Finite Group Databasebeta

← C5×#13381 #11936101

Subgroups — C5×#13381 #11936101

Order 1280.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#466C211yesmaximal, characteristic
2C2×C5×#367C211yesmaximal, characteristic, complemented
3C2×C5×#357C211yesmaximal, characteristic, complemented
4C2×C5×#357C211yesmaximal, characteristic, complemented
5C2×C5×C22⋊C16C211yesmaximal, characteristic, complemented
6C5×#1808C211yesmaximal, characteristic, complemented
7C5×#1380C211yesmaximal, characteristic, complemented
8C22×C5×D8C2211yescharacteristic
9C23×C5×C8C2211yescharacteristic
10C2×C5×#121—21no—
11C22×C5×C16C2211yescharacteristic
12C22×C5×C16C2211yescharacteristic
13C5×#466—22no—
14C5×#466—22no—
15C2×C5×#107C2211yescharacteristic
16C2×C5×#107C2211yescharacteristic
17C2×C5×#107C2211yescharacteristic
18C2×C5×#102—21no—
19C5×#380—21no—
20C5×#367—22no—
21C5×#357—22no—
22C5×#357—22no—
23C5×C22⋊C16—22no—
24C5×#340—21no—
25#13381C511yesmaximal, characteristic, direct product factor, Sylow
26C22×C5×D4—21no—
27C22×C5×D4—41no—
28C23×C4×C5D411yescharacteristic
29C2×C5×D8—21no—
30C2×C5×D8—21no—
31C2×C5×D8—21no—
32C2×C5×D8—21no—
33C2×C5×D8—21no—
34C2×C5×D8D412yes—
35C22×C5×C8—21no—
36C22×C5×C8—21no—
37C22×C5×C8—21no—
38C22×C5×C8—21no—
39C22×C5×C8—21no—
40C22×C5×C8—21no—
41C22×C5×C8D411yescharacteristic
42C22×C5×C8C2311yescharacteristic
43C5×#121—24no—
44C5×#121—42no—
45C2×C5×C4⋊C4—21no—
46C2×C5×C4⋊C4—21no—
47C2×C5×C4⋊C4—21no—
48C2×C5×C22⋊C4—41no—
49C2×C5×C22⋊C4—41no—
50C2×C5×C16—21no—
51C2×C5×C16—21no—
52C2×C5×C16—21no—
53C2×C5×C16—21no—
54C2×C5×C16—21no—
55C2×C5×C16D412yescomplemented
56C5×#107—21no—
57C5×#107—21no—
58C5×#107—21no—
59C5×#107—21no—
60C5×#107—21no—
61C5×#107D412yes—
62C5×#102—24no—
63C5×#95—41no—
64C2×#466C2×C511yescharacteristic
65C2×#367C2×C511yescharacteristic, complemented
66C2×#357C2×C511yescharacteristic, complemented
67C2×#357C2×C511yescharacteristic, complemented
68C2×C22⋊C16C2×C511yescharacteristic, complemented
69#1808C2×C511yescharacteristic, complemented
70#1380C2×C511yescharacteristic, complemented
71C24×C5—41no—
72C24×C5D811yescharacteristic
73C2×C5×D4—22no—
74C2×C5×D4—44no—
75C2×C5×D4—41no—
76C2×C5×D4—41no—
77C2×C5×D4—41no—
78C2×C5×D4—41no—
79C2×C5×D4—41no—
80C2×C5×D4—82no—
81C2×C5×D4—82no—
82C22×C4×C5—21no—
83C22×C4×C5—21no—
84C22×C4×C5—21no—
85C22×C4×C5—21no—
86C22×C4×C5—21no—
87C22×C4×C5—21no—
88C22×C4×C5—41no—
89C22×C4×C5—41no—
90C22×C4×C5—41no—
91C22×C4×C5D811yescharacteristic
92C22×C4×C5C2×D411yescharacteristic
93C5×D8—22no—
94C5×D8—22no—
95C5×D8—41no—
96C5×D8—41no—
97C2×C5×C8—21no—
98C2×C5×C8—21no—
99C2×C5×C8—21no—
100C2×C5×C8—41no—
101C2×C5×C8—41no—
102C2×C5×C8—41no—
103C2×C5×C8—41no—
104C2×C5×C8C2×D411yescharacteristic
105C2×C5×C8C2×D411yescharacteristic
106C2×C5×C8C2×D411yescharacteristic
107C2×C5×C8#4111yescharacteristic
108C2×C5×C8#4111yescharacteristic
109C2×C5×C8#4111yescharacteristic
110C5×C4⋊C4—22no—
