Finite Group Databasebeta

← C5×#34311 #11957031

Subgroups — C5×#34311 #11957031

Order 1280.

2860 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×D4×Q8C211yesmaximal, characteristic
2C2×C5×#519C212yesmaximal, complemented
3C5×#2365C213yesmaximal, complemented
4C5×#2362C216yesmaximal, complemented
5C5×#2354C213yesmaximal
6C5×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
7C5×Q8×C22⋊C4C212yesmaximal, complemented
8C5×#2323C216yesmaximal, complemented
9C4×C5×C4⋊Q8C213yesmaximal, complemented
10C5×#2293C213yesmaximal
11C42×C5×Q8C211yesmaximal, characteristic, complemented
12C23×C5×Q8C2212yes—
13C5×D4×Q8C4116yescomplemented
14C2×C5×C4⋊Q8C2213yes—
15C2×C5×C22⋊Q8C2216yes—
16C5×#519—216no—
17C2×C4×C5×Q8C2211yescharacteristic
18C2×C4×C5×Q8C2216yes—
19C2×C4×C5×Q8C2212yes—
20C2×C4×C5×Q8C2216yes—
21C2×C4×C5×Q8C2211yescharacteristic
22C2×C4×C5×Q8C2211yescharacteristic
23C2×C4×C5×D4C2213yes—
24C2×C5×#117C2216yes—
25C2×C5×#117C2216yes—
26C5×#391C2216yes—
27C5×#391C2216yes—
28C5×#390C2211yescharacteristic
29C5×#390C2213yes—
30C5×#389C2213yes—
31C5×#389C2213yes—
32C5×#389C2216yes—
33C5×#389C2216yes—
34C5×#388C2216yes—
35C5×#388C2216yes—
36C5×#388C22112yes—
37C5×#387C22112yes—
38C5×#384C2213yes—
39C5×#380C2216yes—
40C4×C5×C4⋊C4C2213yes—
41C4×C5×C4⋊C4C2213yes—
42C4×C5×C4⋊C4C2213yes—
43C4×C5×C4⋊C4C2213yes—
44C4×C5×C4⋊C4C2216yes—
45C4×C5×C4⋊C4C2216yes—
46C4×C5×C22⋊C4C2216yes—
47C43×C5C2213yes—
48#34311C511yesmaximal, characteristic, direct product factor, Sylow
49C22×C5×Q8—28no—
50C22×C5×Q8—28no—
51C22×C5×Q8C2×C4116yes—
52C22×C5×Q8C2316yes—
53C22×C5×Q8C2311yescharacteristic
54C22×C5×D4C2311yescharacteristic
55C23×C4×C5C2316yes—
56C5×C4⋊Q8C2×C4124yes—
57C5×C22⋊Q8C2×C4148yes—
58C4×C5×Q8—28no—
59C4×C5×Q8—24no—
60C4×C5×Q8—224no—
61C4×C5×Q8—224no—
62C4×C5×Q8C2×C418yes—
63C4×C5×Q8D418yescomplemented
64C4×C5×D4C2×C4124yes—
65C5×#117—224no—
66C5×#117—224no—
67C2×C5×C4⋊C4C2313yes—
68C2×C5×C4⋊C4C2311yescharacteristic
69C2×C5×C4⋊C4C2313yes—
70C2×C5×C4⋊C4C2316yes—
71C2×C5×C4⋊C4C2316yes—
72C2×C5×C4⋊C4C2316yes—
73C2×C5×C4⋊C4C2316yes—
74C2×C5×C4⋊C4C2313yes—
75C2×C5×C4⋊C4C2313yes—
76C2×C5×C4⋊C4C2316yes—
77C2×C5×C4⋊C4C2313yes—
78C2×C5×C22⋊C4C2312yes—
79C2×C5×C22⋊C4C2316yes—
80C2×C5×C22⋊C4C2312yes—
81C2×C5×C22⋊C4C2316yes—
82C2×C5×C22⋊C4C2316yes—
83C2×C42×C5C2311yescharacteristic
84C2×C42×C5C2313yes—
85C2×C42×C5C2313yes—
86C2×C42×C5C2313yes—
87C2×C42×C5C2316yes—
88C2×C42×C5C23112yes—
89C2×C42×C5C2313yes—
90C2×C42×C5C2313yes—
91C2×C42×C5C2313yes—
92C5×#95C2316yes—
93C5×#95C2316yes—
94C5×#95C2316yes—
95C5×#95C23112yes—
96C5×#95C2316yes—
97C2×D4×Q8C2×C511yescharacteristic
98C2×#519C2×C512yescomplemented
99#2365C2×C513yescomplemented
100#2362C2×C516yescomplemented
101#2354C2×C513yes—
102Q8×C4⋊C4C2×C511yescharacteristic, complemented
