Finite Group Databasebeta

← C5×#3497⋊C3 #14511347

Subgroups — C5×#3497⋊C3 #14511347

Order 1920.

744 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 124 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⋊C3C212yesmaximal, complemented
2C5×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3C5×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C5×#548⋊C3C212yesmaximal, complemented
5C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6C5×#3497C311yesmaximal, characteristic, complemented, Hall
7C5×2−1+4⋊C3—24no—
8C2×C5×#41⋊C3C2212yes—
9C22×C5×SL(2,𝔽3)C2212yes—
10C22×C5×SL(2,𝔽3)C2211yescharacteristic
11C3×C5×#121—41nomaximal
12C4×C5×SL(2,𝔽3)C2211yescharacteristic
13C4×C5×SL(2,𝔽3)C2211yescharacteristic
14#3497⋊C3C511yesmaximal, characteristic, direct product factor, Hall
15C2×C5×2−1+4C2×C312yescomplemented
16C5×#550—31no—
17C5×#550C2×C311yescharacteristic, complemented
18C5×D4×Q8—31no—
19C5×D4×Q8C2×C311yescharacteristic, complemented
20C5×#548—32no—
21C5×#548C2×C312yescomplemented
22C5×#537—32no—
23C5×#532—31no—
24C5×#532—31no—
25C2×C4×C5×Q8C2×C311yescharacteristic, complemented
26C2×C3×C5×D4—42no—
27C2×C3×C5×D4—41no—
28C22×C3×C4×C5—41no—
29C5×#41⋊C3—22no—
30C2×C5×SL(2,𝔽3)—22no—
31C2×C5×SL(2,𝔽3)D412yescomplemented
32C2×C5×SL(2,𝔽3)C2311yescharacteristic
33C3×C5×C4⋊C4—41no—
34C3×C5×C22⋊C4—42no—
35C2×2−1+4⋊C3C2×C512yescomplemented
36C4⋊(#41⋊C3)C2×C511yescharacteristic, complemented
37D4×SL(2,𝔽3)C2×C511yescharacteristic, complemented
38#548⋊C3C2×C512yescomplemented
39C2×C4×SL(2,𝔽3)C2×C511yescharacteristic, complemented
40C5×2−1+4—24no—
41C5×2−1+4—64no—
42C2×C5×#41—32no—
43C2×C5×#41—32no—
44C2×C5×#41—32no—
45C2×C5×#41C22×C312yes—
46C22×C5×Q8—32no—
47C22×C5×Q8C22×C312yes—
48C22×C5×Q8C22×C311yescharacteristic
49C5×C4⋊Q8—31no—
50C5×C4⋊Q8—31no—
51C5×#127—31no—
52C5×#127—31no—
53C5×#124—32no—
54C5×#124—32no—
55C5×C22⋊Q8—34no—
56C5×C22⋊Q8—32no—
57C5×C22⋊Q8—32no—
58C5×#121—32no—
59C5×#121—32no—
60C5×#121—34no—
61C5×#121—32no—
62C5×#121—31no—
63C5×#121—32no—
64C5×#121A411yescharacteristic
65C4×C5×Q8—31no—
66C4×C5×Q8—31no—
67C4×C5×Q8C22×C311yescharacteristic
68C4×C5×Q8C22×C311yescharacteristic
69C4×C5×D4—32no—
70C4×C5×D4—32no—
71C4×C5×D4—31no—
72C4×C5×D4—31no—
73C4×C5×D4—32no—
74C2×C5×C4⋊C4—31no—
75C2×C42×C5—31no—
76#3497C3×C511yescharacteristic, complemented, Sylow
77C23×C3×C5—41no—
78C23×C3×C5—42no—
79C3×C5×D4—82no—
80C3×C5×D4—84no—
81C2×C3×C4×C5—41no—
82C2×C3×C4×C5—41no—
83C2×C3×C4×C5—42no—
84C2×C3×C4×C5—82no—
85C5×SL(2,𝔽3)C2×D411yescharacteristic
862−1+4⋊C3—24no—
87C2×#41⋊C3C22×C512yes—
88C22×SL(2,𝔽3)C22×C512yes—
89C22×SL(2,𝔽3)C22×C511yescharacteristic
90C3×#121—41no—
91C4×SL(2,𝔽3)C22×C511yescharacteristic
92C4×SL(2,𝔽3)C22×C511yescharacteristic
93C5×#41—22no—
94C5×#41—64no—
95C5×#41—64no—
96C5×#41—64no—
97C5×#41—64no—
98C5×#41—64no—
99C5×#41—64no—
100C5×#41—62no—
101C2×C5×Q8—22no—
102C2×C5×Q8—32no—
103C2×C5×Q8—31no—
104C2×C5×Q8—32no—
105C2×C5×Q8—32no—
106C2×C5×Q8—64no—
107C2×C5×Q8—64no—
