Finite Group Databasebeta

← #14322625

Subgroups — #14322625

Order 1920.

1344 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×C3×C22⋊Dic5C211yesmaximal, characteristic
2#11751348C211yesmaximal, characteristic
3C5×#5247C211yesmaximal, characteristic
4#11749506C212yesmaximal, complemented
5#11749486C214yesmaximal, complemented
6#11749485C214yesmaximal, complemented
7Dic5×#217C212yesmaximal, complemented
8#10605551—31nomaximal, Hall
9C25×C3×C5C2211yescharacteristic
10C23×C3×Dic5C2211yescharacteristic
11C23×C3×Dic5C2211yescharacteristic
12C2×#6719C2212yes—
13C2×#6719C2212yes—
14C2×#6719C2212yes—
15C2×C5×#217C2212yes—
16C2×C5×#217C2212yes—
17C2×C5×#217C2212yes—
18C2×C3×C22⋊Dic5C418yescomplemented
19C2×C3×C22⋊Dic5C2212yes—
20C2×C3×C22⋊Dic5C2212yes—
21C2×#6668C2214yescomplemented
22C2×Dic3×Dic5C2214yescomplemented
23#90994C2214yescomplemented
24#90994C2214yescomplemented
25#82093—51nomaximal, Hall
26C22×C22⋊Dic5D311yescharacteristic
27C5×#379—31no—
28#65536—31no—
29#65234—34no—
30#65233—34no—
31#65231—32no—
32C22⋊C4×Dic5—32no—
33C24×C3×C5—24no—
34C24×C3×C5—42no—
35C24×C3×C5—42no—
36C24×C3×C5C2×C414yes—
37C24×C3×C5C2311yescharacteristic
38C24×C3×C5C2311yescharacteristic
39C24×C3×C5C2311yescharacteristic
40C22×Dic15C2314yes—
41C22×C5×Dic3C2314yes—
42C22×C3×Dic5—42no—
43C22×C3×Dic5—42no—
44C22×C3×Dic5C2×C414yes—
45C22×C3×Dic5C2×C414yes—
46C22×C3×Dic5C2312yes—
47C22×C3×Dic5C2312yes—
48#6719—24no—
49#6719—42no—
50#6719—42no—
51C5×#217—24no—
52C5×#217—42no—
53C5×#217—42no—
54C3×C22⋊Dic5C2×C418yes—
55C3×C22⋊Dic5C2×C418yes—
56#6668—44no—
57Dic3×Dic5—44no—
58C22×C3×C22⋊C4—51no—
59#5247—51no—
60#5247D511yescharacteristic
61#5212—52no—
62C4×#217—52no—
63#4951—54no—
64#4951—54no—
65C25×C5C2×D311yescharacteristic
66C23×Dic5C2×D311yescharacteristic
67C23×Dic5C2×D311yescharacteristic
68C2×C5×C22⋊C4—32no—
69C2×C5×C22⋊C4—32no—
70C2×C5×C22⋊C4—32no—
71C2×C22⋊Dic5—32no—
72C2×C22⋊Dic5—32no—
73C2×C22⋊Dic5—32no—
74C2×C22⋊Dic5Dic318yescomplemented
75C2×C22⋊Dic5C2×D312yes—
76C2×C22⋊Dic5C2×D312yes—
77C2×#683—34no—
78C2×C4×Dic5—34no—
79#3963—34no—
80#3963—34no—
81#2312—151noSylow
82C23×C3×C5—22no—
83C23×C3×C5—22no—
84C23×C3×C5—28no—
85C23×C3×C5—22no—
86C23×C3×C5—22no—
87C23×C3×C5—22no—
88C23×C3×C5—22no—
89C23×C3×C5—22no—
90C23×C3×C5—22no—
91C23×C3×C5—44no—
92C23×C3×C5—44no—
93C23×C3×C5—416no—
94C23×C3×C5C22×C412yes—
95C23×C3×C5C22×C412yes—
96C23×C3×C5C22×C412yes—
97C23×C3×C5#4114yes—
98C23×C3×C5C2411yescharacteristic
99C2×Dic15—24no—
100C2×Dic15—24no—
101C2×Dic15—24no—
102C2×C5×Dic3—24no—
103C2×C5×Dic3—24no—
104C2×C5×Dic3—24no—
105C2×C3×Dic5—24no—
106C2×C3×Dic5—24no—
107C2×C3×Dic5—44no—
108C2×C3×Dic5—44no—
109C2×C3×Dic5C22×C414yes—
110C2×C3×Dic5C22×C414yes—
111C25×C3C2×D511yescharacteristic
112C23×C3×C4—51no—
113C23×C3×C4—51no—
114C2×C3×C22⋊C4—58no—
115C2×C3×C22⋊C4—52no—
116C2×C3×C22⋊C4—52no—
117C2×#217—52no—
118C2×#217—52no—
119C2×#217—52no—
120C2×#217C2×D512yes—
121C2×#217C2×D512yes—
122C2×#217C2×D512yes—
123C2×C4⋊Dic3—54no—
124C2×C4×Dic3—54no—
125C3⋊#95—54no—
126C3⋊#95—54no—
127C24×C5—24no—
128C24×C5—42no—
129C24×C5—42no—
130C24×C5C2×Dic314yes—
131C24×C5C22×D311yescharacteristic
