Finite Group Databasebeta

← C5×#3502⋊C3 #14511351

Subgroups — C5×#3502⋊C3 #14511351

Order 1920.

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

normal subgroups only

This group has 508 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
1C5×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C5×#557⋊C3C212yesmaximal, complemented
3C5×#557⋊C3C211yesmaximal, characteristic, complemented
4C5×#519⋊C3C212yesmaximal
5C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic
6C5×#3502C311yesmaximal, characteristic, complemented, Hall
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8C3×C5×C22⋊Q8—41nomaximal
9C4×C5×SL(2,𝔽3)C2211yescharacteristic
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C4×C5×SL(2,𝔽3)C2212yes—
12C4×C5×SL(2,𝔽3)C2212yes—
13#3502⋊C3C511yesmaximal, characteristic, direct product factor, Hall
14C5×Q82—31no—
15C5×Q82C2×C311yescharacteristic, complemented
16C5×#557—31no—
17C5×#557—32no—
18C5×#557C2×C312yescomplemented
19C5×#557C2×C311yescharacteristic, complemented
20C5×#544—32no—
21C5×#533—31no—
22C5×#533—31no—
23C5×#519C2×C312yes—
24C2×C4×C5×Q8C2×C311yescharacteristic
25C2×C3×C5×Q8—41no—
26C22×C3×C4×C5—41no—
27C2×C5×SL(2,𝔽3)Q812yescomplemented
28C2×C5×SL(2,𝔽3)C2311yescharacteristic
29C3×C5×C4⋊C4—42no—
30C3×C5×C4⋊C4—41no—
31C3×C5×C22⋊C4—42no—
32Q8×SL(2,𝔽3)C2×C511yescharacteristic, complemented
33#557⋊C3C2×C512yescomplemented
34#557⋊C3C2×C511yescharacteristic, complemented
35#519⋊C3C2×C512yes—
36C2×C4×SL(2,𝔽3)C2×C511yescharacteristic
37C22×C5×Q8C22×C311yescharacteristic
38C5×C4⋊Q8—31no—
39C5×C4⋊Q8—32no—
40C5×C4⋊Q8—34no—
41C5×C4⋊Q8—31no—
42C5×C4⋊Q8—31no—
43C5×C4⋊Q8—31no—
44C5×C4⋊Q8—32no—
45C5×#125—32no—
46C5×#125—32no—
47C5×#125—32no—
48C5×#125—34no—
49C5×#125—32no—
50C5×C22⋊Q8—32no—
51C5×C22⋊Q8—31no—
52C5×C22⋊Q8—32no—
53C5×C22⋊Q8A411yescharacteristic
54C4×C5×Q8—32no—
55C4×C5×Q8—32no—
56C4×C5×Q8—31no—
57C4×C5×Q8—32no—
58C4×C5×Q8—31no—
59C4×C5×Q8—32no—
60C4×C5×Q8—32no—
61C4×C5×Q8—31no—
62C4×C5×Q8—31no—
63C4×C5×Q8C22×C311yescharacteristic
64C4×C5×Q8C22×C311yescharacteristic
65C4×C5×Q8C22×C312yes—
66C4×C5×Q8C22×C312yes—
67C5×#117—32no—
68C5×#117—32no—
69C2×C5×C4⋊C4—31no—
70C2×C42×C5—31no—
71#3502C3×C511yescharacteristic, complemented, Sylow
72C23×C3×C5—41no—
73C3×C5×Q8—82no—
74C2×C3×C4×C5—41no—
75C2×C3×C4×C5—42no—
76C2×C3×C4×C5—42no—
77C2×C3×C4×C5—41no—
78C2×C3×C4×C5—82no—
79C5×SL(2,𝔽3)C2×Q811yescharacteristic
80C22×SL(2,𝔽3)C22×C511yescharacteristic
81C3×C22⋊Q8—41no—
82C4×SL(2,𝔽3)C22×C511yescharacteristic
83C4×SL(2,𝔽3)C22×C511yescharacteristic
84C4×SL(2,𝔽3)C22×C512yes—
85C4×SL(2,𝔽3)C22×C512yes—
86C2×C5×Q8—32no—
87C2×C5×Q8—31no—
88C2×C5×Q8—32no—
89C2×C5×Q8—31no—
90C2×C5×Q8—32no—
91C2×C5×Q8C3×Q812yescomplemented
92C2×C5×Q8C2×A411yescharacteristic
