Group #22610743 — subgroups

Subgroups — #22610743

Order 9344.

336 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 113 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×#14556515	C₂	1	2	yes	maximal, complemented
2	C₂×#14556512	C₂	1	2	yes	maximal
3	C₂×C₄×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988453	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#124×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10596831	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
8	#14556515	—	2	4	no	—
9	#14556512	—	2	4	no	—
10	C₄×#10596831	—	2	2	no	—
11	#14556498	—	2	2	no	—
12	C₂²×Q₈×C₇₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₇₃	C₂²	1	1	yes	characteristic
14	#124×C₇₃	—	2	2	no	—
15	#124×C₇₃	C₄	1	4	yes	complemented
16	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
17	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²×Dic₇₃	—	2	2	no	—
19	C₂×#10596831	—	2	2	no	—
20	C₂×#10596831	—	2	2	no	—
21	C₂×#10596831	D₄	1	4	yes	—
22	C₄⋊Dic₇₃	—	2	2	no	—
23	C₄⋊Dic₇₃	—	2	2	no	—
24	C₂⁴×C₇₃	—	2	1	no	—
25	C₂×Q₈×C₇₃	—	2	1	no	—
26	C₂×Q₈×C₇₃	—	2	2	no	—
27	C₂×Q₈×C₇₃	—	2	2	no	—
28	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₇₃	—	2	2	no	—
30	C₂×D₄×C₇₃	—	2	1	no	—
31	C₂×D₄×C₇₃	—	2	2	no	—
32	C₂×D₄×C₇₃	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₇₃	—	2	1	no	—
34	C₂²×C₄×C₇₃	D₄	1	2	yes	—
35	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
36	C₂²⋊C₄×C₇₃	—	2	8	no	—
37	C₄²×C₇₃	C₂×C₄	1	2	yes	—
38	C₄²×C₇₃	D₄	1	2	yes	—
39	C₂³×C₇₃	—	2	2	no	—
40	C₂³×C₇₃	—	4	1	no	—
41	C₂³×C₇₃	—	4	1	no	—
42	C₂³×C₇₃	—	4	4	no	—
43	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₇₃	—	2	4	no	—
45	Q₈×C₇₃	—	4	2	no	—
46	D₄×C₇₃	—	2	4	no	—
47	D₄×C₇₃	—	4	2	no	—
48	C₂×C₄×C₇₃	—	2	4	no	—
49	C₂×C₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	—	4	1	no	—
51	C₂×C₄×C₇₃	—	4	1	no	—
52	C₂×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#41	1	2	yes	—
57	C₂×Dic₇₃	—	4	2	no	—
58	C₂×Dic₇₃	—	4	4	no	—
59	#10596831	—	2	4	no	—
60	C₂²×C₇₃	—	4	2	no	—
61	C₂²×C₇₃	—	4	4	no	—
62	C₂²×C₇₃	—	8	2	no	—
63	C₂²×C₇₃	—	8	2	no	—
64	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₇₃	#124	1	1	yes	characteristic
66	C₂²×C₇₃	#127	1	1	yes	characteristic
67	C₂²×C₇₃	#135	1	4	yes	—
68	C₄×C₇₃	—	2	4	no	—
69	C₄×C₇₃	—	4	2	no	—
70	C₄×C₇₃	C₄×D₄	1	2	yes	—
71	C₄×C₇₃	#121	1	2	yes	—
72	Dic₇₃	—	8	2	no	—
73	C₂×C₇₃	—	8	2	no	—
74	C₂×C₇₃	#390	1	1	yes	characteristic
75	C₂×C₇₃	#416	1	2	yes	—
76	C₂×C₇₃	#488	1	2	yes	—
77	C₂×C₇₃	#495	1	2	yes	—
78	#1958	—	73	1	no	maximal, Sylow
79	C₇₃	#1958	1	1	yes	characteristic, complemented, Sylow
80	C₂×#124	D₇₃	1	1	yes	characteristic
81	C₂×#103	—	73	2	no	—
82	C₂×#102	—	73	2	no	—
83	C₂×C₄×C₈	—	73	1	no	—
84	#382	—	73	1	no	—
85	C₂²×Q₈	C₂×D₇₃	1	1	yes	characteristic
86	C₂²×D₄	C₂×D₇₃	1	1	yes	characteristic
87	C₂²×C₈	—	73	2	no	—
88	#124	—	2	2	no	—
89	#124	Dic₇₃	1	4	yes	complemented
90	C₂×C₄⋊C₄	—	73	2	no	—
91	C₂×C₂²⋊C₄	—	2	2	no	—
92	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
93	#103	—	146	4	no	—
94	#102	—	146	4	no	—
95	C₄×C₈	—	146	2	no	—
96	#95	—	146	2	no	—
97	C₂⁴	—	2	1	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×Dic₇₃	1	2	yes	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	C₂×Dic₇₃	1	2	yes	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	146	2	no	—
108	C₂²×C₄	C₂²⋊D₇₃	1	2	yes	—
109	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
110	C₂×C₈	—	73	4	no	—
111	C₂×C₈	—	146	2	no	—
112	C₂×C₈	—	146	2	no	—
113	C₄⋊C₄	—	146	2	no	—
114	C₄⋊C₄	—	146	2	no	—
115	C₂²⋊C₄	—	2	8	no	—
116	C₄²	C₂×Dic₇₃	1	2	yes	—
117	C₄²	C₂²⋊D₇₃	1	2	yes	—
118	C₂³	—	2	2	no	—
119	C₂³	—	4	1	no	—
120	C₂³	—	4	1	no	—
121	C₂³	—	4	4	no	—
122	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
123	Q₈	—	2	4	no	—
124	Q₈	—	4	2	no	—
125	D₄	—	2	4	no	—
126	D₄	—	4	2	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	292	2	no	—
132	C₂×C₄	—	292	4	no	—
133	C₂×C₄	C₂²⋊Dic₇₃	1	4	yes	—
134	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
135	C₂×C₄	D₄×D₇₃	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
137	C₂×C₄	#11924027	1	2	yes	—
138	C₈	—	146	4	no	—
139	C₂²	—	4	2	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	8	2	no	—
142	C₂²	—	8	2	no	—
143	C₂²	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
144	C₂²	#14556578	1	1	yes	characteristic
145	C₂²	#14556587	1	1	yes	characteristic
146	C₂²	#14556611	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	4	2	no	—
149	C₄	—	584	2	no	—
150	C₄	D₄×Dic₇₃	1	2	yes	—
151	C₄	#14556562	1	2	yes	—
152	C₂	—	8	2	no	—
153	C₂	#17988490	1	1	yes	characteristic
154	C₂	#17988583	1	2	yes	—
155	C₂	#17988941	1	2	yes	—
156	C₂	#17988969	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556515

