Finite Group Databasebeta

← C5×D4×GL(2,𝔽3) #14511134

Subgroups — C5×D4×GL(2,𝔽3) #14511134

Order 1920.

766 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 88 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)C212yesmaximal, complemented
2C5×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C5×#5421C212yesmaximal, complemented
4C5×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
5C4×C5×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
6C5×D4×SD16—31nomaximal, Hall
7C2×C5×D3×D4—41nomaximal
8C22×C5×SL(2,𝔽3)C2212yescomplemented
9C2×C5×GL(2,𝔽3)—24no—
10C2×C5×GL(2,𝔽3)C2211yescharacteristic
11C2×C5×GL(2,𝔽3)C2212yescomplemented
12C4×C5×SL(2,𝔽3)C2211yescharacteristic, complemented
13C5×#859C2211yescharacteristic, complemented
14D4×GL(2,𝔽3)C511yesmaximal, characteristic, direct product factor, Hall
15C22×C5×SD16—32no—
16C5×D4×Q8D311yescharacteristic, complemented
17C5×D42—31no—
18C5×#492—31no—
19C5×#465—32no—
20C5×#461—31no—
21C5×#460—31no—
22C5×#450—32no—
23C5×#448—32no—
24C4×C5×SD16—31no—
25C5×C8×D4—31no—
26C23×C5×D3—42no—
27C2×C3×C5×D4—41no—
28C2×C5×#67—42no—
29C5×D3×D4—48no—
30C2×C5×D12—41no—
31C2×C4×C5×D3—41no—
32C2×C5×SL(2,𝔽3)—22no—
33C2×C5×SL(2,𝔽3)C2311yescharacteristic
34C5×GL(2,𝔽3)—22no—
35C5×GL(2,𝔽3)D412yesdirect product factor
36C22×GL(2,𝔽3)C2×C512yescomplemented
37D4×SL(2,𝔽3)C2×C511yescharacteristic, complemented
38#5421C2×C512yescomplemented
39C4⋊GL(2,𝔽3)C2×C511yescharacteristic, complemented
40C4×GL(2,𝔽3)C2×C511yescharacteristic, complemented
41C22×C5×Q8C2×D312yescomplemented
42C22×C5×D4—32no—
43C22×C5×D4—61no—
44C2×C5×SD16—31no—
45C2×C5×SD16—32no—
46C2×C5×SD16—32no—
47C2×C5×SD16—32no—
48C2×C5×SD16—32no—
49C2×C5×SD16—64no—
50C2×C5×SD16—64no—
51C22×C5×C8—32no—
52C5×C4⋊Q8—31no—
53C5×#127—31no—
54C5×C22⋊Q8—32no—
55C5×#121—32no—
56C5×#121—61no—
57C5×#120—62no—
58C4×C5×Q8C2×D311yescharacteristic, complemented
59C4×C5×D4—31no—
60C4×C5×D4—31no—
61C5×#106—31no—
62C5×C4⋊C8—31no—
63C5×#103—32no—
64C5×#103—31no—
65C5×#102—31no—
66C5×#102—32no—
67C5×C22⋊C8—32no—
68C4×C5×C8—31no—
69D4×SD16—31noSylow
70C23×C3×C5—42no—
71C22×C5×D3—41no—
72C22×C5×D3—42no—
73C22×C5×D3—48no—
74C22×C5×D3—84no—
75C22×C5×D3—84no—
76C3×C5×D4—44no—
77C2×C3×C4×C5—41no—
78C5×#67—48no—
79C2×C5×Dic3—41no—
80C5×D12—44no—
81C4×C5×D3—44no—
82C5×SL(2,𝔽3)C2×D411yescharacteristic, complemented
83C2×D3×D4—41no—
84C22×SL(2,𝔽3)C22×C512yescomplemented
85C2×GL(2,𝔽3)—24no—
86C2×GL(2,𝔽3)C22×C511yescharacteristic
87C2×GL(2,𝔽3)C22×C512yescomplemented
88C4×SL(2,𝔽3)C22×C511yescharacteristic, complemented
89#859C22×C511yescharacteristic, complemented
90C24×C5—62no—
91C2×C5×Q8—22no—
92C2×C5×Q8—32no—
93C2×C5×Q8—62no—
