Finite Group Databasebeta

← C2×C5×#5926 #14511428

Subgroups — C2×C5×#5926 #14511428

Order 1920.

1594 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 1594 conjugacy classes of subgroups — too many to draw. Only its 204 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×C5×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C5×#5926C218yesmaximal, direct product factor
3C2×C5×#986C211yesmaximal, characteristic, complemented
4C2×C5×#985C211yesmaximal, characteristic, complemented
5C2×C5×#983C212yesmaximal, complemented
6C22×C5×GL(2,𝔽3)C212yesmaximal, complemented
7C2×C5×#577—31nomaximal, Hall
8C2×C5×D3×D4—41nomaximal
9C5×2−1+4⋊C3C2214yescomplemented
10C2×C5×#41⋊C3C2211yescharacteristic, complemented
11C22×C5×SL(2,𝔽3)C2212yescomplemented
12C5×#986C2214yescomplemented
13C5×#985C2214yescomplemented
14C5×#983C2218yescomplemented
15C2×C5×GL(2,𝔽3)C2211yescharacteristic, complemented
16C2×C5×GL(2,𝔽3)C2212yescomplemented
17C2×C5×GL(2,𝔽3)C2218yescomplemented
18C2×C5×#226C2211yescharacteristic, complemented
19C2×#5926C511yesmaximal, characteristic, direct product factor, Hall
20C2×C5×2−1+4D311yescharacteristic, complemented
21C2×C5×H(5,𝔽2)—31no—
22C5×#577—38no—
23C5×#577—38no—
24C2×C5×#137—31no—
25C2×C5×#137—32no—
26C2×C5×AGL(1,ℤ/8)—32no—
27C2×C5×AGL(1,ℤ/8)—31no—
28C2×C5×#135—32no—
29C2×C5×#135—31no—
30C22×C5×SD16—31no—
31C22×C5×SD16—32no—
32C2×C5×#131—31no—
33C23×C5×D3—42no—
34C2×C3×C5×D4—41no—
35C2×C5×#67—42no—
36C5×D3×D4—48no—
37C2×C5×D12—41no—
38C2×C4×C5×D3—41no—
39C5×#41⋊C3C2312yescomplemented
40C2×C5×SL(2,𝔽3)C2311yescharacteristic
41C2×C5×SL(2,𝔽3)C2314yescomplemented
42C5×GL(2,𝔽3)C2312yescomplemented
43C5×GL(2,𝔽3)C2314yescomplemented
44C5×#226C2312yescomplemented
45C2×2−1+4⋊C3C2×C511yescharacteristic, complemented
46#5926C2×C518yesdirect product factor
47C2×#986C2×C511yescharacteristic, complemented
48C2×#985C2×C511yescharacteristic, complemented
49C2×#983C2×C512yescomplemented
50C22×GL(2,𝔽3)C2×C512yescomplemented
51C5×2−1+4—34no—
52C5×2−1+4C2×D314yescomplemented
53C5×H(5,𝔽2)—38no—
54C5×H(5,𝔽2)—64no—
55C2×C5×#41—31no—
56C2×C5×#41—32no—
57C2×C5×#41—32no—
58C2×C5×#41—31no—
59C2×C5×#41—61no—
60C2×C5×#41C2×D311yescharacteristic, complemented
61C22×C5×Q8—31no—
62C22×C5×Q8C2×D312yescomplemented
63C22×C5×D4—32no—
64C22×C5×D4—31no—
65C22×C5×D4—62no—
66C22×C5×D4—61no—
67C5×#137—38no—
68C5×#137—38no—
69C5×#137—38no—
70C5×AGL(1,ℤ/8)—34no—
