Finite Group Databasebeta

← C22×#11716875 #72242370

Subgroups — C22×#11716875 #72242370

Order 3200.

4288 conjugacy classes of subgroups; only the 486 normal subgroups are shown.

normal subgroups only

This group has 4288 conjugacy classes of subgroups — too many to draw. Only its 486 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×D4×C5⋊D5C211yesmaximal, characteristic
2C22×C4⋊#1043C211yesmaximal, characteristic, complemented
3C22×#87855C211yesmaximal, characteristic, complemented
4C2×#11716875C2112yesmaximal, direct product factor
5C24×C5⋊D5—21no—
6C23×#1043—21no—
7C22×C52×D4C411yescharacteristic, complemented
8C22×C22⋊(C5⋊D5)—21no—
9C2×D4×C5⋊D5—23no—
10C2×D4×C5⋊D5—23no—
11C2×D4×C5⋊D5C416yescomplemented
12C2×D4×C5⋊D5C2216yes—
13C22×C4⋊(C5⋊D5)C411yescharacteristic, complemented
14C22×C4×C5⋊D5C2211yescharacteristic
15C22×#6033—21no—
16C2×C4⋊#1043—23no—
17C2×C4⋊#1043C2216yescomplemented
18C2×#87855—23no—
19C2×#87855C2216yescomplemented
20#11716875C22116yesdirect product factor
21C22×#4007—56nomaximal
22C24×C52—21no—
23C23×C5⋊D5—21no—
24C23×C5⋊D5—26no—
25C23×C5⋊D5—26no—
26C23×C5⋊D5—44no—
27C23×C5⋊D5—44no—
28C23×C5⋊D5D411yescharacteristic
29C22×#1043—26no—
30C22×#1043—44no—
31C2×C52×D4—23no—
32C2×C52×D4C2×C416yescomplemented
33C22×C4×C52C2×C411yescharacteristic
34C2×C22⋊(C5⋊D5)—26no—
35C2×C22⋊(C5⋊D5)—26no—
36C22×#1039D411yescharacteristic
37D4×C5⋊D5—212no—
38D4×C5⋊D5—26no—
39D4×C5⋊D5—26no—
40D4×C5⋊D5C2×C4112yescomplemented
41D4×C5⋊D5C2314yes—
42C2×C4⋊(C5⋊D5)—23no—
43C2×C4⋊(C5⋊D5)C2×C416yescomplemented
44C2×C4×C5⋊D5C2×C413yes—
45C2×C4×C5⋊D5D413yes—
46C2×C4×C5⋊D5D413yes—
47C2×C4×C5⋊D5C2313yes—
48C2×#6033—26no—
49C4⋊#1043—26no—
50C4⋊#1043C2314yescomplemented
51#87855—26no—
52#87855C2314yescomplemented
53C22×D4×D5—56no—
54C22×C4⋊AGL(1,𝔽5)—56no—
55C22×D5⋊C8—56no—
56C2×#4007—572no—
57C23×C52—26no—
58C23×C52—44no—
59C23×C52C22⋊C411yescharacteristic
60C22×C5⋊D5—26no—
61C22×C5⋊D5—212no—
62C22×C5⋊D5—24no—
63C22×C5⋊D5—44no—
64C22×C5⋊D5—424no—
65C22×C5⋊D5—412no—
66C22×C5⋊D5—412no—
67C22×C5⋊D5C22⋊C413yes—
68C22×C5⋊D5D814yes—
69C22×C5⋊D5SD1614yes—
70C22×C5⋊D5C2×D413yes—
71C2×#1043—24no—
72C2×#1043—412no—
73C52×D4—26no—
74C52×D4C22×C414yescomplemented
75C2×C4×C52C22⋊C413yes—
76C2×C4×C52C22×C413yes—
77C22⋊(C5⋊D5)—212no—
78C22⋊(C5⋊D5)—24no—
79C2×#1039C22⋊C413yes—
80C2×#1039C2×D413yes—
81C4⋊(C5⋊D5)—26no—
82C4⋊(C5⋊D5)C22×C414yescomplemented
83C4×C5⋊D5C22⋊C416yes—
84C4×C5⋊D5C22×C413yes—
85C4×C5⋊D5C2×D413yes—
86C4×C5⋊D5C2×D413yes—
87C4×C5⋊D5C2411yescharacteristic
88#6033—24no—
89C24×D5—106no—
90C23×AGL(1,𝔽5)—106no—
91C22×C5×D4AGL(1,𝔽5)16yescomplemented
92C22×C22⋊D5—106no—
93C2×D4×D5—536no—
94C2×D4×D5—536no—
95C2×D4×D5—1018no—
96C2×D4×D5—1018no—
97C22×D20—56no—
98C22×C4×D5—56no—
99C22×#170—106no—
100C2×C4⋊AGL(1,𝔽5)—536no—
101C2×C4⋊AGL(1,𝔽5)—1018no—
102C2×D5⋊C8—536no—
103C2×D5⋊C8—1018no—
104#4007—596no—
105C22×#102—251noSylow
