Finite Group Databasebeta

← C5×#85412 #14511150

Subgroups — C5×#85412 #14511150

Order 1920.

454 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 68 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×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2C5×#5427C212yesmaximal, complemented
3C5×#5426C212yesmaximal, complemented
4C4×C5×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
5C5×#5386C211yesmaximal, characteristic, complemented
6C5×#3332—31nomaximal, Hall
7C2×C5×#41⋊C3C2212yes—
8C2×C5×GL(2,𝔽3)C2211yescharacteristic
9C5×#897—41nomaximal
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C5×#859C2212yes—
12C5×#859C2211yescharacteristic
13#85412C511yesmaximal, characteristic, direct product factor, Hall
14C5×#550D311yescharacteristic, complemented
15C5×#547—31no—
16C5×#486—31no—
17C5×#481—32no—
18C5×#475—31no—
19C5×#469—32no—
20C5×#459—31no—
21C5×#449—32no—
22C4×C5×SD16—31no—
23C5×#436—31no—
24C5×#417—32no—
25C2×C4×C5×D3—41no—
26C2×C5×Dic6—41no—
27C5×#41⋊C3—22no—
28C2×C5×SL(2,𝔽3)C2311yescharacteristic
29C5×GL(2,𝔽3)—21no—
30C3×C5×C4⋊C4—41no—
31C5×(C2×D3)⋊C4—42no—
32C5×C4⋊Dic3—42no—
33C4⋊(#41⋊C3)C2×C511yescharacteristic, complemented
34#5427C2×C512yescomplemented
35#5426C2×C512yescomplemented
36C4×GL(2,𝔽3)C2×C511yescharacteristic, complemented
37#5386C2×C511yescharacteristic, complemented
38C2×C5×#41—61no—
39C2×C5×#41C2×D312yes—
40C2×C5×SD16—31no—
41C2×C5×D8—32no—
42C2×C5×M16—32no—
43C5×C4⋊Q8—31no—
44C5×#127—31no—
45C5×#123—62no—
46C5×C22⋊Q8—61no—
47C5×#121—32no—
48C5×#121—32no—
49C4×C5×Q8C2×D311yescharacteristic
50C4×C5×D4—31no—
51C4×C5×D4—31no—
52C2×C5×C4⋊C4—32no—
53C5×#106—31no—
54C5×#106—32no—
55C5×C4⋊C8—31no—
56C5×#103—31no—
57C5×#103—32no—
58C5×#102—32no—
59C5×#102—32no—
60C5×#102—31no—
61C5×#102—32no—
62C5×C22⋊C8—32no—
63C4×C5×C8—31no—
64#3332—31noSylow
65C22×C5×D3—41no—
66C2×C3×C4×C5—41no—
67C2×C3×C4×C5—42no—
68C2×C5×Dic3—41no—
69C2×C5×Dic3—42no—
70C4×C5×D3—82no—
71C5×Dic6—82no—
72C5×SL(2,𝔽3)#4111yescharacteristic, complemented
73C2×#41⋊C3C22×C512yes—
74C2×GL(2,𝔽3)C22×C511yescharacteristic
75#897—41no—
76C4×SL(2,𝔽3)C22×C511yescharacteristic
77#859C22×C512yes—
78#859C22×C511yescharacteristic
79C5×#41—22no—
80C5×#41—62no—
81C5×#41—124no—
82C2×C5×Q8—61no—
83C2×C5×Q8C22×D311yescharacteristic
84C2×C5×D4—32no—
85C2×C5×D4—31no—
86C2×C5×D4—31no—
87C2×C5×D4—61no—
88C2×C5×D4—62no—
89C22×C4×C5—32no—
90C22×C4×C5—62no—
91C22×C4×C5—61no—
92C5×SD16—61no—
93C5×SD16—61no—
94C5×D8—122no—
95C5×M16—122no—
96C2×C5×C8—31no—
97C2×C5×C8—32no—
98C2×C5×C8—31no—
99C5×C4⋊C4—31no—
100C5×C4⋊C4—31no—
101C5×C4⋊C4—32no—
102C5×C4⋊C4—61no—
103C5×C4⋊C4—62no—
104C5×C4⋊C4S411yescharacteristic
105C5×C22⋊C4—32no—
106C5×C22⋊C4—61no—
107C5×C22⋊C4—62no—
108C42×C5—31no—
109#550C5×D311yescharacteristic, complemented
110#547—31no—
111#486—31no—
112#481—32no—
113#475—31no—
114#469—32no—
115#459—31no—
116#449—32no—
117C4×SD16—31no—
118#436—31no—
119#417—32no—
120C22×C3×C5—41no—
121C2×C5×D3—42no—
122C2×C5×D3—81no—
123C2×C5×D3—81no—
124C3×C4×C5—42no—
125C3×C4×C5—82no—
126C5×Dic3—82no—
127C5×Dic3—81no—
128C2×C4×D3—41no—
129C2×Dic6—41no—
130#41⋊C3—22no—
131C2×SL(2,𝔽3)C23×C511yescharacteristic
132GL(2,𝔽3)—21no—
133C3×C4⋊C4—41no—
134(C2×D3)⋊C4—42no—
135C4⋊Dic3—42no—
136C23×C5—32no—
137C23×C5—61no—
138C5×Q8—31no—
139C5×Q8—122no—
140C5×Q8#23711yescharacteristic
141C5×D4—62no—
142C5×D4—62no—
143C5×D4—61no—
144C5×D4—61no—
145C5×D4—62no—
146C5×D4—124no—
147C5×D4—241no—
148C2×C4×C5—31no—
149C2×C4×C5—31no—
150C2×C4×C5—62no—
151C2×C4×C5—62no—
152C2×C4×C5—61no—
153C2×C4×C5—62no—
154C2×C4×C5—124no—