71C5×AGL(1,ℤ/8)—316no—
72C5×AGL(1,ℤ/8)—34no—
73C5×#135—34no—
74C5×#135—316no—
75C5×#135—34no—
76C2×C5×Q16—32no—
77C2×C5×Q16—31no—
78C2×C5×SD16—32no—
79C2×C5×SD16—32no—
80C2×C5×SD16—32no—
81C2×C5×SD16—38no—
82C2×C5×SD16—38no—
83C2×C5×SD16—31no—
84C2×C5×SD16—31no—
85C2×C5×SD16—32no—
86C2×C5×SD16—38no—
87C2×C5×D8—32no—
88C2×C5×D8—31no—
89C5×#131—38no—
90C2×C5×M16—32no—
91C2×C5×M16—31no—
92C22×C5×C8—32no—
93C22×C5×C8—31no—
94C2×#577—31noSylow
95C23×C3×C5—42no—
96C22×C5×D3—41no—
97C22×C5×D3—42no—
98C22×C5×D3—48no—
99C22×C5×D3—88no—
100C3×C5×D4—44no—
101C2×C3×C4×C5—41no—
102C5×#67—48no—
103C2×C5×Dic3—41no—
104C5×D12—44no—
105C4×C5×D3—44no—
106C5×SL(2,𝔽3)C2411yescharacteristic
107C2×D3×D4—41no—
1082−1+4⋊C3C22×C514yescomplemented
109C2×#41⋊C3C22×C511yescharacteristic, complemented
110C22×SL(2,𝔽3)C22×C512yescomplemented
111#986C22×C514yescomplemented
112#985C22×C514yescomplemented
113#983C22×C518yescomplemented
114C2×GL(2,𝔽3)C22×C511yescharacteristic, complemented
115C2×GL(2,𝔽3)C22×C512yescomplemented
116C2×GL(2,𝔽3)C22×C518yescomplemented
117C2×#226C22×C511yescharacteristic, complemented
118C24×C5—61no—
119C24×C5—62no—
120C5×#41—34no—
121C5×#41—34no—
122C5×#41—32no—
123C5×#41—38no—
124C5×#41—38no—
125C5×#41—34no—
126C5×#41—64no—
127C5×#41—64no—
128C5×#41—62no—
129C5×#41—68no—
130C5×#41C22×D312yescomplemented
131C2×C5×Q8—31no—
132C2×C5×Q8—32no—
133C2×C5×Q8—34no—
134C2×C5×Q8—34no—
135C2×C5×Q8—31no—
136C2×C5×Q8—62no—
137C2×C5×Q8C22×D311yescharacteristic
138C2×C5×Q8C22×D314yescomplemented
139C2×C5×D4—32no—
140C2×C5×D4—31no—
141C2×C5×D4—32no—
142C2×C5×D4—31no—
143C2×C5×D4—38no—
144C2×C5×D4—34no—
145C2×C5×D4—62no—
146C2×C5×D4—64no—
147C2×C5×D4—62no—
148C2×C5×D4—616no—
149C2×C5×D4—61no—
150C2×C5×D4—62no—
151C2×C5×D4—68no—
152C2×C5×D4—61no—
153C2×C5×D4S411yescharacteristic
154C22×C4×C5—32no—
155C22×C4×C5—31no—
156C22×C4×C5—61no—
157C22×C4×C5—62no—
158C22×C4×C5—61no—
159C5×Q16—32no—
160C5×Q16—38no—
161C5×Q16—32no—
162C5×SD16—32no—
163C5×SD16—38no—
164C5×SD16—34no—
165C5×SD16—38no—
166C5×SD16—38no—
167C5×SD16—34no—
168C5×SD16—32no—
169C5×SD16—34no—
170C5×D8—38no—
171C5×D8—34no—
172C5×M16—38no—
173C5×M16—34no—
174C2×C5×C8—31no—
175C2×C5×C8—31no—
176C2×C5×C8—32no—
177C2×C5×C8—38no—
178C2×C5×C8—34no—
179C2×2−1+4C5×D311yescharacteristic, complemented
180C2×H(5,𝔽2)—31no—
181#577—38no—
182#577—38no—
183C2×#137—31no—
184C2×#137—32no—
185C2×AGL(1,ℤ/8)—32no—
