₃	—	4	2	no	—
42	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
44	C₂×C₄×C₇₃	#41	1	2	yes	—
45	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
46	C₂×Dic₇₃	—	2	1	no	—
47	C₂×Dic₇₃	—	2	1	no	—
48	C₂×Dic₇₃	—	2	2	no	—
49	C₂×Dic₇₃	—	4	2	no	—
50	C₂×Dic₇₃	C₂²⋊C₄	1	2	yes	—
51	C₄×D₇₃	—	2	4	no	—
52	C₄×D₇₃	—	2	4	no	—
53	C₈×C₇₃	—	4	2	no	—
54	#10596831	—	4	2	no	—
55	C₂²×C₇₃	—	2	1	no	—
56	C₂²×C₇₃	—	2	1	no	—
57	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₇₃	#117	1	1	yes	characteristic
59	C₂²×C₇₃	#117	1	1	yes	characteristic
60	C₂×D₇₃	—	2	2	no	—
61	C₂×D₇₃	—	2	2	no	—
62	C₂×D₇₃	—	2	2	no	—
63	C₂×D₇₃	—	4	2	no	—
64	C₂×D₇₃	—	4	2	no	—
65	C₄×C₇₃	—	4	2	no	—
66	C₄×C₇₃	#123	1	2	yes	—
67	C₄×C₇₃	#123	1	2	yes	—
68	Dic₇₃	—	2	2	no	—
69	Dic₇₃	—	4	2	no	—
70	C₂×C₇₃	—	2	1	no	—
71	C₂×C₇₃	—	2	1	no	—
72	C₂×C₇₃	#381	1	1	yes	characteristic
73	C₂×C₇₃	#413	1	1	yes	characteristic
74	C₂×C₇₃	#413	1	1	yes	characteristic
75	D₇₃	—	4	2	no	—
76	#1822	—	73	1	no	maximal, Sylow
77	C₇₃	#1822	1	1	yes	characteristic, complemented, Sylow
78	C₂×#117	—	73	1	no	—
79	#407	—	73	2	no	—
80	#343	—	73	2	no	—
81	#343	—	73	1	no	—
82	#343	D₇₃	1	1	yes	characteristic, complemented
83	C₂³×C₄	—	73	1	no	—
84	C₂×M₁₆	—	73	2	no	—
85	C₂×M₁₆	C₂×D₇₃	1	2	yes	—
86	#117	—	146	4	no	—
87	C₂×C₄⋊C₄	—	73	1	no	—
88	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
89	C₂×C₂²⋊C₄	—	73	2	no	—
90	C₂×C₄²	—	73	2	no	—
91	C₂²⋊C₈	—	146	4	no	—
92	C₂⁴	—	73	1	no	—
93	C₂²×C₄	—	73	2	no	—
94	C₂²×C₄	—	73	1	no	—
95	C₂²×C₄	—	73	4	no	—
96	C₂²×C₄	—	146	1	no	—
97	C₂²×C₄	—	146	2	no	—
98	C₂²×C₄	—	146	1	no	—
99	C₂²×C₄	C₄×D₇₃	1	2	yes	—
100	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
101	M₁₆	—	2	2	no	—
102	M₁₆	—	2	2	no	—
103	M₁₆	—	146	2	no	—
104	M₁₆	—	146	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	146	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	146	2	no	—
109	C₂²⋊C₄	—	146	4	no	—
110	C₄²	—	146	4	no	—
111	C₂³	—	292	1	no	—
112	C₂³	—	292	1	no	—
113	C₂³	—	73	2	no	—
114	C₂³	—	146	1	no	—
115	C₂³	—	146	1	no	—
116	C₂³	C₂×C₄×D₇₃	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	292	2	no	—
120	C₂×C₄	—	73	2	no	—
121	C₂×C₄	—	146	2	no	—
122	C₂×C₄	—	146	1	no	—
123	C₂×C₄	—	146	1	no	—
124	C₂×C₄	—	146	4	no	—
125	C₂×C₄	—	146	4	no	—
126	C₂×C₄	D₄×D₇₃	1	2	yes	—
127	C₂×C₄	#11924027	1	1	yes	characteristic
128	C₂×C₄	#11924028	1	2	yes	—
129	C₂×C₄	#11924029	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₈	—	292	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	292	2	no	—
135	C₂²	—	292	2	no	—
136	C₂²	—	146	2	no	—
137	C₂²	—	146	2	no	—
138	C₂²	—	146	2	no	—
139	C₂²	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
140	C₂²	#14556548	1	1	yes	characteristic
141	C₂²	#14556549	1	1	yes	characteristic
142	C₄	—	4	2	no	—
143	C₄	—	292	2	no	—
144	C₄	—	146	2	no	—
145	C₄	#14556573	1	2	yes	—
146	C₄	#14556576	1	2	yes	—
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	—	292	2	no	—
150	C₂	#17988449	1	1	yes	characteristic
151	C₂	#17988568	1	1	yes	characteristic
152	C₂	#17988570	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556549

