Finite Group Databasebeta

← #11754955

Subgroups — #11754955

Order 960.

668 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 136 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×D4×D15C211yesmaximal, characteristic, complemented
2C2×#6771C211yesmaximal, characteristic, complemented
3C5×#938C211yesmaximal, characteristic, complemented
4C3×#4084C211yesmaximal, characteristic, complemented
5#91573C212yesmaximal, complemented
6#91572C212yesmaximal, complemented
7#91567C212yesmaximal, complemented
8#91526C212yesmaximal, complemented
9#91460C211yesmaximal, characteristic, complemented
10#91450C211yesmaximal, characteristic, complemented
11#91347C211yesmaximal, characteristic, complemented
12#65941—31nomaximal, Hall
13C23×D15C2212yes—
14C2×C3×C5×D4C2211yescharacteristic
15C2×#1220C2212yes—
16D4×D15—24no—
17C2×D60C2211yescharacteristic
18C2×C4×D15C2211yescharacteristic
19C2×C5×#67C2212yescomplemented
20C2×C4×C5×D3C2211yescharacteristic, complemented
21C2×C3×C22⋊D5C2212yescomplemented
22C2×C3×C4×D5C2211yescharacteristic, complemented
23C2×#1204C2211yescharacteristic, complemented
24C2×#1203C2211yescharacteristic
25C2×#1203C2212yescomplemented
26C2×#1202C2211yescharacteristic
27C2×#1202C2212yescomplemented
28C2×#1201C2211yescharacteristic, complemented
29#6771—24no—
30C5×#217C2212yescomplemented
31C5×C4⋊Dic3C2211yescharacteristic, complemented
32C3×C22⋊Dic5C2212yescomplemented
33C3×C4⋊Dic5C2211yescharacteristic, complemented
34#6667C2212yes—
35#6667C2212yescomplemented
36#6667C2212yescomplemented
37Dic3×Dic5C2212yescomplemented
38#5607—51nomaximal, Hall
39C2×D4×D5—31no—
40C2×#709—31no—
41C5×#121—31no—
42#4086—32no—
43#4084D311yescharacteristic, complemented
44#4083—32no—
45#4076—31no—
46#4041—31no—
47#4038—31no—
48#4031—32no—
49#4028—32no—
50C23×C3×C5C2312yes—
51C22×D15—24no—
52C22×D15—42no—
53C22×D15—42no—
54C22×D15C2312yes—
55C22×D15C2311yescharacteristic
56C22×C5×D3C2311yescharacteristic
57C22×C3×D5C2311yescharacteristic
58C3×C5×D4—22no—
59C2×C3×C4×C5C2311yescharacteristic
60#1220—24no—
61C2×Dic15C2311yescharacteristic
62D60—22no—
63C4×D15D414yescomplemented
64C5×#67—42no—
65C2×C5×Dic3C2312yes—
66C2×C5×Dic3C2311yescharacteristic
67C4×C5×D3—22no—
68C3×C22⋊D5—42no—
69C2×C3×Dic5C2312yes—
70C2×C3×Dic5C2311yescharacteristic
71C3×C4×D5—22no—
72#1204—22no—
73#1203—22no—
74#1203—42no—
75#1202—22no—
76#1202—42no—
77#1201—22no—
78C2×D3×D4—51no—
79C2×#235—51no—
80C3×#121—51no—
81#940—52no—
82#938D511yescharacteristic, complemented
83#937—52no—
84#930—51no—
85#895—51no—
86#892—51no—
87#885—52no—
88#882—52no—
89C23×D5—32no—
90C2×C5×D4—32no—
91C2×C5×D4C2×D311yescharacteristic
92C22×C4×C5—31no—
93C2×C22⋊D5—32no—
94C2×C22⋊D5—31no—
95C2×C22⋊D5—31no—
96C2×C22⋊D5—32no—
97C2×C22⋊D5C2×D312yescomplemented
98D4×D5—64no—
99#709—64no—
100C2×D20—31no—
101C2×D20—31no—
102C2×D20—32no—
103C2×C4×D5—31no—
104C2×C4×D5C2×D311yescharacteristic, complemented
105C2×Dic10—31no—
106C5×C4⋊C4—31no—
107C5×C22⋊C4—32no—
108C22⋊Dic5C2×D312yescomplemented
109#685—32no—
110#685—32no—
111#685—32no—
112C4⋊Dic5C2×D311yescharacteristic, complemented
113C4×Dic5—32no—
114#535—151noSylow
