Finite Group Databasebeta

← C5×#13317 #11936037

Subgroups — C5×#13317 #11936037

Order 1280.

552 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#510C211yesmaximal, characteristic, complemented
2C5×#3054C211yesmaximal, characteristic
3C2×C5×#470C211yesmaximal, characteristic, complemented
4C2×C5×#360C211yesmaximal, characteristic, complemented
5C5×#2141C211yesmaximal, characteristic, complemented
6C2×C5×#349C211yesmaximal, characteristic, complemented
7C5×#1386C211yesmaximal, characteristic, complemented
8C2×C5×#135C2211yescharacteristic
9C22×C5×Q16C2211yescharacteristic
10C22×C5×M16C2211yescharacteristic
11C5×#525—21no—
12C2×C5×C22⋊Q8—21no—
13C5×#510—22no—
14C5×#510—22no—
15C2×C5×Q32—21no—
16C2×C5×SD32—21no—
17C2×C5×M32C2211yescharacteristic
18C2×C5×M32C2211yescharacteristic
19C5×#470—22no—
20C5×#470—21no—
21C5×#470—22no—
22C5×#469—21no—
23C5×#468—22no—
24C5×#427C2211yescharacteristic
25C2×C5×#106C2211yescharacteristic
26C5×#416—21no—
27C2×C5×#103—21no—
28C5×#360—22no—
29C5×#358—22no—
30C5×#349—22no—
31#13317C511yesmaximal, characteristic, direct product factor, Sylow
32C2×C5×#41—21no—
33C22×C5×Q8—21no—
34C23×C4×C5D411yescharacteristic
35C5×#135—21no—
36C5×#135—21no—
37C5×#135—21no—
38C5×#135—21no—
39C2×C5×Q16—21no—
40C2×C5×Q16—21no—
41C2×C5×Q16—21no—
42C2×C5×Q16—21no—
43C2×C5×Q16—21no—
44C2×C5×Q16D411yescharacteristic
45C2×C5×Q16D412yescomplemented
46C2×C5×SD16—21no—
47C2×C5×D8D411yescharacteristic
48C2×C5×M16—21no—
49C2×C5×M16—21no—
50C2×C5×M16—21no—
51C2×C5×M16—21no—
52C2×C5×M16—21no—
53C2×C5×M16—21no—
54C22×C5×C8D411yescharacteristic
55C22×C5×C8C2311yescharacteristic
56C5×#123—41no—
57C5×C22⋊Q8—22no—
58C5×C22⋊Q8—24no—
59C5×C22⋊Q8—42no—
60C5×#121—22no—
61C5×#120—41no—
62C4×C5×D4—42no—
63C5×#117—21no—
64C2×C5×C4⋊C4—21no—
65C2×C5×C4⋊C4—41no—
66C2×C5×C22⋊C4—41no—
67C5×Q32—24no—
68C5×SD32—24no—
69C5×M32—21no—
70C5×M32—21no—
71C5×M32—21no—
72C5×M32D412yescomplemented
73C2×C5×C16—21no—
74C2×C5×C16—21no—
75C5×#107D412yes—
76C5×#106—21no—
77C5×#106—21no—
78C5×#106D412yescomplemented
79C5×#103—24no—
80C5×#103—22no—
81C5×#102—22no—
82C2×#510C2×C511yescharacteristic, complemented
83#3054C2×C511yescharacteristic
84C2×#470C2×C511yescharacteristic, complemented
85C2×#360C2×C511yescharacteristic, complemented
86#2141C2×C511yescharacteristic, complemented
87C2×#349C2×C511yescharacteristic, complemented
88#1386C2×C511yescharacteristic, complemented
89C24×C5D811yescharacteristic
90C5×#41—41no—
91C5×#41—41no—
92C5×#41—41no—
93C5×#41—41no—
94C2×C5×Q8—21no—
95C2×C5×Q8—22no—
96C2×C5×Q8—41no—
97C2×C5×Q8—41no—
98C2×C5×Q8—41no—
99C2×C5×Q8—41no—
100C2×C5×Q8—41no—
101C2×C5×D4—21no—
102C2×C5×D4—42no—
103C2×C5×D4—41no—
104C22×C4×C5—21no—
105C22×C4×C5—21no—
106C22×C4×C5—21no—
107C22×C4×C5—21no—
108C22×C4×C5—21no—
109C22×C4×C5—21no—
110C22×C4×C5—41no—
111C22×C4×C5—41no—
112C22×C4×C5—41no—
113C22×C4×C5D811yescharacteristic
114C22×C4×C5C2×D411yescharacteristic
115C5×Q16—22no—
116C5×Q16—21no—
117C5×Q16—22no—
118C5×Q16—21no—
119C5×Q16—41no—
120C5×Q16—41no—
121C5×SD16—41no—
122C5×SD16—41no—
123C5×D8—21no—
124C5×D8—21no—
125C5×M16—41no—
126C5×M16—41no—
127C5×M16—41no—
128C5×M16—41no—
129C2×C5×C8—21no—
