Finite Group Databasebeta

← #14288010

Subgroups — #14288010

Order 1920.

512 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 91 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#11756242C211yesmaximal, characteristic
2#11751048C211yesmaximal, characteristic
3#11750840C212yesmaximal, complemented
4C3×C5×#323C211yesmaximal, characteristic
5#11746288C212yesmaximal, complemented
6#10598908—31nomaximal, Hall
7C22×Dic30C2211yescharacteristic
8C2×C3×C5×C22⋊C4C2211yescharacteristic
9C2×#6719—21no—
10C2×#6719—21no—
11#91813—21no—
12#91813C2212yes—
13C2×Dic60—22no—
14#91767—22no—
15#91767—22no—
16C3×C5×C22⋊C8C2212yes—
17#91115—21no—
18#91115C2211yescharacteristic
19#91100—22no—
20#68115—51nomaximal, Hall
21#65817—31no—
22#65236—31no—
23#65028—32no—
24C5×#323D311yescharacteristic
25#64687—32no—
26C24×C3×C5D411yescharacteristic
27C22×C3×C4×C5D411yescharacteristic
28C22×C3×C4×C5C2311yescharacteristic
29C22×Dic15—21no—
30C22×Dic15—21no—
31C22×Dic15—21no—
32C22×Dic15—21no—
33C2×Dic30—22no—
34C2×Dic30—21no—
35C2×Dic30—42no—
36C2×Dic30D412yes—
37C2×C3×C5×C8—22no—
38C3×C5×C22⋊C4—22no—
39#6719—22no—
40#6719—22no—
41#6719—42no—
42C4⋊Dic15—21no—
43C4⋊Dic15D412yes—
44#6712—22no—
45#6712—22no—
46Dic60—44no—
47#5483—51no—
48#4947—51no—
49#4739—52no—
50C3×#323D511yescharacteristic
51#4467—52no—
52C22×Dic10—31no—
53C2×C5×C22⋊C4C2×D311yescharacteristic
54C2×C22⋊Dic5—61no—
55C2×C22⋊Dic5—61no—
56#4070—32no—
57#4070—61no—
58C2×Dic20—62no—
59#4024—62no—
60#4024—62no—
61C5×C22⋊C8C2×D312yes—
62#3963—31no—
63#3963—61no—
64#3948—62no—
65#1602—151noSylow
66C23×C3×C5—21no—
67C23×C3×C5—21no—
68C23×C3×C5—41no—
69C23×C3×C5—41no—
70C23×C3×C5Q1612yescomplemented
71C23×C3×C5C2×D411yescharacteristic
72C2×C3×C4×C5—21no—
73C2×C3×C4×C5—41no—
74C2×C3×C4×C5—41no—
75C2×C3×C4×C5C2×D412yes—
76C2×Dic15—22no—
77C2×Dic15—22no—
78C2×Dic15—22no—
79C2×Dic15—41no—
80C2×Dic15—41no—
81C2×Dic15—41no—
82C2×Dic15—41no—
83C2×Dic15—41no—
84C2×Dic15—81no—
85C2×Dic15—81no—
86Dic30—24no—
87Dic30—42no—
88Dic30—42no—
89Dic30—81no—
90C3×C5×C8—42no—
91C22×Dic6—51no—
92C2×C3×C22⋊C4C2×D511yescharacteristic
93C2×#217—101no—
94C2×#217—101no—
95#924—52no—
96#924—101no—
97C2×Dic12—102no—
98#878—102no—
99#878—102no—
100C3×C22⋊C8C2×D512yes—
101C3⋊#95—51no—
102C3⋊#95—101no—
103#816—102no—
104C24×C5D1211yescharacteristic
105C22×C4×C5D1211yescharacteristic
106C22×C4×C5C22×D311yescharacteristic
107C22×Dic5—61no—
108C22×Dic5—61no—
109C22×Dic5—61no—
110C22×Dic5—61no—
111C2×Dic10—32no—
112C2×Dic10—62no—
113C2×Dic10—61no—
114C2×Dic10—122no—
115C2×C5×C8—22no—
116C5×C22⋊C4—22no—
117C22⋊Dic5—62no—
118C22⋊Dic5—62no—
119C22⋊Dic5—122no—