108C2×C5×Q8—62no—
109C2×C5×Q8C3×D412yescomplemented
110C2×C5×Q8C23×C311yescharacteristic
111C2×C5×D4—32no—
112C2×C5×D4—32no—
113C2×C5×D4—32no—
114C2×C5×D4—32no—
115C2×C5×D4—32no—
116C2×C5×D4—31no—
117C2×C5×D4C2×A411yescharacteristic
118C2×C5×D4C2×A412yes—
119C22×C4×C5—31no—
120C22×C4×C5—34no—
121C22×C4×C5—32no—
122C22×C4×C5—31no—
123C22×C4×C5—32no—
124C22×C4×C5C2×A411yescharacteristic
125C5×C4⋊C4—31no—
126C5×C4⋊C4—31no—
127C5×C4⋊C4—32no—
128C5×C4⋊C4—32no—
129C5×C4⋊C4—31no—
130C5×C4⋊C4—32no—
131C5×C4⋊C4C2×A411yescharacteristic
132C5×C22⋊C4—34no—
133C5×C22⋊C4—32no—
134C5×C22⋊C4—32no—
135C5×C22⋊C4—32no—
136C5×C22⋊C4C2×A412yes—
137C42×C5—31no—
138C42×C5—31no—
139C42×C5—31no—
140C42×C5—31no—
141C2×2−1+4C2×C3×C512yescomplemented
142#550—31no—
143#550C2×C3×C511yescharacteristic, complemented
144D4×Q8—31no—
145D4×Q8C2×C3×C511yescharacteristic, complemented
146#548—32no—
147#548C2×C3×C512yescomplemented
148#537—32no—
149#532—31no—
150#532—31no—
151C2×C4×Q8C2×C3×C511yescharacteristic, complemented
152C22×C3×C5—42no—
153C22×C3×C5—41no—
154C22×C3×C5—81no—
155C22×C3×C5—81no—
156C22×C3×C5—82no—
157C22×C3×C5—82no—
158C22×C3×C5—82no—
159C3×C4×C5—42no—
160C3×C4×C5—82no—
161C3×C4×C5—81no—
162C2×C3×D4—42no—
163C2×C3×D4—41no—
164C22×C3×C4—41no—
165#41⋊C3—22no—
166C2×SL(2,𝔽3)—22no—
167C2×SL(2,𝔽3)C5×D412yescomplemented
168C2×SL(2,𝔽3)C23×C511yescharacteristic
169C3×C4⋊C4—41no—
170C3×C22⋊C4—42no—
171C23×C5—32no—
172C23×C5C22×A412yes—
173C23×C5C22×A411yescharacteristic
174C5×Q8—32no—
175C5×Q8—31no—
176C5×Q8—32no—
177C5×Q8—64no—
178C5×Q8—62no—
179C5×Q8—62no—
180C5×Q8C2×C3×D411yescharacteristic
181C5×D4—24no—
182C5×D4—22no—
183C5×D4—64no—
184C5×D4—62no—
185C5×D4—62no—
186C5×D4—64no—
187C5×D4—62no—
188C5×D4—122no—
189C5×D4—122no—
190C2×C4×C5—22no—
191C2×C4×C5—32no—
192C2×C4×C5—32no—
193C2×C4×C5—31no—
194C2×C4×C5—31no—
195C2×C4×C5—31no—
196C2×C4×C5—31no—
197C2×C4×C5—32no—
198C2×C4×C5—62no—
199C2×C4×C5—62no—
200C2×C4×C5—62no—
201C2×C4×C5—62no—
202C2×C4×C5—68no—
203C2×C4×C5—121no—
204C2×C4×C5C22×A411yescharacteristic
205C2×C4×C5C22×A412yes—
206C2×C4×C5C22×A411yescharacteristic
2072−1+4—24no—
2082−1+4—64no—
209C2×#41—32no—
210C2×#41—32no—
211C2×#41—32no—
212C2×#41C22×C3×C512yes—
213C22×Q8—32no—
214C22×Q8C22×C3×C512yes—
215C22×Q8C22×C3×C511yescharacteristic
216C4⋊Q8—31no—
217C4⋊Q8—31no—
218#127—31no—
219#127—31no—
220#124—32no—
221#124—32no—
222C22⋊Q8—34no—
223C22⋊Q8—32no—
224C22⋊Q8—32no—
225#121—32no—
226#121—32no—
227#121—34no—
228#121—32no—
229#121—31no—
230#121—32no—
231#121C5×A411yescharacteristic
232C4×Q8—31no—
233C4×Q8—31no—
234C4×Q8C22×C3×C511yescharacteristic
235C4×Q8C22×C3×C511yescharacteristic
236C4×D4—32no—
237C4×D4—32no—
238C4×D4—31no—
239C4×D4—31no—
240C4×D4—32no—
241C2×C4⋊C4—31no—
242C2×C42—31no—
243C2×C3×C5—41no—
244C2×C3×C5—41no—
245C2×C3×C5—41no—
246C2×C3×C5—82no—
247C2×C3×C5—162no—
248C23×C3—41no—
249C23×C3—42no—
250C3×D4—82no—
251C3×D4—84no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—82no—
