Finite Group Databasebeta

← C22×#19153844 #37548456

Subgroups — C22×#19153844 #37548456

Order 23168.

1720 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 565 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15668449C211yesmaximal, characteristic, complemented
2C23×D724C211yesmaximal, characteristic, complemented
3C2×#19153844C2124yesmaximal, direct product factor
4C22×#15668438C212yesmaximal, complemented
5C22×D1448C212yesmaximal, complemented
6C22×M16×C181C211yesmaximal, characteristic, complemented
7C24×D181—21no—
8C2×#15668449—26no—
9C2×#15668449C22112yescomplemented
10C22×Dic362C2211yescharacteristic, complemented
11C22×D724—21no—
12C22×D724—26no—
13C22×D724C2212yes—
14C22×D724C22112yescomplemented
15C22×D724C2211yescharacteristic, complemented
16C22×C22⋊D181—21no—
17C22×C4×D181—21no—
18#19153844C22164yesdirect product factor
19C2×#15668438C22124yescomplemented
20C2×D1448C22124yescomplemented
21C23×C4×C181C2211yescharacteristic
22C2×M16×C181C22112yescomplemented
23C22×C8×C181C2212yescomplemented
24C23×D181—21no—
25C23×D181—22no—
26C23×D181—212no—
27C23×D181—48no—
28#15668449—224no—
29#15668449C23116yescomplemented
30C2×Dic362—23no—
31C2×Dic362C2316yescomplemented
32C2×D724—26no—
33C2×D724—212no—
34C2×D724—224no—
35C2×D724—23no—
36C2×D724C23112yes—
37C2×D724C23116yescomplemented
38C2×D724C2316yescomplemented
39C2×C22⋊D181—212no—
40C22×Dic181—21no—
41C2×C4×D181—212no—
42#15668438C23132yescomplemented
43D1448C23132yescomplemented
44C24×C181D411yescharacteristic
45C22×C4×C181D411yescharacteristic
46C22×C4×C181D416yes—
47C22×C4×C181C2311yescharacteristic
48C22×C4×C181C2316yes—
49M16×C181C23116yescomplemented
50C2×C8×C181C23112yescomplemented
51C22×D181—26no—
52C22×D181—212no—
53C22×D181—216no—
54C22×D181—448no—
55C22×D181—44no—
56C22×D181—48no—
57Dic362—26no—
58Dic362C2414yescomplemented
59D724—212no—
60D724—216no—
61D724—26no—
62D724C2418yes—
63D724C2414yescomplemented
64C22⋊D181—216no—
65C2×Dic181—26no—
66C4×D181—216no—
67C23×C181—24no—
68C23×C181C2×D416yes—
69C23×C181C2×D411yescharacteristic
70C2×C4×C181C2×D416yes—
71C2×C4×C181C2×D413yes—
72C2×C4×C181C2×D4112yes—
73C2×C4×C181C2413yes—
74C2×C4×C181C2414yes—
75C8×C181C2418yescomplemented
76C22×C181—212no—
77C22×C181AGL(1,ℤ/8)14yescomplemented
78C22×C181C22×D414yes—
79C22×C181C22×D413yes—
80C2×D181—24no—
81C2×D181—28no—
82C2×D181—432no—
83C2×D181—412no—
84C2×D181—424no—
85C4×C181C22×D414yes—
86C4×C181C22×D413yes—
87C4×C181C2511yescharacteristic
88Dic181—24no—
89C2×C181—24no—
90C2×C181C2×AGL(1,ℤ/8)16yescomplemented
91C2×C181C23×D411yescharacteristic
92D181—44no—
93D181—48no—
94C181C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—1811nomaximal, Sylow
96C22×#41—1811no—
97C23×D4—1811no—
98C2×AGL(1,ℤ/8)—18124no—
99C22×SD16—1812no—
100C22×D8—1812no—
101C22×M16D18111yescharacteristic, complemented
102C25—3621no—
103C2×#41—3626no—
104C2×#41—18112no—
105C22×Q8—1811no—
106C22×D4—3621no—
107C22×D4—3626no—
108C22×D4—3621no—
109C22×D4—1811no—
110C22×D4—1812no—
