Finite Group Databasebeta

← #11753723

Subgroups — #11753723

Order 960.

428 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#91778C211yesmaximal, characteristic, complemented
2C5×#898C211yesmaximal, characteristic, complemented
3C3×D5×C4⋊C4C211yesmaximal, characteristic, complemented
4#91447C211yesmaximal, characteristic
5#91447C211yesmaximal, characteristic
6#91440C211yesmaximal, characteristic
7#91417C211yesmaximal, characteristic, complemented
8#91400C211yesmaximal, characteristic, complemented
9#91358C211yesmaximal, characteristic, complemented
10#91357C211yesmaximal, characteristic, complemented
11#91349C211yesmaximal, characteristic, complemented
12#91346C211yesmaximal, characteristic, complemented
13#91341C211yesmaximal, characteristic, complemented
14#91335C211yesmaximal, characteristic, complemented
15#91324C211yesmaximal, characteristic, complemented
16#66042—31nomaximal, Hall
17C2×C3×C4×D5C2211yescharacteristic
18C2×C3×C4×D5C2211yescharacteristic
19C2×C3×C4×D5C2211yescharacteristic
20C2×#1204C2211yescharacteristic
21C2×#1201C2211yescharacteristic
22C2×D3×Dic5C2211yescharacteristic
23C3×C5×C4⋊C4C2211yescharacteristic
24C4⋊Dic15C2211yescharacteristic
25#6712C2211yescharacteristic
26#6712C2211yescharacteristic
27#6712C2211yescharacteristic
28#6712C2211yescharacteristic
29C4×Dic15C2211yescharacteristic
30C5×(C2×D3)⋊C4C2211yescharacteristic
31C5×(C2×D3)⋊C4C2211yescharacteristic
32C5×(C2×D3)⋊C4C2211yescharacteristic
33C5×C4⋊Dic3C2211yescharacteristic
34C5×Dic3⋊C4C2211yescharacteristic
35C4×C5×Dic3C2211yescharacteristic
36C3×C4⋊Dic5C2211yescharacteristic
37C3×#683C2211yescharacteristic
38C3×#683C2211yescharacteristic
39Dic3⋊Dic5C2211yescharacteristic
40Dic3⋊Dic5C2211yescharacteristic
41#6668C2211yescharacteristic
42#6668C2211yescharacteristic
43#6668C2211yescharacteristic
44(C2×D3)⋊Dic5C2211yescharacteristic
45(C2×D3)⋊Dic5C2211yescharacteristic
46#6665C2211yescharacteristic
47#6665C2211yescharacteristic
48#6665C2211yescharacteristic
49Dic3×Dic5C2211yescharacteristic
50Dic3×Dic5C2211yescharacteristic
51Dic3×Dic5C2211yescharacteristic
52#5513—51nomaximal, Hall
53C5×#126—31no—
54#4042—31no—
55#4038—31no—
56D5×C4⋊C4D311yescharacteristic, complemented
57#4036—31no—
58#4035—31no—
59#4035—31no—
60#4029—31no—
61#4029—31no—
62#4027—31no—
63#4025—31no—
64#4024—31no—
65#4023—31no—
66#4018—31no—
67C4×Dic10—31no—
68C22×C5×D3C2311yescharacteristic
69C22×C3×D5C2311yescharacteristic
70C2×C3×C4×C5C2311yescharacteristic
71C2×C3×C4×C5C2311yescharacteristic
72C2×C3×C4×C5C2311yescharacteristic
73C2×Dic15C2311yescharacteristic
74C2×Dic15C2311yescharacteristic
75C2×Dic15C2311yescharacteristic
76C2×Dic15C2311yescharacteristic
77C2×C5×Dic3C2311yescharacteristic
78C2×C5×Dic3C2311yescharacteristic
79C2×C5×Dic3C2311yescharacteristic
80C2×C3×Dic5C2311yescharacteristic
81C2×C3×Dic5C2311yescharacteristic
82C2×C3×Dic5C2311yescharacteristic
83C3×C4×D5—22no—
84C3×C4×D5—22no—
85C3×C4×D5—22no—
86#1204—22no—
87#1201—22no—
88D3×Dic5—22no—
89C2×C3×C4⋊C4—51no—
90#929—51no—
91#929—51no—
92C4×#67—51no—
93#926—51no—
94#926—51no—
95#924—51no—
96#898—51no—
97#898D511yescharacteristic, complemented
98#897—51no—
99#892—51no—
100#890—51no—
101#889—51no—
102#889—51no—
