Finite Group Databasebeta

← #13088986

Subgroups — #13088986

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#10624229C211yesmaximal, characteristic, complemented
2#10624218C212yesmaximal, complemented
3C5×#3805C211yesmaximal, characteristic, complemented
4C2×#67137C211yesmaximal, characteristic, complemented
5C3×#6782C212yesmaximal, complemented
6#10624055C212yesmaximal, complemented
7#10624052C211yesmaximal, characteristic, complemented
8#10624048C211yesmaximal, characteristic, complemented
9#10624040C211yesmaximal, characteristic, complemented
10#10624035C211yesmaximal, characteristic, complemented
11D3×#1200C212yesmaximal, complemented
12#92024—32nomaximal
13C22×C3⋊D15C2211yescharacteristic
14C2×D3×D15C2212yes—
15C2×C5×C3⋊Dic3C2211yescharacteristic, complemented
16C3×#1220C2212yescomplemented
17C3×C5×#67C2212yescomplemented
18C2×C32×Dic5C2211yescharacteristic, complemented
19#67158C2211yescharacteristic, complemented
20#67157C2211yescharacteristic, complemented
21#67154C2212yescomplemented
22#67152C2212yescomplemented
23#67151C2212yescomplemented
24#67149C2212yes—
25C5×#627C2211yescharacteristic, complemented
26C5×#625C2211yescharacteristic, complemented
27C5×D3×Dic3C2212yescomplemented
28#67137C2211yescharacteristic
29#67137C2211yescharacteristic
30#67137C2211yescharacteristic, complemented
31#67137C2211yescharacteristic, complemented
32C3×#1204C2212yescomplemented
33C3×#1203C2212yescomplemented
34C3×#1200C2212yescomplemented
35C3×D3×Dic5C2212yescomplemented
36#64306—51nomaximal, Hall
37D4×D15—32no—
38C5×#237—32no—
39C3×#711—32no—
40#6790—32no—
41#6790—32no—
42C2×#1200—32no—
43C2×#1200—32no—
44C2×#1200—61no—
45#6782—32no—
46#6782D312yescomplemented
47#6779—32no—
48#6779—32no—
49C4×D3×D5—32no—
50#6773—32no—
51D15⋊Q8—32no—
52#6769—32no—
53C22×C32×C5C2311yescharacteristic
54C2×C3⋊D15—22no—
55C2×C3⋊D15C2311yescharacteristic
56C2×C3⋊D15C2311yescharacteristic
57C2×C3×D15C2312yes—
58C2×C3×C5×D3C2312yes—
59D3×D15—24no—
60C5×C3⋊Dic3C2311yescharacteristic
61C5×C3⋊Dic3C2311yescharacteristic
62C3×Dic15C2312yes—
63C3×C5×Dic3C2312yes—
64C32×Dic5C2311yescharacteristic
65C32×Dic5C2311yescharacteristic
66D5×#41—91noHall
67C2×C4×C3⋊D3—51no—
68C3×#235—52no—
69#3811—52no—
70#3805—51no—
71#3805D511yescharacteristic, complemented
72#3802—51no—
73C4×D32—52no—
74#3798—51no—
75#3798—51no—
76#3797—51no—
77#3796—52no—
78C22×D15—32no—
79C22×D15—32no—
80C22×D15—61no—
81C2×D3×D5—32no—
82C3×C5×D4—32no—
83#1220—32no—
84#1220C2×D312yescomplemented
85D60—32no—
86C4×D15—32no—
87C5×#67—32no—
88C5×#67C2×D312yescomplemented
89C2×C5×Dic3—32no—
90C2×C5×Dic3—32no—
91C2×C5×Dic3—61no—
92C4×C5×D3—32no—
93C5×Dic6—32no—
94C3×C22⋊D5—32no—
95C3×C22⋊D5—32no—
96C2×C3×Dic5—21no—
97C2×C3×Dic5—32no—
98C2×C3×Dic5C2×D312yescomplemented
99C3×C4×D5—32no—
100C3×Dic10—32no—
101#1204—32no—
102#1204C2×D312yescomplemented
103#1203—32no—
104#1203C2×D312yescomplemented
105#1202—32no—
106#1202—32no—
107#1200—32no—
108#1200—32no—
109#1200—32no—
110#1200—32no—
111#1200—32no—
112#1200—32no—
113#1200—61no—
114#1200—61no—
115#1200—61no—
116#1200—61no—
117#1200C2×D312yescomplemented
118D3×Dic5C2×D312yescomplemented
119D5×Dic3—32no—
120D3×#41—152no—
121C2×C32×C5—21no—
122C2×C32×C5C2411yescharacteristic
123C3⋊D15—21no—
124C3⋊D15#4112yescomplemented
125C3×D15—22no—
126C3×C5×D3—22no—
