Finite Group Databasebeta

← C3×#10615841 #14425705

Subgroups — C3×#10615841 #14425705

Order 1920.

1600 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1600 conjugacy classes of subgroups — too many to draw. Only its 322 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×C5×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C3×#66138C213yesmaximal, complemented
3C3×#65533C214yesmaximal, complemented
4C3×#65532C214yesmaximal, complemented
5C2×C3×#3966C213yesmaximal, complemented
6#10615841C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×H(5,𝔽2)—26no—
8C3×C5×H(5,𝔽2)C2214yes—
9C2×C3×C5×#41—23no—
10C22×C3×C5×D4—23no—
11C22×C3×C5×D4C2213yes—
12C22×C3×C22⋊D5C2211yescharacteristic, complemented
13C3×#4099—23no—
14C2×C3×C22⋊Dic5C2213yescomplemented
15C3×#4085—26no—
16C3×#4083C2216yescomplemented
17C3×#4081C2216yescomplemented
18C3×#4075—23no—
19C3×#3966C22112yescomplemented
20C3×#3026—51nomaximal, Hall
21C2×C5×H(5,𝔽2)C2×C311yescharacteristic, complemented
22#66138C2×C313yescomplemented
23#65533C2×C314yescomplemented
24#65532C2×C314yescomplemented
25C2×#3966C2×C313yescomplemented
26C24×C3×C5—21no—
27C24×C3×C5D413yescomplemented
28C24×C3×C5C2311yescharacteristic
29C23×C3×D5C2311yescharacteristic, complemented
30C3×C5×#41—212no—
31C3×C5×#41—212no—
32C2×C3×C5×Q8—21no—
33C2×C3×C5×D4—23no—
34C2×C3×C5×D4—212no—
35C2×C3×C5×D4—26no—
36C2×C3×C5×D4—212no—
37C2×C3×C5×D4—26no—
38C2×C3×C5×D4D416yescomplemented
39C2×C3×C5×D4C2316yes—
40C22×C3×C4×C5—23no—
41C22×C3×C4×C5D413yescomplemented
42C2×C3×C22⋊D5—23no—
43C2×C3×C22⋊D5—23no—
44C2×C3×C22⋊D5—26no—
45C22×C3×Dic5—23no—
46C22×C3×Dic5C2311yescharacteristic, complemented
47C3×C22⋊Dic5—26no—
48C3×C22⋊Dic5—23no—
49C3×C22⋊Dic5C2316yescomplemented
50C3×#685—26no—
51C3×#683—26no—
52C2×C3×H(5,𝔽2)D511yescharacteristic, complemented
53C3×#534—53no—
54C3×#458—54no—
55C3×B(4,𝔽2)—54no—
56C2×C3×#99—53no—
57C5×H(5,𝔽2)—26no—
58C5×H(5,𝔽2)C22×C314yes—
59C2×C5×#41—23no—
60C22×C5×D4—23no—
61C22×C5×D4C22×C313yes—
62C22×C22⋊D5C22×C311yescharacteristic, complemented
63#4099—23no—
64C2×C22⋊Dic5C22×C313yescomplemented
65#4085—26no—
66#4083C22×C316yescomplemented
67#4081C22×C316yescomplemented
68#4075—23no—
69#3966C22×C3112yescomplemented
70#3026—51noSylow
71C23×C3×C5—24no—
72C23×C3×C5—23no—
73C23×C3×C5—23no—
74C23×C3×C5—81no—
75C23×C3×C5—83no—
76C23×C3×C5—82no—
77C23×C3×C5C2×D416yes—
78C23×C3×C5C2×D413yes—
79C23×C3×C5C2×D413yes—
80C23×C3×C5C2411yescharacteristic
81C22×C3×D5—23no—
