Finite Group Databasebeta

← #13085927

Subgroups — #13085927

Order 1440.

410 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 96 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×#67151C211yesmaximal, characteristic
2C2×#67149C211yesmaximal, characteristic, complemented
3C2×C3×#1200C211yesmaximal, characteristic, complemented
4#10623829C211yesmaximal, characteristic, complemented
5C5×Dic32C211yesmaximal, characteristic, complemented
6#10623772C211yesmaximal, characteristic, complemented
7C3×Dic3⋊Dic5C211yesmaximal, characteristic, complemented
8C4×#1202—31nomaximal
9#91428—31nomaximal
10C22×C3⋊D15C2211yescharacteristic
11C22×C3×D15C2211yescharacteristic
12C2×C5×C3⋊Dic3C2211yescharacteristic
13C2×C3×Dic15C2211yescharacteristic
14C2×C3×C5×Dic3C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C32×Dic5C2211yescharacteristic
17#67151C414yescomplemented
18#67149—22no—
19C3×#1200—22no—
20#63904—51nomaximal, Hall
21C2×D60—31no—
22C2×C4×D15—31no—
23C2×C3×C4×D5—31no—
24C2×#1202—31no—
25C2×#1200—31no—
26C2×#1200D311yescharacteristic, complemented
27C4×C5×Dic3—31no—
28C4×C5×Dic3—31no—
29C3×C4⋊Dic5—31no—
30Dic3⋊Dic5D311yescharacteristic, complemented
31#6667—31no—
32#6667—31no—
33#6667—31no—
34#6667—61no—
35#6665—31no—
36C22×C32×C5C2311yescharacteristic
37C2×C3⋊D15—21no—
38C2×C3⋊D15—21no—
39C2×C3⋊D15C2×C412yescomplemented
40C2×C3×D15—21no—
41C2×C3×D15—21no—
42C2×C3×D15C2×C412yescomplemented
43C5×C3⋊Dic3—21no—
44C3×Dic15—21no—
45C3×C5×Dic3C2×C412yescomplemented
46C3×C5×Dic3D412yescomplemented
47C32×Dic5—21no—
48C4×D20—91noHall
49C2×C3×C4×D3—51no—
50C2×#626—51no—
51C2×#624—51no—
52#3752—51no—
53C3×Dic3⋊C4—51no—
54(C2×D3)⋊Dic3—51no—
55Dic32D511yescharacteristic, complemented
56C22×D15—31no—
57C22×D15—31no—
58C22×D15—61no—
59C22×D15C2×D311yescharacteristic
60C22×C3×D5—31no—
61C2×C3×C4×C5—31no—
62C2×C3×C4×C5—31no—
63C2×Dic15C2×D311yescharacteristic
64D60—34no—
65C4×D15—62no—
66C2×C5×Dic3—31no—
67C2×C5×Dic3—31no—
68C2×C5×Dic3—61no—
69C2×C5×Dic3C2×D311yescharacteristic
70C2×C5×Dic3C2×D311yescharacteristic
71C2×C3×Dic5—21no—
72C2×C3×Dic5—31no—
73C2×C3×Dic5C2×D311yescharacteristic
74C2×C3×Dic5C2×D311yescharacteristic
75C3×C4×D5—62no—
76#1202—34no—
77#1200—22no—
78#1200—62no—
79C4×#67—151no—
80#893—151no—
81C2×C32×C5C22×C411yescharacteristic
82C2×C32×C5C2×D411yescharacteristic
83C2×C32×C5#4111yescharacteristic
84C3⋊D15—22no—
85C3×D15—22no—
86C2×D20—91no—
87C2×C4×D5—31no—
88C2×C4×D5—91no—
89C42×C5—91no—
90#685—91no—
91#685—91no—
92C4⋊Dic5—31no—
93C22×C3⋊D3—51no—
94C22×C3×D3—51no—
95C2×C32×C4—51no—
96C2×C3⋊Dic3C2×D511yescharacteristic
97C2×C3×Dic3—51no—
98C2×C3×Dic3C2×D511yescharacteristic
99C2×C3×Dic3C2×D511yescharacteristic
100C3×C4×D3—102no—
101#626—54no—
102#624—102no—
103C22×C3×C5—21no—
