Group #22626241 — subgroups

Subgroups — #22626241

Order 9344.

220 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 64 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	#17989643	C₂	1	1	yes	maximal, characteristic
2	#17989597	C₂	1	1	yes	maximal, characteristic
3	#17989500	C₂	1	4	yes	maximal, complemented
4	C₂×Q₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
5	#14556670	—	2	1	no	—
6	#14556666	—	2	1	no	—
7	#14556665	—	2	1	no	—
8	#14556664	C₂²	1	2	yes	—
9	#14556660	C₂²	1	2	yes	—
10	#14556659	—	2	1	no	—
11	C₂×Q₈×D₇₃	—	2	1	no	—
12	C₂×Dic₂₉₂	C₄	1	1	yes	characteristic, complemented
13	C₂×#11924018	—	2	1	no	—
14	Q₁₆×D₇₃	—	2	1	no	—
15	Q₁₆×D₇₃	—	2	1	no	—
16	Q₁₆×D₇₃	C₄	1	2	yes	complemented
17	Q₁₆×D₇₃	C₂²	1	2	yes	—
18	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
20	C₄⋊#64408	—	2	2	no	—
21	C₂²⋊#64408	—	4	1	no	—
22	C₄×#64408	—	4	1	no	—
23	#11924031	—	2	2	no	—
24	C₂×Dic₁₄₆	—	2	1	no	—
25	C₂×Dic₁₄₆	—	4	1	no	—
26	Q₈×D₇₃	—	2	2	no	—
27	Q₈×D₇₃	—	2	2	no	—
28	Q₈×D₇₃	—	4	1	no	—
29	Q₈×D₇₃	—	4	1	no	—
30	C₂×C₄×D₇₃	—	4	1	no	—
31	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
32	Dic₂₉₂	—	2	1	no	—
33	Dic₂₉₂	C₂×C₄	1	2	yes	—
34	#11924018	—	2	2	no	—
35	#11924018	—	2	2	no	—
36	C₂×#10596831	D₄	1	1	yes	characteristic
37	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
38	C₈×D₇₃	D₄	1	1	yes	characteristic
39	C₈×D₇₃	D₄	1	1	yes	characteristic
40	C₈×D₇₃	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₇₃	—	2	1	no	—
42	Q₁₆×C₇₃	—	2	1	no	—
43	Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₇₃	D₈	1	1	yes	characteristic
46	C₂×#64408	—	4	2	no	—
47	Q₈×C₇₃	—	2	2	no	—
48	Q₈×C₇₃	—	4	1	no	—
49	C₂×C₄×C₇₃	—	4	1	no	—
50	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₇₃	—	4	1	no	—
52	C₂×Dic₇₃	SD₁₆	1	1	yes	characteristic
53	C₄×D₇₃	—	4	2	no	—
54	C₄×D₇₃	—	4	2	no	—
55	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₄×D₇₃	D₈	1	1	yes	characteristic
57	C₄×D₇₃	SD₁₆	1	1	yes	characteristic
58	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
59	Dic₁₄₆	—	2	2	no	—
60	Dic₁₄₆	—	4	1	no	—
61	Dic₁₄₆	—	4	2	no	—
62	Dic₁₄₆	—	4	2	no	—
63	C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	#10596831	C₂²⋊C₄	1	1	yes	characteristic
66	#10596831	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₇₃	#102	1	1	yes	characteristic
68	C₂×D₇₃	—	2	2	no	—
69	C₂×D₇₃	#102	1	1	yes	characteristic
70	C₂×D₇₃	C₂×D₈	1	1	yes	characteristic
71	#64408	—	8	2	no	—
72	C₄×C₇₃	—	4	2	no	—
