Finite Group Databasebeta

← #72231030

Subgroups — #72231030

Order 4800.

2064 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 2064 conjugacy classes of subgroups — too many to draw. Only its 216 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
1C2×(C2×D3)⋊#1043C214yesmaximal, complemented
2C2×D12×C5⋊D5C211yesmaximal, characteristic
3C2×C3×C4⋊#1043C211yesmaximal, characteristic, complemented
4C2×C4×C3⋊#1043C211yesmaximal, characteristic, complemented
5#13104792—31nomaximal, Hall
6C22×D3×C5⋊D5C2212yes—
7C22×C3⋊#1043C2212yes—
8C22×C3×#1043C2212yes—
9C2×C4⋊(C3⋊(C5⋊D5))C411yescharacteristic, complemented
10C2×C52×D12C411yescharacteristic, complemented
11C2×C3×C4×C5⋊D5C2211yescharacteristic
12C2×#10597116C412yescomplemented
13D12×C5⋊D5—22no—
14D12×C5⋊D5—22no—
15C4×C3⋊#1043—22no—
16C3×C4⋊#1043—22no—
17(C2×D3)⋊#1043—28no—
18#11751857—56nomaximal
19C2×D4×C5⋊D5—31no—
20C2×C22⋊#1043—34no—
21C2×C4⋊#1043D311yescharacteristic, complemented
22C2×C4×#1043—31no—
23C22×C3⋊(C5⋊D5)C2×C412yes—
24C22×C52×D3C2×C412yes—
25C22×C3×C5⋊D5C2311yescharacteristic
26C2×D3×C5⋊D5—24no—
27C2×D3×C5⋊D5—22no—
28C2×D3×C5⋊D5—22no—
29C2×D3×C5⋊D5—24no—
30C2×C3⋊#1043—22no—
31C2×C3⋊#1043—22no—
32C2×C3⋊#1043D414yescomplemented
33C2×C3×#1043—24no—
34C2×C3×#1043—22no—
35C2×C3×C4×C52C2×C411yescharacteristic
36C4⋊(C3⋊(C5⋊D5))—22no—
37C52×D12—22no—
38C2×C3×#1039C2×C411yescharacteristic
39C3×C4×C5⋊D5D412yes—
40C3×C4×C5⋊D5D412yes—
41#10597116—24no—
42C2×D5×D12—56no—
43C2×C4×ΓL(1,𝔽16)—56no—
44C2×C3×C4⋊AGL(1,𝔽5)—56no—
45C2×(C2×D3)⋊AGL(1,𝔽5)—524no—
46C23×C5⋊D5—32no—
47C22×#1043—32no—
48C22×#1043C2×D312yes—
49C2×C52×D4—31no—
50C2×C22⋊(C5⋊D5)—32no—
51D4×C5⋊D5—62no—
52D4×C5⋊D5—62no—
53C2×C4⋊(C5⋊D5)—31no—
54C2×C4×C5⋊D5C2×D311yescharacteristic
55C22⋊#1043—68no—
56C4⋊#1043—22no—
57C4×#1043—62no—
58#65780—156no—
59C22×C3×C52C22×C411yescharacteristic
60C2×C3⋊(C5⋊D5)—22no—
61C2×C3⋊(C5⋊D5)—24no—
62C2×C52×D3—22no—
63C2×C52×D3—24no—
64C2×C3×C5⋊D5C2×D411yescharacteristic
65C2×C3×C5⋊D5C2×D412yes—
66C2×C3×C5⋊D5C2×D411yescharacteristic
67C2×C3×C5⋊D5#4112yes—
68D3×C5⋊D5—44no—
69D3×C5⋊D5—42no—
70D3×C5⋊D5—42no—
71C3⋊#1043—24no—
72C3×#1043—42no—
73C3×C4×C52C22⋊C412yescomplemented
74C3×#1039C22⋊C412yescomplemented
75C22×D3×D5—512no—
76C22×ΓL(1,𝔽16)—512no—
77C22×C3×AGL(1,𝔽5)—512no—
78C2×D60—56no—
