Finite Group Databasebeta

← #11754557

Subgroups — #11754557

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×#6817C211yesmaximal, characteristic, complemented
2C2×#6772C211yesmaximal, characteristic, complemented
3C5×#930C211yesmaximal, characteristic, complemented
4C3×C4×C22⋊D5C211yesmaximal, characteristic, complemented
5#91567C212yesmaximal, complemented
6#91558C212yesmaximal, complemented
7#91461C212yesmaximal, complemented
8#91451C211yesmaximal, characteristic, complemented
9#91450C211yesmaximal, characteristic, complemented
10#91406C212yesmaximal, complemented
11#91333C211yesmaximal, characteristic, complemented
12#65933—31nomaximal, Hall
13C22×C3×C4×C5C2211yescharacteristic
14C2×#1220C2212yes—
15#6817—24no—
16C2×D60C2211yescharacteristic
17C2×C4×D15C2212yes—
18C2×Dic30C2211yescharacteristic
19C2×C5×#67C2212yes—
20C2×C5×D12C2211yescharacteristic
21C2×C3×C22⋊D5C2211yescharacteristic
22C2×C3×C4×D5C2211yescharacteristic
23C2×#1203C2212yes—
24C2×#1203C2212yes—
25C2×#1202C2212yes—
26C2×#1201C2212yes—
27C2×D3×Dic5C2212yes—
28#6772—24no—
29C5×(C2×D3)⋊C4C2212yes—
30C5×C4⋊Dic3C2211yescharacteristic
31C3×C22⋊Dic5C2211yescharacteristic
32C3×#685C2211yescharacteristic
33C3×#683C2211yescharacteristic
34C3×C4×Dic5C2211yescharacteristic
35#6669C2212yes—
36#6667C2212yes—
37Dic3×Dic5C2212yes—
38#5553—51nomaximal, Hall
39C2×#711—31no—
40C2×#709—31no—
41C5×#121—31no—
42#4086—32no—
43#4086—31no—
44#4085—32no—
45C4×C22⋊D5D311yescharacteristic, complemented
46#4041—31no—
47#4033—31no—
48#4030—32no—
49#4027—32no—
50C23×C3×C5C2311yescharacteristic
51C22×D15C2312yes—
52C22×C5×D3C2312yes—
53C22×C3×D5C2311yescharacteristic
54C2×C3×C4×C5—22no—
55C2×C3×C4×C5C2311yescharacteristic
56C2×C3×C4×C5C2311yescharacteristic
57#1220—24no—
58C2×Dic15C2312yes—
59D60—22no—
60C4×D15—24no—
61Dic30D414yescomplemented
62C5×#67—42no—
63C2×C5×Dic3C2312yes—
64C5×D12—22no—
65C3×C22⋊D5—22no—
66C2×C3×Dic5C2311yescharacteristic
67C2×C3×Dic5C2311yescharacteristic
68C2×C3×Dic5C2311yescharacteristic
69C3×C4×D5—22no—
70#1203—24no—
71#1203—42no—
72#1202—42no—
73#1201—24no—
74D3×Dic5—24no—
75C2×#235—51no—
76C2×#235—51no—
77C3×C4×D4—51no—
78#940—52no—
79#930—51no—
80#930D511yescharacteristic, complemented
81#893—52no—
82#884—52no—
83#884—52no—
84#874—51no—
85C4×Dic6—51no—
86C2×C5×D4—32no—
87C2×C5×D4—31no—
88C22×C4×C5C2×D311yescharacteristic
89C2×C22⋊D5—32no—
90C2×C22⋊D5—32no—
91C2×C22⋊D5—32no—
92C2×C22⋊D5—32no—
93C2×C22⋊D5C2×D311yescharacteristic
94C22×Dic5—32no—
95#711—64no—
96#709—64no—
97C2×D20—31no—
98C2×D20—32no—
99C2×C4×D5—32no—
100C2×C4×D5C2×D311yescharacteristic
101C2×Dic10—31no—
102C5×C4⋊C4—31no—
103C5×C22⋊C4—32no—
104C22⋊Dic5C2×D311yescharacteristic
105#685—32no—
106#685C2×D311yescharacteristic
107#683—32no—
