Finite Group Databasebeta

← #14300491

Subgroups — #14300491

Order 1920.

816 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
2C2×C4×#1201C211yesmaximal, characteristic
3C2×C3×#3952C211yesmaximal, characteristic, complemented
4C2×(C2×D3)⋊#168C211yesmaximal, characteristic, complemented
5C2×#90955C211yesmaximal, characteristic, complemented
6#11746371C211yesmaximal, characteristic, complemented
7#11745968C211yesmaximal, characteristic, complemented
8#10602789—31nomaximal, Hall
9C22×C4×C5×D3C2211yescharacteristic
10C22×C3×C4×D5C2211yescharacteristic
11C22×#1201C411yescharacteristic
12C22×C3×C5×C8C2211yescharacteristic
13C2×C4×Dic15C2211yescharacteristic
14C22×C3⋊#168C2211yescharacteristic
15C22×C5×C3⋊C8C2211yescharacteristic
16C2×C5×ΓL(1,𝔽25)—22no—
17C2×C5×ΓL(1,𝔽25)—22no—
18C2×C5×ΓL(1,𝔽25)—22no—
19C22×C3×#168C2211yescharacteristic
20C2×#6669C411yescharacteristic
21C2×(C2×D3)⋊Dic5C411yescharacteristic
22C2×#6665C411yescharacteristic
23C4×#1201—22no—
24C4×#1201—22no—
25C3×#3952—22no—
26(C2×D3)⋊#168—22no—
27#90955—22no—
28#69958—51nomaximal, Hall
29C22×C5×M16—31no—
30C2×C4×C22⋊D5—31no—
31C2×C22⋊#168—31no—
32C2×#3952—31no—
33C2×#3952D311yescharacteristic, complemented
34#64770—31no—
35#64770—31no—
36C23×C5×D3C2×C411yescharacteristic
37C23×C3×D5C2×C411yescharacteristic
38C22×C3×C4×C5C2311yescharacteristic
39C22×Dic15C2×C411yescharacteristic
40C22×Dic15C2×C411yescharacteristic
41C22×C5×Dic3C2×C411yescharacteristic
42C2×C4×C5×D3—21no—
43C2×C4×C5×D3—21no—
44C2×C4×C5×D3—21no—
45C2×C4×C5×D3—21no—
46C2×C4×C5×D3D412yes—
47C2×C4×C5×D3D412yes—
48C22×C3×Dic5C2×C411yescharacteristic
49C2×C3×C4×D5—21no—
50C2×C3×C4×D5—21no—
51C2×C3×C4×D5—21no—
52C2×C3×C4×D5—21no—
53C2×C3×C4×D5—21no—
54C2×C3×C4×D5—21no—
55C2×#1201—22no—
56C2×#1201—22no—
57C2×#1201—22no—
58C2×C3×C5×C8—21no—
59C2×C3×C5×C8—21no—
60C2×C3×C5×C8D412yescomplemented
61C4×Dic15—21no—
62C4×Dic15—21no—
63C2×C3⋊#168—21no—
64C2×C3⋊#168—21no—
65C2×C3⋊#168—21no—
66C2×C5×C3⋊C8—21no—
67C2×C5×C3⋊C8—21no—
68C2×C5×C3⋊C8D412yescomplemented
69C5×ΓL(1,𝔽25)—28no—
70C2×C3×#168—21no—
71C2×C3×#168—21no—
72C2×C3×#168—21no—
73#6669—22no—
74(C2×D3)⋊Dic5—22no—
75#6665—22no—
76C2×C4×#67—51no—
77C22×ΓL(1,𝔽25)D511yescharacteristic, complemented
78C2×C3×C22⋊C8—51no—
79C2×#830—51no—
80C2×(C2×D3)⋊C8—51no—
81#4550—51no—
82#4550—51no—
83C23×C4×C5—31no—
84C22×C22⋊D5—31no—
85C22×C4×D5C2×D311yescharacteristic
86C2×C5×M16—62no—
87C2×C5×M16—62no—
88C2×C5×M16—62no—
89C22×C5×C8—31no—
90C22×C5×C8C2×D311yescharacteristic
91C2×C22⋊Dic5—31no—
92C4×C22⋊D5—62no—
93C4×C22⋊D5—62no—
94C2×#685—31no—
95C2×#683—31no—
96C2×C4×Dic5—31no—
97C22×#168—31no—
98C22×#168C2×D311yescharacteristic
99C22⋊#168—62no—
100#3952—22no—
101#3952—62no—
102#1870—151noSylow
103C23×C3×C5C22×C411yescharacteristic
104C22×C5×D3—21no—
105C22×C5×D3—21no—
106C22×C5×D3—21no—
107C22×C5×D3—21no—
108C22×C5×D3—21no—
109C22×C5×D3—21no—
110C22×C5×D3C22⋊C412yes—
