Finite Group Databasebeta

← #13088999

Subgroups — #13088999

Order 1440.

718 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 140 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#10624236C211yesmaximal, characteristic, complemented
2#10624208C212yesmaximal, complemented
3C5×#3811C211yesmaximal, characteristic, complemented
4C2×#67138C211yesmaximal, characteristic, complemented
5C3×D5×#67C212yesmaximal, complemented
6#10624043C212yesmaximal, complemented
7#10624042C212yesmaximal, complemented
8#10624028C212yesmaximal, complemented
9D3×#1201C212yesmaximal, complemented
10#92025—32nomaximal
11C22×C5×C3⋊D3C2211yescharacteristic, complemented
12C22×C32×D5C2211yescharacteristic, complemented
13C2×(C3⋊D3)⋊D5C2211yescharacteristic, complemented
14C2×C5×D32C2211yescharacteristic, complemented
15C2×C3×D3×D5C2212yescomplemented
16C2×C3⋊Dic15C2211yescharacteristic, complemented
17C3×#1220C2212yescomplemented
18C3×C5×#67C2212yescomplemented
19#67156C2212yescomplemented
20#67155C2211yescharacteristic, complemented
21#67153C2212yescomplemented
22#67150C2212yescomplemented
23D3×Dic15C2212yescomplemented
24Dic3×D15C2212yescomplemented
25C5×#626C2212yescomplemented
26C5×#624C2211yescharacteristic, complemented
27#67138C2212yescomplemented
28#67138C2212yescomplemented
29C3×#1202C2212yescomplemented
30C3×#1201C2212yescomplemented
31C3×D5×Dic3C2212yescomplemented
32#64330—51nomaximal, Hall
33#6819—32no—
34C5×D3×D4—32no—
35C3×D4×D5—32no—
36D3×C22⋊D5—32no—
37D3×C22⋊D5—32no—
38D5×#67—32no—
39D5×#67D312yescomplemented
40C2×#1201—32no—
41C2×#1201—32no—
42C2×#1201—61no—
43#6780—32no—
44#6780—32no—
45#6777—32no—
46#6772—32no—
47#6771—32no—
48#6765—32no—
49C22×C32×C5C2311yescharacteristic
50C2×C5×C3⋊D3—41no—
51C2×C5×C3⋊D3C2311yescharacteristic
52C2×C5×C3⋊D3C2311yescharacteristic
53C2×C3×D15C2312yes—
54C2×C3×C5×D3C2312yes—
55C2×C32×D5—41no—
56C2×C32×D5C2312yes—
57(C3⋊D3)⋊D5—41no—
58C5×D32—41no—
59C3×D3×D5—42no—
60C3⋊Dic15C2312yes—
61C3×Dic15C2312yes—
62C3×C5×Dic3C2312yes—
63#4144—91noHall
64C2×#638—51no—
65C2×C3×#67—52no—
66#3811—51no—
67#3811D511yescharacteristic, complemented
68D3×#67—52no—
69D3×#67—52no—
70#3805—52no—
71#3804—52no—
72#3804—52no—
73C22×C5×D3—32no—
74C22×C5×D3—32no—
75C22×C5×D3—61no—
76C22×C3×D5—21no—
77C22×C3×D5—32no—
78C22×C3×D5C2×D312yescomplemented
79C2×D3×D5—32no—
80C2×D3×D5C2×D312yescomplemented
81C3×C5×D4—32no—
82#1220—32no—
83#1220C2×D312yescomplemented
84C2×Dic15—32no—
85C2×Dic15—32no—
86C2×Dic15—61no—
87C4×D15—32no—
88Dic30—32no—
