Finite Group Databasebeta

← C5×#85382 #14511120

Subgroups — C5×#85382 #14511120

Order 1920.

650 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 72 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×C5×#986C211yesmaximal, characteristic, complemented
2C22×C5×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C5×#548⋊C3C211yesmaximal, characteristic, complemented
4C5×#5427C211yesmaximal, characteristic, complemented
5C5×#5421C211yesmaximal, characteristic, complemented
6C5×#5394C211yesmaximal, characteristic, complemented
7C5×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
8C5×#3265—31nomaximal, Hall
9C2×C5×#41⋊C3C2211yescharacteristic, complemented
10C22×C5×SL(2,𝔽3)C2211yescharacteristic, complemented
11C5×#986—22no—
12C2×C5×GL(2,𝔽3)—22no—
13C2×C5×GL(2,𝔽3)C2211yescharacteristic
14C2×C5×GL(2,𝔽3)C2211yescharacteristic, complemented
15C2×C5×GL(2,𝔽3)C2211yescharacteristic, complemented
16C5×#882—41nomaximal
17C4×C5×SL(2,𝔽3)C2211yescharacteristic, complemented
18C5×#859C2211yescharacteristic, complemented
19#85382C511yesmaximal, characteristic, direct product factor, Hall
20C2×C5×AGL(1,ℤ/8)—31no—
21C22×C5×SD16—31no—
22C5×#548D311yescharacteristic, complemented
23C5×D42—31no—
24C5×#496—31no—
25C5×#468—31no—
26C5×#466—31no—
27C5×#463—31no—
28C5×#460—31no—
29C5×#450—31no—
30C5×#449—31no—
31C5×#448—31no—
32C5×#447—31no—
33C5×#440—31no—
34C5×#435—31no—
35C23×C5×D3—41no—
36C2×C5×#67—41no—
37C2×C5×D12—41no—
38C2×C5×D12—41no—
39C5×#41⋊C3—21no—
40C2×C5×SL(2,𝔽3)—21no—
41C2×C5×SL(2,𝔽3)C2311yescharacteristic
42C5×GL(2,𝔽3)—21no—
43C5×GL(2,𝔽3)—21no—
44C5×GL(2,𝔽3)D412yescomplemented
45C3×C5×C22⋊C4—41no—
46C5×(C2×D3)⋊C4—41no—
47C5×(C2×D3)⋊C4—41no—
48C2×#986C2×C511yescharacteristic, complemented
49C22×GL(2,𝔽3)C2×C511yescharacteristic, complemented
50#548⋊C3C2×C511yescharacteristic, complemented
51#5427C2×C511yescharacteristic, complemented
52#5421C2×C511yescharacteristic, complemented
53#5394C2×C511yescharacteristic, complemented
54C4⋊GL(2,𝔽3)C2×C511yescharacteristic, complemented
55C2×C5×#41C2×D311yescharacteristic, complemented
56C22×C5×Q8C2×D311yescharacteristic
57C22×C5×D4—31no—
58C22×C5×D4—31no—
59C22×C5×D4—61no—
60C5×AGL(1,ℤ/8)—62no—
61C5×AGL(1,ℤ/8)—62no—
62C2×C5×SD16—31no—
63C2×C5×SD16—31no—
64C2×C5×SD16—31no—
65C2×C5×SD16—31no—
66C2×C5×SD16—31no—
67C2×C5×SD16—31no—
68C2×C5×SD16—62no—
69C2×C5×SD16—62no—
70C2×C5×D8—31no—
71C2×C5×D8—31no—
72C2×C5×D8—31no—
