Finite Group Databasebeta

← #13087589

Subgroups — #13087589

Order 1440.

358 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 76 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×#67184C211yesmaximal, characteristic
2C2×#67153C211yesmaximal, characteristic, complemented
3C3×#6719C211yesmaximal, characteristic, complemented
4C3×C5×#217C211yesmaximal, characteristic, complemented
5#10623813C211yesmaximal, characteristic, complemented
6#10623813C211yesmaximal, characteristic, complemented
7Dic3×Dic15C211yesmaximal, characteristic, complemented
8#91774—31nomaximal
9#91526—31nomaximal
10C23×C32×C5C2211yescharacteristic
11C22×C3⋊D15C2211yescharacteristic
12#67184—22no—
13C2×C3⋊Dic15C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17C2×C3×C5×Dic3C2211yescharacteristic
18#67153—22no—
19#63953—51nomaximal, Hall
20C2×#1220—31no—
21C2×#1220—31no—
22C2×#1220—61no—
23C2×Dic30—31no—
24C2×#1204—31no—
25C3×C5×C22⋊C4—31no—
26#6719D311yescharacteristic, complemented
27#6714—31no—
28#6714—31no—
29C4×Dic15—31no—
30C5×#217D311yescharacteristic, complemented
31C3×C22⋊Dic5—31no—
32#6667—31no—
33#6667—31no—
34Dic3×Dic5—31no—
35C22×C32×C5—21no—
36C22×C32×C5—21no—
37C22×C32×C5—21no—
38C22×C32×C5C2311yescharacteristic
39C2×C3⋊D15—21no—
40C2×C3⋊D15—21no—
41C2×C3⋊D15—21no—
42C3⋊Dic15D412yes—
43C3×Dic15—21no—
44C3×Dic15—21no—
45C3×C5×Dic3—21no—
46C3×C5×Dic3—21no—
47#4031—91noHall
48C2×#638—51no—
49C2×#627—51no—
50C3×#217—51no—
51C3×#217D511yescharacteristic, complemented
52#3722—51no—
53#3722—51no—
54Dic32—51no—
55C23×C3×C5—21no—
56C23×C3×C5C2×D311yescharacteristic
57C23×C3×C5C2×D311yescharacteristic
58C22×D15—31no—
59C22×D15—31no—
60C22×D15—61no—
61C2×C3×C4×C5—31no—
62C2×C3×C4×C5—31no—
63#1220—62no—
64#1220—62no—
65#1220—64no—
66C2×Dic15—31no—
67C2×Dic15—31no—
68C2×Dic15—61no—
69C2×Dic15C2×D311yescharacteristic
70C2×Dic15C2×D311yescharacteristic
71Dic30—62no—
72C2×C5×Dic3C2×D311yescharacteristic
73C2×C5×Dic3C2×D311yescharacteristic
74C2×C3×Dic5—31no—
75C2×C3×Dic5—31no—
76#1204—62no—
77#885—151no—
78#885—151no—
79C2×C32×C5—41no—
80C2×C32×C5C2×D411yescharacteristic
81C2×C32×C5#4111yescharacteristic
82C2×C32×C5#4111yescharacteristic
83C3⋊D15—41no—
84C2×C22⋊D5—91no—
85C2×Dic10—91no—
86C5×C22⋊C4—31no—
87C22⋊Dic5—31no—
88#685—91no—
89#685—91no—
90C4×Dic5—91no—
91C23×C32C2×D511yescharacteristic
92C22×C3⋊D3—51no—
93#638—102no—
94C2×C3⋊Dic3—51no—
95C2×C3×Dic3—51no—
96C2×C3×Dic3—51no—
97C2×C3×Dic3C2×D511yescharacteristic
98C2×C3×Dic3C2×D511yescharacteristic
99#627—102no—
100C22×C3×C5—22no—
101C22×C3×C5—21no—
102C22×C3×C5—21no—
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—21no—
106C22×C3×C5—21no—
107C22×C3×C5—21no—
108C22×C3×C5—41no—
109C22×C3×C5—41no—
110C22×C3×C5C22×D311yescharacteristic
111C22×C3×C5C22×D311yescharacteristic
112C2×D15—62no—
113C2×D15—61no—
114C2×D15—61no—
115C2×D15—61no—
116C2×D15—61no—
117C2×D15—61no—
118C2×D15—61no—
119C2×D15—121no—
120C2×D15—121no—
121C3×C4×C5—61no—
122C3×C4×C5—61no—
123Dic15—21no—
124Dic15—21no—
125Dic15—32no—
126Dic15—32no—
127Dic15—62no—
128C3×Dic5—61no—
129C3×Dic5—61no—
130C5×Dic3—21no—
131C5×Dic3—21no—
132C2×#67—151no—
133C2×#67—151no—
134C2×#67—301no—
135C2×Dic6—151no—
136C2×Dic6—151no—
137C3×C22⋊C4—31no—
138C3×C22⋊C4—151no—
139#217—51no—
140#217D1511yescharacteristic, complemented
141(C2×D3)⋊C4—151no—
142(C2×D3)⋊C4—151no—
143(C2×D3)⋊C4—151no—
144(C2×D3)⋊C4—151no—
145C4×Dic3—151no—
146C4×Dic3—151no—
147C32×C5#12411yescharacteristic, complemented, Hall
148C23×C5D3211yescharacteristic
149C22×D5—91no—
150C2×C4×C5—31no—
151C2×C4×C5—31no—
152C22⋊D5—182no—
153C2×Dic5—31no—
154C2×Dic5—31no—
155C2×Dic5—91no—
156Dic10—182no—
157C22×C32—21no—
158C22×C32—21no—
159C22×C32—21no—
160C22×C32C22×D511yescharacteristic
161C2×C3⋊D3—101no—
162C2×C3⋊D3—101no—
163C2×C3⋊D3—101no—
164C3⋊Dic3—52no—
165C3×Dic3—21no—
