Finite Group Databasebeta

← #11754340

Subgroups — #11754340

Order 960.

540 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C22×C5×Dic6C211yesmaximal, characteristic, complemented
2C2×#6779C211yesmaximal, characteristic
3#91818C211yesmaximal, characteristic, complemented
4C3×#4075C211yesmaximal, characteristic, complemented
5#91542C211yesmaximal, characteristic, complemented
6#91536C211yesmaximal, characteristic, complemented
7#91529C211yesmaximal, characteristic, complemented
8#91455C212yesmaximal, complemented
9#91424C212yesmaximal, complemented
10#91421C212yesmaximal, complemented
11#91329C212yesmaximal, complemented
12#66150—31nomaximal, Hall
13C22×C3×C4×C5C2211yescharacteristic
14C2×#1220C2211yescharacteristic
15C22×C5×Dic3C2211yescharacteristic
16C22×C5×Dic3C2211yescharacteristic
17C2×C5×Dic6—24no—
18C2×C5×Dic6C2212yes—
19C2×C5×Dic6C2212yes—
20C2×C3×C22⋊D5C2211yescharacteristic
21C2×#1204C2211yescharacteristic
22C2×#1202C2211yescharacteristic
23C2×#1200C2211yescharacteristic
24#6779—24no—
25C2×D5×Dic3C2211yescharacteristic
26#6719C2211yescharacteristic
27#6714C2212yes—
28#6712C2212yes—
29C3×C22⋊Dic5C2211yescharacteristic
30C3×#685C2212yes—
31C3×#683C2212yes—
32Dic3⋊Dic5C2212yes—
33#6669C2212yes—
34#6668C2212yes—
35#6667C2212yes—
36#6665C2212yes—
37Dic3×Dic5C2212yes—
38#5648—51nomaximal, Hall
39C22×C5×Q8—31no—
40C2×#709—31no—
41#4093—32no—
42#4092—32no—
43#4092—32no—
44#4090—32no—
45#4075—31no—
46#4075—31no—
47#4075—31no—
48#4075D311yescharacteristic, complemented
49#4072—31no—
50C23×C3×C5C2311yescharacteristic
51C22×D15C2311yescharacteristic
52C22×C3×D5C2311yescharacteristic
53C2×C3×C4×C5—22no—
54C2×C3×C4×C5C2312yes—
55#1220—22no—
56C2×Dic15C2312yes—
57C2×Dic15C2311yescharacteristic
58C2×C5×Dic3—22no—
59C2×C5×Dic3D414yes—
60C2×C5×Dic3C2312yes—
61C2×C5×Dic3C2311yescharacteristic
62C2×C5×Dic3C2311yescharacteristic
63C5×Dic6—24no—
64C5×Dic6—24no—
65C3×C22⋊D5—22no—
66C2×C3×Dic5C2312yes—
67C2×C3×Dic5C2311yescharacteristic
68#1204—22no—
69#1202—22no—
70#1200—22no—
71D5×Dic3—22no—
72C2×#237—51no—
73C22×Dic6D511yescharacteristic, complemented
74C3×#123—51no—
75#935—51no—
76#929—51no—
77#896—52no—
78#888—52no—
79#886—51no—
80#885—52no—
81#878—52no—
82#877—51no—
83C2×C5×Q8—32no—
84C2×C5×Q8—32no—
85C2×C5×Q8—64no—
86C22×C4×C5—31no—
87C22×C4×C5—31no—
88C22×C4×C5C2×D311yescharacteristic
89C2×C22⋊D5—31no—
90C2×C22⋊D5C2×D311yescharacteristic
91#709—64no—
92C2×D20—31no—
93C2×C4×D5—31no—
94C2×C4×D5—31no—
95C2×Dic10—31no—
96C22⋊Dic5—31no—
97C22⋊Dic5C2×D311yescharacteristic
98#685—32no—
99#685—32no—
100#685—32no—
101#685C2×D312yes—
102C4⋊Dic5—32no—
103#683—32no—
104#683—32no—
105#683—32no—
106#683C2×D312yes—
107C4×Dic5—32no—
108#536—151noSylow
109C22×C3×C5—22no—
110C22×C3×C5C2×D412yes—
111C22×C3×C5C2411yescharacteristic
112C2×D15—21no—
113C2×D15—22no—
114C2×C3×D5—21no—
115C2×C3×D5—22no—
116C3×C4×C5—22no—
117Dic15—21no—
118Dic15—22no—
119C3×Dic5—21no—
120C3×Dic5—22no—
121C5×Dic3—22no—
122C5×Dic3C2×D412yes—
123C5×Dic3C2×D412yes—
124C2×C3×D4—51no—
125C22×C3×C4C2×D511yescharacteristic
126C2×#67—51no—
127C2×#67—51no—
128C22×Dic3—51no—
129C22×Dic3C2×D511yescharacteristic
130C22×Dic3C2×D511yescharacteristic
131#237—104no—
132C2×C4×D3—51no—
133C2×Dic6—24no—
134C2×Dic6—51no—
135C2×Dic6C2×D512yes—
136C2×Dic6C2×D512yes—
137C3×C4⋊C4—52no—
138C3×C22⋊C4—51no—
139C3×C22⋊C4—52no—
140#217—51no—
141#217—52no—
142(C2×D3)⋊C4—52no—
143(C2×D3)⋊C4—52no—
144C4⋊Dic3—52no—
145Dic3⋊C4—52no—
146Dic3⋊C4—52no—
147Dic3⋊C4—52no—
148C4×Dic3—52no—
149C23×C5C22×D311yescharacteristic
150C22×D5—31no—
151C22×D5C22×D311yescharacteristic
152C5×Q8—64no—
153C5×Q8—64no—
154C2×C4×C5—22no—
155C2×C4×C5—34no—
156C2×C4×C5—31no—
