Finite Group Databasebeta

← #11753693

Subgroups — #11753693

Order 960.

624 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6773C211yesmaximal, characteristic, complemented
2C2×#6766C211yesmaximal, characteristic, complemented
3#91784C211yesmaximal, characteristic, complemented
4C5×#887C211yesmaximal, characteristic, complemented
5C3×#4041C211yesmaximal, characteristic, complemented
6#91459C212yesmaximal, complemented
7#91458C212yesmaximal, complemented
8#91458C211yesmaximal, characteristic, complemented
9C4×#1202C211yesmaximal, characteristic, complemented
10#91409C212yesmaximal, complemented
11#91395C212yesmaximal, complemented
12#65911—31nomaximal, Hall
13C2×D60C2211yescharacteristic
14C2×D60C2212yes—
15C2×C4×D15C2211yescharacteristic
16C2×C5×Dic6C2211yescharacteristic
17C2×C3×D20C2211yescharacteristic
18C2×C3×D20C2212yes—
19C2×C3×C4×D5C2211yescharacteristic
20C2×#1202C2212yes—
21C2×#1202C2212yes—
22C2×#1202C2211yescharacteristic
23C2×#1202C2212yes—
24C2×#1202C2212yes—
25C2×#1200C2212yes—
26C2×D5×Dic3C2212yes—
27#6773—24no—
28#6766—24no—
29C3×C5×C4⋊C4C2211yescharacteristic
30#6714C2212yes—
31C5×Dic3⋊C4C2212yes—
32C4×C5×Dic3C2212yes—
33C4×C5×Dic3C2211yescharacteristic
34C3×#685C2212yes—
35Dic3⋊Dic5C2211yescharacteristic
36#6667C2211yescharacteristic
37#6665C2211yescharacteristic
38#5599—51nomaximal, Hall
39C2×#713—31no—
40C2×#713—31no—
41C4×C5×Q8—31no—
42#4041—32no—
43#4041—31no—
44#4041—32no—
45#4041D311yescharacteristic, complemented
46#4020—32no—
47#4020—31no—
48C4×D20—31no—
49C4×D20—32no—
50C22×D15C2311yescharacteristic
51C22×D15C2312yes—
52C22×C3×D5C2311yescharacteristic
53C22×C3×D5C2312yes—
54C2×C3×C4×C5C2312yes—
55C2×C3×C4×C5C2311yescharacteristic
56C2×Dic15C2311yescharacteristic
57D60—22no—
58D60—42no—
59C4×D15—22no—
60C2×C5×Dic3C2311yescharacteristic
61C2×C5×Dic3C2312yes—
62C2×C5×Dic3C2311yescharacteristic
63C5×Dic6D414yescomplemented
64C2×C3×Dic5C2311yescharacteristic
65C3×D20—22no—
66C3×D20—42no—
67C3×C4×D5—22no—
68#1202—24no—
69#1202—24no—
70#1202—24no—
71#1202—22no—
72#1202—42no—
73#1200—24no—
74D5×Dic3—24no—
75C2×#237—51no—
76C2×#235—51no—
77C3×#121—51no—
78#940—52no—
79#940—51no—
80#939—52no—
81C4×#67—51no—
82#895—51no—
83#887D511yescharacteristic, complemented
84#884—52no—
85#67⋊C4—52no—
86C2×C5×Q8—31no—
87#713—64no—
88#713—64no—
89C2×D20—32no—
90C2×D20—32no—
91C2×D20—31no—
92C2×D20—32no—
93C2×D20—31no—
94C2×D20—32no—
95C2×D20—32no—
96C2×D20C2×D311yescharacteristic
97C2×D20C2×D312yes—
98C2×C4×D5—32no—
99C2×C4×D5—31no—
100C2×C4×D5—32no—
101C2×C4×D5C2×D311yescharacteristic
102C5×C4⋊C4—32no—
103C5×C4⋊C4C2×D311yescharacteristic
104C42×C5—32no—
105C42×C5—31no—
106#685—31no—
107#685—32no—
108#685—31no—
109#685C2×D312yes—