127C5×#41—91no—
128#713—91no—
129Q8×D5—91no—
130#711—32no—
131#711—91no—
132D4×D5—91no—
133D4×D5—92no—
134#709—92no—
135#709—91no—
136C2×C4×D5—92no—
137C2×C4×D5—91no—
138C22×C3⋊D3—51no—
139C2×D32—52no—
140C2×C32×C4—51no—
141C2×C3⋊Dic3C2×D511yescharacteristic, complemented
142C4×C3⋊D3—51no—
143C4×C3⋊D3—51no—
144C4×C3⋊D3—51no—
145C4×C3⋊D3—51no—
146C3×#67—52no—
147C3×#67C2×D512yescomplemented
148C3×D12—52no—
149C3×C4×D3—52no—
150C3×C4×D3—52no—
151C3×Dic6—52no—
152#627—51no—
153#627C2×D511yescharacteristic, complemented
154#626—52no—
155#626—52no—
156#625—51no—
157#625C2×D511yescharacteristic, complemented
158#624—52no—
159D3×Dic3—52no—
160D3×Dic3C2×D512yescomplemented
161C22×C3×C5—21no—
162C22×C3×C5—32no—
163C22×C3×C5C22×D312yes—
164C2×D15—32no—
165C2×D15—32no—
166C2×D15—61no—
167C2×D15—61no—
168C2×D15—64no—
169C2×D15—64no—
170C2×D15—64no—
171C2×D15C22×D312yes—
172C2×C5×D3C22×D312yes—
173C2×C3×D5—32no—
174D3×D5—64no—
175C3×C4×C5—32no—
176Dic15C22×D312yes—
177C3×Dic5—21no—
178C3×Dic5—21no—
179C3×Dic5—32no—
180C3×Dic5C22×D312yes—
181C3×Dic5C22×D312yes—
182C5×Dic3—32no—
183C5×Dic3—32no—
184C5×Dic3—61no—
185C5×Dic3—61no—
186C5×Dic3C22×D312yes—
187C3×#41—152no—
188#239—152no—
189D3×Q8—152no—
190#237—32no—
191#237—152no—
192#237—152no—
193D3×D4—152no—
194D3×D4—152no—
195D3×D4—152no—
196#235—52no—
197#235—152no—
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—32no—
208C5×D4—91no—
209C2×C4×C5—92no—
210C2×C4×C5—91no—
211C22⋊D5—32no—
212C22⋊D5—32no—
213C22⋊D5—92no—
214C2×Dic5—32no—
215C2×Dic5D3211yescharacteristic, complemented
216D20—92no—
217D20—91no—
218C4×D5—32no—
219C4×D5—92no—
220C4×D5—92no—
221C4×D5—91no—
222C4×D5—91no—
223C4×D5—91no—
224C4×D5—91no—
225Dic10—32no—
226Dic10—91no—
227C22×C32C22×D511yescharacteristic
228C2×C3⋊D3—51no—
229C2×C3⋊D3—51no—
230C2×C3⋊D3—102no—
231C2×C3×D3—52no—
232C2×C3×D3C22×D512yes—
233D32—104no—
234C32×C4—51no—
235C32×C4—51no—
236C3⋊Dic3C22×D511yescharacteristic
237C3⋊Dic3C22×D511yescharacteristic
238C3×Dic3—52no—
239C3×Dic3C22×D512yes—
240C2×#41—451noSylow
241C2×C3×C5—22no—
242C2×C3×C5—22no—
243C2×C3×C5—21no—
244C2×C3×C5—62no—
245C2×C3×C5C23×D312yes—
246D15—22no—
247D15—34no—
248D15—62no—
249D15—62no—
250D15—62no—
251C3×D5—62no—
252C5×D3—22no—
253C22×D3—152no—
254C22×D3—152no—
255C22×D3—152no—
256C22×D3—301no—
257C3×Q8—152no—
258C3×D4—32no—
259C3×D4—152no—
260C3×D4—152no—
261C2×C3×C4—52no—
262C2×C3×C4—101no—
263C2×C3×C4—152no—
264C2×C3×C4—152no—
265#67—32no—
266#67—52no—
267#67—152no—
268#67—152no—
269#67—152no—
270#67D3×D512yescomplemented
271C2×Dic3—32no—
272C2×Dic3—32no—
273C2×Dic3—61no—
274C2×Dic3—152no—
275D12—52no—
276D12—152no—
277D12—152no—
278C4×D3—32no—
279C4×D3—52no—
280C4×D3—52no—
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×D3—301no—
292Dic6—32no—
293Dic6—52no—
294Dic6—152no—
295C22×C5—32no—
296C22×C5C2×D3211yescharacteristic
297C2×D5—32no—
298C2×D5—91no—
299C2×D5—91no—
300C2×D5—184no—
301C2×D5—182no—
302C4×C5—32no—
303C4×C5—91no—
304C4×C5—91no—
305Dic5—32no—
306Dic5C2×D3211yescharacteristic
307Dic5C2×D3211yescharacteristic
308C2×C32—21no—
309C2×C32C23×D511yescharacteristic
310C3⋊D3—52no—
311C3⋊D3—101no—
312C3×D3—22no—
313C3×D3—102no—
314#41—91no—
315#41—152no—
316#41—451no—
317#41—452no—
318#41—451no—
319#41—451no—
320C2×Q8—451no—
321C2×D4—452no—
322C2×D4—451no—
