Finite Group Databasebeta

← #14326168

Subgroups — #14326168

Order 1920.

1744 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6821C211yesmaximal, characteristic
2C2×C3×C4×C5×Q8C211yesmaximal, characteristic, complemented
3C2×Q8×Dic15C211yesmaximal, characteristic, complemented
4#11751089C213yesmaximal, complemented
5#11751088C213yesmaximal, complemented
6#11751040C213yesmaximal, complemented
7C4×#6714C213yesmaximal, complemented
8#10614188—31nomaximal, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C2×#6821—24no—
11C2×#6821—24no—
12C2×#6821C418yescomplemented
13C22×D60C2213yes—
14C22×C4×D15C2213yes—
15C3×C4×C5×Q8—24no—
16C2×C3×C5×C4⋊C4C2213yes—
17C2×C3×C42×C5C2213yes—
18Q8×Dic15—24no—
19C2×#6714C2213yes—
20C2×#6714C2213yes—
21C2×#6714C2216yes—
22C2×C4⋊Dic15C2213yes—
23C2×C4×Dic15C2213yes—
24#91115C2211yescharacteristic
25#91115C2213yes—
26#82457—51nomaximal, Hall
27C22×#713—31no—
28C2×C4×C5×Q8D311yescharacteristic, complemented
29C2×Q8×Dic5—31no—
30#65277—33no—
31#65276—33no—
32#65228—33no—
33C4×#685—33no—
34C23×D15C2313yes—
35C2×C3×C5×Q8C2×C414yes—
36C2×C3×C5×Q8D414yes—
37C2×C3×C5×Q8D414yes—
38C22×C3×C4×C5C2313yes—
39C22×C3×C4×C5C2311yescharacteristic
40C22×C3×C4×C5C2313yes—
41C22×Dic15C2313yes—
42C22×Dic15C2311yescharacteristic
43C22×Dic15C2311yescharacteristic
44#6821—232no—
45C2×D60—26no—
46C2×D60—26no—
47C2×D60C2×C4112yes—
48C2×C4×D15—26no—
49C2×C4×D15—26no—
50C2×C4×D15C2×C4112yes—
51C3×C5×C4⋊C4—26no—
52C3×C42×C5—26no—
53#6714—26no—
54#6714—26no—
55#6714—46no—
56C4⋊Dic15—26no—
57C4×Dic15—26no—
58C22×#239—51no—
59C2×C3×C4×Q8D511yescharacteristic, complemented
60C2×Q8×Dic3—51no—
61#4988—53no—
62#4987—53no—
63#4939—53no—
64C4×(C2×D3)⋊C4—53no—
65C22×C5×Q8C2×D311yescharacteristic
66C2×#713—38no—
67C2×#713—64no—
68C2×#713—64no—
69C22×D20—33no—
70C22×C4×D5—33no—
71C4×C5×Q8—24no—
72C2×C5×C4⋊C4C2×D313yes—
73C2×C42×C5C2×D313yes—
74Q8×Dic5—64no—
75C2×#685—33no—
76C2×#685—33no—
77C2×#685—36no—
78C2×C4⋊Dic5—33no—
79C2×C4×Dic5—33no—
80#3963—31no—
81#3963—33no—
82#2341—151noSylow
83C23×C3×C5C2411yescharacteristic
84C22×D15—23no—
85C22×D15—23no—
86C22×D15—23no—
87C22×D15—23no—
88C22×D15—23no—
89C22×D15—23no—
90C22×D15C22×C416yes—
91C3×C5×Q8C22⋊C4116yescomplemented
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—23no—
95C2×C3×C4×C5—23no—
96C2×C3×C4×C5—23no—
97C2×C3×C4×C5C22×C416yes—
98C2×C3×C4×C5C2×D416yes—
99C2×C3×C4×C5C2×D416yes—
100C2×C3×C4×C5#4112yescomplemented
101C2×Dic15—21no—
102C2×Dic15—21no—
103C2×Dic15—21no—
104C2×Dic15—21no—
105C2×Dic15—23no—
106C2×Dic15—23no—
107C2×Dic15—23no—
108C2×Dic15C22×C412yes—
109C2×Dic15#4112yescomplemented
110D60—224no—
