Finite Group Databasebeta

← #11753888

Subgroups — #11753888

Order 960.

668 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 140 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
1C2×D3×D20C211yesmaximal, characteristic, complemented
2C2×#6773C211yesmaximal, characteristic, complemented
3#91782C211yesmaximal, characteristic, complemented
4C5×#897C211yesmaximal, characteristic, complemented
5C3×#4041C211yesmaximal, characteristic, complemented
6#91474C212yesmaximal, complemented
7#91473C212yesmaximal, complemented
8#91465C212yesmaximal, complemented
9C4×#1201C211yesmaximal, characteristic, complemented
10#91368C211yesmaximal, characteristic, complemented
11#91356C212yesmaximal, complemented
12#65964—31nomaximal, Hall
13C22×D3×D5C2212yes—
14C2×D60C2211yescharacteristic
15C2×C4×D15C2212yescomplemented
16C2×C4×C5×D3C2211yescharacteristic
17C2×C5×Dic6C2211yescharacteristic, complemented
18C2×C3×D20C2211yescharacteristic
19C2×C3×D20C2212yescomplemented
20C2×C3×C4×D5C2211yescharacteristic, complemented
21C2×#1202C2212yes—
22C2×#1202C2212yescomplemented
23C2×#1201C2211yescharacteristic
24C2×#1200C2212yescomplemented
25D3×D20—24no—
26#6773—24no—
27C3×C5×C4⋊C4C2211yescharacteristic, complemented
28#6714C2212yescomplemented
29C4×Dic15C2211yescharacteristic, complemented
30C5×(C2×D3)⋊C4C2212yes—
31C5×C4⋊Dic3C2212yescomplemented
32C3×#685C2212yescomplemented
33#6669C2211yescharacteristic, complemented
34(C2×D3)⋊Dic5C2211yescharacteristic
35#6665C2211yescharacteristic, complemented
36#6665C2212yescomplemented
37#6665C2212yescomplemented
38#5612—51nomaximal, Hall
39C2×#713—31no—
40C22×D20—31no—
41C5×C22⋊Q8—31no—
42#4093—31no—
43C4×C22⋊D5—31no—
44#4041—32no—
45#4041D311yescharacteristic, complemented
46#4040—32no—
47#4039—31no—
48#4028—32no—
49D5×C22⋊C4—32no—
50C22×D15C2312yes—
51C22×C5×D3C2311yescharacteristic
52C22×C3×D5C2312yes—
53C22×C3×D5C2311yescharacteristic
54C2×D3×D5—24no—
55C2×D3×D5—24no—
56C2×D3×D5—42no—
57C2×C3×C4×C5C2312yes—
58C2×C3×C4×C5C2311yescharacteristic
59C2×Dic15C2311yescharacteristic
60C2×Dic15C2311yescharacteristic
61D60D414yescomplemented
62C4×D15—42no—
63C2×C5×Dic3C2312yes—
64C2×C5×Dic3C2311yescharacteristic
65C4×C5×D3—22no—
66C5×Dic6—22no—
67C2×C3×Dic5C2311yescharacteristic
68C3×D20—22no—
69C3×D20—42no—
70C3×C4×D5—22no—
71#1202—24no—
72#1202—24no—
73#1201—22no—
74#1200—24no—
75C2×D3×D4—51no—
76C2×#235—51no—
77C3×#121—51no—
78#938—52no—
79#937—52no—
80#936—51no—
81C4×#67—51no—
82#897D511yescharacteristic, complemented
83#893—51no—
84#883—52no—
85D3×C22⋊C4—52no—
86C23×D5—32no—
87C2×C5×Q8—31no—
88C22×C4×C5—31no—
89C2×C22⋊D5—31no—
90#713—64no—
91C2×D20—31no—
92C2×D20—32no—
93C2×D20—32no—
94C2×D20—64no—
95C2×D20C2×D311yescharacteristic
96C2×D20C2×D312yescomplemented
97C2×C4×D5—32no—
98C2×C4×D5—32no—
99C2×C4×D5C2×D311yescharacteristic, complemented
100C5×C4⋊C4—32no—
101C5×C4⋊C4C2×D311yescharacteristic, complemented
102C5×C22⋊C4—32no—
103C22⋊Dic5—31no—
104#685—32no—
105#685—32no—
106#685—31no—
107#685—32no—
108#685C2×D312yescomplemented
109#683—31no—
