Finite Group Databasebeta

← C2×C5×#5395 #14511066

Subgroups — C2×C5×#5395 #14511066

Order 1920.

610 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C22×C5×GL(2,𝔽3)C211yesmaximal, characteristic
2C22×C5×#226C211yesmaximal, characteristic, complemented
3C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C5×#5395C214yesmaximal, direct product factor
5C2×C5×#464—31nomaximal, Hall
6C22×C5×SL(2,𝔽3)C2211yescharacteristic
7C2×C5×GL(2,𝔽3)—22no—
8C2×C5×GL(2,𝔽3)C2212yes—
9C2×C5×#226—22no—
10C2×C5×#226C2212yescomplemented
11C2×C5×(C2×D3)⋊C4—41nomaximal
12C4×C5×SL(2,𝔽3)C2212yescomplemented
13C2×#5395C511yesmaximal, characteristic, direct product factor, Hall
14C22×C5×Q16—31no—
15C22×C5×SD16—31no—
16C2×C5×#124—31no—
17C2×C4×C5×Q8D311yescharacteristic, complemented
18C5×#464—34no—
19C5×#464—34no—
20C2×C5×C4⋊C8—31no—
21C2×C5×#103—31no—
22C2×C5×#102—31no—
23C23×C5×D3—41no—
24C22×C3×C4×C5—41no—
25C22×C5×Dic3—41no—
26C2×C5×SL(2,𝔽3)D412yes—
27C2×C5×SL(2,𝔽3)C2311yescharacteristic
28C5×GL(2,𝔽3)—22no—
29C5×#226—22no—
30C5×(C2×D3)⋊C4—44no—
31C22×GL(2,𝔽3)C2×C511yescharacteristic
32C22×#226C2×C511yescharacteristic, complemented
33C2×C4×SL(2,𝔽3)C2×C511yescharacteristic, complemented
34#5395C2×C514yesdirect product factor
35C22×C5×Q8—31no—
36C22×C5×Q8C2×D311yescharacteristic
37C22×C5×D4—31no—
38C2×C5×Q16—32no—
39C2×C5×Q16—32no—
40C2×C5×Q16—62no—
41C2×C5×Q16—62no—
42C2×C5×SD16—32no—
43C2×C5×SD16—32no—
44C2×C5×SD16—62no—
45C2×C5×SD16—62no—
46C22×C5×C8—31no—
47C22×C5×C8—31no—
48C5×#124—34no—
49C5×#124—62no—
50C4×C5×Q8—32no—
51C4×C5×Q8C2×D312yescomplemented
52C2×C5×C4⋊C4—31no—
53C2×C5×C22⋊C4—61no—
54C2×C5×C22⋊C4—61no—
55C2×C42×C5—31no—
56C5×C4⋊C8—34no—
57C5×#103—34no—
58C5×#102—34no—
59C2×#464—31noSylow
60C23×C3×C5—41no—
61C22×C5×D3—42no—
62C22×C5×D3—44no—
63C22×C5×D3—82no—
64C22×C5×D3—82no—
65C2×C3×C4×C5—42no—
66C2×C3×C4×C5—82no—
67C2×C5×Dic3—42no—
68C2×C5×Dic3—82no—
69C5×SL(2,𝔽3)C2×D411yescharacteristic
70C22×SL(2,𝔽3)C22×C511yescharacteristic
71C2×GL(2,𝔽3)—22no—
72C2×GL(2,𝔽3)C22×C512yes—
73C2×#226—22no—
74C2×#226C22×C512yescomplemented
75C2×(C2×D3)⋊C4—41no—
76C4×SL(2,𝔽3)C22×C512yescomplemented
77C24×C5—61no—
78C2×C5×Q8—31no—
79C2×C5×Q8—32no—
80C2×C5×Q8—32no—
81C2×C5×Q8—61no—
82C2×C5×Q8—62no—
83C2×C5×Q8—62no—
84C2×C5×Q8D1212yes—
85C2×C5×Q8C22×D311yescharacteristic
86C2×C5×D4—32no—
87C2×C5×D4—62no—
88C2×C5×D4—62no—
89C2×C5×D4—61no—
90C22×C4×C5—31no—
91C22×C4×C5—31no—
92C22×C4×C5—61no—
93C22×C4×C5S411yescharacteristic
94C5×Q16—64no—
95C5×Q16—62no—
96C5×Q16—62no—
97C5×SD16—64no—
98C5×SD16—62no—