108#683C2×D311yescharacteristic
109C4×Dic5—32no—
110C4×Dic5C2×D311yescharacteristic
111#550—151noSylow
112C22×C3×C5—21no—
113C22×C3×C5—21no—
114C22×C3×C5—21no—
115C22×C3×C5C2411yescharacteristic
116C2×D15—22no—
117C2×D15—22no—
118C2×D15—22no—
119C2×C5×D3—22no—
120C2×C5×D3—22no—
121C2×C5×D3—22no—
122C2×C3×D5—21no—
123C2×C3×D5—21no—
124C2×C3×D5—21no—
125C3×C4×C5—21no—
126C3×C4×C5C2×D412yes—
127Dic15C2×D414yes—
128C3×Dic5—21no—
129C3×Dic5—21no—
130C3×Dic5#4112yescomplemented
131C5×Dic3—22no—
132C2×C3×D4—51no—
133C22×C3×C4—51no—
134C22×C3×C4C2×D511yescharacteristic
135C2×#67—52no—
136C2×#67—52no—
137C2×#67—52no—
138C2×#67C2×D512yes—
139#235—104no—
140#235—104no—
141C2×D12—52no—
142C2×D12—51no—
143C2×D12—52no—
144C2×D12C2×D511yescharacteristic
145C2×C4×D3—52no—
146C2×C4×D3—52no—
147C2×Dic6—51no—
148C3×C4⋊C4—51no—
149C3×C22⋊C4—51no—
150C3×C22⋊C4—51no—
151C3×C42—51no—
152(C2×D3)⋊C4—52no—
153(C2×D3)⋊C4C2×D512yes—
154C4⋊Dic3C2×D511yescharacteristic
155Dic3⋊C4—52no—
156C4×Dic3—52no—
157C23×C5—32no—
158C23×C5C22×D311yescharacteristic
159C22×D5—32no—
160C22×D5C22×D311yescharacteristic
161C5×D4—62no—
162C5×D4—122no—
163C2×C4×C5—22no—
164C2×C4×C5—32no—
165C2×C4×C5C22×D311yescharacteristic
166C2×C4×C5C22×D311yescharacteristic
167C22⋊D5—22no—
168C22⋊D5—64no—
169C22⋊D5—64no—
170C22⋊D5—64no—
171C22⋊D5—122no—
172C2×Dic5—32no—
173C2×Dic5—64no—
174C2×Dic5C22×D311yescharacteristic
175C2×Dic5C22×D311yescharacteristic
176C2×Dic5C22×D311yescharacteristic
177D20—62no—
178D20—122no—
179C4×D5—22no—
180C4×D5—64no—
181Dic10—34no—
182C2×#41—151no—
183C2×#41—151no—
184#127—151no—
185#127—152no—
186#121—31no—
187#121—152no—
188#121—152no—
189#121—151no—
190C4×Q8—151no—
191C4×D4—51no—
192C4×D4—152no—
193C2×C3×C5—41no—
194C2×C3×C5C22×D411yescharacteristic
195C2×C3×C5C2×#4111yescharacteristic
196C2×C3×C5H(5,𝔽2)11yescharacteristic
197D15—42no—
198C3×D5—41no—
199C5×D3—42no—
200C23×C3—51no—
201C23×C3C22×D511yescharacteristic
202C22×D3—52no—
203C22×D3C22×D512yes—
204C3×D4—102no—
205C2×C3×C4—22no—
206C2×C3×C4—51no—
207C2×C3×C4—51no—
208C2×C3×C4—51no—
209C2×C3×C4—102no—
210C2×C3×C4C22×D511yescharacteristic
211C2×C3×C4C22×D511yescharacteristic
212#67—42no—
213#67—104no—
214#67—104no—
215#67—202no—
216C2×Dic3—52no—
217C2×Dic3C22×D512yes—
218D12—22no—
219D12—104no—
220D12—102no—
221D12—202no—
222C4×D3—104no—
223C4×D3—104no—
224Dic6—54no—
225C22×C5—21no—
226C22×C5—21no—
227C22×C5—21no—
228C22×C5—62no—
229C22×C5—62no—
230C22×C5—62no—
231C22×C5C23×D311yescharacteristic
232C2×D5—21no—
233C2×D5—21no—
234C2×D5—21no—
235C2×D5—62no—
236C2×D5—62no—
237C2×D5—62no—
238C4×C5—21no—
239C4×C5—62no—
240C4×C5D3×D412yes—
241Dic5—21no—
242Dic5—21no—
243Dic5—34no—
244Dic5#23512yescomplemented
245#41—304no—
246#41—304no—
247C2×Q8—151no—
