Finite Group Databasebeta

← #11754827

Subgroups — #11754827

Order 960.

580 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 140 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
1C2×#6819C211yesmaximal, characteristic, complemented
2C2×D3×Dic10C211yesmaximal, characteristic, complemented
3C5×#938C211yesmaximal, characteristic, complemented
4C3×#4082C211yesmaximal, characteristic, complemented
5#91576C212yesmaximal, complemented
6#67⋊Dic5C212yesmaximal, complemented
7#91558C212yesmaximal, complemented
8#91526C212yesmaximal, complemented
9C4×#1203C211yesmaximal, characteristic, complemented
10#91390C211yesmaximal, characteristic, complemented
11#91333C211yesmaximal, characteristic, complemented
12#65932—31nomaximal, Hall
13C2×C3×C5×D4C2211yescharacteristic
14C2×#1220C2212yes—
15C22×Dic15C2212yes—
16#6819—24no—
17C2×C4×D15C2211yescharacteristic
18C2×Dic30C2211yescharacteristic
19C2×C5×#67C2212yes—
20C2×C4×C5×D3C2211yescharacteristic
21C2×C3×Dic10C2211yescharacteristic
22C2×#1204C2212yes—
23C2×#1203C2211yescharacteristic
24C2×D3×Dic5C2212yes—
25D3×Dic10—24no—
26C5×#217C2212yes—
27C5×C4⋊Dic3C2211yescharacteristic
28C3×C22⋊Dic5C2212yes—
29C3×C22⋊Dic5C2212yes—
30C3×C4×Dic5C2211yescharacteristic
31Dic3⋊Dic5C2212yes—
32#6669C2212yes—
33#6668C2211yescharacteristic
34#6667C2211yescharacteristic
35#6667C2212yes—
36(C2×D3)⋊Dic5C2211yescharacteristic
37Dic3×Dic5C2212yes—
38#5677—51nomaximal, Hall
39C2×#711—31no—
40C22×Dic10—31no—
41C5×#121—31no—
42#4085—32no—
43#4082D311yescharacteristic, complemented
44D4×Dic5—32no—
45C4×C22⋊D5—31no—
46#4043—31no—
47#4033—31no—
48#4031—32no—
49#4024—32no—
50C23×C3×C5C2312yes—
51C22×D15C2311yescharacteristic
52C22×C5×D3C2311yescharacteristic
53C3×C5×D4—22no—
54C2×C3×C4×C5C2311yescharacteristic
55#1220—24no—
56C2×Dic15—24no—
57C2×Dic15C2311yescharacteristic
58C2×Dic15C2312yes—
59C4×D15—22no—
60Dic30D414yescomplemented
61C5×#67—24no—
62C2×C5×Dic3C2312yes—
63C2×C5×Dic3C2311yescharacteristic
64C4×C5×D3—22no—
65C2×C3×Dic5C2311yescharacteristic
66C2×C3×Dic5C2312yes—
67C2×C3×Dic5C2311yescharacteristic
68C3×Dic10—22no—
69#1204—24no—
70#1203—22no—
71D3×Dic5—24no—
72C2×D3×Q8—51no—
73C2×#237—51no—
74C3×#124—51no—
75#946—51no—
76#938D511yescharacteristic, complemented
77#885—52no—
78#884—52no—
79#67⋊C4—52no—
80#878—52no—
81C4×D12—51no—
82C4×Dic6—51no—
83C2×C5×D4—32no—
84C2×C5×D4C2×D311yescharacteristic
85C22×C4×C5—31no—
86C2×C22⋊D5—32no—
87C2×C22⋊D5—31no—
88C22×Dic5—32no—
89C22×Dic5—32no—
90#711—64no—
91C2×C4×D5—31no—
92C2×Dic10—31no—
93C2×Dic10—32no—
94C2×Dic10—64no—
95C2×Dic10C2×D311yescharacteristic
96C5×C4⋊C4—31no—
97C5×C22⋊C4—32no—