99C5×SD16—62no—
100C2×C5×C8—32no—
101C2×C5×C8—32no—
102C2×C5×C8—62no—
103C2×C5×C8—62no—
104C5×C4⋊C4—32no—
105C5×C4⋊C4—61no—
106C5×C22⋊C4—64no—
107C5×C22⋊C4—64no—
108C42×C5—32no—
109C42×C5—32no—
110C22×Q16—31no—
111C22×SD16—31no—
112C2×#124—31no—
113C2×C4×Q8C5×D311yescharacteristic, complemented
114#464—34no—
115#464—34no—
116C2×C4⋊C8—31no—
117C2×#103—31no—
118C2×#102—31no—
119C22×C3×C5—42no—
120C22×C3×C5—42no—
121C22×C3×C5—42no—
122C22×C3×C5—41no—
123C2×C5×D3—44no—
124C2×C5×D3—82no—
125C2×C5×D3—82no—
126C2×C5×D3—88no—
127C3×C4×C5—82no—
128C5×Dic3—82no—
129C23×D3—41no—
130C22×C3×C4—41no—
131C22×Dic3—41no—
132C2×SL(2,𝔽3)C5×D412yes—
133C2×SL(2,𝔽3)C23×C511yescharacteristic
134GL(2,𝔽3)—22no—
135#226—22no—
136(C2×D3)⋊C4—44no—
137C23×C5—62no—
138C23×C5—122no—
139C23×C5—122no—
140C23×C5—122no—
141C23×C5C2×S411yescharacteristic
142C5×Q8—32no—
143C5×Q8—31no—
144C5×Q8—61no—
145C5×Q8—62no—
146C5×Q8—62no—
147C5×Q8—122no—
148C5×Q8C2×D1211yescharacteristic
149C5×D4—62no—
150C5×D4—62no—
151C5×D4—122no—
152C2×C4×C5—22no—
153C2×C4×C5—32no—
154C2×C4×C5—32no—
155C2×C4×C5—31no—
156C2×C4×C5—31no—
157C2×C4×C5—32no—
158C2×C4×C5—62no—
159C2×C4×C5—62no—
160C2×C4×C5—122no—
161C2×C4×C5C2×S412yes—
162C5×C8—62no—
163C5×C8—62no—
164C22×Q8—31no—
165C22×Q8C2×C5×D311yescharacteristic
166C22×D4—31no—
167C2×Q16—32no—
168C2×Q16—32no—
169C2×Q16—62no—
170C2×Q16—62no—
171C2×SD16—32no—
172C2×SD16—32no—
173C2×SD16—62no—
174C2×SD16—62no—
175C22×C8—31no—
176C22×C8—31no—
177#124—34no—
178#124—62no—
179C4×Q8—32no—
180C4×Q8C2×C5×D312yescomplemented
181C2×C4⋊C4—31no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C42—31no—
185C4⋊C8—34no—
186#103—34no—
187#102—34no—
188C2×C3×C5—44no—
189C2×C3×C5—41no—
190C2×C3×C5—41no—
191C2×C3×C5—41no—
192C5×D3—84no—
193C23×C3—41no—
194C22×D3—42no—
195C22×D3—44no—
196C22×D3—82no—
197C22×D3—82no—
198C2×C3×C4—42no—
199C2×C3×C4—82no—
200C2×Dic3—42no—
201C2×Dic3—82no—
202SL(2,𝔽3)C2×C5×D411yescharacteristic
203C22×C5—122no—
204C22×C5—122no—
205C22×C5—122no—
206C22×C5—244no—
207C22×C5C4⋊S412yes—
208C22×C5#98312yes—
209C22×C5#98512yes—
210C22×C5C22×S411yescharacteristic
211C4×C5—22no—
212C4×C5—31no—
213C4×C5—31no—
214C4×C5—32no—
215C4×C5—62no—
216C4×C5—122no—
217C24—61no—
218C2×Q8—31no—
219C2×Q8—32no—
220C2×Q8—32no—
221C2×Q8—61no—
222C2×Q8—62no—
223C2×Q8—62no—
224C2×Q8C5×D1212yes—
225C2×Q8C22×C5×D311yescharacteristic
226C2×D4—32no—
227C2×D4—62no—
228C2×D4—62no—
229C2×D4—61no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—61no—
233C22×C4C5×S411yescharacteristic
234Q16—64no—
235Q16—62no—