89C5×#67—32no—
90C5×#67C2×D312yescomplemented
91C5×D12—32no—
92C4×C5×D3—32no—
93C3×C22⋊D5—32no—
94C3×C22⋊D5—32no—
95C3×D20—32no—
96C3×C4×D5—32no—
97#1204—32no—
98#1203—32no—
99#1202C2×D312yescomplemented
100#1201—32no—
101#1201—32no—
102#1201—32no—
103#1201—32no—
104#1201—32no—
105#1201—32no—
106#1201—62no—
107#1201—62no—
108#1201C2×D312yescomplemented
109D3×Dic5—32no—
110D3×Dic5—32no—
111D5×Dic3—32no—
112D5×Dic3C2×D312yescomplemented
113#1004—152no—
114C2×C32×C5—21no—
115C2×C32×C5C2411yescharacteristic
116C5×C3⋊D3—21no—
117C5×C3⋊D3—21no—
118C3×D15—22no—
119C3×C5×D3—22no—
120C32×D5—22no—
121C2×C5×D4—91no—
122C2×C22⋊D5—91no—
123C2×C22⋊D5—92no—
124C2×C22⋊D5—91no—
125#711—92no—
126#711—92no—
127D4×D5—32no—
128D4×D5—92no—
129#709—92no—
130C23×C32—51no—
131C22×C3⋊D3C2×D511yescharacteristic, complemented
132C22×C3×D3—52no—
133C2×D32—51no—
134C2×D32C2×D511yescharacteristic, complemented
135#638—52no—
136#638—52no—
137C2×C3⋊Dic3—51no—
138C3×#67—52no—
139C3×#67—52no—
140C3×#67—52no—
141C3×#67C2×D512yescomplemented
142C2×C3×Dic3—52no—
143#627—52no—
144#626—52no—
145#626C2×D512yescomplemented
146#625—52no—
147#624—51no—
148#624C2×D511yescharacteristic, complemented
149D3×Dic3—52no—
150D3×Dic3—52no—
151C22×C3×C5—21no—
152C22×C3×C5—32no—
153C22×C3×C5C22×D312yes—
154C2×D15C22×D312yes—
155C2×C5×D3—32no—
156C2×C5×D3—32no—
157C2×C5×D3—61no—
158C2×C5×D3—61no—
159C2×C5×D3—122no—
160C2×C5×D3—122no—
161C2×C5×D3—122no—
162C2×C5×D3C22×D312yes—
163C2×C3×D5—22no—
164C2×C3×D5—32no—
165C2×C3×D5—42no—
166C2×C3×D5—42no—
167C2×C3×D5—122no—
168C2×C3×D5C22×D312yes—
169C2×C3×D5C22×D312yes—
170D3×D5—42no—
171D3×D5—122no—
172C3×C4×C5—32no—
173Dic15—32no—
174Dic15—32no—
175Dic15—62no—
176Dic15C22×D312yes—
177C3×Dic5—32no—
178C5×Dic3C22×D312yes—
179C2×C3×D4—152no—
180C2×#67—52no—
181C2×#67—152no—
182C2×#67—152no—
183C2×#67—152no—
184C2×#67—301no—
185#237—152no—
186#237—152no—
187#237—152no—
188#237—152no—
189D3×D4—32no—
190D3×D4—152no—
191D3×D4—152no—
192D3×D4—152no—
193#235—152no—
194#235—152no—
195C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
196C23×C5—91no—
197C23×C5—91no—
198C22×D5—32no—
199C22×D5—91no—
200C22×D5D3211yescharacteristic, complemented
201C5×D4—32no—
202C5×D4—92no—
203C2×C4×C5—91no—
204C22⋊D5—32no—
205C22⋊D5—32no—
206C22⋊D5—92no—
207C22⋊D5—92no—
208C22⋊D5—92no—
209C22⋊D5—92no—
210C2×Dic5—91no—
211C2×Dic5—92no—
212C2×Dic5—91no—
213D20—32no—
214C4×D5—32no—
215C4×D5—92no—
216Dic10—92no—
217C22×C32—52no—
218C22×C32—201no—
219C22×C32C22×D511yescharacteristic
