Finite Group Databasebeta

← C5×#85418 #14511156

Subgroups — C5×#85418 #14511156

Order 1920.

414 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 72 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
2C4×C5×GL(2,𝔽3)C211yesmaximal, characteristic
3C4×C5×GL(2,𝔽3)C212yesmaximal
4C5×#5391C211yesmaximal, characteristic, complemented
5C5×#5389C212yesmaximal, complemented
6C5×#3381—31nomaximal, Hall
7C2×C5×GL(2,𝔽3)C2211yescharacteristic
8C5×D3×C4⋊C4—41nomaximal
9C4×C5×SL(2,𝔽3)C2212yes—
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C5×#859C2211yescharacteristic
12C5×#859C2212yes—
13#85418C511yesmaximal, characteristic, direct product factor, Hall
14C5×#557D311yescharacteristic, complemented
15C5×#554—31no—
16C5×#499—32no—
17C5×#498—31no—
18C5×#479—32no—
19C5×#478—32no—
20C5×#477—31no—
21C5×#474—31no—
22C5×C8×Q8—31no—
23C4×C5×SD16—31no—
24C4×C5×SD16—32no—
25C2×C4×C5×D3—41no—
26C2×C4×C5×D3—42no—
27C2×C5×SL(2,𝔽3)C2311yescharacteristic
28C5×GL(2,𝔽3)Q812yescomplemented
29C3×C5×C4⋊C4—41no—
30C5×C4⋊Dic3—41no—
31C5×Dic3⋊C4—42no—
32#557⋊C3C2×C511yescharacteristic, complemented
33C4×GL(2,𝔽3)C2×C511yescharacteristic
34C4×GL(2,𝔽3)C2×C512yes—
35#5391C2×C511yescharacteristic, complemented
36#5389C2×C512yescomplemented
37C2×C5×SD16—31no—
38C5×C4⋊Q8—31no—
39C5×C4⋊Q8—31no—
40C5×#125—32no—
41C5×#125—32no—
42C5×C22⋊Q8—62no—
43C5×C22⋊Q8—61no—
44C4×C5×Q8—31no—
45C4×C5×Q8C2×D311yescharacteristic
46C4×C5×Q8C2×D312yes—
47C4×C5×D4—31no—
48C4×C5×D4—32no—
49C2×C5×C4⋊C4—61no—
50C5×#107—32no—
51C5×#107—32no—
52C5×#106—32no—
53C5×#106—31no—
54C5×#106—32no—
55C5×C4⋊C8—31no—
56C5×C4⋊C8—32no—
57C5×#103—32no—
58C5×#103—31no—
59C5×#102—32no—
60C5×#102—31no—
61C4×C5×C8—31no—
62C4×C5×C8—32no—
63#3381—31noSylow
64C22×C5×D3—41no—
65C2×C3×C4×C5—41no—
66C2×C3×C4×C5—42no—
67C2×C5×Dic3—41no—
68C2×C5×Dic3—42no—
69C4×C5×D3—44no—
70C4×C5×D3—84no—
71C5×SL(2,𝔽3)C2×Q811yescharacteristic
72C2×GL(2,𝔽3)C22×C511yescharacteristic
73D3×C4⋊C4—41no—
74C4×SL(2,𝔽3)C22×C512yes—
75C4×SL(2,𝔽3)C22×C511yescharacteristic
76#859C22×C511yescharacteristic
77#859C22×C512yes—
78C2×C5×Q8—31no—
79C2×C5×Q8—61no—
80C2×C5×Q8C22×D311yescharacteristic
81C2×C5×D4—31no—
82C22×C4×C5—61no—
83C22×C4×C5—62no—
84C5×SD16—32no—
85C5×SD16—32no—
86C2×C5×C8—31no—
87C2×C5×C8—31no—
88C2×C5×C8—32no—
89C5×C4⋊C4—31no—
90C5×C4⋊C4—32no—
91C5×C4⋊C4—32no—
92C5×C4⋊C4—31no—
93C5×C4⋊C4—32no—
94C5×C4⋊C4—61no—
95C5×C4⋊C4—62no—
96C5×C4⋊C4—62no—
97C5×C4⋊C4—61no—
98C5×C4⋊C4S411yescharacteristic
99C5×C22⋊C4—61no—
100C5×C22⋊C4—62no—
101C42×C5—32no—
102C42×C5—31no—
103#557C5×D311yescharacteristic, complemented
104#554—31no—
105#499—32no—
106#498—31no—
107#479—32no—
108#478—32no—
109#477—31no—
110#474—31no—
111C8×Q8—31no—
112C4×SD16—31no—
113C4×SD16—32no—
114C22×C3×C5—41no—
115C2×C5×D3—42no—
116C2×C5×D3—42no—
117C2×C5×D3—42no—
118C3×C4×C5—42no—
119C3×C4×C5—82no—
120C5×Dic3—42no—
121C5×Dic3—82no—
122C2×C4×D3—41no—
123C2×C4×D3—42no—
124C2×SL(2,𝔽3)C23×C511yescharacteristic
125GL(2,𝔽3)C5×Q812yescomplemented
126C3×C4⋊C4—41no—
127C4⋊Dic3—41no—
128Dic3⋊C4—42no—
129C23×C5—61no—
130C5×Q8—31no—
131C5×Q8—62no—
132C5×Q8—241no—
133C5×Q8D3×Q811yescharacteristic
134C5×D4—32no—
135C5×D4—61no—
