Finite Group Databasebeta

← C22×#19149699 #36368289

Subgroups — C22×#19149699 #36368289

Order 22144.

1504 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1504 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×#14583826C211yesmaximal, characteristic, complemented
2C2×#19149699C2124yesmaximal, direct product factor
3C22×#14583816C211yesmaximal, characteristic, complemented
4C22×#14583814C211yesmaximal, characteristic, complemented
5C22×#14583813C211yesmaximal, characteristic, complemented
6C22×#14583811C211yesmaximal, characteristic, complemented
7C23×#13063436C211yesmaximal, characteristic, complemented
8C22×#41×C173C211yesmaximal, characteristic, complemented
9C2×#14583826—26no—
10C2×#14583826C22112yescomplemented
11C22×Dic346C2211yescharacteristic
12C22×D692C2211yescharacteristic
13C22×C22⋊D173—21no—
14C22×C4×D173—21no—
15#19149699C22164yesdirect product factor
16C2×#14583816C22112yescomplemented
17C2×#14583814C22112yescomplemented
18C2×#14583813C22112yescomplemented
19C2×#14583811C22112yescomplemented
20C22×#13063436C2211yescharacteristic
21C22×#13063436C2211yescharacteristic
22C22×#13063436C22112yescomplemented
23C2×#41×C173—26no—
24C2×#41×C173C22112yescomplemented
25C22×Q8×C173C2211yescharacteristic
26C22×D4×C173—21no—
27C22×D4×C173C2211yescharacteristic
28C23×C4×C173—21no—
29C23×C4×C173C2211yescharacteristic
30C23×D173—21no—
31#14583826—224no—
32#14583826C23116yescomplemented
33C2×Dic346—23no—
34C2×Dic346C2316yes—
35C2×D692—23no—
36C2×D692C2316yes—
37C2×C22⋊D173—212no—
38C22×Dic173—21no—
39C2×C4×D173—212no—
40#14583816C23116yescomplemented
41#14583814C23116yescomplemented
42#14583813C23116yescomplemented
43#14583811C23116yescomplemented
44C2×#13063436C2316yes—
45C2×#13063436C2316yes—
46C2×#13063436C23116yescomplemented
47C24×C173—21no—
48C24×C173D411yescharacteristic
49#41×C173—224no—
50#41×C173C23116yescomplemented
51C2×Q8×C173—23no—
52C2×Q8×C173C2316yes—
53C2×D4×C173—212no—
54C2×D4×C173—23no—
55C2×D4×C173C2316yes—
56C22×C4×C173—21no—
57C22×C4×C173—212no—
58C22×C4×C173D416yes—
59C22×C4×C173D411yescharacteristic
60C22×C4×C173C2311yescharacteristic
61C22×C4×C173C2316yes—
62C22×D173—26no—
63C22×D173—44no—
64Dic346—26no—
65Dic346C2414yes—
66D692—26no—
67D692C2414yes—
68C22⋊D173—216no—
69C2×Dic173—26no—
70C4×D173—216no—
71#13063436C2414yes—
72#13063436C2414yes—
73C23×C173—24no—
74C23×C173—26no—
75C23×C173—44no—
76C23×C173C2×D416yes—
77C23×C173C2×D411yescharacteristic
78Q8×C173—26no—
79Q8×C173C2414yes—
80D4×C173—216no—
81D4×C173—26no—
82D4×C173C2414yes—
83C2×C4×C173—26no—
84C2×C4×C173—216no—
85C2×C4×C173C2×D416yes—
86C2×C4×C173C2×D413yes—
87C2×C4×C173C2×D4112yes—
88C2×C4×C173C2413yes—
89C2×C4×C173C2414yes—
90C22×C173—212no—
91C22×C173—24no—
92C22×C173—412no—
93C22×C173#13514yescomplemented
94C22×C173C22×D414yes—
95C22×C173C22×D413yes—
96C2×D173—24no—
97C2×D173—412no—
98C4×C173—24no—
99C4×C173C22×D414yes—
100C4×C173C22×D413yes—
101C4×C173C2511yescharacteristic
102Dic173—24no—
103C2×C173—24no—
104C2×C173—44no—
105C2×C173C2×#13516yescomplemented
106C2×C173C23×D411yescharacteristic
107D173—44no—
108C173C22×#13511yescharacteristic, complemented, Sylow
109C22×#135—1731nomaximal, Sylow
110C22×#41—1731no—
111C22×#41D17311yescharacteristic, complemented
112C2×#135—17324no—
113C22×Q16—1731no—
114C22×SD16—1731no—
115C22×SD16—1731no—
