Finite Group Databasebeta

← C5×#26028 #11948748

Subgroups — C5×#26028 #11948748

Order 1280.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2550C211yesmaximal, characteristic, complemented
2C5×#2357C211yesmaximal, characteristic
3C5×#2067C211yesmaximal, characteristic
4C5×#2065C211yesmaximal, characteristic
5C5×#2025C211yesmaximal, characteristic
6C5×#2023C211yesmaximal, characteristic
7C5×#1989C211yesmaximal, characteristic, complemented
8C5×#1988C211yesmaximal, characteristic, complemented
9C5×#1949C211yesmaximal, characteristic, complemented
10C5×#1939C211yesmaximal, characteristic, complemented
11C5×#1921C211yesmaximal, characteristic, complemented
12C5×#1920C211yesmaximal, characteristic, complemented
13C5×#1835C211yesmaximal, characteristic, complemented
14C5×#1804C211yesmaximal, characteristic
15C4×C5×#102C211yesmaximal, characteristic
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×C5×#125C2211yescharacteristic
18C2×C4×C5×D4C2211yescharacteristic
19C2×C5×#107C2211yescharacteristic
20C2×C5×#106C2211yescharacteristic
21C2×C5×C4⋊C8C2211yescharacteristic
22C2×C5×C4⋊C8C2211yescharacteristic
23C2×C5×C4⋊C8C2211yescharacteristic
24C2×C5×#102C2211yescharacteristic
25C2×C5×#102C2211yescharacteristic
26C2×C5×#102C2211yescharacteristic
27C2×C5×#102C2211yescharacteristic
28C2×C4×C5×C8C2211yescharacteristic
29C5×#398—21no—
30C5×#395—21no—
31C5×#391—21no—
32C5×#390C2211yescharacteristic
33C5×#390C2211yescharacteristic
34C5×#389—21no—
35C5×#389C2211yescharacteristic
36C5×#389C2211yescharacteristic
37C5×#388—21no—
38C5×#388—21no—
39C5×#382C2211yescharacteristic
40C5×#382C2211yescharacteristic
41C5×#381—21no—
42C4×C5×C4⋊C4C2211yescharacteristic
43C4×C5×C4⋊C4C2211yescharacteristic
44C4×C5×C4⋊C4C2211yescharacteristic
45C4×C5×C4⋊C4C2211yescharacteristic
46C4×C5×C22⋊C4—21no—
47C5×#340C2211yescharacteristic
48C5×#340C2211yescharacteristic
49C5×#340C2211yescharacteristic
50C5×#340C2211yescharacteristic
51C5×#340C2211yescharacteristic
52C5×#340C2211yescharacteristic
53C5×#340C2211yescharacteristic
54C5×#340C2211yescharacteristic
55C5×#340C2211yescharacteristic
56C5×#340C2211yescharacteristic
57C5×#336C2211yescharacteristic
58C5×#336C2211yescharacteristic
59#26028C511yesmaximal, characteristic, direct product factor, Sylow
60C22×C5×Q8—21no—
61C22×C5×D4C2311yescharacteristic
62C23×C4×C5—21no—
63C22×C5×C8C2311yescharacteristic
64C22×C5×C8C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C22×C5×C8C2311yescharacteristic
67C5×C4⋊Q8—22no—
68C5×C4⋊Q8—41no—
69C5×#125—22no—
70C5×#125—41no—
71C4×C5×D4—22no—
72C4×C5×D4Q814yescomplemented
73C2×C5×C4⋊C4—21no—
74C2×C5×C4⋊C4—21no—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4D411yescharacteristic
79C2×C5×C4⋊C4D411yescharacteristic
80C2×C5×C4⋊C4C2311yescharacteristic
81C2×C5×C4⋊C4C2311yescharacteristic
82C2×C5×C4⋊C4C2311yescharacteristic
83C2×C5×C4⋊C4C2311yescharacteristic
84C2×C5×C4⋊C4C2311yescharacteristic
85C2×C5×C4⋊C4C2311yescharacteristic
86C2×C5×C4⋊C4C2311yescharacteristic
87C2×C5×C22⋊C4—21no—
88C2×C5×C22⋊C4—21no—
89C2×C5×C22⋊C4—21no—
90C2×C5×C22⋊C4—21no—
91C2×C5×C22⋊C4—21no—
92C2×C42×C5—21no—
93C2×C42×C5—21no—
94C2×C42×C5—21no—
95C2×C42×C5C2311yescharacteristic
96C2×C42×C5C2311yescharacteristic
