Finite Group Databasebeta

← #11754331

Subgroups — #11754331

Order 960.

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

normal subgroups only

This group has 588 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
1C2×#6782C212yesmaximal, complemented
2#91819C211yesmaximal, characteristic, complemented
3C5×#930C211yesmaximal, characteristic, complemented
4C2×C3×#683C211yesmaximal, characteristic, complemented
5#91566C212yesmaximal, complemented
6#91420C212yesmaximal, complemented
7#91419C212yesmaximal, complemented
8#91417C212yesmaximal, complemented
9#91416C212yesmaximal, complemented
10#65931—31nomaximal, Hall
11C22×C3×C4×C5C2211yescharacteristic
12C2×#1220C2212yes—
13C2×D60C2211yescharacteristic
14C2×C5×#67C2212yes—
15C2×C5×D12C2211yescharacteristic
16C22×C3×Dic5C2211yescharacteristic
17C22×C3×Dic5C2211yescharacteristic
18C2×#1204C2212yes—
19C2×#1203C2212yes—
20C2×#1203C2212yes—
21C2×#1200C2212yes—
22#6782—28no—
23C2×D3×Dic5C2212yes—
24#6714C2212yes—
25C4⋊Dic15C2211yescharacteristic
26C5×(C2×D3)⋊C4C2212yes—
27C5×C4⋊Dic3C2211yescharacteristic
28C3×#683C2212yes—
29C3×#683C2212yes—
30#6668C2212yes—
31#6667C2212yes—
32(C2×D3)⋊Dic5C2212yes—
33#5510—51nomaximal, Hall
34C2×#711—32no—
35C5×#121—31no—
36#4085—32no—
37#4085—32no—
38#4076—31no—
39C2×#683D311yescharacteristic, complemented
40#4042—32no—
41#4030—32no—
42#4024—32no—
43C23×C3×C5C2311yescharacteristic
44C22×D15C2312yes—
45C22×C5×D3C2312yes—
46C2×C3×C4×C5—41no—
47C2×C3×C4×C5C2311yescharacteristic
48C2×C3×C4×C5C2311yescharacteristic
49#1220—24no—
50C2×Dic15C2312yes—
51D60—41no—
52C5×#67—24no—
53C2×C5×Dic3C2312yes—
54C5×D12—41no—
55C2×C3×Dic5—41no—
56C2×C3×Dic5D414yescomplemented
57C2×C3×Dic5C2312yes—
58C2×C3×Dic5C2312yes—
59#1204—24no—
60#1203—24no—
61#1203—42no—
62#1200—24no—
63D3×Dic5—24no—
64C2×#235—52no—
65C2×C3×C4⋊C4—51no—
66#930—51no—
67#930—52no—
68#930D511yescharacteristic, complemented
69#897—52no—
70#897—52no—
71#895—52no—
72#895—52no—
73C2×C5×D4—32no—
74C2×C5×D4—31no—
75C22×C4×C5C2×D311yescharacteristic
76C2×C22⋊D5—32no—
77C2×C22⋊D5—32no—
78C2×C22⋊D5—32no—
79C22×Dic5—32no—
80C22×Dic5C2×D311yescharacteristic
81C22×Dic5C2×D311yescharacteristic
82#711—68no—
83C2×D20—31no—
84C2×C4×D5—32no—
85C2×Dic10—32no—
86C5×C4⋊C4—31no—
87C5×C22⋊C4—32no—
88C22⋊Dic5—32no—
89#685—32no—
90#685—32no—
91C4⋊Dic5—31no—
92#683—32no—
93#683C2×D312yes—
94#683C2×D312yes—
95#537—151noSylow
96C22×C3×C5—21no—
97C22×C3×C5—21no—
98C22×C3×C5C2×D412yes—
99C22×C3×C5C2411yescharacteristic
100C2×D15—22no—
101C2×D15—22no—
102C2×D15—22no—
103C2×C5×D3—22no—
104C2×C5×D3—22no—
105C2×C5×D3—22no—
106C3×C4×C5—21no—
107C3×C4×C5—21no—
108Dic15—22no—
109C3×Dic5—22no—
110C3×Dic5C2×D414yes—
111C5×Dic3—22no—
112C22×C3×C4—51no—
113C22×C3×C4—51no—
114C22×C3×C4C2×D511yescharacteristic
115C2×#67—52no—
116C2×#67C2×D512yes—
117#235—108no—
118C2×D12—52no—
119C2×D12—52no—
120C2×D12—51no—
121C2×D12C2×D511yescharacteristic
122C2×C4×D3—52no—
123C2×C4×D3—52no—
124C2×Dic6—52no—
125C3×C4⋊C4—52no—
126C3×C4⋊C4—52no—
127(C2×D3)⋊C4—52no—
128(C2×D3)⋊C4—52no—
129(C2×D3)⋊C4—52no—
130(C2×D3)⋊C4C2×D512yes—
131C4⋊Dic3—51no—
132C4⋊Dic3—52no—
133C4⋊Dic3C2×D511yescharacteristic
134C23×C5—32no—
135C23×C5C22×D311yescharacteristic
136C22×D5—32no—
137C5×D4—64no—
138C5×D4—121no—
139C2×C4×C5—32no—
140C2×C4×C5—41no—
141C2×C4×C5C22×D311yescharacteristic
142C2×C4×C5C22×D311yescharacteristic
143C22⋊D5—64no—
144C22⋊D5—64no—
145C22⋊D5—122no—
146C2×Dic5—32no—
147C2×Dic5—41no—
148C2×Dic5—64no—
149C2×Dic5D1214yescomplemented
150C2×Dic5C22×D312yes—
151C2×Dic5C22×D312yes—
