Finite Group Databasebeta

← C5×#9496 #11932216

Subgroups — C5×#9496 #11932216

Order 1280.

900 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2405C211yesmaximal, characteristic
2C5×#2312C211yesmaximal, characteristic, complemented
3C5×#1819C211yesmaximal, characteristic, complemented
4C5×#1781C212yesmaximal
5C5×#1304C212yesmaximal, complemented
6C23×C5×D4C2211yescharacteristic
7C2×C5×#123—21no—
8C22×C5×C22⋊C4C2211yescharacteristic
9C2×C5×#99—24no—
10C5×#394—22no—
11C5×#388—24no—
12C5×#386—21no—
13C5×#385C412yescomplemented
14C5×#380—21no—
15C5×#379C412yescomplemented
16C5×#379C2212yes—
17C5×#379C2211yescharacteristic
18C5×#379C2212yes—
19C4×C5×C22⋊C4—21no—
20C5×#342—22no—
21#9496C511yesmaximal, characteristic, direct product factor, Sylow
22C25×C5C2×C411yescharacteristic
23C25×C5C2311yescharacteristic
24C22×C5×D4—21no—
25C22×C5×D4—22no—
26C22×C5×D4—21no—
27C22×C5×D4—22no—
28C22×C5×D4—44no—
29C23×C4×C5C2×C411yescharacteristic
30C23×C4×C5D411yescharacteristic
31C23×C4×C5D411yescharacteristic
32C5×#123—44no—
33C2×C5×C4⋊C4—22no—
34C2×C5×C4⋊C4—22no—
35C2×C5×C22⋊C4—22no—
36C2×C5×C22⋊C4—22no—
37C2×C5×C22⋊C4—22no—
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4—22no—
40C2×C5×C22⋊C4—22no—
41C2×C5×C22⋊C4—22no—
42C2×C5×C22⋊C4—22no—
43C2×C5×C22⋊C4—21no—
44C2×C5×C22⋊C4—22no—
45C2×C5×C22⋊C4—22no—
46C2×C5×C22⋊C4C2×C412yes—
47C2×C5×C22⋊C4C2×C412yes—
48C2×C5×C22⋊C4D412yes—
49C2×C5×C22⋊C4D414yes—
50C2×C42×C5—22no—
51C5×#99—48no—
52C5×#95—22no—
53C5×#95—22no—
54C5×#95—22no—
55#2405C2×C511yescharacteristic
56#2312C2×C511yescharacteristic, complemented
57#1819C2×C511yescharacteristic, complemented
58#1781C2×C512yes—
59#1304C2×C512yescomplemented
60C24×C5—21no—
61C24×C5—22no—
62C24×C5—42no—
63C24×C5—42no—
64C24×C5—42no—
65C24×C5—42no—
66C24×C5—81no—
67C24×C5—81no—
68C24×C5C22×C411yescharacteristic
69C24×C5C2×D411yescharacteristic
70C24×C5C2×D411yescharacteristic
71C24×C5C2×D412yes—
72C24×C5#4112yes—
73C2×C5×D4—24no—
74C2×C5×D4—28no—
75C2×C5×D4—42no—
76C2×C5×D4—44no—
77C2×C5×D4—42no—
78C2×C5×D4—44no—
79C2×C5×D4—42no—
80C2×C5×D4—44no—
81C2×C5×D4—41no—
82C2×C5×D4—42no—
83C2×C5×D4—41no—
84C22×C4×C5—21no—
85C22×C4×C5—22no—
86C22×C4×C5—21no—
87C22×C4×C5—21no—
88C22×C4×C5—21no—
89C22×C4×C5—21no—
90C22×C4×C5—22no—
91C22×C4×C5—24no—
92C22×C4×C5—42no—
93C22×C4×C5—42no—
94C22×C4×C5—42no—
95C22×C4×C5—81no—
96C22×C4×C5C22⋊C412yes—
97C22×C4×C5C22⋊C412yes—
98C5×C4⋊C4—44no—
99C5×C4⋊C4—82no—
100C5×C22⋊C4—24no—
101C5×C22⋊C4—24no—
102C5×C22⋊C4—44no—
103C5×C22⋊C4—44no—
104C5×C22⋊C4—42no—
105C5×C22⋊C4—42no—
106C5×C22⋊C4—44no—
107C5×C22⋊C4—44no—
108C5×C22⋊C4—44no—
109C5×C22⋊C4—82no—
110C5×C22⋊C4—81no—
111C5×C22⋊C4—82no—
