Finite Group Databasebeta

← #11754966

Subgroups — #11754966

Order 960.

472 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 132 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
1#91827C211yesmaximal, characteristic, complemented
2C5×#938C211yesmaximal, characteristic, complemented
3C3×D4×Dic5C211yesmaximal, characteristic, complemented
4#91566C212yesmaximal, complemented
5#91528C212yesmaximal, complemented
6#91526C212yesmaximal, complemented
7#91525C212yesmaximal, complemented
8#91468C211yesmaximal, characteristic, complemented
9C4×#1203C211yesmaximal, characteristic, complemented
10#91347C211yesmaximal, characteristic, complemented
11#91338C211yesmaximal, characteristic, complemented
12#65950—31nomaximal, Hall
13C2×C3×C5×D4C2211yescharacteristic
14C2×#1220C2212yes—
15C2×C4×D15C2211yescharacteristic
16C2×C5×#67C2212yes—
17C2×C4×C5×D3C2211yescharacteristic
18C22×C3×Dic5C2212yes—
19C2×#1204C2212yes—
20C2×#1203C2211yescharacteristic
21#6719C2212yes—
22C4⋊Dic15C2211yescharacteristic
23C5×#217C2212yes—
24C5×C4⋊Dic3C2211yescharacteristic
25C3×C22⋊Dic5C2212yes—
26C3×C4⋊Dic5C2211yescharacteristic
27C3×C4×Dic5C2211yescharacteristic
28#6668C2211yescharacteristic
29#6668C2212yes—
30#6667C2211yescharacteristic
31#6667C2212yes—
32(C2×D3)⋊Dic5C2211yescharacteristic
33(C2×D3)⋊Dic5C2212yes—
34Dic3×Dic5C2212yes—
35Dic3×Dic5C2212yes—
36#5563—51nomaximal, Hall
37C5×#121—31no—
38#4085—32no—
39#4084—31no—
40#4082—32no—
41D4×Dic5D311yescharacteristic, complemented
42C4×C22⋊D5—31no—
43#4072—31no—
44#4038—31no—
45#4034—31no—
46#4031—32no—
47#4023—32no—
48C23×C3×C5C2312yes—
49C22×D15C2311yescharacteristic
50C22×C5×D3C2311yescharacteristic
51C3×C5×D4—22no—
52C2×C3×C4×C5C2311yescharacteristic
53#1220—42no—
54C2×Dic15C2311yescharacteristic
55C2×Dic15C2312yes—
56C4×D15—22no—
57C5×#67—42no—
58C2×C5×Dic3C2311yescharacteristic
59C2×C5×Dic3C2312yes—
60C4×C5×D3—22no—
61C2×C3×Dic5—24no—
62C2×C3×Dic5C2312yes—
63C2×C3×Dic5C2311yescharacteristic
64C2×C3×Dic5C2311yescharacteristic
65#1204—42no—
66#1203—22no—
67C3×C4×D4—51no—
68#938—51no—
69#938D511yescharacteristic, complemented
70#930—52no—
71#926—52no—
72#892—51no—
73#892—51no—
74#885—52no—
75#885—52no—
76C4×D12—51no—
77#869—51no—
78C2×C5×D4—32no—
79C2×C5×D4C2×D311yescharacteristic
80C22×C4×C5—31no—
81C2×C22⋊D5—32no—
82C2×C22⋊D5—31no—
83C22×Dic5C2×D312yes—
84C2×C4×D5—31no—
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—32no—
99C4×Dic5—32no—
100C4×Dic5C2×D311yescharacteristic
101#555—151noSylow
102C22×C3×C5—22no—
103C22×C3×C5—22no—
104C22×C3×C5—22no—
105C22×C3×C5C2411yescharacteristic
106C2×D15—21no—
107C2×D15—21no—
108C2×D15—21no—
109C2×C5×D3—21no—
110C2×C5×D3—21no—
111C2×C5×D3—21no—
112C3×C4×C5#4112yes—
113Dic15—21no—
114Dic15—22no—
115C3×Dic5—22no—
116C3×Dic5—21no—
117C3×Dic5#4112yes—
118C5×Dic3—21no—
119C5×Dic3—22no—
120C2×C3×D4C2×D511yescharacteristic
121C22×C3×C4—52no—
122C2×#67—52no—
123C2×#67C2×D512yes—
124C2×D12—51no—
125C2×C4×D3—51no—
126C2×C4×D3C2×D511yescharacteristic
127C2×Dic6—52no—
128C3×C4⋊C4—51no—
129C3×C22⋊C4—52no—
130C3×C42—51no—
131#217—52no—
132#217C2×D512yes—
133(C2×D3)⋊C4—51no—
134(C2×D3)⋊C4—52no—
135(C2×D3)⋊C4—51no—
136(C2×D3)⋊C4—52no—
137C4⋊Dic3—51no—
138C4⋊Dic3—52no—
139C4⋊Dic3—51no—
140C4⋊Dic3C2×D511yescharacteristic
141C4×Dic3—52no—
142C4×Dic3—52no—
143C23×C5—31no—
144C23×C5C22×D312yes—
145C22×D5—31no—
146C5×D4—22no—
147C5×D4—122no—
