Finite Group Databasebeta

← #13088991

Subgroups — #13088991

Order 1440.

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

normal subgroups only

This group has 704 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
1C2×#67157C211yesmaximal, characteristic, complemented
2#67×D15C212yesmaximal, complemented
3C5×#3805C211yesmaximal, characteristic, complemented
4#10624162C211yesmaximal, characteristic, complemented
5C3×#6790C212yesmaximal, complemented
6#10624053C212yesmaximal, complemented
7#10624052C211yesmaximal, characteristic, complemented
8#10624047C211yesmaximal, characteristic, complemented
9#10624035C211yesmaximal, characteristic, complemented
10D3×#1202C212yesmaximal, complemented
11D5×#625C211yesmaximal, characteristic, complemented
12#92026—32nomaximal
13C22×C3×D15C2212yescomplemented
14C2×D3×D15C2212yescomplemented
15C2×C3×D3×D5C2212yescomplemented
16#67184C2211yescharacteristic, complemented
17C2×C5×C3⋊Dic3C2211yescharacteristic, complemented
18C3×C5×#67C2212yescomplemented
19C32×C22⋊D5C2211yescharacteristic, complemented
20#67157C2211yescharacteristic, complemented
21#67157C2212yescomplemented
22#67157C2211yescharacteristic, complemented
23#67151C2212yescomplemented
24#67150C2212yescomplemented
25Dic3×D15C2212yescomplemented
26C5×#627C2211yescharacteristic, complemented
27C5×#625C2211yescharacteristic, complemented
28C5×D3×Dic3C2212yescomplemented
29#67141C2211yescharacteristic, complemented
30#67139C2211yescharacteristic, complemented
31#67137C2211yescharacteristic, complemented
32D5×C3⋊Dic3C2211yescharacteristic, complemented
33C3×#1203C2212yescomplemented
34C3×#1202C2212yescomplemented
35C3×#1200C2212yescomplemented
36#64313—51nomaximal, Hall
37D4×D15—32no—
38C5×#237—32no—
39C3×D4×D5—32no—
40#6790—32no—
41#6790D312yescomplemented
42D5×#67—32no—
43D5×#67—32no—
44C2×#1202—32no—
45C2×#1202—32no—
46#6779—32no—
47#6779—32no—
48#6779—61no—
49D3×D20—32no—
50#6773—32no—
51#6771—32no—
52#6766—32no—
53C22×C32×C5C2311yescharacteristic
54C2×C3⋊D15C2311yescharacteristic
55C2×C3×D15—42no—
56C2×C3×D15C2312yes—
57C2×C3×D15C2312yes—
58C2×C3×C5×D3C2312yes—
59C2×C32×D5C2311yescharacteristic
60D3×D15—42no—
61C3×D3×D5—42no—
62C3⋊Dic15C2311yescharacteristic
63C5×C3⋊Dic3C2311yescharacteristic
64C5×C3⋊Dic3C2311yescharacteristic
65C3×C5×Dic3C2312yes—
66C32×Dic5C2311yescharacteristic
67#4149—91noHall
68#3830—51no—
69C3×D3×D4—52no—
70D3×#67—52no—
71D3×#67—52no—
72C2×#625—51no—
73C2×#625—51no—
74#3805—51no—
75#3805D511yescharacteristic, complemented
76#3802—51no—
77#3798—51no—
78#3795—52no—
79C22×D15—32no—
80C22×D15C2×D312yescomplemented
81C22×C3×D5—32no—