110C4⋊Dic5—31no—
111#550—151noSylow
112C22×C3×C5C2411yescharacteristic
113C2×D15—22no—
114C2×D15—21no—
115C2×D15—21no—
116C2×D15—22no—
117C2×D15—22no—
118C2×D15—21no—
119C2×C3×D5—22no—
120C2×C3×D5—21no—
121C2×C3×D5—21no—
122C2×C3×D5—22no—
123C2×C3×D5—22no—
124C2×C3×D5—21no—
125C3×C4×C5—22no—
126C3×C4×C5C2×D412yes—
127Dic15—21no—
128C3×Dic5—21no—
129C5×Dic3—21no—
130C5×Dic3C2×D414yes—
131C5×Dic3#4112yescomplemented
132C2×C3×D4—51no—
133C2×C3×D4—52no—
134C22×C3×C4—51no—
135C2×#67—52no—
136C2×#67—52no—
137C2×#67—51no—
138C2×#67—52no—
139C2×#67—52no—
140C22×Dic3—52no—
141#237—104no—
142#235—104no—
143C2×D12—51no—
144C2×D12—52no—
145C2×C4×D3—51no—
146C2×C4×D3—52no—
147C2×Dic6C2×D511yescharacteristic
148C3×C4⋊C4C2×D511yescharacteristic
149C3×C22⋊C4—52no—
150#217—51no—
151(C2×D3)⋊C4—52no—
152(C2×D3)⋊C4—51no—
153Dic3⋊C4—51no—
154Dic3⋊C4C2×D512yes—
155C4×Dic3C2×D512yes—
156C4×Dic3C2×D511yescharacteristic
157C22×D5—32no—
158C22×D5—31no—
159C22×D5C22×D311yescharacteristic
160C22×D5C22×D312yes—
161C5×Q8—34no—
162C2×C4×C5—31no—
163C2×C4×C5—32no—
164C2×C4×C5—31no—
165C2×C4×C5C22×D312yes—
166C2×C4×C5C22×D311yescharacteristic
167C2×Dic5—31no—
168C2×Dic5C22×D311yescharacteristic
169D20—22no—
170D20—42no—
171D20—62no—
172D20—64no—
173D20—64no—
174D20—64no—
175D20—62no—
176D20—122no—
177D20—122no—
178C4×D5—22no—
179C4×D5—64no—
180C4×D5—64no—
181C4×D5—62no—
182C2×#41—151no—
183C2×#41—151no—
184#127—151no—
185#127—152no—
186#121—51no—
187#121—152no—
188#121—152no—
189#121—151no—
190C4×Q8—31no—
191C4×D4—152no—
192C4×D4—151no—
193C2×C3×C5C22×D411yescharacteristic
194C2×C3×C5C2×#4111yescharacteristic
195C2×C3×C5H(5,𝔽2)11yescharacteristic
196D15—42no—
197D15—41no—
198C3×D5—42no—
199C3×D5—41no—
200C23×C3—51no—
201C23×C3—52no—
202C22×D3—52no—
203C22×D3—51no—
204C3×D4—102no—
205C3×D4—202no—
206C2×C3×C4—51no—
207C2×C3×C4—102no—
208C2×C3×C4C22×D512yes—
209C2×C3×C4C22×D511yescharacteristic
210#67—104no—
211#67—104no—
212#67—104no—
213#67—102no—
214#67—202no—
215C2×Dic3—51no—
216C2×Dic3—104no—
217C2×Dic3C22×D511yescharacteristic
218C2×Dic3C22×D512yes—
219C2×Dic3C22×D511yescharacteristic
220D12—102no—
221D12—202no—
222C4×D3—102no—
223C4×D3—104no—
224Dic6D2014yescomplemented
225C22×C5C23×D311yescharacteristic
226C2×D5—21no—
227C2×D5—22no—
228C2×D5—21no—
229C2×D5—22no—
230C2×D5—21no—
231C2×D5—22no—
232C2×D5—62no—
233C2×D5—61no—
234C2×D5—62no—
235C2×D5—61no—
236C2×D5—61no—
237C2×D5—62no—
238C4×C5—22no—
239C4×C5—34no—
240C4×C5—32no—
241C4×C5—61no—
242C4×C5D3×D412yes—
243Dic5—21no—
244Dic5—61no—
245#41—304no—
246#41—304no—
247C2×Q8—31no—
248C2×D4—51no—
249C2×D4—52no—
250C2×D4—152no—
251C2×D4—151no—
252C2×D4—152no—
253C2×D4—152no—
254C2×D4—152no—
255C2×D4—151no—
