Finite Group Databasebeta

← #13086109

Subgroups — #13086109

Order 1440.

366 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 74 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×#67156C211yesmaximal, characteristic
2C2×C3⋊D3×Dic5C211yesmaximal, characteristic, complemented
3Dic3⋊Dic15C211yesmaximal, characteristic, complemented
4#10623811C211yesmaximal, characteristic, complemented
5C5×#3722C211yesmaximal, characteristic, complemented
6C3×Dic3⋊Dic5C211yesmaximal, characteristic, complemented
7C3×#6667C211yesmaximal, characteristic, complemented
8#91457—31nomaximal
9#91427—31nomaximal
10C22×C5×C3⋊D3C2211yescharacteristic
11C22×C3×D15C2211yescharacteristic
12C2×C3⋊Dic15C2211yescharacteristic
13C2×C3×Dic15C2211yescharacteristic
14C2×C3×C5×Dic3C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C32×Dic5C2211yescharacteristic
17#67156—22no—
18C3⋊D3×Dic5—22no—
19#63853—51nomaximal, Hall
20C2×#1203—31no—
21C2×#1201—31no—
22C2×D3×Dic5—31no—
23C2×D3×Dic5—31no—
24C2×D3×Dic5—61no—
25#6714—31no—
26C4⋊Dic15—31no—
27C5×(C2×D3)⋊C4—31no—
28C5×(C2×D3)⋊C4—31no—
29C3×#685—31no—
30C3×C4⋊Dic5—31no—
31Dic3⋊Dic5—31no—
32Dic3⋊Dic5D311yescharacteristic, complemented
33#6667D311yescharacteristic, complemented
34#6665—31no—
35C22×C32×C5C2311yescharacteristic
36C2×C5×C3⋊D3—21no—
37C2×C5×C3⋊D3—21no—
38C2×C5×C3⋊D3D412yes—
39C2×C3×D15—21no—
40C2×C3×D15—21no—
41C2×C3×D15—21no—
42C3⋊Dic15—21no—
43C3×Dic15—21no—
44C3×C5×Dic3—21no—
45C3×C5×Dic3—21no—
46C32×Dic5—21no—
47#4032—91noHall
48C2×C4×C3⋊D3—51no—
49C2×#626—51no—
50C3×(C2×D3)⋊C4—51no—
51C3×Dic3⋊C4—51no—
52Dic3⋊Dic3—51no—
53#3722D511yescharacteristic, complemented
54(C2×D3)⋊Dic3—51no—
55C22×D15C2×D311yescharacteristic
56C22×C5×D3—31no—
57C22×C5×D3—31no—
58C22×C5×D3—61no—
59C22×C3×D5—31no—
60C2×C3×C4×C5—31no—
61C2×C3×C4×C5—31no—
62C2×Dic15—31no—
63C2×Dic15—31no—
64C2×Dic15—61no—
65C2×Dic15C2×D311yescharacteristic
66C2×C5×Dic3C2×D311yescharacteristic
67C2×C5×Dic3C2×D311yescharacteristic
68C2×C3×Dic5—21no—
69C2×C3×Dic5—31no—
70C2×C3×Dic5C2×D311yescharacteristic
71C2×C3×Dic5C2×D311yescharacteristic
72#1203—62no—
73#1201—62no—
74D3×Dic5—62no—
75D3×Dic5—62no—
76D3×Dic5—64no—
77#894—151no—
78#883—151no—
79C2×C32×C5C2×D411yescharacteristic
80C2×C32×C5#4111yescharacteristic
81C2×C32×C5#4111yescharacteristic
82C5×C3⋊D3—22no—
83C3×D15—41no—
84C2×C22⋊D5—91no—
85C22×Dic5—91no—
86C5×C22⋊C4—91no—
87#685—31no—
88#685—91no—
89C4⋊Dic5—31no—
90C4⋊Dic5—91no—
91C22×C3⋊D3C2×D511yescharacteristic
92C22×C3×D3—51no—
93C2×C32×C4—51no—
94C2×C3⋊Dic3—51no—
95C4×C3⋊D3—102no—
96C2×C3×Dic3—51no—
