Finite Group Databasebeta

← #11754958

Subgroups — #11754958

Order 960.

668 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 138 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×#6773C211yesmaximal, characteristic, complemented
3C5×#936C211yesmaximal, characteristic, complemented
4C3×#4084C211yesmaximal, characteristic, complemented
5#91572C212yesmaximal, complemented
6#91562C212yesmaximal, complemented
7#91561C212yesmaximal, complemented
8#91533C212yesmaximal, complemented
9C4×#1202C211yesmaximal, characteristic, complemented
10#91428C211yesmaximal, characteristic, complemented
11#91383C211yesmaximal, characteristic, complemented
12#66010—31nomaximal, Hall
13C23×D15C2212yes—
14C2×C3×C5×D4C2211yescharacteristic
15C2×#1220C2212yes—
16D4×D15—24no—
17C2×D60C2211yescharacteristic
18C2×C4×D15C2211yescharacteristic
19C2×C5×Dic6C2211yescharacteristic, complemented
20C2×C3×C22⋊D5C2212yescomplemented
21C2×C3×C4×D5C2211yescharacteristic, complemented
22C2×#1202C2211yescharacteristic
23C2×#1202C2212yescomplemented
24C2×#1200C2212yescomplemented
25C2×#1200C2212yescomplemented
26#6773—24no—
27C5×#217C2212yescomplemented
28C5×#217C2212yescomplemented
29C4×C5×Dic3C2211yescharacteristic, complemented
30C3×C22⋊Dic5C2212yescomplemented
31C3×C4⋊Dic5C2211yescharacteristic, complemented
32Dic3⋊Dic5C2211yescharacteristic, complemented
33#6668C2212yescomplemented
34#6667C2211yescharacteristic
35#6667C2212yes—
36#6667C2212yescomplemented
37#6665C2211yescharacteristic, complemented
38#5612—51nomaximal, Hall
39C2×#713—31no—
40C2×D4×D5—31no—
41C5×#124—31no—
42#4092—31no—
43#4084D311yescharacteristic, complemented
44#4030—32no—
45#4029—32no—
46#4028—32no—
47D5×C22⋊C4—32no—
48C4×D20—31no—
49C4×D20—31no—
50C23×C3×C5C2312yes—
51C22×D15—24no—
52C22×D15—24no—
53C22×D15—42no—
54C22×D15C2312yes—
55C22×D15C2311yescharacteristic
56C22×C3×D5C2311yescharacteristic
57C3×C5×D4—22no—
58C2×C3×C4×C5C2311yescharacteristic
59#1220—24no—
60C2×Dic15C2311yescharacteristic
61D60D414yescomplemented
62C4×D15—22no—
63C2×C5×Dic3C2311yescharacteristic
64C2×C5×Dic3C2312yes—
65C2×C5×Dic3C2311yescharacteristic
66C5×Dic6—22no—
67C3×C22⋊D5—42no—
68C2×C3×Dic5C2312yes—
69C2×C3×Dic5C2311yescharacteristic
70C3×C4×D5—22no—
71#1202—24no—
72#1202—22no—
73#1200—24no—
74#1200—42no—
75C2×D3×D4—51no—
76C2×#235—51no—
77C3×#121—51no—
78#938—52no—
79#937—52no—
80#936D511yescharacteristic, complemented
81C4×#67—51no—
82#897—51no—
83#893—51no—
84#883—52no—
85D3×C22⋊C4—52no—
86C23×D5—32no—
87C2×C5×Q8—31no—
88C2×C5×D4C2×D311yescharacteristic
89C2×C22⋊D5—32no—
90C2×C22⋊D5C2×D312yescomplemented
91#713—64no—
92D4×D5—64no—
93C2×D20—31no—
94C2×D20—32no—
95C2×D20—31no—
96C2×C4×D5—31no—
97C2×C4×D5—32no—
98C2×C4×D5—32no—
99C2×C4×D5C2×D311yescharacteristic, complemented
100C5×C22⋊C4—32no—
101C5×C22⋊C4—32no—
102C42×C5—31no—
103C22⋊Dic5C2×D312yescomplemented
104#685—31no—
105#685—31no—
106#685—32no—
107#685—32no—
108C4⋊Dic5—31no—
109C4⋊Dic5C2×D311yescharacteristic, complemented