106C22×C52—24no—
107C22×C52—412no—
108C22×C52#10214yescomplemented
109C22×C52C2×C22⋊C413yes—
110C2×C5⋊D5—24no—
111C2×C5⋊D5—412no—
112C2×C5⋊D5—44no—
113C2×C5⋊D5—44no—
114C2×C5⋊D5—424no—
115C2×C5⋊D5#102112yescomplemented
116C2×C5⋊D5C2×C22⋊C413yes—
117C2×C5⋊D5C2×D816yes—
118C2×C5⋊D5C2×SD1616yes—
119C2×C5⋊D5C22×D411yescharacteristic
120#1043—44no—
121C4×C52C2×C22⋊C413yes—
122C4×C52C23×C411yescharacteristic
123#1039C2×C22⋊C413yes—
124#1039C22×D411yescharacteristic
125C24×C5—26no—
126C23×D5—56no—
127C23×D5—1036no—
128C23×D5—1036no—
129C23×D5—106no—
130C23×D5—2024no—
131C23×D5—2024no—
132C22×AGL(1,𝔽5)—1036no—
133C22×AGL(1,𝔽5)—2024no—
134C2×C5×D4—218no—
135C2×C5×D4C2×AGL(1,𝔽5)136yescomplemented
136C22×C4×C5C2×AGL(1,𝔽5)16yes—
137C2×C22⋊D5—1036no—
138C2×C22⋊D5—1036no—
139C22×Dic5—56no—
140D4×D5—572no—
141D4×D5—524no—
142D4×D5—1072no—
143D4×D5—1036no—
144D4×D5—1036no—
145C2×D20—536no—
146C2×D20—1018no—
147C2×C4×D5—518no—
148C2×C4×D5—518no—
149C2×C4×D5—518no—
150C2×C4×D5—518no—
151C2×#170—1036no—
152C4⋊AGL(1,𝔽5)—524no—
153C4⋊AGL(1,𝔽5)—1036no—
154D5⋊C8—524no—
155D5⋊C8—1036no—
156C23×D4—251no—
157C23×C8—251no—
158C22×C4⋊C4—251no—
159C2×#102—2512no—
160C2×C52—44no—
161C2×C52C2×#10216yescomplemented
162C2×C52C22×C22⋊C411yescharacteristic
163C5⋊D5—44no—
164C5⋊D5C2×#10216yescomplemented
165C5⋊D5C22×D811yescharacteristic
166C5⋊D5C22×SD1611yescharacteristic
167C23×C5—236no—
168C23×C5—424no—
169C23×C5C22⋊AGL(1,𝔽5)16yes—
170C22×D5—518no—
171C22×D5—518no—
172C22×D5—524no—
173C22×D5—524no—
174C22×D5—1072no—
175C22×D5—1024no—
176C22×D5—1036no—
177C22×D5—20144no—
178C22×D5—2072no—
179C22×D5—2072no—
180C22×D5—2024no—
181C2×AGL(1,𝔽5)—1024no—
182C2×AGL(1,𝔽5)—2072no—
183C5×D4—236no—
184C5×D4C22×AGL(1,𝔽5)124yescomplemented
185C2×C4×C5C22⋊AGL(1,𝔽5)118yes—
186C2×C4×C5C22×AGL(1,𝔽5)118yes—
187C22⋊D5—1072no—
188C22⋊D5—1024no—
189C2×Dic5—518no—
190C2×Dic5—518no—
191D20—524no—
192D20—1036no—
193C4×D5—536no—
194C4×D5—518no—
195C4×D5—518no—
196C4×D5—518no—
197C4×D5—56no—
198#170—1024no—
199C25—501no—
200C22×D4—256no—
201C22×D4—256no—
202C22×D4—251no—
203C22×D4—503no—
204C22×D4—503no—
205C22×D4—501no—
206C22×D4#104311yescharacteristic, complemented
207C23×C4—251no—
208C23×C4—501no—
209C22×C8—256no—
210C22×C8—503no—
211C22×C8—501no—
212C2×C4⋊C4—256no—
213C2×C4⋊C4—503no—
214#102—2516no—
215C52C22×#10211yescharacteristic, complemented, Sylow
216C22×C5—224no—
217C22×C5—472no—
218C22×C5#4007124yescomplemented
219C22×C5C2×C22⋊AGL(1,𝔽5)118yes—
220C2×D5—518no—
221C2×D5—56no—
222C2×D5—572no—
223C2×D5—536no—
224C2×D5—536no—
225C2×D5—1024no—
226C2×D5—2024no—
227C2×D5—2024no—
228C2×D5—20144no—
229C2×D5—2072no—
230AGL(1,𝔽5)—2024no—
231C4×C5C2×C22⋊AGL(1,𝔽5)118yes—
232C4×C5C23×AGL(1,𝔽5)16yes—
233Dic5—518no—
234Dic5—56no—
235C24—21no—
236C24—251no—
237C24—506no—
238C24—506no—
239C24—501no—
240C24—1004no—
241C24—1004no—
242C2×D4—23no—
243C2×D4—254no—
244C2×D4—2512no—
