Finite Group Databasebeta

← C5×#27027 #11949747

Subgroups — C5×#27027 #11949747

Order 1280.

690 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2454C211yesmaximal, characteristic
2C5×#2370C211yesmaximal, characteristic, complemented
3C5×#2067C211yesmaximal, characteristic
4C5×#2059C211yesmaximal, characteristic, complemented
5C5×#2059C211yesmaximal, characteristic, complemented
6C5×#2025C211yesmaximal, characteristic
7C5×#1989C211yesmaximal, characteristic, complemented
8C5×#1944C211yesmaximal, characteristic, complemented
9C5×#1926C211yesmaximal, characteristic
10C5×#1925C211yesmaximal, characteristic
11C5×#1919C211yesmaximal, characteristic, complemented
12C5×#1918C211yesmaximal, characteristic, complemented
13C5×#1917C211yesmaximal, characteristic, complemented
14C5×#1864C211yesmaximal, characteristic
15C5×#1762C211yesmaximal, characteristic
16C2×C5×#125C2211yescharacteristic
17C2×C5×#125C2211yescharacteristic
18C2×C4×C5×D4C2211yescharacteristic
19C2×C5×#107C2211yescharacteristic
20C2×C5×#107C2211yescharacteristic
21C2×C5×#106C2211yescharacteristic
22C2×C5×#106C2211yescharacteristic
23C2×C5×C4⋊C8C2211yescharacteristic
24C2×C5×#102C2211yescharacteristic
25C2×C5×#102C2211yescharacteristic
26C2×C5×#102C2211yescharacteristic
27C2×C5×#102C2211yescharacteristic
28C2×C5×#97C2211yescharacteristic
29C5×#399—21no—
30C5×#396—21no—
31C5×#390C2211yescharacteristic
32C5×#390C2211yescharacteristic
33C5×#389—21no—
34C5×#389C2211yescharacteristic
35C5×#389C2211yescharacteristic
36C5×#389C2211yescharacteristic
37C5×#389C2211yescharacteristic
38C5×#387—21no—
39C5×#387—21no—
40C5×#385—21no—
41C5×#380—21no—
42C4×C5×C4⋊C4—21no—
43C4×C5×C4⋊C4C2211yescharacteristic
44C4×C5×C4⋊C4C2211yescharacteristic
45C4×C5×C4⋊C4C2211yescharacteristic
46C4×C5×C4⋊C4C2211yescharacteristic
47C5×#340C2211yescharacteristic
48C5×#340C2211yescharacteristic
49C5×#340C2211yescharacteristic
50C5×#340C2211yescharacteristic
51C5×#340C2211yescharacteristic
52C5×#340C2211yescharacteristic
53C5×#340C2211yescharacteristic
54C5×#340C2211yescharacteristic
55C5×#336C2211yescharacteristic
56C5×#336C2211yescharacteristic
57C5×#336C2211yescharacteristic
58C5×#336C2211yescharacteristic
59#27027C511yesmaximal, characteristic, direct product factor, Sylow
60C22×C5×D4C2311yescharacteristic
61C23×C4×C5—21no—
62C22×C5×C8C2311yescharacteristic
63C22×C5×C8C2311yescharacteristic
64C22×C5×C8C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C5×#125—22no—
67C5×#125—22no—
68C5×#125—22no—
69C5×#125—41no—
70C4×C5×D4—22no—
71C4×C5×D4—41no—
72C2×C5×C4⋊C4—21no—
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⋊C4—21no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4D411yescharacteristic
81C2×C5×C4⋊C4D411yescharacteristic
82C2×C5×C4⋊C4C2311yescharacteristic
83C2×C5×C4⋊C4C2311yescharacteristic
84C2×C5×C4⋊C4C2311yescharacteristic
85C2×C5×C4⋊C4C2311yescharacteristic
86C2×C5×C4⋊C4C2311yescharacteristic
87C2×C5×C4⋊C4C2311yescharacteristic
88C2×C5×C4⋊C4C2311yescharacteristic
89C2×C5×C22⋊C4—21no—
90C2×C5×C22⋊C4—21no—
91C2×C5×C22⋊C4—21no—
92C2×C5×C22⋊C4—21no—
93C2×C5×C22⋊C4—21no—
94C2×C42×C5—21no—
95C2×C42×C5—21no—
96C2×C42×C5—21no—
97C2×C42×C5C2311yescharacteristic
98C2×C42×C5C2311yescharacteristic
99C2×C42×C5C2311yescharacteristic
100C5×#107—22no—
101C5×#107—41no—
102C5×#106—22no—
103C5×#106—22no—
104C5×C4⋊C8—41no—
105C5×#102—22no—
106C5×#102—22no—
107C5×#102—41no—
108C5×#102Q814yes—
109C5×#97—41no—
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#2454C2×C511yescharacteristic
120#2370C2×C511yescharacteristic, complemented
121#2067C2×C511yescharacteristic
122#2059C2×C511yescharacteristic, complemented
123#2059C2×C511yescharacteristic, complemented
124#2025C2×C511yescharacteristic
125#1989C2×C511yescharacteristic, complemented
126#1944C2×C511yescharacteristic, complemented
127#1926C2×C511yescharacteristic
128#1925C2×C511yescharacteristic
129#1919C2×C511yescharacteristic, complemented