82C22×C3×D5—32no—
83C2×D3×D5—32no—
84C2×D3×D5C2×D312yescomplemented
85C3×C5×D4—32no—
86#1220—32no—
87#1220—32no—
88#1220—61no—
89D60—32no—
90C4×D15—32no—
91C5×#67—32no—
92C5×#67C2×D312yescomplemented
93C2×C5×Dic3—32no—
94C2×C5×Dic3—32no—
95C2×C5×Dic3—61no—
96C4×C5×D3—32no—
97C5×Dic6—32no—
98C3×C22⋊D5—21no—
99C3×C22⋊D5—32no—
100C3×C22⋊D5C2×D312yescomplemented
101C3×D20—32no—
102C3×C4×D5—32no—
103#1204—32no—
104#1204—61no—
105#1203C2×D312yescomplemented
106#1202—32no—
107#1202—32no—
108#1202—32no—
109#1202—32no—
110#1202—32no—
111#1202—32no—
112#1202—61no—
113#1202C2×D312yescomplemented
114#1201—32no—
115#1200—32no—
116#1200—61no—
117#1200C2×D312yescomplemented
118D5×Dic3—32no—
119D5×Dic3—32no—
120D5×Dic3—61no—
121#1004—152no—
122C2×C32×C5—21no—
123C2×C32×C5C2411yescharacteristic
124C3⋊D15—21no—
125C3×D15—22no—
126C3×D15—22no—
127C3×C5×D3—22no—
128C32×D5—21no—
129C5×#41—91no—
130#713—91no—
131#713—91no—
132D4×D5—32no—
133D4×D5—91no—
134D4×D5—92no—
135D4×D5—91no—
136#709—92no—
137#709—91no—
138C2×D20—92no—
139C2×D20—91no—
140C22×C3×D3—52no—
141C22×C3×D3—52no—
142C2×D32—52no—
143C32×D4—51no—
144#638—51no—
145#638—51no—
146C2×C3⋊Dic3—51no—
147C2×C3⋊Dic3C2×D511yescharacteristic, complemented
148C4×C3⋊D3—51no—
149#634—51no—
150C3×#67—52no—
151C3×#67C2×D512yescomplemented
152C3×D12—52no—
153C3×C4×D3—52no—
154#627C2×D511yescharacteristic, complemented
155#626—52no—
156#625—52no—
157#625—52no—
158#625—51no—
159#625—51no—
160#625C2×D511yescharacteristic, complemented
161D3×Dic3—52no—
162D3×Dic3C2×D512yescomplemented
163C22×C3×C5—21no—
164C22×C3×C5—32no—
165C22×C3×C5C22×D312yes—
166C2×D15—32no—
167C2×D15—42no—
168C2×D15—61no—
169C2×D15—122no—
170C2×D15C22×D312yes—
171C2×D15C22×D312yes—
172C2×C5×D3C22×D312yes—
173C2×C3×D5—21no—
174C2×C3×D5—32no—
175C2×C3×D5—32no—
176C2×C3×D5—122no—
177C2×C3×D5—122no—
178C2×C3×D5C22×D312yes—
179D3×D5—42no—
180D3×D5—122no—
181C3×C4×C5—32no—
182Dic15—32no—
183Dic15—61no—
184C3×Dic5—21no—
185C3×Dic5C22×D312yes—
186C5×Dic3—32no—
187C5×Dic3—32no—
188C5×Dic3—61no—
189C5×Dic3—61no—
190C5×Dic3C22×D312yes—
191C2×C3×D4—152no—
192C2×#67—152no—
193C2×#67—152no—
194C2×#67—152no—
195C2×#67—152no—
196#237—32no—
197#237—152no—
198#237—152no—
199#237—152no—
200#237—301no—
201D3×D4—52no—
202D3×D4—152no—
203D3×D4—152no—
204D3×D4—152no—
205#235—152no—
206#235—152no—
207C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
208C22×D5—32no—
209C22×D5—32no—
210C22×D5—92no—
211C5×Q8—91no—
212C5×D4—32no—
213C5×D4—91no—
214C2×C4×C5—92no—
215C2×C4×C5—91no—
216C22⋊D5—32no—
217C22⋊D5—92no—
218C22⋊D5—91no—
219C22⋊D5D3211yescharacteristic, complemented
