Finite Group Databasebeta

← C5×#69698 #14287717

Subgroups — C5×#69698 #14287717

Order 1920.

632 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C5×#5483C211yesmaximal, characteristic, complemented
2C3×C5×#392C211yesmaximal, characteristic
3C5×#5218C212yesmaximal, complemented
4C5×#4525C211yesmaximal, characteristic, complemented
5C5×#4452C212yesmaximal, complemented
6C5×#1596—31nomaximal, Hall
7C22×C3×C5×D4—21no—
8C22×C3×C5×D4C2211yescharacteristic
9C2×C3×C5×C22⋊C4—21no—
10C2×C3×C5×C22⋊C4C2211yescharacteristic
11C2×C5×#217—21no—
12C2×C5×#214—22no—
13C5×#924—21no—
14C5×#924C2212yes—
15C5×#878—22no—
16C5×#878—22no—
17C3×C5×#95C2211yescharacteristic
18C5×#832—22no—
19C5×#830C2212yes—
20#69698C511yesmaximal, characteristic, direct product factor, Hall
21C5×#543—31no—
22C5×#450—32no—
23C5×#392D311yescharacteristic
24C5×#328—32no—
25C5×#323—31no—
26C24×C3×C5—21no—
27C24×C3×C5D411yescharacteristic
28C2×C3×C5×D4—22no—
29C2×C3×C5×D4—42no—
30C2×C3×C5×D4—42no—
31C2×C3×C5×D4—42no—
32C2×C3×C5×D4—42no—
33C22×C3×C4×C5—21no—
34C22×C3×C4×C5D411yescharacteristic
35C22×C3×C4×C5C2311yescharacteristic
36C22×C5×Dic3—21no—
37C22×C5×Dic3—21no—
38C2×C5×Dic6—21no—
39C2×C5×Dic6D412yescomplemented
40C3×C5×C22⋊C4—22no—
41C3×C5×C22⋊C4—42no—
42C5×#217—22no—
43C5×#217—22no—
44C5×#214—44no—
45C5×C4⋊Dic3—21no—
46C5×C4⋊Dic3D412yescomplemented
47C5×Dic3⋊C4—22no—
48C5×Dic3⋊C4—22no—
49C2×C5×C3⋊C8—22no—
50#5483C2×C511yescharacteristic, complemented
51C3×#392C2×C511yescharacteristic
52#5218C2×C512yescomplemented
53#4525C2×C511yescharacteristic, complemented
54#4452C2×C512yescomplemented
55C22×C5×D4—21no—
56C22×C5×D4C2×D311yescharacteristic
57C2×C5×SD16—62no—
58C5×C22⋊Q8—32no—
59C5×C22⋊Q8—62no—
60C5×C22⋊Q8—62no—
61C5×C22⋊Q8—61no—
62C2×C5×C22⋊C4—21no—
63C2×C5×C22⋊C4—61no—
64C2×C5×C22⋊C4C2×D311yescharacteristic
65C5×#102—62no—
66C5×C22⋊C8—32no—
67C5×#95C2×D311yescharacteristic
68#1596—31noSylow
69C23×C3×C5—21no—
70C23×C3×C5—21no—
71C23×C3×C5—22no—
72C23×C3×C5—41no—
73C23×C3×C5—41no—
74C23×C3×C5—41no—
75C23×C3×C5—41no—
76C23×C3×C5—41no—
77C23×C3×C5—41no—
78C23×C3×C5—41no—
79C23×C3×C5—41no—
80C23×C3×C5SD1612yescomplemented
81C23×C3×C5C2×D411yescharacteristic
82C3×C5×D4—24no—
83C3×C5×D4—42no—
84C3×C5×D4—48no—
85C2×C3×C4×C5—22no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—41no—
88C2×C3×C4×C5—41no—
89C2×C3×C4×C5—41no—
90C2×C3×C4×C5—41no—
91C2×C3×C4×C5C2×D412yes—
92C2×C5×Dic3—22no—
93C2×C5×Dic3—22no—
94C2×C5×Dic3—81no—
95C2×C5×Dic3—81no—
96C5×Dic6—42no—
97C5×Dic6—81no—
98C5×C3⋊C8—42no—
99C22×C3×D4—21no—
100C22×C3×D4C22×C511yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4C22×C511yescharacteristic
103C2×#217—21no—
104C2×#214—22no—
105#924—21no—
106#924C22×C512yes—
107#878—22no—
108#878—22no—
109C3×#95C22×C511yescharacteristic
110#832—22no—
111#830C22×C512yes—
112C24×C5—21no—
113C24×C5#6711yescharacteristic
114C2×C5×Q8—32no—
115C2×C5×Q8—61no—
116C2×C5×D4—22no—
117C2×C5×D4—42no—
118C2×C5×D4—42no—
119C2×C5×D4—42no—
120C2×C5×D4—42no—
121C22×C4×C5—21no—
122C22×C4×C5—61no—
123C22×C4×C5—61no—
124C22×C4×C5#6711yescharacteristic
125C22×C4×C5C22×D311yescharacteristic
