Finite Group Databasebeta

← #14512794

Subgroups — #14512794

Order 1920.

554 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 59 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
1SL(2,𝔽3)×D20C211yesmaximal, characteristic, complemented
2#11752105C211yesmaximal, characteristic, complemented
3C4×C5×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#11751613C212yesmaximal, complemented
5D5×#859C212yesmaximal, complemented
6#10611409—31nomaximal, Hall
7C2×#6765—41nomaximal
8C2×D5×SL(2,𝔽3)C2212yes—
9C2×C5×GL(2,𝔽3)C2211yescharacteristic
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11#91184C2212yes—
12C5×#859C2211yescharacteristic
13#85397—51nomaximal, Hall
14Q8×D20D311yescharacteristic, complemented
15#65885—31no—
16C4×C5×SD16—31no—
17#65459—32no—
18#65309—31no—
19#65308—31no—
20#65151—32no—
21D5×#103—32no—
22#65067—32no—
23C8×D20—31no—
24#64970—31no—
25C2×Dic30—41no—
26C2×C4×C5×D3—41no—
27C2×C3×D20—41no—
28C2×C5×SL(2,𝔽3)C2311yescharacteristic
29C2×#1201—42no—
30C2×D5×Dic3—42no—
31#6765—48no—
32D5×SL(2,𝔽3)—22no—
33C5×GL(2,𝔽3)—21no—
34D4×SL(2,𝔽3)—51no—
35#5421—52no—
36C2×#859—52no—
37C4×GL(2,𝔽3)D511yescharacteristic, complemented
38#5386—51no—
39C2×Q8×D5C2×D312yes—
40C2×#709—61no—
41C2×C5×SD16—31no—
42C4×C5×Q8C2×D311yescharacteristic
43C4×C5×D4—31no—
44#4084—32no—
45C2×#687—32no—
46#4070—61no—
47C2×C8×D5—32no—
48#4043—32no—
49D5×C4⋊C4—32no—
50#4029—62no—
51C4×D20—31no—
52#4015—31no—
53#4015—31no—
54C5×#106—31no—
55C5×#103—31no—
56C5×#102—31no—
57C4×C5×C8—31no—
58#3967—32no—
59#3952—32no—
60#3948—32no—
61#3948—32no—
62#3942—32no—
63#3940—32no—
64C4⋊#168—31no—
65#3298—151noSylow
66C22×C5×D3—41no—
67C22×C3×D5—42no—
68C2×C3×C4×C5—41no—
69C2×Dic15—42no—
70Dic30—44no—
71C2×C5×Dic3—41no—
72C4×C5×D3—44no—
73C3×D20—44no—
74C5×SL(2,𝔽3)#4111yescharacteristic, complemented
75#1201—48no—
76D5×Dic3—48no—
77C2×#237—201no—
78C22×SL(2,𝔽3)—52no—
79C2×GL(2,𝔽3)C2×D511yescharacteristic
80C4×SL(2,𝔽3)C2×D511yescharacteristic
81#859—52no—
82#859C2×D511yescharacteristic
83C2×C5×Q8C22×D311yescharacteristic
84C2×C5×D4—31no—
85C22×C4×C5—61no—
86C2×C22⋊D5—62no—
87Q8×D5—22no—
88Q8×D5—62no—
89#709—124no—
90C2×D20S411yescharacteristic
91C2×C4×D5—32no—
92C2×C4×D5—62no—
93C2×Dic10—32no—
94C2×Dic10—61no—
95C5×SD16—61no—
96C5×SD16—61no—
97C2×C5×C8—31no—
98C2×C5×C8—31no—
99C5×C4⋊C4—31no—
100C5×C4⋊C4—31no—
101C5×C22⋊C4—61no—
102C42×C5—31no—
103C22⋊Dic5—62no—
104#687—122no—
105#685—32no—
106C4⋊Dic5—31no—
107C4⋊Dic5—32no—
108C4⋊Dic5—61no—
109C4⋊Dic5—61no—
110#683—62no—
111C2×#168—32no—
112C8×D5—122no—
113D4×Q8—51no—
114#547—151no—
115#487—151no—
116#483—152no—
117#475—151no—
118#465—152no—
119#461—151no—
120#448—152no—
121C4×SD16—31no—
122C8×D4—151no—
123C2×#103—152no—
124C22×C3×C5—41no—
125C2×C5×D3—42no—
126C2×C5×D3—81no—
127C2×C5×D3—81no—
128C2×C3×D5—44no—
129C2×C3×D5—82no—
130C2×C3×D5—82no—
131C3×C4×C5—42no—
132Dic15—44no—
133C5×Dic3—42no—
134C2×C3×D4—201no—
135C2×#67—202no—
136C22×Dic3—202no—
137#237—208no—
138C2×C4×D3—41no—
139C2×Dic6—201no—
140C2×SL(2,𝔽3)—102no—
141C2×SL(2,𝔽3)C22×D511yescharacteristic
142GL(2,𝔽3)—21no—
143C23×C5—61no—
144C22×D5C2×S412yes—
145C5×Q8—31no—
146C5×Q8#23711yescharacteristic, complemented
147C5×D4—61no—
148C5×D4—61no—
149C2×C4×C5—31no—
150C2×C4×C5—31no—
151C2×C4×C5—61no—
152C2×C4×C5—122no—
153C2×C4×C5C2×S411yescharacteristic
154C22⋊D5—124no—
155C2×Dic5—32no—
156C2×Dic5—62no—
157D20GL(2,𝔽3)14yescomplemented
158C4×D5—62no—
159C4×D5—62no—
