Finite Group Databasebeta

← #14326204

Subgroups — #14326204

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×C5×#239C211yesmaximal, characteristic
2C2×Q8×Dic15C211yesmaximal, characteristic, complemented
3C2×C3×Q8×Dic5C211yesmaximal, characteristic, complemented
4#11748797C213yesmaximal, complemented
5#11748786C213yesmaximal, complemented
6#11748764C213yesmaximal, complemented
7C4×(C2×D3)⋊Dic5C213yesmaximal, complemented
8#10615820—31nomaximal, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C2×C5×#239—24no—
11C2×C5×#239—24no—
12C2×C5×#239C418yescomplemented
13C22×C5×D12C2213yes—
14C22×C4×C5×D3C2213yes—
15Q8×Dic15—24no—
16C2×C4⋊Dic15C2213yes—
17C2×C4×Dic15C2213yes—
18C3×Q8×Dic5—24no—
19C2×C3×C4⋊Dic5C2213yes—
20C2×C3×C4×Dic5C2213yes—
21C2×(C2×D3)⋊Dic5C2213yes—
22C2×(C2×D3)⋊Dic5C2213yes—
23C2×(C2×D3)⋊Dic5C2216yes—
24#90994C2211yescharacteristic
25#90994C2213yes—
26#82457—51nomaximal, Hall
27C22×C5×#41—31no—
28C2×Q8×Dic5—31no—
29C2×Q8×Dic5D311yescharacteristic, complemented
30#65510—33no—
31#65510—33no—
32#65501—33no—
33C4×C22⋊Dic5—33no—
34C23×C5×D3C2313yes—
35C2×C3×C5×Q8C2×C414yes—
36C2×C3×C5×Q8D414yes—
37C2×C3×C5×Q8D414yes—
38C22×C3×C4×C5C2313yes—
39C22×Dic15C2313yes—
40C22×Dic15C2311yescharacteristic
41C22×C5×Dic3C2311yescharacteristic
42C5×#239—232no—
43C2×C5×D12—26no—
44C2×C5×D12—26no—
45C2×C5×D12C2×C4112yes—
46C2×C4×C5×D3—26no—
47C2×C4×C5×D3—26no—
48C2×C4×C5×D3C2×C4112yes—
49C22×C3×Dic5C2313yes—
50C22×C3×Dic5C2311yescharacteristic
51C4⋊Dic15—26no—
52C4×Dic15—26no—
53C3×C4⋊Dic5—26no—
54C3×C4×Dic5—26no—
55(C2×D3)⋊Dic5—26no—
56(C2×D3)⋊Dic5—26no—
57(C2×D3)⋊Dic5—46no—
58C22×#239D511yescharacteristic
59C2×C3×C4×Q8—51no—
60C2×Q8×Dic3—51no—
61#4988—53no—
62#4987—53no—
63#4939—53no—
64C4×(C2×D3)⋊C4—53no—
65C2×C5×#41—38no—
66C2×C5×#41—64no—
67C2×C5×#41—64no—
68C22×C5×Q8C2×D311yescharacteristic
69C22×C5×D4—33no—
70C23×C4×C5—33no—
71C2×C22⋊Dic5—33no—
72C2×C22⋊Dic5—33no—
73C2×C22⋊Dic5—36no—
74Q8×Dic5—24no—
75Q8×Dic5—64no—
76C2×C4⋊Dic5—33no—
77C2×C4⋊Dic5C2×D313yes—
78C2×C4×Dic5—33no—
79C2×C4×Dic5C2×D313yes—
80#3963—31no—
81#3963—33no—
82#2341—151noSylow
83C23×C3×C5C2411yescharacteristic
84C22×C5×D3—23no—
85C22×C5×D3—23no—
86C22×C5×D3—23no—
87C22×C5×D3—23no—
88C22×C5×D3—23no—
89C22×C5×D3—23no—
90C22×C5×D3C22×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×Dic15—23no—
98C2×Dic15—23no—
99C2×Dic15—23no—
100C2×Dic15#4112yescomplemented
101C2×C5×Dic3—21no—
102C2×C5×Dic3—21no—
103C2×C5×Dic3C22×C412yes—
104C5×D12—224no—
105C4×C5×D3—224no—
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—24no—
114C2×#239—24no—
115C2×#239Dic518yescomplemented
116C22×D12C2×D513yes—
117C22×C4×D3C2×D513yes—
118C3×C4×Q8—104no—
119C2×C3×C4⋊C4—53no—
120C2×C3×C42—53no—
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—
129C24×C5—33no—
130C5×#41—632no—
131C2×C5×Q8C4×D314yes—
132C2×C5×Q8D1214yes—
133C2×C5×Q8#6714yes—
134C2×C5×D4—312no—
135C2×C5×D4—66no—
136C2×C5×D4—66no—
137C22×C4×C5—31no—
138C22×C4×C5—312no—
139C22×C4×C5—66no—
140C22×C4×C5—66no—
141C22×C4×C5C22×D313yes—
142C22×Dic5—31no—
143C22×Dic5—33no—
144C22×Dic5C22×D313yes—
145C22×Dic5C22×D311yescharacteristic
146C22⋊Dic5—66no—
147C22⋊Dic5—66no—
148C22⋊Dic5—126no—
149C4⋊Dic5—26no—
