Finite Group Databasebeta

← #11754463

Subgroups — #11754463

Order 960.

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

normal subgroups only

This group has 428 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
2C5×#924C211yesmaximal, characteristic, complemented
3C3×#4072C211yesmaximal, characteristic, complemented
4#91562C212yesmaximal
5#91531C212yesmaximal, complemented
6#91437C211yesmaximal, characteristic, complemented
7#91402C212yesmaximal, complemented
8#91384C211yesmaximal, characteristic, complemented
9#91383C211yesmaximal, characteristic, complemented
10#91378C212yesmaximal, complemented
11#91344C211yesmaximal, characteristic, complemented
12#65973—31nomaximal, Hall
13C22×C3×C4×C5C2211yescharacteristic
14C2×#1220C2211yescharacteristic
15C2×C4×D15C2211yescharacteristic
16C2×C5×Dic6C2211yescharacteristic
17C2×#1200C2212yes—
18#6719C2211yescharacteristic
19#6714C2211yescharacteristic
20#6712C2211yescharacteristic
21C4×Dic15C2211yescharacteristic
22C5×#217C2212yes—
23C5×C4⋊Dic3C2211yescharacteristic
24C5×Dic3⋊C4C2212yes—
25C3×C22⋊Dic5C2212yes—
26C3×C4⋊Dic5C2211yescharacteristic
27C3×C4⋊Dic5C2211yescharacteristic
28C3×C4×Dic5C2212yes—
29Dic3⋊Dic5C2212yes—
30Dic3⋊Dic5C2212yes—
31#6669C2212yes—
32#6668C2212yes—
33#6668C2212yes—
34#6667C2212yes—
35Dic3×Dic5C2212yes—
36#5529—51nomaximal, Hall
37C5×C22⋊Q8—31no—
38#4092—31no—
39Q8×Dic5—31no—
40C4×C22⋊D5—31no—
41#4072D311yescharacteristic, complemented
42#4038—32no—
43D5×C4⋊C4—31no—
44#4036—31no—
45#4035—32no—
46#4029—32no—
47#4025—32no—
48C23×C3×C5C2311yescharacteristic
49C22×D15C2311yescharacteristic
50C2×C3×C4×C5—22no—
51C2×C3×C4×C5C2311yescharacteristic
52C2×C3×C4×C5C2311yescharacteristic
53#1220—22no—
54C2×Dic15C2311yescharacteristic
55C2×Dic15C2311yescharacteristic
56C2×Dic15C2311yescharacteristic
57C4×D15—22no—
58C2×C5×Dic3C2312yes—
59C2×C5×Dic3C2312yes—
60C5×Dic6—22no—
61C2×C3×Dic5C2312yes—
62C2×C3×Dic5C2312yes—
63#1200—24no—
64C3×#117—51no—
65C4×#67—51no—
66#924D511yescharacteristic, complemented
67#897—51no—
68D3×C4⋊C4—51no—
69#890—51no—
70#887—51no—
71#883—52no—
72#879—52no—
73#872—52no—
74#870—52no—
75C2×C5×Q8—31no—
76C22×C4×C5C2×D311yescharacteristic
77C2×C22⋊D5—31no—
78C2×C4×D5—31no—
79C2×C4×D5—32no—
80C5×C4⋊C4—32no—
81C5×C4⋊C4—31no—
82C5×C22⋊C4—32no—
83C22⋊Dic5—31no—
84C22⋊Dic5C2×D312yes—
85#685—31no—
86#685—32no—
87C4⋊Dic5—32no—
88C4⋊Dic5—32no—
89C4⋊Dic5C2×D311yescharacteristic
90C4⋊Dic5C2×D311yescharacteristic
91#683—31no—
92#683—32no—
93#683—32no—
94#683—32no—
95C4×Dic5—31no—
96C4×Dic5—32no—
97C4×Dic5C2×D312yes—
98#552—151noSylow
99C22×C3×C5—21no—
100C22×C3×C5—21no—
101C22×C3×C5—21no—
102C22×C3×C5C2411yescharacteristic
103C2×D15—21no—
104C2×D15—21no—
105C2×D15#4112yes—
106C3×C4×C5—21no—
107C3×C4×C5#4112yes—
108Dic15—21no—
109Dic15—21no—
110Dic15—21no—
111C3×Dic5—22no—
112C3×Dic5—22no—
113C5×Dic3—22no—
114C5×Dic3—22no—
115C22×C3×C4C2×D511yescharacteristic
116C2×#67—51no—
117C2×C4×D3—51no—
118C2×C4×D3—52no—
119C2×Dic6C2×D511yescharacteristic
120C3×C4⋊C4—51no—
121C3×C4⋊C4—51no—
122C3×C22⋊C4—52no—
123C3×C42—52no—
124#217—51no—
125#217C2×D512yes—
126(C2×D3)⋊C4—51no—
127(C2×D3)⋊C4—52no—
128C4⋊Dic3—52no—
129C4⋊Dic3—52no—
130C4⋊Dic3C2×D511yescharacteristic
131Dic3⋊C4—51no—
132Dic3⋊C4—52no—
133Dic3⋊C4—52no—
134Dic3⋊C4—52no—
135Dic3⋊C4C2×D512yes—
136C4×Dic3—51no—
137C4×Dic3—52no—
138C23×C5C22×D311yescharacteristic
139C22×D5—31no—
140C5×Q8—62no—
141C2×C4×C5—22no—
142C2×C4×C5—32no—
143C2×C4×C5—32no—
144C2×C4×C5C22×D311yescharacteristic