94C2×C5×Q8C22×D311yescharacteristic
95C2×C5×D4—31no—
96C2×C5×D4—32no—
97C2×C5×D4—62no—
98C2×C5×D4—61no—
99C2×C5×D4—64no—
100C2×C5×D4—62no—
101C2×C5×D4—61no—
102C2×C5×D4—122no—
103C2×C5×D4—124no—
104C2×C5×D4S411yescharacteristic
105C22×C4×C5—32no—
106C22×C4×C5—61no—
107C5×SD16—32no—
108C5×SD16—32no—
109C5×SD16—64no—
110C5×SD16—64no—
111C5×SD16—62no—
112C5×SD16—62no—
113C5×SD16—122no—
114C2×C5×C8—31no—
115C2×C5×C8—32no—
116C2×C5×C8—31no—
117C2×C5×C8—64no—
118C5×C4⋊C4—31no—
119C5×C4⋊C4—31no—
120C5×C4⋊C4—31no—
121C5×C4⋊C4—61no—
122C5×C22⋊C4—32no—
123C5×C22⋊C4—61no—
124C5×C22⋊C4—62no—
125C42×C5—31no—
126C22×SD16—32no—
127D4×Q8C5×D311yescharacteristic, complemented
128D42—31no—
129#492—31no—
130#465—32no—
131#461—31no—
132#460—31no—
133#450—32no—
134#448—32no—
135C4×SD16—31no—
136C8×D4—31no—
137C22×C3×C5—44no—
138C22×C3×C5—41no—
139C22×C3×C5—82no—
140C22×C3×C5—82no—
141C2×C5×D3—42no—
142C2×C5×D3—42no—
143C2×C5×D3—42no—
144C2×C5×D3—44no—
145C2×C5×D3—82no—
146C2×C5×D3—82no—
147C2×C5×D3—816no—
148C3×C4×C5—42no—
149C5×Dic3—42no—
150C23×D3—42no—
151C2×C3×D4—41no—
152C2×#67—42no—
153D3×D4—48no—
154C2×D12—41no—
155C2×C4×D3—41no—
156C2×SL(2,𝔽3)—22no—
157C2×SL(2,𝔽3)C23×C511yescharacteristic
158GL(2,𝔽3)—22no—
159GL(2,𝔽3)C5×D412yesdirect product factor
160C23×C5—61no—
161C23×C5—62no—
162C23×C5—124no—
163C23×C5—124no—
164C23×C5—242no—
165C23×C5C2×S412yes—
166C5×Q8—31no—
167C5×Q8—62no—
168C5×Q8—62no—
169C5×Q8D3×D411yescharacteristic, complemented
170C5×D4—32no—
171C5×D4—61no—
172C5×D4—62no—
173C5×D4—62no—
174C5×D4—124no—
175C5×D4—124no—
176C5×D4—122no—
177C5×D4—241no—
178C5×D4GL(2,𝔽3)14yesdirect product factor
179C2×C4×C5—31no—
180C2×C4×C5—31no—
181C2×C4×C5—32no—
182C2×C4×C5—62no—
183C2×C4×C5—62no—
184C2×C4×C5—61no—
185C2×C4×C5—122no—
186C2×C4×C5C2×S411yescharacteristic
187C5×C8—32no—
188C5×C8—62no—
189C5×C8—61no—
190C22×Q8C2×C5×D312yescomplemented
191C22×D4—32no—
192C22×D4—61no—
193C2×SD16—31no—
194C2×SD16—32no—
195C2×SD16—32no—
196C2×SD16—32no—
197C2×SD16—32no—
198C2×SD16—64no—
199C2×SD16—64no—
200C22×C8—32no—
201C4⋊Q8—31no—
202#127—31no—
203C22⋊Q8—32no—
204#121—32no—
205#121—61no—
206#120—62no—
207C4×Q8C2×C5×D311yescharacteristic, complemented
208C4×D4—31no—
209C4×D4—31no—
210#106—31no—
211C4⋊C8—31no—
212#103—32no—
213#103—31no—
214#102—31no—
215#102—32no—
216C22⋊C8—32no—
217C4×C8—31no—
218C2×C3×C5—41no—
219C2×C3×C5—41no—
220C2×C3×C5—41no—
221C2×C3×C5—84no—
222C5×D3—44no—
223C5×D3—84no—
224C23×C3—42no—
225C22×D3—41no—
226C22×D3—42no—
227C22×D3—48no—
228C22×D3—84no—
229C22×D3—84no—
230C3×D4—44no—
231C2×C3×C4—41no—
232#67—48no—
233C2×Dic3—41no—
234D12—44no—
235C4×D3—44no—