73	C₄×C₇₃	#102	1	1	yes	characteristic
74	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	Dic₇₃	—	4	2	no	—
76	Dic₇₃	#102	1	1	yes	characteristic
77	Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₇₃	—	2	1	no	—
79	C₂×C₇₃	C₂×#102	1	1	yes	characteristic
80	D₇₃	—	2	1	no	—
81	D₇₃	#510	1	2	yes	—
82	#2141	—	73	1	no	maximal, Sylow
83	C₇₃	#2141	1	1	yes	characteristic, complemented, Sylow
84	C₂²×Q₁₆	—	73	1	no	—
85	C₂×M₃₂	—	73	1	no	—
86	#427	—	73	1	no	—
87	#358	—	73	4	no	—
88	C₂²×Q₈	—	146	1	no	—
89	C₂×Q₁₆	—	73	2	no	—
90	C₂×Q₁₆	—	73	2	no	—
91	C₂×Q₁₆	—	73	1	no	—
92	C₂×Q₁₆	—	146	1	no	—
93	C₂×Q₁₆	—	146	1	no	—
94	C₂×Q₁₆	—	146	1	no	—
95	C₂×Q₁₆	#64408	1	1	yes	characteristic, complemented
96	C₂²×C₈	—	73	1	no	—
97	#117	—	146	1	no	—
98	M₃₂	—	146	1	no	—
99	M₃₂	—	146	1	no	—
100	C₂×C₁₆	—	73	2	no	—
101	#107	—	73	2	no	—
102	#106	—	146	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	292	1	no	—
105	C₂×Q₈	—	292	1	no	—
106	C₂×Q₈	—	292	1	no	—
107	C₂×Q₈	—	146	2	no	—
108	C₂×Q₈	—	146	2	no	—
109	C₂×Q₈	—	146	1	no	—
110	C₂²×C₄	—	292	1	no	—
111	C₂²×C₄	—	73	1	no	—
112	Q₁₆	—	2	1	no	—
113	Q₁₆	—	73	2	no	—
114	Q₁₆	—	146	2	no	—
115	Q₁₆	—	146	2	no	—
116	Q₁₆	—	146	1	no	—
117	Q₁₆	C₂×#64408	1	2	yes	—
118	C₂×C₈	—	73	1	no	—
119	C₂×C₈	—	73	1	no	—
120	C₂×C₈	—	73	1	no	—
121	C₂×C₈	—	73	1	no	—
122	C₂×C₈	—	73	1	no	—
123	C₂×C₈	C₂×#64408	1	1	yes	characteristic
124	C₄⋊C₄	—	146	2	no	—
125	C₂²⋊C₄	—	292	1	no	—
126	C₄²	—	292	1	no	—
127	C₁₆	—	146	2	no	—
128	C₂³	—	73	1	no	—
129	Q₈	—	2	2	no	—
130	Q₈	—	4	1	no	—
131	Q₈	—	292	2	no	—
132	Q₈	—	292	2	no	—
133	Q₈	—	292	1	no	—
134	Q₈	—	146	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	292	2	no	—
137	C₂×C₄	—	292	2	no	—
138	C₂×C₄	—	292	1	no	—
139	C₂×C₄	—	292	2	no	—
140	C₂×C₄	—	73	1	no	—
141	C₂×C₄	—	73	1	no	—
142	C₂×C₄	—	73	1	no	—
143	C₂×C₄	—	73	1	no	—
144	C₂×C₄	—	73	1	no	—
145	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
146	C₈	—	73	1	no	—
147	C₈	—	73	1	no	—
148	C₈	C₂²⋊#64408	1	1	yes	characteristic
149	C₈	C₂²×#64408	1	1	yes	characteristic
150	C₂²	—	73	1	no	—
151	C₂²	—	73	1	no	—
152	C₂²	—	146	2	no	—
153	C₂²	#14556653	1	1	yes	characteristic
154	C₄	—	4	2	no	—
155	C₄	—	292	2	no	—
156	C₄	—	584	2	no	—
157	C₄	—	73	1	no	—
158	C₄	—	73	1	no	—
159	C₄	#14556653	1	1	yes	characteristic
160	C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	73	2	no	—
163	C₂	—	146	1	no	—
164	C₂	C₂×#14556653	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989643

