Finite Group Databasebeta

← #11754312

Subgroups — #11754312

Order 960.

588 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×#6772C211yesmaximal, characteristic, complemented
3#91813C211yesmaximal, characteristic, complemented
4C5×#930C211yesmaximal, characteristic, complemented
5#91565C212yesmaximal, complemented
6#91549C212yesmaximal, complemented
7#91449C211yesmaximal, characteristic, complemented
8D5×C4⋊Dic3C211yesmaximal, characteristic, complemented
9#91410C212yesmaximal, complemented
10#91408C212yesmaximal, complemented
11#91360C211yesmaximal, characteristic, complemented
12#65942—31nomaximal, Hall
13C22×C3×C4×C5C2211yescharacteristic
14C2×Dic30C2211yescharacteristic
15C2×C5×#67C2212yes—
16C2×C5×D12C2211yescharacteristic
17C2×C3×C22⋊D5C2211yescharacteristic
18C2×C3×C22⋊D5C2211yescharacteristic
19C3×#709—24no—
20C2×C3×D20C2211yescharacteristic
21C2×C3×C4×D5C2211yescharacteristic
22C2×C3×C4×D5C2211yescharacteristic
23C2×C3×Dic10C2211yescharacteristic
24C2×#1201C2212yes—
25C2×#1201C2212yes—
26C2×D3×Dic5C2212yes—
27C2×D5×Dic3C2212yes—
28#6772—24no—
29#6719C2212yes—
30C4⋊Dic15C2211yescharacteristic
31#6712C2212yes—
32C5×(C2×D3)⋊C4C2212yes—
33C5×C4⋊Dic3C2211yescharacteristic
34Dic3⋊Dic5C2212yes—
35#6668C2212yes—
36(C2×D3)⋊Dic5C2212yes—
37#6665C2212yes—
38#5833—51nomaximal, Hall
39C2×#711—31no—
40C2×#709D311yescharacteristic, complemented
41C5×#121—31no—
42#4085—32no—
43#4084—32no—
44#4084—31no—
45#4070—31no—
46#4043—31no—
47D5×C4⋊C4—31no—
48#4030—32no—
49#4024—32no—
50C23×C3×C5C2311yescharacteristic
51C22×C5×D3C2312yes—
52C22×C3×D5C2311yescharacteristic
53C22×C3×D5C2311yescharacteristic
54C2×C3×C4×C5—22no—
55C2×C3×C4×C5C2311yescharacteristic
56C2×C3×C4×C5C2311yescharacteristic
57C2×Dic15C2312yes—
58C2×Dic15C2312yes—
59Dic30—22no—
60C5×#67—42no—
61C2×C5×Dic3C2312yes—
62C5×D12—22no—
63C3×C22⋊D5—22no—
64C3×C22⋊D5—22no—
65C2×C3×Dic5C2311yescharacteristic
66C2×C3×Dic5C2311yescharacteristic
67C3×D20—22no—
68C3×C4×D5—22no—
69C3×C4×D5D414yescomplemented
70C3×Dic10—22no—
71#1201—24no—
72#1201—42no—
73D3×Dic5—24no—
74D5×Dic3—42no—
75C2×C3×#41—51no—
76C2×#235—51no—
77#946—52no—
78#939—52no—
79#939—52no—
80#938—52no—
81#930—51no—
82#930D511yescharacteristic, complemented
83C2×C4⋊Dic3—51no—
84#924—51no—
85#924—51no—
86C2×C5×D4—32no—
87C2×C5×D4—31no—
88C22×C4×C5C2×D311yescharacteristic
89C2×C22⋊D5—32no—
90C2×C22⋊D5—32no—
91C2×C22⋊D5C2×D311yescharacteristic
92C2×C22⋊D5C2×D311yescharacteristic
93C22×Dic5—32no—
94#711—64no—
95#709—24no—
96C2×D20C2×D311yescharacteristic
97C2×C4×D5—32no—
98C2×C4×D5C2×D311yescharacteristic
99C2×C4×D5C2×D311yescharacteristic