103#887—51no—
104C2×C22⋊D5—31no—
105C22×Dic5—31no—
106C2×C4×D5C2×D311yescharacteristic
107C2×C4×D5C2×D311yescharacteristic
108C2×C4×D5C2×D311yescharacteristic
109C2×Dic10—31no—
110C5×C4⋊C4—31no—
111C5×C4⋊C4—31no—
112C5×C4⋊C4C2×D311yescharacteristic
113C5×C22⋊C4—31no—
114C5×C22⋊C4—31no—
115C5×C22⋊C4—31no—
116C42×C5—31no—
117C22⋊Dic5—31no—
118C22⋊Dic5—31no—
119#685—31no—
120#685—31no—
121#685—31no—
122C4⋊Dic5—31no—
123C4⋊Dic5—31no—
124C4⋊Dic5—31no—
125C4⋊Dic5C2×D311yescharacteristic
126#683—31no—
127#683—31no—
128#683—31no—
129#683—31no—
130#683—31no—
131#683—31no—
132#683—31no—
133#683C2×D311yescharacteristic
134#683C2×D311yescharacteristic
135C4×Dic5—31no—
136C4×Dic5—31no—
137C4×Dic5—31no—
138C4×Dic5—31no—
139#552—151noSylow
140C22×C3×C5C2411yescharacteristic
141C2×C5×D3—21no—
142C2×C5×D3—21no—
143C2×C5×D3—21no—
144C2×C3×D5—21no—
145C2×C3×D5—21no—
146C2×C3×D5#4112yes—
147C3×C4×C5—21no—
148C3×C4×C5—21no—
149C3×C4×C5—21no—
150Dic15—21no—
151Dic15—21no—
152Dic15—21no—
153Dic15—21no—
154C3×Dic5—21no—
155C3×Dic5—21no—
156C3×Dic5#4112yes—
157C5×Dic3—21no—
158C5×Dic3—21no—
159C5×Dic3—21no—
160C22×C3×C4—51no—
161C22×C3×C4—51no—
162C22×C3×C4—51no—
163C2×#67—51no—
164C2×C4×D3—51no—
165C2×Dic6—51no—
166C3×C4⋊C4—51no—
167C3×C4⋊C4—51no—
168C3×C4⋊C4—51no—
169C3×C4⋊C4C2×D511yescharacteristic
170#217—51no—
171#217—51no—
172#217—51no—
173(C2×D3)⋊C4—51no—
174(C2×D3)⋊C4—51no—
175(C2×D3)⋊C4C2×D511yescharacteristic
176(C2×D3)⋊C4C2×D511yescharacteristic
177(C2×D3)⋊C4C2×D511yescharacteristic
178C4⋊Dic3—51no—
179C4⋊Dic3—51no—
180C4⋊Dic3—51no—
181C4⋊Dic3—51no—
182C4⋊Dic3C2×D511yescharacteristic
183Dic3⋊C4—51no—
184Dic3⋊C4—51no—
185Dic3⋊C4—51no—
186Dic3⋊C4—51no—
187Dic3⋊C4—51no—
188Dic3⋊C4—51no—
189Dic3⋊C4C2×D511yescharacteristic
190C4×Dic3—51no—
191C4×Dic3—51no—
192C4×Dic3—51no—
193C4×Dic3—51no—
194C4×Dic3C2×D511yescharacteristic
195C23×C5—31no—
196C22×D5C22×D311yescharacteristic
197C2×C4×C5—31no—
198C2×C4×C5—31no—
199C2×C4×C5—31no—
200C2×C4×C5C22×D311yescharacteristic
201C2×C4×C5C22×D311yescharacteristic
202C2×C4×C5C22×D311yescharacteristic
203C22⋊D5—62no—
204C2×Dic5—31no—
205C2×Dic5—31no—
206C2×Dic5—31no—
207C2×Dic5—31no—
208C2×Dic5—62no—
209C2×Dic5C22×D311yescharacteristic
210C2×Dic5C22×D311yescharacteristic
211C2×Dic5C22×D311yescharacteristic
212C4×D5—22no—
213C4×D5—22no—
214C4×D5—22no—
215Dic10—62no—
216#126—31no—
217#126—151no—
218#125—151no—
219#125—151no—
220#125—151no—
221#123—151no—
222#123—151no—
223C22⋊Q8—151no—
224C22⋊Q8—151no—
225C4×Q8—151no—
226C4×D4—151no—
227#117—151no—
228#117—151no—
229#117—151no—
230C2×C4⋊C4—51no—
231C2×C3×C5C2×#4111yescharacteristic
232C2×C3×C5C2×#4111yescharacteristic
233C2×C3×C52−1+411yescharacteristic
234C3×D5—22no—
235C5×D3—41no—
236C23×C3—51no—
237C22×D3C22×D511yescharacteristic
238C2×C3×C4—51no—
239C2×C3×C4—51no—
240C2×C3×C4—51no—
241C2×C3×C4—102no—
242C2×C3×C4—102no—
243C2×C3×C4—102no—
244C2×C3×C4C22×D511yescharacteristic
245C2×C3×C4C22×D511yescharacteristic
246C2×C3×C4C22×D511yescharacteristic
247#67—102no—
248C2×Dic3—51no—
249C2×Dic3—51no—
250C2×Dic3—51no—
251C2×Dic3—51no—
252C2×Dic3C22×D511yescharacteristic
253C2×Dic3C22×D511yescharacteristic
254C2×Dic3C22×D511yescharacteristic
255C4×D3—102no—
256Dic6—102no—
