Finite Group Databasebeta

← #11754800

Subgroups — #11754800

Order 960.

660 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#711C211yesmaximal, characteristic, complemented
2C22×#1200C211yesmaximal, characteristic, complemented
3#91826C211yesmaximal, characteristic, complemented
4C5×#935C211yesmaximal, characteristic, complemented
5#91572C211yesmaximal, characteristic, complemented
6#91552C212yesmaximal, complemented
7#91536C211yesmaximal, characteristic, complemented
8#91533C212yesmaximal, complemented
9#91528C211yesmaximal, characteristic, complemented
10#91444C212yesmaximal, complemented
11#91411C212yesmaximal, complemented
12#65990—31nomaximal, Hall
13C23×D15C2211yescharacteristic
14C2×C3×C5×D4C2211yescharacteristic
15C2×#1220C2212yes—
16C22×C5×Dic3C2211yescharacteristic
17C2×C3×C22⋊D5C2211yescharacteristic
18C2×C3×C22⋊D5C2211yescharacteristic, complemented
19C22×C3×Dic5C2211yescharacteristic
20C22×C3×Dic5C2211yescharacteristic, complemented
21C3×#711—24no—
22C2×C3×C4×D5C2211yescharacteristic, complemented
23C2×C3×Dic10C2211yescharacteristic, complemented
24C2×#1202C2212yes—
25C2×#1200—24no—
26C2×#1200C2212yes—
27C2×#1200C2212yes—
28#6719C2211yescharacteristic
29#6714C2212yes—
30C5×#217C2211yescharacteristic
31C5×#217C2212yescomplemented
32C5×Dic3⋊C4C2212yescomplemented
33#6669C2212yescomplemented
34#6668C2212yescomplemented
35#6667C2212yes—
36#6665C2212yescomplemented
37Dic3×Dic5C2212yescomplemented
38#5828—51nomaximal, Hall
39C2×#711D311yescharacteristic, complemented
40C22×C4×D5—31no—
41C5×#123—31no—
42#4083—31no—
43#4075—31no—
44#4042—32no—
45#4039—32no—
46#4030—32no—
47#4028—31no—
48#4027—32no—
49#4023—31no—
50C23×C3×C5C2311yescharacteristic
51C23×C3×C5C2311yescharacteristic
52C22×D15—22no—
53C22×D15—24no—
54C22×D15C2312yes—
55C22×C3×D5C2311yescharacteristic
56C3×C5×D4—22no—
57C2×C3×C4×C5C2311yescharacteristic
58#1220—24no—
59C2×Dic15C2312yes—
60C2×C5×Dic3—22no—
61C2×C5×Dic3C2312yes—
62C2×C5×Dic3C2312yes—
63C3×C22⋊D5—22no—
64C3×C22⋊D5—22no—
65C2×C3×Dic5—22no—
66C2×C3×Dic5D414yescomplemented
67C2×C3×Dic5C2311yescharacteristic
68C2×C3×Dic5C2311yescharacteristic
69C2×C3×Dic5C2311yescharacteristic
70C3×C4×D5—22no—
71C3×Dic10—22no—
72#1202—24no—
73#1200—24no—
74#1200—24no—
75C2×C3×#41—51no—
76C22×C4×D3—51no—
77#946—52no—
78#938—52no—
79#937—51no—
80#937—51no—
81#935—51no—
82#935D511yescharacteristic, complemented
83C4×#67—52no—
84C4×#67—52no—
85#926—51no—
86C23×D5—31no—
87C2×C5×D4C2×D311yescharacteristic
88C22×C4×C5—31no—
89C2×C22⋊D5—32no—
90C2×C22⋊D5C2×D311yescharacteristic
91C2×C22⋊D5C2×D311yescharacteristic, complemented
92C22×Dic5C2×D311yescharacteristic
93C22×Dic5C2×D311yescharacteristic, complemented
94#711—24no—
95C2×D20—32no—
96C2×C4×D5—32no—
97C2×C4×D5—32no—
98C2×C4×D5—64no—
99C2×C4×D5C2×D311yescharacteristic, complemented
100C2×Dic10C2×D311yescharacteristic, complemented
