Finite Group Databasebeta

← C5×#3459⋊C3 #14511389

Subgroups — C5×#3459⋊C3 #14511389

Order 1920.

500 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 46 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×#543⋊C3C211yesmaximal, characteristic, complemented
2C5×#3459C311yesmaximal, characteristic, complemented, Hall
3C2×C5×#41⋊C3—42nomaximal
4C2×C4×C5×A4—41nomaximal
5C4×C5×SL(2,𝔽3)—42nomaximal
6#3459⋊C3C511yesmaximal, characteristic, direct product factor, Hall
7C5×#543C2×C311yescharacteristic
8C5×#539—32no—
9C5×#538—32no—
10C5×#534—31no—
11C5×#529—32no—
12C5×#525—32no—
13C2×C4×C5×D4—31no—
14C22×C5×A4—41no—
15C5×#41⋊C3—44no—
16C2×C5×SL(2,𝔽3)—42no—
17C2×C5×SL(2,𝔽3)—42no—
18C4×C5×A4—42no—
19#543⋊C3C2×C511yescharacteristic, complemented
20C2×C5×#41A412yescomplemented
21C22×C5×D4—31no—
22C23×C4×C5—31no—
23C23×C4×C5A411yescharacteristic
24C5×#126—34no—
25C5×#125—32no—
26C5×#124—32no—
27C5×#123—34no—
28C5×#123—34no—
29C5×C22⋊Q8—32no—
30C5×C22⋊Q8—32no—
31C5×C22⋊Q8—32no—
32C5×#121—32no—
33C5×#121—32no—
34C5×#121—32no—
35C5×#120—34no—
36C4×C5×Q8A412yescomplemented
37C4×C5×D4—34no—
38C4×C5×D4—34no—
39C4×C5×D4—32no—
40C5×#117—32no—
41C2×C5×C4⋊C4—31no—
42C2×C5×C22⋊C4—31no—
43C2×C5×C22⋊C4—31no—
44C2×C42×C5—31no—
45#3459C3×C511yescharacteristic, complemented, Sylow
46C2×C5×A4—41no—
47C2×C5×A4—41no—
48C2×C5×A4—41no—
49C2×C3×C4×C5—161no—
50C5×SL(2,𝔽3)—42no—
51C5×SL(2,𝔽3)—42no—
52C2×#41⋊C3—42no—
53C2×C4×A4—41no—
54C4×SL(2,𝔽3)—42no—
55C24×C5—31no—
56C24×C5C2×A411yescharacteristic
57C5×#41—44no—
58C2×C5×Q8C2×A412yes—
59C2×C5×Q8C2×A412yes—
60C2×C5×D4—34no—
61C2×C5×D4—32no—
62C2×C5×D4—32no—
63C2×C5×D4—62no—
64C2×C5×D4—61no—
65C2×C5×D4—61no—
66C22×C4×C5—31no—
67C22×C4×C5—31no—
68C22×C4×C5—31no—
69C22×C4×C5—32no—
70C22×C4×C5—31no—
71C22×C4×C5—32no—
72C22×C4×C5—31no—
73C22×C4×C5—42no—
74C22×C4×C5—122no—
75C5×C4⋊C4—32no—
76C5×C4⋊C4—34no—
77C5×C4⋊C4—32no—
78C5×C4⋊C4—32no—
79C5×C4⋊C4—32no—
80C5×C22⋊C4—34no—
81C5×C22⋊C4—34no—
82C5×C22⋊C4—34no—
83C5×C22⋊C4—34no—
84C42×C5—32no—
85C42×C5—32no—
86#543C2×C3×C511yescharacteristic
87#539—32no—
88#538—32no—
89#534—31no—
90#529—32no—
91#525—32no—
92C2×C4×D4—31no—
93C22×C3×C5—161no—
94C5×A4—41no—
95C3×C4×C5—162no—
96C22×A4—41no—
97#41⋊C3—44no—
98C2×SL(2,𝔽3)—42no—
99C2×SL(2,𝔽3)—42no—
100C4×A4—42no—
101C23×C5—31no—
102C23×C5—31no—
103C23×C5—32no—
104C23×C5—41no—
105C23×C5—41no—
106C23×C5—61no—
107C23×C5—61no—
108C23×C5—121no—
109C23×C5—121no—
110C23×C5#41⋊C311yescharacteristic, complemented
111C5×Q8—42no—
112C5×Q8—42no—
113C5×D4—64no—
114C5×D4—64no—
115C5×D4—122no—
116C5×D4—122no—
117C2×C4×C5—32no—
118C2×C4×C5—32no—
119C2×C4×C5—31no—
120C2×C4×C5—32no—
121C2×C4×C5—61no—
122C2×C4×C5—61no—
123C2×C4×C5—62no—
124C2×C4×C5—64no—
125C2×C4×C5—121no—
126C2×C4×C5—122no—
127C2×C4×C5—121no—
128C2×C4×C5—121no—
129C2×C4×C5C22⋊A411yescharacteristic
130C2×#41C5×A412yescomplemented
131C22×D4—31no—
132C23×C4—31no—
133C23×C4C5×A411yescharacteristic
134#126—34no—
135#125—32no—
136#124—32no—
137#123—34no—
138#123—34no—
139C22⋊Q8—32no—
140C22⋊Q8—32no—
141C22⋊Q8—32no—
142#121—32no—
143#121—32no—
