Finite Group Databasebeta

← #13087768

Subgroups — #13087768

Order 1440.

430 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 79 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×C5×#638C211yesmaximal, characteristic, complemented
2C2×C3×#1220C211yesmaximal, characteristic, complemented
3C2×#67156C211yesmaximal, characteristic, complemented
4C2×#67155C211yesmaximal, characteristic, complemented
5C3×#6719C211yesmaximal, characteristic, complemented
6#10623831C211yesmaximal, characteristic, complemented
7#10623829C211yesmaximal, characteristic, complemented
8#91565—31nomaximal
9#91558—31nomaximal
10C23×C32×C5C2211yescharacteristic
11C22×C5×C3⋊D3C2211yescharacteristic
12C22×C3×D15C2211yescharacteristic
13C5×#638—22no—
14C2×C5×C3⋊Dic3C2211yescharacteristic
15C3×#1220—22no—
16C2×C3×Dic15C2211yescharacteristic
17C2×C3×Dic15C2211yescharacteristic
18C2×C3×Dic15C2211yescharacteristic
19#67156—22no—
20#67155—22no—
21#63958—51nomaximal, Hall
22C2×#1220D311yescharacteristic, complemented
23C2×C5×#67—31no—
24C2×C5×#67—31no—
25C2×C5×#67—61no—
26C2×C3×C22⋊D5—31no—
27C2×#1203—31no—
28C2×#1201—31no—
29C2×D3×Dic5—31no—
30C2×D3×Dic5—31no—
31#6719D311yescharacteristic, complemented
32C3×C22⋊Dic5—31no—
33#6669—31no—
34#6668—31no—
35#6667—31no—
36#6665—31no—
37C22×C32×C5—21no—
38C22×C32×C5—21no—
39C22×C32×C5—21no—
40C22×C32×C5C2311yescharacteristic
41C2×C5×C3⋊D3—21no—
42C2×C5×C3⋊D3—21no—
43C2×C5×C3⋊D3D412yescomplemented
44C2×C3×D15—21no—
45C2×C3×D15—21no—
46C2×C3×D15—21no—
47C5×C3⋊Dic3—21no—
48C3×Dic15—21no—
49C3×Dic15—21no—
50C3×Dic15D412yescomplemented
51#4085—91noHall
52C2×#638D511yescharacteristic, complemented
53C2×C3×#67—51no—
54C2×#626—51no—
55C2×#624—51no—
56C3×#217—51no—
57Dic3⋊Dic3—51no—
58(C2×D3)⋊Dic3—51no—
59C23×C3×C5—21no—
60C23×C3×C5C2×D311yescharacteristic
61C23×C3×C5C2×D311yescharacteristic
62C22×D15C2×D311yescharacteristic
63C22×C5×D3—31no—
64C22×C5×D3—31no—
65C22×C5×D3—61no—
66C22×C3×D5—31no—
67#1220—22no—
68C2×Dic15C2×D311yescharacteristic
69C2×Dic15C2×D311yescharacteristic
70C2×Dic15C2×D311yescharacteristic
71C5×#67—62no—
72C5×#67—62no—
73C5×#67—64no—
74C2×C5×Dic3—31no—
75C2×C5×Dic3—31no—
76C2×C5×Dic3—61no—
77C3×C22⋊D5—62no—
78C2×C3×Dic5—31no—
79C2×C3×Dic5—31no—
80C2×C3×Dic5—31no—
81#1203—62no—
82#1201—62no—
83D3×Dic5—62no—
84D3×Dic5—62no—
85#938—151no—
86#884—151no—
87C2×C32×C5—41no—
88C2×C32×C5C2×D411yescharacteristic
89C2×C32×C5C2×D411yescharacteristic
90C2×C32×C5#4111yescharacteristic
91C5×C3⋊D3—22no—
92C3×D15—41no—
93C2×C5×D4—91no—
94C2×C22⋊D5—31no—
95C2×C22⋊D5—91no—
96C22×Dic5—91no—
97C22⋊Dic5—31no—
98#685—91no—
99#683—91no—
100C23×C32C2×D511yescharacteristic
101C22×C3⋊D3C2×D511yescharacteristic
102C22×C3×D3—51no—
103#638—22no—
104C2×C3⋊Dic3C2×D511yescharacteristic
105C3×#67—102no—
106C2×C3×Dic3—51no—
107C2×C3×Dic3—51no—
108C2×C3×Dic3—51no—