82C22×C3×D5—81no—
83C3×C5×Q8—24no—
84C3×C5×D4—212no—
85C3×C5×D4—212no—
86C3×C5×D4—212no—
87C2×C3×C4×C5—212no—
88C2×C3×C4×C5—23no—
89C2×C3×C4×C5—23no—
90C2×C3×C4×C5C2×D416yes—
91C3×C22⋊D5—41no—
92C3×C22⋊D5—43no—
93C3×C22⋊D5—86no—
94C2×C3×Dic5—23no—
95C2×C3×Dic5—26no—
96C2×C3×Dic5—83no—
97C3×H(5,𝔽2)—26no—
98C3×H(5,𝔽2)C2×D514yes—
99C2×C3×#41—23no—
100C22×C3×D4—23no—
101C22×C3×D4—51no—
102C22×C3×D4C2×D513yes—
103C3×#123—56no—
104C3×#123—103no—
105C3×#121—106no—
106C3×#120—56no—
107C3×#120—103no—
108C2×C3×C22⋊C4—53no—
109C3×#99—512no—
110C24×C5—21no—
111C24×C5C3×D413yescomplemented
112C24×C5C23×C311yescharacteristic
113C23×D5C23×C311yescharacteristic, complemented
114C5×#41—212no—
115C5×#41—212no—
116C2×C5×Q8—21no—
117C2×C5×D4—23no—
118C2×C5×D4—212no—
119C2×C5×D4—26no—
120C2×C5×D4—212no—
121C2×C5×D4—26no—
122C2×C5×D4C3×D416yescomplemented
123C2×C5×D4C23×C316yes—
124C22×C4×C5—23no—
125C22×C4×C5C3×D413yescomplemented
126C2×C22⋊D5—23no—
127C2×C22⋊D5—23no—
128C2×C22⋊D5—26no—
129C22×Dic5—23no—
130C22×Dic5C23×C311yescharacteristic, complemented
131C22⋊Dic5—23no—
132C22⋊Dic5—26no—
133C22⋊Dic5C23×C316yescomplemented
134#685—26no—
135#683—26no—
136C2×H(5,𝔽2)C3×D511yescharacteristic, complemented
137#534—53no—
138#458—54no—
139B(4,𝔽2)—54no—
140C2×#99—53no—
141C22×C3×C5—26no—
142C22×C3×C5—43no—
143C22×C3×C5—41no—
144C22×C3×C5—43no—
145C22×C3×C5—46no—
146C22×C3×C5—46no—
147C22×C3×C5—83no—
148C22×C3×C5—84no—
149C22×C3×C5#12011yescharacteristic
150C22×C3×C5#12013yes—
151C22×C3×C5C22×D413yes—
152C2×C3×D5—46no—
153C2×C3×D5—41no—
154C3×C4×C5—26no—
155C3×Dic5—46no—
156C3×Dic5—41no—
157C24×C3—21no—
158C24×C3—51no—
159C24×C3C22⋊D513yescomplemented
160C24×C3C22×D511yescharacteristic
161C3×#41—212no—
162C3×#41—212no—
163C2×C3×Q8—21no—
164C2×C3×D4—23no—
165C2×C3×D4—212no—
166C2×C3×D4—26no—
167C2×C3×D4—212no—
168C2×C3×D4—26no—
169C2×C3×D4—103no—
170C2×C3×D4—103no—
171C2×C3×D4—106no—
172C2×C3×D4C22⋊D516yescomplemented
173C2×C3×D4C22×D516yes—
174C22×C3×C4—23no—
175C22×C3×C4—51no—
176C22×C3×C4—103no—
177C22×C3×C4C22⋊D513yescomplemented
178C3×C4⋊C4—106no—
179C3×C22⋊C4—56no—
180C3×C22⋊C4—103no—
181C3×C22⋊C4—106no—
182C3×C22⋊C4—106no—
183C23×C5—24no—
184C23×C5—23no—
185C23×C5—23no—
186C23×C5—81no—
187C23×C5—83no—
188C23×C5—82no—
189C23×C5C2×C3×D416yes—
190C23×C5C2×C3×D413yes—
191C23×C5C2×C3×D413yes—
192C23×C5C24×C311yescharacteristic
193C22×D5—23no—
194C22×D5—81no—
195C5×Q8—24no—
196C5×D4—212no—
197C5×D4—212no—
198C5×D4—212no—
199C2×C4×C5—212no—