130C2×C5×C8—21no—
131C2×C5×C8—21no—
132C2×C5×C8C2×D411yescharacteristic
133C2×C5×C8C2×D411yescharacteristic
134C2×C5×C8C2×D411yescharacteristic
135C2×C5×C8C2×D411yescharacteristic
136C2×C5×C8C2×D411yescharacteristic
137C2×C5×C8#4111yescharacteristic
138C5×C4⋊C4—22no—
139C5×C4⋊C4—22no—
140C5×C4⋊C4—44no—
141C5×C4⋊C4—42no—
142C5×C4⋊C4—41no—
143C5×C22⋊C4—42no—
144C5×C22⋊C4—44no—
145C5×C22⋊C4—42no—
146C5×C22⋊C4—41no—
147C42×C5—41no—
148C5×C16—22no—
149C5×C16—41no—
150C2×#135C22×C511yescharacteristic
151C22×Q16C22×C511yescharacteristic
152C22×M16C22×C511yescharacteristic
153#525—21no—
154C2×C22⋊Q8—21no—
155#510—22no—
156#510—22no—
157C2×Q32—21no—
158C2×SD32—21no—
159C2×M32C22×C511yescharacteristic
160C2×M32C22×C511yescharacteristic
161#470—21no—
162#470—22no—
163#470—22no—
164#469—21no—
165#468—22no—
166#427C22×C511yescharacteristic
167C2×#106C22×C511yescharacteristic
168#416—21no—
169C2×#103—21no—
170#360—22no—
171#358—22no—
172#349—22no—
173C23×C5—21no—
174C23×C5—21no—
175C23×C5—21no—
176C23×C5—42no—
177C23×C5—41no—
178C23×C5C2×D811yescharacteristic
179C5×Q8—41no—
180C5×Q8—42no—
181C5×Q8—42no—
182C5×Q8—41no—
183C5×Q8—81no—
184C5×Q8—81no—
185C5×D4—41no—
186C5×D4—41no—
187C5×D4—84no—
188C5×D4—81no—
189C5×D4—81no—
190C2×C4×C5—21no—
191C2×C4×C5—21no—
192C2×C4×C5—21no—
193C2×C4×C5—41no—
194C2×C4×C5—41no—
195C2×C4×C5—41no—
196C2×C4×C5—41no—
197C2×C4×C5—42no—
198C2×C4×C5—42no—
199C2×C4×C5—42no—
200C2×C4×C5—41no—
201C2×C4×C5—82no—
202C2×C4×C5—81no—
203C2×C4×C5—81no—
204C2×C4×C5—81no—
205C2×C4×C5—81no—
206C2×C4×C5#12011yescharacteristic
207C2×C4×C5#12011yescharacteristic
208C2×C4×C5#12111yescharacteristic
209C2×C4×C5#13511yescharacteristic
210C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
211C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
212C5×C8—41no—
213C5×C8#12011yescharacteristic
214C5×C8#12111yescharacteristic
215C5×C8#12111yescharacteristic
216C5×C8#12711yescharacteristic
217C2×#41—21no—
218C22×Q8—21no—
219C23×C4C5×D411yescharacteristic
220#135—21no—
221#135—21no—
222#135—21no—
223#135—21no—
224C2×Q16—21no—
225C2×Q16—21no—
226C2×Q16—21no—
227C2×Q16—21no—
228C2×Q16—21no—
229C2×Q16C5×D411yescharacteristic
230C2×Q16C5×D412yescomplemented
231C2×SD16—21no—
232C2×D8C5×D411yescharacteristic
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C22×C8C5×D411yescharacteristic
240C22×C8C23×C511yescharacteristic
241#123—41no—
242C22⋊Q8—24no—
243C22⋊Q8—22no—
244C22⋊Q8—42no—
245#121—22no—
246#120—41no—
247C4×D4—42no—
248#117—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—41no—
251C2×C22⋊C4—41no—
252Q32—24no—
253SD32—24no—
254M32—21no—
255M32—21no—
256M32—21no—
257M32C5×D412yescomplemented
258C2×C16—21no—
259C2×C16—21no—
260#107C5×D412yes—
261#106—21no—
262#106—21no—
263#106C5×D412yescomplemented
264#103—24no—
265#103—22no—
266#102—22no—
267C22×C5—22no—
268C22×C5—42no—
269C22×C5—42no—
270C22×C5—41no—
271C22×C5—81no—
272C22×C5—82no—
273C22×C5—81no—
274C22×C5#44711yescharacteristic
275C22×C5#44711yescharacteristic
276C22×C5#46611yescharacteristic
277C4×C5—41no—
278C4×C5—82no—
279C4×C5—82no—
280C4×C5—81no—
281C4×C5—82no—
282C4×C5#39211yescharacteristic
283C4×C5#44911yescharacteristic
