Finite Group Databasebeta

← #11753808

Subgroups — #11753808

Order 960.

476 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 476 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⋊C4×D15C211yesmaximal, characteristic, complemented
2C5×#892C211yesmaximal, characteristic, complemented
3C3×#4039C211yesmaximal, characteristic, complemented
4#91465C211yesmaximal, characteristic, complemented
5#91461C212yesmaximal, complemented
6#91456C211yesmaximal, characteristic, complemented
7#91447C212yesmaximal, complemented
8C4×#1203C211yesmaximal, characteristic, complemented
9C4×#1202C212yesmaximal, complemented
10#91388C211yesmaximal, characteristic, complemented
11#91363C212yesmaximal, complemented
12#65864—31nomaximal, Hall
13C2×C4×D15C2211yescharacteristic
14C2×C4×D15C2212yes—
15C2×C4×C5×D3C2211yescharacteristic
16C2×C3×D20C2211yescharacteristic
17C2×C3×C4×D5C2212yes—
18C2×#1203C2211yescharacteristic
19C2×#1202C2212yes—
20C2×#1201C2212yes—
21C3×C5×C4⋊C4C2211yescharacteristic
22C4⋊Dic15C2211yescharacteristic
23#6712C2212yes—
24C5×(C2×D3)⋊C4C2212yes—
25C5×C4⋊Dic3C2211yescharacteristic
26C4×C5×Dic3C2212yes—
27C3×#685C2212yes—
28C3×C4×Dic5C2211yescharacteristic
29Dic3⋊Dic5C2212yes—
30#6669C2212yes—
31#6668C2211yescharacteristic
32#6667C2211yescharacteristic
33(C2×D3)⋊Dic5C2211yescharacteristic
34#6665C2212yes—
35#6665C2212yes—
36#5558—51nomaximal, Hall
37C5×#117—31no—
38#4076—31no—
39C4×C22⋊D5—31no—
40#4041—31no—
41#4039D311yescharacteristic, complemented
42D5×C4⋊C4—31no—
43#4036—31no—
44#4030—32no—
45#4029—32no—
46#4022—32no—
47C4×D20—32no—
48C22×D15C2311yescharacteristic
49C22×C5×D3C2311yescharacteristic
50C22×C3×D5C2312yes—
51C2×C3×C4×C5C2312yes—
52C2×C3×C4×C5C2311yescharacteristic
53C2×Dic15C2311yescharacteristic
54C2×Dic15C2312yes—
55C4×D15—22no—
56C4×D15—42no—
57C2×C5×Dic3C2311yescharacteristic
58C2×C5×Dic3C2312yes—
59C4×C5×D3—22no—
60C2×C3×Dic5C2311yescharacteristic
61C2×C3×Dic5C2311yescharacteristic
62C3×D20—22no—
63C3×C4×D5—42no—
64#1203—22no—
65#1202—24no—
66#1201—42no—
67C3×C4×D4—51no—
68#938—51no—
69#938—51no—
70#929—52no—
71C4×#67—52no—
72#892D511yescharacteristic, complemented
73D3×C4⋊C4—51no—
74#884—52no—
75#879—52no—
76C4×D12—51no—
77#870—51no—
78C22×C4×C5—31no—
79C2×C22⋊D5—32no—
80C2×C22⋊D5—31no—
81C2×D20—32no—
82C2×D20C2×D311yescharacteristic
83C2×C4×D5—31no—
84C2×C4×D5—32no—
85C2×C4×D5C2×D312yes—
86C5×C4⋊C4—31no—
87C5×C4⋊C4C2×D311yescharacteristic
88C5×C22⋊C4—32no—
89C42×C5—32no—
90C22⋊Dic5—31no—
91#685—32no—
92#685—32no—
93#685—31no—
94#685C2×D312yes—
95C4⋊Dic5—31no—
96C4⋊Dic5—32no—
97#683—32no—
98#683—31no—
99#683—32no—
100C4×Dic5C2×D311yescharacteristic
101#553—151noSylow
102C22×C3×C5C2411yescharacteristic
103C2×D15—21no—
104C2×D15—21no—
105C2×D15#4112yes—
106C2×C5×D3—21no—
107C2×C5×D3—21no—
108C2×C5×D3—21no—
109C2×C3×D5—22no—
110C2×C3×D5—22no—
111C2×C3×D5—22no—
112C3×C4×C5—22no—
113C3×C4×C5#4112yes—
114Dic15—21no—
115Dic15—22no—
116C3×Dic5—21no—
117C3×Dic5—21no—
118C5×Dic3—22no—
119C5×Dic3—21no—
120C2×C3×D4—51no—
121C22×C3×C4—52no—
122C2×#67—52no—
123C2×#67—52no—
124C2×D12—51no—
125C2×C4×D3—51no—
126C2×C4×D3—52no—
127C2×C4×D3C2×D511yescharacteristic
128C3×C4⋊C4C2×D511yescharacteristic
129C3×C22⋊C4—52no—
130C3×C42—51no—
131#217—52no—
132#217—52no—
133(C2×D3)⋊C4—51no—
134(C2×D3)⋊C4—51no—
135(C2×D3)⋊C4C2×D512yes—
136C4⋊Dic3—51no—
137C4⋊Dic3—51no—
138C4⋊Dic3C2×D511yescharacteristic
139Dic3⋊C4—52no—
140Dic3⋊C4—52no—
141Dic3⋊C4—52no—
142C4×Dic3C2×D512yes—
143C23×C5—31no—
144C22×D5—31no—
145C22×D5C22×D312yes—
146C2×C4×C5—32no—
147C2×C4×C5—31no—
148C2×C4×C5—62no—
149C2×C4×C5C22×D312yes—
150C2×C4×C5C22×D311yescharacteristic
151C22⋊D5—62no—
152C22⋊D5—122no—
153C2×Dic5—31no—