323C22×C4—451no—
324C22×C4—452no—
325C3×C5—21no—
326C3×C5D3×#4112yescomplemented
327C22×C3—21no—
328C22×C3—32no—
329C22×C3—152no—
330C22×C3C2×D3×D512yes—
331C2×D3—52no—
332C2×D3—152no—
333C2×D3—152no—
334C2×D3—301no—
335C2×D3—301no—
336C2×D3—304no—
337C2×D3—304no—
338C2×D3—304no—
339C2×D3—304no—
340C2×D3C2×D3×D512yes—
341C3×C4—32no—
342C3×C4—52no—
343C3×C4—52no—
344C3×C4—101no—
345C3×C4—101no—
346C3×C4—152no—
347Dic3—32no—
348Dic3—32no—
349Dic3—52no—
350Dic3—61no—
351Dic3—61no—
352Dic3C2×D3×D512yes—
353C2×C5—21no—
354C2×C5—62no—
355C2×C5C22×D3211yescharacteristic
356D5—62no—
357D5—92no—
358D5—181no—
359C32D5×#4111yescharacteristic, complemented, Sylow
360C23—452no—
361C23—451no—
362Q8—91no—
363Q8—152no—
364Q8—451no—
365D4—32no—
366D4—91no—
367D4—152no—
368D4—152no—
369D4—452no—
370D4—451no—
371D4—452no—
372C2×C4—51no—
373C2×C4—92no—
374C2×C4—91no—
375C2×C4—152no—
376C2×C4—152no—
377C2×C4—452no—
378C2×C4—451no—
379C2×C4—451no—
380C2×C4—451no—
381C2×C4—451no—
382C2×C4—452no—
383C2×C3—22no—
384C2×C3—22no—
385C2×C3—21no—
386C2×C3—62no—
387C2×C3—302no—
388C2×C3C22×D3×D512yes—
389D3—22no—
390D3—102no—
391D3—154no—
392D3—302no—
393D3—302no—
394D3—302no—
395C5#6430611yescharacteristic, complemented, Sylow
396C22—32no—
397C22—152no—
398C22—451no—
399C22—451no—
400C22—902no—
401C22—904no—
402C22D32×D511yescharacteristic
403C4—32no—
404C4—51no—
405C4—51no—
406C4—91no—
407C4—91no—
408C4—152no—
409C3—21no—
410C3#9202412yescomplemented
411C2—21no—
412C2—62no—
413C2—302no—
414C2—452no—
415C2—901no—
416C2C2×D32×D511yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×D5
commutator, socle, cosocleC2×C32×C5C24
FittingC22×C32×C5C23
hypercenterC22D32×D5
hypocenterC32×C5C2×#41
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #10624229C2semidirect
2. #10624218C2semidirect
3. C5×#3805C2semidirect
4. C2×#67137C2semidirect
5. C3×#6782C2semidirect
6. #10624055C2semidirect
7. #10624052C2semidirect
8. #10624048C2semidirect
9. #10624040C2semidirect
10. #10624035C2semidirect
11. D3×#1200C2semidirect
15. C2×C5×C3⋊Dic3C22semidirect
16. C3×#1220C22semidirect
17. C3×C5×#67C22semidirect
18. C2×C32×Dic5C22semidirect
19. #67158C22semidirect
20. #67157C22semidirect
21. #67154C22semidirect
22. #67152C22semidirect
23. #67151C22semidirect
25. C5×#627C22semidirect
26. C5×#625C22semidirect
27. C5×D3×Dic3C22semidirect
30. #67137C22semidirect
31. #67137C22semidirect
32. C3×#1204C22semidirect
33. C3×#1203C22semidirect
34. C3×#1200C22semidirect
35. C3×D3×Dic5C22semidirect
46. #6782D3semidirect
71. #3805D5semidirect
84. #1220C2×D3semidirect
88. C5×#67C2×D3semidirect
98. C2×C3×Dic5C2×D3semidirect
102. #1204C2×D3semidirect
104. #1203C2×D3semidirect
117. #1200C2×D3semidirect
118. D3×Dic5C2×D3semidirect
124. C3⋊D15#41semidirect
141. C2×C3⋊Dic3C2×D5semidirect
147. C3×#67C2×D5semidirect
153. #627C2×D5semidirect
157. #625C2×D5semidirect
160. D3×Dic3C2×D5semidirect
203. C32×C5C2×#41semidirect
215. C2×Dic5D32semidirect
270. #67D3×D5semidirect
326. C3×C5D3×#41semidirect
359. C32D5×#41semidirect
395. C5#64306semidirect
410. C3#92024semidirect

Maximal subgroups

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

subgroupindex
#106242292
#106242182
C5×#38052
C2×#671372
C3×#67822
#106240552
#106240522
#106240482
#106240402
#106240352
D3×#12002
#920243
#643065