97C2×C42×C5C2311yescharacteristic
98C5×#107—22no—
99C5×#106—22no—
100C5×C4⋊C8—22no—
101C5×C4⋊C8—41no—
102C5×C4⋊C8—41no—
103C5×#102—22no—
104C5×#102—22no—
105C5×#102—41no—
106C5×#102—41no—
107C4×C5×C8—22no—
108C5×#95—21no—
109C5×#95—21no—
110C5×#95—21no—
111C5×#95—21no—
112C5×#95—21no—
113C5×#95—21no—
114C5×#95—21no—
115C5×#95—21no—
116C5×#95—21no—
117C5×#95D411yescharacteristic
118C5×#95D411yescharacteristic
119#2550C2×C511yescharacteristic, complemented
120#2357C2×C511yescharacteristic
121#2067C2×C511yescharacteristic
122#2065C2×C511yescharacteristic
123#2025C2×C511yescharacteristic
124#2023C2×C511yescharacteristic
125#1989C2×C511yescharacteristic, complemented
126#1988C2×C511yescharacteristic, complemented
127#1949C2×C511yescharacteristic, complemented
128#1939C2×C511yescharacteristic, complemented
129#1921C2×C511yescharacteristic, complemented
130#1920C2×C511yescharacteristic, complemented
131#1835C2×C511yescharacteristic, complemented
132#1804C2×C511yescharacteristic
133C4×#102C2×C511yescharacteristic
134C24×C5—21no—
135C2×C5×Q8—42no—
136C2×C5×Q8—42no—
137C2×C5×Q8—81no—
138C2×C5×D4—21no—
139C2×C5×D4—21no—
140C2×C5×D4—21no—
141C2×C5×D4—21no—
142C2×C5×D4—21no—
143C2×C5×D4C2×Q812yes—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—24no—
148C22×C4×C5—21no—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5—81no—
153C22×C4×C5—81no—
154C22×C4×C5C2×D411yescharacteristic
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2411yescharacteristic
161C2×C5×C8—21no—
162C2×C5×C8—21no—
163C2×C5×C8—21no—
164C2×C5×C8—21no—
165C2×C5×C8—21no—
166C2×C5×C8—21no—
167C2×C5×C8—21no—
168C2×C5×C8—21no—
169C2×C5×C8—21no—
170C2×C5×C8—21no—
171C2×C5×C8—21no—
172C2×C5×C8—21no—
173C5×C4⋊C4—22no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—21no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4—21no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4—21no—
185C5×C4⋊C4—41no—
186C5×C4⋊C4—42no—
187C5×C4⋊C4—42no—
188C5×C4⋊C4—42no—
189C5×C4⋊C4—42no—
190C5×C4⋊C4—81no—
191C5×C4⋊C4C2×Q812yes—
192C5×C4⋊C4#4112yes—
193C5×C4⋊C4#4112yes—
194C5×C22⋊C4—24no—
195C5×C22⋊C4—42no—
196C5×C22⋊C4—42no—
197C5×C22⋊C4—81no—
198C5×C22⋊C4—81no—
199C42×C5—21no—
200C42×C5—21no—
201C42×C5—21no—
202C42×C5—21no—
203C42×C5—81no—
204C42×C5—81no—
205C42×C5—81no—
206C42×C5C2×Q812yes—
207C42×C5#4112yes—
208C2×C4⋊Q8C22×C511yescharacteristic
209C2×#125C22×C511yescharacteristic
210C2×C4×D4C22×C511yescharacteristic
211C2×#107C22×C511yescharacteristic
212C2×#106C22×C511yescharacteristic
213C2×C4⋊C8C22×C511yescharacteristic
214C2×C4⋊C8C22×C511yescharacteristic
215C2×C4⋊C8C22×C511yescharacteristic
216C2×#102C22×C511yescharacteristic
217C2×#102C22×C511yescharacteristic
218C2×#102C22×C511yescharacteristic
219C2×#102C22×C511yescharacteristic
220C2×C4×C8C22×C511yescharacteristic
221#398—21no—
222#395—21no—
223#391—21no—
224#390C22×C511yescharacteristic
225#390C22×C511yescharacteristic
226#389—21no—
227#389C22×C511yescharacteristic
228#389C22×C511yescharacteristic
229#388—21no—
230#388—21no—
231#382C22×C511yescharacteristic
232#382C22×C511yescharacteristic
233#381—21no—
234C4×C4⋊C4C22×C511yescharacteristic
235C4×C4⋊C4C22×C511yescharacteristic
236C4×C4⋊C4C22×C511yescharacteristic
237C4×C4⋊C4C22×C511yescharacteristic
238C4×C22⋊C4—21no—
