Finite Group Databasebeta

← #11756358

Subgroups — #11756358

Order 960.

624 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6819C211yesmaximal, characteristic, complemented
2C2×#6817C211yesmaximal, characteristic, complemented
3C3×C5×#121C211yesmaximal, characteristic, complemented
4#91829C212yesmaximal, complemented
5#91829C211yesmaximal, characteristic, complemented
6#91828C212yesmaximal, complemented
7C4×#1220C211yesmaximal, characteristic, complemented
8#91784C211yesmaximal, characteristic, complemented
9#91776C211yesmaximal, characteristic, complemented
10#91773C212yesmaximal, complemented
11#91770C212yesmaximal, complemented
12#65933—31nomaximal, Hall
13C2×C3×C5×D4C2211yescharacteristic
14C2×C3×C5×D4C2212yes—
15C22×C3×C4×C5C2211yescharacteristic
16C2×#1220C2212yes—
17C2×#1220C2212yes—
18C2×#1220C2211yescharacteristic
19C2×#1220C2212yes—
20C2×#1220C2212yes—
21C22×Dic15C2212yes—
22#6819—24no—
23#6817—24no—
24C2×D60C2211yescharacteristic
25C2×D60C2212yes—
26C2×C4×D15C2212yes—
27C2×C4×D15C2211yescharacteristic
28C2×Dic30C2211yescharacteristic
29C3×C5×C4⋊C4C2211yescharacteristic
30C3×C5×C22⋊C4C2212yes—
31#6719C2211yescharacteristic
32#6714C2212yes—
33#6714C2211yescharacteristic
34#6712C2211yescharacteristic
35#6712C2212yes—
36C4×Dic15C2212yes—
37C4×Dic15C2211yescharacteristic
38#5599—51nomaximal, Hall
39C2×#711—31no—
40C2×#709—31no—
41C5×#121D311yescharacteristic, complemented
42#4086—32no—
43#4086—31no—
44#4085—32no—
45C4×C22⋊D5—31no—
46#4041—31no—
47#4033—31no—
48#4030—32no—
49#4027—32no—
50C23×C3×C5C2311yescharacteristic
51C23×C3×C5C2312yes—
52C22×D15C2312yes—
53C22×D15C2311yescharacteristic
54C3×C5×D4—22no—
55C3×C5×D4—42no—
56C2×C3×C4×C5—22no—
57C2×C3×C4×C5C2312yes—
58C2×C3×C4×C5C2311yescharacteristic
59C2×C3×C4×C5C2311yescharacteristic
60#1220—24no—
61#1220—24no—
62#1220—24no—
63#1220—22no—
64#1220—42no—
65C2×Dic15—24no—
66C2×Dic15C2311yescharacteristic
67C2×Dic15C2311yescharacteristic
68C2×Dic15C2312yes—
69C2×Dic15C2311yescharacteristic
70D60—22no—
71D60—42no—
72C4×D15—22no—
73C4×D15—24no—
74Dic30D414yescomplemented
75C2×#237—51no—
76C2×#235—51no—
77C3×#121D511yescharacteristic, complemented
78#940—52no—
79#940—51no—
80#939—52no—
81C4×#67—51no—
82#895—51no—
83#887—51no—
84#884—52no—
85#67⋊C4—52no—
86C2×C5×D4C2×D311yescharacteristic
87C2×C5×D4C2×D312yes—
88C22×C4×C5C2×D311yescharacteristic
89C2×C22⋊D5—32no—
90C2×C22⋊D5—32no—
91C2×C22⋊D5—32no—
92C2×C22⋊D5—31no—
93C2×C22⋊D5—32no—
94C22×Dic5—32no—
95#711—64no—
96#709—64no—
97C2×D20—31no—
98C2×D20—32no—
99C2×C4×D5—32no—
100C2×C4×D5—31no—
101C2×Dic10—31no—
102C5×C4⋊C4C2×D311yescharacteristic
103C5×C22⋊C4C2×D312yes—
104C22⋊Dic5—31no—
105#685—32no—
106#685—31no—
107#683—31no—
108#683—32no—
109C4×Dic5—32no—
110C4×Dic5—31no—
111#550—151noSylow
