Finite Group Databasebeta

← #11755399

Subgroups — #11755399

Order 960.

660 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×Dic15C211yesmaximal, characteristic, complemented
2C2×#6780C211yesmaximal, characteristic, complemented
3C5×#953C211yesmaximal, characteristic, complemented
4C3×#4099C211yesmaximal, characteristic, complemented
5#91576C212yesmaximal, complemented
6#91571C212yesmaximal, complemented
7#67⋊Dic5C212yesmaximal, complemented
8#91559C212yesmaximal, complemented
9#91543C211yesmaximal, characteristic, complemented
10#91537C211yesmaximal, characteristic, complemented
11#91532C211yesmaximal, characteristic, complemented
12#65920—31nomaximal, Hall
13C24×C3×C5C2211yescharacteristic
14C22×Dic15—24no—
15C22×Dic15C2211yescharacteristic
16C22×Dic15C2212yes—
17C22×Dic15C2212yes—
18C22×Dic15C2211yescharacteristic
19C2×C5×#67C2211yescharacteristic
20C2×C5×#67C2212yes—
21C2×C3×C22⋊D5C2211yescharacteristic
22C2×C3×C22⋊D5C2212yes—
23C2×#1204C2211yescharacteristic, complemented
24C2×#1201C2211yescharacteristic, complemented
25C2×D3×Dic5C2211yescharacteristic, complemented
26#6780—24no—
27C2×D5×Dic3C2211yescharacteristic, complemented
28C5×#217C2211yescharacteristic
29C5×#217C2212yes—
30C3×C22⋊Dic5C2211yescharacteristic
31C3×C22⋊Dic5C2212yes—
32Dic3⋊Dic5C2212yescomplemented
33#6669C2212yescomplemented
34#6668C2212yescomplemented
35(C2×D3)⋊Dic5C2212yescomplemented
36#6665C2212yescomplemented
37Dic3×Dic5C2212yescomplemented
38#5586—51nomaximal, Hall
39C23×Dic5—31no—
40C2×#711—31no—
41C5×#120—31no—
42#4099D311yescharacteristic, complemented
43#4085—32no—
44#4081—31no—
45D4×Dic5—32no—
46#4032—31no—
47#4027—32no—
48#4024—32no—
49#4023—31no—
50C23×C3×C5—22no—
51C23×C3×C5—24no—
52C23×C3×C5C2311yescharacteristic
53C23×C3×C5C2312yes—
54C22×C5×D3C2311yescharacteristic
55C22×C3×D5C2311yescharacteristic
56C2×Dic15—22no—
57C2×Dic15—24no—
58C2×Dic15—24no—
59C2×Dic15D414yescomplemented
60C2×Dic15C2312yes—
61C2×Dic15C2311yescharacteristic
62C2×Dic15C2311yescharacteristic
63C5×#67—22no—
64C5×#67—24no—
65C2×C5×Dic3C2312yes—
66C2×C5×Dic3C2311yescharacteristic
67C3×C22⋊D5—22no—
68C3×C22⋊D5—24no—
69C2×C3×Dic5C2312yes—
70C2×C3×Dic5C2311yescharacteristic
71#1204—22no—
72#1201—22no—
73D3×Dic5—22no—
74D5×Dic3—22no—
75C23×Dic3—51no—
76C2×#237—51no—
77C3×#120—51no—
78#953D511yescharacteristic, complemented
79#939—52no—
80#935—51no—
81D4×Dic3—52no—
82#886—51no—
83#67⋊C4—52no—
84#878—52no—
85#877—51no—
86C24×C5C2×D311yescharacteristic
87C2×C5×D4—31no—
88C2×C5×D4—32no—
89C2×C22⋊D5—31no—
90C2×C22⋊D5C2×D311yescharacteristic
91C2×C22⋊D5C2×D312yes—
92C22×Dic5—31no—
93C22×Dic5—32no—
94C22×Dic5—32no—
95C22×Dic5—31no—
96C22×Dic5—31no—
97C22×Dic5—64no—
98#711—64no—
99C2×C4×D5—31no—
100C2×Dic10—31no—
101C5×C22⋊C4—31no—
