Finite Group Databasebeta

← #13086106

Subgroups — #13086106

Order 1440.

366 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 72 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×#67152C211yesmaximal, characteristic
2C2×#67137C211yesmaximal, characteristic, complemented
3#10623831C211yesmaximal, characteristic, complemented
4#10623813C211yesmaximal, characteristic, complemented
5C5×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
6C3×Dic3⋊Dic5C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊Dic5C211yesmaximal, characteristic, complemented
8#91562—31nomaximal
9#91438—31nomaximal
10C22×C3⋊D15C2211yescharacteristic
11C22×C3×C5×D3C2211yescharacteristic
12C2×C5×C3⋊Dic3C2211yescharacteristic
13C2×C3×Dic15C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C32×Dic5C2211yescharacteristic
17#67152—22no—
18#67137—22no—
19#63853—51nomaximal, Hall
20C2×#1220—31no—
21C2×#1203—31no—
22C2×#1200—31no—
23C2×#1200—31no—
24C2×#1200—61no—
25#6714—31no—
26C5×#217—31no—
27C5×(C2×D3)⋊C4—31no—
28C3×C22⋊Dic5—31no—
29C3×C4⋊Dic5—31no—
30Dic3⋊Dic5D311yescharacteristic, complemented
31#6669—31no—
32#6668—31no—
33#6667—31no—
34(C2×D3)⋊Dic5D311yescharacteristic, complemented
35C22×C32×C5C2311yescharacteristic
36C2×C3⋊D15—21no—
37C2×C3⋊D15—21no—
38C2×C3⋊D15D412yes—
39C2×C3×C5×D3—21no—
40C2×C3×C5×D3—21no—
41C2×C3×C5×D3—21no—
42C5×C3⋊Dic3—21no—
43C3×Dic15—21no—
44C3×Dic15—21no—
45C3×C5×Dic3—21no—
46C32×Dic5—21no—
47#4029—91noHall
48C2×C4×C3⋊D3—51no—
49C2×#626—51no—
50C3×(C2×D3)⋊C4—51no—
51C3×Dic3⋊C4—51no—
52Dic3⋊Dic3—51no—
53#3722—51no—
54(C2×D3)⋊Dic3D511yescharacteristic, complemented
55C23×C3×C5—31no—
56C22×D15—31no—
57C22×D15—31no—
58C22×D15—61no—
59C22×C5×D3C2×D311yescharacteristic
60C2×C3×C4×C5—31no—
61#1220—62no—
62C2×Dic15C2×D311yescharacteristic
63C2×Dic15C2×D311yescharacteristic
64C2×C5×Dic3—31no—
65C2×C5×Dic3—31no—
66C2×C5×Dic3—61no—
67C2×C5×Dic3C2×D311yescharacteristic
68C2×C3×Dic5—21no—
69C2×C3×Dic5—31no—
70C2×C3×Dic5—31no—
71C2×C3×Dic5C2×D311yescharacteristic
72C2×C3×Dic5C2×D311yescharacteristic
73#1203—62no—
74#1200—62no—
75#1200—62no—
76#1200—64no—
77#894—151no—
78#883—151no—
79C2×C32×C5C2×D411yescharacteristic
80C2×C32×C5#4111yescharacteristic
81C2×C32×C5#4111yescharacteristic
82C3⋊D15—22no—
83C3×C5×D3—41no—
84C2×C22⋊D5—91no—
85C2×C4×D5—91no—
86C5×C22⋊C4—91no—
87C22⋊Dic5—31no—
88#685—91no—
89C4⋊Dic5—31no—
90#683—91no—
91C22×C3⋊D3—51no—
92C22×C3×D3C2×D511yescharacteristic
93C2×C32×C4—51no—
94C2×C3⋊Dic3C2×D511yescharacteristic
95C4×C3⋊D3—102no—
96C2×C3×Dic3—51no—
97C2×C3×Dic3—51no—
98C2×C3×Dic3C2×D511yescharacteristic
99#626—102no—
100C22×C3×C5—21no—
101C22×C3×C5—61no—
102C22×C3×C5—61no—
103C22×C3×C5—61no—
104C22×C3×C5C22×D311yescharacteristic
105C22×C3×C5C22×D311yescharacteristic
106C2×D15—32no—
107C2×D15—32no—
108C2×D15—62no—
109C2×D15—62no—
110C2×D15—62no—
111C2×D15—61no—
112C2×D15—61no—
113C2×D15—61no—
114C2×D15—61no—
115C2×C5×D3—21no—
116C2×C5×D3—21no—
117C2×C5×D3—21no—
118C3×C4×C5—61no—
119Dic15—21no—
120Dic15—21no—
121C3×Dic5—22no—
122C3×Dic5—21no—
123C3×Dic5—21no—
124C3×Dic5—61no—
