Finite Group Databasebeta

← #14513941

Subgroups — #14513941

Order 1920.

680 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 65 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×C5×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C22×C5⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#11752146C212yesmaximal, complemented
4#11752140C212yesmaximal, complemented
5#11752139C211yesmaximal, characteristic, complemented
6#10615844—31nomaximal, Hall
7C5×2−1+4⋊C3—22no—
8C2×C5×#41⋊C3C2211yescharacteristic, complemented
9C22×C5×SL(2,𝔽3)C2211yescharacteristic
10C22×C5×SL(2,𝔽3)C2211yescharacteristic, complemented
11C2×C5⋊GL(2,𝔽3)—22no—
12C2×C5⋊GL(2,𝔽3)C2212yes—
13#91826—41nomaximal
14#91184C2212yescomplemented
15#85537—51nomaximal, Hall
16C2×C5×2−1+4D311yescharacteristic, complemented
17#66164—31no—
18C2×#4095—31no—
19C22×#688—31no—
20#65527—32no—
21#65527—32no—
22#65523—31no—
23#65522—31no—
24#65516—32no—
25#65516—32no—
26#65514—31no—
27C23×D15—41no—
28C2×C3×C5×D4—41no—
29C2×#1220—42no—
30C5×#41⋊C3—21no—
31C2×C5×SL(2,𝔽3)—21no—
32C2×C5×SL(2,𝔽3)D412yescomplemented
33C2×C5×SL(2,𝔽3)C2311yescharacteristic
34C5⋊GL(2,𝔽3)—22no—
35#6719—41no—
36#6714—42no—
37C2×2−1+4⋊C3D511yescharacteristic, complemented
38C22×GL(2,𝔽3)—51no—
39#5427—52no—
40#5421—52no—
41#5420—51no—
42C5×2−1+4—22no—
43C5×2−1+4—62no—
44C2×C5×#41—31no—
45C2×C5×#41—31no—
46C2×C5×#41—31no—
47C2×C5×#41C2×D311yescharacteristic, complemented
48C22×C5×Q8—31no—
49C22×C5×Q8C2×D311yescharacteristic
50C22×C5×Q8C2×D311yescharacteristic, complemented
51C22×D20—31no—
52#4095—64no—
53#4093—61no—
54C2×#688—32no—
55C2×#688—32no—
56C2×#688—32no—
57C2×#688—62no—
58C2×#688—62no—
59#4086—61no—
60#4083—61no—
61#4083—61no—
62C2×#686—32no—
63#4076—32no—
64#4076—32no—
65#4072—31no—
66C2×#681—31no—
67C22×#168—31no—
68#3967—32no—
69#3967—32no—
70#3967—32no—
71#3964—32no—
72C22⋊#168—32no—
73C22⋊#168—32no—
74#3010—151noSylow
75C23×C3×C5—41no—
76C23×C3×C5—41no—
77C22×D15—42no—
78C22×D15—84no—
79C22×D15—82no—
80C3×C5×D4—82no—
81C2×C3×C4×C5—41no—
82#1220—84no—
83C2×Dic15—42no—
84C5×SL(2,𝔽3)C2×D411yescharacteristic
852−1+4⋊C3—22no—
86C2×#41⋊C3C2×D511yescharacteristic, complemented
87C22×SL(2,𝔽3)C2×D511yescharacteristic
88C22×SL(2,𝔽3)C2×D511yescharacteristic, complemented
89C2×GL(2,𝔽3)—52no—
90C2×GL(2,𝔽3)—102no—
91#937—201no—
92#859—52no—
93C23×D5—61no—
94C5×#41—21no—
95C5×#41—34no—
96C5×#41—62no—
97C5×#41—62no—
98C5×#41—62no—
99C5×#41—62no—
100C5×#41—62no—
101C5×#41—61no—
102C2×C5×Q8—21no—
103C2×C5×Q8—31no—
104C2×C5×Q8—31no—
105C2×C5×Q8—31no—
106C2×C5×Q8—32no—
107C2×C5×Q8—62no—
108C2×C5×Q8—62no—
109C2×C5×Q8—61no—
110C2×C5×Q8#6712yescomplemented
111C2×C5×Q8C22×D311yescharacteristic
112C2×C5×D4—31no—
113C2×C5×D4—31no—
114C2×C5×D4—31no—
115C2×C5×D4S411yescharacteristic
116C22×C4×C5—31no—
117C22×C4×C5—31no—
118C22×C4×C5—31no—
119C22×C4×C5—61no—
120C2×C22⋊D5—62no—
121C2×C22⋊D5—62no—
122C2×D20—32no—
123C2×D20—62no—
124C2×D20—62no—