130#1918C2×C511yescharacteristic, complemented
131#1917C2×C511yescharacteristic, complemented
132#1864C2×C511yescharacteristic
133#1762C2×C511yescharacteristic
134C24×C5—21no—
135C2×C5×D4—21no—
136C2×C5×D4—21no—
137C2×C5×D4—21no—
138C2×C5×D4—21no—
139C2×C5×D4—21no—
140C2×C5×D4C2×Q812yes—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—42no—
149C22×C4×C5—42no—
150C22×C4×C5—81no—
151C22×C4×C5C2×D411yescharacteristic
152C22×C4×C5C2×D411yescharacteristic
153C22×C4×C5C2×D411yescharacteristic
154C22×C4×C5C2×D411yescharacteristic
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2411yescharacteristic
158C2×C5×C8—21no—
159C2×C5×C8—21no—
160C2×C5×C8—21no—
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×C8C2×Q812yes—
170C5×C4⋊C4—24no—
171C5×C4⋊C4—24no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4—21no—
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—41no—
183C5×C4⋊C4—41no—
184C5×C4⋊C4—42no—
185C5×C4⋊C4—42no—
186C5×C4⋊C4—42no—
187C5×C4⋊C4—81no—
188C5×C4⋊C4—81no—
189C5×C4⋊C4—81no—
190C5×C4⋊C4C2×Q812yes—
191C5×C4⋊C4#4112yes—
192C5×C4⋊C4#4112yes—
193C5×C4⋊C4#4112yes—
194C5×C22⋊C4—42no—
195C5×C22⋊C4—42no—
196C5×C22⋊C4—81no—
197C5×C22⋊C4—81no—
198C5×C22⋊C4—81no—
199C42×C5—21no—
200C42×C5—21no—
201C42×C5—21no—
202C42×C5—21no—
203C42×C5—21no—
204C42×C5—21no—
205C42×C5—42no—
206C42×C5—81no—
207C42×C5—81no—
208C2×#125C22×C511yescharacteristic
209C2×#125C22×C511yescharacteristic
210C2×C4×D4C22×C511yescharacteristic
211C2×#107C22×C511yescharacteristic
212C2×#107C22×C511yescharacteristic
213C2×#106C22×C511yescharacteristic
214C2×#106C22×C511yescharacteristic
215C2×C4⋊C8C22×C511yescharacteristic
216C2×#102C22×C511yescharacteristic
217C2×#102C22×C511yescharacteristic
218C2×#102C22×C511yescharacteristic
219C2×#102C22×C511yescharacteristic
220C2×#97C22×C511yescharacteristic
221#399—21no—
222#396—21no—
223#390C22×C511yescharacteristic
224#390C22×C511yescharacteristic
225#389—21no—
226#389C22×C511yescharacteristic
227#389C22×C511yescharacteristic
228#389C22×C511yescharacteristic
229#389C22×C511yescharacteristic
230#387—21no—
231#387—21no—
232#385—21no—
233#380—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4C22×C511yescharacteristic
236C4×C4⋊C4C22×C511yescharacteristic
237C4×C4⋊C4C22×C511yescharacteristic
238C4×C4⋊C4C22×C511yescharacteristic
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#336C22×C511yescharacteristic
248#336C22×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×D4—22no—
260C5×D4—22no—
261C5×D4—41no—
262C5×D4—41no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—21no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
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—21no—
283C2×C4×C5—21no—
284C2×C4×C5—21no—
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—41no—
301C2×C4×C5—41no—
302C2×C4×C5—84no—
303C2×C4×C5C22×Q811yescharacteristic
304C2×C4×C5C2×#4111yescharacteristic
305C2×C4×C5C2×#4111yescharacteristic
306C2×C4×C5C2×#4111yescharacteristic
307C2×C4×C5H(5,𝔽2)11yescharacteristic
308C2×C4×C52−1+411yescharacteristic
309C5×C8—22no—
310C5×C8—41no—
311C5×C8—41no—
312C5×C8—41no—
313C22×D4C23×C511yescharacteristic
314C23×C4—21no—
315C22×C8C23×C511yescharacteristic
316C22×C8C23×C511yescharacteristic
317C22×C8C23×C511yescharacteristic
318C22×C8C23×C511yescharacteristic
319#125—22no—
320#125—22no—
321#125—22no—
322#125—41no—
323C4×D4—22no—
324C4×D4—41no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4C5×D411yescharacteristic
334C2×C4⋊C4C5×D411yescharacteristic
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×C4⋊C4C23×C511yescharacteristic
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42C23×C511yescharacteristic
351C2×C42C23×C511yescharacteristic
352C2×C42C23×C511yescharacteristic
353#107—22no—
354#107—41no—
355#106—22no—
356#106—22no—
357C4⋊C8—41no—
358#102—22no—
359#102—22no—
360#102—41no—
361#102C5×Q814yes—
