Finite Group Databasebeta

← C3×#91550 #13086709

Subgroups — C3×#91550 #13086709

Order 1440.

430 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 100 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×#1220C211yesmaximal, characteristic, complemented
2C2×C32×C22⋊D5C211yesmaximal, characteristic, complemented
3C2×C3×#1202C211yesmaximal, characteristic, complemented
4C2×C3×D5×Dic3C211yesmaximal, characteristic, complemented
5C3×C5×#217C211yesmaximal, characteristic, complemented
6C3×#6669C211yesmaximal, characteristic, complemented
7C3×#6667C211yesmaximal, characteristic, complemented
8C3×#4030—31nomaximal
9#91550C311yesmaximal, characteristic, direct product factor
10C23×C32×C5C2211yescharacteristic
11C22×C3×D15C2211yescharacteristic
12C22×C32×D5C2211yescharacteristic
13C3×#1220—22no—
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17C32×C22⋊D5—22no—
18C2×C32×Dic5C2211yescharacteristic
19C3×#1202—22no—
20C3×D5×Dic3—22no—
21C3×#939—51nomaximal, Hall
22C2×#1220C2×C311yescharacteristic, complemented
23C2×C3×C22⋊D5—21no—
24C2×C3×C22⋊D5—31no—
25C2×C3×C22⋊D5D311yescharacteristic, complemented
26C2×C3×C22⋊D5C2×C311yescharacteristic, complemented
27C2×C3×D20—31no—
28C2×C3×C4×D5—31no—
29C2×#1202C2×C311yescharacteristic, complemented
30C2×D5×Dic3C2×C311yescharacteristic, complemented
31C3×C5×C22⋊C4—31no—
32C5×#217C2×C311yescharacteristic, complemented
33C3×#685—31no—
34C3×#683—31no—
35#6669C2×C311yescharacteristic, complemented
36#6667C2×C311yescharacteristic, complemented
37C22×C32×C5—21no—
38C22×C32×C5—21no—
39C22×C32×C5—21no—
40C22×C32×C5C2311yescharacteristic
41C2×C3×D15—21no—
42C2×C3×D15—21no—
43C2×C3×D15—21no—
44C2×C32×D5—21no—
45C2×C32×D5—21no—
46C2×C32×D5D412yescomplemented
47C3×Dic15D412yescomplemented
48C3×C5×Dic3—21no—
49C3×C5×Dic3—21no—
50C32×Dic5—21no—
51#4030—31noHall
52C2×C32×D4—51no—
53C2×C3×#67—51no—
54C2×C3×#67—51no—
55C22×C3×Dic3—51no—
56C3×#217D511yescharacteristic, complemented
57C3×(C2×D3)⋊C4—51no—
58C3×Dic3⋊C4—51no—
59C23×C3×C5—21no—
60C23×C3×C5C2×D311yescharacteristic
61C23×C3×C5C22×C311yescharacteristic
62C22×D15C22×C311yescharacteristic
63C22×C3×D5—21no—
64C22×C3×D5—31no—
65C22×C3×D5C2×D311yescharacteristic
66C22×C3×D5C22×C311yescharacteristic
67C2×C3×C4×C5—31no—
68C2×C3×C4×C5—31no—
69#1220—22no—
70C2×Dic15C22×C311yescharacteristic
71C2×C5×Dic3C22×C311yescharacteristic
72C2×C5×Dic3C22×C311yescharacteristic
73C3×C22⋊D5—24no—
74C3×C22⋊D5—22no—
75C3×C22⋊D5—22no—
76C3×C22⋊D5—62no—
77C2×C3×Dic5—21no—
78C2×C3×Dic5—31no—
79C2×C3×Dic5C2×D311yescharacteristic
80C2×C3×Dic5C22×C311yescharacteristic
81C3×D20—62no—
82C3×C4×D5—62no—
83#1202—22no—
84D5×Dic3—22no—
85C3×#121—151no—
86#939—51no—
87C2×C32×C5—41no—
88C2×C32×C5C2×D411yescharacteristic
89C2×C32×C5C2×D411yescharacteristic
90C2×C32×C5#4111yescharacteristic
91C3×D15—41no—
92C32×D5—22no—
93C2×C22⋊D5—31no—
94C2×C22⋊D5C3×D311yescharacteristic, complemented
95C2×D20—31no—
96C2×C4×D5—31no—
97C5×C22⋊C4—31no—
98#685—31no—
99#683—31no—
100C23×C32—51no—
101C23×C32C2×D511yescharacteristic
102C22×C3×D3—51no—
103C32×D4—102no—
