Finite Group Databasebeta

← #13088953

Subgroups — #13088953

Order 1440.

710 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 142 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#10624223C211yesmaximal, characteristic, complemented
2C4×D3×D15C212yesmaximal, complemented
3C5×#3798C211yesmaximal, characteristic, complemented
4C4×D5×C3⋊D3C211yesmaximal, characteristic, complemented
5C3×#6771C212yesmaximal, complemented
6#10624056C212yesmaximal, complemented
7#10624047C211yesmaximal, characteristic, complemented
8#10624046C211yesmaximal, characteristic, complemented
9#10624040C211yesmaximal, characteristic, complemented
10#10624035C211yesmaximal, characteristic, complemented
11#10624030C212yesmaximal, complemented
12#92016—32nomaximal
13C2×D3×D15C2212yes—
14C2×D5×C3⋊D3C2211yescharacteristic
15C4×C3⋊D15C2211yescharacteristic
16C4×C5×C3⋊D3C2211yescharacteristic, complemented
17C3×C4×D15C2212yescomplemented
18C3×C4×C5×D3C2212yescomplemented
19C32×C4×D5C2211yescharacteristic, complemented
20#67158C2211yescharacteristic, complemented
21#67157C2211yescharacteristic, complemented
22#67156C2212yescomplemented
23#67149C2212yes—
24D3×Dic15C2212yescomplemented
25Dic3×D15C2212yescomplemented
26C5×#627C2211yescharacteristic, complemented
27C5×#626C2212yescomplemented
28C5×#625C2211yescharacteristic, complemented
29#67137C2211yescharacteristic
30C3⋊D3×Dic5C2211yescharacteristic, complemented
31D5×C3⋊Dic3C2211yescharacteristic, complemented
32C3×#1204C2212yescomplemented
33C3×#1203C2212yescomplemented
34C3×#1202C2212yescomplemented
35C3×#1201C2212yescomplemented
36#64306—51nomaximal, Hall
37C2×C4×D15—32no—
38C5×#235—32no—
39C3×#709—32no—
40#6790—32no—
41#6790—32no—
42#6780—32no—
43#6780—32no—
44#6777—32no—
45C4×D3×D5—32no—
46C4×D3×D5—32no—
47C4×D3×D5—61no—
48#6771—32no—
49#6771D312yescomplemented
50D15⋊Q8—32no—
51#6768—32no—
52#6766—32no—
53C2×C3⋊D15C2311yescharacteristic
54C2×C5×C3⋊D3C2311yescharacteristic
55C2×C3×D15C2312yes—
56C2×C3×C5×D3C2312yes—
57C2×C32×D5C2311yescharacteristic
58D3×D15—24no—
59D5×C3⋊D3—22no—
60C32×C4×C5C2311yescharacteristic
61C3⋊Dic15C2311yescharacteristic
62C5×C3⋊Dic3C2311yescharacteristic
63C3×Dic15C2312yes—
64C3×C5×Dic3C2312yes—
65C32×Dic5C2311yescharacteristic
66D5×#41—91noHall
67C2×C4×C3⋊D3—51no—
68C3×#235—52no—
69#3811—52no—
70#3805—51no—
71#3805—51no—
72#3802—51no—
73C4×D32—52no—
74#3798—51no—
75#3798D511yescharacteristic, complemented
76#3797—51no—
77#3796—52no—
78C22×D15—32no—
79C2×D3×D5—32no—
80C2×D3×D5—32no—
81C2×D3×D5—61no—
82C2×C3×C4×C5—32no—
83C2×Dic15—32no—
84C4×D15—32no—