109#626—102no—
110#624—102no—
111C22×C3×C5—22no—
112C22×C3×C5—21no—
113C22×C3×C5—21no—
114C22×C3×C5—21no—
115C22×C3×C5—21no—
116C22×C3×C5—21no—
117C22×C3×C5—21no—
118C22×C3×C5—21no—
119C22×C3×C5—41no—
120C22×C3×C5—41no—
121C22×C3×C5C22×D311yescharacteristic
122C22×C3×C5C22×D311yescharacteristic
123C2×D15—21no—
124C2×D15—21no—
125C2×D15—21no—
126C2×C5×D3—32no—
127C2×C5×D3—32no—
128C2×C5×D3—62no—
129C2×C5×D3—61no—
130C2×C5×D3—61no—
131C2×C5×D3—61no—
132C2×C5×D3—61no—
133C2×C5×D3—121no—
134C2×C5×D3—121no—
135C2×C3×D5—61no—
136C2×C3×D5—61no—
137C2×C3×D5—61no—
138Dic15—21no—
139Dic15—21no—
140Dic15#6712yescomplemented
141C3×Dic5—32no—
142C3×Dic5—61no—
143C3×Dic5—61no—
144C5×Dic3—62no—
145C5×Dic3—61no—
146C5×Dic3—61no—
147C2×C3×D4—151no—
148C2×#67—31no—
149C2×#67—31no—
150C2×#67—51no—
151C2×#67—61no—
152C2×#67—151no—
153C2×D12—151no—
154C2×C4×D3—151no—
155C2×C4×D3—151no—
156C3×C22⋊C4—151no—
157#217—51no—
158#217—151no—
159(C2×D3)⋊C4—151no—
160C4⋊Dic3—151no—
161Dic3⋊C4—151no—
162C32×C5#12111yescharacteristic, complemented, Hall
163C23×C5—91no—
164C23×C5D3211yescharacteristic
165C22×D5—31no—
166C5×D4—182no—
167C2×C4×C5—91no—
168C22⋊D5—62no—
169C22⋊D5—182no—
170C2×Dic5—31no—
171C2×Dic5—31no—
172C2×Dic5—31no—
173C2×Dic5—182no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32C22×D511yescharacteristic
178C2×C3⋊D3—21no—
179C2×C3⋊D3—21no—
180C2×C3⋊D3C22⋊D512yescomplemented
181C2×C3×D3—101no—
182C2×C3×D3—101no—
183C2×C3×D3—101no—
184C3⋊Dic3—21no—
185C3×Dic3—52no—
186C3×Dic3—101no—
187C3×Dic3—101no—
188#121—451noSylow
189C2×C3×C5—21no—
190C2×C3×C5—21no—
191C2×C3×C5—21no—
192C2×C3×C5—42no—
193C2×C3×C5—41no—
194C2×C3×C5—41no—
195C2×C3×C5#23511yescharacteristic
196C2×C3×C5D3×D411yescharacteristic
197C2×C3×C5D3×D411yescharacteristic
198C2×C3×C5D3×D411yescharacteristic
199C2×C3×C5#23711yescharacteristic
200C2×C3×C5C2×#6711yescharacteristic
201D15—41no—
202C3×D5—121no—
203C5×D3—62no—
204C5×D3—62no—
205C5×D3—122no—
206C23×C3—21no—
207C23×C3—151no—
208C23×C3D3×D511yescharacteristic
209C23×C3D3×D511yescharacteristic
210C22×D3—31no—
211C22×D3—31no—
212C22×D3—51no—
213C22×D3—61no—
214C3×D4—302no—
215C2×C3×C4—151no—
216C2×C3×C4—151no—
217C2×C3×C4—151no—
218#67—62no—
219#67—62no—
220#67—64no—
221#67—102no—
222#67—302no—
223C2×Dic3—31no—
224C2×Dic3—31no—
225C2×Dic3—51no—
226C2×Dic3—51no—
227C2×Dic3—51no—
228C2×Dic3—61no—
229D12—302no—
230C4×D3—302no—
231C4×D3—302no—
232C22×C5—21no—
233C22×C5—21no—
234C22×C5—21no—
235C22×C5—92no—
236C22×C5—181no—
237C22×C5—181no—
238C22×C5C2×D3211yescharacteristic
239C2×D5—61no—
240C2×D5—61no—
241C2×D5—61no—
242C4×C5—181no—
243Dic5—32no—
244Dic5—61no—
245Dic5—61no—
246C2×C32—41no—
247C2×C32D4×D511yescharacteristic
248C2×C32#71111yescharacteristic
249C2×C32C2×C22⋊D511yescharacteristic
250C3⋊D3—22no—
251C3×D3—201no—
252C2×D4—91no—
253C2×D4—151no—
254C2×D4—451no—
