Finite Group Databasebeta

← C5×#68971 #14308983

Subgroups — C5×#68971 #14308983

Order 1920.

932 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 134 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
2C5×#4955C211yesmaximal, characteristic
3C5×#4955C212yesmaximal
4C3×C5×#342C211yesmaximal, characteristic, complemented
5C5×#4527C212yesmaximal, complemented
6C5×#2029—31nomaximal, Hall
7C24×C5×D3C2211yescharacteristic
8C2×C3×C5×C22⋊C4C2212yes—
9C2×C3×C5×C22⋊C4C2211yescharacteristic
10C2×C5×#217C2211yescharacteristic
11C2×C5×#217C2212yes—
12C2×C5×(C2×D3)⋊C4—22no—
13C2×C5×(C2×D3)⋊C4—24no—
14C5×#924—22no—
15C5×#878—24no—
16#68971C511yesmaximal, characteristic, direct product factor, Hall
17C5×#543—31no—
18C5×#379—32no—
19C5×#379—31no—
20C5×#342—32no—
21C5×#342D311yescharacteristic, complemented
22C24×C3×C5C2311yescharacteristic
23C23×C5×D3—21no—
24C23×C5×D3—22no—
25C23×C5×D3—22no—
26C23×C5×D3—21no—
27C23×C5×D3—44no—
28C23×C5×D3Q812yescomplemented
29C22×C3×C4×C5—22no—
30C22×C3×C4×C5D412yes—
31C22×C5×Dic3—21no—
32C22×C5×Dic3D414yes—
33C2×C5×Dic6—22no—
34C3×C5×C22⋊C4—22no—
35C3×C5×C22⋊C4—22no—
36C3×C5×C22⋊C4—22no—
37C5×#217—22no—
38C5×#217—22no—
39C5×#217—22no—
40C5×(C2×D3)⋊C4—48no—
41C5×(C2×D3)⋊C4—44no—
42C5×C4⋊Dic3—22no—
43C5×Dic3⋊C4—24no—
44#5483C2×C511yescharacteristic, complemented
45#4955C2×C511yescharacteristic
46#4955C2×C512yes—
47C3×#342C2×C511yescharacteristic, complemented
48#4527C2×C512yescomplemented
49C25×C5—31no—
50C5×C22⋊Q8—64no—
51C5×C22⋊Q8—62no—
52C2×C5×C22⋊C4—32no—
53C2×C5×C22⋊C4—31no—
54C2×C5×C22⋊C4—64no—
55C2×C5×C22⋊C4—62no—
56C2×C5×C22⋊C4C2×D312yes—
57C2×C5×C22⋊C4C2×D311yescharacteristic
58#2029—31noSylow
59C23×C3×C5—42no—
60C23×C3×C5C2×D411yescharacteristic
61C23×C3×C5C2×D412yes—
62C23×C3×C5C2×Q811yescharacteristic
63C23×C3×C5#4112yes—
64C23×C3×C5#4111yescharacteristic
65C22×C5×D3—24no—
66C22×C5×D3—28no—
67C22×C5×D3—24no—
68C22×C5×D3—22no—
69C22×C5×D3—44no—
70C22×C5×D3—48no—
71C22×C5×D3—44no—
72C22×C5×D3—42no—
73C22×C5×D3—44no—
74C22×C5×D3—41no—
75C22×C5×D3—41no—
76C22×C5×D3—42no—
77C2×C3×C4×C5—22no—
78C2×C3×C4×C5—42no—
79C2×C3×C4×C5—44no—
80C2×C3×C4×C5—42no—
81C2×C5×Dic3—24no—
82C2×C5×Dic3—44no—
83C2×C5×Dic3—42no—
84C2×C5×Dic3—41no—
85C5×Dic6—82no—
86C24×D3C22×C511yescharacteristic
87C2×C3×C22⋊C4C22×C512yes—
88C2×C3×C22⋊C4C22×C511yescharacteristic
89C2×#217C22×C511yescharacteristic
90C2×#217C22×C512yes—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—24no—
93#924—22no—
94#878—24no—
95C24×C5—32no—
96C24×C5—61no—
97C24×C5—62no—
98C24×C5—62no—
99C24×C5—61no—
100C24×C5—124no—
101C24×C5C22×D311yescharacteristic
102C2×C5×Q8—62no—
103C22×C4×C5—22no—
104C22×C4×C5—34no—
105C22×C4×C5—61no—
106C22×C4×C5D1212yes—
107C5×C4⋊C4—64no—
108C5×C4⋊C4—62no—
109C5×C22⋊C4—22no—
110C5×C22⋊C4—22no—
111C5×C22⋊C4—22no—
112C5×C22⋊C4—62no—
113C5×C22⋊C4—62no—
114C5×C22⋊C4—62no—
115C5×C22⋊C4—124no—
116C5×C22⋊C4—128no—
117#543—31no—
