Group #22609764 — subgroups

Subgroups — #22609764

Order 9344.

324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556514	C₂	1	2	yes	maximal
3	C₂×#14556502	C₂	1	1	yes	maximal, characteristic
4	#17988349	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
7	C₂²×#10596831	C₂²	1	2	yes	—
8	C₄×Dic₁₄₆	—	2	2	no	—
9	C₄×Dic₁₄₆	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×#11924007	—	2	1	no	—
12	C₂×C₄×Dic₇₃	—	2	1	no	—
13	#14556514	C₄	1	8	yes	complemented
14	#14556502	—	2	2	no	—
15	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₇₃	—	2	1	no	—
17	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	#95×C₇₃	—	2	1	no	—
19	C₂×Dic₁₄₆	—	2	1	no	—
20	C₂×Dic₁₄₆	—	2	2	no	—
21	C₂×Dic₁₄₆	C₂×C₄	1	4	yes	—
22	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
23	C₂²×Dic₇₃	—	2	1	no	—
24	C₂²×Dic₇₃	—	2	1	no	—
25	C₂×#10596831	—	2	2	no	—
26	C₂×#10596831	—	2	2	no	—
27	C₂×#10596831	C₂×C₄	1	4	yes	—
28	C₄⋊Dic₇₃	—	2	1	no	—
29	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
30	#11924007	—	2	4	no	—
31	C₄×Dic₇₃	—	2	4	no	—
32	C₂²×C₄×C₇₃	—	2	2	no	—
33	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₇₃	—	2	2	no	—
37	C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
38	C₄²×C₇₃	—	4	2	no	—
39	C₄²×C₇₃	C₂×C₄	1	2	yes	—
40	C₄²×C₇₃	D₄	1	2	yes	—
41	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	—	2	1	no	—
43	C₂×C₄×C₇₃	—	2	2	no	—
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	2	4	no	—
46	C₂×C₄×C₇₃	—	4	2	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	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
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	C₂×Dic₇₃	—	2	2	no	—
57	C₂×Dic₇₃	—	4	1	no	—
58	C₂×Dic₇₃	—	4	1	no	—
59	Dic₁₄₆	—	2	2	no	—
60	Dic₁₄₆	—	2	4	no	—
61	Dic₁₄₆	C₄²	1	4	yes	complemented
62	#10596831	—	2	4	no	—
63	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₇₃	C₄×D₄	1	2	yes	—
65	C₂²×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₇₃	#137	1	2	yes	—
67	C₂²×C₇₃	#137	1	1	yes	characteristic
68	C₄×C₇₃	—	2	2	no	—
69	C₄×C₇₃	—	2	2	no	—
70	C₄×C₇₃	—	4	4	no	—
71	C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
72	C₄×C₇₃	#117	1	1	yes	characteristic
73	C₄×C₇₃	C₄×D₄	1	2	yes	—
74	Dic₇₃	—	2	4	no	—
75	Dic₇₃	—	4	2	no	—
76	C₂×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₇₃	#417	1	1	yes	characteristic
78	C₂×C₇₃	#419	1	1	yes	characteristic
79	C₂×C₇₃	#440	1	2	yes	—
80	C₂×C₇₃	#441	1	2	yes	—
81	#1763	—	73	1	no	maximal, Sylow
82	C₇₃	#1763	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄×Q₈	—	73	1	no	—
84	C₂×#103	—	73	2	no	—
85	C₂×#97	—	73	1	no	—
86	C₄×C₄⋊C₄	D₇₃	1	1	yes	characteristic
87	#340	—	73	2	no	—
88	C₂²×Q₈	—	73	1	no	—
89	C₂²×C₈	—	73	2	no	—
90	C₄×Q₈	—	73	4	no	—
91	C₄×Q₈	—	146	2	no	—
92	C₂×C₄⋊C₄	—	73	1	no	—
93	C₂×C₄⋊C₄	—	146	1	no	—
94	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
95	C₂×C₄²	—	2	1	no	—
96	C₂×C₄²	—	146	1	no	—
97	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
98	#103	—	73	8	no	—
99	#97	—	146	2	no	—
100	#95	—	2	1	no	—
101	C₂×Q₈	—	73	4	no	—
102	C₂×Q₈	—	73	2	no	—
103	C₂×Q₈	—	146	1	no	—
104	C₂×Q₈	—	146	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	146	1	no	—
107	C₂²×C₄	—	146	1	no	—
108	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
111	C₂×C₈	—	73	4	no	—
112	C₂×C₈	—	146	2	no	—
113	C₂×C₈	—	146	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	73	2	no	—
116	C₄⋊C₄	—	146	4	no	—
117	C₄⋊C₄	—	146	1	no	—
118	C₄⋊C₄	C₄×D₇₃	1	4	yes	—
119	C₄²	—	4	2	no	—
120	C₄²	—	146	4	no	—
121	C₄²	C₄×D₇₃	1	2	yes	—
122	C₄²	D₂₉₂	1	2	yes	—
123	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
124	Q₈	—	73	4	no	—
125	Q₈	—	146	2	no	—
126	Q₈	—	146	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	292	1	no	—
134	C₂×C₄	—	292	1	no	—
135	C₂×C₄	—	146	2	no	—
136	C₂×C₄	—	146	2	no	—
137	C₂×C₄	—	146	4	no	—
138	C₂×C₄	#11924005	1	2	yes	—
139	C₂×C₄	#11924005	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₇₃	1	2	yes	—
142	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
143	C₂×C₄	#11924028	1	2	yes	—
144	C₈	—	146	4	no	—
145	C₂²	C₂×#11924005	1	1	yes	characteristic
146	C₂²	C₄×C₂²⋊D₇₃	1	2	yes	—
147	C₂²	#14556617	1	1	yes	characteristic
148	C₂²	#14556621	1	2	yes	—
149	C₂²	#14556622	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	—	292	2	no	—
154	C₄	—	146	4	no	—
155	C₄	C₄²×D₇₃	1	1	yes	characteristic
156	C₄	#14556548	1	1	yes	characteristic
157	C₄	C₄×D₂₉₂	1	2	yes	—
158	C₂	C₄×#11924005	1	1	yes	characteristic
159	C₂	#17988586	1	1	yes	characteristic
160	C₂	#17988595	1	1	yes	characteristic
161	C₂	#17988687	1	2	yes	—
162	C₂	#17988692	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₄₆