144#121—32no—
145#120—34no—
146C4×Q8C5×A412yescomplemented
147C4×D4—34no—
148C4×D4—34no—
149C4×D4—32no—
150#117—32no—
151C2×C4⋊C4—31no—
152C2×C22⋊C4—31no—
153C2×C22⋊C4—31no—
154C2×C42—31no—
155C2×C3×C5—161no—
156C2×C3×C5—161no—
157C2×C3×C5—161no—
158C2×A4—41no—
159C2×A4—41no—
160C2×A4—41no—
161C2×C3×C4—161no—
162SL(2,𝔽3)—42no—
163SL(2,𝔽3)—42no—
164C22×C5—31no—
165C22×C5—31no—
166C22×C5—41no—
167C22×C5—61no—
168C22×C5—61no—
169C22×C5—62no—
170C22×C5—62no—
171C22×C5—62no—
172C22×C5—121no—
173C22×C5—122no—
174C22×C5—122no—
175C22×C5C2×C22⋊A411yescharacteristic
176C4×C5—62no—
177C4×C5—61no—
178C4×C5—62no—
179C4×C5—62no—
180C4×C5#99712yes—
181C24—31no—
182C24C2×C5×A411yescharacteristic
183#41—44no—
184C2×Q8C2×C5×A412yes—
185C2×Q8C2×C5×A412yes—
186C2×D4—34no—
187C2×D4—32no—
188C2×D4—32no—
189C2×D4—61no—
190C2×D4—62no—
191C2×D4—61no—
192C22×C4—31no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—32no—
196C22×C4—31no—
197C22×C4—32no—
198C22×C4—31no—
199C22×C4—42no—
200C22×C4—122no—
201C4⋊C4—32no—
202C4⋊C4—34no—
203C4⋊C4—32no—
204C4⋊C4—32no—
205C4⋊C4—32no—
206C22⋊C4—34no—
207C22⋊C4—34no—
208C22⋊C4—34no—
209C22⋊C4—34no—
210C42—32no—
211C42—32no—
212C3×C5—161noHall
213C22×C3—161no—
214A4—41no—
215C3×C4—162no—
216C2×C5—61no—
217C2×C5—61no—
218C2×C5—122no—
219C2×C5#594811yescharacteristic
220C2×C5C2×#99711yescharacteristic
221C2×C5#595011yescharacteristic
222C23—31no—
223C23—31no—
224C23—32no—
225C23—41no—
226C23—41no—
227C23—61no—
228C23—61no—
229C23—121no—
230C23—121no—
231C23C5×#41⋊C311yescharacteristic, complemented
232Q8—42no—
233Q8—42no—
234D4—64no—
235D4—64no—
236D4—122no—
237D4—122no—
238C2×C4—32no—
239C2×C4—32no—
240C2×C4—31no—
241C2×C4—32no—
242C2×C4—61no—
243C2×C4—61no—
244C2×C4—62no—
245C2×C4—64no—
246C2×C4—122no—
247C2×C4—121no—
248C2×C4—121no—
249C2×C4—121no—
250C2×C4C5×C22⋊A411yescharacteristic
251C2×C3—161no—
252C2×C3—161no—
253C2×C3—161no—
254C5#3459⋊C311yescharacteristic, direct product factor, Sylow
255C22—31no—
256C22—31no—
257C22—41no—
258C22—61no—
259C22—61no—
260C22—62no—
261C22—62no—
262C22—62no—
263C22—121no—
264C22—122no—
265C22—122no—
266C22C2×C5×C22⋊A411yescharacteristic
267C4—62no—
268C4—61no—
269C4—62no—
270C4—62no—
271C4C5×#99712yes—
272C3—161noSylow
273C2—61no—
274C2—61no—
275C2—122no—
276C2C5×#594811yescharacteristic
277C2C2×C5×#99711yescharacteristic
278C2C5×#595011yescharacteristic
subgroupG/H
center, hypercenterC2×C4×C5C22⋊A4
commutator, cosocle, hypocenter#543C2×C3×C5
FrattiniC22C2×C5×C22⋊A4
FittingC5×#3459C3
socleC22×C5C2×C22⋊A4
LCSG ▷ #543
UCSC1 ◁ C2×C4×C5

Complements

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

NH ≅ G/Ntype
1. C5×#543⋊C3C2semidirect
2. C5×#3459C3semidirect
6. #3459⋊C3C5direct (N × H)
19. #543⋊C3C2×C5semidirect
20. C2×C5×#41A4semidirect
36. C4×C5×Q8A4semidirect
45. #3459C3×C5semidirect
110. C23×C5#41⋊C3semidirect
130. C2×#41C5×A4semidirect
146. C4×Q8C5×A4semidirect
231. C23C5×#41⋊C3semidirect
254. C5#3459⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#543⋊C32
C5×#34593
C2×C5×#41⋊C34
C2×C4×C5×A44
C4×C5×SL(2,𝔽3)4
#3459⋊C35