Finite Group Databasebeta

← #14281608

Subgroups — #14281608

Order 1920.

552 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11755200C211yesmaximal, characteristic
2C4×#6738C211yesmaximal, characteristic, complemented
3#11751777C211yesmaximal, characteristic, complemented
4#11746801C211yesmaximal, characteristic, complemented
5C3×Q8⋊#170C211yesmaximal, characteristic, complemented
6#11746491C211yesmaximal, characteristic, complemented
7#11746490C211yesmaximal, characteristic, complemented
8#10604436—31nomaximal, Hall
9C2×#6821C411yescharacteristic
10C2×#6738—21no—
11C5×Q8×Dic3C411yescharacteristic
12C3×Q8×Dic5C2211yescharacteristic
13#91428—21no—
14#91428C411yescharacteristic
15#91418—21no—
16#91418C2211yescharacteristic
17C4×#1200—21no—
18C4×#1200C2211yescharacteristic
19#91347—21no—
20#91347C411yescharacteristic
21C4⋊(C3⋊#170)C2211yescharacteristic
22C4×C3⋊#170C2211yescharacteristic
23C3×C4⋊#170C2211yescharacteristic
24C3×C4×#170C2211yescharacteristic
25#91171—21no—
26Dic3×#170—21no—
27Dic3×#170—21no—
28#68229—51nomaximal, Hall
29C4×#713—31no—
30#65680—31no—
31C4×D5⋊C8—31no—
32Q8⋊#170—31no—
33Q8⋊#170D311yescharacteristic, complemented
34#64871—31no—
35#64871—31no—
36C2×C3×C5×Q8C2×C411yescharacteristic
37#6821—22no—
38#6821C814yescomplemented
39C2×D60—21no—
40C2×D60C2×C411yescharacteristic
41C2×C4×D15—21no—
42C2×C4×D15C2×C411yescharacteristic
43C2×#1200—21no—
44C2×#1200—21no—
45C2×#1200D411yescharacteristic
46C2×#1200D411yescharacteristic
47C2×C3⋊#170—21no—
48C2×C3⋊#170—21no—
49C2×C3×#170—21no—
50C2×C3×#170—21no—
51#6738—24no—
52C5×C4⋊Dic3—21no—
53C5×C4⋊Dic3C2×C411yescharacteristic
54C4×C5×Dic3—21no—
55C4×C5×Dic3C2×C411yescharacteristic
56C3×C4⋊Dic5—21no—
57C3×C4⋊Dic5C2×C411yescharacteristic
58C3×C4×Dic5—21no—
59C3×C4×Dic5C2311yescharacteristic
60#6667—21no—
61#6667—21no—
62#6667—21no—
63Dic3×Dic5—21no—
64Dic3×Dic5D411yescharacteristic
65Dic3×Dic5D411yescharacteristic
66C4×#239—51no—
67#4833—51no—
68C4×D3×C8—51no—
69C3×Q8⋊C8—51no—
70#4535—51no—
71#4483—51no—
72#4459—51no—
73C2×#713—31no—
74C2×D5⋊C8—61no—
75C4×C5×Q8—31no—
76Q8×Dic5C2×D311yescharacteristic
77#4039—31no—
78#4039—61no—
79#4038—31no—
80#4038—61no—
81C4×D20—31no—
82C4×D20—61no—
83C42×D5—31no—
84C42×D5—61no—
85#4003—61no—
86C4⋊#170—31no—
87C4⋊#170C2×D311yescharacteristic
88C4×#170—31no—
89C4×#170—61no—
90C4×#170—61no—
91C4×#170C2×D311yescharacteristic
92#1476—151noSylow
93C22×D15—21no—
94C22×D15C22⋊C411yescharacteristic
95C3×C5×Q8—21no—
96C3×C5×Q8C2×C812yescomplemented
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5C22×C411yescharacteristic
99C2×Dic15C22⋊C411yescharacteristic
100D60—24no—
101D60—21no—
102D60C2×C812yescomplemented
103C4×D15—24no—
104C4×D15C2×C812yes—
105C4×D15M1612yes—
106C2×C5×Dic3—21no—
107C2×C5×Dic3C22⋊C411yescharacteristic
108C2×C5×Dic3C22⋊C411yescharacteristic
109C2×C3×Dic5—21no—
110C2×C3×Dic5C2×D411yescharacteristic
111C2×C3×Dic5C2×D411yescharacteristic
112#1200—21no—
113#1200—21no—
114#1200—22no—
115#1200—41no—
116#1200—41no—
117#1200—42no—
118C3⋊#170—22no—
119C3⋊#170—41no—
120C3×#170—22no—
121C3×#170—41no—
122C2×#239—51no—
123C3×C4×Q8—51no—
124Q8×Dic3AGL(1,𝔽5)11yescharacteristic
125C2×D3×C8—101no—
126#893—51no—
127#893—101no—
128#892—51no—
129#892—101no—
130C4×D12—51no—
131C4×D12—101no—
132C42×D3—51no—
133C42×D3—101no—
134C3×C4⋊C8—51no—
135C3×C4×C8—51no—
136(C2×D3)⋊C8—101no—
137C8×Dic3—101no—
138C8×Dic3—101no—