111C5×C4⋊C4—41no—
112C5×C4⋊C4—41no—
113C5×C4⋊C4—41no—
114C5×C4⋊C4—41no—
115C5×C4⋊C4—41no—
116C5×C22⋊C4—44no—
117C5×C22⋊C4—82no—
118C5×C16—22no—
119C5×C16—41no—
120C22×D8C22×C511yescharacteristic
121C23×C8C22×C511yescharacteristic
122C2×#121—21no—
123C22×C16C22×C511yescharacteristic
124C22×C16C22×C511yescharacteristic
125#466—22no—
126#466—22no—
127C2×#107C22×C511yescharacteristic
128C2×#107C22×C511yescharacteristic
129C2×#107C22×C511yescharacteristic
130C2×#102—21no—
131#380—21no—
132#367—22no—
133#357—22no—
134#357—22no—
135C22⋊C16—22no—
136#340—21no—
137C23×C5—21no—
138C23×C5—21no—
139C23×C5—21no—
140C23×C5—21no—
141C23×C5—21no—
142C23×C5—21no—
143C23×C5—42no—
144C23×C5—81no—
145C23×C5—81no—
146C23×C5—81no—
147C23×C5—81no—
148C23×C5—81no—
149C23×C5—81no—
150C23×C5SD3212yes—
151C23×C5C2×D811yescharacteristic
152C5×D4—42no—
153C5×D4—42no—
154C5×D4—88no—
155C5×D4—81no—
156C5×D4—81no—
157C2×C4×C5—21no—
158C2×C4×C5—21no—
159C2×C4×C5—21no—
160C2×C4×C5—41no—
161C2×C4×C5—41no—
162C2×C4×C5—41no—
163C2×C4×C5—41no—
164C2×C4×C5—42no—
165C2×C4×C5—81no—
166C2×C4×C5—81no—
167C2×C4×C5—81no—
168C2×C4×C5—81no—
169C2×C4×C5—81no—
170C2×C4×C5—81no—
171C2×C4×C5—81no—
172C2×C4×C5—81no—
173C2×C4×C5#12011yescharacteristic
174C2×C4×C5#12111yescharacteristic
175C2×C4×C5#12311yescharacteristic
176C2×C4×C5#13511yescharacteristic
177C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C5#13711yescharacteristic
179C5×C8—41no—
180C5×C8#12111yescharacteristic
181C5×C8#12111yescharacteristic
182C5×C8#12311yescharacteristic
183C5×C8#12411yescharacteristic
184C22×D4—21no—
185C22×D4—41no—
186C23×C4C5×D411yescharacteristic
187C2×D8—21no—
188C2×D8—21no—
189C2×D8—21no—
190C2×D8—21no—
191C2×D8—21no—
192C2×D8C5×D412yes—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8C5×D411yescharacteristic
200C22×C8C23×C511yescharacteristic
201#121—24no—
202#121—42no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16C5×D412yescomplemented
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107C5×D412yes—
220#102—24no—
221#95—41no—
222C22×C5—22no—
223C22×C5—22no—
224C22×C5—22no—
225C22×C5—41no—
226C22×C5—41no—
227C22×C5—41no—
228C22×C5—41no—
229C22×C5—82no—
230C22×C5—82no—
231C22×C5—82no—
232C22×C5—161no—
233C22×C5—161no—
234C22×C5—161no—
235C22×C5—161no—
236C22×C5#44711yescharacteristic
237C22×C5#46611yescharacteristic
238C22×C5#48011yescharacteristic
239C22×C5C2×SD3211yescharacteristic
240C22×C5#50811yescharacteristic
241C22×C5#50911yescharacteristic
242C22×C5#50911yescharacteristic
243C4×C5—41no—
244C4×C5—82no—
245C4×C5—161no—
246C4×C5—161no—
247C4×C5#39411yescharacteristic
248C4×C5#45211yescharacteristic
249C4×C5#46811yescharacteristic
250C4×C5#48211yescharacteristic
251C24—41no—
252C24C5×D811yescharacteristic
253C2×D4—22no—
254C2×D4—44no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—82no—
261C2×D4—82no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4C5×D811yescharacteristic