200C2×C4×C5—23no—
201C2×C4×C5—23no—
202C2×C4×C5C2×C3×D416yes—
203C22⋊D5—41no—
204C22⋊D5—43no—
205C22⋊D5—86no—
206C2×Dic5—23no—
207C2×Dic5—26no—
208C2×Dic5—83no—
209H(5,𝔽2)—26no—
210H(5,𝔽2)C2×C3×D514yes—
211C2×#41—23no—
212C22×D4—23no—
213C22×D4—51no—
214C22×D4C2×C3×D513yes—
215#123—56no—
216#123—103no—
217#121—106no—
218#120—56no—
219#120—103no—
220C2×C22⋊C4—53no—
221#99—512no—
222C2×C3×C5—23no—
223C2×C3×C5—21no—
224C2×C3×C5—23no—
225C2×C3×C5—46no—
226C2×C3×C5C2×#12011yescharacteristic
227C3×D5—81no—
228C23×C3—24no—
229C23×C3—23no—
230C23×C3—23no—
231C23×C3—81no—
232C23×C3—83no—
233C23×C3—82no—
234C23×C3—103no—
235C23×C3—401no—
236C23×C3C2×C22⋊D516yes—
237C23×C3C2×C22⋊D513yes—
238C23×C3C2×C22⋊D513yes—
239C23×C3C23×D511yescharacteristic
240C3×Q8—24no—
241C3×D4—212no—
242C3×D4—212no—
243C3×D4—212no—
244C3×D4—201no—
245C3×D4—203no—
246C3×D4—406no—
247C2×C3×C4—212no—
248C2×C3×C4—23no—
249C2×C3×C4—23no—
250C2×C3×C4—106no—
251C2×C3×C4—103no—
252C2×C3×C4—403no—
253C2×C3×C4C2×C22⋊D516yes—
254C22×C5—26no—
255C22×C5—43no—
256C22×C5—41no—
257C22×C5—43no—
258C22×C5—46no—
259C22×C5—46no—
260C22×C5—83no—
261C22×C5—84no—
262C22×C5C3×#12011yescharacteristic
263C22×C5C3×#12013yes—
264C22×C5C22×C3×D413yes—
265C2×D5—46no—
266C2×D5—41no—
267C4×C5—26no—
268Dic5—46no—
269Dic5—41no—
270C24—21no—
271C24—51no—
272C24C3×C22⋊D513yescomplemented
273C24C22×C3×D511yescharacteristic
274#41—212no—
275#41—212no—
276C2×Q8—21no—
277C2×D4—23no—
278C2×D4—212no—
279C2×D4—26no—
280C2×D4—212no—
281C2×D4—26no—
282C2×D4—103no—
283C2×D4—103no—
284C2×D4—106no—
285C2×D4C3×C22⋊D516yescomplemented
286C2×D4C22×C3×D516yes—
287C22×C4—23no—
288C22×C4—51no—
289C22×C4—103no—
290C22×C4C3×C22⋊D513yescomplemented
291C4⋊C4—106no—
292C22⋊C4—56no—
293C22⋊C4—103no—
294C22⋊C4—106no—
295C22⋊C4—106no—
296C3×C5#302611yescharacteristic, complemented, Hall
297C22×C3—26no—
298C22×C3—43no—
299C22×C3—43no—
300C22×C3—41no—
301C22×C3—46no—
302C22×C3—46no—
303C22×C3—83no—
304C22×C3—84no—
305C22×C3—206no—
306C22×C3—201no—
307C22×C3#409911yescharacteristic
308C22×C3#409913yes—
309C22×C3C22×C22⋊D513yes—
310C3×C4—26no—
311C3×C4—206no—
312C3×C4—201no—
313C2×C5—23no—
314C2×C5—21no—
315C2×C5—23no—
316C2×C5—46no—
317C2×C5C2×C3×#12011yescharacteristic
318D5—81no—
319C23—24no—
320C23—23no—
321C23—23no—
322C23—81no—
323C23—83no—
324C23—82no—
325C23—103no—
326C23—401no—
327C23C2×C3×C22⋊D516yes—
328C23C2×C3×C22⋊D513yes—
329C23C2×C3×C22⋊D513yes—
330C23C23×C3×D511yescharacteristic