160C4×D5—124no—
161Dic10—62no—
162Dic10—62no—
163Dic10—122no—
164C5×C8—61no—
165C5×C8—61no—
166#168—62no—
167C2×#41—301no—
168C22×Q8—52no—
169C2×SD16—31no—
170C2×SD16—152no—
171C22×C8—152no—
172C4⋊Q8—151no—
173C4⋊Q8—151no—
174#123—302no—
175C22⋊Q8—152no—
176C22⋊Q8—301no—
177#121—152no—
178C4×Q8D3×D511yescharacteristic
179C4×D4—31no—
180C4×D4—151no—
181C2×C4⋊C4—152no—
182#106—31no—
183#106—152no—
184C4⋊C8—151no—
185#103—31no—
186#103—152no—
187#103—152no—
188#103—152no—
189#103—152no—
190#102—31no—
191C22⋊C8—152no—
192C4×C8—31no—
193C2×C3×C5—41no—
194C2×C3×C5—41no—
195C2×C3×C5—41no—
196C3×D5—84no—
197C5×D3—82no—
198C23×C3—202no—
199C22×D3—41no—
200C3×D4—204no—
201C2×C3×C4—41no—
202#67—208no—
203C2×Dic3—41no—
204C2×Dic3—202no—
205C2×Dic3—208no—
206C4×D3—44no—
207Dic6—204no—
208SL(2,𝔽3)#70911yescharacteristic, complemented
209C22×C5—121no—
210C22×C5—121no—
211C22×C5—121no—
212C22×C5C22×S411yescharacteristic
213C2×D5—22no—
214C2×D5—22no—
215C2×D5C2×GL(2,𝔽3)14yes—
216C4×C5—31no—
217C4×C5—31no—
218C4×C5—61no—
219C4×C5—121no—
220C4×C5C2×GL(2,𝔽3)12yes—
221Dic5—62no—
222Dic5—122no—
223#41—604no—
224C2×Q8—102no—
225C2×Q8—152no—
226C2×Q8—301no—
227C2×Q8—302no—
228C2×Q8C2×D3×D511yescharacteristic
229C2×D4—31no—
230C2×D4—51no—
231C2×D4—302no—
232C22×C4—61no—
233C22×C4—152no—
234C22×C4—302no—
235SD16—61no—
236SD16—61no—
237SD16—602no—
238C2×C8—31no—
239C2×C8—31no—
240C2×C8—152no—
241C2×C8—602no—
242C4⋊C4—31no—
243C4⋊C4—31no—
244C4⋊C4—152no—
245C4⋊C4—151no—
246C4⋊C4—301no—
247C4⋊C4—301no—
248C4⋊C4—302no—
249C22⋊C4—61no—
250C22⋊C4—152no—
251C22⋊C4—302no—
252C42—31no—
253C3×C5—41noHall
254C22×C3—41no—
255C22×C3—204no—
256C22×C3—402no—
257C22×C3—402no—
258C2×D3—42no—
259C2×D3—81no—
260C2×D3—81no—
261C3×C4—42no—
262Dic3—42no—
263Dic3—204no—
264C2×C5—241no—
265C2×C5#591411yescharacteristic
266C2×C5C22×GL(2,𝔽3)11yescharacteristic
267C2×C5#592711yescharacteristic
268D5—24no—
269C23—52no—
270C23—61no—
271Q8—31no—
272Q8—302no—
273Q8—302no—
274Q8—602no—
275Q8#676511yescharacteristic, complemented
276D4—54no—
277D4—61no—
278D4—61no—
279D4—604no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—61no—
283C2×C4—122no—
284C2×C4—152no—
285C2×C4—302no—
286C2×C4—302no—
287C2×C4—302no—
288C2×C4—604no—
289C2×C4D5×S411yescharacteristic
290C8—61no—
291C8—61no—
292C8—302no—
293C2×C3—41no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—404no—
297D3—82no—
298C5#8539711yescharacteristic, complemented, Sylow
299C22—54no—
300C22—102no—
301C22—102no—
302C22—121no—
303C22—121no—
304C22—121no—
305C22C2×D5×S411yescharacteristic
306C4—31no—
307C4—31no—
308C4—61no—
309C4—121no—
310C4—302no—
311C4—602no—
312C4D5×GL(2,𝔽3)12yes—
313C3—41noSylow
314C2—104no—
315C2—241no—
316C2#1175675011yescharacteristic
317C2C2×D5×GL(2,𝔽3)11yescharacteristic
318C2#1175677811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FittingC4×C5×Q8C2×D3
socleC22×C5C22×S4
hypercenterC2×C4D5×S4
hypocenterC5×SL(2,𝔽3)#41
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. SL(2,𝔽3)×D20C2semidirect
2. #11752105C2semidirect
3. C4×C5×GL(2,𝔽3)C2semidirect
4. #11751613C2semidirect
5. D5×#859C2semidirect
14. Q8×D20D3semidirect
37. C4×GL(2,𝔽3)D5semidirect
74. C5×SL(2,𝔽3)#41semidirect
146. C5×Q8#237semidirect
157. D20GL(2,𝔽3)semidirect
208. SL(2,𝔽3)#709semidirect
275. Q8#6765semidirect
298. C5#85397semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×D202
#117521052
C4×C5×GL(2,𝔽3)2
#117516132
D5×#8592
#106114093
C2×#67654
#853975