Finite Group Databasebeta

← #11754413

Subgroups — #11754413

Order 960.

476 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#1220C211yesmaximal, characteristic, complemented
2C5×#930C211yesmaximal, characteristic, complemented
3C2×C3×#683C211yesmaximal, characteristic, complemented
4#91566C211yesmaximal, characteristic, complemented
5#91566C211yesmaximal, characteristic, complemented
6Dic5×#67C211yesmaximal, characteristic, complemented
7Dic5×#67C211yesmaximal, characteristic, complemented
8#91423C211yesmaximal, characteristic, complemented
9#91416C211yesmaximal, characteristic, complemented
10#91406C211yesmaximal, characteristic, complemented
11#91400C211yesmaximal, characteristic, complemented
12#91367C211yesmaximal, characteristic, complemented
13#91355C211yesmaximal, characteristic, complemented
14#91337C211yesmaximal, characteristic, complemented
15#91335C211yesmaximal, characteristic, complemented
16#65935—31nomaximal, Hall
17C22×C3×C4×C5C2211yescharacteristic
18C2×#1220C2211yescharacteristic
19C2×C4×D15C2211yescharacteristic
20C2×C5×#67C2211yescharacteristic
21C2×C5×#67C2211yescharacteristic
22C2×C5×D12C2211yescharacteristic
23C22×C3×Dic5C2211yescharacteristic
24C22×C3×Dic5C2211yescharacteristic
25C2×#1203C2211yescharacteristic
26C2×#1203C2211yescharacteristic
27C2×D3×Dic5C2211yescharacteristic
28C2×D3×Dic5C2211yescharacteristic
29#6719C2211yescharacteristic
30#6714C2211yescharacteristic
31#6712C2211yescharacteristic
32C4×Dic15C2211yescharacteristic
33C5×(C2×D3)⋊C4C2211yescharacteristic
34C5×(C2×D3)⋊C4C2211yescharacteristic
35C5×C4⋊Dic3C2211yescharacteristic
36C3×#683C2211yescharacteristic
37C3×#683C2211yescharacteristic
38C3×#683C2211yescharacteristic
39C3×#683C2211yescharacteristic
40Dic3⋊Dic5C2211yescharacteristic
41Dic3⋊Dic5C2211yescharacteristic
42#6668C2211yescharacteristic
43#6668C2211yescharacteristic
44#6667C2211yescharacteristic
45#6667C2211yescharacteristic
46(C2×D3)⋊Dic5C2211yescharacteristic
47(C2×D3)⋊Dic5C2211yescharacteristic
48(C2×D3)⋊Dic5C2211yescharacteristic
49(C2×D3)⋊Dic5C2211yescharacteristic
50Dic3×Dic5C2211yescharacteristic
51Dic3×Dic5C2211yescharacteristic
52#5514—51nomaximal, Hall
53C5×#121—31no—
54#4085—31no—
55#4085—31no—
56#4085—31no—
57#4081—31no—
58D4×Dic5—31no—
59D4×Dic5—31no—
60C4×C22⋊D5—31no—
61C2×#683D311yescharacteristic, complemented
62#4044—31no—
63#4035—31no—
64#4029—31no—
65#4027—31no—
66#4025—31no—
67#4023—31no—
68C23×C3×C5C2311yescharacteristic
69C22×D15C2311yescharacteristic
70C22×C5×D3C2311yescharacteristic
71C22×C5×D3C2311yescharacteristic
72C2×C3×C4×C5—41no—
73C2×C3×C4×C5C2311yescharacteristic
