Finite Group Databasebeta

← #11754529

Subgroups — #11754529

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×C3×#709C211yesmaximal, characteristic, complemented
2C2×#6768C211yesmaximal, characteristic, complemented
3C4×#1220C211yesmaximal, characteristic, complemented
4C5×#930C211yesmaximal, characteristic, complemented
5#91567C212yesmaximal, complemented
6#91565C212yesmaximal, complemented
7#91465C211yesmaximal, characteristic, complemented
8#91463C211yesmaximal, characteristic, complemented
9#91448C212yesmaximal, complemented
10#91440C212yesmaximal, complemented
11#91331C211yesmaximal, characteristic, complemented
12#65933—31nomaximal, Hall
13C22×C3×C4×C5C2211yescharacteristic
14C2×#1220C2211yescharacteristic
15C2×C4×D15C2211yescharacteristic
16C2×C5×#67C2212yes—
17C2×C5×D12C2211yescharacteristic
18C2×C3×C22⋊D5C2212yes—
19C3×#709—24no—
20C2×C3×D20C2211yescharacteristic
21C2×C3×C4×D5C2212yes—
22C2×C3×Dic10C2211yescharacteristic
23C2×#1203C2212yes—
24C2×#1202C2212yes—
25C2×#1201C2212yes—
26C2×#1201C2212yes—
27C2×D3×Dic5C2212yes—
28#6768—24no—
29#6719C2211yescharacteristic
30#6714C2211yescharacteristic
31#6712C2211yescharacteristic
32C4×Dic15C2211yescharacteristic
33C5×(C2×D3)⋊C4C2212yes—
34C5×C4⋊Dic3C2211yescharacteristic
35#6668C2212yes—
36#6665C2212yes—
37Dic3×Dic5C2212yes—
38#5834—51nomaximal, Hall
39C2×#711—31no—
40C2×#709D311yescharacteristic, complemented
41C5×#121—31no—
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
51C22×D15C2311yescharacteristic
52C22×C5×D3C2312yes—
53C22×C3×D5C2312yes—
54C2×C3×C4×C5—22no—
55C2×C3×C4×C5C2311yescharacteristic
56C2×C3×C4×C5C2311yescharacteristic
57#1220—22no—
58C2×Dic15C2311yescharacteristic
59C2×Dic15C2311yescharacteristic
60C2×Dic15C2311yescharacteristic
61C4×D15—22no—
62C5×#67—42no—
63C2×C5×Dic3C2312yes—
64C5×D12—22no—
65C3×C22⋊D5—24no—
66C2×C3×Dic5C2312yes—
67C3×D20—22no—
68C3×C4×D5—24no—
69C3×Dic10D414yescomplemented
70#1203—24no—
71#1202—42no—
72#1201—24no—
73#1201—42no—
74D3×Dic5—24no—
75C2×C3×#41—51no—
76C2×#239—51no—
77Q8×Dic3—51no—
78#940—52no—
79#940—51no—
80#938—52no—
81#938—51no—
82#930—52no—
83#930D511yescharacteristic, complemented
84C4×#67—51no—
85C4×#67—52no—
86C2×C5×D4—32no—
87C2×C5×D4—31no—
88C22×C4×C5C2×D311yescharacteristic
89C2×C22⋊D5—31no—
90C2×C22⋊D5—32no—
91C2×C22⋊D5—32no—
92C2×C22⋊D5—32no—
93C2×C22⋊D5C2×D312yes—
94C22×Dic5—32no—
95#711—64no—
96#709—24no—
97C2×D20—32no—
98C2×D20C2×D311yescharacteristic
99C2×C4×D5—31no—
100C2×C4×D5C2×D312yes—
101C2×Dic10C2×D311yescharacteristic
102C5×C4⋊C4—31no—
103C5×C22⋊C4—32no—
104C22⋊Dic5—31no—
105#685—32no—
106#685—31no—
107#683—31no—
108#683—32no—
109C4×Dic5—31no—
110C4×Dic5—32no—