256SL(2,𝔽3)C2×C5×D411yescharacteristic
257C22×C5—21no—
258C22×C5—21no—
259C22×C5—22no—
260C22×C5—22no—
261C22×C5—22no—
262C22×C5—62no—
263C22×C5—62no—
264C22×C5—62no—
265C22×C5D4×A412yes—
266C22×C5C23×A411yescharacteristic
267C4×C5—22no—
268C4×C5—21no—
269C4×C5—32no—
270C4×C5—31no—
271C4×C5—31no—
272C4×C5—62no—
273C4×C5—61no—
274C4×C5—61no—
275C4×C52−1+4⋊C312yescomplemented
276#41—22no—
277#41—64no—
278#41—64no—
279#41—64no—
280#41—64no—
281#41—64no—
282#41—64no—
283#41—62no—
284C2×Q8—22no—
285C2×Q8—32no—
286C2×Q8—31no—
287C2×Q8—32no—
288C2×Q8—32no—
289C2×Q8—64no—
290C2×Q8—64no—
291C2×Q8—62no—
292C2×Q8C3×C5×D412yescomplemented
293C2×Q8C23×C3×C511yescharacteristic
294C2×D4—32no—
295C2×D4—32no—
296C2×D4—32no—
297C2×D4—32no—
298C2×D4—32no—
299C2×D4—31no—
300C2×D4C2×C5×A411yescharacteristic
301C2×D4C2×C5×A412yes—
302C22×C4—31no—
303C22×C4—34no—
304C22×C4—32no—
305C22×C4—31no—
306C22×C4—32no—
307C22×C4C2×C5×A411yescharacteristic
308C4⋊C4—31no—
309C4⋊C4—31no—
310C4⋊C4—32no—
311C4⋊C4—32no—
312C4⋊C4—31no—
313C4⋊C4—32no—
314C4⋊C4C2×C5×A411yescharacteristic
315C22⋊C4—34no—
316C22⋊C4—32no—
317C22⋊C4—32no—
318C22⋊C4—32no—
319C22⋊C4C2×C5×A412yes—
320C42—31no—
321C42—31no—
322C42—31no—
323C42—31no—
324C3×C5—41noHall
325C22×C3—42no—
326C22×C3—41no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—82no—
330C22×C3—82no—
331C22×C3—82no—
332C3×C4—42no—
333C3×C4—81no—
334C3×C4—82no—
335C2×C5—22no—
336C2×C5—42no—
337C2×C5—122no—
338C2×C5C2×D4×A411yescharacteristic
339C2×C5C2×2−1+4⋊C311yescharacteristic
340C2×C5#594511yescharacteristic
341C23—32no—
342C23C22×C5×A412yes—
343C23C22×C5×A411yescharacteristic
344Q8—32no—
345Q8—31no—
346Q8—32no—
347Q8—64no—
348Q8—62no—
349Q8—62no—
350Q8C2×C3×C5×D411yescharacteristic
351D4—24no—
352D4—22no—
353D4—64no—
354D4—62no—
355D4—62no—
356D4—64no—
357D4—62no—
358D4—122no—
359D4—122no—
360C2×C4—22no—
361C2×C4—32no—
362C2×C4—32no—
363C2×C4—31no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—32no—
368C2×C4—62no—
369C2×C4—62no—
370C2×C4—62no—
371C2×C4—62no—
372C2×C4—68no—
373C2×C4—121no—
374C2×C4C22×C5×A411yescharacteristic
375C2×C4C22×C5×A412yes—
376C2×C4C22×C5×A411yescharacteristic
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—82no—
381C2×C3—162no—
382C5#3497⋊C311yescharacteristic, direct product factor, Sylow
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—62no—
389C22—62no—
390C22—62no—
391C22C5×D4×A412yes—
392C22C23×C5×A411yescharacteristic
393C4—22no—
394C4—21no—
395C4—31no—
396C4—32no—
397C4—31no—
398C4—62no—
399C4—61no—
400C4—61no—
401C4C5×2−1+4⋊C312yescomplemented
402C3—41noSylow
403C2—22no—
404C2—42no—
405C2—122no—
406C2C2×C5×D4×A411yescharacteristic
407C2C2×C5×2−1+4⋊C311yescharacteristic
408C2C5×#594511yescharacteristic
subgroupG/H
center, socleC22×C5C23×A4
commutator, cosocleC2×Q8C23×C3×C5
FrattiniC22C23×C5×A4
FittingC5×#3497C3
hypercenterC5×#121A4
hypocenterQ8C2×C3×C5×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C5 ◁ C5×#121

