Finite Group Databasebeta

← #13087451

Subgroups — #13087451

Order 1440.

338 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 82 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×C5×D3C211yesmaximal, characteristic, complemented
2C2×#67150C211yesmaximal, characteristic
3#10623906C211yesmaximal, characteristic, complemented
4C3×#6714C211yesmaximal, characteristic, complemented
5#10623814C211yesmaximal, characteristic, complemented
6(C2×D3)⋊Dic15C211yesmaximal, characteristic, complemented
7#10623811C211yesmaximal, characteristic, complemented
8#91818—31nomaximal
9#91452—31nomaximal
10C22×C3×D15C2211yescharacteristic
11C22×C3×C5×D3C2211yescharacteristic
12C2×C32×C4×C5C2211yescharacteristic
13C2×C3⋊Dic15C2211yescharacteristic
14C2×C3⋊Dic15C2211yescharacteristic
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17C3×C4×C5×D3—22no—
18#67150—22no—
19#63905—51nomaximal, Hall
20C22×C3×C4×C5—31no—
21C2×#1220—31no—
22C2×C4×C5×D3D311yescharacteristic, complemented
23C2×#1201—31no—
24#6719—31no—
25#6714—31no—
26#6714D311yescharacteristic, complemented
27#6712—31no—
28#6712—31no—
29#6712—31no—
30#6712—61no—
31C3×#685—31no—
32#6668—31no—
33(C2×D3)⋊Dic5—31no—
34#6665—31no—
35C22×C32×C5C2311yescharacteristic
36C2×C3×D15—21no—
37C2×C3×D15—21no—
38C2×C3×D15—21no—
39C2×C3×C5×D3—21no—
40C2×C3×C5×D3—21no—
41C2×C3×C5×D3D412yes—
42C32×C4×C5—21no—
43C3⋊Dic15—21no—
44C3⋊Dic15—21no—
45C3×Dic15—21no—
46C3×C5×Dic3—21no—
47#4075—91noHall
48C2×C3×C4×D3D511yescharacteristic, complemented
49C2×#625—51no—
50#3750—51no—
51C3×(C2×D3)⋊C4—51no—
52Dic3⋊Dic3—51no—
53(C2×D3)⋊Dic3—51no—
54(C2×D3)⋊Dic3—51no—
55C23×C3×C5—31no—
56C22×D15C2×D311yescharacteristic
57C22×C5×D3C2×D311yescharacteristic
58C22×C3×D5—31no—
59C2×C3×C4×C5—21no—
60C2×C3×C4×C5—31no—
61C2×C3×C4×C5—62no—
62C2×C3×C4×C5C2×D311yescharacteristic
63C2×C3×C4×C5C2×D311yescharacteristic
64#1220—62no—
65C2×Dic15—31no—
66C2×Dic15—31no—
67C2×Dic15—31no—
68C2×Dic15—31no—
69C2×Dic15—61no—
70C2×Dic15—61no—
71C2×Dic15C2×D311yescharacteristic
72C2×C5×Dic3C2×D311yescharacteristic
73C4×C5×D3—22no—
74C2×C3×Dic5—31no—
75#1201—62no—
76#929—151no—
77#883—151no—
78C2×C32×C5C2×D411yescharacteristic
79C2×C32×C5#4111yescharacteristic
80C2×C32×C5#4111yescharacteristic
81C3×D15—41no—
82C3×C5×D3—22no—
83C22×C4×C5—31no—
84C2×C22⋊D5—91no—
85C22⋊Dic5—91no—
86#685—31no—
87#685—91no—
88#683—91no—
89#683—91no—
90C22×C3×D3—51no—
91C22×C3×D3C2×D511yescharacteristic
92C2×C32×C4C2×D511yescharacteristic
93C2×C3⋊Dic3—51no—
94C2×C3⋊Dic3—51no—
95C2×C3×Dic3—51no—
96C2×C3×Dic3C2×D511yescharacteristic