93C2×C5×Q8C23×C311yescharacteristic
94C22×C4×C5—31no—
95C22×C4×C5—31no—
96C22×C4×C5C2×A411yescharacteristic
97C5×C4⋊C4—31no—
98C5×C4⋊C4—31no—
99C5×C4⋊C4—32no—
100C5×C4⋊C4—32no—
101C5×C4⋊C4—32no—
102C5×C4⋊C4—32no—
103C5×C4⋊C4—34no—
104C5×C4⋊C4—32no—
105C5×C4⋊C4—32no—
106C5×C4⋊C4—32no—
107C5×C4⋊C4—31no—
108C5×C4⋊C4—32no—
109C5×C4⋊C4—32no—
110C5×C4⋊C4—32no—
111C5×C4⋊C4C2×A412yes—
112C5×C4⋊C4C2×A411yescharacteristic
113C5×C22⋊C4—32no—
114C5×C22⋊C4C2×A412yes—
115C42×C5—34no—
116C42×C5—31no—
117C42×C5—31no—
118C42×C5—31no—
119C42×C5—31no—
120C42×C5—32no—
121C42×C5—32no—
122Q82—31no—
123Q82C2×C3×C511yescharacteristic, complemented
124#557—31no—
125#557—32no—
126#557C2×C3×C512yescomplemented
127#557C2×C3×C511yescharacteristic, complemented
128#544—32no—
129#533—31no—
130#533—31no—
131#519C2×C3×C512yes—
132C2×C4×Q8C2×C3×C511yescharacteristic
133C22×C3×C5—42no—
134C22×C3×C5—41no—
135C22×C3×C5—81no—
136C22×C3×C5—81no—
137C3×C4×C5—42no—
138C3×C4×C5—82no—
139C3×C4×C5—81no—
140C3×C4×C5—82no—
141C2×C3×Q8—41no—
142C22×C3×C4—41no—
143C2×SL(2,𝔽3)C5×Q812yescomplemented
144C2×SL(2,𝔽3)C23×C511yescharacteristic
145C3×C4⋊C4—42no—
146C3×C4⋊C4—41no—
147C3×C22⋊C4—42no—
148C23×C5C22×A411yescharacteristic
149C5×Q8—22no—
150C5×Q8—32no—
151C5×Q8—31no—
152C5×Q8—32no—
153C5×Q8—62no—
154C5×Q8—122no—
155C5×Q8—122no—
156C5×Q8C2×C3×Q811yescharacteristic
157C2×C4×C5—22no—
158C2×C4×C5—32no—
159C2×C4×C5—32no—
160C2×C4×C5—31no—
161C2×C4×C5—31no—
162C2×C4×C5—32no—
163C2×C4×C5—32no—
164C2×C4×C5—31no—
165C2×C4×C5—31no—
166C2×C4×C5—121no—
167C2×C4×C5C22×A411yescharacteristic
168C2×C4×C5C22×A412yes—
169C2×C4×C5C22×A412yes—
170C2×C4×C5C22×A411yescharacteristic
171C22×Q8C22×C3×C511yescharacteristic
172C4⋊Q8—31no—
173C4⋊Q8—32no—
174C4⋊Q8—34no—
175C4⋊Q8—31no—
176C4⋊Q8—31no—
177C4⋊Q8—31no—
178C4⋊Q8—32no—
179#125—32no—
180#125—32no—
181#125—32no—
182#125—34no—
183#125—32no—
184C22⋊Q8—32no—
185C22⋊Q8—31no—
186C22⋊Q8—32no—
187C22⋊Q8C5×A411yescharacteristic
188C4×Q8—32no—
189C4×Q8—32no—
190C4×Q8—31no—
191C4×Q8—32no—
192C4×Q8—31no—
193C4×Q8—32no—
194C4×Q8—32no—
195C4×Q8—31no—
196C4×Q8—31no—
197C4×Q8C22×C3×C511yescharacteristic
198C4×Q8C22×C3×C511yescharacteristic
199C4×Q8C22×C3×C512yes—
200C4×Q8C22×C3×C512yes—
201#117—32no—
202#117—32no—
203C2×C4⋊C4—31no—
204C2×C42—31no—
205C2×C3×C5—41no—
206C2×C3×C5—41no—
207C2×C3×C5—41no—
208C2×C3×C5—82no—
209C23×C3—41no—
210C3×Q8—82no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—82no—
216SL(2,𝔽3)C2×C5×Q811yescharacteristic
217C22×C5—21no—
218C22×C5—21no—
219C22×C5Q8×A412yes—
220C22×C5C23×A411yescharacteristic
221C4×C5—21no—
222C4×C5—22no—
223C4×C5—22no—
224C4×C5—31no—
225C4×C5—32no—
226C4×C5—31no—
227C4×C5—62no—
228C4×C5—61no—
229C4×C5—61no—
230C4×C5—62no—
231C4×C52−1+4⋊C312yes—
232C2×Q8—32no—