125C2×D20—61no—
126C22⋊Dic5—61no—
127#688—64no—
128#688—62no—
129#688—64no—
130#688—62no—
131#686—64no—
132#685—62no—
133#685—62no—
134C4⋊Dic5—32no—
135C4×Dic5—61no—
136#681—62no—
137C2×#168—32no—
138C2×#168—32no—
139C2×#168—62no—
140C2×2−1+4D1511yescharacteristic, complemented
141C2×AGL(1,ℤ/8)—151no—
142C22×SD16—151no—
143#535—151no—
144#449—152no—
145#449—152no—
146#448—152no—
147#448—152no—
148#419—151no—
149#416—151no—
150#408—151no—
151C22×C3×C5—42no—
152C22×C3×C5—41no—
153C22×C3×C5—82no—
154C22×C3×C5—81no—
155C22×C3×C5—82no—
156C2×D15—44no—
157C2×D15—82no—
158C2×D15—82no—
159C2×D15—88no—
160C3×C4×C5—81no—
161Dic15—82no—
162C23×D3—201no—
163C2×C3×D4—41no—
164C2×#67—202no—
165#41⋊C3—21no—
166C2×SL(2,𝔽3)—21no—
167C2×SL(2,𝔽3)C22⋊D512yescomplemented
168C2×SL(2,𝔽3)C22×D511yescharacteristic
169GL(2,𝔽3)—102no—
170#217—201no—
171(C2×D3)⋊C4—202no—
172C23×C5—31no—
173C23×C5C2×S411yescharacteristic
174C23×C5C2×S411yescharacteristic
175C22×D5—62no—
176C22×D5—122no—
177C22×D5—242no—
178C5×Q8—32no—
179C5×Q8—31no—
180C5×Q8—32no—
181C5×Q8—62no—
182C5×Q8—61no—
183C5×Q8—61no—
184C5×Q8C2×#6711yescharacteristic
185C5×D4—22no—
186C5×D4—32no—
187C5×D4—62no—
188C5×D4—61no—
189C5×D4—62no—
190C2×C4×C5—32no—
191C2×C4×C5—32no—
192C2×C4×C5—31no—
193C2×C4×C5—31no—
194C2×C4×C5—31no—
195C2×C4×C5—61no—
196C2×C4×C5—61no—
197C2×C4×C5—61no—
198C2×C4×C5—61no—
199C2×C4×C5—64no—
200C2×C4×C5C2×S411yescharacteristic
201C22⋊D5—242no—
202C22⋊D5—242no—
203C2×Dic5—62no—
204D20—62no—
205D20—62no—
206D20—122no—
207#168—62no—
208#168—62no—
2092−1+4—22no—
2102−1+4—62no—
211C2×#41—31no—
212C2×#41—31no—
213C2×#41—31no—
214C2×#41C2×D1511yescharacteristic, complemented
215C22×Q8—31no—
216C22×Q8C2×D1511yescharacteristic
217C22×Q8C2×D1511yescharacteristic, complemented
218C22×D4—151no—
219AGL(1,ℤ/8)—304no—
220C2×SD16—152no—
221C2×SD16—152no—
222C2×SD16—152no—
223C2×SD16—302no—
224C2×SD16—302no—
225C2×D8—152no—
226C2×M16—151no—
227C22×C8—151no—
228#127—301no—
229#124—301no—
230#121—152no—
231#121—152no—
232#120—301no—
233#120—301no—
234#117—151no—
235#103—152no—
236#103—152no—
237#103—152no—
238#102—152no—
239C22⋊C8—152no—
240C22⋊C8—152no—
241C2×C3×C5—42no—
242C2×C3×C5—41no—
243C2×C3×C5—82no—
244C2×C3×C5—161no—
245D15—84no—
246C23×C3—41no—
247C23×C3—41no—
248C22×D3—202no—
249C22×D3—402no—
250C22×D3—404no—
251C3×D4—82no—
252C2×C3×C4—41no—
253#67—404no—
254C2×Dic3—202no—
255SL(2,𝔽3)C2×C22⋊D511yescharacteristic
256C22×C5—22no—
257C22×C5—22no—
258C22×C5—21no—
259C22×C5—62no—
260C22×C5—61no—
261C22×C5GL(2,ℤ/4)12yes—
262C22×C5C22×S411yescharacteristic
263C2×D5—122no—
264C2×D5—122no—
265C2×D5—122no—
266C2×D5—244no—
267C4×C5—21no—
268C4×C5—31no—
269C4×C5—31no—
270C4×C5—32no—
271C4×C5—61no—
272Dic5—122no—
273C24—301no—
274#41—21no—
275#41—34no—
276#41—62no—
277#41—62no—
278#41—62no—
279#41—62no—
280#41—62no—
281#41—61no—
282C2×Q8—21no—
283C2×Q8—31no—
284C2×Q8—31no—
285C2×Q8—31no—
286C2×Q8—32no—
287C2×Q8—62no—
