Finite Group Databasebeta

← #11754978

Subgroups — #11754978

Order 960.

472 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 130 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
1D4×Dic15C211yesmaximal, characteristic, complemented
2C5×#938C211yesmaximal, characteristic, complemented
3C3×#4084C211yesmaximal, characteristic, complemented
4#91571C212yesmaximal, complemented
5#91534C212yesmaximal, complemented
6#91532C212yesmaximal, complemented
7#91525C212yesmaximal, complemented
8#91470C211yesmaximal, characteristic, complemented
9C4×#1201C211yesmaximal, characteristic, complemented
10#91358C211yesmaximal, characteristic, complemented
11#91338C211yesmaximal, characteristic, complemented
12#65950—31nomaximal, Hall
13C2×C3×C5×D4C2211yescharacteristic
14C22×Dic15C2212yes—
15C2×C5×#67C2212yes—
16C2×C4×C5×D3C2211yescharacteristic
17C2×C3×C22⋊D5C2212yes—
18C2×C3×C4×D5C2211yescharacteristic
19C2×#1204C2212yes—
20C2×#1201C2211yescharacteristic
21#6719C2212yes—
22C4⋊Dic15C2211yescharacteristic
23C4×Dic15C2211yescharacteristic
24C5×#217C2212yes—
25C5×C4⋊Dic3C2211yescharacteristic
26C3×C22⋊Dic5C2212yes—
27C3×C4⋊Dic5C2211yescharacteristic
28#6669C2211yescharacteristic
29#6669C2212yes—
30(C2×D3)⋊Dic5C2211yescharacteristic
31(C2×D3)⋊Dic5C2212yes—
32#6665C2211yescharacteristic
33#6665C2212yes—
34Dic3×Dic5C2212yes—
35Dic3×Dic5C2212yes—
36#5616—51nomaximal, Hall
37C5×#121—31no—
38#4085—32no—
39#4084D311yescharacteristic, complemented
40#4082—32no—
41D4×Dic5—31no—
42C4×C22⋊D5—31no—
43#4072—31no—
44#4038—31no—
45#4034—31no—
46#4031—32no—
47#4023—32no—
48C23×C3×C5C2312yes—
49C22×C5×D3C2311yescharacteristic
50C22×C3×D5C2311yescharacteristic
51C3×C5×D4—22no—
52C2×C3×C4×C5C2311yescharacteristic
53C2×Dic15—24no—
54C2×Dic15C2312yes—
55C2×Dic15C2311yescharacteristic
56C2×Dic15C2311yescharacteristic
57C5×#67—42no—
58C2×C5×Dic3C2311yescharacteristic
59C2×C5×Dic3C2312yes—
60C4×C5×D3—22no—
61C3×C22⋊D5—42no—
62C2×C3×Dic5C2311yescharacteristic
63C2×C3×Dic5C2312yes—
64C3×C4×D5—22no—
65#1204—42no—
66#1201—22no—
67C3×#121—51no—
68#939—52no—
69#938D511yescharacteristic, complemented
70#936—52no—
71D4×Dic3—51no—
72C4×#67—51no—
73#926—51no—
74#892—51no—
75#888—51no—
76#885—52no—
77#877—52no—
78C2×C5×D4—32no—
79C2×C5×D4C2×D311yescharacteristic
80C22×C4×C5—31no—
81C2×C22⋊D5—31no—
82C2×C22⋊D5C2×D312yes—
83C22×Dic5—32no—
84C2×C4×D5C2×D311yescharacteristic
85C2×Dic10—32no—
86C5×C4⋊C4—31no—
87C5×C22⋊C4—32no—
88C22⋊Dic5—32no—
89C22⋊Dic5—32no—
90C22⋊Dic5—31no—
91C22⋊Dic5C2×D312yes—
92#685—31no—
93#685—32no—
94C4⋊Dic5—31no—
95C4⋊Dic5C2×D311yescharacteristic
96#683—31no—
97#683—32no—
98C4×Dic5—31no—
99C4×Dic5—32no—
100C4×Dic5—32no—
101#555—151noSylow
102C22×C3×C5—22no—
103C22×C3×C5—22no—
104C22×C3×C5—22no—
105C22×C3×C5C2411yescharacteristic
106C2×C5×D3—21no—
107C2×C5×D3—21no—
108C2×C5×D3—21no—
109C2×C3×D5—21no—
110C2×C3×D5—21no—
111C2×C3×D5—21no—
112C3×C4×C5#4112yes—
113Dic15—22no—
114Dic15—21no—
115Dic15#4112yes—
116C3×Dic5—21no—
117C3×Dic5—22no—
118C5×Dic3—21no—
119C5×Dic3—22no—
120C2×C3×D4—52no—
121C2×C3×D4C2×D511yescharacteristic
122C22×C3×C4—51no—
123C2×#67—51no—
124C2×#67C2×D512yes—
125C22×Dic3—52no—
126C2×C4×D3C2×D511yescharacteristic
127C2×Dic6—52no—
128C3×C4⋊C4—51no—
129C3×C22⋊C4—52no—
130#217—52no—
131#217—52no—
132#217—51no—
133#217C2×D512yes—
134(C2×D3)⋊C4—51no—
135(C2×D3)⋊C4—52no—
136C4⋊Dic3—51no—
137C4⋊Dic3C2×D511yescharacteristic
138Dic3⋊C4—51no—
139Dic3⋊C4—52no—
140C4×Dic3—51no—
141C4×Dic3—52no—
142C4×Dic3—52no—
143C23×C5—31no—
144C23×C5C22×D312yes—
145C22×D5C22×D311yescharacteristic
146C5×D4—22no—
147C5×D4—122no—
148C2×C4×C5—31no—
149C2×C4×C5—32no—
150C2×C4×C5—62no—
151C2×C4×C5C22×D311yescharacteristic
152C22⋊D5—42no—
