Finite Group Databasebeta

← #14282315

Subgroups — #14282315

Order 1920.

632 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 98 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
1D3×#4039C211yesmaximal, characteristic
2C4×#6739C211yesmaximal, characteristic, complemented
3#11751807C211yesmaximal, characteristic, complemented
4#11746742C211yesmaximal, characteristic, complemented
5C3×#64871C211yesmaximal, characteristic, complemented
6#11746491C211yesmaximal, characteristic, complemented
7#11746490C211yesmaximal, characteristic, complemented
8#10604274—31nomaximal, Hall
9C2×D3×D20C411yescharacteristic
10C2×C4×D3×D5—21no—
11C2×#6739—21no—
12#91782C411yescharacteristic
13C5×D3×C4⋊C4C411yescharacteristic
14C3×#4039C2211yescharacteristic
15D3×#685—21no—
16#91444—21no—
17#91434C411yescharacteristic
18#91418C2211yescharacteristic
19C4×D3×Dic5C2211yescharacteristic
20#91396—21no—
21C4⋊(C3⋊#170)C2211yescharacteristic, complemented
22C4×C3⋊#170C2211yescharacteristic, complemented
23C3×C4⋊#170C2211yescharacteristic, complemented
24C3×C4×#170C2211yescharacteristic, complemented
25(C2×D3)⋊#170—21no—
26#91169—21no—
27#91169—21no—
28#68175—51nomaximal, Hall
29C2×#4039—31no—
30#65759—31no—
31C4×#704—31no—
32#64871—31no—
33#64871—31no—
34#64871—31no—
35#64871D311yescharacteristic, complemented
36C22×D3×D5—21no—
37C2×D60C2×C411yescharacteristic
38C2×C4×D15—21no—
39C2×C4×C5×D3—21no—
40C2×C4×C5×D3C2×C411yescharacteristic
41C2×C3×D20C2×C411yescharacteristic
42C2×C3×C4×D5—21no—
43C2×#1202—21no—
44C2×#1200—21no—
45C2×D3×Dic5D411yescharacteristic
46C2×D3×Dic5D411yescharacteristic
47C2×D5×Dic3—21no—
48D3×D20—22no—
49D3×D20—22no—
50C4×D3×D5—42no—
51C4×D3×D5—42no—
52C2×C3⋊#170—21no—
53C2×C3⋊#170—21no—
54C2×C3×#170—21no—
55C2×C3×#170—21no—
56#6739—24no—
57C3×C5×C4⋊C4C2×C411yescharacteristic
58#6714—21no—
59C4×Dic15C2×C411yescharacteristic
60C5×C4⋊Dic3C2×C411yescharacteristic
61C5×Dic3⋊C4—21no—
62C3×#685—21no—
63C3×C4×Dic5C2311yescharacteristic
64#6667—21no—
65#6665—21no—
66Dic3×Dic5D411yescharacteristic
67Dic3×Dic5D411yescharacteristic
68C4×D3×D4—51no—
69#4835—51no—
70C4×ΓL(1,𝔽25)—51no—
71C3×D4⋊C8—51no—
72D4⋊(C3⋊C8)—51no—
73#4483—51no—
74#4459—51no—
75C22×D20—31no—
76C22×C4×D5—61no—
77C2×#704—61no—
78C2×C5×C4⋊C4—31no—
79C2×#685—61no—
80C2×C4×Dic5—31no—
81#4039—31no—
82#4039—31no—
83#4039—31no—
84#4039—61no—
85#4039—61no—
86#4039C2×D311yescharacteristic
87C22⋊#170—61no—
88#4002—61no—
89#4002—61no—
90C4⋊#170—31no—
91C4⋊#170C2×D311yescharacteristic, complemented
92C4×#170—31no—
93C4×#170C2×D311yescharacteristic, complemented
94#1490—151noSylow
95C22×D15—21no—
96C22×C5×D3C22⋊C411yescharacteristic
97C22×C3×D5—21no—
98C2×D3×D5—24no—
99C2×D3×D5—42no—
100C2×D3×D5—42no—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5C22×C411yescharacteristic
