Finite Group Databasebeta

← #11754795

Subgroups — #11754795

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×D3×Dic5C211yesmaximal, characteristic, complemented
3#91824C211yesmaximal, characteristic, complemented
4C5×#937C211yesmaximal, characteristic, complemented
5#91568C211yesmaximal, characteristic, complemented
6#91565C212yesmaximal, complemented
7Dic5×#67C212yesmaximal, complemented
8#91537C211yesmaximal, characteristic, complemented
9#91528C211yesmaximal, characteristic, complemented
10#91440C212yesmaximal, complemented
11#91410C212yesmaximal, complemented
12#65920—31nomaximal, Hall
13C23×C5×D3C2211yescharacteristic
14C2×C3×C5×D4C2211yescharacteristic
15C22×Dic15C2211yescharacteristic
16C2×C5×#67C2212yes—
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×#1201C2212yes—
25C2×D3×Dic5—24no—
26C2×D3×Dic5C2212yes—
27C2×D3×Dic5C2212yes—
28#6719C2211yescharacteristic
29#6719C2212yescomplemented
30#6712C2212yescomplemented
31C5×#217C2211yescharacteristic
32C5×(C2×D3)⋊C4C2212yes—
33Dic3⋊Dic5C2212yescomplemented
34#6668C2212yescomplemented
35(C2×D3)⋊Dic5C2212yes—
36#6665C2212yescomplemented
37Dic3×Dic5C2212yescomplemented
38#5828—51nomaximal, Hall
39C23×Dic5—31no—
40C2×#711D311yescharacteristic, complemented
41C5×#120—31no—
42#4099—31no—
43#4085—32no—
44#4081—31no—
45D4×Dic5—32no—
46#4032—31no—
47#4027—32no—
48#4024—32no—
49#4023—31no—
50C23×C3×C5C2311yescharacteristic
51C23×C3×C5C2311yescharacteristic
52C22×C5×D3—22no—
53C22×C5×D3—24no—
54C22×C5×D3C2312yes—
55C22×C3×D5C2311yescharacteristic
56C3×C5×D4—22no—
57C2×C3×C4×C5C2311yescharacteristic
58C2×Dic15—22no—
59C2×Dic15C2312yes—
60C2×Dic15C2312yes—
61C5×#67—24no—
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#1201—24no—
73D3×Dic5—24no—
74D3×Dic5—24no—
75C2×C3×#41—51no—
76C22×C4×D3—51no—
77#946—52no—
78#938—52no—
79#937—51no—
80#937D511yescharacteristic, complemented
81#935—51no—
82#935—51no—
83C4×#67—52no—
84C4×#67—52no—
85#926—51no—
86C24×C5—31no—
87C2×C5×D4—32no—
88C2×C5×D4C2×D311yescharacteristic
89C2×C22⋊D5—32no—
90C2×C22⋊D5C2×D311yescharacteristic
91C2×C22⋊D5C2×D311yescharacteristic, complemented
92C22×Dic5—31no—
93C22×Dic5—32no—
94C22×Dic5—32no—
95C22×Dic5—64no—
96C22×Dic5C2×D311yescharacteristic
97C22×Dic5C2×D311yescharacteristic, complemented
98#711—24no—
99C2×C4×D5C2×D311yescharacteristic, complemented
100C2×Dic10C2×D311yescharacteristic, complemented
101C5×C22⋊C4—31no—
102C5×C22⋊C4—32no—
103C22⋊Dic5—31no—
104C22⋊Dic5—32no—
105C22⋊Dic5—32no—
106#685—32no—
107C4⋊Dic5—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×C5×D3—28no—
118C2×C5×D3—22no—
119C2×C5×D3—22no—
120C2×C5×D3#4114yescomplemented
121C2×C3×D5—21no—
122C2×C3×D5—22no—
123C3×C4×C5—21no—
124Dic15—22no—
125Dic15—22no—
126C3×Dic5—21no—
127C3×Dic5C2×D412yes—
128C3×Dic5C2×D412yes—
129C5×Dic3—22no—
130C23×D3C2×D511yescharacteristic
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×#67C2×D512yes—
141C22×Dic3—51no—
142C2×C4×D3—52no—
143C2×C4×D3—52no—
144C2×C4×D3—104no—
145#217—51no—
146#217—52no—
147#217—52no—
148#217C2×D511yescharacteristic
149(C2×D3)⋊C4—52no—
150(C2×D3)⋊C4C2×D512yes—
151C4⋊Dic3—52no—
152Dic3⋊C4—52no—
153Dic3⋊C4—52no—
154C4×Dic3—52no—
155C23×C5—32no—
156C23×C5—64no—
157C23×C5—62no—
158C23×C5C22×D311yescharacteristic
159C23×C5C22×D311yescharacteristic
160C22×D5C22×D311yescharacteristic
161C5×D4—22no—
162C5×D4—64no—
163C2×C4×C5—32no—
164C2×C4×C5C22×D311yescharacteristic
165C22⋊D5—22no—
166C22⋊D5—22no—
167C22⋊D5—64no—
168C2×Dic5—22no—
169C2×Dic5—32no—
170C2×Dic5—32no—
171C2×Dic5—62no—
172C2×Dic5—64no—
173C2×Dic5—64no—
174C2×Dic5#6714yescomplemented