111C4×D15—224no—
112C22×C3×Q8C2×D511yescharacteristic
113C2×#239—58no—
114C2×#239—104no—
115C2×#239—104no—
116C22×D12—53no—
117C22×C4×D3—53no—
118C3×C4×Q8—24no—
119C2×C3×C4⋊C4C2×D513yes—
120C2×C3×C42C2×D513yes—
121Q8×Dic3—104no—
122C2×(C2×D3)⋊C4—53no—
123C2×(C2×D3)⋊C4—53no—
124C2×(C2×D3)⋊C4—56no—
125C2×C4⋊Dic3—53no—
126C2×C4×Dic3—53no—
127C3⋊#95—51no—
128C3⋊#95—53no—
129C23×D5—33no—
130C2×C5×Q8C4×D314yes—
131C2×C5×Q8D1214yes—
132C2×C5×Q8#6714yes—
133C22×C4×C5C22×D313yes—
134C22×C4×C5C22×D311yescharacteristic
135C22×C4×C5C22×D313yes—
136C22×Dic5—31no—
137C22×Dic5—31no—
138C22×Dic5—33no—
139#713—632no—
140C2×D20—312no—
141C2×D20—66no—
142C2×D20—66no—
143C2×C4×D5—312no—
144C2×C4×D5—66no—
145C2×C4×D5—66no—
146C5×C4⋊C4—26no—
147C42×C5—26no—
148#685—66no—
149#685—66no—
150#685—126no—
151C4⋊Dic5—66no—
152C4×Dic5—66no—
153C22×#41—151no—
154C2×C4×Q8—151no—
155C2×C4×Q8D1511yescharacteristic, complemented
156#390—153no—
157#390—153no—
158#380—153no—
159C4×C22⋊C4—153no—
160C22×C3×C5C23×C411yescharacteristic
161C22×C3×C5C22×D411yescharacteristic
162C22×C3×C5C22×D411yescharacteristic
163C22×C3×C5C2×#4111yescharacteristic
164C22×C3×C5C2×#4111yescharacteristic
165C22×C3×C5H(5,𝔽2)11yescharacteristic
166C22×C3×C52−1+411yescharacteristic
167C2×D15—26no—
168C2×D15—26no—
169C2×D15—26no—
170C2×D15—43no—
171C2×D15—43no—
172C2×D15—43no—
173C2×D15—43no—
174C3×C4×C5—22no—
175C3×C4×C5—43no—
176C3×C4×C5C2×C22⋊C4112yes—
177Dic15—22no—
178Dic15—22no—
179Dic15—43no—
180C23×D3—53no—
181C2×C3×Q8C4×D514yes—
182C2×C3×Q8D2014yes—
183C2×C3×Q8C22⋊D514yes—
184C22×C3×C4C22×D513yes—
185C22×C3×C4C22×D511yescharacteristic
186C22×C3×C4C22×D513yes—
187C22×Dic3—51no—
188C22×Dic3—51no—
189C22×Dic3—53no—
190#239—1032no—
191C2×D12—512no—
192C2×D12—106no—
193C2×D12—106no—
194C2×C4×D3—512no—
195C2×C4×D3—106no—
196C2×C4×D3—106no—
197C3×C4⋊C4—26no—
198C3×C42—26no—
199(C2×D3)⋊C4—106no—
200(C2×D3)⋊C4—106no—
201(C2×D3)⋊C4—206no—
202C4⋊Dic3—106no—
203C4×Dic3—106no—
204C23×C5C23×D311yescharacteristic
205C22×D5—36no—
206C22×D5—63no—
207C22×D5—63no—
208C22×D5—63no—
209C22×D5—63no—
210C22×D5—63no—
211C22×D5—63no—
212C5×Q8(C2×D3)⋊C4116yescomplemented
213C2×C4×C5—21no—
214C2×C4×C5—21no—
215C2×C4×C5—23no—
216C2×C4×C5—23no—
217C2×C4×C5—23no—
218C2×C4×C5C2×C4×D316yes—
219C2×C4×C5C2×D1216yes—
220C2×C4×C5#23912yescomplemented
221C2×C4×C5C2×#6716yes—
222C2×Dic5—32no—
223C2×Dic5—32no—
224C2×Dic5—61no—
225C2×Dic5—61no—
226C2×Dic5—61no—
227C2×Dic5—61no—
228C2×Dic5—63no—
229C2×Dic5—63no—
230C2×Dic5—63no—
231D20—624no—
232C4×D5—624no—
233C2×#41—158no—
234C2×#41—304no—
235C2×#41—304no—
236C22×Q8C2×D1511yescharacteristic
237C22×D4—153no—
238C23×C4—153no—