104C22×C3×C5C22×D311yescharacteristic
105C22×C3×C5C22×D311yescharacteristic
106C2×D15—21no—
107C2×D15—21no—
108C2×D15—32no—
109C2×D15—32no—
110C2×D15—62no—
111C2×D15—61no—
112C2×D15—61no—
113C2×D15—61no—
114C2×D15—61no—
115C2×D15—121no—
116C2×D15—121no—
117C2×D15C4×D312yescomplemented
118C2×C3×D5—32no—
119C2×C3×D5—61no—
120C2×C3×D5—61no—
121C3×C4×C5—32no—
122C3×C4×C5—32no—
123Dic15—21no—
124C3×Dic5—21no—
125C3×Dic5—21no—
126C3×Dic5—41no—
127C3×Dic5—61no—
128C5×Dic3—61no—
129C5×Dic3—61no—
130C5×Dic3—121no—
131C5×Dic3C4×D312yescomplemented
132C5×Dic3#6712yescomplemented
133C22×C3×C4—151no—
134C2×#67—151no—
135C2×D12—151no—
136C2×C4×D3—51no—
137C2×C4×D3—151no—
138C2×C4×D3—151no—
139C3×C4⋊C4—151no—
140#217—151no—
141(C2×D3)⋊C4—151no—
142(C2×D3)⋊C4—151no—
143(C2×D3)⋊C4—151no—
144(C2×D3)⋊C4—301no—
145Dic3⋊C4—51no—
146C4×Dic3—31no—
147C4×Dic3—31no—
148C32×C5C4×D411yescharacteristic, complemented, Hall
149C22×D5—31no—
150C22×D5—91no—
151C2×C4×C5—31no—
152C2×C4×C5—31no—
153C2×C4×C5—91no—
154C2×Dic5—31no—
155C2×Dic5D3211yescharacteristic
156D20—94no—
157C4×D5—62no—
158C4×D5—182no—
159C22×C32C22×D511yescharacteristic
160C2×C3⋊D3—52no—
161C2×C3⋊D3—101no—
162C2×C3⋊D3—101no—
163C2×C3×D3—52no—
164C2×C3×D3—101no—
165C2×C3×D3—101no—
166C32×C4—101no—
167C3⋊Dic3—21no—
168C3×Dic3—101no—
169C3×Dic3C4×D512yescomplemented
170C3×Dic3D2012yescomplemented
171C4×D4—451noSylow
172C2×C3×C5—21no—
173C2×C3×C5—21no—
174C2×C3×C5—21no—
175C2×C3×C5C2×C4×D311yescharacteristic
176C2×C3×C5C2×C4×D311yescharacteristic
177C2×C3×C5#23511yescharacteristic
178C2×C3×C5D3×D411yescharacteristic
179C2×C3×C5#23911yescharacteristic
180C2×C3×C5C2×#6711yescharacteristic
181D15—22no—
182D15—62no—
183D15—62no—
184D15—122no—
185C3×D5—62no—
186C23×C3—151no—
187C22×D3—51no—
188C22×D3—151no—
189C22×D3—151no—
190C22×D3—301no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—51no—
194C2×C3×C4—51no—
195C2×C3×C4—101no—
196C2×C3×C4—151no—
197C2×C3×C4—302no—
198#67—154no—
199C2×Dic3—31no—
200C2×Dic3—31no—
201C2×Dic3—51no—
202C2×Dic3—61no—
203C2×Dic3D3×D511yescharacteristic
204C2×Dic3D3×D511yescharacteristic
205D12—154no—
206C4×D3—102no—
207C4×D3—302no—
208C4×D3—302no—
209C22×C5C2×D3211yescharacteristic
210C2×D5—32no—
211C2×D5—61no—
212C2×D5—61no—
213C2×D5—92no—
214C2×D5—181no—
215C2×D5—181no—
216C4×C5—32no—
217C4×C5—32no—
218C4×C5—181no—
219Dic5—21no—
220Dic5—61no—
221C2×C32C2×C4×D511yescharacteristic
222C2×C32C2×D2011yescharacteristic
223C2×C32#70911yescharacteristic
224C3⋊D3—102no—
225C3×D3—102no—
226C2×D4—451no—
227C22×C4—151no—
228C22×C4—451no—
229C4⋊C4—151no—
230C22⋊C4—451no—
231C22⋊C4—451no—
232C42—91no—
233C3×C5—21no—
234C3×C5#89311yescharacteristic, complemented
235C3×C5C4×#6711yescharacteristic, complemented
236C22×C3—21no—
237C22×C3—152no—
238C22×C3—301no—