220D20—32no—
221D20—92no—
222D20—92no—
223D20—91no—
224D20—91no—
225D20—91no—
226C4×D5—32no—
227C4×D5—92no—
228C4×D5—91no—
229C4×D5—91no—
230Dic10—91no—
231C22×C32—51no—
232C22×C32C22×D511yescharacteristic
233C2×C3⋊D3—51no—
234C2×C3×D3—52no—
235C2×C3×D3—52no—
236C2×C3×D3—202no—
237C2×C3×D3—202no—
238C2×C3×D3C22×D512yes—
239D32—202no—
240C32×C4—51no—
241C3⋊Dic3—51no—
242C3⋊Dic3C22×D511yescharacteristic
243C3⋊Dic3C22×D511yescharacteristic
244C3×Dic3C22×D512yes—
245H(5,𝔽2)—451noSylow
246C2×C3×C5—22no—
247C2×C3×C5—21no—
248C2×C3×C5—41no—
249C2×C3×C5—62no—
250C2×C3×C5C23×D312yes—
251D15—22no—
252D15—22no—
253D15—62no—
254D15—121no—
255C3×D5—22no—
256C3×D5—41no—
257C3×D5—62no—
258C3×D5—62no—
259C5×D3—22no—
260C23×C3—152no—
261C23×C3—152no—
262C22×D3—52no—
263C22×D3—52no—
264C22×D3—152no—
265C22×D3—152no—
266C3×D4—32no—
267C3×D4—52no—
268C3×D4—101no—
269C3×D4—152no—
270C3×D4—152no—
271C2×C3×C4—152no—
272#67—32no—
273#67—52no—
274#67—152no—
275#67—152no—
276#67—152no—
277#67—152no—
278#67—152no—
279#67—152no—
280#67—152no—
281#67—152no—
282#67—152no—
283#67—301no—
284#67—301no—
285#67D3×D512yescomplemented
286C2×Dic3—32no—
287C2×Dic3—32no—
288C2×Dic3—61no—
289C2×Dic3—152no—
290C2×Dic3—152no—
291C2×Dic3—301no—
292D12—52no—
293D12—152no—
294C4×D3—32no—
295C4×D3—52no—
296C4×D3—152no—
297C4×D3—152no—
298C4×D3—301no—
299Dic6—32no—
300Dic6—152no—
301Dic6—301no—
302C22×C5—32no—
303C22×C5C2×D3211yescharacteristic
304C2×D5—32no—
305C2×D5—32no—
306C2×D5—91no—
307C2×D5—122no—
308C2×D5—122no—
309C2×D5—362no—
310C2×D5C2×D3211yescharacteristic
311C4×C5—32no—
312C4×C5—91no—
313C4×C5—91no—
314Dic5—91no—
315Dic5C2×D3211yescharacteristic
316C2×C32—21no—
317C2×C32—101no—
318C2×C32C23×D511yescharacteristic
319C3⋊D3—101no—
320C3×D3—22no—
321C3×D3—102no—
322C3×D3—102no—
323#41—91no—
324#41—452no—
325#41—451no—
326#41—451no—
327#41—451no—
328C2×D4—152no—
329C2×D4—452no—
330C2×D4—451no—
331C2×D4—452no—
332C2×D4—451no—
333C2×D4—451no—
334C3×C5—21no—
335C3×C5#100412yescomplemented
336C22×C3—21no—
337C22×C3—32no—
338C22×C3—52no—
339C22×C3—101no—
340C22×C3—152no—
341C22×C3—152no—
342C22×C3—602no—
343C22×C3—602no—
344C22×C3C2×D3×D512yes—
345C2×D3—52no—
346C2×D3—52no—
347C2×D3—152no—
348C2×D3—202no—
349C2×D3—202no—
350C2×D3—301no—
351C2×D3—602no—
352C2×D3—602no—
353C2×D3C2×D3×D512yes—
354C3×C4—32no—
355C3×C4—52no—
356C3×C4—101no—
357Dic3—32no—
358Dic3—32no—
359Dic3—61no—
360Dic3—61no—
361Dic3—152no—
362Dic3—301no—
363Dic3C2×D3×D512yes—
364C2×C5—21no—
365C2×C5—62no—
366C2×C5C22×D3211yescharacteristic
367D5—21no—
368D5—62no—
369D5—62no—
370D5—181no—
371C32#414911yescharacteristic, complemented, Sylow
