Finite Group Databasebeta

← C2×C3×C5×GL(2,𝔽5) #72219125

Subgroups — C2×C3×C5×GL(2,𝔽5) #72219125

Order 14400.

712 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×GU(2,𝔽3)—51nomaximal
2C2×C3×GL(2,𝔽5)C511yesmaximal, characteristic, direct product factor
3C2×C5×GL(2,𝔽5)C311yesmaximal, characteristic, direct product factor
4C3×C5×GL(2,𝔽5)C212yesmaximal, direct product factor
5C5×GL(2,𝔽5)C2×C312yesdirect product factor
6C2×C3×C5×#6732C211yesmaximal, characteristic
7C2×C5×#6732C2×C311yescharacteristic
8C2×C3×C5×SL(2,𝔽5)C411yescharacteristic, complemented
9C3×C5×#6732C411yescharacteristic, complemented
10C3×C5×#6732C2211yescharacteristic
11C2×C3×C4×C5×AGL(1,𝔽5)—61nomaximal
12C3×C5×SL(2,𝔽5)C2×C411yescharacteristic, complemented
13C2×C3×C5×#41⋊C3—51no—
14C2×C3×#6732C2×C511yescharacteristic
15C2×C3×C5×ΓL(1,𝔽25)—101nomaximal
16C3×C5×GU(2,𝔽3)—52no—
17C3×GL(2,𝔽5)C2×C512yesdirect product factor
18C22×C3×C5×AGL(1,𝔽5)—62no—
19C2×C3×C4×C5×D5—61no—
20C3×C4×C5×AGL(1,𝔽5)—62no—
21C3×C4×C5×AGL(1,𝔽5)—62no—
22C2×C5×SL(2,𝔽5)C3×C411yescharacteristic, complemented
23C5×#6732C3×C411yescharacteristic, complemented
24C5×#6732C22×C311yescharacteristic
25C2×C5×GU(2,𝔽3)—51no—
26C2×GL(2,𝔽5)C3×C511yescharacteristic, direct product factor
27C2×C3×C5×#104—151no—
28C2×C4×C5×AGL(1,𝔽5)—61no—
29C2×C3×C4×C5×D3—101no—
30C3×C5×#41⋊C3—51no—
31C3×C5×#41⋊C3—51no—
32C2×C3×C5×SL(2,𝔽3)—51no—
33C2×C3×SL(2,𝔽5)C4×C511yescharacteristic, complemented
34C3×#6732C4×C511yescharacteristic, complemented
35C3×#6732C22×C511yescharacteristic
36C2×C32×C5×C8—101no—
37C2×C3×C5×C3⋊C8—101no—
38C3×C5×ΓL(1,𝔽25)—102no—
39C3×C5×ΓL(1,𝔽25)—102no—
40C22×C3×C5×D5—61no—
41C2×C3×C5×AGL(1,𝔽5)—68no—
42C2×C3×C5×AGL(1,𝔽5)—64no—
43C2×C3×C4×C52—61no—
44C2×C3×C5×Dic5—61no—
45C3×C4×C5×D5—61no—
46C3×C4×C5×D5—61no—
47C3×C4×C5×D5—61no—
48C3×C4×C5×D5—61no—
49C5×SL(2,𝔽5)C2×C3×C411yescharacteristic, complemented
50C2×C3×GU(2,𝔽3)—51noHall
51C2×C3×C5×#41—51no—
52C2×C5×#41⋊C3—51no—
53C2×C5×#41⋊C3—101no—
54C2×C3×C4×AGL(1,𝔽5)—61no—
55C2×#6732C2×C3×C511yescharacteristic
56C2×C3×C5×M16—151no—
57C2×C3×C42×C5—151no—
58C2×C5×ΓL(1,𝔽25)—101no—
59C5×GU(2,𝔽3)—52no—
60GL(2,𝔽5)C2×C3×C512yesdirect product factor
61C3×C5×#104—152no—
62C3×C5×#104—152no—
63C22×C5×AGL(1,𝔽5)—62no—
64C2×C4×C5×D5—61no—
65C4×C5×AGL(1,𝔽5)—62no—
66C4×C5×AGL(1,𝔽5)—62no—
67C22×C3×C5×D3—101no—
68C2×C32×C4×C5—101no—
69C2×C3×C5×Dic3—101no—
70C3×C4×C5×D3—101no—
71C3×C4×C5×D3—101no—
72C3×C4×C5×D3—101no—
73C3×C4×C5×D3—101no—
74C3×C5×SL(2,𝔽3)—51no—
75C3×SL(2,𝔽5)C2×C4×C511yescharacteristic, complemented