73C2×C5×M16—31no—
74C22×C5×C8—31no—
75C5×#127—31no—
76C5×#124—31no—
77C5×C22⋊Q8—31no—
78C5×#121—31no—
79C5×#121—31no—
80C5×#121—31no—
81C5×#121—61no—
82C5×#120—61no—
83C5×#120—61no—
84C4×C5×Q8C2×D311yescharacteristic, complemented
85C4×C5×D4—31no—
86C4×C5×D4—31no—
87C5×#107—31no—
88C5×C4⋊C8—31no—
89C5×#103—31no—
90C5×#103—31no—
91C5×#102—31no—
92C5×#102—31no—
93C5×#102—31no—
94C5×#102—31no—
95C5×C22⋊C8—31no—
96C5×C22⋊C8—31no—
97C5×#97—31no—
98#3265—31noSylow
99C23×C3×C5—41no—
100C22×C5×D3—41no—
101C22×C5×D3—41no—
102C22×C5×D3—41no—
103C22×C5×D3—82no—
104C22×C5×D3—82no—
105C22×C5×D3—82no—
106C2×C3×C4×C5—41no—
107C2×C3×C4×C5—41no—
108C5×#67—82no—
109C2×C5×Dic3—41no—
110C5×D12—82no—
111C5×D12—82no—
112C5×SL(2,𝔽3)C2×D411yescharacteristic, complemented
113C2×#41⋊C3C22×C511yescharacteristic, complemented
114C22×SL(2,𝔽3)C22×C511yescharacteristic, complemented
115#986—22no—
116C2×GL(2,𝔽3)—22no—
117C2×GL(2,𝔽3)C22×C511yescharacteristic
118C2×GL(2,𝔽3)C22×C511yescharacteristic, complemented
119C2×GL(2,𝔽3)C22×C511yescharacteristic, complemented
120#882—41no—
121C4×SL(2,𝔽3)C22×C511yescharacteristic, complemented
122#859C22×C511yescharacteristic, complemented
123C24×C5—61no—
124C24×C5—61no—
125C5×#41—21no—
126C5×#41—61no—
127C2×C5×Q8—21no—
128C2×C5×Q8—31no—
129C2×C5×Q8—61no—
130C2×C5×Q8C22×D311yescharacteristic
131C2×C5×D4—31no—
132C2×C5×D4—31no—
133C2×C5×D4—31no—
134C2×C5×D4—31no—
135C2×C5×D4—31no—
136C2×C5×D4—61no—
137C2×C5×D4—61no—
138C2×C5×D4—62no—
139C2×C5×D4—62no—
140C2×C5×D4—61no—
141C2×C5×D4—61no—
142C2×C5×D4—61no—
143C2×C5×D4—61no—
144C2×C5×D4—121no—
145C2×C5×D4—121no—
146C2×C5×D4—124no—
147C22×C4×C5—31no—
148C22×C4×C5—31no—
149C22×C4×C5—61no—
150C5×SD16—32no—
151C5×SD16—32no—
152C5×SD16—62no—
153C5×SD16—62no—
154C5×SD16—61no—
155C5×SD16—61no—
156C5×SD16—61no—
157C5×SD16—61no—
158C5×SD16—121no—
159C5×D8—62no—
160C5×D8—62no—
161C5×D8—121no—
162C5×M16—62no—
163C2×C5×C8—31no—
164C2×C5×C8—31no—
165C2×C5×C8—31no—
166C2×C5×C8—31no—
167C2×C5×C8—62no—
168C5×C4⋊C4—31no—
169C5×C4⋊C4—31no—
170C5×C4⋊C4—31no—
171C5×C22⋊C4—31no—
172C5×C22⋊C4—61no—
173C5×C22⋊C4—61no—
174C5×C22⋊C4—61no—
175C5×C22⋊C4—61no—
176C5×C22⋊C4S411yescharacteristic
177C42×C5—31no—
178C2×AGL(1,ℤ/8)—31no—
179C22×SD16—31no—
180#548C5×D311yescharacteristic, complemented
181D42—31no—
182#496—31no—
183#468—31no—
184#466—31no—
185#463—31no—
186#460—31no—
187#450—31no—
188#449—31no—
189#448—31no—
190#447—31no—