233C2×Q8—31no—
234C2×Q8—32no—
235C2×Q8—31no—
236C2×Q8—32no—
237C2×Q8C3×C5×Q812yescomplemented
238C2×Q8C2×C5×A411yescharacteristic
239C2×Q8C23×C3×C511yescharacteristic
240C22×C4—31no—
241C22×C4—31no—
242C22×C4C2×C5×A411yescharacteristic
243C4⋊C4—31no—
244C4⋊C4—31no—
245C4⋊C4—32no—
246C4⋊C4—32no—
247C4⋊C4—32no—
248C4⋊C4—32no—
249C4⋊C4—34no—
250C4⋊C4—32no—
251C4⋊C4—32no—
252C4⋊C4—32no—
253C4⋊C4—31no—
254C4⋊C4—32no—
255C4⋊C4—32no—
256C4⋊C4—32no—
257C4⋊C4C2×C5×A412yes—
258C4⋊C4C2×C5×A411yescharacteristic
259C22⋊C4—32no—
260C22⋊C4C2×C5×A412yes—
261C42—34no—
262C42—31no—
263C42—31no—
264C42—31no—
265C42—31no—
266C42—32no—
267C42—32no—
268C3×C5—41noHall
269C22×C3—42no—
270C22×C3—41no—
271C22×C3—81no—
272C22×C3—81no—
273C3×C4—42no—
274C3×C4—82no—
275C3×C4—81no—
276C3×C4—82no—
277C2×C5—22no—
278C2×C5C2×Q8×A411yescharacteristic
279C2×C5C2×2−1+4⋊C311yescharacteristic
280C2×C5#594511yescharacteristic
281C23C22×C5×A411yescharacteristic
282Q8—22no—
283Q8—32no—
284Q8—31no—
285Q8—32no—
286Q8—62no—
287Q8—122no—
288Q8—122no—
289Q8C2×C3×C5×Q811yescharacteristic
290C2×C4—22no—
291C2×C4—32no—
292C2×C4—32no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—32no—
296C2×C4—32no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—121no—
300C2×C4C22×C5×A411yescharacteristic
301C2×C4C22×C5×A412yes—
302C2×C4C22×C5×A412yes—
303C2×C4C22×C5×A411yescharacteristic
304C2×C3—41no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—82no—
308C5#3502⋊C311yescharacteristic, direct product factor, Sylow
309C22—21no—
310C22—21no—
311C22C5×Q8×A412yes—
312C22C23×C5×A411yescharacteristic
313C4—21no—
314C4—22no—
315C4—22no—
316C4—31no—
317C4—32no—
318C4—31no—
319C4—62no—
320C4—61no—
321C4—61no—
322C4—62no—
323C4C5×2−1+4⋊C312yes—
324C3—41noSylow
325C2—22no—
326C2C2×C5×Q8×A411yescharacteristic
327C2C2×C5×2−1+4⋊C311yescharacteristic
328C2C5×#594511yescharacteristic
subgroupG/H
center, socleC22×C5C23×A4
commutator, cosocleC2×Q8C23×C3×C5
FrattiniC22C23×C5×A4
FittingC5×#3502C3
prime radicalC22×C5×SL(2,𝔽3)C22
hypercenterC5×C22⋊Q8A4
hypocenterQ8C2×C3×C5×Q8
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×#557⋊C3C2semidirect
3. C5×#557⋊C3C2semidirect
6. C5×#3502C3semidirect
13. #3502⋊C3C5direct (N × H)
15. C5×Q82C2×C3semidirect
18. C5×#557C2×C3semidirect
19. C5×#557C2×C3semidirect
27. C2×C5×SL(2,𝔽3)Q8semidirect
32. Q8×SL(2,𝔽3)C2×C5semidirect
33. #557⋊C3C2×C5semidirect
34. #557⋊C3C2×C5semidirect
71. #3502C3×C5semidirect
91. C2×C5×Q8C3×Q8semidirect
123. Q82C2×C3×C5semidirect
126. #557C2×C3×C5semidirect
127. #557C2×C3×C5semidirect
143. C2×SL(2,𝔽3)C5×Q8semidirect
237. C2×Q8C3×C5×Q8semidirect
308. C5#3502⋊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×#557⋊C32
C5×#519⋊C32
C2×C4×C5×SL(2,𝔽3)2
C5×#35023
C3×C5×C22⋊Q84
#3502⋊C35