Finite Group Databasebeta

← C22×#19153837 #37548448

Subgroups — C22×#19153837 #37548448

Order 23168.

1504 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

normal subgroups only

This group has 1504 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×#15668448C211yesmaximal, characteristic, complemented
3C2×#19153837C2124yesmaximal, direct product factor
4C22×#15668439C211yesmaximal, characteristic, complemented
5C22×#15668438C211yesmaximal, characteristic, complemented
6C22×#15668434C211yesmaximal, characteristic, complemented
7C22×C8×D181C211yesmaximal, characteristic, complemented
8C22×SD16×C181C211yesmaximal, characteristic, complemented
9C2×#15668450—26no—
10C2×#15668450C22112yescomplemented
11C2×#15668448—26no—
12C2×#15668448C22112yescomplemented
13C22×Dic362C2211yescharacteristic
14C22×D724—21no—
15C22×D724C2211yescharacteristic
16C22×C22⋊D181—21no—
17C23×Dic181—21no—
18C22×C4×D181—21no—
19C22×C4×D181C2211yescharacteristic
20#19153837C22164yesdirect product factor
21C2×#15668439C22112yescomplemented
22C2×#15668438C22112yescomplemented
23C2×#15668434C22112yescomplemented
24C22×#13089678C2211yescharacteristic
25C2×C8×D181C22112yescomplemented
26C22×Q8×C181C2211yescharacteristic
27C22×D4×C181C2211yescharacteristic
28C2×SD16×C181C22112yescomplemented
29C22×C8×C181C2211yescharacteristic
30C23×D181—21no—
31C23×D181D411yescharacteristic
32#15668450—224no—
33#15668450C23116yescomplemented
34#15668448—224no—
35#15668448C23116yescomplemented
36C2×Dic362—23no—
37C2×Dic362C2316yes—
38C2×D724—212no—
39C2×D724—23no—
40C2×D724C2316yes—
41C2×C22⋊D181—212no—
42C22×Dic181—212no—
43C22×Dic181—21no—
44C22×Dic181D411yescharacteristic
45C2×C4×D181—212no—
46C2×C4×D181D416yes—
47C2×C4×D181C2316yes—
48#15668439C23116yescomplemented
49#15668438C23116yescomplemented
50#15668434C23116yescomplemented
51C2×#13089678C2316yes—
52C8×D181C23116yescomplemented
53C24×C181—21no—
54C2×Q8×C181—23no—
55C2×Q8×C181C2316yes—
56C2×D4×C181—23no—
57C2×D4×C181C2316yes—
58C22×C4×C181—21no—
59C22×C4×C181C2311yescharacteristic
60SD16×C181C23116yescomplemented
61C2×C8×C181C2316yes—
62C22×D181—24no—
63C22×D181—26no—
64C22×D181—44no—
65C22×D181C2×D416yes—
66Dic362—26no—
67Dic362C2414yes—
68D724—216no—
69D724—26no—
70D724C2414yes—
71C22⋊D181—216no—
72C2×Dic181—26no—
73C2×Dic181—216no—
74C2×Dic181C2×D416yes—
75C4×D181—216no—
76C4×D181C2×D4112yes—
77C4×D181C2414yes—
78#13089678C2414yes—
79C23×C181—26no—
80C23×C181—44no—
81C23×C181C2×D411yescharacteristic
82Q8×C181—26no—
83Q8×C181C2414yes—
84D4×C181—26no—
85D4×C181C2414yes—
86C2×C4×C181—26no—
87C2×C4×C181C2×D413yes—
88C2×C4×C181C2413yes—
89C8×C181C2414yes—
90C22×C181—24no—
91C22×C181—412no—
92C22×C181#13514yescomplemented
93C22×C181C22×D413yes—
94C2×D181—212no—
95C2×D181—24no—
96C2×D181—412no—
97C2×D181C22×D414yes—
98C4×C181—24no—
99C4×C181C22×D413yes—
100C4×C181C2511yescharacteristic
101Dic181—24no—
102Dic181C22×D414yes—
103C2×C181—44no—
104C2×C181C2×#13516yescomplemented
105C2×C181C23×D411yescharacteristic
106D181—24no—
107D181—44no—
108C181C22×#13511yescharacteristic, complemented, Sylow
109C22×#135—1811nomaximal, Sylow
110C22×#41—1811no—
111C22×#41—1811no—
112C2×#135—18124no—
113C22×Q16—1811no—
114C22×SD16—1811no—
