Finite Group Databasebeta

← C3×#10615856 #14425662

Subgroups — C3×#10615856 #14425662

Order 1920.

1532 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1532 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
1C3×C5×#585C211yesmaximal, characteristic, complemented
2C3×#66162C213yesmaximal, complemented
3C3×#65533C214yesmaximal, complemented
4C3×#65532C214yesmaximal, complemented
5C3×#65529C213yesmaximal, complemented
6#10615856C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×2−1+4—23no—
8C3×C5×H(5,𝔽2)—23no—
9C3×C5×H(5,𝔽2)C2214yes—
10C2×C3×C5×#41—26no—
11C2×C3×C5×#41C2213yes—
12C22×C3×C4×D5C2211yescharacteristic, complemented
13C3×#4092—23no—
14C3×#4084—23no—
15C3×#4083C2216yescomplemented
16C3×#4081C2216yescomplemented
17C3×C4×C22⋊D5—26no—
18C3×#4072C2213yescomplemented
19C3×#3966C22112yescomplemented
20C3×#3025—51nomaximal, Hall
21C5×#585C2×C311yescharacteristic, complemented
22#66162C2×C313yescomplemented
23#65533C2×C314yescomplemented
24#65532C2×C314yescomplemented
25#65529C2×C313yescomplemented
26C23×C3×D5C2311yescharacteristic
27C3×C5×#41—24no—
28C3×C5×#41—212no—
29C3×C5×#41—26no—
30C3×C5×#41—212no—
31C3×C5×#41—26no—
32C2×C3×C5×Q8—26no—
33C2×C3×C5×Q8D413yescomplemented
34C2×C3×C5×D4—26no—
35C2×C3×C5×D4—212no—
36C2×C3×C5×D4D413yescomplemented
37C2×C3×C5×D4C2316yes—
38C22×C3×C4×C5—24no—
39C22×C3×C4×C5D416yescomplemented
40C22×C3×C4×C5C2311yescharacteristic
41C2×C3×C22⋊D5—26no—
42C22×C3×Dic5C2311yescharacteristic
43C2×C3×C4×D5—23no—
44C2×C3×C4×D5—23no—
45C3×C22⋊Dic5—26no—
46C3×C22⋊Dic5C2316yes—
47C3×#685—26no—
48C3×C4⋊Dic5—23no—
49C3×#683—26no—
50C3×C4×Dic5—23no—
51C3×#585D511yescharacteristic, complemented
52C3×#525—53no—
53C3×#458—54no—
54C3×B(4,𝔽2)—54no—
55C3×#410—53no—
56C5×2−1+4—23no—
57C5×H(5,𝔽2)—23no—
58C5×H(5,𝔽2)C22×C314yes—
59C2×C5×#41—26no—
60C2×C5×#41C22×C313yes—
61C22×C4×D5C22×C311yescharacteristic, complemented
62#4092—23no—
63#4084—23no—
64#4083C22×C316yescomplemented
65#4081C22×C316yescomplemented
66C4×C22⋊D5—26no—
67#4072C22×C313yescomplemented
68#3966C22×C3112yescomplemented
69#3025—51noSylow
70C23×C3×C5—24no—
71C23×C3×C5C2×D416yes—
72C23×C3×C5C2411yescharacteristic
73C22×C3×D5—23no—
74C22×C3×D5—42no—
75C3×C5×Q8—24no—
76C3×C5×Q8—26no—
77C3×C5×D4—212no—
78C3×C5×D4—212no—
79C3×C5×D4—26no—
80C2×C3×C4×C5—212no—
81C2×C3×C4×C5—26no—
82C2×C3×C4×C5—26no—
83C2×C3×C4×C5C2×D416yes—
84C2×C3×C4×C5C2×D413yes—
85C2×C3×C4×C5C2×D413yes—
86C3×C22⋊D5—412no—
87C2×C3×Dic5—26no—
88C2×C3×Dic5—23no—
89C3×C4×D5—43no—
90C3×C4×D5—41no—
91C3×2−1+4—23no—
92C3×H(5,𝔽2)—23no—
93C3×H(5,𝔽2)C2×D514yes—
94C2×C3×#41—26no—
95C2×C3×#41C2×D513yes—
96C23×C3×C4—51no—
97C3×#123—56no—
98C3×C22⋊Q8—103no—
99C3×#121—103no—
100C3×#120—56no—
101C3×C4×D4—106no—
102C3×#117—53no—
103C3×#99—512no—
104C23×D5C23×C311yescharacteristic
105C5×#41—24no—
106C5×#41—212no—
107C5×#41—26no—
108C5×#41—212no—
109C5×#41—26no—
110C2×C5×Q8—26no—
111C2×C5×Q8C3×D413yescomplemented
112C2×C5×D4—26no—
113C2×C5×D4—212no—
114C2×C5×D4C3×D413yescomplemented
115C2×C5×D4C23×C316yes—
116C22×C4×C5—24no—
117C22×C4×C5C3×D416yescomplemented
118C22×C4×C5C23×C311yescharacteristic
119C2×C22⋊D5—26no—
120C22×Dic5C23×C311yescharacteristic
121C2×C4×D5—23no—
122C2×C4×D5—23no—
123C22⋊Dic5—26no—
124C22⋊Dic5C23×C316yes—
125#685—26no—
126C4⋊Dic5—23no—
127#683—26no—
128C4×Dic5—23no—
129#585C3×D511yescharacteristic, complemented
130#525—53no—
131#458—54no—
132B(4,𝔽2)—54no—
133#410—53no—
134C22×C3×C5—26no—
135C22×C3×C5—41no—
136C22×C3×C5—46no—
137C22×C3×C5—84no—
138C22×C3×C5C22×D413yes—
139C2×C3×D5—26no—
140C2×C3×D5—43no—
141C2×C3×D5—41no—
142C3×C4×C5—26no—
143C3×C4×C5#12013yes—
144C3×C4×C5#12011yescharacteristic
145C3×Dic5—46no—
146C3×Dic5—41no—
147C24×C3—51no—
148C3×#41—24no—
149C3×#41—212no—
150C3×#41—26no—
151C3×#41—212no—
