Finite Group Databasebeta

← C5×#3523⋊C3 #14511358

Subgroups — C5×#3523⋊C3 #14511358

Order 1920.

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

normal subgroups only

This group has 554 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×#557⋊C3C211yesmaximal, characteristic, complemented
2C5×D4×SL(2,𝔽3)C212yesmaximal, complemented
3C5×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C5×#519⋊C3C212yesmaximal, complemented
5C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6C5×#3523C311yesmaximal, characteristic, complemented, Hall
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8C22×C5×SL(2,𝔽3)C2212yes—
9C3×C5×#121—41nomaximal
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C4×C5×SL(2,𝔽3)C2211yescharacteristic
12C4×C5×SL(2,𝔽3)C2212yes—
13#3523⋊C3C511yesmaximal, characteristic, direct product factor, Hall
14C5×#557—31no—
15C5×#557C2×C311yescharacteristic, complemented
16C5×#556—31no—
17C5×#552—32no—
18C5×D4×Q8C2×C312yescomplemented
19C5×D4×Q8C2×C311yescharacteristic, complemented
20C5×#541—32no—
21C5×#533—31no—
22C5×#529—31no—
23C5×#519C2×C312yescomplemented
24C2×C4×C5×Q8C2×C311yescharacteristic, complemented
25C2×C3×C5×D4—41no—
26C2×C3×C5×D4—42no—
27C22×C3×C4×C5—41no—
28C2×C5×SL(2,𝔽3)—21no—
29C2×C5×SL(2,𝔽3)—22no—
30C2×C5×SL(2,𝔽3)C2311yescharacteristic
31C3×C5×C4⋊C4—41no—
32C3×C5×C22⋊C4—42no—
33#557⋊C3C2×C511yescharacteristic, complemented
34D4×SL(2,𝔽3)C2×C511yescharacteristic, complemented
35D4×SL(2,𝔽3)C2×C512yescomplemented
36#519⋊C3C2×C512yescomplemented
37C2×C4×SL(2,𝔽3)C2×C511yescharacteristic, complemented
38C22×C5×Q8C22×C311yescharacteristic
39C22×C5×Q8C22×C312yes—
40C5×C4⋊Q8—31no—
41C5×C4⋊Q8—31no—
42C5×C4⋊Q8—32no—
43C5×#126—34no—
44C5×#126—32no—
45C5×#125—32no—
46C5×#125—32no—
47C5×#125—34no—
48C5×#125—31no—
49C5×#124—31no—
50C5×#123—32no—
51C5×C22⋊Q8—31no—
52C5×C22⋊Q8—32no—
53C5×C22⋊Q8—32no—
54C5×C22⋊Q8—32no—
55C5×#121A411yescharacteristic
56C4×C5×Q8—32no—
57C4×C5×Q8—31no—
58C4×C5×Q8—32no—
59C4×C5×Q8—32no—
60C4×C5×Q8—31no—
61C4×C5×Q8—31no—
62C4×C5×Q8C22×C311yescharacteristic
63C4×C5×Q8C22×C311yescharacteristic
64C4×C5×Q8C22×C312yes—
65C4×C5×D4—32no—
66C4×C5×D4—31no—
67C5×#117—32no—
68C5×#117—32no—
69C5×#117—32no—
70C2×C5×C4⋊C4—31no—
71C2×C42×C5—31no—
72#3523C3×C511yescharacteristic, complemented, Sylow
73C23×C3×C5—41no—
74C23×C3×C5—42no—
75C3×C5×D4—82no—
76C3×C5×D4—84no—
77C2×C3×C4×C5—41no—
78C2×C3×C4×C5—41no—
79C2×C3×C4×C5—42no—
80C2×C3×C4×C5—82no—
81C5×SL(2,𝔽3)#4111yescharacteristic
82C22×SL(2,𝔽3)C22×C511yescharacteristic
83C22×SL(2,𝔽3)C22×C512yes—
84C3×#121—41no—
85C4×SL(2,𝔽3)C22×C511yescharacteristic
86C4×SL(2,𝔽3)C22×C511yescharacteristic
87C4×SL(2,𝔽3)C22×C512yes—
88C2×C5×Q8—21no—
89C2×C5×Q8—22no—
90C2×C5×Q8—31no—
91C2×C5×Q8—32no—
92C2×C5×Q8—31no—
93C2×C5×Q8—61no—
94C2×C5×Q8—62no—
95C2×C5×Q8C23×C311yescharacteristic
96C2×C5×D4C2×A412yes—
97C2×C5×D4C2×A411yescharacteristic
98C22×C4×C5—31no—
99C22×C4×C5—31no—
100C22×C4×C5—32no—
101C22×C4×C5C2×A411yescharacteristic
102C5×C4⋊C4—31no—
103C5×C4⋊C4—31no—
104C5×C4⋊C4—32no—
105C5×C4⋊C4—32no—
106C5×C4⋊C4—32no—
107C5×C4⋊C4—32no—
108C5×C4⋊C4—34no—
109C5×C4⋊C4—32no—
110C5×C4⋊C4—32no—
111C5×C4⋊C4—31no—
112C5×C4⋊C4—32no—
113C5×C4⋊C4—32no—
114C5×C4⋊C4C2×A411yescharacteristic
115C5×C22⋊C4—32no—
116C5×C22⋊C4—32no—
117C5×C22⋊C4—32no—
118C5×C22⋊C4—32no—
119C5×C22⋊C4C2×A412yes—
120C42×C5—34no—
121C42×C5—31no—
122C42×C5—31no—
123C42×C5—31no—
124C42×C5—31no—
125C42×C5—32no—
126#557—31no—
127#557C2×C3×C511yescharacteristic, complemented
128#556—31no—
129#552—32no—
130D4×Q8C2×C3×C512yescomplemented
131D4×Q8C2×C3×C511yescharacteristic, complemented
132#541—32no—
133#533—31no—
134#529—31no—
135#519C2×C3×C512yescomplemented
136C2×C4×Q8C2×C3×C511yescharacteristic, complemented
137C22×C3×C5—42no—
138C22×C3×C5—41no—
139C22×C3×C5—81no—
140C22×C3×C5—81no—
141C22×C3×C5—82no—
142C22×C3×C5—82no—
143C22×C3×C5—82no—
144C3×C4×C5—42no—
145C3×C4×C5—81no—
146C3×C4×C5—82no—
147C2×C3×D4—42no—
148C2×C3×D4—41no—
149C22×C3×C4—41no—
