Finite Group Databasebeta

← #13089010

Subgroups — #13089010

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#10624218C212yesmaximal, complemented
3C5×#3811C211yesmaximal, characteristic, complemented
4C2×#67140C211yesmaximal, characteristic, complemented
5C3×#6782C212yesmaximal, complemented
6#10624054C212yesmaximal, complemented
7#10624051C212yesmaximal, complemented
8#10624033C212yesmaximal, complemented
9D3×#1203C212yesmaximal, complemented
10#92027—32nomaximal
11C22×C3⋊D15C2211yescharacteristic, complemented
12C22×C5×C3⋊D3C2211yescharacteristic, complemented
13C2×(C3⋊D3)⋊D5C2211yescharacteristic, complemented
14C2×D3×D15C2212yescomplemented
15C2×C5×D32C2211yescharacteristic, complemented
16C3×#1220C2212yescomplemented
17C3×C5×#67C2212yescomplemented
18C2×C32×Dic5C2211yescharacteristic, complemented
19#67156C2212yescomplemented
20#67155C2211yescharacteristic, complemented
21#67152C2212yescomplemented
22#67151C2212yescomplemented
23#67149C2212yescomplemented
24C5×#626C2212yescomplemented
25C5×#624C2211yescharacteristic, complemented
26#67140C2212yescomplemented
27#67140C2212yescomplemented
28C3×#1204C2212yescomplemented
29C3×#1203C2212yescomplemented
30C3×#1200C2212yescomplemented
31C3×D3×Dic5C2212yescomplemented
32#64308—51nomaximal, Hall
33D4×D15—32no—
34C5×D3×D4—32no—
35C3×#711—32no—
36#6790—32no—
37#6790—32no—
38D3×C22⋊D5—32no—
39D3×C22⋊D5—32no—
40C2×#1203—32no—
41C2×#1203—32no—
42C2×#1203—61no—
43#6782—32no—
44#6782D312yescomplemented
45D5×D12—32no—
46#6771—32no—
47#6769—32no—
48#6768—32no—
49C22×C32×C5C2311yescharacteristic
50C2×C3⋊D15—41no—
51C2×C3⋊D15C2312yes—
52C2×C5×C3⋊D3—41no—
53C2×C5×C3⋊D3C2311yescharacteristic
54C2×C5×C3⋊D3C2311yescharacteristic
55C2×C3×D15C2312yes—
56C2×C3×C5×D3C2312yes—
57(C3⋊D3)⋊D5—41no—
58D3×D15—42no—
59C5×D32—41no—
60C3×Dic15C2312yes—
61C3×C5×Dic3C2312yes—
62C32×Dic5C2312yes—
63#4144—91noHall
64C2×C4⋊(C3⋊D3)—51no—
65C3×#235—52no—
66#3811—51no—
67#3811—52no—
68#3811D511yescharacteristic, complemented
69D3×D12—52no—
70D3×D12—52no—
71#3799—52no—
72#3799—52no—
73C22×D15—32no—
74C22×D15—32no—
75C22×D15—61no—
76C22×C5×D3—32no—
77C22×C5×D3—32no—
78C22×C5×D3—61no—
79C2×D3×D5—32no—
80C2×D3×D5—32no—
81C3×C5×D4—32no—
82#1220—32no—
83#1220C2×D312yescomplemented
84D60—32no—
85C4×D15—32no—
86C5×#67—32no—
87C5×#67C2×D312yescomplemented
88C5×D12—32no—
89C4×C5×D3—32no—
90C3×C22⋊D5—32no—
91C3×C22⋊D5—32no—
92C2×C3×Dic5—21no—
93C2×C3×Dic5—32no—
94C2×C3×Dic5C2×D312yescomplemented
95C3×C4×D5—32no—
96C3×Dic10—32no—
97#1204C2×D312yescomplemented
98#1203—32no—
99#1203—32no—
100#1203—32no—
101#1203—32no—
102#1203—32no—
103#1203—32no—
104#1203—62no—
105#1203—62no—
106#1203C2×D312yescomplemented
107#1202—32no—
108#1201—32no—
109#1200—32no—
110#1200C2×D312yescomplemented
111D3×Dic5—32no—
112D3×Dic5C2×D312yescomplemented
113#1009—152no—
114C2×C32×C5—21no—
115C2×C32×C5C2411yescharacteristic
116C3⋊D15—22no—
117C5×C3⋊D3—21no—
118C5×C3⋊D3—21no—
119C3×D15—22no—
120C3×C5×D3—22no—
121C2×C5×D4—91no—
122C2×C22⋊D5—92no—
123C2×C22⋊D5—91no—
124C2×C22⋊D5—91no—
125#711—32no—
126#711—92no—
127D4×D5—92no—
128D4×D5—92no—
129#709—92no—
130C22×C3⋊D3—51no—