112C5×C22⋊C4—82no—
113C42×C5—82no—
114C23×D4C22×C511yescharacteristic
115C2×#123—21no—
116C22×C22⋊C4C22×C511yescharacteristic
117C2×#99—24no—
118#394—22no—
119#388—24no—
120#386—21no—
121#385C4×C512yescomplemented
122#380—21no—
123#379C4×C512yescomplemented
124#379C22×C512yes—
125#379C22×C511yescharacteristic
126#379C22×C512yes—
127C4×C22⋊C4—21no—
128#342—22no—
129C23×C5—22no—
130C23×C5—22no—
131C23×C5—22no—
132C23×C5—22no—
133C23×C5—24no—
134C23×C5—22no—
135C23×C5—24no—
136C23×C5—41no—
137C23×C5—41no—
138C23×C5—41no—
139C23×C5—42no—
140C23×C5—41no—
141C23×C5—42no—
142C23×C5—41no—
143C23×C5—42no—
144C23×C5—42no—
145C23×C5—42no—
146C23×C5—42no—
147C23×C5—41no—
148C23×C5—42no—
149C23×C5—41no—
150C23×C5—42no—
151C23×C5—82no—
152C23×C5—82no—
153C23×C5—88no—
154C23×C5—88no—
155C23×C5C2×C22⋊C411yescharacteristic
156C23×C5C4×D412yes—
157C23×C5#12011yescharacteristic
158C23×C5#12112yes—
159C23×C5#12311yescharacteristic
160C5×D4—48no—
161C5×D4—44no—
162C5×D4—44no—
163C2×C4×C5—22no—
164C2×C4×C5—24no—
165C2×C4×C5—22no—
166C2×C4×C5—22no—
167C2×C4×C5—42no—
168C2×C4×C5—41no—
169C2×C4×C5—42no—
170C2×C4×C5—41no—
171C2×C4×C5—42no—
172C2×C4×C5—42no—
173C2×C4×C5—42no—
174C2×C4×C5—42no—
175C2×C4×C5—44no—
176C2×C4×C5—42no—
177C2×C4×C5—44no—
178C2×C4×C5—42no—
179C2×C4×C5—44no—
180C2×C4×C5—81no—
181C2×C4×C5—81no—
182C2×C4×C5—82no—
183C2×C4×C5—82no—
184C2×C4×C5—82no—
185C25C2×C4×C511yescharacteristic
186C25C23×C511yescharacteristic
187C22×D4—21no—
188C22×D4—22no—
189C22×D4—21no—
190C22×D4—22no—
191C22×D4—44no—
192C23×C4C2×C4×C511yescharacteristic
193C23×C4C5×D411yescharacteristic
194C23×C4C5×D411yescharacteristic
195#123—44no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4C2×C4×C512yes—
210C2×C22⋊C4C2×C4×C512yes—
211C2×C22⋊C4C5×D412yes—
212C2×C22⋊C4C5×D414yes—
213C2×C42—22no—
214#99—48no—
215#95—22no—
216#95—22no—
217#95—22no—
218C22×C5—21no—
219C22×C5—21no—
220C22×C5—21no—
221C22×C5—21no—
222C22×C5—22no—
223C22×C5—22no—
224C22×C5—21no—
225C22×C5—22no—
226C22×C5—22no—
227C22×C5—21no—
228C22×C5—22no—
229C22×C5—21no—
230C22×C5—22no—
231C22×C5—44no—
232C22×C5—44no—
233C22×C5—44no—
234C22×C5—42no—
235C22×C5—84no—
236C22×C5—82no—
237C22×C5—82no—
238C22×C5—84no—
239C22×C5—84no—
240C22×C5—84no—
241C22×C5—81no—
242C22×C5—81no—
243C22×C5#38611yescharacteristic
244C22×C5#41012yes—
245C22×C5B(4,𝔽2)12yes—
246C4×C5—42no—
247C4×C5—42no—
248C4×C5—42no—
249C4×C5—84no—
250C4×C5—82no—
251C4×C5—162no—
252C24—21no—
253C24—22no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24—81no—
259C24—81no—
260C24C22×C4×C511yescharacteristic
261C24C2×C5×D411yescharacteristic
262C24C2×C5×D411yescharacteristic
263C24C2×C5×D412yes—
264C24C5×#4112yes—
265C2×D4—28no—