102C5×C22⋊C4—32no—
103C22⋊Dic5—32no—
104C22⋊Dic5C2×D311yescharacteristic
105C22⋊Dic5C2×D312yes—
106#685—32no—
107C4⋊Dic5—32no—
108#683—32no—
109#683—32no—
110C4×Dic5—32no—
111#525—151noSylow
112C22×C3×C5—28no—
113C22×C3×C5—22no—
114C22×C3×C5—22no—
115C22×C3×C5—22no—
116C22×C3×C5C2×D412yes—
117C22×C3×C5#4114yescomplemented
118C22×C3×C5C2411yescharacteristic
119C2×C5×D3—21no—
120C2×C5×D3—22no—
121C2×C3×D5—21no—
122C2×C3×D5—22no—
123Dic15—22no—
124Dic15C2×D412yes—
125Dic15C2×D412yes—
126C3×Dic5—21no—
127C3×Dic5—22no—
128C5×Dic3—21no—
129C5×Dic3—22no—
130C24×C3C2×D511yescharacteristic
131C2×C3×D4—52no—
132C2×C3×D4—51no—
133C2×#67—51no—
134C2×#67C2×D512yes—
135C2×#67C2×D511yescharacteristic
136C22×Dic3—51no—
137C22×Dic3—52no—
138C22×Dic3—52no—
139C22×Dic3—51no—
140C22×Dic3—51no—
141C22×Dic3—104no—
142#237—104no—
143C2×C4×D3—51no—
144C2×Dic6—51no—
145C3×C22⋊C4—51no—
146C3×C22⋊C4—52no—
147#217—52no—
148#217C2×D511yescharacteristic
149#217C2×D512yes—
150(C2×D3)⋊C4—52no—
151C4⋊Dic3—52no—
152Dic3⋊C4—52no—
153Dic3⋊C4—52no—
154C4×Dic3—52no—
155C23×C5—22no—
156C23×C5—24no—
157C23×C5—31no—
158C23×C5C22×D311yescharacteristic
159C23×C5C22×D312yes—
160C22×D5C22×D311yescharacteristic
161C5×D4—62no—
162C5×D4—64no—
163C2×C4×C5—32no—
164C2×C4×C5—31no—
165C22⋊D5—22no—
166C22⋊D5—24no—
167C22⋊D5—62no—
168C2×Dic5—34no—
169C2×Dic5—31no—
170C2×Dic5—31no—
171C2×Dic5—32no—
172C2×Dic5—62no—
173C2×Dic5—64no—
174C2×Dic5—64no—
175C2×Dic5—62no—
176C2×Dic5C22×D312yes—
177C2×Dic5C22×D311yescharacteristic
178C4×D5—62no—
179Dic10—62no—
180C2×#41—151no—
181C23×C4—151no—
182#123—151no—
183#123—151no—
184C22⋊Q8—152no—
185#121—152no—
186#120—31no—
187#120—51no—
188C4×D4—152no—
189C4×D4—152no—
190#117—151no—
191C2×C3×C5—22no—
192C2×C3×C5—24no—
193C2×C3×C5C22×D411yescharacteristic
194C2×C3×C5C2×#4112yes—
195C3×D5—41no—
196C5×D3—41no—
197C23×C3—22no—
198C23×C3—24no—
199C23×C3—51no—
200C23×C3C22×D511yescharacteristic
201C23×C3C22×D512yes—
202C22×D3C22×D511yescharacteristic
203C3×D4—102no—
204C3×D4—104no—
205C2×C3×C4—52no—
206C2×C3×C4—51no—
207#67—22no—
208#67—24no—
209#67—102no—
210C2×Dic3—54no—
211C2×Dic3—51no—
212C2×Dic3—51no—
213C2×Dic3—52no—
214C2×Dic3—102no—
215C2×Dic3—104no—
216C2×Dic3—104no—
217C2×Dic3—102no—
218C2×Dic3C22×D512yes—
219C2×Dic3C22×D511yescharacteristic
220C4×D3—102no—
221Dic6—102no—
222C22×C5—28no—
223C22×C5—22no—
224C22×C5—22no—
225C22×C5—22no—
226C22×C5—61no—
227C22×C5—62no—
228C22×C5D3×D412yes—
229C22×C5#23714yescomplemented
230C22×C5C23×D311yescharacteristic
231C2×D5—21no—
232C2×D5—22no—
233C4×C5—61no—
234C4×C5—62no—
235Dic5—21no—
236Dic5—22no—
237Dic5—32no—
238Dic5—32no—
239Dic5—62no—
240C24D3×D511yescharacteristic
241#41—304no—
242C2×Q8—151no—
243C2×D4—32no—
244C2×D4—31no—
245C2×D4—52no—
246C2×D4—51no—
247C2×D4—151no—