186C2×AGL(1,ℤ/8)—31no—
187C2×#135—32no—
188C2×#135—31no—
189C22×SD16—31no—
190C22×SD16—32no—
191C2×#131—31no—
192C22×C3×C5—44no—
193C22×C3×C5—41no—
194C22×C3×C5—84no—
195C2×C5×D3—42no—
196C2×C5×D3—44no—
197C2×C5×D3—44no—
198C2×C5×D3—84no—
199C2×C5×D3—816no—
200C3×C4×C5—42no—
201C5×Dic3—42no—
202C23×D3—42no—
203C2×C3×D4—41no—
204C2×#67—42no—
205D3×D4—48no—
206C2×D12—41no—
207C2×C4×D3—41no—
208#41⋊C3C23×C512yescomplemented
209C2×SL(2,𝔽3)C23×C511yescharacteristic
210C2×SL(2,𝔽3)C23×C514yescomplemented
211GL(2,𝔽3)C23×C512yescomplemented
212GL(2,𝔽3)C23×C514yescomplemented
213#226C23×C512yescomplemented
214C23×C5—31no—
215C23×C5—64no—
216C23×C5—61no—
217C23×C5—62no—
218C23×C5—68no—
219C23×C5—244no—
220C23×C5—242no—
221C23×C5C2×S412yes—
222C5×Q8—31no—
223C5×Q8—34no—
224C5×Q8—34no—
225C5×Q8—32no—
226C5×Q8—62no—
227C5×Q8—61no—
228C5×Q8C23×D311yescharacteristic
229C5×D4—32no—
230C5×D4—38no—
231C5×D4—34no—
232C5×D4—32no—
233C5×D4—61no—
234C5×D4—61no—
235C5×D4—62no—
236C5×D4—68no—
237C5×D4—64no—
238C5×D4—68no—
239C5×D4—64no—
240C5×D4C2×S414yes—
241C2×C4×C5—34no—
242C2×C4×C5—34no—
243C2×C4×C5—31no—
244C2×C4×C5—32no—
245C2×C4×C5—32no—
246C2×C4×C5—32no—
247C2×C4×C5—64no—
248C2×C4×C5—68no—
249C2×C4×C5—61no—
250C2×C4×C5—64no—
251C2×C4×C5C2×S411yescharacteristic
252C5×C8—32no—
253C5×C8—32no—
254C5×C8—34no—
2552−1+4—34no—
2562−1+4C2×C5×D314yescomplemented
257H(5,𝔽2)—38no—
258H(5,𝔽2)—64no—
259C2×#41—31no—
260C2×#41—32no—
261C2×#41—32no—
262C2×#41—31no—
263C2×#41—61no—
264C2×#41C2×C5×D311yescharacteristic, complemented
265C22×Q8—31no—
266C22×Q8C2×C5×D312yescomplemented
267C22×D4—32no—
268C22×D4—31no—
269C22×D4—62no—
270C22×D4—61no—
271#137—38no—
272#137—38no—
273#137—38no—
274AGL(1,ℤ/8)—34no—
275AGL(1,ℤ/8)—316no—
276AGL(1,ℤ/8)—34no—
277#135—34no—
278#135—316no—
279#135—34no—
280C2×Q16—32no—
281C2×Q16—31no—
282C2×SD16—32no—
283C2×SD16—32no—
284C2×SD16—32no—
285C2×SD16—38no—
286C2×SD16—38no—
287C2×SD16—31no—
288C2×SD16—31no—
289C2×SD16—32no—
290C2×SD16—38no—
291C2×D8—32no—
292C2×D8—31no—
293#131—38no—
294C2×M16—32no—
295C2×M16—31no—
296C22×C8—32no—
297C22×C8—31no—
298C2×C3×C5—42no—
299C2×C3×C5—41no—
300C2×C3×C5—84no—
301C5×D3—44no—
302C5×D3—84no—
303C23×C3—42no—
304C22×D3—41no—
305C22×D3—42no—
306C22×D3—48no—
307C22×D3—88no—
308C3×D4—44no—
309C2×C3×C4—41no—
310#67—48no—
311C2×Dic3—41no—
312D12—44no—
313C4×D3—44no—
314SL(2,𝔽3)C24×C511yescharacteristic