245C2×D4—256no—
246C2×D4—506no—
247C2×D4—506no—
248C2×D4—5012no—
249C2×D4—506no—
250C2×D4—506no—
251C2×D4—503no—
252C2×D4C2×#104316yescomplemented
253C22×C4—251no—
254C22×C4—253no—
255C22×C4—253no—
256C22×C4—253no—
257C22×C4—253no—
258C22×C4—506no—
259C22×C4—1004no—
260C22×C4C2×#104311yescharacteristic
261C2×C8—254no—
262C2×C8—506no—
263C2×C8—506no—
264C4⋊C4—254no—
265C4⋊C4—506no—
266C2×C5—424no—
267C2×C5C2×#4007136yescomplemented
268C2×C5C22×C22⋊AGL(1,𝔽5)16yes—
269D5—56no—
270D5—56no—
271D5—536no—
272D5—2024no—
273C23—26no—
274C23—44no—
275C23—254no—
276C23—254no—
277C23—253no—
278C23—253no—
279C23—504no—
280C23—5012no—
281C23—506no—
282C23—10012no—
283C23—10024no—
284C23—10012no—
285C23—1004no—
286C23C22⋊#104311yescharacteristic
287D4—26no—
288D4—254no—
289D4—504no—
290D4—5012no—
291D4—506no—
292D4C22×#104314yescomplemented
293C2×C4—253no—
294C2×C4—253no—
295C2×C4—253no—
296C2×C4—253no—
297C2×C4—256no—
298C2×C4—251no—
299C2×C4—253no—
300C2×C4—504no—
301C2×C4—10012no—
302C2×C4C22⋊#104313yes—
303C2×C4C22×#104313yes—
304C8—504no—
305C5C22×#400716yescomplemented
306C22—24no—
307C22—412no—
308C22—251no—
309C22—253no—
310C22—256no—
311C22—2512no—
312C22—256no—
313C22—504no—
314C22—1004no—
315C22—1004no—
316C22—10024no—
317C22—10012no—
318C22#1171687514yesdirect product factor
319C22C2×C22⋊#104313yes—
320C4—251no—
321C4—253no—
322C4—1004no—
323C4C2×C22⋊#104313yes—
324C4C23×#104311yescharacteristic
325C2—44no—
326C2—251no—
327C2—251no—
328C2—256no—
329C2—1004no—
330C2C2×#1171687516yesdirect product factor
331C2C22×C22⋊#104311yescharacteristic
subgroupG/H
centerC23C22⋊#1043
commutatorC4×C52C23×C4
FrattiniC4C23×#1043
FittingC22×C52×D4C4
socleC23×C52C22⋊C4
prime radicalC22×D4×C5⋊D5C2
cosocleC4×C5⋊D5C24
hypercenterC22×D4#1043
hypocenterC52C22×#102
LCSG ▷ C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊#1043C2semidirect
3. C22×#87855C2semidirect
4. C2×#11716875C2direct (N × H)
7. C22×C52×D4C4semidirect
11. C2×D4×C5⋊D5C4semidirect
13. C22×C4⋊(C5⋊D5)C4semidirect
17. C2×C4⋊#1043C22semidirect
19. C2×#87855C22semidirect
20. #11716875C22direct (N × H)
32. C2×C52×D4C2×C4semidirect
40. D4×C5⋊D5C2×C4semidirect
43. C2×C4⋊(C5⋊D5)C2×C4semidirect
50. C4⋊#1043C23semidirect
52. #87855C23semidirect
74. C52×D4C22×C4semidirect
82. C4⋊(C5⋊D5)C22×C4semidirect
91. C22×C5×D4AGL(1,𝔽5)semidirect
108. C22×C52#102semidirect
115. C2×C5⋊D5#102semidirect
135. C2×C5×D4C2×AGL(1,𝔽5)semidirect
161. C2×C52C2×#102semidirect
164. C5⋊D5C2×#102semidirect
184. C5×D4C22×AGL(1,𝔽5)semidirect
206. C22×D4#1043semidirect
215. C52C22×#102semidirect
218. C22×C5#4007semidirect
252. C2×D4C2×#1043semidirect
267. C2×C5C2×#4007semidirect
292. D4C22×#1043semidirect
305. C5C22×#4007semidirect
318. C22#11716875direct (N × H)
330. C2C2×#11716875direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×C5⋊D52
C22×C4⋊#10432
C22×#878552
C2×#117168752
C22×#40075