Finite Group Databasebeta

← #13087453

Subgroups — #13087453

Order 1440.

410 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 90 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×C3×C4×C5×D3C211yesmaximal, characteristic, complemented
2C2×#67152C211yesmaximal, characteristic, complemented
3C2×#67151C211yesmaximal, characteristic, complemented
4C2×#67150C211yesmaximal, characteristic, complemented
5#10623908C211yesmaximal, characteristic, complemented
6C3×#6714C211yesmaximal, characteristic, complemented
7Dic3⋊Dic15C211yesmaximal, characteristic, complemented
8#91819—31nomaximal
9#91454—31nomaximal
10C22×C3⋊D15C2211yescharacteristic
11C22×C3×D15C2211yescharacteristic
12C22×C3×C5×D3C2211yescharacteristic
13C2×C32×C4×C5C2211yescharacteristic
14C2×C3⋊Dic15C2211yescharacteristic
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17C3×C4×C5×D3—22no—
18#67152—22no—
19#67151—22no—
20#67150—22no—
21#63906—51nomaximal, Hall
22C22×C3×C4×C5—31no—
23C2×#1220—31no—
24C2×#1220—31no—
25C2×D60—31no—
26C2×C4×C5×D3D311yescharacteristic, complemented
27C2×#1203—31no—
28C2×#1202—31no—
29C2×#1201—31no—
30#6714—31no—
31#6714—31no—
32#6714—61no—
33#6714D311yescharacteristic, complemented
34C4⋊Dic15—31no—
35C3×#685—31no—
36Dic3⋊Dic5—31no—
37C22×C32×C5C2311yescharacteristic
38C2×C3⋊D15—21no—
39C2×C3⋊D15—21no—
40C2×C3⋊D15—21no—
41C2×C3×D15—21no—
42C2×C3×D15—21no—
43C2×C3×D15—21no—
44C2×C3×C5×D3—21no—
45C2×C3×C5×D3—21no—
46C2×C3×C5×D3D412yescomplemented
47C32×C4×C5—21no—
48C3⋊Dic15—21no—
49C3×Dic15—21no—
50C3×C5×Dic3D412yescomplemented
51#4076—91noHall
52C2×C3×C4×D3D511yescharacteristic, complemented
53C2×#626—51no—
54C2×#626—51no—
55C2×#625—51no—
56#3752—51no—
57C3×(C2×D3)⋊C4—51no—
58Dic3⋊Dic3—51no—
59C23×C3×C5—31no—
60C22×D15—31no—
61C22×D15—31no—
62C22×D15—61no—
63C22×D15C2×D311yescharacteristic
64C22×C5×D3C2×D311yescharacteristic
65C22×C3×D5—31no—
66C2×C3×C4×C5—21no—
67C2×C3×C4×C5—31no—
68C2×C3×C4×C5—62no—
69C2×C3×C4×C5C2×D311yescharacteristic
70C2×C3×C4×C5C2×D311yescharacteristic
71#1220—62no—
72#1220—62no—
73C2×Dic15—31no—
74C2×Dic15—31no—
75C2×Dic15—61no—
76C2×Dic15C2×D311yescharacteristic
77D60—62no—
78C2×C5×Dic3C2×D311yescharacteristic
79C4×C5×D3—22no—
80C2×C3×Dic5—31no—
81#1203—62no—
82#1202—62no—
83#1201—62no—
84#930—151no—
85#884—151no—
86C2×C32×C5C2×D411yescharacteristic
87C2×C32×C5C2×D411yescharacteristic
88C2×C32×C5#4111yescharacteristic
89C3⋊D15—41no—
90C3×D15—41no—
91C3×C5×D3—22no—
92C22×C4×C5—31no—
93C2×C22⋊D5—91no—
94C2×C22⋊D5—91no—
95C2×D20—91no—
96#685—31no—
97#685—91no—
98C4⋊Dic5—91no—
99C22×C3⋊D3—51no—
100C22×C3×D3—51no—
101C22×C3×D3C2×D511yescharacteristic
102C2×C32×C4C2×D511yescharacteristic
103C2×C3⋊Dic3—51no—
104C2×C3×Dic3—51no—
105C2×C3×Dic3C2×D511yescharacteristic
106C3×C4×D3—22no—
107#626—102no—
108#626—102no—
109#625—102no—
110C22×C3×C5—21no—
111C22×C3×C5—32no—
112C22×C3×C5—61no—
113C22×C3×C5—61no—