139C4⋊(C3⋊C8)—51no—
140C4×C3⋊C8—51no—
141C2×C5×Q8C4×D311yescharacteristic
142#713—34no—
143#713—62no—
144C2×D20—31no—
145C2×D20—61no—
146C2×C4×D5—31no—
147C2×C4×D5—31no—
148C2×C4×D5—31no—
149C2×C4×D5—61no—
150C2×C4×D5—61no—
151C2×C4×D5—61no—
152C2×#170—21no—
153C2×#170—21no—
154C2×#170—61no—
155C2×#170—61no—
156D5⋊C8—64no—
157C5×C4⋊C4—31no—
158C5×C4⋊C4—61no—
159C42×C5—31no—
160C42×C5—61no—
161#685—61no—
162#685—61no—
163#685—61no—
164C4⋊Dic5—21no—
165C4⋊Dic5C4×D311yescharacteristic
166C4×Dic5—21no—
167C4×Dic5—31no—
168C4×Dic5—31no—
169C4×Dic5—61no—
170C4×Dic5C22×D311yescharacteristic
171C4×#41—151no—
172#431—151no—
173C2×C4×C8—151no—
174Q8⋊C8—51no—
175Q8⋊C8—151no—
176D4⋊C8—151no—
177D4⋊C8—151no—
178C22×C3×C5C2×C22⋊C411yescharacteristic
179C2×D15—21no—
180C2×D15—21no—
181C2×D15—22no—
182C2×D15—41no—
183C2×D15—41no—
184C2×D15C22⋊C812yes—
185C3×C4×C5—22no—
186C3×C4×C5C22×C811yescharacteristic
187C3×C4×C5C2×M1611yescharacteristic
188Dic15C22⋊C812yes—
189C3×Dic5—21no—
190C3×Dic5—41no—
191C3×Dic5#13511yescharacteristic
192C3×Dic5#13511yescharacteristic
193C5×Dic3—21no—
194C5×Dic3—41no—
195C5×Dic3#10412yes—
196C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
197#239—54no—
198#239—102no—
199C2×D12—51no—
200C2×D12—101no—
201C2×C4×D3—51no—
202C2×C4×D3—51no—
203C2×C4×D3—51no—
204C2×C4×D3—101no—
205C2×C4×D3—101no—
206C2×C4×D3—101no—
207C2×C3×C8—101no—
208C2×C3×C8—101no—
209C3×C4⋊C4—51no—
210C3×C4⋊C4—101no—
211C3×C42—51no—
212C3×C42—101no—
213(C2×D3)⋊C4—101no—
214(C2×D3)⋊C4—101no—
215(C2×D3)⋊C4—101no—
216C4⋊Dic3—21no—
217C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
218C4×Dic3—21no—
219C4×Dic3—51no—
220C4×Dic3—51no—
221C4×Dic3—101no—
222C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
223C2×C3⋊C8—101no—
224C2×C3⋊C8—101no—
225D3×C8—104no—
226C22×D5—31no—
227C22×D5—61no—
228C5×Q8—21no—
229C5×Q8D3×C812yescomplemented
230C2×C4×C5—21no—
231C2×C4×C5—31no—
232C2×C4×C5—31no—
233C2×C4×C5—61no—
234C2×C4×C5C2×C4×D311yescharacteristic
235C2×Dic5—21no—
236C2×Dic5—31no—
237C2×Dic5D3×D411yescharacteristic
238C2×Dic5D3×D411yescharacteristic
239D20—32no—
240D20—64no—
241D20—61no—
242C4×D5—32no—
243C4×D5—32no—
244C4×D5—62no—
245C4×D5—64no—
246C4×D5—61no—
247C4×D5—61no—
248C4×D5—122no—
249C4×D5—121no—
250C4×D5—121no—
251#170—22no—
252#170—41no—
253#170—62no—
254#170—121no—
255C2×#41—151no—
256C22×C8—301no—
257C4×Q8—31no—
258C4×Q8—51no—
259C4×D4—151no—
260C4×D4—151no—
261C4×D4—301no—
262C4×D4—301no—
263#117—151no—
264#117—301no—
265C2×C42—151no—
266C2×C42—301no—
267C4⋊C8—51no—
268C4⋊C8—151no—
269C22⋊C8—301no—
270C4×C8—51no—
271C4×C8—151no—
272C4×C8—301no—
273C4×C8—301no—
274C2×C3×C5C2×C22⋊C811yescharacteristic
275C2×C3×C5#41611yescharacteristic
276C2×C3×C5C2×#10411yescharacteristic
277D15—22no—
278D15—42no—
279C22×D3—51no—
280C22×D3—101no—
281C3×Q8—21no—
282C3×Q8D5⋊C812yescomplemented
283C2×C3×C4—21no—
284C2×C3×C4—51no—
285C2×C3×C4—51no—
286C2×C3×C4—101no—
287C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
288C2×Dic3—21no—
289C2×Dic3—51no—
290C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
291C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
292D12—52no—
293D12—104no—
294D12—101no—
295C4×D3—52no—
296C4×D3—52no—
297C4×D3—101no—
298C4×D3—101no—
299C4×D3—104no—
300C4×D3—102no—
301C4×D3—201no—
302C4×D3—201no—
303C4×D3—202no—
304C3×C8—102no—
305C3×C8—201no—
306C3⋊C8—102no—
307C3⋊C8—201no—
308C22×C5D3×C22⋊C411yescharacteristic