236SL(2,𝔽3)C2×C5×D411yescharacteristic, complemented
237C22×C5—22no—
238C22×C5—22no—
239C22×C5—62no—
240C22×C5—121no—
241C22×C5—121no—
242C22×C5—122no—
243C22×C5—122no—
244C22×C5—122no—
245C22×C5—248no—
246C22×C5C2×GL(2,𝔽3)14yescomplemented
247C22×C5C22×S411yescharacteristic
248C4×C5—31no—
249C4×C5—31no—
250C4×C5—62no—
251C4×C5—61no—
252C4×C5—121no—
253C4×C5C2×GL(2,𝔽3)12yescomplemented
254C24—62no—
255C2×Q8—22no—
256C2×Q8—32no—
257C2×Q8—62no—
258C2×Q8C22×C5×D311yescharacteristic
259C2×D4—31no—
260C2×D4—32no—
261C2×D4—62no—
262C2×D4—61no—
263C2×D4—64no—
264C2×D4—62no—
265C2×D4—61no—
266C2×D4—122no—
267C2×D4—124no—
268C2×D4C5×S411yescharacteristic
269C22×C4—32no—
270C22×C4—61no—
271SD16—32no—
272SD16—32no—
273SD16—64no—
274SD16—64no—
275SD16—62no—
276SD16—62no—
277SD16—122no—
278C2×C8—31no—
279C2×C8—31no—
280C2×C8—32no—
281C2×C8—64no—
282C4⋊C4—31no—
283C4⋊C4—31no—
284C4⋊C4—31no—
285C4⋊C4—61no—
286C22⋊C4—32no—
287C22⋊C4—61no—
288C22⋊C4—62no—
289C42—31no—
290C3×C5—41noHall
291C22×C3—44no—
292C22×C3—41no—
293C22×C3—82no—
294C22×C3—82no—
295C2×D3—42no—
296C2×D3—42no—
297C2×D3—42no—
298C2×D3—44no—
299C2×D3—82no—
300C2×D3—82no—
301C2×D3—816no—
302C3×C4—42no—
303Dic3—42no—
304C2×C5—24no—
305C2×C5—122no—
306C2×C5—242no—
307C2×C5D4×S411yescharacteristic
308C2×C5C22×GL(2,𝔽3)11yescharacteristic
309C2×C5#592611yescharacteristic
310C23—61no—
311C23—62no—
312C23—124no—
313C23—124no—
314C23—242no—
315C23C2×C5×S412yes—
316Q8—31no—
317Q8—62no—
318Q8—62no—
319Q8C5×D3×D411yescharacteristic, complemented
320D4—32no—
321D4—61no—
322D4—62no—
323D4—62no—
324D4—124no—
325D4—124no—
326D4—122no—
327D4—241no—
328D4C5×GL(2,𝔽3)14yesdirect product factor
329C2×C4—31no—
330C2×C4—31no—
331C2×C4—32no—
332C2×C4—62no—
333C2×C4—62no—
334C2×C4—61no—
335C2×C4—122no—
336C2×C4C2×C5×S411yescharacteristic
337C8—32no—
338C8—62no—
339C8—61no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—84no—
344D3—44no—
345D3—84no—
346C5D4×GL(2,𝔽3)11yescharacteristic, direct product factor, Sylow
347C22—22no—
348C22—22no—
349C22—62no—
350C22—121no—
351C22—121no—
352C22—122no—
353C22—122no—
354C22—122no—
355C22—248no—
356C22C2×C5×GL(2,𝔽3)14yescomplemented
357C22C22×C5×S411yescharacteristic
358C4—31no—
359C4—31no—
360C4—62no—
361C4—61no—
362C4—121no—
363C4C2×C5×GL(2,𝔽3)12yescomplemented
364C3—41noSylow
365C2—24no—
366C2—122no—
367C2—242no—
368C2C5×D4×S411yescharacteristic
369C2C22×C5×GL(2,𝔽3)11yescharacteristic
370C2C5×#592611yescharacteristic
subgroupG/H
center, socleC22×C5C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC5×D4×Q8D3
hypercenterC2×C5×D4S4
hypocenterSL(2,𝔽3)C2×C5×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C5 ◁ C2×C5×D4