103Q8×C22⋊C4C2×C512yescomplemented
104#2323C2×C516yescomplemented
105C4×C4⋊Q8C2×C513yescomplemented
106#2293C2×C513yes—
107C42×Q8C2×C511yescharacteristic, complemented
108C24×C5C2412yes—
109C2×C5×Q8—264no—
110C2×C5×Q8—212no—
111C2×C5×Q8—212no—
112C2×C5×Q8C22×C4124yes—
113C2×C5×Q8C22×C414yes—
114C2×C5×Q8C2×D414yes—
115C2×C5×Q8#4114yes—
116C2×C5×D4—22no—
117C2×C5×D4—22no—
118C2×C5×D4C22×C414yes—
119C22×C4×C5—212no—
120C22×C4×C5—212no—
121C22×C4×C5C22×C4124yes—
122C22×C4×C5C2416yes—
123C22×C4×C5C2412yes—
124C22×C4×C5C2413yes—
125C22×C4×C5C2413yes—
126C22×C4×C5C2416yes—
127C22×C4×C5C2411yescharacteristic
128C22×C4×C5C2413yes—
129C22×C4×C5C2411yescharacteristic
130C22×C4×C5C2413yes—
131C22×C4×C5C2411yescharacteristic
132C5×C4⋊C4—212no—
133C5×C4⋊C4—212no—
134C5×C4⋊C4—212no—
135C5×C4⋊C4—26no—
136C5×C4⋊C4—22no—
137C5×C4⋊C4—26no—
138C5×C4⋊C4—46no—
139C5×C4⋊C4C22×C4112yes—
140C5×C4⋊C4C22×C4112yes—
141C5×C4⋊C4C22×C4124yes—
142C5×C4⋊C4C2×D4112yes—
143C5×C22⋊C4—212no—
144C5×C22⋊C4—24no—
145C5×C22⋊C4—24no—
146C5×C22⋊C4—46no—
147C5×C22⋊C4C22×C4124yes—
148C42×C5—224no—
149C42×C5—212no—
150C42×C5—26no—
151C42×C5—26no—
152C42×C5—22no—
153C42×C5—26no—
154C42×C5—43no—
155C42×C5C22×C4112yes—
156C42×C5C2×D4112yes—
157C23×Q8C22×C512yes—
158D4×Q8C4×C5116yescomplemented
159C2×C4⋊Q8C22×C513yes—
160C2×C22⋊Q8C22×C516yes—
161#519—216no—
162C2×C4×Q8C22×C516yes—
163C2×C4×Q8C22×C511yescharacteristic
164C2×C4×Q8C22×C512yes—
165C2×C4×Q8C22×C516yes—
166C2×C4×Q8C22×C511yescharacteristic
167C2×C4×Q8C22×C511yescharacteristic
168C2×C4×D4C22×C513yes—
169C2×#117C22×C516yes—
170C2×#117C22×C516yes—
171#391C22×C516yes—
172#391C22×C516yes—
173#390C22×C513yes—
174#390C22×C511yescharacteristic
175#389C22×C513yes—
176#389C22×C516yes—
177#389C22×C516yes—
178#389C22×C513yes—
179#388C22×C516yes—
180#388C22×C5112yes—
181#388C22×C516yes—
182#387C22×C5112yes—
183#384C22×C513yes—
184#380C22×C516yes—
185C4×C4⋊C4C22×C516yes—
186C4×C4⋊C4C22×C516yes—
187C4×C4⋊C4C22×C513yes—
188C4×C4⋊C4C22×C513yes—
189C4×C4⋊C4C22×C513yes—
190C4×C4⋊C4C22×C513yes—
191C4×C22⋊C4C22×C516yes—
192C43C22×C513yes—
193C23×C5—22no—
194C23×C5—22no—
195C23×C5—22no—
196C23×C5—22no—
197C23×C5—22no—
198C23×C5—22no—
199C23×C5C23×C414yes—
200C23×C5C2511yescharacteristic
201C5×Q8—248no—
202C5×Q8C4×D4116yes—
203C5×D4—28no—
204C2×C4×C5—26no—
205C2×C4×C5—21no—
206C2×C4×C5—26no—
207C2×C4×C5—23no—
208C2×C4×C5—21no—
209C2×C4×C5—23no—
210C2×C4×C5—23no—
211C2×C4×C5—21no—
212C2×C4×C5—21no—
213C2×C4×C5—23no—
214C2×C4×C5—248no—
215C2×C4×C5—26no—
216C2×C4×C5—22no—
217C2×C4×C5—23no—
218C2×C4×C5—23no—
219C2×C4×C5—22no—
220C2×C4×C5—26no—
221C2×C4×C5—22no—
222C2×C4×C5—23no—
223C2×C4×C5—26no—
224C2×C4×C5C23×C416yes—
225C2×C4×C5C23×C4112yes—
226C2×C4×C5C23×C416yes—
227C2×C4×C5C23×C412yes—
228C2×C4×C5C22×D416yes—
229C2×C4×C5C22×D412yes—
