Finite Group Databasebeta

← #11754956

Subgroups — #11754956

Order 960.

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

normal subgroups only

This group has 624 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×#6819C211yesmaximal, characteristic, complemented
2C2×#6772C211yesmaximal, characteristic, complemented
3C5×#940C211yesmaximal, characteristic, complemented
4C3×#4084C211yesmaximal, characteristic, complemented
5#91571C212yesmaximal, complemented
6#91567C212yesmaximal, complemented
7#67⋊Dic5C212yesmaximal, complemented
8#91558C212yesmaximal, complemented
9#91466C211yesmaximal, characteristic, complemented
10C4×#1202C211yesmaximal, characteristic, complemented
11#91340C211yesmaximal, characteristic, complemented
12#66052—31nomaximal, Hall
13C2×C3×C5×D4C2211yescharacteristic
14C2×#1220C2212yes—
15C22×Dic15C2212yes—
16#6819—24no—
17C2×C4×D15C2211yescharacteristic
18C2×Dic30C2211yescharacteristic
19C2×C5×#67C2212yes—
20C2×C5×#67C2212yes—
21C2×C5×D12C2211yescharacteristic
22C2×C3×C22⋊D5C2212yes—
23C2×C3×C4×D5C2211yescharacteristic
24C2×#1203C2212yes—
25C2×#1202C2211yescharacteristic
26C2×#1201C2212yes—
27C2×D3×Dic5C2212yes—
28#6772—24no—
29C4×C5×Dic3C2211yescharacteristic
30C3×C22⋊Dic5C2212yes—
31C3×C4⋊Dic5C2211yescharacteristic
32Dic3⋊Dic5C2211yescharacteristic
33#6669C2212yes—
34#6667C2211yescharacteristic
35(C2×D3)⋊Dic5C2212yes—
36#6665C2211yescharacteristic
37Dic3×Dic5C2212yes—
38#5599—51nomaximal, Hall
39C2×#711—31no—
40C2×#711—31no—
41C5×#127—31no—
42#4086—32no—
43#4085—32no—
44#4085—32no—
45#4084—31no—
46#4084D311yescharacteristic, complemented
47D4×Dic5—32no—
48C4×D20—31no—
49C4×Dic10—31no—
50C23×C3×C5C2312yes—
51C22×D15C2311yescharacteristic
52C22×C5×D3C2312yes—
53C22×C3×D5C2311yescharacteristic
54C3×C5×D4—22no—
55C2×C3×C4×C5C2311yescharacteristic
56#1220—24no—
57C2×Dic15—24no—
58C2×Dic15C2311yescharacteristic
59C2×Dic15C2312yes—
60C4×D15—22no—
61Dic30D414yescomplemented
62C5×#67—24no—
63C5×#67—42no—
64C2×C5×Dic3C2311yescharacteristic
65C2×C5×Dic3C2311yescharacteristic
66C5×D12—22no—
67C3×C22⋊D5—42no—
68C2×C3×Dic5C2312yes—
69C2×C3×Dic5C2311yescharacteristic
70C3×C4×D5—22no—
71#1203—42no—
72#1202—22no—
73#1201—24no—
74D3×Dic5—24no—
75C2×#237—51no—
76C2×#235—51no—
77C3×#121—51no—
78#940—52no—
79#940D511yescharacteristic, complemented
80#939—52no—
81C4×#67—51no—
82#895—51no—
83#887—51no—
84#884—52no—
85#67⋊C4—52no—
86C2×C5×D4—32no—
87C2×C5×D4—32no—
88C2×C5×D4—31no—
89C2×C5×D4C2×D311yescharacteristic
90C2×C22⋊D5—32no—
91C2×C22⋊D5—32no—
92C2×C22⋊D5—32no—
93C2×C22⋊D5C2×D312yes—
94C22×Dic5—32no—
95C22×Dic5—32no—
96#711—64no—
97#711—64no—
98C2×D20—31no—
99C2×C4×D5—31no—
100C2×C4×D5C2×D311yescharacteristic
101C2×Dic10—31no—