100C2×Dic10—31no—
101C2×Dic10C2×D311yescharacteristic
102C5×C4⋊C4—31no—
103C5×C22⋊C4—32no—
104C22⋊Dic5—32no—
105C22⋊Dic5—32no—
106#685—32no—
107C4⋊Dic5—31no—
108C4⋊Dic5—32no—
109#683—32no—
110#683—32no—
111#537—151noSylow
112C22×C3×C5—21no—
113C22×C3×C5—21no—
114C22×C3×C5—21no—
115C22×C3×C5C2411yescharacteristic
116C2×C5×D3—22no—
117C2×C5×D3—22no—
118C2×C5×D3—22no—
119C2×C3×D5—21no—
120C2×C3×D5—21no—
121C2×C3×D5—21no—
122C2×C3×D5—21no—
123C2×C3×D5—21no—
124C2×C3×D5C2×D412yes—
125C3×C4×C5—21no—
126C3×C4×C5C2×D412yes—
127Dic15—22no—
128Dic15—22no—
129C3×Dic5—21no—
130C3×Dic5C2×D412yes—
131C5×Dic3—22no—
132C3×#41—104no—
133C2×C3×Q8—51no—
134C2×C3×D4—51no—
135C2×C3×D4—51no—
136C2×C3×D4—51no—
137C22×C3×C4—51no—
138C22×C3×C4—51no—
139C22×C3×C4C2×D511yescharacteristic
140C2×#67—52no—
141C2×#67—52no—
142C2×#67C2×D512yes—
143C22×Dic3—52no—
144#235—104no—
145C2×D12C2×D511yescharacteristic
146C2×C4×D3—52no—
147C2×Dic6—51no—
148#217—52no—
149#217—52no—
150(C2×D3)⋊C4—52no—
151(C2×D3)⋊C4C2×D512yes—
152C4⋊Dic3—51no—
153C4⋊Dic3—52no—
154C4⋊Dic3C2×D511yescharacteristic
155Dic3⋊C4—52no—
156Dic3⋊C4—52no—
157C23×C5—32no—
158C23×C5C22×D311yescharacteristic
159C22×D5C22×D311yescharacteristic
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—22no—
169C22⋊D5—64no—
170C22⋊D5—122no—
171C2×Dic5—32no—
172C2×Dic5—32no—
173C2×Dic5—64no—
174C2×Dic5C22×D311yescharacteristic
175C2×Dic5C22×D311yescharacteristic
176D20—22no—
177C4×D5—22no—
178C4×D5—122no—
179C4×D5#6714yescomplemented
180Dic10—22no—
181Dic10—62no—
182C2×#41—51no—
183C2×#41—151no—
184C22⋊Q8—152no—
185C22⋊Q8—151no—
186C22⋊Q8—151no—
187#121—31no—
188#121—152no—
189#121—151no—
190#121—152no—
191#121—152no—
192C2×C4⋊C4—151no—
193C2×C3×C5—41no—
194C2×C3×C5C22×D411yescharacteristic
195C2×C3×C5H(5,𝔽2)11yescharacteristic
196C2×C3×C52−1+411yescharacteristic
197C3×D5—22no—
198C3×D5—41no—
199C5×D3—42no—
200C23×C3—51no—
201C23×C3—51no—
202C23×C3C22×D511yescharacteristic
203C22×D3C22×D512yes—
204C3×Q8—102no—
205C3×D4—102no—
206C3×D4—102no—
207C3×D4—102no—
208C2×C3×C4—22no—
209C2×C3×C4—51no—
210C2×C3×C4—54no—
211C2×C3×C4—51no—
212C2×C3×C4—102no—
213C2×C3×C4C22×D511yescharacteristic
214C2×C3×C4C22×D511yescharacteristic
215#67—42no—
216#67—104no—
217#67—202no—
218C2×Dic3—52no—
219C2×Dic3—52no—
220C2×Dic3—202no—
221C2×Dic3C22×D512yes—
222D12—22no—
223C4×D3—104no—
224Dic6—102no—
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—21no—
236C2×D5—21no—
237C2×D5C2×#6712yes—
238C4×C5—21no—
239C4×C5—62no—
240C4×C5C2×#6712yes—
241Dic5—21no—
242Dic5—62no—
243Dic5—62no—
244Dic5C2×#6712yes—
245#41—104no—
246#41—304no—
247C2×Q8—51no—
248C2×Q8—151no—
