Finite Group Databasebeta

← #11755009

Subgroups — #11755009

Order 960.

540 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 136 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×#6819C211yesmaximal, characteristic
2C22×#1204C211yesmaximal, characteristic, complemented
3C5×#935C211yesmaximal, characteristic, complemented
4C3×#4081C211yesmaximal, characteristic, complemented
5#91576C212yesmaximal, complemented
6#91546C211yesmaximal, characteristic, complemented
7#91542C211yesmaximal, characteristic, complemented
8#91532C211yesmaximal, characteristic, complemented
9#91526C212yesmaximal, complemented
10#91377C212yesmaximal, complemented
11#91328C212yesmaximal, complemented
12#65982—31nomaximal, Hall
13C2×C3×C5×D4C2211yescharacteristic
14C2×#1220C2211yescharacteristic
15C2×#1220C2211yescharacteristic
16C22×Dic15C2211yescharacteristic
17C22×Dic15C2211yescharacteristic
18#6819—24no—
19C2×C4×D15C2211yescharacteristic
20C2×Dic30C2211yescharacteristic
21C22×C5×Dic3C2211yescharacteristic
22C22×C3×Dic5C2211yescharacteristic
23C2×#1204—24no—
24C2×#1204C2212yes—
25C2×#1204C2212yes—
26C5×#217C2212yes—
27C5×#217C2211yescharacteristic
28C5×Dic3⋊C4C2212yes—
29C3×C22⋊Dic5C2212yes—
30C3×C22⋊Dic5C2211yescharacteristic
31C3×#683C2212yes—
32Dic3⋊Dic5C2212yes—
33#6669C2212yes—
34#6668C2212yes—
35#6667C2212yes—
36#6667C2212yes—
37Dic3×Dic5C2212yes—
38#5648—51nomaximal, Hall
39C2×#711—31no—
40C22×Dic10—31no—
41C5×#123—31no—
42#4081D311yescharacteristic, complemented
43#4075—31no—
44#4042—32no—
45#4034—32no—
46#4032—31no—
47#4031—32no—
48#4024—32no—
49#4023—31no—
50C23×C3×C5C2311yescharacteristic
51C23×C3×C5C2311yescharacteristic
52C22×D15C2311yescharacteristic
53C3×C5×D4—22no—
54C2×C3×C4×C5C2311yescharacteristic
55#1220—22no—
56#1220—22no—
57C2×Dic15—22no—
58C2×Dic15D414yes—
59C2×Dic15C2311yescharacteristic
60C2×Dic15C2311yescharacteristic
61C2×Dic15C2311yescharacteristic
62C4×D15—22no—
63Dic30—22no—
64C2×C5×Dic3—22no—
65C2×C5×Dic3C2312yes—
66C2×C5×Dic3C2312yes—
67C2×C3×Dic5—22no—
68C2×C3×Dic5C2312yes—
69C2×C3×Dic5C2312yes—
70#1204—24no—
71#1204—24no—
72C2×#237—51no—
73C22×Dic6—51no—
74C3×#123—51no—
75#935D511yescharacteristic, complemented
76#929—51no—
77#896—52no—
78#888—52no—
79#886—51no—
80#885—52no—
81#878—52no—
82#877—51no—
83C2×C5×D4C2×D311yescharacteristic
84C22×C4×C5—31no—
85C2×C22⋊D5—31no—
86C2×C22⋊D5—31no—
87C22×Dic5—31no—
88C22×Dic5—31no—
89C22×Dic5C2×D311yescharacteristic
90#711—64no—
91C2×C4×D5—31no—
92C2×Dic10—31no—
93C2×Dic10—32no—
94C2×Dic10—32no—
95C2×Dic10—64no—
96C5×C4⋊C4—32no—
97C5×C22⋊C4—31no—
98C5×C22⋊C4—32no—
99C22⋊Dic5C2×D311yescharacteristic
100C22⋊Dic5C2×D312yes—
101#685—32no—
102#685—32no—
103C4⋊Dic5—32no—
104#683—32no—
105#683—32no—
106#683C2×D312yes—
107C4×Dic5—32no—
108#536—151noSylow
109C22×C3×C5—22no—
110C22×C3×C5—21no—
111C22×C3×C5—22no—
112C22×C3×C5C2×D412yes—
113C22×C3×C5C2411yescharacteristic
114C2×D15—21no—
115C2×D15—22no—
116C3×C4×C5—21no—
117Dic15—21no—
118Dic15C2×D412yes—
119Dic15C2×D412yes—
120C3×Dic5—22no—
121C3×Dic5—22no—
122C5×Dic3—22no—
123C5×Dic3—22no—
124C2×C3×D4C2×D511yescharacteristic
125C22×C3×C4—51no—
126C2×#67—51no—
127C2×#67—51no—
128C22×Dic3—51no—
129C22×Dic3—51no—
130C22×Dic3C2×D511yescharacteristic
131#237—104no—
132C2×C4×D3—51no—
133C2×Dic6—51no—
134C2×Dic6—52no—
135C2×Dic6—52no—
136C2×Dic6—104no—
137C3×C4⋊C4—52no—
138C3×C22⋊C4—51no—
139C3×C22⋊C4—52no—
140#217C2×D511yescharacteristic
141#217C2×D512yes—
142(C2×D3)⋊C4—52no—
143(C2×D3)⋊C4—52no—
144C4⋊Dic3—52no—
145Dic3⋊C4—52no—
146Dic3⋊C4—52no—
147Dic3⋊C4C2×D512yes—
148C4×Dic3—52no—
149C23×C5C22×D311yescharacteristic
150C23×C5C22×D311yescharacteristic
151C22×D5—31no—
152C5×D4—22no—
153C2×C4×C5—32no—
154C2×C4×C5—32no—