362#97—41no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95C5×D411yescharacteristic
371#95C5×D411yescharacteristic
372C22×C5—42no—
373C22×C5—42no—
374C22×C5—42no—
375C22×C5—81no—
376C22×C5—81no—
377C22×C5—81no—
378C22×C5—81no—
379C22×C5#52511yescharacteristic
380C22×C5#54611yescharacteristic
381C22×C5D4×Q811yescharacteristic
382C22×C5#57511yescharacteristic
383C22×C5#57511yescharacteristic
384C22×C5#57611yescharacteristic
385C22×C5#57711yescharacteristic
386C4×C5—41no—
387C4×C5—42no—
388C4×C5—41no—
389C4×C5—42no—
390C4×C5—42no—
391C4×C5—41no—
392C4×C5—41no—
393C4×C5—41no—
394C4×C5—41no—
395C4×C5—42no—
396C4×C5—81no—
397C4×C5—81no—
398C4×C5—81no—
399C4×C5#53311yescharacteristic
400C4×C5#54211yescharacteristic
401C4×C5#55411yescharacteristic
402C4×C5#55611yescharacteristic
403C24—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4C2×C5×Q812yes—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—81no—
420C22×C4C2×C5×D411yescharacteristic
421C22×C4C2×C5×D411yescharacteristic
422C22×C4C2×C5×D411yescharacteristic
423C22×C4C2×C5×D411yescharacteristic
424C22×C4C2×C5×D411yescharacteristic
425C22×C4C2×C5×D411yescharacteristic
426C22×C4C24×C511yescharacteristic
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8C2×C5×Q812yes—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4C2×C5×Q812yes—
460C4⋊C4C5×#4112yes—
461C4⋊C4C5×#4112yes—
462C4⋊C4C5×#4112yes—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—81no—
466C22⋊C4—81no—
467C22⋊C4—81no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—42no—
475C42—81no—
476C42—81no—
477C2×C5—82no—
478C2×C5#246511yescharacteristic
479C2×C5#311111yescharacteristic
480C2×C5#311911yescharacteristic
481C2×C5#330911yescharacteristic
482C2×C5#332311yescharacteristic
483C2×C5#338811yescharacteristic
484C2×C5#338911yescharacteristic
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C22×C5×D411yescharacteristic
493D4—22no—
494D4—22no—
495D4—41no—
496D4—41no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
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—41no—
536C2×C4—84no—
537C2×C4C22×C5×Q811yescharacteristic
538C2×C4C2×C5×#4111yescharacteristic
539C2×C4C2×C5×#4111yescharacteristic
540C2×C4C2×C5×#4111yescharacteristic
541C2×C4C5×H(5,𝔽2)11yescharacteristic
542C2×C4C5×2−1+411yescharacteristic
543C8—22no—
544C8—41no—
545C8—41no—
546C8—41no—
547C5#2702711yescharacteristic, direct product factor, Sylow
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22C5×#52511yescharacteristic
556C22C5×#54611yescharacteristic
557C22C5×D4×Q811yescharacteristic
558C22C5×#57511yescharacteristic
559C22C5×#57511yescharacteristic
560C22C5×#57611yescharacteristic
561C22C5×#57711yescharacteristic
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—42no—
567C4—41no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—42no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4C5×#53311yescharacteristic
576C4C5×#54211yescharacteristic
577C4C5×#55411yescharacteristic
578C4C5×#55611yescharacteristic
579C2—82no—
580C2C5×#246511yescharacteristic
581C2C5×#311111yescharacteristic
582C2C5×#311911yescharacteristic
583C2C5×#330911yescharacteristic
584C2C5×#332311yescharacteristic
585C2C5×#338811yescharacteristic
586C2C5×#338911yescharacteristic
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
2. C5×#2370C2semidirect
4. C5×#2059C2semidirect
5. C5×#2059C2semidirect
7. C5×#1989C2semidirect
8. C5×#1944C2semidirect
11. C5×#1919C2semidirect
12. C5×#1918C2semidirect
13. C5×#1917C2semidirect
59. #27027C5direct (N × H)
120. #2370C2×C5semidirect
122. #2059C2×C5semidirect
123. #2059C2×C5semidirect
125. #1989C2×C5semidirect
126. #1944C2×C5semidirect
129. #1919C2×C5semidirect
130. #1918C2×C5semidirect
131. #1917C2×C5semidirect
547. C5#27027direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#24542
C5×#23702
C5×#20672
C5×#20592
C5×#20252
C5×#19892
C5×#19442
C5×#19262
C5×#19252
C5×#19192
C5×#19182
C5×#19172
C5×#18642
C5×#17622
#270275