157C2×C4×C5—31no—
158C2×C4×C5—32no—
159C2×C4×C5—62no—
160C2×C4×C5C22×D312yes—
161C22⋊D5—22no—
162C22⋊D5—62no—
163C2×Dic5—32no—
164C2×Dic5—31no—
165C2×Dic5C22×D312yes—
166C2×Dic5C22×D311yescharacteristic
167D20—62no—
168C4×D5—62no—
169C4×D5—62no—
170Dic10—62no—
171C2×#41—151no—
172C22×Q8—31no—
173C4⋊Q8—152no—
174#124—152no—
175#123—51no—
176#123—151no—
177#123—151no—
178#123—151no—
179C22⋊Q8—152no—
180C22⋊Q8—152no—
181#117—151no—
182C2×C3×C5—22no—
183C2×C3×C5C22×D411yescharacteristic
184C2×C3×C52−1+412yes—
185D15—41no—
186C3×D5—41no—
187C23×C3—51no—
188C23×C3C22×D511yescharacteristic
189C22×D3—51no—
190C3×D4—102no—
191C2×C3×C4—22no—
192C2×C3×C4—52no—
193C2×C3×C4—51no—
194C2×C3×C4C22×D512yes—
195#67—102no—
196#67—102no—
197C2×Dic3—22no—
198C2×Dic3—52no—
199C2×Dic3—51no—
200C2×Dic3—102no—
201C2×Dic3C22⋊D514yes—
202C2×Dic3C22×D512yes—
203C2×Dic3C22×D511yescharacteristic
204C2×Dic3C22×D511yescharacteristic
205C4×D3—102no—
206Dic6—24no—
207Dic6—24no—
208Dic6—102no—
209C22×C5—22no—
210C22×C5D3×D412yes—
211C22×C5C23×D311yescharacteristic
212C2×D5—21no—
213C2×D5—22no—
214C2×D5—61no—
215C2×D5—62no—
216C4×C5—22no—
217C4×C5—32no—
218C4×C5—32no—
219C4×C5—62no—
220Dic5—21no—
221Dic5—22no—
222Dic5—61no—
223Dic5—62no—
224#41—304no—
225C2×Q8—32no—
226C2×Q8—32no—
227C2×Q8—64no—
228C2×Q8—151no—
229C2×D4—51no—
230C2×D4—151no—
231C2×D4—151no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—151no—
235C22×C4—151no—
236C22×C4D3×D511yescharacteristic
237C4⋊C4—52no—
238C4⋊C4—152no—
239C4⋊C4—152no—
240C4⋊C4—152no—
241C4⋊C4—152no—
242C22⋊C4—51no—
243C22⋊C4—52no—
244C22⋊C4—152no—
245C22⋊C4—152no—
246C22⋊C4—151no—
247C22⋊C4—152no—
248C42—152no—
249C3×C5#53611yescharacteristic, complemented, Hall
250C22×C3—22no—
251C22×C3—102no—
252C22×C3—101no—
253C22×C3C2×C22⋊D512yes—
254C22×C3C23×D511yescharacteristic
255C2×D3—102no—
256C2×D3—101no—
257C3×C4—22no—
258C3×C4—102no—
259C3×C4—101no—
260Dic3—22no—
261Dic3—102no—
262Dic3—101no—
263Dic3C2×C22⋊D512yes—
264Dic3C2×C22⋊D512yes—
265C2×C5—22no—
266C2×C5C2×D3×D411yescharacteristic
267C2×C5#101012yes—
268D5—41no—
269D5—121no—
270C23—51no—
271C23—151no—
272C23C2×D3×D511yescharacteristic
273Q8—64no—
274Q8—64no—
275Q8—302no—
276D4—102no—
277D4—302no—
278D4—302no—
279C2×C4—22no—
280C2×C4—34no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—32no—
284C2×C4—52no—
285C2×C4—51no—
286C2×C4—62no—
287C2×C4—152no—
288C2×C4—151no—
289C2×C4—302no—
290C2×C4—302no—
291C2×C4C2×D3×D512yes—
292C2×C3—22no—
293C2×C3—201no—
294C2×C3#414712yes—
295C2×C3C22×C22⋊D511yescharacteristic
296D3—201no—
297C5#564811yescharacteristic, complemented, Sylow
298C22—22no—
299C22—102no—
300C22—101no—
301C22—302no—
302C22—301no—
303C22D3×C22⋊D512yes—
304C22C22×D3×D511yescharacteristic
305C4—22no—
306C4—32no—
307C4—32no—
308C4—62no—
309C4—102no—
310C4—101no—
311C4—302no—
312C4—301no—
313C3#6615011yescharacteristic, complemented, Sylow
314C2—22no—
315C2—201no—
316C2—601no—
317C2#9200012yes—
318C2C2×D3×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
prime radicalC2×#6779C2
hypercenterC22×C4D3×D5
hypocenterC3×C5#536
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. C22×C5×Dic6C2semidirect
3. #91818C2semidirect
4. C3×#4075C2semidirect
5. #91542C2semidirect
6. #91536C2semidirect
7. #91529C2semidirect
8. #91455C2semidirect
9. #91424C2semidirect
10. #91421C2semidirect
11. #91329C2semidirect
48. #4075D3semidirect
73. C22×Dic6D5semidirect
249. C3×C5#536semidirect
297. C5#5648semidirect
313. C3#66150semidirect

Maximal subgroups

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

subgroupindex
C22×C5×Dic62
C2×#67792
#918182
C3×#40752
#915422
#915362
#915292
#914552
#914242
#914212
#913292
#661503
#56485