153C22⋊D5—62no—
154C2×Dic5—31no—
155C2×Dic5—32no—
156C2×Dic5—31no—
157C2×Dic5—64no—
158C2×Dic5C22×D311yescharacteristic
159C2×Dic5C22×D312yes—
160C4×D5—22no—
161Dic10—122no—
162C4⋊Q8—151no—
163#124—152no—
164#124—152no—
165#121—31no—
166#121—51no—
167#121—152no—
168C4×D4—151no—
169C4×D4—151no—
170#117—152no—
171#117—151no—
172#117—151no—
173C2×C3×C5—42no—
174C2×C3×C5C2×#4111yescharacteristic
175C2×C3×C5C2×#4111yescharacteristic
176C2×C3×C5H(5,𝔽2)11yescharacteristic
177C3×D5—41no—
178C5×D3—41no—
179C23×C3—51no—
180C23×C3C22×D512yes—
181C22×D3C22×D511yescharacteristic
182C3×D4—22no—
183C3×D4—202no—
184C2×C3×C4—51no—
185C2×C3×C4—52no—
186C2×C3×C4—102no—
187C2×C3×C4C22×D511yescharacteristic
188#67—42no—
189#67—102no—
190C2×Dic3—51no—
191C2×Dic3—52no—
192C2×Dic3—51no—
193C2×Dic3—104no—
194C2×Dic3C22×D511yescharacteristic
195C2×Dic3C22×D512yes—
196C4×D3—22no—
197Dic6—202no—
198C22×C5—22no—
199C22×C5—22no—
200C22×C5—22no—
201C22×C5—61no—
202C22×C5—61no—
203C22×C5—61no—
204C22×C5C23×D311yescharacteristic
205C2×D5—21no—
206C2×D5—21no—
207C2×D5—21no—
208C4×C5—61no—
209C4×C5—62no—
210C4×C5#23712yes—
211Dic5—21no—
212Dic5—22no—
213Dic5—32no—
214Dic5—61no—
215Dic5—62no—
216C2×Q8—152no—
217C2×D4—32no—
218C2×D4—52no—
219C2×D4—151no—
220C2×D4D3×D511yescharacteristic
221C22×C4—31no—
222C22×C4—51no—
223C22×C4—152no—
224C4⋊C4—31no—
225C4⋊C4—51no—
226C4⋊C4—151no—
227C4⋊C4—152no—
228C4⋊C4—151no—
229C22⋊C4—32no—
230C22⋊C4—52no—
231C22⋊C4—152no—
232C22⋊C4—151no—
233C22⋊C4—152no—
234C22⋊C4—151no—
235C22⋊C4—152no—
236C42—152no—
237C42—152no—
238C42—151no—
239C3×C5#55511yescharacteristic, complemented, Hall
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—101no—
244C22×C3—101no—
245C22×C3—101no—
246C22×C3C23×D511yescharacteristic
247C2×D3—21no—
248C2×D3—21no—
249C2×D3—21no—
250C3×C4—101no—
251C3×C4—102no—
252C3×C4#71112yes—
253Dic3—21no—
254Dic3—22no—
255Dic3—52no—
256Dic3—101no—
257Dic3—102no—
258C2×C5—42no—
259C2×C5—121no—
260C2×C5C2×#23711yescharacteristic
261C2×C5#100411yescharacteristic
262C2×C5D3×#4111yescharacteristic
263D5—41no—
264C23—31no—
265C23—51no—
266C23C2×D3×D512yes—
267Q8—602no—
268D4—22no—
269D4—122no—
270D4—202no—
271D4—302no—
272C2×C4—31no—
273C2×C4—32no—
274C2×C4—51no—
275C2×C4—52no—
276C2×C4—62no—
277C2×C4—102no—
278C2×C4—151no—
279C2×C4—152no—
280C2×C4—151no—
281C2×C4—304no—
282C2×C4C2×D3×D511yescharacteristic
283C2×C3—42no—
284C2×C3—201no—
285C2×C3C2×#71111yescharacteristic
286C2×C3#414411yescharacteristic
287C2×C3D5×#4111yescharacteristic
288D3—41no—
289C5#561611yescharacteristic, complemented, Sylow
290C22—22no—
291C22—22no—
292C22—22no—
293C22—61no—
294C22—61no—
295C22—61no—
296C22—101no—
297C22—101no—
298C22—101no—
299C22C22×D3×D511yescharacteristic
300C4—62no—
301C4—61no—
302C4—102no—
303C4—101no—
304C4—152no—
305C4—302no—
306C4—301no—
307C4#678012yes—
308C3#6595011yescharacteristic, complemented, Sylow
309C2—42no—
310C2—121no—
311C2—201no—
312C2#9202411yescharacteristic
313C2#9202511yescharacteristic
314C2C2×#678011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#555
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. D4×Dic15C2semidirect
2. C5×#938C2semidirect
3. C3×#4084C2semidirect
4. #91571C2semidirect
5. #91534C2semidirect
6. #91532C2semidirect
7. #91525C2semidirect
8. #91470C2semidirect
9. C4×#1201C2semidirect
10. #91358C2semidirect
11. #91338C2semidirect
39. #4084D3semidirect
69. #938D5semidirect
239. C3×C5#555semidirect
289. C5#5616semidirect
308. C3#65950semidirect

Maximal subgroups

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

subgroupindex
D4×Dic152
C5×#9382
C3×#40842
#915712
#915342
#915322
#915252
#914702
C4×#12012
#913582
#913382
#659503
#56165