111C22×C3×D5—21no—
112C22×C3×D5—21no—
113C22×C3×D5—21no—
114C22×C3×D5—21no—
115C22×C3×D5—21no—
116C22×C3×D5—21no—
117C22×C3×D5M1612yescomplemented
118C2×C3×C4×C5C2×D411yescharacteristic
119C2×C3×C4×C5C2×D411yescharacteristic
120C2×C3×C4×C5C2×D411yescharacteristic
121C2×C3×C4×C5C2×D411yescharacteristic
122C2×C3×C4×C5#4111yescharacteristic
123C2×C3×C4×C5#4111yescharacteristic
124C2×Dic15—21no—
125C2×Dic15—21no—
126C2×Dic15—21no—
127C2×Dic15—21no—
128C2×Dic15—21no—
129C2×Dic15M1612yescomplemented
130C2×C5×Dic3—21no—
131C2×C5×Dic3—21no—
132C2×C5×Dic3C22⋊C412yes—
133C4×C5×D3—22no—
134C4×C5×D3—22no—
135C4×C5×D3—22no—
136C4×C5×D3—22no—
137C2×C3×Dic5—21no—
138C2×C3×Dic5—21no—
139C2×C3×Dic5—21no—
140C3×C4×D5—41no—
141C3×C4×D5—41no—
142C3×C4×D5—41no—
143C3×C4×D5—41no—
144#1201—28no—
145C3×C5×C8—22no—
146C3⋊#168—41no—
147C3×#168—41no—
148C5×C3⋊C8—22no—
149C23×C3×C4—51no—
150C22×#67—51no—
151C22×C4×D3C2×D511yescharacteristic
152C22×C3×C8—51no—
153C22×C3×C8C2×D511yescharacteristic
154C2×#217—51no—
155C4×#67—102no—
156C4×#67—102no—
157C2×(C2×D3)⋊C4—51no—
158C2×Dic3⋊C4—51no—
159C2×C4×Dic3—51no—
160C22×C3⋊C8—51no—
161C22×C3⋊C8C2×D511yescharacteristic
162C2×ΓL(1,𝔽25)—22no—
163C2×ΓL(1,𝔽25)—22no—
164C2×ΓL(1,𝔽25)—22no—
165C3×C22⋊C8—102no—
166#830—102no—
167(C2×D3)⋊C8—102no—
168C24×C5—31no—
169C23×D5C4×D311yescharacteristic
170C22×C4×C5—31no—
171C22×C4×C5—32no—
172C22×C4×C5—32no—
173C22×C4×C5—61no—
174C22×C4×C5—61no—
175C22×C4×C5—61no—
176C22×C4×C5—61no—
177C22×C4×C5C22×D311yescharacteristic
178C2×C22⋊D5—62no—
179C2×C22⋊D5—62no—
180C2×C22⋊D5—62no—
181C22×Dic5—31no—
182C22×Dic5—31no—
183C22×Dic5C4×D311yescharacteristic
184C2×C4×D5—21no—
185C2×C4×D5—21no—
186C2×C4×D5—21no—
187C2×C4×D5—21no—
188C2×C4×D5—21no—
189C2×C4×D5—21no—
190C5×M16—68no—
191C2×C5×C8—21no—
192C2×C5×C8—21no—
193C2×C5×C8—32no—
194C2×C5×C8—61no—
195C2×C5×C8—61no—
196C2×C5×C8#6712yescomplemented
197C22⋊Dic5—62no—
198#685—62no—
199#683—62no—
200C4×Dic5—61no—
201C4×Dic5—61no—
202C2×#168—21no—
203C2×#168—21no—
204C2×#168—21no—
205C2×#168—61no—
206C2×#168—61no—
207C2×#168—61no—
208C22×M16—31no—
209C2×C4×D4—151no—
210C2×C22⋊C8—51no—
211C2×C22⋊C8—151no—
212C2×C22⋊C8—151no—
213#336—151no—
214#336—151no—
215C22×C3×C5C2×C22⋊C411yescharacteristic
216C22×C3×C5C4×D411yescharacteristic
217C22×C3×C5C4×D411yescharacteristic
218C22×C3×C5C2×M1611yescharacteristic
219C22×C3×C5C2×M1611yescharacteristic
220C22×C3×C5#13111yescharacteristic
221C22×C3×C5#13111yescharacteristic
222C2×C5×D3—22no—
223C2×C5×D3—22no—
224C2×C5×D3—22no—
225C2×C5×D3—41no—
226C2×C5×D3—41no—
227C2×C5×D3—41no—
228C2×C5×D3—41no—
229C2×C3×D5—22no—
230C2×C3×D5—22no—
231C2×C3×D5—22no—
232C2×C3×D5—41no—
233C2×C3×D5—41no—
234C2×C3×D5—41no—
235C2×C3×D5—41no—
236C3×C4×C5#12011yescharacteristic
237C3×C4×C5#12111yescharacteristic
238C3×C4×C5#12111yescharacteristic
239C3×C4×C5#12311yescharacteristic
240Dic15—22no—
241Dic15—41no—
242C3×Dic5—41no—
