Finite Group Databasebeta

← C4×#67150 #13087429

Subgroups — C4×#67150 #13087429

Order 1440.

406 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 108 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, complemented
2C2×C3×C4×C5×D3C211yesmaximal, characteristic, complemented
3C2×#67150C211yesmaximal, characteristic
4C4×C3⋊Dic15C211yesmaximal, characteristic, complemented
5#10623815C211yesmaximal, characteristic, complemented
6(C2×D3)⋊Dic15C211yesmaximal, characteristic, complemented
7#10623811C211yesmaximal, characteristic, complemented
8C4×#1220—31nomaximal
9C4×#1201—31nomaximal
10C22×C3×D15C2211yescharacteristic
11C22×C3×C5×D3C2211yescharacteristic
12C2×C32×C4×C5C2211yescharacteristic
13C2×C3⋊Dic15C2211yescharacteristic
14C2×C3⋊Dic15C2211yescharacteristic
15C2×C3×Dic15C2211yescharacteristic
16C3×C4×D15—22no—
17C2×C3×C5×Dic3C2211yescharacteristic
18C3×C4×C5×D3—22no—
19#67150C414yesdirect product factor
20C4×#625—51nomaximal, Hall
21C22×C3×C4×C5—31no—
22C2×#1220—31no—
23C2×C4×D15D311yescharacteristic, complemented
24C2×C4×C5×D3D311yescharacteristic, complemented
25C2×C3×C4×D5—31no—
26C2×#1201—31no—
27#6719—31no—
28#6714—31no—
29#6712—31no—
30C4×Dic15—31no—
31C4×Dic15—31no—
32C4×Dic15—61no—
33#6669—31no—
34(C2×D3)⋊Dic5—31no—
35#6665—31no—
36C22×C32×C5C2311yescharacteristic
37C2×C3×D15—21no—
38C2×C3×D15—21no—
39C2×C3×D15C2×C412yescomplemented
40C2×C3×C5×D3—21no—
41C2×C3×C5×D3—21no—
42C2×C3×C5×D3C2×C412yescomplemented
43C32×C4×C5D412yescomplemented
44C3⋊Dic15—21no—
45C3⋊Dic15C2×C412yescomplemented
46C3×Dic15—21no—
47C3×C5×Dic3—21no—
48C4×C22⋊D5—91noHall
49C2×C3×C4×D3—51no—
50C2×C3×C4×D3D511yescharacteristic, complemented
51C2×#625—51no—
52C4×C3⋊Dic3—51no—
53#3724—51no—
54(C2×D3)⋊Dic3—51no—
55(C2×D3)⋊Dic3—51no—
56C23×C3×C5—31no—
57C22×D15C2×D311yescharacteristic
58C22×C5×D3C2×D311yescharacteristic
59C22×C3×D5—31no—
60C2×C3×C4×C5—21no—
61C2×C3×C4×C5—31no—
62C2×C3×C4×C5—62no—
63C2×C3×C4×C5C2×D311yescharacteristic
64C2×C3×C4×C5C2×D311yescharacteristic
65#1220—34no—
66C2×Dic15—31no—
67C2×Dic15—31no—
68C2×Dic15—31no—
69C2×Dic15—31no—
70C2×Dic15—61no—
71C2×Dic15—61no—
72C2×Dic15C2×D311yescharacteristic
73C4×D15—22no—
74C2×C5×Dic3C2×D311yescharacteristic
75C4×C5×D3—22no—
76C2×C3×Dic5—31no—
77C3×C4×D5—62no—
78#1201—34no—
79C4×#67—151no—
80C4×#67—151no—
81C2×C32×C5C22×C411yescharacteristic
82C2×C32×C5C2×D411yescharacteristic
83C2×C32×C5#4111yescharacteristic
84C3×D15—22no—
85C3×C5×D3—22no—
86C22×C4×C5—31no—
87C2×C22⋊D5—91no—
88C2×C4×D5—31no—
89C22⋊Dic5—91no—
90#685—91no—
91#683—91no—
92C4×Dic5—91no—
93C22×C3×D3—51no—
94C22×C3×D3C2×D511yescharacteristic
95C2×C32×C4C2×D511yescharacteristic
96C2×C3⋊Dic3—51no—
97C2×C3⋊Dic3—51no—
98C2×C3×Dic3—51no—
99C2×C3×Dic3C2×D511yescharacteristic
100C3×C4×D3—22no—
101C3×C4×D3—102no—
102#625—54no—
103C22×C3×C5—21no—
104C22×C3×C5—32no—
105C22×C3×C5—61no—