74C2×C3×C4×C5C2311yescharacteristic
75#1220—22no—
76C2×Dic15C2311yescharacteristic
77C2×Dic15C2311yescharacteristic
78C2×Dic15C2311yescharacteristic
79C4×D15—41no—
80C5×#67—22no—
81C5×#67—22no—
82C2×C5×Dic3C2311yescharacteristic
83C2×C5×Dic3C2311yescharacteristic
84C5×D12—41no—
85C2×C3×Dic5—22no—
86C2×C3×Dic5—41no—
87C2×C3×Dic5C2311yescharacteristic
88C2×C3×Dic5C2311yescharacteristic
89C2×C3×Dic5C2311yescharacteristic
90C2×C3×Dic5C2311yescharacteristic
91#1203—22no—
92#1203—41no—
93D3×Dic5—22no—
94D3×Dic5—41no—
95C2×C3×C4⋊C4—51no—
96#930—51no—
97#930—51no—
98#930D511yescharacteristic, complemented
99C4×#67—51no—
100C4×#67—51no—
101C4×#67—51no—
102#898—51no—
103#898—51no—
104#895—51no—
105#894—51no—
106#894—51no—
107#893—51no—
108#892—51no—
109#890—51no—
110C2×C5×D4—31no—
111C2×C5×D4—31no—
112C2×C5×D4—31no—
113C22×C4×C5C2×D311yescharacteristic
114C2×C22⋊D5—31no—
115C2×C22⋊D5—31no—
116C2×C22⋊D5—31no—
117C22×Dic5—31no—
118C22×Dic5—31no—
119C22×Dic5C2×D311yescharacteristic
120C22×Dic5C2×D311yescharacteristic
121C2×C4×D5—31no—
122C5×C4⋊C4—31no—
123C5×C22⋊C4—31no—
124C5×C22⋊C4—31no—
125C22⋊Dic5—31no—
126C22⋊Dic5—31no—
127C22⋊Dic5—31no—
128C22⋊Dic5—31no—
129C22⋊Dic5—31no—
130#685—31no—
131#685—31no—
132#685—31no—
133C4⋊Dic5—31no—
134C4⋊Dic5—31no—
135#683—31no—
136#683—31no—
137#683—31no—
138#683C2×D311yescharacteristic
139#683C2×D311yescharacteristic
140#683C2×D311yescharacteristic
141#683C2×D311yescharacteristic
142C4×Dic5—31no—
143C4×Dic5—31no—
144C4×Dic5—31no—
145#553—151noSylow
146C22×C3×C5—21no—
147C22×C3×C5—21no—
148C22×C3×C5#4112yes—
149C22×C3×C5C2411yescharacteristic
150C2×D15—21no—
151C2×D15—21no—
152C2×D15—21no—
153C2×C5×D3—21no—
154C2×C5×D3—21no—
155C2×C5×D3—21no—
156C2×C5×D3—21no—
157C2×C5×D3—21no—
158C2×C5×D3—21no—
159C3×C4×C5—21no—
160C3×C4×C5—21no—
161Dic15—21no—
162Dic15—21no—
163Dic15—21no—
164C3×Dic5—21no—
165C3×Dic5—21no—
166C3×Dic5—21no—
167C3×Dic5#4112yes—
168C5×Dic3—21no—
169C5×Dic3—21no—
170C22×C3×C4—51no—
171C22×C3×C4—51no—
172C22×C3×C4C2×D511yescharacteristic
173C2×#67—51no—
174C2×#67C2×D511yescharacteristic
175C2×#67C2×D511yescharacteristic
176C2×D12—51no—
177C2×D12—51no—
178C2×D12C2×D511yescharacteristic
179C2×C4×D3—51no—
180C2×C4×D3—51no—
181C2×C4×D3—51no—
182C3×C4⋊C4—51no—
183C3×C4⋊C4—51no—
184C3×C4⋊C4—51no—
185C3×C4⋊C4—51no—
186#217—51no—
187(C2×D3)⋊C4—51no—
188(C2×D3)⋊C4—51no—
189(C2×D3)⋊C4—51no—
190(C2×D3)⋊C4—51no—
191(C2×D3)⋊C4—51no—
192(C2×D3)⋊C4—51no—
193(C2×D3)⋊C4—51no—
194(C2×D3)⋊C4C2×D511yescharacteristic
195(C2×D3)⋊C4C2×D511yescharacteristic
196C4⋊Dic3—51no—
197C4⋊Dic3—51no—
198C4⋊Dic3C2×D511yescharacteristic
199Dic3⋊C4—51no—