C₂

yes

maximal, characteristic

C₂×#14556514

C₂

yes

maximal

C₂×#14556502

C₂

yes

maximal, characteristic

#17988349

C₂

yes

maximal

C₄×C₄⋊C₄×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

characteristic

C₂²×#10596831

C₂²

yes

—

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

C₄

yes

complemented

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×#11924007

—

C₂×C₄×Dic₇₃

—

#14556514

C₄

yes

complemented

#14556502

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

#95×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

C₂×C₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

D₄

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×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

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

C₄²

yes

complemented

#10596831

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

C₄×D₄

yes

—

C₂²×C₇₃

AGL(1,ℤ/8)

yes

characteristic

C₂²×C₇₃

#137

yes

—

C₂²×C₇₃

#137

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄²

yes

characteristic

C₄×C₇₃

#117

yes

characteristic

C₄×C₇₃

C₄×D₄

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

C₄×C₂²⋊C₄

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#441

yes

—

#1763

—

maximal, Sylow

C₇₃

#1763

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₂×#103

—

C₂×#97

—

C₄×C₄⋊C₄

D₇₃

yes

characteristic

#340

—

C₂²×Q₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

#103

—

#97

—

146

—

100

#95

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

146

—

104

C₂×Q₈

—

146

—

105

C₂²×C₄

—

106

C₂²×C₄

—

146

—

107

C₂²×C₄

—

146

—

108

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

109

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

110

C₂²×C₄

C₂²×D₇₃

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

146

—

113

C₂×C₈

—

146

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

146

—

117

C₄⋊C₄

—

146

—

118

C₄⋊C₄

C₄×D₇₃

yes

—

119

C₄²

—

120

C₄²

—

146

—

121

C₄²

C₄×D₇₃

yes

—

122

C₄²

D₂₉₂

yes

—

123

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

124

Q₈

—

125

Q₈

—

146

—

126

Q₈

—

146

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

292

—

134

C₂×C₄

—

292

—

135

C₂×C₄

—

146

—

136

C₂×C₄

—

146

—

137

C₂×C₄

—

146

—

138

C₂×C₄

#11924005

yes

—

139

C₂×C₄

#11924005

yes

—

140

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

141

C₂×C₄

C₂×C₄×D₇₃

yes

—

142

C₂×C₄

C₂×D₂₉₂

yes

characteristic

143

C₂×C₄

#11924028

yes

—

144

C₈

—

146

—

145

C₂²

C₂×#11924005

yes

characteristic

146

C₂²

C₄×C₂²⋊D₇₃

yes

—

147

C₂²

#14556617

yes

characteristic

148

C₂²

#14556621

yes

—

149

C₂²

#14556622

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

292

—

154

C₄

—

146

—

155

C₄

C₄²×D₇₃

yes

characteristic

156

C₄

#14556548

yes

characteristic

157

C₄

C₄×D₂₉₂

yes

—

158

C₂

C₄×#11924005

yes

characteristic

159

C₂

#17988586

yes

characteristic

160

C₂

#17988595

yes

characteristic

161

C₂

#17988687

yes

—

162

C₂

#17988692

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₃
commutator	C₄×C₇₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₄×C₄⋊C₄×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₄×C₄⋊C₄	D₇₃
hypocenter	C₇₃	#1763
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₃

commutator

C₄×C₇₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₄×C₄⋊C₄×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

hypercenter

C₄×C₄⋊C₄

D₇₃

hypocenter

C₇₃

#1763

LCS

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
9. C₄×Dic₁₄₆	C₄	semidirect
13. #14556514	C₄	semidirect
61. Dic₁₄₆	C₄²	semidirect
82. C₇₃	#1763	semidirect

H ≅ G/N

type

9. C₄×Dic₁₄₆

C₄

semidirect

13. #14556514

C₄

semidirect

61. Dic₁₄₆

C₄²

semidirect

82. C₇₃

#1763

semidirect

Subgroups — #22609764

Conjugacy classes of subgroups

Complements