103C2×Dic15C22⋊C411yescharacteristic
104C2×Dic15C22⋊C411yescharacteristic
105D60—21no—
106D60M1612yescomplemented
107C4×D15—42no—
108C2×C5×Dic3—21no—
109C2×C5×Dic3C22⋊C411yescharacteristic
110C4×C5×D3—21no—
111C4×C5×D3—21no—
112C4×C5×D3—42no—
113C2×C3×Dic5C2×D411yescharacteristic
114C2×C3×Dic5C2×D411yescharacteristic
115C3×D20—21no—
116C3×D20M1612yescomplemented
117C3×C4×D5—41no—
118C3×C4×D5—41no—
119#1202—24no—
120#1200—41no—
121#1200—41no—
122D3×Dic5—22no—
123D3×Dic5—21no—
124D3×Dic5—21no—
125D5×Dic3—42no—
126C3⋊#170—22no—
127C3⋊#170—41no—
128C3×#170—22no—
129C3×#170—41no—
130C2×D3×D4—51no—
131C22×C4×D3—101no—
132C3×C4×D4—51no—
133D4×Dic3—51no—
134C4×#67—101no—
135C2×ΓL(1,𝔽25)—101no—
136#893—51no—
137D3×C4⋊C4AGL(1,𝔽5)11yescharacteristic
138#67⋊C4—101no—
139D3×C22⋊C4—101no—
140C4×D12—51no—
141C42×D3—51no—
142C3×C4⋊C8—51no—
143C3×C4×C8—51no—
144(C2×D3)⋊C8—101no—
145#815—101no—
146#815—101no—
147C4⋊(C3⋊C8)—51no—
148C4×C3⋊C8—51no—
149C23×D5—61no—
150C22×C4×C5—31no—
151C22×C4×C5—61no—
152C22×Dic5—31no—
153C22×Dic5—31no—
154C2×D20—31no—
155C2×D20—62no—
156C2×D20—61no—
157C2×D20—62no—
158C2×D20C4×D311yescharacteristic
159C2×C4×D5—21no—
160C2×C4×D5—61no—
161C2×C4×D5—61no—
162C2×C4×D5—61no—
163C2×C4×D5—122no—
164C2×C4×D5—122no—
165#704—64no—
166C2×#170—21no—
167C2×#170—21no—
168C2×#170—61no—
169C2×#170—61no—
170C5×C4⋊C4—31no—
171C5×C4⋊C4—61no—
172C5×C4⋊C4C4×D311yescharacteristic
173#685—21no—
174#685—61no—
175#685—61no—
176#685—61no—
177C4×Dic5—31no—
178C4×Dic5—31no—
179C4×Dic5—31no—
180C4×Dic5C22×D311yescharacteristic
181C2×C4×D4—151no—
182#432—151no—
183C4×M16—151no—
184D4⋊C8—51no—
185D4⋊C8—151no—
186D4⋊C8—151no—
187D4⋊C8—151no—
188C22×C3×C5C2×C22⋊C411yescharacteristic
189C2×D15—22no—
190C2×D15—41no—
191C2×D15—41no—
192C2×C5×D3—21no—
193C2×C5×D3—21no—
194C2×C5×D3#10412yes—
195C2×C3×D5—22no—
196C2×C3×D5—41no—
197C2×C3×D5—41no—
198D3×D5—48no—
199C3×C4×C5—41no—
200C3×C4×C5C2×M1611yescharacteristic
201C3×C4×C5C2×M1611yescharacteristic
202Dic15—21no—
203Dic15—21no—
204C3×Dic5—21no—
205C3×Dic5AGL(1,ℤ/8)11yescharacteristic
206C3×Dic5AGL(1,ℤ/8)11yescharacteristic
207C5×Dic3—21no—
208C5×Dic3—41no—
209C23×D3—101no—
210C2×C3×D4—51no—
211C22×C3×C4—101no—
212C2×#67—101no—
213C22×Dic3—101no—
214D3×D4—102no—
215D3×D4—102no—
216C2×D12—51no—
217C2×C4×D3—21no—
218C2×C4×D3—51no—
219C2×C4×D3—51no—
220C2×C4×D3—101no—
221C2×C4×D3—101no—
222C2×C4×D3—202no—
223C2×C4×D3—202no—
224C2×C4×D3C2×AGL(1,𝔽5)11yescharacteristic
225C2×C3×C8—101no—
226C2×C3×C8—101no—
227C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
228C3×C22⋊C4—101no—
229C3×C42—51no—
230#217—101no—
231(C2×D3)⋊C4—101no—
232(C2×D3)⋊C4—101no—
233C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
234Dic3⋊C4—21no—
235C4×Dic3—51no—
236C4×Dic3—51no—
237C4×Dic3—51no—
238C2×C3⋊C8—101no—
239C2×C3⋊C8—101no—
240ΓL(1,𝔽25)—104no—