104C2×C32×C4—51no—
105C3×#67—102no—
106C3×#67—102no—
107C2×C3×Dic3—51no—
108C2×C3×Dic3—102no—
109C2×C3×Dic3C2×D511yescharacteristic
110C2×C3×Dic3C2×D511yescharacteristic
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×C5C23×C311yescharacteristic
123C2×D15—21no—
124C2×D15—21no—
125C2×D15—21no—
126C2×C3×D5—21no—
127C2×C3×D5—21no—
128C2×C3×D5—21no—
129C2×C3×D5—21no—
130C2×C3×D5—22no—
131C2×C3×D5—41no—
132C2×C3×D5—41no—
133C2×C3×D5—61no—
134C2×C3×D5—61no—
135C2×C3×D5—61no—
136C2×C3×D5#6712yescomplemented
137C2×C3×D5C3×D412yescomplemented
138C3×C4×C5—61no—
139C3×C4×C5—61no—
140Dic15C3×D412yescomplemented
141C3×Dic5—22no—
142C3×Dic5—21no—
143C3×Dic5—21no—
144C3×Dic5—32no—
145C5×Dic3—21no—
146C5×Dic3—21no—
147C2×C3×D4—51no—
148C2×C3×D4—51no—
149C2×C3×D4—101no—
150C2×C3×D4—151no—
151C2×C3×D4—151no—
152C22×C3×C4—151no—
153C2×#67—51no—
154C2×#67—51no—
155C22×Dic3—51no—
156C3×C4⋊C4—151no—
157C3×C22⋊C4—31no—
158C3×C22⋊C4—151no—
159#217C3×D511yescharacteristic, complemented
160(C2×D3)⋊C4—51no—
161Dic3⋊C4—51no—
162C32×C5#12111yescharacteristic, complemented, Hall
163C23×C5C2×C3×D311yescharacteristic
164C22×D5—31no—
165C22×D5C2×C3×D311yescharacteristic
166C2×C4×C5—31no—
167C2×C4×C5—31no—
168C22⋊D5—22no—
169C22⋊D5—62no—
170C2×Dic5—31no—
171C2×Dic5C2×C3×D311yescharacteristic
172D20—62no—
173C4×D5—62no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—52no—
178C22×C32—101no—
179C22×C32—101no—
180C22×C32C22×D511yescharacteristic
181C2×C3×D3—101no—
182C2×C3×D3—101no—
183C2×C3×D3—101no—
184C32×C4—101no—
185C3×Dic3—21no—
186C3×Dic3—21no—
187C3×Dic3—52no—
188#121—151noSylow
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×C5D3×D411yescharacteristic
196C2×C3×C5#23711yescharacteristic
197C2×C3×C5C2×#6711yescharacteristic
198C2×C3×C5C2×C3×D411yescharacteristic
199C2×C3×C5C2×C3×D411yescharacteristic
200C2×C3×C5C3×#4111yescharacteristic
201D15—41no—
202C3×D5—22no—
203C3×D5—22no—
204C3×D5—42no—
205C3×D5—121no—
206C23×C3—21no—
207C23×C3—51no—
208C23×C3—51no—
209C23×C3—101no—
210C23×C3—151no—
211C23×C3D3×D511yescharacteristic
212C23×C3C2×C3×D511yescharacteristic
213C22×D3—51no—
214C3×D4—102no—
215C3×D4—102no—
216C3×D4—104no—
217C3×D4—302no—
218C3×D4—302no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—51no—
222C2×C3×C4—51no—
223C2×C3×C4—101no—
224C2×C3×C4—151no—
225C2×C3×C4—302no—
226#67—102no—
227#67—102no—
228C2×Dic3—51no—
229C2×Dic3—102no—
230C2×Dic3C2×C3×D511yescharacteristic
231C2×Dic3C2×C3×D511yescharacteristic
232C22×C5—21no—
233C22×C5—21no—
234C22×C5—21no—
235C22×C5C22×C3×D311yescharacteristic
236C2×D5—21no—
237C2×D5—21no—
238C2×D5—61no—
239C2×D5—61no—
240C2×D5—61no—
241C2×D5C3×#6712yescomplemented
242C4×C5—61no—
243C4×C5—61no—
244Dic5—21no—
245Dic5—32no—
246C2×C32—41no—
247C2×C32—102no—
248C2×C32#70911yescharacteristic
249C2×C32D4×D511yescharacteristic
250C2×C32D4×D511yescharacteristic
251C3×D3—201no—
252C2×D4—51no—
253C2×D4—151no—
254C2×D4—151no—
255C22×C4—151no—
256C4⋊C4—151no—
257C22⋊C4—31no—
258C22⋊C4—151no—
259C3×C5—21no—
260C3×C5#93911yescharacteristic, complemented