191#440—31no—
192#435—31no—
193C22×C3×C5—42no—
194C22×C3×C5—41no—
195C22×C3×C5—81no—
196C22×C3×C5—81no—
197C2×C5×D3—42no—
198C2×C5×D3—42no—
199C2×C5×D3—81no—
200C2×C5×D3—81no—
201C2×C5×D3—81no—
202C2×C5×D3—81no—
203C2×C5×D3—81no—
204C2×C5×D3—81no—
205C2×C5×D3—81no—
206C2×C5×D3—88no—
207C3×C4×C5—81no—
208C3×C4×C5—81no—
209C5×Dic3—81no—
210C23×D3—41no—
211C2×#67—41no—
212C2×D12—41no—
213C2×D12—41no—
214#41⋊C3—21no—
215C2×SL(2,𝔽3)—21no—
216C2×SL(2,𝔽3)C23×C511yescharacteristic
217GL(2,𝔽3)—21no—
218GL(2,𝔽3)—21no—
219GL(2,𝔽3)C5×D412yescomplemented
220C3×C22⋊C4—41no—
221(C2×D3)⋊C4—41no—
222(C2×D3)⋊C4—41no—
223C23×C5—31no—
224C23×C5—61no—
225C23×C5—61no—
226C23×C5—61no—
227C23×C5—122no—
228C23×C5—122no—
229C23×C5—122no—
230C23×C5—122no—
231C23×C5—241no—
232C23×C5—241no—
233C23×C5C2×S411yescharacteristic
234C5×Q8—31no—
235C5×Q8—61no—
236C5×Q8—61no—
237C5×Q8D3×D411yescharacteristic
238C5×D4—32no—
239C5×D4—61no—
240C5×D4—61no—
241C5×D4—61no—
242C5×D4—61no—
243C5×D4—61no—
244C5×D4—61no—
245C5×D4—61no—
246C5×D4—62no—
247C5×D4—122no—
248C5×D4—122no—
249C5×D4—122no—
250C5×D4—122no—
251C5×D4—122no—
252C5×D4—241no—
253C2×C4×C5—31no—
254C2×C4×C5—31no—
255C2×C4×C5—31no—
256C2×C4×C5—61no—
257C2×C4×C5—61no—
258C2×C4×C5—61no—
259C2×C4×C5—61no—
260C2×C4×C5—61no—
261C2×C4×C5—122no—
262C2×C4×C5C2×S411yescharacteristic
263C2×C4×C5C2×S411yescharacteristic
264C5×C8—32no—
265C5×C8—61no—
266C5×C8—61no—
267C5×C8—61no—
268C2×#41C2×C5×D311yescharacteristic, complemented
269C22×Q8C2×C5×D311yescharacteristic
270C22×D4—31no—
271C22×D4—31no—
272C22×D4—61no—
273AGL(1,ℤ/8)—62no—
274AGL(1,ℤ/8)—62no—
275C2×SD16—31no—
276C2×SD16—31no—
277C2×SD16—31no—
278C2×SD16—31no—
279C2×SD16—31no—
280C2×SD16—31no—
281C2×SD16—62no—
282C2×SD16—62no—
283C2×D8—31no—
284C2×D8—31no—
285C2×D8—31no—
286C2×M16—31no—
287C22×C8—31no—
288#127—31no—
289#124—31no—
290C22⋊Q8—31no—
291#121—31no—
292#121—31no—
293#121—31no—
294#121—61no—
295#120—61no—
296#120—61no—
297C4×Q8C2×C5×D311yescharacteristic, complemented
298C4×D4—31no—
299C4×D4—31no—
300#107—31no—
301C4⋊C8—31no—
302#103—31no—
303#103—31no—
304#102—31no—
305#102—31no—
306#102—31no—
307#102—31no—
308C22⋊C8—31no—
309C22⋊C8—31no—
310#97—31no—
311C2×C3×C5—41no—
312C2×C3×C5—41no—
313C2×C3×C5—41no—
314C2×C3×C5—82no—
315C5×D3—82no—
316C5×D3—82no—
317C5×D3—161no—
318C23×C3—41no—
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—82no—
323C22×D3—82no—