126C5×SD16—124no—
127C2×C5×C8—62no—
128C5×C4⋊C4—32no—
129C5×C4⋊C4—62no—
130C5×C4⋊C4—61no—
131C5×C4⋊C4—62no—
132C5×C22⋊C4—22no—
133C5×C22⋊C4—42no—
134C5×C22⋊C4—62no—
135C5×C22⋊C4—62no—
136#543—31no—
137#450—32no—
138#392C5×D311yescharacteristic
139#328—32no—
140#323—31no—
141C22×C3×C5—21no—
142C22×C3×C5—21no—
143C22×C3×C5—22no—
144C22×C3×C5—42no—
145C22×C3×C5—42no—
146C22×C3×C5—42no—
147C22×C3×C5—41no—
148C22×C3×C5—41no—
149C22×C3×C5—42no—
150C22×C3×C5—81no—
151C22×C3×C5—81no—
152C22×C3×C5—81no—
153C22×C3×C5—81no—
154C22×C3×C5—81no—
155C22×C3×C5#12011yescharacteristic
156C22×C3×C5C2×SD1611yescharacteristic
157C22×C3×C5#13711yescharacteristic
158C3×C4×C5—22no—
159C3×C4×C5—42no—
160C3×C4×C5—41no—
161C5×Dic3—42no—
162C5×Dic3—42no—
163C24×C3—21no—
164C24×C3C5×D411yescharacteristic
165C2×C3×D4—22no—
166C2×C3×D4—42no—
167C2×C3×D4—42no—
168C2×C3×D4—42no—
169C2×C3×D4—42no—
170C22×C3×C4—21no—
171C22×C3×C4C5×D411yescharacteristic
172C22×C3×C4C23×C511yescharacteristic
173C22×Dic3—21no—
174C22×Dic3—21no—
175C2×Dic6—21no—
176C2×Dic6C5×D412yescomplemented
177C3×C22⋊C4—22no—
178C3×C22⋊C4—42no—
179#217—22no—
180#217—22no—
181#214—44no—
182C4⋊Dic3—21no—
183C4⋊Dic3C5×D412yescomplemented
184Dic3⋊C4—22no—
185Dic3⋊C4—22no—
186C2×C3⋊C8—22no—
187C23×C5—21no—
188C23×C5—21no—
189C23×C5—22no—
190C23×C5—41no—
191C23×C5—41no—
192C23×C5—41no—
193C23×C5—41no—
194C23×C5—41no—
195C23×C5—41no—
196C23×C5—41no—
197C23×C5—41no—
198C23×C5#21412yescomplemented
199C23×C5C2×#6711yescharacteristic
200C5×Q8—122no—
201C5×Q8—241no—
202C5×D4—24no—
203C5×D4—42no—
204C5×D4—48no—
205C2×C4×C5—22no—
206C2×C4×C5—21no—
207C2×C4×C5—41no—
208C2×C4×C5—41no—
209C2×C4×C5—41no—
210C2×C4×C5—41no—
211C2×C4×C5—62no—
212C2×C4×C5—62no—
213C2×C4×C5—241no—
214C2×C4×C5—241no—
215C2×C4×C5D3×D412yes—
216C5×C8—122no—
217C22×D4—21no—
218C22×D4C2×C5×D311yescharacteristic
219C2×SD16—62no—
220C22⋊Q8—32no—
221C22⋊Q8—62no—
222C22⋊Q8—62no—
223C22⋊Q8—61no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—61no—
226C2×C22⋊C4C2×C5×D311yescharacteristic
227#102—62no—
228C22⋊C8—32no—
229#95C2×C5×D311yescharacteristic
230C2×C3×C5—21no—
231C2×C3×C5—21no—
232C2×C3×C5—42no—
233C2×C3×C5—82no—
234C2×C3×C5#44811yescharacteristic
235C2×C3×C5#45311yescharacteristic
236C2×C3×C5B(4,𝔽2)11yescharacteristic
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—22no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3C5×SD1612yescomplemented
249C23×C3C2×C5×D411yescharacteristic
250C3×D4—24no—
251C3×D4—42no—
252C3×D4—48no—
253C2×C3×C4—22no—
254C2×C3×C4—21no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4C2×C5×D412yes—
260C2×Dic3—22no—
261C2×Dic3—22no—
262C2×Dic3—81no—
263C2×Dic3—81no—
264Dic6—42no—
265Dic6—81no—
266C3⋊C8—42no—
267C22×C5—21no—
268C22×C5—21no—
269C22×C5—22no—
270C22×C5—42no—
271C22×C5—42no—
272C22×C5—42no—
273C22×C5—41no—
274C22×C5—41no—
275C22×C5—42no—
276C22×C5—81no—
277C22×C5—81no—
278C22×C5—81no—
279C22×C5—81no—
280C22×C5—81no—
281C22×C5C2×#21411yescharacteristic
282C22×C5#93711yescharacteristic
283C22×C5#95111yescharacteristic
284C4×C5—22no—
285C4×C5—42no—
286C4×C5—41no—
287C4×C5—122no—
288C4×C5—122no—