102C42×C5—31no—
103C22⋊Dic5—32no—
104C22⋊Dic5C2×D312yes—
105#685—31no—
106#685—31no—
107C4⋊Dic5—31no—
108C4⋊Dic5C2×D311yescharacteristic
109#683—32no—
110C4×Dic5—32no—
111#550—151noSylow
112C22×C3×C5—22no—
113C22×C3×C5—22no—
114C22×C3×C5—22no—
115C22×C3×C5C2411yescharacteristic
116C2×D15—21no—
117C2×D15—21no—
118C2×D15—21no—
119C2×C5×D3—22no—
120C2×C5×D3—22no—
121C2×C5×D3—22no—
122C2×C3×D5—21no—
123C2×C3×D5—21no—
124C2×C3×D5—21no—
125C3×C4×C5C2×D412yes—
126Dic15—21no—
127Dic15C2×D414yes—
128C3×Dic5—21no—
129C3×Dic5—22no—
130C5×Dic3—21no—
131C5×Dic3#4112yescomplemented
132C2×C3×D4—52no—
133C2×C3×D4C2×D511yescharacteristic
134C22×C3×C4—51no—
135C2×#67—52no—
136C2×#67—52no—
137C2×#67—51no—
138C2×#67C2×D512yes—
139C2×#67C2×D512yes—
140C22×Dic3—52no—
141#237—104no—
142#235—104no—
143C2×D12—52no—
144C2×D12C2×D511yescharacteristic
145C2×C4×D3—52no—
146C2×C4×D3—51no—
147C2×Dic6—51no—
148C3×C4⋊C4—51no—
149C3×C22⋊C4—52no—
150#217—51no—
151(C2×D3)⋊C4—52no—
152(C2×D3)⋊C4—51no—
153Dic3⋊C4—52no—
154Dic3⋊C4—51no—
155C4×Dic3—52no—
156C4×Dic3C2×D511yescharacteristic
157C23×C5—32no—
158C23×C5C22×D312yes—
159C22×D5—31no—
160C22×D5C22×D311yescharacteristic
161C5×D4—22no—
162C5×D4—64no—
163C5×D4—62no—
164C5×D4—122no—
165C2×C4×C5—31no—
166C2×C4×C5—31no—
167C2×C4×C5C22×D311yescharacteristic
168C22⋊D5—42no—
169C22⋊D5—64no—
170C22⋊D5—64no—
171C22⋊D5—122no—
172C2×Dic5—32no—
173C2×Dic5—31no—
174C2×Dic5—64no—
175C2×Dic5—64no—
176C2×Dic5C22×D312yes—
177C2×Dic5C22×D311yescharacteristic
178D20—62no—
179C4×D5—22no—
180C4×D5—62no—
181Dic10—34no—
182C2×#41—151no—
183C2×#41—151no—
184#127—31no—
185#127—152no—
186#121—51no—
187#121—152no—
188#121—151no—
189#121—152no—
190C4×Q8—151no—
191C4×D4—151no—
192C4×D4—152no—
193C2×C3×C5—42no—
194C2×C3×C5C22×D411yescharacteristic
195C2×C3×C5C2×#4111yescharacteristic
196C2×C3×C5H(5,𝔽2)11yescharacteristic
197D15—41no—
198C3×D5—41no—
199C5×D3—42no—
200C23×C3—51no—
201C23×C3C22×D512yes—
202C22×D3—51no—
203C22×D3C22×D512yes—
204C3×D4—22no—
205C3×D4—202no—
206C2×C3×C4—52no—
207C2×C3×C4—51no—
208C2×C3×C4—102no—
209C2×C3×C4C22×D511yescharacteristic
210#67—24no—
211#67—42no—
212#67—104no—
213#67—104no—
214#67—102no—
215C2×Dic3—52no—
216C2×Dic3—51no—
217C2×Dic3—104no—
218C2×Dic3C22×D511yescharacteristic
219C2×Dic3C22×D511yescharacteristic
220D12—22no—
221D12—202no—
222C4×D3—104no—
223C4×D3—102no—
224Dic6—54no—
225C22×C5—22no—
226C22×C5—22no—
227C22×C5—22no—
228C22×C5—62no—
229C22×C5—62no—
230C22×C5—62no—
231C22×C5C23×D311yescharacteristic
232C2×D5—21no—
233C2×D5—21no—
234C2×D5—21no—
235C2×D5—61no—
236C2×D5—61no—
237C2×D5—61no—
238C4×C5—32no—
239C4×C5—61no—
240C4×C5D3×D412yes—
241Dic5—21no—
242Dic5—22no—
243Dic5—34no—
244Dic5—61no—
245#41—304no—