115C22×C3×C5—22no—
116C22×C3×C5—22no—
117C22×C3×C5—22no—
118C22×C3×C5C2411yescharacteristic
119C2×D15—21no—
120C2×D15—21no—
121C2×D15—22no—
122C2×D15—22no—
123C2×D15—22no—
124C2×D15—48no—
125C2×D15C2×D412yes—
126C2×C5×D3—21no—
127C2×C5×D3—21no—
128C2×C5×D3—21no—
129C2×C3×D5—21no—
130C2×C3×D5—21no—
131C2×C3×D5—21no—
132C3×C4×C5C2×D412yes—
133Dic15C2×D412yes—
134C3×Dic5—21no—
135C3×Dic5—22no—
136C5×Dic3—21no—
137C5×Dic3—22no—
138C23×D3—52no—
139C2×C3×D4—52no—
140C2×C3×D4C2×D511yescharacteristic
141C22×C3×C4—51no—
142C2×#67—52no—
143C2×#67—51no—
144C2×#67—51no—
145C2×#67—52no—
146C2×#67C2×D512yescomplemented
147D3×D4—104no—
148#235—104no—
149C2×D12—51no—
150C2×D12—51no—
151C2×D12—52no—
152C2×C4×D3—51no—
153C2×C4×D3C2×D511yescharacteristic, complemented
154C2×Dic6—51no—
155C3×C4⋊C4—51no—
156C3×C22⋊C4—52no—
157#217C2×D512yescomplemented
158(C2×D3)⋊C4—52no—
159(C2×D3)⋊C4—52no—
160(C2×D3)⋊C4—52no—
161C4⋊Dic3C2×D511yescharacteristic, complemented
162C4×Dic3—52no—
163C23×C5—31no—
164C23×C5C22×D312yes—
165C22×D5—31no—
166C22×D5—32no—
167C22×D5—64no—
168C22×D5—122no—
169C22×D5—122no—
170C22×D5C22×D311yescharacteristic
171C5×D4—22no—
172C5×D4—122no—
173C2×C4×C5—32no—
174C2×C4×C5—31no—
175C2×C4×C5—62no—
176C2×C4×C5C22×D311yescharacteristic
177C22⋊D5—42no—
178C22⋊D5—62no—
179C22⋊D5—62no—
180C22⋊D5—64no—
181C22⋊D5—122no—
182C2×Dic5—31no—
183C2×Dic5C22×D312yes—
184C2×Dic5C22×D311yescharacteristic
185D20—62no—
186D20—62no—
187D20—122no—
188C4×D5—22no—
189C4×D5—34no—
190Dic10—62no—
191C2×#41—151no—
192C22×D4—151no—
193#127—152no—
194#124—152no—
195#121—31no—
196#121—51no—
197#121—151no—
198#121—151no—
199#120—152no—
200#120—152no—
201#117—151no—
202C2×C3×C5—42no—
203C2×C3×C5C22×D411yescharacteristic
204C2×C3×C5H(5,𝔽2)11yescharacteristic
205C2×C3×C5H(5,𝔽2)11yescharacteristic
206D15—22no—
207D15—42no—
208C3×D5—41no—
209C5×D3—41no—
210C23×C3—51no—
211C23×C3C22×D512yes—
212C22×D3—51no—
213C22×D3—52no—
214C22×D3—104no—
215C22×D3—202no—
216C22×D3—202no—
217C22×D3C22×D511yescharacteristic
218C3×D4—22no—
219C3×D4—202no—
220C2×C3×C4—52no—
221C2×C3×C4—51no—
222C2×C3×C4—102no—
223C2×C3×C4C22×D511yescharacteristic
224#67—42no—
225#67—102no—
226#67—102no—
227#67—104no—
228#67—202no—
229C2×Dic3—51no—
230C2×Dic3C22×D512yes—
231C2×Dic3C22×D511yescharacteristic
232D12—102no—
233D12—102no—
234D12—202no—
235C4×D3—22no—
236C4×D3—54no—
237Dic6—102no—
238C22×C5—22no—
239C22×C5—22no—
240C22×C5—22no—
241C22×C5—61no—
242C22×C5—61no—
243C22×C5—61no—
244C22×C5C23×D311yescharacteristic
245C2×D5—21no—
246C2×D5—21no—
247C2×D5—21no—
248C2×D5—32no—
249C2×D5—61no—
250C2×D5—61no—
251C2×D5—62no—
252C2×D5—62no—
253C2×D5—62no—
254C2×D5—128no—
255C4×C5—61no—
256C4×C5—62no—
257C4×C5D3×D412yes—
258Dic5—21no—
259Dic5—22no—
260Dic5—32no—
261C24—152no—
262#41—304no—
263C2×Q8—151no—
264C2×D4—32no—
265C2×D4—52no—
266C2×D4—151no—
267C2×D4—152no—
268C2×D4—151no—
269C2×D4—152no—
270C2×D4—151no—
271C2×D4—152no—
272C2×D4—151no—
273C2×D4—304no—
274C2×D4D3×D511yescharacteristic
275C22×C4—31no—
276C22×C4—51no—
277C22×C4—151no—
278C4⋊C4—31no—
279C4⋊C4—51no—
280C22⋊C4—32no—
281C22⋊C4—52no—
282C22⋊C4—152no—
