Finite Group Databasebeta

← C5×#34263 #11956983

Subgroups — C5×#34263 #11956983

Order 1280.

2390 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#520C211yesmaximal, characteristic
2C5×#2357C213yesmaximal, complemented
3C5×#2354C213yesmaximal, complemented
4C5×#2351C211yesmaximal, characteristic, complemented
5C5×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
6C5×#2345C213yesmaximal, complemented
7C5×#2344C213yesmaximal, complemented
8C5×#2341C211yesmaximal, characteristic
9C5×#2341C211yesmaximal, characteristic
10C5×#2319C213yesmaximal, complemented
11C5×#2314C213yesmaximal, complemented
12C5×#2298C213yesmaximal, complemented
13C5×#2293C213yesmaximal, complemented
14C5×#2276C211yesmaximal, characteristic, complemented
15C42×C5×Q8C211yesmaximal, characteristic, complemented
16C22×C5×#41C2211yescharacteristic
17C5×#520C4116yescomplemented
18C2×C4×C5×Q8C2211yescharacteristic
19C2×C4×C5×Q8C2211yescharacteristic
20C2×C4×C5×Q8C2211yescharacteristic
21C2×C4×C5×Q8C2211yescharacteristic
22C2×C4×C5×Q8C2211yescharacteristic
23C2×C4×C5×Q8C2211yescharacteristic
24C2×C4×C5×D4C2213yes—
25C2×C4×C5×D4C2213yes—
26C2×C5×#117C2216yes—
27C2×C5×#117C2213yes—
28C22×C5×C4⋊C4C2213yes—
29C5×#390C2213yes—
30C5×#390C2213yes—
31C5×#390C2213yes—
32C5×#390C2213yes—
33C5×#389C2213yes—
34C5×#389C2213yes—
35C5×#389C2213yes—
36C5×#389C2213yes—
37C5×#388C2213yes—
38C5×#388C2216yes—
39C5×#388C2213yes—
40C5×#388C2213yes—
41C5×#388C2213yes—
42C5×#388C2216yes—
43C5×#387C2211yescharacteristic
44C5×#387C2213yes—
45C5×#387C2213yes—
46C5×#387C2213yes—
47C5×#387C2216yes—
48C5×#384C2213yes—
49C5×#382C2213yes—
50C5×#381C2216yes—
51C5×#380C2213yes—
52C5×#380C2213yes—
53C5×#380C2216yes—
54C4×C5×C4⋊C4C2213yes—
55C4×C5×C4⋊C4C2213yes—
56C4×C5×C4⋊C4C2213yes—
57C4×C5×C4⋊C4C2213yes—
58C4×C5×C4⋊C4C2213yes—
59C4×C5×C4⋊C4C2213yes—
60C4×C5×C4⋊C4C2213yes—
61C4×C5×C4⋊C4C2213yes—
62C4×C5×C22⋊C4C2216yes—
63C4×C5×C22⋊C4C2213yes—
64C4×C5×C22⋊C4C2213yes—
65C5×#376C2213yes—
66C43×C5C2213yes—
67#34263C511yesmaximal, characteristic, direct product factor, Sylow
68C2×C5×#41—24no—
69C2×C5×#41—24no—
70C2×C5×#41C2×C418yes—
71C22×C5×Q8C2311yescharacteristic
72C22×C5×D4C2313yes—
73C23×C4×C5C2313yes—
74C23×C4×C5C2316yes—
75C4×C5×Q8—24no—
76C4×C5×Q8—24no—
77C4×C5×Q8—24no—
78C4×C5×Q8—24no—
79C4×C5×Q8C2×C418yes—
80C4×C5×Q8C2×C418yes—
81C4×C5×D4C2×C4124yes—
82C4×C5×D4C2×C4124yes—
83C5×#117—412no—
84C5×#117C2×C4124yes—
85C2×C5×C4⋊C4C2×C4124yes—
86C2×C5×C4⋊C4C2311yescharacteristic
87C2×C5×C4⋊C4C2313yes—
88C2×C5×C4⋊C4C2313yes—
89C2×C5×C4⋊C4C2313yes—
90C2×C5×C4⋊C4C2313yes—
91C2×C5×C4⋊C4C2313yes—
92C2×C5×C4⋊C4C2313yes—
93C2×C5×C4⋊C4C2313yes—
94C2×C5×C4⋊C4C2316yes—
95C2×C5×C4⋊C4C2313yes—
96C2×C5×C4⋊C4C2313yes—
97C2×C5×C4⋊C4C2311yescharacteristic
98C2×C5×C4⋊C4C2313yes—
99C2×C5×C22⋊C4C2313yes—
100C2×C5×C22⋊C4C2313yes—