106C22×C3×C5—61no—
107C22×C3×C5C22×D311yescharacteristic
108C22×C3×C5C22×D311yescharacteristic
109C2×D15—21no—
110C2×D15—21no—
111C2×D15C4×D312yescomplemented
112C2×C5×D3—21no—
113C2×C5×D3—21no—
114C2×C5×D3C4×D312yescomplemented
115C2×C3×D5—32no—
116C2×C3×D5—61no—
117C2×C3×D5—61no—
118C3×C4×C5—22no—
119C3×C4×C5—61no—
120C3×C4×C5#6712yescomplemented
121C3×C4×C5#6712yescomplemented
122Dic15—21no—
123Dic15—32no—
124Dic15—32no—
125Dic15—62no—
126Dic15—62no—
127Dic15—61no—
128Dic15—61no—
129C3×Dic5—61no—
130C5×Dic3—21no—
131C22×C3×C4—31no—
132C22×C3×C4—151no—
133C2×#67—151no—
134C2×#67—151no—
135C2×C4×D3—51no—
136C2×C4×D3D1511yescharacteristic, complemented
137#217—151no—
138#217—151no—
139(C2×D3)⋊C4—151no—
140(C2×D3)⋊C4—151no—
141Dic3⋊C4—151no—
142Dic3⋊C4—151no—
143C4×Dic3—151no—
144C4×Dic3—151no—
145C4×Dic3—301no—
146C32×C5C4×D411yescharacteristic, complemented, Hall
147C23×C5—31no—
148C22×D5—31no—
149C2×C4×C5—31no—
150C2×C4×C5—62no—
151C2×C4×C5D3211yescharacteristic
152C22⋊D5—94no—
153C2×Dic5—31no—
154C2×Dic5—91no—
155C2×Dic5—91no—
156C4×D5—62no—
157C22×C32C22×D511yescharacteristic
158C2×C3×D3—21no—
159C2×C3×D3—21no—
160C2×C3×D3—52no—
161C2×C3×D3—101no—
162C2×C3×D3—101no—
163C2×C3×D3C4×D512yescomplemented
164C32×C4C22⋊D512yescomplemented
165C3⋊Dic3—52no—
166C3⋊Dic3—101no—
167C3×Dic3—21no—
168C3×Dic3—101no—
169C4×D4—451noSylow
170C2×C3×C5—21no—
171C2×C3×C5—21no—
172C2×C3×C5—21no—
173C2×C3×C5—62no—
174C2×C3×C5C2×C4×D311yescharacteristic
175C2×C3×C5C2×C4×D311yescharacteristic
176C2×C3×C5#23511yescharacteristic
177C2×C3×C5#23511yescharacteristic
178C2×C3×C5C2×#6711yescharacteristic
179C2×C3×C5C2×#6711yescharacteristic
180D15—22no—
181C3×D5—62no—
182C5×D3—22no—
183C23×C3—31no—
184C23×C3—151no—
185C22×D3—51no—
186C22×D3C2×D1511yescharacteristic
187C2×C3×C4—21no—
188C2×C3×C4—31no—
189C2×C3×C4—62no—
190C2×C3×C4—151no—
191C2×C3×C4—302no—
192C2×C3×C4D3×D511yescharacteristic
193C2×C3×C4C2×D1511yescharacteristic
194#67—154no—
195#67—154no—
196C2×Dic3—51no—
197C2×Dic3—151no—
198C2×Dic3—151no—
199C2×Dic3—151no—
200C2×Dic3—151no—
201C2×Dic3—301no—
202C2×Dic3—301no—
203C2×Dic3C2×D1511yescharacteristic
204C4×D3—22no—
205C4×D3—102no—
206C22×C5—32no—
207C22×C5—61no—
208C22×C5—61no—
209C22×C5C2×D3211yescharacteristic
210C2×D5—32no—
211C2×D5—61no—
212C2×D5—61no—
213C4×C5—61no—
214C4×C5#62512yescomplemented
215Dic5—61no—
216Dic5—92no—
217Dic5—181no—
218C2×C32C2×C4×D511yescharacteristic
219C2×C32#70911yescharacteristic
220C2×C32C2×C22⋊D511yescharacteristic
221C3×D3—22no—
222C3×D3—102no—
223C2×D4—451no—
224C22×C4—31no—
225C22×C4—151no—
226C4⋊C4—451no—
227C22⋊C4—451no—
228C22⋊C4—451no—
229C42—451no—
230C3×C5—21no—
231C3×C5C4×#6711yescharacteristic, complemented
232C3×C5C4×#6711yescharacteristic, complemented
233C22×C3—21no—
234C22×C3—32no—
235C22×C3—61no—
236C22×C3—61no—
237C22×C3—152no—
238C22×C3—301no—
239C22×C3—301no—
