Finite Group Databasebeta

← #13087684

Subgroups — #13087684

Order 1440.

334 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 79 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×C5×#634C211yesmaximal, characteristic, complemented
2C2×#67154C211yesmaximal, characteristic
3C3×#6714C211yesmaximal, characteristic
4C3×#6712C211yesmaximal, characteristic, complemented
5#10623832C211yesmaximal, characteristic, complemented
6#10623831C211yesmaximal, characteristic, complemented
7#10623829C211yesmaximal, characteristic
8#91424—31nomaximal
9#91421—31nomaximal
10C22×C3×D15C2211yescharacteristic
11C2×C32×C4×C5C2211yescharacteristic
12C2×C5×C3⋊Dic3C2211yescharacteristic
13C2×C5×C3⋊Dic3C2211yescharacteristic
14C5×#634—22no—
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×Dic15C2211yescharacteristic
17C2×C3×Dic15C2211yescharacteristic
18#67154—22no—
19#63896—51nomaximal, Hall
20C2×C5×Dic6—31no—
21C2×C5×Dic6—31no—
22C2×C5×Dic6—61no—
23C2×#1200—31no—
24C2×D5×Dic3—31no—
25#6714D311yescharacteristic
26#6712D311yescharacteristic, complemented
27C3×#685—31no—
28C3×#683—31no—
29Dic3⋊Dic5—31no—
30Dic3⋊Dic5—31no—
31#6669—31no—
32#6668—31no—
33#6667—31no—
34#6665—31no—
35C22×C32×C5C2311yescharacteristic
36C2×C3×D15—21no—
37C2×C3×D15—21no—
38C2×C3×D15Q812yescomplemented
39C32×C4×C5—21no—
40C5×C3⋊Dic3—21no—
41C5×C3⋊Dic3D412yes—
42C3×Dic15—21no—
43C3×Dic15—21no—
44C3×Dic15—21no—
45#4092—91noHall
46C2×#634D511yescharacteristic, complemented
47C2×D3×Dic3—51no—
48C3×(C2×D3)⋊C4—51no—
49C3×Dic3⋊C4—51no—
50#3724—51no—
51Dic3⋊Dic3—51no—
52(C2×D3)⋊Dic3—51no—
53C22×D15C2×D311yescharacteristic
54C22×C3×D5—31no—
55C2×C3×C4×C5—21no—
56C2×C3×C4×C5C2×D311yescharacteristic
57C2×C3×C4×C5C2×D311yescharacteristic
58C2×Dic15C2×D311yescharacteristic
59C2×Dic15C2×D311yescharacteristic
60C2×Dic15C2×D311yescharacteristic
61C2×C5×Dic3—31no—
62C2×C5×Dic3—31no—
63C2×C5×Dic3—31no—
64C2×C5×Dic3—31no—
65C2×C5×Dic3—61no—
66C2×C5×Dic3—61no—
67C5×Dic6—62no—
68C5×Dic6—62no—
69C5×Dic6—64no—
70C2×C3×Dic5—31no—
71C2×C3×Dic5—31no—
72C2×C3×Dic5—31no—
73#1200—62no—
74D5×Dic3—62no—
75#896—151no—
76#878—151no—
77C2×C32×C5C2×D411yescharacteristic
78C2×C32×C5C2×Q811yescharacteristic
79C2×C32×C5#4111yescharacteristic
80C3×D15—22no—
81C2×C5×Q8—91no—
82C2×C4×D5—91no—
83#685—31no—
84#685—91no—
85C4⋊Dic5—91no—
86#683—31no—
87#683—91no—
88C22×C3×D3—51no—
89C2×C32×C4C2×D511yescharacteristic
90C2×C3⋊Dic3C2×D511yescharacteristic
91C2×C3⋊Dic3C2×D511yescharacteristic
92#634—22no—
93C2×C3×Dic3—51no—
94C2×C3×Dic3—51no—
95C2×C3×Dic3—51no—
96D3×Dic3—102no—
97C22×C3×C5—21no—
98C22×C3×C5C22×D311yescharacteristic
99C22×C3×C5C22×D311yescharacteristic
100C2×D15—21no—
101C2×D15—21no—
102C2×D15Dic612yescomplemented
103C2×C3×D5—32no—
104C2×C3×D5—61no—
105C2×C3×D5—61no—
106C3×C4×C5—22no—
107C3×C4×C5—21no—
108C3×C4×C5—21no—
109Dic15—21no—
110Dic15—21no—
111Dic15—21no—
112C3×Dic5—61no—
113C3×Dic5—61no—
114C3×Dic5—61no—
115C5×Dic3—32no—
116C5×Dic3—32no—
117C5×Dic3—62no—
118C5×Dic3—62no—
119C5×Dic3—61no—
120C5×Dic3—61no—
121C22×Dic3—151no—
122C2×C4×D3—151no—
123C2×Dic6—31no—
124C2×Dic6—31no—
125C2×Dic6—61no—
126C3×C4⋊C4—151no—
127C3×C22⋊C4—151no—
128#217—151no—
129(C2×D3)⋊C4—51no—
130(C2×D3)⋊C4—151no—
131C4⋊Dic3—151no—
132Dic3⋊C4—51no—
133Dic3⋊C4—151no—
134Dic3⋊C4—151no—
135Dic3⋊C4—151no—
136C32×C5C22⋊Q811yescharacteristic, complemented, Hall
137C22×D5—31no—
138C5×Q8—182no—
139C2×C4×C5—91no—
140C2×C4×C5—91no—
141C2×C4×C5D3211yescharacteristic
142C2×Dic5—31no—
143C2×Dic5—31no—
144C2×Dic5—31no—
145C4×D5—182no—
146C22×C32C22×D511yescharacteristic
147C2×C3×D3—52no—
148C2×C3×D3—101no—