C₂

yes

maximal, complemented

C₂×#14556512

C₂

yes

maximal

C₂×C₄×#10596831

C₂

yes

maximal, characteristic, complemented

#17988453

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₇₃

C₂

yes

maximal, characteristic

C₂²×#10596831

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

—

—

—

—

C₂²×Q₈×C₇₃

C₂²

yes

characteristic

C₂²×D₄×C₇₃

C₂²

yes

characteristic

#124×C₇₃

—

#124×C₇₃

C₄

yes

complemented

C₂×C₂²⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

D₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂⁴×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

D₄

yes

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×D₄

yes

—

C₄×C₇₃

#121

yes

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#390

yes

characteristic

C₂×C₇₃

#416

yes

—

C₂×C₇₃

#488

yes

—

C₂×C₇₃

#495

yes

—

#1958

—

maximal, Sylow

C₇₃

#1958

yes

characteristic, complemented, Sylow

C₂×#124

D₇₃

yes

characteristic

C₂×#103

—

C₂×#102

—

C₂×C₄×C₈

—

#382

—

C₂²×Q₈

C₂×D₇₃

yes

characteristic

C₂²×D₄

C₂×D₇₃

yes

characteristic

C₂²×C₈

—

#124

—

#124

Dic₇₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

#103

—

146

—

#102

—

146

—

C₄×C₈

—

146

—

#95

—

146

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×Dic₇₃

yes

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

C₂×Dic₇₃

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

146

—

108

C₂²×C₄

C₂²⋊D₇₃

yes

—

109

C₂²×C₄

C₂²×D₇₃

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

146

—

112

C₂×C₈

—

146

—

113

C₄⋊C₄

—

146

—

114

C₄⋊C₄

—

146

—

115

C₂²⋊C₄

—

116

C₄²

C₂×Dic₇₃

yes

—

117

C₄²

C₂²⋊D₇₃

yes

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

123

Q₈

—

124

Q₈

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

292

—

132

C₂×C₄

—

292

—

133

C₂×C₄

C₂²⋊Dic₇₃

yes

—

134

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

135

C₂×C₄

D₄×D₇₃

yes

—

136

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

137

C₂×C₄

#11924027

yes

—

138

C₈

—

146

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×C₂²⋊Dic₇₃

yes

characteristic

144

C₂²

#14556578

yes

characteristic

145

C₂²

#14556587

yes

characteristic

146

C₂²

#14556611

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

584

—

150

C₄

D₄×Dic₇₃

yes

—

151

C₄

#14556562

yes

—

152

C₂

—

153

C₂

#17988490

yes

characteristic

154

C₂

#17988583

yes

—

155

C₂

#17988941

yes

—

156

C₂

#17988969

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×#124×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×#124	D₇₃
hypocenter	C₇₃	#1958
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×#124×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×#14556515	C₂	semidirect
3. C₂×C₄×#10596831	C₂	semidirect
4. #17988453	C₂	semidirect
15. #124×C₇₃	C₄	semidirect
79. C₇₃	#1958	semidirect
89. #124	Dic₇₃	semidirect

H ≅ G/N

type

1. C₂×#14556515

C₂

semidirect

3. C₂×C₄×#10596831

C₂

semidirect

4. #17988453

C₂

semidirect

15. #124×C₇₃

C₄

semidirect

79. C₇₃

#1958

semidirect

89. #124

Dic₇₃

semidirect

Subgroups — #22610743

Conjugacy classes of subgroups

Complements