283C22⋊C4—152no—
284C22⋊C4—152no—
285C42—152no—
286C3×C5#53511yescharacteristic, complemented, Hall
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—101no—
291C22×C3—101no—
292C22×C3—101no—
293C22×C3C23×D511yescharacteristic
294C2×D3—21no—
295C2×D3—21no—
296C2×D3—21no—
297C2×D3—52no—
298C2×D3—101no—
299C2×D3—101no—
300C2×D3—102no—
301C2×D3—102no—
302C2×D3—102no—
303C2×D3—208no—
304C3×C4—101no—
305C3×C4—102no—
306C3×C4D4×D512yes—
307Dic3—21no—
308Dic3—22no—
309Dic3—52no—
310C2×C5—42no—
311C2×C5—121no—
312C2×C5C2×D3×D411yescharacteristic
313C2×C5#100411yescharacteristic
314C2×C5#100911yescharacteristic
315D5—41no—
316D5—62no—
317D5—122no—
318C23—31no—
319C23—51no—
320C23—151no—
321C23—152no—
322C23—304no—
323C23—602no—
324C23—602no—
325C23C2×D3×D512yes—
326Q8—302no—
327D4—22no—
328D4—122no—
329D4—202no—
330D4—302no—
331D4—302no—
332D4—304no—
333D4—302no—
334D4—302no—
335D4—602no—
336D4—602no—
337C2×C4—32no—
338C2×C4—31no—
339C2×C4—52no—
340C2×C4—51no—
341C2×C4—62no—
342C2×C4—102no—
343C2×C4—151no—
344C2×C4—154no—
345C2×C4C2×D3×D511yescharacteristic
346C2×C3—42no—
347C2×C3—201no—
348C2×C3C2×D4×D511yescharacteristic
349C2×C3#414411yescharacteristic
350C2×C3#414911yescharacteristic
351D3—41no—
352D3—102no—
353D3—202no—
354C5#560711yescharacteristic, complemented, Sylow
355C22—22no—
356C22—22no—
357C22—22no—
358C22—61no—
359C22—61no—
360C22—61no—
361C22—101no—
362C22—101no—
363C22—101no—
364C22—152no—
365C22—301no—
366C22—301no—
367C22—302no—
368C22—302no—
369C22—302no—
370C22—608no—
371C22C22×D3×D511yescharacteristic
372C4—62no—
373C4—61no—
374C4—102no—
375C4—101no—
376C4—152no—
377C4#679012yes—
378C3#6594111yescharacteristic, complemented, Sylow
379C2—42no—
380C2—121no—
381C2—201no—
382C2—302no—
383C2—602no—
384C2#9202611yescharacteristic
385C2#9202711yescharacteristic
386C2C2×#679011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#535
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×D4×D15C2semidirect
2. C2×#6771C2semidirect
3. C5×#938C2semidirect
4. C3×#4084C2semidirect
5. #91573C2semidirect
6. #91572C2semidirect
7. #91567C2semidirect
8. #91526C2semidirect
9. #91460C2semidirect
10. #91450C2semidirect
11. #91347C2semidirect
19. C2×C5×#67C22semidirect
20. C2×C4×C5×D3C22semidirect
21. C2×C3×C22⋊D5C22semidirect
22. C2×C3×C4×D5C22semidirect
23. C2×#1204C22semidirect
25. C2×#1203C22semidirect
27. C2×#1202C22semidirect
28. C2×#1201C22semidirect
30. C5×#217C22semidirect
31. C5×C4⋊Dic3C22semidirect
32. C3×C22⋊Dic5C22semidirect
33. C3×C4⋊Dic5C22semidirect
35. #6667C22semidirect
36. #6667C22semidirect
37. Dic3×Dic5C22semidirect
43. #4084D3semidirect
63. C4×D15D4semidirect
82. #938D5semidirect
97. C2×C22⋊D5C2×D3semidirect
104. C2×C4×D5C2×D3semidirect
108. C22⋊Dic5C2×D3semidirect
112. C4⋊Dic5C2×D3semidirect
146. C2×#67C2×D5semidirect
153. C2×C4×D3C2×D5semidirect
157. #217C2×D5semidirect
161. C4⋊Dic3C2×D5semidirect
286. C3×C5#535semidirect
354. C5#5607semidirect
378. C3#65941semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D152
C2×#67712
C5×#9382
C3×#40842
#915732
#915722
#915672
#915262
#914602
#914502
#913472
#659413
#56075