149C2×C3×D3—101no—
150C32×C4—21no—
151C3⋊Dic3—21no—
152C3⋊Dic3C22⋊D512yes—
153C3×Dic3—101no—
154C3×Dic3—101no—
155C3×Dic3—101no—
156C22⋊Q8—451noSylow
157C2×C3×C5—21no—
158C2×C3×C5—21no—
159C2×C3×C5—21no—
160C2×C3×C5C2×Dic611yescharacteristic
161C2×C3×C5#23511yescharacteristic
162C2×C3×C5D3×D411yescharacteristic
163C2×C3×C5D3×D411yescharacteristic
164C2×C3×C5#23711yescharacteristic
165C2×C3×C5D3×Q811yescharacteristic
166D15—22no—
167C3×D5—62no—
168C23×C3—151no—
169C22×D3—51no—
170C2×C3×C4—21no—
171C2×C3×C4—151no—
172C2×C3×C4—151no—
173C2×C3×C4—151no—
174C2×C3×C4D3×D511yescharacteristic
175C2×C3×C4D3×D511yescharacteristic
176C2×Dic3—31no—
177C2×Dic3—31no—
178C2×Dic3—31no—
179C2×Dic3—31no—
180C2×Dic3—51no—
181C2×Dic3—51no—
182C2×Dic3—51no—
183C2×Dic3—61no—
184C2×Dic3—61no—
185C2×Dic3—302no—
186C4×D3—302no—
187Dic6—62no—
188Dic6—62no—
189Dic6—64no—
190C22×C5C2×D3211yescharacteristic
191C2×D5—32no—
192C2×D5—61no—
193C2×D5—61no—
194C4×C5—21no—
195C4×C5—92no—
196C4×C5—181no—
197Dic5—61no—
198Dic5—61no—
199Dic5—61no—
200C2×C32Q8×D511yescharacteristic
201C2×C32#71311yescharacteristic
202C2×C32C2×C22⋊D511yescharacteristic
203C3×D3—102no—
204C2×Q8—91no—
205C22×C4—451no—
206C4⋊C4—151no—
207C4⋊C4—451no—
208C4⋊C4—451no—
209C22⋊C4—151no—
210C22⋊C4—451no—
211C3×C5—21no—
212C3×C5#87811yescharacteristic, complemented
213C3×C5#89611yescharacteristic, complemented
214C22×C3—21no—
215C22×C3—152no—
216C22×C3—301no—
217C22×C3—301no—
218C22×C3C2×D3×D511yescharacteristic
219C22×C3C2×D3×D511yescharacteristic
220C2×D3—52no—
221C2×D3—101no—
222C2×D3—101no—
223C3×C4—22no—
224C3×C4—21no—
225C3×C4—21no—
226C3×C4—301no—
227C3×C4—301no—
228C3×C4—301no—
229Dic3—32no—
230Dic3—32no—
231Dic3—62no—
232Dic3—62no—
233Dic3—61no—
234Dic3—61no—
235Dic3—101no—
236Dic3—101no—
237Dic3—101no—
238C2×C5D3×Dic611yescharacteristic
239C2×C5#379511yescharacteristic
240C2×C5#381111yescharacteristic
241D5—62no—
242C32#409211yescharacteristic, complemented, Sylow
243C23—151no—
244Q8—182no—
245C2×C4—91no—
246C2×C4—91no—
247C2×C4—151no—
248C2×C4—151no—
249C2×C4—151no—
250C2×C4—902no—
251C2×C4(C3⋊D3)⋊D511yescharacteristic
252C2×C3—21no—
253C2×C3—21no—
254C2×C3—21no—
255C2×C3—302no—
256C2×C3D5×Dic611yescharacteristic
257C2×C3#676611yescharacteristic
258C2×C3D15⋊Q811yescharacteristic
259C2×C3#677311yescharacteristic
260C2×C3D3×C22⋊D511yescharacteristic
261C2×C3D3×C22⋊D511yescharacteristic
262D3—102no—
263C5#6389611yescharacteristic, complemented, Sylow
264C22—152no—
265C22—301no—
266C22—301no—
267C22C2×(C3⋊D3)⋊D511yescharacteristic
268C4—21no—
269C4—92no—
270C4—181no—
271C4—301no—
272C4—301no—
273C4—301no—
274C3—21no—
275C3#9142111yescharacteristic, complemented
276C3#9142411yescharacteristic, complemented
277C2—302no—
278C2#1062421911yescharacteristic
279C2#1062422111yescharacteristic
280C2#1062423611yescharacteristic
subgroupG/H
center, FrattiniC22C2×(C3⋊D3)⋊D5
commutator, socle, cosocleC22×C32×C5C23
FittingC2×C32×C4×C5C22
prime radicalC22×C3×D15C22
hypercenterC2×C4(C3⋊D3)⋊D5
hypocenterC32×C5C22⋊Q8
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C5×#634C2semidirect
4. C3×#6712C2semidirect
5. #10623832C2semidirect
6. #10623831C2semidirect
26. #6712D3semidirect
38. C2×C3×D15Q8semidirect
46. C2×#634D5semidirect
102. C2×D15Dic6semidirect
136. C32×C5C22⋊Q8semidirect
212. C3×C5#878semidirect
213. C3×C5#896semidirect
242. C32#4092semidirect
263. C5#63896semidirect
275. C3#91421semidirect
276. C3#91424semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#6342
C2×#671542
C3×#67142
C3×#67122
#106238322
#106238312
#106238292
#914243
#914213
#638965