150C4⋊Dic5—66no—
151C4×Dic5—26no—
152C4×Dic5—66no—
153C22×#41—31no—
154C2×C4×Q8—51no—
155C2×C4×Q8—151no—
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×C5×D3—26no—
168C2×C5×D3—26no—
169C2×C5×D3—26no—
170C2×C5×D3—43no—
171C2×C5×D3—43no—
172C2×C5×D3—43no—
173C2×C5×D3—43no—
174C3×C4×C5C2×C22⋊C4112yes—
175Dic15—22no—
176Dic15—43no—
177C3×Dic5—22no—
178C3×Dic5—43no—
179C5×Dic3—22no—
180C23×D3C22×D513yes—
181C2×C3×Q8C2×Dic514yes—
182C2×C3×Q8C22⋊D514yes—
183C2×C3×Q8C22⋊D514yes—
184C22×C3×C4—51no—
185C22×C3×C4—53no—
186C22×C3×C4C22×D513yes—
187C22×Dic3—51no—
188C22×Dic3—53no—
189C22×Dic3C22×D511yescharacteristic
190#239—232no—
191C2×D12—26no—
192C2×D12—26no—
193C2×D12C2×Dic5112yes—
194C2×C4×D3—26no—
195C2×C4×D3—26no—
196C2×C4×D3C2×Dic5112yes—
197C3×C4⋊C4—106no—
198C3×C42—106no—
199(C2×D3)⋊C4—106no—
200(C2×D3)⋊C4—106no—
201(C2×D3)⋊C4—206no—
202C4⋊Dic3—106no—
203C4×Dic3—106no—
204C23×C5—36no—
205C23×C5—63no—
206C23×C5—63no—
207C23×C5—63no—
208C23×C5—63no—
209C23×C5—63no—
210C23×C5—63no—
211C23×C5C23×D311yescharacteristic
212C5×Q8(C2×D3)⋊C4116yescomplemented
213C5×D4—624no—
214C2×C4×C5—32no—
215C2×C4×C5—61no—
216C2×C4×C5—61no—
217C2×C4×C5—624no—
218C2×C4×C5C2×C4×D316yes—
219C2×C4×C5C2×D1216yes—
220C2×C4×C5C2×#6716yes—
221C2×Dic5—21no—
222C2×Dic5—21no—
223C2×Dic5—23no—
224C2×Dic5—23no—
225C2×Dic5—23no—
226C2×Dic5—32no—
227C2×Dic5—61no—
228C2×Dic5—61no—
229C2×Dic5—63no—
230C2×Dic5—63no—
231C2×Dic5—63no—
232C2×Dic5#23912yescomplemented
233C2×#41—38no—
234C2×#41—64no—
235C2×#41—64no—
236C22×Q8D3×D511yescharacteristic
237C22×D4—33no—
238C23×C4—33no—
239C4×Q8—104no—
240C4×Q8—304no—
241C2×C4⋊C4—53no—
242C2×C4⋊C4—153no—
243C2×C22⋊C4—153no—
244C2×C22⋊C4—153no—
245C2×C22⋊C4—156no—
246C2×C42—53no—
247C2×C42—153no—
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
257C5×D3—46no—
258C23×C3C23×D511yescharacteristic
259C22×D3—23no—
260C22×D3—23no—
261C22×D3—23no—
262C22×D3—23no—
263C22×D3—23no—
264C22×D3—23no—
265C22×D3C22×Dic516yes—
266C3×Q8C22⋊Dic5116yescomplemented
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×C4C22×Dic516yes—
274C2×C3×C4C2×C22⋊D516yes—
275C2×C3×C4C2×C22⋊D516yes—
276C2×Dic3—21no—
277C2×Dic3—21no—
278C2×Dic3—52no—
279C2×Dic3—101no—
280C2×Dic3—101no—
281C2×Dic3—103no—
282C2×Dic3—103no—
283C2×Dic3—103no—
284C2×Dic3C22×Dic512yes—
285D12—224no—
286C4×D3—224no—
287C22×C5—66no—
288C22×C5—66no—
289C22×C5—66no—
290C22×C5—123no—
291C22×C5—123no—
292C22×C5—123no—
293C22×C5—123no—
294C22×C5C22×C4×D311yescharacteristic
295C22×C5C22×D1211yescharacteristic
296C22×C5C2×#23911yescharacteristic
297C22×C5#100711yescharacteristic
298C22×C5D3×#4111yescharacteristic
299C22×C5#100911yescharacteristic
300C22×C5C22×#6711yescharacteristic
301C4×C5—62no—
302C4×C5C2×(C2×D3)⋊C4112yes—
303Dic5—22no—
304Dic5—43no—
305Dic5—62no—
306Dic5—123no—
307C24—33no—
308#41—632no—
309C2×Q8D3×Dic514yes—
310C2×Q8#120114yes—
311C2×Q8#120314yes—
312C2×D4—312no—
313C2×D4—66no—
314C2×D4—66no—
315C22×C4—31no—
316C22×C4—312no—
317C22×C4—51no—
318C22×C4—53no—
319C22×C4—66no—
320C22×C4—66no—
321C22×C4—151no—
322C22×C4—153no—
323C22×C4C2×D3×D513yes—
324C4⋊C4—106no—
325C4⋊C4—306no—
326C22⋊C4—306no—
327C22⋊C4—306no—
328C22⋊C4—606no—
329C42—106no—
330C42—306no—
