Finite Group Databasebeta

← #72230290

Subgroups — #72230290

Order 2880.

1588 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1588 conjugacy classes of subgroups — too many to draw. Only its 235 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×D5×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C3×#91989C212yesmaximal, complemented
3#13088746C214yesmaximal, complemented
4#13088732C218yesmaximal, complemented
5#11756837—32nomaximal
6C22×D5×C3⋊D3C2212yes—
7C2×ΓL(1,𝔽16)⋊D3C2218yescomplemented
8C2×C4⋊(C3⋊D15)C411yescharacteristic, complemented
9C2×C5×C4⋊(C3⋊D3)C411yescharacteristic, complemented
10C2×C32×C4×D5C2211yescharacteristic
11C2×#67140C412yescomplemented
12D5×C4⋊(C3⋊D3)C414yescomplemented
13D5×C4⋊(C3⋊D3)C2214yes—
14C3×#6763C2214yes—
15C3×C4⋊ΓL(1,𝔽16)C2214yescomplemented
16C3×C4×ΓL(1,𝔽16)C2214yescomplemented
17#10623978C2218yescomplemented
18#10594996—51nomaximal, Hall
19C2×D5×D12—32no—
20C2×D5×D12—61no—
21#91989D312yescomplemented
22C3×#4130—32no—
23D3×C22⋊AGL(1,𝔽5)—38no—
24#91921—38no—
25AGL(1,𝔽5)×D12—38no—
26C22×C3⋊D15C2×C412yescomplemented
27C22×C5×C3⋊D3C2×C412yescomplemented
28C22×C32×D5C2311yescharacteristic
29C2×D5×C3⋊D3—44no—
30C2×D5×C3⋊D3C2×C414yescomplemented
31C2×D5×C3⋊D3C2314yes—
32C2×C3×ΓL(1,𝔽16)C2318yes—
33ΓL(1,𝔽16)⋊D3—48no—
34C2×C32×C4×C5C2×C411yescharacteristic, complemented
35C4⋊(C3⋊D15)C2×C414yescomplemented
36C5×C4⋊(C3⋊D3)C2×C414yescomplemented
37C2×C32×Dic5C2×C411yescharacteristic, complemented
38C32×C4×D5C2×C412yescomplemented
39C32×C4×D5C2312yes—
40#67140C2×C418yescomplemented
41#66258—91noHall
42C22×C4⋊(C3⋊D3)—51no—
43C3×#926—52no—
44#63974—54no—
45#63894—58no—
46C22×D3×D5—34no—
47C22×D3×D5—62no—
48C2×D3×AGL(1,𝔽5)—316no—
49C2×D60—32no—
50C2×D60—61no—
51C2×C5×D12—32no—
52C2×C5×D12—61no—
53C2×C3×C4×D5—21no—
54C2×C3×C4×D5C2×D312yes—
55C2×#1203—34no—
56C2×#1203—62no—
57D5×D12—34no—
58D5×D12—34no—
59D5×D12—34no—
60D5×D12—34no—
61D5×D12—64no—
62D5×D12—64no—
63#6763C2×D314yes—
64C4⋊ΓL(1,𝔽16)C2×D314yescomplemented
65C4×ΓL(1,𝔽16)C2×D314yescomplemented
66C3×C22⋊AGL(1,𝔽5)—34no—
67C3×C4⋊AGL(1,𝔽5)—34no—
68C3×C4×AGL(1,𝔽5)—34no—
69(C2×D3)⋊AGL(1,𝔽5)—316no—
70#5521—152no—
71C22×C32×C5C22×C411yescharacteristic
72C2×C3⋊D15—42no—
73C2×C3⋊D15C22×C414yes—
74C2×C5×C3⋊D3—42no—
75C2×C5×C3⋊D3C22×C414yes—
76C2×C32×D5—22no—
77C2×C32×D5C22×C411yescharacteristic
78C2×C32×D5C2411yescharacteristic
79D5×C3⋊D3—28no—
80D5×C3⋊D3—44no—
81C3×ΓL(1,𝔽16)—28no—
82C32×C4×C5C22×C412yes—
83C32×Dic5C22×C412yes—
84C2×D4×D5—91no—
85C2×C22⋊AGL(1,𝔽5)—94no—
86D4×AGL(1,𝔽5)—98no—
87#4130—32no—
88C23×C3⋊D3—52no—
89C22×C32×C4—51no—
90C2×C4⋊(C3⋊D3)—52no—
91C2×C4⋊(C3⋊D3)—51no—
92C2×C4⋊(C3⋊D3)—54no—
93C2×C4⋊(C3⋊D3)—54no—
94C2×C4⋊(C3⋊D3)AGL(1,𝔽5)11yescharacteristic, complemented
95C2×#624—58no—
96C3×#217—54no—