239#340C22×C511yescharacteristic
240#340C22×C511yescharacteristic
241#340C22×C511yescharacteristic
242#340C22×C511yescharacteristic
243#340C22×C511yescharacteristic
244#340C22×C511yescharacteristic
245#340C22×C511yescharacteristic
246#340C22×C511yescharacteristic
247#340C22×C511yescharacteristic
248#340C22×C511yescharacteristic
249#336C22×C511yescharacteristic
250#336C22×C511yescharacteristic
251C23×C5—22no—
252C23×C5—41no—
253C23×C5—41no—
254C23×C5—41no—
255C23×C5—41no—
256C23×C5—41no—
257C23×C5—41no—
258C23×C5C22×D411yescharacteristic
259C5×Q8—84no—
260C5×D4—22no—
261C5×D4—22no—
262C5×D4—41no—
263C5×D4—41no—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—21no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—22no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5—21no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—41no—
289C2×C4×C5—41no—
290C2×C4×C5—41no—
291C2×C4×C5—41no—
292C2×C4×C5—41no—
293C2×C4×C5—41no—
294C2×C4×C5—41no—
295C2×C4×C5—41no—
296C2×C4×C5—41no—
297C2×C4×C5—41no—
298C2×C4×C5—41no—
299C2×C4×C5—41no—
300C2×C4×C5—84no—
301C2×C4×C5C22⋊Q812yes—
302C2×C4×C5C22⋊Q812yes—
303C2×C4×C5#13512yes—
304C2×C4×C5C22×Q811yescharacteristic
305C2×C4×C5C2×#4111yescharacteristic
306C2×C4×C5C2×#4111yescharacteristic
307C2×C4×C5C2×#4111yescharacteristic
308C2×C4×C5H(5,𝔽2)11yescharacteristic
309C2×C4×C52−1+411yescharacteristic
310C5×C8—41no—
311C5×C8—41no—
312C5×C8—41no—
313C5×C8—41no—
314C22×Q8—21no—
315C22×D4C23×C511yescharacteristic
316C23×C4—21no—
317C22×C8C23×C511yescharacteristic
318C22×C8C23×C511yescharacteristic
319C22×C8C23×C511yescharacteristic
320C22×C8C23×C511yescharacteristic
321C4⋊Q8—22no—
322C4⋊Q8—41no—
323#125—22no—
324#125—41no—
325C4×D4—22no—
326C4×D4C5×Q814yescomplemented
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4C5×D411yescharacteristic
333C2×C4⋊C4C5×D411yescharacteristic
334C2×C4⋊C4C23×C511yescharacteristic
335C2×C4⋊C4C23×C511yescharacteristic
336C2×C4⋊C4C23×C511yescharacteristic
337C2×C4⋊C4C23×C511yescharacteristic
338C2×C4⋊C4C23×C511yescharacteristic
339C2×C4⋊C4C23×C511yescharacteristic
340C2×C4⋊C4C23×C511yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42C23×C511yescharacteristic
350C2×C42C23×C511yescharacteristic
351C2×C42C23×C511yescharacteristic
352#107—22no—
353#106—22no—
354C4⋊C8—22no—
355C4⋊C8—41no—
356C4⋊C8—41no—
357#102—22no—
358#102—22no—
359#102—41no—
360#102—41no—
361C4×C8—22no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95C5×D411yescharacteristic
372#95C5×D411yescharacteristic
373C22×C5—42no—
374C22×C5—42no—
375C22×C5—42no—
376C22×C5—81no—
377C22×C5—81no—
378C22×C5—81no—
379C22×C5—81no—
380C22×C5C2×C22⋊Q811yescharacteristic
381C22×C5#54611yescharacteristic
382C22×C5#54811yescharacteristic
383C22×C5C2×#13511yescharacteristic
384C22×C5#57511yescharacteristic
385C22×C5#57611yescharacteristic
386C22×C5#57711yescharacteristic
387C4×C5—22no—
388C4×C5—22no—
389C4×C5—22no—
390C4×C5—42no—
391C4×C5—42no—
392C4×C5—41no—
393C4×C5—42no—
394C4×C5—41no—
395C4×C5—41no—
396C4×C5—81no—
397C4×C5—81no—
398C4×C5—81no—
399C4×C5—81no—
400C4×C5#53311yescharacteristic
401C4×C5#54411yescharacteristic
402C4×C5#55511yescharacteristic
403C4×C5#55611yescharacteristic
