Finite Group Databasebeta

← #11754671

Subgroups — #11754671

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×#6817C211yesmaximal, characteristic, complemented
2C2×#6777C211yesmaximal, characteristic, complemented
3C5×#930C211yesmaximal, characteristic, complemented
4C3×#4076C211yesmaximal, characteristic, complemented
5#91577C212yesmaximal, complemented
6#91567C212yesmaximal, complemented
7#91476C212yesmaximal, complemented
8#91466C211yesmaximal, characteristic, complemented
9#91465C211yesmaximal, characteristic, complemented
10#91462C212yesmaximal, complemented
11#91347C211yesmaximal, characteristic, complemented
12#65941—31nomaximal, Hall
13C22×D3×D5C2212yes—
14C22×C3×C4×C5C2211yescharacteristic, complemented
15C2×#1220C2211yescharacteristic
16C2×#1220C2211yescharacteristic, complemented
17#6817—24no—
18C2×D60C2211yescharacteristic
19C2×C4×D15C2211yescharacteristic
20C2×C4×D15C2211yescharacteristic, complemented
21C2×Dic30C2211yescharacteristic, complemented
22C2×C5×#67C2212yescomplemented
23C2×C5×D12C2211yescharacteristic
24C2×C3×C22⋊D5C2212yescomplemented
25C2×C3×D20C2211yescharacteristic
26C2×#1203C2212yescomplemented
27C2×#1202C2212yescomplemented
28C2×#1201C2212yes—
29#6777—24no—
30C5×(C2×D3)⋊C4C2212yescomplemented
31C5×C4⋊Dic3C2211yescharacteristic, complemented
32C3×#685C2212yescomplemented
33C3×C4⋊Dic5C2211yescharacteristic, complemented
34#6667C2212yes—
35(C2×D3)⋊Dic5C2212yescomplemented
36#6665C2212yescomplemented
37Dic3×Dic5C2212yescomplemented
38#5607—51nomaximal, Hall
39C2×D4×D5—31no—
40C2×#709—31no—
41C5×#121—31no—
42#4086—32no—
43#4084—31no—
44#4083—32no—
45#4076D311yescharacteristic, complemented
46#4041—31no—
47#4038—31no—
48#4031—32no—
49#4028—32no—
50C23×C3×C5C2311yescharacteristic
51C22×D15C2311yescharacteristic
52C22×D15C2311yescharacteristic
53C22×C5×D3C2312yes—
54C22×C3×D5C2312yes—
55C2×D3×D5—24no—
56C2×D3×D5—42no—
57C2×D3×D5—42no—
58C2×C3×C4×C5—22no—
59C2×C3×C4×C5C2311yescharacteristic
60C2×C3×C4×C5C2311yescharacteristic
61#1220—22no—
62#1220—22no—
63C2×Dic15C2311yescharacteristic
64C2×Dic15C2311yescharacteristic
65D60—22no—
66C4×D15—22no—
67C4×D15D414yescomplemented
68Dic30—22no—
69C5×#67—42no—
70C2×C5×Dic3C2312yes—
71C5×D12—22no—
72C3×C22⋊D5—42no—
73C2×C3×Dic5C2312yes—
74C3×D20—22no—
75#1203—42no—
76#1202—42no—
77#1201—24no—
78C2×D3×D4—51no—
79C2×#235—51no—
80C3×#121—51no—
81#940—52no—
82#938—51no—
83#937—52no—
84#930D511yescharacteristic, complemented
85#895—51no—
86#892—51no—
87#885—52no—
88#882—52no—
89C23×D5—32no—
90C2×C5×D4—32no—
91C2×C5×D4—31no—
92C22×C4×C5C2×D311yescharacteristic, complemented
93C2×C22⋊D5—31no—
94C2×C22⋊D5—31no—
95C2×C22⋊D5—32no—
96C2×C22⋊D5—32no—
97C2×C22⋊D5C2×D312yescomplemented
98D4×D5—64no—
99#709—64no—
100C2×D20—31no—
101C2×D20—32no—
102C2×D20C2×D311yescharacteristic
103C2×C4×D5—31no—
104C2×C4×D5—31no—
105C2×Dic10—31no—
106C5×C4⋊C4—31no—
107C5×C22⋊C4—32no—
108C22⋊Dic5—32no—
109#685—32no—
110#685—32no—
111#685C2×D312yescomplemented
112C4⋊Dic5C2×D311yescharacteristic, complemented
