Finite Group Databasebeta

← #14288388

Subgroups — #14288388

Order 1920.

508 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C3×C5×#537C211yesmaximal, characteristic
2#11751134C212yesmaximal, complemented
3#11751082C211yesmaximal, characteristic, complemented
4#11746358C212yesmaximal, complemented
5#11746349C211yesmaximal, characteristic, complemented
6#10602667—31nomaximal, Hall
7C2×C3×C5×#41—21no—
8C3×C5×C22⋊Q8—22no—
9C3×C5×#121—22no—
10C3×C5×#121—21no—
11C3×C5×#121C2212yes—
12C2×C3×C5×C4⋊C4C2211yescharacteristic
13C2×C4⋊Dic15C2211yescharacteristic
14C2×#6712—21no—
15#91116—22no—
16#91115—21no—
17#91115C2211yescharacteristic
18#91114C2212yes—
19#91091—22no—
20#71874—51nomaximal, Hall
21C5×#537D311yescharacteristic
22#65322—32no—
23#65272—31no—
24#64757—32no—
25#64748—31no—
26C3×C5×#41—42no—
27C3×C5×#41—42no—
28C2×C3×C5×Q8—21no—
29C2×C3×C5×D4—21no—
30C2×C3×C5×D4—22no—
31C2×C3×C5×D4—21no—
32C2×C3×C5×D4D412yes—
33C22×C3×C4×C5—22no—
34C22×C3×C4×C5D411yescharacteristic
35C22×C3×C4×C5D411yescharacteristic
36C22×C3×C4×C5C2311yescharacteristic
37C22×Dic15—21no—
38C22×Dic15—21no—
39C3×C5×C4⋊C4—21no—
40C3×C5×C4⋊C4—22no—
41C3×C5×C4⋊C4D412yes—
42C3×C5×C22⋊C4—22no—
43C3×C5×C22⋊C4—22no—
44C4⋊Dic15—22no—
45#6712—42no—
46C2×C3⋊#168—22no—
47C3×#537D511yescharacteristic
48#5033—52no—
49#4981—51no—
50#4537—52no—
51#4528—51no—
52C2×C5×#41—21no—
53C5×C22⋊Q8—22no—
54C5×#121—22no—
55C5×#121—21no—
56C5×#121C2×D312yes—
57C2×C5×C4⋊C4C2×D311yescharacteristic
58C2×C4⋊Dic5—31no—
59C2×#683—61no—
60#3964—62no—
61#3963—31no—
62#3963—61no—
63C22⋊#168—32no—
64#3939—62no—
65#1610—151noSylow
66C23×C3×C5—22no—
67C23×C3×C5C2×D411yescharacteristic
68C3×C5×Q8—42no—
69C3×C5×D4—42no—
70C3×C5×D4—42no—
71C3×C5×D4—42no—
72C3×C5×D4—42no—
73C3×C5×D4—81no—
74C2×C3×C4×C5—21no—
75C2×C3×C4×C5—21no—
76C2×C3×C4×C5—21no—
77C2×C3×C4×C5—22no—
78C2×C3×C4×C5—41no—
79C2×C3×C4×C5—44no—
80C2×C3×C4×C5—41no—
81C2×C3×C4×C5D812yes—
82C2×C3×C4×C5C2×D412yes—
83C2×Dic15—22no—
84C2×Dic15—41no—
85C2×Dic15—41no—
86C2×Dic15—41no—
87C2×Dic15—41no—
88C2×Dic15—41no—
89C3⋊#168—42no—
90C2×C3×#41—21no—
91C3×C22⋊Q8—22no—
92C3×#121—22no—
93C3×#121—21no—
94C3×#121C2×D512yes—
95C2×C3×C4⋊C4C2×D511yescharacteristic
96C2×C4⋊Dic3—51no—
97C2×Dic3⋊C4—101no—
98#832—102no—
99C3⋊#95—51no—
100C3⋊#95—101no—
101#830—52no—
102#807—102no—
103C5×#41—42no—
104C5×#41—42no—
105C2×C5×Q8—21no—
106C2×C5×D4—21no—
107C2×C5×D4—22no—
108C2×C5×D4—21no—
109C2×C5×D4#6712yes—
110C22×C4×C5—22no—
111C22×C4×C5#6711yescharacteristic
112C22×C4×C5#6711yescharacteristic
113C22×C4×C5C22×D311yescharacteristic
114C22×Dic5—61no—
115C22×Dic5—61no—
116C5×C4⋊C4—21no—
117C5×C4⋊C4—22no—
118C5×C4⋊C4#6712yes—
119C5×C22⋊C4—22no—
120C5×C22⋊C4—22no—
121C4⋊Dic5—62no—
122#683—122no—
123C2×#168—62no—
124#537D1511yescharacteristic
125#480—152no—
126#398—151no—
127#327—152no—
128#324—151no—
129C22×C3×C5—21no—
130C22×C3×C5—21no—
131C22×C3×C5—42no—
132C22×C3×C5—42no—
133C22×C3×C5—42no—
134C22×C3×C5#12011yescharacteristic
135C22×C3×C5C2×D811yescharacteristic
136C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
137C3×C4×C5—22no—
138C3×C4×C5—22no—
139C3×C4×C5—42no—
140C3×C4×C5—41no—
141Dic15—42no—
142Dic15—81no—
143C3×#41—42no—
144C3×#41—42no—
145C2×C3×Q8—21no—
146C2×C3×D4—21no—
147C2×C3×D4—22no—
148C2×C3×D4—21no—
149C2×C3×D4C22⋊D512yes—
150C22×C3×C4—22no—
151C22×C3×C4C22⋊D511yescharacteristic
152C22×C3×C4C22⋊D511yescharacteristic
153C22×C3×C4C22×D511yescharacteristic
154C22×Dic3—101no—
155C22×Dic3—101no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4C22⋊D512yes—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—22no—
161C4⋊Dic3—102no—
162Dic3⋊C4—202no—
163C2×C3⋊C8—102no—
164C23×C5—22no—
165C23×C5C2×#6711yescharacteristic
166C5×Q8—42no—
167C5×D4—42no—
168C5×D4—42no—
169C5×D4—42no—
170C5×D4—42no—
171C5×D4—81no—
172C2×C4×C5—21no—