120C4⋊Dic5—32no—
121C4⋊Dic5—61no—
122#683—62no—
123#683—62no—
124Dic20—124no—
125#543—151no—
126#451—152no—
127#393—151no—
128#328—152no—
129#323D1511yescharacteristic
130C22×C3×C5—21no—
131C22×C3×C5—21no—
132C22×C3×C5—22no—
133C22×C3×C5—42no—
134C22×C3×C5—41no—
135C22×C3×C5—41no—
136C22×C3×C5—81no—
137C22×C3×C5#12011yescharacteristic
138C22×C3×C5C2×Q1611yescharacteristic
139C22×C3×C5#13711yescharacteristic
140C3×C4×C5—22no—
141C3×C4×C5—41no—
142Dic15—42no—
143Dic15—42no—
144Dic15—42no—
145Dic15—81no—
146C24×C3D2011yescharacteristic
147C22×C3×C4D2011yescharacteristic
148C22×C3×C4C22×D511yescharacteristic
149C22×Dic3—101no—
150C22×Dic3—101no—
151C22×Dic3—101no—
152C22×Dic3—101no—
153C2×Dic6—52no—
154C2×Dic6—102no—
155C2×Dic6—101no—
156C2×Dic6—202no—
157C2×C3×C8—22no—
158C3×C22⋊C4—22no—
159#217—102no—
160#217—102no—
161#217—202no—
162C4⋊Dic3—52no—
163C4⋊Dic3—101no—
164Dic3⋊C4—102no—
165Dic3⋊C4—102no—
166Dic12—204no—
167C23×C5—21no—
168C23×C5—21no—
169C23×C5—41no—
170C23×C5—41no—
171C23×C5Dic1212yescomplemented
172C23×C5C2×D1211yescharacteristic
173C2×C4×C5—21no—
174C2×C4×C5—41no—
175C2×C4×C5—41no—
176C2×C4×C5D3×D412yes—
177C2×Dic5—62no—
178C2×Dic5—62no—
179C2×Dic5—62no—
180C2×Dic5—121no—
181C2×Dic5—121no—
182C2×Dic5—121no—
183C2×Dic5—121no—
184C2×Dic5—121no—
185C2×Dic5—241no—
186C2×Dic5—241no—
187Dic10—64no—
188Dic10—122no—
189Dic10—122no—
190Dic10—241no—
191C5×C8—42no—
192C22×Q8—151no—
193C2×Q16—302no—
194C22⋊Q8—152no—
195C22⋊Q8—302no—
196C22⋊Q8—302no—
197C22⋊Q8—301no—
198C2×C22⋊C4—301no—
199C2×C22⋊C4—301no—
200C2×C22⋊C4C2×D1511yescharacteristic
201#103—302no—
202C22⋊C8C2×D1512yes—
203#95—151no—
204#95—301no—
205C2×C3×C5—21no—
206C2×C3×C5—21no—
207C2×C3×C5—42no—
208C2×C3×C5#45111yescharacteristic
209C2×C3×C5#45511yescharacteristic
210C2×C3×C5B(4,𝔽2)11yescharacteristic
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3Dic2012yescomplemented
216C23×C3C2×D2011yescharacteristic
217C2×C3×C4—21no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4D4×D512yes—
221C2×Dic3—102no—
222C2×Dic3—102no—
223C2×Dic3—102no—
224C2×Dic3—201no—
225C2×Dic3—201no—
226C2×Dic3—201no—
227C2×Dic3—201no—
228C2×Dic3—201no—
229C2×Dic3—401no—
230C2×Dic3—401no—
231Dic6—104no—
232Dic6—202no—
233Dic6—202no—
234Dic6—401no—
235C3×C8—42no—
236C22×C5—21no—
237C22×C5—21no—
238C22×C5—22no—
239C22×C5—42no—
240C22×C5—41no—
241C22×C5—41no—
242C22×C5—81no—
243C22×C5#88211yescharacteristic
244C22×C5C2×Dic1211yescharacteristic
245C22×C5#90911yescharacteristic
246C4×C5—22no—
247C4×C5—41no—
248Dic5—122no—
249Dic5—122no—
250Dic5—122no—
251Dic5—241no—
252C24D6011yescharacteristic
253C2×Q8—152no—