114C22×C3×C5C22×D311yescharacteristic
115C22×C3×C5C22×D311yescharacteristic
116C2×D15—21no—
117C2×D15—21no—
118C2×D15—21no—
119C2×D15—62no—
120C2×D15—61no—
121C2×D15—61no—
122C2×D15—61no—
123C2×D15—61no—
124C2×D15—61no—
125C2×D15—61no—
126C2×D15—121no—
127C2×D15—121no—
128C2×C5×D3—21no—
129C2×C5×D3—21no—
130C2×C5×D3D1212yescomplemented
131C2×C3×D5—61no—
132C2×C3×D5—61no—
133C2×C3×D5—61no—
134C3×C4×C5—21no—
135C3×C4×C5—21no—
136C3×C4×C5—32no—
137C3×C4×C5—41no—
138Dic15—21no—
139Dic15—61no—
140Dic15—61no—
141Dic15—121no—
142C3×Dic5—61no—
143C5×Dic3#6712yescomplemented
144C22×C3×C4—31no—
145C2×#67—151no—
146C2×#67—151no—
147C2×#67—151no—
148C2×#67—151no—
149C2×D12—151no—
150C2×D12—151no—
151C2×C4×D3D1511yescharacteristic, complemented
152C3×C22⋊C4—151no—
153(C2×D3)⋊C4—51no—
154(C2×D3)⋊C4—151no—
155(C2×D3)⋊C4—151no—
156(C2×D3)⋊C4—301no—
157C4⋊Dic3—151no—
158Dic3⋊C4—151no—
159C32×C5#12111yescharacteristic, complemented, Hall
160C23×C5—31no—
161C22×D5—31no—
162C22×D5—91no—
163C2×C4×C5—31no—
164C2×C4×C5—62no—
165C2×C4×C5D3211yescharacteristic
166C22⋊D5—182no—
167C22⋊D5—182no—
168C2×Dic5—31no—
169C2×Dic5—91no—
170D20—182no—
171C22×C32C22×D511yescharacteristic
172C2×C3⋊D3—101no—
173C2×C3⋊D3—101no—
174C2×C3⋊D3—101no—
175C2×C3×D3—21no—
176C2×C3×D3—21no—
177C2×C3×D3—101no—
178C2×C3×D3—101no—
179C2×C3×D3—101no—
180C2×C3×D3D2012yescomplemented
181C32×C4—21no—
182C3⋊Dic3—101no—
183C3×Dic3—101no—
184C3×Dic3C22⋊D512yescomplemented
185#121—451noSylow
186C2×C3×C5—21no—
187C2×C3×C5—21no—
188C2×C3×C5—21no—
189C2×C3×C5—62no—
190C2×C3×C5C2×D1211yescharacteristic
191C2×C3×C5#23511yescharacteristic
192C2×C3×C5#23511yescharacteristic
193C2×C3×C5D3×D411yescharacteristic
194C2×C3×C5D3×D411yescharacteristic
195C2×C3×C5C2×#6711yescharacteristic
196D15—41no—
197D15—122no—
198D15—121no—
199D15—121no—
200C3×D5—121no—
201C5×D3—22no—
202C23×C3—31no—
203C23×C3—151no—
204C22×D3—51no—
205C22×D3—151no—
206C22×D3—151no—
207C22×D3—301no—
208C22×D3C2×D1511yescharacteristic
209C2×C3×C4—21no—
210C2×C3×C4—31no—
211C2×C3×C4—62no—
212C2×C3×C4—151no—
213C2×C3×C4D3×D511yescharacteristic
214C2×C3×C4C2×D1511yescharacteristic
215#67—302no—
216#67—302no—
217#67—302no—
218#67—302no—
219C2×Dic3—51no—
220C2×Dic3—151no—
221C2×Dic3—151no—
222C2×Dic3—301no—
223C2×Dic3C2×D1511yescharacteristic
224D12—302no—
225D12—302no—
226C4×D3—22no—
227C22×C5—32no—
228C22×C5—61no—
229C22×C5—61no—
230C22×C5C2×D3211yescharacteristic
231C2×D5—61no—
232C2×D5—61no—
233C2×D5—61no—
234C2×D5—181no—
235C2×D5—181no—
236C2×D5—181no—
237C4×C5—21no—
238C4×C5—32no—
239Dic5—61no—
240Dic5—181no—
241C2×C32C2×D2011yescharacteristic
242C2×C32#70911yescharacteristic
243C2×C32C2×C22⋊D511yescharacteristic
244C3⋊D3—201no—
245C3×D3—22no—
246C3×D3—201no—
247C2×D4—451no—
248C2×D4—451no—
249C2×D4—451no—
250C22×C4—31no—
251C4⋊C4—451no—
252C22⋊C4—151no—
253C22⋊C4—451no—
254C3×C5—21no—
255C3×C5#88411yescharacteristic, complemented