230C2×C4×C5C22×D416yes—
231C2×C4×C5C2×#4116yes—
232C2×C4×C52−1+412yes—
233C2×C4×C52−1+412yes—
234C2×C4×C52−1+412yes—
235C22×Q8—28no—
236C22×Q8—28no—
237C22×Q8C2×C4×C5116yes—
238C22×Q8C23×C516yes—
239C22×Q8C23×C511yescharacteristic
240C22×D4C23×C511yescharacteristic
241C23×C4C23×C516yes—
242C4⋊Q8C2×C4×C5124yes—
243C22⋊Q8C2×C4×C5148yes—
244C4×Q8—28no—
245C4×Q8—24no—
246C4×Q8—224no—
247C4×Q8—224no—
248C4×Q8C2×C4×C518yes—
249C4×Q8C5×D418yescomplemented
250C4×D4C2×C4×C5124yes—
251#117—224no—
252#117—224no—
253C2×C4⋊C4C23×C516yes—
254C2×C4⋊C4C23×C516yes—
255C2×C4⋊C4C23×C516yes—
256C2×C4⋊C4C23×C513yes—
257C2×C4⋊C4C23×C513yes—
258C2×C4⋊C4C23×C511yescharacteristic
259C2×C4⋊C4C23×C516yes—
260C2×C4⋊C4C23×C513yes—
261C2×C4⋊C4C23×C513yes—
262C2×C4⋊C4C23×C516yes—
263C2×C4⋊C4C23×C513yes—
264C2×C22⋊C4C23×C512yes—
265C2×C22⋊C4C23×C516yes—
266C2×C22⋊C4C23×C512yes—
267C2×C22⋊C4C23×C516yes—
268C2×C22⋊C4C23×C516yes—
269C2×C42C23×C516yes—
270C2×C42C23×C5112yes—
271C2×C42C23×C513yes—
272C2×C42C23×C511yescharacteristic
273C2×C42C23×C513yes—
274C2×C42C23×C513yes—
275C2×C42C23×C513yes—
276C2×C42C23×C513yes—
277C2×C42C23×C513yes—
278#95C23×C516yes—
279#95C23×C516yes—
280#95C23×C516yes—
281#95C23×C5112yes—
282#95C23×C516yes—
283C22×C5—24no—
284C22×C5—24no—
285C22×C5—24no—
286C22×C5—42no—
287C22×C5—42no—
288C22×C5—42no—
289C22×C5—42no—
290C22×C5C24×C411yescharacteristic
291C22×C5C23×D411yescharacteristic
292C22×C5C22×#4111yescharacteristic
293C22×C5C2×2−1+411yescharacteristic
294C22×C5C2×2−1+411yescharacteristic
295C22×C5C2×2−1+411yescharacteristic
296C22×C5#58511yescharacteristic
297C4×C5—212no—
298C4×C5—26no—
299C4×C5—22no—
300C4×C5—26no—
301C4×C5—22no—
302C4×C5—46no—
303C4×C5—42no—
304C4×C5—43no—
305C4×C5C2×C4×D4112yes—
306C4×C5#51914yescomplemented
307C24C24×C512yes—
308C2×Q8—264no—
309C2×Q8—212no—
310C2×Q8—212no—
311C2×Q8C22×C4×C5124yes—
312C2×Q8C22×C4×C514yes—
313C2×Q8C2×C5×D414yes—
314C2×Q8C5×#4114yes—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4C22×C4×C514yes—
318C22×C4—212no—
319C22×C4—212no—
320C22×C4C22×C4×C5124yes—
321C22×C4C24×C513yes—
322C22×C4C24×C512yes—
323C22×C4C24×C516yes—
324C22×C4C24×C513yes—
325C22×C4C24×C516yes—
326C22×C4C24×C513yes—
327C22×C4C24×C511yescharacteristic
328C22×C4C24×C511yescharacteristic
329C22×C4C24×C513yes—
330C22×C4C24×C511yescharacteristic
331C4⋊C4—212no—
332C4⋊C4—212no—
333C4⋊C4—212no—
334C4⋊C4—26no—
335C4⋊C4—22no—
336C4⋊C4—26no—
337C4⋊C4—46no—
338C4⋊C4C22×C4×C5112yes—
339C4⋊C4C22×C4×C5112yes—
340C4⋊C4C22×C4×C5124yes—
341C4⋊C4C2×C5×D4112yes—
342C22⋊C4—212no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—46no—
346C22⋊C4C22×C4×C5124yes—
347C42—224no—
348C42—212no—
349C42—26no—
350C42—26no—
351C42—22no—
352C42—26no—
353C42—43no—
354C42C22×C4×C5112yes—
355C42C2×C5×D4112yes—
356C2×C5—44no—