97C3×C4×D3—22no—
98#625—102no—
99C22×C3×C5—21no—
100C22×C3×C5—32no—
101C22×C3×C5—61no—
102C22×C3×C5—61no—
103C22×C3×C5C22×D311yescharacteristic
104C22×C3×C5C22×D311yescharacteristic
105C2×D15—21no—
106C2×D15—21no—
107C2×D15—21no—
108C2×C5×D3—21no—
109C2×C5×D3—21no—
110C2×C5×D3#6712yes—
111C2×C3×D5—61no—
112C2×C3×D5—61no—
113C2×C3×D5—61no—
114C3×C4×C5—21no—
115C3×C4×C5—21no—
116C3×C4×C5—41no—
117C3×C4×C5—61no—
118Dic15—21no—
119Dic15—62no—
120Dic15—61no—
121Dic15—61no—
122Dic15—61no—
123Dic15—61no—
124Dic15—121no—
125C3×Dic5—61no—
126C5×Dic3—21no—
127C22×C3×C4—31no—
128C2×#67—151no—
129C2×#67—151no—
130C2×C4×D3D1511yescharacteristic, complemented
131C3×C22⋊C4—151no—
132#217—151no—
133#217—151no—
134(C2×D3)⋊C4—51no—
135(C2×D3)⋊C4—151no—
136(C2×D3)⋊C4—151no—
137C4⋊Dic3—151no—
138Dic3⋊C4—151no—
139Dic3⋊C4—151no—
140Dic3⋊C4—151no—
141Dic3⋊C4—301no—
142C32×C5#12311yescharacteristic, complemented, Hall
143C23×C5—31no—
144C22×D5—31no—
145C2×C4×C5—31no—
146C2×C4×C5—62no—
147C2×C4×C5D3211yescharacteristic
148C22⋊D5—182no—
149C2×Dic5—31no—
150C2×Dic5—91no—
151C2×Dic5—91no—
152C22×C32C22×D511yescharacteristic
153C2×C3×D3—21no—
154C2×C3×D3—21no—
155C2×C3×D3—101no—
156C2×C3×D3—101no—
157C2×C3×D3—101no—
158C2×C3×D3C22⋊D512yes—
159C32×C4—21no—
160C3⋊Dic3—101no—
161C3⋊Dic3—101no—
162C3×Dic3—21no—
163C3×Dic3—101no—
164#123—451noSylow
165C2×C3×C5—21no—
166C2×C3×C5—21no—
167C2×C3×C5—21no—
168C2×C3×C5—62no—
169C2×C3×C5#23511yescharacteristic
170C2×C3×C5#23511yescharacteristic
171C2×C3×C5#23511yescharacteristic
172C2×C3×C5D3×D411yescharacteristic
173C2×C3×C5#23711yescharacteristic
174C2×C3×C5C2×#6711yescharacteristic
175D15—41no—
176C3×D5—121no—
177C5×D3—22no—
178C23×C3—31no—
179C23×C3—151no—
180C22×D3—51no—
181C22×D3C2×D1511yescharacteristic
182C2×C3×C4—21no—
183C2×C3×C4—31no—
184C2×C3×C4—62no—
185C2×C3×C4—151no—
186C2×C3×C4D3×D511yescharacteristic
187C2×C3×C4C2×D1511yescharacteristic
188#67—302no—
189#67—302no—
190C2×Dic3—51no—
191C2×Dic3—151no—
192C2×Dic3—151no—
193C2×Dic3—151no—
194C2×Dic3—151no—
195C2×Dic3—301no—
196C2×Dic3—301no—
197C2×Dic3C2×D1511yescharacteristic
198C4×D3—22no—
199C22×C5—32no—
200C22×C5—61no—
201C22×C5—61no—
202C22×C5C2×D3211yescharacteristic
203C2×D5—61no—
204C2×D5—61no—
205C2×D5—61no—
206C4×C5—21no—
207C4×C5—61no—
208Dic5—61no—
209Dic5—181no—
210Dic5—181no—
211C2×C32#70911yescharacteristic
212C2×C32#70911yescharacteristic
213C2×C32C2×C22⋊D511yescharacteristic
214C3×D3—22no—
215C3×D3—201no—
216C2×D4—451no—
217C22×C4—31no—
218C4⋊C4—451no—
219C4⋊C4—451no—
220C22⋊C4—151no—
221C22⋊C4—451no—