C₂

yes

maximal, characteristic

#17988550

C₂

yes

maximal

#17988356

C₂

yes

maximal, characteristic, complemented

#17988355

C₂

yes

maximal, complemented

#343×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₇₃

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

—

#14556549

—

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×#11924005

C₄

yes

—

C₂×C₄×Dic₇₃

C₄

yes

—

#14556503

—

C₂×M₁₆×C₇₃

C₂²

yes

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂³×D₇₃

C₂×C₄

yes

characteristic

C₂²×Dic₇₃

C₂×C₄

yes

characteristic

C₂²×Dic₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

D₄

yes

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

#11924005

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂³×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#123

yes

—

C₄×C₇₃

#123

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1822

—

maximal, Sylow

C₇₃

#1822

yes

characteristic, complemented, Sylow

—

—

—

—

D₇₃

yes

characteristic, complemented

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₃

yes

—

#117

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

146

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂²×C₄

—

146

—

C₂²×C₄

—

146

—

C₂²×C₄

C₄×D₇₃

yes

—

100

C₂²×C₄

C₂²×D₇₃

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

146

—

104

M₁₆

—

146

—

105

C₂×C₈

—

106

C₂×C₈

—

146

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

146

—

109

C₂²⋊C₄

—

146

—

110

C₄²

—

146

—

111

C₂³

—

292

—

112

C₂³

—

292

—

113

C₂³

—

114

C₂³

—

146

—

115

C₂³

—

146

—

116

C₂³

C₂×C₄×D₇₃

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

292

—

120

C₂×C₄

—

121

C₂×C₄

—

146

—

122

C₂×C₄

—

146

—

123

C₂×C₄

—

146

—

124

C₂×C₄

—

146

—

125

C₂×C₄

—

146

—

126

C₂×C₄

D₄×D₇₃

yes

—

127

C₂×C₄

#11924027

yes

characteristic

128

C₂×C₄

#11924028

yes

—

129

C₂×C₄

#11924029

yes

characteristic

130

C₈

—

131

C₈

—

292

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

292

—

135

C₂²

—

292

—

136

C₂²

—

146

—

137

C₂²

—

146

—

138

C₂²

—

146

—

139

C₂²

C₂²⋊C₄×D₇₃

yes

characteristic

140

C₂²

#14556548

yes

characteristic

141

C₂²

#14556549

yes

characteristic

142

C₄

—

143

C₄

—

292

—

144

C₄

—

146

—

145

C₄

#14556573

yes

—

146

C₄

#14556576

yes

—

147

C₂

—

148

C₂

—

149

C₂

—

292

—

150

C₂

#17988449

yes

characteristic

151

C₂

#17988568

yes

characteristic

152

C₂

#17988570

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₇₃
commutator	C₂³×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#343×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	#343	D₇₃
hypocenter	C₇₃	#1822
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₇₃

commutator

C₂³×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#343×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
3. #17988356	C₂	semidirect
4. #17988355	C₂	semidirect
5. #343×C₇₃	C₂	semidirect
77. C₇₃	#1822	semidirect
82. #343	D₇₃	semidirect

H ≅ G/N

type

3. #17988356

C₂

semidirect

4. #17988355

C₂

semidirect

5. #343×C₇₃

C₂

semidirect

77. C₇₃

#1822

semidirect

82. #343

D₇₃

semidirect

Subgroups — #22610013

Conjugacy classes of subgroups

Complements