79C2×C5×D12AGL(1,𝔽5)16yescomplemented
80C2×C3×C4×D5—56no—
81C2×#1203—512no—
82D5×D12—1012no—
83D5×D12—1012no—
84C4×ΓL(1,𝔽16)—1012no—
85C3×C4⋊AGL(1,𝔽5)—1012no—
86(C2×D3)⋊AGL(1,𝔽5)—1048no—
87C23×C52—32no—
88C22×C5⋊D5—32no—
89C22×C5⋊D5—62no—
90C22×C5⋊D5—64no—
91C22×C5⋊D5—64no—
92C22×C5⋊D5—62no—
93C22×C5⋊D5C22×D311yescharacteristic
94C2×#1043—24no—
95C2×#1043—22no—
96C2×#1043—34no—
97C2×#1043—62no—
98C2×#1043—62no—
99C52×D4—62no—
100C2×C4×C52C4×D311yescharacteristic
101C22⋊(C5⋊D5)—64no—
102C2×#1039C4×D311yescharacteristic
103C4⋊(C5⋊D5)—62no—
104C4×C5⋊D5D1212yes—
105C4×C5⋊D5#6712yes—
106#4988—251noHall
107C2×D4×D5—156no—
108C2×C22⋊AGL(1,𝔽5)—1524no—
109C2×C4⋊AGL(1,𝔽5)—56no—
110C2×C4×AGL(1,𝔽5)—156no—
111C2×C3×C52C2×C22⋊C411yescharacteristic
112C2×C3×C52C4×D412yes—
113C3⋊(C5⋊D5)—42no—
114C52×D3—42no—
115C3×C5⋊D5#12112yes—
116C3×C5⋊D5#12411yescharacteristic
117C3×C5⋊D5#12711yescharacteristic
118C22×D15—512no—
119C22×C5×D3C2×AGL(1,𝔽5)112yes—
120C22×C3×D5—56no—
121C2×D3×D5—1024no—
122C2×D3×D5—1012no—
123C2×D3×D5—1012no—
124C2×D3×D5—1024no—
125C2×ΓL(1,𝔽16)—524no—
126C2×ΓL(1,𝔽16)—1012no—
127C2×ΓL(1,𝔽16)—1012no—
128C2×C3×AGL(1,𝔽5)—1024no—
129C2×C3×AGL(1,𝔽5)—1012no—
130C2×C3×C4×C5C2×AGL(1,𝔽5)16yes—
131D60—1012no—
132C5×D12—212no—
133C2×C3×Dic5—56no—
134C3×C4×D5—512no—
135C3×C4×D5—512no—
136#1203—1024no—
137C22×C52—64no—
138C22×C52—62no—
139C22×C52C2×C4×D311yescharacteristic
140C2×C5⋊D5—64no—
141C2×C5⋊D5—62no—
142C2×C5⋊D5—124no—
143C2×C5⋊D5—122no—
144C2×C5⋊D5—122no—
145C2×C5⋊D5C2×D1211yescharacteristic
146C2×C5⋊D5D3×D412yes—
147C2×C5⋊D5#23912yes—
148C2×C5⋊D5C2×#6711yescharacteristic
149#1043—42no—
150#1043—64no—
151C4×C52(C2×D3)⋊C412yescomplemented
152#1039(C2×D3)⋊C412yescomplemented
153C22×D12—251no—
154C2×C3×C4⋊C4—251no—
155C2×(C2×D3)⋊C4—254no—
156C2×C4×Dic3—251no—
157C23×D5—1512no—
158C22×AGL(1,𝔽5)—512no—
159C22×AGL(1,𝔽5)—1512no—
160C2×C5×D4—36no—
161C2×C22⋊D5—1512no—
162D4×D5—3012no—
163D4×D5—3012no—
164C2×D20—156no—
165C2×C4×D5—56no—
166C22⋊AGL(1,𝔽5)—3048no—
167C4⋊AGL(1,𝔽5)—1012no—
168C4×AGL(1,𝔽5)—3012no—
169C3×C52#39011yescharacteristic, complemented, Hall
170#390—751noSylow
171C22×C3×C5C22×AGL(1,𝔽5)16yes—
172C2×D15—1024no—
173C2×D15—1012no—
174C2×C5×D3—224no—
175C2×C5×D3—212no—
176C2×C3×D5—512no—
177C2×C3×D5—512no—
178C2×C3×D5—56no—
179C2×C3×D5—56no—
180D3×D5—2024no—