101C2×C5×C22⋊C4C2316yes—
102C2×C5×C22⋊C4C2313yes—
103C2×C5×C22⋊C4C2313yes—
104C2×C5×C22⋊C4C2316yes—
105C2×C5×C22⋊C4C2313yes—
106C2×C5×C22⋊C4C2313yes—
107C2×C42×C5C2311yescharacteristic
108C2×C42×C5C2313yes—
109C2×C42×C5C2313yes—
110C2×C42×C5C2313yes—
111C2×C42×C5C2313yes—
112C2×C42×C5C2313yes—
113C2×C42×C5C2313yes—
114C2×C42×C5C2313yes—
115C2×C42×C5C2316yes—
116C2×C42×C5C2313yes—
117C2×C42×C5C2313yes—
118C5×#95C2311yescharacteristic
119C5×#95C2313yes—
120C5×#95C2313yes—
121C5×#95C2313yes—
122C5×#95C2311yescharacteristic
123C5×#95C2311yescharacteristic
124C5×#95C2313yes—
125C5×#95C2313yes—
126C5×#95C2313yes—
127C5×#95C2316yes—
128C5×#95C2316yes—
129C5×#95C2311yescharacteristic
130C5×#95C2313yes—
131C5×#95C2313yes—
132C2×#520C2×C511yescharacteristic
133#2357C2×C513yescomplemented
134#2354C2×C513yescomplemented
135#2351C2×C511yescharacteristic, complemented
136Q8×C4⋊C4C2×C511yescharacteristic, complemented
137#2345C2×C513yescomplemented
138#2344C2×C513yescomplemented
139#2341C2×C511yescharacteristic
140#2341C2×C511yescharacteristic
141#2319C2×C513yescomplemented
142#2314C2×C513yescomplemented
143#2298C2×C513yescomplemented
144#2293C2×C513yescomplemented
145#2276C2×C511yescharacteristic, complemented
146C42×Q8C2×C511yescharacteristic, complemented
147C24×C5C2413yes—
148C5×#41—232no—
149C2×C5×Q8C22×C414yes—
150C2×C5×Q8#4114yes—
151C2×C5×Q8#4114yes—
152C2×C5×D4—26no—
153C2×C5×D4—26no—
154C2×C5×D4C22×C4112yes—
155C22×C4×C5—26no—
156C22×C4×C5—26no—
157C22×C4×C5—46no—
158C22×C4×C5—46no—
159C22×C4×C5C22×C4112yes—
160C22×C4×C5C22×C4124yes—
161C22×C4×C5C2413yes—
162C22×C4×C5C2413yes—
163C22×C4×C5C2411yescharacteristic
164C22×C4×C5C2413yes—
165C22×C4×C5C2413yes—
166C22×C4×C5C2411yescharacteristic
167C22×C4×C5C2411yescharacteristic
168C22×C4×C5C2413yes—
169C22×C4×C5C2411yescharacteristic
170C22×C4×C5C2413yes—
171C22×C4×C5C2411yescharacteristic
172C22×C4×C5C2411yescharacteristic
173C22×C4×C5C2411yescharacteristic
174C22×C4×C5C2413yes—
175C5×C4⋊C4—26no—
176C5×C4⋊C4—26no—
177C5×C4⋊C4—26no—
178C5×C4⋊C4—26no—
179C5×C4⋊C4—26no—
180C5×C4⋊C4—22no—
181C5×C4⋊C4—43no—
182C5×C4⋊C4—43no—
183C5×C4⋊C4—46no—
184C5×C4⋊C4C22×C4112yes—
185C5×C4⋊C4C22×C4112yes—
186C5×C4⋊C4C22×C4112yes—
187C5×C4⋊C4C22×C414yes—
188C5×C22⋊C4—26no—
189C5×C22⋊C4—26no—
190C5×C22⋊C4—46no—
191C5×C22⋊C4—43no—
192C5×C22⋊C4—43no—
193C5×C22⋊C4—46no—
194C5×C22⋊C4C22×C4112yes—
195C5×C22⋊C4C22×C4112yes—
196C42×C5—26no—
197C42×C5—26no—
198C42×C5—26no—
199C42×C5—26no—
200C42×C5—26no—
201C42×C5—22no—
202C42×C5—43no—
203C42×C5—43no—
204C42×C5—46no—
205C42×C5C22×C4112yes—
206C42×C5C22×C4112yes—
207C22×#41C22×C511yescharacteristic
208#520C4×C5116yescomplemented
209C2×C4×Q8C22×C511yescharacteristic
210C2×C4×Q8C22×C511yescharacteristic
211C2×C4×Q8C22×C511yescharacteristic
212C2×C4×Q8C22×C511yescharacteristic
