Finite Group Databasebeta

← C5×#71893 #14288002

Subgroups — C5×#71893 #14288002

Order 1920.

512 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 114 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×#543C211yesmaximal, characteristic
2C5×#5050C212yesmaximal, complemented
3C5×#4947C211yesmaximal, characteristic
4C5×#4538C212yesmaximal, complemented
5C5×#4525C211yesmaximal, characteristic
6C5×#1602—31nomaximal, Hall
7C22×C5×Dic6C2211yescharacteristic
8C3×C5×C22⋊Q8—22no—
9C3×C5×C22⋊Q8—21no—
10C3×C5×C22⋊Q8—22no—
11C3×C5×C22⋊Q8C2212yes—
12C2×C3×C5×C22⋊C4—21no—
13C2×C3×C5×C22⋊C4C2211yescharacteristic
14C2×C5×#217—21no—
15C2×C5×#216—22no—
16C5×C3⋊#95—21no—
17C5×C3⋊#95C2211yescharacteristic
18C5×#830C2212yes—
19C5×#810—22no—
20#71893C511yesmaximal, characteristic, direct product factor, Hall
21C5×#543D311yescharacteristic
22C5×#451—32no—
23C5×#393—31no—
24C5×#328—32no—
25C5×#323—31no—
26C24×C3×C5D411yescharacteristic
27C2×C3×C5×Q8—21no—
28C2×C3×C5×Q8D412yes—
29C22×C3×C4×C5—21no—
30C22×C3×C4×C5—21no—
31C22×C3×C4×C5D411yescharacteristic
32C22×C3×C4×C5C2311yescharacteristic
33C22×C5×Dic3—21no—
34C22×C5×Dic3—21no—
35C2×C5×Dic6—22no—
36C2×C5×Dic6—42no—
37C3×C5×C4⋊C4—21no—
38C3×C5×C4⋊C4—22no—
39C3×C5×C4⋊C4—22no—
40C3×C5×C4⋊C4D412yes—
41C3×C5×C22⋊C4—22no—
42C3×C5×C22⋊C4—22no—
43C3×C5×C22⋊C4—22no—
44C5×#217—42no—
45C5×#216—44no—
46C2×C5×C3⋊C8—22no—
47C3×#543C2×C511yescharacteristic
48#5050C2×C512yescomplemented
49#4947C2×C511yescharacteristic
50#4538C2×C512yescomplemented
51#4525C2×C511yescharacteristic
52C22×C5×Q8—31no—
53C2×C5×Q16—62no—
54C5×C22⋊Q8—22no—
55C5×C22⋊Q8—21no—
56C5×C22⋊Q8—22no—
57C5×C22⋊Q8C2×D312yes—
58C2×C5×C22⋊C4—21no—
59C2×C5×C22⋊C4—61no—
60C2×C5×C22⋊C4C2×D311yescharacteristic
61C5×#103—62no—
62C5×C22⋊C8—32no—
63C5×#95—31no—
64C5×#95—61no—
65#1602—31noSylow
66C23×C3×C5—21no—
67C23×C3×C5—21no—
68C23×C3×C5—41no—
69C23×C3×C5—41no—
70C23×C3×C5Q1612yescomplemented
71C23×C3×C5C2×D411yescharacteristic
72C3×C5×Q8—42no—
73C3×C5×Q8—81no—
74C2×C3×C4×C5—22no—
75C2×C3×C4×C5—21no—
76C2×C3×C4×C5—22no—
77C2×C3×C4×C5—41no—
78C2×C3×C4×C5—41no—
79C2×C3×C4×C5—81no—
80C2×C3×C4×C5—81no—
81C2×C3×C4×C5C2×D412yes—
82C2×C5×Dic3—22no—
83C2×C5×Dic3—41no—
84C2×C5×Dic3—41no—
85C2×C5×Dic3—41no—
86C2×C5×Dic3—41no—
87C2×C5×Dic3—41no—
88C5×Dic6—24no—
89C5×Dic6—42no—
90C5×C3⋊C8—42no—
91C22×Dic6C22×C511yescharacteristic
92C3×C22⋊Q8—22no—
93C3×C22⋊Q8—21no—
94C3×C22⋊Q8—22no—
95C3×C22⋊Q8C22×C512yes—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4C22×C511yescharacteristic
98C2×#217—21no—
99C2×#216—22no—
100C3⋊#95—21no—
101C3⋊#95C22×C511yescharacteristic
102#830C22×C512yes—
103#810—22no—
104C24×C5#6711yescharacteristic
105C2×C5×Q8—21no—
106C2×C5×Q8—62no—
107C2×C5×Q8—122no—
