Finite Group Databasebeta

← #13087560

Subgroups — #13087560

Order 1440.

430 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 102 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
2C22×C3×Dic15C211yesmaximal, characteristic, complemented
3C2×#67149C211yesmaximal, characteristic, complemented
4C3×C5×#217C211yesmaximal, characteristic, complemented
5#10623815C211yesmaximal, characteristic, complemented
6#10623813C211yesmaximal, characteristic, complemented
7Dic3×Dic15C211yesmaximal, characteristic, complemented
8#91770—31nomaximal
9#91552—31nomaximal
10C23×C32×C5C2211yescharacteristic
11C22×C3⋊D15C2211yescharacteristic
12#67184C414yescomplemented
13C2×C3⋊Dic15C2211yescharacteristic
14C2×C3×Dic15—22no—
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×Dic15C2211yescharacteristic
17C2×C3×C5×Dic3C2211yescharacteristic
18C2×C3×C5×Dic3C2211yescharacteristic
19#67149—22no—
20#63952—51nomaximal, Hall
21C2×#1220—31no—
22C2×#1220—31no—
23C2×#1220—61no—
24C22×Dic15D311yescharacteristic, complemented
25C2×C4×D15—31no—
26C22×C3×Dic5—31no—
27C2×#1200—31no—
28C3×C5×C22⋊C4—31no—
29#6714—31no—
30#6712—31no—
31C4×Dic15—31no—
32C5×#217D311yescharacteristic, complemented
33#6669—31no—
34#6667—31no—
35Dic3×Dic5—31no—
36C22×C32×C5—21no—
37C22×C32×C5—21no—
38C22×C32×C5C2×C412yescomplemented
39C22×C32×C5C2311yescharacteristic
40C2×C3⋊D15—21no—
41C2×C3⋊D15—21no—
42C2×C3⋊D15C2×C412yescomplemented
43C3⋊Dic15C2×C412yescomplemented
44C3×Dic15—21no—
45C3×Dic15D412yescomplemented
46C3×C5×Dic3—21no—
47C3×C5×Dic3—21no—
48#4027—91noHall
49C2×#638—51no—
50C22×C3×Dic3—51no—
51C2×#624—51no—
52C3×#217D511yescharacteristic, complemented
53#3724—51no—
54#3722—51no—
55Dic32—51no—
56C23×C3×C5—21no—
57C23×C3×C5C2×D311yescharacteristic
58C23×C3×C5C2×D311yescharacteristic
59C22×D15—31no—
60C22×D15—31no—
61C22×D15—61no—
62C2×C3×C4×C5—31no—
63C2×C3×C4×C5—31no—
64#1220—34no—
65#1220—34no—
66#1220—64no—
67C2×Dic15—22no—
68C2×Dic15—31no—
69C2×Dic15—31no—
70C2×Dic15—61no—
71C2×Dic15C2×D311yescharacteristic
72C2×Dic15C2×D311yescharacteristic
73C4×D15—62no—
74C2×C5×Dic3C2×D311yescharacteristic
75C2×C5×Dic3C2×D311yescharacteristic
76C2×C3×Dic5—31no—
77C2×C3×Dic5—31no—
78C2×C3×Dic5—62no—
79#1200—62no—
80C4×#67—151no—
81#67⋊C4—151no—
82C2×C32×C5—22no—
83C2×C32×C5C22×C411yescharacteristic
84C2×C32×C5C2×D411yescharacteristic
85C2×C32×C5#4111yescharacteristic
86C3⋊D15—22no—
87C2×C22⋊D5—91no—
88C22×Dic5—31no—
89C2×C4×D5—91no—
90C5×C22⋊C4—31no—
91#685—91no—
92#683—91no—
93C4×Dic5—91no—
94C23×C32C2×D511yescharacteristic
95C22×C3⋊D3—51no—
96#638—54no—
97C2×C3⋊Dic3—51no—
98C2×C3×Dic3—51no—
99C2×C3×Dic3—51no—
100C2×C3×Dic3—102no—
101C2×C3×Dic3C2×D511yescharacteristic
102C2×C3×Dic3C2×D511yescharacteristic
103#624—102no—
104C22×C3×C5—21no—
105C22×C3×C5—21no—
106C22×C3×C5—21no—
107C22×C3×C5—21no—
108C22×C3×C5—22no—
109C22×C3×C5—21no—
110C22×C3×C5—41no—
