Finite Group Databasebeta

← C5×#85319 #14511057

Subgroups — C5×#85319 #14511057

Order 1920.

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

normal subgroups only

This group has 418 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
1C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic
2C5×#5419C212yesmaximal
3C5×C4⋊#226C211yesmaximal, characteristic, complemented
4C5×#5388C212yesmaximal, complemented
5C5×#5386C211yesmaximal, characteristic, complemented
6C5×#3113—31nomaximal, Hall
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8C2×C5×#226C2212yes—
9C5×#924—41nomaximal
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C4×C5×SL(2,𝔽3)C2211yescharacteristic
12C5×#859C2212yes—
13#85319C511yesmaximal, characteristic, direct product factor, Hall
14C5×#533—31no—
15C2×C4×C5×Q8D311yescharacteristic
16C5×#483—32no—
17C5×#475—31no—
18C5×#475—31no—
19C5×#462—31no—
20C5×#462—31no—
21C5×#451—32no—
22C5×#441—32no—
23C5×#441—32no—
24C5×#432—31no—
25C22×C3×C4×C5—41no—
26C2×C5×Dic6—41no—
27C2×C5×SL(2,𝔽3)—21no—
28C2×C5×SL(2,𝔽3)C2311yescharacteristic
29C5×#226—22no—
30C5×#217—42no—
31C5×C4⋊Dic3—41no—
32C5×Dic3⋊C4—42no—
33C2×C4×SL(2,𝔽3)C2×C511yescharacteristic
34#5419C2×C512yes—
35C4⋊#226C2×C511yescharacteristic, complemented
36#5388C2×C512yescomplemented
37#5386C2×C511yescharacteristic, complemented
38C22×C5×Q8C2×D311yescharacteristic
39C2×C5×Q16—32no—
40C2×C5×Q16—32no—
41C5×C4⋊Q8—31no—
42C5×C4⋊Q8—31no—
43C5×#125—61no—
44C5×C22⋊Q8—32no—
45C5×C22⋊Q8—61no—
46C4×C5×Q8—31no—
47C4×C5×Q8—31no—
48C4×C5×Q8—32no—
49C4×C5×Q8—61no—
50C4×C5×Q8C2×D311yescharacteristic
51C4×C5×Q8C2×D311yescharacteristic
52C5×#117—61no—
53C2×C5×C4⋊C4—31no—
54C2×C42×C5—31no—
55C5×#106—32no—
56C5×#106—32no—
57C5×C4⋊C8—31no—
58C5×C4⋊C8—31no—
59C5×#103—32no—
60C5×#103—32no—
61C5×#103—32no—
62C5×#103—32no—
63C5×C22⋊C8—32no—
64C5×#97—32no—
65#3113—31noSylow
66C23×C3×C5—41no—
67C2×C3×C4×C5—41no—
68C2×C3×C4×C5—41no—
69C2×C3×C4×C5—82no—
70C2×C5×Dic3—42no—
71C2×C5×Dic3—42no—
72C5×Dic6—82no—
73C5×SL(2,𝔽3)#4111yescharacteristic
74C22×SL(2,𝔽3)C22×C511yescharacteristic
75C2×#226C22×C512yes—
76#924—41no—
77C4×SL(2,𝔽3)C22×C511yescharacteristic
78C4×SL(2,𝔽3)C22×C511yescharacteristic
79#859C22×C512yes—
80C2×C5×Q8—21no—
81C2×C5×Q8—31no—
82C2×C5×Q8—32no—
83C2×C5×Q8—61no—
84C2×C5×Q8—61no—
85C2×C5×Q8C22×D311yescharacteristic
86C22×C4×C5—31no—
87C22×C4×C5—31no—
88C22×C4×C5S411yescharacteristic
89C5×Q16—62no—
90C5×Q16—64no—
91C5×Q16—62no—
92C2×C5×C8—32no—
93C2×C5×C8—32no—
94C5×C4⋊C4—31no—
95C5×C4⋊C4—31no—
96C5×C4⋊C4—32no—
97C5×C4⋊C4—61no—
98C5×C4⋊C4—61no—
99C5×C4⋊C4—61no—
100C5×C4⋊C4—62no—
101C5×C4⋊C4—62no—
102C5×C4⋊C4—61no—
103C5×C22⋊C4—62no—
104C42×C5—31no—
105C42×C5—31no—
106C42×C5—31no—
107C42×C5—31no—
108C42×C5—62no—
109#533—31no—
110C2×C4×Q8C5×D311yescharacteristic
111#483—32no—
112#475—31no—
113#475—31no—
114#462—31no—
115#462—31no—
116#451—32no—
117#441—32no—
118#441—32no—
119#432—31no—
120C22×C3×C5—42no—
121C22×C3×C5—41no—
122C22×C3×C5—81no—
123C22×C3×C5—81no—
124C3×C4×C5—42no—
125C3×C4×C5—81no—
126C5×Dic3—82no—
127C5×Dic3—82no—
128C22×C3×C4—41no—
129C2×Dic6—41no—
130C2×SL(2,𝔽3)—21no—
131C2×SL(2,𝔽3)C23×C511yescharacteristic
132#226—22no—
133#217—42no—
134C4⋊Dic3—41no—
135Dic3⋊C4—42no—
136C23×C5C2×S411yescharacteristic
137C5×Q8—31no—
138C5×Q8—32no—
139C5×Q8—61no—
140C5×Q8—62no—
141C5×Q8—62no—
142C5×Q8—122no—
143C5×Q8#23511yescharacteristic
144C2×C4×C5—22no—