243C5×Dic3—22no—
244C24×C3—51no—
245C23×D3C4×D511yescharacteristic
246C22×C3×C4—51no—
247C22×C3×C4—101no—
248C22×C3×C4—101no—
249C22×C3×C4—101no—
250C22×C3×C4—101no—
251C22×C3×C4—101no—
252C22×C3×C4—101no—
253C22×C3×C4C22×D511yescharacteristic
254C2×#67—102no—
255C2×#67—102no—
256C2×#67—102no—
257C22×Dic3—51no—
258C22×Dic3—51no—
259C22×Dic3C4×D511yescharacteristic
260C2×C4×D3—21no—
261C2×C4×D3—21no—
262C2×C4×D3—21no—
263C2×C4×D3—21no—
264C2×C4×D3D2012yes—
265C2×C4×D3C22⋊D512yes—
266C2×C3×C8—21no—
267C2×C3×C8—21no—
268C2×C3×C8—101no—
269C2×C3×C8—101no—
270C2×C3×C8—101no—
271C2×C3×C8C22⋊D512yescomplemented
272#217—102no—
273(C2×D3)⋊C4—102no—
274Dic3⋊C4—102no—
275C4×Dic3—101no—
276C4×Dic3—101no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—101no—
280C2×C3⋊C8—101no—
281C2×C3⋊C8—101no—
282C2×C3⋊C8C22⋊D512yescomplemented
283ΓL(1,𝔽25)—28no—
284C23×C5—32no—
285C23×C5—61no—
286C23×C5—61no—
287C23×C5—61no—
288C23×C5—61no—
289C23×C5—61no—
290C23×C5—61no—
291C23×C5C2×C4×D311yescharacteristic
292C22×D5—21no—
293C22×D5—21no—
294C22×D5—21no—
295C22×D5—21no—
296C22×D5—21no—
297C22×D5—21no—
298C22×D5ΓL(1,𝔽25)12yescomplemented
299C2×C4×C5—32no—
300C2×C4×C5—61no—
301C2×C4×C5—61no—
302C2×C4×C5—62no—
303C2×C4×C5—62no—
304C2×C4×C5—62no—
305C2×C4×C5—62no—
306C2×C4×C5#23511yescharacteristic
307C2×C4×C5D3×D411yescharacteristic
308C2×C4×C5D3×D411yescharacteristic
309C2×C4×C5D3×D411yescharacteristic
310C2×C4×C5#23711yescharacteristic
311C2×C4×C5C2×#6711yescharacteristic
312C22⋊D5—68no—
313C2×Dic5—21no—
314C2×Dic5—21no—
315C2×Dic5—21no—
316C2×Dic5—32no—
317C2×Dic5—61no—
318C2×Dic5—61no—
319C2×Dic5—61no—
320C2×Dic5—61no—
321C2×Dic5—61no—
322C4×D5—41no—
323C4×D5—41no—
324C4×D5—41no—
325C4×D5—41no—
326C5×C8—22no—
327C5×C8—62no—
328#168—41no—
329#168—121no—
330C22×D4—151no—
331C23×C4—31no—
332C23×C4—51no—
333C2×M16—62no—
334C2×M16—62no—
335C2×M16—62no—
336C22×C8—31no—
337C22×C8—51no—
338C22×C8—151no—
339C22×C8D3×D511yescharacteristic
340C4×D4—302no—
341C4×D4—302no—
342C2×C4⋊C4—151no—
343C2×C22⋊C4—151no—
344C2×C22⋊C4—151no—
345C2×C42—151no—
346C22⋊C8—102no—
347C22⋊C8—302no—
348C22⋊C8—302no—
349C2×C3×C5#38611yescharacteristic
350C2×C3×C5#40711yescharacteristic
351C2×C3×C5#40811yescharacteristic
352C2×C3×C5#43511yescharacteristic
353C2×C3×C5#43511yescharacteristic
354C2×C3×C5#43611yescharacteristic
355C2×C3×C5#43611yescharacteristic
356C3×D5—42no—
357C5×D3—42no—
358C23×C3—52no—
359C23×C3—101no—
360C23×C3—101no—
361C23×C3—101no—
362C23×C3—101no—
363C23×C3—101no—
364C23×C3—101no—
365C23×C3C2×C4×D511yescharacteristic
366C22×D3—21no—
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3—21no—
371C22×D3—21no—
372C22×D3#68512yes—
373C2×C3×C4—101no—
374C2×C3×C4—101no—
375C2×C3×C4—101no—
376C2×C3×C4—201no—
377C2×C3×C4—201no—
378C2×C3×C4—201no—
379C2×C3×C4—201no—
380C2×C3×C4C2×D2011yescharacteristic
381C2×C3×C4#70911yescharacteristic
382C2×C3×C4#70911yescharacteristic