112C22×C3×C5—22no—
113C22×C3×C5—21no—
114C22×C3×C5—21no—
115C22×C3×C5—22no—
116C22×C3×C5—22no—
117C22×C3×C5—21no—
118C22×C3×C5C2411yescharacteristic
119C2×D15—22no—
120C2×D15—21no—
121C2×D15—22no—
122C2×D15—21no—
123C2×D15—21no—
124C2×D15—22no—
125C3×C4×C5—21no—
126C3×C4×C5—22no—
127C3×C4×C5C2×D412yes—
128Dic15—21no—
129Dic15—21no—
130Dic15C2×D414yes—
131Dic15#4112yescomplemented
132C2×C3×D4C2×D511yescharacteristic
133C2×C3×D4C2×D512yes—
134C22×C3×C4C2×D511yescharacteristic
135C2×#67—52no—
136C2×#67—52no—
137C2×#67—51no—
138C2×#67—52no—
139C2×#67—52no—
140C22×Dic3—52no—
141#237—104no—
142#235—104no—
143C2×D12—51no—
144C2×D12—52no—
145C2×C4×D3—52no—
146C2×C4×D3—51no—
147C2×Dic6—51no—
148C3×C4⋊C4C2×D511yescharacteristic
149C3×C22⋊C4C2×D512yes—
150#217—51no—
151(C2×D3)⋊C4—52no—
152(C2×D3)⋊C4—51no—
153Dic3⋊C4—51no—
154Dic3⋊C4—52no—
155C4×Dic3—52no—
156C4×Dic3—51no—
157C23×C5C22×D311yescharacteristic
158C23×C5C22×D312yes—
159C22×D5—32no—
160C22×D5—31no—
161C5×D4—22no—
162C5×D4—42no—
163C2×C4×C5—22no—
164C2×C4×C5C22×D312yes—
165C2×C4×C5C22×D311yescharacteristic
166C2×C4×C5C22×D311yescharacteristic
167C22⋊D5—64no—
168C22⋊D5—64no—
169C22⋊D5—64no—
170C22⋊D5—62no—
171C22⋊D5—122no—
172C2×Dic5—31no—
173C2×Dic5—32no—
174C2×Dic5—31no—
175C2×Dic5—31no—
176C2×Dic5—64no—
177D20—62no—
178D20—122no—
179C4×D5—64no—
180C4×D5—62no—
181Dic10—34no—
182C2×#41—151no—
183C2×#41—151no—
184#127—152no—
185#127—151no—
186#121—152no—
187#121—151no—
188#121—152no—
189#121D1511yescharacteristic, complemented
190C4×Q8—151no—
191C4×D4—151no—
192C4×D4—152no—
193C2×C3×C5—42no—
194C2×C3×C5—41no—
195C2×C3×C5C22×D411yescharacteristic
196C2×C3×C5C2×#4111yescharacteristic
197C2×C3×C5H(5,𝔽2)11yescharacteristic
198D15—42no—
199D15—41no—
200C23×C3C22×D511yescharacteristic
201C23×C3C22×D512yes—
202C22×D3—52no—
203C22×D3—51no—
204C3×D4—22no—
205C3×D4—42no—
206C2×C3×C4—22no—
207C2×C3×C4C22×D512yes—
208C2×C3×C4C22×D511yescharacteristic
209C2×C3×C4C22×D511yescharacteristic
210#67—104no—
211#67—104no—
212#67—104no—
213#67—102no—
214#67—202no—
215C2×Dic3—51no—
216C2×Dic3—52no—
217C2×Dic3—51no—
218C2×Dic3—51no—
219C2×Dic3—104no—
220D12—102no—
221D12—202no—
222C4×D3—104no—
223C4×D3—102no—
224Dic6—54no—
225C22×C5—22no—
226C22×C5—21no—
227C22×C5—22no—
228C22×C5—21no—
229C22×C5—21no—
230C22×C5—22no—
231C22×C5C23×D311yescharacteristic
232C2×D5—62no—
233C2×D5—61no—
234C2×D5—61no—
235C2×D5—62no—
236C2×D5—62no—
237C2×D5—61no—
238C4×C5—21no—
239C4×C5—22no—
240C4×C5D3×D412yes—
241Dic5—34no—
242Dic5—32no—
243Dic5—61no—
244Dic5—61no—
245#41—304no—
246#41—304no—
247C2×Q8—151no—
248C2×D4—151no—
249C2×D4—152no—
250C2×D4—152no—
251C2×D4—152no—
252C2×D4—152no—
253C2×D4—151no—
254C2×D4—152no—
255C2×D4C2×D1511yescharacteristic
256C2×D4C2×D1512yes—
257C22×C4—152no—