118#379—32no—
119#379—31no—
120#342—32no—
121#342C5×D311yescharacteristic, complemented
122C22×C3×C5—22no—
123C22×C3×C5—24no—
124C22×C3×C5—41no—
125C22×C3×C5—42no—
126C22×C3×C5—41no—
127C22×C3×C5#12011yescharacteristic
128C22×C3×C5C22⋊Q812yes—
129C22×C3×C5C22⋊Q811yescharacteristic
130C22×C3×C5#12411yescharacteristic
131C22×C3×C5#12412yes—
132C2×C5×D3—28no—
133C2×C5×D3—24no—
134C2×C5×D3—44no—
135C2×C5×D3—48no—
136C2×C5×D3—44no—
137C2×C5×D3—42no—
138C2×C5×D3—42no—
139C2×C5×D3—44no—
140C3×C4×C5—42no—
141C3×C4×C5—82no—
142C5×Dic3—44no—
143C5×Dic3—81no—
144C24×C3C23×C511yescharacteristic
145C23×D3—21no—
146C23×D3—22no—
147C23×D3—22no—
148C23×D3—21no—
149C23×D3—44no—
150C23×D3C5×Q812yescomplemented
151C22×C3×C4—22no—
152C22×C3×C4C5×D412yes—
153C22×Dic3—21no—
154C22×Dic3C5×D414yes—
155C2×Dic6—22no—
156C3×C22⋊C4—22no—
157C3×C22⋊C4—22no—
158C3×C22⋊C4—22no—
159#217—22no—
160#217—22no—
161#217—22no—
162(C2×D3)⋊C4—48no—
163(C2×D3)⋊C4—44no—
164C4⋊Dic3—22no—
165Dic3⋊C4—24no—
166C23×C5—42no—
167C23×C5—64no—
168C23×C5—68no—
169C23×C5—64no—
170C23×C5—62no—
171C23×C5—122no—
172C23×C5—124no—
173C23×C5—124no—
174C23×C5—128no—
175C23×C5—124no—
176C23×C5—121no—
177C23×C5—121no—
178C23×C5—122no—
179C23×C5C2×D1211yescharacteristic
180C23×C5#23512yes—
181C23×C5D3×D412yes—
182C23×C5#23711yescharacteristic
183C23×C5D3×Q811yescharacteristic
184C5×Q8—242no—
185C2×C4×C5—22no—
186C2×C4×C5—42no—
187C2×C4×C5—44no—
188C2×C4×C5—42no—
189C2×C4×C5—64no—
190C2×C4×C5—124no—
191C2×C4×C5—122no—
192C2×C4×C5—121no—
193C25—31no—
194C22⋊Q8—64no—
195C22⋊Q8—62no—
196C2×C22⋊C4—32no—
197C2×C22⋊C4—31no—
198C2×C22⋊C4—64no—
199C2×C22⋊C4—62no—
200C2×C22⋊C4C2×C5×D312yes—
201C2×C22⋊C4C2×C5×D311yescharacteristic
202C2×C3×C5—21no—
203C2×C3×C5—22no—
204C2×C3×C5—22no—
205C2×C3×C5—21no—
206C2×C3×C5#39311yescharacteristic
207C2×C3×C5B(4,𝔽2)12yes—
208C5×D3—44no—
209C23×C3—42no—
210C23×C3C2×C5×D411yescharacteristic
211C23×C3C2×C5×D412yes—
212C23×C3C2×C5×Q811yescharacteristic
213C23×C3C5×#4112yes—
214C23×C3C5×#4111yescharacteristic
215C22×D3—24no—
216C22×D3—28no—
217C22×D3—24no—
218C22×D3—22no—
219C22×D3—44no—
220C22×D3—48no—
221C22×D3—44no—
222C22×D3—42no—
223C22×D3—44no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—42no—
227C2×C3×C4—22no—
228C2×C3×C4—42no—
229C2×C3×C4—44no—
230C2×C3×C4—42no—
231C2×Dic3—24no—
232C2×Dic3—44no—
233C2×Dic3—42no—
234C2×Dic3—41no—
235Dic6—82no—
236C22×C5—22no—
237C22×C5—24no—
238C22×C5—42no—
239C22×C5—41no—
240C22×C5—41no—
241C22×C5—68no—
242C22×C5—64no—
243C22×C5—128no—
244C22×C5—124no—
245C22×C5—124no—
246C22×C5—122no—
247C22×C5—122no—
248C22×C5—124no—
249C22×C5#87511yescharacteristic
250C22×C5#88211yescharacteristic
251C22×C5#88512yes—
252C22×C5#89612yes—
253C22×C5#89711yescharacteristic
254C4×C5—42no—
255C4×C5—82no—
256C4×C5—124no—
257C4×C5—241no—
258C24—32no—
259C24—61no—
260C24—62no—
261C24—62no—
262C24—61no—
263C24—124no—
264C24C22×C5×D311yescharacteristic