113C4×Dic5—32no—
114#535—151noSylow
115C22×C3×C5—21no—
116C22×C3×C5—21no—
117C22×C3×C5—21no—
118C22×C3×C5C2411yescharacteristic
119C2×D15—21no—
120C2×D15—21no—
121C2×D15—21no—
122C2×D15—21no—
123C2×D15—21no—
124C2×D15C2×D412yes—
125C2×C5×D3—22no—
126C2×C5×D3—22no—
127C2×C5×D3—22no—
128C2×C3×D5—22no—
129C2×C3×D5—22no—
130C2×C3×D5—22no—
131D3×D5—48no—
132C3×C4×C5—21no—
133C3×C4×C5C2×D412yes—
134Dic15—21no—
135Dic15C2×D412yes—
136C3×Dic5—22no—
137C5×Dic3—22no—
138C23×D3—52no—
139C2×C3×D4—52no—
140C2×C3×D4—51no—
141C22×C3×C4C2×D511yescharacteristic, complemented
142C2×#67—51no—
143C2×#67—51no—
144C2×#67—52no—
145C2×#67—52no—
146C2×#67C2×D512yescomplemented
147D3×D4—104no—
148#235—104no—
149C2×D12—51no—
150C2×D12—52no—
151C2×D12C2×D511yescharacteristic
152C2×C4×D3—51no—
153C2×C4×D3—51no—
154C2×Dic6—51no—
155C3×C4⋊C4—51no—
156C3×C22⋊C4—52no—
157#217—52no—
158(C2×D3)⋊C4—52no—
159(C2×D3)⋊C4—52no—
160(C2×D3)⋊C4C2×D512yescomplemented
161C4⋊Dic3C2×D511yescharacteristic, complemented
162C4×Dic3—52no—
163C23×C5—32no—
164C23×C5C22×D311yescharacteristic
165C22×D5—31no—
166C22×D5—31no—
167C22×D5—64no—
168C22×D5—122no—
169C22×D5—122no—
170C22×D5C22×D312yes—
171C5×D4—62no—
172C5×D4—122no—
173C2×C4×C5—22no—
174C2×C4×C5—32no—
175C2×C4×C5C22×D311yescharacteristic
176C2×C4×C5C22×D311yescharacteristic
177C22⋊D5—42no—
178C22⋊D5—64no—
179C22⋊D5—62no—
180C22⋊D5—62no—
181C22⋊D5—122no—
182C2×Dic5—31no—
183C2×Dic5—31no—
184C2×Dic5C22×D312yes—
185D20—22no—
186D20—62no—
187D20—122no—
188C4×D5—34no—
189C4×D5—62no—
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—41no—
203C2×C3×C5C22×D411yescharacteristic
204C2×C3×C5H(5,𝔽2)11yescharacteristic
205C2×C3×C5H(5,𝔽2)11yescharacteristic
206D15—22no—
207D15—41no—
208C3×D5—42no—
209C5×D3—42no—
210C23×C3—52no—
211C23×C3C22×D511yescharacteristic
212C22×D3—51no—
213C22×D3—51no—
214C22×D3—104no—
215C22×D3—202no—
216C22×D3—202no—
217C22×D3C22×D512yes—
218C3×D4—102no—
219C3×D4—202no—
220C2×C3×C4—22no—
221C2×C3×C4—52no—
222C2×C3×C4C22×D511yescharacteristic
223C2×C3×C4C22×D511yescharacteristic
224#67—42no—
225#67—104no—
226#67—102no—
227#67—102no—
228#67—202no—
229C2×Dic3—51no—
230C2×Dic3—51no—
231C2×Dic3C22×D512yes—
232D12—22no—
233D12—102no—
234D12—202no—
235C4×D3—54no—
236C4×D3—102no—
237Dic6—102no—
238C22×C5—21no—
239C22×C5—21no—
240C22×C5—21no—
241C22×C5—62no—
242C22×C5—62no—
243C22×C5—62no—
244C22×C5C23×D311yescharacteristic
245C2×D5—22no—
246C2×D5—22no—
247C2×D5—22no—
248C2×D5—32no—
249C2×D5—61no—
250C2×D5—61no—
251C2×D5—61no—
252C2×D5—61no—
253C2×D5—61no—
254C2×D5—128no—
255C4×C5—21no—
256C4×C5—62no—
257C4×C5D3×D412yes—
258Dic5—22no—
259Dic5—32no—
260Dic5—61no—
261C24—152no—
262#41—304no—
263C2×Q8—151no—
264C2×D4—32no—
265C2×D4—31no—
266C2×D4—52no—
267C2×D4—51no—
268C2×D4—152no—
269C2×D4—152no—
270C2×D4—152no—
271C2×D4—151no—
272C2×D4—151no—
273C2×D4—151no—
274C2×D4—304no—
275C22×C4—151no—
276C22×C4—151no—
277C22×C4D3×D511yescharacteristic, complemented