150C2×SL(2,𝔽3)—21no—
151C2×SL(2,𝔽3)—22no—
152C2×SL(2,𝔽3)C23×C511yescharacteristic
153C3×C4⋊C4—41no—
154C3×C22⋊C4—42no—
155C23×C5C22×A412yes—
156C23×C5C22×A411yescharacteristic
157C5×Q8—31no—
158C5×Q8—61no—
159C5×Q8—62no—
160C5×Q8—62no—
161C5×Q8—61no—
162C5×Q8—62no—
163C5×Q8C3×#4111yescharacteristic
164C5×D4—24no—
165C5×D4—22no—
166C2×C4×C5—22no—
167C2×C4×C5—31no—
168C2×C4×C5—31no—
169C2×C4×C5—32no—
170C2×C4×C5—31no—
171C2×C4×C5—32no—
172C2×C4×C5—31no—
173C2×C4×C5—61no—
174C2×C4×C5—62no—
175C2×C4×C5—61no—
176C2×C4×C5—62no—
177C2×C4×C5—62no—
178C2×C4×C5C22×A411yescharacteristic
179C2×C4×C5C22×A412yes—
180C2×C4×C5C22×A411yescharacteristic
181C22×Q8C22×C3×C511yescharacteristic
182C22×Q8C22×C3×C512yes—
183C4⋊Q8—31no—
184C4⋊Q8—31no—
185C4⋊Q8—32no—
186#126—34no—
187#126—32no—
188#125—32no—
189#125—32no—
190#125—34no—
191#125—31no—
192#124—31no—
193#123—32no—
194C22⋊Q8—31no—
195C22⋊Q8—32no—
196C22⋊Q8—32no—
197C22⋊Q8—32no—
198#121C5×A411yescharacteristic
199C4×Q8—32no—
200C4×Q8—31no—
201C4×Q8—32no—
202C4×Q8—32no—
203C4×Q8—31no—
204C4×Q8—31no—
205C4×Q8C22×C3×C511yescharacteristic
206C4×Q8C22×C3×C511yescharacteristic
207C4×Q8C22×C3×C512yes—
208C4×D4—32no—
209C4×D4—31no—
210#117—32no—
211#117—32no—
212#117—32no—
213C2×C4⋊C4—31no—
214C2×C42—31no—
215C2×C3×C5—41no—
216C2×C3×C5—41no—
217C2×C3×C5—41no—
218C2×C3×C5—82no—
219C2×C3×C5—162no—
220C23×C3—41no—
221C23×C3—42no—
222C3×D4—82no—
223C3×D4—84no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—82no—
228SL(2,𝔽3)C5×#4111yescharacteristic
229C22×C5—21no—
230C22×C5—21no—
231C22×C5—22no—
232C22×C5—22no—
233C22×C5—22no—
234C22×C52−1+4⋊C312yes—
235C22×C5C23×A411yescharacteristic
236C4×C5—21no—
237C4×C5—22no—
238C4×C5—31no—
239C4×C5—31no—
240C4×C5—62no—
241C4×C5—61no—
242C4×C5—61no—
243C4×C5—61no—
244C4×C5—62no—
245C4×C52−1+4⋊C312yes—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—31no—
249C2×Q8—32no—
250C2×Q8—31no—
251C2×Q8—61no—
252C2×Q8—62no—
253C2×Q8C23×C3×C511yescharacteristic
254C2×D4C2×C5×A412yes—
255C2×D4C2×C5×A411yescharacteristic
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—32no—
259C22×C4C2×C5×A411yescharacteristic
260C4⋊C4—31no—
261C4⋊C4—31no—
262C4⋊C4—32no—
263C4⋊C4—32no—
264C4⋊C4—32no—
265C4⋊C4—32no—
266C4⋊C4—34no—
267C4⋊C4—32no—
268C4⋊C4—32no—
269C4⋊C4—31no—
270C4⋊C4—32no—
271C4⋊C4—32no—
272C4⋊C4C2×C5×A411yescharacteristic
273C22⋊C4—32no—
274C22⋊C4—32no—
275C22⋊C4—32no—
276C22⋊C4—32no—
277C22⋊C4C2×C5×A412yes—
278C42—34no—
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—
289C22×C3—82no—
290C22×C3—82no—
291C22×C3—82no—
292C3×C4—42no—
293C3×C4—81no—
294C3×C4—82no—
295C2×C5—22no—
296C2×C5—42no—
297C2×C5A4×#4111yescharacteristic
298C2×C5C2×2−1+4⋊C311yescharacteristic
299C2×C5C2×2−1+4⋊C311yescharacteristic
300C23C22×C5×A412yes—
301C23C22×C5×A411yescharacteristic
302Q8—31no—
303Q8—61no—
304Q8—62no—
305Q8—62no—
306Q8—61no—
307Q8—62no—
308Q8C3×C5×#4111yescharacteristic
309D4—24no—
310D4—22no—
311C2×C4—22no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—32no—
315C2×C4—31no—
316C2×C4—32no—
317C2×C4—31no—
318C2×C4—61no—
319C2×C4—62no—
320C2×C4—61no—
321C2×C4—62no—
322C2×C4—62no—
323C2×C4C22×C5×A411yescharacteristic
324C2×C4C22×C5×A412yes—
325C2×C4C22×C5×A411yescharacteristic
326C2×C3—41no—
327C2×C3—41no—
328C2×C3—41no—
329C2×C3—82no—
330C2×C3—162no—
331C5#3523⋊C311yescharacteristic, direct product factor, Sylow
332C22—21no—
333C22—21no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22C5×2−1+4⋊C312yes—
338C22C23×C5×A411yescharacteristic
339C4—21no—
340C4—22no—
341C4—31no—
342C4—31no—
343C4—62no—
344C4—61no—
345C4—61no—
346C4—61no—
347C4—62no—
348C4C5×2−1+4⋊C312yes—
349C3—41noSylow
350C2—22no—
351C2—42no—
352C2C5×A4×#4111yescharacteristic
353C2C2×C5×2−1+4⋊C311yescharacteristic
354C2C2×C5×2−1+4⋊C311yescharacteristic
subgroupG/H
center, socleC22×C5C23×A4
commutator, cosocleC2×Q8C23×C3×C5
FrattiniC22C23×C5×A4
FittingC5×#3523C3
hypercenterC5×#121A4
hypocenterQ8C3×C5×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C5 ◁ C5×#121