101C5×C4⋊C4—32no—
102C5×C22⋊C4—31no—
103C5×C22⋊C4—32no—
104C22⋊Dic5—31no—
105#685—32no—
106#685—32no—
107#685—32no—
108#683—32no—
109#683—32no—
110C4×Dic5—32no—
111#525—151noSylow
112C22×C3×C5—22no—
113C22×C3×C5—21no—
114C22×C3×C5—22no—
115C22×C3×C5C2×D412yes—
116C22×C3×C5C2411yescharacteristic
117C2×D15—28no—
118C2×D15—22no—
119C2×D15—22no—
120C2×D15#4114yescomplemented
121C2×C3×D5—21no—
122C2×C3×D5—22no—
123C3×C4×C5—21no—
124Dic15—22no—
125C3×Dic5—21no—
126C3×Dic5C2×D412yes—
127C3×Dic5C2×D412yes—
128C5×Dic3—22no—
129C5×Dic3—22no—
130C23×D3—51no—
131C3×#41—104no—
132C2×C3×Q8—51no—
133C2×C3×D4—51no—
134C2×C3×D4—51no—
135C2×C3×D4C2×D511yescharacteristic
136C22×C3×C4—51no—
137C22×C3×C4—51no—
138C22×C3×C4—51no—
139C2×#67—52no—
140C2×#67—52no—
141C22×Dic3C2×D511yescharacteristic
142C2×C4×D3—52no—
143C2×C4×D3—52no—
144C2×C4×D3—104no—
145#217—51no—
146#217—52no—
147#217C2×D511yescharacteristic
148#217C2×D512yescomplemented
149(C2×D3)⋊C4—52no—
150(C2×D3)⋊C4—52no—
151C4⋊Dic3—52no—
152Dic3⋊C4—52no—
153Dic3⋊C4C2×D512yescomplemented
154C4×Dic3—52no—
155C23×C5C22×D311yescharacteristic
156C23×C5C22×D311yescharacteristic
157C22×D5—32no—
158C22×D5—64no—
159C22×D5—62no—
160C22×D5C22×D311yescharacteristic
161C5×D4—22no—
162C2×C4×C5—32no—
163C2×C4×C5—32no—
164C2×C4×C5—62no—
165C2×C4×C5C22×D311yescharacteristic
166C22⋊D5—22no—
167C22⋊D5—22no—
168C22⋊D5—64no—
169C2×Dic5—22no—
170C2×Dic5—32no—
171C2×Dic5#6714yescomplemented
172C2×Dic5C22×D311yescharacteristic
173C2×Dic5C22×D311yescharacteristic
174C2×Dic5C22×D311yescharacteristic
175D20—64no—
176C4×D5—22no—
177C4×D5—64no—
178C4×D5—64no—
179Dic10—22no—
180C2×#41—51no—
181C23×C4—151no—
182#123—31no—
183#123—151no—
184C22⋊Q8—152no—
185#121—152no—
186#120—151no—
187#120—151no—
188C4×D4—152no—
189C4×D4—152no—
190#117—151no—
191C2×C3×C5—22no—
192C2×C3×C5—41no—
193C2×C3×C5C22×D411yescharacteristic
194C2×C3×C5C2×#4112yes—
195D15—24no—
196C3×D5—41no—
197C23×C3—51no—
198C23×C3C22×D511yescharacteristic
199C23×C3C22×D511yescharacteristic
200C22×D3—52no—
201C22×D3—104no—
202C22×D3—102no—
203C3×Q8—102no—
204C3×D4—22no—
205C3×D4—102no—
206C3×D4—102no—
207C2×C3×C4—54no—
208C2×C3×C4—51no—
209C2×C3×C4—51no—
210C2×C3×C4—51no—
211C2×C3×C4—102no—
212C2×C3×C4—102no—
213C2×C3×C4C22×D511yescharacteristic
214#67—104no—
215#67—104no—
216C2×Dic3—22no—
217C2×Dic3—52no—
218C2×Dic3C22×D512yes—
219C2×Dic3C22×D512yes—
220C4×D3—104no—
221C4×D3—104no—
222C22×C5—22no—
223C22×C5—22no—
224C22×C5—21no—
225C22×C5C2×#6712yes—
226C22×C5C23×D311yescharacteristic
227C2×D5—22no—
228C2×D5—21no—
229C2×D5—34no—
230C2×D5—68no—
231C2×D5—62no—
232C2×D5—62no—
233C4×C5—21no—
234C4×C5—62no—
235C4×C5—62no—
236Dic5—21no—
237Dic5—62no—
238Dic5C2×#6712yes—
239Dic5C2×#6712yes—
240C24—151no—
241#41—104no—
242C2×Q8—51no—
243C2×D4—51no—
244C2×D4—51no—
245C2×D4—152no—
246C2×D4—152no—
247C2×D4D3×D511yescharacteristic
248C22×C4—31no—
