Finite Group Databasebeta

← #13087716

Subgroups — #13087716

Order 1440.

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

normal subgroups only

This group has 322 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×C3×C4×D15C211yesmaximal, characteristic
2C2×#67158C211yesmaximal, characteristic, complemented
3C5×#3750C211yesmaximal, characteristic, complemented
4C3×#6712C211yesmaximal, characteristic, complemented
5#10623831C211yesmaximal, characteristic, complemented
6#10623829C211yesmaximal, characteristic
7#10623829C211yesmaximal, characteristic
8#91377—31nomaximal
9#91349—31nomaximal
10C22×C3×D15C2211yescharacteristic
11C2×C32×C4×C5C2211yescharacteristic
12C2×C5×C3⋊Dic3C2211yescharacteristic
13C2×C5×C3⋊Dic3C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×Dic15C2211yescharacteristic
17C3×C4×D15—22no—
18#67158—22no—
19#63851—51nomaximal, Hall
20C2×C4×D15D311yescharacteristic
21C2×C3×C4×D5—31no—
22C2×#1204—31no—
23C2×#1204—31no—
24#6712D311yescharacteristic, complemented
25C5×Dic3⋊C4—31no—
26C5×Dic3⋊C4—31no—
27C5×Dic3⋊C4—61no—
28C3×#683—31no—
29#6669—31no—
30#6668—31no—
31#6667—31no—
32#6667—31no—
33#6665—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⋊Dic3—21no—
42C3×Dic15—21no—
43C3×Dic15—21no—
44C3×Dic15D412yes—
45#4042—91noHall
46C2×C3×C4×D3—51no—
47C2×#627—51no—
48#3750D511yescharacteristic, complemented
49C3×Dic3⋊C4—51no—
50Dic3⋊Dic3—51no—
51(C2×D3)⋊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
61C4×D15—22no—
62C2×C5×Dic3—31no—
63C2×C5×Dic3—31no—
64C2×C5×Dic3—31no—
65C2×C5×Dic3—31no—
66C2×C5×Dic3—61no—
67C2×C5×Dic3—61no—
68C2×C3×Dic5—31no—
69C2×C3×Dic5—31no—
70C2×C3×Dic5—31no—
71C3×C4×D5—62no—
72#1204—62no—
73#1204—62no—
74#924—151no—
75#896—151no—
76C2×C32×C5C2×D411yescharacteristic
77C2×C32×C5C2×Q811yescharacteristic
78C2×C32×C5#4111yescharacteristic
79C3×D15—22no—
80C2×C4×D5—31no—
81C2×Dic10—91no—
82C5×C4⋊C4—91no—
83#685—91no—
84#685—91no—
85#683—31no—
86#683—91no—
87C22×C3×D3—51no—
88C2×C32×C4C2×D511yescharacteristic
89C2×C3⋊Dic3C2×D511yescharacteristic
90C2×C3⋊Dic3C2×D511yescharacteristic
91C2×C3×Dic3—51no—
92C2×C3×Dic3—51no—
93C2×C3×Dic3—51no—
94C3×C4×D3—102no—
95#627—102no—
96C22×C3×C5—21no—
97C22×C3×C5C22×D311yescharacteristic
98C22×C3×C5C22×D311yescharacteristic
99C2×D15—21no—
100C2×D15—21no—
101C2×D15Dic612yescomplemented
102C2×C3×D5—32no—
103C2×C3×D5—61no—
104C2×C3×D5—61no—
105C3×C4×C5—21no—
106C3×C4×C5—21no—
107C3×C4×C5—41no—
108Dic15—21no—
109Dic15—21no—
110Dic15#6712yes—
111C3×Dic5—32no—
112C3×Dic5—61no—
113C3×Dic5—61no—
114C5×Dic3—62no—
115C5×Dic3—61no—
116C5×Dic3—61no—
117C5×Dic3—61no—
118C5×Dic3—61no—
119C5×Dic3—121no—
120C22×C3×C4—151no—
121C2×C4×D3—51no—
122C2×Dic6—151no—
123C2×Dic6—151no—
124C3×C4⋊C4—151no—
125#217—151no—
126#217—151no—
127(C2×D3)⋊C4—151no—
128(C2×D3)⋊C4—151no—
129C4⋊Dic3—151no—
130Dic3⋊C4—31no—
131Dic3⋊C4—31no—
132Dic3⋊C4—51no—
133Dic3⋊C4—61no—
134Dic3⋊C4—151no—
135C32×C5C22⋊Q811yescharacteristic, complemented, Hall
136C22×D5—31no—
137C2×C4×C5—91no—
138C2×C4×C5—91no—
139C2×C4×C5D3211yescharacteristic
140C2×Dic5—31no—
141C2×Dic5—31no—
142C2×Dic5—31no—
143C4×D5—62no—
144Dic10—182no—
145C22×C32C22×D511yescharacteristic
146C2×C3×D3—52no—