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—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—102no—
291C22×C3—102no—
292C22×C3—102no—
293C22×C3C23×D511yescharacteristic
294C2×D3—22no—
295C2×D3—22no—
296C2×D3—22no—
297C2×D3—52no—
298C2×D3—101no—
299C2×D3—101no—
300C2×D3—101no—
301C2×D3—101no—
302C2×D3—101no—
303C2×D3—208no—
304C3×C4—21no—
305C3×C4—102no—
306C3×C4D4×D512yes—
307Dic3—22no—
308Dic3—52no—
309Dic3—101no—
310C2×C5—41no—
311C2×C5—122no—
312C2×C5C2×D3×D411yescharacteristic
313C2×C5#100411yescharacteristic
314C2×C5#100911yescharacteristic
315D5—42no—
316D5—62no—
317D5—121no—
318C23—32no—
319C23—52no—
320C23—151no—
321C23—151no—
322C23—304no—
323C23—602no—
324C23—602no—
325C23C2×D3×D511yescharacteristic
326Q8—302no—
327D4—62no—
328D4—102no—
329D4—122no—
330D4—202no—
331D4—302no—
332D4—302no—
333D4—304no—
334D4—302no—
335D4—602no—
336D4—602no—
337C2×C4—22no—
338C2×C4—32no—
339C2×C4—52no—
340C2×C4—151no—
341C2×C4—154no—
342C2×C4—151no—
343C2×C4—302no—
344C2×C4C2×D3×D511yescharacteristic
345C2×C4C2×D3×D511yescharacteristic
346C2×C3—41no—
347C2×C3—202no—
348C2×C3C2×D4×D511yescharacteristic
349C2×C3#414411yescharacteristic
350C2×C3#414911yescharacteristic
351D3—42no—
352D3—102no—
353D3—201no—
354C5#560711yescharacteristic, complemented, Sylow
355C22—21no—
356C22—21no—
357C22—21no—
358C22—62no—
359C22—62no—
360C22—62no—
361C22—102no—
362C22—102no—
363C22—102no—
364C22—152no—
365C22—301no—
366C22—301no—
367C22—301no—
368C22—301no—
369C22—301no—
370C22—608no—
371C22C22×D3×D511yescharacteristic
372C4—21no—
373C4—62no—
374C4—102no—
375C4—152no—
376C4—301no—
377C4#679012yes—
378C3#6594111yescharacteristic, complemented, Sylow
379C2—41no—
380C2—122no—
381C2—202no—
382C2—302no—
383C2—601no—
384C2#9201711yescharacteristic
385C2#9201811yescharacteristic
386C2C2×#679011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#535
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#6817C2semidirect
2. C2×#6777C2semidirect
3. C5×#930C2semidirect
4. C3×#4076C2semidirect
5. #91577C2semidirect
6. #91567C2semidirect
7. #91476C2semidirect
8. #91466C2semidirect
9. #91465C2semidirect
10. #91462C2semidirect
11. #91347C2semidirect
14. C22×C3×C4×C5C22semidirect
16. C2×#1220C22semidirect
20. C2×C4×D15C22semidirect
21. C2×Dic30C22semidirect
22. C2×C5×#67C22semidirect
24. C2×C3×C22⋊D5C22semidirect
26. C2×#1203C22semidirect
27. C2×#1202C22semidirect
30. C5×(C2×D3)⋊C4C22semidirect
31. C5×C4⋊Dic3C22semidirect
32. C3×#685C22semidirect
33. C3×C4⋊Dic5C22semidirect
35. (C2×D3)⋊Dic5C22semidirect
36. #6665C22semidirect
37. Dic3×Dic5C22semidirect
45. #4076D3semidirect
67. C4×D15D4semidirect
84. #930D5semidirect
92. C22×C4×C5C2×D3semidirect
97. C2×C22⋊D5C2×D3semidirect
111. #685C2×D3semidirect
112. C4⋊Dic5C2×D3semidirect
141. C22×C3×C4C2×D5semidirect
146. C2×#67C2×D5semidirect
160. (C2×D3)⋊C4C2×D5semidirect
161. C4⋊Dic3C2×D5semidirect
277. C22×C4D3×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×#68172
C2×#67772
C5×#9302
C3×#40762
#915772
#915672
#914762
#914662
#914652
#914622
#913472
#659413
#56075