155C2×C4×C5—122no—
156C2×C4×C5C2×S412yes—
157C2×C4×C5C2×S411yescharacteristic
158C5×C8—61no—
159C5×C8—62no—
160C5×C8—61no—
161C2×#41—61no—
162C2×#41C2×C5×D312yes—
163C2×SD16—31no—
164C2×D8—32no—
165C2×M16—32no—
166C4⋊Q8—31no—
167#127—31no—
168#123—62no—
169C22⋊Q8—61no—
170#121—32no—
171#121—32no—
172C4×Q8C2×C5×D311yescharacteristic
173C4×D4—31no—
174C4×D4—31no—
175C2×C4⋊C4—32no—
176#106—31no—
177#106—32no—
178C4⋊C8—31no—
179#103—31no—
180#103—32no—
181#102—32no—
182#102—32no—
183#102—31no—
184#102—32no—
185C22⋊C8—32no—
186C4×C8—31no—
187C2×C3×C5—41no—
188C2×C3×C5—41no—
189C2×C3×C5—41no—
190C5×D3—82no—
191C22×D3—41no—
192C2×C3×C4—41no—
193C2×C3×C4—42no—
194C2×Dic3—41no—
195C2×Dic3—42no—
196C4×D3—82no—
197Dic6—82no—
198SL(2,𝔽3)C5×#4111yescharacteristic, complemented
199C22×C5—62no—
200C22×C5—62no—
201C22×C5—62no—
202C22×C5—121no—
203C22×C5—121no—
204C22×C5—121no—
205C22×C5C22×S411yescharacteristic
206C4×C5—22no—
207C4×C5—31no—
208C4×C5—31no—
209C4×C5—61no—
210C4×C5—121no—
211C4×C5—122no—
212C4×C5#98612yes—
213#41—22no—
214#41—62no—
215#41—124no—
216C2×Q8—61no—
217C2×Q8C22×C5×D311yescharacteristic
218C2×D4—32no—
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—61no—
222C2×D4—62no—
223C22×C4—32no—
224C22×C4—62no—
225C22×C4—61no—
226SD16—61no—
227SD16—61no—
228D8—122no—
229M16—122no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8—32no—
233C4⋊C4—31no—
234C4⋊C4—31no—
235C4⋊C4—32no—
236C4⋊C4—61no—
237C4⋊C4—62no—
238C4⋊C4C5×S411yescharacteristic
239C22⋊C4—32no—
240C22⋊C4—61no—
241C22⋊C4—62no—
242C42—31no—
243C3×C5—41noHall
244C22×C3—41no—
245C2×D3—42no—
246C2×D3—81no—
247C2×D3—81no—
248C3×C4—42no—
249C3×C4—82no—
250Dic3—82no—
251Dic3—81no—
252C2×C5—122no—
253C2×C5—241no—
254C2×C5#591411yescharacteristic
255C2×C5C2×#98611yescharacteristic
256C2×C5#592411yescharacteristic
257C23—32no—
258C23—61no—
259Q8—31no—
260Q8—122no—
261Q8C5×#23711yescharacteristic
262D4—62no—
263D4—62no—
264D4—61no—
265D4—61no—
266D4—62no—
267D4—124no—
268D4—241no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—62no—
272C2×C4—62no—
273C2×C4—61no—
274C2×C4—62no—
275C2×C4—124no—
276C2×C4—122no—
277C2×C4C2×C5×S412yes—
278C2×C4C2×C5×S411yescharacteristic
279C8—61no—
280C8—62no—
281C8—61no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—41no—
285D3—82no—
286C5#8541211yescharacteristic, direct product factor, Sylow
287C22—62no—
288C22—62no—
289C22—62no—
290C22—121no—
291C22—121no—
292C22—121no—
293C22C22×C5×S411yescharacteristic
294C4—22no—
295C4—31no—
296C4—31no—
297C4—61no—
298C4—121no—
299C4—122no—
300C4C5×#98612yes—
301C3—41noSylow
302C2—122no—
303C2—241no—
304C2C5×#591411yescharacteristic
305C2C2×C5×#98611yescharacteristic
306C2C5×#592411yescharacteristic
subgroupG/H
center, socleC22×C5C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC5×#550D3
hypercenterC5×C4⋊C4S4
hypocenterSL(2,𝔽3)C5×#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C5 ◁ C5×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C5×C4⋊(#41⋊C3)C2semidirect
2. C5×#5427C2semidirect
3. C5×#5426C2semidirect
4. C4×C5×GL(2,𝔽3)C2semidirect
5. C5×#5386C2semidirect
13. #85412C5direct (N × H)
14. C5×#550D3semidirect
33. C4⋊(#41⋊C3)C2×C5semidirect
34. #5427C2×C5semidirect
35. #5426C2×C5semidirect
36. C4×GL(2,𝔽3)C2×C5semidirect
37. #5386C2×C5semidirect
72. C5×SL(2,𝔽3)#41semidirect
109. #550C5×D3semidirect
198. SL(2,𝔽3)C5×#41semidirect
286. C5#85412direct (N × H)

Maximal subgroups

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

subgroupindex
C5×C4⋊(#41⋊C3)2
C5×#54272
C5×#54262
C4×C5×GL(2,𝔽3)2
C5×#53862
C5×#33323
C5×#8974
#854125