Finite Group Databasebeta

← C22×#22548736 #44565092

Subgroups — C22×#22548736 #44565092

Order 32128.

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×#16860811C211yesmaximal, characteristic, complemented
2C22×#16860809C211yesmaximal, characteristic, complemented
3C2×#22548736C2124yesmaximal, direct product factor
4C22×#16860800C211yesmaximal, characteristic, complemented
5C22×#16860799C211yesmaximal, characteristic, complemented
6C22×#16860795C211yesmaximal, characteristic, complemented
7C22×C8×D251C211yesmaximal, characteristic, complemented
8C22×SD16×C251C211yesmaximal, characteristic, complemented
9C2×#16860811—26no—
10C2×#16860811C22112yescomplemented
11C2×#16860809—26no—
12C2×#16860809C22112yescomplemented
13C22×Dic502C2211yescharacteristic
14C22×D1004—21no—
15C22×D1004C2211yescharacteristic
16C22×C22⋊D251—21no—
17C23×Dic251—21no—
18C22×C4×D251—21no—
19C22×C4×D251C2211yescharacteristic
20#22548736C22164yesdirect product factor
21C2×#16860800C22112yescomplemented
22C2×#16860799C22112yescomplemented
23C2×#16860795C22112yescomplemented
24C22×C251⋊C8C2211yescharacteristic
25C2×C8×D251C22112yescomplemented
26C22×Q8×C251C2211yescharacteristic
27C22×D4×C251C2211yescharacteristic
28C2×SD16×C251C22112yescomplemented
29C22×C8×C251C2211yescharacteristic
30C23×D251—21no—
31C23×D251D411yescharacteristic
32#16860811—224no—
33#16860811C23116yescomplemented
34#16860809—224no—
35#16860809C23116yescomplemented
36C2×Dic502—23no—
37C2×Dic502C2316yes—
38C2×D1004—212no—
39C2×D1004—23no—
40C2×D1004C2316yes—
41C2×C22⋊D251—212no—
42C22×Dic251—212no—
43C22×Dic251—21no—
44C22×Dic251D411yescharacteristic
45C2×C4×D251—212no—
46C2×C4×D251D416yes—
47C2×C4×D251C2316yes—
48#16860800C23116yescomplemented
49#16860799C23116yescomplemented
50#16860795C23116yescomplemented
51C2×C251⋊C8C2316yes—
52C8×D251C23116yescomplemented
53C24×C251—21no—
54C2×Q8×C251—23no—
55C2×Q8×C251C2316yes—
56C2×D4×C251—23no—
57C2×D4×C251C2316yes—
58C22×C4×C251—21no—
59C22×C4×C251C2311yescharacteristic
60SD16×C251C23116yescomplemented
61C2×C8×C251C2316yes—
62C22×D251—24no—
63C22×D251—26no—
64C22×D251—44no—
65C22×D251C2×D416yes—
66Dic502—26no—
67Dic502C2414yes—
68D1004—216no—
69D1004—26no—
70D1004C2414yes—
71C22⋊D251—216no—
72C2×Dic251—26no—
73C2×Dic251—216no—
74C2×Dic251C2×D416yes—
75C4×D251—216no—
76C4×D251C2×D4112yes—
77C4×D251C2414yes—
78C251⋊C8C2414yes—
79C23×C251—26no—
80C23×C251—44no—
81C23×C251C2×D411yescharacteristic
82Q8×C251—26no—
83Q8×C251C2414yes—
84D4×C251—26no—
85D4×C251C2414yes—
86C2×C4×C251—26no—
87C2×C4×C251C2×D413yes—
88C2×C4×C251C2413yes—
89C8×C251C2414yes—
90C22×C251—24no—
91C22×C251—412no—
92C22×C251#13514yescomplemented
93C22×C251C22×D413yes—
94C2×D251—212no—
95C2×D251—24no—
96C2×D251—412no—
97C2×D251C22×D414yes—
98C4×C251—24no—
99C4×C251C22×D413yes—
100C4×C251C2511yescharacteristic
101Dic251—24no—
102Dic251C22×D414yes—
103C2×C251—44no—
104C2×C251C2×#13516yescomplemented
105C2×C251C23×D411yescharacteristic
106D251—24no—
107D251—44no—
108C251C22×#13511yescharacteristic, complemented, Sylow
109C22×#135—2511nomaximal, Sylow
110C22×#41—2511no—
111C22×#41—2511no—
112C2×#135—25124no—
113C22×Q16—2511no—
114C22×SD16—2511no—
115C22×SD16D25111yescharacteristic, complemented