181D3×D5—2012no—
182D3×D5—2012no—
183ΓL(1,𝔽16)—1024no—
184C3×AGL(1,𝔽5)—2012no—
185C3×C4×C5C22⋊AGL(1,𝔽5)112yescomplemented
186C3×Dic5—512no—
187C2×C52—122no—
188C2×C52#89312yes—
189C2×C52C2×(C2×D3)⋊C411yescharacteristic
190C5⋊D5—122no—
191C5⋊D5#89512yes—
192C5⋊D5#94011yescharacteristic
193C5⋊D5#94711yescharacteristic
194C23×D3—252no—
195C22×C3×C4—251no—
196C22×C3×C4—252no—
197C22×Dic3—252no—
198C2×D12—252no—
199C2×D12—251no—
200C2×D12—502no—
201C2×D12—502no—
202C2×D12#104311yescharacteristic, complemented
203C3×C4⋊C4—502no—
204(C2×D3)⋊C4—508no—
205C4×Dic3—502no—
206C23×C5—312no—
207C22×D5—56no—
208C22×D5—1512no—
209C22×D5—3024no—
210C22×D5—3012no—
211C22×D5—3012no—
212C22×D5—3024no—
213C2×AGL(1,𝔽5)—1012no—
214C2×AGL(1,𝔽5)—1024no—
215C2×AGL(1,𝔽5)—1524no—
216C2×AGL(1,𝔽5)—3012no—
217C2×AGL(1,𝔽5)—3012no—
218C5×D4—612no—
219C2×C4×C5D3×AGL(1,𝔽5)16yes—
220C22⋊D5—3024no—
221C2×Dic5—56no—
222D20—3012no—
223C4×D5—512no—
224C4×D5—512no—
225C22×D4—751no—
226C2×C4⋊C4—251no—
227C2×C22⋊C4—754no—
228C2×C42—751no—
229C2×C3×C5D4×AGL(1,𝔽5)112yes—
230C2×C3×C5C2×C22⋊AGL(1,𝔽5)16yes—
231D15—2012no—
232C3×D5—512no—
233C3×D5—56no—
234C3×D5—56no—
235C5×D3—412no—
236C52#498811yescharacteristic, complemented, Sylow
237C23×C3—251no—
238C22×D3—252no—
239C22×D3—504no—
240C22×D3—502no—
241C22×D3—502no—
242C22×D3—504no—
243C22×D3C2×#104312yes—
244C2×C3×C4—251no—
245C2×C3×C4—252no—
246C2×C3×C4—252no—
247C2×C3×C4—504no—
248C2×C3×C4—502no—
249C2×C3×C4C2×#104311yescharacteristic
250C2×Dic3—254no—
251C2×Dic3—502no—
252C2×Dic3—502no—
253D12—22no—
254D12—504no—
255D12—502no—
256C22×C5—624no—
257C22×C5—612no—
258C22×C5C2×D3×AGL(1,𝔽5)16yes—
259C2×D5—512no—
260C2×D5—512no—
261C2×D5—56no—
262C2×D5—56no—
263C2×D5—3024no—
264C2×D5—3012no—
265C2×D5—6024no—
266C2×D5—6012no—
267C2×D5—6012no—
268AGL(1,𝔽5)—2012no—
269AGL(1,𝔽5)—3024no—
270C4×C5(C2×D3)⋊AGL(1,𝔽5)112yescomplemented
271Dic5—512no—
272C24—752no—
273C2×D4—31no—
274C2×D4—752no—
275C2×D4—751no—
276C2×D4—1502no—
277C2×D4—1502no—
278C22×C4—251no—
279C22×C4—252no—
280C22×C4—752no—
281C4⋊C4—502no—
282C22⋊C4—1508no—
283C42—1502no—
284C3×C5#6578016yescomplemented
285C22×C3—252no—
286C22×C3—252no—
287C22×C3—251no—
288C22×C3—251no—
289C22×C3C22×#104311yescharacteristic
290C2×D3—24no—
291C2×D3—22no—
292C2×D3—504no—
293C2×D3—502no—
294C2×D3—1004no—
295C2×D3—1002no—
