Finite Group Databasebeta

← #217×#1043 #72260155

Subgroups — #217×#1043 #72260155

Order 4800.

2064 conjugacy classes of subgroups; only the 252 normal subgroups are shown.

normal subgroups only

This group has 2064 conjugacy classes of subgroups — too many to draw. Only its 252 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
1#72231415C212yesmaximal, complemented
2#217×C5⋊D5C211yesmaximal, characteristic
3C2×#11925533C211yesmaximal, characteristic
4C2×Dic3×#1043C212yesmaximal, complemented
5C23×C3×#1043C211yesmaximal, characteristic
6C22⋊C4×#1043—31nomaximal, Hall
7C23×C3×C5⋊D5C2211yescharacteristic
8C22×C3⋊#1043C2212yes—
9C22×C3×#1043—24no—
10C22×C3×#1043C414yescomplemented
11C22×C3×#1043C2211yescharacteristic
12C22×C3×#1043C2211yescharacteristic
13C2×Dic3×C5⋊D5C2212yes—
14#11925533C414yescomplemented
15Dic3×#1043—24no—
16#11925254C411yescharacteristic, complemented
17C52×#217C411yescharacteristic, complemented
18#11925118C412yescomplemented
19AGL(1,𝔽5)×#217—56nomaximal
20C23×#1043D311yescharacteristic
21C2×C22⋊#1043—31no—
22C2×C4×#1043—32no—
23C22⋊C4×C5⋊D5—31no—
24#11716874—32no—
25C23×C3×C52C2×C411yescharacteristic
26C22×C3×C5⋊D5—22no—
27C22×C3×C5⋊D5—22no—
28C22×C3×C5⋊D5C2×C411yescharacteristic
29C22×C3×C5⋊D5C2×C412yes—
30C22×C3×C5⋊D5C2×C412yes—
31C22×C3×C5⋊D5C2311yescharacteristic
32C2×C3⋊#1043—22no—
33C2×C3⋊#1043—22no—
34C2×C3⋊#1043C2×C414yescomplemented
35C2×C3×#1043—22no—
36C2×C3×#1043—28no—
37C2×C3×#1043C2×C412yes—
38C2×C3×#1043C2×C412yes—
39C2×C3×#1043D414yes—
40C2×C3⋊#1039C2×C412yescomplemented
41C2×C52×Dic3C2×C412yescomplemented
42Dic3×C5⋊D5—22no—
43Dic3×C5⋊D5—22no—
44C23×C3×AGL(1,𝔽5)—56no—
45C2×#6763—56no—
46C2×Dic3×AGL(1,𝔽5)—512no—
47D5×#217—56no—
48(C2×AGL(1,𝔽5))⋊Dic3—512no—
49C23×C5⋊D5C2×D311yescharacteristic
50C22×#1043—24no—
51C22×#1043—32no—
52C22×#1043Dic314yescomplemented
53C22×#1043C2×D311yescharacteristic
54C22×#1043C2×D311yescharacteristic
55C2×C4×C5⋊D5—32no—
56C22⋊#1043—34no—
57C4×#1043—64no—
58C52×C22⋊C4—31no—
59C22⋊#1039—31no—
60#87808—32no—
61C22⋊C4×AGL(1,𝔽5)—156no—
62C22×C3×C52—22no—
63C22×C3×C52C4212yescomplemented
64C22×C3×C52C22×C411yescharacteristic
65C2×C3×C5⋊D5—22no—
66C2×C3×C5⋊D5—22no—
67C2×C3×C5⋊D5—22no—
68C2×C3×C5⋊D5—24no—
69C2×C3×C5⋊D5C4212yescomplemented
70C2×C3×C5⋊D5C22×C411yescharacteristic
71C2×C3×C5⋊D5C22×C411yescharacteristic
72C2×C3×C5⋊D5C2×D412yes—
73C2×C3×C5⋊D5#4112yes—
74C3⋊#1043—24no—
75C3×#1043—22no—
76C3×#1043C22⋊C414yescomplemented
77C3⋊#1039—22no—
78C52×Dic3—22no—
79C23×C3×D5—56no—
