Finite Group Databasebeta

← C2×C3×#65532 #14425631

Subgroups — C2×C3×#65532 #14425631

Order 1920.

1588 conjugacy classes of subgroups; only the 330 normal subgroups are shown.

normal subgroups only

This group has 1588 conjugacy classes of subgroups — too many to draw. Only its 330 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
2C2×C3×#4081C213yesmaximal, complemented
3C3×#65532C218yesmaximal, direct product factor
4C2×C3×#3966C213yesmaximal, complemented
5C2×#65532C311yesmaximal, characteristic, direct product factor, Hall
6C3×C5×H(5,𝔽2)—26no—
7C3×C5×H(5,𝔽2)C2214yes—
8C2×C3×C5×#41—23no—
9C22×C3×C5×D4—23no—
10C22×C3×C5×D4C2213yes—
11C23×C3×Dic5C2211yescharacteristic, complemented
12C2×C3×C22⋊Dic5—23no—
13C2×C3×C22⋊Dic5C2213yescomplemented
14C3×#4081—26no—
15C3×#4081C22112yescomplemented
16C2×C3×#683—23no—
17C3×#3966C22112yescomplemented
18C2×C3×#458—51nomaximal, Hall
19C2×C5×H(5,𝔽2)C2×C311yescharacteristic
20C2×#4081C2×C313yescomplemented
21#65532C2×C318yesdirect product factor
22C2×#3966C2×C313yescomplemented
23C24×C3×C5—21no—
24C24×C3×C5D413yes—
25C24×C3×C5C2311yescharacteristic
26C3×C5×#41—212no—
27C3×C5×#41—212no—
28C2×C3×C5×Q8—21no—
29C2×C3×C5×D4—23no—
30C2×C3×C5×D4—212no—
31C2×C3×C5×D4—26no—
32C2×C3×C5×D4—212no—
33C2×C3×C5×D4—26no—
34C2×C3×C5×D4D416yes—
35C2×C3×C5×D4C2316yes—
36C22×C3×C4×C5—23no—
37C22×C3×C4×C5D413yes—
38C22×C3×Dic5—23no—
39C22×C3×Dic5—23no—
40C22×C3×Dic5—23no—
41C22×C3×Dic5C2312yescomplemented
42C3×C22⋊Dic5—23no—
43C3×C22⋊Dic5—212no—
44C3×C22⋊Dic5C2316yescomplemented
45C3×#683—212no—
46C2×C3×H(5,𝔽2)D511yescharacteristic
47C2×C3×#123—53no—
48C3×#458—58no—
49C2×C3×#99—53no—
50C5×H(5,𝔽2)—26no—
51C5×H(5,𝔽2)C22×C314yes—
52C2×C5×#41—23no—
53C22×C5×D4—23no—
54C22×C5×D4C22×C313yes—
55C23×Dic5C22×C311yescharacteristic, complemented
56C2×C22⋊Dic5—23no—
57C2×C22⋊Dic5C22×C313yescomplemented
58#4081—26no—
59#4081C22×C3112yescomplemented
60C2×#683—23no—
61#3966C22×C3112yescomplemented
62C2×#458—51noSylow
63C23×C3×C5—24no—
64C23×C3×C5—23no—
65C23×C3×C5—23no—
66C23×C3×C5—42no—
67C23×C3×C5—46no—
68C23×C3×C5—82no—
69C23×C3×C5C2×D413yes—
70C23×C3×C5C2×D416yes—
71C23×C3×C5C2×D413yes—
72C23×C3×C5C2411yescharacteristic
73C3×C5×Q8—24no—
74C3×C5×D4—212no—
75C3×C5×D4—212no—
76C3×C5×D4—212no—
77C2×C3×C4×C5—212no—
78C2×C3×C4×C5—23no—
79C2×C3×C4×C5—23no—
80C2×C3×C4×C5C2×D416yes—
81C2×C3×Dic5—26no—
82C2×C3×Dic5—26no—
83C2×C3×Dic5—46no—
84C2×C3×Dic5—43no—
85C2×C3×Dic5—41no—
86C3×H(5,𝔽2)—26no—
87C3×H(5,𝔽2)C2×D514yes—
88C2×C3×#41—23no—
89C22×C3×D4—23no—
90C22×C3×D4C2×D513yes—
91C23×C3×C4—51no—
92C3×#123—512no—
93C3×#123—106no—
94C2×C3×C4⋊C4—103no—
95C2×C3×C22⋊C4—53no—
96C2×C3×C22⋊C4—103no—
97C3×#99—512no—
98C24×C5—21no—
99C24×C5C3×D413yes—
100C24×C5C23×C311yescharacteristic
101C5×#41—212no—
102C5×#41—212no—
103C2×C5×Q8—21no—
104C2×C5×D4—23no—