296C2×D3—1002no—
297C3×C4—252no—
298C3×C4—1002no—
299C3×C4C22⋊#104312yescomplemented
300Dic3—504no—
301C2×C5—1212no—
302C2×C5#91921112yes—
303C2×C5C2×(C2×D3)⋊AGL(1,𝔽5)16yes—
304D5—512no—
305D5—56no—
306D5—56no—
307D5—6012no—
308C23—32no—
309C23—251no—
310C23—752no—
311C23—1504no—
312C23—1502no—
313C23—1502no—
314C23—1504no—
315D4—62no—
316D4—1504no—
317D4—1502no—
318C2×C4—251no—
319C2×C4—252no—
320C2×C4—252no—
321C2×C4—504no—
322C2×C4—502no—
323C2×C4—754no—
324C2×C4—1502no—
325C2×C4—1502no—
326C2×C4D3×#104311yescharacteristic
327C2×C3—252no—
328C2×C3—251no—
329C2×C3—251no—
330C2×C3D4×#104312yes—
331C2×C3C2×C22⋊#104311yescharacteristic
332D3—42no—
333D3—1002no—
334C5#1175185716yescomplemented
335C22—64no—
336C22—62no—
337C22—252no—
338C22—252no—
339C22—251no—
340C22—251no—
341C22—3002no—
342C22—3004no—
343C22—3002no—
344C22—1504no—
345C22—1502no—
346C22C2×D3×#104311yescharacteristic
347C4—252no—
348C4—1002no—
349C4—1504no—
350C4(C2×D3)⋊#104312yescomplemented
351C3#1310479211yescharacteristic, complemented, Sylow
352C2—122no—
353C2—251no—
354C2—252no—
355C2—251no—
356C2—3002no—
357C2#7222953112yes—
358C2C2×(C2×D3)⋊#104311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#1043
commutator, socleC22×C3×C52C22×C4
FittingC2×C3×C4×C52C2×C4
prime radicalC2×D12×C5⋊D5C2
cosocleC22×C3×C5⋊D5C23
hypercenterC2×C4D3×#1043
hypocenterC3×C52#390
LCSG ▷ C22×C3×C52 ▷ C3×C52
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×(C2×D3)⋊#1043C2semidirect
3. C2×C3×C4⋊#1043C2semidirect
4. C2×C4×C3⋊#1043C2semidirect
9. C2×C4⋊(C3⋊(C5⋊D5))C4semidirect
10. C2×C52×D12C4semidirect
12. C2×#10597116C4semidirect
21. C2×C4⋊#1043D3semidirect
32. C2×C3⋊#1043D4semidirect
73. C3×C4×C52C22⋊C4semidirect
74. C3×#1039C22⋊C4semidirect
79. C2×C5×D12AGL(1,𝔽5)semidirect
151. C4×C52(C2×D3)⋊C4semidirect
152. #1039(C2×D3)⋊C4semidirect
169. C3×C52#390semidirect
185. C3×C4×C5C22⋊AGL(1,𝔽5)semidirect
202. C2×D12#1043semidirect
236. C52#4988semidirect
270. C4×C5(C2×D3)⋊AGL(1,𝔽5)semidirect
284. C3×C5#65780semidirect
299. C3×C4C22⋊#1043semidirect
334. C5#11751857semidirect
350. C4(C2×D3)⋊#1043semidirect
351. C3#13104792semidirect

Maximal subgroups

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

subgroupindex
C2×(C2×D3)⋊#10432
C2×D12×C5⋊D52
C2×C3×C4⋊#10432
C2×C4×C3⋊#10432
#131047923
#117518575