166C3×Dic3—21no—
167C3×Dic3—101no—
168C3×Dic3—101no—
169#124—451noSylow
170C2×C3×C5—21no—
171C2×C3×C5—21no—
172C2×C3×C5—21no—
173C2×C3×C5—42no—
174C2×C3×C5—41no—
175C2×C3×C5—41no—
176C2×C3×C5#23511yescharacteristic
177C2×C3×C5#23511yescharacteristic
178C2×C3×C5D3×D411yescharacteristic
179C2×C3×C5D3×D411yescharacteristic
180C2×C3×C5#23711yescharacteristic
181C2×C3×C5#23711yescharacteristic
182D15—122no—
183D15—121no—
184D15—121no—
185C23×C3—21no—
186C23×C3D3×D511yescharacteristic
187C23×C3C2×D1511yescharacteristic
188C22×D3—151no—
189C22×D3—151no—
190C22×D3—301no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—151no—
194C2×C3×C4—151no—
195#67—302no—
196#67—302no—
197#67—304no—
198C2×Dic3—51no—
199C2×Dic3—51no—
200C2×Dic3—151no—
201C2×Dic3—151no—
202C2×Dic3—301no—
203C2×Dic3C2×D1511yescharacteristic
204C2×Dic3C2×D1511yescharacteristic
205Dic6—302no—
206Dic6—302no—
207C22×C5—21no—
208C22×C5—21no—
209C22×C5—21no—
210C22×C5C2×D3211yescharacteristic
211C2×D5—181no—
212C2×D5—181no—
213C2×D5—181no—
214C4×C5—61no—
215C4×C5—61no—
216Dic5—61no—
217Dic5—61no—
218Dic5—92no—
219C2×C32—41no—
220C2×C32#70911yescharacteristic
221C2×C32D4×D511yescharacteristic
222C2×C32#71111yescharacteristic
223C3⋊D3—201no—
224C2×Q8—451no—
225C2×D4—451no—
226C22⋊C4—31no—
227C22⋊C4—151no—
228C22⋊C4—451no—
229C22⋊C4—451no—
230C42—451no—
231C3×C5—21no—
232C3×C5#88511yescharacteristic, complemented
233C3×C5#88511yescharacteristic, complemented
234C22×C3—22no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3C2×D3×D511yescharacteristic
245C22×C3C22×D1511yescharacteristic
246C2×D3—302no—
247C2×D3—301no—
248C2×D3—301no—
249C2×D3—301no—
250C2×D3—301no—
251C2×D3—301no—
252C2×D3—301no—
253C2×D3—601no—
254C2×D3—601no—
255C3×C4—61no—
256C3×C4—61no—
257C3×C4—301no—
258C3×C4—301no—
259Dic3—21no—
260Dic3—21no—
261Dic3—101no—
262Dic3—101no—
263Dic3—152no—
264Dic3—152no—
265Dic3—302no—
266C2×C5—41no—
267C2×C5#380411yescharacteristic
268C2×C5#380411yescharacteristic
269C2×C5#381111yescharacteristic
270D5—361no—
271C32#403111yescharacteristic, complemented, Sylow
272C23—451no—
273C23D3×D1511yescharacteristic
274Q8—902no—
275D4—902no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—151no—
279C2×C4—151no—
280C2×C4—451no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—42no—
285C2×C3—41no—
286C2×C3—41no—
287C2×C3#677911yescharacteristic
288C2×C3#678211yescharacteristic
289C2×C3#679011yescharacteristic
290C2×C3#681711yescharacteristic
291C2×C3D4×D1511yescharacteristic
292C2×C3#681911yescharacteristic
293D3—602no—
294D3—601no—
295D3—601no—
296C5#6395311yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—21no—
300C22—901no—
301C22—901no—
302C22—901no—
303C22C2×D3×D1511yescharacteristic
304C4—61no—
305C4—61no—
306C4—301no—
307C4—301no—
308C4—452no—
309C3—21no—
310C3#9152611yescharacteristic, complemented
311C3#9177411yescharacteristic, complemented
312C2—41no—
313C2—1801no—
314C2#1062420711yescharacteristic
315C2#1062421011yescharacteristic
316C2#1062421811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×D15
commutator, socle, cosocleC22×C32×C5C23
FittingC23×C32×C5C22
prime radicalC2×#67184C2
hypercenterC23D3×D15
hypocenterC32×C5#124
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#67153C2semidirect
3. C3×#6719C2semidirect
4. C3×C5×#217C2semidirect
5. #10623813C2semidirect
6. #10623813C2semidirect
7. Dic3×Dic15C2semidirect
26. #6719D3semidirect
30. C5×#217D3semidirect
51. C3×#217D5semidirect
140. #217D15semidirect
147. C32×C5#124semidirect
232. C3×C5#885semidirect
233. C3×C5#885semidirect
271. C32#4031semidirect
296. C5#63953semidirect
310. C3#91526semidirect
311. C3#91774semidirect

Maximal subgroups

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

subgroupindex
C2×#671842
C2×#671532
C3×#67192
C3×C5×#2172
#106238132
Dic3×Dic152
#917743
#915263
#639535