213C2×C4×Q8C22×C511yescharacteristic
214C2×C4×Q8C22×C511yescharacteristic
215C2×C4×D4C22×C513yes—
216C2×C4×D4C22×C513yes—
217C2×#117C22×C516yes—
218C2×#117C22×C513yes—
219C22×C4⋊C4C22×C513yes—
220#390C22×C513yes—
221#390C22×C513yes—
222#390C22×C513yes—
223#390C22×C513yes—
224#389C22×C513yes—
225#389C22×C513yes—
226#389C22×C513yes—
227#389C22×C513yes—
228#388C22×C516yes—
229#388C22×C516yes—
230#388C22×C513yes—
231#388C22×C513yes—
232#388C22×C513yes—
233#388C22×C513yes—
234#387C22×C513yes—
235#387C22×C516yes—
236#387C22×C513yes—
237#387C22×C513yes—
238#387C22×C511yescharacteristic
239#384C22×C513yes—
240#382C22×C513yes—
241#381C22×C516yes—
242#380C22×C513yes—
243#380C22×C513yes—
244#380C22×C516yes—
245C4×C4⋊C4C22×C513yes—
246C4×C4⋊C4C22×C513yes—
247C4×C4⋊C4C22×C513yes—
248C4×C4⋊C4C22×C513yes—
249C4×C4⋊C4C22×C513yes—
250C4×C4⋊C4C22×C513yes—
251C4×C4⋊C4C22×C513yes—
252C4×C4⋊C4C22×C513yes—
253C4×C22⋊C4C22×C513yes—
254C4×C22⋊C4C22×C513yes—
255C4×C22⋊C4C22×C516yes—
256#376C22×C513yes—
257C43C22×C513yes—
258C23×C5—23no—
259C23×C5—23no—
260C23×C5—23no—
261C23×C5—23no—
262C23×C5—23no—
263C23×C5—23no—
264C23×C5C23×C416yes—
265C23×C5C2511yescharacteristic
266C5×Q8#117116yescomplemented
267C5×D4—224no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5—23no—
273C2×C4×C5—23no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—23no—
280C2×C4×C5—21no—
281C2×C4×C5—23no—
282C2×C4×C5—224no—
283C2×C4×C5—21no—
284C2×C4×C5—21no—
285C2×C4×C5—23no—
286C2×C4×C5—23no—
287C2×C4×C5—23no—
288C2×C4×C5—23no—
289C2×C4×C5—23no—
290C2×C4×C5—23no—
291C2×C4×C5—23no—
292C2×C4×C5—23no—
293C2×C4×C5—21no—
294C2×C4×C5—23no—
295C2×C4×C5—23no—
296C2×C4×C5—23no—
297C2×C4×C5—21no—
298C2×C4×C5—21no—
299C2×C4×C5—23no—
300C2×C4×C5—21no—
301C2×C4×C5—424no—
302C2×C4×C5C23×C412yes—
303C2×C4×C5C23×C412yes—
304C2×C4×C5C23×C416yes—
305C2×C4×C5C23×C416yes—
306C2×C4×C5C23×C412yes—
307C2×C4×C5C23×C416yes—
308C2×C4×C5C2×#4116yes—
309C2×C4×C5C2×#4116yes—
310C2×C4×C5H(5,𝔽2)12yes—
311C2×C4×C52−1+412yes—
312C2×#41—24no—
313C2×#41—24no—
314C2×#41C2×C4×C518yes—
315C22×Q8C23×C511yescharacteristic
316C22×D4C23×C513yes—
317C23×C4C23×C513yes—
318C23×C4C23×C516yes—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—24no—
323C4×Q8C2×C4×C518yes—
324C4×Q8C2×C4×C518yes—
325C4×D4C2×C4×C5124yes—
326C4×D4C2×C4×C5124yes—
327#117—412no—
328#117C2×C4×C5124yes—
329C2×C4⋊C4C2×C4×C5124yes—
330C2×C4⋊C4C23×C513yes—
331C2×C4⋊C4C23×C513yes—
332C2×C4⋊C4C23×C513yes—
333C2×C4⋊C4C23×C513yes—
334C2×C4⋊C4C23×C513yes—
335C2×C4⋊C4C23×C511yescharacteristic
336C2×C4⋊C4C23×C516yes—
337C2×C4⋊C4C23×C513yes—
338C2×C4⋊C4C23×C513yes—
339C2×C4⋊C4C23×C513yes—
340C2×C4⋊C4C23×C513yes—
341C2×C4⋊C4C23×C513yes—
342C2×C4⋊C4C23×C511yescharacteristic
343C2×C22⋊C4C23×C513yes—
344C2×C22⋊C4C23×C513yes—
345C2×C22⋊C4C23×C516yes—
346C2×C22⋊C4C23×C513yes—
