Finite Group Databasebeta

← #13089002

Subgroups — #13089002

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#10624229C211yesmaximal, characteristic, complemented
2#10624218C212yesmaximal, complemented
3C5×#3805C211yesmaximal, characteristic, complemented
4C2×#67139C211yesmaximal, characteristic, complemented
5C3×D5×#67C212yesmaximal, complemented
6#10624044C212yesmaximal, complemented
7#10624041C212yesmaximal, complemented
8#10624027C212yesmaximal, complemented
9D3×#1202C212yesmaximal, complemented
10#92026—32nomaximal
11C22×C3⋊D15C2211yescharacteristic, complemented
12C22×C32×D5C2211yescharacteristic, complemented
13C2×D3×D15C2212yescomplemented
14C2×C3×D3×D5C2212yescomplemented
15C2×C5×C3⋊Dic3C2211yescharacteristic, complemented
16C3×#1220C2212yescomplemented
17C3×C5×#67C2212yescomplemented
18#67158C2211yescharacteristic, complemented
19#67157C2211yescharacteristic, complemented
20#67154C2212yescomplemented
21#67152C2212yescomplemented
22#67151C2212yescomplemented
23#67149C2212yescomplemented
24C5×#627C2211yescharacteristic, complemented
25C5×#625C2211yescharacteristic, complemented
26C5×D3×Dic3C2212yescomplemented
27#67139C2212yescomplemented
28#67139C2212yescomplemented
29C3×#1202C2212yescomplemented
30C3×#1201C2212yescomplemented
31C3×D5×Dic3C2212yescomplemented
32#64330—51nomaximal, Hall
33D4×D15—32no—
34C5×#237—32no—
35C3×D4×D5—32no—
36#6790—32no—
37#6790—32no—
38D5×#67—32no—
39D5×#67D312yescomplemented
40C2×#1202—32no—
41C2×#1202—32no—
42C2×#1202—61no—
43#6779—32no—
44#6779—32no—
45D3×D20—32no—
46#6773—32no—
47#6771—32no—
48#6766—32no—
49C22×C32×C5C2311yescharacteristic
50C2×C3⋊D15—41no—
51C2×C3⋊D15C2312yes—
52C2×C3×D15C2312yes—
53C2×C3×C5×D3C2312yes—
54C2×C32×D5—41no—
55C2×C32×D5C2312yes—
56D3×D15—42no—
57C3×D3×D5—42no—
58C5×C3⋊Dic3C2311yescharacteristic
59C5×C3⋊Dic3C2311yescharacteristic
60C3×Dic15C2312yes—
61C3×C5×Dic3C2312yes—
62#4149—91noHall
63C2×#638—51no—
64C2×C3×#67—52no—
65#3811—52no—
66D3×#67—52no—
67D3×#67—52no—
68#3805—51no—
69#3805D511yescharacteristic, complemented
70#3804—52no—
71#3804—52no—
72C22×D15—32no—
73C22×D15—32no—
74C22×D15—61no—
75C22×C3×D5—21no—
76C22×C3×D5—32no—
77C22×C3×D5C2×D312yescomplemented
78C2×D3×D5—32no—
79C2×D3×D5C2×D312yescomplemented
80C3×C5×D4—32no—
81#1220—32no—
82#1220C2×D312yescomplemented
83D60—32no—
84C4×D15—32no—
85C5×#67—32no—
86C5×#67C2×D312yescomplemented
87C2×C5×Dic3—32no—
88C2×C5×Dic3—32no—
89C2×C5×Dic3—61no—
90C4×C5×D3—32no—
91C5×Dic6—32no—
92C3×C22⋊D5—32no—
93C3×C22⋊D5—32no—
94C3×D20—32no—
95C3×C4×D5—32no—
96#1204—32no—
97#1203—32no—
98#1202—32no—
99#1202—32no—
100#1202—32no—
101#1202—32no—
102#1202—32no—
103#1202—32no—
104#1202—62no—
105#1202—62no—
106#1202C2×D312yescomplemented
107#1201C2×D312yescomplemented
108#1200—32no—
109#1200—32no—
110D5×Dic3—32no—
111D5×Dic3C2×D312yescomplemented
112#1004—152no—
113C2×C32×C5—21no—
114C2×C32×C5C2411yescharacteristic
115C3⋊D15—22no—
116C3×D15—22no—
117C3×C5×D3—22no—
118C32×D5—22no—
119C5×#41—91no—
120#713—92no—
121D4×D5—32no—
122D4×D5—92no—
123D4×D5—92no—
124#709—92no—
125#709—91no—
126C2×D20—92no—
127C2×D20—91no—
128C23×C32—51no—
129C22×C3⋊D3—51no—
130C22×C3×D3—52no—