110#683—32no—
111#546—151noSylow
112C22×C3×C5—22no—
113C22×C3×C5—22no—
114C22×C3×C5—22no—
115C22×C3×C5C2411yescharacteristic
116C2×D15—21no—
117C2×D15—22no—
118C2×D15—22no—
119C2×D15—21no—
120C2×D15—48no—
121C2×D15C2×D414yes—
122C2×D15#4112yescomplemented
123C2×C3×D5—21no—
124C2×C3×D5—21no—
125C2×C3×D5—21no—
126C3×C4×C5C2×D412yes—
127Dic15—21no—
128C3×Dic5—21no—
129C3×Dic5—22no—
130C5×Dic3—22no—
131C5×Dic3—21no—
132C5×Dic3—21no—
133C23×D3—52no—
134C2×C3×D4—52no—
135C2×C3×D4C2×D511yescharacteristic
136C22×C3×C4—51no—
137C2×#67—52no—
138C2×#67—51no—
139C2×#67—52no—
140D3×D4—104no—
141#235—104no—
142C2×D12—51no—
143C2×C4×D3—51no—
144C2×C4×D3—52no—
145C2×C4×D3—52no—
146C2×Dic6C2×D511yescharacteristic, complemented
147C3×C4⋊C4—51no—
148C3×C22⋊C4—52no—
149#217—51no—
150#217C2×D512yescomplemented
151#217C2×D512yescomplemented
152(C2×D3)⋊C4—51no—
153(C2×D3)⋊C4—52no—
154(C2×D3)⋊C4—52no—
155C4⋊Dic3—52no—
156Dic3⋊C4—51no—
157C4×Dic3C2×D511yescharacteristic, complemented
158C23×C5C22×D312yes—
159C22×D5—31no—
160C22×D5—32no—
161C22×D5—64no—
162C22×D5—64no—
163C22×D5—122no—
164C22×D5C22×D311yescharacteristic
165C5×Q8—62no—
166C5×D4—22no—
167C2×C4×C5—32no—
168C2×C4×C5—31no—
169C2×C4×C5—31no—
170C2×C4×C5C22×D311yescharacteristic
171C22⋊D5—42no—
172C22⋊D5—64no—
173C2×Dic5—31no—
174C2×Dic5C22×D312yes—
175C2×Dic5C22×D311yescharacteristic
176D20—34no—
177D20—62no—
178D20—64no—
179C4×D5—22no—
180C4×D5—64no—
181C4×D5—62no—
182C4×D5—122no—
183C2×#41—151no—
184C22×D4—151no—
185#124—31no—
186#123—152no—
187C22⋊Q8—151no—
188#121—51no—
189#121—152no—
190#120—152no—
191C4×D4—151no—
192C4×D4—151no—
193C2×C22⋊C4—152no—
194C2×C3×C5—42no—
195C2×C3×C5C22×D411yescharacteristic
196C2×C3×C5C2×#4111yescharacteristic
197C2×C3×C5H(5,𝔽2)11yescharacteristic
198D15—22no—
199D15—24no—
200C3×D5—41no—
201C23×C3—51no—
202C23×C3C22×D512yes—
203C22×D3—51no—
204C22×D3—52no—
205C22×D3—104no—
206C22×D3—104no—
207C22×D3—202no—
208C3×D4—22no—
209C3×D4—202no—
210C2×C3×C4—51no—
211C2×C3×C4—52no—
212C2×C3×C4—102no—
213C2×C3×C4C22×D511yescharacteristic
214#67—104no—
215#67—102no—
216#67—104no—
217C2×Dic3—51no—
218C2×Dic3C22×D511yescharacteristic
219C2×Dic3C22×D511yescharacteristic
220C2×Dic3C22×D512yes—
221D12—54no—
222C4×D3—104no—
223C4×D3—102no—
224C4×D3—202no—
225Dic6—22no—
226C22×C5—22no—
227C22×C5—22no—
228C22×C5—22no—
229C22×C5C23×D311yescharacteristic
230C2×D5—21no—
231C2×D5—21no—
232C2×D5—21no—
233C2×D5—34no—
234C2×D5—32no—
235C2×D5—61no—
236C2×D5—62no—
237C2×D5—61no—
238C2×D5—62no—
239C2×D5—128no—
240C4×C5—62no—
241C4×C5—61no—
242C4×C5—61no—
243C4×C5D3×D412yes—
244Dic5—21no—
245Dic5—22no—
246Dic5—61no—
247C24—152no—
248#41—304no—
249C2×Q8—31no—
250C2×D4—52no—
251C2×D4—151no—
252C2×D4—152no—
253C2×D4—152no—
254C2×D4—151no—
255C2×D4—304no—
256C2×D4D3×D511yescharacteristic
257C22×C4—51no—
258C22×C4—152no—
259C22×C4—152no—
260C22×C4—151no—
261C4⋊C4—51no—
262C4⋊C4—151no—
263C4⋊C4—152no—
264C22⋊C4—32no—
265C22⋊C4—32no—
266C22⋊C4—52no—
267C22⋊C4—151no—
268C22⋊C4—152no—