347C2×C22⋊C4C23×C516yes—
348C2×C22⋊C4C23×C513yes—
349C2×C22⋊C4C23×C513yes—
350C2×C22⋊C4C23×C513yes—
351C2×C42C23×C513yes—
352C2×C42C23×C513yes—
353C2×C42C23×C513yes—
354C2×C42C23×C513yes—
355C2×C42C23×C513yes—
356C2×C42C23×C511yescharacteristic
357C2×C42C23×C516yes—
358C2×C42C23×C513yes—
359C2×C42C23×C513yes—
360C2×C42C23×C513yes—
361C2×C42C23×C513yes—
362#95C23×C511yescharacteristic
363#95C23×C513yes—
364#95C23×C513yes—
365#95C23×C513yes—
366#95C23×C511yescharacteristic
367#95C23×C511yescharacteristic
368#95C23×C513yes—
369#95C23×C513yes—
370#95C23×C513yes—
371#95C23×C516yes—
372#95C23×C516yes—
373#95C23×C511yescharacteristic
374#95C23×C513yes—
375#95C23×C513yes—
376C22×C5—26no—
377C22×C5—26no—
378C22×C5—26no—
379C22×C5—43no—
380C22×C5—43no—
381C22×C5—43no—
382C22×C5—43no—
383C22×C5C24×C411yescharacteristic
384C22×C5C22×#4111yescharacteristic
385C22×C5C22×#4111yescharacteristic
386C22×C5C2×H(5,𝔽2)11yescharacteristic
387C22×C5C2×2−1+411yescharacteristic
388C22×C5#58511yescharacteristic
389C22×C5#58511yescharacteristic
390C4×C5—22no—
391C4×C5—26no—
392C4×C5—26no—
393C4×C5—22no—
394C4×C5—22no—
395C4×C5—22no—
396C4×C5—22no—
397C4×C5—41no—
398C4×C5—43no—
399C4×C5—43no—
400C4×C5—43no—
401C4×C5—41no—
402C4×C5—43no—
403C4×C5C2×#117112yes—
404C24C24×C513yes—
405#41—232no—
406C2×Q8C22×C4×C514yes—
407C2×Q8C5×#4114yes—
408C2×Q8C5×#4114yes—
409C2×D4—26no—
410C2×D4—26no—
411C2×D4C22×C4×C5112yes—
412C22×C4—26no—
413C22×C4—26no—
414C22×C4—46no—
415C22×C4—46no—
416C22×C4C22×C4×C5112yes—
417C22×C4C22×C4×C5124yes—
418C22×C4C24×C511yescharacteristic
419C22×C4C24×C511yescharacteristic
420C22×C4C24×C513yes—
421C22×C4C24×C513yes—
422C22×C4C24×C513yes—
423C22×C4C24×C513yes—
424C22×C4C24×C511yescharacteristic
425C22×C4C24×C511yescharacteristic
426C22×C4C24×C513yes—
427C22×C4C24×C511yescharacteristic
428C22×C4C24×C513yes—
429C22×C4C24×C511yescharacteristic
430C22×C4C24×C511yescharacteristic
431C22×C4C24×C513yes—
432C4⋊C4—26no—
433C4⋊C4—26no—
434C4⋊C4—26no—
435C4⋊C4—26no—
436C4⋊C4—26no—
437C4⋊C4—22no—
438C4⋊C4—46no—
439C4⋊C4—43no—
440C4⋊C4—43no—
441C4⋊C4C22×C4×C5112yes—
442C4⋊C4C22×C4×C5112yes—
443C4⋊C4C22×C4×C5112yes—
444C4⋊C4C22×C4×C514yes—
445C22⋊C4—26no—
446C22⋊C4—26no—
447C22⋊C4—46no—
448C22⋊C4—46no—
449C22⋊C4—43no—
450C22⋊C4—43no—
451C22⋊C4C22×C4×C5112yes—
452C22⋊C4C22×C4×C5112yes—
453C42—26no—
454C42—26no—
455C42—26no—
456C42—26no—
457C42—26no—
458C42—22no—
459C42—46no—
460C42—43no—
461C42—43no—
462C42C22×C4×C5112yes—
463C42C22×C4×C5112yes—
464C2×C5—46no—
465C2×C5C22×#11711yescharacteristic
466C2×C5C4×H(5,𝔽2)11yescharacteristic
467C2×C5C4×2−1+411yescharacteristic
468C2×C5#351611yescharacteristic
469C2×C5#351711yescharacteristic
470C2×C5#352211yescharacteristic
471C2×C5#352411yescharacteristic
472C23—23no—
473C23—23no—
474C23—23no—
475C23—23no—
476C23—23no—