85C4×D15—32no—
86C4×D15—32no—
87C4×D15—61no—
88C4×D15C2×D312yescomplemented
89C5×#67—32no—
90C5×#67—32no—
91C5×D12—32no—
92C4×C5×D3—32no—
93C4×C5×D3—61no—
94C4×C5×D3C2×D312yescomplemented
95C5×Dic6—32no—
96C3×C22⋊D5—32no—
97C3×C22⋊D5—32no—
98C3×D20—32no—
99C3×C4×D5—21no—
100C3×C4×D5—32no—
101C3×C4×D5C2×D312yescomplemented
102C3×Dic10—32no—
103#1204—32no—
104#1204C2×D312yescomplemented
105#1203—32no—
106#1203C2×D312yescomplemented
107#1202—32no—
108#1202C2×D312yescomplemented
109#1201—32no—
110#1201C2×D312yescomplemented
111#1200—32no—
112#1200—32no—
113#1200—61no—
114D3×Dic5—32no—
115D3×Dic5—32no—
116D3×Dic5—61no—
117D5×Dic3—32no—
118D5×Dic3—32no—
119D5×Dic3—61no—
120D3×#41—152no—
121C2×C32×C5C2411yescharacteristic
122C3⋊D15#4112yescomplemented
123C5×C3⋊D3—21no—
124C3×D15—22no—
125C3×C5×D3—22no—
126C32×D5—21no—
127C5×#41—91no—
128#713—91no—
129Q8×D5—91no—
130#711—92no—
131#711—91no—
132D4×D5—92no—
133D4×D5—91no—
134#709—32no—
135#709—91no—
136C2×C4×D5—92no—
137C2×C4×D5—91no—
138C22×C3⋊D3—51no—
139C2×D32—52no—
140C2×C32×C4—51no—
141C2×C3⋊Dic3—51no—
142C4×C3⋊D3—51no—
143C4×C3⋊D3—51no—
144C4×C3⋊D3—51no—
145C4×C3⋊D3C2×D511yescharacteristic, complemented
146C3×#67—52no—
147C3×#67—52no—
148C3×D12—52no—
149C3×C4×D3—52no—
150C3×C4×D3C2×D512yescomplemented
151C3×Dic6—52no—
152#627—51no—
153#627C2×D511yescharacteristic, complemented
154#626—52no—
155#626C2×D512yescomplemented
156#625—51no—
157#625C2×D511yescharacteristic, complemented
158#624—52no—
159D3×Dic3—52no—
160D3×Dic3—52no—
161C22×C3×C5—32no—
162C2×D15—32no—
163C2×D15—61no—
164C2×D15—64no—
165C2×D15C22×D312yes—
166C2×C5×D3—32no—
167C2×C5×D3—61no—
168C2×C5×D3C22×D312yes—
169C2×C3×D5—21no—
170C2×C3×D5—32no—
171C2×C3×D5C22×D312yes—
172D3×D5—64no—
173D3×D5—64no—
174D3×D5—64no—
175C3×C4×C5—21no—
176C3×C4×C5—32no—
177C3×C4×C5C22×D312yes—
178Dic15—32no—
179Dic15—61no—
180Dic15C22×D312yes—
181C3×Dic5—21no—
182C3×Dic5—32no—
183C3×Dic5C22×D312yes—
184C5×Dic3—32no—
185C5×Dic3—61no—
186C5×Dic3C22×D312yes—
187C3×#41—152no—
188#239—152no—
189D3×Q8—152no—
190#237—152no—
191#237—152no—
192#237—152no—
193D3×D4—152no—
194D3×D4—152no—
195D3×D4—152no—
196#235—32no—
197#235—52no—
198#235—152no—
199C2×C4×D3—152no—
200C2×C4×D3—152no—
201C2×C4×D3—152no—
202C2×C4×D3—301no—
203C32×C5C2×#4111yescharacteristic, complemented, Hall
204C22×D5—92no—
205C22×D5—91no—
206C5×Q8—91no—
207C5×D4—92no—
208C5×D4—91no—
209C2×C4×C5—32no—
210C2×C4×C5—91no—
211C22⋊D5—32no—
212C22⋊D5—32no—
213C22⋊D5—92no—
214C2×Dic5—92no—
215C2×Dic5—91no—
216D20—32no—
217D20—91no—
218C4×D5—32no—
219C4×D5—92no—