249C22×C4—51no—
250C22×C4—51no—
251C22×C4—51no—
252C22×C4—152no—
253C22×C4—152no—
254C22×C4—304no—
255C4⋊C4—32no—
256C4⋊C4—152no—
257C4⋊C4—152no—
258C22⋊C4—31no—
259C22⋊C4—32no—
260C22⋊C4—152no—
261C22⋊C4—152no—
262C22⋊C4—151no—
263C22⋊C4—152no—
264C42—152no—
265C3×C5#52511yescharacteristic, complemented, Hall
266C22×C3—22no—
267C22×C3—21no—
268C22×C3—22no—
269C22×C3—101no—
270C22×C3—102no—
271C22×C3D4×D512yes—
272C22×C3C23×D511yescharacteristic
273C2×D3—54no—
274C2×D3—108no—
275C2×D3—102no—
276C2×D3—102no—
277C3×C4—21no—
278C3×C4—52no—
279C3×C4—52no—
280C3×C4—101no—
281Dic3—22no—
282Dic3—22no—
283Dic3—102no—
284C2×C5—22no—
285C2×C5—41no—
286C2×C5D3×#4112yes—
287C2×C5C22×#6711yescharacteristic
288D5—41no—
289D5—64no—
290C23—51no—
291C23—152no—
292C23—304no—
293C23—302no—
294C23C2×D3×D511yescharacteristic
295C23C2×D3×D511yescharacteristic
296Q8—102no—
297D4—22no—
298D4—102no—
299D4—102no—
300D4—304no—
301D4—304no—
302C2×C4—32no—
303C2×C4—32no—
304C2×C4—54no—
305C2×C4—51no—
306C2×C4—51no—
307C2×C4—51no—
308C2×C4—62no—
309C2×C4—102no—
310C2×C4—102no—
311C2×C4—152no—
312C2×C4—304no—
313C2×C4—304no—
314C2×C4C2×D3×D511yescharacteristic
315C2×C3—22no—
316C2×C3—41no—
317C2×C3—201no—
318C2×C3C2×D4×D511yescharacteristic
319C2×C3D5×#4112yes—
320D3—104no—
321C5#582811yescharacteristic, complemented, Sylow
322C22—22no—
323C22—22no—
324C22—21no—
325C22—102no—
326C22—101no—
327C22—154no—
328C22—308no—
329C22—302no—
330C22—302no—
331C22D5×#6712yes—
332C22C22×D3×D511yescharacteristic
333C4—21no—
334C4—52no—
335C4—52no—
336C4—62no—
337C4—62no—
338C4—101no—
339C4—302no—
340C3#6599011yescharacteristic, complemented, Sylow
341C2—22no—
342C2—41no—
343C2—201no—
344C2—304no—
345C2#9202412yes—
346C2C2×D5×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#525
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#711C2semidirect
2. C22×#1200C2semidirect
3. #91826C2semidirect
4. C5×#935C2semidirect
5. #91572C2semidirect
6. #91552C2semidirect
7. #91536C2semidirect
8. #91533C2semidirect
9. #91528C2semidirect
10. #91444C2semidirect
11. #91411C2semidirect
18. C2×C3×C22⋊D5C22semidirect
20. C22×C3×Dic5C22semidirect
22. C2×C3×C4×D5C22semidirect
23. C2×C3×Dic10C22semidirect
31. C5×#217C22semidirect
32. C5×Dic3⋊C4C22semidirect
33. #6669C22semidirect
34. #6668C22semidirect
36. #6665C22semidirect
37. Dic3×Dic5C22semidirect
39. C2×#711D3semidirect
66. C2×C3×Dic5D4semidirect
82. #935D5semidirect
91. C2×C22⋊D5C2×D3semidirect
93. C22×Dic5C2×D3semidirect
99. C2×C4×D5C2×D3semidirect
100. C2×Dic10C2×D3semidirect
120. C2×D15#41semidirect
148. #217C2×D5semidirect
153. Dic3⋊C4C2×D5semidirect
171. C2×Dic5#67semidirect
265. C3×C5#525semidirect
321. C5#5828semidirect
340. C3#65990semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#7112
C22×#12002
#918262
C5×#9352
#915722
#915522
#915362
#915332
#915282
#914442
#914112
#659903
#58285