145C2×C4×C5—31no—
146C2×C4×C5—31no—
147C2×C4×C5—31no—
148C2×C4×C5—31no—
149C2×C4×C5—61no—
150C2×C4×C5—61no—
151C2×C4×C5—62no—
152C2×C4×C5—62no—
153C2×C4×C5—62no—
154C2×C4×C5C2×S411yescharacteristic
155C2×C4×C5C2×S411yescharacteristic
156C5×C8—62no—
157C5×C8—62no—
158C22×Q8C2×C5×D311yescharacteristic
159C2×Q16—32no—
160C2×Q16—32no—
161C4⋊Q8—31no—
162C4⋊Q8—31no—
163#125—61no—
164C22⋊Q8—32no—
165C22⋊Q8—61no—
166C4×Q8—31no—
167C4×Q8—31no—
168C4×Q8—32no—
169C4×Q8—61no—
170C4×Q8C2×C5×D311yescharacteristic
171C4×Q8C2×C5×D311yescharacteristic
172#117—61no—
173C2×C4⋊C4—31no—
174C2×C42—31no—
175#106—32no—
176#106—32no—
177C4⋊C8—31no—
178C4⋊C8—31no—
179#103—32no—
180#103—32no—
181#103—32no—
182#103—32no—
183C22⋊C8—32no—
184#97—32no—
185C2×C3×C5—41no—
186C2×C3×C5—41no—
187C2×C3×C5—41no—
188C2×C3×C5—82no—
189C23×C3—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—82no—
193C2×Dic3—42no—
194C2×Dic3—42no—
195Dic6—82no—
196SL(2,𝔽3)C5×#4111yescharacteristic
197C22×C5—21no—
198C22×C5—21no—
199C22×C5#98412yes—
200C22×C5C22×S411yescharacteristic
201C4×C5—21no—
202C4×C5—31no—
203C4×C5—31no—
204C4×C5—61no—
205C4×C5—61no—
206C4×C5—61no—
207C4×C5—122no—
208C4×C5—122no—
209C4×C5#98312yes—
210C2×Q8—21no—
211C2×Q8—31no—
212C2×Q8—32no—
213C2×Q8—61no—
214C2×Q8—61no—
215C2×Q8C22×C5×D311yescharacteristic
216C22×C4—31no—
217C22×C4—31no—
218C22×C4C5×S411yescharacteristic
219Q16—62no—
220Q16—64no—
221Q16—62no—
222C2×C8—32no—
223C2×C8—32no—
224C4⋊C4—31no—
225C4⋊C4—31no—
226C4⋊C4—32no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—62no—
231C4⋊C4—62no—
232C4⋊C4—61no—
233C22⋊C4—62no—
234C42—31no—
235C42—31no—
236C42—31no—
237C42—31no—
238C42—62no—
239C3×C5—41noHall
240C22×C3—42no—
241C22×C3—41no—
242C22×C3—81no—
243C22×C3—81no—
244C3×C4—42no—
245C3×C4—81no—
246Dic3—82no—
247Dic3—82no—
248C2×C5—22no—
249C2×C5#591211yescharacteristic
250C2×C5C2×#98311yescharacteristic
251C2×C5C2×#98411yescharacteristic
252C23C2×C5×S411yescharacteristic
253Q8—31no—
254Q8—32no—
255Q8—61no—
256Q8—62no—
257Q8—62no—
258Q8—122no—
259Q8C5×#23511yescharacteristic
260C2×C4—22no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—31no—
264C2×C4—31no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4C2×C5×S411yescharacteristic
271C2×C4C2×C5×S411yescharacteristic
272C8—62no—
273C8—62no—
274C2×C3—41no—
275C2×C3—41no—
276C2×C3—41no—
277C2×C3—82no—
278C5#8531911yescharacteristic, direct product factor, Sylow
279C22—21no—
280C22—21no—
281C22C5×#98412yes—
282C22C22×C5×S411yescharacteristic
283C4—21no—
284C4—31no—
285C4—31no—
286C4—61no—
287C4—61no—
288C4—61no—
289C4—122no—
290C4—122no—
291C4C5×#98312yes—
292C3—41noSylow
293C2—22no—
294C2C5×#591211yescharacteristic
295C2C2×C5×#98311yescharacteristic
296C2C2×C5×#98411yescharacteristic
subgroupG/H
center, socleC22×C5C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC2×C4×C5×Q8D3
prime radicalC22×C5×SL(2,𝔽3)C22
hypercenterC22×C4×C5S4
hypocenterSL(2,𝔽3)C5×#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C5 ◁ C22×C4×C5

Complements

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

NH ≅ G/Ntype
3. C5×C4⋊#226C2semidirect
4. C5×#5388C2semidirect
5. C5×#5386C2semidirect
13. #85319C5direct (N × H)
35. C4⋊#226C2×C5semidirect
36. #5388C2×C5semidirect
37. #5386C2×C5semidirect
278. C5#85319direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C5×SL(2,𝔽3)2
C5×#54192
C5×C4⋊#2262
C5×#53882
C5×#53862
C5×#31133
C5×#9244
#853195