152C3×#41—26no—
153C2×C3×Q8—26no—
154C2×C3×Q8C22⋊D513yescomplemented
155C2×C3×D4—26no—
156C2×C3×D4—212no—
157C2×C3×D4—106no—
158C2×C3×D4C22⋊D513yescomplemented
159C2×C3×D4C22×D516yes—
160C22×C3×C4—24no—
161C22×C3×C4—51no—
162C22×C3×C4—103no—
163C22×C3×C4—103no—
164C22×C3×C4C22⋊D516yescomplemented
165C22×C3×C4C22×D511yescharacteristic
166C3×C4⋊C4—106no—
167C3×C4⋊C4—103no—
168C3×C22⋊C4—56no—
169C3×C22⋊C4—106no—
170C3×C22⋊C4—106no—
171C3×C42—103no—
172C23×C5—24no—
173C23×C5C2×C3×D416yes—
174C23×C5C24×C311yescharacteristic
175C22×D5—23no—
176C22×D5—42no—
177C5×Q8—24no—
178C5×Q8—26no—
179C5×D4—212no—
180C5×D4—212no—
181C5×D4—26no—
182C2×C4×C5—212no—
183C2×C4×C5—26no—
184C2×C4×C5—26no—
185C2×C4×C5C2×C3×D416yes—
186C2×C4×C5C2×C3×D413yes—
187C2×C4×C5C2×C3×D413yes—
188C22⋊D5—412no—
189C2×Dic5—26no—
190C2×Dic5—23no—
191C4×D5—41no—
192C4×D5—43no—
1932−1+4—23no—
194H(5,𝔽2)—23no—
195H(5,𝔽2)C2×C3×D514yes—
196C2×#41—26no—
197C2×#41C2×C3×D513yes—
198C23×C4—51no—
199#123—56no—
200C22⋊Q8—103no—
201#121—103no—
202#120—56no—
203C4×D4—106no—
204#117—53no—
205#99—512no—
206C2×C3×C5—23no—
207C2×C3×C5—46no—
208C2×C3×C5C2×#12011yescharacteristic
209C3×D5—42no—
210C23×C3—24no—
211C23×C3—103no—
212C23×C3—202no—
213C23×C3C2×C22⋊D516yes—
214C23×C3C23×D511yescharacteristic
215C3×Q8—24no—
216C3×Q8—26no—
217C3×D4—212no—
218C3×D4—212no—
219C3×D4—26no—
220C3×D4—2012no—
221C2×C3×C4—212no—
222C2×C3×C4—26no—
223C2×C3×C4—26no—
224C2×C3×C4—106no—
225C2×C3×C4—103no—
226C2×C3×C4—203no—
227C2×C3×C4—201no—
228C2×C3×C4C2×C22⋊D516yes—
229C2×C3×C4C2×C22⋊D513yes—
230C2×C3×C4C2×C22⋊D513yes—
231C22×C5—26no—
232C22×C5—41no—
233C22×C5—46no—
234C22×C5—84no—
235C22×C5C22×C3×D413yes—
236C2×D5—26no—
237C2×D5—43no—
238C2×D5—41no—
239C4×C5—26no—
240C4×C5C3×#12011yescharacteristic
241C4×C5C3×#12013yes—
242Dic5—46no—
243Dic5—41no—
244C24—51no—
245#41—24no—
246#41—212no—
247#41—26no—
248#41—212no—
249#41—26no—
250C2×Q8—26no—
251C2×Q8C3×C22⋊D513yescomplemented
252C2×D4—26no—
253C2×D4—212no—
254C2×D4—106no—
255C2×D4C3×C22⋊D513yescomplemented
256C2×D4C22×C3×D516yes—
257C22×C4—24no—
258C22×C4—51no—
259C22×C4—103no—
260C22×C4—103no—
261C22×C4C3×C22⋊D516yescomplemented
262C22×C4C22×C3×D511yescharacteristic
263C4⋊C4—103no—
264C4⋊C4—106no—
265C22⋊C4—56no—
266C22⋊C4—106no—
267C22⋊C4—106no—
268C42—103no—
269C3×C5#302511yescharacteristic, complemented, Hall
270C22×C3—26no—
271C22×C3—41no—
272C22×C3—46no—
273C22×C3—84no—
274C22×C3—106no—
275C22×C3—203no—
276C22×C3—201no—
277C22×C3C22×C22⋊D513yes—
278C3×C4—26no—
279C3×C4—206no—
280C3×C4—201no—
281C3×C4#409913yes—
282C3×C4#409911yescharacteristic
283C2×C5—23no—
284C2×C5—46no—
285C2×C5C2×C3×#12011yescharacteristic
286D5—42no—
287C23—24no—
288C23—103no—
289C23—202no—
290C23C2×C3×C22⋊D516yes—
291C23C23×C3×D511yescharacteristic
292Q8—24no—
293Q8—26no—
294D4—212no—
295D4—212no—
296D4—26no—
297D4—2012no—
298C2×C4—212no—
299C2×C4—26no—
300C2×C4—26no—
301C2×C4—106no—
302C2×C4—103no—
303C2×C4—201no—
304C2×C4—203no—
305C2×C4C2×C3×C22⋊D516yes—
306C2×C4C2×C3×C22⋊D513yes—
307C2×C4C2×C3×C22⋊D513yes—
308C2×C3—23no—
309C2×C3—46no—
310C2×C3—202no—
311C2×C3C2×#409911yescharacteristic
312C5C3×#302511yescharacteristic, complemented, Sylow
313C22—26no—
314C22—41no—
315C22—46no—
316C22—84no—
317C22—106no—
318C22—203no—
319C22—201no—
320C22C22×C3×C22⋊D513yes—
321C4—26no—
322C4—206no—
323C4—201no—
324C4C3×#409911yescharacteristic
325C4C3×#409913yes—
326C3#1061585611yescharacteristic, direct product factor, Sylow
327C2—23no—
328C2—46no—
329C2—202no—
330C2C2×C3×#409911yescharacteristic
subgroupG/H
centerC3×C4#4099
commutator, cosocleC23×C5C24×C3
FrattiniC23C23×C3×D5
FittingC3×C5×#585C2
socleC2×C3×C5C2×#120
hypercenterC3×#585D5
hypocenterC5C3×#3025
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#585

