Finite Group Databasebeta

← C5×#3522⋊C3 #14511357

Subgroups — C5×#3522⋊C3 #14511357

Order 1920.

558 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C5×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C5×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3C5×#548⋊C3C212yesmaximal, complemented
4C5×#520⋊C3C212yesmaximal, complemented
5C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6C5×#3522C311yesmaximal, characteristic, complemented, Hall
7C2×C5×#41⋊C3C2212yes—
8C22×C5×SL(2,𝔽3)C2211yescharacteristic
9C3×C5×C22⋊Q8—41nomaximal
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C4×C5×SL(2,𝔽3)C2211yescharacteristic
12C4×C5×SL(2,𝔽3)C2212yes—
13#3522⋊C3C511yesmaximal, characteristic, direct product factor, Hall
14C5×#559—31no—
15C5×Q82C2×C311yescharacteristic, complemented
16C5×#554—31no—
17C5×#550C2×C311yescharacteristic, complemented
18C5×#548C2×C312yes—
19C5×#547—32no—
20C5×#542—32no—
21C5×#532—31no—
22C5×#529—31no—
23C5×#520C2×C312yescomplemented
24C2×C4×C5×Q8C2×C311yescharacteristic
25C2×C3×C5×Q8—41no—
26C22×C3×C4×C5—41no—
27C5×#41⋊C3—22no—
28C2×C5×SL(2,𝔽3)—21no—
29C2×C5×SL(2,𝔽3)C2311yescharacteristic
30C3×C5×C4⋊C4—42no—
31C3×C5×C4⋊C4—41no—
32C3×C5×C22⋊C4—42no—
33Q8×SL(2,𝔽3)C2×C511yescharacteristic, complemented
34C4⋊(#41⋊C3)C2×C511yescharacteristic, complemented
35#548⋊C3C2×C512yescomplemented
36#520⋊C3C2×C512yescomplemented
37C2×C4×SL(2,𝔽3)C2×C511yescharacteristic, complemented
38C2×C5×#41—32no—
39C2×C5×#41C22×C312yes—
40C22×C5×Q8C22×C311yescharacteristic
41C5×C4⋊Q8—31no—
42C5×C4⋊Q8—32no—
43C5×#127—31no—
44C5×#126—32no—
45C5×#125—31no—
46C5×#124—32no—
47C5×#124—31no—
48C5×#124—32no—
49C5×#123—34no—
50C5×#123—32no—
51C5×C22⋊Q8—34no—
52C5×C22⋊Q8—32no—
53C5×C22⋊Q8A411yescharacteristic
54C5×#121—32no—
55C5×#121—31no—
56C5×#121—32no—
57C4×C5×Q8—31no—
58C4×C5×Q8—31no—
59C4×C5×Q8—31no—
60C4×C5×Q8C22×C311yescharacteristic
61C4×C5×Q8C22×C311yescharacteristic
62C4×C5×Q8C22×C312yes—
63C4×C5×D4—32no—
64C4×C5×D4—31no—
65C4×C5×D4—32no—
66C4×C5×D4—32no—
67C4×C5×D4—32no—
68C5×#117—32no—
69C2×C5×C4⋊C4—31no—
70C2×C5×C4⋊C4—32no—
71C2×C42×C5—31no—
72#3522C3×C511yescharacteristic, complemented, Sylow
73C23×C3×C5—41no—
74C3×C5×Q8—82no—
75C2×C3×C4×C5—41no—
76C2×C3×C4×C5—42no—
77C2×C3×C4×C5—42no—
78C2×C3×C4×C5—41no—
79C2×C3×C4×C5—82no—
80C5×SL(2,𝔽3)#4111yescharacteristic, complemented
81C2×#41⋊C3C22×C512yes—
82C22×SL(2,𝔽3)C22×C511yescharacteristic
83C3×C22⋊Q8—41no—
84C4×SL(2,𝔽3)C22×C511yescharacteristic
85C4×SL(2,𝔽3)C22×C511yescharacteristic
86C4×SL(2,𝔽3)C22×C512yes—
87C5×#41—22no—
88C5×#41—62no—
89C5×#41—124no—
90C2×C5×Q8—21no—
91C2×C5×Q8—32no—
92C2×C5×Q8—31no—
93C2×C5×Q8—61no—
94C2×C5×Q8C2×A411yescharacteristic
95C2×C5×Q8C23×C311yescharacteristic
96C2×C5×D4—32no—
97C2×C5×D4—32no—
98C2×C5×D4—32no—
99C2×C5×D4—31no—
100C22×C4×C5—31no—
101C22×C4×C5—31no—
102C22×C4×C5—34no—
103C22×C4×C5—32no—
104C22×C4×C5C2×A411yescharacteristic