76C32×C5×C8—102no—
77C3×C5×C3⋊C8—102no—
78C2×C5×#104—151no—
79C22×C3×C52—61no—
80C2×C3×C5×D5—61no—
81C2×C3×C5×D5—61no—
82C2×C3×C5×D5—64no—
83C3×C5×AGL(1,𝔽5)—68no—
84C3×C4×C52—61no—
85C3×C4×C52—61no—
86C3×C5×Dic5—61no—
87C3×C5×Dic5—61no—
88C2×C3×#41⋊C3—51no—
89C2×C3×ΓL(1,𝔽25)—101no—
90C3×GU(2,𝔽3)—52no—
91C22×C3×AGL(1,𝔽5)—62no—
92C3×C5×#41—51no—
93C3×C5×#41—51no—
94C3×C5×#41—151no—
95C3×C5×#41—151no—
96C2×C3×C5×Q8—51no—
97C2×C3×C5×D4—151no—
98C22×C3×C4×C5—151no—
99C22×C3×C4×C5—301no—
100C2×C4×C5×D3—101no—
101C2×C3×C4×D5—61no—
102C5×#41⋊C3—51no—
103C5×#41⋊C3—51no—
104C5×#41⋊C3—101no—
105C5×#41⋊C3—101no—
106C2×C5×SL(2,𝔽3)—51no—
107C2×C5×SL(2,𝔽3)—101no—
108C3×C4×AGL(1,𝔽5)—62no—
109C3×C4×AGL(1,𝔽5)—62no—
110C2×SL(2,𝔽5)C3×C4×C511yescharacteristic, complemented
111#6732C3×C4×C511yescharacteristic, complemented
112#6732C22×C3×C511yescharacteristic
113C3×C5×M16—152no—
114C3×C5×M16—301no—
115C2×C3×C5×C8—101no—
116C2×C3×C5×C8—101no—
117C2×C3×C5×C8—201no—
118C3×C42×C5—152no—
119C3×C42×C5—301no—
120C2×C5×C3⋊C8—101no—
121C5×ΓL(1,𝔽25)—102no—
122C5×ΓL(1,𝔽25)—102no—
123C22×C5×D5—61no—
124C2×C5×AGL(1,𝔽5)—64no—
125C2×C5×AGL(1,𝔽5)—68no—
126C2×C4×C52—61no—
127C2×C5×Dic5—61no—
128C4×C5×D5—61no—
129C4×C5×D5—61no—
130C4×C5×D5—61no—
131C4×C5×D5—61no—
132C2×GU(2,𝔽3)—51no—
133C2×C3×#104—151no—
134C22×C32×C5—101no—
135C2×C3×C5×D3—101no—
136C2×C3×C5×D3—101no—
137C2×C3×C5×D3—202no—
138C32×C4×C5—101no—
139C32×C4×C5—101no—
140C3×C5×Dic3—101no—
141C3×C5×Dic3—101no—
142C2×C5×#41—51no—
143C2×C4×AGL(1,𝔽5)—61no—
144C2×C5×M16—151no—
145C2×C42×C5—151no—
146C5×#104—152no—
147C5×#104—152no—
148C2×C3×C52—62no—
149C2×C3×C52—61no—
150C3×C5×D5—62no—
151C3×C5×D5—62no—
152C2×C3×C4×D3—101no—
153C3×#41⋊C3—51no—
154C3×#41⋊C3—51no—
155C2×C3×SL(2,𝔽3)—51no—
156C2×C32×C8—101no—
157C2×C3×C3⋊C8—101no—
158C3×ΓL(1,𝔽25)—102no—
159C3×ΓL(1,𝔽25)—102no—
160C23×C3×C5—151no—
161C22×C5×D3—101no—
162C22×C3×D5—61no—
163C2×C3×AGL(1,𝔽5)—64no—
164C2×C3×AGL(1,𝔽5)—68no—
165C3×C5×Q8—51no—
166C3×C5×Q8—151no—
167C3×C5×D4—151no—
168C3×C5×D4—151no—
169C3×C5×D4—301no—
170C2×C3×C4×C5—61no—
171C2×C3×C4×C5—101no—
172C2×C3×C4×C5—151no—
173C2×C3×C4×C5—151no—
174C2×C3×C4×C5—151no—
175C2×C3×C4×C5—151no—
176C2×C3×C4×C5—151no—
177C2×C3×C4×C5—201no—
178C2×C3×C4×C5—241no—
179C2×C3×C4×C5—302no—
180C2×C3×C4×C5—304no—
181C2×C3×C4×C5S511yescharacteristic
182C2×C5×Dic3—101no—
183C4×C5×D3—101no—
184C4×C5×D3—101no—
185C4×C5×D3—101no—
186C4×C5×D3—101no—
187C2×C3×Dic5—61no—
188C3×C4×D5—61no—
189C3×C4×D5—61no—
190C3×C4×D5—61no—
191C3×C4×D5—61no—
