Group #22610690 — subgroups

Subgroups — #22610690

Order 9344.

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

normal subgroups only

This group has 256 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×#14556525	C₂	1	2	yes	maximal
2	#17988457	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	#17988350	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₇₃	—	2	1	no	—
7	C₂×C₄⋊Dic₇₃	—	2	1	no	—
8	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₇₃	—	2	1	no	—
12	#14556525	—	2	4	no	—
13	#14556498	—	2	1	no	—
14	C₂²×C₈×C₇₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₇₃	C₄	1	4	yes	complemented
17	C₂²×Dic₇₃	—	2	1	no	—
18	C₂²×Dic₇₃	—	2	1	no	—
19	C₂²×Dic₇₃	—	2	2	no	—
20	C₄⋊Dic₇₃	—	2	1	no	—
21	C₄⋊Dic₇₃	—	2	1	no	—
22	C₄⋊Dic₇₃	—	2	2	no	—
23	C₄⋊Dic₇₃	—	2	2	no	—
24	C₄⋊Dic₇₃	—	4	2	no	—
25	C₄⋊Dic₇₃	—	4	2	no	—
26	C₄⋊Dic₇₃	D₄	1	2	yes	—
27	C₄⋊Dic₇₃	Q₈	1	2	yes	—
28	C₄×Dic₇₃	—	4	2	no	—
29	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₇₃	—	2	2	no	—
33	C₂×C₈×C₇₃	—	2	2	no	—
34	C₂×C₈×C₇₃	C₂×C₄	1	4	yes	—
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	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₇₃	D₈	1	2	yes	—
43	C₂×C₄×C₇₃	Q₁₆	1	2	yes	—
44	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
50	C₂×Dic₇₃	—	2	2	no	—
51	C₂×Dic₇₃	—	2	2	no	—
52	C₂×Dic₇₃	—	4	1	no	—
53	C₂×Dic₇₃	—	4	1	no	—
54	C₂×Dic₇₃	—	4	1	no	—
55	C₂×Dic₇₃	—	4	1	no	—
56	C₂×Dic₇₃	—	4	2	no	—
57	C₂×Dic₇₃	—	4	2	no	—
58	C₂×Dic₇₃	—	4	2	no	—
59	C₈×C₇₃	—	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×D₈	1	1	yes	characteristic
64	C₂²×C₇₃	C₂×Q₁₆	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	#107	1	4	yes	complemented
69	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
70	C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
71	C₄×C₇₃	#125	1	1	yes	characteristic
72	C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
73	Dic₇₃	—	4	2	no	—
74	Dic₇₃	—	4	2	no	—
75	Dic₇₃	—	8	2	no	—
76	C₂×C₇₃	#389	1	1	yes	characteristic
77	C₂×C₇₃	C₂×#107	1	1	yes	characteristic
78	C₂×C₇₃	#428	1	1	yes	characteristic
79	C₂×C₇₃	#459	1	1	yes	characteristic
80	C₂×C₇₃	#462	1	1	yes	characteristic
81	C₂×C₇₃	#474	1	1	yes	characteristic
82	C₂×C₇₃	#477	1	1	yes	characteristic
83	#1947	—	73	1	no	maximal, Sylow
84	C₇₃	#1947	1	1	yes	characteristic, complemented, Sylow
85	C₂×#107	—	73	2	no	—
86	C₂×C₄⋊C₈	D₇₃	1	1	yes	characteristic
87	#384	—	73	1	no	—
88	C₄×C₄⋊C₄	—	73	1	no	—
89	#340	—	73	2	no	—
90	C₂²×C₈	C₂×D₇₃	1	2	yes	—
91	C₂×C₄⋊C₄	—	73	1	no	—
92	C₂×C₄⋊C₄	—	73	1	no	—
93	C₂×C₄⋊C₄	—	73	2	no	—
94	C₂×C₄⋊C₄	—	146	1	no	—
95	C₂×C₄⋊C₄	—	146	1	no	—
96	C₂×C₄²	—	146	1	no	—
97	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
98	#107	—	146	4	no	—
99	C₄⋊C₈	Dic₇₃	1	4	yes	complemented
100	#95	—	146	1	no	—
101	C₂²×C₄	—	146	2	no	—
102	C₂²×C₄	—	146	1	no	—
103	C₂²×C₄	—	146	1	no	—
104	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
105	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	C₂×Dic₇₃	1	4	yes	—
110	C₄⋊C₄	—	292	2	no	—
111	C₄⋊C₄	—	292	2	no	—
112	C₄⋊C₄	—	73	2	no	—
113	C₄⋊C₄	—	73	2	no	—
114	C₄⋊C₄	—	146	2	no	—
115	C₄⋊C₄	—	146	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	C₂×Dic₇₃	1	2	yes	—
121	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	292	1	no	—
125	C₂×C₄	—	292	1	no	—
126	C₂×C₄	—	292	1	no	—
127	C₂×C₄	—	292	1	no	—
128	C₂×C₄	—	292	2	no	—
129	C₂×C₄	—	292	2	no	—
130	C₂×C₄	—	292	2	no	—
131	C₂×C₄	—	146	2	no	—
132	C₂×C₄	—	146	2	no	—
133	C₂×C₄	C₄⋊Dic₇₃	1	2	yes	—
134	C₂×C₄	C₄⋊Dic₇₃	1	2	yes	—
135	C₂×C₄	D₅₈₄	1	2	yes	—
136	C₂×C₄	Dic₂₉₂	1	2	yes	—
137	C₂×C₄	C₂²×Dic₇₃	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	—	2	4	no	—
144	C₂²	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
145	C₂²	#14556566	1	1	yes	characteristic
146	C₂²	#14556571	1	1	yes	characteristic
147	C₂²	C₂×D₅₈₄	1	1	yes	characteristic
148	C₂²	C₂×Dic₂₉₂	1	1	yes	characteristic
149	C₂²	#14556619	1	1	yes	characteristic
150	C₂²	#14556624	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	292	2	no	—
153	C₄	—	292	2	no	—
154	C₄	—	584	2	no	—
155	C₄	#14556525	1	4	yes	complemented
156	C₄	D₄×Dic₇₃	1	1	yes	characteristic
157	C₄	Q₈×Dic₇₃	1	1	yes	characteristic
158	C₄	#14556582	1	1	yes	characteristic
159	C₄	#14556590	1	1	yes	characteristic
160	C₂	#17988486	1	1	yes	characteristic
161	C₂	C₂×#14556525	1	1	yes	characteristic
162	C₂	#17988629	1	1	yes	characteristic
163	C₂	#17988783	1	1	yes	characteristic
164	C₂	#17988798	1	1	yes	characteristic
165	C₂	#17988868	1	1	yes	characteristic
166	C₂	#17988883	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556525