220C4×D5—92no—
221C4×D5—91no—
222C4×D5—91no—
223C4×D5—91no—
224C4×D5D3211yescharacteristic, complemented
225Dic10—32no—
226Dic10—91no—
227C22×C32—51no—
228C2×C3⋊D3—51no—
229C2×C3⋊D3—102no—
230C2×C3⋊D3C22×D511yescharacteristic
231C2×C3×D3—52no—
232C2×C3×D3C22×D512yes—
233D32—104no—
234C32×C4—51no—
235C32×C4C22×D511yescharacteristic
236C3⋊Dic3—51no—
237C3⋊Dic3C22×D511yescharacteristic
238C3×Dic3—52no—
239C3×Dic3C22×D512yes—
240C2×#41—451noSylow
241C2×C3×C5—21no—
242C2×C3×C5—62no—
243C2×C3×C5C23×D312yes—
244D15—22no—
245D15—34no—
246D15—62no—
247C3×D5—22no—
248C3×D5—22no—
249C3×D5—62no—
250C5×D3—22no—
251C5×D3—62no—
252C5×D3—62no—
253C22×D3—152no—
254C22×D3—152no—
255C22×D3—152no—
256C22×D3—301no—
257C3×Q8—152no—
258C3×D4—152no—
259C3×D4—152no—
260C3×D4—152no—
261C2×C3×C4—32no—
262C2×C3×C4—52no—
263C2×C3×C4—101no—
264C2×C3×C4—152no—
265#67—32no—
266#67—32no—
267#67—52no—
268#67—52no—
269#67—152no—
270#67—152no—
271C2×Dic3—152no—
272C2×Dic3—152no—
273C2×Dic3—152no—
274C2×Dic3—301no—
275D12—32no—
276D12—52no—
277D12—152no—
278C4×D3—32no—
279C4×D3—52no—
280C4×D3—61no—
281C4×D3—152no—
282C4×D3—152no—
283C4×D3—152no—
284C4×D3—152no—
285C4×D3—152no—
286C4×D3—152no—
287C4×D3—152no—
288C4×D3—301no—
289C4×D3—301no—
290C4×D3—301no—
291C4×D3D3×D512yescomplemented
292Dic6—32no—
293Dic6—52no—
294Dic6—152no—
295C22×C5—32no—
296C22×C5—91no—
297C2×D5—32no—
298C2×D5—91no—
299C2×D5—182no—
300C2×D5—184no—
301C2×D5C2×D3211yescharacteristic
302C4×C5—32no—
303C4×C5—91no—
304C4×C5C2×D3211yescharacteristic
305Dic5—32no—
306Dic5—91no—
307Dic5C2×D3211yescharacteristic
308C2×C32—101no—
309C2×C32C23×D511yescharacteristic
310C3⋊D3—21no—
311C3⋊D3—52no—
312C3×D3—22no—
313C3×D3—102no—
314#41—91no—
315#41—152no—
316#41—451no—
317#41—451no—
318#41—452no—
319#41—451no—
320C2×Q8—451no—
321C2×D4—452no—
322C2×D4—451no—
323C22×C4—452no—
324C22×C4—451no—
325C3×C5—21no—
326C3×C5D3×#4112yescomplemented
327C22×C3—32no—
328C22×C3—52no—
329C22×C3—101no—
330C22×C3—152no—
331C2×D3—32no—
332C2×D3—52no—
333C2×D3—61no—
334C2×D3—152no—
335C2×D3—301no—
336C2×D3—304no—
337C2×D3—304no—
338C2×D3—304no—
339C2×D3—304no—
340C2×D3C2×D3×D512yes—
341C3×C4—21no—
342C3×C4—32no—
343C3×C4—52no—
344C3×C4—101no—
345C3×C4—152no—
346C3×C4C2×D3×D512yes—
347Dic3—32no—
348Dic3—52no—
349Dic3—61no—
350Dic3—152no—
351Dic3—301no—
352Dic3C2×D3×D512yes—
353C2×C5—62no—
354C2×C5—181no—
355C2×C5C22×D3211yescharacteristic
356D5—21no—
357D5—62no—
358D5—92no—
359C32D5×#4111yescharacteristic, complemented, Sylow
360C23—452no—
361C23—451no—
362Q8—91no—
363Q8—152no—
364Q8—451no—
