Finite Group Databasebeta

← C22×#19149700 #36368290

Subgroups — C22×#19149700 #36368290

Order 22144.

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×#14583827C211yesmaximal, characteristic, complemented
2C22×#14583825C211yesmaximal, characteristic, complemented
3C2×#19149700C2124yesmaximal, direct product factor
4C22×#14583816C211yesmaximal, characteristic, complemented
5C22×#14583815C211yesmaximal, characteristic, complemented
6C22×#14583811C211yesmaximal, characteristic, complemented
7C22×C8×D173C211yesmaximal, characteristic, complemented
8C22×SD16×C173C211yesmaximal, characteristic, complemented
9C2×#14583827—26no—
10C2×#14583827C22112yescomplemented
11C2×#14583825—26no—
12C2×#14583825C22112yescomplemented
13C22×Dic346C2211yescharacteristic
14C22×D692—21no—
15C22×D692C2211yescharacteristic
16C22×C22⋊D173—21no—
17C23×Dic173—21no—
18C22×C4×D173—21no—
19C22×C4×D173C2211yescharacteristic
20#19149700C22164yesdirect product factor
21C2×#14583816C22112yescomplemented
22C2×#14583815C22112yescomplemented
23C2×#14583811C22112yescomplemented
24C22×#13063436C2211yescharacteristic
25C2×C8×D173C22112yescomplemented
26C22×Q8×C173C2211yescharacteristic
27C22×D4×C173C2211yescharacteristic
28C2×SD16×C173C22112yescomplemented
29C22×C8×C173C2211yescharacteristic
30C23×D173—21no—
31C23×D173D411yescharacteristic
32#14583827—224no—
33#14583827C23116yescomplemented
34#14583825—224no—
35#14583825C23116yescomplemented
36C2×Dic346—23no—
37C2×Dic346C2316yes—
38C2×D692—212no—
39C2×D692—23no—
40C2×D692C2316yes—
41C2×C22⋊D173—212no—
42C22×Dic173—212no—
43C22×Dic173—21no—
44C22×Dic173D411yescharacteristic
45C2×C4×D173—212no—
46C2×C4×D173D416yes—
47C2×C4×D173C2316yes—
48#14583816C23116yescomplemented
49#14583815C23116yescomplemented
50#14583811C23116yescomplemented
51C2×#13063436C2316yes—
52C8×D173C23116yescomplemented
53C24×C173—21no—
54C2×Q8×C173—23no—
55C2×Q8×C173C2316yes—
56C2×D4×C173—23no—
57C2×D4×C173C2316yes—
58C22×C4×C173—21no—
59C22×C4×C173C2311yescharacteristic
60SD16×C173C23116yescomplemented
61C2×C8×C173C2316yes—
62C22×D173—24no—
63C22×D173—26no—
64C22×D173—44no—
65C22×D173C2×D416yes—
66Dic346—26no—
67Dic346C2414yes—
68D692—216no—
69D692—26no—
70D692C2414yes—
71C22⋊D173—216no—
72C2×Dic173—26no—
73C2×Dic173—216no—
74C2×Dic173C2×D416yes—
75C4×D173—216no—
76C4×D173C2×D4112yes—
77C4×D173C2414yes—
78#13063436C2414yes—
79C23×C173—26no—
80C23×C173—44no—
81C23×C173C2×D411yescharacteristic
82Q8×C173—26no—
83Q8×C173C2414yes—
84D4×C173—26no—
85D4×C173C2414yes—
86C2×C4×C173—26no—
87C2×C4×C173C2×D413yes—
88C2×C4×C173C2413yes—
89C8×C173C2414yes—
90C22×C173—24no—
91C22×C173—412no—
92C22×C173#13514yescomplemented
93C22×C173C22×D413yes—
94C2×D173—212no—
95C2×D173—24no—
96C2×D173—412no—
97C2×D173C22×D414yes—
98C4×C173—24no—
99C4×C173C22×D413yes—
100C4×C173C2511yescharacteristic
101Dic173—24no—
102Dic173C22×D414yes—
103C2×C173—44no—
104C2×C173C2×#13516yescomplemented
105C2×C173C23×D411yescharacteristic
106D173—24no—
107D173—44no—
108C173C22×#13511yescharacteristic, complemented, Sylow
109C22×#135—1731nomaximal, Sylow
110C22×#41—1731no—
111C22×#41—1731no—
112C2×#135—17324no—
113C22×Q16—1731no—
114C22×SD16—1731no—
115C22×SD16D17311yescharacteristic, complemented