220C2×C3⋊D3—41no—
221C2×C3⋊D3C22×D511yescharacteristic
222C2×C3⋊D3C22×D511yescharacteristic
223C2×C3×D3—52no—
224C2×C3×D3—202no—
225C2×C3×D3C22×D512yes—
226D32—41no—
227D32—201no—
228C3⋊Dic3—52no—
229C3×Dic3—52no—
230C3×Dic3C22×D512yes—
231H(5,𝔽2)—451noSylow
232C2×C3×C5—22no—
233C2×C3×C5—22no—
234C2×C3×C5—21no—
235C2×C3×C5—62no—
236C2×C3×C5C23×D312yes—
237D15—22no—
238C3×D5—22no—
239C3×D5—22no—
240C3×D5—42no—
241C3×D5—62no—
242C5×D3—22no—
243C5×D3—62no—
244C5×D3—62no—
245C5×D3—121no—
246C5×D3—121no—
247C23×C3—52no—
248C23×C3—101no—
249C23×C3—152no—
250C22×D3—32no—
251C22×D3—32no—
252C22×D3—52no—
253C22×D3—61no—
254C22×D3—152no—
255C3×D4—32no—
256C3×D4—152no—
257C3×D4—152no—
258C3×D4—152no—
259C2×C3×C4—152no—
260#67—32no—
261#67—52no—
262#67—52no—
263#67—52no—
264#67—152no—
265#67—152no—
266#67—152no—
267#67—152no—
268#67—152no—
269#67—152no—
270#67—152no—
271#67—302no—
272#67—302no—
273#67D3×D512yescomplemented
274C2×Dic3—52no—
275C2×Dic3—152no—
276C2×Dic3—152no—
277C2×Dic3—152no—
278C2×Dic3—301no—
279D12—32no—
280D12—152no—
281C4×D3—32no—
282C4×D3—152no—
283C4×D3—152no—
284C4×D3—152no—
285Dic6—152no—
286Dic6—152no—
287C22×C5—32no—
288C22×C5—91no—
289C22×C5—91no—
290C22×C5—361no—
291C22×C5—361no—
292C22×C5C2×D3211yescharacteristic
293C2×D5—32no—
294C2×D5—41no—
295C2×D5—122no—
296C2×D5—361no—
297C2×D5C2×D3212yes—
298C4×C5—32no—
299Dic5—32no—
300Dic5—92no—
301C2×C32—21no—
302C2×C32—102no—
303C2×C32C23×D511yescharacteristic
304C3⋊D3—21no—
305C3⋊D3—21no—
306C3×D3—22no—
307C3×D3—102no—
308#41—452no—
309#41—452no—
310#41—452no—
311C2×D4—91no—
312C2×D4—152no—
313C2×D4—452no—
314C2×D4—451no—
315C2×D4—452no—
316C2×D4—451no—
317C3×C5—21no—
318C3×C5#100412yescomplemented
319C22×C3—21no—
320C22×C3—32no—
321C22×C3—52no—
322C22×C3—52no—
323C22×C3—102no—
324C22×C3—152no—
325C22×C3—202no—
326C22×C3—202no—
327C22×C3—602no—
328C22×C3C2×D3×D512yes—
329C2×D3—32no—
330C2×D3—32no—
331C2×D3—52no—
332C2×D3—61no—
333C2×D3—61no—
334C2×D3—122no—
335C2×D3—122no—
336C2×D3—122no—
337C2×D3—202no—
338C2×D3—602no—
339C2×D3C2×D3×D512yes—
340C3×C4—32no—
341C3×C4—152no—
342Dic3—52no—
343Dic3—152no—
344Dic3—152no—
345Dic3—302no—
346Dic3C2×D3×D512yes—
347C2×C5—21no—
348C2×C5—62no—
349C2×C5—181no—
350C2×C5—181no—
351C2×C5C22×D3211yescharacteristic
352D5—22no—
353D5—62no—
354C32#414411yescharacteristic, complemented, Sylow
355C23—51no—
356C23—91no—
357C23—91no—
358C23—152no—
359C23—451no—
360Q8—452no—
361D4—32no—
362D4—92no—
363D4—152no—
364D4—152no—
365D4—152no—
366D4—452no—