324C22×D3—82no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327#67—82no—
328C2×Dic3—41no—
329D12—82no—
330D12—82no—
331SL(2,𝔽3)C2×C5×D411yescharacteristic, complemented
332C22×C5—21no—
333C22×C5—21no—
334C22×C5—62no—
335C22×C5—61no—
336C22×C5—61no—
337C22×C5—61no—
338C22×C5—121no—
339C22×C5—121no—
340C22×C5—121no—
341C22×C5—121no—
342C22×C5—121no—
343C22×C5—121no—
344C22×C5—121no—
345C22×C5—121no—
346C22×C5—244no—
347C22×C5—244no—
348C22×C5#98612yescomplemented
349C22×C5C22×S411yescharacteristic
350C4×C5—21no—
351C4×C5—21no—
352C4×C5—31no—
353C4×C5—31no—
354C4×C5—61no—
355C4×C5—61no—
356C4×C5—121no—
357C24—61no—
358C24—61no—
359#41—21no—
360#41—61no—
361C2×Q8—21no—
362C2×Q8—31no—
363C2×Q8—61no—
364C2×Q8C22×C5×D311yescharacteristic
365C2×D4—31no—
366C2×D4—31no—
367C2×D4—31no—
368C2×D4—31no—
369C2×D4—31no—
370C2×D4—61no—
371C2×D4—61no—
372C2×D4—62no—
373C2×D4—62no—
374C2×D4—61no—
375C2×D4—61no—
376C2×D4—61no—
377C2×D4—61no—
378C2×D4—121no—
379C2×D4—121no—
380C2×D4—124no—
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—61no—
384SD16—32no—
385SD16—32no—
386SD16—62no—
387SD16—62no—
388SD16—61no—
389SD16—61no—
390SD16—61no—
391SD16—61no—
392SD16—121no—
393D8—62no—
394D8—62no—
395D8—121no—
396M16—62no—
397C2×C8—31no—
398C2×C8—31no—
399C2×C8—31no—
400C2×C8—31no—
401C2×C8—62no—
402C4⋊C4—31no—
403C4⋊C4—31no—
404C4⋊C4—31no—
405C22⋊C4—31no—
406C22⋊C4—61no—
407C22⋊C4—61no—
408C22⋊C4—61no—
409C22⋊C4—61no—
410C22⋊C4C5×S411yescharacteristic
411C42—31no—
412C3×C5—41noHall
413C22×C3—42no—
414C22×C3—41no—
415C22×C3—81no—
416C22×C3—81no—
417C2×D3—42no—
418C2×D3—42no—
419C2×D3—81no—
420C2×D3—81no—
421C2×D3—81no—
422C2×D3—81no—
423C2×D3—81no—
424C2×D3—81no—
425C2×D3—81no—
426C2×D3—88no—
427C3×C4—81no—
428C3×C4—81no—
429Dic3—81no—
430C2×C5—22no—
431C2×C5—121no—
432C2×C5—122no—
433C2×C5—241no—
434C2×C5—241no—
435C2×C5D4×S411yescharacteristic
436C2×C5C2×#98611yescharacteristic
437C2×C5#592611yescharacteristic
438C23—31no—
439C23—61no—
440C23—61no—
441C23—61no—
442C23—122no—
443C23—122no—
444C23—122no—
445C23—122no—
446C23—241no—
447C23—241no—
448C23C2×C5×S411yescharacteristic
449Q8—31no—
450Q8—61no—
451Q8—61no—
452Q8C5×D3×D411yescharacteristic
453D4—32no—
454D4—61no—
455D4—61no—
456D4—61no—
457D4—61no—
458D4—61no—
459D4—61no—
460D4—61no—
461D4—62no—
462D4—122no—
463D4—122no—
464D4—122no—
465D4—122no—
466D4—122no—
467D4—241no—
468C2×C4—31no—
469C2×C4—31no—
470C2×C4—31no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—122no—