105C2×C5×D4—212no—
106C2×C5×D4—26no—
107C2×C5×D4—212no—
108C2×C5×D4—26no—
109C2×C5×D4C3×D416yes—
110C2×C5×D4C23×C316yes—
111C22×C4×C5—23no—
112C22×C4×C5C3×D413yes—
113C22×Dic5—23no—
114C22×Dic5—23no—
115C22×Dic5—23no—
116C22×Dic5C23×C312yescomplemented
117C22⋊Dic5—212no—
118C22⋊Dic5—23no—
119C22⋊Dic5C23×C316yescomplemented
120#683—212no—
121C2×H(5,𝔽2)C3×D511yescharacteristic
122C2×#123—53no—
123#458—58no—
124C2×#99—53no—
125C22×C3×C5—26no—
126C22×C3×C5—26no—
127C22×C3×C5—43no—
128C22×C3×C5—41no—
129C22×C3×C5—46no—
130C22×C3×C5—46no—
131C22×C3×C5—83no—
132C22×C3×C5—84no—
133C22×C3×C5#12013yes—
134C22×C3×C5#12011yescharacteristic
135C22×C3×C5C22×D413yes—
136C3×C4×C5—26no—
137C3×Dic5—46no—
138C3×Dic5—42no—
139C24×C3—21no—
140C24×C3C22⋊D513yes—
141C24×C3C22×D511yescharacteristic
142C3×#41—212no—
143C3×#41—212no—
144C2×C3×Q8—21no—
145C2×C3×D4—23no—
146C2×C3×D4—212no—
147C2×C3×D4—26no—
148C2×C3×D4—212no—
149C2×C3×D4—26no—
150C2×C3×D4C22⋊D516yes—
151C2×C3×D4C22×D516yes—
152C22×C3×C4—23no—
153C22×C3×C4—52no—
154C22×C3×C4—103no—
155C22×C3×C4—103no—
156C22×C3×C4—103no—
157C22×C3×C4C22⋊D513yes—
158C3×C4⋊C4—1012no—
159C3×C22⋊C4—56no—
160C3×C22⋊C4—103no—
161C3×C22⋊C4—1012no—
162C23×C5—24no—
163C23×C5—23no—
164C23×C5—23no—
165C23×C5—42no—
166C23×C5—46no—
167C23×C5—82no—
168C23×C5C2×C3×D413yes—
169C23×C5C2×C3×D416yes—
170C23×C5C2×C3×D413yes—
171C23×C5C24×C311yescharacteristic
172C5×Q8—24no—
173C5×D4—212no—
174C5×D4—212no—
175C5×D4—212no—
176C2×C4×C5—212no—
177C2×C4×C5—23no—
178C2×C4×C5—23no—
179C2×C4×C5C2×C3×D416yes—
180C2×Dic5—26no—
181C2×Dic5—26no—
182C2×Dic5—46no—
183C2×Dic5—43no—
184C2×Dic5—41no—
185H(5,𝔽2)—26no—
186H(5,𝔽2)C2×C3×D514yes—
187C2×#41—23no—
188C22×D4—23no—
189C22×D4C2×C3×D513yes—
190C23×C4—51no—
191#123—512no—
192#123—106no—
193C2×C4⋊C4—103no—
194C2×C22⋊C4—53no—
195C2×C22⋊C4—103no—
196#99—512no—
197C2×C3×C5—23no—
198C2×C3×C5—23no—
199C2×C3×C5—46no—
200C2×C3×C5#45812yescomplemented
201C2×C3×C5C2×#12011yescharacteristic
202C23×C3—24no—
203C23×C3—23no—
204C23×C3—23no—
205C23×C3—42no—
206C23×C3—46no—
207C23×C3—82no—
208C23×C3C2×C22⋊D513yes—
209C23×C3C2×C22⋊D516yes—
210C23×C3C2×C22⋊D513yes—
211C23×C3C23×D511yescharacteristic
212C3×Q8—24no—
213C3×D4—212no—
214C3×D4—212no—
215C3×D4—212no—
216C2×C3×C4—212no—
217C2×C3×C4—23no—
218C2×C3×C4—23no—
219C2×C3×C4—106no—
220C2×C3×C4—106no—
221C2×C3×C4—206no—
222C2×C3×C4—203no—
223C2×C3×C4—201no—
224C2×C3×C4C2×C22⋊D516yes—
225C22×C5—26no—
226C22×C5—26no—
227C22×C5—43no—
228C22×C5—41no—
229C22×C5—46no—
230C22×C5—46no—
231C22×C5—83no—
232C22×C5—84no—
233C22×C5C3×#12013yes—
234C22×C5C3×#12011yescharacteristic
235C22×C5C22×C3×D413yes—
236C4×C5—26no—
237Dic5—46no—
238Dic5—42no—
239C24—21no—
240C24C3×C22⋊D513yes—
241C24C22×C3×D511yescharacteristic
242#41—212no—
243#41—212no—
244C2×Q8—21no—
245C2×D4—23no—
246C2×D4—212no—
247C2×D4—26no—
