Finite Group Databasebeta

← #11753700

Subgroups — #11753700

Order 960.

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

normal subgroups only

This group has 624 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×#6772C211yesmaximal, characteristic, complemented
2C2×#6765C211yesmaximal, characteristic, complemented
3#91776C211yesmaximal, characteristic, complemented
4C5×#895C211yesmaximal, characteristic, complemented
5C3×#4041C211yesmaximal, characteristic, complemented
6#91463C211yesmaximal, characteristic, complemented
7#91463C212yesmaximal, complemented
8#91453C212yesmaximal, complemented
9#91442C212yesmaximal, complemented
10C4×#1201C211yesmaximal, characteristic, complemented
11#91408C212yesmaximal, complemented
12#65933—31nomaximal, Hall
13C2×Dic30C2211yescharacteristic
14C2×C5×D12C2211yescharacteristic
15C2×C5×D12C2212yes—
16C2×C4×C5×D3C2211yescharacteristic
17C2×C3×D20C2211yescharacteristic
18C2×C3×D20C2212yes—
19C2×C3×C4×D5C2211yescharacteristic
20C2×#1201C2212yes—
21C2×#1201C2211yescharacteristic
22C2×#1201C2212yes—
23C2×#1201C2212yes—
24C2×#1201C2212yes—
25C2×D3×Dic5C2212yes—
26C2×D5×Dic3C2212yes—
27#6772—24no—
28#6765—24no—
29C3×C5×C4⋊C4C2211yescharacteristic
30#6712C2212yes—
31C4×Dic15C2211yescharacteristic
32C4×Dic15C2212yes—
33C5×(C2×D3)⋊C4C2212yes—
34C3×#685C2212yes—
35#6669C2211yescharacteristic
36(C2×D3)⋊Dic5C2211yescharacteristic
37#6665C2211yescharacteristic
38#5599—51nomaximal, Hall
39C2×#711—31no—
40C2×#709—31no—
41C5×#121—31no—
42#4086—32no—
43#4086—31no—
44#4085—32no—
45C4×C22⋊D5—31no—
46#4041D311yescharacteristic, complemented
47#4033—31no—
48#4030—32no—
49#4027—32no—
50C22×C5×D3C2311yescharacteristic
51C22×C5×D3C2312yes—
52C22×C3×D5C2311yescharacteristic
53C22×C3×D5C2312yes—
54C2×C3×C4×C5C2312yes—
55C2×C3×C4×C5C2311yescharacteristic
56C2×Dic15C2311yescharacteristic
57C2×Dic15C2312yes—
58C2×Dic15C2311yescharacteristic
59Dic30D414yescomplemented
60C2×C5×Dic3C2311yescharacteristic
61C5×D12—22no—
62C5×D12—42no—
63C4×C5×D3—22no—
64C2×C3×Dic5C2311yescharacteristic
65C3×D20—22no—
66C3×D20—42no—
67C3×C4×D5—22no—
68#1201—24no—
69#1201—24no—
70#1201—22no—
71#1201—24no—
72#1201—42no—
73D3×Dic5—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#895D511yescharacteristic, complemented
83#887—51no—
84#884—52no—
85#67⋊C4—52no—
86C2×C5×D4—31no—
87C2×C5×D4—32no—
88C22×C4×C5—31no—
89C2×C22⋊D5—32no—
90C2×C22⋊D5—32no—
91C2×C22⋊D5—32no—
92C2×C22⋊D5—31no—
93C2×C22⋊D5—32no—
94C22×Dic5—32no—
95#711—64no—
96#709—64no—
97C2×D20C2×D311yescharacteristic
98C2×D20C2×D312yes—
99C2×C4×D5—32no—
100C2×C4×D5C2×D311yescharacteristic
101C2×Dic10—31no—
102C5×C4⋊C4C2×D311yescharacteristic
103C5×C22⋊C4—32no—
104C22⋊Dic5—31no—
105#685—31no—
106#685C2×D312yes—
107#683—32no—
108#683—31no—
109C4×Dic5—32no—
110C4×Dic5—31no—
111#550—151noSylow
112C22×C3×C5C2411yescharacteristic
113C2×C5×D3—22no—
114C2×C5×D3—21no—
115C2×C5×D3—21no—
116C2×C5×D3—22no—
117C2×C5×D3—22no—
118C2×C5×D3—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—
128Dic15C2×D414yes—
129Dic15#4112yescomplemented
130C3×Dic5—21no—
131C5×Dic3—21no—
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×D12C2×D511yescharacteristic
144C2×D12C2×D512yes—
145C2×C4×D3—52no—
146C2×C4×D3C2×D511yescharacteristic
147C2×Dic6—51no—
148C3×C4⋊C4C2×D511yescharacteristic
149C3×C22⋊C4—52no—
150#217—51no—
151(C2×D3)⋊C4—51no—
152(C2×D3)⋊C4C2×D512yes—
153Dic3⋊C4—52no—
154Dic3⋊C4—51no—
155C4×Dic3—52no—
156C4×Dic3—51no—
157C23×C5—31no—
158C23×C5—32no—
159C22×D5C22×D311yescharacteristic
160C22×D5C22×D312yes—
161C5×D4—62no—
162C5×D4—122no—
163C2×C4×C5—31no—
164C2×C4×C5—62no—
165C2×C4×C5C22×D312yes—
166C2×C4×C5C22×D311yescharacteristic
167C22⋊D5—64no—
168C22⋊D5—64no—
169C22⋊D5—64no—
170C22⋊D5—62no—
171C22⋊D5—122no—
172C2×Dic5—31no—
173C2×Dic5—32no—
174C2×Dic5—31no—