239C4×Q8—24no—
240C4×Q8—304no—
241C2×C4⋊C4—153no—
242C2×C4⋊C4C2×D1513yes—
243C2×C22⋊C4—153no—
244C2×C22⋊C4—153no—
245C2×C22⋊C4—156no—
246C2×C42—153no—
247C2×C42C2×D1513yes—
248#95—151no—
249#95—153no—
250C2×C3×C5C22×C22⋊C411yescharacteristic
251C2×C3×C5C4×#4111yescharacteristic
252C2×C3×C5#52011yescharacteristic
253C2×C3×C5#54811yescharacteristic
254C2×C3×C5#54811yescharacteristic
255C2×C3×C5#55011yescharacteristic
256C2×C3×C5#55011yescharacteristic
257D15—46no—
258C23×C3C23×D511yescharacteristic
259C22×D3—56no—
260C22×D3—103no—
261C22×D3—103no—
262C22×D3—103no—
263C22×D3—103no—
264C22×D3—103no—
265C22×D3—103no—
266C3×Q8#685116yescomplemented
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—23no—
270C2×C3×C4—23no—
271C2×C3×C4—23no—
272C2×C3×C4C2×C4×D516yes—
273C2×C3×C4C2×D2016yes—
274C2×C3×C4#71312yescomplemented
275C2×C3×C4C2×C22⋊D516yes—
276C2×Dic3—52no—
277C2×Dic3—52no—
278C2×Dic3—101no—
279C2×Dic3—101no—
280C2×Dic3—101no—
281C2×Dic3—101no—
282C2×Dic3—103no—
283C2×Dic3—103no—
284C2×Dic3—103no—
285D12—1024no—
286C4×D3—1024no—
287C22×C5C22×C4×D311yescharacteristic
288C22×C5C22×D1211yescharacteristic
289C22×C5C2×#23911yescharacteristic
290C22×C5#100711yescharacteristic
291C22×C5D3×#4111yescharacteristic
292C22×C5#100911yescharacteristic
293C22×C5C22×#6711yescharacteristic
294C2×D5—66no—
295C2×D5—66no—
296C2×D5—66no—
297C2×D5—123no—
298C2×D5—123no—
299C2×D5—123no—
300C2×D5—123no—
301C4×C5—22no—
302C4×C5—43no—
303C4×C5C2×(C2×D3)⋊C4112yes—
304Dic5—62no—
305Dic5—62no—
306Dic5—123no—
307C24—153no—
308#41—3032no—
309C2×Q8C4×D1514yes—
310C2×Q8D6014yes—
311C2×Q8#122014yes—
312C2×D4—1512no—
313C2×D4—306no—
314C2×D4—306no—
315C22×C4—151no—
316C22×C4—151no—
317C22×C4—1512no—
318C22×C4—153no—
319C22×C4—306no—
320C22×C4—306no—
321C22×C4C22×D1513yes—
322C22×C4C22×D1511yescharacteristic
323C22×C4C22×D1513yes—
324C4⋊C4—26no—
325C4⋊C4—306no—
326C22⋊C4—306no—
327C22⋊C4—306no—
328C22⋊C4—606no—
329C42—26no—
330C42—306no—
331C3×C5#234111yescharacteristic, complemented, Hall
332C22×C3C22×C4×D511yescharacteristic
333C22×C3C22×D2011yescharacteristic
334C22×C3C2×#71311yescharacteristic
335C22×C3#414711yescharacteristic
336C22×C3D5×#4111yescharacteristic
337C22×C3#414911yescharacteristic
338C22×C3C22×C22⋊D511yescharacteristic
339C2×D3—106no—
340C2×D3—106no—
341C2×D3—106no—
342C2×D3—203no—
343C2×D3—203no—
344C2×D3—203no—
345C2×D3—203no—
346C3×C4—22no—
347C3×C4—43no—
348C3×C4C2×#685112yes—
349Dic3—102no—
350Dic3—102no—
351Dic3—203no—
352C2×C5C4×#23911yescharacteristic
353C2×C5#557411yescharacteristic
354C2×C5#557611yescharacteristic
355C2×C5#557711yescharacteristic
356C2×C5C22×(C2×D3)⋊C411yescharacteristic