248C2×D4—32no—
249C2×D4—31no—
250C2×D4—51no—
251C2×D4—152no—
252C2×D4—152no—
253C2×D4—152no—
254C2×D4—152no—
255C2×D4—152no—
256C2×D4—151no—
257C22×C4—51no—
258C22×C4—152no—
259C22×C4—152no—
260C22×C4D3×D511yescharacteristic
261C4⋊C4—31no—
262C4⋊C4—51no—
263C4⋊C4—152no—
264C22⋊C4—32no—
265C22⋊C4—51no—
266C22⋊C4—51no—
267C22⋊C4—152no—
268C42—51no—
269C42—152no—
270C3×C5#55011yescharacteristic, complemented, Hall
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—101no—
275C22×C3—101no—
276C22×C3—101no—
277C22×C3C23×D511yescharacteristic
278C2×D3—22no—
279C2×D3—22no—
280C2×D3—22no—
281C2×D3—102no—
282C2×D3—102no—
283C2×D3—102no—
284C3×C4—21no—
285C3×C4—52no—
286C3×C4—101no—
287C3×C4—101no—
288C3×C4D4×D512yes—
289Dic3—22no—
290Dic3—54no—
291C2×C5—41no—
292C2×C5—122no—
293C2×C5C2×#23511yescharacteristic
294C2×C5C2×D3×D411yescharacteristic
295C2×C5#100911yescharacteristic
296D5—41no—
297D5—122no—
298C23—32no—
299C23—51no—
300C23—152no—
301C23C2×D3×D511yescharacteristic
302Q8—154no—
303D4—62no—
304D4—102no—
305D4—122no—
306D4—304no—
307D4—304no—
308D4—304no—
309D4—302no—
310D4—602no—
311D4—602no—
312C2×C4—22no—
313C2×C4—32no—
314C2×C4—51no—
315C2×C4—51no—
316C2×C4—51no—
317C2×C4—102no—
318C2×C4—152no—
319C2×C4—304no—
320C2×C4—304no—
321C2×C4C2×D3×D511yescharacteristic
322C2×C4C2×D3×D511yescharacteristic
323C2×C3—41no—
324C2×C3—201no—
325C2×C3C2×D4×D511yescharacteristic
326C2×C3#414411yescharacteristic
327C2×C3D5×#4111yescharacteristic
328D3—42no—
329D3—202no—
330C5#555311yescharacteristic, complemented, Sylow
331C22—21no—
332C22—21no—
333C22—21no—
334C22—62no—
335C22—62no—
336C22—62no—
337C22—101no—
338C22—101no—
339C22—101no—
340C22—302no—
341C22—302no—
342C22—302no—
343C22C22×D3×D511yescharacteristic
344C4—21no—
345C4—52no—
346C4—62no—
347C4—101no—
348C4—101no—
349C4—154no—
350C4#679012yes—
351C3#6593311yescharacteristic, complemented, Sylow
352C2—41no—
353C2—122no—
354C2—201no—
355C2—602no—
356C2D5×#23511yescharacteristic
357C2#9201811yescharacteristic
358C2C2×#679011yescharacteristic
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×#6817C2semidirect
2. C2×#6772C2semidirect
3. C5×#930C2semidirect
4. C3×C4×C22⋊D5C2semidirect
5. #91567C2semidirect
6. #91558C2semidirect
7. #91461C2semidirect
8. #91451C2semidirect
9. #91450C2semidirect
10. #91406C2semidirect
11. #91333C2semidirect
45. C4×C22⋊D5D3semidirect
61. Dic30D4semidirect
80. #930D5semidirect
130. C3×Dic5#41semidirect
244. Dic5#235semidirect
270. C3×C5#550semidirect
330. C5#5553semidirect
351. C3#65933semidirect

Maximal subgroups

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

subgroupindex
C2×#68172
C2×#67722
C5×#9302
C3×C4×C22⋊D52
#915672
#915582
#914612
#914512
#914502
#914062
#913332
#659333
#55535