111#550—151noSylow
112C22×C3×C5—21no—
113C22×C3×C5—21no—
114C22×C3×C5—21no—
115C22×C3×C5C2411yescharacteristic
116C2×D15—21no—
117C2×D15—21no—
118C2×D15—21no—
119C2×C5×D3—22no—
120C2×C5×D3—22no—
121C2×C5×D3—22no—
122C2×C3×D5—22no—
123C2×C3×D5—22no—
124C2×C3×D5—22no—
125C3×C4×C5—21no—
126C3×C4×C5C2×D412yes—
127Dic15—21no—
128Dic15—21no—
129Dic15#4112yescomplemented
130C3×Dic5C2×D414yes—
131C5×Dic3—22no—
132C3×#41—104no—
133C2×C3×Q8—51no—
134C2×C3×D4—52no—
135C2×C3×D4—51no—
136C22×C3×C4—52no—
137C22×C3×C4C2×D511yescharacteristic
138C2×#67—51no—
139C2×#67—52no—
140C2×#67—52no—
141C2×#67—52no—
142C2×#67C2×D512yes—
143#239—104no—
144C2×D12—52no—
145C2×D12C2×D511yescharacteristic
146C2×C4×D3—52no—
147C2×C4×D3—51no—
148#217—51no—
149#217—52no—
150(C2×D3)⋊C4—51no—
151(C2×D3)⋊C4C2×D512yes—
152C4⋊Dic3—52no—
153C4⋊Dic3C2×D511yescharacteristic
154Dic3⋊C4—51no—
155C4×Dic3—51no—
156C4×Dic3—52no—
157C23×C5—32no—
158C23×C5C22×D311yescharacteristic
159C22×D5—31no—
160C22×D5C22×D312yes—
161C5×D4—62no—
162C5×D4—122no—
163C2×C4×C5—22no—
164C2×C4×C5—32no—
165C2×C4×C5C22×D311yescharacteristic
166C2×C4×C5C22×D311yescharacteristic
167C22⋊D5—24no—
168C22⋊D5—64no—
169C22⋊D5—62no—
170C22⋊D5—64no—
171C22⋊D5—122no—
172C2×Dic5—31no—
173C2×Dic5—31no—
174C2×Dic5—31no—
175C2×Dic5—64no—
176C2×Dic5C22×D312yes—
177D20—22no—
178D20—122no—
179C4×D5—24no—
180C4×D5—62no—
181Dic10#6714yescomplemented
182C2×#41—51no—
183C2×#41—151no—
184#127—152no—
185#127—151no—
186#121—31no—
187#121—152no—
188#121—151no—
189#121—152no—
190C4×Q8—151no—
191C4×D4—152no—
192C4×D4—151no—
193C2×C3×C5—41no—
194C2×C3×C5C22×D411yescharacteristic
195C2×C3×C5C2×#4111yescharacteristic
196C2×C3×C5H(5,𝔽2)11yescharacteristic
197D15—41no—
198C3×D5—42no—
199C5×D3—42no—
200C23×C3—52no—
201C23×C3C22×D511yescharacteristic
202C22×D3—51no—
203C22×D3C22×D512yes—
204C3×Q8—54no—
205C3×D4—104no—
206C3×D4—102no—
207C2×C3×C4—22no—
208C2×C3×C4—52no—
209C2×C3×C4—104no—
210C2×C3×C4C22×D511yescharacteristic
211C2×C3×C4C22×D511yescharacteristic
212#67—42no—
213#67—104no—
214#67—102no—
215#67—202no—
216#67—202no—
217C2×Dic3—51no—
218C2×Dic3—51no—
219C2×Dic3—51no—
220C2×Dic3C22×D512yes—
221D12—22no—
222D12—104no—
223C4×D3—104no—
224C4×D3—102no—
225C22×C5—21no—
226C22×C5—21no—
227C22×C5—21no—
228C22×C5—62no—
229C22×C5—62no—
230C22×C5—62no—
231C22×C5C23×D311yescharacteristic
232C2×D5—22no—
233C2×D5—22no—
234C2×D5—22no—
235C2×D5—61no—
236C2×D5—61no—
237C2×D5—61no—
238C4×C5—21no—
239C4×C5—62no—
240C4×C5C2×#6712yes—
241Dic5—32no—
242Dic5—61no—
243Dic5—61no—
244Dic5C2×#6714yes—
245#41—104no—
246#41—304no—
247C2×Q8—51no—
248C2×D4—32no—
249C2×D4—31no—
250C2×D4—52no—
251C2×D4—51no—