97C2×C3×Dic3C2×D511yescharacteristic
98C2×C3×Dic3C2×D511yescharacteristic
99#626—102no—
100C22×C3×C5—21no—
101C22×C3×C5C22×D311yescharacteristic
102C22×C3×C5C22×D311yescharacteristic
103C2×D15—21no—
104C2×D15—21no—
105C2×D15—21no—
106C2×C5×D3—32no—
107C2×C5×D3—32no—
108C2×C5×D3—62no—
109C2×C5×D3—62no—
110C2×C5×D3—62no—
111C2×C5×D3—61no—
112C2×C5×D3—61no—
113C2×C5×D3—61no—
114C2×C5×D3—61no—
115C2×C3×D5—61no—
116C2×C3×D5—61no—
117C2×C3×D5—61no—
118C3×C4×C5—61no—
119C3×C4×C5—61no—
120Dic15—21no—
121Dic15—62no—
122Dic15—61no—
123Dic15—61no—
124C3×Dic5—22no—
125C3×Dic5—21no—
126C3×Dic5—21no—
127C3×Dic5—61no—
128C5×Dic3—21no—
129C5×Dic3—21no—
130C2×#67—151no—
131C2×D12—151no—
132C2×C4×D3—151no—
133C2×C4×D3—151no—
134C2×C4×D3—301no—
135C3×C4⋊C4—151no—
136C3×C22⋊C4—151no—
137#217—151no—
138(C2×D3)⋊C4—31no—
139(C2×D3)⋊C4—31no—
140(C2×D3)⋊C4—51no—
141(C2×D3)⋊C4—151no—
142C4⋊Dic3—151no—
143Dic3⋊C4—51no—
144Dic3⋊C4—151no—
145C32×C5#12311yescharacteristic, complemented, Hall
146C23×C5—91no—
147C22×D5—31no—
148C2×C4×C5—31no—
149C2×C4×C5—31no—
150C22⋊D5—182no—
151C2×Dic5—31no—
152C2×Dic5—91no—
153C2×Dic5—182no—
154C2×Dic5D3211yescharacteristic
155C22×C32C22×D511yescharacteristic
156C2×C3⋊D3—21no—
157C2×C3⋊D3—21no—
158C2×C3⋊D3D2012yes—
159C2×C3×D3—101no—
160C2×C3×D3—101no—
161C2×C3×D3—101no—
162C32×C4—101no—
163C3⋊Dic3—101no—
164C3×Dic3—21no—
165C3×Dic3—21no—
166C3×Dic3—101no—
167#123—451noSylow
168C2×C3×C5—21no—
169C2×C3×C5—21no—
170C2×C3×C5—21no—
171C2×C3×C5#23511yescharacteristic
172C2×C3×C5#23511yescharacteristic
173C2×C3×C5D3×D411yescharacteristic
174C2×C3×C5D3×D411yescharacteristic
175C2×C3×C5#23711yescharacteristic
176C2×C3×C5#23911yescharacteristic
177D15—41no—
178C3×D5—121no—
179C5×D3—64no—
180C5×D3—62no—
181C5×D3—62no—
182C23×C3—151no—
183C22×D3—31no—
184C22×D3—31no—
185C22×D3—51no—
186C22×D3—61no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—51no—
190C2×C3×C4—51no—
191C2×C3×C4—101no—
192C2×C3×C4—151no—
193#67—302no—
194C2×Dic3—51no—
195C2×Dic3—151no—
196C2×Dic3—151no—
197C2×Dic3—301no—
198C2×Dic3D3×D511yescharacteristic
199C2×Dic3D3×D511yescharacteristic
200D12—302no—
201C4×D3—302no—
202C4×D3—302no—
203C4×D3—304no—
204C22×C5—92no—
205C22×C5—181no—
206C22×C5—181no—
207C22×C5C2×D3211yescharacteristic
208C2×D5—61no—
209C2×D5—61no—
210C2×D5—61no—
211C4×C5—61no—
212C4×C5—61no—
213Dic5—21no—
214Dic5—61no—
215Dic5—181no—
216C2×C32C2×D2011yescharacteristic
217C2×C32#71111yescharacteristic
218C2×C32#71111yescharacteristic
219C3⋊D3—22no—
220C3×D3—201no—
221C2×D4—451no—
222C22×C4—451no—
223C4⋊C4—151no—