265C2×Q8—62no—
266C22×C4—22no—
267C22×C4—34no—
268C22×C4—61no—
269C22×C4C5×D1212yes—
270C4⋊C4—64no—
271C4⋊C4—62no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—62no—
276C22⋊C4—62no—
277C22⋊C4—62no—
278C22⋊C4—124no—
279C22⋊C4—128no—
280C3×C5#202911yescharacteristic, complemented, Hall
281C22×C3—22no—
282C22×C3—24no—
283C22×C3—41no—
284C22×C3—42no—
285C22×C3—41no—
286C22×C3C5×#12011yescharacteristic
287C22×C3C5×C22⋊Q812yes—
288C22×C3C5×C22⋊Q811yescharacteristic
289C22×C3C5×#12411yescharacteristic
290C22×C3C5×#12412yes—
291C2×D3—28no—
292C2×D3—24no—
293C2×D3—44no—
294C2×D3—48no—
295C2×D3—44no—
296C2×D3—42no—
297C2×D3—42no—
298C2×D3—44no—
299C3×C4—42no—
300C3×C4—82no—
301Dic3—44no—
302Dic3—81no—
303C2×C5—22no—
304C2×C5—21no—
305C2×C5—22no—
306C2×C5—21no—
307C2×C5—124no—
308C2×C5#467311yescharacteristic
309C2×C5#474112yes—
310C23—42no—
311C23—64no—
312C23—68no—
313C23—64no—
314C23—62no—
315C23—122no—
316C23—124no—
317C23—124no—
318C23—128no—
319C23—124no—
320C23—121no—
321C23—121no—
322C23—122no—
323C23C2×C5×D1211yescharacteristic
324C23C5×#23512yes—
325C23C5×D3×D412yes—
326C23C5×#23711yescharacteristic
327C23C5×D3×Q811yescharacteristic
328Q8—242no—
329C2×C4—22no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—64no—
334C2×C4—124no—
335C2×C4—122no—
336C2×C4—121no—
337C2×C3—21no—
338C2×C3—22no—
339C2×C3—22no—
340C2×C3—21no—
341C2×C3C5×#39311yescharacteristic
342C2×C3C5×B(4,𝔽2)12yes—
343D3—44no—
344C5#6897111yescharacteristic, direct product factor, Sylow
345C22—22no—
346C22—24no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—68no—
351C22—64no—
352C22—128no—
353C22—124no—
354C22—124no—
355C22—122no—
356C22—122no—
357C22—124no—
358C22C5×#87511yescharacteristic
359C22C5×#88211yescharacteristic
360C22C5×#88512yes—
361C22C5×#89612yes—
362C22C5×#89711yescharacteristic
363C4—42no—
364C4—82no—
365C4—124no—
366C4—241no—
367C3C5×#202911yescharacteristic, complemented, Sylow
368C2—22no—
369C2—21no—
370C2—22no—
371C2—21no—
372C2—124no—
373C2C5×#467311yescharacteristic
374C2C5×#474112yes—
subgroupG/H
centerC22×C5#882
commutator, cosocleC24×C3C23×C5
FrattiniC24C22×C5×D3
FittingC3×C5×#342C2
socleC22×C3×C5#120
prime radicalC24×C5×D3C22
hypercenterC5×#342D3
hypocenterC3C5×#2029
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C24×C5 ◁ C5×#342

Complements

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

NH ≅ G/Ntype
1. C5×#5483C2semidirect
4. C3×C5×#342C2semidirect
5. C5×#4527C2semidirect
16. #68971C5direct (N × H)
21. C5×#342D3semidirect
28. C23×C5×D3Q8semidirect
44. #5483C2×C5semidirect
47. C3×#342C2×C5semidirect
48. #4527C2×C5semidirect
121. #342C5×D3semidirect
150. C23×D3C5×Q8semidirect
280. C3×C5#2029semidirect
344. C5#68971direct (N × H)
367. C3C5×#2029semidirect

Maximal subgroups

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

subgroupindex
C5×#54832
C5×#49552
C3×C5×#3422
C5×#45272
C5×#20293
#689715