309C2×D5—32no—
310C2×D5—61no—
311C2×D5—61no—
312C2×D5—62no—
313C2×D5—121no—
314C2×D5—121no—
315C4×C5—22no—
316C4×C5—32no—
317C4×C5—61no—
318C4×C5—121no—
319C4×C5C2×D3×C811yescharacteristic
320C4×C5D3×M1611yescharacteristic
321Dic5—21no—
322Dic5—32no—
323Dic5—41no—
324Dic5#91611yescharacteristic
325Dic5#91911yescharacteristic
326#41—154no—
327#41—302no—
328C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
329C2×D4—151no—
330C2×D4—301no—
331C22×C4—151no—
332C22×C4—151no—
333C22×C4—151no—
334C22×C4—301no—
335C22×C4—301no—
336C22×C4—301no—
337C2×C8—101no—
338C2×C8—101no—
339C2×C8—304no—
340C2×C8—301no—
341C2×C8—301no—
342C4⋊C4—31no—
343C4⋊C4—51no—
344C4⋊C4—61no—
345C4⋊C4—101no—
346C22⋊C4—301no—
347C22⋊C4—301no—
348C22⋊C4—301no—
349C42—31no—
350C42—51no—
351C42—61no—
352C42—101no—
353C42—151no—
354C42—151no—
355C42—301no—
356C3×C5#147611yescharacteristic, complemented, Hall
357C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
358C2×D3—52no—
359C2×D3—101no—
360C2×D3—101no—
361C2×D3—102no—
362C2×D3—201no—
363C2×D3—201no—
364C3×C4—22no—
365C3×C4—51no—
366C3×C4—51no—
367C3×C4—101no—
368C3×C4—201no—
369C3×C4C2×D5⋊C811yescharacteristic
370C3×C4D5⋊M1611yescharacteristic
371Dic3—21no—
372Dic3—41no—
373Dic3—52no—
374Dic3#401012yes—
375C2×C5D3×C22⋊C811yescharacteristic
376C2×C5#480411yescharacteristic
377C2×C5D3×#10411yescharacteristic
378D5—62no—
379D5—122no—
380C23—151no—
381C23—301no—
382Q8—21no—
383Q8#673812yescomplemented
384D4—152no—
385D4—304no—
386D4—301no—
387C2×C4—21no—
388C2×C4—31no—
389C2×C4—31no—
390C2×C4—51no—
391C2×C4—51no—
392C2×C4—61no—
393C2×C4—101no—
394C2×C4—151no—
395C2×C4—152no—
396C2×C4—152no—
397C2×C4—304no—
398C2×C4—302no—
399C2×C4—301no—
400C2×C4—301no—
401C2×C4—602no—
402C2×C4—601no—
403C2×C4—601no—
404C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
405C8—102no—
406C8—201no—
407C8—302no—
408C8—601no—
409C2×C3D5⋊(C22⋊C8)11yescharacteristic
410C2×C3C2×#401011yescharacteristic
411C2×C3#6579411yescharacteristic
412D3—102no—
413D3—202no—
414C5#6822911yescharacteristic, complemented, Sylow
415C22—152no—
416C22—301no—
417C22—301no—
418C22—302no—
419C22—601no—
420C22—601no—
421C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
422C4—22no—
423C4—32no—
424C4—51no—
425C4—51no—
426C4—61no—
427C4—101no—
428C4—121no—
429C4—152no—
430C4—201no—
431C4C2×#673811yescharacteristic
432C4#9193111yescharacteristic
433C3#1060443611yescharacteristic, complemented, Sylow
434C2—302no—
435C2—602no—
436C2#1175197111yescharacteristic
437C2D3×#401011yescharacteristic
438C2#1175204211yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC3×C4×C5C22×C8
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×#6821C4
cosocleC3×C4×Dic5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1476
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C4×#6738C2semidirect
3. #11751777C2semidirect
4. #11746801C2semidirect
5. C3×Q8⋊#170C2semidirect
6. #11746491C2semidirect
7. #11746490C2semidirect
33. Q8⋊#170D3semidirect
38. #6821C8semidirect
96. C3×C5×Q8C2×C8semidirect
102. D60C2×C8semidirect
229. C5×Q8D3×C8semidirect
282. C3×Q8D5⋊C8semidirect
356. C3×C5#1476semidirect
383. Q8#6738semidirect
414. C5#68229semidirect
433. C3#10604436semidirect

Maximal subgroups

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

subgroupindex
#117552002
C4×#67382
#117517772
#117468012
C3×Q8⋊#1702
#117464912
#117464902
#106044363
#682295