80C22×ΓL(1,𝔽16)—512no—
81C22×C3×AGL(1,𝔽5)—56no—
82C22×C3×AGL(1,𝔽5)—56no—
83C22×C3×AGL(1,𝔽5)—524no—
84C22×C3×AGL(1,𝔽5)—1024no—
85C2×D5×Dic3—512no—
86#6763—524no—
87Dic3×AGL(1,𝔽5)—1024no—
88#6719—56no—
89C5×#217AGL(1,𝔽5)16yescomplemented
90#6665—512no—
91C23×C52C4×D311yescharacteristic
92C22×C5⋊D5—22no—
93C22×C5⋊D5—22no—
94C22×C5⋊D5C4×D311yescharacteristic
95C22×C5⋊D5C4×D312yes—
96C22×C5⋊D5C2×Dic312yes—
97C22×C5⋊D5C22×D311yescharacteristic
98C2×#1043—22no—
99C2×#1043—28no—
100C2×#1043—34no—
101C2×#1043—62no—
102C2×#1043—62no—
103C2×#1043C2×Dic312yes—
104C2×#1043C2×Dic312yes—
105C2×#1043#6714yes—
106C2×C4×C52—32no—
107C2×#1039—32no—
108C4×C5⋊D5—62no—
109C4×C5⋊D5—62no—
110C4×#217—251noHall
111C23×AGL(1,𝔽5)—56no—
112C2×C22⋊AGL(1,𝔽5)—156no—
113C2×C4×AGL(1,𝔽5)—1512no—
114D5×C22⋊C4—156no—
115(C2×AGL(1,𝔽5))⋊C4—1512no—
116C2×C3×C52—22no—
117C2×C3×C52C2×C4211yescharacteristic
118C2×C3×C52C4×D412yes—
119C3×C5⋊D5—22no—
120C3×C5⋊D5C2×C22⋊C411yescharacteristic
121C3×C5⋊D5#11711yescharacteristic
122C3×C5⋊D5C4×D412yes—
123C23×C3×C5C2×AGL(1,𝔽5)16yes—
124C22×C3×D5—56no—
125C22×C3×D5—512no—
126C22×C3×D5—512no—
127C22×C3×D5—56no—
128C22×C3×D5—1012no—
129C22×C3×D5—1012no—
130C2×ΓL(1,𝔽16)—524no—
131C2×ΓL(1,𝔽16)—1012no—
132C2×ΓL(1,𝔽16)—1012no—
133C2×C3×AGL(1,𝔽5)—524no—
134C2×C3×AGL(1,𝔽5)—512no—
135C2×C3×AGL(1,𝔽5)—512no—
136C2×C3×AGL(1,𝔽5)—1048no—
137C2×C3×AGL(1,𝔽5)—1012no—
138C2×Dic15—512no—
139C2×C5×Dic3C2×AGL(1,𝔽5)112yescomplemented
140D5×Dic3—1012no—
141D5×Dic3—1012no—
142C22×C52—22no—
143C22×C52C4×Dic312yescomplemented
144C22×C52C2×C4×D311yescharacteristic
145C2×C5⋊D5—22no—
146C2×C5⋊D5—22no—
147C2×C5⋊D5—24no—
148C2×C5⋊D5—22no—
149C2×C5⋊D5C4×Dic312yescomplemented
150C2×C5⋊D5C2×C4×D311yescharacteristic
151C2×C5⋊D5#23512yes—
152C2×C5⋊D5C22×Dic311yescharacteristic
153C2×C5⋊D5C2×#6712yes—
154#1043—22no—
155#1043—64no—
156#1043#21714yesdirect product factor
157C4×C52—62no—
158#1039—62no—
159C23×C3×C4—251no—
160C2×#217—251no—
161C2×#217—251no—
162C2×C4×Dic3—252no—
163C3⋊#95—252no—
164C23×D5—56no—
165C22×AGL(1,𝔽5)—56no—
166C22×AGL(1,𝔽5)—524no—
167C22×AGL(1,𝔽5)—56no—
168C22×AGL(1,𝔽5)—1024no—
169C22×AGL(1,𝔽5)—1512no—
170C2×C4×D5—1512no—
171C22⋊AGL(1,𝔽5)—1524no—
172C4×AGL(1,𝔽5)—3024no—
173C5×C22⋊C4—36no—
174C22⋊Dic5—156no—
175#685—1512no—
176C3×C52C4×C22⋊C411yescharacteristic, complemented, Hall
177C4×C22⋊C4—751noSylow
178C22×C3×C5—212no—
179C22×C3×C5C4×AGL(1,𝔽5)112yescomplemented
180C22×C3×C5C22×AGL(1,𝔽5)16yes—
181C2×C3×D5—56no—
182C2×C3×D5—56no—
183C2×C3×D5—512no—
184C2×C3×D5—512no—