116C22×D8—1731no—
117C23×C8—1731no—
118C2×#41—3466no—
119C2×#41—3466no—
120C2×#41—17312no—
121C2×#41—17312no—
122C22×Q8—1731no—
123C22×Q8C2×D17311yescharacteristic
124C22×D4—3461no—
125C22×D4—3461no—
126C22×D4—1731no—
127C22×D4C2×D17311yescharacteristic
128C23×C4—3461no—
129C23×C4—3461no—
130C23×C4—1731no—
131#135—17364no—
132C2×Q16—17312no—
133C2×SD16—17312no—
134C2×SD16C2×D173112yescomplemented
135C2×D8—17312no—
136C22×C8—1731no—
137C22×C8—17312no—
138C22×C8C2×D17311yescharacteristic
139C24—21no—
140C24—3461no—
141C24—1731no—
142#41—34624no—
143#41—34624no—
144#41—17316no—
145#41—17316no—
146C2×Q8—23no—
147C2×Q8—3463no—
148C2×Q8—1736no—
149C2×Q8C22×D17316yes—
150C2×D4—23no—
151C2×D4—34612no—
152C2×D4—34612no—
153C2×D4—3463no—
154C2×D4—1736no—
155C2×D4C22×D17316yes—
156C22×C4—21no—
157C22×C4—34612no—
158C22×C4—3461no—
159C22×C4—34612no—
160C22×C4—1731no—
161C22×C4—1736no—
162C22×C4—1736no—
163C22×C4C22×D17311yescharacteristic
164Q16—17316no—
165SD16—17316no—
166SD16C22×D173116yescomplemented
167D8—17316no—
168C2×C8—1736no—
169C2×C8—17316no—
170C2×C8C22×D17316yes—
171C23—26no—
172C23—44no—
173C23—3464no—
174C23—3466no—
175C23—1736no—
176C23—6924no—
177C23D4×D17311yescharacteristic
178Q8—26no—
179Q8—3466no—
180Q8—1734no—
181Q8C23×D17314yes—
182D4—26no—
183D4—34616no—
184D4—34616no—
185D4—3466no—
186D4—1734no—
187D4C23×D17314yes—
188C2×C4—26no—
189C2×C4—34616no—
190C2×C4—3466no—
191C2×C4—34616no—
192C2×C4—1736no—
193C2×C4—17312no—
194C2×C4—1734no—
195C2×C4D4×D17313yes—
196C2×C4C23×D17313yes—
197C8—1734no—
198C8C23×D17314yes—
199C22—24no—
200C22—412no—
201C22—34612no—
202C22—3464no—
203C22—1734no—
204C22—69212no—
205C22#1914970014yesdirect product factor
206C22C2×D4×D17313yes—
207C4—24no—
208C4—3464no—
209C4—1734no—
210C4C2×D4×D17313yes—
211C4C24×D17311yescharacteristic
212C2—44no—
213C2—3464no—
214C2—6924no—
215C2C2×#1914970016yesdirect product factor
216C2C22×D4×D17311yescharacteristic
subgroupG/H
centerC23D4×D173
commutator, cosocleC4×C173C25
FrattiniC4C24×D173
FittingC22×SD16×C173C2
socleC23×C173C2×D4
hypercenterC22×SD16D173
hypocenterC173C22×#135
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#14583827C2semidirect
2. C22×#14583825C2semidirect
3. C2×#19149700C2direct (N × H)
4. C22×#14583816C2semidirect
5. C22×#14583815C2semidirect
6. C22×#14583811C2semidirect
7. C22×C8×D173C2semidirect
8. C22×SD16×C173C2semidirect
10. C2×#14583827C22semidirect
12. C2×#14583825C22semidirect
20. #19149700C22direct (N × H)
21. C2×#14583816C22semidirect
22. C2×#14583815C22semidirect
23. C2×#14583811C22semidirect
25. C2×C8×D173C22semidirect
28. C2×SD16×C173C22semidirect
33. #14583827C23semidirect
35. #14583825C23semidirect
48. #14583816C23semidirect
49. #14583815C23semidirect
50. #14583811C23semidirect
52. C8×D173C23semidirect
60. SD16×C173C23semidirect
92. C22×C173#135semidirect
104. C2×C173C2×#135semidirect
108. C173C22×#135semidirect
115. C22×SD16D173semidirect
134. C2×SD16C2×D173semidirect
166. SD16C22×D173semidirect
205. C22#19149700direct (N × H)
215. C2C2×#19149700direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145838272
C22×#145838252
C2×#191497002
C22×#145838162
C22×#145838152
C22×#145838112
C22×C8×D1732
C22×SD16×C1732
C22×#135173