256C3×C5#93011yescharacteristic, complemented
257C22×C3—21no—
258C22×C3—32no—
259C22×C3—61no—
260C22×C3—61no—
261C22×C3—301no—
262C22×C3—301no—
263C22×C3—301no—
264C22×C3C2×D3×D511yescharacteristic
265C22×C3C22×D1511yescharacteristic
266C2×D3—21no—
267C2×D3—21no—
268C2×D3—101no—
269C2×D3—101no—
270C2×D3—101no—
271C2×D3—302no—
272C2×D3—301no—
273C2×D3—301no—
274C2×D3—301no—
275C2×D3—301no—
276C2×D3—301no—
277C2×D3—301no—
278C2×D3—601no—
279C2×D3—601no—
280C2×D3D6012yescomplemented
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—32no—
284C3×C4—41no—
285C3×C4—301no—
286Dic3—101no—
287Dic3—301no—
288Dic3—301no—
289Dic3—601no—
290Dic3#122012yescomplemented
291C2×C5—62no—
292C2×C5#379811yescharacteristic
293C2×C5D3×D1211yescharacteristic
294C2×C5D3×#6711yescharacteristic
295D5—121no—
296D5—361no—
297C32#407611yescharacteristic, complemented, Sylow
298C23—31no—
299C23—151no—
300C23—451no—
301D4—902no—
302D4—902no—
303D4—902no—
304C2×C4—31no—
305C2×C4—62no—
306C2×C4—151no—
307C2×C4—451no—
308C2×C4D3×D1511yescharacteristic
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—62no—
313C2×C3—601no—
314C2×C3#677111yescharacteristic
315C2×C3D3×D2011yescharacteristic
316C2×C3D3×C22⋊D511yescharacteristic
317C2×C3C2×D6011yescharacteristic
318C2×C3#681711yescharacteristic
319C2×C3C2×#122011yescharacteristic
320D3—22no—
321D3—201no—
322D3—602no—
323D3—601no—
324D3—601no—
325C5#6390611yescharacteristic, complemented, Sylow
326C22—32no—
327C22—61no—
328C22—61no—
329C22—301no—
330C22—301no—
331C22—301no—
332C22—901no—
333C22—901no—
334C22—901no—
335C22C2×D3×D1511yescharacteristic
336C4—21no—
337C4—32no—
338C4—301no—
339C4—901no—
340C3—21no—
341C3#9145411yescharacteristic, complemented
342C3#9181911yescharacteristic, complemented
343C2—62no—
344C2—601no—
345C2—1801no—
346C2#1062419911yescharacteristic
347C2D3×D6011yescharacteristic
348C2D3×#122011yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×D15
commutator, socle, cosocleC22×C32×C5C23
FittingC2×C32×C4×C5C22
hypercenterC2×C4D3×D15
hypocenterC32×C5#121
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C5×D3C2semidirect
2. C2×#67152C2semidirect
3. C2×#67151C2semidirect
4. C2×#67150C2semidirect
5. #10623908C2semidirect
6. C3×#6714C2semidirect
7. Dic3⋊Dic15C2semidirect
26. C2×C4×C5×D3D3semidirect
33. #6714D3semidirect
46. C2×C3×C5×D3D4semidirect
50. C3×C5×Dic3D4semidirect
52. C2×C3×C4×D3D5semidirect
130. C2×C5×D3D12semidirect
143. C5×Dic3#67semidirect
151. C2×C4×D3D15semidirect
159. C32×C5#121semidirect
180. C2×C3×D3D20semidirect
184. C3×Dic3C22⋊D5semidirect
255. C3×C5#884semidirect
256. C3×C5#930semidirect
280. C2×D3D60semidirect
290. Dic3#1220semidirect
297. C32#4076semidirect
325. C5#63906semidirect
341. C3#91454semidirect
342. C3#91819semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C5×D32
C2×#671522
C2×#671512
C2×#671502
#106239082
C3×#67142
Dic3⋊Dic152
#918193
#914543
#639065