Finite Group Databasebeta

← #11756423

Subgroups — #11756423

Order 960.

472 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C3×C5×#123C211yesmaximal, characteristic, complemented
2#91829C211yesmaximal, characteristic
3#91824C211yesmaximal, characteristic, complemented
4C4×#1220C212yesmaximal, complemented
5#91783C212yesmaximal, complemented
6#91776C212yesmaximal, complemented
7#91774C212yesmaximal, complemented
8#91774C212yesmaximal, complemented
9#91770C212yesmaximal, complemented
10#65998—31nomaximal, Hall
11C2×C3×C5×D4C2211yescharacteristic
12C22×C3×C4×C5C2211yescharacteristic
13C2×#1220C2212yes—
14C2×#1220C2212yes—
15C22×Dic15C2211yescharacteristic
16C2×D60C2211yescharacteristic
17C2×C4×D15C2212yes—
18C2×Dic30C2212yes—
19C3×C5×C4⋊C4C2212yes—
20C3×C5×C22⋊C4C2212yes—
21C3×C5×C22⋊C4C2211yescharacteristic
22#6719C2211yescharacteristic
23#6719C2212yes—
24#6714C2212yes—
25#6714C2212yes—
26#6714C2212yes—
27#6712C2212yes—
28#6712C2212yes—
29C4×Dic15C2212yes—
30C4×Dic15C2212yes—
31C4×Dic15C2211yescharacteristic
32#5664—51nomaximal, Hall
33C5×#123D311yescharacteristic, complemented
34#4086—31no—
35#4081—31no—
36C4×C22⋊D5—32no—
37#4040—32no—
38#4033—32no—
39#4031—32no—
40#4031—32no—
41#4027—32no—
42C23×C3×C5C2311yescharacteristic
43C23×C3×C5C2311yescharacteristic
44C22×D15C2312yes—
45C3×C5×D4—41no—
46C2×C3×C4×C5—41no—
47C2×C3×C4×C5C2311yescharacteristic
48C2×C3×C4×C5C2312yes—
49C2×C3×C4×C5C2312yes—
50#1220—24no—
51#1220—24no—
52C2×Dic15—41no—
53C2×Dic15C2312yes—
54C2×Dic15C2312yes—
55C2×Dic15C2312yes—
56D60—41no—
57C4×D15—42no—
58Dic30—24no—
59C3×#123D511yescharacteristic, complemented
60#940—51no—
61#935—51no—
62C4×#67—52no—
63#894—52no—
64#887—52no—
65#885—52no—
66#885—52no—
67#67⋊C4—52no—
68C2×C5×D4C2×D311yescharacteristic
69C22×C4×C5C2×D311yescharacteristic
70C2×C22⋊D5—32no—
71C2×C22⋊D5—32no—
72C22×Dic5—31no—
73C2×D20—31no—
74C2×C4×D5—32no—
75C2×Dic10—32no—
76C5×C4⋊C4C2×D312yes—
77C5×C22⋊C4C2×D312yes—
78C5×C22⋊C4C2×D311yescharacteristic
79C22⋊Dic5—31no—
80C22⋊Dic5—32no—
81#685—32no—
82#685—32no—
83#685—32no—
84#683—32no—
85#683—32no—
86C4×Dic5—32no—
87C4×Dic5—32no—
88C4×Dic5—31no—
89#559—151noSylow
90C22×C3×C5—22no—
91C22×C3×C5—22no—
92C22×C3×C5—21no—
93C22×C3×C5—21no—
94C22×C3×C5C2411yescharacteristic
95C2×D15—22no—
96C2×D15—22no—
97C2×D15—22no—
98C3×C4×C5—22no—
99C3×C4×C5—22no—
100C3×C4×C5—21no—
101Dic15—22no—
102Dic15—22no—
103Dic15#4114yes—
104C2×C3×D4C2×D511yescharacteristic
105C22×C3×C4C2×D511yescharacteristic
106C2×#67—52no—
107C2×#67—52no—
108C22×Dic3—51no—
109C2×D12—51no—
110C2×C4×D3—52no—
111C2×Dic6—52no—
112C3×C4⋊C4C2×D512yes—
113C3×C22⋊C4C2×D512yes—
114C3×C22⋊C4C2×D511yescharacteristic
115#217—51no—
116#217—52no—
117(C2×D3)⋊C4—52no—
118(C2×D3)⋊C4—52no—
119(C2×D3)⋊C4—52no—
120Dic3⋊C4—52no—
121Dic3⋊C4—52no—
122C4×Dic3—52no—
123C4×Dic3—52no—
124C4×Dic3—51no—
125C23×C5C22×D311yescharacteristic
126C23×C5C22×D311yescharacteristic
127C22×D5—32no—
128C5×D4—41no—
129C2×C4×C5—41no—
130C2×C4×C5C22×D311yescharacteristic
131C2×C4×C5C22×D312yes—
132C2×C4×C5C22×D312yes—
133C22⋊D5—64no—
134C22⋊D5—64no—
135C2×Dic5—32no—
136C2×Dic5—32no—
137C2×Dic5—32no—
138C2×Dic5—121no—