98C22⋊Dic5—31no—
99C22⋊Dic5C2×D312yes—
100C22⋊Dic5C2×D312yes—
101#685—31no—
102#685—32no—
103C4⋊Dic5—32no—
104#683—31no—
105#683—32no—
106C4×Dic5—32no—
107C4×Dic5C2×D311yescharacteristic
108#548—151noSylow
109C22×C3×C5—22no—
110C22×C3×C5—22no—
111C22×C3×C5—22no—
112C22×C3×C5C2411yescharacteristic
113C2×D15—21no—
114C2×D15—21no—
115C2×D15—21no—
116C2×C5×D3—21no—
117C2×C5×D3—21no—
118C2×C5×D3#4112yescomplemented
119C3×C4×C5C2×D412yes—
120Dic15—21no—
121Dic15C2×D414yes—
122C3×Dic5—22no—
123C3×Dic5—21no—
124C3×Dic5—21no—
125C5×Dic3—21no—
126C5×Dic3—22no—
127C2×C3×Q8—51no—
128C2×C3×D4C2×D511yescharacteristic
129C2×#67—52no—
130C2×#67C2×D512yes—
131C22×Dic3—52no—
132D3×Q8—104no—
133#237—104no—
134C2×D12—51no—
135C2×C4×D3—52no—
136C2×C4×D3—51no—
137C2×C4×D3C2×D511yescharacteristic
138C2×Dic6—51no—
139C2×Dic6—52no—
140C3×C22⋊C4—52no—
141C3×C22⋊C4—52no—
142C3×C42—51no—
143#217C2×D512yes—
144(C2×D3)⋊C4—51no—
145(C2×D3)⋊C4—51no—
146(C2×D3)⋊C4—52no—
147C4⋊Dic3—51no—
148C4⋊Dic3C2×D511yescharacteristic
149Dic3⋊C4—52no—
150Dic3⋊C4—52no—
151C4×Dic3—52no—
152C23×C5—31no—
153C23×C5C22×D312yes—
154C22×D5—31no—
155C5×D4—22no—
156C5×D4—64no—
157C2×C4×C5—32no—
158C2×C4×C5—31no—
159C2×C4×C5—62no—
160C2×C4×C5C22×D311yescharacteristic
161C22⋊D5—62no—
162C22⋊D5—64no—
163C2×Dic5—32no—
164C2×Dic5—31no—
165C2×Dic5—64no—
166C2×Dic5—64no—
167C2×Dic5C22×D311yescharacteristic
168C2×Dic5C22×D311yescharacteristic
169C2×Dic5C22×D312yes—
170C4×D5—62no—
171Dic10—22no—
172Dic10—34no—
173Dic10—64no—
174C2×#41—151no—
175C22×Q8—151no—
176#124—51no—
177#124—152no—
178C22⋊Q8—152no—
179C22⋊Q8—151no—
180#121—31no—
181#121—152no—
182C4×Q8—151no—
183C4×D4—151no—
184C4×D4—152no—
185C2×C3×C5—42no—
186C2×C3×C5C22×D411yescharacteristic
187C2×C3×C5C2×#4111yescharacteristic
188C2×C3×C52−1+411yescharacteristic
189D15—41no—
190C5×D3—22no—
191C23×C3C22×D512yes—
192C22×D3—51no—
193C22×D3C22×D511yescharacteristic
194C3×Q8—102no—
195C3×D4—22no—
196C2×C3×C4—51no—
197C2×C3×C4—51no—
198C2×C3×C4—52no—
199C2×C3×C4C22×D511yescharacteristic
200#67—24no—
201#67—104no—
202C2×Dic3—52no—
203C2×Dic3—51no—
204C2×Dic3—104no—
205C2×Dic3C22×D512yes—
206C2×Dic3C22×D511yescharacteristic
207D12—102no—
208C4×D3—22no—
209C4×D3—104no—
210C4×D3—102no—
211Dic6—54no—
212Dic6—104no—
213C22×C5—22no—
214C22×C5—22no—
215C22×C5—22no—
216C22×C5—32no—
217C22×C5—61no—
218C22×C5—61no—
219C22×C5C23×D311yescharacteristic
220C2×D5—61no—
221C2×D5—61no—
222C2×D5—61no—
223C4×C5—61no—
224C4×C5—62no—
225C4×C5D3×D412yes—
226Dic5—22no—
227Dic5—21no—
228Dic5—21no—
229Dic5—34no—
230Dic5—61no—
231#41—304no—
232C2×Q8—51no—
233C2×Q8—152no—
234C2×Q8—151no—