115C22×SD16D18111yescharacteristic, complemented
116C22×D8—1811no—
117C23×C8—1811no—
118C2×#41—3626no—
119C2×#41—3626no—
120C2×#41—18112no—
121C2×#41—18112no—
122C22×Q8—1811no—
123C22×Q8C2×D18111yescharacteristic
124C22×D4—3621no—
125C22×D4—3621no—
126C22×D4—1811no—
127C22×D4C2×D18111yescharacteristic
128C23×C4—3621no—
129C23×C4—3621no—
130C23×C4—1811no—
131#135—18164no—
132C2×Q16—18112no—
133C2×SD16—18112no—
134C2×SD16C2×D181112yescomplemented
135C2×D8—18112no—
136C22×C8—1811no—
137C22×C8—18112no—
138C22×C8C2×D18111yescharacteristic
139C24—21no—
140C24—3621no—
141C24—1811no—
142#41—36224no—
143#41—36224no—
144#41—18116no—
145#41—18116no—
146C2×Q8—23no—
147C2×Q8—3623no—
148C2×Q8—1816no—
149C2×Q8C22×D18116yes—
150C2×D4—23no—
151C2×D4—36212no—
152C2×D4—36212no—
153C2×D4—3623no—
154C2×D4—1816no—
155C2×D4C22×D18116yes—
156C22×C4—21no—
157C22×C4—36212no—
158C22×C4—3621no—
159C22×C4—36212no—
160C22×C4—1811no—
161C22×C4—1816no—
162C22×C4—1816no—
163C22×C4C22×D18111yescharacteristic
164Q16—18116no—
165SD16—18116no—
166SD16C22×D181116yescomplemented
167D8—18116no—
168C2×C8—1816no—
169C2×C8—18116no—
170C2×C8C22×D18116yes—
171C23—26no—
172C23—44no—
173C23—3624no—
174C23—3626no—
175C23—1816no—
176C23—7244no—
177C23D4×D18111yescharacteristic
178Q8—26no—
179Q8—3626no—
180Q8—1814no—
181Q8C23×D18114yes—
182D4—26no—
183D4—36216no—
184D4—36216no—
185D4—3626no—
186D4—1814no—
187D4C23×D18114yes—
188C2×C4—26no—
189C2×C4—36216no—
190C2×C4—3626no—
191C2×C4—36216no—
192C2×C4—1816no—
193C2×C4—18112no—
194C2×C4—1814no—
195C2×C4D4×D18113yes—
196C2×C4C23×D18113yes—
197C8—1814no—
198C8C23×D18114yes—
199C22—24no—
200C22—412no—
201C22—36212no—
202C22—3624no—
203C22—1814no—
204C22—72412no—
205C22#1915383714yesdirect product factor
206C22C2×D4×D18113yes—
207C4—24no—
208C4—3624no—
209C4—1814no—
210C4C2×D4×D18113yes—
211C4C24×D18111yescharacteristic
212C2—44no—
213C2—3624no—
214C2—7244no—
215C2C2×#1915383716yesdirect product factor
216C2C22×D4×D18111yescharacteristic
subgroupG/H
centerC23D4×D181
commutator, cosocleC4×C181C25
FrattiniC4C24×D181
FittingC22×SD16×C181C2
socleC23×C181C2×D4
hypercenterC22×SD16D181
hypocenterC181C22×#135
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×#15668448C2semidirect
3. C2×#19153837C2direct (N × H)
4. C22×#15668439C2semidirect
5. C22×#15668438C2semidirect
6. C22×#15668434C2semidirect
7. C22×C8×D181C2semidirect
8. C22×SD16×C181C2semidirect
10. C2×#15668450C22semidirect
12. C2×#15668448C22semidirect
20. #19153837C22direct (N × H)
21. C2×#15668439C22semidirect
22. C2×#15668438C22semidirect
23. C2×#15668434C22semidirect
25. C2×C8×D181C22semidirect
28. C2×SD16×C181C22semidirect
33. #15668450C23semidirect
35. #15668448C23semidirect
48. #15668439C23semidirect
49. #15668438C23semidirect
50. #15668434C23semidirect
52. C8×D181C23semidirect
60. SD16×C181C23semidirect
92. C22×C181#135semidirect
104. C2×C181C2×#135semidirect
108. C181C22×#135semidirect
115. C22×SD16D181semidirect
134. C2×SD16C2×D181semidirect
166. SD16C22×D181semidirect
205. C22#19153837direct (N × H)
215. C2C2×#19153837direct (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×#156684482
C2×#191538372
C22×#156684392
C22×#156684382
C22×#156684342
C22×C8×D1812
C22×SD16×C1812
C22×#135181