315C22×C5—24no—
316C22×C5—32no—
317C22×C5—62no—
318C22×C5—62no—
319C22×C5—64no—
320C22×C5—122no—
321C22×C5—124no—
322C22×C5—248no—
323C22×C5—244no—
324C22×C5C22×S414yes—
325C22×C5C22×S411yescharacteristic
326C4×C5—31no—
327C4×C5—31no—
328C4×C5—34no—
329C4×C5—62no—
330C4×C5C22×S412yes—
331C24—61no—
332C24—62no—
333#41—34no—
334#41—34no—
335#41—32no—
336#41—38no—
337#41—38no—
338#41—34no—
339#41—64no—
340#41—64no—
341#41—62no—
342#41—68no—
343#41C22×C5×D312yescomplemented
344C2×Q8—31no—
345C2×Q8—32no—
346C2×Q8—34no—
347C2×Q8—34no—
348C2×Q8—31no—
349C2×Q8—62no—
350C2×Q8C22×C5×D311yescharacteristic
351C2×Q8C22×C5×D314yescomplemented
352C2×D4—32no—
353C2×D4—31no—
354C2×D4—32no—
355C2×D4—31no—
356C2×D4—38no—
357C2×D4—34no—
358C2×D4—62no—
359C2×D4—64no—
360C2×D4—62no—
361C2×D4—616no—
362C2×D4—61no—
363C2×D4—62no—
364C2×D4—68no—
365C2×D4—61no—
366C2×D4C5×S411yescharacteristic
367C22×C4—32no—
368C22×C4—31no—
369C22×C4—61no—
370C22×C4—62no—
371C22×C4—61no—
372Q16—32no—
373Q16—38no—
374Q16—32no—
375SD16—32no—
376SD16—38no—
377SD16—34no—
378SD16—38no—
379SD16—38no—
380SD16—34no—
381SD16—32no—
382SD16—34no—
383D8—38no—
384D8—34no—
385M16—38no—
386M16—34no—
387C2×C8—31no—
388C2×C8—31no—
389C2×C8—32no—
390C2×C8—38no—
391C2×C8—34no—
392C3×C5—41noHall
393C22×C3—41no—
394C22×C3—44no—
395C22×C3—84no—
396C2×D3—42no—
397C2×D3—44no—
398C2×D3—44no—
399C2×D3—84no—
400C2×D3—816no—
401C3×C4—42no—
402Dic3—42no—
403C2×C5—24no—
404C2×C5—62no—
405C2×C5—122no—
406C2×C5—124no—
407C2×C5#592612yesdirect product factor
408C2×C5C23×S411yescharacteristic
409C23—31no—
410C23—64no—
411C23—61no—
412C23—62no—
413C23—68no—
414C23—244no—
415C23—242no—
416C23C2×C5×S412yes—
417Q8—31no—
418Q8—34no—
419Q8—34no—
420Q8—32no—
421Q8—62no—
422Q8—61no—
423Q8C23×C5×D311yescharacteristic
424D4—32no—
425D4—38no—
426D4—34no—
427D4—32no—
428D4—61no—
429D4—61no—
430D4—62no—
431D4—68no—
432D4—64no—
433D4—68no—
434D4—64no—
435D4C2×C5×S414yes—
436C2×C4—31no—
437C2×C4—34no—
438C2×C4—32no—
439C2×C4—34no—
440C2×C4—32no—
441C2×C4—32no—
442C2×C4—64no—
443C2×C4—68no—
444C2×C4—61no—
445C2×C4—64no—
446C2×C4C2×C5×S411yescharacteristic
447C8—32no—
448C8—32no—
449C8—34no—
450C2×C3—42no—
451C2×C3—41no—
452C2×C3—84no—
453D3—44no—
454D3—84no—
455C5C2×#592611yescharacteristic, direct product factor, Sylow
456C22—24no—
457C22—32no—
458C22—62no—
459C22—62no—
460C22—64no—
461C22—122no—
462C22—124no—
463C22—248no—