365D4—92no—
366D4—91no—
367D4—152no—
368D4—152no—
369D4—152no—
370D4—451no—
371D4—452no—
372C2×C4—32no—
373C2×C4—51no—
374C2×C4—91no—
375C2×C4—152no—
376C2×C4—452no—
377C2×C4—452no—
378C2×C4—451no—
379C2×C4—451no—
380C2×C4—451no—
381C2×C4—452no—
382C2×C4—451no—
383C2×C3—21no—
384C2×C3—62no—
385C2×C3—102no—
386C2×C3—102no—
387C2×C3—302no—
388C2×C3C22×D3×D512yes—
389D3—22no—
390D3—62no—
391D3—62no—
392D3—102no—
393D3—154no—
394D3—302no—
395C5#6430611yescharacteristic, complemented, Sylow
396C22—32no—
397C22—51no—
398C22—91no—
399C22—152no—
400C22—451no—
401C22—904no—
402C22—902no—
403C4—32no—
404C4—51no—
405C4—91no—
406C4—152no—
407C4—451no—
408C4D32×D511yescharacteristic
409C3—21no—
410C3#9201612yescomplemented
411C2—62no—
412C2—101no—
413C2—181no—
414C2—302no—
415C2—452no—
416C2C2×D32×D511yescharacteristic
subgroupG/H
center, hypercenterC4D32×D5
commutator, socle, cosocleC2×C32×C5C24
FrattiniC2C2×D32×D5
FittingC32×C4×C5C23
hypocenterC32×C5C2×#41
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #10624223C2semidirect
2. C4×D3×D15C2semidirect
3. C5×#3798C2semidirect
4. C4×D5×C3⋊D3C2semidirect
5. C3×#6771C2semidirect
6. #10624056C2semidirect
7. #10624047C2semidirect
8. #10624046C2semidirect
9. #10624040C2semidirect
10. #10624035C2semidirect
11. #10624030C2semidirect
16. C4×C5×C3⋊D3C22semidirect
17. C3×C4×D15C22semidirect
18. C3×C4×C5×D3C22semidirect
19. C32×C4×D5C22semidirect
20. #67158C22semidirect
21. #67157C22semidirect
22. #67156C22semidirect
24. D3×Dic15C22semidirect
25. Dic3×D15C22semidirect
26. C5×#627C22semidirect
27. C5×#626C22semidirect
28. C5×#625C22semidirect
30. C3⋊D3×Dic5C22semidirect
31. D5×C3⋊Dic3C22semidirect
32. C3×#1204C22semidirect
33. C3×#1203C22semidirect
34. C3×#1202C22semidirect
35. C3×#1201C22semidirect
49. #6771D3semidirect
75. #3798D5semidirect
88. C4×D15C2×D3semidirect
94. C4×C5×D3C2×D3semidirect
101. C3×C4×D5C2×D3semidirect
104. #1204C2×D3semidirect
106. #1203C2×D3semidirect
108. #1202C2×D3semidirect
110. #1201C2×D3semidirect
122. C3⋊D15#41semidirect
145. C4×C3⋊D3C2×D5semidirect
150. C3×C4×D3C2×D5semidirect
153. #627C2×D5semidirect
155. #626C2×D5semidirect
157. #625C2×D5semidirect
203. C32×C5C2×#41semidirect
224. C4×D5D32semidirect
291. C4×D3D3×D5semidirect
326. C3×C5D3×#41semidirect
359. C32D5×#41semidirect
395. C5#64306semidirect
410. C3#92016semidirect

Maximal subgroups

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

subgroupindex
#106242232
C4×D3×D152
C5×#37982
C4×D5×C3⋊D32
C3×#67712
#106240562
#106240472
#106240462
#106240402
#106240352
#106240302
#920163
#643065