Finite Group Databasebeta

← #14289438

Subgroups — #14289438

Order 1920.

512 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 90 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#11752545C211yesmaximal, characteristic
2#11751185C211yesmaximal, characteristic, complemented
3C5×#5091C211yesmaximal, characteristic, complemented
4C3×#65374C211yesmaximal, characteristic, complemented
5#11745852C211yesmaximal, characteristic, complemented
6#11745843C211yesmaximal, characteristic, complemented
7#11745839C211yesmaximal, characteristic, complemented
8#10601953—31nomaximal, Hall
9C2×#6766—21no—
10C3×C5×C4⋊Q8C2211yescharacteristic
11C4×D60C2211yescharacteristic
12C4×C5×Dic6C2211yescharacteristic
13C2×C3×#688—22no—
14C3×#4020C2211yescharacteristic
15#91465—21no—
16#91458—21no—
17Dic3×D20—21no—
18#91409—22no—
19#91093—21no—
20#91092—21no—
21C4⋊(C3⋊#168)C2211yescharacteristic
22C5×#810—21no—
23C5×#807—21no—
24C5×C4⋊(C3⋊C8)C2211yescharacteristic
25C3×C4×#168C2211yescharacteristic
26#71271—51nomaximal, Hall
27#65911—31no—
28C5×#477—31no—
29#65374D311yescharacteristic, complemented
30#65373—31no—
31Q8⋊#168—31no—
32#64705—31no—
33#64703—31no—
34C2×C3×C5×Q8—21no—
35C2×D60D411yescharacteristic
36C2×C4×D15—21no—
37C2×C5×Dic6D411yescharacteristic
38C2×C3×D20—21no—
39C2×C3×D20—21no—
40C2×C3×D20D412yes—
41C2×#1202—22no—
42C2×#1202—22no—
43C2×D5×Dic3—22no—
44#6766—42no—
45#6766—42no—
46C3×C5×C4⋊C4—21no—
47C3×C5×C4⋊C4—21no—
48C3×C42×C5C2311yescharacteristic
49#6714—21no—
50C4⋊Dic15D411yescharacteristic
51C2×C3⋊#168—21no—
52C5×C4⋊Dic3D411yescharacteristic
53C5×Dic3⋊C4—21no—
54C4×C5×Dic3—21no—
55C2×C5×C3⋊C8—21no—
56C3×#688—44no—
57C2×C3×#168—21no—
58#6665—22no—
59#5718—51no—
60C3×#492—51no—
61#5091D511yescharacteristic, complemented
62#5084—51no—
63#4544—51no—
64#4459—51no—
65#4456—51no—
66C2×#713—61no—
67C5×C4⋊Q8C2×D311yescharacteristic
68C4×C5×Q8—31no—
69C2×#688—22no—
70#4041—61no—
71#4041—62no—
72#4020—61no—
73#4020C2×D311yescharacteristic
74C4×D20—31no—
75C4×D20—61no—
76C5×#107—61no—
77C5×C4⋊C8—31no—
78C5×#103—61no—
79#3941—61no—
80#3940—61no—
81C4⋊#168—31no—
82C4×#168C2×D311yescharacteristic
83#1631—151noSylow
84C22×D15—21no—
85C22×C3×D5—22no—
86C3×C5×Q8—42no—
87C2×C3×C4×C5—41no—
88C2×C3×C4×C5C2×D411yescharacteristic
89C2×C3×C4×C5C2×D411yescharacteristic
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×Dic15—21no—
92D60—21no—
93D60—21no—
94C4×D15—41no—
95C4×D15—41no—
96C2×C5×Dic3—21no—
97C2×C5×Dic3—21no—
98C5×Dic6—21no—
99C5×Dic6SD1612yescomplemented
100C3×D20—42no—
101C3×D20—41no—
102C3×D20—41no—
103C3×D20—41no—
104C3×D20—41no—
105#1202—44no—
106#1202—82no—
107D5×Dic3—44no—
108C3⋊#168—41no—
109C3×#168—22no—
110C5×C3⋊C8—41no—
111C2×#237—101no—
112C2×C3×SD16—102no—
113C3×C4⋊Q8C2×D511yescharacteristic
114C3×#127—51no—
115#940—101no—
116#939—102no—
117#938—101no—
118D4×Dic3—101no—
119C4×D12—51no—
120C4×Dic6C2×D511yescharacteristic
121C3×C4×C8—51no—
122#810—21no—
123#809—101no—
124#808—101no—
125#807—21no—
126C4⋊(C3⋊C8)—51no—
127C4⋊(C3⋊C8)C2×D511yescharacteristic