192C5×SL(2,𝔽3)—51no—
193C5×SL(2,𝔽3)—101no—
194SL(2,𝔽5)C2×C3×C4×C511yescharacteristic, complemented
195C3×C5×C8—102no—
196C3×C5×C8—102no—
197C3×C5×C8—202no—
198C5×C3⋊C8—102no—
199C22×C52—61no—
200C2×C5×D5—61no—
201C2×C5×D5—61no—
202C2×C5×D5—64no—
203C5×AGL(1,𝔽5)—68no—
204C4×C52—61no—
205C4×C52—61no—
206C5×Dic5—61no—
207C5×Dic5—61no—
208C2×C3×#41—51no—
209C2×#41⋊C3—51no—
210C2×#41⋊C3—101no—
211C2×C3×M16—151no—
212C2×C3×C42—151no—
213C2×ΓL(1,𝔽25)—101no—
214GU(2,𝔽3)—52no—
215C3×#104—152no—
216C3×#104—152no—
217C2×C32×C5—102no—
218C2×C32×C5—101no—
219C3×C5×D3—201no—
220C3×C5×D3—201no—
221C22×AGL(1,𝔽5)—62no—
222C5×#41—51no—
223C5×#41—51no—
224C5×#41—151no—
225C5×#41—151no—
226C2×C5×Q8—51no—
227C2×C5×D4—151no—
228C22×C4×C5—151no—
229C22×C4×C5—301no—
230C2×C4×D5—61no—
231C4×AGL(1,𝔽5)—62no—
232C4×AGL(1,𝔽5)—62no—
233C5×M16—152no—
234C5×M16—301no—
235C2×C5×C8—101no—
236C42×C5—152no—
237C42×C5—301no—
238C3×C52—61no—
239C22×C3×D3—101no—
240C2×C32×C4—101no—
241C2×C3×Dic3—101no—
242C3×C4×D3—101no—
243C3×C4×D3—101no—
244C3×C4×D3—101no—
245C3×C4×D3—101no—
246C3×SL(2,𝔽3)—51no—
247C32×C8—102no—
248C3×C3⋊C8—102no—
249C2×#104—151noSylow
250C22×C3×C5—61no—
251C22×C3×C5—101no—
252C22×C3×C5—151no—
253C22×C3×C5—151no—
254C22×C3×C5—201no—
255C22×C3×C5—241no—
256C22×C3×C5—302no—
257C22×C3×C5A5⋊C411yescharacteristic
258C2×C5×D3—101no—
259C2×C5×D3—101no—
260C2×C5×D3—202no—
261C2×C3×D5—61no—
262C2×C3×D5—61no—
263C2×C3×D5—64no—
264C3×AGL(1,𝔽5)—68no—
265C3×C4×C5—61no—
266C3×C4×C5—61no—
267C3×C4×C5—101no—
268C3×C4×C5—101no—
269C3×C4×C5—151no—
270C3×C4×C5—151no—
271C3×C4×C5—201no—
272C3×C4×C5—201no—
273C3×C4×C5—241no—
274C3×C4×C5—241no—
275C3×C4×C5—304no—
276C3×C4×C5A5⋊C411yescharacteristic
277C3×C4×C5C2×S511yescharacteristic
278C3×Dic5—61no—
279C3×Dic5—61no—
280C5×Dic3—101no—
281C5×Dic3—101no—
282C2×C52—61no—
283C2×C52—62no—
284C5×D5—62no—
285C5×D5—62no—
286C3×#41—51no—
287C3×#41—51no—
288C3×#41—151no—
289C3×#41—151no—
290C2×C3×Q8—51no—
291C2×C3×D4—151no—
292C22×C3×C4—151no—
293C22×C3×C4—301no—
294C2×C4×D3—101no—
295#41⋊C3—51no—
296#41⋊C3—51no—
297#41⋊C3—101no—
298#41⋊C3—101no—
299C2×SL(2,𝔽3)—51no—
300C2×SL(2,𝔽3)—101no—
301C3×M16—152no—
302C3×M16—301no—
303C2×C3×C8—101no—
304C2×C3×C8—101no—
305C2×C3×C8—201no—
306C3×C42—152no—
307C3×C42—301no—
308C2×C3⋊C8—101no—
309ΓL(1,𝔽25)—102no—
310ΓL(1,𝔽25)—102no—
311C32×C5—101no—
312C23×C5—151no—
313C22×D5—61no—
314C2×AGL(1,𝔽5)—64no—
315C2×AGL(1,𝔽5)—68no—
316C5×Q8—51no—
317C5×Q8—151no—
318C5×D4—151no—
319C5×D4—151no—
320C5×D4—301no—
321C2×C4×C5—61no—
