Finite Group Databasebeta

← #14289894

Subgroups — #14289894

Order 1920.

616 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 92 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
1#11752745C211yesmaximal, characteristic
2#11751188C211yesmaximal, characteristic, complemented
3C5×#5083C211yesmaximal, characteristic, complemented
4C3×#65377C211yesmaximal, characteristic, complemented
5#11745852C211yesmaximal, characteristic, complemented
6#11745839C211yesmaximal, characteristic, complemented
7#11745837C211yesmaximal, characteristic, complemented
8#10601837—31nomaximal, Hall
9C2×#6777—21no—
10C2×D3×D20—21no—
11C3×C5×C4⋊Q8C2211yescharacteristic, complemented
12C4×D60C2211yescharacteristic
13C2×C5×#215—21no—
14C4×C5×D12C2211yescharacteristic
15C3×#4020C2211yescharacteristic
16#91474—22no—
17#91465—21no—
18#91456—21no—
19#91093—21no—
20#91091—21no—
21C4⋊(C3⋊#168)C2211yescharacteristic, complemented
22C5×#835—21no—
23C5×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
24C3×#3941—22no—
25C3×C4×#168C2211yescharacteristic, complemented
26#71198—51nomaximal, Hall
27D4×D20—31no—
28C5×#461—31no—
29#65377D311yescharacteristic, complemented
30#65376—31no—
31D4⋊#168—31no—
32#64705—31no—
33#64703—31no—
34C22×D3×D5—22no—
35C2×C3×C5×Q8—21no—
36C2×D60D411yescharacteristic
37C2×C4×D15—21no—
38C2×C5×D12D411yescharacteristic
39C2×C4×C5×D3—21no—
40C2×C3×D20—21no—
41C2×C3×D20—21no—
42C2×C3×D20D412yes—
43C2×#1202—22no—
44C2×#1201—22no—
45#6777—42no—
46#6777—42no—
47D3×D20—42no—
48D3×D20—42no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4—21no—
51C3×C42×C5C2311yescharacteristic
52#6714—21no—
53C4⋊Dic15D411yescharacteristic
54C2×C3⋊#168—21no—
55C5×#215—42no—
56C5×(C2×D3)⋊C4—21no—
57C5×C4⋊Dic3D411yescharacteristic
58C2×C5×C3⋊C8—21no—
59C2×C3×#168—21no—
60#6665—22no—
61#5717—51no—
62C3×#486—51no—
63#5087—51no—
64#5083D511yescharacteristic, complemented
65#4544—51no—
66#4459—51no—
67#4459—51no—
68C2×D4×D5—61no—
69C22×D20—61no—
70C2×C5×SD16—61no—
71C5×C4⋊Q8C2×D311yescharacteristic, complemented
72C4×C5×D4—31no—
73#4076—61no—
74#4041—61no—
75#4028—62no—
76#4020C2×D311yescharacteristic
77C4×D20—31no—
78C5×C4⋊C8—31no—
79C5×#103—61no—
80#3941—22no—
81#3941—61no—
82#3939—61no—
83C4⋊#168—31no—
84C4×#168C2×D311yescharacteristic, complemented
85#1639—151noSylow
86C22×D15—21no—
87C22×C5×D3—21no—
88C22×C3×D5—22no—
89C2×D3×D5—44no—
90C2×D3×D5—44no—
91C2×D3×D5—82no—
92C3×C5×Q8—41no—
93C3×C5×Q8—41no—
94C2×C3×C4×C5—41no—
95C2×C3×C4×C5C2×D411yescharacteristic
96C2×C3×C4×C5C2×D411yescharacteristic
97C2×C3×C4×C5C2×D411yescharacteristic
98C2×Dic15—21no—
99D60—21no—
100D60SD1612yescomplemented
101C4×D15—41no—
102C4×D15—41no—
103C2×C5×Dic3—21no—
104C5×D12—21no—
105C5×D12D812yes—
106C4×C5×D3—41no—
107C4×C5×D3—41no—
108C3×D20—42no—
109C3×D20—41no—
110C3×D20—41no—
111C3×D20—41no—
112C3×D20—41no—
113#1202—44no—
114#1201—44no—
115C3⋊#168—41no—
116C3×#168—41no—
117C5×C3⋊C8—41no—
118C2×D3×D4—101no—
119C2×D3×D4—101no—
120C3×C4⋊Q8C2×D511yescharacteristic, complemented
121C3×#127—51no—
122C2×#215—21no—
123#938—101no—
124#938—101no—
125#937—102no—
126C4×D12—51no—
127C4×D12C2×D511yescharacteristic
128C3×#102—102no—
129C3×C4×C8—51no—
130#835—21no—
131#809—101no—
132#807—101no—
133C4⋊(C3⋊C8)—51no—
134C4⋊(C3⋊C8)C2×D511yescharacteristic, complemented
135C23×D5—62no—
136C2×C5×Q8—21no—
137C2×C5×D4—31no—
138C22×C4×C5—61no—
139C2×C22⋊D5—62no—
140D4×D5—122no—