110C4×Dic5—31no—
111#546—151noSylow
112C22×C3×C5C2411yescharacteristic
113C2×D15—22no—
114C2×D15—22no—
115C2×D15C2×D414yes—
116C2×C5×D3—21no—
117C2×C5×D3—21no—
118C2×C5×D3#4112yescomplemented
119C2×C3×D5—21no—
120C2×C3×D5—22no—
121C2×C3×D5—21no—
122C2×C3×D5—22no—
123C2×C3×D5—21no—
124C2×C3×D5—22no—
125D3×D5—48no—
126C3×C4×C5—22no—
127C3×C4×C5C2×D412yes—
128Dic15—21no—
129Dic15—21no—
130C3×Dic5—21no—
131C5×Dic3—22no—
132C5×Dic3—21no—
133C23×D3—52no—
134C2×C3×D4—51no—
135C2×C3×D4—52no—
136C22×C3×C4—51no—
137C2×#67—51no—
138C2×#67—52no—
139C2×#67—52no—
140D3×D4—104no—
141#235—104no—
142C2×D12—51no—
143C2×C4×D3—52no—
144C2×C4×D3—52no—
145C2×C4×D3C2×D511yescharacteristic
146C2×Dic6C2×D511yescharacteristic, complemented
147C3×C4⋊C4C2×D511yescharacteristic, complemented
148C3×C22⋊C4—52no—
149#217—52no—
150#217—52no—
151#217—51no—
152(C2×D3)⋊C4—51no—
153(C2×D3)⋊C4—52no—
154(C2×D3)⋊C4C2×D512yes—
155C4⋊Dic3C2×D512yescomplemented
156Dic3⋊C4—51no—
157C4×Dic3—51no—
158C23×C5—31no—
159C22×D5—32no—
160C22×D5—64no—
161C22×D5—64no—
162C22×D5—122no—
163C22×D5C22×D312yes—
164C22×D5C22×D311yescharacteristic
165C5×Q8—62no—
166C2×C4×C5—31no—
167C2×C4×C5—32no—
168C2×C4×C5—62no—
169C2×C4×C5C22×D312yes—
170C2×C4×C5C22×D311yescharacteristic
171C22⋊D5—62no—
172C2×Dic5—31no—
173C2×Dic5—31no—
174C2×Dic5C22×D311yescharacteristic
175D20—22no—
176D20—34no—
177D20—42no—
178D20—64no—
179D20—64no—
180C4×D5—22no—
181C4×D5—64no—
182C4×D5—122no—
183C2×#41—151no—
184C22×D4—151no—
185#124—151no—
186#123—152no—
187C22⋊Q8—31no—
188#121—51no—
189#121—152no—
190#120—152no—
191C4×D4—151no—
192C4×D4—151no—
193C2×C22⋊C4—152no—
194C2×C3×C5C22×D411yescharacteristic
195C2×C3×C5C2×#4111yescharacteristic
196C2×C3×C5H(5,𝔽2)11yescharacteristic
197D15—24no—
198C3×D5—42no—
199C3×D5—41no—
200C5×D3—22no—
201C23×C3—52no—
202C23×C3—51no—
203C22×D3—52no—
204C22×D3—104no—
205C22×D3—104no—
206C22×D3—202no—
207C22×D3C22×D511yescharacteristic
208C3×D4—102no—
209C3×D4—202no—
210C2×C3×C4—51no—
211C2×C3×C4—102no—
212C2×C3×C4C22×D512yes—
213C2×C3×C4C22×D511yescharacteristic
214#67—104no—
215#67—104no—
216#67—102no—
217C2×Dic3—51no—
218C2×Dic3—51no—
219C2×Dic3C22×D511yescharacteristic
220C2×Dic3C22×D512yes—
221D12—54no—
222C4×D3—22no—
223C4×D3—104no—
224C4×D3—202no—
225Dic6—22no—
226C22×C5—32no—
227C22×C5—61no—
228C22×C5—61no—
229C22×C5C23×D311yescharacteristic
230C2×D5—21no—
231C2×D5—22no—
232C2×D5—21no—
233C2×D5—22no—
234C2×D5—21no—
235C2×D5—22no—
236C2×D5—34no—
237C2×D5—62no—
238C2×D5—62no—
239C2×D5—128no—
240C4×C5—22no—
241C4×C5—62no—
242C4×C5—61no—
243C4×C5D3×D412yes—
244Dic5—21no—
245Dic5—61no—
246Dic5—61no—
247C24—152no—
248#41—304no—
249C2×Q8—31no—
250C2×D4—51no—
251C2×D4—52no—
252C2×D4—152no—
253C2×D4—152no—
254C2×D4—151no—
255C2×D4—151no—
256C2×D4—304no—
257C22×C4—31no—
258C22×C4—51no—
259C22×C4—152no—
260C22×C4—152no—
261C4⋊C4—32no—
262C4⋊C4—151no—
263C4⋊C4D3×D511yescharacteristic, complemented
264C22⋊C4—32no—
265C22⋊C4—52no—
266C22⋊C4—152no—
267C22⋊C4—151no—
268C22⋊C4—152no—
269C22⋊C4—151no—
270C22⋊C4—152no—