284C4×C5#44911yescharacteristic
285C4×C5#46911yescharacteristic
286C24C5×D811yescharacteristic
287#41—41no—
288#41—41no—
289#41—41no—
290#41—41no—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×D4—21no—
299C2×D4—42no—
300C2×D4—41no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4C5×D811yescharacteristic
311C22×C4C2×C5×D411yescharacteristic
312Q16—22no—
313Q16—21no—
314Q16—22no—
315Q16—21no—
316Q16—41no—
317Q16—41no—
318SD16—41no—
319SD16—41no—
320D8—21no—
321D8—21no—
322M16—41no—
323M16—41no—
324M16—41no—
325M16—41no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8C2×C5×D411yescharacteristic
330C2×C8C2×C5×D411yescharacteristic
331C2×C8C2×C5×D411yescharacteristic
332C2×C8C2×C5×D411yescharacteristic
333C2×C8C2×C5×D411yescharacteristic
334C2×C8C5×#4111yescharacteristic
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C22⋊C4—42no—
341C22⋊C4—44no—
342C22⋊C4—42no—
343C22⋊C4—41no—
344C42—41no—
345C16—22no—
346C16—41no—
347C2×C5—21no—
348C2×C5—21no—
349C2×C5—81no—
350C2×C5—161no—
351C2×C5#199911yescharacteristic
352C2×C5#219112yes—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—42no—
357C23—41no—
358C23C2×C5×D811yescharacteristic
359Q8—41no—
360Q8—42no—
361Q8—42no—
362Q8—41no—
363Q8—81no—
364Q8—81no—
365D4—41no—
366D4—41no—
367D4—84no—
368D4—81no—
369D4—81no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4C5×#12011yescharacteristic
387C2×C4C5×#12011yescharacteristic
388C2×C4C5×#12111yescharacteristic
389C2×C4C5×#13511yescharacteristic
390C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
391C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
392C8—41no—
393C8C5×#12011yescharacteristic
394C8C5×#12111yescharacteristic
395C8C5×#12111yescharacteristic
396C8C5×#12711yescharacteristic
397C5#1331711yescharacteristic, direct product factor, Sylow
398C22—22no—
399C22—42no—
400C22—42no—
401C22—41no—
402C22—81no—
403C22—82no—
404C22—81no—
405C22C5×#44711yescharacteristic
406C22C5×#44711yescharacteristic
407C22C5×#46611yescharacteristic
408C4—41no—
409C4—82no—
410C4—82no—
411C4—81no—
412C4—82no—
413C4C5×#39211yescharacteristic
414C4C5×#44911yescharacteristic
415C4C5×#44911yescharacteristic
416C4C5×#46911yescharacteristic
417C2—21no—
418C2—21no—
419C2—81no—
420C2—161no—
421C2C5×#199911yescharacteristic
422C2C5×#219112yes—
subgroupG/H
center, socleC22×C5#447
commutator, Frattini, cosocleC22×C8C23×C5
prime radicalC5×#3054C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C22×C5×M16 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C5×#510C2semidirect
3. C2×C5×#470C2semidirect
4. C2×C5×#360C2semidirect
5. C5×#2141C2semidirect
6. C2×C5×#349C2semidirect
7. C5×#1386C2semidirect
31. #13317C5direct (N × H)
45. C2×C5×Q16D4semidirect
72. C5×M32D4semidirect
78. C5×#106D4semidirect
82. C2×#510C2×C5semidirect
84. C2×#470C2×C5semidirect
85. C2×#360C2×C5semidirect
86. #2141C2×C5semidirect
87. C2×#349C2×C5semidirect
88. #1386C2×C5semidirect
230. C2×Q16C5×D4semidirect
257. M32C5×D4semidirect
263. #106C5×D4semidirect
397. C5#13317direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5102
C5×#30542
C2×C5×#4702
C2×C5×#3602
C5×#21412
C2×C5×#3492
C5×#13862
#133175