152D20—121no—
153C4×D5—64no—
154Dic10—64no—
155C2×#41—152no—
156C22⋊Q8—152no—
157C22⋊Q8—152no—
158#121—31no—
159#121—152no—
160#121—152no—
161#121—152no—
162#121—151no—
163C2×C4⋊C4—51no—
164C2×C3×C5—22no—
165C2×C3×C5C22×D411yescharacteristic
166C2×C3×C5H(5,𝔽2)11yescharacteristic
167C2×C3×C52−1+411yescharacteristic
168D15—42no—
169C5×D3—42no—
170C23×C3C22×D511yescharacteristic
171C22×D3—52no—
172C22×D3C22×D512yes—
173C2×C3×C4—41no—
174C2×C3×C4—54no—
175C2×C3×C4—52no—
176C2×C3×C4—52no—
177C2×C3×C4—201no—
178C2×C3×C4C22×D511yescharacteristic
179C2×C3×C4C22×D511yescharacteristic
180#67—24no—
181#67—104no—
182C2×Dic3—52no—
183C2×Dic3C22×D512yes—
184D12—41no—
185D12—104no—
186D12—201no—
187D12—202no—
188C4×D3—104no—
189C4×D3—104no—
190Dic6—104no—
191C22×C5—21no—
192C22×C5—21no—
193C22×C5—62no—
194C22×C5—62no—
195C22×C5—62no—
196C22×C5C2×D1212yes—
197C22×C5C23×D311yescharacteristic
198C2×D5—62no—
199C2×D5—62no—
200C2×D5—62no—
201C4×C5—21no—
202C4×C5—21no—
203C4×C5—62no—
204Dic5—22no—
205Dic5—62no—
206Dic5C2×D1214yes—
207#41—308no—
208C2×Q8—152no—
209C2×D4—32no—
210C2×D4—31no—
211C2×D4—152no—
212C2×D4—152no—
213C2×D4—152no—
214C2×D4—151no—
215C22×C4—51no—
216C22×C4—51no—
217C22×C4—152no—
218C22×C4—152no—
219C22×C4D3×D511yescharacteristic
220C4⋊C4—31no—
221C4⋊C4—52no—
222C4⋊C4—52no—
223C4⋊C4—152no—
224C4⋊C4—151no—
225C22⋊C4—32no—
226C22⋊C4—152no—
227C22⋊C4—152no—
228C22⋊C4—152no—
229C3×C5#53711yescharacteristic, complemented, Hall
230C22×C3—21no—
231C22×C3—21no—
232C22×C3D4×D512yes—
233C22×C3C23×D511yescharacteristic
234C2×D3—22no—
235C2×D3—22no—
236C2×D3—22no—
237C2×D3—102no—
238C2×D3—102no—
239C2×D3—102no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—54no—
243C3×C4—102no—
244Dic3—22no—
245Dic3—102no—
246C2×C5—22no—
247C2×C5—122no—
248C2×C5C22×D1211yescharacteristic
249C2×C5#100411yescharacteristic
250C2×C5#100711yescharacteristic
251D5—122no—
252C23—32no—
253C23—152no—
254C23C2×D3×D511yescharacteristic
255Q8—304no—
256D4—64no—
257D4—121no—
258D4—304no—
259D4—304no—
260D4—601no—
261D4—602no—
262C2×C4—32no—
263C2×C4—41no—
264C2×C4—54no—
265C2×C4—52no—
266C2×C4—52no—
267C2×C4—152no—
268C2×C4—201no—
269C2×C4—304no—
270C2×C4—304no—
271C2×C4C2×D3×D511yescharacteristic
272C2×C4C2×D3×D511yescharacteristic
273C2×C3—22no—
274C2×C3C2×D4×D511yescharacteristic
275C2×C3#414411yescharacteristic
276C2×C3#415011yescharacteristic
277D3—42no—
278D3—202no—
279C5#551011yescharacteristic, complemented, Sylow
280C22—21no—
281C22—21no—
282C22—62no—
283C22—62no—
284C22—62no—
285C22—302no—
286C22—302no—
287C22—302no—
288C22D5×D1212yes—
289C22C22×D3×D511yescharacteristic
290C4—21no—
291C4—21no—
292C4—54no—
293C4—62no—
294C4—102no—
295C4—302no—
296C3#6593111yescharacteristic, complemented, Sylow
297C2—22no—
298C2—122no—
299C2—602no—
300C2#9200411yescharacteristic
301C2C2×D5×D1211yescharacteristic
302C2#9204911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#537
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. C2×#6782C2semidirect
2. #91819C2semidirect
3. C5×#930C2semidirect
4. C2×C3×#683C2semidirect
5. #91566C2semidirect
6. #91420C2semidirect
7. #91419C2semidirect
8. #91417C2semidirect
9. #91416C2semidirect
39. C2×#683D3semidirect
56. C2×C3×Dic5D4semidirect
68. #930D5semidirect
149. C2×Dic5D12semidirect
229. C3×C5#537semidirect
279. C5#5510semidirect
296. C3#65931semidirect

Maximal subgroups

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

subgroupindex
C2×#67822
#918192
C5×#9302
C2×C3×#6832
#915662
#914202
#914192
#914172
#914162
#659313
#55105