477C2×C4C2×C5×S411yescharacteristic
478C2×C4C2×C5×S411yescharacteristic
479C8—32no—
480C8—61no—
481C8—61no—
482C8—61no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—82no—
487D3—82no—
488D3—82no—
489D3—161no—
490C5#8538211yescharacteristic, direct product factor, Sylow
491C22—21no—
492C22—21no—
493C22—61no—
494C22—62no—
495C22—61no—
496C22—61no—
497C22—121no—
498C22—121no—
499C22—121no—
500C22—121no—
501C22—121no—
502C22—121no—
503C22—121no—
504C22—121no—
505C22—244no—
506C22—244no—
507C22C5×#98612yescomplemented
508C22C22×C5×S411yescharacteristic
509C4—21no—
510C4—21no—
511C4—31no—
512C4—31no—
513C4—61no—
514C4—61no—
515C4—121no—
516C3—41noSylow
517C2—22no—
518C2—121no—
519C2—122no—
520C2—241no—
521C2—241no—
522C2C5×D4×S411yescharacteristic
523C2C2×C5×#98611yescharacteristic
524C2C5×#592611yescharacteristic
subgroupG/H
center, socleC22×C5C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC5×#548D3
hypercenterC5×C22⋊C4S4
hypocenterSL(2,𝔽3)C2×C5×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C5 ◁ C5×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C5×#986C2semidirect
2. C22×C5×GL(2,𝔽3)C2semidirect
3. C5×#548⋊C3C2semidirect
4. C5×#5427C2semidirect
5. C5×#5421C2semidirect
6. C5×#5394C2semidirect
7. C5×C4⋊GL(2,𝔽3)C2semidirect
9. C2×C5×#41⋊C3C22semidirect
10. C22×C5×SL(2,𝔽3)C22semidirect
14. C2×C5×GL(2,𝔽3)C22semidirect
15. C2×C5×GL(2,𝔽3)C22semidirect
17. C4×C5×SL(2,𝔽3)C22semidirect
18. C5×#859C22semidirect
19. #85382C5direct (N × H)
22. C5×#548D3semidirect
44. C5×GL(2,𝔽3)D4semidirect
48. C2×#986C2×C5semidirect
49. C22×GL(2,𝔽3)C2×C5semidirect
50. #548⋊C3C2×C5semidirect
51. #5427C2×C5semidirect
52. #5421C2×C5semidirect
53. #5394C2×C5semidirect
54. C4⋊GL(2,𝔽3)C2×C5semidirect
55. C2×C5×#41C2×D3semidirect
84. C4×C5×Q8C2×D3semidirect
112. C5×SL(2,𝔽3)C2×D4semidirect
113. C2×#41⋊C3C22×C5semidirect
114. C22×SL(2,𝔽3)C22×C5semidirect
118. C2×GL(2,𝔽3)C22×C5semidirect
119. C2×GL(2,𝔽3)C22×C5semidirect
121. C4×SL(2,𝔽3)C22×C5semidirect
122. #859C22×C5semidirect
180. #548C5×D3semidirect
219. GL(2,𝔽3)C5×D4semidirect
268. C2×#41C2×C5×D3semidirect
297. C4×Q8C2×C5×D3semidirect
331. SL(2,𝔽3)C2×C5×D4semidirect
348. C22×C5#986semidirect
490. C5#85382direct (N × H)
507. C22C5×#986semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9862
C22×C5×GL(2,𝔽3)2
C5×#548⋊C32
C5×#54272
C5×#54212
C5×#53942
C5×C4⋊GL(2,𝔽3)2
C5×#32653
C5×#8824
#853825