255C22×C4—451no—
256C4⋊C4—451no—
257C22⋊C4—151no—
258C22⋊C4—451no—
259C3×C5—21no—
260C3×C5#88411yescharacteristic, complemented
261C3×C5#93811yescharacteristic, complemented
262C22×C3—22no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—301no—
273C22×C3—301no—
274C22×C3—301no—
275C22×C3C2×D3×D511yescharacteristic
276C22×C3C2×D3×D511yescharacteristic
277C2×D3—32no—
278C2×D3—32no—
279C2×D3—62no—
280C2×D3—61no—
281C2×D3—61no—
282C2×D3—61no—
283C2×D3—61no—
284C2×D3—101no—
285C2×D3—101no—
286C2×D3—101no—
287C2×D3—121no—
288C2×D3—121no—
289C3×C4—152no—
290C3×C4—301no—
291C3×C4—301no—
292Dic3—52no—
293Dic3—62no—
294Dic3—61no—
295Dic3—61no—
296Dic3—101no—
297Dic3—101no—
298C2×C5—41no—
299C2×C5—182no—
300C2×C5#380411yescharacteristic
301C2×C5D3×#6711yescharacteristic
302C2×C5#381111yescharacteristic
303D5—121no—
304C32#408511yescharacteristic, complemented, Sylow
305C23—91no—
306C23—151no—
307C23(C3⋊D3)⋊D511yescharacteristic
308D4—182no—
309D4—302no—
310D4—902no—
311C2×C4—91no—
312C2×C4—151no—
313C2×C4—151no—
314C2×C4—151no—
315C2×C4—902no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—42no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—601no—
323C2×C3#678011yescharacteristic
324C2×C3#678211yescharacteristic
325C2×C3D5×#6711yescharacteristic
326C2×C3D3×C22⋊D511yescharacteristic
327C2×C3D3×C22⋊D511yescharacteristic
328C2×C3#679011yescharacteristic
329D3—62no—
330D3—62no—
331D3—122no—
332D3—201no—
333C5#6395811yescharacteristic, complemented, Sylow
334C22—21no—
335C22—21no—
336C22—21no—
337C22—92no—
338C22—181no—
339C22—181no—
340C22—301no—
341C22—301no—
342C22—301no—
343C22C2×(C3⋊D3)⋊D511yescharacteristic
344C4—152no—
345C4—181no—
346C4—301no—
347C4—301no—
348C3—21no—
349C3#9155811yescharacteristic, complemented
350C3#9156511yescharacteristic, complemented
351C2—41no—
352C2—182no—
353C2—601no—
354C2#1062423011yescharacteristic
355C2#1062423511yescharacteristic
356C2#1062423611yescharacteristic
subgroupG/H
center, FrattiniC22C2×(C3⋊D3)⋊D5
commutator, socle, cosocleC22×C32×C5C23
FittingC23×C32×C5C22
hypercenterC23(C3⋊D3)⋊D5
hypocenterC32×C5#121
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×C5×#638C2semidirect
2. C2×C3×#1220C2semidirect
3. C2×#67156C2semidirect
4. C2×#67155C2semidirect
5. C3×#6719C2semidirect
6. #10623831C2semidirect
7. #10623829C2semidirect
22. C2×#1220D3semidirect
31. #6719D3semidirect
43. C2×C5×C3⋊D3D4semidirect
50. C3×Dic15D4semidirect
52. C2×#638D5semidirect
140. Dic15#67semidirect
162. C32×C5#121semidirect
180. C2×C3⋊D3C22⋊D5semidirect
260. C3×C5#884semidirect
261. C3×C5#938semidirect
304. C32#4085semidirect
333. C5#63958semidirect
349. C3#91558semidirect
350. C3#91565semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#6382
C2×C3×#12202
C2×#671562
C2×#671552
C3×#67192
#106238312
#106238292
#915653
#915583
#639585