Finite Group Databasebeta

← #11755136

Subgroups — #11755136

Order 960.

392 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#91835C211yesmaximal, characteristic, complemented
2C5×#946C211yesmaximal, characteristic, complemented
3C3×#4090C211yesmaximal, characteristic, complemented
4#91387C211yesmaximal, characteristic, complemented
5#91377C211yesmaximal, characteristic, complemented
6#91372C211yesmaximal, characteristic, complemented
7#91355C211yesmaximal, characteristic
8#91352C211yesmaximal, characteristic, complemented
9#91351C211yesmaximal, characteristic
10#91343C211yesmaximal, characteristic, complemented
11#91342C211yesmaximal, characteristic, complemented
12#91337C211yesmaximal, characteristic, complemented
13#91336C211yesmaximal, characteristic, complemented
14#91327C211yesmaximal, characteristic, complemented
15#91326C211yesmaximal, characteristic
16#65980—31nomaximal, Hall
17C2×C3×C5×Q8C2211yescharacteristic
18C2×C4×D15C2211yescharacteristic
19C2×C4×C5×D3C2211yescharacteristic
20C2×C3×Dic10C2211yescharacteristic
21C2×#1204C2211yescharacteristic
22C2×#1203C2211yescharacteristic
23#6714C2211yescharacteristic
24#6714C2211yescharacteristic
25C4⋊Dic15C2211yescharacteristic
26#6712C2211yescharacteristic
27#6712C2211yescharacteristic
28C5×(C2×D3)⋊C4C2211yescharacteristic
29C5×(C2×D3)⋊C4C2211yescharacteristic
30C5×C4⋊Dic3C2211yescharacteristic
31C5×Dic3⋊C4C2211yescharacteristic
32C5×Dic3⋊C4C2211yescharacteristic
33C3×#683C2211yescharacteristic
34C3×#683C2211yescharacteristic
35C3×#683C2211yescharacteristic
36C3×#683C2211yescharacteristic
37C3×C4×Dic5C2211yescharacteristic
38Dic3⋊Dic5C2211yescharacteristic
39Dic3⋊Dic5C2211yescharacteristic
40Dic3⋊Dic5C2211yescharacteristic
41#6669C2211yescharacteristic
42#6669C2211yescharacteristic
43#6669C2211yescharacteristic
44#6667C2211yescharacteristic
45#6667C2211yescharacteristic
46#6667C2211yescharacteristic
47(C2×D3)⋊Dic5C2211yescharacteristic
48(C2×D3)⋊Dic5C2211yescharacteristic
49(C2×D3)⋊Dic5C2211yescharacteristic
50Dic3×Dic5C2211yescharacteristic
51Dic3×Dic5C2211yescharacteristic
52#5735—51nomaximal, Hall
53C5×C22⋊Q8—31no—
54#4092—31no—
55#4090D311yescharacteristic, complemented
56#4075—31no—
57#4070—31no—
58#4044—31no—
59#4044—31no—
60#4042—31no—
61#4036—31no—
62#4035—31no—
63#4034—31no—
64#4031—31no—
65#4029—31no—
66#4025—31no—
67#4025—31no—
68C22×D15C2311yescharacteristic
69C22×C5×D3C2311yescharacteristic
70C3×C5×Q8—41no—
71C2×C3×C4×C5C2311yescharacteristic
72C2×C3×C4×C5C2311yescharacteristic
73C2×C3×C4×C5C2311yescharacteristic
74C2×Dic15C2311yescharacteristic
75C2×Dic15C2311yescharacteristic
76C2×Dic15C2311yescharacteristic
77C4×D15—41no—
78C2×C5×Dic3C2311yescharacteristic
79C2×C5×Dic3C2311yescharacteristic
80C2×C5×Dic3C2311yescharacteristic
81C4×C5×D3—41no—
82C2×C3×Dic5C2311yescharacteristic
83C2×C3×Dic5C2311yescharacteristic
84C2×C3×Dic5C2311yescharacteristic
85C2×C3×Dic5C2311yescharacteristic
86C3×Dic10—41no—
87#1204—41no—
88#1203—41no—
89C3×C4⋊Q8—51no—
90#947—51no—
91#946—51no—
92#946D511yescharacteristic, complemented
93#944—51no—
94#898—51no—
95#898—51no—
96#896—51no—
97#896—51no—
98#894—51no—
