Finite Group Databasebeta

← C5×#10551 #11933271

Subgroups — C5×#10551 #11933271

Order 1280.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3141C211yesmaximal, characteristic
2C5×#1950C211yesmaximal, characteristic, complemented
3C5×#1549C211yesmaximal, characteristic, complemented
4C5×#1547C211yesmaximal, characteristic
5C2×C5×#333C211yesmaximal, characteristic, complemented
6C5×#1480C211yesmaximal, characteristic
7C5×#1283C211yesmaximal, characteristic, complemented
8C5×#532C2211yescharacteristic
9C2×C5×C4⋊Q8C2211yescharacteristic
10C5×#489—22no—
11C5×#487—21no—
12C5×#486—21no—
13C5×#431C2211yescharacteristic
14C2×C5×#108—21no—
15C5×#423C2211yescharacteristic
16C2×C5×C4⋊C8C2211yescharacteristic
17C2×C5×C4⋊C8C2211yescharacteristic
18C5×#417—21no—
19C5×#417—21no—
20C4×C5×M16C2211yescharacteristic
21C5×#341—21no—
22C5×#333—22no—
23C5×#332—21no—
24C5×#332—22no—
25C5×#331—21no—
26C5×D4⋊C8—22no—
27#10551C511yesmaximal, characteristic, direct product factor, Sylow
28C2×C5×#41—21no—
29C22×C5×Q8—21no—
30C2×C5×M16—21no—
31C2×C5×M16D411yescharacteristic
32C2×C5×M16D411yescharacteristic
33C22×C5×C8—21no—
34C22×C5×C8—21no—
35C5×C4⋊Q8—21no—
36C5×C4⋊Q8—21no—
37C5×C4⋊Q8—21no—
38C5×C4⋊Q8D411yescharacteristic
39C5×C4⋊Q8D412yes—
40C5×#127D411yescharacteristic
41C5×#124—21no—
42C5×#121—21no—
43C5×#121—21no—
44C4×C5×D4—21no—
45C4×C5×D4—21no—
46C2×C5×C4⋊C4—21no—
47C2×C5×C4⋊C4—21no—
48C2×C42×C5C2311yescharacteristic
49C5×#108—42no—
50C5×C4⋊C8—21no—
51C5×C4⋊C8—21no—
52C5×C4⋊C8—21no—
53C5×C4⋊C8—21no—
54C5×C4⋊C8—21no—
55C5×C4⋊C8—21no—
56C5×C4⋊C8D412yes—
57C5×#103—22no—
58C5×#103—22no—
59C5×#102—22no—
60C5×#102—22no—
61C5×C22⋊C8—21no—
62C5×#97—21no—
63C4×C5×C8—21no—
64C4×C5×C8—21no—
65#3141C2×C511yescharacteristic
66#1950C2×C511yescharacteristic, complemented
67#1549C2×C511yescharacteristic, complemented
68#1547C2×C511yescharacteristic
69C2×#333C2×C511yescharacteristic, complemented
70#1480C2×C511yescharacteristic
71#1283C2×C511yescharacteristic, complemented
72C5×#41—41no—
73C5×#41—41no—
74C5×#41—41no—
75C5×#41—41no—
76C2×C5×Q8—21no—
77C2×C5×Q8—22no—
78C2×C5×Q8—41no—
79C2×C5×Q8—82no—
80C2×C5×D4—21no—
81C2×C5×D4—21no—
82C2×C5×D4—21no—
83C2×C5×D4—41no—
84C22×C4×C5—41no—
85C22×C4×C5—41no—
86C22×C4×C5C2×D411yescharacteristic
87C22×C4×C5#4111yescharacteristic
88C22×C4×C5#4111yescharacteristic
89C5×M16—21no—
90C5×M16—21no—
91C5×M16—21no—
92C5×M16—21no—
93C5×M16—41no—
94C5×M16—41no—
95C2×C5×C8—21no—
96C2×C5×C8—21no—
97C2×C5×C8—22no—
98C2×C5×C8—41no—
99C2×C5×C8—41no—
100C2×C5×C8—41no—
101C2×C5×C8—41no—
102C2×C5×C8—41no—
103C2×C5×C8—81no—
104C5×C4⋊C4—22no—
105C5×C4⋊C4—21no—
106C5×C4⋊C4—22no—
107C5×C4⋊C4—21no—
108C5×C4⋊C4—41no—
109C5×C4⋊C4—41no—
110C5×C22⋊C4—41no—
111C5×C22⋊C4—41no—
112C42×C5C2×D411yescharacteristic
113C42×C5C2×D411yescharacteristic
114C42×C5C2×D411yescharacteristic
115C42×C5#4111yescharacteristic
116#532C22×C511yescharacteristic
117C2×C4⋊Q8C22×C511yescharacteristic
118#489—22no—
119#487—21no—
120#486—21no—
121#431C22×C511yescharacteristic
122C2×#108—21no—
123#423C22×C511yescharacteristic
124C2×C4⋊C8C22×C511yescharacteristic
125C2×C4⋊C8C22×C511yescharacteristic
126#417—21no—
127#417—21no—
128C4×M16C22×C511yescharacteristic
129#341—21no—
130#333—22no—
131#332—21no—
132#332—22no—
133#331—21no—
134D4⋊C8—22no—
135C23×C5—41no—
136C23×C5#12411yescharacteristic
137C5×Q8—41no—
138C5×Q8—41no—
139C5×Q8—82no—
140C5×Q8—82no—
141C5×D4—41no—
142C5×D4—41no—
143C5×D4—41no—
144C5×D4—41no—
145C5×D4—81no—
146C5×D4—81no—
147C5×D4—81no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
150C2×C4×C5—41no—
151C2×C4×C5—41no—
152C2×C4×C5—42no—
153C2×C4×C5—82no—
154C2×C4×C5—81no—
155C2×C4×C5—81no—
156C2×C4×C5#12011yescharacteristic
157C2×C4×C5#12111yescharacteristic
158C2×C4×C5#12111yescharacteristic
159C2×C4×C5#12111yescharacteristic
160C2×C4×C5#12311yescharacteristic
161C2×C4×C5#12311yescharacteristic
162C2×C4×C5#13511yescharacteristic