175C2×Dic5—64no—
176C2×Dic5C22×D311yescharacteristic
177D20—22no—
178D20—42no—
179C4×D5—22no—
180C4×D5—64no—
181Dic10—34no—
182C2×#41—151no—
183C2×#41—151no—
184#127—151no—
185#127—152no—
186#121—31no—
187#121—51no—
188#121—152no—
189#121—152no—
190C4×Q8—151no—
191C4×D4—152no—
192C4×D4—151no—
193C2×C3×C5C22×D411yescharacteristic
194C2×C3×C5C2×#4111yescharacteristic
195C2×C3×C5H(5,𝔽2)11yescharacteristic
196C3×D5—42no—
197C3×D5—41no—
198C5×D3—42no—
199C5×D3—41no—
200C23×C3—51no—
201C23×C3—52no—
202C22×D3C22×D511yescharacteristic
203C22×D3C22×D512yes—
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—52no—
217C2×Dic3—51no—
218C2×Dic3—104no—
219C2×Dic3C22×D511yescharacteristic
220D12—22no—
221D12—42no—
222C4×D3—22no—
223C4×D3—104no—
224Dic6—54no—
225C22×C5—62no—
226C22×C5—61no—
227C22×C5—62no—
228C22×C5—61no—
229C22×C5—61no—
230C22×C5—62no—
231C22×C5C23×D311yescharacteristic
232C2×D5—21no—
233C2×D5—22no—
234C2×D5—21no—
235C2×D5—22no—
236C2×D5—21no—
237C2×D5—22no—
238C4×C5—22no—
239C4×C5—61no—
240C4×C5D3×D412yes—
241Dic5—21no—
242Dic5—34no—
243Dic5—32no—
244Dic5—61no—
245#41—304no—
246#41—304no—
247C2×Q8—151no—
248C2×D4—31no—
249C2×D4—32no—
250C2×D4—51no—
251C2×D4—52no—
252C2×D4—152no—
253C2×D4—151no—
254C2×D4—152no—
255C2×D4—152no—
256C2×D4—152no—
257C22×C4—31no—
258C22×C4—51no—
259C22×C4—152no—
260C22×C4—152no—
261C4⋊C4—151no—
262C4⋊C4—152no—
263C4⋊C4D3×D511yescharacteristic
264C22⋊C4—32no—
265C22⋊C4—52no—
266C22⋊C4—151no—
267C22⋊C4—151no—
268C42—152no—
269C42—151no—
270C3×C5#55011yescharacteristic, complemented, Hall
271C22×C3—102no—
272C22×C3—101no—
273C22×C3—102no—
274C22×C3—101no—
275C22×C3—101no—
276C22×C3—102no—
277C22×C3C23×D511yescharacteristic
278C2×D3—21no—
279C2×D3—22no—
280C2×D3—21no—
281C2×D3—22no—
282C2×D3—21no—
283C2×D3—22no—
284C3×C4—22no—
285C3×C4—101no—
286C3×C4D4×D512yes—
287Dic3—21no—
288Dic3—54no—
289Dic3—52no—
290Dic3—101no—
291C2×C5—122no—
292C2×C5—121no—
293C2×C5C2×D3×D411yescharacteristic
294C2×C5#100411yescharacteristic
295C2×C5D3×#4111yescharacteristic
296D5—41no—
297D5—42no—
298C23—32no—
299C23—31no—
300C23—52no—
301C23—51no—
302Q8—154no—
303D4—62no—
304D4—102no—
305D4—122no—
306D4—202no—
307D4—304no—
308D4—304no—
309D4—304no—
310D4—302no—
311D4—602no—
312C2×C4—31no—
313C2×C4—51no—
314C2×C4—62no—
315C2×C4—102no—
316C2×C4—151no—
317C2×C4—152no—
318C2×C4—151no—
319C2×C4—304no—
320C2×C4—304no—
321C2×C4C2×D3×D512yes—
322C2×C4C2×D3×D511yescharacteristic
323C2×C3—202no—
324C2×C3—201no—
325C2×C3C2×D4×D511yescharacteristic
326C2×C3#414411yescharacteristic
327C2×C3D5×#4111yescharacteristic
328D3—41no—
329D3—42no—
330C5#559911yescharacteristic, complemented, Sylow
331C22—61no—
332C22—62no—
333C22—61no—
334C22—62no—
335C22—62no—
336C22—61no—
337C22—101no—
338C22—102no—
339C22—101no—
340C22—102no—
341C22—102no—
342C22—101no—
343C22C22×D3×D511yescharacteristic
344C4—22no—
345C4—61no—
346C4—101no—
347C4—154no—
348C4—152no—
349C4—301no—
350C4#677712yes—
351C3#6593311yescharacteristic, complemented, Sylow
352C2—121no—
353C2—122no—
354C2—201no—
355C2—202no—
356C2C2×#677711yescharacteristic
357C2#9202511yescharacteristic
358C2#9203411yescharacteristic
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×#6772C2semidirect
2. C2×#6765C2semidirect
3. #91776C2semidirect
4. C5×#895C2semidirect
5. C3×#4041C2semidirect
6. #91463C2semidirect
7. #91463C2semidirect
8. #91453C2semidirect
9. #91442C2semidirect
10. C4×#1201C2semidirect
11. #91408C2semidirect
46. #4041D3semidirect
59. Dic30D4semidirect
82. #895D5semidirect
129. Dic15#41semidirect
270. C3×C5#550semidirect
330. C5#5599semidirect
351. C3#65933semidirect

Maximal subgroups

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

subgroupindex
C2×#67722
C2×#67652
#917762
C5×#8952
C3×#40412
#914632
#914532
#914422
C4×#12012
#914082
#659333
#55995