404C24—21no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—81no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4C2×C5×Q812yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4C2×C5×D411yescharacteristic
425C22×C4C2×C5×D411yescharacteristic
426C22×C4C2×C5×D411yescharacteristic
427C22×C4C2×C5×D411yescharacteristic
428C22×C4C2×C5×D411yescharacteristic
429C22×C4C2×C5×D411yescharacteristic
430C22×C4C24×C511yescharacteristic
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—41no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4C2×C5×Q812yes—
462C4⋊C4C5×#4112yes—
463C4⋊C4C5×#4112yes—
464C22⋊C4—24no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—81no—
474C42—81no—
475C42—81no—
476C42C2×C5×Q812yes—
477C42C5×#4112yes—
478C2×C5—82no—
479C2×C5#243911yescharacteristic
480C2×C5#307611yescharacteristic
481C2×C5#308111yescharacteristic
482C2×C5#329711yescharacteristic
483C2×C5#332611yescharacteristic
484C2×C5#336511yescharacteristic
485C2×C5#337011yescharacteristic
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23C22×C5×D411yescharacteristic
494Q8—84no—
495D4—22no—
496D4—22no—
497D4—41no—
498D4—41no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4C5×C22⋊Q812yes—
537C2×C4C5×C22⋊Q812yes—
538C2×C4C5×#13512yes—
539C2×C4C22×C5×Q811yescharacteristic
540C2×C4C2×C5×#4111yescharacteristic
541C2×C4C2×C5×#4111yescharacteristic
542C2×C4C2×C5×#4111yescharacteristic
543C2×C4C5×H(5,𝔽2)11yescharacteristic
544C2×C4C5×2−1+411yescharacteristic
545C8—41no—
546C8—41no—
547C8—41no—
548C8—41no—
549C5#2602811yescharacteristic, direct product factor, Sylow
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×C5×C22⋊Q811yescharacteristic
558C22C5×#54611yescharacteristic
559C22C5×#54811yescharacteristic
560C22C2×C5×#13511yescharacteristic
561C22C5×#57511yescharacteristic
562C22C5×#57611yescharacteristic
563C22C5×#57711yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—41no—
568C4—42no—
569C4—41no—
570C4—41no—
571C4—42no—
572C4—42no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4C5×#53311yescharacteristic
578C4C5×#54411yescharacteristic
579C4C5×#55511yescharacteristic
580C4C5×#55611yescharacteristic
581C2—82no—
582C2C5×#243911yescharacteristic
583C2C5×#307611yescharacteristic
584C2C5×#308111yescharacteristic
585C2C5×#329711yescharacteristic
586C2C5×#332611yescharacteristic
587C2C5×#336511yescharacteristic
588C2C5×#337011yescharacteristic
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC22×C5×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C4×C5×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#2550C2semidirect
7. C5×#1989C2semidirect
8. C5×#1988C2semidirect
9. C5×#1949C2semidirect
10. C5×#1939C2semidirect
11. C5×#1921C2semidirect
12. C5×#1920C2semidirect
13. C5×#1835C2semidirect
59. #26028C5direct (N × H)
72. C4×C5×D4Q8semidirect
119. #2550C2×C5semidirect
125. #1989C2×C5semidirect
126. #1988C2×C5semidirect
127. #1949C2×C5semidirect
128. #1939C2×C5semidirect
129. #1921C2×C5semidirect
130. #1920C2×C5semidirect
131. #1835C2×C5semidirect
326. C4×D4C5×Q8semidirect
549. C5#26028direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#25502
C5×#23572
C5×#20672
C5×#20652
C5×#20252
C5×#20232
C5×#19892
C5×#19882
C5×#19492
C5×#19392
C5×#19212
C5×#19202
C5×#18352
C5×#18042
C4×C5×#1022
#260285