322C2×C4×C5—151no—
323C2×C4×C5—151no—
324C2×C4×C5—151no—
325C2×C4×C5—151no—
326C2×C4×C5—151no—
327C2×C4×C5—241no—
328C2×C4×C5—302no—
329C2×C4×C5—304no—
330C2×C4×C5C3×S511yescharacteristic
331C2×Dic5—61no—
332C4×D5—61no—
333C4×D5—61no—
334C4×D5—61no—
335C4×D5—61no—
336C5×C8—102no—
337C22×C32—101no—
338C2×C3×D3—101no—
339C2×C3×D3—101no—
340C2×C3×D3—202no—
341C32×C4—101no—
342C32×C4—101no—
343C3×Dic3—101no—
344C3×Dic3—101no—
345C2×#41—51no—
346C2×M16—151no—
347C2×C42—151no—
348#104—152no—
349#104—152no—
350C2×C3×C5—62no—
351C2×C3×C5—61no—
352C2×C3×C5—101no—
353C2×C3×C5—102no—
354C2×C3×C5—201no—
355C2×C3×C5—202no—
356C2×C3×C5—242no—
357C2×C3×C5—241no—
358C2×C3×C5—301no—
359C2×C3×C5—301no—
360C2×C3×C5GL(2,𝔽5)12yesdirect product factor
361C2×C3×C5C2×A5⋊C411yescharacteristic
362C3×D5—62no—
363C3×D5—62no—
364C5×D3—201no—
365C5×D3—201no—
366C52—61noSylow
367C23×C3—151no—
368C22×D3—101no—
369C3×Q8—51no—
370C3×Q8—151no—
371C3×D4—151no—
372C3×D4—151no—
373C3×D4—301no—
374C2×C3×C4—101no—
375C2×C3×C4—151no—
376C2×C3×C4—151no—
377C2×C3×C4—151no—
378C2×C3×C4—151no—
379C2×C3×C4—151no—
380C2×C3×C4—201no—
381C2×C3×C4—302no—
382C2×C3×C4—304no—
383C2×C3×C4C5×S511yescharacteristic
384C2×Dic3—101no—
385C4×D3—101no—
386C4×D3—101no—
387C4×D3—101no—
388C4×D3—101no—
389SL(2,𝔽3)—51no—
390SL(2,𝔽3)—101no—
391C3×C8—102no—
392C3×C8—102no—
393C3×C8—202no—
394C3⋊C8—102no—
395C22×C5—61no—
396C22×C5—151no—
397C22×C5—151no—
398C22×C5—241no—
399C22×C5—302no—
400C22×C5C3×A5⋊C411yescharacteristic
401C2×D5—61no—
402C2×D5—61no—
403C2×D5—64no—
404AGL(1,𝔽5)—68no—
405C4×C5—61no—
406C4×C5—61no—
407C4×C5—151no—
408C4×C5—151no—
409C4×C5—241no—
410C4×C5—241no—
411C4×C5—304no—
412C4×C5C3×A5⋊C411yescharacteristic
413C4×C5C2×C3×S511yescharacteristic
414Dic5—61no—
415Dic5—61no—
416C2×C32—102no—
417C2×C32—101no—
418C3×D3—201no—
419C3×D3—201no—
420#41—51no—
421#41—51no—
422#41—151no—
423#41—151no—
424C2×Q8—51no—
425C2×D4—151no—
426C22×C4—151no—
427C22×C4—301no—
428M16—152no—
429M16—301no—
430C2×C8—101no—
431C42—152no—
432C42—301no—
433C3×C5—61no—
434C3×C5—101no—
435C3×C5—201no—
436C3×C5—241no—
437C3×C5C2×GL(2,𝔽5)11yescharacteristic, direct product factor
438C22×C3—101no—
439C22×C3—151no—
440C22×C3—151no—
441C22×C3—201no—
442C22×C3—302no—
443C22×C3C5×A5⋊C411yescharacteristic
444C2×D3—101no—
445C2×D3—101no—
446C2×D3—202no—
447C3×C4—101no—
448C3×C4—101no—
449C3×C4—151no—
450C3×C4—151no—
451C3×C4—201no—
452C3×C4—201no—
453C3×C4—304no—
454C3×C4C5×A5⋊C411yescharacteristic
455C3×C4C2×C5×S511yescharacteristic
456Dic3—101no—