Complements

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

NH ≅ G/Ntype
1. C5×#557⋊C3C2semidirect
2. C5×D4×SL(2,𝔽3)C2semidirect
3. C5×D4×SL(2,𝔽3)C2semidirect
4. C5×#519⋊C3C2semidirect
5. C2×C4×C5×SL(2,𝔽3)C2semidirect
6. C5×#3523C3semidirect
13. #3523⋊C3C5direct (N × H)
15. C5×#557C2×C3semidirect
18. C5×D4×Q8C2×C3semidirect
19. C5×D4×Q8C2×C3semidirect
23. C5×#519C2×C3semidirect
24. C2×C4×C5×Q8C2×C3semidirect
33. #557⋊C3C2×C5semidirect
34. D4×SL(2,𝔽3)C2×C5semidirect
35. D4×SL(2,𝔽3)C2×C5semidirect
36. #519⋊C3C2×C5semidirect
37. C2×C4×SL(2,𝔽3)C2×C5semidirect
72. #3523C3×C5semidirect
127. #557C2×C3×C5semidirect
130. D4×Q8C2×C3×C5semidirect
131. D4×Q8C2×C3×C5semidirect
135. #519C2×C3×C5semidirect
136. C2×C4×Q8C2×C3×C5semidirect
331. C5#3523⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#557⋊C32
C5×D4×SL(2,𝔽3)2
C5×#519⋊C32
C2×C4×C5×SL(2,𝔽3)2
C5×#35233
C3×C5×#1214
#3523⋊C35