Finite Group Databasebeta

← C5×#69704 #14287858

Subgroups — C5×#69704 #14287858

Order 1920.

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

normal subgroups only

This group has 532 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×#5500C211yesmaximal, characteristic, complemented
2C3×C5×#394C211yesmaximal, characteristic
3C5×#5218C212yesmaximal
4C5×#4528C211yesmaximal, characteristic, complemented
5C5×#4452C212yesmaximal, complemented
6C5×#1599—31nomaximal, Hall
7C22×C3×C5×D4C2211yescharacteristic
8C2×C3×C5×C4⋊C4C2211yescharacteristic
9C2×C3×C5×C22⋊C4—21no—
10C2×C3×C5×C22⋊C4—21no—
11C2×C5×#214—22no—
12C5×#926—21no—
13C5×#924—21no—
14C5×#924C2212yes—
15C5×#890—22no—
16C5×#888—22no—
17C3×C5×#95C2211yescharacteristic
18C5×#832—22no—
19C5×#830C2212yes—
20#69704C511yesmaximal, characteristic, direct product factor, Hall
21C5×#544—31no—
22C5×#450—32no—
23C5×#394D311yescharacteristic
24C5×#328—32no—
25C5×#324—31no—
26C24×C3×C5—21no—
27C2×C3×C5×D4—22no—
28C2×C3×C5×D4—42no—
29C22×C3×C4×C5—21no—
30C22×C3×C4×C5D411yescharacteristic
31C22×C3×C4×C5D411yescharacteristic
32C22×C3×C4×C5C2311yescharacteristic
33C2×C5×Dic6—21no—
34C2×C5×Dic6D412yes—
35C3×C5×C4⋊C4—22no—
36C3×C5×C22⋊C4—42no—
37C3×C5×C22⋊C4—42no—
38C5×#217—22no—
39C5×#214—44no—
40C5×C4⋊Dic3—21no—
41C5×C4⋊Dic3—21no—
42C5×C4⋊Dic3—21no—
43C5×C4⋊Dic3D412yes—
44C5×Dic3⋊C4—22no—
45C5×Dic3⋊C4—22no—
46C4×C5×Dic3—22no—
47C2×C5×C3⋊C8—22no—
48#5500C2×C511yescharacteristic, complemented
49C3×#394C2×C511yescharacteristic
50#5218C2×C512yes—
51#4528C2×C511yescharacteristic, complemented
52#4452C2×C512yescomplemented
53C22×C5×D4C2×D311yescharacteristic
54C2×C5×SD16—62no—
55C5×C4⋊Q8—62no—
56C5×#125—62no—
57C5×C22⋊Q8—32no—
58C5×C22⋊Q8—61no—
59C5×#117—61no—
60C2×C5×C4⋊C4C2×D311yescharacteristic
61C2×C5×C22⋊C4—21no—
62C2×C5×C22⋊C4—21no—
63C5×#102—62no—
64C5×C22⋊C8—32no—
65C5×#95C2×D311yescharacteristic
66#1599—31noSylow
67C23×C3×C5—22no—
68C23×C3×C5—41no—
69C23×C3×C5—41no—
70C23×C3×C5—41no—
71C23×C3×C5—41no—
72C23×C3×C5—41no—
73C23×C3×C5—41no—
74C23×C3×C5C2×D411yescharacteristic
75C3×C5×D4—24no—
76C3×C5×D4—42no—
77C2×C3×C4×C5—21no—
78C2×C3×C4×C5—21no—
79C2×C3×C4×C5—21no—
80C2×C3×C4×C5—41no—
81C2×C3×C4×C5—41no—
82C2×C3×C4×C5—41no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5—41no—
85C2×C3×C4×C5SD1612yescomplemented
86C2×C3×C4×C5C2×D412yes—
87C2×C5×Dic3—22no—
88C2×C5×Dic3—22no—
89C5×Dic6—42no—
90C5×Dic6—81no—
91C5×C3⋊C8—42no—
92C22×C3×D4C22×C511yescharacteristic
93C2×C3×C4⋊C4C22×C511yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×#214—22no—
97#926—21no—
98#924—21no—
99#924C22×C512yes—
100#890—22no—
101#888—22no—
102C3×#95C22×C511yescharacteristic
103#832—22no—
104#830C22×C512yes—
105C24×C5—21no—
106C2×C5×Q8—32no—
107C2×C5×Q8—61no—
108C2×C5×D4—22no—