372C23—152no—
373C23—152no—
374C23—452no—
375Q8—91no—
376Q8—451no—
377D4—32no—
378D4—51no—
379D4—91no—
380D4—152no—
381D4—152no—
382D4—452no—
383D4—451no—
384D4—452no—
385D4—451no—
386D4—451no—
387D4—452no—
388D4—451no—
389C2×C4—92no—
390C2×C4—91no—
391C2×C4—152no—
392C2×C4—452no—
393C2×C4—451no—
394C2×C4—451no—
395C2×C3—22no—
396C2×C3—21no—
397C2×C3—41no—
398C2×C3—62no—
399C2×C3—102no—
400C2×C3—201no—
401C2×C3—302no—
402C2×C3—302no—
403C2×C3C22×D3×D512yes—
404D3—22no—
405D3—102no—
406D3—102no—
407D3—302no—
408D3—601no—
409C5#6431311yescharacteristic, complemented, Sylow
410C22—32no—
411C22—51no—
412C22—152no—
413C22—152no—
414C22—451no—
415C22—602no—
416C22—602no—
417C22—1802no—
418C22D32×D511yescharacteristic
419C4—32no—
420C4—51no—
421C4—91no—
422C4—91no—
423C4—451no—
424C3—21no—
425C3#9202612yescomplemented
426C2—21no—
427C2—62no—
428C2—101no—
429C2—302no—
430C2—302no—
431C2—901no—
432C2C2×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. C2×#67157C2semidirect
2. #67×D15C2semidirect
3. C5×#3805C2semidirect
4. #10624162C2semidirect
5. C3×#6790C2semidirect
6. #10624053C2semidirect
7. #10624052C2semidirect
8. #10624047C2semidirect
9. #10624035C2semidirect
10. D3×#1202C2semidirect
11. D5×#625C2semidirect
13. C22×C3×D15C22semidirect
14. C2×D3×D15C22semidirect
15. C2×C3×D3×D5C22semidirect
16. #67184C22semidirect
17. C2×C5×C3⋊Dic3C22semidirect
18. C3×C5×#67C22semidirect
19. C32×C22⋊D5C22semidirect
20. #67157C22semidirect
21. #67157C22semidirect
22. #67157C22semidirect
23. #67151C22semidirect
24. #67150C22semidirect
25. Dic3×D15C22semidirect
26. C5×#627C22semidirect
27. C5×#625C22semidirect
28. C5×D3×Dic3C22semidirect
29. #67141C22semidirect
30. #67139C22semidirect
31. #67137C22semidirect
32. D5×C3⋊Dic3C22semidirect
33. C3×#1203C22semidirect
34. C3×#1202C22semidirect
35. C3×#1200C22semidirect
41. #6790D3semidirect
75. #3805D5semidirect
80. C22×D15C2×D3semidirect
84. C2×D3×D5C2×D3semidirect
92. C5×#67C2×D3semidirect
100. C3×C22⋊D5C2×D3semidirect
105. #1203C2×D3semidirect
113. #1202C2×D3semidirect
117. #1200C2×D3semidirect
147. C2×C3⋊Dic3C2×D5semidirect
151. C3×#67C2×D5semidirect
154. #627C2×D5semidirect
160. #625C2×D5semidirect
162. D3×Dic3C2×D5semidirect
207. C32×C5H(5,𝔽2)semidirect
219. C22⋊D5D32semidirect
285. #67D3×D5semidirect
335. C3×C5#1004semidirect
371. C32#4149semidirect
409. C5#64313semidirect
425. C3#92026semidirect

Maximal subgroups

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

subgroupindex
C2×#671572
#67×D152
C5×#38052
#106241622
C3×#67902
#106240532
#106240522
#106240472
#106240352
D3×#12022
D5×#6252
#920263
#643135