Finite Group Databasebeta

← #11754959

Subgroups — #11754959

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×#6769C211yesmaximal, characteristic, complemented
3C5×#938C211yesmaximal, characteristic, complemented
4C3×#4082C211yesmaximal, characteristic, complemented
5#91573C212yesmaximal, complemented
6#91562C212yesmaximal, complemented
7#91561C212yesmaximal, complemented
8#91557C212yesmaximal, complemented
9C4×#1203C211yesmaximal, characteristic, complemented
10#91418C211yesmaximal, characteristic, complemented
11#91390C211yesmaximal, characteristic, complemented
12#65946—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×Dic10C2211yescharacteristic, complemented
22C2×#1203C2211yescharacteristic
23C2×#1203C2212yescomplemented
24C2×#1200C2212yescomplemented
25C2×#1200C2212yescomplemented
26#6769—24no—
27C5×#217C2212yescomplemented
28C5×C4⋊Dic3C2211yescharacteristic, complemented
29C3×C22⋊Dic5C2212yescomplemented
30C3×C22⋊Dic5C2212yescomplemented
31C3×C4×Dic5C2211yescharacteristic, complemented
32Dic3⋊Dic5C2212yescomplemented
33#6668C2211yescharacteristic, complemented
34#6667C2211yescharacteristic
35#6667C2212yes—
36#6667C2212yescomplemented
37(C2×D3)⋊Dic5C2211yescharacteristic, complemented
38#5676—51nomaximal, Hall
39C2×D4×D5—31no—
40C2×#709—31no—
41C5×#121—31no—
42#4084—32no—
43#4083—32no—
44#4082D311yescharacteristic, complemented
45C4×C22⋊D5—31no—
46#4043—31no—
47#4039—31no—
48#4029—32no—
49D5×C22⋊C4—32no—
50C23×C3×C5C2312yes—
51C22×D15—24no—
52C22×D15—24no—
53C22×D15—42no—
54C22×D15C2312yes—
55C22×D15C2311yescharacteristic
56C22×C5×D3C2311yescharacteristic
57C3×C5×D4—22no—
58C2×C3×C4×C5C2311yescharacteristic
59#1220—24no—
60C2×Dic15C2311yescharacteristic
61D60D414yescomplemented
62C4×D15—22no—
63C5×#67—42no—
64C2×C5×Dic3C2312yes—
65C2×C5×Dic3C2311yescharacteristic
66C4×C5×D3—22no—
67C2×C3×Dic5C2311yescharacteristic
68C2×C3×Dic5C2312yes—
69C2×C3×Dic5C2311yescharacteristic
70C3×Dic10—22no—
71#1203—24no—
72#1203—22no—
73#1200—24no—
74#1200—42no—
75C2×#239—51no—
76C2×D3×D4—51no—
77C3×#124—51no—
78#946—51no—
79#938D511yescharacteristic, complemented
80#884—52no—
81#883—52no—
82#882—52no—
83D3×C22⋊C4—52no—
84C4×D12—51no—
85C4×D12—51no—
86C23×D5—32no—
87C2×C5×D4—32no—
88C2×C5×D4C2×D311yescharacteristic
89C22×C4×C5—31no—
90C2×C22⋊D5—32no—
91C2×C22⋊D5—32no—
92C2×C22⋊D5—31no—
93D4×D5—64no—
94#709—64no—
95C2×D20—31no—
96C2×C4×D5—31no—
97C2×C4×D5—32no—
98C2×C4×D5—32no—
99C2×Dic10C2×D311yescharacteristic, complemented
100C5×C4⋊C4—31no—
101C5×C22⋊C4—32no—
102C22⋊Dic5—31no—
103C22⋊Dic5C2×D312yescomplemented
104C22⋊Dic5C2×D312yescomplemented
105#685—31no—
106#685—32no—
107#685—32no—
108C4⋊Dic5—32no—
109#683—31no—
110C4×Dic5C2×D311yescharacteristic, complemented