261C3×C5C3×#12111yescharacteristic, 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—52no—
273C22×C3—52no—
274C22×C3—102no—
275C22×C3—101no—
276C22×C3—101no—
277C22×C3—101no—
278C22×C3—101no—
279C22×C3—201no—
280C22×C3—201no—
281C22×C3—301no—
282C22×C3—301no—
283C22×C3—301no—
284C22×C3C2×D3×D511yescharacteristic
285C22×C3C22×C3×D511yescharacteristic
286C2×D3—101no—
287C2×D3—101no—
288C2×D3—101no—
289C3×C4—61no—
290C3×C4—61no—
291C3×C4—102no—
292C3×C4—101no—
293C3×C4—101no—
294C3×C4—152no—
295Dic3—21no—
296Dic3—21no—
297Dic3—52no—
298C2×C5—41no—
299C2×C5C3×D3×D411yescharacteristic
300C2×C5C3×#23711yescharacteristic
301C2×C5C2×C3×#6711yescharacteristic
302D5—22no—
303D5—121no—
304C32#403011yescharacteristic, complemented, Sylow
305C23—51no—
306C23—151no—
307C23C3×D3×D511yescharacteristic
308D4—102no—
309D4—302no—
310D4—302no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—51no—
314C2×C4—151no—
315C2×C4—302no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—42no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—102no—
323C2×C3—102no—
324C2×C3—202no—
325C2×C3—601no—
326C2×C3#677911yescharacteristic
327C2×C3D5×#6711yescharacteristic
328C2×C3#679011yescharacteristic
329C2×C3C3×#70911yescharacteristic
330C2×C3C3×D4×D511yescharacteristic
331C2×C3C3×D4×D511yescharacteristic
332D3—201no—
333C5C3×#93911yescharacteristic, complemented, Sylow
334C22—21no—
335C22—21no—
336C22—21no—
337C22—52no—
338C22—101no—
339C22—101no—
340C22—301no—
341C22—301no—
342C22—301no—
343C22C2×C3×D3×D511yescharacteristic
344C4—61no—
345C4—61no—
346C4—101no—
347C4—152no—
348C3—21no—
349C3#9155011yescharacteristic, direct product factor
350C3C3×#403011yescharacteristic, complemented
351C2—41no—
352C2—102no—
353C2—601no—
354C2C3×#677911yescharacteristic
355C2C3×D5×#6711yescharacteristic
356C2C3×#679011yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D5
commutator, cosocleC22×C3×C5C23×C3
FrattiniC22C2×C3×D3×D5
FittingC23×C32×C5C22
socleC22×C32×C5C23
hypercenterC23×C3D3×D5
hypocenterC3×C5C3×#121
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1220C2semidirect
2. C2×C32×C22⋊D5C2semidirect
3. C2×C3×#1202C2semidirect
4. C2×C3×D5×Dic3C2semidirect
5. C3×C5×#217C2semidirect
6. C3×#6669C2semidirect
7. C3×#6667C2semidirect
9. #91550C3direct (N × H)
22. C2×#1220C2×C3semidirect
25. C2×C3×C22⋊D5D3semidirect
26. C2×C3×C22⋊D5C2×C3semidirect
29. C2×#1202C2×C3semidirect
30. C2×D5×Dic3C2×C3semidirect
32. C5×#217C2×C3semidirect
35. #6669C2×C3semidirect
36. #6667C2×C3semidirect
46. C2×C32×D5D4semidirect
47. C3×Dic15D4semidirect
56. C3×#217D5semidirect
94. C2×C22⋊D5C3×D3semidirect
136. C2×C3×D5#67semidirect
137. C2×C3×D5C3×D4semidirect
140. Dic15C3×D4semidirect
159. #217C3×D5semidirect
162. C32×C5#121semidirect
241. C2×D5C3×#67semidirect
260. C3×C5#939semidirect
261. C3×C5C3×#121semidirect
304. C32#4030semidirect
333. C5C3×#939semidirect
349. C3#91550direct (N × H)
350. C3C3×#4030semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#12202
C2×C32×C22⋊D52
C2×C3×#12022
C2×C3×D5×Dic32
C3×C5×#2172
C3×#66692
C3×#66672
C3×#40303
#915503
C3×#9395