Group #22610692 — subgroups

Subgroups — #22610692

Order 9344.

256 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 121 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×#14556522	C₂	1	2	yes	maximal
2	C₂×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	#17988349	C₂	1	2	yes	maximal
5	#384×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10596831	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₇₃	—	2	1	no	—
10	#14556522	—	2	4	no	—
11	C₄⋊#10596831	C₄	1	4	yes	complemented
12	#14556498	—	2	1	no	—
13	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
14	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
15	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₇₃	—	2	1	no	—
18	C₂²×Dic₇₃	—	2	1	no	—
19	C₂×#10596831	—	2	2	no	—
20	C₂×#10596831	—	2	2	no	—
21	C₂×#10596831	C₂×C₄	1	4	yes	—
22	C₄⋊Dic₇₃	—	2	1	no	—
23	C₄⋊Dic₇₃	—	2	1	no	—
24	C₄⋊Dic₇₃	D₄	1	2	yes	—
25	C₄⋊Dic₇₃	Q₈	1	2	yes	—
26	C₄×Dic₇₃	—	4	2	no	—
27	C₂²×C₄×C₇₃	—	2	2	no	—
28	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₇₃	—	2	2	no	—
32	C₄⋊C₄×C₇₃	—	2	2	no	—
33	C₄⋊C₄×C₇₃	—	4	2	no	—
34	C₄⋊C₄×C₇₃	—	4	2	no	—
35	C₄²×C₇₃	C₂×C₄	1	2	yes	—
36	C₄²×C₇₃	Q₈	1	2	yes	—
37	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₃	—	2	1	no	—
39	C₂×C₄×C₇₃	—	2	1	no	—
40	C₂×C₄×C₇₃	—	4	2	no	—
41	C₂×C₄×C₇₃	—	4	2	no	—
42	C₂×C₄×C₇₃	—	4	2	no	—
43	C₂×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₇₃	SD₁₆	1	2	yes	—
46	C₂×C₄×C₇₃	SD₁₆	1	2	yes	—
47	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
53	C₂×Dic₇₃	—	2	2	no	—
54	C₂×Dic₇₃	—	2	2	no	—
55	C₂×Dic₇₃	—	4	1	no	—
56	C₂×Dic₇₃	—	4	1	no	—
57	C₂×Dic₇₃	—	4	1	no	—
58	C₂×Dic₇₃	—	4	1	no	—
59	#10596831	—	2	4	no	—
60	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₇₃	#121	1	1	yes	characteristic
62	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂²×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₇₃	#137	1	1	yes	characteristic
67	C₄×C₇₃	—	2	2	no	—
68	C₄×C₇₃	—	8	2	no	—
69	C₄×C₇₃	#106	1	4	yes	complemented
70	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
71	C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
72	C₄×C₇₃	#125	1	1	yes	characteristic
73	C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
74	Dic₇₃	—	4	2	no	—
75	Dic₇₃	—	4	2	no	—
76	C₂×C₇₃	#389	1	1	yes	characteristic
77	C₂×C₇₃	C₂×#106	1	1	yes	characteristic
78	C₂×C₇₃	#428	1	1	yes	characteristic
79	C₂×C₇₃	#460	1	1	yes	characteristic
80	C₂×C₇₃	#461	1	1	yes	characteristic
81	C₂×C₇₃	#475	1	1	yes	characteristic
82	C₂×C₇₃	#476	1	1	yes	characteristic
83	#1948	—	73	1	no	maximal, Sylow
84	C₇₃	#1948	1	1	yes	characteristic, complemented, Sylow
85	C₂×#106	—	73	2	no	—
86	C₂×C₄⋊C₈	—	73	1	no	—
87	#384	D₇₃	1	1	yes	characteristic
88	C₄×C₄⋊C₄	—	73	1	no	—
89	#340	—	73	2	no	—
90	C₂²×C₈	—	73	2	no	—
91	C₂×C₄⋊C₄	—	2	1	no	—
92	C₂×C₄⋊C₄	—	2	1	no	—
93	C₂×C₄⋊C₄	—	73	1	no	—
94	C₂×C₄⋊C₄	—	73	1	no	—
95	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
96	C₂×C₄²	—	146	1	no	—
97	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
98	#106	—	146	4	no	—
99	C₄⋊C₈	—	73	4	no	—
100	#95	—	146	1	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	146	1	no	—
103	C₂²×C₄	—	146	1	no	—
104	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
107	C₂×C₈	—	73	4	no	—
108	C₂×C₈	—	146	2	no	—
109	C₂×C₈	—	146	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	73	2	no	—
115	C₄⋊C₄	—	73	2	no	—
116	C₄⋊C₄	—	146	1	no	—
117	C₄⋊C₄	—	146	1	no	—
118	C₄²	—	292	2	no	—
119	C₄²	Dic₁₄₆	1	2	yes	—
120	C₄²	C₄×D₇₃	1	2	yes	—
121	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	292	1	no	—
128	C₂×C₄	—	292	1	no	—
129	C₂×C₄	—	292	1	no	—
130	C₂×C₄	—	292	1	no	—
131	C₂×C₄	—	146	2	no	—
132	C₂×C₄	—	146	2	no	—
133	C₂×C₄	#11924007	1	2	yes	—
134	C₂×C₄	#11924007	1	2	yes	—
135	C₂×C₄	#11924015	1	2	yes	—
136	C₂×C₄	#11924016	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
138	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
139	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
140	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
141	C₂×C₄	#11924027	1	1	yes	characteristic
142	C₂×C₄	#11924029	1	1	yes	characteristic
143	C₈	—	146	4	no	—
144	C₂²	C₂×#11924007	1	1	yes	characteristic
145	C₂²	#14556562	1	1	yes	characteristic
146	C₂²	#14556567	1	1	yes	characteristic
147	C₂²	C₂×#11924015	1	1	yes	characteristic
148	C₂²	C₂×#11924016	1	1	yes	characteristic
149	C₂²	#14556617	1	1	yes	characteristic
150	C₂²	#14556622	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	8	2	no	—
153	C₄	—	292	2	no	—
154	C₄	—	292	2	no	—
155	C₄	#14556522	1	4	yes	complemented
156	C₄	#14556553	1	1	yes	characteristic
157	C₄	#14556556	1	1	yes	characteristic
158	C₄	#14556582	1	1	yes	characteristic
159	C₄	#14556590	1	1	yes	characteristic
160	C₂	#17988483	1	1	yes	characteristic
161	C₂	C₂×#14556522	1	1	yes	characteristic
162	C₂	#17988627	1	1	yes	characteristic
163	C₂	#17988784	1	1	yes	characteristic
164	C₂	#17988789	1	1	yes	characteristic
165	C₂	#17988869	1	1	yes	characteristic
166	C₂	#17988874	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556522