C₂

yes

maximal

#17988457

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₇₃

C₂

yes

maximal, characteristic

#17988350

C₂

yes

maximal

C₂×C₄⋊C₈×C₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄×Dic₇₃

—

#14556525

—

#14556498

—

C₂²×C₈×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₄

yes

complemented

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

—

C₄⋊Dic₇₃

Q₈

yes

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

Q₈

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

D₈

yes

—

C₂×C₄×C₇₃

Q₁₆

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

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₃

#121

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₂²×C₇₃

C₂×Q₁₆

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

#107

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

—

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×#107

—

C₂×C₄⋊C₈

D₇₃

yes

characteristic

#384

—

C₄×C₄⋊C₄

—

#340

—

C₂²×C₈

C₂×D₇₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

—

146

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

#107

—

146

—

C₄⋊C₈

Dic₇₃

yes

complemented

100

#95

—

146

—

101

C₂²×C₄

—

146

—

102

C₂²×C₄

—

146

—

103

C₂²×C₄

—

146

—

104

C₂²×C₄

D₂₉₂

yes

characteristic

105

C₂²×C₄

D₂₉₂

yes

characteristic

106

C₂²×C₄

C₂²×D₇₃

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂×Dic₇₃

yes

—

110

C₄⋊C₄

—

292

—

111

C₄⋊C₄

—

292

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

146

—

115

C₄⋊C₄

—

146

—

116

C₄⋊C₄

—

146

—

117

C₄⋊C₄

—

146

—

118

C₄²

—

292

—

119

C₄²

Dic₁₄₆

yes

—

120

C₄²

C₂×Dic₇₃

yes

—

121

C₂³

C₂×D₂₉₂

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

292

—

125

C₂×C₄

—

292

—

126

C₂×C₄

—

292

—

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

C₄⋊Dic₇₃

yes

—

134

C₂×C₄

C₄⋊Dic₇₃

yes

—

135

C₂×C₄

D₅₈₄

yes

—

136

C₂×C₄

Dic₂₉₂

yes

—

137

C₂×C₄

C₂²×Dic₇₃

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

—

144

C₂²

C₂×C₄⋊Dic₇₃

yes

characteristic

145

C₂²

#14556566

yes

characteristic

146

C₂²

#14556571

yes

characteristic

147

C₂²

C₂×D₅₈₄

yes

characteristic

148

C₂²

C₂×Dic₂₉₂

yes

characteristic

149

C₂²

#14556619

yes

characteristic

150

C₂²

#14556624

yes

characteristic

151

C₄

—

152

C₄

—

292

—

153

C₄

—

292

—

154

C₄

—

584

—

155

C₄

#14556525

yes

complemented

156

C₄

D₄×Dic₇₃

yes

characteristic

157

C₄

Q₈×Dic₇₃

yes

characteristic

158

C₄

#14556582

yes

characteristic

159

C₄

#14556590

yes

characteristic

160

C₂

#17988486

yes

characteristic

161

C₂

C₂×#14556525

yes

characteristic

162

C₂

#17988629

yes

characteristic

163

C₂

#17988783

yes

characteristic

164

C₂

#17988798

yes

characteristic

165

C₂

#17988868

yes

characteristic

166

C₂

#17988883

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×C₄⋊C₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₇₃
hypocenter	C₇₃	#1947
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×C₄⋊C₈×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₇₃

hypocenter

C₇₃

#1947

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
16. C₄⋊C₈×C₇₃	C₄	semidirect
68. C₄×C₇₃	#107	semidirect
84. C₇₃	#1947	semidirect
99. C₄⋊C₈	Dic₇₃	semidirect
155. C₄	#14556525	semidirect

H ≅ G/N

type

16. C₄⋊C₈×C₇₃

C₄

semidirect

68. C₄×C₇₃

#107

semidirect

84. C₇₃

#1947

semidirect

99. C₄⋊C₈

Dic₇₃

semidirect

155. C₄

#14556525

semidirect

Subgroups — #22610690

Conjugacy classes of subgroups

Complements