Group #22610778 — subgroups

Subgroups — #22610778

Order 9344.

312 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14556579	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556509	C₂	1	2	yes	maximal, complemented
3	C₂×#14556507	C₂	1	2	yes	maximal
4	C₄×#11924012	C₂	1	1	yes	maximal, characteristic, complemented
5	#423×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
7	C₂²×D₂₉₂	C₂²	1	1	yes	characteristic
8	C₂×#11924012	C₂²	1	2	yes	—
9	#14556579	—	2	1	no	—
10	#14556579	—	2	1	no	—
11	#14556579	—	2	2	no	—
12	C₂×#11924005	C₄	1	4	yes	—
13	#14556509	—	2	4	no	—
14	#14556507	—	2	4	no	—
15	#14556502	—	2	1	no	—
16	C₄×#10596831	—	2	1	no	—
17	C₂×M₁₆×C₇₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₇₃	—	2	2	no	—
20	C₂³×D₇₃	C₂×C₄	1	2	yes	—
21	C₂×Dic₁₄₆	—	2	2	no	—
22	C₂×Dic₁₄₆	—	2	1	no	—
23	C₂×Dic₁₄₆	—	2	1	no	—
24	C₂×Dic₁₄₆	—	2	2	no	—
25	C₂×D₂₉₂	—	2	2	no	—
26	C₂×D₂₉₂	—	2	1	no	—
27	C₂×D₂₉₂	—	2	1	no	—
28	C₂×D₂₉₂	—	2	2	no	—
29	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
30	#11924012	—	2	2	no	—
31	#11924012	D₄	1	4	yes	—
32	C₂×#10596831	—	2	2	no	—
33	#11924005	—	2	4	no	—
34	#11924005	—	2	4	no	—
35	C₂²×C₄×C₇₃	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
37	M₁₆×C₇₃	—	2	2	no	—
38	M₁₆×C₇₃	—	2	2	no	—
39	C₂×C₈×C₇₃	—	2	2	no	—
40	C₄²×C₇₃	D₄	1	1	yes	characteristic
41	C₄²×C₇₃	D₄	1	1	yes	characteristic
42	C₄²×C₇₃	D₄	1	2	yes	—
43	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×D₇₃	—	2	2	no	—
45	C₂²×D₇₃	—	2	4	no	—
46	C₂²×D₇₃	—	4	4	no	—
47	C₂×C₄×C₇₃	—	4	2	no	—
48	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#41	1	2	yes	—
54	C₂×Dic₇₃	—	2	2	no	—
55	C₂×Dic₇₃	—	2	4	no	—
56	D₂₉₂	—	4	2	no	—
57	D₂₉₂	—	4	1	no	—
58	D₂₉₂	—	4	1	no	—
59	Dic₁₄₆	—	4	2	no	—
60	Dic₁₄₆	—	4	1	no	—
61	Dic₁₄₆	—	4	1	no	—
62	C₈×C₇₃	—	4	2	no	—
63	#10596831	—	2	4	no	—
64	C₂²×C₇₃	—	2	2	no	—
65	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₇₃	C₄×D₄	1	2	yes	—
67	C₂×D₇₃	—	4	2	no	—
68	C₂×D₇₃	—	4	4	no	—
69	C₂×D₇₃	—	8	4	no	—
70	C₄×C₇₃	—	2	2	no	—
71	C₄×C₇₃	—	2	2	no	—
72	C₄×C₇₃	#121	1	2	yes	—
73	C₄×C₇₃	#124	1	1	yes	characteristic
74	C₄×C₇₃	#127	1	1	yes	characteristic
75	Dic₇₃	—	4	2	no	—
76	C₂×C₇₃	—	2	2	no	—
77	C₂×C₇₃	#390	1	1	yes	characteristic
78	C₂×C₇₃	#413	1	2	yes	—
79	D₇₃	—	8	2	no	—
80	#1967	—	73	1	no	maximal, Sylow
81	C₇₃	#1967	1	1	yes	characteristic, complemented, Sylow
82	C₂×#124	—	73	1	no	—
83	#423	D₇₃	1	1	yes	characteristic, complemented
84	C₂×#101	—	73	2	no	—
85	C₂×#100	—	73	2	no	—
86	C₄×M₁₆	—	73	1	no	—
87	C₂²×Q₈	—	73	1	no	—
88	C₂²×D₄	—	73	1	no	—
89	C₂×M₁₆	—	73	2	no	—
90	C₂×M₁₆	C₂×D₇₃	1	2	yes	—
91	#124	—	146	1	no	—
92	#124	—	146	1	no	—
93	#124	—	146	2	no	—
94	C₂×C₂²⋊C₄	—	73	4	no	—
95	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
96	C₄⋊C₈	—	2	2	no	—
97	#101	—	146	4	no	—
98	#100	—	146	4	no	—
99	#97	—	146	1	no	—
100	C₄×C₈	—	146	1	no	—
101	C₂⁴	—	73	2	no	—
102	C₂×Q₈	—	146	2	no	—
103	C₂×Q₈	—	146	1	no	—
104	C₂×Q₈	—	146	1	no	—
105	C₂×Q₈	—	146	2	no	—
106	C₂×D₄	—	146	2	no	—
107	C₂×D₄	—	146	1	no	—
108	C₂×D₄	—	146	1	no	—
109	C₂×D₄	—	146	2	no	—
110	C₂²×C₄	—	73	2	no	—
111	C₂²×C₄	C₄×D₇₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	73	4	no	—
116	M₁₆	—	146	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	146	2	no	—
119	C₂²⋊C₄	—	146	4	no	—
120	C₂²⋊C₄	—	146	4	no	—
121	C₄²	D₂₉₂	1	1	yes	characteristic
122	C₄²	D₂₉₂	1	1	yes	characteristic
123	C₄²	C₂²⋊D₇₃	1	2	yes	—
124	C₂³	—	292	4	no	—
125	C₂³	—	146	2	no	—
126	C₂³	—	146	4	no	—
127	C₂³	C₂×C₄×D₇₃	1	1	yes	characteristic
128	Q₈	—	292	2	no	—
129	Q₈	—	292	1	no	—
130	Q₈	—	292	1	no	—
131	D₄	—	292	2	no	—
132	D₄	—	292	1	no	—
133	D₄	—	292	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	146	2	no	—
136	C₂×C₄	—	146	4	no	—
137	C₂×C₄	#11924005	1	2	yes	—
138	C₂×C₄	#11924005	1	2	yes	—
139	C₂×C₄	D₄×D₇₃	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
141	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
142	C₂×C₄	#11924029	1	2	yes	—
143	C₈	—	4	2	no	—
144	C₈	—	146	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	292	2	no	—
147	C₂²	—	292	4	no	—
148	C₂²	—	584	4	no	—
149	C₂²	C₂×#11924005	1	1	yes	characteristic
150	C₂²	#14556553	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	292	2	no	—
154	C₄	#14556566	1	2	yes	—
155	C₄	#14556580	1	1	yes	characteristic
156	C₄	#14556587	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	584	2	no	—
159	C₂	#17988488	1	1	yes	characteristic
160	C₂	#17988569	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556579