175C2×Dic5C22×D311yescharacteristic
176C2×Dic5C22×D311yescharacteristic
177C2×Dic5C22×D311yescharacteristic
178C4×D5—22no—
179Dic10—22no—
180C2×#41—51no—
181C23×C4—151no—
182#123—151no—
183#123—151no—
184C22⋊Q8—152no—
185#121—152no—
186#120—31no—
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—
195C3×D5—41no—
196C5×D3—24no—
197C23×C3—51no—
198C23×C3C22×D511yescharacteristic
199C23×C3C22×D511yescharacteristic
200C22×D3—22no—
201C22×D3—24no—
202C22×D3C22×D512yes—
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—24no—
215#67—104no—
216C2×Dic3—52no—
217C2×Dic3—52no—
218C2×Dic3—102no—
219C2×Dic3C22×D512yes—
220C4×D3—104no—
221C4×D3—104no—
222C22×C5—22no—
223C22×C5—22no—
224C22×C5—21no—
225C22×C5—34no—
226C22×C5—68no—
227C22×C5—62no—
228C22×C5—62no—
229C22×C5C2×#6712yes—
230C22×C5C23×D311yescharacteristic
231C2×D5—22no—
232C2×D5—21no—
233C4×C5—21no—
234C4×C5—62no—
235Dic5—21no—
236Dic5—62no—
237Dic5—62no—
238Dic5C2×#6712yes—
239Dic5C2×#6712yes—
240C24—31no—
241#41—104no—
242C2×Q8—51no—
243C2×D4—32no—
244C2×D4—51no—
245C2×D4—51no—
246C2×D4—152no—
247C2×D4D3×D511yescharacteristic
248C22×C4—51no—
249C22×C4—51no—
250C22×C4—51no—
251C22×C4—152no—
252C22×C4—152no—
253C22×C4—151no—
254C22×C4—304no—
255C4⋊C4—152no—
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—28no—
274C2×D3—22no—
275C2×D3—22no—
276C2×D3#71114yescomplemented
277C3×C4—21no—
278C3×C4—52no—
279C3×C4—52no—
280C3×C4—101no—
281Dic3—22no—
282Dic3—102no—
283Dic3—102no—
284C2×C5—22no—
285C2×C5—41no—
286C2×C5—64no—
287C2×C5D3×#4112yes—
288C2×C5C22×#6711yescharacteristic
289D5—41no—
290C23—32no—
291C23—51no—
292C23—64no—
293C23—62no—
294C23C2×D3×D511yescharacteristic
295C23C2×D3×D511yescharacteristic
296Q8—102no—
297D4—22no—
298D4—64no—
299D4—102no—
300D4—102no—
301D4—304no—
302C2×C4—32no—
303C2×C4—54no—
304C2×C4—51no—
305C2×C4—51no—
306C2×C4—51no—
307C2×C4—102no—
308C2×C4—102no—
309C2×C4—152no—
310C2×C4—152no—
311C2×C4—304no—
312C2×C4—304no—
313C2×C4—302no—
314C2×C4C2×D3×D511yescharacteristic
315C2×C3—22no—
316C2×C3—41no—
317C2×C3—201no—
318C2×C3C2×D4×D511yescharacteristic
319C2×C3C2×#71112yes—
320D3—24no—
321C5#582811yescharacteristic, complemented, Sylow
322C22—22no—
323C22—22no—
324C22—21no—
325C22—34no—
326C22—68no—
327C22—62no—
328C22—62no—
329C22—102no—
330C22—101no—
331C22D5×#6712yes—
332C22C22×D3×D511yescharacteristic
333C4—21no—
334C4—52no—
335C4—52no—
336C4—62no—
337C4—101no—
338C4—302no—
339C4—302no—
340C3#6592011yescharacteristic, complemented, Sylow
341C2—22no—
342C2—41no—
343C2—64no—
344C2—201no—
345C2D3×#71112yes—
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×D3×Dic5C2semidirect
3. #91824C2semidirect
4. C5×#937C2semidirect
5. #91568C2semidirect
6. #91565C2semidirect
7. Dic5×#67C2semidirect
8. #91537C2semidirect
9. #91528C2semidirect
10. #91440C2semidirect
11. #91410C2semidirect
18. C2×C3×C22⋊D5C22semidirect
20. C22×C3×Dic5C22semidirect
22. C2×C3×C4×D5C22semidirect
23. C2×C3×Dic10C22semidirect
29. #6719C22semidirect
30. #6712C22semidirect
33. Dic3⋊Dic5C22semidirect
34. #6668C22semidirect
36. #6665C22semidirect
37. Dic3×Dic5C22semidirect
40. C2×#711D3semidirect
66. C2×C3×Dic5D4semidirect
80. #937D5semidirect
91. C2×C22⋊D5C2×D3semidirect
97. C22×Dic5C2×D3semidirect
99. C2×C4×D5C2×D3semidirect
100. C2×Dic10C2×D3semidirect
120. C2×C5×D3#41semidirect
174. C2×Dic5#67semidirect
265. C3×C5#525semidirect
276. C2×D3#711semidirect
321. C5#5828semidirect
340. C3#65920semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#7112
C22×D3×Dic52
#918242
C5×#9372
#915682
#915652
Dic5×#672
#915372
#915282
#914402
#914102
#659203
#58285