367D4—452no—
368D4—452no—
369D4—452no—
370C2×C4—91no—
371C2×C4—152no—
372C2×C4—452no—
373C2×C4—451no—
374C2×C4—452no—
375C2×C4—451no—
376C2×C3—22no—
377C2×C3—22no—
378C2×C3—21no—
379C2×C3—62no—
380C2×C3—102no—
381C2×C3—102no—
382C2×C3—202no—
383C2×C3—302no—
384C2×C3C22×D3×D512yes—
385D3—22no—
386D3—62no—
387D3—62no—
388D3—102no—
389D3—121no—
390D3—121no—
391C5#6433011yescharacteristic, complemented, Sylow
392C22—32no—
393C22—52no—
394C22—91no—
395C22—91no—
396C22—152no—
397C22—201no—
398C22—361no—
399C22—361no—
400C22—602no—
401C22—1801no—
402C22D32×D511yescharacteristic
403C4—32no—
404C4—152no—
405C4—452no—
406C3—21no—
407C3#9202512yescomplemented
408C2—21no—
409C2—62no—
410C2—102no—
411C2—181no—
412C2—181no—
413C2—302no—
414C2C2×D32×D511yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×D5
commutator, socle, cosocleC2×C32×C5C24
FittingC22×C32×C5C23
hypercenterC22D32×D5
hypocenterC32×C5H(5,𝔽2)
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #10624236C2semidirect
2. #10624208C2semidirect
3. C5×#3811C2semidirect
4. C2×#67138C2semidirect
5. C3×D5×#67C2semidirect
6. #10624043C2semidirect
7. #10624042C2semidirect
8. #10624028C2semidirect
9. D3×#1201C2semidirect
11. C22×C5×C3⋊D3C22semidirect
12. C22×C32×D5C22semidirect
13. C2×(C3⋊D3)⋊D5C22semidirect
14. C2×C5×D32C22semidirect
15. C2×C3×D3×D5C22semidirect
16. C2×C3⋊Dic15C22semidirect
17. C3×#1220C22semidirect
18. C3×C5×#67C22semidirect
19. #67156C22semidirect
20. #67155C22semidirect
21. #67153C22semidirect
22. #67150C22semidirect
23. D3×Dic15C22semidirect
24. Dic3×D15C22semidirect
25. C5×#626C22semidirect
26. C5×#624C22semidirect
27. #67138C22semidirect
28. #67138C22semidirect
29. C3×#1202C22semidirect
30. C3×#1201C22semidirect
31. C3×D5×Dic3C22semidirect
39. D5×#67D3semidirect
67. #3811D5semidirect
78. C22×C3×D5C2×D3semidirect
80. C2×D3×D5C2×D3semidirect
83. #1220C2×D3semidirect
90. C5×#67C2×D3semidirect
99. #1202C2×D3semidirect
108. #1201C2×D3semidirect
112. D5×Dic3C2×D3semidirect
131. C22×C3⋊D3C2×D5semidirect
134. C2×D32C2×D5semidirect
141. C3×#67C2×D5semidirect
145. #626C2×D5semidirect
148. #624C2×D5semidirect
195. C32×C5H(5,𝔽2)semidirect
200. C22×D5D32semidirect
273. #67D3×D5semidirect
318. C3×C5#1004semidirect
354. C32#4144semidirect
391. C5#64330semidirect
407. C3#92025semidirect

Maximal subgroups

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

subgroupindex
#106242362
#106242082
C5×#38112
C2×#671382
C3×D5×#672
#106240432
#106240422
#106240282
D3×#12012
#920253
#643305