Finite Group Databasebeta

← #11754317

Subgroups — #11754317

Order 960.

540 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C22×C3×Dic10C211yesmaximal, characteristic, complemented
2C2×#6782C211yesmaximal, characteristic
3#91818C211yesmaximal, characteristic, complemented
4C5×#929C211yesmaximal, characteristic, complemented
5#91546C211yesmaximal, characteristic, complemented
6#91539C211yesmaximal, characteristic, complemented
7#91528C211yesmaximal, characteristic, complemented
8#91411C212yesmaximal, complemented
9#91410C212yesmaximal, complemented
10#91327C212yesmaximal, complemented
11#91326C212yesmaximal, complemented
12#65982—31nomaximal, Hall
13C22×C3×C4×C5C2211yescharacteristic
14C2×#1220C2211yescharacteristic
15C2×C5×#67C2211yescharacteristic
16C22×C3×Dic5C2211yescharacteristic
17C22×C3×Dic5C2211yescharacteristic
18C2×C3×Dic10—24no—
19C2×C3×Dic10C2212yes—
20C2×C3×Dic10C2212yes—
21C2×#1204C2211yescharacteristic
22C2×#1203C2211yescharacteristic
23C2×#1200C2211yescharacteristic
24#6782—24no—
25C2×D3×Dic5C2211yescharacteristic
26#6719C2211yescharacteristic
27#6714C2212yes—
28#6712C2212yes—
29C5×#217C2211yescharacteristic
30C5×(C2×D3)⋊C4C2212yes—
31C5×Dic3⋊C4C2212yes—
32Dic3⋊Dic5C2212yes—
33#6669C2212yes—
34#6668C2212yes—
35#6667C2212yes—
36(C2×D3)⋊Dic5C2212yes—
37Dic3×Dic5C2212yes—
38#5816—51nomaximal, Hall
39C2×#711—31no—
40C22×Dic10D311yescharacteristic, complemented
41C5×#123—31no—
42#4081—31no—
43#4075—31no—
44#4042—32no—
45#4034—32no—
46#4032—31no—
47#4031—32no—
48#4024—32no—
49#4023—31no—
50C23×C3×C5C2311yescharacteristic
51C22×D15C2311yescharacteristic
52C22×C5×D3C2311yescharacteristic
53C2×C3×C4×C5—22no—
54C2×C3×C4×C5C2312yes—
55#1220—22no—
56C2×Dic15C2312yes—
57C2×Dic15C2311yescharacteristic
58C5×#67—22no—
59C2×C5×Dic3C2312yes—
60C2×C5×Dic3C2311yescharacteristic
61C2×C3×Dic5—22no—
62C2×C3×Dic5D414yes—
63C2×C3×Dic5C2312yes—
64C2×C3×Dic5C2311yescharacteristic
65C2×C3×Dic5C2311yescharacteristic
66C3×Dic10—24no—
67C3×Dic10—24no—
68#1204—22no—
69#1203—22no—
70#1200—22no—
71D3×Dic5—22no—
72C22×C3×Q8—51no—
73C2×#235—51no—
74#947—52no—
75#946—52no—
76#946—52no—
77#944—52no—
78#929—51no—
79#929—51no—
80#929—51no—
81#929D511yescharacteristic, complemented
82#926—51no—
83C2×C5×D4—31no—
84C22×C4×C5C2×D311yescharacteristic
85C2×C22⋊D5—31no—
86C2×C22⋊D5—31no—
87C22×Dic5—31no—
88C22×Dic5C2×D311yescharacteristic
89C22×Dic5C2×D311yescharacteristic
90#711—64no—
91C2×C4×D5—31no—
92C2×Dic10—24no—
93C2×Dic10—31no—
94C2×Dic10C2×D312yes—
95C2×Dic10C2×D312yes—
96C5×C4⋊C4—32no—
97C5×C22⋊C4—31no—
98C5×C22⋊C4—32no—
99C22⋊Dic5—31no—
100C22⋊Dic5—32no—
101#685—32no—
102#685—32no—
103C4⋊Dic5—32no—
104#683—32no—
105#683—32no—
106#683—32no—
107C4×Dic5—32no—
108#536—151noSylow
109C22×C3×C5—22no—
110C22×C3×C5C2×D412yes—
111C22×C3×C5C2411yescharacteristic
112C2×D15—21no—
113C2×D15—22no—
114C2×C5×D3—21no—
115C2×C5×D3—22no—
116C3×C4×C5—22no—
117Dic15—21no—
118Dic15—22no—
119C3×Dic5—22no—
120C3×Dic5C2×D412yes—
121C3×Dic5C2×D412yes—
122C5×Dic3—21no—
123C5×Dic3—22no—
124C2×C3×Q8—52no—
125C2×C3×Q8—52no—
126C2×C3×Q8—104no—
127C22×C3×C4—51no—
128C22×C3×C4—51no—
129C22×C3×C4C2×D511yescharacteristic
130C2×#67—51no—
131C2×#67C2×D511yescharacteristic
132#235—104no—
133C2×D12—51no—
134C2×C4×D3—51no—
135C2×C4×D3—51no—
136C2×Dic6—51no—
137#217—51no—
138#217C2×D511yescharacteristic
139(C2×D3)⋊C4—52no—
140(C2×D3)⋊C4—52no—
141(C2×D3)⋊C4—52no—
142(C2×D3)⋊C4C2×D512yes—
143C4⋊Dic3—52no—
144Dic3⋊C4—52no—
145Dic3⋊C4—52no—
146Dic3⋊C4—52no—
147Dic3⋊C4C2×D512yes—
148C4×Dic3—52no—
149C23×C5—31no—
150C23×C5C22×D311yescharacteristic
151C22×D5—31no—
152C5×D4—62no—
153C2×C4×C5—22no—