272C22×C4C2×C5×D411yescharacteristic
273D8—22no—
274D8—22no—
275D8—41no—
276D8—41no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8C2×C5×D411yescharacteristic
285C2×C8C2×C5×D411yescharacteristic
286C2×C8C2×C5×D411yescharacteristic
287C2×C8C5×#4111yescharacteristic
288C2×C8C5×#4111yescharacteristic
289C2×C8C5×#4111yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C22⋊C4—44no—
297C22⋊C4—82no—
298C16—22no—
299C16—41no—
300C2×C5—42no—
301C2×C5—162no—
302C2×C5#203311yescharacteristic
303C2×C5#218611yescharacteristic
304C2×C5#218711yescharacteristic
305C2×C5#221711yescharacteristic
306C2×C5#221811yescharacteristic
307C2×C5#223311yescharacteristic
308C2×C5#223411yescharacteristic
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—42no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23C5×SD3212yes—
323C23C2×C5×D811yescharacteristic
324D4—42no—
325D4—42no—
326D4—88no—
327D4—81no—
328D4—81no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4C5×#12011yescharacteristic
346C2×C4C5×#12111yescharacteristic
347C2×C4C5×#12311yescharacteristic
348C2×C4C5×#13511yescharacteristic
349C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
350C2×C4C5×#13711yescharacteristic
351C8—41no—
352C8C5×#12111yescharacteristic
353C8C5×#12111yescharacteristic
354C8C5×#12311yescharacteristic
355C8C5×#12411yescharacteristic
356C5#1338111yescharacteristic, direct product factor, Sylow
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22—161no—
368C22—161no—
369C22—161no—
370C22—161no—
371C22C5×#44711yescharacteristic
372C22C5×#46611yescharacteristic
373C22C5×#48011yescharacteristic
374C22C2×C5×SD3211yescharacteristic
375C22C5×#50811yescharacteristic
376C22C5×#50911yescharacteristic
377C22C5×#50911yescharacteristic
378C4—41no—
379C4—82no—
380C4—161no—
381C4—161no—
382C4C5×#39411yescharacteristic
383C4C5×#45211yescharacteristic
384C4C5×#46811yescharacteristic
385C4C5×#48211yescharacteristic
386C2—42no—
387C2—162no—
388C2C5×#203311yescharacteristic
389C2C5×#218611yescharacteristic
390C2C5×#218711yescharacteristic
391C2C5×#221711yescharacteristic
392C2C5×#221811yescharacteristic
393C2C5×#223311yescharacteristic
394C2C5×#223411yescharacteristic
subgroupG/H
center, socleC23×C5C2×D8
commutator, Frattini, cosocleC22×C8C23×C5
prime radicalC2×C5×#466C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#367C2semidirect
3. C2×C5×#357C2semidirect
4. C2×C5×#357C2semidirect
5. C2×C5×C22⋊C16C2semidirect
6. C5×#1808C2semidirect
7. C5×#1380C2semidirect
25. #13381C5direct (N × H)
55. C2×C5×C16D4semidirect
65. C2×#367C2×C5semidirect
66. C2×#357C2×C5semidirect
67. C2×#357C2×C5semidirect
68. C2×C22⋊C16C2×C5semidirect
69. #1808C2×C5semidirect
70. #1380C2×C5semidirect
213. C2×C16C5×D4semidirect
356. C5#13381direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#4662
C2×C5×#3672
C2×C5×#3572
C2×C5×C22⋊C162
C5×#18082
C5×#13802
#133815