185C2×C3×D5—512no—
186C2×C3×D5—1012no—
187C2×C3×D5—1012no—
188C2×C3×D5—1024no—
189C2×C3×D5—1012no—
190ΓL(1,𝔽16)—1024no—
191C3×AGL(1,𝔽5)—524no—
192C3×AGL(1,𝔽5)—1012no—
193Dic15—1012no—
194C5×Dic3—212no—
195C2×C52—22no—
196C2×C52C2×C4×Dic311yescharacteristic
197C2×C52C4×#6712yes—
198C5⋊D5—22no—
199C5⋊D5#92611yescharacteristic
200C5⋊D5C4×#6712yes—
201C5⋊D5C2×#21711yescharacteristic
202C24×C3—251no—
203C22×C3×C4—251no—
204C22×C3×C4—251no—
205C22×C3×C4—254no—
206C22×C3×C4—504no—
207C22×Dic3—252no—
208C22×Dic3—252no—
209#217—254no—
210#217—251no—
211#217—252no—
212#217#104311yescharacteristic, direct product factor
213C4×Dic3—504no—
214C23×C5D3×AGL(1,𝔽5)16yes—
215C22×D5—56no—
216C22×D5—512no—
217C22×D5—512no—
218C22×D5—56no—
219C22×D5—1012no—
220C22×D5—1012no—
221C2×AGL(1,𝔽5)—512no—
222C2×AGL(1,𝔽5)—524no—
223C2×AGL(1,𝔽5)—512no—
224C2×AGL(1,𝔽5)—1048no—
225C2×AGL(1,𝔽5)—1012no—
226C2×AGL(1,𝔽5)—1524no—
227C2×AGL(1,𝔽5)—3012no—
228C2×AGL(1,𝔽5)—3012no—
229C2×C4×C5—312no—
230C2×Dic5—1512no—
231C4×D5—3012no—
232C4×D5—3012no—
233C23×C4—251no—
234C2×C22⋊C4—751no—
235C2×C22⋊C4—751no—
236C2×C42—752no—
237#95—752no—
238C2×C3×C5—212no—
239C2×C3×C5C2×C4×AGL(1,𝔽5)16yes—
240C2×C3×C5D4×AGL(1,𝔽5)112yes—
241C3×D5—56no—
242C3×D5—56no—
243C3×D5—512no—
244C3×D5—1012no—
245C52C4×#21711yescharacteristic, complemented, Sylow
246C23×C3—251no—
247C23×C3—252no—
248C23×C3—252no—
249C23×C3—251no—
250C23×C3—502no—
251C23×C3—502no—
252C23×C3C2×#104311yescharacteristic
253C2×C3×C4—254no—
254C2×C3×C4—252no—
255C2×C3×C4—252no—
256C2×C3×C4—508no—
257C2×C3×C4—502no—
258C2×Dic3—254no—
259C2×Dic3—252no—
260C2×Dic3—502no—
261C2×Dic3—502no—
262C2×Dic3—502no—
263C2×Dic3—502no—
264C2×Dic3C2×#104312yescomplemented
265C22×C5—212no—
266C22×C5Dic3×AGL(1,𝔽5)112yescomplemented
267C22×C5C2×D3×AGL(1,𝔽5)16yes—
268C2×D5—56no—
269C2×D5—56no—
270C2×D5—512no—
271C2×D5—512no—
272C2×D5—512no—
273C2×D5—1012no—
274C2×D5—1012no—
275C2×D5—1024no—
276C2×D5—1012no—
277AGL(1,𝔽5)—524no—
278AGL(1,𝔽5)—1012no—
279AGL(1,𝔽5)—3024no—
280C4×C5—612no—
281Dic5—3012no—
282C24—251no—
283C22×C4—251no—
284C22×C4—251no—
285C22×C4—254no—
286C22×C4—504no—
287C22×C4—752no—
288C22×C4—752no—
289C22⋊C4—31no—
290C22⋊C4—752no—
291C22⋊C4—754no—
292C22⋊C4—751no—
293C42—1504no—
294C3×C5C22⋊C4×AGL(1,𝔽5)16yescomplemented
295C22×C3—22no—
296C22×C3—251no—
297C22×C3—251no—
298C22×C3—252no—
299C22×C3—252no—
300C22×C3—252no—
301C22×C3—502no—
302C22×C3—504no—
303C22×C3—502no—
304C22×C3—502no—
305C22×C3C4×#104312yescomplemented
306C22×C3C22×#104311yescharacteristic