254C2×Q8—301no—
255C2×Q8—302no—
256C2×Q8—602no—
257C22×C4—301no—
258C22×C4—301no—
259C22×C4—301no—
260C22×C4—301no—
261C22×C4D6011yescharacteristic
262C22×C4C22×D1511yescharacteristic
263Q16—604no—
264C2×C8—22no—
265C4⋊C4—152no—
266C4⋊C4—302no—
267C4⋊C4—301no—
268C4⋊C4—302no—
269C22⋊C4—22no—
270C22⋊C4—302no—
271C22⋊C4—302no—
272C22⋊C4—602no—
273C3×C5#160211yescharacteristic, complemented, Hall
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—22no—
277C22×C3—42no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—81no—
281C22×C3#402811yescharacteristic
282C22×C3C2×Dic2011yescharacteristic
283C22×C3#405511yescharacteristic
284C3×C4—22no—
285C3×C4—41no—
286Dic3—202no—
287Dic3—202no—
288Dic3—202no—
289Dic3—401no—
290C2×C5—21no—
291C2×C5—21no—
292C2×C5—42no—
293C2×C5#473911yescharacteristic
294C2×C5#474111yescharacteristic
295C2×C5#474611yescharacteristic
296C23—21no—
297C23—21no—
298C23—41no—
299C23—41no—
300C23Dic6012yescomplemented
301C23C2×D6011yescharacteristic
302Q8—304no—
303Q8—602no—
304Q8—602no—
305Q8—1201no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—302no—
310C2×C4—302no—
311C2×C4—302no—
312C2×C4—601no—
313C2×C4—601no—
314C2×C4—601no—
315C2×C4—601no—
316C2×C4—601no—
317C2×C4—1201no—
318C2×C4—1201no—
319C2×C4D4×D1512yes—
320C8—42no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—42no—
324C2×C3#6502811yescharacteristic
325C2×C3#6503011yescharacteristic
326C2×C3#6503511yescharacteristic
327C5#6811511yescharacteristic, complemented, Sylow
328C22—21no—
329C22—21no—
330C22—22no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22#9177111yescharacteristic
336C22C2×Dic6011yescharacteristic
337C22#9179811yescharacteristic
338C4—22no—
339C4—41no—
340C4—602no—
341C4—602no—
342C4—602no—
343C4—1201no—
344C3#1059890811yescharacteristic, complemented, Sylow
345C2—21no—
346C2—21no—
347C2—42no—
348C2#1175084011yescharacteristic
349C2#1175084211yescharacteristic
350C2#1175084711yescharacteristic
subgroupG/H
centerC22#91771
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C22×D15
FittingC3×C5×#323C2
socleC22×C3×C5#120
prime radicalC24×C3×C5D4
hypercenter#323D15
hypocenterC3×C5#1602
LCSG ▷ C22×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
3. #11750840C2semidirect
5. #11746288C2semidirect
70. C23×C3×C5Q16semidirect
171. C23×C5Dic12semidirect
215. C23×C3Dic20semidirect
273. C3×C5#1602semidirect
300. C23Dic60semidirect
327. C5#68115semidirect
344. C3#10598908semidirect

Maximal subgroups

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

subgroupindex
#117562422
#117510482
#117508402
C3×C5×#3232
#117462882
#105989083
#681155