269C22⋊C4—152no—
270C22⋊C4—151no—
271C42—31no—
272C3×C5#54611yescharacteristic, complemented, Hall
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—101no—
277C22×C3—101no—
278C22×C3—101no—
279C22×C3C23×D511yescharacteristic
280C2×D3—54no—
281C2×D3—52no—
282C2×D3—101no—
283C2×D3—102no—
284C2×D3—102no—
285C2×D3—101no—
286C2×D3—208no—
287C3×C4—101no—
288C3×C4—102no—
289C3×C4D4×D512yes—
290Dic3—22no—
291Dic3—21no—
292Dic3—21no—
293Dic3—101no—
294C2×C5—42no—
295C2×C5C2×D3×D411yescharacteristic
296C2×C5#100411yescharacteristic
297C2×C5D3×#4111yescharacteristic
298D5—41no—
299D5—62no—
300D5—64no—
301C23—51no—
302C23—151no—
303C23—152no—
304C23—304no—
305C23—304no—
306C23—602no—
307C23C2×D3×D512yes—
308Q8—62no—
309D4—22no—
310D4—154no—
311D4—202no—
312D4—304no—
313D4—302no—
314D4—304no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—32no—
318C2×C4—51no—
319C2×C4—52no—
320C2×C4—102no—
321C2×C4—151no—
322C2×C4—304no—
323C2×C4—302no—
324C2×C4—602no—
325C2×C4C2×D3×D511yescharacteristic
326C2×C3—42no—
327C2×C3—201no—
328C2×C3C2×D4×D511yescharacteristic
329C2×C3D5×#4111yescharacteristic
330C2×C3#414911yescharacteristic
331D3—102no—
332D3—104no—
333C5#561211yescharacteristic, complemented, Sylow
334C22—22no—
335C22—22no—
336C22—22no—
337C22—101no—
338C22—101no—
339C22—101no—
340C22—152no—
341C22—154no—
342C22—302no—
343C22—302no—
344C22—301no—
345C22—301no—
346C22—608no—
347C22C22×D3×D511yescharacteristic
348C4—61no—
349C4—61no—
350C4—62no—
351C4—102no—
352C4—101no—
353C4—301no—
354C4#679012yes—
355C3#6601011yescharacteristic, complemented, Sylow
356C2—42no—
357C2—201no—
358C2—302no—
359C2—304no—
360C2#9202411yescharacteristic
361C2#9202611yescharacteristic
362C2C2×#679011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#546
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×#6773C2semidirect
3. C5×#936C2semidirect
4. C3×#4084C2semidirect
5. #91572C2semidirect
6. #91562C2semidirect
7. #91561C2semidirect
8. #91533C2semidirect
9. C4×#1202C2semidirect
10. #91428C2semidirect
11. #91383C2semidirect
19. C2×C5×Dic6C22semidirect
20. C2×C3×C22⋊D5C22semidirect
21. C2×C3×C4×D5C22semidirect
23. C2×#1202C22semidirect
24. C2×#1200C22semidirect
25. C2×#1200C22semidirect
27. C5×#217C22semidirect
28. C5×#217C22semidirect
29. C4×C5×Dic3C22semidirect
30. C3×C22⋊Dic5C22semidirect
31. C3×C4⋊Dic5C22semidirect
32. Dic3⋊Dic5C22semidirect
33. #6668C22semidirect
36. #6667C22semidirect
37. #6665C22semidirect
43. #4084D3semidirect
61. D60D4semidirect
80. #936D5semidirect
90. C2×C22⋊D5C2×D3semidirect
99. C2×C4×D5C2×D3semidirect
103. C22⋊Dic5C2×D3semidirect
109. C4⋊Dic5C2×D3semidirect
122. C2×D15#41semidirect
146. C2×Dic6C2×D5semidirect
150. #217C2×D5semidirect
151. #217C2×D5semidirect
157. C4×Dic3C2×D5semidirect
272. C3×C5#546semidirect
333. C5#5612semidirect
355. C3#66010semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D152
C2×#67732
C5×#9362
C3×#40842
#915722
#915622
#915612
#915332
C4×#12022
#914282
#913832
#660103
#56125