108C2×C5×Q8#6712yes—
109C22×C4×C5—21no—
110C22×C4×C5—21no—
111C22×C4×C5—61no—
112C22×C4×C5—61no—
113C22×C4×C5#6711yescharacteristic
114C22×C4×C5C22×D311yescharacteristic
115C5×Q16—124no—
116C2×C5×C8—62no—
117C5×C4⋊C4—21no—
118C5×C4⋊C4—22no—
119C5×C4⋊C4—22no—
120C5×C4⋊C4#6712yes—
121C5×C22⋊C4—22no—
122C5×C22⋊C4—22no—
123C5×C22⋊C4—22no—
124C5×C22⋊C4—122no—
125#543C5×D311yescharacteristic
126#451—32no—
127#393—31no—
128#328—32no—
129#323—31no—
130C22×C3×C5—21no—
131C22×C3×C5—21no—
132C22×C3×C5—22no—
133C22×C3×C5—42no—
134C22×C3×C5—41no—
135C22×C3×C5—41no—
136C22×C3×C5—81no—
137C22×C3×C5#12011yescharacteristic
138C22×C3×C5C2×Q1611yescharacteristic
139C22×C3×C5#13711yescharacteristic
140C3×C4×C5—22no—
141C3×C4×C5—41no—
142C3×C4×C5—42no—
143C3×C4×C5—42no—
144C5×Dic3—42no—
145C5×Dic3—81no—
146C24×C3C5×D411yescharacteristic
147C2×C3×Q8—21no—
148C2×C3×Q8C5×D412yes—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4C5×D411yescharacteristic
152C22×C3×C4C23×C511yescharacteristic
153C22×Dic3—21no—
154C22×Dic3—21no—
155C2×Dic6—22no—
156C2×Dic6—42no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4C5×D412yes—
161C3×C22⋊C4—22no—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—22no—
164#217—42no—
165#216—44no—
166C2×C3⋊C8—22no—
167C23×C5—21no—
168C23×C5—21no—
169C23×C5—41no—
170C23×C5—41no—
171C23×C5#21612yescomplemented
172C23×C5C2×#6711yescharacteristic
173C5×Q8—42no—
174C5×Q8—64no—
175C5×Q8—81no—
176C5×Q8—122no—
177C2×C4×C5—22no—
178C2×C4×C5—21no—
179C2×C4×C5—22no—
180C2×C4×C5—41no—
181C2×C4×C5—41no—
182C2×C4×C5—62no—
183C2×C4×C5—81no—
184C2×C4×C5—81no—
185C2×C4×C5—121no—
186C2×C4×C5—121no—
187C2×C4×C5—121no—
188C2×C4×C5—121no—
189C2×C4×C5—121no—
190C2×C4×C5C2×#6712yes—
191C5×C8—122no—
192C22×Q8—31no—
193C2×Q16—62no—
194C22⋊Q8—22no—
195C22⋊Q8—21no—
196C22⋊Q8—22no—
197C22⋊Q8C2×C5×D312yes—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4C2×C5×D311yescharacteristic
201#103—62no—
202C22⋊C8—32no—
203#95—31no—
204#95—61no—
205C2×C3×C5—21no—
206C2×C3×C5—21no—
207C2×C3×C5—42no—
208C2×C3×C5#45111yescharacteristic
209C2×C3×C5#45511yescharacteristic
210C2×C3×C5B(4,𝔽2)11yescharacteristic
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3C5×Q1612yescomplemented
216C23×C3C2×C5×D411yescharacteristic
217C3×Q8—42no—
218C3×Q8—81no—
219C2×C3×C4—22no—
220C2×C3×C4—21no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4C2×C5×D412yes—
227C2×Dic3—22no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233Dic6—24no—
234Dic6—42no—
235C3⋊C8—42no—
236C22×C5—21no—
237C22×C5—21no—
238C22×C5—22no—
239C22×C5—42no—
240C22×C5—41no—
241C22×C5—41no—
242C22×C5—81no—
243C22×C5#94211yescharacteristic
244C22×C5C2×#21611yescharacteristic
245C22×C5#95311yescharacteristic
246C4×C5—22no—
247C4×C5—41no—