131C2×D32—52no—
132#638—52no—
133#638—52no—
134C2×C3⋊Dic3C2×D511yescharacteristic, complemented
135C3×#67—52no—
136C3×#67—52no—
137C3×#67—52no—
138C3×#67C2×D512yescomplemented
139C2×C3×Dic3—52no—
140#627—51no—
141#627C2×D511yescharacteristic, complemented
142#626—52no—
143#626—52no—
144#625—51no—
145#625C2×D511yescharacteristic, complemented
146#624—52no—
147D3×Dic3—52no—
148D3×Dic3C2×D512yescomplemented
149C22×C3×C5—21no—
150C22×C3×C5—32no—
151C22×C3×C5C22×D312yes—
152C2×D15—32no—
153C2×D15—32no—
154C2×D15—62no—
155C2×D15—122no—
156C2×D15—122no—
157C2×D15—122no—
158C2×D15C22×D312yes—
159C2×C5×D3C22×D312yes—
160C2×C3×D5—22no—
161C2×C3×D5—32no—
162C2×C3×D5—42no—
163C2×C3×D5—42no—
164C2×C3×D5—122no—
165C2×C3×D5C22×D312yes—
166C2×C3×D5C22×D312yes—
167D3×D5—42no—
168D3×D5—122no—
169C3×C4×C5—32no—
170Dic15C22×D312yes—
171C3×Dic5—32no—
172C5×Dic3—32no—
173C5×Dic3—32no—
174C5×Dic3—61no—
175C5×Dic3—61no—
176C5×Dic3C22×D312yes—
177C2×C3×D4—152no—
178C2×#67—52no—
179C2×#67—152no—
180C2×#67—152no—
181C2×#67—152no—
182C2×#67—301no—
183#237—32no—
184#237—152no—
185#237—152no—
186#237—152no—
187D3×D4—152no—
188D3×D4—152no—
189D3×D4—152no—
190D3×D4—152no—
191#235—152no—
192#235—152no—
193C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
194C22×D5—32no—
195C22×D5—92no—
196C22×D5—91no—
197C22×D5D3211yescharacteristic, complemented
198C5×Q8—91no—
199C5×D4—32no—
200C5×D4—91no—
201C2×C4×C5—92no—
202C2×C4×C5—91no—
203C22⋊D5—32no—
204C22⋊D5—32no—
205C22⋊D5—92no—
206D20—32no—
207D20—92no—
208D20—91no—
209D20—92no—
210D20—92no—
211C4×D5—32no—
212C4×D5—92no—
213C4×D5—92no—
214Dic10—91no—
215C22×C32—52no—
216C22×C32—201no—
217C22×C32C22×D511yescharacteristic
218C2×C3⋊D3—52no—
219C2×C3⋊D3—201no—
220C2×C3×D3—52no—
221C2×C3×D3—202no—
222C2×C3×D3C22×D512yes—
223D32—202no—
224C3⋊Dic3C22×D511yescharacteristic
225C3⋊Dic3C22×D511yescharacteristic
226C3×Dic3—52no—
227C3×Dic3C22×D512yes—
228H(5,𝔽2)—451noSylow
229C2×C3×C5—22no—
230C2×C3×C5—22no—
231C2×C3×C5—21no—
232C2×C3×C5—62no—
233C2×C3×C5C23×D312yes—
234D15—22no—
235D15—62no—
236D15—62no—
237D15—122no—
238C3×D5—22no—
239C3×D5—22no—
240C3×D5—42no—
241C3×D5—62no—
242C5×D3—22no—
243C23×C3—52no—
244C23×C3—101no—
245C23×C3—152no—
246C22×D3—52no—
247C22×D3—152no—
248C22×D3—152no—
249C22×D3—152no—
250C22×D3—301no—
251C3×D4—32no—
252C3×D4—152no—
253C3×D4—152no—
254C3×D4—152no—
255C2×C3×C4—152no—
256#67—32no—
257#67—52no—
258#67—52no—
259#67—52no—
260#67—152no—
261#67—152no—
262#67—152no—
263#67—152no—
264#67—152no—
265#67—152no—
266#67—152no—
267#67—302no—
268#67—302no—
269#67D3×D512yescomplemented
270C2×Dic3—32no—
271C2×Dic3—32no—
272C2×Dic3—52no—
273C2×Dic3—61no—
274C2×Dic3—152no—
275D12—152no—
276D12—152no—
277C4×D3—32no—
278C4×D3—152no—
279C4×D3—152no—
280C4×D3—152no—
281Dic6—32no—
282Dic6—152no—
283C22×C5—32no—
284C22×C5C2×D3211yescharacteristic
285C2×D5—32no—
286C2×D5—41no—
287C2×D5—92no—
288C2×D5—122no—
289C2×D5—362no—
290C2×D5—361no—
291C2×D5C2×D3212yes—
292C4×C5—32no—
293C4×C5—91no—
294C4×C5—91no—
295Dic5—32no—
296C2×C32—21no—
297C2×C32—102no—
298C2×C32C23×D511yescharacteristic
299C3⋊D3—102no—
300C3×D3—22no—
301C3×D3—102no—
302#41—91no—
303#41—452no—