173C2×C4×C5—21no—
174C2×C4×C5—21no—
175C2×C4×C5—22no—
176C2×C4×C5—41no—
177C2×C4×C5—44no—
178C2×C4×C5—41no—
179C2×C4×C5#21312yes—
180C2×C4×C5C2×#6712yes—
181C2×Dic5—62no—
182C2×Dic5—121no—
183C2×Dic5—121no—
184C2×Dic5—121no—
185C2×Dic5—121no—
186C2×Dic5—121no—
187#168—122no—
188C2×#41—21no—
189C22⋊Q8—22no—
190#121—22no—
191#121—21no—
192#121C2×D1512yes—
193C2×C4⋊C4—151no—
194C2×C4⋊C4—301no—
195C2×C4⋊C4C2×D1511yescharacteristic
196#107—302no—
197#102—302no—
198C22⋊C8—152no—
199#95—151no—
200#95—301no—
201C2×C3×C5—21no—
202C2×C3×C5—21no—
203C2×C3×C5—82no—
204C2×C3×C5#44711yescharacteristic
205C2×C3×C5#45611yescharacteristic
206C2×C3×C5#45811yescharacteristic
207C23×C3—22no—
208C23×C3C2×C22⋊D511yescharacteristic
209C3×Q8—42no—
210C3×D4—42no—
211C3×D4—42no—
212C3×D4—42no—
213C3×D4—42no—
214C3×D4—81no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—44no—
221C2×C3×C4—41no—
222C2×C3×C4#68612yes—
223C2×C3×C4C2×C22⋊D512yes—
224C2×Dic3—102no—
225C2×Dic3—201no—
226C2×Dic3—201no—
227C2×Dic3—201no—
228C2×Dic3—201no—
229C2×Dic3—201no—
230C3⋊C8—202no—
231C22×C5—21no—
232C22×C5—21no—
233C22×C5—42no—
234C22×C5—42no—
235C22×C5—42no—
236C22×C5C2×#21311yescharacteristic
237C22×C5#93211yescharacteristic
238C22×C5#95311yescharacteristic
239C4×C5—22no—
240C4×C5—22no—
241C4×C5—42no—
242C4×C5—41no—
243Dic5—122no—
244Dic5—241no—
245#41—42no—
246#41—42no—
247C2×Q8—21no—
248C2×D4—21no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4#122012yes—
252C22×C4—22no—
253C22×C4—301no—
254C22×C4—301no—
255C22×C4#122011yescharacteristic
256C22×C4#122011yescharacteristic
257C22×C4C22×D1511yescharacteristic
258C2×C8—302no—
259C4⋊C4—21no—
260C4⋊C4—22no—
261C4⋊C4—302no—
262C4⋊C4—602no—
263C4⋊C4#122012yes—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C3×C5#161011yescharacteristic, complemented, Hall
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—42no—
270C22×C3—42no—
271C22×C3—42no—
272C22×C3C2×#68611yescharacteristic
273C22×C3#407811yescharacteristic
274C22×C3#409911yescharacteristic
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—42no—
278C3×C4—41no—
279Dic3—202no—
280Dic3—401no—
281C2×C5—21no—
282C2×C5—21no—
283C2×C5—82no—
284C2×C5#521711yescharacteristic
285C2×C5#524311yescharacteristic
286C2×C5#524611yescharacteristic
287C23—22no—
288C23C2×#122011yescharacteristic
289Q8—42no—
290D4—42no—
291D4—42no—
292D4—42no—
293D4—42no—
294D4—81no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—41no—
302C2×C4—302no—
303C2×C4—601no—
304C2×C4—601no—
305C2×C4—601no—
306C2×C4—601no—
307C2×C4—601no—
308C2×C4#671512yes—
309C2×C4C2×#122012yes—
310C8—602no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—82no—
314C2×C3#6550611yescharacteristic
315C2×C3#6553211yescharacteristic
316C2×C3#6553511yescharacteristic
317C5#7187411yescharacteristic, complemented, Sylow
318C22—21no—
319C22—21no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22C2×#671511yescharacteristic
324C22#9182111yescharacteristic
325C22#9184211yescharacteristic
326C4—22no—
327C4—22no—
328C4—42no—
329C4—41no—
330C4—602no—
331C4—1201no—
332C3#1060266711yescharacteristic, complemented, Sylow
333C2—21no—
334C2—21no—
335C2—82no—
336C2#1175131811yescharacteristic
337C2#1175134411yescharacteristic
338C2#1175134711yescharacteristic
subgroupG/H
centerC22#91842
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C22×D15
Fitting, prime radicalC3×C5×#537C2
socleC22×C3×C5#120
hypercenter#537D15
hypocenterC3×C5#1610
LCSG ▷ C22×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
2. #11751134C2semidirect
3. #11751082C2semidirect
4. #11746358C2semidirect
5. #11746349C2semidirect
266. C3×C5#1610semidirect
317. C5#71874semidirect
332. C3#10602667semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5372
#117511342
#117510822
#117463582
#117463492
#106026673
#718745