256C2×D4—152no—
257C22×C4—51no—
258C22×C4—152no—
259C22×C4—152no—
260C22×C4—151no—
261C4⋊C4—32no—
262C4⋊C4—151no—
263C4⋊C4D3×D511yescharacteristic
264C22⋊C4—52no—
265C22⋊C4—151no—
266C22⋊C4—151no—
267C22⋊C4—152no—
268C42—32no—
269C42—31no—
270C3×C5#55011yescharacteristic, complemented, Hall
271C22×C3—102no—
272C22×C3—101no—
273C22×C3—101no—
274C22×C3—102no—
275C22×C3—102no—
276C22×C3—101no—
277C22×C3C23×D511yescharacteristic
278C2×D3—102no—
279C2×D3—101no—
280C2×D3—102no—
281C2×D3—101no—
282C2×D3—101no—
283C2×D3—102no—
284C3×C4—22no—
285C3×C4—101no—
286C3×C4C2×D2012yes—
287Dic3—21no—
288Dic3—101no—
289Dic3C2×D2014yes—
290Dic3#71312yescomplemented
291C2×C5C2×D3×D411yescharacteristic
292C2×C5#100411yescharacteristic
293C2×C5D3×#4111yescharacteristic
294D5—41no—
295D5—42no—
296D5—122no—
297D5—121no—
298C23—52no—
299C23—51no—
300C23—151no—
301C23—152no—
302Q8—34no—
303D4—102no—
304D4—202no—
305D4—304no—
306D4—304no—
307D4—304no—
308D4—302no—
309D4—302no—
310D4—602no—
311D4—602no—
312C2×C4—31no—
313C2×C4—32no—
314C2×C4—31no—
315C2×C4—51no—
316C2×C4—102no—
317C2×C4—151no—
318C2×C4—304no—
319C2×C4—304no—
320C2×C4—302no—
321C2×C4C2×D3×D512yes—
322C2×C4C2×D3×D511yescharacteristic
323C2×C3—202no—
324C2×C3—201no—
325C2×C3C22×D2011yescharacteristic
326C2×C3C2×#71311yescharacteristic
327C2×C3#414911yescharacteristic
328D3—202no—
329D3—201no—
330C5#559911yescharacteristic, complemented, Sylow
331C22—101no—
332C22—102no—
333C22—101no—
334C22—102no—
335C22—102no—
336C22—101no—
337C22—302no—
338C22—301no—
339C22—301no—
340C22—302no—
341C22—301no—
342C22—302no—
343C22C22×D3×D511yescharacteristic
344C4—22no—
345C4—34no—
346C4—32no—
347C4—61no—
348C4—101no—
349C4—301no—
350C4D3×D2012yes—
351C3#6591111yescharacteristic, complemented, Sylow
352C2—201no—
353C2—202no—
354C2—602no—
355C2—601no—
356C2C2×D3×D2011yescharacteristic
357C2#9202611yescharacteristic
358C2D3×#71311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C4⋊C4C22
hypercenterC4⋊C4D3×D5
hypocenterC3×C5#550
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#6773C2semidirect
2. C2×#6766C2semidirect
3. #91784C2semidirect
4. C5×#887C2semidirect
5. C3×#4041C2semidirect
6. #91459C2semidirect
7. #91458C2semidirect
8. #91458C2semidirect
9. C4×#1202C2semidirect
10. #91409C2semidirect
11. #91395C2semidirect
45. #4041D3semidirect
63. C5×Dic6D4semidirect
83. #887D5semidirect
131. C5×Dic3#41semidirect
224. Dic6D20semidirect
270. C3×C5#550semidirect
290. Dic3#713semidirect
330. C5#5599semidirect
351. C3#65911semidirect

Maximal subgroups

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

subgroupindex
C2×#67732
C2×#67662
#917842
C5×#8872
C3×#40412
#914592
#914582
C4×#12022
#914092
#913952
#659113
#55995