Finite Group Databasebeta

← #11754447

Subgroups — #11754447

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
1C4×#1220C211yesmaximal, characteristic, complemented
2C5×#926C211yesmaximal, characteristic, complemented
3C3×#4076C211yesmaximal, characteristic, complemented
4#91550C212yesmaximal, complemented
5#91534C212yesmaximal, complemented
6#91470C211yesmaximal, characteristic, complemented
7#91465C211yesmaximal, characteristic, complemented
8#91434C211yesmaximal, characteristic, complemented
9#91393C212yesmaximal, complemented
10#91353C212yesmaximal, complemented
11#91347C211yesmaximal, characteristic, complemented
12#65866—31nomaximal, Hall
13C22×C3×C4×C5C2211yescharacteristic
14C2×#1220C2211yescharacteristic
15C2×C4×D15C2211yescharacteristic
16C2×C3×C22⋊D5C2212yes—
17C2×C3×D20C2211yescharacteristic
18C2×#1204C2212yes—
19C2×#1202C2212yes—
20C2×D5×Dic3C2212yes—
21#6719C2211yescharacteristic
22#6714C2211yescharacteristic
23#6712C2211yescharacteristic
24C4×Dic15C2211yescharacteristic
25C5×#217C2212yes—
26C5×C4⋊Dic3C2211yescharacteristic
27C5×C4⋊Dic3C2211yescharacteristic
28C4×C5×Dic3C2212yes—
29C3×#685C2212yes—
30C3×C4⋊Dic5C2211yescharacteristic
31#6669C2212yes—
32#6667C2212yes—
33#6665C2212yes—
34#6665C2212yes—
35Dic3×Dic5C2212yes—
36#5616—51nomaximal, Hall
37C5×#117—31no—
38#4076D311yescharacteristic, complemented
39C4×C22⋊D5—31no—
40#4041—31no—
41#4039—31no—
42#4038—31no—
43#4034—31no—
44#4031—32no—
45#4030—32no—
46#4021—32no—
47#4018—32no—
48C23×C3×C5C2311yescharacteristic
49C22×D15C2311yescharacteristic
50C22×C3×D5C2312yes—
51C2×C3×C4×C5—22no—
52C2×C3×C4×C5C2311yescharacteristic
53C2×C3×C4×C5C2311yescharacteristic
54#1220—22no—
55C2×Dic15C2311yescharacteristic
56C2×Dic15C2311yescharacteristic
57C2×Dic15C2311yescharacteristic
58C4×D15—22no—
59C2×C5×Dic3C2312yes—
60C2×C5×Dic3C2312yes—
61C3×C22⋊D5—42no—
62C2×C3×Dic5C2312yes—
63C3×D20—22no—
64#1204—42no—
65#1202—42no—
66D5×Dic3—24no—
67C3×#121—51no—
68#939—52no—
69#938—51no—
70#936—52no—
71D4×Dic3—51no—
72C4×#67—51no—
73#926D511yescharacteristic, complemented
74#892—51no—
75#888—51no—
76#885—52no—
77#877—52no—
78C22×C4×C5C2×D311yescharacteristic
79C2×C22⋊D5—31no—
80C2×C22⋊D5C2×D312yes—
81C2×D20—32no—
82C2×D20C2×D311yescharacteristic
83C2×C4×D5—32no—
84C2×C4×D5—31no—
85C2×Dic10—32no—
86C5×C4⋊C4—31no—
87C5×C4⋊C4—31no—
88C5×C22⋊C4—32no—
89C42×C5—32no—
90C22⋊Dic5—31no—
91#685—32no—
92#685—32no—
93#685—31no—
94#685—32no—
95#685C2×D312yes—
96C4⋊Dic5C2×D311yescharacteristic
97#683—31no—
98#683—32no—
99C4×Dic5—31no—
100C4×Dic5—32no—
101#555—151noSylow
102C22×C3×C5—21no—
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5C2411yescharacteristic
106C2×D15—21no—
107C2×D15—21no—
108C2×D15—21no—
109C2×C3×D5—22no—
110C2×C3×D5—22no—
111C2×C3×D5—22no—
112C3×C4×C5—21no—
113C3×C4×C5#4112yes—
114Dic15—21no—
115Dic15—21no—
116Dic15#4112yes—
117C3×Dic5—22no—
118C5×Dic3—22no—
119C5×Dic3—22no—
120C2×C3×D4—52no—
121C2×C3×D4—51no—
122C22×C3×C4C2×D511yescharacteristic
123C2×#67—51no—
124C2×#67—52no—
125C22×Dic3—52no—
126C2×C4×D3—51no—
127C2×Dic6—52no—
128C3×C4⋊C4—51no—
129C3×C22⋊C4—52no—
130#217—51no—
131#217—52no—
132#217—52no—
133#217C2×D512yes—
134(C2×D3)⋊C4—51no—
135(C2×D3)⋊C4—52no—
136C4⋊Dic3C2×D511yescharacteristic
137C4⋊Dic3C2×D511yescharacteristic
138Dic3⋊C4—51no—
139Dic3⋊C4—52no—
140C4×Dic3—51no—
141C4×Dic3—52no—
142C4×Dic3C2×D512yes—
143C23×C5C22×D311yescharacteristic
144C22×D5—31no—
145C22×D5C22×D312yes—
146C2×C4×C5—22no—
147C2×C4×C5—32no—
148C2×C4×C5—32no—
149C2×C4×C5C22×D311yescharacteristic
150C2×C4×C5C22×D311yescharacteristic
151C22⋊D5—42no—
152C22⋊D5—62no—
153C2×Dic5—31no—
154C2×Dic5—31no—