477C23—23no—
478C23C23×C4×C516yes—
479C23C25×C511yescharacteristic
480Q8C5×#117116yescomplemented
481D4—224no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—23no—
487C2×C4—23no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—23no—
494C2×C4—21no—
495C2×C4—23no—
496C2×C4—224no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—23no—
500C2×C4—23no—
501C2×C4—23no—
502C2×C4—23no—
503C2×C4—23no—
504C2×C4—23no—
505C2×C4—23no—
506C2×C4—23no—
507C2×C4—21no—
508C2×C4—23no—
509C2×C4—23no—
510C2×C4—23no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—23no—
514C2×C4—21no—
515C2×C4—424no—
516C2×C4C23×C4×C512yes—
517C2×C4C23×C4×C512yes—
518C2×C4C23×C4×C516yes—
519C2×C4C23×C4×C516yes—
520C2×C4C23×C4×C512yes—
521C2×C4C23×C4×C516yes—
522C2×C4C2×C5×#4116yes—
523C2×C4C2×C5×#4116yes—
524C2×C4C5×H(5,𝔽2)12yes—
525C2×C4C5×2−1+412yes—
526C5#3426311yescharacteristic, direct product factor, Sylow
527C22—26no—
528C22—26no—
529C22—26no—
530C22—43no—
531C22—43no—
532C22—43no—
533C22—43no—
534C22C24×C4×C511yescharacteristic
535C22C22×C5×#4111yescharacteristic
536C22C22×C5×#4111yescharacteristic
537C22C2×C5×H(5,𝔽2)11yescharacteristic
538C22C2×C5×2−1+411yescharacteristic
539C22C5×#58511yescharacteristic
540C22C5×#58511yescharacteristic
541C4—22no—
542C4—26no—
543C4—26no—
544C4—22no—
545C4—22no—
546C4—22no—
547C4—22no—
548C4—41no—
549C4—43no—
550C4—43no—
551C4—43no—
552C4—41no—
553C4—43no—
554C4C2×C5×#117112yes—
555C2—46no—
556C2C22×C5×#11711yescharacteristic
557C2C4×C5×H(5,𝔽2)11yescharacteristic
558C2C4×C5×2−1+411yescharacteristic
559C2C5×#351611yescharacteristic
560C2C5×#351711yescharacteristic
561C2C5×#352211yescharacteristic
562C2C5×#352411yescharacteristic
subgroupG/H
center, socleC23×C5C25
commutatorC22C24×C4×C5
Frattini, cosocleC23C25×C5
prime radicalC22×C5×#41C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2357C2semidirect
3. C5×#2354C2semidirect
4. C5×#2351C2semidirect
5. C5×Q8×C4⋊C4C2semidirect
6. C5×#2345C2semidirect
7. C5×#2344C2semidirect
10. C5×#2319C2semidirect
11. C5×#2314C2semidirect
12. C5×#2298C2semidirect
13. C5×#2293C2semidirect
14. C5×#2276C2semidirect
15. C42×C5×Q8C2semidirect
17. C5×#520C4semidirect
67. #34263C5direct (N × H)
133. #2357C2×C5semidirect
134. #2354C2×C5semidirect
135. #2351C2×C5semidirect
136. Q8×C4⋊C4C2×C5semidirect
137. #2345C2×C5semidirect
138. #2344C2×C5semidirect
141. #2319C2×C5semidirect
142. #2314C2×C5semidirect
143. #2298C2×C5semidirect
144. #2293C2×C5semidirect
145. #2276C2×C5semidirect
146. C42×Q8C2×C5semidirect
208. #520C4×C5semidirect
266. C5×Q8#117semidirect
480. Q8C5×#117semidirect
526. C5#34263direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5202
C5×#23572
C5×#23542
C5×#23512
C5×Q8×C4⋊C42
C5×#23452
C5×#23442
C5×#23412
C5×#23192
C5×#23142
C5×#22982
C5×#22932
C5×#22762
C42×C5×Q82
#342635