239C22×C3—301no—
240C22×C3C2×D3×D511yescharacteristic
241C22×C3C2×D3×D511yescharacteristic
242C2×D3—52no—
243C2×D3—101no—
244C2×D3—101no—
245C2×D3—152no—
246C2×D3—152no—
247C2×D3—302no—
248C2×D3—301no—
249C2×D3—301no—
250C2×D3—301no—
251C2×D3—301no—
252C2×D3—601no—
253C2×D3—601no—
254C3×C4—32no—
255C3×C4—32no—
256C3×C4—101no—
257C3×C4—101no—
258C3×C4—201no—
259C3×C4—301no—
260Dic3—61no—
261Dic3—61no—
262Dic3—101no—
263Dic3—121no—
264Dic3#120012yescomplemented
265Dic3#120212yescomplemented
266C2×C5#379911yescharacteristic
267C2×C5C4×D3211yescharacteristic
268C2×C5D3×#6711yescharacteristic
269D5—62no—
270D5—182no—
271C32C4×D2011yescharacteristic, complemented, Sylow
272C23—151no—
273C23—451no—
274D4—454no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—51no—
278C2×C4—91no—
279C2×C4—151no—
280C2×C4—302no—
281C2×C4—902no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—302no—
286C2×C3#676911yescharacteristic
287C2×C3#677111yescharacteristic
288C2×C3C4×D3×D511yescharacteristic
289C2×C3D3×D2011yescharacteristic
290C2×C3C2×#120011yescharacteristic
291C2×C3C2×#120211yescharacteristic
292D3—102no—
293D3—302no—
294D3—302no—
295D3—602no—
296C5#6390411yescharacteristic, complemented, Sylow
297C22—152no—
298C22—301no—
299C22—301no—
300C22—452no—
301C22—901no—
302C22—901no—
303C22D32×D511yescharacteristic
304C4—32no—
305C4—32no—
306C4—101no—
307C4—181no—
308C4—301no—
309C3—21no—
310C3#9142811yescharacteristic, complemented
311C3C4×#120211yescharacteristic, complemented
312C2—302no—
313C2—902no—
314C2D3×#120011yescharacteristic
315C2D3×#120211yescharacteristic
316C2#1062405411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×D5
commutatorC2×C32×C5C22×C4
Fitting, socle, cosocleC22×C32×C5C23
prime radicalC2×#67151C2
hypocenterC32×C5C4×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×#67149C2semidirect
3. C2×C3×#1200C2semidirect
4. #10623829C2semidirect
5. C5×Dic32C2semidirect
6. #10623772C2semidirect
7. C3×Dic3⋊Dic5C2semidirect
17. #67151C4semidirect
26. C2×#1200D3semidirect
30. Dic3⋊Dic5D3semidirect
39. C2×C3⋊D15C2×C4semidirect
42. C2×C3×D15C2×C4semidirect
45. C3×C5×Dic3C2×C4semidirect
46. C3×C5×Dic3D4semidirect
55. Dic32D5semidirect
117. C2×D15C4×D3semidirect
131. C5×Dic3C4×D3semidirect
132. C5×Dic3#67semidirect
148. C32×C5C4×D4semidirect
169. C3×Dic3C4×D5semidirect
170. C3×Dic3D20semidirect
234. C3×C5#893semidirect
235. C3×C5C4×#67semidirect
264. Dic3#1200semidirect
265. Dic3#1202semidirect
271. C32C4×D20semidirect
296. C5#63904semidirect
310. C3#91428semidirect
311. C3C4×#1202semidirect

Maximal subgroups

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

subgroupindex
C2×#671512
C2×#671492
C2×C3×#12002
#106238292
C5×Dic322
#106237722
C3×Dic3⋊Dic52
C4×#12023
#914283
#639045