Finite Group Databasebeta

← C5×#3487⋊C3 #14511344

Subgroups — C5×#3487⋊C3 #14511344

Order 1920.

732 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 120 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
2C22×C5×#41⋊C3C211yesmaximal, characteristic, complemented
3C5×C4⋊(#41⋊C3)C212yesmaximal, complemented
4C5×#548⋊C3C212yesmaximal, complemented
5C5×#519⋊C3C211yesmaximal, characteristic, complemented
6C5×#3487C311yesmaximal, characteristic, complemented, Hall
7C5×2−1+4⋊C3—22no—
8C2×C5×#41⋊C3—22no—
9C2×C5×#41⋊C3C2212yes—
10C2×C5×#41⋊C3C2211yescharacteristic, complemented
11C22×C5×SL(2,𝔽3)C2211yescharacteristic
12C22×C5×SL(2,𝔽3)C2211yescharacteristic, complemented
13C3×C5×#123—41nomaximal
14C4×C5×SL(2,𝔽3)C2212yescomplemented
15#3487⋊C3C511yesmaximal, characteristic, direct product factor, Hall
16C2×C5×2−1+4C2×C311yescharacteristic
17C22×C5×#41C2×C311yescharacteristic, complemented
18C5×#550—32no—
19C5×#550C2×C312yescomplemented
20C5×#548—32no—
21C5×#548C2×C312yescomplemented
22C5×#535—31no—
23C5×#535—31no—
24C5×#525—31no—
25C5×#525—31no—
26C5×#519C2×C311yescharacteristic, complemented
27C2×C3×C5×D4—41no—
28C22×C3×C4×C5—41no—
29C5×#41⋊C3—22no—
30C5×#41⋊C3—21no—
31C2×C5×SL(2,𝔽3)—21no—
32C2×C5×SL(2,𝔽3)D412yescomplemented
33C2×C5×SL(2,𝔽3)C2311yescharacteristic
34C3×C5×C4⋊C4—42no—
35C3×C5×C22⋊C4—41no—
36C3×C5×C22⋊C4—42no—
37C2×2−1+4⋊C3C2×C511yescharacteristic, complemented
38C22×#41⋊C3C2×C511yescharacteristic, complemented
39C4⋊(#41⋊C3)C2×C512yescomplemented
40#548⋊C3C2×C512yescomplemented
41#519⋊C3C2×C511yescharacteristic, complemented
42C5×2−1+4—22no—
43C5×2−1+4—62no—
44C2×C5×#41—22no—
45C2×C5×#41—31no—
46C2×C5×#41—31no—
47C2×C5×#41—31no—
48C2×C5×#41—32no—
49C2×C5×#41—62no—
50C2×C5×#41C22×C312yes—
51C2×C5×#41C22×C311yescharacteristic
52C22×C5×Q8—31no—
53C22×C5×Q8C22×C311yescharacteristic
54C22×C5×Q8C22×C311yescharacteristic
55C22×C5×D4—31no—
56C23×C4×C5—31no—
57C5×#127—32no—
58C5×#127—32no—
59C5×#124—32no—
60C5×#124—32no—
61C5×#123—31no—
62C5×#123A411yescharacteristic
63C5×C22⋊Q8—32no—
64C5×C22⋊Q8—32no—
65C5×#121—32no—
66C5×#121—34no—
67C5×#121—32no—
68C5×#121—32no—
69C5×#121—32no—
70C5×#120—32no—
71C5×#120—31no—
72C5×#120—31no—
73C4×C5×Q8—32no—
74C4×C5×Q8C22×C312yes—
75C4×C5×D4—34no—
76C4×C5×D4—32no—
77C4×C5×D4—32no—
78C5×#117—31no—
79C5×#117—31no—
80#3487C3×C511yescharacteristic, complemented, Sylow
81C23×C3×C5—41no—
82C23×C3×C5—41no—
83C3×C5×D4—82no—
84C2×C3×C4×C5—42no—
85C2×C3×C4×C5—42no—
86C2×C3×C4×C5—41no—
87C2×C3×C4×C5—82no—
88C5×SL(2,𝔽3)C2×D411yescharacteristic
892−1+4⋊C3—22no—
90C2×#41⋊C3—22no—
91C2×#41⋊C3C22×C512yes—
92C2×#41⋊C3C22×C511yescharacteristic, complemented
93C22×SL(2,𝔽3)C22×C511yescharacteristic
94C22×SL(2,𝔽3)C22×C511yescharacteristic, complemented
95C3×#123—41no—
96C4×SL(2,𝔽3)C22×C512yescomplemented
97C24×C5—31no—
98C5×#41—22no—
99C5×#41—21no—
100C5×#41—62no—
101C5×#41—62no—
102C5×#41—62no—
103C5×#41—62no—
104C5×#41—64no—
105C5×#41—62no—
106C5×#41—62no—
107C5×#41—64no—
108C5×#41—62no—
109C5×#41—61no—
110C2×C5×Q8—21no—
111C2×C5×Q8—31no—
112C2×C5×Q8—31no—