307C3×C4—254no—
308C3×C4—502no—
309Dic3—22no—
310Dic3—504no—
311Dic3—502no—
312C2×C5—212no—
313C2×C5C2×Dic3×AGL(1,𝔽5)16yes—
314C2×C5AGL(1,𝔽5)×#67112yes—
315D5—56no—
316D5—512no—
317D5—56no—
318D5—1012no—
319C23—251no—
320C23—251no—
321C23—252no—
322C23—252no—
323C23—502no—
324C23—502no—
325C23D3×#104311yescharacteristic
326C2×C4—32no—
327C2×C4—254no—
328C2×C4—252no—
329C2×C4—252no—
330C2×C4—508no—
331C2×C4—502no—
332C2×C4—754no—
333C2×C4—752no—
334C2×C4—1502no—
335C2×C4—1502no—
336C2×C4—1502no—
337C2×C4—1502no—
338C2×C3—22no—
339C2×C3—252no—
340C2×C3—251no—
341C2×C3—251no—
342C2×C3—502no—
343C2×C3C2×C4×#104311yescharacteristic
344C2×C3D4×#104312yes—
345C5AGL(1,𝔽5)×#21716yescomplemented
346C22—22no—
347C22—251no—
348C22—251no—
349C22—252no—
350C22—252no—
351C22—252no—
352C22—502no—
353C22—504no—
354C22—502no—
355C22—502no—
356C22Dic3×#104312yescomplemented
357C22C2×D3×#104311yescharacteristic
358C4—62no—
359C4—254no—
360C4—502no—
361C4—1504no—
362C4—1502no—
363C3C22⋊C4×#104311yescharacteristic, complemented, Sylow
364C2—22no—
365C2—251no—
366C2—252no—
367C2—251no—
368C2—502no—
369C2C2×Dic3×#104311yescharacteristic
370C2#67×#104312yes—
subgroupG/H
center, FrattiniC22C2×D3×#1043
commutatorC2×C3×C52C2×C42
FittingC23×C3×C52C2×C4
socleC22×C3×C52C22×C4
prime radicalC23×C3×C5⋊D5C22
cosocleC22×C3×C5⋊D5C23
hypercenterC23D3×#1043
hypocenterC3×C52C4×C22⋊C4
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. #72231415C2semidirect
4. C2×Dic3×#1043C2semidirect
10. C22×C3×#1043C4semidirect
14. #11925533C4semidirect
16. #11925254C4semidirect
17. C52×#217C4semidirect
18. #11925118C4semidirect
34. C2×C3⋊#1043C2×C4semidirect
40. C2×C3⋊#1039C2×C4semidirect
41. C2×C52×Dic3C2×C4semidirect
52. C22×#1043Dic3semidirect
63. C22×C3×C52C42semidirect
69. C2×C3×C5⋊D5C42semidirect
76. C3×#1043C22⋊C4semidirect
89. C5×#217AGL(1,𝔽5)semidirect
139. C2×C5×Dic3C2×AGL(1,𝔽5)semidirect
143. C22×C52C4×Dic3semidirect
149. C2×C5⋊D5C4×Dic3semidirect
156. #1043#217direct (N × H)
176. C3×C52C4×C22⋊C4semidirect
179. C22×C3×C5C4×AGL(1,𝔽5)semidirect
212. #217#1043direct (N × H)
245. C52C4×#217semidirect
264. C2×Dic3C2×#1043semidirect
266. C22×C5Dic3×AGL(1,𝔽5)semidirect
294. C3×C5C22⋊C4×AGL(1,𝔽5)semidirect
305. C22×C3C4×#1043semidirect
345. C5AGL(1,𝔽5)×#217semidirect
356. C22Dic3×#1043semidirect
363. C3C22⋊C4×#1043semidirect

Maximal subgroups

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

subgroupindex
#722314152
#217×C5⋊D52
C2×#119255332
C2×Dic3×#10432
C23×C3×#10432
C22⋊C4×#10433
AGL(1,𝔽5)×#2175