132C24×C5C22×D311yescharacteristic
133C24×C5C22×D311yescharacteristic
134C22×C4×C5—34no—
135C22×Dic5—34no—
136C22×Dic5—42no—
137C22×Dic5—42no—
138C22×Dic5C2×Dic314yes—
139C22×Dic5C2×Dic314yes—
140C22×Dic5C22×D312yes—
141C22×Dic5C22×D312yes—
142C5×C22⋊C4—64no—
143C5×C22⋊C4—122no—
144C5×C22⋊C4—122no—
145C22⋊Dic5—64no—
146C22⋊Dic5—122no—
147C22⋊Dic5—122no—
148C22⋊Dic5C2×Dic318yes—
149C22⋊Dic5C2×Dic318yes—
150#683—124no—
151C4×Dic5—124no—
152C22×C22⋊C4—51no—
153#388—154no—
154#388—154no—
155#380—152no—
156#379—31no—
157#379—151no—
158C4×C22⋊C4—152no—
159C22×C3×C5—22no—
160C22×C3×C5—22no—
161C22×C3×C5—28no—
162C22×C3×C5—22no—
163C22×C3×C5—22no—
164C22×C3×C5—22no—
165C22×C3×C5—22no—
166C22×C3×C5—22no—
167C22×C3×C5—22no—
168C22×C3×C5—416no—
169C22×C3×C5—44no—
170C22×C3×C5—44no—
171C22×C3×C5#11714yes—
172C22×C3×C5C23×C411yescharacteristic
173C22×C3×C5C2×#4112yes—
174C22×C3×C5H(5,𝔽2)12yes—
175C22×C3×C5H(5,𝔽2)12yes—
176Dic15—44no—
177C3×Dic5—24no—
178C3×Dic5—24no—
179C5×Dic3—44no—
180C24×C3—24no—
181C24×C3—42no—
182C24×C3—42no—
183C24×C3C4×D514yes—
184C24×C3C22×D511yescharacteristic
185C24×C3C22×D511yescharacteristic
186C24×C3C22×D511yescharacteristic
187C22×C3×C4—52no—
188C22×C3×C4—52no—
189C22×C3×C4—54no—
190C22×C3×C4—54no—
191C22×C3×C4—202no—
192C22×C3×C4—202no—
193C22×Dic3—54no—
194C22×Dic3C22×D514yes—
195C3×C22⋊C4—58no—
196C3×C22⋊C4—58no—
197#217—24no—
198#217—42no—
199#217—42no—
200#217—104no—
201#217—202no—
202#217—202no—
203C4⋊Dic3—204no—
204C4×Dic3—204no—
205C23×C5—22no—
206C23×C5—22no—
207C23×C5—28no—
208C23×C5—22no—
209C23×C5—22no—
210C23×C5—22no—
211C23×C5—22no—
212C23×C5—22no—
213C23×C5—22no—
214C23×C5—44no—
215C23×C5—44no—
216C23×C5—416no—
217C23×C5#23514yes—
218C23×C5C22×Dic312yes—
219C23×C5C22×Dic312yes—
220C23×C5C22×Dic312yes—
221C23×C5C23×D311yescharacteristic
222C2×C4×C5—64no—
223C2×C4×C5—64no—
224C2×C4×C5—64no—
225C2×Dic5—24no—
226C2×Dic5—24no—
227C2×Dic5—44no—
228C2×Dic5—44no—
229C2×Dic5—64no—
230C2×Dic5—64no—
231C2×Dic5—64no—
232C2×Dic5C22×Dic314yes—
233C2×Dic5C22×Dic314yes—
234C25D3×D511yescharacteristic
235C23×C4—51no—
236C23×C4—51no—
237C2×C4⋊C4—154no—
238C2×C22⋊C4—32no—
239C2×C22⋊C4—32no—
240C2×C22⋊C4—32no—
241C2×C22⋊C4—58no—
242C2×C22⋊C4—52no—
243C2×C22⋊C4—52no—
244C2×C22⋊C4—152no—
245C2×C22⋊C4—152no—
246C2×C22⋊C4—152no—
247C2×C42—154no—
248#95—154no—
249#95—154no—
250C2×C3×C5—24no—
251C2×C3×C5—42no—
252C2×C3×C5—42no—
253C2×C3×C5C2×#11711yescharacteristic
254C2×C3×C5#51811yescharacteristic
255C2×C3×C5#51811yescharacteristic
256C2×C3×C5#53814yes—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—28no—
266C23×C3—44no—
267C23×C3—44no—
268C23×C3—416no—
269C23×C3C2×C4×D512yes—
270C23×C3C2×C4×D512yes—
271C23×C3C2×C4×D512yes—
272C23×C3#71114yes—
273C23×C3C23×D511yescharacteristic
274C2×C3×C4—54no—
275C2×C3×C4—54no—
276C2×C3×C4—104no—
277C2×C3×C4—104no—
278C2×C3×C4—204no—
279C2×C3×C4—204no—
280C2×Dic3—24no—
281C2×Dic3—24no—
282C2×Dic3—24no—
283C2×Dic3—104no—
284C2×Dic3—104no—
285C2×Dic3—104no—
286C22×C5—28no—
287C22×C5—22no—
288C22×C5—22no—
289C22×C5—22no—
290C22×C5—22no—
291C22×C5—22no—
292C22×C5—22no—
293C22×C5—22no—
294C22×C5—22no—
295C22×C5—416no—
296C22×C5—44no—
297C22×C5—44no—