Complements

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

NH ≅ G/Ntype
1. C22×C5×GL(2,𝔽3)C2semidirect
2. C5×D4×SL(2,𝔽3)C2semidirect
3. C5×#5421C2semidirect
4. C5×C4⋊GL(2,𝔽3)C2semidirect
5. C4×C5×GL(2,𝔽3)C2semidirect
8. C22×C5×SL(2,𝔽3)C22semidirect
11. C2×C5×GL(2,𝔽3)C22semidirect
12. C4×C5×SL(2,𝔽3)C22semidirect
13. C5×#859C22semidirect
14. D4×GL(2,𝔽3)C5direct (N × H)
16. C5×D4×Q8D3semidirect
35. C5×GL(2,𝔽3)D4direct (N × H)
36. C22×GL(2,𝔽3)C2×C5semidirect
37. D4×SL(2,𝔽3)C2×C5semidirect
38. #5421C2×C5semidirect
39. C4⋊GL(2,𝔽3)C2×C5semidirect
40. C4×GL(2,𝔽3)C2×C5semidirect
41. C22×C5×Q8C2×D3semidirect
58. C4×C5×Q8C2×D3semidirect
82. C5×SL(2,𝔽3)C2×D4semidirect
84. C22×SL(2,𝔽3)C22×C5semidirect
87. C2×GL(2,𝔽3)C22×C5semidirect
88. C4×SL(2,𝔽3)C22×C5semidirect
89. #859C22×C5semidirect
127. D4×Q8C5×D3semidirect
159. GL(2,𝔽3)C5×D4direct (N × H)
169. C5×Q8D3×D4semidirect
178. C5×D4GL(2,𝔽3)direct (N × H)
190. C22×Q8C2×C5×D3semidirect
207. C4×Q8C2×C5×D3semidirect
236. SL(2,𝔽3)C2×C5×D4semidirect
246. C22×C5C2×GL(2,𝔽3)semidirect
253. C4×C5C2×GL(2,𝔽3)semidirect
319. Q8C5×D3×D4semidirect
328. D4C5×GL(2,𝔽3)direct (N × H)
346. C5D4×GL(2,𝔽3)direct (N × H)
356. C22C2×C5×GL(2,𝔽3)semidirect
363. C4C2×C5×GL(2,𝔽3)semidirect

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
C5×D4×SL(2,𝔽3)2
C5×#54212
C5×C4⋊GL(2,𝔽3)2
C4×C5×GL(2,𝔽3)2
C5×D4×SD163
C2×C5×D3×D44
D4×GL(2,𝔽3)5