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×C5×D3—21no—
124C2×C5×D3—21no—
125C2×C5×D3—21no—
126C3×C4×C5C2×D412yes—
127Dic15—21no—
128C3×Dic5—22no—
129C3×Dic5—21no—
130C3×Dic5—21no—
131C5×Dic3—21no—
132C5×Dic3—22no—
133C23×D3—52no—
134C2×C3×Q8—51no—
135C2×C3×D4C2×D511yescharacteristic
136C2×#67—52no—
137C2×#67C2×D512yescomplemented
138#239—104no—
139D3×D4—104no—
140C2×D12—51no—
141C2×D12—51no—
142C2×D12—52no—
143C2×C4×D3—51no—
144C2×C4×D3—52no—
145C2×C4×D3—52no—
146C2×C4×D3C2×D511yescharacteristic, complemented
147C3×C22⋊C4—52no—
148C3×C22⋊C4—52no—
149C3×C42—51no—
150#217C2×D512yescomplemented
151(C2×D3)⋊C4—51no—
152(C2×D3)⋊C4—51no—
153(C2×D3)⋊C4—52no—
154(C2×D3)⋊C4—52no—
155C4⋊Dic3—51no—
156C4⋊Dic3C2×D511yescharacteristic, complemented
157Dic3⋊C4—52no—
158C23×C5—31no—
159C23×C5C22×D312yes—
160C22×D5—31no—
161C22×D5—32no—
162C22×D5—64no—
163C22×D5—64no—
164C22×D5—122no—
165C5×D4—22no—
166C5×D4—122no—
167C2×C4×C5—31no—
168C2×C4×C5—32no—
169C2×C4×C5—62no—
170C2×C4×C5C22×D311yescharacteristic
171C22⋊D5—64no—
172C22⋊D5—62no—
173C22⋊D5—64no—
174C2×Dic5—31no—
175C2×Dic5C22×D311yescharacteristic
176C2×Dic5C22×D311yescharacteristic
177C2×Dic5C22×D312yes—
178D20—34no—
179C4×D5—64no—
180C4×D5—62no—
181C4×D5—122no—
182Dic10—22no—
183C2×#41—151no—
184C22×D4—151no—
185#124—51no—
186#123—152no—
187C22⋊Q8—151no—
188#121—31no—
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—
200C5×D3—41no—
201C23×C3C22×D512yes—
202C22×D3—51no—
203C22×D3—52no—
204C22×D3—104no—
205C22×D3—104no—
206C22×D3—202no—
207C22×D3C22×D511yescharacteristic
208C3×Q8—102no—
209C3×D4—22no—
210C2×C3×C4—52no—
211C2×C3×C4—51no—
212C2×C3×C4—51no—
213C2×C3×C4C22×D511yescharacteristic
214#67—42no—
215#67—104no—
216C2×Dic3—51no—
217C2×Dic3C22×D512yes—
218C2×Dic3C22×D511yescharacteristic
219D12—54no—
220D12—102no—
221D12—104no—
222C4×D3—22no—
223C4×D3—104no—
224C4×D3—102no—
225C4×D3—202no—
226C22×C5—22no—
227C22×C5—22no—
228C22×C5—22no—
229C22×C5—61no—
230C22×C5—61no—
231C22×C5—61no—
232C22×C5C23×D311yescharacteristic
233C2×D5—34no—
234C2×D5—32no—
235C2×D5—61no—
236C2×D5—62no—
237C2×D5—62no—
238C2×D5—61no—
239C2×D5—128no—
240C4×C5—61no—
241C4×C5—62no—
242C4×C5D3×D412yes—
243Dic5—22no—
244Dic5—21no—
245Dic5—21no—
246Dic5—61no—
247C24—152no—
248#41—304no—
249C2×Q8—51no—
250C2×D4—32no—
251C2×D4—151no—
252C2×D4—152no—
253C2×D4—152no—
254C2×D4—151no—
255C2×D4—304no—
256C2×D4D3×D511yescharacteristic
257C22×C4—31no—
258C22×C4—152no—
259C22×C4—152no—
260C22×C4—151no—
261C4⋊C4—31no—
262C4⋊C4—151no—
263C4⋊C4—152no—
264C22⋊C4—32no—
265C22⋊C4—52no—
266C22⋊C4—52no—
267C22⋊C4—151no—
268C22⋊C4—152no—
269C22⋊C4—152no—
270C22⋊C4—151no—
271C42—51no—
272C3×C5#54611yescharacteristic, complemented, Hall
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3C23×D511yescharacteristic