97C3×C4⋊Dic3—54no—
98C3×C4×Dic3—54no—
99#3722—58no—
100C22×D15—34no—
101C22×D15—62no—
102C22×C5×D3—34no—
103C22×C5×D3—62no—
104C22×C3×D5—21no—
105C22×C3×D5C22×D312yes—
106C2×D3×D5—38no—
107C2×D3×D5—38no—
108C2×D3×D5—64no—
109C2×D3×D5—64no—
110C2×D3×D5—128no—
111C2×D3×D5—124no—
112C2×D3×D5—124no—
113C2×ΓL(1,𝔽16)C22×D318yes—
114C2×C3×AGL(1,𝔽5)—38no—
115D3×AGL(1,𝔽5)—1216no—
116C2×C3×C4×C5—21no—
117C2×C3×C4×C5C4×D312yescomplemented
118D60—34no—
119D60—34no—
120D60—64no—
121C5×D12—34no—
122C5×D12—34no—
123C5×D12—64no—
124C2×C3×Dic5—21no—
125C2×C3×Dic5C4×D312yescomplemented
126C3×C4×D5—22no—
127C3×C4×D5—22no—
128C3×C4×D5C4×D312yescomplemented
129C3×C4×D5C4×D312yescomplemented
130C3×C4×D5C22×D312yes—
131C3×C4×D5C22×D312yes—
132#1203—38no—
133#1203—38no—
134#1203—68no—
135C22×D12—152no—
136C22×D12—301no—
137C3×#117—152no—
138#926—52no—
139#893—158no—
140D3×C22⋊C4—158no—
141C4×D12—158no—
142C2×C32×C5—21no—
143C2×C32×C5C23×C411yescharacteristic
144C3⋊D15—24no—
145C5×C3⋊D3—24no—
146C32×D5—21no—
147C32×D5H(5,𝔽2)12yes—
148C23×D5—92no—
149C22×AGL(1,𝔽5)—98no—
150C2×C5×D4—91no—
151C2×C22⋊D5—92no—
152D4×D5—94no—
153D4×D5—94no—
154C2×D20—91no—
155C2×C4×D5D3211yescharacteristic
156C22⋊AGL(1,𝔽5)—34no—
157C22⋊AGL(1,𝔽5)—98no—
158C4⋊AGL(1,𝔽5)—34no—
159C4×AGL(1,𝔽5)—34no—
160C23×C32—51no—
161C22×C3⋊D3—52no—
162C22×C3⋊D3—54no—
163C22×C3⋊D3—54no—
164C22×C3⋊D3—204no—
165C22×C3⋊D3C2×AGL(1,𝔽5)12yescomplemented
166C2×C32×C4—51no—
167C2×C32×C4—52no—
168C2×C32×C4—52no—
169C2×C32×C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
170C4⋊(C3⋊D3)—58no—
171C4⋊(C3⋊D3)—54no—
172C4⋊(C3⋊D3)C2×AGL(1,𝔽5)14yescomplemented
173C2×C3×Dic3—58no—
174#624—208no—
175#518—451noSylow
176C22×C3×C5—21no—
177C22×C3×C5C2×C4×D312yes—
178C2×D15—38no—
179C2×D15—64no—
180C2×D15—124no—
181C2×D15—122no—
182C2×D15—122no—
183C2×C5×D3—38no—
184C2×C5×D3—64no—
185C2×C5×D3—124no—
186C2×C5×D3—122no—
187C2×C5×D3—122no—
188C2×C3×D5—22no—
189C2×C3×D5—22no—
190C2×C3×D5—24no—
191C2×C3×D5—21no—
192C2×C3×D5—21no—
193C2×C3×D5C2×C4×D312yes—
194C2×C3×D5C23×D312yes—
195D3×D5—616no—
196D3×D5—128no—
197D3×D5—128no—
198D3×D5—124no—
199D3×D5—124no—
200ΓL(1,𝔽16)—28no—
201C3×AGL(1,𝔽5)—68no—
202C3×C4×C5—22no—
203C3×C4×C5C2×C4×D312yes—
204C3×C4×C5C2×C4×D312yes—
205C3×Dic5—22no—
206C3×Dic5C2×C4×D312yes—
207C3×Dic5C2×C4×D312yes—
208C23×D3—154no—
209C23×D3—302no—
210C22×C3×C4—52no—
211C22×C3×C4—101no—
212C2×D12—32no—
213C2×D12—61no—
214C2×D12—154no—
215C2×D12—154no—
216C2×D12—154no—
217C2×D12—154no—
218C2×D12—154no—
219C2×D12—152no—
220C2×D12—304no—
221C2×D12—304no—
222C2×D12—301no—
223C2×D12—302no—
224C2×C4×D3—1516no—
225C3×C4⋊C4—154no—
226C3×C22⋊C4—154no—
227C3×C42—154no—
228#217—54no—
229(C2×D3)⋊C4—1516no—
230C4⋊Dic3—54no—
231C4×Dic3—54no—