C₂

yes

maximal

C₂×C₄⋊#10596831

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₇₃

C₂

yes

maximal, characteristic

#17988349

C₂

yes

maximal

#384×C₇₃

C₂

yes

maximal, characteristic

C₂²×#10596831

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄×Dic₇₃

—

#14556522

—

C₄⋊#10596831

C₄

yes

complemented

#14556498

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

C₂×C₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

—

C₄⋊Dic₇₃

Q₈

yes

—

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

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

Q₈

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

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

#10596831

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₃

#121

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₇₃

C₂×SD₁₆

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#106

yes

complemented

C₄×C₇₃

C₄×D₄

yes

characteristic

C₄×C₇₃

C₄×Q₈

yes

characteristic

C₄×C₇₃

#125

yes

characteristic

C₄×C₇₃

C₄⋊Q₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×#106

—

C₂×C₄⋊C₈

—

#384

D₇₃

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

C₂×C₄²

—

146

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

#106

—

146

—

C₄⋊C₈

—

100

#95

—

146

—

101

C₂²×C₄

—

102

C₂²×C₄

—

146

—

103

C₂²×C₄

—

146

—

104

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

105

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

106

C₂²×C₄

C₂²×D₇₃

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

146

—

109

C₂×C₈

—

146

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

146

—

117

C₄⋊C₄

—

146

—

118

C₄²

—

292

—

119

C₄²

Dic₁₄₆

yes

—

120

C₄²

C₄×D₇₃

yes

—

121

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

292

—

128

C₂×C₄

—

292

—

129

C₂×C₄

—

292

—

130

C₂×C₄

—

292

—

131

C₂×C₄

—

146

—

132

C₂×C₄

—

146

—

133

C₂×C₄

#11924007

yes

—

134

C₂×C₄

#11924007

yes

—

135

C₂×C₄

#11924015

yes

—

136

C₂×C₄

#11924016

yes

—

137

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

138

C₂×C₄

D₄×D₇₃

yes

characteristic

139

C₂×C₄

Q₈×D₇₃

yes

characteristic

140

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

141

C₂×C₄

#11924027

yes

characteristic

142

C₂×C₄

#11924029

yes

characteristic

143

C₈

—

146

—

144

C₂²

C₂×#11924007

yes

characteristic

145

C₂²

#14556562

yes

characteristic

146

C₂²

#14556567

yes

characteristic

147

C₂²

C₂×#11924015

yes

characteristic

148

C₂²

C₂×#11924016

yes

characteristic

149

C₂²

#14556617

yes

characteristic

150

C₂²

#14556622

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

292

—

154

C₄

—

292

—

155

C₄

#14556522

yes

complemented

156

C₄

#14556553

yes

characteristic

157

C₄

#14556556

yes

characteristic

158

C₄

#14556582

yes

characteristic

159

C₄

#14556590

yes

characteristic

160

C₂

#17988483

yes

characteristic

161

C₂

C₂×#14556522

yes

characteristic

162

C₂

#17988627

yes

characteristic

163

C₂

#17988784

yes

characteristic

164

C₂

#17988789

yes

characteristic

165

C₂

#17988869

yes

characteristic

166

C₂

#17988874

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#384×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	#384	D₇₃
hypocenter	C₇₃	#1948
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#384×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. C₄⋊#10596831	C₄	semidirect
69. C₄×C₇₃	#106	semidirect
84. C₇₃	#1948	semidirect
155. C₄	#14556522	semidirect

H ≅ G/N

type

11. C₄⋊#10596831

C₄

semidirect

69. C₄×C₇₃

#106

semidirect

84. C₇₃

#1948

semidirect

155. C₄

#14556522

semidirect

Subgroups — #22610692

Conjugacy classes of subgroups

Complements