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—21no—
280C2×D3—54no—
281C2×D3—52no—
282C2×D3—101no—
283C2×D3—102no—
284C2×D3—101no—
285C2×D3—102no—
286C2×D3—208no—
287C3×C4—102no—
288C3×C4—101no—
289C3×C4—101no—
290C3×C4D4×D512yes—
291Dic3—21no—
292Dic3—22no—
293Dic3—101no—
294C2×C5—42no—
295C2×C5—121no—
296C2×C5C2×D3×D411yescharacteristic
297C2×C5D3×#4111yescharacteristic
298C2×C5#100911yescharacteristic
299D5—62no—
300D5—64no—
301C23—31no—
302C23—151no—
303C23—152no—
304C23—304no—
305C23—304no—
306C23—602no—
307C23C2×D3×D512yes—
308Q8—102no—
309D4—22no—
310D4—122no—
311D4—154no—
312D4—304no—
313D4—302no—
314D4—304no—
315C2×C4—31no—
316C2×C4—32no—
317C2×C4—51no—
318C2×C4—51no—
319C2×C4—52no—
320C2×C4—62no—
321C2×C4—151no—
322C2×C4—304no—
323C2×C4—302no—
324C2×C4—602no—
325C2×C4C2×D3×D511yescharacteristic
326C2×C3—42no—
327C2×C3C2×D4×D511yescharacteristic
328C2×C3#414411yescharacteristic
329C2×C3D5×#4111yescharacteristic
330D3—41no—
331D3—102no—
332D3—104no—
333C5#567611yescharacteristic, complemented, Sylow
334C22—22no—
335C22—22no—
336C22—22no—
337C22—61no—
338C22—61no—
339C22—61no—
340C22—152no—
341C22—154no—
342C22—302no—
343C22—302no—
344C22—301no—
345C22—301no—
346C22—608no—
347C22C22×D3×D511yescharacteristic
348C4—62no—
349C4—61no—
350C4—101no—
351C4—101no—
352C4—102no—
353C4—301no—
354C4#679012yes—
355C3#6594611yescharacteristic, complemented, Sylow
356C2—42no—
357C2—121no—
358C2—302no—
359C2—304no—
360C2#9202411yescharacteristic
361C2#9202711yescharacteristic
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×#6769C2semidirect
3. C5×#938C2semidirect
4. C3×#4082C2semidirect
5. #91573C2semidirect
6. #91562C2semidirect
7. #91561C2semidirect
8. #91557C2semidirect
9. C4×#1203C2semidirect
10. #91418C2semidirect
11. #91390C2semidirect
19. C2×C5×#67C22semidirect
20. C2×C4×C5×D3C22semidirect
21. C2×C3×Dic10C22semidirect
23. C2×#1203C22semidirect
24. C2×#1200C22semidirect
25. C2×#1200C22semidirect
27. C5×#217C22semidirect
28. C5×C4⋊Dic3C22semidirect
29. C3×C22⋊Dic5C22semidirect
30. C3×C22⋊Dic5C22semidirect
31. C3×C4×Dic5C22semidirect
32. Dic3⋊Dic5C22semidirect
33. #6668C22semidirect
36. #6667C22semidirect
37. (C2×D3)⋊Dic5C22semidirect
44. #4082D3semidirect
61. D60D4semidirect
79. #938D5semidirect
99. C2×Dic10C2×D3semidirect
103. C22⋊Dic5C2×D3semidirect
104. C22⋊Dic5C2×D3semidirect
110. C4×Dic5C2×D3semidirect
122. C2×D15#41semidirect
137. C2×#67C2×D5semidirect
146. C2×C4×D3C2×D5semidirect
150. #217C2×D5semidirect
156. C4⋊Dic3C2×D5semidirect
272. C3×C5#546semidirect
333. C5#5676semidirect
355. C3#65946semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D152
C2×#67692
C5×#9382
C3×#40822
#915732
#915622
#915612
#915572
C4×#12032
#914182
#913902
#659463
#56765