232C32×C5#51811yescharacteristic, complemented, Hall
233C23×C5—92no—
234C22×D5—92no—
235C22×D5—94no—
236C22×D5—94no—
237C22×D5—364no—
238C22×D5C2×D3211yescharacteristic
239C2×AGL(1,𝔽5)—38no—
240C2×AGL(1,𝔽5)—368no—
241C5×D4—94no—
242C2×C4×C5#62411yescharacteristic, complemented
243C22⋊D5—98no—
244C2×Dic5#62411yescharacteristic, complemented
245D20—94no—
246C4×D5#62412yescomplemented
247C4×D5C2×D3212yes—
248C22×C32—51no—
249C22×C32—51no—
250C22×C32—102no—
251C22×C32C22×AGL(1,𝔽5)11yescharacteristic
252C2×C3⋊D3—42no—
253C2×C3⋊D3—54no—
254C2×C3⋊D3—108no—
255C2×C3⋊D3—204no—
256C2×C3⋊D3—202no—
257C2×C3⋊D3C22×AGL(1,𝔽5)14yes—
258C32×C4—52no—
259C32×C4C22×AGL(1,𝔽5)12yes—
260C3×Dic3—108no—
261C22×D4—451no—
262C4×D4—458no—
263#117—152no—
264C2×C22⋊C4—454no—
265C2×C3×C5—21no—
266C2×C3×C5—21no—
267C2×C3×C5—22no—
268C2×C3×C5—21no—
269C2×C3×C5C22×C4×D312yes—
270D15—68no—
271D15—124no—
272C3×D5—21no—
273C3×D5—21no—
274C3×D5—22no—
275C3×D5—22no—
276C3×D5#100914yes—
277C5×D3—68no—
278C5×D3—124no—
279C23×C3—52no—
280C23×C3—101no—
281C22×D3—34no—
282C22×D3—62no—
283C22×D3—158no—
284C22×D3—158no—
285C22×D3—154no—
286C22×D3—304no—
287C22×D3—304no—
288C22×D3—302no—
289C22×D3—608no—
290C22×D3—604no—
291C22×D3—604no—
292C2×C3×C4—21no—
293C2×C3×C4—52no—
294C2×C3×C4—52no—
295C2×C3×C4—52no—
296C2×C3×C4—52no—
297C2×C3×C4—52no—
298C2×C3×C4—102no—
299C2×C3×C4—102no—
300C2×C3×C4—101no—
301C2×C3×C4—158no—
302C2×C3×C4D3×AGL(1,𝔽5)12yescomplemented
303C2×Dic3—58no—
304D12—34no—
305D12—34no—
306D12—64no—
307D12—158no—
308D12—158no—
309D12—154no—
310D12—154no—
311D12—304no—
312D12—308no—
313C4×D3—6016no—
314C22×C5—94no—
315C22×C5—362no—
316C22×C5C2×#62411yescharacteristic
317C2×D5—22no—
318C2×D5—94no—
319C2×D5—188no—
320C2×D5—364no—
321C2×D5—362no—
322C2×D5C2×#62411yescharacteristic
323C2×D5C22×D3211yescharacteristic
324AGL(1,𝔽5)—68no—
325C4×C5C2×#62412yes—
326Dic5C2×#62412yes—
327C2×C32—21no—
328C2×C32—52no—
329C2×C32—101no—
330C2×C32C23×AGL(1,𝔽5)11yescharacteristic
331C3⋊D3—24no—
332C3⋊D3—104no—
333C24—452no—
334C2×D4—91no—
335C2×D4—454no—
336C2×D4—454no—
337C2×D4—452no—
338C2×D4—451no—
339C22×C4—51no—
340C22×C4—458no—
341C4⋊C4—154no—
342C22⋊C4—154no—
343C22⋊C4—458no—
344C42—154no—
345C3×C5—21no—
346C3×C5#552112yescomplemented
347C22×C3—21no—
348C22×C3—52no—
349C22×C3—52no—
350C22×C3—101no—
351C22×C3—101no—
352C22×C3—104no—
353C22×C3—102no—
354C22×C3—102no—
355C22×C3C2×D3×AGL(1,𝔽5)12yes—
356C2×D3—38no—
357C2×D3—64no—
358C2×D3—124no—
359C2×D3—122no—
360C2×D3—122no—
361C2×D3—158no—
362C2×D3—304no—
363C2×D3—3016no—
364C2×D3—608no—
365C2×D3—608no—
366C2×D3—604no—
367C2×D3—604no—
368C2×D3—604no—
369C2×D3—602no—
370C2×D3—602no—
371C3×C4—22no—
372C3×C4—52no—
373C3×C4—52no—
374C3×C4—102no—
375C3×C4—308no—
376C3×C4C2×D3×AGL(1,𝔽5)12yes—
377C3×C4C2×D3×AGL(1,𝔽5)12yes—