99#894—51no—
100#889—51no—
101#889—51no—
102#876—51no—
103#876—51no—
104C2×C5×Q8C2×D311yescharacteristic
105C22×C4×C5—31no—
106C2×C22⋊D5—31no—
107C2×C4×D5—31no—
108C2×Dic10—31no—
109C2×Dic10C2×D311yescharacteristic
110C5×C4⋊C4—31no—
111C5×C4⋊C4—31no—
112C5×C4⋊C4—31no—
113C5×C22⋊C4—31no—
114C5×C22⋊C4—31no—
115C22⋊Dic5—31no—
116C22⋊Dic5—31no—
117C22⋊Dic5—31no—
118#685—31no—
119#685—31no—
120#685—31no—
121#685—31no—
122#685—31no—
123C4⋊Dic5—31no—
124C4⋊Dic5—31no—
125C4⋊Dic5—31no—
126C4⋊Dic5—31no—
127#683—31no—
128#683—31no—
129#683—31no—
130#683—31no—
131#683—31no—
132#683C2×D311yescharacteristic
133#683C2×D311yescharacteristic
134#683C2×D311yescharacteristic
135#683C2×D311yescharacteristic
136C4×Dic5—31no—
137C4×Dic5—31no—
138C4×Dic5C2×D311yescharacteristic
139#563—151noSylow
140C22×C3×C5C2411yescharacteristic
141C2×D15—21no—
142C2×D15—21no—
143C2×D15—21no—
144C2×C5×D3—21no—
145C2×C5×D3—21no—
146C2×C5×D3—21no—
147C3×C4×C5—21no—
148C3×C4×C5—21no—
149C3×C4×C5—21no—
150Dic15—21no—
151Dic15—21no—
152Dic15—21no—
153C3×Dic5—21no—
154C3×Dic5—21no—
155C3×Dic5—21no—
156C3×Dic5—21no—
157C5×Dic3—21no—
158C5×Dic3—21no—
159C5×Dic3—21no—
160C2×C3×Q8—51no—
161C2×C3×Q8C2×D511yescharacteristic
162C2×D12—51no—
163C2×C4×D3—51no—
164C2×C4×D3C2×D511yescharacteristic
165C2×Dic6—51no—
166C3×C4⋊C4—51no—
167C3×C4⋊C4—51no—
168C3×C4⋊C4—51no—
169C3×C4⋊C4—51no—
170C3×C42—51no—
171(C2×D3)⋊C4—51no—
172(C2×D3)⋊C4—51no—
173(C2×D3)⋊C4—51no—
174(C2×D3)⋊C4—51no—
175(C2×D3)⋊C4—51no—
176(C2×D3)⋊C4—51no—
177(C2×D3)⋊C4—51no—
178(C2×D3)⋊C4—51no—
179(C2×D3)⋊C4C2×D511yescharacteristic
180(C2×D3)⋊C4C2×D511yescharacteristic
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⋊C4—51no—
190Dic3⋊C4—51no—
191Dic3⋊C4C2×D511yescharacteristic
192Dic3⋊C4C2×D511yescharacteristic
193C4×Dic3—51no—
194C4×Dic3—51no—
195C23×C5—31no—
196C22×D5—31no—
197C5×Q8—41no—
198C2×C4×C5—31no—
199C2×C4×C5—31no—
200C2×C4×C5—31no—
201C2×C4×C5—121no—
202C2×C4×C5C22×D311yescharacteristic
203C2×C4×C5C22×D311yescharacteristic
204C2×C4×C5C22×D311yescharacteristic
205C22⋊D5—121no—
206C2×Dic5—31no—
207C2×Dic5—31no—
208C2×Dic5—31no—
209C2×Dic5C22×D311yescharacteristic
210C2×Dic5C22×D311yescharacteristic
211C2×Dic5C22×D311yescharacteristic
212C2×Dic5C22×D311yescharacteristic
213C4×D5—121no—
214Dic10—41no—
215Dic10—121no—
216C4⋊Q8—51no—
217C4⋊Q8—151no—
218#126—151no—
219#126—151no—
220#126—151no—
221#126—151no—
222#125—151no—
223#125—151no—
224#124—151no—
225#123—151no—
226#123—151no—
227C22⋊Q8—31no—
228C22⋊Q8—151no—
229C22⋊Q8—151no—
230C22⋊Q8—151no—
231C2×C3×C5H(5,𝔽2)11yescharacteristic
232C2×C3×C52−1+411yescharacteristic
233C2×C3×C52−1+411yescharacteristic
234D15—41no—
235C5×D3—41no—
236C22×D3—51no—
237C22×D3C22×D511yescharacteristic
238C3×Q8—41no—
239C3×Q8—201no—
240C2×C3×C4—51no—
241C2×C3×C4—51no—
242C2×C3×C4—51no—
243C2×C3×C4—51no—
244C2×C3×C4C22×D511yescharacteristic
245C2×C3×C4C22×D511yescharacteristic
246C2×C3×C4C22×D511yescharacteristic
247C2×Dic3—51no—
248C2×Dic3—51no—
249C2×Dic3—51no—
250C2×Dic3C22×D511yescharacteristic
251C2×Dic3C22×D511yescharacteristic
252C2×Dic3C22×D511yescharacteristic
253D12—201no—
254C4×D3—41no—