241C23×C5—31no—
242C22×D5—21no—
243C22×D5—64no—
244C22×D5—61no—
245C22×D5—122no—
246C22×D5—122no—
247C2×C4×C5—21no—
248C2×C4×C5—31no—
249C2×C4×C5—61no—
250C2×C4×C5—61no—
251C2×C4×C5—61no—
252C2×C4×C5—122no—
253C2×C4×C5C2×C4×D311yescharacteristic
254C2×Dic5—31no—
255C2×Dic5—31no—
256C2×Dic5—62no—
257C2×Dic5—61no—
258C2×Dic5—61no—
259C2×Dic5D3×D411yescharacteristic
260C2×Dic5D3×D411yescharacteristic
261D20—21no—
262D20—32no—
263D20—64no—
264D20—61no—
265D20ΓL(1,𝔽25)12yescomplemented
266C4×D5—41no—
267C4×D5—41no—
268C4×D5—122no—
269C4×D5—122no—
270C4×D5—121no—
271C4×D5—121no—
272#170—22no—
273#170—41no—
274#170—62no—
275#170—121no—
276C22×D4—151no—
277C23×C4—301no—
278C2×M16—301no—
279C4×D4—51no—
280C4×D4—151no—
281C4×D4—151no—
282C4×D4—151no—
283C4×D4—301no—
284C4×D4—301no—
285C2×C4⋊C4—31no—
286C2×C22⋊C4—301no—
287C2×C42—151no—
288C4⋊C8—51no—
289C4⋊C8—151no—
290C22⋊C8—301no—
291#97—301no—
292#97—301no—
293C4×C8—51no—
294C4×C8—151no—
295C2×C3×C5#40711yescharacteristic
296C2×C3×C5#41811yescharacteristic
297C2×C3×C5C2×#10411yescharacteristic
298D15—42no—
299C3×D5—42no—
300C5×D3—22no—
301C23×C3—101no—
302C22×D3—104no—
303C22×D3—101no—
304C22×D3—202no—
305C22×D3—202no—
306C22×D3C22⋊AGL(1,𝔽5)11yescharacteristic
307C3×D4—52no—
308C3×D4—101no—
309C2×C3×C4—21no—
310C2×C3×C4—51no—
311C2×C3×C4—51no—
312C2×C3×C4—201no—
313C2×C3×C4—201no—
314C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
315#67—104no—
316C2×Dic3—21no—
317C2×Dic3—51no—
318C2×Dic3—51no—
319C2×Dic3—202no—
320C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
321D12—52no—
322D12—101no—
323C4×D3—21no—
324C4×D3—21no—
325C4×D3—42no—
326C4×D3—101no—
327C4×D3—101no—
328C4×D3—102no—
329C4×D3—201no—
330C4×D3—201no—
331C4×D3—202no—
332C3×C8—102no—
333C3×C8—201no—
334C3⋊C8—102no—
335C3⋊C8—201no—
336C22×C5—32no—
337C22×C5—61no—
338C22×C5—61no—
339C22×C5D3×C22⋊C411yescharacteristic
340C2×D5—22no—
341C2×D5—41no—
342C2×D5—41no—
343C2×D5—62no—
344C2×D5—128no—
345C2×D5—121no—
346C2×D5—121no—
347C4×C5—41no—
348C4×C5—61no—
349C4×C5—121no—
350C4×C5C2×ΓL(1,𝔽25)11yescharacteristic
351C4×C5D3×M1611yescharacteristic
352Dic5—21no—
353Dic5—61no—
354Dic5—61no—
355Dic5#91111yescharacteristic
356Dic5#91411yescharacteristic
357C24—301no—
358C2×D4—51no—
359C2×D4—151no—
360C2×D4—301no—
361C2×D4—302no—
362C2×D4—302no—
363C22×C4—31no—
364C22×C4—61no—
365C22×C4—101no—
366C22×C4—151no—
367C22×C4—151no—
368C22×C4—301no—
369C22×C4—301no—
370C22×C4—301no—
371C22×C4—602no—
372C22×C4—602no—
373M16—304no—
374C2×C8—101no—
375C2×C8—101no—
376C2×C8—301no—
377C2×C8—301no—
378C4⋊C4—31no—
379C4⋊C4—61no—
380C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
381C22⋊C4—101no—
382C22⋊C4—301no—
383C22⋊C4—301no—
384C22⋊C4—301no—
385C42—51no—
386C42—151no—
387C42—151no—
388C42—151no—
389C3×C5#149011yescharacteristic, complemented, Hall