147C2×C3×D3—101no—
148C2×C3×D3—101no—
149C32×C4—21no—
150C3⋊Dic3—21no—
151C3⋊Dic3—21no—
152C3×Dic3—52no—
153C3×Dic3—101no—
154C3×Dic3—101no—
155C22⋊Q8—451noSylow
156C2×C3×C5—21no—
157C2×C3×C5—21no—
158C2×C3×C5—21no—
159C2×C3×C5C2×Dic611yescharacteristic
160C2×C3×C5#23511yescharacteristic
161C2×C3×C5#23511yescharacteristic
162C2×C3×C5D3×D411yescharacteristic
163C2×C3×C5D3×Q811yescharacteristic
164C2×C3×C5C2×#6711yescharacteristic
165D15—22no—
166C3×D5—62no—
167C23×C3—151no—
168C22×D3—51no—
169C2×C3×C4—21no—
170C2×C3×C4—151no—
171C2×C3×C4—151no—
172C2×C3×C4—151no—
173C2×C3×C4—302no—
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—
185C4×D3—102no—
186Dic6—302no—
187Dic6—302no—
188C22×C5C2×D3211yescharacteristic
189C2×D5—32no—
190C2×D5—61no—
191C2×D5—61no—
192C4×C5—21no—
193C4×C5—181no—
194C4×C5—181no—
195Dic5—32no—
196Dic5—61no—
197Dic5—61no—
198C2×C32#70911yescharacteristic
199C2×C32D4×D511yescharacteristic
200C2×C32Q8×D511yescharacteristic
201C3×D3—102no—
202C2×Q8—451no—
203C22×C4—151no—
204C4⋊C4—91no—
205C4⋊C4—151no—
206C4⋊C4—451no—
207C22⋊C4—451no—
208C22⋊C4—451no—
209C3×C5—21no—
210C3×C5#89611yescharacteristic, complemented
211C3×C5#92411yescharacteristic, complemented
212C22×C3—21no—
213C22×C3—152no—
214C22×C3—301no—
215C22×C3—301no—
216C22×C3C2×D3×D511yescharacteristic
217C22×C3C2×D3×D511yescharacteristic
218C2×D3—52no—
219C2×D3—101no—
220C2×D3—101no—
221C3×C4—21no—
222C3×C4—21no—
223C3×C4—41no—
224C3×C4—152no—
225C3×C4—301no—
226C3×C4—301no—
227Dic3—52no—
228Dic3—62no—
229Dic3—61no—
230Dic3—61no—
231Dic3—61no—
232Dic3—61no—
233Dic3—101no—
234Dic3—101no—
235Dic3—121no—
236C2×C5D3×Dic611yescharacteristic
237C2×C5#379811yescharacteristic
238C2×C5D3×#6711yescharacteristic
239D5—62no—
240C32#404211yescharacteristic, complemented, Sylow
241C23—151no—
242Q8—902no—
243C2×C4—91no—
244C2×C4—91no—
245C2×C4—151no—
246C2×C4—151no—
247C2×C4—151no—
248C2×C4—302no—
249C2×C4(C3⋊D3)⋊D511yescharacteristic
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—302no—
254C2×C3D5×Dic611yescharacteristic
255C2×C3D15⋊Q811yescharacteristic
256C2×C3#677111yescharacteristic
257C2×C3#677111yescharacteristic
258C2×C3D5×#6711yescharacteristic
259C2×C3#679011yescharacteristic
260D3—102no—
261C5#6385111yescharacteristic, complemented, Sylow
262C22—152no—
263C22—301no—
264C22—301no—
265C22C2×(C3⋊D3)⋊D511yescharacteristic
266C4—21no—
267C4—152no—
268C4—181no—
269C4—181no—
270C4—301no—
271C4—301no—
272C3—21no—
273C3#9134911yescharacteristic, complemented
274C3#9137711yescharacteristic, complemented
275C2—302no—
276C2#1062421911yescharacteristic
277C2#1062422311yescharacteristic
278C2#1062423511yescharacteristic
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
2. C2×#67158C2semidirect
3. C5×#3750C2semidirect
4. C3×#6712C2semidirect
5. #10623831C2semidirect
24. #6712D3semidirect
38. C2×C3×D15Q8semidirect
48. #3750D5semidirect
101. C2×D15Dic6semidirect
135. C32×C5C22⋊Q8semidirect
210. C3×C5#896semidirect
211. C3×C5#924semidirect
240. C32#4042semidirect
261. C5#63851semidirect
273. C3#91349semidirect
274. C3#91377semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D152
C2×#671582
C5×#37502
C3×#67122
#106238312
#106238292
#913773
#913493
#638515