113C2×C5×Q8—32no—
114C2×C5×Q8—31no—
115C2×C5×Q8—62no—
116C2×C5×Q8—62no—
117C2×C5×Q8—61no—
118C2×C5×Q8C3×D412yes—
119C2×C5×Q8C23×C311yescharacteristic
120C2×C5×D4—32no—
121C2×C5×D4—31no—
122C2×C5×D4—32no—
123C2×C5×D4—31no—
124C2×C5×D4—31no—
125C2×C5×D4—34no—
126C2×C5×D4—32no—
127C2×C5×D4—32no—
128C2×C5×D4—62no—
129C2×C5×D4—62no—
130C2×C5×D4C2×A411yescharacteristic
131C22×C4×C5—31no—
132C22×C4×C5—32no—
133C22×C4×C5—31no—
134C22×C4×C5—31no—
135C22×C4×C5—32no—
136C22×C4×C5—32no—
137C22×C4×C5—62no—
138C22×C4×C5—62no—
139C22×C4×C5C2×A411yescharacteristic
140C5×C4⋊C4—32no—
141C5×C4⋊C4—32no—
142C5×C4⋊C4—32no—
143C5×C4⋊C4C2×A412yes—
144C5×C22⋊C4—34no—
145C5×C22⋊C4—31no—
146C5×C22⋊C4—32no—
147C5×C22⋊C4—32no—
148C5×C22⋊C4—32no—
149C5×C22⋊C4C2×A411yescharacteristic
150C5×C22⋊C4C2×A412yes—
151C42×C5—32no—
152C42×C5—32no—
153C2×2−1+4C2×C3×C511yescharacteristic
154C22×#41C2×C3×C511yescharacteristic, complemented
155#550—32no—
156#550C2×C3×C512yescomplemented
157#548—32no—
158#548C2×C3×C512yescomplemented
159#535—31no—
160#535—31no—
161#525—31no—
162#525—31no—
163#519C2×C3×C511yescharacteristic, complemented
164C22×C3×C5—42no—
165C22×C3×C5—41no—
166C22×C3×C5—82no—
167C22×C3×C5—81no—
168C22×C3×C5—82no—
169C3×C4×C5—82no—
170C3×C4×C5—81no—
171C3×C4×C5—82no—
172C2×C3×D4—41no—
173C22×C3×C4—41no—
174#41⋊C3—22no—
175#41⋊C3—21no—
176C2×SL(2,𝔽3)—21no—
177C2×SL(2,𝔽3)C5×D412yescomplemented
178C2×SL(2,𝔽3)C23×C511yescharacteristic
179C3×C4⋊C4—42no—
180C3×C22⋊C4—41no—
181C3×C22⋊C4—42no—
182C23×C5—32no—
183C23×C5—31no—
184C23×C5—62no—
185C23×C5—122no—
186C23×C5C22×A411yescharacteristic
187C23×C5C22×A411yescharacteristic
188C5×Q8—32no—
189C5×Q8—31no—
190C5×Q8—32no—
191C5×Q8—62no—
192C5×Q8—61no—
193C5×Q8—61no—
194C5×Q8C2×C3×D411yescharacteristic
195C5×D4—22no—
196C5×D4—62no—
197C5×D4—62no—
198C5×D4—62no—
199C5×D4—61no—
200C5×D4—61no—
201C5×D4—64no—
202C5×D4—62no—
203C5×D4—122no—
204C5×D4—122no—
205C5×D4—124no—
206C2×C4×C5—22no—
207C2×C4×C5—32no—
208C2×C4×C5—32no—
209C2×C4×C5—31no—
210C2×C4×C5—31no—
211C2×C4×C5—32no—
212C2×C4×C5—31no—
213C2×C4×C5—62no—
214C2×C4×C5—64no—
215C2×C4×C5—62no—
216C2×C4×C5—61no—
217C2×C4×C5—61no—
218C2×C4×C5—61no—
219C2×C4×C5—61no—
220C2×C4×C5—64no—
221C2×C4×C5C22×A412yes—
222C2×C4×C5C22×A411yescharacteristic
223C2×C4×C5C22×A412yes—
2242−1+4—22no—
2252−1+4—62no—
226C2×#41—22no—
227C2×#41—31no—
228C2×#41—31no—
229C2×#41—31no—
230C2×#41—32no—
231C2×#41—62no—
232C2×#41C22×C3×C512yes—
233C2×#41C22×C3×C511yescharacteristic
234C22×Q8—31no—
235C22×Q8C22×C3×C511yescharacteristic
236C22×Q8C22×C3×C511yescharacteristic
237C22×D4—31no—
238C23×C4—31no—
239#127—32no—
240#127—32no—
241#124—32no—
242#124—32no—
243#123—31no—
244#123C5×A411yescharacteristic
245C22⋊Q8—32no—
246C22⋊Q8—32no—
247#121—32no—
248#121—34no—
249#121—32no—
250#121—32no—
251#121—32no—
252#120—32no—
253#120—31no—
254#120—31no—
255C4×Q8—32no—
256C4×Q8C22×C3×C512yes—
257C4×D4—34no—
258C4×D4—32no—
259C4×D4—32no—
260#117—31no—
261#117—31no—
262C2×C3×C5—41no—
263C2×C3×C5—42no—
264C2×C3×C5—82no—
265C2×C3×C5—161no—
266C23×C3—41no—
267C23×C3—41no—
