Finite Group Databasebeta

← C5×#10185 #11932905

Subgroups — C5×#10185 #11932905

Order 1280.

550 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3226C211yesmaximal, characteristic
2C5×#3037C211yesmaximal, characteristic
3C2×C5×#462C211yesmaximal, characteristic, complemented
4C5×#1546C211yesmaximal, characteristic, complemented
5C5×#1480C211yesmaximal, characteristic
6C5×#1461C211yesmaximal, characteristic, complemented
7C5×#1301C211yesmaximal, characteristic, complemented
8C2×C5×#137—21no—
9C22×C5×Q16—21no—
10C5×#532C2211yescharacteristic
11C2×C5×C4⋊Q8C2211yescharacteristic
12C2×C4×C5×Q8C2211yescharacteristic
13C5×#500—21no—
14C5×#486—21no—
15C5×#480—22no—
16C5×#474—22no—
17C5×#464—22no—
18C5×#462—22no—
19C5×#462—22no—
20C5×#462—21no—
21C5×#460—21no—
22C5×#431C2211yescharacteristic
23C5×#431C2211yescharacteristic
24C5×#423C2211yescharacteristic
25C2×C5×C4⋊C8C2211yescharacteristic
26C5×#417—21no—
27C2×C5×#103—21no—
28C5×#332—22no—
29C5×D4⋊C8—22no—
30C5×#324—22no—
31#10185C511yesmaximal, characteristic, direct product factor, Sylow
32C2×C5×#41—21no—
33C22×C5×Q8—21no—
34C22×C5×Q8D411yescharacteristic
35C5×#137—42no—
36C5×#137—42no—
37C2×C5×Q16—24no—
38C2×C5×Q16—22no—
39C2×C5×Q16—42no—
40C2×C5×Q16—42no—
41C2×C5×SD16—22no—
42C2×C5×M16—21no—
43C22×C5×C8—21no—
44C5×C4⋊Q8—21no—
45C5×C4⋊Q8—21no—
46C5×C4⋊Q8—21no—
47C5×C4⋊Q8D412yes—
48C5×C4⋊Q8D411yescharacteristic
49C5×#127D411yescharacteristic
50C5×#124—21no—
51C5×#121—21no—
52C5×#121—21no—
53C4×C5×Q8—21no—
54C4×C5×Q8—21no—
55C4×C5×Q8—21no—
56C4×C5×Q8—21no—
57C4×C5×D4—21no—
58C4×C5×D4—21no—
59C2×C5×C4⋊C4—21no—
60C2×C5×C4⋊C4—21no—
61C2×C5×C4⋊C4—21no—
62C2×C5×C4⋊C4D411yescharacteristic
63C2×C42×C5—21no—
64C2×C42×C5C2311yescharacteristic
65C5×#107—24no—
66C5×C4⋊C8—21no—
67C5×C4⋊C8—21no—
68C5×C4⋊C8—21no—
69C5×C4⋊C8—21no—
70C5×C4⋊C8D412yes—
71C5×C4⋊C8D412yes—
72C5×#103—24no—
73C5×#103—22no—
74C5×#102—22no—
75C5×#102—24no—
76C5×C22⋊C8—21no—
77C5×C22⋊C8D412yes—
78C4×C5×C8—21no—
79C4×C5×C8—21no—
80#3226C2×C511yescharacteristic
81#3037C2×C511yescharacteristic
82C2×#462C2×C511yescharacteristic, complemented
83#1546C2×C511yescharacteristic, complemented
84#1480C2×C511yescharacteristic
85#1461C2×C511yescharacteristic, complemented
86#1301C2×C511yescharacteristic, complemented
87C5×#41—41no—
88C5×#41—41no—
89C5×#41—41no—
90C5×#41—41no—
91C2×C5×Q8—21no—
92C2×C5×Q8—21no—
93C2×C5×Q8—21no—
94C2×C5×Q8—21no—
95C2×C5×Q8—22no—
96C2×C5×Q8—21no—
97C2×C5×Q8—21no—
98C2×C5×Q8—41no—
99C2×C5×Q8—41no—
100C2×C5×Q8—41no—
101C2×C5×Q8—41no—
102C2×C5×Q8—41no—
103C2×C5×Q8D812yes—
104C2×C5×D4—21no—
105C2×C5×D4—21no—
106C2×C5×D4—21no—
107C2×C5×D4—41no—
108C22×C4×C5—21no—
109C22×C4×C5—21no—
110C22×C4×C5—41no—
111C22×C4×C5—41no—
112C22×C4×C5C2×D411yescharacteristic
113C22×C4×C5C2×D411yescharacteristic
114C22×C4×C5#4111yescharacteristic
115C5×Q16—48no—
116C5×Q16—44no—
117C5×SD16—44no—
118C5×M16—41no—
119C5×M16—41no—
120C2×C5×C8—22no—