357C2×C5C22×C4×D411yescharacteristic
358C2×C5C2×#51911yescharacteristic
359C2×C5C4×2−1+411yescharacteristic
360C2×C5#349711yescharacteristic
361C2×C5#349911yescharacteristic
362C2×C5#351811yescharacteristic
363C2×C5#352311yescharacteristic
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23C23×C4×C514yes—
371C23C25×C511yescharacteristic
372Q8—248no—
373Q8C4×C5×D4116yes—
374D4—28no—
375C2×C4—26no—
376C2×C4—21no—
377C2×C4—26no—
378C2×C4—23no—
379C2×C4—21no—
380C2×C4—23no—
381C2×C4—23no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—23no—
385C2×C4—248no—
386C2×C4—26no—
387C2×C4—22no—
388C2×C4—23no—
389C2×C4—23no—
390C2×C4—22no—
391C2×C4—26no—
392C2×C4—22no—
393C2×C4—23no—
394C2×C4—26no—
395C2×C4C23×C4×C516yes—
396C2×C4C23×C4×C5112yes—
397C2×C4C23×C4×C516yes—
398C2×C4C23×C4×C512yes—
399C2×C4C22×C5×D416yes—
400C2×C4C22×C5×D412yes—
401C2×C4C22×C5×D416yes—
402C2×C4C2×C5×#4116yes—
403C2×C4C5×2−1+412yes—
404C2×C4C5×2−1+412yes—
405C2×C4C5×2−1+412yes—
406C5#3431111yescharacteristic, direct product factor, Sylow
407C22—24no—
408C22—24no—
409C22—24no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22C24×C4×C511yescharacteristic
415C22C23×C5×D411yescharacteristic
416C22C22×C5×#4111yescharacteristic
417C22C2×C5×2−1+411yescharacteristic
418C22C2×C5×2−1+411yescharacteristic
419C22C2×C5×2−1+411yescharacteristic
420C22C5×#58511yescharacteristic
421C4—212no—
422C4—26no—
423C4—22no—
424C4—26no—
425C4—22no—
426C4—46no—
427C4—42no—
428C4—43no—
429C4C2×C4×C5×D4112yes—
430C4C5×#51914yescomplemented
431C2—44no—
432C2C22×C4×C5×D411yescharacteristic
433C2C2×C5×#51911yescharacteristic
434C2C4×C5×2−1+411yescharacteristic
435C2C5×#349711yescharacteristic
436C2C5×#349911yescharacteristic
437C2C5×#351811yescharacteristic
438C2C5×#352311yescharacteristic
subgroupG/H
center, socleC23×C5C25
commutatorC22C24×C4×C5
Frattini, cosocleC23C25×C5
prime radicalC22×C5×D4C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#519C2semidirect
3. C5×#2365C2semidirect
4. C5×#2362C2semidirect
6. C5×Q8×C4⋊C4C2semidirect
7. C5×Q8×C22⋊C4C2semidirect
8. C5×#2323C2semidirect
9. C4×C5×C4⋊Q8C2semidirect
11. C42×C5×Q8C2semidirect
13. C5×D4×Q8C4semidirect
48. #34311C5direct (N × H)
63. C4×C5×Q8D4semidirect
98. C2×#519C2×C5semidirect
99. #2365C2×C5semidirect
100. #2362C2×C5semidirect
102. Q8×C4⋊C4C2×C5semidirect
103. Q8×C22⋊C4C2×C5semidirect
104. #2323C2×C5semidirect
105. C4×C4⋊Q8C2×C5semidirect
107. C42×Q8C2×C5semidirect
158. D4×Q8C4×C5semidirect
249. C4×Q8C5×D4semidirect
306. C4×C5#519semidirect
406. C5#34311direct (N × H)
430. C4C5×#519semidirect

Maximal subgroups

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

subgroupindex
C2×C5×D4×Q82
C2×C5×#5192
C5×#23652
C5×#23622
C5×#23542
C5×Q8×C4⋊C42
C5×Q8×C22⋊C42
C5×#23232
C4×C5×C4⋊Q82
C5×#22932
C42×C5×Q82
#343115