236Q16—62no—
237SD16—64no—
238SD16—62no—
239SD16—62no—
240C2×C8—32no—
241C2×C8—32no—
242C2×C8—62no—
243C2×C8—62no—
244C4⋊C4—32no—
245C4⋊C4—61no—
246C22⋊C4—64no—
247C22⋊C4—64no—
248C42—32no—
249C42—32no—
250C3×C5—41noHall
251C22×C3—42no—
252C22×C3—42no—
253C22×C3—42no—
254C22×C3—41no—
255C2×D3—44no—
256C2×D3—82no—
257C2×D3—82no—
258C2×D3—88no—
259C3×C4—82no—
260Dic3—82no—
261C2×C5—242no—
262C2×C5#539514yesdirect product factor
263C2×C5C2×C4⋊S411yescharacteristic
264C2×C5C2×#98311yescharacteristic
265C2×C5C2×#98511yescharacteristic
266C23—62no—
267C23—122no—
268C23—122no—
269C23—122no—
270C23C2×C5×S411yescharacteristic
271Q8—32no—
272Q8—31no—
273Q8—61no—
274Q8—62no—
275Q8—62no—
276Q8—122no—
277Q8C2×C5×D1211yescharacteristic
278D4—62no—
279D4—62no—
280D4—122no—
281C2×C4—22no—
282C2×C4—32no—
283C2×C4—32no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—32no—
287C2×C4—62no—
288C2×C4—62no—
289C2×C4—122no—
290C2×C4C2×C5×S412yes—
291C8—62no—
292C8—62no—
293C2×C3—41no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—44no—
297D3—84no—
298C5C2×#539511yescharacteristic, direct product factor, Sylow
299C22—122no—
300C22—122no—
301C22—122no—
302C22—244no—
303C22C5×C4⋊S412yes—
304C22C5×#98312yes—
305C22C5×#98512yes—
306C22C22×C5×S411yescharacteristic
307C4—22no—
308C4—31no—
309C4—31no—
310C4—32no—
311C4—62no—
312C4—122no—
313C3—41noSylow
314C2—242no—
315C2C5×#539514yesdirect product factor
316C2C2×C5×C4⋊S411yescharacteristic
317C2C2×C5×#98311yescharacteristic
318C2C2×C5×#98511yescharacteristic
subgroupG/H
center, socleC23×C5C2×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC2×C4×C5×Q8D3
prime radicalC22×C5×GL(2,𝔽3)C2
hypercenterC22×C4×C5S4
hypocenterSL(2,𝔽3)C2×C5×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23×C5 ◁ C22×C4×C5

Complements

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

NH ≅ G/Ntype
2. C22×C5×#226C2semidirect
3. C2×C4×C5×SL(2,𝔽3)C2semidirect
4. C5×#5395C2direct (N × H)
10. C2×C5×#226C22semidirect
12. C4×C5×SL(2,𝔽3)C22semidirect
13. C2×#5395C5direct (N × H)
17. C2×C4×C5×Q8D3semidirect
32. C22×#226C2×C5semidirect
33. C2×C4×SL(2,𝔽3)C2×C5semidirect
34. #5395C2×C5direct (N × H)
51. C4×C5×Q8C2×D3semidirect
74. C2×#226C22×C5semidirect
76. C4×SL(2,𝔽3)C22×C5semidirect
113. C2×C4×Q8C5×D3semidirect
180. C4×Q8C2×C5×D3semidirect
262. C2×C5#5395direct (N × H)
298. C5C2×#5395direct (N × H)
315. C2C5×#5395direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×GL(2,𝔽3)2
C22×C5×#2262
C2×C4×C5×SL(2,𝔽3)2
C5×#53952
C2×C5×#4643
C2×C5×(C2×D3)⋊C44
C2×#53955