378Dic3—108no—
379C2×C5—21no—
380C2×C5—184no—
381C2×C5C22×#62411yescharacteristic
382D5—21no—
383D5—184no—
384D5#6430812yes—
385C32#6625811yescharacteristic, complemented, Sylow
386C23—51no—
387C23—92no—
388C23—454no—
389C23—454no—
390C23—452no—
391C23—1804no—
392D4—94no—
393D4—458no—
394D4—454no—
395C2×C4—51no—
396C2×C4—52no—
397C2×C4—52no—
398C2×C4—158no—
399C2×C4—1808no—
400C2×C4ΓL(1,𝔽16)⋊D311yescharacteristic, complemented
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—22no—
404C2×C3—21no—
405C2×C3—54no—
406C2×C3—102no—
407C2×C3—101no—
408C2×C3—101no—
409C2×C3—102no—
410C2×C3C22×D3×AGL(1,𝔽5)12yes—
411D3—68no—
412D3—124no—
413D3—308no—
414D3—604no—
415C5#1059499611yescharacteristic, complemented, Sylow
416C22—51no—
417C22—51no—
418C22—94no—
419C22—102no—
420C22—362no—
421C22—454no—
422C22—908no—
423C22—1802no—
424C22—1804no—
425C22C2×ΓL(1,𝔽16)⋊D311yescharacteristic
426C4—52no—
427C4—308no—
428C4C2×ΓL(1,𝔽16)⋊D312yes—
429C3—21no—
430C3#1175683712yescomplemented
431C2—21no—
432C2—52no—
433C2—101no—
434C2—184no—
435C2—904no—
436C2C22×ΓL(1,𝔽16)⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×ΓL(1,𝔽16)⋊D3
commutator, socleC2×C32×C5C23×C4
FittingC2×C32×C4×C5C2×C4
prime radicalC2×D5×C4⋊(C3⋊D3)C2
cosocleC2×C32×D5C24
hypercenterC2×C4ΓL(1,𝔽16)⋊D3
hypocenterC32×C5#518
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#91989C2semidirect
3. #13088746C2semidirect
4. #13088732C2semidirect
7. C2×ΓL(1,𝔽16)⋊D3C22semidirect
8. C2×C4⋊(C3⋊D15)C4semidirect
9. C2×C5×C4⋊(C3⋊D3)C4semidirect
11. C2×#67140C4semidirect
12. D5×C4⋊(C3⋊D3)C4semidirect
15. C3×C4⋊ΓL(1,𝔽16)C22semidirect
16. C3×C4×ΓL(1,𝔽16)C22semidirect
17. #10623978C22semidirect
21. #91989D3semidirect
26. C22×C3⋊D15C2×C4semidirect
27. C22×C5×C3⋊D3C2×C4semidirect
30. C2×D5×C3⋊D3C2×C4semidirect
34. C2×C32×C4×C5C2×C4semidirect
35. C4⋊(C3⋊D15)C2×C4semidirect
36. C5×C4⋊(C3⋊D3)C2×C4semidirect
37. C2×C32×Dic5C2×C4semidirect
38. C32×C4×D5C2×C4semidirect
40. #67140C2×C4semidirect
64. C4⋊ΓL(1,𝔽16)C2×D3semidirect
65. C4×ΓL(1,𝔽16)C2×D3semidirect
94. C2×C4⋊(C3⋊D3)AGL(1,𝔽5)semidirect
117. C2×C3×C4×C5C4×D3semidirect
125. C2×C3×Dic5C4×D3semidirect
128. C3×C4×D5C4×D3semidirect
129. C3×C4×D5C4×D3semidirect
165. C22×C3⋊D3C2×AGL(1,𝔽5)semidirect
169. C2×C32×C4C2×AGL(1,𝔽5)semidirect
172. C4⋊(C3⋊D3)C2×AGL(1,𝔽5)semidirect
232. C32×C5#518semidirect
242. C2×C4×C5#624semidirect
244. C2×Dic5#624semidirect
246. C4×D5#624semidirect
302. C2×C3×C4D3×AGL(1,𝔽5)semidirect
346. C3×C5#5521semidirect
385. C32#66258semidirect
400. C2×C4ΓL(1,𝔽16)⋊D3semidirect
415. C5#10594996semidirect
430. C3#11756837semidirect

Maximal subgroups

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

subgroupindex
C2×D5×C4⋊(C3⋊D3)2
C3×#919892
#130887462
#130887322
#117568373
#105949965