155C2×Dic5—31no—
156C2×Dic5C22×D312yes—
157D20—22no—
158D20—122no—
159C4×D5—64no—
160C4×D5—62no—
161Dic10—122no—
162C4⋊Q8—151no—
163#124—152no—
164#124—152no—
165#121—51no—
166#121—152no—
167#121—151no—
168C4×D4—151no—
169C4×D4—151no—
170#117—31no—
171#117—152no—
172#117—151no—
173C2×C3×C5—41no—
174C2×C3×C5C2×#4111yescharacteristic
175C2×C3×C5C2×#4111yescharacteristic
176C2×C3×C5H(5,𝔽2)11yescharacteristic
177D15—41no—
178C3×D5—42no—
179C23×C3—52no—
180C23×C3C22×D511yescharacteristic
181C22×D3—51no—
182C3×D4—102no—
183C3×D4—202no—
184C2×C3×C4—22no—
185C2×C3×C4—52no—
186C2×C3×C4C22×D511yescharacteristic
187C2×C3×C4C22×D511yescharacteristic
188#67—102no—
189#67—202no—
190C2×Dic3—51no—
191C2×Dic3—51no—
192C2×Dic3—51no—
193C2×Dic3—104no—
194C2×Dic3C22×D512yes—
195C2×Dic3C22×D512yes—
196C4×D3—102no—
197Dic6—202no—
198C22×C5—21no—
199C22×C5—21no—
200C22×C5—21no—
201C22×C5C23×D311yescharacteristic
202C2×D5—22no—
203C2×D5—22no—
204C2×D5—22no—
205C2×D5—61no—
206C2×D5—61no—
207C2×D5—61no—
208C4×C5—21no—
209C4×C5—62no—
210C4×C5—62no—
211C4×C5#23712yes—
212Dic5—22no—
213Dic5—32no—
214Dic5—61no—
215Dic5—61no—
216C2×Q8—152no—
217C2×D4—52no—
218C2×D4—51no—
219C2×D4—152no—
220C2×D4—151no—
221C22×C4—152no—
222C22×C4—151no—
223C22×C4D3×D511yescharacteristic
224C4⋊C4—31no—
225C4⋊C4—31no—
226C4⋊C4—51no—
227C4⋊C4—152no—
228C4⋊C4—151no—
229C22⋊C4—32no—
230C22⋊C4—52no—
231C22⋊C4—152no—
232C22⋊C4—152no—
233C22⋊C4—152no—
234C22⋊C4—151no—
235C22⋊C4—151no—
236C42—32no—
237C42—152no—
238C42—151no—
239C3×C5#55511yescharacteristic, complemented, Hall
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—102no—
244C22×C3—102no—
245C22×C3—102no—
246C22×C3C23×D511yescharacteristic
247C2×D3—101no—
248C2×D3—101no—
249C2×D3—101no—
250C3×C4—21no—
251C3×C4—102no—
252C3×C4#70912yes—
253Dic3—22no—
254Dic3—22no—
255Dic3—52no—
256Dic3—101no—
257Dic3—101no—
258C2×C5—41no—
259C2×C5C2×#23711yescharacteristic
260C2×C5#100411yescharacteristic
261C2×C5D3×#4111yescharacteristic
262D5—42no—
263D5—121no—
264C23—52no—
265C23—151no—
266C23C2×D3×D511yescharacteristic
267Q8—602no—
268D4—102no—
269D4—202no—
270D4—302no—
271D4—602no—
272C2×C4—22no—
273C2×C4—32no—
274C2×C4—32no—
275C2×C4—52no—
276C2×C4—151no—
277C2×C4—151no—
278C2×C4—151no—
279C2×C4—304no—
280C2×C4—302no—
281C2×C4C2×D3×D511yescharacteristic
282C2×C4C2×D3×D511yescharacteristic
283C2×C3—41no—
284C2×C3—202no—
285C2×C3C2×#70911yescharacteristic
286C2×C3D5×#4111yescharacteristic
287C2×C3#414911yescharacteristic
288D3—201no—
289C5#561611yescharacteristic, complemented, Sylow
290C22—21no—
291C22—21no—
292C22—21no—
293C22—102no—
294C22—102no—
295C22—102no—
296C22—301no—
297C22—301no—
298C22—301no—
299C22C22×D3×D511yescharacteristic
300C4—21no—
301C4—62no—
302C4—62no—
303C4—102no—
304C4—152no—
305C4—301no—
306C4—301no—
307C4#677912yes—
308C3#6586611yescharacteristic, complemented, Sylow
309C2—41no—
310C2—202no—
311C2—601no—
312C2#9201611yescharacteristic
313C2#9201711yescharacteristic
314C2C2×#677911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#555
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C4×#1220C2semidirect
2. C5×#926C2semidirect
3. C3×#4076C2semidirect
4. #91550C2semidirect
5. #91534C2semidirect
6. #91470C2semidirect
7. #91465C2semidirect
8. #91434C2semidirect
9. #91393C2semidirect
10. #91353C2semidirect
11. #91347C2semidirect
38. #4076D3semidirect
73. #926D5semidirect
239. C3×C5#555semidirect
289. C5#5616semidirect
308. C3#65866semidirect

Maximal subgroups

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

subgroupindex
C4×#12202
C5×#9262
C3×#40762
#915502
#915342
#914702
#914652
#914342
#913932
#913532
#913472
#658663
#56165