298C22×C5#92614yes—
299C22×C5C2×#23512yes—
300C22×C5#100412yes—
301C22×C5#100412yes—
302C22×C5C23×Dic311yescharacteristic
303C4×C5—124no—
304Dic5—24no—
305Dic5—24no—
306Dic5—124no—
307C24—24no—
308C24—42no—
309C24—42no—
310C24D5×Dic314yes—
311C24C2×D3×D511yescharacteristic
312C24C2×D3×D511yescharacteristic
313C24C2×D3×D511yescharacteristic
314C22×C4—34no—
315C22×C4—52no—
316C22×C4—52no—
317C22×C4—54no—
318C22×C4—54no—
319C22×C4—154no—
320C22×C4—202no—
321C22×C4—202no—
322C4⋊C4—604no—
323C22⋊C4—58no—
324C22⋊C4—58no—
325C22⋊C4—64no—
326C22⋊C4—122no—
327C22⋊C4—122no—
328C22⋊C4—304no—
329C22⋊C4—602no—
330C22⋊C4—602no—
331C42—604no—
332C3×C5#231211yescharacteristic, complemented, Hall
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—28no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—416no—
343C22×C3—44no—
344C22×C3—44no—
345C22×C3#402314yes—
346C22×C3C22×C4×D511yescharacteristic
347C22×C3C2×#71112yes—
348C22×C3#414412yes—
349C22×C3#414412yes—
350C3×C4—104no—
351C3×C4—104no—
352Dic3—44no—
353Dic3—204no—
354C2×C5—24no—
355C2×C5—42no—
356C2×C5—42no—
357C2×C5#549414yes—
358C2×C5C2×#92611yescharacteristic
359C2×C5#579811yescharacteristic
360C2×C5#579811yescharacteristic
361C23—22no—
362C23—22no—
363C23—28no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—44no—
371C23—44no—
372C23—416no—
373C23C2×D5×Dic312yes—
374C23C2×D5×Dic312yes—
375C23C2×D5×Dic312yes—
376C23#678214yes—
377C23C22×D3×D511yescharacteristic
378C2×C4—54no—
379C2×C4—54no—
380C2×C4—64no—
381C2×C4—64no—
382C2×C4—64no—
383C2×C4—104no—
384C2×C4—104no—
385C2×C4—204no—
386C2×C4—204no—
387C2×C4—304no—
388C2×C4—304no—
389C2×C4—304no—
390C2×C3—24no—
391C2×C3—42no—
392C2×C3—42no—
393C2×C3C2×#402311yescharacteristic
394C2×C3#6582011yescharacteristic
395C2×C3#6582011yescharacteristic
396C2×C3#6592714yes—
397C5#8209311yescharacteristic, complemented, Sylow
398C22—22no—
399C22—22no—
400C22—28no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—416no—
408C22—44no—
409C22—44no—
410C22#9152814yes—
411C22C22×D5×Dic311yescharacteristic
412C22C2×#678212yes—
413C22#9204912yes—
414C22#9204912yes—
415C4—104no—
416C4—104no—
417C4—124no—
418C4—604no—
419C3#1060555111yescharacteristic, complemented, Sylow
420C2—24no—
421C2—42no—
422C2—42no—
423C2C2×#9152811yescharacteristic
424C2#1175540611yescharacteristic
425C2#1175540611yescharacteristic
426C2#1175541114yes—
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
Fitting, prime radicalC25×C3×C5C22
socle, cosocleC23×C3×C5C24
hypercenterC25D3×D5
hypocenterC3×C5#2312
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C25

Complements

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

NH ≅ G/Ntype
4. #11749506C2semidirect
5. #11749486C2semidirect
6. #11749485C2semidirect
7. Dic5×#217C2semidirect
18. C2×C3×C22⋊Dic5C4semidirect
21. C2×#6668C22semidirect
22. C2×Dic3×Dic5C22semidirect
23. #90994C22semidirect
24. #90994C22semidirect
74. C2×C22⋊Dic5Dic3semidirect
332. C3×C5#2312semidirect
397. C5#82093semidirect
419. C3#10605551semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Dic52
#117513482
C5×#52472
#117495062
#117494862
#117494852
Dic5×#2172
#106055513
#820935