383C2×C3×C4C2×C22⋊D511yescharacteristic
384C2×C3×C4C2×C22⋊D511yescharacteristic
385C2×C3×C4C2×C22⋊D511yescharacteristic
386#67—108no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—52no—
390C2×Dic3—101no—
391C2×Dic3—101no—
392C2×Dic3—101no—
393C2×Dic3—101no—
394C2×Dic3—101no—
395C2×Dic3#68512yes—
396C4×D3—22no—
397C4×D3—22no—
398C4×D3—22no—
399C4×D3—22no—
400C3×C8—22no—
401C3×C8—201no—
402C3⋊C8—22no—
403C3⋊C8—201no—
404C22×C5—62no—
405C22×C5—62no—
406C22×C5—62no—
407C22×C5—121no—
408C22×C5—121no—
409C22×C5—121no—
410C22×C5—121no—
411C22×C5D3×C22⋊C411yescharacteristic
412C22×C5#67⋊C411yescharacteristic
413C22×C5C2×ΓL(1,𝔽25)11yescharacteristic
414C22×C5#90111yescharacteristic
415C22×C5D3×M1611yescharacteristic
416C22×C5#90711yescharacteristic
417C22×C5C4×#6711yescharacteristic
418C2×D5—22no—
419C2×D5—22no—
420C2×D5—22no—
421C2×D5—41no—
422C2×D5—41no—
423C2×D5—41no—
424C2×D5—41no—
425C4×C5—62no—
426C4×C5#88311yescharacteristic
427C4×C5#88411yescharacteristic
428C4×C5#93711yescharacteristic
429C4×C5#93811yescharacteristic
430Dic5—41no—
431Dic5—62no—
432Dic5—121no—
433C24—31no—
434C24—51no—
435C2×D4—302no—
436C2×D4—302no—
437C2×D4—302no—
438C22×C4—31no—
439C22×C4—32no—
440C22×C4—32no—
441C22×C4—51no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—101no—
447C22×C4—101no—
448C22×C4—101no—
449C22×C4—101no—
450C22×C4—101no—
451C22×C4—101no—
452C22×C4—151no—
453C22×C4—151no—
454C22×C4C2×D3×D511yescharacteristic
455M16—68no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—32no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—101no—
462C2×C8—101no—
463C2×C8—101no—
464C2×C8—301no—
465C2×C8—301no—
466C2×C8—301no—
467C2×C8#120112yescomplemented
468C4⋊C4—302no—
469C22⋊C4—302no—
470C22⋊C4—302no—
471C42—301no—
472C42—301no—
473C3×C5#187011yescharacteristic, complemented, Hall
474C22×C3—102no—
475C22×C3—102no—
476C22×C3—102no—
477C22×C3—201no—
478C22×C3—201no—
479C22×C3—201no—
480C22×C3—201no—
481C22×C3D5×M1611yescharacteristic
482C22×C3D5×M1611yescharacteristic
483C22×C3#405311yescharacteristic
484C22×C3#405311yescharacteristic
485C22×C3C2×#68511yescharacteristic
486C22×C3C4×C22⋊D511yescharacteristic
487C22×C3C4×C22⋊D511yescharacteristic
488C2×D3—22no—
489C2×D3—22no—
490C2×D3—22no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—41no—
494C2×D3—41no—
495C3×C4—201no—
496C3×C4#407511yescharacteristic
497C3×C4#407611yescharacteristic
498C3×C4#407611yescharacteristic
499C3×C4#409911yescharacteristic
500Dic3—22no—
501Dic3—102no—
502Dic3—201no—
503C2×C5—122no—
504C2×C5#472611yescharacteristic
505C2×C5#472711yescharacteristic
506C2×C5#472911yescharacteristic
507C2×C5#473011yescharacteristic
508C2×C5#495611yescharacteristic
509C2×C5#511111yescharacteristic
510C2×C5#512811yescharacteristic
511D5—42no—
512C23—32no—
513C23—52no—
514C23—61no—
515C23—61no—
516C23—61no—
517C23—61no—
518C23—61no—
519C23—61no—
520C23—101no—
521C23—101no—