248C2×D4—212no—
249C2×D4—26no—
250C2×D4C3×C22⋊D516yes—
251C2×D4C22×C3×D516yes—
252C22×C4—23no—
253C22×C4—52no—
254C22×C4—103no—
255C22×C4—103no—
256C22×C4—103no—
257C22×C4C3×C22⋊D513yes—
258C4⋊C4—1012no—
259C22⋊C4—56no—
260C22⋊C4—103no—
261C22⋊C4—1012no—
262C3×C5C2×#45811yescharacteristic, complemented, Hall
263C22×C3—26no—
264C22×C3—26no—
265C22×C3—43no—
266C22×C3—41no—
267C22×C3—46no—
268C22×C3—46no—
269C22×C3—83no—
270C22×C3—84no—
271C22×C3#409913yes—
272C22×C3#409911yescharacteristic
273C22×C3C22×C22⋊D513yes—
274C3×C4—26no—
275C3×C4—206no—
276C3×C4—202no—
277C2×C5—23no—
278C2×C5—23no—
279C2×C5—46no—
280C2×C5C3×#45812yescomplemented
281C2×C5C2×C3×#12011yescharacteristic
282C23—24no—
283C23—23no—
284C23—23no—
285C23—42no—
286C23—46no—
287C23—82no—
288C23C2×C3×C22⋊D513yes—
289C23C2×C3×C22⋊D516yes—
290C23C2×C3×C22⋊D513yes—
291C23C23×C3×D511yescharacteristic
292Q8—24no—
293D4—212no—
294D4—212no—
295D4—212no—
296C2×C4—212no—
297C2×C4—23no—
298C2×C4—23no—
299C2×C4—106no—
300C2×C4—106no—
301C2×C4—206no—
302C2×C4—203no—
303C2×C4—201no—
304C2×C4C2×C3×C22⋊D516yes—
305C2×C3—23no—
306C2×C3—23no—
307C2×C3—46no—
308C2×C3#6553212yesdirect product factor
309C2×C3C2×#409911yescharacteristic
310C5C2×C3×#45811yescharacteristic, complemented, Sylow
311C22—26no—
312C22—26no—
313C22—43no—
314C22—41no—
315C22—46no—
316C22—46no—
317C22—83no—
318C22—84no—
319C22C3×#409913yes—
320C22C3×#409911yescharacteristic
321C22C22×C3×C22⋊D513yes—
322C4—26no—
323C4—206no—
324C4—202no—
325C3C2×#6553211yescharacteristic, direct product factor, Sylow
326C2—23no—
327C2—23no—
328C2—46no—
329C2C3×#6553212yesdirect product factor
330C2C2×C3×#409911yescharacteristic
subgroupG/H
centerC22×C3#4099
commutator, cosocleC23×C5C24×C3
FrattiniC23C23×C3×D5
Fitting, prime radicalC2×C3×C5×H(5,𝔽2)C2
socleC22×C3×C5#120
hypercenterC2×C3×H(5,𝔽2)D5
hypocenterC5C2×C3×#458
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C2×C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4081C2semidirect
3. C3×#65532C2direct (N × H)
4. C2×C3×#3966C2semidirect
5. C2×#65532C3direct (N × H)
11. C23×C3×Dic5C22semidirect
13. C2×C3×C22⋊Dic5C22semidirect
15. C3×#4081C22semidirect
17. C3×#3966C22semidirect
20. C2×#4081C2×C3semidirect
21. #65532C2×C3direct (N × H)
22. C2×#3966C2×C3semidirect
41. C22×C3×Dic5C23semidirect
44. C3×C22⋊Dic5C23semidirect
55. C23×Dic5C22×C3semidirect
57. C2×C22⋊Dic5C22×C3semidirect
59. #4081C22×C3semidirect
61. #3966C22×C3semidirect
116. C22×Dic5C23×C3semidirect
119. C22⋊Dic5C23×C3semidirect
200. C2×C3×C5#458semidirect
262. C3×C5C2×#458semidirect
280. C2×C5C3×#458semidirect
308. C2×C3#65532direct (N × H)
310. C5C2×C3×#458semidirect
325. C3C2×#65532direct (N × H)
329. C2C3×#65532direct (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
C2×C3×#40812
C3×#655322
C2×C3×#39662
C2×#655323
C2×C3×#4585