141D4×D5—122no—
142C2×D20—21no—
143C2×D20—21no—
144C2×D20—31no—
145C2×D20—62no—
146C2×D20—122no—
147C2×D20—122no—
148C2×D20#6712yes—
149C2×C4×D5—61no—
150C5×SD16—122no—
151C2×C5×C8—61no—
152C5×C4⋊C4—21no—
153C5×C4⋊C4—21no—
154C5×C4⋊C4—31no—
155C5×C22⋊C4—61no—
156C42×C5C22×D311yescharacteristic
157#685—62no—
158#685—61no—
159C4⋊Dic5—31no—
160C2×#168—21no—
161C2×#168—61no—
162D42—151no—
163#486—51no—
164#474—151no—
165#461—31no—
166#331—151no—
167D4⋊C8—151no—
168D4⋊C8—151no—
169C22×C3×C5#12011yescharacteristic
170C2×D15—22no—
171C2×D15—41no—
172C2×D15—41no—
173C2×C5×D3—22no—
174C2×C5×D3—41no—
175C2×C5×D3—41no—
176C2×C3×D5—42no—
177C2×C3×D5—42no—
178C2×C3×D5—42no—
179D3×D5—88no—
180C3×C4×C5—21no—
181C3×C4×C5—81no—
182C3×C4×C5C2×D811yescharacteristic
183C3×C4×C5C2×SD1611yescharacteristic
184C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
185C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
186Dic15—41no—
187C5×Dic3—41no—
188C23×D3—102no—
189C2×C3×Q8—21no—
190C2×C3×D4—52no—
191C2×C3×D4—101no—
192C2×C3×D4—101no—
193C2×#67—102no—
194C2×#67—102no—
195D3×D4—202no—
196D3×D4—202no—
197D3×D4—202no—
198D3×D4—202no—
199C2×D12—51no—
200C2×D12C22⋊D511yescharacteristic
201C2×C4×D3—21no—
202C2×C4×D3—101no—
203C2×C3×C8—101no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C42C22×D511yescharacteristic
207#217—102no—
208#215—42no—
209(C2×D3)⋊C4—21no—
210(C2×D3)⋊C4—101no—
211C4⋊Dic3—51no—
212C4⋊Dic3C22⋊D511yescharacteristic
213C2×C3⋊C8—21no—
214C2×C3⋊C8—101no—
215C23×C5—61no—
216C22×D5—22no—
217C22×D5—61no—
218C22×D5—124no—
219C22×D5—124no—
220C22×D5—242no—
221C5×Q8—41no—
222C5×Q8—41no—
223C5×D4—32no—
224C5×D4—61no—
225C2×C4×C5—41no—
226C2×C4×C5—61no—
227C2×C4×C5—121no—
228C2×C4×C5—121no—
229C2×C4×C5D3×D411yescharacteristic
230C2×C4×C5D3×D411yescharacteristic
231C2×C4×C5C2×#6711yescharacteristic
232C22⋊D5—124no—
233C2×Dic5—61no—
234D20—32no—
235D20—42no—
236D20—41no—
237D20—41no—
238D20—41no—
239D20—41no—
240D20—61no—
241D20—124no—
242C4×D5—121no—
243C4×D5—121no—
244C5×C8—121no—
245#168—41no—
246#168—121no—
247C22×D4—301no—
248C22×D4—301no—
249C2×SD16—61no—
250C4⋊Q8D3×D511yescharacteristic, complemented
251#127—51no—
252#121—301no—
253#121—301no—
254#120—302no—
255C4×D4—31no—
256C4×D4—151no—
257#107—301no—
258C4⋊C8—31no—
259C4⋊C8—151no—
260#103—61no—
261#102—102no—
262#102—301no—
263C4×C8—51no—
264C2×C3×C5#44711yescharacteristic
265C2×C3×C5#45011yescharacteristic
266C2×C3×C5#45511yescharacteristic
267D15—42no—
268C3×D5—82no—
269C5×D3—42no—
270C23×C3—102no—
271C22×D3—21no—
272C22×D3—101no—
273C22×D3—204no—
274C22×D3—204no—
275C22×D3—402no—
276C3×Q8—41no—
277C3×Q8—41no—
278C3×D4—202no—
279C3×D4—201no—
280C3×D4—201no—
281C3×D4—201no—
282C3×D4—201no—
283C2×C3×C4—41no—
284C2×C3×C4D4×D511yescharacteristic
285C2×C3×C4D4×D511yescharacteristic
286C2×C3×C4C2×C22⋊D511yescharacteristic
287#67—204no—
288#67—204no—
289C2×Dic3—21no—
290C2×Dic3—101no—
291D12—21no—
292D12—52no—
293D12—101no—
294D12#68612yes—
295C4×D3—41no—
296C4×D3—41no—
297C4×D3—201no—
298C4×D3—201no—
299C3×C8—201no—
300C3⋊C8—41no—
301C3⋊C8—201no—
302C22×C5—62no—
303C22×C5—121no—
304C22×C5—121no—
305C22×C5#93711yescharacteristic
306C2×D5—42no—
307C2×D5—42no—
308C2×D5—42no—
309C2×D5—62no—
310C2×D5—121no—
311C2×D5—121no—
312C2×D5—248no—
313C4×C5—21no—
314C4×C5—81no—
315C4×C5—121no—
316C4×C5D3×D811yescharacteristic
317C4×C5#91111yescharacteristic
318C4×C5D3×SD1611yescharacteristic