131C22×C3⋊D3C2×D511yescharacteristic, complemented
132C2×D32—51no—
133C2×D32—52no—
134C2×D32C2×D511yescharacteristic, complemented
135C2×C32×C4—51no—
136C4⋊(C3⋊D3)—52no—
137C4⋊(C3⋊D3)—52no—
138C3×#67—52no—
139C3×#67C2×D512yescomplemented
140C3×D12—52no—
141C3×C4×D3—52no—
142C3×C4×D3—52no—
143C3×Dic6—52no—
144#626—52no—
145#626—52no—
146#626—52no—
147#626C2×D512yescomplemented
148#624—51no—
149#624—52no—
150#624C2×D511yescharacteristic, complemented
151C22×C3×C5—21no—
152C22×C3×C5—32no—
153C22×C3×C5C22×D312yes—
154C2×D15—32no—
155C2×D15—32no—
156C2×D15—62no—
157C2×D15—122no—
158C2×D15—122no—
159C2×D15—122no—
160C2×D15C22×D312yes—
161C2×C5×D3—32no—
162C2×C5×D3—32no—
163C2×C5×D3—61no—
164C2×C5×D3—61no—
165C2×C5×D3—122no—
166C2×C5×D3—122no—
167C2×C5×D3—122no—
168C2×C5×D3C22×D312yes—
169C2×C3×D5—32no—
170D3×D5—122no—
171D3×D5—122no—
172C3×C4×C5—32no—
173Dic15C22×D312yes—
174C3×Dic5—22no—
175C3×Dic5—32no—
176C3×Dic5C22×D312yes—
177C3×Dic5C22×D312yes—
178C5×Dic3C22×D312yes—
179C3×#41—152no—
180#239—152no—
181#239—152no—
182D3×D4—32no—
183D3×D4—152no—
184D3×D4—152no—
185D3×D4—152no—
186D3×D4—152no—
187D3×D4—152no—
188#235—52no—
189#235—152no—
190#235—152no—
191C2×D12—152no—
192C2×D12—152no—
193C2×D12—152no—
194C2×D12—301no—
195C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
196C23×C5—91no—
197C23×C5—91no—
198C22×D5—91no—
199C22×D5—92no—
200C22×D5—91no—
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—32no—
211C2×Dic5—91no—
212C2×Dic5D3211yescharacteristic, complemented
213D20—92no—
214C4×D5—32no—
215C4×D5—92no—
216Dic10—32no—
217C22×C32C22×D511yescharacteristic
218C2×C3⋊D3—41no—
219C2×C3⋊D3—52no—
220C2×C3⋊D3—201no—
221C2×C3⋊D3C22×D511yescharacteristic
222C2×C3⋊D3C22×D511yescharacteristic
223C2×C3×D3—52no—
224C2×C3×D3C22×D512yes—
225D32—41no—
226D32—202no—
227D32—201no—
228C32×C4—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—
238D15—62no—
239D15—62no—
240D15—122no—
241C3×D5—62no—
242C5×D3—22no—
243C5×D3—62no—
244C5×D3—62no—
245C5×D3—121no—
246C5×D3—121no—
247C22×D3—32no—
248C22×D3—32no—
249C22×D3—61no—
250C22×D3—152no—
251C22×D3—152no—
252C22×D3—152no—
253C22×D3—152no—
254C22×D3—301no—
255C3×Q8—152no—
256C3×D4—32no—
257C3×D4—152no—
258C3×D4—152no—
259C2×C3×C4—52no—
260C2×C3×C4—101no—
261C2×C3×C4—152no—
262C2×C3×C4—152no—
263#67—32no—
264#67—52no—
265#67—152no—
266#67—152no—
267#67—152no—
268#67D3×D512yescomplemented
269D12—32no—
270D12—52no—
271D12—152no—
272D12—152no—
273D12—152no—
274D12—152no—
275D12—152no—
276D12—152no—
277D12—152no—
278D12—302no—
279D12—302no—
280C4×D3—32no—
281C4×D3—52no—
282C4×D3—52no—
283C4×D3—152no—
284C4×D3—152no—
285C4×D3—152no—
286Dic6—52no—
287C22×C5—32no—
288C22×C5—91no—
289C22×C5—91no—
290C22×C5—361no—
291C22×C5—361no—
292C22×C5C2×D3211yescharacteristic
293C2×D5—32no—
294C2×D5—92no—
295C2×D5—361no—
296C2×D5—362no—
297C2×D5—361no—
298C4×C5—32no—
299Dic5—32no—
300Dic5C2×D3212yes—
301C2×C32—21no—
302C2×C32C23×D511yescharacteristic
303C3⋊D3—21no—
304C3⋊D3—21no—
305C3⋊D3—102no—
306C3×D3—22no—
307C3×D3—102no—
308#41—152no—
309#41—452no—
310#41—452no—
311C2×D4—91no—
312C2×D4—451no—
313C2×D4—452no—
314C2×D4—452no—
315C2×D4—452no—