257C22×C5—61no—
258C22×C5—61no—
259C22×C5—61no—
260C22×C5C23×D311yescharacteristic
261C2×D5—21no—
262C2×D5—21no—
263C2×D5#23712yes—
264C4×C5—21no—
265C4×C5—21no—
266C4×C5—21no—
267C4×C5—61no—
268C4×C5—61no—
269C4×C5—61no—
270Dic5—21no—
271Dic5—21no—
272Dic5—61no—
273Dic5—61no—
274Dic5—61no—
275Dic5—61no—
276Dic5#23512yes—
277C2×Q8—151no—
278C2×D4—151no—
279C22×C4—51no—
280C22×C4—51no—
281C22×C4—51no—
282C22×C4—151no—
283C4⋊C4—31no—
284C4⋊C4—31no—
285C4⋊C4—51no—
286C4⋊C4—51no—
287C4⋊C4—51no—
288C4⋊C4—151no—
289C4⋊C4—151no—
290C4⋊C4—151no—
291C4⋊C4—151no—
292C4⋊C4—151no—
293C4⋊C4—151no—
294C4⋊C4—151no—
295C4⋊C4—151no—
296C4⋊C4—151no—
297C4⋊C4—151no—
298C4⋊C4D3×D511yescharacteristic
299C22⋊C4—31no—
300C22⋊C4—31no—
301C22⋊C4—31no—
302C22⋊C4—151no—
303C22⋊C4—151no—
304C22⋊C4—151no—
305C22⋊C4—151no—
306C22⋊C4—151no—
307C42—31no—
308C42—151no—
309C42—151no—
310C42—151no—
311C42—151no—
312C3×C5#55211yescharacteristic, complemented, Hall
313C22×C3—52no—
314C22×C3—101no—
315C22×C3—101no—
316C22×C3C23×D511yescharacteristic
317C2×D3—21no—
318C2×D3—21no—
319C2×D3—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—52no—
324C3×C4—101no—
325C3×C4—101no—
326Dic3—21no—
327Dic3—21no—
328Dic3—21no—
329Dic3—101no—
330Dic3—101no—
331Dic3—101no—
332Dic3—101no—
333C2×C5—121no—
334C2×C5C2×#23511yescharacteristic
335C2×C5C2×#23711yescharacteristic
336C2×C5#100711yescharacteristic
337D5—22no—
338C23—31no—
339C23—51no—
340Q8—302no—
341D4—302no—
342C2×C4—31no—
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—51no—
346C2×C4—51no—
347C2×C4—51no—
348C2×C4—102no—
349C2×C4—102no—
350C2×C4—102no—
351C2×C4—151no—
352C2×C4—151no—
353C2×C4—151no—
354C2×C4—151no—
355C2×C4—302no—
356C2×C4C2×D3×D511yescharacteristic
357C2×C4C2×D3×D511yescharacteristic
358C2×C4C2×D3×D511yescharacteristic
359C2×C3—102no—
360C2×C3D5×#4111yescharacteristic
361C2×C3D5×#4111yescharacteristic
362C2×C3#415011yescharacteristic
363D3—41no—
364C5#551311yescharacteristic, complemented, Sylow
365C22—52no—
366C22—61no—
367C22—61no—
368C22—61no—
369C22—101no—
370C22—101no—
371C22C22×D3×D511yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—52no—
376C4—61no—
377C4—61no—
378C4—61no—
379C4—101no—
380C4—101no—
381C4—301no—
382C4—301no—
383C4—301no—
384C4—301no—
385C3#6604211yescharacteristic, complemented, Sylow
386C2—102no—
387C2—121no—
388C2D5×#23511yescharacteristic
389C2D5×#23711yescharacteristic
390C2#9203011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C4⋊C4C22
prime radicalC2×#1201C22
hypercenterC4⋊C4D3×D5
hypocenterC3×C5#552
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #91778C2semidirect
2. C5×#898C2semidirect
3. C3×D5×C4⋊C4C2semidirect
7. #91417C2semidirect
8. #91400C2semidirect
9. #91358C2semidirect
10. #91357C2semidirect
11. #91349C2semidirect
12. #91346C2semidirect
13. #91341C2semidirect
14. #91335C2semidirect
15. #91324C2semidirect
56. D5×C4⋊C4D3semidirect
97. #898D5semidirect
312. C3×C5#552semidirect
364. C5#5513semidirect
385. C3#66042semidirect

Maximal subgroups

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

subgroupindex
#917782
C5×#8982
C3×D5×C4⋊C42
#914472
#914402
#914172
#914002
#913582
#913572
#913492
#913462
#913412
#913352
#913242
#660423
#55135