357C2×C5#581711yescharacteristic
358C2×C5#583411yescharacteristic
359D5—126no—
360C23—156no—
361C23—303no—
362C23—303no—
363C23—303no—
364C23—303no—
365C23—303no—
366C23—303no—
367C23C23×D1511yescharacteristic
368Q8#6714116yescomplemented
369D4—3024no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—23no—
373C2×C4—23no—
374C2×C4—23no—
375C2×C4—152no—
376C2×C4—152no—
377C2×C4—301no—
378C2×C4—301no—
379C2×C4—301no—
380C2×C4—301no—
381C2×C4—3024no—
382C2×C4—303no—
383C2×C4—303no—
384C2×C4—303no—
385C2×C4C2×C4×D1516yes—
386C2×C4C2×D6016yes—
387C2×C4#682112yescomplemented
388C2×C4C2×#122016yes—
389C2×C3C4×#71311yescharacteristic
390C2×C3#6590811yescharacteristic
391C2×C3#6591011yescharacteristic
392C2×C3#6591111yescharacteristic
393C2×C3C22×#68511yescharacteristic
394C2×C3#6615111yescharacteristic
395C2×C3#6616811yescharacteristic
396D3—206no—
397C5#8245711yescharacteristic, complemented, Sylow
398C22—306no—
399C22—306no—
400C22—306no—
401C22—603no—
402C22—603no—
403C22—603no—
404C22—603no—
405C22C22×C4×D1511yescharacteristic
406C22C22×D6011yescharacteristic
407C22C2×#682111yescharacteristic
408C22#9209811yescharacteristic
409C22#41×D1511yescharacteristic
410C22#9210011yescharacteristic
411C22C22×#122011yescharacteristic
412C4—22no—
413C4—43no—
414C4—302no—
415C4—302no—
416C4—603no—
417C4C2×#6714112yes—
418C3#1061418811yescharacteristic, complemented, Sylow
419C2—606no—
420C2C4×#682111yescharacteristic
421C2#1175633311yescharacteristic
422C2#1175633511yescharacteristic
423C2#1175633611yescharacteristic
424C2C22×#671411yescharacteristic
425C2#1175657611yescharacteristic
426C2#1175659311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC2×C3×C4×C5×Q8C2
socle, cosocleC23×C3×C5C24
prime radicalC22×#6821C2
hypercenterC2×C4×Q8D15
hypocenterC3×C5#2341
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×C5×Q8C2semidirect
3. C2×Q8×Dic15C2semidirect
4. #11751089C2semidirect
5. #11751088C2semidirect
6. #11751040C2semidirect
7. C4×#6714C2semidirect
12. C2×#6821C4semidirect
28. C2×C4×C5×Q8D3semidirect
59. C2×C3×C4×Q8D5semidirect
91. C3×C5×Q8C22⋊C4semidirect
100. C2×C3×C4×C5#41semidirect
109. C2×Dic15#41semidirect
155. C2×C4×Q8D15semidirect
212. C5×Q8(C2×D3)⋊C4semidirect
220. C2×C4×C5#239semidirect
266. C3×Q8#685semidirect
274. C2×C3×C4#713semidirect
331. C3×C5#2341semidirect
368. Q8#6714semidirect
387. C2×C4#6821semidirect
397. C5#82457semidirect
418. C3#10614188semidirect

Maximal subgroups

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

subgroupindex
C22×#68212
C2×C3×C4×C5×Q82
C2×Q8×Dic152
#117510892
#117510882
#117510402
C4×#67142
#106141883
#824575