319C4×C5#91411yescharacteristic
320Dic5—121no—
321C24—302no—
322C2×Q8—21no—
323C2×D4—31no—
324C2×D4—52no—
325C2×D4—101no—
326C2×D4—101no—
327C2×D4—151no—
328C2×D4—302no—
329C2×D4—302no—
330C2×D4—602no—
331C2×D4—602no—
332C2×D4—602no—
333C2×D4—602no—
334C22×C4—61no—
335C22×C4—301no—
336SD16—122no—
337C2×C8—61no—
338C2×C8—101no—
339C2×C8—301no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—31no—
343C4⋊C4—151no—
344C22⋊C4—61no—
345C22⋊C4—302no—
346C22⋊C4—301no—
347C42C2×D3×D511yescharacteristic
348C3×C5#163911yescharacteristic, complemented, Hall
349C22×C3—202no—
350C22×C3—202no—
351C22×C3—202no—
352C22×C3#408311yescharacteristic
353C2×D3—22no—
354C2×D3—41no—
355C2×D3—41no—
356C2×D3—102no—
357C2×D3—201no—
358C2×D3—201no—
359C2×D3—408no—
360C3×C4—21no—
361C3×C4—81no—
362C3×C4D5×SD1611yescharacteristic
363C3×C4#406011yescharacteristic
364C3×C4C2×#68611yescharacteristic
365C3×C4#409511yescharacteristic
366Dic3—41no—
367Dic3—201no—
368C2×C5—122no—
369C2×C5#506111yescharacteristic
370C2×C5#515611yescharacteristic
371C2×C5#516911yescharacteristic
372D5—82no—
373D5—122no—
374C23—61no—
375C23—102no—
376C23—301no—
377C23—604no—
378C23—604no—
379C23—1202no—
380Q8—41no—
381Q8—41no—
382D4—32no—
383D4—61no—
384D4—152no—
385D4—202no—
386D4—201no—
387D4—201no—
388D4—201no—
389D4—201no—
390D4—301no—
391D4—604no—
392D4—604no—
393C2×C4—41no—
394C2×C4—61no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—301no—
398C2×C4—601no—
399C2×C4—601no—
400C2×C4D5×#6711yescharacteristic
401C2×C4D3×C22⋊D511yescharacteristic
402C2×C4#679011yescharacteristic
403C8—121no—
404C8—201no—
405C8—601no—
406C2×C3—402no—
407C2×C3#6532511yescharacteristic
408C2×C3#6545811yescharacteristic
409C2×C3#6549111yescharacteristic
410D3—42no—
411D3—202no—
412C5#7119811yescharacteristic, complemented, Sylow
413C22—62no—
414C22—121no—
415C22—121no—
416C22—202no—
417C22—202no—
418C22—202no—
419C22—302no—
420C22—601no—
421C22—601no—
422C22—1208no—
423C22#9157711yescharacteristic
424C4—21no—
425C4—81no—
426C4—121no—
427C4—601no—
428C4D3×#68611yescharacteristic
429C4#9148011yescharacteristic
430C4#9150511yescharacteristic
431C4#9150611yescharacteristic
432C3#1060183711yescharacteristic, complemented, Sylow
433C2—122no—
434C2—402no—
435C2—602no—
436C2#1174915211yescharacteristic
437C2#1174939911yescharacteristic
438C2#1174940911yescharacteristic
subgroupG/H
centerC22#91577
commutator, cosocleC3×C42×C5C23
FrattiniC42C2×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5#120
prime radical#11752745C2
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#1639
LCSG ▷ C3×C42×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #11751188C2semidirect
3. C5×#5083C2semidirect
4. C3×#65377C2semidirect
5. #11745852C2semidirect
6. #11745839C2semidirect
7. #11745837C2semidirect
11. C3×C5×C4⋊Q8C22semidirect
21. C4⋊(C3⋊#168)C22semidirect
23. C5×C4⋊(C3⋊C8)C22semidirect
25. C3×C4×#168C22semidirect
29. #65377D3semidirect
64. #5083D5semidirect
71. C5×C4⋊Q8C2×D3semidirect
84. C4×#168C2×D3semidirect
100. D60SD16semidirect
120. C3×C4⋊Q8C2×D5semidirect
134. C4⋊(C3⋊C8)C2×D5semidirect
250. C4⋊Q8D3×D5semidirect
348. C3×C5#1639semidirect
412. C5#71198semidirect
432. C3#10601837semidirect

Maximal subgroups

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

subgroupindex
#117527452
#117511882
C5×#50832
C3×#653772
#117458522
#117458392
#117458372
#106018373
#711985