Finite Group Databasebeta

← #14326203

Subgroups — #14326203

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×C5×Q8×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×Q8×Dic5C211yesmaximal, characteristic, complemented
4#11748799C213yesmaximal, complemented
5#11748789C213yesmaximal, complemented
6#11748768C213yesmaximal, complemented
7C4×#6667C213yesmaximal, complemented
8#10614188—31nomaximal, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C2×#6821—24no—
11C2×#6821—24no—
12C2×#6821C418yescomplemented
13C22×D60C2213yes—
14C22×C4×D15C2213yes—
15C5×Q8×Dic3—24no—
16C2×C5×C4⋊Dic3C2213yes—
17C2×C4×C5×Dic3C2213yes—
18C3×Q8×Dic5—24no—
19C2×C3×C4⋊Dic5C2213yes—
20C2×C3×C4×Dic5C2213yes—
21C2×#6667C2213yes—
22C2×#6667C2213yes—
23C2×#6667C2216yes—
24#90994C2211yescharacteristic
25#90994C2213yes—
26#82457—51nomaximal, Hall
27C22×#713—31no—
28C2×C4×C5×Q8—31no—
29C2×Q8×Dic5D311yescharacteristic, complemented
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×Dic15C2311yescharacteristic
40#6821—232no—
41C2×D60—26no—
42C2×D60—26no—
43C2×D60C2×C4112yes—
44C2×C4×D15—26no—
45C2×C4×D15—26no—
46C2×C4×D15C2×C4112yes—
47C22×C5×Dic3C2313yes—
48C22×C5×Dic3C2311yescharacteristic
49C22×C3×Dic5C2313yes—
50C22×C3×Dic5C2311yescharacteristic
51C5×C4⋊Dic3—26no—
52C4×C5×Dic3—26no—
53C3×C4⋊Dic5—26no—
54C3×C4×Dic5—26no—
55#6667—26no—
56#6667—26no—
57#6667—46no—
58C22×#239—51no—
59C2×C3×C4×Q8—51no—
60C2×Q8×Dic3D511yescharacteristic, complemented
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—64no—
72C2×C5×C4⋊C4—33no—
73C2×C42×C5—33no—
74Q8×Dic5—24no—
75C2×#685—33no—
76C2×#685—33no—
77C2×#685—36no—
78C2×C4⋊Dic5C2×D313yes—
79C2×C4×Dic5C2×D313yes—
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×C5C22×C416yes—
93C2×C3×C4×C5C2×D416yes—
94C2×C3×C4×C5C2×D416yes—
95C2×Dic15—21no—
96C2×Dic15—21no—
97C2×Dic15C22×C412yes—
98D60—224no—
99C4×D15—224no—
100C2×C5×Dic3—21no—
101C2×C5×Dic3—21no—
102C2×C5×Dic3—23no—
103C2×C5×Dic3—23no—
104C2×C5×Dic3—23no—
105C2×C5×Dic3#4112yescomplemented
106C2×C3×Dic5—21no—
107C2×C3×Dic5—21no—
108C2×C3×Dic5—23no—
109C2×C3×Dic5—23no—
110C2×C3×Dic5—23no—
111C2×C3×Dic5#4112yescomplemented
112C22×C3×Q8C2×D511yescharacteristic
113C2×#239—58no—
114C2×#239—104no—
115C2×#239—104no—
116C22×D12—53no—
117C22×C4×D3—53no—
118C3×C4×Q8—104no—
119C2×C3×C4⋊C4—53no—
120C2×C3×C42—53no—
121Q8×Dic3—24no—
122C2×(C2×D3)⋊C4—53no—
123C2×(C2×D3)⋊C4—53no—
124C2×(C2×D3)⋊C4—56no—
125C2×C4⋊Dic3C2×D513yes—
126C2×C4×Dic3C2×D513yes—
127C3⋊#95—51no—
128C3⋊#95—53no—
129C23×D5—33no—
130C2×C5×Q8C4×D314yes—
131C2×C5×Q8D1214yes—
132C2×C5×Q8#6714yes—
133C22×C4×C5—31no—
134C22×C4×C5—33no—
135C22×C4×C5C22×D313yes—
136C22×Dic5—31no—
137C22×Dic5C22×D313yes—
138C22×Dic5C22×D311yescharacteristic
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—66no—
147C42×C5—66no—
148#685—66no—
149#685—66no—
150#685—126no—
151C4⋊Dic5—26no—
152C4×Dic5—26no—
153C22×#41—151no—
154C2×C4×Q8—31no—
155C2×C4×Q8—51no—
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×C5C2×C22⋊C4112yes—
175Dic15—22no—
176C3×Dic5—22no—
177C3×Dic5—43no—
178C5×Dic3—22no—
179C5×Dic3—43no—
180C23×D3—53no—
181C2×C3×Q8C4×D514yes—
182C2×C3×Q8D2014yes—
183C2×C3×Q8C22⋊D514yes—
184C22×C3×C4—51no—
185C22×C3×C4—53no—
186C22×C3×C4C22×D513yes—
187C22×Dic3—51no—
188C22×Dic3C22×D513yes—
189C22×Dic3C22×D511yescharacteristic
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—106no—
198C3×C42—106no—
199(C2×D3)⋊C4—106no—
200(C2×D3)⋊C4—106no—
201(C2×D3)⋊C4—206no—
202C4⋊Dic3—26no—
203C4×Dic3—26no—
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—32no—
214C2×C4×C5—61no—
215C2×C4×C5—61no—
216C2×C4×C5—63no—
217C2×C4×C5—63no—
218C2×C4×C5—63no—