163C2×C4×C5#13511yescharacteristic
164C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C5#13711yescharacteristic
166C5×C8—41no—
167C5×C8—41no—
168C5×C8—42no—
169C5×C8—81no—
170C5×C8—81no—
171C5×C8—81no—
172C2×#41—21no—
173C22×Q8—21no—
174C2×M16—21no—
175C2×M16C5×D411yescharacteristic
176C2×M16C5×D411yescharacteristic
177C22×C8—21no—
178C22×C8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8C5×D412yes—
183C4⋊Q8C5×D411yescharacteristic
184#127C5×D411yescharacteristic
185#124—21no—
186#121—21no—
187#121—21no—
188C4×D4—21no—
189C4×D4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C42C23×C511yescharacteristic
193#108—42no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8C5×D412yes—
201#103—22no—
202#103—22no—
203#102—22no—
204#102—22no—
205C22⋊C8—21no—
206#97—21no—
207C4×C8—21no—
208C4×C8—21no—
209C22×C5—21no—
210C22×C5—21no—
211C22×C5—81no—
212C22×C5—81no—
213C22×C5—81no—
214C22×C5#39411yescharacteristic
215C22×C5#48811yescharacteristic
216C22×C5#48911yescharacteristic
217C4×C5—21no—
218C4×C5—21no—
219C4×C5—21no—
220C4×C5—82no—
221C4×C5—81no—
222C4×C5#44911yescharacteristic
223C4×C5#45211yescharacteristic
224C4×C5#46311yescharacteristic
225C4×C5#46411yescharacteristic
226C4×C5#47111yescharacteristic
227C4×C5#47311yescharacteristic
228#41—41no—
229#41—41no—
230#41—41no—
231#41—41no—
232C2×Q8—21no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—82no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4C2×C5×D411yescharacteristic
243C22×C4C5×#4111yescharacteristic
244C22×C4C5×#4111yescharacteristic
245M16—21no—
246M16—21no—
247M16—21no—
248M16—21no—
249M16—41no—
250M16—41no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—22no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—81no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C42C2×C5×D411yescharacteristic
269C42C2×C5×D411yescharacteristic
270C42C2×C5×D411yescharacteristic
271C42C5×#4111yescharacteristic
272C2×C5—21no—
273C2×C5—21no—
274C2×C5—161no—
275C2×C5#204211yescharacteristic
276C2×C5#204811yescharacteristic
277C2×C5#205211yescharacteristic
278C23—41no—
279C23C5×#12411yescharacteristic
280Q8—41no—
281Q8—41no—
282Q8—82no—
283Q8—82no—
284D4—41no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—81no—
289D4—81no—
290D4—81no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4C5×#12011yescharacteristic
300C2×C4C5×#12111yescharacteristic
301C2×C4C5×#12111yescharacteristic
302C2×C4C5×#12111yescharacteristic
303C2×C4C5×#12311yescharacteristic
304C2×C4C5×#12311yescharacteristic
305C2×C4C5×#13511yescharacteristic
306C2×C4C5×#13511yescharacteristic
307C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
308C2×C4C5×#13711yescharacteristic
309C8—41no—
310C8—41no—
311C8—42no—
312C8—81no—
313C8—81no—
314C8—81no—
315C5#1055111yescharacteristic, direct product factor, Sylow
316C22—21no—
317C22—21no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22C5×#39411yescharacteristic
322C22C5×#48811yescharacteristic
323C22C5×#48911yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—81no—
328C4—82no—
329C4C5×#44911yescharacteristic
330C4C5×#45211yescharacteristic
331C4C5×#46311yescharacteristic
332C4C5×#46411yescharacteristic
333C4C5×#47111yescharacteristic
334C4C5×#47311yescharacteristic
335C2—21no—
336C2—21no—
337C2—161no—
338C2C5×#204211yescharacteristic
339C2C5×#204811yescharacteristic
340C2C5×#205211yescharacteristic
subgroupG/H
center, socleC22×C5#394
commutator, Frattini, cosocleC2×C42C23×C5
prime radicalC5×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#1950C2semidirect
3. C5×#1549C2semidirect
5. C2×C5×#333C2semidirect
7. C5×#1283C2semidirect
27. #10551C5direct (N × H)
66. #1950C2×C5semidirect
67. #1549C2×C5semidirect
69. C2×#333C2×C5semidirect
71. #1283C2×C5semidirect
315. C5#10551direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#31412
C5×#19502
C5×#15492
C5×#15472
C2×C5×#3332
C5×#14802
C5×#12832
#105515