136C2×C4×C5—31no—
137C2×C4×C5—31no—
138C2×C4×C5—32no—
139C2×C4×C5—61no—
140C2×C4×C5—62no—
141C2×C4×C5—122no—
142C2×C4×C5—242no—
143C2×C4×C5C2×S412yes—
144C2×C4×C5C2×S411yescharacteristic
145C5×C8—32no—
146C5×C8—61no—
147C5×C8—62no—
148C2×SD16—31no—
149C4⋊Q8—31no—
150C4⋊Q8—31no—
151#125—32no—
152#125—32no—
153C22⋊Q8—62no—
154C22⋊Q8—61no—
155C4×Q8—31no—
156C4×Q8C2×C5×D311yescharacteristic
157C4×Q8C2×C5×D312yes—
158C4×D4—31no—
159C4×D4—32no—
160C2×C4⋊C4—61no—
161#107—32no—
162#107—32no—
163#106—32no—
164#106—31no—
165#106—32no—
166C4⋊C8—31no—
167C4⋊C8—32no—
168#103—32no—
169#103—31no—
170#102—32no—
171#102—31no—
172C4×C8—31no—
173C4×C8—32no—
174C2×C3×C5—41no—
175C2×C3×C5—41no—
176C2×C3×C5—41no—
177C5×D3—44no—
178C22×D3—41no—
179C2×C3×C4—41no—
180C2×C3×C4—42no—
181C2×Dic3—41no—
182C2×Dic3—42no—
183C4×D3—44no—
184C4×D3—84no—
185SL(2,𝔽3)C2×C5×Q811yescharacteristic
186C22×C5—62no—
187C22×C5—121no—
188C22×C5—121no—
189C22×C5C22×S411yescharacteristic
190C4×C5—22no—
191C4×C5—31no—
192C4×C5—31no—
193C4×C5—62no—
194C4×C5—61no—
195C4×C5—121no—
196C4×C5—122no—
197C4×C5#98512yes—
198C2×Q8—31no—
199C2×Q8—61no—
200C2×Q8C22×C5×D311yescharacteristic
201C2×D4—31no—
202C22×C4—61no—
203C22×C4—62no—
204SD16—32no—
205SD16—32no—
206C2×C8—31no—
207C2×C8—31no—
208C2×C8—32no—
209C4⋊C4—32no—
210C4⋊C4—31no—
211C4⋊C4—32no—
212C4⋊C4—32no—
213C4⋊C4—31no—
214C4⋊C4—61no—
215C4⋊C4—62no—
216C4⋊C4—62no—
217C4⋊C4—61no—
218C4⋊C4C5×S411yescharacteristic
219C22⋊C4—61no—
220C22⋊C4—62no—
221C42—32no—
222C42—31no—
223C3×C5—41noHall
224C22×C3—41no—
225C2×D3—42no—
226C2×D3—42no—
227C2×D3—42no—
228C3×C4—42no—
229C3×C4—82no—
230Dic3—42no—
231Dic3—82no—
232C2×C5—122no—
233C2×C5Q8×S411yescharacteristic
234C2×C5C2×#98511yescharacteristic
235C2×C5#592611yescharacteristic
236C23—61no—
237Q8—31no—
238Q8—62no—
239Q8—241no—
240Q8C5×D3×Q811yescharacteristic
241D4—32no—
242D4—61no—
243C2×C4—31no—
244C2×C4—31no—
245C2×C4—32no—
246C2×C4—61no—
247C2×C4—62no—
248C2×C4—122no—
249C2×C4—242no—
250C2×C4C2×C5×S412yes—
251C2×C4C2×C5×S411yescharacteristic
252C8—32no—
253C8—61no—
254C8—62no—
255C2×C3—41no—
256C2×C3—41no—
257C2×C3—41no—
258D3—44no—
259C5#8541811yescharacteristic, direct product factor, Sylow
260C22—62no—
261C22—121no—
262C22—121no—
263C22C22×C5×S411yescharacteristic
264C4—22no—
265C4—31no—
266C4—31no—
267C4—62no—
268C4—61no—
269C4—121no—
270C4—122no—
271C4C5×#98512yes—
272C3—41noSylow
273C2—122no—
274C2C5×Q8×S411yescharacteristic
275C2C2×C5×#98511yescharacteristic
276C2C5×#592611yescharacteristic
subgroupG/H
center, socleC22×C5C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC5×#557D3
prime radicalC2×C5×GL(2,𝔽3)C22
hypercenterC5×C4⋊C4S4
hypocenterSL(2,𝔽3)C2×C5×Q8
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×#557⋊C3C2semidirect
4. C5×#5391C2semidirect
5. C5×#5389C2semidirect
13. #85418C5direct (N × H)
14. C5×#557D3semidirect
28. C5×GL(2,𝔽3)Q8semidirect
32. #557⋊C3C2×C5semidirect
35. #5391C2×C5semidirect
36. #5389C2×C5semidirect
103. #557C5×D3semidirect
125. GL(2,𝔽3)C5×Q8semidirect
259. C5#85418direct (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
C4×C5×GL(2,𝔽3)2
C5×#53912
C5×#53892
C5×#33813
C5×D3×C4⋊C44
#854185