Finite Group Databasebeta

← C5×#85397 #14511135

Subgroups — C5×#85397 #14511135

Order 1920.

554 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 84 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C5×#5421C212yesmaximal, complemented
3C2×C5×#859C212yesmaximal, complemented
4C4×C5×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
5C5×#5386C211yesmaximal, characteristic, complemented
6C5×#3298—31nomaximal, Hall
7C2×C5×#237—41nomaximal
8C22×C5×SL(2,𝔽3)C2212yes—
9C2×C5×GL(2,𝔽3)C2211yescharacteristic
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C5×#859C2212yes—
12C5×#859C2211yescharacteristic
13#85397C511yesmaximal, characteristic, direct product factor, Hall
14C5×D4×Q8D311yescharacteristic, complemented
15C5×#547—31no—
16C5×#487—31no—
17C5×#483—32no—
18C5×#475—31no—
19C5×#465—32no—
20C5×#461—31no—
21C5×#448—32no—
22C4×C5×SD16—31no—
23C5×C8×D4—31no—
24C2×C5×#103—32no—
25C2×C3×C5×D4—41no—
26C2×C5×#67—42no—
27C22×C5×Dic3—42no—
28C5×#237—48no—
29C2×C4×C5×D3—41no—
30C2×C5×Dic6—41no—
31C2×C5×SL(2,𝔽3)—22no—
32C2×C5×SL(2,𝔽3)C2311yescharacteristic
33C5×GL(2,𝔽3)—21no—
34D4×SL(2,𝔽3)C2×C511yescharacteristic, complemented
35#5421C2×C512yescomplemented
36C2×#859C2×C512yescomplemented
37C4×GL(2,𝔽3)C2×C511yescharacteristic, complemented
38#5386C2×C511yescharacteristic, complemented
39C2×C5×#41—61no—
40C22×C5×Q8C2×D312yes—
41C2×C5×SD16—31no—
42C2×C5×SD16—32no—
43C22×C5×C8—32no—
44C5×C4⋊Q8—31no—
45C5×C4⋊Q8—31no—
46C5×#123—62no—
47C5×C22⋊Q8—32no—
48C5×C22⋊Q8—61no—
49C5×#121—32no—
50C4×C5×Q8C2×D311yescharacteristic
51C4×C5×D4—31no—
52C4×C5×D4—31no—
53C2×C5×C4⋊C4—32no—
54C5×#106—31no—
55C5×#106—32no—
56C5×C4⋊C8—31no—
57C5×#103—32no—
58C5×#103—32no—
59C5×#103—32no—
60C5×#103—31no—
61C5×#103—32no—
62C5×#102—31no—
63C5×C22⋊C8—32no—
64C4×C5×C8—31no—
65#3298—31noSylow
66C23×C3×C5—42no—
67C22×C5×D3—41no—
68C3×C5×D4—44no—
69C2×C3×C4×C5—41no—
70C5×#67—48no—
71C2×C5×Dic3—42no—
72C2×C5×Dic3—41no—
73C2×C5×Dic3—48no—
74C4×C5×D3—44no—
75C5×Dic6—44no—
76C5×SL(2,𝔽3)#4111yescharacteristic, complemented
77C2×#237—41no—
78C22×SL(2,𝔽3)C22×C512yes—
79C2×GL(2,𝔽3)C22×C511yescharacteristic
80C4×SL(2,𝔽3)C22×C511yescharacteristic
81#859C22×C512yes—
82#859C22×C511yescharacteristic
83C5×#41—124no—
84C2×C5×Q8—22no—
85C2×C5×Q8—32no—
86C2×C5×Q8—61no—
87C2×C5×Q8—62no—
88C2×C5×Q8C22×D311yescharacteristic
89C2×C5×D4—31no—
90C2×C5×D4—62no—
91C2×C5×D4S411yescharacteristic
92C22×C4×C5—32no—
93C22×C4×C5—62no—
94C22×C4×C5—61no—
95C5×SD16—61no—