224C4⋊C4—451no—
225C22⋊C4—91no—
226C22⋊C4—151no—
227C22⋊C4—451no—
228C3×C5—21no—
229C3×C5#88311yescharacteristic, complemented
230C3×C5#89411yescharacteristic, complemented
231C22×C3—21no—
232C22×C3—301no—
233C22×C3—301no—
234C22×C3—301no—
235C22×C3C2×D3×D511yescharacteristic
236C22×C3C2×D3×D511yescharacteristic
237C2×D3—32no—
238C2×D3—32no—
239C2×D3—62no—
240C2×D3—62no—
241C2×D3—62no—
242C2×D3—61no—
243C2×D3—61no—
244C2×D3—61no—
245C2×D3—61no—
246C2×D3—101no—
247C2×D3—101no—
248C2×D3—101no—
249C3×C4—61no—
250C3×C4—61no—
251C3×C4—102no—
252C3×C4—101no—
253C3×C4—101no—
254C3×C4—301no—
255Dic3—21no—
256Dic3—21no—
257Dic3—101no—
258Dic3—302no—
259Dic3—301no—
260Dic3—301no—
261C2×C5—182no—
262C2×C5#379611yescharacteristic
263C2×C5#379811yescharacteristic
264C2×C5#381111yescharacteristic
265D5—121no—
266C32#403211yescharacteristic, complemented, Sylow
267C23—91no—
268C23—151no—
269D4—902no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—51no—
273C2×C4—151no—
274C2×C4—451no—
275C2×C4—902no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—601no—
280C2×C3#676811yescharacteristic
281C2×C3#677211yescharacteristic
282C2×C3D3×D2011yescharacteristic
283C2×C3D3×D2011yescharacteristic
284C2×C3#678011yescharacteristic
285C2×C3#678211yescharacteristic
286D3—64no—
287D3—62no—
288D3—62no—
289D3—201no—
290C5#6385311yescharacteristic, complemented, Sylow
291C22—92no—
292C22—181no—
293C22—181no—
294C22—301no—
295C22—301no—
296C22—301no—
297C22D32×D511yescharacteristic
298C4—61no—
299C4—61no—
300C4—101no—
301C4—301no—
302C4—901no—
303C3—21no—
304C3#9142711yescharacteristic, complemented
305C3#9145711yescharacteristic, complemented
306C2—182no—
307C2—601no—
308C2#1062403111yescharacteristic
309C2#1062404911yescharacteristic
310C2#1062405611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×D5
commutator, Fitting, socle, cosocleC22×C32×C5C23
prime radicalC2×#67156C2
hypocenterC32×C5#123
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊D3×Dic5C2semidirect
3. Dic3⋊Dic15C2semidirect
4. #10623811C2semidirect
5. C5×#3722C2semidirect
6. C3×Dic3⋊Dic5C2semidirect
7. C3×#6667C2semidirect
32. Dic3⋊Dic5D3semidirect
33. #6667D3semidirect
53. #3722D5semidirect
145. C32×C5#123semidirect
229. C3×C5#883semidirect
230. C3×C5#894semidirect
266. C32#4032semidirect
290. C5#63853semidirect
304. C3#91427semidirect
305. C3#91457semidirect

Maximal subgroups

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

subgroupindex
C2×#671562
C2×C3⋊D3×Dic52
Dic3⋊Dic152
#106238112
C5×#37222
C3×Dic3⋊Dic52
C3×#66672
#914573
#914273
#638535