105C5×C4⋊C4—31no—
106C5×C4⋊C4—32no—
107C5×C4⋊C4—31no—
108C5×C4⋊C4—32no—
109C5×C4⋊C4—32no—
110C5×C4⋊C4—31no—
111C5×C4⋊C4—32no—
112C5×C4⋊C4—32no—
113C5×C4⋊C4C2×A412yes—
114C5×C4⋊C4C2×A411yescharacteristic
115C5×C22⋊C4—34no—
116C5×C22⋊C4—32no—
117C5×C22⋊C4—32no—
118C5×C22⋊C4—32no—
119C5×C22⋊C4—32no—
120C5×C22⋊C4C2×A412yes—
121C42×C5—31no—
122C42×C5—31no—
123C42×C5—31no—
124C42×C5—31no—
125C42×C5—32no—
126#559—31no—
127Q82C2×C3×C511yescharacteristic, complemented
128#554—31no—
129#550C2×C3×C511yescharacteristic, complemented
130#548C2×C3×C512yes—
131#547—32no—
132#542—32no—
133#532—31no—
134#529—31no—
135#520C2×C3×C512yescomplemented
136C2×C4×Q8C2×C3×C511yescharacteristic
137C22×C3×C5—42no—
138C22×C3×C5—41no—
139C22×C3×C5—81no—
140C22×C3×C5—81no—
141C3×C4×C5—42no—
142C3×C4×C5—82no—
143C3×C4×C5—81no—
144C3×C4×C5—82no—
145C2×C3×Q8—41no—
146C22×C3×C4—41no—
147#41⋊C3—22no—
148C2×SL(2,𝔽3)—21no—
149C2×SL(2,𝔽3)C23×C511yescharacteristic
150C3×C4⋊C4—42no—
151C3×C4⋊C4—41no—
152C3×C22⋊C4—42no—
153C23×C5—32no—
154C23×C5C22×A411yescharacteristic
155C5×Q8—22no—
156C5×Q8—31no—
157C5×Q8—61no—
158C5×Q8—61no—
159C5×Q8—122no—
160C5×Q8C3×#4111yescharacteristic
161C5×D4—62no—
162C5×D4—62no—
163C5×D4—64no—
164C5×D4—62no—
165C5×D4—122no—
166C2×C4×C5—22no—
167C2×C4×C5—31no—
168C2×C4×C5—31no—
169C2×C4×C5—31no—
170C2×C4×C5—31no—
171C2×C4×C5—32no—
172C2×C4×C5—62no—
173C2×C4×C5—62no—
174C2×C4×C5—61no—
175C2×C4×C5—61no—
176C2×C4×C5—62no—
177C2×C4×C5—124no—
178C2×C4×C5C22×A411yescharacteristic
179C2×C4×C5C22×A412yes—
180C2×C4×C5C22×A412yes—
181C2×C4×C5C22×A411yescharacteristic
182C2×#41—32no—
183C2×#41C22×C3×C512yes—
184C22×Q8C22×C3×C511yescharacteristic
185C4⋊Q8—31no—
186C4⋊Q8—32no—
187#127—31no—
188#126—32no—
189#125—31no—
190#124—32no—
191#124—31no—
192#124—32no—
193#123—34no—
194#123—32no—
195C22⋊Q8—34no—
196C22⋊Q8—32no—
197C22⋊Q8C5×A411yescharacteristic
198#121—32no—
199#121—31no—
200#121—32no—
201C4×Q8—31no—
202C4×Q8—31no—
203C4×Q8—31no—
204C4×Q8C22×C3×C511yescharacteristic
205C4×Q8C22×C3×C511yescharacteristic
206C4×Q8C22×C3×C512yes—
207C4×D4—32no—
208C4×D4—31no—
209C4×D4—32no—
210C4×D4—32no—
211C4×D4—32no—
212#117—32no—
213C2×C4⋊C4—31no—
214C2×C4⋊C4—32no—
215C2×C42—31no—
216C2×C3×C5—41no—
217C2×C3×C5—41no—
218C2×C3×C5—41no—
219C2×C3×C5—82no—
220C23×C3—41no—
221C3×Q8—82no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—82no—
227SL(2,𝔽3)C5×#4111yescharacteristic, complemented
228C22×C5—21no—
229C22×C5—21no—
230C22×C5—62no—
231C22×C5—62no—
232C22×C5—62no—
233C22×C5#99412yes—
234C22×C5C23×A411yescharacteristic
235C4×C5—22no—
236C4×C5—21no—
237C4×C5—22no—
238C4×C5—31no—
239C4×C5—31no—
240C4×C5—61no—
241C4×C5—61no—
242C4×C5—61no—
243C4×C5—62no—
244C4×C52−1+4⋊C312yes—
245#41—22no—
246#41—62no—
247#41—124no—
248C2×Q8—21no—
249C2×Q8—32no—
250C2×Q8—31no—
251C2×Q8—61no—
252C2×Q8C2×C5×A411yescharacteristic
253C2×Q8C23×C3×C511yescharacteristic
254C2×D4—32no—