464C22—244no—
465C22C22×C5×S414yes—
466C22C22×C5×S411yescharacteristic
467C4—31no—
468C4—31no—
469C4—34no—
470C4—62no—
471C4C22×C5×S412yes—
472C3—41noSylow
473C2—24no—
474C2—62no—
475C2—122no—
476C2—124no—
477C2C5×#592612yesdirect product factor
478C2C23×C5×S411yescharacteristic
subgroupG/H
center, socleC22×C5C22×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C24×C5
FrattiniC2C23×C5×S4
FittingC2×C5×2−1+4D3
hypercenterC2×C5×D4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C5 ◁ C2×C5×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×2−1+4⋊C3C2semidirect
2. C5×#5926C2direct (N × H)
3. C2×C5×#986C2semidirect
4. C2×C5×#985C2semidirect
5. C2×C5×#983C2semidirect
6. C22×C5×GL(2,𝔽3)C2semidirect
9. C5×2−1+4⋊C3C22semidirect
10. C2×C5×#41⋊C3C22semidirect
11. C22×C5×SL(2,𝔽3)C22semidirect
12. C5×#986C22semidirect
13. C5×#985C22semidirect
14. C5×#983C22semidirect
15. C2×C5×GL(2,𝔽3)C22semidirect
16. C2×C5×GL(2,𝔽3)C22semidirect
17. C2×C5×GL(2,𝔽3)C22semidirect
18. C2×C5×#226C22semidirect
19. C2×#5926C5direct (N × H)
20. C2×C5×2−1+4D3semidirect
39. C5×#41⋊C3C23semidirect
41. C2×C5×SL(2,𝔽3)C23semidirect
42. C5×GL(2,𝔽3)C23semidirect
43. C5×GL(2,𝔽3)C23semidirect
44. C5×#226C23semidirect
45. C2×2−1+4⋊C3C2×C5semidirect
46. #5926C2×C5direct (N × H)
47. C2×#986C2×C5semidirect
48. C2×#985C2×C5semidirect
49. C2×#983C2×C5semidirect
50. C22×GL(2,𝔽3)C2×C5semidirect
52. C5×2−1+4C2×D3semidirect
60. C2×C5×#41C2×D3semidirect
62. C22×C5×Q8C2×D3semidirect
108. 2−1+4⋊C3C22×C5semidirect
109. C2×#41⋊C3C22×C5semidirect
110. C22×SL(2,𝔽3)C22×C5semidirect
111. #986C22×C5semidirect
112. #985C22×C5semidirect
113. #983C22×C5semidirect
114. C2×GL(2,𝔽3)C22×C5semidirect
115. C2×GL(2,𝔽3)C22×C5semidirect
116. C2×GL(2,𝔽3)C22×C5semidirect
117. C2×#226C22×C5semidirect
130. C5×#41C22×D3semidirect
138. C2×C5×Q8C22×D3semidirect
179. C2×2−1+4C5×D3semidirect
208. #41⋊C3C23×C5semidirect
210. C2×SL(2,𝔽3)C23×C5semidirect
211. GL(2,𝔽3)C23×C5semidirect
212. GL(2,𝔽3)C23×C5semidirect
213. #226C23×C5semidirect
256. 2−1+4C2×C5×D3semidirect
264. C2×#41C2×C5×D3semidirect
266. C22×Q8C2×C5×D3semidirect
343. #41C22×C5×D3semidirect
351. C2×Q8C22×C5×D3semidirect
407. C2×C5#5926direct (N × H)
455. C5C2×#5926direct (N × H)
477. C2C5×#5926direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×2−1+4⋊C32
C5×#59262
C2×C5×#9862
C2×C5×#9852
C2×C5×#9832
C22×C5×GL(2,𝔽3)2
C2×C5×#5773
C2×C5×D3×D44
C2×#59265