266C2×D4—24no—
267C2×D4—42no—
268C2×D4—44no—
269C2×D4—42no—
270C2×D4—44no—
271C2×D4—42no—
272C2×D4—44no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—41no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—81no—
288C22×C4C5×C22⋊C412yes—
289C22×C4C5×C22⋊C412yes—
290C4⋊C4—44no—
291C4⋊C4—82no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—44no—
295C22⋊C4—42no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—42no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4—82no—
302C22⋊C4—81no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C42—82no—
306C2×C5—22no—
307C2×C5—22no—
308C2×C5—42no—
309C2×C5—41no—
310C2×C5—42no—
311C2×C5—82no—
312C2×C5#188911yescharacteristic
313C2×C5#189911yescharacteristic
314C2×C5#191311yescharacteristic
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—24no—
320C23—22no—
321C23—24no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—42no—
326C23—41no—
327C23—42no—
328C23—41no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—41no—
334C23—42no—
335C23—41no—
336C23—42no—
337C23—82no—
338C23—82no—
339C23—88no—
340C23—88no—
341C23C2×C5×C22⋊C411yescharacteristic
342C23C4×C5×D412yes—
343C23C5×#12011yescharacteristic
344C23C5×#12112yes—
345C23C5×#12311yescharacteristic
346D4—48no—
347D4—44no—
348D4—44no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C5#949611yescharacteristic, direct product factor, Sylow
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—22no—
377C22—22no—
378C22—21no—
379C22—22no—
380C22—22no—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—22no—
385C22—44no—
386C22—44no—
387C22—44no—
388C22—42no—
389C22—84no—
390C22—82no—
391C22—82no—
392C22—84no—
393C22—84no—
394C22—84no—
395C22—81no—
396C22—81no—
397C22C5×#38611yescharacteristic
398C22C5×#41012yes—
399C22C5×B(4,𝔽2)12yes—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—84no—
404C4—82no—
405C4—162no—
406C2—22no—
407C2—22no—
408C2—42no—
409C2—42no—
410C2—41no—
411C2—82no—
412C2C5×#188911yescharacteristic
413C2C5×#189911yescharacteristic
414C2C5×#191311yescharacteristic
subgroupG/H
center, socleC22×C5#386
commutatorC24C22×C4×C5
Frattini, cosocleC25C23×C5
prime radicalC23×C5×D4C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C25×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2312C2semidirect
3. C5×#1819C2semidirect
5. C5×#1304C2semidirect
13. C5×#385C4semidirect
15. C5×#379C4semidirect
21. #9496C5direct (N × H)
56. #2312C2×C5semidirect
57. #1819C2×C5semidirect
59. #1304C2×C5semidirect
121. #385C4×C5semidirect
123. #379C4×C5semidirect
371. C5#9496direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#24052
C5×#23122
C5×#18192
C5×#17812
C5×#13042
#94965