258C22×C4—152no—
259C22×C4—151no—
260C22×C4C2×D1511yescharacteristic
261C4⋊C4—151no—
262C4⋊C4—152no—
263C4⋊C4C2×D1511yescharacteristic
264C22⋊C4—151no—
265C22⋊C4—151no—
266C22⋊C4—152no—
267C22⋊C4C2×D1512yes—
268C42—152no—
269C42—151no—
270C3×C5#55011yescharacteristic, complemented, Hall
271C22×C3—22no—
272C22×C3—21no—
273C22×C3—22no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—22no—
277C22×C3C23×D511yescharacteristic
278C2×D3—102no—
279C2×D3—101no—
280C2×D3—101no—
281C2×D3—102no—
282C2×D3—102no—
283C2×D3—101no—
284C3×C4—21no—
285C3×C4—22no—
286C3×C4D4×D512yes—
287Dic3—54no—
288Dic3—52no—
289Dic3—101no—
290Dic3—101no—
291C2×C5—42no—
292C2×C5—41no—
293C2×C5C2×D3×D411yescharacteristic
294C2×C5#100411yescharacteristic
295C2×C5D3×#4111yescharacteristic
296D5—122no—
297D5—121no—
298C23—151no—
299C23—152no—
300C23C22×D1511yescharacteristic
301C23C22×D1512yes—
302Q8—154no—
303D4—22no—
304D4—42no—
305D4—304no—
306D4—304no—
307D4—302no—
308D4—304no—
309D4—302no—
310D4—602no—
311D4—602no—
312C2×C4—22no—
313C2×C4—151no—
314C2×C4—152no—
315C2×C4—151no—
316C2×C4—151no—
317C2×C4—304no—
318C2×C4—304no—
319C2×C4—302no—
320C2×C4C22×D1512yes—
321C2×C4C22×D1511yescharacteristic
322C2×C4C22×D1511yescharacteristic
323C2×C3—42no—
324C2×C3—41no—
325C2×C3C2×D4×D511yescharacteristic
326C2×C3#414411yescharacteristic
327C2×C3D5×#4111yescharacteristic
328D3—202no—
329D3—201no—
330C5#559911yescharacteristic, complemented, Sylow
331C22—21no—
332C22—22no—
333C22—21no—
334C22—22no—
335C22—21no—
336C22—22no—
337C22—302no—
338C22—301no—
339C22—301no—
340C22—302no—
341C22—301no—
342C22—302no—
343C22C23×D1511yescharacteristic
344C4—21no—
345C4—22no—
346C4—154no—
347C4—152no—
348C4—301no—
349C4—301no—
350C4D4×D1512yes—
351C3#6593311yescharacteristic, complemented, Sylow
352C2—41no—
353C2—42no—
354C2—602no—
355C2—601no—
356C2C2×D4×D1511yescharacteristic
357C2#9209511yescharacteristic
358C2#41×D1511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D15
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×#121C2
hypercenter#121D15
hypocenterC3×C5#550
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#6819C2semidirect
2. C2×#6817C2semidirect
3. C3×C5×#121C2semidirect
4. #91829C2semidirect
5. #91829C2semidirect
6. #91828C2semidirect
7. C4×#1220C2semidirect
8. #91784C2semidirect
9. #91776C2semidirect
10. #91773C2semidirect
11. #91770C2semidirect
41. C5×#121D3semidirect
74. Dic30D4semidirect
77. C3×#121D5semidirect
131. Dic15#41semidirect
189. #121D15semidirect
270. C3×C5#550semidirect
330. C5#5599semidirect
351. C3#65933semidirect

Maximal subgroups

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

subgroupindex
C2×#68192
C2×#68172
C3×C5×#1212
#918292
#918282
C4×#12202
#917842
#917762
#917732
#917702
#659333
#55995