219C2×C4×C5C2×C4×D316yes—
220C2×C4×C5C2×D1216yes—
221C2×C4×C5C2×#6716yes—
222C2×Dic5—21no—
223C2×Dic5—21no—
224C2×Dic5—23no—
225C2×Dic5—23no—
226C2×Dic5—23no—
227C2×Dic5—32no—
228C2×Dic5—61no—
229C2×Dic5—61no—
230C2×Dic5#23912yescomplemented
231D20—624no—
232C4×D5—624no—
233C2×#41—158no—
234C2×#41—304no—
235C2×#41—304no—
236C22×Q8D3×D511yescharacteristic
237C22×D4—153no—
238C23×C4—153no—
239C4×Q8—64no—
240C4×Q8—104no—
241C2×C4⋊C4—33no—
242C2×C4⋊C4—53no—
243C2×C22⋊C4—153no—
244C2×C22⋊C4—153no—
245C2×C22⋊C4—156no—
246C2×C42—33no—
247C2×C42—53no—
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—52no—
268C2×C3×C4—101no—
269C2×C3×C4—101no—
270C2×C3×C4—103no—
271C2×C3×C4—103no—
272C2×C3×C4—103no—
273C2×C3×C4C2×C4×D516yes—
274C2×C3×C4C2×D2016yes—
275C2×C3×C4C2×C22⋊D516yes—
276C2×Dic3—21no—
277C2×Dic3—21no—
278C2×Dic3—23no—
279C2×Dic3—23no—
280C2×Dic3—23no—
281C2×Dic3—52no—
282C2×Dic3—101no—
283C2×Dic3—101no—
284C2×Dic3#71312yescomplemented
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—62no—
302C4×C5—123no—
303C4×C5C2×(C2×D3)⋊C4112yes—
304Dic5—22no—
305Dic5—43no—
306Dic5—62no—
307C24—153no—
308#41—3032no—
309C2×Q8#120014yes—
310C2×Q8#120214yes—
311C2×Q8#120314yes—
312C2×D4—1512no—
313C2×D4—306no—
314C2×D4—306no—
315C22×C4—31no—
316C22×C4—33no—
317C22×C4—51no—
318C22×C4—53no—
319C22×C4—151no—
320C22×C4—1512no—
321C22×C4—306no—
322C22×C4—306no—
323C22×C4C2×D3×D513yes—
324C4⋊C4—66no—
325C4⋊C4—106no—
326C22⋊C4—306no—
327C22⋊C4—306no—
328C22⋊C4—606no—
329C42—66no—
330C42—106no—
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—102no—
347C3×C4—203no—
348C3×C4C2×#685112yes—
349Dic3—22no—
350Dic3—43no—
351Dic3—102no—
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—
367C23C22×D3×D511yescharacteristic
368Q8#6667116yescomplemented
369D4—3024no—
370C2×C4—32no—
371C2×C4—52no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—63no—
375C2×C4—63no—
376C2×C4—63no—
377C2×C4—101no—
378C2×C4—101no—
379C2×C4—103no—
380C2×C4—103no—
381C2×C4—103no—
382C2×C4—152no—
383C2×C4—301no—
384C2×C4—301no—
385C2×C4—3024no—
386C2×C4C2×#120016yes—
387C2×C4C2×#120216yes—
388C2×C4C2×#120316yes—
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—
405C22#9202911yescharacteristic
406C22D5×#23911yescharacteristic
407C22D3×#71311yescharacteristic
408C22#9203511yescharacteristic
409C22C22×#120011yescharacteristic
410C22C22×#120211yescharacteristic
411C22C22×#120311yescharacteristic
412C4—62no—
413C4—102no—
414C4—123no—
415C4—203no—
416C4—302no—
417C4C2×#6667112yes—
418C3#1061418811yescharacteristic, complemented, Sylow
419C2—606no—
420C2#1175512811yescharacteristic
421C2#1175513111yescharacteristic
422C2#1175514911yescharacteristic
423C2#1175520011yescharacteristic
424C2#1175524511yescharacteristic
425C2#1175525111yescharacteristic
426C2C22×#666711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×Q8C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#6821C2
hypercenterC22×Q8D3×D5
hypocenterC3×C5#2341
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C5×Q8×Dic3C2semidirect
3. C2×C3×Q8×Dic5C2semidirect
4. #11748799C2semidirect
5. #11748789C2semidirect
6. #11748768C2semidirect
7. C4×#6667C2semidirect
12. C2×#6821C4semidirect
29. C2×Q8×Dic5D3semidirect
60. C2×Q8×Dic3D5semidirect
91. C3×C5×Q8C22⋊C4semidirect
105. C2×C5×Dic3#41semidirect
111. C2×C3×Dic5#41semidirect
212. C5×Q8(C2×D3)⋊C4semidirect
230. C2×Dic5#239semidirect
266. C3×Q8#685semidirect
284. C2×Dic3#713semidirect
331. C3×C5#2341semidirect
368. Q8#6667semidirect
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×C5×Q8×Dic32
C2×C3×Q8×Dic52
#117487992
#117487892
#117487682
C4×#66672
#106141883
#824575