271C42—151no—
272C3×C5#54611yescharacteristic, complemented, Hall
273C22×C3—101no—
274C22×C3—102no—
275C22×C3—101no—
276C22×C3—102no—
277C22×C3—101no—
278C22×C3—102no—
279C22×C3C23×D511yescharacteristic
280C2×D3—21no—
281C2×D3—21no—
282C2×D3—54no—
283C2×D3—102no—
284C2×D3—102no—
285C2×D3—208no—
286C2×D3#71312yescomplemented
287C3×C4—22no—
288C3×C4—101no—
289C3×C4D4×D512yes—
290Dic3—22no—
291Dic3—21no—
292Dic3—101no—
293Dic3—101no—
294C2×C5—62no—
295C2×C5C2×D3×D411yescharacteristic
296C2×C5#100411yescharacteristic
297C2×C5D3×#4111yescharacteristic
298D5—42no—
299D5—41no—
300D5—64no—
301C23—31no—
302C23—52no—
303C23—51no—
304C23—152no—
305C23—304no—
306C23—304no—
307C23—602no—
308Q8—62no—
309D4—102no—
310D4—154no—
311D4—202no—
312D4—304no—
313D4—304no—
314D4—302no—
315C2×C4—32no—
316C2×C4—31no—
317C2×C4—51no—
318C2×C4—62no—
319C2×C4—102no—
320C2×C4—151no—
321C2×C4—151no—
322C2×C4—304no—
323C2×C4—602no—
324C2×C4C2×D3×D512yes—
325C2×C4C2×D3×D511yescharacteristic
326C2×C3—202no—
327C2×C3—201no—
328C2×C3C2×D4×D511yescharacteristic
329C2×C3C2×#71311yescharacteristic
330C2×C3#414911yescharacteristic
331D3—22no—
332D3—104no—
333C5#561211yescharacteristic, complemented, Sylow
334C22—32no—
335C22—61no—
336C22—61no—
337C22—101no—
338C22—102no—
339C22—101no—
340C22—102no—
341C22—101no—
342C22—102no—
343C22—154no—
344C22—302no—
345C22—302no—
346C22—608no—
347C22C22×D3×D511yescharacteristic
348C4—22no—
349C4—61no—
350C4—62no—
351C4—101no—
352C4—301no—
353C4—301no—
354C4#677712yes—
355C3#6596411yescharacteristic, complemented, Sylow
356C2—62no—
357C2—202no—
358C2—201no—
359C2—304no—
360C2C2×#677711yescharacteristic
361C2#9202611yescharacteristic
362C2D3×#71311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C4⋊C4C22
hypercenterC4⋊C4D3×D5
hypocenterC3×C5#546
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×D20C2semidirect
2. C2×#6773C2semidirect
3. #91782C2semidirect
4. C5×#897C2semidirect
5. C3×#4041C2semidirect
6. #91474C2semidirect
7. #91473C2semidirect
8. #91465C2semidirect
9. C4×#1201C2semidirect
10. #91368C2semidirect
11. #91356C2semidirect
15. C2×C4×D15C22semidirect
17. C2×C5×Dic6C22semidirect
19. C2×C3×D20C22semidirect
20. C2×C3×C4×D5C22semidirect
22. C2×#1202C22semidirect
24. C2×#1200C22semidirect
27. C3×C5×C4⋊C4C22semidirect
28. #6714C22semidirect
29. C4×Dic15C22semidirect
31. C5×C4⋊Dic3C22semidirect
32. C3×#685C22semidirect
33. #6669C22semidirect
35. #6665C22semidirect
36. #6665C22semidirect
37. #6665C22semidirect
45. #4041D3semidirect
61. D60D4semidirect
82. #897D5semidirect
96. C2×D20C2×D3semidirect
99. C2×C4×D5C2×D3semidirect
101. C5×C4⋊C4C2×D3semidirect
108. #685C2×D3semidirect
118. C2×C5×D3#41semidirect
146. C2×Dic6C2×D5semidirect
147. C3×C4⋊C4C2×D5semidirect
155. C4⋊Dic3C2×D5semidirect
263. C4⋊C4D3×D5semidirect
272. C3×C5#546semidirect
286. C2×D3#713semidirect
333. C5#5612semidirect
355. C3#65964semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D202
C2×#67732
#917822
C5×#8972
C3×#40412
#914742
#914732
#914652
C4×#12012
#913682
#913562
#659643
#56125