331Q8—24no—
332D4—212no—
333D4—212no—
334D4—212no—
335D4—201no—
336D4—203no—
337D4—406no—
338C2×C4—212no—
339C2×C4—23no—
340C2×C4—23no—
341C2×C4—106no—
342C2×C4—103no—
343C2×C4—403no—
344C2×C4C2×C3×C22⋊D516yes—
345C2×C3—23no—
346C2×C3—21no—
347C2×C3—23no—
348C2×C3—46no—
349C2×C3—401no—
350C2×C3C2×#409911yescharacteristic
351C5C3×#302611yescharacteristic, complemented, Sylow
352C22—26no—
353C22—43no—
354C22—43no—
355C22—41no—
356C22—46no—
357C22—46no—
358C22—83no—
359C22—84no—
360C22—206no—
361C22—201no—
362C22C3×#409911yescharacteristic
363C22C3×#409913yes—
364C22C22×C3×C22⋊D513yes—
365C4—26no—
366C4—206no—
367C4—201no—
368C3#1061584111yescharacteristic, direct product factor, Sylow
369C2—23no—
370C2—21no—
371C2—23no—
372C2—46no—
373C2—401no—
374C2C2×C3×#409911yescharacteristic
subgroupG/H
centerC2×C3C2×#4099
commutator, cosocleC23×C5C24×C3
FrattiniC23C23×C3×D5
FittingC2×C3×C5×H(5,𝔽2)C2
socleC2×C3×C5C2×#120
hypercenterC2×C3×H(5,𝔽2)D5
hypocenterC5C3×#3026
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C24×C3 ◁ C2×C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×H(5,𝔽2)C2semidirect
2. C3×#66138C2semidirect
3. C3×#65533C2semidirect
4. C3×#65532C2semidirect
5. C2×C3×#3966C2semidirect
6. #10615841C3direct (N × H)
12. C22×C3×C22⋊D5C22semidirect
14. C2×C3×C22⋊Dic5C22semidirect
16. C3×#4083C22semidirect
17. C3×#4081C22semidirect
19. C3×#3966C22semidirect
21. C2×C5×H(5,𝔽2)C2×C3semidirect
22. #66138C2×C3semidirect
23. #65533C2×C3semidirect
24. #65532C2×C3semidirect
25. C2×#3966C2×C3semidirect
27. C24×C3×C5D4semidirect
29. C23×C3×D5C23semidirect
38. C2×C3×C5×D4D4semidirect
41. C22×C3×C4×C5D4semidirect
46. C22×C3×Dic5C23semidirect
49. C3×C22⋊Dic5C23semidirect
52. C2×C3×H(5,𝔽2)D5semidirect
62. C22×C22⋊D5C22×C3semidirect
64. C2×C22⋊Dic5C22×C3semidirect
66. #4083C22×C3semidirect
67. #4081C22×C3semidirect
69. #3966C22×C3semidirect
111. C24×C5C3×D4semidirect
113. C23×D5C23×C3semidirect
122. C2×C5×D4C3×D4semidirect
125. C22×C4×C5C3×D4semidirect
130. C22×Dic5C23×C3semidirect
133. C22⋊Dic5C23×C3semidirect
136. C2×H(5,𝔽2)C3×D5semidirect
159. C24×C3C22⋊D5semidirect
172. C2×C3×D4C22⋊D5semidirect
177. C22×C3×C4C22⋊D5semidirect
272. C24C3×C22⋊D5semidirect
285. C2×D4C3×C22⋊D5semidirect
290. C22×C4C3×C22⋊D5semidirect
296. C3×C5#3026semidirect
351. C5C3×#3026semidirect
368. C3#10615841direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×H(5,𝔽2)2
C3×#661382
C3×#655332
C3×#655322
C2×C3×#39662
#106158413
C3×#30265