316C2×D4—451no—
317C3×C5—21no—
318C3×C5#100912yescomplemented
319C22×C3—21no—
320C22×C3—32no—
321C22×C3—152no—
322C22×C3C2×D3×D512yes—
323C2×D3—32no—
324C2×D3—32no—
325C2×D3—52no—
326C2×D3—61no—
327C2×D3—61no—
328C2×D3—122no—
329C2×D3—122no—
330C2×D3—122no—
331C2×D3—152no—
332C2×D3—152no—
333C2×D3—302no—
334C2×D3—602no—
335C2×D3—602no—
336C2×D3—602no—
337C2×D3—602no—
338C2×D3—602no—
339C2×D3C2×D3×D512yes—
340C3×C4—32no—
341C3×C4—52no—
342C3×C4—52no—
343C3×C4—102no—
344C3×C4—152no—
345Dic3—52no—
346Dic3C2×D3×D512yes—
347C2×C5—21no—
348C2×C5—62no—
349C2×C5—181no—
350C2×C5—181no—
351C2×C5C22×D3211yescharacteristic
352D5—62no—
353D5—182no—
354C32#414411yescharacteristic, complemented, Sylow
355C23—91no—
356C23—91no—
357C23—452no—
358C23—451no—
359C23—451no—
360Q8—152no—
361D4—32no—
362D4—92no—
363D4—152no—
364D4—152no—
365D4—452no—
366D4—452no—
367D4—452no—
368D4—452no—
369D4—452no—
370C2×C4—51no—
371C2×C4—91no—
372C2×C4—152no—
373C2×C4—152no—
374C2×C4—452no—
375C2×C4—451no—
376C2×C3—22no—
377C2×C3—22no—
378C2×C3—21no—
379C2×C3—62no—
380C2×C3—302no—
381C2×C3C22×D3×D512yes—
382D3—22no—
383D3—62no—
384D3—62no—
385D3—102no—
386D3—121no—
387D3—121no—
388D3—302no—
389D3—302no—
390D3—602no—
391C5#6430811yescharacteristic, complemented, Sylow
392C22—32no—
393C22—91no—
394C22—91no—
395C22—152no—
396C22—361no—
397C22—361no—
398C22—452no—
399C22—1801no—
400C22—1802no—
401C22—1801no—
402C22D32×D511yescharacteristic
403C4—32no—
404C4—52no—
405C4—152no—
406C3—21no—
407C3#9202712yescomplemented
408C2—21no—
409C2—62no—
410C2—181no—
411C2—181no—
412C2—302no—
413C2—902no—
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. #10624218C2semidirect
3. C5×#3811C2semidirect
4. C2×#67140C2semidirect
5. C3×#6782C2semidirect
6. #10624054C2semidirect
7. #10624051C2semidirect
8. #10624033C2semidirect
9. D3×#1203C2semidirect
11. C22×C3⋊D15C22semidirect
12. C22×C5×C3⋊D3C22semidirect
13. C2×(C3⋊D3)⋊D5C22semidirect
14. C2×D3×D15C22semidirect
15. C2×C5×D32C22semidirect
16. C3×#1220C22semidirect
17. C3×C5×#67C22semidirect
18. C2×C32×Dic5C22semidirect
19. #67156C22semidirect
20. #67155C22semidirect
21. #67152C22semidirect
22. #67151C22semidirect
23. #67149C22semidirect
24. C5×#626C22semidirect
25. C5×#624C22semidirect
26. #67140C22semidirect
27. #67140C22semidirect
28. C3×#1204C22semidirect
29. C3×#1203C22semidirect
30. C3×#1200C22semidirect
31. C3×D3×Dic5C22semidirect
44. #6782D3semidirect
68. #3811D5semidirect
83. #1220C2×D3semidirect
87. C5×#67C2×D3semidirect
94. C2×C3×Dic5C2×D3semidirect
97. #1204C2×D3semidirect
106. #1203C2×D3semidirect
110. #1200C2×D3semidirect
112. D3×Dic5C2×D3semidirect
131. C22×C3⋊D3C2×D5semidirect
134. C2×D32C2×D5semidirect
139. C3×#67C2×D5semidirect
147. #626C2×D5semidirect
150. #624C2×D5semidirect
195. C32×C5H(5,𝔽2)semidirect
212. C2×Dic5D32semidirect
268. #67D3×D5semidirect
318. C3×C5#1009semidirect
354. C32#4144semidirect
391. C5#64308semidirect
407. C3#92027semidirect

Maximal subgroups

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

subgroupindex
#106242362
#106242182
C5×#38112
C2×#671402
C3×#67822
#106240542
#106240512
#106240332
D3×#12032
#920273
#643085