249C2×D4—32no—
250C2×D4—31no—
251C2×D4—51no—
252C2×D4—51no—
253C2×D4—51no—
254C2×D4—152no—
255C2×D4—152no—
256C22×C4—51no—
257C22×C4—51no—
258C22×C4—152no—
259C22×C4—152no—
260C22×C4D3×D511yescharacteristic
261C4⋊C4—31no—
262C4⋊C4—152no—
263C4⋊C4—152no—
264C4⋊C4—151no—
265C4⋊C4—152no—
266C22⋊C4—32no—
267C22⋊C4—152no—
268C22⋊C4—152no—
269C22⋊C4—152no—
270C3×C5#53711yescharacteristic, complemented, Hall
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—52no—
275C22×C3—101no—
276C22×C3—101no—
277C22×C3—101no—
278C22×C3—101no—
279C22×C3—101no—
280C22×C3C23×D511yescharacteristic
281C2×D3—22no—
282C2×D3—22no—
283C2×D3—22no—
284C3×C4—21no—
285C3×C4—52no—
286C3×C4—101no—
287C3×C4D4×D512yes—
288Dic3—22no—
289Dic3—102no—
290Dic3—102no—
291C2×C5—41no—
292C2×C5—122no—
293C2×C5#100911yescharacteristic
294C2×C5#101011yescharacteristic
295C2×C5C22×#6711yescharacteristic
296D5—22no—
297D5—41no—
298C23—32no—
299C23—51no—
300C23—51no—
301C23C2×D3×D511yescharacteristic
302Q8—102no—
303Q8—302no—
304D4—62no—
305D4—102no—
306D4—102no—
307D4—102no—
308D4—122no—
309D4—304no—
310D4—602no—
311C2×C4—22no—
312C2×C4—32no—
313C2×C4—51no—
314C2×C4—54no—
315C2×C4—51no—
316C2×C4—102no—
317C2×C4—152no—
318C2×C4—152no—
319C2×C4—304no—
320C2×C4—602no—
321C2×C4C2×D3×D511yescharacteristic
322C2×C4C2×D3×D511yescharacteristic
323C2×C3—41no—
324C2×C3—102no—
325C2×C3—201no—
326C2×C3C2×D4×D511yescharacteristic
327C2×C3#414411yescharacteristic
328C2×C3#415011yescharacteristic
329D3—42no—
330C5#583311yescharacteristic, complemented, Sylow
331C22—21no—
332C22—21no—
333C22—21no—
334C22—52no—
335C22—62no—
336C22—62no—
337C22—62no—
338C22—101no—
339C22—101no—
340C22—101no—
341C22—101no—
342C22—101no—
343C22C22×D3×D511yescharacteristic
344C4—21no—
345C4—52no—
346C4—62no—
347C4—101no—
348C4—302no—
349C4—302no—
350C4D5×#6712yes—
351C3#6594211yescharacteristic, complemented, Sylow
352C2—41no—
353C2—102no—
354C2—122no—
355C2—201no—
356C2#9200111yescharacteristic
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#537
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×#6772C2semidirect
3. #91813C2semidirect
4. C5×#930C2semidirect
5. #91565C2semidirect
6. #91549C2semidirect
7. #91449C2semidirect
8. D5×C4⋊Dic3C2semidirect
9. #91410C2semidirect
10. #91408C2semidirect
11. #91360C2semidirect
40. C2×#709D3semidirect
69. C3×C4×D5D4semidirect
82. #930D5semidirect
179. C4×D5#67semidirect
270. C3×C5#537semidirect
330. C5#5833semidirect
351. C3#65942semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#7092
C2×#67722
#918132
C5×#9302
#915652
#915492
#914492
D5×C4⋊Dic32
#914102
#914082
#913602
#659423
#58335