288C2×Q8—62no—
289C2×Q8—61no—
290C2×Q8#122012yescomplemented
291C2×Q8C22×D1511yescharacteristic
292C2×D4—31no—
293C2×D4—31no—
294C2×D4—31no—
295C2×D4—152no—
296C2×D4—302no—
297C2×D4—302no—
298C2×D4—301no—
299C2×D4—302no—
300C2×D4—302no—
301C2×D4C5⋊S411yescharacteristic
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—61no—
306SD16—304no—
307SD16—302no—
308SD16—304no—
309SD16—302no—
310D8—304no—
311M16—302no—
312C2×C8—152no—
313C2×C8—152no—
314C2×C8—302no—
315C4⋊C4—152no—
316C22⋊C4—302no—
317C22⋊C4—301no—
318C22⋊C4—302no—
319C42—301no—
320C3×C5—41noHall
321C22×C3—42no—
322C22×C3—41no—
323C22×C3—82no—
324C22×C3—82no—
325C22×C3—81no—
326C2×D3—204no—
327C2×D3—402no—
328C2×D3—402no—
329C2×D3—408no—
330C3×C4—81no—
331Dic3—402no—
332C2×C5—22no—
333C2×C5—41no—
334C2×C5—121no—
335C2×C5#592612yes—
336C2×C5C2×GL(2,ℤ/4)11yescharacteristic
337D5—242no—
338C23—31no—
339C23—302no—
340C23—602no—
341C23—1202no—
342C23C2×C5⋊S411yescharacteristic
343C23C2×C5⋊S411yescharacteristic
344Q8—32no—
345Q8—31no—
346Q8—32no—
347Q8—62no—
348Q8—61no—
349Q8—61no—
350Q8C2×#122011yescharacteristic
351D4—22no—
352D4—32no—
353D4—62no—
354D4—61no—
355D4—62no—
356D4—302no—
357D4—302no—
358D4—602no—
359D4—1202no—
360D4—1202no—
361C2×C4—32no—
362C2×C4—32no—
363C2×C4—31no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—64no—
371C2×C4—302no—
372C2×C4C2×C5⋊S411yescharacteristic
373C8—302no—
374C8—302no—
375C2×C3—42no—
376C2×C3—41no—
377C2×C3—82no—
378C2×C3—161no—
379D3—404no—
380C5#8553711yescharacteristic, complemented, Sylow
381C22—22no—
382C22—22no—
383C22—21no—
384C22—62no—
385C22—61no—
386C22—602no—
387C22—602no—
388C22—602no—
389C22—1204no—
390C22#9195812yes—
391C22C22×C5⋊S411yescharacteristic
392C4—21no—
393C4—31no—
394C4—31no—
395C4—32no—
396C4—61no—
397C4—602no—
398C3—41noSylow
399C2—22no—
400C2—41no—
401C2—121no—
402C2—1202no—
403C2#1175691012yes—
404C2C2×#9195811yescharacteristic
subgroupG/H
center, FrattiniC22C22×C5⋊S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FittingC2×C5×2−1+4D3
socleC22×C5C22×S4
hypercenterC2×D4C5⋊S4
hypocenterC5×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×2−1+4⋊C3C2semidirect
2. C22×C5⋊GL(2,𝔽3)C2semidirect
3. #11752146C2semidirect
4. #11752140C2semidirect
5. #11752139C2semidirect
8. C2×C5×#41⋊C3C22semidirect
10. C22×C5×SL(2,𝔽3)C22semidirect
14. #91184C22semidirect
16. C2×C5×2−1+4D3semidirect
32. C2×C5×SL(2,𝔽3)D4semidirect
37. C2×2−1+4⋊C3D5semidirect
47. C2×C5×#41C2×D3semidirect
50. C22×C5×Q8C2×D3semidirect
86. C2×#41⋊C3C2×D5semidirect
88. C22×SL(2,𝔽3)C2×D5semidirect
110. C2×C5×Q8#67semidirect
140. C2×2−1+4D15semidirect
167. C2×SL(2,𝔽3)C22⋊D5semidirect
214. C2×#41C2×D15semidirect
217. C22×Q8C2×D15semidirect
290. C2×Q8#1220semidirect
380. C5#85537semidirect

Maximal subgroups

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

subgroupindex
C2×C5×2−1+4⋊C32
C22×C5⋊GL(2,𝔽3)2
#117521462
#117521402
#117521392
#106158443
#918264
#855375