121C2×C5×C8—22no—
122C2×C5×C8—22no—
123C2×C5×C8—41no—
124C2×C5×C8—41no—
125C2×C5×C8—41no—
126C5×C4⋊C4—22no—
127C5×C4⋊C4—21no—
128C5×C4⋊C4—21no—
129C5×C4⋊C4—21no—
130C5×C4⋊C4—22no—
131C5×C4⋊C4—21no—
132C5×C4⋊C4—41no—
133C5×C4⋊C4—41no—
134C5×C4⋊C4—41no—
135C5×C4⋊C4—41no—
136C5×C22⋊C4—41no—
137C5×C22⋊C4—41no—
138C42×C5—41no—
139C42×C5—41no—
140C42×C5C2×D411yescharacteristic
141C42×C5C2×D411yescharacteristic
142C42×C5C2×D411yescharacteristic
143C42×C5C2×D411yescharacteristic
144C2×#137—21no—
145C22×Q16—21no—
146#532C22×C511yescharacteristic
147C2×C4⋊Q8C22×C511yescharacteristic
148C2×C4×Q8C22×C511yescharacteristic
149#500—21no—
150#486—21no—
151#480—22no—
152#474—22no—
153#464—22no—
154#462—22no—
155#462—21no—
156#462—22no—
157#460—21no—
158#431C22×C511yescharacteristic
159#431C22×C511yescharacteristic
160#423C22×C511yescharacteristic
161C2×C4⋊C8C22×C511yescharacteristic
162#417—21no—
163C2×#103—21no—
164#332—22no—
165D4⋊C8—22no—
166#324—22no—
167C23×C5—41no—
168C23×C5#12111yescharacteristic
169C5×Q8—22no—
170C5×Q8—22no—
171C5×Q8—41no—
172C5×Q8—41no—
173C5×Q8—42no—
174C5×Q8—41no—
175C5×Q8—42no—
176C5×Q8—41no—
177C5×Q8—81no—
178C5×Q8—81no—
179C5×D4—42no—
180C5×D4—41no—
181C5×D4—41no—
182C5×D4—42no—
183C5×D4—81no—
184C5×D4—81no—
185C2×C4×C5—22no—
186C2×C4×C5—22no—
187C2×C4×C5—41no—
188C2×C4×C5—41no—
189C2×C4×C5—41no—
190C2×C4×C5—41no—
191C2×C4×C5—41no—
192C2×C4×C5—42no—
193C2×C4×C5—81no—
194C2×C4×C5—81no—
195C2×C4×C5—81no—
196C2×C4×C5—81no—
197C2×C4×C5—81no—
198C2×C4×C5#12011yescharacteristic
199C2×C4×C5#12011yescharacteristic
200C2×C4×C5#12011yescharacteristic
201C2×C4×C5#12111yescharacteristic
202C2×C4×C5#12111yescharacteristic
203C2×C4×C5#12711yescharacteristic
204C2×C4×C5C2×D811yescharacteristic
205C2×C4×C5#13511yescharacteristic
206C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
207C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
208C5×C8—42no—
209C5×C8—42no—
210C5×C8—42no—
211C5×C8—81no—
212C2×#41—21no—
213C22×Q8—21no—
214C22×Q8C5×D411yescharacteristic
215#137—42no—
216#137—42no—
217C2×Q16—24no—
218C2×Q16—22no—
219C2×Q16—42no—
220C2×Q16—42no—
221C2×SD16—22no—
222C2×M16—21no—
223C22×C8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8C5×D412yes—
228C4⋊Q8C5×D411yescharacteristic
229#127C5×D411yescharacteristic
230#124—21no—
231#121—21no—
232#121—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×D4—21no—
238C4×D4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C5×D411yescharacteristic
243C2×C42—21no—
244C2×C42C23×C511yescharacteristic
245#107—24no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8C5×D412yes—
251C4⋊C8C5×D412yes—
252#103—24no—
253#103—22no—
254#102—22no—
255#102—24no—
256C22⋊C8—21no—
257C22⋊C8C5×D412yes—
258C4×C8—21no—
259C4×C8—21no—
260C22×C5—21no—
261C22×C5—21no—
262C22×C5—81no—
263C22×C5—81no—
264C22×C5—81no—
265C22×C5#39211yescharacteristic
266C22×C5#46611yescharacteristic
267C22×C5#46911yescharacteristic