139D20—121no—
140C4×D5—122no—
141Dic10—64no—
142#127—151no—
143#124—152no—
144#124—152no—
145#123—151no—
146#123—152no—
147#123D1511yescharacteristic, complemented
148C4×Q8—152no—
149C4×D4—152no—
150C4×D4—152no—
151C2×C3×C5—41no—
152C2×C3×C5—41no—
153C2×C3×C5C2×#4112yes—
154C2×C3×C5H(5,𝔽2)11yescharacteristic
155D15—42no—
156C23×C3C22×D511yescharacteristic
157C23×C3C22×D511yescharacteristic
158C22×D3—52no—
159C3×D4—41no—
160C2×C3×C4—41no—
161C2×C3×C4C22×D511yescharacteristic
162C2×C3×C4C22×D512yes—
163C2×C3×C4C22×D512yes—
164#67—104no—
165#67—104no—
166C2×Dic3—52no—
167C2×Dic3—52no—
168C2×Dic3—52no—
169C2×Dic3—201no—
170D12—201no—
171C4×D3—202no—
172Dic6—104no—
173C22×C5—21no—
174C22×C5—21no—
175C22×C5—22no—
176C22×C5—22no—
177C22×C5C23×D311yescharacteristic
178C2×D5—62no—
179C2×D5—62no—
180C2×D5—62no—
181C4×C5—22no—
182C4×C5—21no—
183C4×C5—22no—
184Dic5—34no—
185Dic5—62no—
186Dic5—62no—
187C2×Q8—152no—
188C2×D4—152no—
189C2×D4—152no—
190C2×D4—151no—
191C2×D4C2×D1511yescharacteristic
192C22×C4—151no—
193C22×C4—152no—
194C22×C4C2×D1511yescharacteristic
195C4⋊C4—152no—
196C4⋊C4—152no—
197C4⋊C4C2×D1512yes—
198C22⋊C4—152no—
199C22⋊C4—151no—
200C22⋊C4—152no—
201C22⋊C4—152no—
202C22⋊C4—152no—
203C22⋊C4C2×D1512yes—
204C22⋊C4C2×D1511yescharacteristic
205C42—152no—
206C42—152no—
207C42—151no—
208C3×C5#55911yescharacteristic, complemented, Hall
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3C23×D511yescharacteristic
214C2×D3—102no—
215C2×D3—102no—
216C2×D3—102no—
217C3×C4—22no—
218C3×C4—21no—
219C3×C4—22no—
220Dic3—54no—
221Dic3—102no—
222Dic3—102no—
223C2×C5—41no—
224C2×C5—41no—
225C2×C5#100411yescharacteristic
226C2×C5D3×#4112yes—
227D5—122no—
228C23—152no—
229C23C22×D1511yescharacteristic
230C23C22×D1511yescharacteristic
231Q8—304no—
232D4—41no—
233D4—304no—
234D4—304no—
235D4—601no—
236C2×C4—41no—
237C2×C4—152no—
238C2×C4—152no—
239C2×C4—152no—
240C2×C4—602no—
241C2×C4—601no—
242C2×C4C22×D1512yes—
243C2×C4C22×D1512yes—
244C2×C4C22×D1511yescharacteristic
245C2×C3—41no—
246C2×C3—41no—
247C2×C3#414411yescharacteristic
248C2×C3D5×#4112yes—
249D3—202no—
250C5#566411yescharacteristic, complemented, Sylow
251C22—21no—
252C22—21no—
253C22—22no—
254C22—22no—
255C22—302no—
256C22—302no—
257C22—302no—
258C22C23×D1511yescharacteristic
259C4—22no—
260C4—21no—
261C4—22no—
262C4—154no—
263C4—302no—
264C4—302no—
265C3#6599811yescharacteristic, complemented, Sylow
266C2—41no—
267C2—41no—
268C2—602no—
269C2#9209511yescharacteristic
270C2#41×D1512yes—
subgroupG/H
center, FrattiniC22C23×D15
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×#123C2
prime radical#91829C2
hypercenter#123D15
hypocenterC3×C5#559
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C3×C5×#123C2semidirect
3. #91824C2semidirect
4. C4×#1220C2semidirect
5. #91783C2semidirect
6. #91776C2semidirect
7. #91774C2semidirect
8. #91774C2semidirect
9. #91770C2semidirect
33. C5×#123D3semidirect
59. C3×#123D5semidirect
147. #123D15semidirect
208. C3×C5#559semidirect
250. C5#5664semidirect
265. C3#65998semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#1232
#918292
#918242
C4×#12202
#917832
#917762
#917742
#917702
#659983
#56645