248C4×C5—42no—
249C4×C5—42no—
250C4×C5—122no—
251C4×C5—241no—
252C24C5×#6711yescharacteristic
253C2×Q8—21no—
254C2×Q8—62no—
255C2×Q8—122no—
256C2×Q8C5×#6712yes—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4C5×#6711yescharacteristic
262C22×C4C22×C5×D311yescharacteristic
263Q16—124no—
264C2×C8—62no—
265C4⋊C4—21no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4C5×#6712yes—
269C22⋊C4—22no—
270C22⋊C4—22no—
271C22⋊C4—22no—
272C22⋊C4—122no—
273C3×C5#160211yescharacteristic, complemented, Hall
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—22no—
277C22×C3—42no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—81no—
281C22×C3C5×#12011yescharacteristic
282C22×C3C2×C5×Q1611yescharacteristic
283C22×C3C5×#13711yescharacteristic
284C3×C4—22no—
285C3×C4—41no—
286C3×C4—42no—
287C3×C4—42no—
288Dic3—42no—
289Dic3—81no—
290C2×C5—21no—
291C2×C5—21no—
292C2×C5—42no—
293C2×C5#522811yescharacteristic
294C2×C5#524211yescharacteristic
295C2×C5#524411yescharacteristic
296C23—21no—
297C23—21no—
298C23—41no—
299C23—41no—
300C23C5×#21612yescomplemented
301C23C2×C5×#6711yescharacteristic
302Q8—42no—
303Q8—64no—
304Q8—81no—
305Q8—122no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—62no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4C2×C5×#6712yes—
320C8—122no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—42no—
324C2×C3C5×#45111yescharacteristic
325C2×C3C5×#45511yescharacteristic
326C2×C3C5×B(4,𝔽2)11yescharacteristic
327C5#7189311yescharacteristic, direct product factor, Sylow
328C22—21no—
329C22—21no—
330C22—22no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22C5×#94211yescharacteristic
336C22C2×C5×#21611yescharacteristic
337C22C5×#95311yescharacteristic
338C4—22no—
339C4—41no—
340C4—42no—
341C4—42no—
342C4—122no—
343C4—241no—
344C3C5×#160211yescharacteristic, complemented, Sylow
345C2—21no—
346C2—21no—
347C2—42no—
348C2C5×#522811yescharacteristic
349C2C5×#524211yescharacteristic
350C2C5×#524411yescharacteristic
subgroupG/H
centerC22×C5#953
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#543C2
socleC22×C3×C5#120
prime radicalC24×C3×C5D4
hypercenterC5×#543D3
hypocenterC3C5×#1602
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#543

Complements

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

NH ≅ G/Ntype
2. C5×#5050C2semidirect
4. C5×#4538C2semidirect
20. #71893C5direct (N × H)
48. #5050C2×C5semidirect
50. #4538C2×C5semidirect
70. C23×C3×C5Q16semidirect
171. C23×C5#216semidirect
215. C23×C3C5×Q16semidirect
273. C3×C5#1602semidirect
300. C23C5×#216semidirect
327. C5#71893direct (N × H)
344. C3C5×#1602semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5432
C5×#50502
C5×#49472
C5×#45382
C5×#45252
C5×#16023
#718935