128C2×C5×Q8—21no—
129C2×C5×Q8—31no—
130#713—122no—
131#713—122no—
132C2×D20—21no—
133C2×D20—21no—
134C2×D20—31no—
135C2×D20—62no—
136C2×D20—62no—
137C2×D20#6712yes—
138C2×C4×D5—62no—
139C2×C4×D5—61no—
140C2×C5×C8—61no—
141C5×C4⋊C4—21no—
142C5×C4⋊C4—21no—
143C5×C4⋊C4—31no—
144C5×C4⋊C4—61no—
145C42×C5—61no—
146C42×C5C22×D311yescharacteristic
147#688—44no—
148#685—62no—
149#685—61no—
150C4⋊Dic5—31no—
151C2×#168—21no—
152C2×#168—61no—
153#550—151no—
154#492—51no—
155#477—31no—
156#476—151no—
157#331—151no—
158Q8⋊C8—151no—
159D4⋊C8—151no—
160C22×C3×C5#12011yescharacteristic
161C2×D15—41no—
162C2×D15—41no—
163C2×D15—41no—
164C2×C3×D5—42no—
165C2×C3×D5—42no—
166C2×C3×D5—42no—
167C3×C4×C5—21no—
168C3×C4×C5—81no—
169C3×C4×C5C2×SD1611yescharacteristic
170C3×C4×C5#13511yescharacteristic
171C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
172C3×C4×C5#13711yescharacteristic
173Dic15—41no—
174C5×Dic3—22no—
175C5×Dic3—41no—
176C2×C3×Q8—21no—
177C2×C3×D4—52no—
178C2×C3×D4—101no—
179C2×C3×D4—101no—
180C2×#67—102no—
181C2×#67—102no—
182C22×Dic3—102no—
183#237—202no—
184#237—202no—
185C2×D12—51no—
186C2×C4×D3—101no—
187C2×Dic6C22⋊D511yescharacteristic
188C3×SD16—204no—
189C2×C3×C8—101no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C42C22×D511yescharacteristic
193#217—102no—
194(C2×D3)⋊C4—101no—
195C4⋊Dic3—51no—
196C4⋊Dic3C22⋊D511yescharacteristic
197Dic3⋊C4—21no—
198C4×Dic3—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—101no—
201C22×D5—22no—
202C22×D5—61no—
203C5×Q8—32no—
204C5×Q8—42no—
205C5×Q8—61no—
206C2×C4×C5—41no—
207C2×C4×C5—61no—
208C2×C4×C5—61no—
209C2×C4×C5D3×D411yescharacteristic
210C2×C4×C5D3×D411yescharacteristic
211C2×C4×C5C2×#6711yescharacteristic
212C2×Dic5—61no—
213D20—42no—
214D20—41no—
215D20—41no—
216D20—41no—
217D20—41no—
218D20—61no—
219D20—61no—
220D20—124no—
221D20—242no—
222C4×D5—124no—
223C4×D5—121no—
224C4×D5—121no—
225C5×C8—121no—
226#168—22no—
227#168—121no—
228C2×#41—301no—
229C2×SD16—102no—
230C4⋊Q8D3×D511yescharacteristic
231#127—51no—
232#127—301no—
233#121—301no—
234#121—302no—
235C4×Q8—31no—
236C4×D4—151no—
237C4×D4—301no—
238#107—61no—
239#106—301no—
240C4⋊C8—31no—
241C4⋊C8—151no—
242#103—61no—
243#102—301no—
244C4×C8—51no—
245C2×C3×C5#45011yescharacteristic
246C2×C3×C5#45211yescharacteristic
247C2×C3×C5#45311yescharacteristic
248D15—81no—
249C3×D5—82no—
250C23×C3—102no—
251C22×D3—101no—
252C3×Q8—42no—
253C3×D4—202no—
254C3×D4—201no—
255C3×D4—201no—
256C3×D4—201no—
257C3×D4—201no—
258C2×C3×C4—41no—
259C2×C3×C4D4×D511yescharacteristic
260C2×C3×C4D4×D511yescharacteristic
261C2×C3×C4C2×C22⋊D511yescharacteristic
262#67—204no—
263#67—402no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—101no—
267C2×Dic3—204no—
268D12—101no—
269D12—101no—
270C4×D3—201no—
271C4×D3—201no—
272Dic6—21no—
273Dic6#68812yescomplemented
274C3×C8—102no—
275C3⋊C8—41no—
276C3⋊C8—201no—
277C22×C5#93711yescharacteristic
278C2×D5—42no—
279C2×D5—42no—
280C2×D5—42no—
281C2×D5—121no—
282C2×D5—121no—
283C2×D5—121no—
284C4×C5—21no—
285C4×C5—62no—