252C2×D4—152no—
253C2×D4—152no—
254C2×D4—152no—
255C2×D4—151no—
256C2×D4—152no—
257C22×C4—52no—
258C22×C4—152no—
259C22×C4—151no—
260C22×C4D3×D511yescharacteristic
261C4⋊C4—31no—
262C4⋊C4—152no—
263C4⋊C4—151no—
264C22⋊C4—32no—
265C22⋊C4—152no—
266C22⋊C4—151no—
267C22⋊C4—151no—
268C42—152no—
269C42—151no—
270C3×C5#55011yescharacteristic, complemented, Hall
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—102no—
275C22×C3—102no—
276C22×C3—102no—
277C22×C3C23×D511yescharacteristic
278C2×D3—22no—
279C2×D3—22no—
280C2×D3—22no—
281C2×D3—101no—
282C2×D3—101no—
283C2×D3—101no—
284C3×C4—21no—
285C3×C4—54no—
286C3×C4D4×D512yes—
287Dic3—22no—
288Dic3—52no—
289Dic3—101no—
290Dic3—101no—
291C2×C5—41no—
292C2×C5—122no—
293C2×C5D3×#4111yescharacteristic
294C2×C5#100911yescharacteristic
295C2×C5C22×#6711yescharacteristic
296D5—42no—
297D5—121no—
298C23—32no—
299C23—52no—
300C23—151no—
301C23C2×D3×D511yescharacteristic
302Q8—54no—
303D4—62no—
304D4—104no—
305D4—102no—
306D4—122no—
307D4—304no—
308D4—302no—
309D4—304no—
310D4—602no—
311D4—602no—
312C2×C4—22no—
313C2×C4—32no—
314C2×C4—52no—
315C2×C4—104no—
316C2×C4—151no—
317C2×C4—151no—
318C2×C4—151no—
319C2×C4—304no—
320C2×C4—302no—
321C2×C4C2×D3×D511yescharacteristic
322C2×C4C2×D3×D511yescharacteristic
323C2×C3—41no—
324C2×C3—202no—
325C2×C3C2×D4×D511yescharacteristic
326C2×C3#414411yescharacteristic
327C2×C3D5×#4111yescharacteristic
328D3—42no—
329D3—201no—
330C5#583411yescharacteristic, complemented, Sylow
331C22—21no—
332C22—21no—
333C22—21no—
334C22—62no—
335C22—62no—
336C22—62no—
337C22—102no—
338C22—102no—
339C22—102no—
340C22—301no—
341C22—301no—
342C22—301no—
343C22C22×D3×D511yescharacteristic
344C4—21no—
345C4—54no—
346C4—62no—
347C4—152no—
348C4—301no—
349C4—301no—
350C4D5×#6712yes—
351C3#6593311yescharacteristic, complemented, Sylow
352C2—41no—
353C2—122no—
354C2—202no—
355C2—601no—
356C2#9201611yescharacteristic
357C2#9201811yescharacteristic
358C2C2×D5×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#550
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. C2×C3×#709C2semidirect
2. C2×#6768C2semidirect
3. C4×#1220C2semidirect
4. C5×#930C2semidirect
5. #91567C2semidirect
6. #91565C2semidirect
7. #91465C2semidirect
8. #91463C2semidirect
9. #91448C2semidirect
10. #91440C2semidirect
11. #91331C2semidirect
40. C2×#709D3semidirect
69. C3×Dic10D4semidirect
83. #930D5semidirect
129. Dic15#41semidirect
181. Dic10#67semidirect
270. C3×C5#550semidirect
330. C5#5834semidirect
351. C3#65933semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#7092
C2×#67682
C4×#12202
C5×#9302
#915672
#915652
#914652
#914632
#914482
#914402
#913312
#659333
#58345