Finite Group Databasebeta

← C22×#19153843 #37548455

Subgroups — C22×#19153843 #37548455

Order 23168.

1720 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 525 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×#15668450C211yesmaximal, characteristic, complemented
2C22×D4×D181C211yesmaximal, characteristic, complemented
3C2×#19153843C2124yesmaximal, direct product factor
4C22×#15668436C211yesmaximal, characteristic, complemented
5C22×D1448C211yesmaximal, characteristic, complemented
6C22×#15668434C211yesmaximal, characteristic, complemented
7C22×#15668432C211yesmaximal, characteristic, complemented
8C22×SD16×C181C211yesmaximal, characteristic, complemented
9C24×D181—21no—
10C2×#15668450—26no—
11C2×#15668450C22112yescomplemented
12C22×D724—21no—
13C22×D724C2211yescharacteristic
14C22×D724C2211yescharacteristic, complemented
15C22×C22⋊D181—21no—
16C2×D4×D181—26no—
17C2×D4×D181C22112yescomplemented
18C22×C4×D181—21no—
19C22×C4×D181C2211yescharacteristic
20#19153843C22164yesdirect product factor
21C2×#15668436C22112yescomplemented
22C2×D1448C22112yescomplemented
23C2×#15668434C22112yescomplemented
24C2×#15668432C22112yescomplemented
25C22×#13089678C2211yescharacteristic, complemented
26C22×Q8×C181C2211yescharacteristic, complemented
27C22×D4×C181C2211yescharacteristic
28C2×SD16×C181C22112yescomplemented
29C22×C8×C181C2211yescharacteristic, complemented
30C23×D181—212no—
31C23×D181—21no—
32C23×D181—21no—
33C23×D181—48no—
34C23×D181D411yescharacteristic
35#15668450—224no—
36#15668450C23116yescomplemented
37C2×D724—212no—
38C2×D724—23no—
39C2×D724—23no—
40C2×D724C2316yes—
41C2×D724C2316yescomplemented
42C2×C22⋊D181—212no—
43D4×D181—224no—
44D4×D181C23116yescomplemented
45C22×Dic181D411yescharacteristic
46C2×C4×D181—212no—
47C2×C4×D181D416yes—
48C2×C4×D181C2316yes—
49#15668436C23116yescomplemented
50D1448C23116yescomplemented
51#15668434C23116yescomplemented
52#15668432C23116yescomplemented
53C2×#13089678C2316yescomplemented
54C24×C181—21no—
55C2×Q8×C181—23no—
56C2×Q8×C181C2316yescomplemented
57C2×D4×C181—23no—
58C2×D4×C181C2316yes—
59C22×C4×C181—21no—
60C22×C4×C181C2311yescharacteristic
61SD16×C181C23116yescomplemented
62C2×C8×C181C2316yescomplemented
63C22×D181—24no—
64C22×D181—26no—
65C22×D181—26no—
66C22×D181—216no—
67C22×D181—448no—
68C22×D181—44no—
69C22×D181—44no—
70C22×D181C2×D416yes—
71D724—216no—
72D724—26no—
73D724—26no—
74D724C2414yes—
75D724C2414yescomplemented
76C22⋊D181—216no—
77C2×Dic181C2×D416yes—
78C4×D181—216no—
79C4×D181C2×D4112yes—
80C4×D181C2414yes—
81#13089678C2414yescomplemented
82C23×C181—26no—
83C23×C181—44no—
84C23×C181C2×D411yescharacteristic
85Q8×C181—26no—
86Q8×C181C2414yescomplemented
87D4×C181—26no—
88D4×C181C2414yes—
89C2×C4×C181—26no—
90C2×C4×C181C2×D413yes—
91C2×C4×C181C2413yes—
92C8×C181C2414yescomplemented
93C22×C181—24no—
94C22×C181—412no—
95C22×C181AGL(1,ℤ/8)14yescomplemented
96C22×C181C22×D413yes—
97C2×D181—212no—
98C2×D181—24no—
99C2×D181—24no—
100C2×D181—432no—
101C2×D181—412no—
102C2×D181—412no—
103C2×D181C22×D414yes—
104C4×C181—24no—
105C4×C181C22×D413yes—
106C4×C181C2511yescharacteristic
107Dic181C22×D414yes—
108C2×C181—44no—
109C2×C181C2×AGL(1,ℤ/8)16yescomplemented
110C2×C181C23×D411yescharacteristic
111D181—24no—
112D181—44no—
113D181—44no—
114C181C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
115C22×AGL(1,ℤ/8)—1811nomaximal, Sylow
116C22×#41—1811no—
117C23×D4—1811no—
118C2×AGL(1,ℤ/8)—18124no—
119C22×SD16—1811no—
120C22×SD16D18111yescharacteristic, complemented
121C22×D8—1811no—
122C22×D8—1811no—
123C22×M16—1811no—
124C25—3621no—
125C2×#41—3626no—
126C2×#41—18112no—
127C22×Q8C2×D18111yescharacteristic, complemented
128C22×D4—3621no—
129C22×D4—3626no—