111C22×D4—18112no—
112C23×C4—3621no—
113C23×C4C2×D18111yescharacteristic
114AGL(1,ℤ/8)—18164no—
115C2×SD16—18124no—
116C2×D8—18124no—
117C2×M16C2×D181112yescomplemented
118C22×C8C2×D18112yescomplemented
119C24—36212no—
120C24—3621no—
121C24—3622no—
122C24—7248no—
123C24D72411yescharacteristic
124#41—36224no—
125#41—18116no—
126C2×Q8—3623no—
127C2×Q8—1816no—
128C2×D4—36212no—
129C2×D4—36212no—
130C2×D4—3626no—
131C2×D4—36224no—
132C2×D4—3623no—
133C2×D4—1816no—
134C2×D4—18112no—
135C2×D4—18116no—
136C22×C4—3621no—
137C22×C4—36212no—
138C22×C4D72416yes—
139C22×C4D72411yescharacteristic
140C22×C4C22×D18116yes—
141C22×C4C22×D18111yescharacteristic
142SD16—18132no—
143D8—18132no—
144M16C22×D181116yescomplemented
145C2×C8C22×D181112yescomplemented
146C23—24no—
147C23—3626no—
148C23—36216no—
149C23—36212no—
150C23—72448no—
151C23—7248no—
152C23—7244no—
153C23C2×D72416yes—
154C23C2×D72411yescharacteristic
155Q8—3626no—
156Q8—1814no—
157D4—36216no—
158D4—36216no—
159D4—36212no—
160D4—3626no—
161D4—1814no—
162D4—1818no—
163C2×C4—3626no—
164C2×C4—36216no—
165C2×C4C2×D72416yes—
166C2×C4C2×D72413yes—
167C2×C4C2×D724112yes—
168C2×C4C23×D18113yes—
169C2×C4C23×D18114yes—
170C8C23×D18118yescomplemented
171C22—212no—
172C22—3624no—
173C22—3628no—
174C22—72432no—
175C22—72424no—
176C22—72412no—
177C22#1915384414yesdirect product factor
178C22C22×D72414yes—
179C22C22×D72413yes—
180C4—3624no—
181C4C22×D72414yes—
182C4C22×D72413yes—
183C4C24×D18111yescharacteristic
184C2—24no—
185C2—7248no—
186C2—7244no—
187C2C2×#1915384416yesdirect product factor
188C2C23×D72411yescharacteristic
subgroupG/H
centerC23C2×D724
commutator, cosocleC4×C181C25
FrattiniC4C24×D181
FittingC22×M16×C181C2
socleC23×C181C2×D4
hypercenterC22×M16D181
hypocenterC181C22×AGL(1,ℤ/8)
LCSG ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#15668449C2semidirect
2. C23×D724C2semidirect
3. C2×#19153844C2direct (N × H)
4. C22×#15668438C2semidirect
5. C22×D1448C2semidirect
6. C22×M16×C181C2semidirect
9. C2×#15668449C22semidirect
10. C22×Dic362C22semidirect
14. C22×D724C22semidirect
15. C22×D724C22semidirect
18. #19153844C22direct (N × H)
19. C2×#15668438C22semidirect
20. C2×D1448C22semidirect
22. C2×M16×C181C22semidirect
23. C22×C8×C181C22semidirect
29. #15668449C23semidirect
31. C2×Dic362C23semidirect
37. C2×D724C23semidirect
38. C2×D724C23semidirect
42. #15668438C23semidirect
43. D1448C23semidirect
49. M16×C181C23semidirect
50. C2×C8×C181C23semidirect
58. Dic362C24semidirect
63. D724C24semidirect
75. C8×C181C24semidirect
77. C22×C181AGL(1,ℤ/8)semidirect
90. C2×C181C2×AGL(1,ℤ/8)semidirect
94. C181C22×AGL(1,ℤ/8)semidirect
101. C22×M16D181semidirect
117. C2×M16C2×D181semidirect
118. C22×C8C2×D181semidirect
144. M16C22×D181semidirect
145. C2×C8C22×D181semidirect
170. C8C23×D181semidirect
177. C22#19153844direct (N × H)
187. C2C2×#19153844direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#156684492
C23×D7242
C2×#191538442
C22×#156684382
C22×D14482
C22×M16×C1812
C22×AGL(1,ℤ/8)181