268C3×D4—82no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—82no—
273SL(2,𝔽3)C2×C5×D411yescharacteristic
274C22×C5—22no—
275C22×C5—21no—
276C22×C5—22no—
277C22×C5—61no—
278C22×C5—62no—
279C22×C5—62no—
280C22×C5—62no—
281C22×C5—62no—
282C22×C5—124no—
283C22×C5D4×A412yes—
284C22×C5C23×A411yescharacteristic
285C4×C5—22no—
286C4×C5—21no—
287C4×C5—22no—
288C4×C5—32no—
289C4×C5—31no—
290C4×C5—31no—
291C4×C5—61no—
292C4×C5—62no—
293C24—31no—
294#41—22no—
295#41—21no—
296#41—62no—
297#41—62no—
298#41—62no—
299#41—62no—
300#41—64no—
301#41—62no—
302#41—62no—
303#41—64no—
304#41—62no—
305#41—61no—
306C2×Q8—21no—
307C2×Q8—31no—
308C2×Q8—31no—
309C2×Q8—32no—
310C2×Q8—31no—
311C2×Q8—62no—
312C2×Q8—62no—
313C2×Q8—61no—
314C2×Q8C3×C5×D412yes—
315C2×Q8C23×C3×C511yescharacteristic
316C2×D4—32no—
317C2×D4—31no—
318C2×D4—32no—
319C2×D4—31no—
320C2×D4—31no—
321C2×D4—34no—
322C2×D4—32no—
323C2×D4—32no—
324C2×D4—62no—
325C2×D4—62no—
326C2×D4C2×C5×A411yescharacteristic
327C22×C4—31no—
328C22×C4—32no—
329C22×C4—31no—
330C22×C4—31no—
331C22×C4—32no—
332C22×C4—32no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4C2×C5×A411yescharacteristic
336C4⋊C4—32no—
337C4⋊C4—32no—
338C4⋊C4—32no—
339C4⋊C4C2×C5×A412yes—
340C22⋊C4—34no—
341C22⋊C4—31no—
342C22⋊C4—32no—
343C22⋊C4—32no—
344C22⋊C4—32no—
345C22⋊C4C2×C5×A411yescharacteristic
346C22⋊C4C2×C5×A412yes—
347C42—32no—
348C42—32no—
349C3×C5—41noHall
350C22×C3—42no—
351C22×C3—41no—
352C22×C3—82no—
353C22×C3—81no—
354C22×C3—82no—
355C3×C4—82no—
356C3×C4—82no—
357C3×C4—81no—
358C2×C5—22no—
359C2×C5—41no—
360C2×C5—121no—
361C2×C5—122no—
362C2×C5C2×D4×A411yescharacteristic
363C2×C5#594512yes—
364C23—32no—
365C23—31no—
366C23—62no—
367C23—122no—
368C23C22×C5×A411yescharacteristic
369C23C22×C5×A411yescharacteristic
370Q8—32no—
371Q8—31no—
372Q8—32no—
373Q8—62no—
374Q8—61no—
375Q8—61no—
376Q8C2×C3×C5×D411yescharacteristic
377D4—22no—
378D4—62no—
379D4—62no—
380D4—62no—
381D4—61no—
382D4—61no—
383D4—64no—
384D4—62no—
385D4—122no—
386D4—122no—
387D4—124no—
388C2×C4—22no—
389C2×C4—32no—
390C2×C4—32no—
391C2×C4—31no—
392C2×C4—31no—
393C2×C4—32no—
394C2×C4—31no—
395C2×C4—62no—
396C2×C4—62no—
397C2×C4—64no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—64no—
403C2×C4C22×C5×A412yes—
404C2×C4C22×C5×A411yescharacteristic
405C2×C4C22×C5×A412yes—
406C2×C3—41no—
407C2×C3—42no—
408C2×C3—82no—
409C2×C3—161no—
410C5#3487⋊C311yescharacteristic, direct product factor, Sylow
411C22—22no—
412C22—21no—
413C22—22no—
414C22—61no—
415C22—62no—
416C22—62no—
417C22—62no—
418C22—62no—
419C22—124no—
420C22C5×D4×A412yes—
421C22C23×C5×A411yescharacteristic
422C4—22no—
423C4—21no—
424C4—22no—
425C4—31no—
426C4—32no—
427C4—31no—
428C4—61no—
429C4—62no—
430C3—41noSylow
431C2—22no—
432C2—41no—
433C2—122no—
434C2—121no—
435C2C2×C5×D4×A411yescharacteristic
436C2C5×#594512yes—
subgroupG/H
center, socleC22×C5C23×A4
commutator, cosocleC2×Q8C23×C3×C5
FrattiniC22C23×C5×A4
FittingC5×#3487C3
hypercenterC5×#123A4
hypocenterQ8C2×C3×C5×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C5 ◁ C5×#123