130C22×D4—3621no—
131C22×D4—18112no—
132C22×D4—1811no—
133C22×D4—1811no—
134C22×D4C2×D18111yescharacteristic
135C23×C4—3621no—
136C23×C4—1811no—
137AGL(1,ℤ/8)—18164no—
138C2×SD16—18112no—
139C2×SD16C2×D181112yescomplemented
140C2×D8—18112no—
141C2×D8—18112no—
142C2×M16—18112no—
143C22×C8—1811no—
144C22×C8C2×D18111yescharacteristic, complemented
145C24—21no—
146C24—36212no—
147C24—3621no—
148C24—3621no—
149C24—1811no—
150C24—7248no—
151#41—36224no—
152#41—18116no—
153C2×Q8—23no—
154C2×Q8C22×D18116yescomplemented
155C2×D4—23no—
156C2×D4—36212no—
157C2×D4—36224no—
158C2×D4—36212no—
159C2×D4—3623no—
160C2×D4—3623no—
161C2×D4—18116no—
162C2×D4—1816no—
163C2×D4—1816no—
164C2×D4C22×D18116yes—
165C22×C4—21no—
166C22×C4—36212no—
167C22×C4—1811no—
168C22×C4—1816no—
169C22×C4—1816no—
170C22×C4C22×D18111yescharacteristic
171SD16—18116no—
172SD16C22×D181116yescomplemented
173D8—18116no—
174D8—18116no—
175M16—18116no—
176C2×C8—1816no—
177C2×C8C22×D18116yescomplemented
178C23—26no—
179C23—44no—
180C23—36216no—
181C23—3624no—
182C23—3626no—
183C23—3626no—
184C23—1816no—
185C23—72448no—
186C23—7244no—
187C23—7244no—
188C23D4×D18111yescharacteristic
189Q8—26no—
190Q8C23×D18114yescomplemented
191D4—26no—
192D4—36216no—
193D4—36216no—
194D4—3626no—
195D4—3626no—
196D4—1814no—
197D4—1814no—
198D4C23×D18114yes—
199C2×C4—26no—
200C2×C4—36216no—
201C2×C4—1816no—
202C2×C4—18112no—
203C2×C4—1814no—
204C2×C4D4×D18113yes—
205C2×C4C23×D18113yes—
206C8—1814no—
207C8C23×D18114yescomplemented
208C22—24no—
209C22—412no—
210C22—36212no—
211C22—3624no—
212C22—3624no—
213C22—1814no—
214C22—72432no—
215C22—72412no—
216C22—72412no—
217C22#1915384314yesdirect product factor
218C22C2×D4×D18113yes—
219C4—24no—
220C4—1814no—
221C4C2×D4×D18113yes—
222C4C24×D18111yescharacteristic
223C2—44no—
224C2—3624no—
225C2—7244no—
226C2—7244no—
227C2C2×#1915384316yesdirect product factor
228C2C22×D4×D18111yescharacteristic
subgroupG/H
centerC23D4×D181
commutator, cosocleC4×C181C25
FrattiniC4C24×D181
FittingC22×SD16×C181C2
socleC23×C181C2×D4
hypercenterC22×SD16D181
hypocenterC181C22×AGL(1,ℤ/8)
LCSG ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#15668450C2semidirect
2. C22×D4×D181C2semidirect
3. C2×#19153843C2direct (N × H)
4. C22×#15668436C2semidirect
5. C22×D1448C2semidirect
6. C22×#15668434C2semidirect
7. C22×#15668432C2semidirect
8. C22×SD16×C181C2semidirect
11. C2×#15668450C22semidirect
14. C22×D724C22semidirect
17. C2×D4×D181C22semidirect
20. #19153843C22direct (N × H)
21. C2×#15668436C22semidirect
22. C2×D1448C22semidirect
23. C2×#15668434C22semidirect
24. C2×#15668432C22semidirect
25. C22×#13089678C22semidirect
26. C22×Q8×C181C22semidirect
28. C2×SD16×C181C22semidirect
29. C22×C8×C181C22semidirect
36. #15668450C23semidirect
41. C2×D724C23semidirect
44. D4×D181C23semidirect
49. #15668436C23semidirect
50. D1448C23semidirect
51. #15668434C23semidirect
52. #15668432C23semidirect
53. C2×#13089678C23semidirect
56. C2×Q8×C181C23semidirect
61. SD16×C181C23semidirect
62. C2×C8×C181C23semidirect
75. D724C24semidirect
81. #13089678C24semidirect
86. Q8×C181C24semidirect
92. C8×C181C24semidirect
95. C22×C181AGL(1,ℤ/8)semidirect
109. C2×C181C2×AGL(1,ℤ/8)semidirect
114. C181C22×AGL(1,ℤ/8)semidirect
120. C22×SD16D181semidirect
127. C22×Q8C2×D181semidirect
139. C2×SD16C2×D181semidirect
144. C22×C8C2×D181semidirect
154. C2×Q8C22×D181semidirect
172. SD16C22×D181semidirect
177. C2×C8C22×D181semidirect
190. Q8C23×D181semidirect
207. C8C23×D181semidirect
217. C22#19153843direct (N × H)
227. C2C2×#19153843direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156684502
C22×D4×D1812
C2×#191538432
C22×#156684362
C22×D14482
C22×#156684342
C22×#156684322
C22×SD16×C1812
C22×AGL(1,ℤ/8)181