248C22×C4—151no—
249C22×C4—151no—
250C22×C4—151no—
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—51no—
261C22⋊C4—52no—
262C22⋊C4—152no—
263C22⋊C4—152no—
264C42—152no—
265C3×C5#52511yescharacteristic, complemented, Hall
266C22×C3—28no—
267C22×C3—22no—
268C22×C3—22no—
269C22×C3—22no—
270C22×C3—101no—
271C22×C3—102no—
272C22×C3D4×D512yes—
273C22×C3#71114yescomplemented
274C22×C3C23×D511yescharacteristic
275C2×D3—21no—
276C2×D3—22no—
277C3×C4—101no—
278C3×C4—102no—
279Dic3—21no—
280Dic3—22no—
281Dic3—52no—
282Dic3—52no—
283Dic3—102no—
284C2×C5—22no—
285C2×C5—24no—
286C2×C5—121no—
287C2×C5C2×D3×D411yescharacteristic
288C2×C5C2×#23712yes—
289D5—41no—
290C23—24no—
291C23—22no—
292C23—31no—
293C23—51no—
294C23C2×D3×D512yes—
295C23C2×D3×D511yescharacteristic
296Q8—302no—
297D4—64no—
298D4—62no—
299D4—104no—
300D4—102no—
301D4—302no—
302C2×C4—32no—
303C2×C4—31no—
304C2×C4—52no—
305C2×C4—51no—
306C2×C4—154no—
307C2×C4—151no—
308C2×C4—151no—
309C2×C4—152no—
310C2×C4—302no—
311C2×C4—302no—
312C2×C4—304no—
313C2×C4—304no—
314C2×C4—302no—
315C2×C3—22no—
316C2×C3—24no—
317C2×C3—201no—
318C2×C3C2×D4×D511yescharacteristic
319C2×C3C2×#71112yes—
320D3—41no—
321C5#558611yescharacteristic, complemented, Sylow
322C22—28no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—62no—
327C22—61no—
328C22—102no—
329C22—101no—
330C22#678014yescomplemented
331C22#679012yes—
332C22C22×D3×D511yescharacteristic
333C4—62no—
334C4—61no—
335C4—102no—
336C4—101no—
337C4—152no—
338C4—152no—
339C4—302no—
340C3#6592011yescharacteristic, complemented, Sylow
341C2—24no—
342C2—22no—
343C2—121no—
344C2—201no—
345C2C2×#678012yes—
346C2C2×#679011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC24×C3×C5C22
hypercenterC24D3×D5
hypocenterC3×C5#525
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×Dic15C2semidirect
2. C2×#6780C2semidirect
3. C5×#953C2semidirect
4. C3×#4099C2semidirect
5. #91576C2semidirect
6. #91571C2semidirect
7. #67⋊Dic5C2semidirect
8. #91559C2semidirect
9. #91543C2semidirect
10. #91537C2semidirect
11. #91532C2semidirect
23. C2×#1204C22semidirect
24. C2×#1201C22semidirect
25. C2×D3×Dic5C22semidirect
27. C2×D5×Dic3C22semidirect
32. Dic3⋊Dic5C22semidirect
33. #6669C22semidirect
34. #6668C22semidirect
35. (C2×D3)⋊Dic5C22semidirect
36. #6665C22semidirect
37. Dic3×Dic5C22semidirect
42. #4099D3semidirect
59. C2×Dic15D4semidirect
78. #953D5semidirect
117. C22×C3×C5#41semidirect
229. C22×C5#237semidirect
265. C3×C5#525semidirect
273. C22×C3#711semidirect
321. C5#5586semidirect
330. C22#6780semidirect
340. C3#65920semidirect

Maximal subgroups

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

subgroupindex
C23×Dic152
C2×#67802
C5×#9532
C3×#40992
#915762
#915712
#67⋊Dic52
#915592
#915432
#915372
#915322
#659203
#55865