200Dic3⋊C4—51no—
201Dic3⋊C4—51no—
202C4×Dic3—51no—
203C4×Dic3—51no—
204C4×Dic3—51no—
205C23×C5—31no—
206C23×C5—31no—
207C23×C5C22×D311yescharacteristic
208C22×D5—31no—
209C5×D4—62no—
210C5×D4—62no—
211C5×D4—121no—
212C2×C4×C5—31no—
213C2×C4×C5—31no—
214C2×C4×C5—41no—
215C2×C4×C5C22×D311yescharacteristic
216C2×C4×C5C22×D311yescharacteristic
217C22⋊D5—62no—
218C22⋊D5—62no—
219C22⋊D5—121no—
220C2×Dic5—22no—
221C2×Dic5—31no—
222C2×Dic5—31no—
223C2×Dic5—31no—
224C2×Dic5—41no—
225C2×Dic5—62no—
226C2×Dic5—121no—
227C2×Dic5C22×D311yescharacteristic
228C2×Dic5C22×D311yescharacteristic
229C2×Dic5C22×D311yescharacteristic
230C2×Dic5C22×D311yescharacteristic
231C4×D5—121no—
232#126—151no—
233#126—151no—
234#125—151no—
235#123—151no—
236#123—151no—
237#121—31no—
238#121—151no—
239#121—151no—
240#121—151no—
241C4×D4—151no—
242C4×D4—151no—
243C4×D4—151no—
244C4×D4—151no—
245#117—151no—
246C2×C4⋊C4—51no—
247C2×C3×C5—22no—
248C2×C3×C5C2×#4111yescharacteristic
249C2×C3×C5C2×#4111yescharacteristic
250C2×C3×C5H(5,𝔽2)11yescharacteristic
251D15—41no—
252C5×D3—41no—
253C5×D3—41no—
254C23×C3C22×D511yescharacteristic
255C22×D3—51no—
256C22×D3C22×D511yescharacteristic
257C22×D3C22×D511yescharacteristic
258C2×C3×C4—41no—
259C2×C3×C4—51no—
260C2×C3×C4—51no—
261C2×C3×C4—51no—
262C2×C3×C4—51no—
263C2×C3×C4—102no—
264C2×C3×C4—201no—
265C2×C3×C4C22×D511yescharacteristic
266C2×C3×C4C22×D511yescharacteristic
267#67—22no—
268#67—22no—
269#67—102no—
270C2×Dic3—51no—
271C2×Dic3—51no—
272C2×Dic3—51no—
273C2×Dic3C22×D511yescharacteristic
274C2×Dic3C22×D511yescharacteristic
275D12—41no—
276D12—102no—
277D12—201no—
278C4×D3—102no—
279C4×D3—201no—
280C4×D3—201no—
281C22×C5—21no—
282C22×C5—21no—
283C22×C5—61no—
284C22×C5—61no—
285C22×C5—61no—
286C22×C5—61no—
287C22×C5—61no—
288C22×C5—61no—
289C22×C5#23912yes—
290C22×C5C23×D311yescharacteristic
291C2×D5—61no—
292C2×D5—61no—
293C2×D5—61no—
294C4×C5—21no—
295C4×C5—21no—
296C4×C5—61no—
297C4×C5—61no—
298Dic5—21no—
299Dic5—21no—
300Dic5—21no—
301Dic5—61no—
302Dic5—61no—
303Dic5—61no—
304Dic5#23512yes—
305C2×D4—31no—
306C2×D4—31no—
307C2×D4—31no—
308C2×D4—151no—
309C2×D4—151no—
310C2×D4—151no—
311C22×C4—51no—
312C22×C4—51no—
313C22×C4—151no—
314C22×C4—151no—
315C22×C4—151no—
316C22×C4D3×D511yescharacteristic
317C4⋊C4—31no—
318C4⋊C4—51no—
319C4⋊C4—51no—
320C4⋊C4—51no—
321C4⋊C4—51no—
322C4⋊C4—151no—
323C4⋊C4—151no—
324C4⋊C4—151no—
325C4⋊C4—151no—
326C4⋊C4—151no—
327C22⋊C4—31no—
328C22⋊C4—31no—
329C22⋊C4—151no—
330C22⋊C4—151no—
331C22⋊C4—151no—
332C22⋊C4—151no—
333C22⋊C4—151no—
334C22⋊C4—151no—
335C22⋊C4—151no—
336C22⋊C4—151no—
337C42—151no—
338C42—151no—
339C42—151no—