304#41—451no—
305#41—452no—
306C2×D4—152no—
307C2×D4—452no—
308C2×D4—451no—
309C2×D4—452no—
310C2×D4—452no—
311C3×C5—21no—
312C3×C5#100412yescomplemented
313C22×C3—21no—
314C22×C3—32no—
315C22×C3—52no—
316C22×C3—52no—
317C22×C3—102no—
318C22×C3—152no—
319C22×C3—202no—
320C22×C3—202no—
321C22×C3—602no—
322C22×C3C2×D3×D512yes—
323C2×D3—52no—
324C2×D3—152no—
325C2×D3—152no—
326C2×D3—202no—
327C2×D3—302no—
328C2×D3—602no—
329C2×D3—602no—
330C2×D3—602no—
331C2×D3—602no—
332C2×D3C2×D3×D512yes—
333C3×C4—32no—
334C3×C4—152no—
335Dic3—32no—
336Dic3—32no—
337Dic3—52no—
338Dic3—61no—
339Dic3—61no—
340Dic3C2×D3×D512yes—
341C2×C5—21no—
342C2×C5—62no—
343C2×C5C22×D3211yescharacteristic
344D5—22no—
345D5—62no—
346D5—182no—
347C32#414911yescharacteristic, complemented, Sylow
348C23—51no—
349C23—152no—
350C23—452no—
351C23—451no—
352Q8—91no—
353Q8—451no—
354D4—32no—
355D4—91no—
356D4—152no—
357D4—152no—
358D4—152no—
359D4—452no—
360D4—452no—
361D4—452no—
362D4—451no—
363D4—452no—
364C2×C4—92no—
365C2×C4—91no—
366C2×C4—152no—
367C2×C4—452no—
368C2×C4—452no—
369C2×C3—22no—
370C2×C3—22no—
371C2×C3—21no—
372C2×C3—62no—
373C2×C3—102no—
374C2×C3—102no—
375C2×C3—202no—
376C2×C3—302no—
377C2×C3C22×D3×D512yes—
378D3—22no—
379D3—102no—
380D3—302no—
381D3—302no—
382D3—602no—
383C5#6433011yescharacteristic, complemented, Sylow
384C22—32no—
385C22—52no—
386C22—152no—
387C22—201no—
388C22—452no—
389C22—602no—
390C22—1801no—
391C22—1802no—
392C22D32×D511yescharacteristic
393C4—32no—
394C4—91no—
395C4—91no—
396C4—152no—
397C3—21no—
398C3#9202612yescomplemented
399C2—21no—
400C2—62no—
401C2—102no—
402C2—302no—
403C2—902no—
404C2C2×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. #10624229C2semidirect
2. #10624218C2semidirect
3. C5×#3805C2semidirect
4. C2×#67139C2semidirect
5. C3×D5×#67C2semidirect
6. #10624044C2semidirect
7. #10624041C2semidirect
8. #10624027C2semidirect
9. D3×#1202C2semidirect
11. C22×C3⋊D15C22semidirect
12. C22×C32×D5C22semidirect
13. C2×D3×D15C22semidirect
14. C2×C3×D3×D5C22semidirect
15. C2×C5×C3⋊Dic3C22semidirect
16. C3×#1220C22semidirect
17. C3×C5×#67C22semidirect
18. #67158C22semidirect
19. #67157C22semidirect
20. #67154C22semidirect
21. #67152C22semidirect
22. #67151C22semidirect
23. #67149C22semidirect
24. C5×#627C22semidirect
25. C5×#625C22semidirect
26. C5×D3×Dic3C22semidirect
27. #67139C22semidirect
28. #67139C22semidirect
29. C3×#1202C22semidirect
30. C3×#1201C22semidirect
31. C3×D5×Dic3C22semidirect
39. D5×#67D3semidirect
69. #3805D5semidirect
77. C22×C3×D5C2×D3semidirect
79. C2×D3×D5C2×D3semidirect
82. #1220C2×D3semidirect
86. C5×#67C2×D3semidirect
106. #1202C2×D3semidirect
107. #1201C2×D3semidirect
111. D5×Dic3C2×D3semidirect
134. C2×C3⋊Dic3C2×D5semidirect
138. C3×#67C2×D5semidirect
141. #627C2×D5semidirect
145. #625C2×D5semidirect
148. D3×Dic3C2×D5semidirect
193. C32×C5H(5,𝔽2)semidirect
197. C22×D5D32semidirect
269. #67D3×D5semidirect
312. C3×C5#1004semidirect
347. C32#4149semidirect
383. C5#64330semidirect
398. C3#92026semidirect

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×#671392
C3×D5×#672
#106240442
#106240412
#106240272
D3×#12022
#920263
#643305