116C22×D8—2511no—
117C23×C8—2511no—
118C2×#41—5026no—
119C2×#41—5026no—
120C2×#41—25112no—
121C2×#41—25112no—
122C22×Q8—2511no—
123C22×Q8C2×D25111yescharacteristic
124C22×D4—5021no—
125C22×D4—5021no—
126C22×D4—2511no—
127C22×D4C2×D25111yescharacteristic
128C23×C4—5021no—
129C23×C4—5021no—
130C23×C4—2511no—
131#135—25164no—
132C2×Q16—25112no—
133C2×SD16—25112no—
134C2×SD16C2×D251112yescomplemented
135C2×D8—25112no—
136C22×C8—2511no—
137C22×C8—25112no—
138C22×C8C2×D25111yescharacteristic
139C24—21no—
140C24—5021no—
141C24—2511no—
142#41—50224no—
143#41—50224no—
144#41—25116no—
145#41—25116no—
146C2×Q8—23no—
147C2×Q8—5023no—
148C2×Q8—2516no—
149C2×Q8C22×D25116yes—
150C2×D4—23no—
151C2×D4—50212no—
152C2×D4—50212no—
153C2×D4—5023no—
154C2×D4—2516no—
155C2×D4C22×D25116yes—
156C22×C4—21no—
157C22×C4—50212no—
158C22×C4—5021no—
159C22×C4—50212no—
160C22×C4—2511no—
161C22×C4—2516no—
162C22×C4—2516no—
163C22×C4C22×D25111yescharacteristic
164Q16—25116no—
165SD16—25116no—
166SD16C22×D251116yescomplemented
167D8—25116no—
168C2×C8—2516no—
169C2×C8—25116no—
170C2×C8C22×D25116yes—
171C23—26no—
172C23—44no—
173C23—10044no—
174C23—5026no—
175C23—5024no—
176C23—2516no—
177C23D4×D25111yescharacteristic
178Q8—26no—
179Q8—5026no—
180Q8—2514no—
181Q8C23×D25114yes—
182D4—26no—
183D4—50216no—
184D4—50216no—
185D4—5026no—
186D4—2514no—
187D4C23×D25114yes—
188C2×C4—26no—
189C2×C4—50216no—
190C2×C4—5026no—
191C2×C4—50216no—
192C2×C4—2516no—
193C2×C4—25112no—
194C2×C4—2514no—
195C2×C4D4×D25113yes—
196C2×C4C23×D25113yes—
197C8—2514no—
198C8C23×D25114yes—
199C22—24no—
200C22—412no—
201C22—100412no—
202C22—5024no—
203C22—50212no—
204C22—2514no—
205C22#2254873614yesdirect product factor
206C22C2×D4×D25113yes—
207C4—24no—
208C4—5024no—
209C4—2514no—
210C4C2×D4×D25113yes—
211C4C24×D25111yescharacteristic
212C2—44no—
213C2—10044no—
214C2—5024no—
215C2C2×#2254873616yesdirect product factor
216C2C22×D4×D25111yescharacteristic
subgroupG/H
centerC23D4×D251
commutator, cosocleC4×C251C25
FrattiniC4C24×D251
FittingC22×SD16×C251C2
socleC23×C251C2×D4
hypercenterC22×SD16D251
hypocenterC251C22×#135
LCSG ▷ C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#16860811C2semidirect
2. C22×#16860809C2semidirect
3. C2×#22548736C2direct (N × H)
4. C22×#16860800C2semidirect
5. C22×#16860799C2semidirect
6. C22×#16860795C2semidirect
7. C22×C8×D251C2semidirect
8. C22×SD16×C251C2semidirect
10. C2×#16860811C22semidirect
12. C2×#16860809C22semidirect
20. #22548736C22direct (N × H)
21. C2×#16860800C22semidirect
22. C2×#16860799C22semidirect
23. C2×#16860795C22semidirect
25. C2×C8×D251C22semidirect
28. C2×SD16×C251C22semidirect
33. #16860811C23semidirect
35. #16860809C23semidirect
48. #16860800C23semidirect
49. #16860799C23semidirect
50. #16860795C23semidirect
52. C8×D251C23semidirect
60. SD16×C251C23semidirect
92. C22×C251#135semidirect
104. C2×C251C2×#135semidirect
108. C251C22×#135semidirect
115. C22×SD16D251semidirect
134. C2×SD16C2×D251semidirect
166. SD16C22×D251semidirect
205. C22#22548736direct (N × H)
215. C2C2×#22548736direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#168608112
C22×#168608092
C2×#225487362
C22×#168608002
C22×#168607992
C22×#168607952
C22×C8×D2512
C22×SD16×C2512
C22×#135251