155C2×C4×C5—62no—
156C2×C4×C5C22×D311yescharacteristic
157C22⋊D5—62no—
158C22⋊D5—62no—
159C2×Dic5—22no—
160C2×Dic5—34no—
161C2×Dic5—31no—
162C2×Dic5—31no—
163C2×Dic5—31no—
164C2×Dic5—62no—
165C2×Dic5C22×D312yes—
166C2×Dic5C22×D312yes—
167C4×D5—62no—
168Dic10—64no—
169Dic10—64no—
170Dic10—62no—
171C2×#41—151no—
172C22×Q8—151no—
173C4⋊Q8—152no—
174#124—152no—
175#123—31no—
176#123—51no—
177#123—151no—
178#123—151no—
179C22⋊Q8—152no—
180C22⋊Q8—152no—
181#117—151no—
182C2×C3×C5—22no—
183C2×C3×C5—41no—
184C2×C3×C5C22×D411yescharacteristic
185C2×C3×C52−1+412yes—
186D15—41no—
187C23×C3C22×D511yescharacteristic
188C23×C3C22×D511yescharacteristic
189C22×D3—51no—
190C3×D4—22no—
191C2×C3×C4—52no—
192C2×C3×C4—52no—
193C2×C3×C4—102no—
194C2×C3×C4C22×D511yescharacteristic
195#67—102no—
196#67—102no—
197C2×Dic3—22no—
198C2×Dic3—54no—
199C2×Dic3—51no—
200C2×Dic3—51no—
201C2×Dic3—51no—
202C2×Dic3—102no—
203C2×Dic3C22×D512yes—
204C2×Dic3C22×D512yes—
205C4×D3—102no—
206Dic6—104no—
207Dic6—104no—
208Dic6—102no—
209C22×C5—22no—
210C22×C5—21no—
211C22×C5—22no—
212C22×C5D3×D412yes—
213C22×C5C23×D311yescharacteristic
214C2×D5—61no—
215C2×D5—62no—
216C4×C5—21no—
217C4×C5—62no—
218C4×C5—62no—
219Dic5—22no—
220Dic5—22no—
221Dic5—32no—
222Dic5—32no—
223Dic5—61no—
224#41—304no—
225C2×Q8—152no—
226C2×Q8—152no—
227C2×Q8—151no—
228C2×Q8—304no—
229C2×D4—151no—
230C2×D4—151no—
231C2×D4D3×D511yescharacteristic
232C22×C4—31no—
233C22×C4—51no—
234C22×C4—151no—
235C22×C4—151no—
236C22×C4—151no—
237C4⋊C4—32no—
238C4⋊C4—52no—
239C4⋊C4—152no—
240C4⋊C4—152no—
241C4⋊C4—152no—
242C22⋊C4—31no—
243C22⋊C4—32no—
244C22⋊C4—51no—
245C22⋊C4—52no—
246C22⋊C4—152no—
247C22⋊C4—152no—
248C42—152no—
249C3×C5#53611yescharacteristic, complemented, Hall
250C22×C3—22no—
251C22×C3—21no—
252C22×C3—22no—
253C22×C3D4×D512yes—
254C22×C3C23×D511yescharacteristic
255C2×D3—101no—
256C2×D3—102no—
257C3×C4—21no—
258C3×C4—102no—
259C3×C4—102no—
260Dic3—22no—
261Dic3—22no—
262Dic3—52no—
263Dic3—52no—
264Dic3—101no—
265C2×C5—22no—
266C2×C5—41no—
267C2×C5C2×D3×D411yescharacteristic
268C2×C5#101012yes—
269D5—121no—
270C23—151no—
271C23C2×D3×D511yescharacteristic
272C23C2×D3×D511yescharacteristic
273Q8—302no—
274Q8—304no—
275Q8—304no—
276D4—22no—
277D4—302no—
278D4—302no—
279C2×C4—32no—
280C2×C4—32no—
281C2×C4—52no—
282C2×C4—52no—
283C2×C4—62no—
284C2×C4—102no—
285C2×C4—154no—
286C2×C4—151no—
287C2×C4—151no—
288C2×C4—151no—
289C2×C4—302no—
290C2×C4—302no—
291C2×C4C2×D3×D511yescharacteristic
292C2×C3—22no—
293C2×C3—41no—
294C2×C3C2×D4×D511yescharacteristic
295C2×C3#415012yes—
296D3—201no—
297C5#564811yescharacteristic, complemented, Sylow
298C22—22no—
299C22—22no—
300C22—21no—
301C22—302no—
302C22—301no—
303C22#679012yes—
304C22C22×D3×D511yescharacteristic
305C4—21no—
306C4—62no—
307C4—62no—
308C4—102no—
309C4—102no—
310C4—152no—
311C4—152no—
312C4—301no—
313C3#6598211yescharacteristic, complemented, Sylow
314C2—22no—
315C2—41no—
316C2—601no—
317C2#9202012yes—
318C2C2×#679011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
prime radicalC2×#6819C2
hypercenterC2×D4D3×D5
hypocenterC3×C5#536
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×#1204C2semidirect
3. C5×#935C2semidirect
4. C3×#4081C2semidirect
5. #91576C2semidirect
6. #91546C2semidirect
7. #91542C2semidirect
8. #91532C2semidirect
9. #91526C2semidirect
10. #91377C2semidirect
11. #91328C2semidirect
42. #4081D3semidirect
75. #935D5semidirect
249. C3×C5#536semidirect
297. C5#5648semidirect
313. C3#65982semidirect

Maximal subgroups

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

subgroupindex
C2×#68192
C22×#12042
C5×#9352
C3×#40812
#915762
#915462
#915422
#915322
#915262
#913772
#913282
#659823
#56485