154C2×C4×C5—32no—
155C2×C4×C5—31no—
156C2×C4×C5C22×D312yes—
157C22⋊D5—62no—
158C22⋊D5—62no—
159C2×Dic5—22no—
160C2×Dic5—32no—
161C2×Dic5—31no—
162C2×Dic5—62no—
163C2×Dic5#6714yes—
164C2×Dic5C22×D312yes—
165C2×Dic5C22×D311yescharacteristic
166C2×Dic5C22×D311yescharacteristic
167C4×D5—62no—
168Dic10—24no—
169Dic10—24no—
170Dic10—62no—
171C2×#41—151no—
172C22×Q8—51no—
173C4⋊Q8—152no—
174#124—152no—
175#123—31no—
176#123—151no—
177#123—151no—
178#123—151no—
179C22⋊Q8—152no—
180C22⋊Q8—152no—
181#117—151no—
182C2×C3×C5—22no—
183C2×C3×C5C22×D411yescharacteristic
184C2×C3×C52−1+412yes—
185D15—41no—
186C5×D3—41no—
187C23×C3C22×D511yescharacteristic
188C22×D3—51no—
189C22×D3C22×D511yescharacteristic
190C3×Q8—104no—
191C3×Q8—104no—
192C2×C3×C4—22no—
193C2×C3×C4—54no—
194C2×C3×C4—51no—
195C2×C3×C4—51no—
196C2×C3×C4—52no—
197C2×C3×C4—102no—
198C2×C3×C4C22×D512yes—
199#67—22no—
200#67—102no—
201C2×Dic3—52no—
202C2×Dic3—51no—
203C2×Dic3C22×D512yes—
204C2×Dic3C22×D511yescharacteristic
205D12—102no—
206C4×D3—102no—
207C4×D3—102no—
208Dic6—102no—
209C22×C5—22no—
210C22×C5—62no—
211C22×C5—61no—
212C22×C5C2×#6712yes—
213C22×C5C23×D311yescharacteristic
214C2×D5—62no—
215C2×D5—61no—
216C4×C5—22no—
217C4×C5—62no—
218C4×C5—61no—
219Dic5—22no—
220Dic5—62no—
221Dic5—61no—
222Dic5C2×#6712yes—
223Dic5C2×#6712yes—
224#41—304no—
225C2×Q8—52no—
226C2×Q8—52no—
227C2×Q8—104no—
228C2×Q8—151no—
229C2×D4—31no—
230C2×D4—151no—
231C2×D4—151no—
232C22×C4—51no—
233C22×C4—51no—
234C22×C4—151no—
235C22×C4—151no—
236C22×C4D3×D511yescharacteristic
237C4⋊C4—32no—
238C4⋊C4—152no—
239C4⋊C4—152no—
240C4⋊C4—152no—
241C4⋊C4—152no—
242C22⋊C4—31no—
243C22⋊C4—32no—
244C22⋊C4—152no—
245C22⋊C4—152no—
246C22⋊C4—151no—
247C22⋊C4—152no—
248C42—152no—
249C3×C5#53611yescharacteristic, complemented, Hall
250C22×C3—22no—
251C22×C3D4×D512yes—
252C22×C3C23×D511yescharacteristic
253C2×D3—21no—
254C2×D3—22no—
255C2×D3—101no—
256C2×D3—102no—
257C3×C4—22no—
258C3×C4—52no—
259C3×C4—52no—
260C3×C4—102no—
261Dic3—21no—
262Dic3—22no—
263Dic3—101no—
264Dic3—102no—
265C2×C5—22no—
266C2×C5—121no—
267C2×C5#100712yes—
268C2×C5C22×#6711yescharacteristic
269D5—121no—
270C23—31no—
271C23—151no—
272C23C2×D3×D511yescharacteristic
273Q8—104no—
274Q8—104no—
275Q8—302no—
276D4—62no—
277D4—302no—
278D4—302no—
279C2×C4—22no—
280C2×C4—32no—
281C2×C4—31no—
282C2×C4—54no—
283C2×C4—51no—
284C2×C4—51no—
285C2×C4—52no—
286C2×C4—102no—
287C2×C4—152no—
288C2×C4—151no—
289C2×C4—302no—
290C2×C4—302no—
291C2×C4C2×D3×D512yes—
292C2×C3—22no—
293C2×C3C2×D4×D511yescharacteristic
294C2×C3#415012yes—
295D3—41no—
296D3—201no—
297C5#581611yescharacteristic, complemented, Sylow
298C22—22no—
299C22—62no—
300C22—61no—
301C22—302no—
302C22—301no—
303C22D5×#6712yes—
304C22C22×D3×D511yescharacteristic
305C4—22no—
306C4—52no—
307C4—52no—
308C4—62no—
309C4—61no—
310C4—102no—
311C4—302no—
312C4—301no—
313C3#6598211yescharacteristic, complemented, Sylow
314C2—22no—
315C2—121no—
316C2—601no—
317C2#9200412yes—
318C2C2×D5×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
prime radicalC2×#6782C2
hypercenterC22×C4D3×D5
hypocenterC3×C5#536
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×Dic10C2semidirect
3. #91818C2semidirect
4. C5×#929C2semidirect
5. #91546C2semidirect
6. #91539C2semidirect
7. #91528C2semidirect
8. #91411C2semidirect
9. #91410C2semidirect
10. #91327C2semidirect
11. #91326C2semidirect
40. C22×Dic10D3semidirect
81. #929D5semidirect
249. C3×C5#536semidirect
297. C5#5816semidirect
313. C3#65982semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic102
C2×#67822
#918182
C5×#9292
#915462
#915392
#915282
#914112
#914102
#913272
#913262
#659823
#58165