Finite Group Databasebeta

← C5×#18287 #11941007

Subgroups — C5×#18287 #11941007

Order 1280.

1064 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#543C211yesmaximal, characteristic
2C5×#3189C212yesmaximal, complemented
3C5×#2626C211yesmaximal, characteristic
4C5×#1614C214yesmaximal, complemented
5C5×#1602C214yesmaximal, complemented
6C5×#1520C212yesmaximal, complemented
7C2×C5×#323C211yesmaximal, characteristic
8C5×#543—24no—
9C5×#543—22no—
10C5×#543C2214yes—
11C2×C5×C22⋊Q8—22no—
12C2×C5×C22⋊Q8—21no—
13C2×C5×C22⋊Q8—22no—
14C2×C5×C22⋊Q8C2212yes—
15C2×C5×C22⋊Q8C2211yescharacteristic
16C22×C5×C22⋊C4—21no—
17C22×C5×C22⋊C4C2211yescharacteristic
18C5×#484—22no—
19C5×#484C2214yescomplemented
20C5×#467—24no—
21C5×#451—22no—
22C5×#451C2214yescomplemented
23C2×C5×C22⋊C8C2212yes—
24C5×#399C2212yes—
25C5×#396—22no—
26C5×#393C2212yes—
27C5×#385—21no—
28C5×#380C2211yescharacteristic
29C5×#379—21no—
30C5×#328C2218yescomplemented
31C5×#323C2214yes—
32#18287C511yesmaximal, characteristic, direct product factor, Sylow
33C25×C5D411yescharacteristic
34C22×C5×Q8—21no—
35C22×C5×Q8D412yes—
36C22×C5×Q8C2311yescharacteristic
37C23×C4×C5—21no—
38C23×C4×C5—21no—
39C23×C4×C5D411yescharacteristic
40C23×C4×C5C2311yescharacteristic
41C2×C5×Q16—24no—
42C22×C5×C8—22no—
43C5×C22⋊Q8—28no—
44C5×C22⋊Q8—28no—
45C5×C22⋊Q8—24no—
46C5×C22⋊Q8—28no—
47C5×C22⋊Q8—24no—
48C5×C22⋊Q8—22no—
49C5×C22⋊Q8—22no—
50C5×C22⋊Q8—28no—
51C5×C22⋊Q8—22no—
52C5×C22⋊Q8—22no—
53C5×C22⋊Q8D414yes—
54C5×C22⋊Q8C2314yes—
55C2×C5×C4⋊C4—21no—
56C2×C5×C4⋊C4—22no—
57C2×C5×C4⋊C4—22no—
58C2×C5×C4⋊C4—21no—
59C2×C5×C4⋊C4—21no—
60C2×C5×C4⋊C4—22no—
61C2×C5×C4⋊C4D412yes—
62C2×C5×C4⋊C4C2311yescharacteristic
63C2×C5×C22⋊C4—22no—
64C2×C5×C22⋊C4—24no—
65C2×C5×C22⋊C4—22no—
66C2×C5×C22⋊C4—24no—
67C2×C5×C22⋊C4—22no—
68C2×C5×C22⋊C4—22no—
69C2×C5×C22⋊C4—21no—
70C2×C5×C22⋊C4—21no—
71C2×C5×C22⋊C4—22no—
72C2×C5×C22⋊C4—22no—
73C2×C5×C22⋊C4D412yes—
74C2×C5×C22⋊C4C2312yes—
75C5×#107—24no—
76C5×#103—24no—
77C5×#103—24no—
78C5×C22⋊C8—22no—
79C5×C22⋊C8C2314yes—
80C5×#95—22no—
81C5×#95C2312yes—
82C2×#543C2×C511yescharacteristic
83#3189C2×C512yescomplemented
84#2626C2×C511yescharacteristic
85#1614C2×C514yescomplemented
86#1602C2×C514yescomplemented
87#1520C2×C512yescomplemented
88C2×#323C2×C511yescharacteristic
89C24×C5—21no—
90C24×C5—21no—
91C24×C5—22no—
92C24×C5—81no—
93C24×C5—81no—
94C24×C5Q1614yescomplemented
95C24×C5C2×D412yes—
96C24×C5C2×D411yescharacteristic
97C2×C5×Q8—24no—
98C2×C5×Q8—22no—
99C2×C5×Q8—22no—
100C2×C5×Q8—81no—
101C2×C5×Q8—82no—
102C2×C5×Q8—82no—
103C2×C5×Q8C2×D414yes—
104C22×C4×C5—22no—
105C22×C4×C5—21no—
106C22×C4×C5—22no—
107C22×C4×C5—24no—
108C22×C4×C5—24no—
109C22×C4×C5—21no—
110C22×C4×C5—21no—
111C22×C4×C5—22no—
112C22×C4×C5—21no—
113C22×C4×C5—81no—
114C22×C4×C5—81no—
115C22×C4×C5—82no—
116C22×C4×C5—82no—
117C22×C4×C5C2×D412yes—
118C22×C4×C5C2×D411yescharacteristic
119C22×C4×C5C2×D412yes—
120C22×C4×C5C2×D412yes—
121C22×C4×C5C2411yescharacteristic
122C5×Q16—84no—