111C22×C3×C5—41no—
112C22×C3×C5C4×D312yescomplemented
113C22×C3×C5C4×D312yescomplemented
114C22×C3×C5C22×D311yescharacteristic
115C22×C3×C5C22×D311yescharacteristic
116C2×D15—32no—
117C2×D15—32no—
118C2×D15—62no—
119C2×D15—61no—
120C2×D15—61no—
121C2×D15—61no—
122C2×D15—61no—
123C2×D15—121no—
124C2×D15—121no—
125C3×C4×C5—61no—
126C3×C4×C5—61no—
127Dic15—21no—
128Dic15—32no—
129Dic15—32no—
130Dic15—62no—
131Dic15#6712yescomplemented
132C3×Dic5—32no—
133C3×Dic5—61no—
134C5×Dic3—21no—
135C5×Dic3—21no—
136C22×C3×C4—151no—
137C2×#67—151no—
138C2×#67—151no—
139C2×#67—301no—
140C22×Dic3—51no—
141C2×C4×D3—151no—
142C2×C4×D3—151no—
143C3×C22⋊C4—31no—
144#217D1511yescharacteristic, complemented
145(C2×D3)⋊C4—151no—
146(C2×D3)⋊C4—151no—
147Dic3⋊C4—151no—
148Dic3⋊C4—151no—
149C4×Dic3—151no—
150C4×Dic3—151no—
151C32×C5C4×D411yescharacteristic, complemented, Hall
152C23×C5D3211yescharacteristic
153C22×D5—91no—
154C2×C4×C5—31no—
155C2×C4×C5—31no—
156C22⋊D5—94no—
157C2×Dic5—31no—
158C2×Dic5—31no—
159C2×Dic5—62no—
160C2×Dic5—91no—
161C4×D5—182no—
162C22×C32—21no—
163C22×C32—21no—
164C22×C32C4×D512yescomplemented
165C22×C32C22×D511yescharacteristic
166C2×C3⋊D3—52no—
167C2×C3⋊D3—101no—
168C2×C3⋊D3—101no—
169C3⋊Dic3—52no—
170C3×Dic3—21no—
171C3×Dic3—21no—
172C3×Dic3—52no—
173C3×Dic3—101no—
174C4×D4—451noSylow
175C2×C3×C5—22no—
176C2×C3×C5—22no—
177C2×C3×C5—21no—
178C2×C3×C5—21no—
179C2×C3×C5—21no—
180C2×C3×C5—42no—
181C2×C3×C5C2×C4×D311yescharacteristic
182C2×C3×C5C2×C4×D311yescharacteristic
183C2×C3×C5#23511yescharacteristic
184C2×C3×C5D3×D411yescharacteristic
185C2×C3×C5#23711yescharacteristic
186C2×C3×C5C2×#6711yescharacteristic
187D15—62no—
188D15—62no—
189D15—122no—
190C23×C3—21no—
191C23×C3D3×D511yescharacteristic
192C23×C3C2×D1511yescharacteristic
193C22×D3—151no—
194C22×D3—151no—
195C22×D3—301no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—151no—
199C2×C3×C4—151no—
200C2×C3×C4—302no—
201#67—154no—
202#67—154no—
203#67—304no—
204C2×Dic3—51no—
205C2×Dic3—51no—
206C2×Dic3—102no—
207C2×Dic3—151no—
208C2×Dic3—151no—
209C2×Dic3—301no—
210C2×Dic3C2×D1511yescharacteristic
211C2×Dic3C2×D1511yescharacteristic
212C4×D3—302no—
213C4×D3—302no—
214C22×C5—21no—
215C22×C5—21no—
216C22×C5#62412yescomplemented
217C22×C5C2×D3211yescharacteristic
218C2×D5—92no—
219C2×D5—181no—
220C2×D5—181no—
221C4×C5—61no—
222C4×C5—61no—
223Dic5—32no—
224Dic5—61no—
225Dic5—92no—
226C2×C32—22no—
227C2×C32C2×C4×D511yescharacteristic
228C2×C32D4×D511yescharacteristic
229C2×C32#71111yescharacteristic
230C3⋊D3—102no—
231C2×D4—451no—
232C22×C4—151no—
233C22×C4—451no—
234C4⋊C4—451no—
235C22⋊C4—31no—
236C22⋊C4—451no—
237C42—451no—
238C3×C5—21no—
239C3×C5#67⋊C411yescharacteristic, complemented
240C3×C5C4×#6711yescharacteristic, complemented
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—22no—