C₂

yes

maximal, characteristic

#17989597

C₂

yes

maximal, characteristic

#17989500

C₂

yes

maximal, complemented

C₂×Q₁₆×D₇₃

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#14556660

C₂²

yes

—

#14556659

—

C₂×Q₈×D₇₃

—

C₂×Dic₂₉₂

C₄

yes

characteristic, complemented

C₂×#11924018

—

Q₁₆×D₇₃

—

Q₁₆×D₇₃

—

Q₁₆×D₇₃

C₄

yes

complemented

Q₁₆×D₇₃

C₂²

yes

—

C₂×C₈×D₇₃

C₂²

yes

characteristic

C₂×Q₁₆×C₇₃

C₄

yes

characteristic, complemented

—

—

—

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

Q₈×D₇₃

—

Q₈×D₇₃

—

Q₈×D₇₃

—

Q₈×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

D₄

yes

characteristic

Dic₂₉₂

—

Dic₂₉₂

C₂×C₄

yes

—

#11924018

—

#11924018

—

C₂×#10596831

D₄

yes

characteristic

C₈×D₇₃

C₂×C₄

yes

characteristic

C₈×D₇₃

D₄

yes

characteristic

C₈×D₇₃

D₄

yes

characteristic

C₈×D₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

D₈

yes

characteristic

C₂×#64408

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

SD₁₆

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂²⋊C₄

yes

characteristic

C₄×D₇₃

D₈

yes

characteristic

C₄×D₇₃

SD₁₆

yes

characteristic

C₄×D₇₃

C₂×D₄

yes

characteristic

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₈×C₇₃

C₂²×C₄

yes

characteristic

#10596831

C₂²⋊C₄

yes

characteristic

#10596831

C₂×D₄

yes

characteristic

C₂²×C₇₃

#102

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

#102

yes

characteristic

C₂×D₇₃

C₂×D₈

yes

characteristic

#64408

—

C₄×C₇₃

—

C₄×C₇₃

#102

yes

characteristic

C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

Dic₇₃

—

Dic₇₃

#102

yes

characteristic

Dic₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

C₂×#102

yes

characteristic

D₇₃

—

D₇₃

#510

yes

—

#2141

—

maximal, Sylow

C₇₃

#2141

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×M₃₂

—

#427

—

#358

—

C₂²×Q₈

—

146

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₂×Q₁₆

#64408

yes

characteristic, complemented

C₂²×C₈

—

#117

—

146

—

M₃₂

—

146

—

M₃₂

—

146

—

100

C₂×C₁₆

—

101

#107

—

102

#106

—

146

—

103

C₂×Q₈

—

104

C₂×Q₈

—

292

—

105

C₂×Q₈

—

292

—

106

C₂×Q₈

—

292

—

107

C₂×Q₈

—

146

—

108

C₂×Q₈

—

146

—

109

C₂×Q₈

—

146

—

110

C₂²×C₄

—

292

—

111

C₂²×C₄

—

112

Q₁₆

—

113

Q₁₆

—

114

Q₁₆

—

146

—

115

Q₁₆

—

146

—

116

Q₁₆

—

146

—

117

Q₁₆

C₂×#64408

yes

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

C₂×#64408

yes

characteristic

124

C₄⋊C₄

—

146

—

125

C₂²⋊C₄

—

292

—

126

C₄²

—

292

—

127

C₁₆

—

146

—

128

C₂³

—

129

Q₈

—

130

Q₈

—

131

Q₈

—

292

—

132

Q₈

—

292

—

133

Q₈

—

292

—

134

Q₈

—

146

—

135

C₂×C₄

—

136

C₂×C₄

—

292

—

137

C₂×C₄

—

292

—

138

C₂×C₄

—

292

—

139

C₂×C₄

—

292

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

C₂²⋊#64408

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

C₂²⋊#64408

yes

characteristic

149

C₈

C₂²×#64408

yes

characteristic

150

C₂²

—

151

C₂²

—

152

C₂²

—

146

—

153

C₂²

#14556653

yes

characteristic

154

C₄

—

155

C₄

—

292

—

156

C₄

—

584

—

157

C₄

—

158

C₄

—

159

C₄

#14556653

yes

characteristic

160

C₄

C₂×C₂²⋊#64408

yes

characteristic

161

C₂

—

162

C₂

—

163

C₂

—

146

—

164

C₂

C₂×#14556653

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#14556653
commutator	C₈×C₇₃	C₂²×C₄
Frattini	C₈	C₂²×#64408
Fitting	C₂×Q₁₆×C₇₃	C₄
socle	C₂×C₇₃	C₂×#102
hypercenter	C₂×Q₁₆	#64408
hypocenter	C₇₃	#2141
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂

C₂×#14556653

commutator

C₈×C₇₃

C₂²×C₄

Frattini

C₈

C₂²×#64408

Fitting

C₂×Q₁₆×C₇₃

C₄

C₂×C₇₃

C₂×Q₁₆

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

N	H ≅ G/N	type
3. #17989500	C₂	semidirect
12. C₂×Dic₂₉₂	C₄	semidirect
16. Q₁₆×D₇₃	C₄	semidirect
19. C₂×Q₁₆×C₇₃	C₄	semidirect
83. C₇₃	#2141	semidirect
95. C₂×Q₁₆	#64408	semidirect

H ≅ G/N

type

3. #17989500

C₂

semidirect

12. C₂×Dic₂₉₂

C₄

semidirect

16. Q₁₆×D₇₃

C₄

semidirect

19. C₂×Q₁₆×C₇₃

C₄

semidirect

83. C₇₃

#2141

semidirect

95. C₂×Q₁₆

#64408

semidirect

Subgroups — #22626241

Conjugacy classes of subgroups

Complements