123C2×C5×C8—24no—
124C2×C5×C8—82no—
125C5×C4⋊C4—24no—
126C5×C4⋊C4—28no—
127C5×C4⋊C4—24no—
128C5×C4⋊C4—22no—
129C5×C4⋊C4—24no—
130C5×C4⋊C4—22no—
131C5×C4⋊C4—42no—
132C5×C4⋊C4—81no—
133C5×C4⋊C4—82no—
134C5×C4⋊C4C2×D414yes—
135C5×C22⋊C4—28no—
136C5×C22⋊C4—24no—
137C5×C22⋊C4—24no—
138C5×C22⋊C4—24no—
139C5×C22⋊C4—24no—
140C5×C22⋊C4—41no—
141C5×C22⋊C4—41no—
142C5×C22⋊C4—42no—
143C5×C22⋊C4—82no—
144C5×C22⋊C4—82no—
145#543—24no—
146#543—22no—
147#543C22×C514yes—
148C2×C22⋊Q8—22no—
149C2×C22⋊Q8—21no—
150C2×C22⋊Q8—22no—
151C2×C22⋊Q8C22×C512yes—
152C2×C22⋊Q8C22×C511yescharacteristic
153C22×C22⋊C4—21no—
154C22×C22⋊C4C22×C511yescharacteristic
155#484—22no—
156#484C22×C514yescomplemented
157#467—24no—
158#451—22no—
159#451C22×C514yescomplemented
160C2×C22⋊C8C22×C512yes—
161#399C22×C512yes—
162#396—22no—
163#393C22×C512yes—
164#385—21no—
165#380C22×C511yescharacteristic
166#379—21no—
167#328C22×C518yescomplemented
168#323C22×C514yes—
169C23×C5—22no—
170C23×C5—24no—
171C23×C5—22no—
172C23×C5—24no—
173C23×C5—41no—
174C23×C5—41no—
175C23×C5—42no—
176C23×C5—42no—
177C23×C5—42no—
178C23×C5—42no—
179C23×C5—81no—
180C23×C5—81no—
181C23×C5—84no—
182C23×C5—84no—
183C23×C5#12011yescharacteristic
184C23×C5#12011yescharacteristic
185C23×C5C2×Q1614yes—
186C23×C5C2×Q1612yes—
187C23×C5#13712yes—
188C23×C5C22×D411yescharacteristic
189C5×Q8—44no—
190C5×Q8—44no—
191C5×Q8—82no—
192C5×Q8—84no—
193C2×C4×C5—24no—
194C2×C4×C5—22no—
195C2×C4×C5—24no—
196C2×C4×C5—41no—
197C2×C4×C5—42no—
198C2×C4×C5—42no—
199C2×C4×C5—41no—
200C2×C4×C5—42no—
201C2×C4×C5—44no—
202C2×C4×C5—44no—
203C2×C4×C5—42no—
204C2×C4×C5—42no—
205C2×C4×C5—42no—
206C2×C4×C5—42no—
207C2×C4×C5—42no—
208C2×C4×C5—41no—
209C2×C4×C5—41no—
210C2×C4×C5—42no—
211C2×C4×C5—81no—
212C2×C4×C5—84no—
213C2×C4×C5—84no—
214C2×C4×C5#12012yes—
215C2×C4×C5C22×D412yes—
216C5×C8—44no—
217C25C5×D411yescharacteristic
218C22×Q8—21no—
219C22×Q8C5×D412yes—
220C22×Q8C23×C511yescharacteristic
221C23×C4—21no—
222C23×C4—21no—
223C23×C4C5×D411yescharacteristic
224C23×C4C23×C511yescharacteristic
225C2×Q16—24no—
226C22×C8—22no—
227C22⋊Q8—28no—
228C22⋊Q8—28no—
229C22⋊Q8—24no—
230C22⋊Q8—28no—
231C22⋊Q8—24no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—28no—
235C22⋊Q8—22no—
236C22⋊Q8—22no—
237C22⋊Q8C5×D414yes—
238C22⋊Q8C23×C514yes—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4C5×D412yes—
246C2×C4⋊C4C23×C511yescharacteristic
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4C5×D412yes—
258C2×C22⋊C4C23×C512yes—
259#107—24no—
260#103—24no—
261#103—24no—
262C22⋊C8—22no—
263C22⋊C8C23×C514yes—
264#95—22no—
265#95C23×C512yes—
266C22×C5—21no—
267C22×C5—21no—
268C22×C5—21no—
269C22×C5—21no—
270C22×C5—21no—
271C22×C5—21no—
272C22×C5—24no—
273C22×C5—42no—
274C22×C5—42no—
275C22×C5—44no—
276C22×C5—48no—
277C22×C5—42no—
278C22×C5—42no—
279C22×C5—82no—
280C22×C5—84no—
281C22×C5#45112yes—
282C22×C5#45112yes—
283C22×C5C2×#12011yescharacteristic
284C22×C5C22×Q1611yescharacteristic
285C22×C5C2×#13711yescharacteristic