246C22×C3—21no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3#120012yescomplemented
250C22×C3C4×D1512yescomplemented
251C22×C3C2×D3×D511yescharacteristic
252C22×C3C22×D1511yescharacteristic
253C2×D3—152no—
254C2×D3—152no—
255C2×D3—302no—
256C2×D3—301no—
257C2×D3—301no—
258C2×D3—301no—
259C2×D3—301no—
260C2×D3—601no—
261C2×D3—601no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4—152no—
265C3×C4—301no—
266Dic3—21no—
267Dic3—21no—
268Dic3—52no—
269Dic3—101no—
270Dic3—152no—
271Dic3—152no—
272Dic3—302no—
273C2×C5—22no—
274C2×C5#380411yescharacteristic
275C2×C5C2×#62411yescharacteristic
276C2×C5D3×#6711yescharacteristic
277D5—182no—
278C32#402711yescharacteristic, complemented, Sylow
279C23—451no—
280C23D3×D1511yescharacteristic
281D4—454no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—151no—
285C2×C4—151no—
286C2×C4—302no—
287C2×C4—451no—
288C2×C4—902no—
289C2×C3—22no—
290C2×C3—22no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—42no—
295C2×C3#678211yescharacteristic
296C2×C3C2×#120011yescharacteristic
297C2×C3D5×#6711yescharacteristic
298C2×C3C2×C4×D1511yescharacteristic
299C2×C3D4×D1511yescharacteristic
300C2×C3#681911yescharacteristic
301D3—302no—
302D3—302no—
303D3—602no—
304C5#6395211yescharacteristic, complemented, Sylow
305C22—21no—
306C22—21no—
307C22—452no—
308C22—901no—
309C22—901no—
310C22#6714912yescomplemented
311C22C2×D3×D1511yescharacteristic
312C4—61no—
313C4—61no—
314C4—152no—
315C4—301no—
316C4—452no—
317C3—21no—
318C3#9155211yescharacteristic, complemented
319C3#9177011yescharacteristic, complemented
320C2—22no—
321C2—902no—
322C2#1062421011yescharacteristic
323C2C2×#6714911yescharacteristic
324C2#67×D1511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×D15
commutatorC2×C32×C5C22×C4
FittingC23×C32×C5C22
socle, cosocleC22×C32×C5C23
prime radicalC2×#67184C2
hypercenterC23D3×D15
hypocenterC32×C5C4×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3×Dic15C2semidirect
3. C2×#67149C2semidirect
4. C3×C5×#217C2semidirect
5. #10623815C2semidirect
6. #10623813C2semidirect
7. Dic3×Dic15C2semidirect
12. #67184C4semidirect
24. C22×Dic15D3semidirect
32. C5×#217D3semidirect
38. C22×C32×C5C2×C4semidirect
42. C2×C3⋊D15C2×C4semidirect
43. C3⋊Dic15C2×C4semidirect
45. C3×Dic15D4semidirect
52. C3×#217D5semidirect
112. C22×C3×C5C4×D3semidirect
113. C22×C3×C5C4×D3semidirect
131. Dic15#67semidirect
144. #217D15semidirect
151. C32×C5C4×D4semidirect
164. C22×C32C4×D5semidirect
216. C22×C5#624semidirect
239. C3×C5#67⋊C4semidirect
240. C3×C5C4×#67semidirect
249. C22×C3#1200semidirect
250. C22×C3C4×D15semidirect
278. C32#4027semidirect
304. C5#63952semidirect
310. C22#67149semidirect
318. C3#91552semidirect
319. C3#91770semidirect

Maximal subgroups

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

subgroupindex
C2×#671842
C22×C3×Dic152
C2×#671492
C3×C5×#2172
#106238152
#106238132
Dic3×Dic152
#917703
#915523
#639525