96C5×SD16—61no—
97C5×SD16—122no—
98C2×C5×C8—31no—
99C2×C5×C8—32no—
100C2×C5×C8—31no—
101C2×C5×C8—122no—
102C5×C4⋊C4—31no—
103C5×C4⋊C4—31no—
104C5×C4⋊C4—31no—
105C5×C4⋊C4—32no—
106C5×C4⋊C4—61no—
107C5×C4⋊C4—62no—
108C5×C4⋊C4—61no—
109C5×C22⋊C4—32no—
110C5×C22⋊C4—61no—
111C5×C22⋊C4—62no—
112C42×C5—31no—
113D4×Q8C5×D311yescharacteristic, complemented
114#547—31no—
115#487—31no—
116#483—32no—
117#475—31no—
118#465—32no—
119#461—31no—
120#448—32no—
121C4×SD16—31no—
122C8×D4—31no—
123C2×#103—32no—
124C22×C3×C5—44no—
125C22×C3×C5—41no—
126C22×C3×C5—82no—
127C22×C3×C5—82no—
128C2×C5×D3—42no—
129C2×C5×D3—81no—
130C2×C5×D3—81no—
131C3×C4×C5—42no—
132C5×Dic3—44no—
133C5×Dic3—42no—
134C2×C3×D4—41no—
135C2×#67—42no—
136C22×Dic3—42no—
137#237—48no—
138C2×C4×D3—41no—
139C2×Dic6—41no—
140C2×SL(2,𝔽3)—22no—
141C2×SL(2,𝔽3)C23×C511yescharacteristic
142GL(2,𝔽3)—21no—
143C23×C5—61no—
144C23×C5C2×S412yes—
145C5×Q8—31no—
146C5×Q8—62no—
147C5×Q8—62no—
148C5×Q8—122no—
149C5×Q8#23711yescharacteristic, complemented
150C5×D4—61no—
151C5×D4—61no—
152C5×D4—124no—
153C5×D4GL(2,𝔽3)14yescomplemented
154C2×C4×C5—31no—
155C2×C4×C5—31no—
156C2×C4×C5—32no—
157C2×C4×C5—62no—
158C2×C4×C5—62no—
159C2×C4×C5—61no—
160C2×C4×C5—62no—
161C2×C4×C5—124no—
162C2×C4×C5—122no—
163C2×C4×C5C2×S411yescharacteristic
164C5×C8—61no—
165C5×C8—62no—
166C5×C8—61no—
167C2×#41—61no—
168C22×Q8C2×C5×D312yes—
169C2×SD16—31no—
170C2×SD16—32no—
171C22×C8—32no—
172C4⋊Q8—31no—
173C4⋊Q8—31no—
174#123—62no—
175C22⋊Q8—32no—
176C22⋊Q8—61no—
177#121—32no—
178C4×Q8C2×C5×D311yescharacteristic
179C4×D4—31no—
180C4×D4—31no—
181C2×C4⋊C4—32no—
182#106—31no—
183#106—32no—
184C4⋊C8—31no—
185#103—32no—
186#103—32no—
187#103—32no—
188#103—31no—
189#103—32no—
190#102—31no—
191C22⋊C8—32no—
192C4×C8—31no—
193C2×C3×C5—41no—
194C2×C3×C5—41no—
195C2×C3×C5—41no—
196C2×C3×C5—84no—
197C5×D3—82no—
198C23×C3—42no—
199C22×D3—41no—
200C3×D4—44no—
201C2×C3×C4—41no—
202#67—48no—
203C2×Dic3—42no—
204C2×Dic3—41no—
205C2×Dic3—48no—
206C4×D3—44no—
207Dic6—44no—
208SL(2,𝔽3)C5×#4111yescharacteristic, complemented
209C22×C5—22no—
210C22×C5—22no—
211C22×C5—121no—
212C22×C5—121no—
213C22×C5—121no—
214C22×C5C2×GL(2,𝔽3)14yes—
215C22×C5C22×S411yescharacteristic
216C4×C5—31no—
217C4×C5—31no—
218C4×C5—62no—
219C4×C5—61no—
220C4×C5—121no—
221C4×C5—122no—
222C4×C5C2×GL(2,𝔽3)12yes—
223#41—124no—
224C2×Q8—22no—
225C2×Q8—32no—
226C2×Q8—61no—
227C2×Q8—62no—
228C2×Q8C22×C5×D311yescharacteristic
229C2×D4—31no—
230C2×D4—62no—