286C4×C5—22no—
287C4×C5—22no—
288C4×C5—42no—
289C4×C5—44no—
290C4×C5—44no—
291C4×C5—82no—
292C4×C5—81no—
293C4×C5—81no—
294C24—21no—
295C24—21no—
296C24—22no—
297C24—81no—
298C24—81no—
299C24C5×Q1614yescomplemented
300C24C2×C5×D412yes—
301C24C2×C5×D411yescharacteristic
302C2×Q8—24no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—81no—
306C2×Q8—82no—
307C2×Q8—82no—
308C2×Q8C2×C5×D414yes—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—81no—
319C22×C4—81no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4C2×C5×D412yes—
323C22×C4C2×C5×D411yescharacteristic
324C22×C4C2×C5×D412yes—
325C22×C4C2×C5×D412yes—
326C22×C4C24×C511yescharacteristic
327Q16—84no—
328C2×C8—24no—
329C2×C8—82no—
330C4⋊C4—24no—
331C4⋊C4—28no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—42no—
337C4⋊C4—81no—
338C4⋊C4—82no—
339C4⋊C4C2×C5×D414yes—
340C22⋊C4—28no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—42no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C2×C5—21no—
351C2×C5—21no—
352C2×C5—22no—
353C2×C5—22no—
354C2×C5—44no—
355C2×C5C2×#45111yescharacteristic
356C2×C5#301911yescharacteristic
357C2×C5#302611yescharacteristic
358C23—24no—
359C23—22no—
360C23—24no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—81no—
369C23—81no—
370C23—84no—
371C23—84no—
372C23C5×#12011yescharacteristic
373C23C5×#12011yescharacteristic
374C23C2×C5×Q1614yes—
375C23C2×C5×Q1612yes—
376C23C5×#13712yes—
377C23C22×C5×D411yescharacteristic
378Q8—44no—
379Q8—44no—
380Q8—82no—
381Q8—84no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—81no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4C5×#12012yes—
404C2×C4C22×C5×D412yes—
405C8—44no—
406C5#1828711yescharacteristic, direct product factor, Sylow
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—24no—
414C22—42no—
415C22—42no—
416C22—44no—
417C22—48no—
418C22—42no—
419C22—42no—
420C22—82no—
421C22—84no—
422C22C5×#45112yes—
423C22C5×#45112yes—
424C22C2×C5×#12011yescharacteristic
425C22C22×C5×Q1611yescharacteristic
426C22C2×C5×#13711yescharacteristic
427C4—22no—
428C4—22no—
429C4—42no—
430C4—44no—
431C4—44no—
432C4—82no—
433C4—81no—
434C4—81no—
435C2—21no—
436C2—22no—
437C2—21no—
438C2—22no—
439C2—44no—
440C2C2×C5×#45111yescharacteristic
441C2C5×#301911yescharacteristic
442C2C5×#302611yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC25×C5D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#3189C2semidirect
4. C5×#1614C2semidirect
5. C5×#1602C2semidirect
6. C5×#1520C2semidirect
19. C5×#484C22semidirect
22. C5×#451C22semidirect
30. C5×#328C22semidirect
32. #18287C5direct (N × H)
83. #3189C2×C5semidirect
85. #1614C2×C5semidirect
86. #1602C2×C5semidirect
87. #1520C2×C5semidirect
94. C24×C5Q16semidirect
156. #484C22×C5semidirect
159. #451C22×C5semidirect
167. #328C22×C5semidirect
299. C24C5×Q16semidirect
406. C5#18287direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5432
C5×#31892
C5×#26262
C5×#16142
C5×#16022
C5×#15202
C2×C5×#3232
#182875