Complements

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

NH ≅ G/Ntype
1. C2×C5×2−1+4⋊C3C2semidirect
2. C22×C5×#41⋊C3C2semidirect
3. C5×C4⋊(#41⋊C3)C2semidirect
4. C5×#548⋊C3C2semidirect
5. C5×#519⋊C3C2semidirect
6. C5×#3487C3semidirect
10. C2×C5×#41⋊C3C22semidirect
12. C22×C5×SL(2,𝔽3)C22semidirect
14. C4×C5×SL(2,𝔽3)C22semidirect
15. #3487⋊C3C5direct (N × H)
17. C22×C5×#41C2×C3semidirect
19. C5×#550C2×C3semidirect
21. C5×#548C2×C3semidirect
26. C5×#519C2×C3semidirect
32. C2×C5×SL(2,𝔽3)D4semidirect
37. C2×2−1+4⋊C3C2×C5semidirect
38. C22×#41⋊C3C2×C5semidirect
39. C4⋊(#41⋊C3)C2×C5semidirect
40. #548⋊C3C2×C5semidirect
41. #519⋊C3C2×C5semidirect
80. #3487C3×C5semidirect
92. C2×#41⋊C3C22×C5semidirect
94. C22×SL(2,𝔽3)C22×C5semidirect
96. C4×SL(2,𝔽3)C22×C5semidirect
154. C22×#41C2×C3×C5semidirect
156. #550C2×C3×C5semidirect
158. #548C2×C3×C5semidirect
163. #519C2×C3×C5semidirect
177. C2×SL(2,𝔽3)C5×D4semidirect
410. C5#3487⋊C3direct (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
C22×C5×#41⋊C32
C5×C4⋊(#41⋊C3)2
C5×#548⋊C32
C5×#519⋊C32
C5×#34873
C3×C5×#1234
#3487⋊C35