340C3×C5#55311yescharacteristic, complemented, Hall
341C22×C3—21no—
342C22×C3—21no—
343C22×C3#71112yes—
344C22×C3C23×D511yescharacteristic
345C2×D3—21no—
346C2×D3—21no—
347C2×D3—21no—
348C2×D3—21no—
349C2×D3—21no—
350C2×D3—21no—
351C2×D3—101no—
352C2×D3—101no—
353C2×D3—101no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—52no—
357C3×C4—101no—
358C3×C4—101no—
359C3×C4—101no—
360Dic3—21no—
361Dic3—21no—
362Dic3—101no—
363Dic3—101no—
364Dic3—101no—
365C2×C5—22no—
366C2×C5—121no—
367C2×C5—121no—
368C2×C5C2×#23511yescharacteristic
369C2×C5#100411yescharacteristic
370C2×C5C2×#23911yescharacteristic
371D5—121no—
372C23—31no—
373C23—31no—
374C23—151no—
375C23C2×D3×D511yescharacteristic
376D4—62no—
377D4—62no—
378D4—121no—
379D4—302no—
380D4—302no—
381D4—601no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—41no—
385C2×C4—51no—
386C2×C4—51no—
387C2×C4—51no—
388C2×C4—51no—
389C2×C4—102no—
390C2×C4—151no—
391C2×C4—151no—
392C2×C4—151no—
393C2×C4—201no—
394C2×C4—302no—
395C2×C4—601no—
396C2×C4—601no—
397C2×C4C2×D3×D511yescharacteristic
398C2×C4C2×D3×D511yescharacteristic
399C2×C3—22no—
400C2×C3C2×#71111yescharacteristic
401C2×C3#414411yescharacteristic
402C2×C3D5×#4111yescharacteristic
403D3—41no—
404D3—41no—
405D3—201no—
406C5#551411yescharacteristic, complemented, Sylow
407C22—21no—
408C22—21no—
409C22—61no—
410C22—61no—
411C22—61no—
412C22—61no—
413C22—61no—
414C22—61no—
415C22—301no—
416C22—301no—
417C22—301no—
418C22#676812yes—
419C22C22×D3×D511yescharacteristic
420C4—21no—
421C4—21no—
422C4—52no—
423C4—61no—
424C4—61no—
425C4—101no—
426C4—101no—
427C4—101no—
428C4—301no—
429C4—301no—
430C4—301no—
431C3#6593511yescharacteristic, complemented, Sylow
432C2—22no—
433C2—121no—
434C2—121no—
435C2—601no—
436C2C2×#676811yescharacteristic
437C2D5×#23511yescharacteristic
438C2#9204911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#553
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. C4×#1220C2semidirect
2. C5×#930C2semidirect
3. C2×C3×#683C2semidirect
4. #91566C2semidirect
5. #91566C2semidirect
6. Dic5×#67C2semidirect
7. Dic5×#67C2semidirect
8. #91423C2semidirect
9. #91416C2semidirect
10. #91406C2semidirect
11. #91400C2semidirect
12. #91367C2semidirect
13. #91355C2semidirect
14. #91337C2semidirect
15. #91335C2semidirect
61. C2×#683D3semidirect
98. #930D5semidirect
340. C3×C5#553semidirect
406. C5#5514semidirect
431. C3#65935semidirect

Maximal subgroups

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

subgroupindex
C4×#12202
C5×#9302
C2×C3×#6832
#915662
Dic5×#672
#914232
#914162
#914062
#914002
#913672
#913552
#913372
#913352
#659353
#55145