255C4×D3—201no—
256Dic6—201no—
257C22×C5—61no—
258C22×C5—61no—
259C22×C5—61no—
260C22×C5C23×D311yescharacteristic
261C2×D5—61no—
262C2×D5—61no—
263C2×D5—61no—
264C4×C5—21no—
265C4×C5—21no—
266C4×C5—21no—
267C4×C5—61no—
268C4×C5—61no—
269C4×C5—61no—
270Dic5—21no—
271Dic5—21no—
272Dic5—21no—
273Dic5—21no—
274Dic5—61no—
275Dic5—61no—
276Dic5—61no—
277C2×Q8—51no—
278C2×Q8—151no—
279C2×Q8D3×D511yescharacteristic
280C2×D4—151no—
281C22×C4—31no—
282C22×C4—151no—
283C4⋊C4—31no—
284C4⋊C4—31no—
285C4⋊C4—31no—
286C4⋊C4—51no—
287C4⋊C4—51no—
288C4⋊C4—51no—
289C4⋊C4—51no—
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⋊C4—151no—
299C22⋊C4—31no—
300C22⋊C4—31no—
301C22⋊C4—151no—
302C22⋊C4—151no—
303C22⋊C4—151no—
304C22⋊C4—151no—
305C22⋊C4—151no—
306C22⋊C4—151no—
307C22⋊C4—151no—
308C22⋊C4—151no—
309C42—51no—
310C42—151no—
311C42—151no—
312C3×C5#56311yescharacteristic, complemented, Hall
313C22×C3C23×D511yescharacteristic
314C2×D3—21no—
315C2×D3—21no—
316C2×D3—21no—
317C2×D3—101no—
318C2×D3—101no—
319C2×D3—101no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—101no—
324C3×C4—101no—
325C3×C4—101no—
326C3×C4—101no—
327Dic3—21no—
328Dic3—21no—
329Dic3—21no—
330Dic3—101no—
331Dic3—101no—
332Dic3—101no—
333C2×C5—121no—
334C2×C5#100411yescharacteristic
335C2×C5#100711yescharacteristic
336C2×C5#100711yescharacteristic
337D5—121no—
338C23—31no—
339C23—151no—
340Q8—41no—
341Q8—201no—
342Q8—601no—
343D4—601no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—31no—
347C2×C4—51no—
348C2×C4—51no—
349C2×C4—51no—
350C2×C4—51no—
351C2×C4—121no—
352C2×C4—151no—
353C2×C4—151no—
354C2×C4—151no—
355C2×C4—601no—
356C2×C4C2×D3×D511yescharacteristic
357C2×C4C2×D3×D511yescharacteristic
358C2×C4C2×D3×D511yescharacteristic
359C2×C3#414411yescharacteristic
360C2×C3#414711yescharacteristic
361C2×C3#415011yescharacteristic
362D3—41no—
363D3—201no—
364C5#573511yescharacteristic, complemented, Sylow
365C22—61no—
366C22—61no—
367C22—61no—
368C22—301no—
369C22—301no—
370C22—301no—
371C22C22×D3×D511yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—61no—
376C4—61no—
377C4—61no—
378C4—101no—
379C4—101no—
380C4—101no—
381C4—101no—
382C4—301no—
383C4—301no—
384C4—301no—
385C3#6598011yescharacteristic, complemented, Sylow
386C2—121no—
387C2—601no—
388C2#9202911yescharacteristic
389C2#9203011yescharacteristic
390C2#9204911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×Q8C22
prime radicalC2×#1203C22
hypercenterC2×Q8D3×D5
hypocenterC3×C5#563
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #91835C2semidirect
2. C5×#946C2semidirect
3. C3×#4090C2semidirect
4. #91387C2semidirect
5. #91377C2semidirect
6. #91372C2semidirect
8. #91352C2semidirect
10. #91343C2semidirect
11. #91342C2semidirect
12. #91337C2semidirect
13. #91336C2semidirect
14. #91327C2semidirect
55. #4090D3semidirect
92. #946D5semidirect
312. C3×C5#563semidirect
364. C5#5735semidirect
385. C3#65980semidirect

Maximal subgroups

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

subgroupindex
#918352
C5×#9462
C3×#40902
#913872
#913772
#913722
#913552
#913522
#913512
#913432
#913422
#913372
#913362
#913272
#913262
#659803
#57355