148C2×C4×C5—31no—
149C2×C4×C5—32no—
150C2×C4×C5—62no—
151C2×C4×C5C22×D311yescharacteristic
152C22⋊D5—62no—
153C22⋊D5—122no—
154C2×Dic5—24no—
155C2×Dic5—31no—
156C2×Dic5—32no—
157C2×Dic5C22×D312yes—
158C2×Dic5C22×D311yescharacteristic
159C2×Dic5C22×D311yescharacteristic
160C4×D5—62no—
161Dic10—122no—
162C4⋊Q8—151no—
163#124—152no—
164#124—152no—
165#121—31no—
166#121—152no—
167#121—151no—
168C4×D4—51no—
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
177D15—41no—
178C5×D3—41no—
179C23×C3C22×D512yes—
180C22×D3—51no—
181C22×D3C22×D511yescharacteristic
182C3×D4—22no—
183C2×C3×C4—51no—
184C2×C3×C4—52no—
185C2×C3×C4—51no—
186C2×C3×C4—104no—
187C2×C3×C4C22×D511yescharacteristic
188#67—42no—
189#67—202no—
190C2×Dic3—51no—
191C2×Dic3—52no—
192C2×Dic3C22×D511yescharacteristic
193C2×Dic3C22×D512yes—
194D12—102no—
195C4×D3—22no—
196C4×D3—102no—
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—61no—
206C2×D5—61no—
207C2×D5—61no—
208C4×C5—61no—
209C4×C5—62no—
210C4×C5#23512yes—
211Dic5—22no—
212Dic5—21no—
213Dic5—61no—
214Dic5—62no—
215Dic5#23712yes—
216C2×Q8—152no—
217C2×D4—32no—
218C2×D4—151no—
219C2×D4—152no—
220C2×D4D3×D511yescharacteristic
221C22×C4—31no—
222C22×C4—52no—
223C22×C4—151no—
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—51no—
237C42—152no—
238C42—152no—
239C3×C5#55511yescharacteristic, complemented, Hall
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3C23×D511yescharacteristic
244C2×D3—21no—
245C2×D3—21no—
246C2×D3—21no—
247C2×D3—101no—
248C2×D3—101no—
249C2×D3—101no—
250C3×C4—52no—
251C3×C4—101no—
252C3×C4—102no—
253C3×C4#71112yes—
254Dic3—21no—
255Dic3—22no—
256Dic3—101no—
257Dic3—102no—
258C2×C5—42no—
259C2×C5—121no—
260C2×C5C2×#23511yescharacteristic
261C2×C5C2×#23711yescharacteristic
262C2×C5#100911yescharacteristic
263D5—121no—
264C23—31no—
265C23—151no—
266C23C2×D3×D512yes—
267Q8—602no—
268D4—22no—
269D4—122no—
270D4—302no—
271D4—602no—
272C2×C4—31no—
273C2×C4—32no—
274C2×C4—51no—
275C2×C4—51no—
276C2×C4—52no—
277C2×C4—62no—
278C2×C4—104no—
279C2×C4—151no—
280C2×C4—152no—
281C2×C4—302no—
282C2×C4C2×D3×D511yescharacteristic
283C2×C3—42no—
284C2×C3C2×#71111yescharacteristic
285C2×C3#414411yescharacteristic
286C2×C3D5×#4111yescharacteristic
287D3—41no—
288D3—201no—
289C5#556311yescharacteristic, complemented, Sylow
290C22—22no—
291C22—22no—
292C22—22no—
293C22—61no—
294C22—61no—
295C22—61no—
296C22—301no—
297C22—301no—
298C22—301no—
299C22C22×D3×D511yescharacteristic
300C4—52no—
301C4—62no—
302C4—61no—
303C4—102no—
304C4—101no—
305C4—302no—
306C4—301no—
307C4#678212yes—
308C3#6595011yescharacteristic, complemented, Sylow
309C2—42no—
310C2—121no—
311C2—601no—
312C2D5×#23711yescharacteristic
313C2#9202711yescharacteristic
314C2C2×#678211yescharacteristic
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. #91827C2semidirect
2. C5×#938C2semidirect
3. C3×D4×Dic5C2semidirect
4. #91566C2semidirect
5. #91528C2semidirect
6. #91526C2semidirect
7. #91525C2semidirect
8. #91468C2semidirect
9. C4×#1203C2semidirect
10. #91347C2semidirect
11. #91338C2semidirect
41. D4×Dic5D3semidirect
69. #938D5semidirect
239. C3×C5#555semidirect
289. C5#5563semidirect
308. C3#65950semidirect

Maximal subgroups

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

subgroupindex
#918272
C5×#9382
C3×D4×Dic52
#915662
#915282
#915262
#915252
#914682
C4×#12032
#913472
#913382
#659503
#55635