Complements

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

NH ≅ G/Ntype
1. C2×C5×2−1+4⋊C3C2semidirect
2. C5×C4⋊(#41⋊C3)C2semidirect
3. C5×D4×SL(2,𝔽3)C2semidirect
4. C5×#548⋊C3C2semidirect
5. C2×C4×C5×SL(2,𝔽3)C2semidirect
6. C5×#3497C3semidirect
14. #3497⋊C3C5direct (N × H)
15. C2×C5×2−1+4C2×C3semidirect
17. C5×#550C2×C3semidirect
19. C5×D4×Q8C2×C3semidirect
21. C5×#548C2×C3semidirect
25. C2×C4×C5×Q8C2×C3semidirect
31. C2×C5×SL(2,𝔽3)D4semidirect
35. C2×2−1+4⋊C3C2×C5semidirect
36. C4⋊(#41⋊C3)C2×C5semidirect
37. D4×SL(2,𝔽3)C2×C5semidirect
38. #548⋊C3C2×C5semidirect
39. C2×C4×SL(2,𝔽3)C2×C5semidirect
76. #3497C3×C5semidirect
109. C2×C5×Q8C3×D4semidirect
141. C2×2−1+4C2×C3×C5semidirect
143. #550C2×C3×C5semidirect
145. D4×Q8C2×C3×C5semidirect
147. #548C2×C3×C5semidirect
151. C2×C4×Q8C2×C3×C5semidirect
167. C2×SL(2,𝔽3)C5×D4semidirect
275. C4×C52−1+4⋊C3semidirect
292. C2×Q8C3×C5×D4semidirect
382. C5#3497⋊C3direct (N × H)
401. C4C5×2−1+4⋊C3semidirect

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×C4⋊(#41⋊C3)2
C5×D4×SL(2,𝔽3)2
C5×#548⋊C32
C2×C4×C5×SL(2,𝔽3)2
C5×#34973
C3×C5×#1214
#3497⋊C35