145C2×C4×C5C22×D311yescharacteristic
146C22⋊D5—62no—
147C2×Dic5—31no—
148C2×Dic5—31no—
149C2×Dic5—31no—
150C2×Dic5C22×D312yes—
151C2×Dic5C22×D312yes—
152C4×D5—64no—
153C4×D5—62no—
154#126—152no—
155#125—152no—
156#125—151no—
157#123—152no—
158C22⋊Q8—31no—
159C22⋊Q8—151no—
160C4×Q8—151no—
161C4×D4—151no—
162#117—51no—
163#117—152no—
164C2×C4⋊C4—151no—
165C2×C3×C5—41no—
166C2×C3×C5C2×#4111yescharacteristic
167C2×C3×C5C2×#4111yescharacteristic
168C2×C3×C52−1+411yescharacteristic
169D15—22no—
170C23×C3C22×D511yescharacteristic
171C22×D3—51no—
172C2×C3×C4—22no—
173C2×C3×C4—52no—
174C2×C3×C4—52no—
175C2×C3×C4C22×D511yescharacteristic
176C2×C3×C4C22×D511yescharacteristic
177#67—102no—
178C2×Dic3—51no—
179C2×Dic3—51no—
180C2×Dic3—51no—
181C2×Dic3C22×D512yes—
182C2×Dic3C22×D512yes—
183C4×D3—104no—
184C4×D3—102no—
185Dic6—22no—
186C22×C5—21no—
187C22×C5—21no—
188C22×C5—21no—
189C22×C5C23×D311yescharacteristic
190C2×D5—32no—
191C2×D5—61no—
192C2×D5—61no—
193C4×C5—21no—
194C4×C5—62no—
195C4×C5—62no—
196C4×C5#23512yes—
197Dic5—22no—
198Dic5—22no—
199Dic5—61no—
200Dic5—61no—
201Dic5—61no—
202C2×Q8—31no—
203C2×D4—151no—
204C22×C4—152no—
205C22×C4—151no—
206C22×C4D3×D511yescharacteristic
207C4⋊C4—31no—
208C4⋊C4—32no—
209C4⋊C4—51no—
210C4⋊C4—51no—
211C4⋊C4—152no—
212C4⋊C4—152no—
213C4⋊C4—152no—
214C4⋊C4—152no—
215C4⋊C4—152no—
216C4⋊C4—151no—
217C22⋊C4—32no—
218C22⋊C4—52no—
219C22⋊C4—152no—
220C22⋊C4—151no—
221C22⋊C4—151no—
222C42—52no—
223C42—152no—
224C42—151no—
225C3×C5#55211yescharacteristic, complemented, Hall
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3C23×D511yescharacteristic
230C2×D3—52no—
231C2×D3—101no—
232C2×D3—101no—
233C3×C4—21no—
234C3×C4—102no—
235C3×C4—102no—
236C3×C4#71112yes—
237Dic3—22no—
238Dic3—22no—
239Dic3—101no—
240Dic3—101no—
241Dic3—101no—
242C2×C5—41no—
243C2×C5C2×#23511yescharacteristic
244C2×C5D3×#4111yescharacteristic
245C2×C5#101011yescharacteristic
246D5—62no—
247C23—151no—
248C23C2×D3×D511yescharacteristic
249Q8—62no—
250D4—302no—
251C2×C4—22no—
252C2×C4—32no—
253C2×C4—32no—
254C2×C4—52no—
255C2×C4—52no—
256C2×C4—151no—
257C2×C4—151no—
258C2×C4—151no—
259C2×C4—304no—
260C2×C4—302no—
261C2×C4C2×D3×D511yescharacteristic
262C2×C4C2×D3×D511yescharacteristic
263C2×C3—41no—
264C2×C3C2×#71111yescharacteristic
265C2×C3#414711yescharacteristic
266C2×C3D5×#4111yescharacteristic
267D3—102no—
268C5#552911yescharacteristic, complemented, Sylow
269C22—21no—
270C22—21no—
271C22—21no—
272C22—152no—
273C22—301no—
274C22—301no—
275C22C22×D3×D511yescharacteristic
276C4—21no—
277C4—62no—
278C4—62no—
279C4—102no—
280C4—102no—
281C4—301no—
282C4—301no—
283C4—301no—
284C4#678212yes—
285C3#6597311yescharacteristic, complemented, Sylow
286C2—41no—
287C2—302no—
288C2#9200011yescharacteristic
289C2#9201611yescharacteristic
290C2C2×#678211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
prime radicalC2×#1220C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#552
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C5×#924C2semidirect
3. C3×#4072C2semidirect
5. #91531C2semidirect
6. #91437C2semidirect
7. #91402C2semidirect
8. #91384C2semidirect
9. #91383C2semidirect
10. #91378C2semidirect
11. #91344C2semidirect
41. #4072D3semidirect
66. #924D5semidirect
225. C3×C5#552semidirect
268. C5#5529semidirect
285. C3#65973semidirect

Maximal subgroups

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

subgroupindex
C4×#12202
C5×#9242
C3×#40722
#915622
#915312
#914372
#914022
#913842
#913832
#913782
#913442
#659733
#55295