116C22×D8—1731no—
117C23×C8—1731no—
118C2×#41—26no—
119C2×#41—3466no—
120C2×#41—17312no—
121C2×#41C2×D173112yescomplemented
122C22×Q8—1731no—
123C22×Q8C2×D17311yescharacteristic
124C22×D4—21no—
125C22×D4—3461no—
126C22×D4—1731no—
127C22×D4C2×D17311yescharacteristic
128C23×C4—21no—
129C23×C4—3461no—
130C23×C4C2×D17311yescharacteristic
131#135—17364no—
132C2×Q16—17312no—
133C2×SD16—17312no—
134C2×SD16—17312no—
135C2×D8—17312no—
136C22×C8—1731no—
137C22×C8—1731no—
138C22×C8—17312no—
139C24—21no—
140C24—3461no—
141C24C22⋊D17311yescharacteristic
142#41—224no—
143#41—34624no—
144#41—17316no—
145#41C22×D173116yescomplemented
146C2×Q8—23no—
147C2×Q8—3463no—
148C2×Q8—1736no—
149C2×Q8C22×D17316yes—
150C2×D4—212no—
151C2×D4—23no—
152C2×D4—34612no—
153C2×D4—3463no—
154C2×D4—1736no—
155C2×D4C22×D17316yes—
156C22×C4—21no—
157C22×C4—212no—
158C22×C4—3461no—
159C22×C4—34612no—
160C22×C4C22⋊D17316yes—
161C22×C4C22⋊D17311yescharacteristic
162C22×C4C22×D17311yescharacteristic
163C22×C4C22×D17316yes—
164Q16—17316no—
165SD16—17316no—
166SD16—17316no—
167D8—17316no—
168C2×C8—1736no—
169C2×C8—1736no—
170C2×C8—17316no—
171C23—24no—
172C23—26no—
173C23—44no—
174C23—3466no—
175C23—6924no—
176C23C2×C22⋊D17316yes—
177C23C2×C22⋊D17311yescharacteristic
178Q8—26no—
179Q8—3466no—
180Q8—1734no—
181Q8C23×D17314yes—
182D4—216no—
183D4—26no—
184D4—34616no—
185D4—3466no—
186D4—1734no—
187D4C23×D17314yes—
188C2×C4—26no—
189C2×C4—216no—
190C2×C4—3466no—
191C2×C4—34616no—
192C2×C4C2×C22⋊D17316yes—
193C2×C4C2×C22⋊D17313yes—
194C2×C4C2×C22⋊D173112yes—
195C2×C4C23×D17313yes—
196C2×C4C23×D17314yes—
197C8—1734no—
198C8—1734no—
199C22—212no—
200C22—24no—
201C22—412no—
202C22—3464no—
203C22—69212no—
204C22#1914969914yesdirect product factor
205C22C22×C22⋊D17314yes—
206C22C22×C22⋊D17313yes—
207C4—24no—
208C4—3464no—
209C4C22×C22⋊D17314yes—
210C4C22×C22⋊D17313yes—
211C4C24×D17311yescharacteristic
212C2—24no—
213C2—44no—
214C2—6924no—
215C2C2×#1914969916yesdirect product factor
216C2C23×C22⋊D17311yescharacteristic
subgroupG/H
centerC22×C4C22⋊D173
commutator, cosocleC4×C173C25
FrattiniC4C24×D173
FittingC22×#41×C173C2
socleC23×C173C2×D4
hypercenterC22×#41D173
hypocenterC173C22×#135
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×#14583826C2semidirect
2. C2×#19149699C2direct (N × H)
3. C22×#14583816C2semidirect
4. C22×#14583814C2semidirect
5. C22×#14583813C2semidirect
6. C22×#14583811C2semidirect
7. C23×#13063436C2semidirect
8. C22×#41×C173C2semidirect
10. C2×#14583826C22semidirect
15. #19149699C22direct (N × H)
16. C2×#14583816C22semidirect
17. C2×#14583814C22semidirect
18. C2×#14583813C22semidirect
19. C2×#14583811C22semidirect
22. C22×#13063436C22semidirect
24. C2×#41×C173C22semidirect
32. #14583826C23semidirect
40. #14583816C23semidirect
41. #14583814C23semidirect
42. #14583813C23semidirect
43. #14583811C23semidirect
46. C2×#13063436C23semidirect
50. #41×C173C23semidirect
93. C22×C173#135semidirect
105. C2×C173C2×#135semidirect
108. C173C22×#135semidirect
111. C22×#41D173semidirect
121. C2×#41C2×D173semidirect
145. #41C22×D173semidirect
204. C22#19149699direct (N × H)
215. C2C2×#19149699direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145838262
C2×#191496992
C22×#145838162
C22×#145838142
C22×#145838132
C22×#145838112
C23×#130634362
C22×#41×C1732
C22×#135173