331C3×C5#234111yescharacteristic, complemented, Hall
332C22×C3D5×#4111yescharacteristic
333C22×C3D5×#4111yescharacteristic
334C22×C3#414911yescharacteristic
335C22×C3#415011yescharacteristic
336C22×C3C23×Dic511yescharacteristic
337C22×C3C22×C22⋊D511yescharacteristic
338C22×C3C22×C22⋊D511yescharacteristic
339C2×D3—26no—
340C2×D3—26no—
341C2×D3—26no—
342C2×D3—43no—
343C2×D3—43no—
344C2×D3—43no—
345C2×D3—43no—
346C3×C4—102no—
347C3×C4—203no—
348C3×C4C2×C22⋊Dic5112yes—
349Dic3—22no—
350Dic3—102no—
351Dic3—203no—
352C2×C5—126no—
353C2×C5C4×#23911yescharacteristic
354C2×C5#557411yescharacteristic
355C2×C5#557611yescharacteristic
356C2×C5#557711yescharacteristic
357C2×C5C22×(C2×D3)⋊C411yescharacteristic
358C2×C5#581711yescharacteristic
359C2×C5#583411yescharacteristic
360C23—36no—
361C23—63no—
362C23—63no—
363C23—63no—
364C23—63no—
365C23—63no—
366C23—63no—
367C23C22×D3×D511yescharacteristic
368Q8(C2×D3)⋊Dic5116yescomplemented
369D4—624no—
370C2×C4—32no—
371C2×C4—52no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—624no—
375C2×C4—101no—
376C2×C4—101no—
377C2×C4—103no—
378C2×C4—103no—
379C2×C4—103no—
380C2×C4—152no—
381C2×C4—301no—
382C2×C4—301no—
383C2×C4—303no—
384C2×C4—303no—
385C2×C4—303no—
386C2×C4C2×D3×Dic516yes—
387C2×C4C2×#120116yes—
388C2×C4C2×#120316yes—
389C2×C3#41×Dic511yescharacteristic
390C2×C3#6616111yescharacteristic
391C2×C3#6616511yescharacteristic
392C2×C3#6616511yescharacteristic
393C2×C3#6616811yescharacteristic
394C2×C3#6616811yescharacteristic
395C2×C3C22×C22⋊Dic511yescharacteristic
396D3—46no—
397C5#8245711yescharacteristic, complemented, Sylow
398C22—66no—
399C22—66no—
400C22—66no—
401C22—123no—
402C22—123no—
403C22—123no—
404C22—123no—
405C22#9203011yescharacteristic
406C22D5×#23911yescharacteristic
407C22#9203411yescharacteristic
408C22#9203511yescharacteristic
409C22C22×D3×Dic511yescharacteristic
410C22C22×#120111yescharacteristic
411C22C22×#120311yescharacteristic
412C4—62no—
413C4—102no—
414C4—203no—
415C4—302no—
416C4—603no—
417C4C2×(C2×D3)⋊Dic5112yes—
418C3#1061582011yescharacteristic, complemented, Sylow
419C2—126no—
420C2#1175513411yescharacteristic
421C2Dic5×#23911yescharacteristic
422C2#1175522011yescharacteristic
423C2#1175524411yescharacteristic
424C2#1175524511yescharacteristic
425C2#1175525411yescharacteristic
426C2C22×(C2×D3)⋊Dic511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×Q8C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C5×#239C2
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×Q8×Dic15C2semidirect
3. C2×C3×Q8×Dic5C2semidirect
4. #11748797C2semidirect
5. #11748786C2semidirect
6. #11748764C2semidirect
7. C4×(C2×D3)⋊Dic5C2semidirect
12. C2×C5×#239C4semidirect
29. C2×Q8×Dic5D3semidirect
91. C3×C5×Q8C22⋊C4semidirect
100. C2×Dic15#41semidirect
111. C2×C3×Dic5#41semidirect
115. C2×#239Dic5semidirect
212. C5×Q8(C2×D3)⋊C4semidirect
232. C2×Dic5#239semidirect
266. C3×Q8C22⋊Dic5semidirect
331. C3×C5#2341semidirect
368. Q8(C2×D3)⋊Dic5semidirect
397. C5#82457semidirect
418. C3#10615820semidirect

Maximal subgroups

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

subgroupindex
C22×C5×#2392
C2×Q8×Dic152
C2×C3×Q8×Dic52
#117487972
#117487862
#117487642
C4×(C2×D3)⋊Dic52
#106158203
#824575