286C4×C5—81no—
287C4×C5—121no—
288C4×C5D3×SD1611yescharacteristic
289C4×C5#91411yescharacteristic
290C4×C5#91511yescharacteristic
291C4×C5#91611yescharacteristic
292Dic5—121no—
293#41—602no—
294#41—602no—
295C2×Q8—21no—
296C2×Q8—31no—
297C2×D4—52no—
298C2×D4—101no—
299C2×D4—101no—
300C2×D4—151no—
301C2×D4—302no—
302C2×D4—302no—
303C22×C4—302no—
304C22×C4—301no—
305SD16—204no—
306C2×C8—61no—
307C2×C8—101no—
308C2×C8—301no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—31no—
312C4⋊C4—61no—
313C4⋊C4—151no—
314C22⋊C4—302no—
315C22⋊C4—301no—
316C42—61no—
317C42C2×D3×D511yescharacteristic
318C3×C5#163111yescharacteristic, complemented, Hall
319C22×C3—202no—
320C22×C3—202no—
321C22×C3—202no—
322C22×C3#408311yescharacteristic
323C2×D3—201no—
324C2×D3—201no—
325C2×D3—201no—
326C3×C4—21no—
327C3×C4—81no—
328C3×C4#406411yescharacteristic
329C3×C4#406511yescharacteristic
330C3×C4C2×#68811yescharacteristic
331C3×C4#409511yescharacteristic
332Dic3—22no—
333Dic3—41no—
334Dic3—201no—
335C2×C5#507711yescharacteristic
336C2×C5#516911yescharacteristic
337C2×C5#517311yescharacteristic
338D5—82no—
339D5—241no—
340C23—102no—
341C23—301no—
342Q8—32no—
343Q8—42no—
344Q8—61no—
345D4—202no—
346D4—201no—
347D4—201no—
348D4—201no—
349D4—201no—
350D4—301no—
351D4—301no—
352D4—604no—
353D4—1202no—
354C2×C4—41no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—301no—
358C2×C4—604no—
359C2×C4—601no—
360C2×C4—601no—
361C2×C4D5×#6711yescharacteristic
362C2×C4D3×C22⋊D511yescharacteristic
363C2×C4#679011yescharacteristic
364C8—102no—
365C8—121no—
366C8—601no—
367C2×C3—402no—
368C2×C3#6533511yescharacteristic
369C2×C3#6547611yescharacteristic
370C2×C3#6548711yescharacteristic
371D3—401no—
372C5#7127111yescharacteristic, complemented, Sylow
373C22—202no—
374C22—202no—
375C22—202no—
376C22—601no—
377C22—601no—
378C22—601no—
379C22#9157711yescharacteristic
380C4—21no—
381C4—62no—
382C4—81no—
383C4—121no—
384C4—601no—
385C4D3×#68811yescharacteristic
386C4#9150411yescharacteristic
387C4#9152111yescharacteristic
388C4#9152211yescharacteristic
389C3#1060195311yescharacteristic, complemented, Sylow
390C2—402no—
391C2—1201no—
392C2#1174916311yescharacteristic
393C2#1174939811yescharacteristic
394C2#1174940411yescharacteristic
subgroupG/H
centerC22#91577
commutator, cosocleC3×C42×C5C23
FrattiniC42C2×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5#120
prime radical#11752545C2
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#1631
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. #11751185C2semidirect
3. C5×#5091C2semidirect
4. C3×#65374C2semidirect
5. #11745852C2semidirect
6. #11745843C2semidirect
7. #11745839C2semidirect
29. #65374D3semidirect
61. #5091D5semidirect
99. C5×Dic6SD16semidirect
273. Dic6#688semidirect
318. C3×C5#1631semidirect
372. C5#71271semidirect
389. C3#10601953semidirect

Maximal subgroups

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

subgroupindex
#117525452
#117511852
C5×#50912
C3×#653742
#117458522
#117458432
#117458392
#106019533
#712715