222C22⋊C4—451no—
223C3×C5—21no—
224C3×C5#88311yescharacteristic, complemented
225C3×C5#92911yescharacteristic, complemented
226C22×C3—21no—
227C22×C3—32no—
228C22×C3—61no—
229C22×C3—61no—
230C22×C3—301no—
231C22×C3—301no—
232C22×C3—301no—
233C22×C3C2×D3×D511yescharacteristic
234C22×C3C22×D1511yescharacteristic
235C2×D3—21no—
236C2×D3—21no—
237C2×D3—101no—
238C2×D3—101no—
239C2×D3—101no—
240C2×D3#122012yes—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—41no—
244C3×C4—61no—
245C3×C4—301no—
246Dic3—21no—
247Dic3—101no—
248Dic3—302no—
249Dic3—301no—
250Dic3—301no—
251Dic3—301no—
252Dic3—301no—
253Dic3—601no—
254C2×C5—62no—
255C2×C5#379511yescharacteristic
256C2×C5#379811yescharacteristic
257C2×C5D3×#6711yescharacteristic
258D5—121no—
259C32#407511yescharacteristic, complemented, Sylow
260C23—31no—
261C23—151no—
262D4—902no—
263C2×C4—31no—
264C2×C4—62no—
265C2×C4—151no—
266C2×C4—451no—
267C2×C4—451no—
268C2×C4D3×D1511yescharacteristic
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—21no—
272C2×C3—62no—
273C2×C3—601no—
274C2×C3#676511yescharacteristic
275C2×C3#677111yescharacteristic
276C2×C3D3×C22⋊D511yescharacteristic
277C2×C3#681711yescharacteristic
278C2×C3#681711yescharacteristic
279C2×C3C2×#122011yescharacteristic
280D3—22no—
281D3—201no—
282C5#6390511yescharacteristic, complemented, Sylow
283C22—32no—
284C22—61no—
285C22—61no—
286C22—301no—
287C22—301no—
288C22—301no—
289C22C2×D3×D1511yescharacteristic
290C4—21no—
291C4—61no—
292C4—301no—
293C4—901no—
294C4—901no—
295C3—21no—
296C3#9145211yescharacteristic, complemented
297C3#9181811yescharacteristic, complemented
298C2—62no—
299C2—601no—
300C2#1062419311yescharacteristic
301C2#1062419911yescharacteristic
302C2D3×#122011yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×D15
commutator, socle, cosocleC22×C32×C5C23
FittingC2×C32×C4×C5C22
prime radicalC2×#67150C2
hypercenterC2×C4D3×D15
hypocenterC32×C5#123
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×C3×C4×C5×D3C2semidirect
3. #10623906C2semidirect
4. C3×#6714C2semidirect
5. #10623814C2semidirect
6. (C2×D3)⋊Dic15C2semidirect
7. #10623811C2semidirect
22. C2×C4×C5×D3D3semidirect
26. #6714D3semidirect
48. C2×C3×C4×D3D5semidirect
130. C2×C4×D3D15semidirect
142. C32×C5#123semidirect
224. C3×C5#883semidirect
225. C3×C5#929semidirect
259. C32#4075semidirect
282. C5#63905semidirect
296. C3#91452semidirect
297. C3#91818semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C5×D32
C2×#671502
#106239062
C3×#67142
#106238142
(C2×D3)⋊Dic152
#106238112
#918183
#914523
#639055