268C4×C5—21no—
269C4×C5—21no—
270C4×C5—21no—
271C4×C5—42no—
272C4×C5—81no—
273C4×C5—82no—
274C4×C5—81no—
275C4×C5#44711yescharacteristic
276C4×C5#44711yescharacteristic
277C4×C5#44911yescharacteristic
278C4×C5#44911yescharacteristic
279C4×C5#47312yes—
280#41—41no—
281#41—41no—
282#41—41no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—22no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8C5×D812yes—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—41no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4C2×C5×D411yescharacteristic
306C22×C4C2×C5×D411yescharacteristic
307C22×C4C5×#4111yescharacteristic
308Q16—48no—
309Q16—44no—
310SD16—44no—
311M16—41no—
312M16—41no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C4⋊C4—21no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C42—41no—
332C42—41no—
333C42C2×C5×D411yescharacteristic
334C42C2×C5×D411yescharacteristic
335C42C2×C5×D411yescharacteristic
336C42C2×C5×D411yescharacteristic
337C2×C5—21no—
338C2×C5—21no—
339C2×C5—161no—
340C2×C5#199911yescharacteristic
341C2×C5#200911yescharacteristic
342C2×C5#202011yescharacteristic
343C23—41no—
344C23C5×#12111yescharacteristic
345Q8—22no—
346Q8—22no—
347Q8—41no—
348Q8—41no—
349Q8—42no—
350Q8—41no—
351Q8—42no—
352Q8—41no—
353Q8—81no—
354Q8—81no—
355D4—42no—
356D4—41no—
357D4—41no—
358D4—42no—
359D4—81no—
360D4—81no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4C5×#12011yescharacteristic
375C2×C4C5×#12011yescharacteristic
376C2×C4C5×#12011yescharacteristic
377C2×C4C5×#12111yescharacteristic
378C2×C4C5×#12111yescharacteristic
379C2×C4C5×#12711yescharacteristic
380C2×C4C2×C5×D811yescharacteristic
381C2×C4C5×#13511yescharacteristic
382C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
383C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
384C8—42no—
385C8—42no—
386C8—42no—
387C8—81no—
388C5#1018511yescharacteristic, direct product factor, Sylow
389C22—21no—
390C22—21no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22C5×#39211yescharacteristic
395C22C5×#46611yescharacteristic
396C22C5×#46911yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—81no—
402C4—82no—
403C4—81no—
404C4C5×#44711yescharacteristic
405C4C5×#44711yescharacteristic
406C4C5×#44911yescharacteristic
407C4C5×#44911yescharacteristic
408C4C5×#47312yes—
409C2—21no—
410C2—21no—
411C2—161no—
412C2C5×#199911yescharacteristic
413C2C5×#200911yescharacteristic
414C2C5×#202011yescharacteristic
subgroupG/H
center, socleC22×C5#392
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
3. C2×C5×#462C2semidirect
4. C5×#1546C2semidirect
6. C5×#1461C2semidirect
7. C5×#1301C2semidirect
31. #10185C5direct (N × H)
82. C2×#462C2×C5semidirect
83. #1546C2×C5semidirect
85. #1461C2×C5semidirect
86. #1301C2×C5semidirect
388. C5#10185direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#32262
C5×#30372
C2×C5×#4622
C5×#15462
C5×#14802
C5×#14612
C5×#13012
#101855