289C24—21no—
290C24C5×#6711yescharacteristic
291C2×Q8—32no—
292C2×Q8—61no—
293C2×D4—22no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C22×C4—21no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4C5×#6711yescharacteristic
302C22×C4C22×C5×D311yescharacteristic
303SD16—124no—
304C2×C8—62no—
305C4⋊C4—32no—
306C4⋊C4—62no—
307C4⋊C4—61no—
308C4⋊C4—62no—
309C22⋊C4—22no—
310C22⋊C4—42no—
311C22⋊C4—62no—
312C22⋊C4—62no—
313C3×C5#159611yescharacteristic, complemented, Hall
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—22no—
317C22×C3—42no—
318C22×C3—42no—
319C22×C3—42no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—42no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3C5×#12011yescharacteristic
329C22×C3C2×C5×SD1611yescharacteristic
330C22×C3C5×#13711yescharacteristic
331C3×C4—22no—
332C3×C4—42no—
333C3×C4—41no—
334Dic3—42no—
335Dic3—42no—
336C2×C5—21no—
337C2×C5—21no—
338C2×C5—42no—
339C2×C5—82no—
340C2×C5#503211yescharacteristic
341C2×C5#504011yescharacteristic
342C2×C5#507711yescharacteristic
343C23—21no—
344C23—21no—
345C23—22no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23C5×#21412yescomplemented
355C23C2×C5×#6711yescharacteristic
356Q8—122no—
357Q8—241no—
358D4—24no—
359D4—42no—
360D4—48no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—62no—
368C2×C4—62no—
369C2×C4—241no—
370C2×C4—241no—
371C2×C4C5×D3×D412yes—
372C8—122no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—42no—
376C2×C3—82no—
377C2×C3C5×#44811yescharacteristic
378C2×C3C5×#45311yescharacteristic
379C2×C3C5×B(4,𝔽2)11yescharacteristic
380C5#6969811yescharacteristic, direct product factor, Sylow
381C22—21no—
382C22—21no—
383C22—22no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—42no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22C2×C5×#21411yescharacteristic
396C22C5×#93711yescharacteristic
397C22C5×#95111yescharacteristic
398C4—22no—
399C4—42no—
400C4—41no—
401C4—122no—
402C4—122no—
403C3C5×#159611yescharacteristic, complemented, Sylow
404C2—21no—
405C2—21no—
406C2—42no—
407C2—82no—
408C2C5×#503211yescharacteristic
409C2C5×#504011yescharacteristic
410C2C5×#507711yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
Fitting, prime radicalC3×C5×#392C2
socleC22×C3×C5#120
hypercenterC5×#392D3
hypocenterC3C5×#1596
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#392

Complements

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

NH ≅ G/Ntype
1. C5×#5483C2semidirect
3. C5×#5218C2semidirect
4. C5×#4525C2semidirect
5. C5×#4452C2semidirect
20. #69698C5direct (N × H)
39. C2×C5×Dic6D4semidirect
46. C5×C4⋊Dic3D4semidirect
50. #5483C2×C5semidirect
52. #5218C2×C5semidirect
53. #4525C2×C5semidirect
54. #4452C2×C5semidirect
80. C23×C3×C5SD16semidirect
176. C2×Dic6C5×D4semidirect
183. C4⋊Dic3C5×D4semidirect
198. C23×C5#214semidirect
248. C23×C3C5×SD16semidirect
313. C3×C5#1596semidirect
354. C23C5×#214semidirect
380. C5#69698direct (N × H)
403. C3C5×#1596semidirect

Maximal subgroups

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

subgroupindex
C5×#54832
C3×C5×#3922
C5×#52182
C5×#45252
C5×#44522
C5×#15963
#696985