255C2×D4—32no—
256C2×D4—32no—
257C2×D4—31no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—34no—
261C22×C4—32no—
262C22×C4C2×C5×A411yescharacteristic
263C4⋊C4—31no—
264C4⋊C4—32no—
265C4⋊C4—31no—
266C4⋊C4—32no—
267C4⋊C4—32no—
268C4⋊C4—31no—
269C4⋊C4—32no—
270C4⋊C4—32no—
271C4⋊C4C2×C5×A412yes—
272C4⋊C4C2×C5×A411yescharacteristic
273C22⋊C4—34no—
274C22⋊C4—32no—
275C22⋊C4—32no—
276C22⋊C4—32no—
277C22⋊C4—32no—
278C22⋊C4C2×C5×A412yes—
279C42—31no—
280C42—31no—
281C42—31no—
282C42—31no—
283C42—32no—
284C3×C5—41noHall
285C22×C3—42no—
286C22×C3—41no—
287C22×C3—81no—
288C22×C3—81no—
289C3×C4—42no—
290C3×C4—81no—
291C3×C4—82no—
292C3×C4—82no—
293C2×C5—22no—
294C2×C5—122no—
295C2×C5A4×#4111yescharacteristic
296C2×C5C2×#99411yescharacteristic
297C2×C5C2×2−1+4⋊C311yescharacteristic
298C23—32no—
299C23C22×C5×A411yescharacteristic
300Q8—22no—
301Q8—31no—
302Q8—61no—
303Q8—61no—
304Q8—122no—
305Q8C3×C5×#4111yescharacteristic
306D4—62no—
307D4—62no—
308D4—64no—
309D4—62no—
310D4—122no—
311C2×C4—22no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—31no—
316C2×C4—32no—
317C2×C4—62no—
318C2×C4—62no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—62no—
322C2×C4—124no—
323C2×C4C22×C5×A411yescharacteristic
324C2×C4C22×C5×A412yes—
325C2×C4C22×C5×A412yes—
326C2×C4C22×C5×A411yescharacteristic
327C2×C3—41no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—82no—
331C5#3522⋊C311yescharacteristic, direct product factor, Sylow
332C22—21no—
333C22—21no—
334C22—62no—
335C22—62no—
336C22—62no—
337C22C5×#99412yes—
338C22C23×C5×A411yescharacteristic
339C4—22no—
340C4—21no—
341C4—22no—
342C4—31no—
343C4—31no—
344C4—61no—
345C4—61no—
346C4—61no—
347C4—62no—
348C4C5×2−1+4⋊C312yes—
349C3—41noSylow
350C2—22no—
351C2—122no—
352C2C5×A4×#4111yescharacteristic
353C2C2×C5×#99411yescharacteristic
354C2C2×C5×2−1+4⋊C311yescharacteristic
subgroupG/H
center, socleC22×C5C23×A4
commutator, cosocleC2×Q8C23×C3×C5
FrattiniC22C23×C5×A4
FittingC5×#3522C3
hypercenterC5×C22⋊Q8A4
hypocenterQ8C3×C5×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C5 ◁ C5×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C5×Q8×SL(2,𝔽3)C2semidirect
2. C5×C4⋊(#41⋊C3)C2semidirect
3. C5×#548⋊C3C2semidirect
4. C5×#520⋊C3C2semidirect
5. C2×C4×C5×SL(2,𝔽3)C2semidirect
6. C5×#3522C3semidirect
13. #3522⋊C3C5direct (N × H)
15. C5×Q82C2×C3semidirect
17. C5×#550C2×C3semidirect
23. C5×#520C2×C3semidirect
33. Q8×SL(2,𝔽3)C2×C5semidirect
34. C4⋊(#41⋊C3)C2×C5semidirect
35. #548⋊C3C2×C5semidirect
36. #520⋊C3C2×C5semidirect
37. C2×C4×SL(2,𝔽3)C2×C5semidirect
72. #3522C3×C5semidirect
80. C5×SL(2,𝔽3)#41semidirect
127. Q82C2×C3×C5semidirect
129. #550C2×C3×C5semidirect
135. #520C2×C3×C5semidirect
227. SL(2,𝔽3)C5×#41semidirect
331. C5#3522⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C5×Q8×SL(2,𝔽3)2
C5×C4⋊(#41⋊C3)2
C5×#548⋊C32
C5×#520⋊C32
C2×C4×C5×SL(2,𝔽3)2
C5×#35223
C3×C5×C22⋊Q84
#3522⋊C35