457Dic3—101no—
458C2×C5—62no—
459C2×C5—61no—
460C2×C5—242no—
461C2×C5—241no—
462C2×C5—301no—
463C2×C5—301no—
464C2×C5C3×GL(2,𝔽5)12yesdirect product factor
465C2×C5C2×C3×A5⋊C411yescharacteristic
466D5—62no—
467D5—62no—
468C32—101noSylow
469C23—151no—
470Q8—51no—
471Q8—151no—
472D4—151no—
473D4—151no—
474D4—301no—
475C2×C4—151no—
476C2×C4—151no—
477C2×C4—151no—
478C2×C4—151no—
479C2×C4—151no—
480C2×C4—302no—
481C2×C4—304no—
482C2×C4C3×C5×S511yescharacteristic
483C8—102no—
484C2×C3—102no—
485C2×C3—101no—
486C2×C3—202no—
487C2×C3—201no—
488C2×C3—301no—
489C2×C3—301no—
490C2×C3C5×GL(2,𝔽5)12yesdirect product factor
491C2×C3C2×C5×A5⋊C411yescharacteristic
492D3—201no—
493D3—201no—
494C5—61no—
495C5—241no—
496C5C2×C3×GL(2,𝔽5)11yescharacteristic, direct product factor
497C22—151no—
498C22—151no—
499C22—302no—
500C22C3×C5×A5⋊C411yescharacteristic
501C4—151no—
502C4—151no—
503C4—304no—
504C4C3×C5×A5⋊C411yescharacteristic
505C4C2×C3×C5×S511yescharacteristic
506C3—101no—
507C3—201no—
508C3C2×C5×GL(2,𝔽5)11yescharacteristic, direct product factor
509C2—301no—
510C2—301no—
511C2C3×C5×GL(2,𝔽5)12yesdirect product factor
512C2C2×C3×C5×A5⋊C411yescharacteristic
subgroupG/H
center, Fitting, hypercenterC2×C3×C4×C5S5
commutator, hypocenterSL(2,𝔽5)C2×C3×C4×C5
FrattiniC4C2×C3×C5×S5
socleC22×C3×C5A5⋊C4
prime radicalC2×C3×C5×#6732C2
cosocle#6732C22×C3×C5
LCSG ▷ SL(2,𝔽5)
UCSC1 ◁ C2×C3×C4×C5

Complements

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

NH ≅ G/Ntype
2. C2×C3×GL(2,𝔽5)C5direct (N × H)
3. C2×C5×GL(2,𝔽5)C3direct (N × H)
4. C3×C5×GL(2,𝔽5)C2direct (N × H)
5. C5×GL(2,𝔽5)C2×C3direct (N × H)
8. C2×C3×C5×SL(2,𝔽5)C4semidirect
9. C3×C5×#6732C4semidirect
12. C3×C5×SL(2,𝔽5)C2×C4semidirect
17. C3×GL(2,𝔽5)C2×C5direct (N × H)
22. C2×C5×SL(2,𝔽5)C3×C4semidirect
23. C5×#6732C3×C4semidirect
26. C2×GL(2,𝔽5)C3×C5direct (N × H)
33. C2×C3×SL(2,𝔽5)C4×C5semidirect
34. C3×#6732C4×C5semidirect
49. C5×SL(2,𝔽5)C2×C3×C4semidirect
60. GL(2,𝔽5)C2×C3×C5direct (N × H)
75. C3×SL(2,𝔽5)C2×C4×C5semidirect
110. C2×SL(2,𝔽5)C3×C4×C5semidirect
111. #6732C3×C4×C5semidirect
194. SL(2,𝔽5)C2×C3×C4×C5semidirect
360. C2×C3×C5GL(2,𝔽5)direct (N × H)
437. C3×C5C2×GL(2,𝔽5)direct (N × H)
464. C2×C5C3×GL(2,𝔽5)direct (N × H)
490. C2×C3C5×GL(2,𝔽5)direct (N × H)
496. C5C2×C3×GL(2,𝔽5)direct (N × H)
508. C3C2×C5×GL(2,𝔽5)direct (N × H)
511. C2C3×C5×GL(2,𝔽5)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×GU(2,𝔽3)5
C2×C3×GL(2,𝔽5)5
C2×C5×GL(2,𝔽5)3
C3×C5×GL(2,𝔽5)2
C2×C3×C5×#67322
C2×C3×C4×C5×AGL(1,𝔽5)6
C2×C3×C5×ΓL(1,𝔽25)10