154C2×Dic5—32no—
155C2×Dic5C22×D311yescharacteristic
156C2×Dic5C22×D311yescharacteristic
157D20—22no—
158D20—64no—
159C4×D5—42no—
160C4×D5—62no—
161C4×D5—122no—
162#126—152no—
163#125—151no—
164#123—152no—
165#121—152no—
166#121—151no—
167#121—151no—
168C4×D4—51no—
169C4×D4—152no—
170C4×D4—151no—
171#117—31no—
172C2×C4⋊C4—151no—
173C2×C3×C5C2×#4111yescharacteristic
174C2×C3×C5C2×#4111yescharacteristic
175C2×C3×C5H(5,𝔽2)11yescharacteristic
176D15—22no—
177C3×D5—42no—
178C5×D3—41no—
179C23×C3—52no—
180C22×D3—51no—
181C22×D3C22×D511yescharacteristic
182C3×D4—102no—
183C2×C3×C4—51no—
184C2×C3×C4—51no—
185C2×C3×C4—202no—
186C2×C3×C4C22×D512yes—
187C2×C3×C4C22×D511yescharacteristic
188#67—104no—
189#67—202no—
190C2×Dic3—52no—
191C2×Dic3—51no—
192C2×Dic3C22×D511yescharacteristic
193C2×Dic3C22×D512yes—
194D12—102no—
195C4×D3—22no—
196C4×D3—102no—
197C4×D3—202no—
198C22×C5—61no—
199C22×C5—61no—
200C22×C5—61no—
201C22×C5C23×D311yescharacteristic
202C2×D5—22no—
203C2×D5—22no—
204C2×D5—22no—
205C2×D5—32no—
206C2×D5—61no—
207C2×D5—61no—
208C4×C5—22no—
209C4×C5—61no—
210C4×C5—62no—
211C4×C5#23512yes—
212Dic5—21no—
213Dic5—21no—
214Dic5—61no—
215Dic5—62no—
216C2×D4—51no—
217C2×D4—152no—
218C2×D4—151no—
219C2×D4—152no—
220C22×C4—31no—
221C22×C4—52no—
222C22×C4—151no—
223C22×C4—152no—
224C4⋊C4—31no—
225C4⋊C4—152no—
226C4⋊C4—151no—
227C4⋊C4—152no—
228C4⋊C4—151no—
229C4⋊C4—152no—
230C4⋊C4D3×D511yescharacteristic
231C22⋊C4—32no—
232C22⋊C4—52no—
233C22⋊C4—151no—
234C22⋊C4—152no—
235C22⋊C4—152no—
236C22⋊C4—151no—
237C42—32no—
238C42—51no—
239C3×C5#55311yescharacteristic, complemented, Hall
240C22×C3—102no—
241C22×C3—102no—
242C22×C3—102no—
243C22×C3C23×D511yescharacteristic
244C2×D3—21no—
245C2×D3—21no—
246C2×D3—21no—
247C2×D3—52no—
248C2×D3—101no—
249C2×D3—101no—
250C3×C4—22no—
251C3×C4—101no—
252C3×C4—101no—
253C3×C4#70912yes—
254Dic3—21no—
255Dic3—22no—
256Dic3—101no—
257Dic3—102no—
258C2×C5—121no—
259C2×C5C2×#23511yescharacteristic
260C2×C5#100411yescharacteristic
261C2×C5D3×#4111yescharacteristic
262D5—42no—
263D5—62no—
264C23—31no—
265C23—52no—
266C23—151no—
267D4—102no—
268D4—304no—
269D4—302no—
270D4—602no—
271C2×C4—31no—
272C2×C4—32no—
273C2×C4—51no—
274C2×C4—51no—
275C2×C4—62no—
276C2×C4—151no—
277C2×C4—152no—
278C2×C4—202no—
279C2×C4—302no—
280C2×C4—602no—
281C2×C4C2×D3×D512yes—
282C2×C4C2×D3×D511yescharacteristic
283C2×C3—202no—
284C2×C3C2×#70911yescharacteristic
285C2×C3D5×#4111yescharacteristic
286C2×C3#414911yescharacteristic
287D3—41no—
288D3—102no—
289C5#555811yescharacteristic, complemented, Sylow
290C22—61no—
291C22—61no—
292C22—61no—
293C22—102no—
294C22—102no—
295C22—102no—
296C22—152no—
297C22—301no—
298C22—301no—
299C22C22×D3×D511yescharacteristic
300C4—22no—
301C4—61no—
302C4—62no—
303C4—101no—
304C4—101no—
305C4—301no—
306C4—302no—
307C4#677112yes—
308C3#6586411yescharacteristic, complemented, Sylow
309C2—121no—
310C2—202no—
311C2—302no—
312C2C2×#677111yescharacteristic
313C2#9202611yescharacteristic
314C2#9203411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C4⋊C4C22
hypercenterC4⋊C4D3×D5
hypocenterC3×C5#553
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C4⋊C4×D15C2semidirect
2. C5×#892C2semidirect
3. C3×#4039C2semidirect
4. #91465C2semidirect
5. #91461C2semidirect
6. #91456C2semidirect
7. #91447C2semidirect
8. C4×#1203C2semidirect
9. C4×#1202C2semidirect
10. #91388C2semidirect
11. #91363C2semidirect
41. #4039D3semidirect
72. #892D5semidirect
239. C3×C5#553semidirect
289. C5#5558semidirect
308. C3#65864semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×D152
C5×#8922
C3×#40392
#914652
#914612
#914562
#914472
C4×#12032
C4×#12022
#913882
#913632
#658643
#55585