235C2×Q8—304no—
236C2×D4—32no—
237C2×D4—151no—
238C2×D4—152no—
239C2×D4D3×D511yescharacteristic
240C22×C4—31no—
241C22×C4—152no—
242C22×C4—152no—
243C22×C4—151no—
244C4⋊C4—31no—
245C4⋊C4—151no—
246C4⋊C4—152no—
247C4⋊C4—152no—
248C22⋊C4—32no—
249C22⋊C4—52no—
250C22⋊C4—52no—
251C22⋊C4—151no—
252C22⋊C4—152no—
253C22⋊C4—151no—
254C42—51no—
255C42—152no—
256C3×C5#54811yescharacteristic, complemented, Hall
257C22×C3—22no—
258C22×C3—22no—
259C22×C3—22no—
260C22×C3C23×D511yescharacteristic
261C2×D3—21no—
262C2×D3—21no—
263C2×D3—101no—
264C2×D3—101no—
265C2×D3—101no—
266C2×D3#71112yescomplemented
267C3×C4—102no—
268C3×C4—101no—
269C3×C4—101no—
270C3×C4D4×D512yes—
271Dic3—21no—
272Dic3—22no—
273Dic3—54no—
274Dic3—101no—
275C2×C5—42no—
276C2×C5—62no—
277C2×C5C2×D3×D411yescharacteristic
278C2×C5D3×#4111yescharacteristic
279C2×C5#101011yescharacteristic
280D5—121no—
281C23—31no—
282C23—151no—
283C23C2×D3×D512yes—
284Q8—102no—
285Q8—154no—
286Q8—304no—
287D4—22no—
288D4—64no—
289D4—302no—
290D4—304no—
291C2×C4—32no—
292C2×C4—31no—
293C2×C4—51no—
294C2×C4—51no—
295C2×C4—52no—
296C2×C4—62no—
297C2×C4—152no—
298C2×C4—151no—
299C2×C4—304no—
300C2×C4—304no—
301C2×C4—302no—
302C2×C4C2×D3×D511yescharacteristic
303C2×C3—42no—
304C2×C3C2×D4×D511yescharacteristic
305C2×C3C2×#71111yescharacteristic
306C2×C3#415011yescharacteristic
307D3—22no—
308D3—201no—
309C5#567711yescharacteristic, complemented, Sylow
310C22—22no—
311C22—22no—
312C22—22no—
313C22—32no—
314C22—61no—
315C22—61no—
316C22—301no—
317C22—301no—
318C22—301no—
319C22C22×D3×D511yescharacteristic
320C4—62no—
321C4—61no—
322C4—101no—
323C4—101no—
324C4—102no—
325C4—154no—
326C4—301no—
327C4#679012yes—
328C3#6593211yescharacteristic, complemented, Sylow
329C2—42no—
330C2—62no—
331C2—601no—
332C2#9202011yescharacteristic
333C2D3×#71111yescharacteristic
334C2C2×#679011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#548
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#6819C2semidirect
2. C2×D3×Dic10C2semidirect
3. C5×#938C2semidirect
4. C3×#4082C2semidirect
5. #91576C2semidirect
6. #67⋊Dic5C2semidirect
7. #91558C2semidirect
8. #91526C2semidirect
9. C4×#1203C2semidirect
10. #91390C2semidirect
11. #91333C2semidirect
43. #4082D3semidirect
60. Dic30D4semidirect
76. #938D5semidirect
118. C2×C5×D3#41semidirect
256. C3×C5#548semidirect
266. C2×D3#711semidirect
309. C5#5677semidirect
328. C3#65932semidirect

Maximal subgroups

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

subgroupindex
C2×#68192
C2×D3×Dic102
C5×#9382
C3×#40822
#915762
#67⋊Dic52
#915582
#915262
C4×#12032
#913902
#913332
#659323
#56775