C₂

yes

maximal, characteristic

C₂×#14556509

C₂

yes

maximal, complemented

C₂×#14556507

C₂

yes

maximal

C₄×#11924012

C₂

yes

maximal, characteristic, complemented

#423×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₄₆

C₂²

yes

characteristic

C₂²×D₂₉₂

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

—

—

—

—

C₄

yes

—

—

—

—

—

C₂×M₁₆×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂³×D₇₃

C₂×C₄

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂²×Dic₇₃

C₂×C₄

yes

—

#11924012

—

#11924012

D₄

yes

—

C₂×#10596831

—

#11924005

—

#11924005

—

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

—

C₂³×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

C₄×D₄

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#121

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#390

yes

characteristic

C₂×C₇₃

#413

yes

—

D₇₃

—

#1967

—

maximal, Sylow

C₇₃

#1967

yes

characteristic, complemented, Sylow

C₂×#124

—

#423

D₇₃

yes

characteristic, complemented

C₂×#101

—

C₂×#100

—

C₄×M₁₆

—

C₂²×Q₈

—

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₃

yes

—

#124

—

146

—

#124

—

146

—

#124

—

146

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

#101

—

146

—

#100

—

146

—

#97

—

146

—

100

C₄×C₈

—

146

—

101

C₂⁴

—

102

C₂×Q₈

—

146

—

103

C₂×Q₈

—

146

—

104

C₂×Q₈

—

146

—

105

C₂×Q₈

—

146

—

106

C₂×D₄

—

146

—

107

C₂×D₄

—

146

—

108

C₂×D₄

—

146

—

109

C₂×D₄

—

146

—

110

C₂²×C₄

—

111

C₂²×C₄

C₄×D₇₃

yes

—

112

C₂²×C₄

C₂²×D₇₃

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

146

—

117

C₂×C₈

—

118

C₂×C₈

—

146

—

119

C₂²⋊C₄

—

146

—

120

C₂²⋊C₄

—

146

—

121

C₄²

D₂₉₂

yes

characteristic

122

C₄²

D₂₉₂

yes

characteristic

123

C₄²

C₂²⋊D₇₃

yes

—

124

C₂³

—

292

—

125

C₂³

—

146

—

126

C₂³

—

146

—

127

C₂³

C₂×C₄×D₇₃

yes

characteristic

128

Q₈

—

292

—

129

Q₈

—

292

—

130

Q₈

—

292

—

131

D₄

—

292

—

132

D₄

—

292

—

133

D₄

—

292

—

134

C₂×C₄

—

135

C₂×C₄

—

146

—

136

C₂×C₄

—

146

—

137

C₂×C₄

#11924005

yes

—

138

C₂×C₄

#11924005

yes

—

139

C₂×C₄

D₄×D₇₃

yes

—

140

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

141

C₂×C₄

C₂×D₂₉₂

yes

characteristic

142

C₂×C₄

#11924029

yes

—

143

C₈

—

144

C₈

—

146

—

145

C₂²

—

146

C₂²

—

292

—

147

C₂²

—

292

—

148

C₂²

—

584

—

149

C₂²

C₂×#11924005

yes

characteristic

150

C₂²

#14556553

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

292

—

154

C₄

#14556566

yes

—

155

C₄

#14556580

yes

characteristic

156

C₄

#14556587

yes

characteristic

157

C₂

—

158

C₂

—

584

—

159

C₂

#17988488

yes

characteristic

160

C₂

#17988569

yes

—

	subgroup	G/H
center	C₂²	C₂×#11924005
commutator	C₂³×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#423×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	#423	D₇₃
hypocenter	C₇₃	#1967
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#11924005

commutator

C₂³×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#423×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#14556509	C₂	semidirect
4. C₄×#11924012	C₂	semidirect
5. #423×C₇₃	C₂	semidirect
81. C₇₃	#1967	semidirect
83. #423	D₇₃	semidirect

H ≅ G/N

type

2. C₂×#14556509

C₂

semidirect

4. C₄×#11924012

C₂

semidirect

5. #423×C₇₃

C₂

semidirect

81. C₇₃

#1967

semidirect

83. #423

D₇₃

semidirect

Subgroups — #22610778

Conjugacy classes of subgroups

Complements