231C2×D4C5×S411yescharacteristic
232C22×C4—32no—
233C22×C4—62no—
234C22×C4—61no—
235SD16—61no—
236SD16—61no—
237SD16—122no—
238C2×C8—31no—
239C2×C8—31no—
240C2×C8—32no—
241C2×C8—122no—
242C4⋊C4—31no—
243C4⋊C4—31no—
244C4⋊C4—31no—
245C4⋊C4—32no—
246C4⋊C4—61no—
247C4⋊C4—62no—
248C4⋊C4—61no—
249C22⋊C4—32no—
250C22⋊C4—61no—
251C22⋊C4—62no—
252C42—31no—
253C3×C5—41noHall
254C22×C3—44no—
255C22×C3—41no—
256C22×C3—82no—
257C22×C3—82no—
258C2×D3—42no—
259C2×D3—81no—
260C2×D3—81no—
261C3×C4—42no—
262Dic3—44no—
263Dic3—42no—
264C2×C5—24no—
265C2×C5—241no—
266C2×C5#591411yescharacteristic
267C2×C5C22×GL(2,𝔽3)11yescharacteristic
268C2×C5#592711yescharacteristic
269C23—61no—
270C23C2×C5×S412yes—
271Q8—31no—
272Q8—62no—
273Q8—62no—
274Q8—122no—
275Q8C5×#23711yescharacteristic, complemented
276D4—61no—
277D4—61no—
278D4—124no—
279D4C5×GL(2,𝔽3)14yescomplemented
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—32no—
283C2×C4—62no—
284C2×C4—62no—
285C2×C4—61no—
286C2×C4—62no—
287C2×C4—124no—
288C2×C4—122no—
289C2×C4C2×C5×S411yescharacteristic
290C8—61no—
291C8—62no—
292C8—61no—
293C2×C3—41no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—84no—
297D3—82no—
298C5#8539711yescharacteristic, direct product factor, Sylow
299C22—22no—
300C22—22no—
301C22—121no—
302C22—121no—
303C22—121no—
304C22C2×C5×GL(2,𝔽3)14yes—
305C22C22×C5×S411yescharacteristic
306C4—31no—
307C4—31no—
308C4—62no—
309C4—61no—
310C4—121no—
311C4—122no—
312C4C2×C5×GL(2,𝔽3)12yes—
313C3—41noSylow
314C2—24no—
315C2—241no—
316C2C5×#591411yescharacteristic
317C2C22×C5×GL(2,𝔽3)11yescharacteristic
318C2C5×#592711yescharacteristic
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)C5×#41
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. C5×D4×SL(2,𝔽3)C2semidirect
2. C5×#5421C2semidirect
3. C2×C5×#859C2semidirect
4. C4×C5×GL(2,𝔽3)C2semidirect
5. C5×#5386C2semidirect
13. #85397C5direct (N × H)
14. C5×D4×Q8D3semidirect
34. D4×SL(2,𝔽3)C2×C5semidirect
35. #5421C2×C5semidirect
36. C2×#859C2×C5semidirect
37. C4×GL(2,𝔽3)C2×C5semidirect
38. #5386C2×C5semidirect
76. C5×SL(2,𝔽3)#41semidirect
113. D4×Q8C5×D3semidirect
149. C5×Q8#237semidirect
153. C5×D4GL(2,𝔽3)semidirect
208. SL(2,𝔽3)C5×#41semidirect
275. Q8C5×#237semidirect
279. D4C5×GL(2,𝔽3)semidirect
298. C5#85397direct (N × H)

Maximal subgroups

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

subgroupindex
C5×D4×SL(2,𝔽3)2
C5×#54212
C2×C5×#8592
C4×C5×GL(2,𝔽3)2
C5×#53862
C5×#32983
C2×C5×#2374
#853975