246#41—304no—
247C2×Q8—151no—
248C2×D4—32no—
249C2×D4—32no—
250C2×D4—31no—
251C2×D4—52no—
252C2×D4—151no—
253C2×D4—152no—
254C2×D4—152no—
255C2×D4—152no—
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—52no—
265C22⋊C4—151no—
266C22⋊C4—151no—
267C22⋊C4—152no—
268C42—31no—
269C42—152no—
270C3×C5#55011yescharacteristic, complemented, Hall
271C22×C3—22no—
272C22×C3—22no—
273C22×C3—22no—
274C22×C3—101no—
275C22×C3—101no—
276C22×C3—101no—
277C22×C3C23×D511yescharacteristic
278C2×D3—22no—
279C2×D3—22no—
280C2×D3—22no—
281C2×D3—101no—
282C2×D3—101no—
283C2×D3—101no—
284C3×C4—101no—
285C3×C4—102no—
286C3×C4D4×D512yes—
287Dic3—21no—
288Dic3—54no—
289Dic3—101no—
290Dic3#71112yescomplemented
291C2×C5—42no—
292C2×C5—122no—
293C2×C5C2×D3×D411yescharacteristic
294C2×C5#100411yescharacteristic
295C2×C5D3×#4111yescharacteristic
296D5—41no—
297D5—121no—
298C23—32no—
299C23—51no—
300C23—151no—
301C23C2×D3×D512yes—
302Q8—154no—
303D4—22no—
304D4—64no—
305D4—62no—
306D4—122no—
307D4—202no—
308D4—302no—
309D4—304no—
310D4—304no—
311D4—602no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—52no—
315C2×C4—51no—
316C2×C4—102no—
317C2×C4—152no—
318C2×C4—151no—
319C2×C4—304no—
320C2×C4—304no—
321C2×C4—302no—
322C2×C4C2×D3×D511yescharacteristic
323C2×C3—42no—
324C2×C3—201no—
325C2×C3C2×D4×D511yescharacteristic
326C2×C3C2×#71111yescharacteristic
327C2×C3#414411yescharacteristic
328D3—42no—
329D3—201no—
330C5#559911yescharacteristic, complemented, Sylow
331C22—22no—
332C22—22no—
333C22—22no—
334C22—62no—
335C22—62no—
336C22—62no—
337C22—101no—
338C22—101no—
339C22—101no—
340C22—301no—
341C22—301no—
342C22—301no—
343C22C22×D3×D511yescharacteristic
344C4—32no—
345C4—61no—
346C4—102no—
347C4—101no—
348C4—154no—
349C4—301no—
350C4#679012yes—
351C3#6605211yescharacteristic, complemented, Sylow
352C2—42no—
353C2—122no—
354C2—201no—
355C2—601no—
356C2D3×#71111yescharacteristic
357C2#9202511yescharacteristic
358C2C2×#679011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#550
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×#6819C2semidirect
2. C2×#6772C2semidirect
3. C5×#940C2semidirect
4. C3×#4084C2semidirect
5. #91571C2semidirect
6. #91567C2semidirect
7. #67⋊Dic5C2semidirect
8. #91558C2semidirect
9. #91466C2semidirect
10. C4×#1202C2semidirect
11. #91340C2semidirect
46. #4084D3semidirect
61. Dic30D4semidirect
79. #940D5semidirect
131. C5×Dic3#41semidirect
270. C3×C5#550semidirect
290. Dic3#711semidirect
330. C5#5599semidirect
351. C3#66052semidirect

Maximal subgroups

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

subgroupindex
C2×#68192
C2×#67722
C5×#9402
C3×#40842
#915712
#915672
#67⋊Dic52
#915582
#914662
C4×#12022
#913402
#660523
#55995