240C22×C3C2×D3×D511yescharacteristic
241C22×C3C22×D1511yescharacteristic
242C2×D3—21no—
243C2×D3—21no—
244C2×D3—52no—
245C2×D3—101no—
246C2×D3—101no—
247C2×D3C4×D1512yescomplemented
248C3×C4—22no—
249C3×C4—61no—
250C3×C4—301no—
251C3×C4#120112yescomplemented
252C3×C4#122012yescomplemented
253Dic3—21no—
254Dic3—101no—
255Dic3—152no—
256Dic3—152no—
257Dic3—302no—
258Dic3—302no—
259Dic3—301no—
260Dic3—301no—
261C2×C5—62no—
262C2×C5#379811yescharacteristic
263C2×C5C4×D3211yescharacteristic
264C2×C5C2×#62511yescharacteristic
265D5—62no—
266C32C4×C22⋊D511yescharacteristic, complemented, Sylow
267C23—31no—
268C23—151no—
269D4—454no—
270C2×C4—31no—
271C2×C4—62no—
272C2×C4—151no—
273C2×C4—302no—
274C2×C4—451no—
275C2×C4—451no—
276C2×C4D3×D1511yescharacteristic
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—62no—
281C2×C3—302no—
282C2×C3#677111yescharacteristic
283C2×C3C4×D3×D511yescharacteristic
284C2×C3C2×#120111yescharacteristic
285C2×C3C2×C4×D1511yescharacteristic
286C2×C3#681711yescharacteristic
287C2×C3C2×#122011yescharacteristic
288D3—22no—
289D3—102no—
290C5C4×#62511yescharacteristic, complemented, Sylow
291C22—32no—
292C22—61no—
293C22—61no—
294C22—152no—
295C22—301no—
296C22—301no—
297C22C2×D3×D1511yescharacteristic
298C4—61no—
299C4—301no—
300C4—452no—
301C4—901no—
302C4#6715012yesdirect product factor
303C3—21no—
304C3C4×#120111yescharacteristic, complemented
305C3C4×#122011yescharacteristic, complemented
306C2—62no—
307C2—302no—
308C2#1062419911yescharacteristic
309C2C4×D3×D1511yescharacteristic
310C2C2×#6715011yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×D15
commutatorC2×C32×C5C22×C4
FrattiniC22C2×D3×D15
FittingC2×C32×C4×C5C22
socle, cosocleC22×C32×C5C23
prime radicalC2×#67150C2
hypocenterC32×C5C4×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D15C2semidirect
2. C2×C3×C4×C5×D3C2semidirect
4. C4×C3⋊Dic15C2semidirect
5. #10623815C2semidirect
6. (C2×D3)⋊Dic15C2semidirect
7. #10623811C2semidirect
19. #67150C4direct (N × H)
23. C2×C4×D15D3semidirect
24. C2×C4×C5×D3D3semidirect
39. C2×C3×D15C2×C4semidirect
42. C2×C3×C5×D3C2×C4semidirect
43. C32×C4×C5D4semidirect
45. C3⋊Dic15C2×C4semidirect
50. C2×C3×C4×D3D5semidirect
111. C2×D15C4×D3semidirect
114. C2×C5×D3C4×D3semidirect
120. C3×C4×C5#67semidirect
121. C3×C4×C5#67semidirect
136. C2×C4×D3D15semidirect
146. C32×C5C4×D4semidirect
163. C2×C3×D3C4×D5semidirect
164. C32×C4C22⋊D5semidirect
214. C4×C5#625semidirect
231. C3×C5C4×#67semidirect
232. C3×C5C4×#67semidirect
247. C2×D3C4×D15semidirect
251. C3×C4#1201semidirect
252. C3×C4#1220semidirect
266. C32C4×C22⋊D5semidirect
290. C5C4×#625semidirect
302. C4#67150direct (N × H)
304. C3C4×#1201semidirect
305. C3C4×#1220semidirect

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×C3×C4×C5×D32
C2×#671502
C4×C3⋊Dic152
#106238152
(C2×D3)⋊Dic152
#106238112
C4×#12203
C4×#12013
C4×#6255