522C23—101no—
523C23—101no—
524C23—101no—
525C23—101no—
526C23C4×D3×D511yescharacteristic
527D4—308no—
528C2×C4—32no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—101no—
536C2×C4—101no—
537C2×C4—101no—
538C2×C4—152no—
539C2×C4—201no—
540C2×C4—201no—
541C2×C4—201no—
542C2×C4—201no—
543C2×C4—301no—
544C2×C4—301no—
545C2×C4—301no—
546C2×C4—301no—
547C2×C4—301no—
548C2×C4#676511yescharacteristic
549C2×C4#677111yescharacteristic
550C2×C4D3×D2011yescharacteristic
551C2×C4C2×#120111yescharacteristic
552C2×C4D3×C22⋊D511yescharacteristic
553C2×C4D3×C22⋊D511yescharacteristic
554C8—22no—
555C8—62no—
556C8—201no—
557C8—601no—
558C2×C3—202no—
559C2×C3#6541511yescharacteristic
560C2×C3#6541611yescharacteristic
561C2×C3#6541611yescharacteristic
562C2×C3#6541711yescharacteristic
563C2×C3#6541711yescharacteristic
564C2×C3#6541811yescharacteristic
565C2×C3#6550211yescharacteristic
566D3—42no—
567C5#6995811yescharacteristic, complemented, Sylow
568C22—62no—
569C22—62no—
570C22—62no—
571C22—102no—
572C22—102no—
573C22—102no—
574C22—121no—
575C22—121no—
576C22—121no—
577C22—121no—
578C22—201no—
579C22—201no—
580C22—201no—
581C22—201no—
582C22D5×ΓL(1,𝔽25)11yescharacteristic
583C22#9124611yescharacteristic
584C22#9126611yescharacteristic
585C22#9126811yescharacteristic
586C22C4×#120111yescharacteristic
587C22#9144111yescharacteristic
588C22D3×#68511yescharacteristic
589C4—62no—
590C4—201no—
591C4—302no—
592C4—601no—
593C4#9145211yescharacteristic
594C4#9145411yescharacteristic
595C4#9145611yescharacteristic
596C4#9156811yescharacteristic
597C3#1060278911yescharacteristic, complemented, Sylow
598C2—122no—
599C2—202no—
600C2#1174770111yescharacteristic
601C2#1174770311yescharacteristic
602C2#1174772111yescharacteristic
603C2#1174772411yescharacteristic
604C2#1174810211yescharacteristic
605C2#1174814011yescharacteristic
606C2#1174891011yescharacteristic
subgroupG/H
center, FrattiniC22×C4C2×D3×D5
commutator, socleC23×C3×C5C22×C4
FittingC22×C3×C5×C8C22
prime radicalC22×#1201C4
cosocleC22×C3×C4×C5C23
hypercenterC22×C8D3×D5
hypocenterC3×C5#1870
LCSG ▷ C23×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
1. C22×C5×ΓL(1,𝔽25)C2semidirect
3. C2×C3×#3952C2semidirect
4. C2×(C2×D3)⋊#168C2semidirect
5. C2×#90955C2semidirect
6. #11746371C2semidirect
7. #11745968C2semidirect
33. C2×#3952D3semidirect
60. C2×C3×C5×C8D4semidirect
68. C2×C5×C3⋊C8D4semidirect
77. C22×ΓL(1,𝔽25)D5semidirect
117. C22×C3×D5M16semidirect
129. C2×Dic15M16semidirect
196. C2×C5×C8#67semidirect
271. C2×C3×C8C22⋊D5semidirect
282. C2×C3⋊C8C22⋊D5semidirect
298. C22×D5ΓL(1,𝔽25)semidirect
467. C2×C8#1201semidirect
473. C3×C5#1870semidirect
567. C5#69958semidirect
597. C3#10602789semidirect

Maximal subgroups

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

subgroupindex
C22×C5×ΓL(1,𝔽25)2
C2×C4×#12012
C2×C3×#39522
C2×(C2×D3)⋊#1682
C2×#909552
#117463712
#117459682
#106027893
#699585