109C2×C5×D4—42no—
110C22×C4×C5—21no—
111C22×C4×C5#6711yescharacteristic
112C22×C4×C5#6711yescharacteristic
113C22×C4×C5C22×D311yescharacteristic
114C5×SD16—124no—
115C2×C5×C8—62no—
116C5×C4⋊C4—22no—
117C5×C4⋊C4—32no—
118C5×C4⋊C4—61no—
119C5×C4⋊C4—61no—
120C5×C4⋊C4—61no—
121C5×C4⋊C4—62no—
122C5×C4⋊C4—62no—
123C5×C22⋊C4—42no—
124C5×C22⋊C4—42no—
125C5×C22⋊C4—62no—
126C42×C5—62no—
127#544—31no—
128#450—32no—
129#394C5×D311yescharacteristic
130#328—32no—
131#324—31no—
132C22×C3×C5—21no—
133C22×C3×C5—21no—
134C22×C3×C5—42no—
135C22×C3×C5—42no—
136C22×C3×C5—42no—
137C22×C3×C5—81no—
138C22×C3×C5—81no—
139C22×C3×C5—81no—
140C22×C3×C5—81no—
141C22×C3×C5#12011yescharacteristic
142C22×C3×C5C2×SD1611yescharacteristic
143C22×C3×C5#13711yescharacteristic
144C3×C4×C5—22no—
145C3×C4×C5—22no—
146C3×C4×C5—41no—
147C3×C4×C5—81no—
148C5×Dic3—42no—
149C5×Dic3—42no—
150C24×C3—21no—
151C2×C3×D4—22no—
152C2×C3×D4—42no—
153C22×C3×C4—21no—
154C22×C3×C4C5×D411yescharacteristic
155C22×C3×C4C5×D411yescharacteristic
156C22×C3×C4C23×C511yescharacteristic
157C2×Dic6—21no—
158C2×Dic6C5×D412yes—
159C3×C4⋊C4—22no—
160C3×C22⋊C4—42no—
161C3×C22⋊C4—42no—
162#217—22no—
163#214—44no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3C5×D412yes—
168Dic3⋊C4—22no—
169Dic3⋊C4—22no—
170C4×Dic3—22no—
171C2×C3⋊C8—22no—
172C23×C5—22no—
173C23×C5—41no—
174C23×C5—41no—
175C23×C5—41no—
176C23×C5—41no—
177C23×C5—41no—
178C23×C5—41no—
179C23×C5C2×#6711yescharacteristic
180C5×Q8—122no—
181C5×Q8—241no—
182C5×D4—24no—
183C5×D4—42no—
184C2×C4×C5—21no—
185C2×C4×C5—21no—
186C2×C4×C5—21no—
187C2×C4×C5—41no—
188C2×C4×C5—41no—
189C2×C4×C5—41no—
190C2×C4×C5—41no—
191C2×C4×C5—41no—
192C2×C4×C5—62no—
193C2×C4×C5—62no—
194C2×C4×C5#21412yescomplemented
195C2×C4×C5D3×D412yes—
196C5×C8—122no—
197C22×D4C2×C5×D311yescharacteristic
198C2×SD16—62no—
199C4⋊Q8—62no—
200#125—62no—
201C22⋊Q8—32no—
202C22⋊Q8—61no—
203#117—61no—
204C2×C4⋊C4C2×C5×D311yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207#102—62no—
208C22⋊C8—32no—
209#95C2×C5×D311yescharacteristic
210C2×C3×C5—21no—
211C2×C3×C5—21no—
212C2×C3×C5—82no—
213C2×C3×C5#44811yescharacteristic
214C2×C3×C5#45411yescharacteristic
215C2×C3×C5B(4,𝔽2)11yescharacteristic
216C23×C3—22no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3C2×C5×D411yescharacteristic
224C3×D4—24no—
225C3×D4—42no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4C5×SD1612yescomplemented
235C2×C3×C4C2×C5×D412yes—
236C2×Dic3—22no—
237C2×Dic3—22no—
238Dic6—42no—
239Dic6—81no—
240C3⋊C8—42no—
241C22×C5—21no—
242C22×C5—21no—
243C22×C5—42no—
244C22×C5—42no—
245C22×C5—42no—
246C22×C5—81no—
247C22×C5—81no—
248C22×C5—81no—
249C22×C5—81no—
250C22×C5C2×#21411yescharacteristic
251C22×C5#93711yescharacteristic
252C22×C5#95111yescharacteristic
253C4×C5—22no—
254C4×C5—22no—