Complements

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

NH ≅ G/Ntype
1. C3×C5×#585C2semidirect
2. C3×#66162C2semidirect
3. C3×#65533C2semidirect
4. C3×#65532C2semidirect
5. C3×#65529C2semidirect
6. #10615856C3direct (N × H)
12. C22×C3×C4×D5C22semidirect
15. C3×#4083C22semidirect
16. C3×#4081C22semidirect
18. C3×#4072C22semidirect
19. C3×#3966C22semidirect
21. C5×#585C2×C3semidirect
22. #66162C2×C3semidirect
23. #65533C2×C3semidirect
24. #65532C2×C3semidirect
25. #65529C2×C3semidirect
33. C2×C3×C5×Q8D4semidirect
36. C2×C3×C5×D4D4semidirect
39. C22×C3×C4×C5D4semidirect
51. C3×#585D5semidirect
61. C22×C4×D5C22×C3semidirect
64. #4083C22×C3semidirect
65. #4081C22×C3semidirect
67. #4072C22×C3semidirect
68. #3966C22×C3semidirect
111. C2×C5×Q8C3×D4semidirect
114. C2×C5×D4C3×D4semidirect
117. C22×C4×C5C3×D4semidirect
129. #585C3×D5semidirect
154. C2×C3×Q8C22⋊D5semidirect
158. C2×C3×D4C22⋊D5semidirect
164. C22×C3×C4C22⋊D5semidirect
251. C2×Q8C3×C22⋊D5semidirect
255. C2×D4C3×C22⋊D5semidirect
261. C22×C4C3×C22⋊D5semidirect
269. C3×C5#3025semidirect
312. C5C3×#3025semidirect
326. C3#10615856direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5852
C3×#661622
C3×#655332
C3×#655322
C3×#655292
#106158563
C3×#30255