390C22×C3—102no—
391C22×C3—201no—
392C22×C3—201no—
393C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
394C2×D3—21no—
395C2×D3—21no—
396C2×D3—102no—
397C2×D3—208no—
398C2×D3—201no—
399C2×D3—201no—
400C2×D3#401012yes—
401C3×C4—41no—
402C3×C4—51no—
403C3×C4—51no—
404C3×C4—101no—
405C3×C4D5⋊M1611yescharacteristic
406C3×C4D5⋊M1611yescharacteristic
407Dic3—21no—
408Dic3—41no—
409Dic3—101no—
410Dic3—101no—
411C2×C5—62no—
412C2×C5#472611yescharacteristic
413C2×C5#477011yescharacteristic
414C2×C5D3×#10411yescharacteristic
415D5—42no—
416D5—122no—
417C23—31no—
418C23—101no—
419C23—304no—
420C23—301no—
421C23—602no—
422C23—602no—
423D4—52no—
424D4—101no—
425D4—152no—
426D4—304no—
427D4—301no—
428C2×C4—21no—
429C2×C4—31no—
430C2×C4—51no—
431C2×C4—51no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—122no—
436C2×C4—151no—
437C2×C4—151no—
438C2×C4—201no—
439C2×C4—201no—
440C2×C4—301no—
441C2×C4—301no—
442C2×C4—302no—
443C2×C4—602no—
444C2×C4—601no—
445C2×C4—601no—
446C2×C4—602no—
447C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
448C8—102no—
449C8—201no—
450C8—302no—
451C8—601no—
452C2×C3—202no—
453C2×C3#6575511yescharacteristic
454C2×C3#6576711yescharacteristic
455C2×C3C2×#401011yescharacteristic
456D3—22no—
457D3—202no—
458C5#6817511yescharacteristic, complemented, Sylow
459C22—32no—
460C22—61no—
461C22—61no—
462C22—102no—
463C22—201no—
464C22—201no—
465C22—302no—
466C22—608no—
467C22—601no—
468C22—601no—
469C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
470C4—41no—
471C4—51no—
472C4—51no—
473C4—61no—
474C4—101no—
475C4—121no—
476C4—301no—
477C4—301no—
478C4#9191511yescharacteristic
479C4D5⋊ΓL(1,𝔽25)11yescharacteristic
480C3#1060427411yescharacteristic, complemented, Sylow
481C2—62no—
482C2—202no—
483C2—602no—
484C2#1175191911yescharacteristic
485C2#1175193811yescharacteristic
486C2D3×#401011yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C5×C4⋊C4C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×D3×D20C4
cosocleC3×C4×Dic5C23
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1490
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4×#6739C2semidirect
3. #11751807C2semidirect
4. #11746742C2semidirect
5. C3×#64871C2semidirect
6. #11746491C2semidirect
7. #11746490C2semidirect
21. C4⋊(C3⋊#170)C22semidirect
22. C4×C3⋊#170C22semidirect
23. C3×C4⋊#170C22semidirect
24. C3×C4×#170C22semidirect
35. #64871D3semidirect
91. C4⋊#170C2×D3semidirect
93. C4×#170C2×D3semidirect
106. D60M16semidirect
116. C3×D20M16semidirect
265. D20ΓL(1,𝔽25)semidirect
389. C3×C5#1490semidirect
458. C5#68175semidirect
480. C3#10604274semidirect

Maximal subgroups

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

subgroupindex
D3×#40392
C4×#67392
#117518072
#117467422
C3×#648712
#117464912
#117464902
#106042743
#681755