255C4×C5—41no—
256C4×C5—81no—
257C4×C5—122no—
258C4×C5—122no—
259C24—21no—
260C2×Q8—32no—
261C2×Q8—61no—
262C2×D4—22no—
263C2×D4—42no—
264C22×C4—21no—
265C22×C4C5×#6711yescharacteristic
266C22×C4C5×#6711yescharacteristic
267C22×C4C22×C5×D311yescharacteristic
268SD16—124no—
269C2×C8—62no—
270C4⋊C4—22no—
271C4⋊C4—32no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—62no—
276C4⋊C4—62no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—62no—
280C42—62no—
281C3×C5#159911yescharacteristic, complemented, Hall
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—42no—
285C22×C3—42no—
286C22×C3—42no—
287C22×C3—81no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3C5×#12011yescharacteristic
292C22×C3C2×C5×SD1611yescharacteristic
293C22×C3C5×#13711yescharacteristic
294C3×C4—22no—
295C3×C4—22no—
296C3×C4—41no—
297C3×C4—81no—
298Dic3—42no—
299Dic3—42no—
300C2×C5—21no—
301C2×C5—21no—
302C2×C5—82no—
303C2×C5#503211yescharacteristic
304C2×C5#504011yescharacteristic
305C2×C5#506211yescharacteristic
306C23—22no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23C2×C5×#6711yescharacteristic
314Q8—122no—
315Q8—241no—
316D4—24no—
317D4—42no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—62no—
327C2×C4—62no—
328C2×C4C5×#21412yescomplemented
329C2×C4C5×D3×D412yes—
330C8—122no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—82no—
334C2×C3C5×#44811yescharacteristic
335C2×C3C5×#45411yescharacteristic
336C2×C3C5×B(4,𝔽2)11yescharacteristic
337C5#6970411yescharacteristic, direct product factor, Sylow
338C22—21no—
339C22—21no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22C2×C5×#21411yescharacteristic
348C22C5×#93711yescharacteristic
349C22C5×#95111yescharacteristic
350C4—22no—
351C4—22no—
352C4—41no—
353C4—81no—
354C4—122no—
355C4—122no—
356C3C5×#159911yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—82no—
360C2C5×#503211yescharacteristic
361C2C5×#504011yescharacteristic
362C2C5×#506211yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#394C2
socleC22×C3×C5#120
prime radicalC22×C3×C5×D4C22
hypercenterC5×#394D3
hypocenterC3C5×#1599
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#394

Complements

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

NH ≅ G/Ntype
1. C5×#5500C2semidirect
4. C5×#4528C2semidirect
5. C5×#4452C2semidirect
20. #69704C5direct (N × H)
48. #5500C2×C5semidirect
51. #4528C2×C5semidirect
52. #4452C2×C5semidirect
85. C2×C3×C4×C5SD16semidirect
194. C2×C4×C5#214semidirect
234. C2×C3×C4C5×SD16semidirect
281. C3×C5#1599semidirect
328. C2×C4C5×#214semidirect
337. C5#69704direct (N × H)
356. C3C5×#1599semidirect

Maximal subgroups

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

subgroupindex
C5×#55002
C3×C5×#3942
C5×#52182
C5×#45282
C5×#44522
C5×#15993
#697045