125C3×Dic5—61no—
126C5×Dic3—21no—
127C5×Dic3—62no—
128C5×Dic3—61no—
129C5×Dic3—61no—
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—31no—
138(C2×D3)⋊C4—31no—
139(C2×D3)⋊C4—51no—
140(C2×D3)⋊C4—151no—
141(C2×D3)⋊C4—151no—
142C4⋊Dic3—151no—
143Dic3⋊C4—51no—
144Dic3⋊C4—151no—
145C32×C5#12311yescharacteristic, complemented, Hall
146C23×C5—31no—
147C22×D5—91no—
148C2×C4×C5—31no—
149C2×C4×C5—91no—
150C22⋊D5—182no—
151C2×Dic5—31no—
152C2×Dic5—31no—
153C2×Dic5D3211yescharacteristic
154C4×D5—182no—
155C22×C32C22×D511yescharacteristic
156C2×C3⋊D3—52no—
157C2×C3⋊D3—101no—
158C2×C3⋊D3—101no—
159C2×C3×D3—21no—
160C2×C3×D3—21no—
161C2×C3×D3—21no—
162C32×C4—101no—
163C3⋊Dic3—21no—
164C3×Dic3—21no—
165C3×Dic3—101no—
166C3×Dic3—101no—
167#123—451noSylow
168C2×C3×C5—21no—
169C2×C3×C5—21no—
170C2×C3×C5—21no—
171C2×C3×C5—121no—
172C2×C3×C5#23511yescharacteristic
173C2×C3×C5#23511yescharacteristic
174C2×C3×C5D3×D411yescharacteristic
175C2×C3×C5D3×D411yescharacteristic
176C2×C3×C5#23711yescharacteristic
177C2×C3×C5#23911yescharacteristic
178D15—64no—
179D15—62no—
180D15—62no—
181C5×D3—41no—
182C23×C3—31no—
183C22×D3—151no—
184C22×D3—151no—
185C22×D3—301no—
186C22×D3D3×D511yescharacteristic
187C2×C3×C4—31no—
188C2×C3×C4—51no—
189C2×C3×C4—51no—
190C2×C3×C4—101no—
191C2×C3×C4—151no—
192C2×C3×C4—151no—
193#67—302no—
194C2×Dic3—31no—
195C2×Dic3—31no—
196C2×Dic3—51no—
197C2×Dic3—51no—
198C2×Dic3—61no—
199C2×Dic3D3×D511yescharacteristic
200D12—302no—
201C4×D3—302no—
202C4×D3—302no—
203C4×D3—304no—
204C22×C5—61no—
205C22×C5—61no—
206C22×C5—61no—
207C22×C5C2×D3211yescharacteristic
208C2×D5—92no—
209C2×D5—181no—
210C2×D5—181no—
211C4×C5—61no—
212C4×C5—181no—
213Dic5—21no—
214Dic5—61no—
215Dic5—61no—
216C2×C32#70911yescharacteristic
217C2×C32D4×D511yescharacteristic
218C2×C32#71111yescharacteristic
219C3⋊D3—102no—
220C3×D3—41no—
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—61no—
233C22×C3—61no—
234C22×C3—61no—
235C22×C3C2×D3×D511yescharacteristic
236C22×C3C2×D3×D511yescharacteristic
237C2×D3—21no—
238C2×D3—21no—
239C2×D3—21no—
240C2×D3—152no—
241C2×D3—152no—
242C2×D3—302no—
243C2×D3—302no—
244C2×D3—302no—
245C2×D3—301no—
246C2×D3—301no—
247C2×D3—301no—
248C2×D3—301no—
249C3×C4—61no—
250C3×C4—102no—
251C3×C4—101no—
252C3×C4—101no—
253C3×C4—301no—
254C3×C4—301no—
255Dic3—21no—
256Dic3—62no—
257Dic3—61no—
258Dic3—61no—
259Dic3—101no—
260Dic3—101no—
261C2×C5—121no—
262C2×C5#379611yescharacteristic
263C2×C5#379811yescharacteristic
264C2×C5#381111yescharacteristic
265D5—182no—
266C32#402911yescharacteristic, complemented, Sylow
267C23—31no—
268C23—451no—
269D4—902no—
270C2×C4—31no—
271C2×C4—51no—
272C2×C4—91no—
273C2×C4—151no—
274C2×C4—151no—
275C2×C4—902no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—121no—
280C2×C3#676911yescharacteristic
281C2×C3#677711yescharacteristic
282C2×C3#677911yescharacteristic
283C2×C3#678211yescharacteristic
284C2×C3#678211yescharacteristic
285C2×C3#679011yescharacteristic
286D3—41no—
287D3—304no—
288D3—302no—
289D3—302no—
290C5#6385311yescharacteristic, complemented, Sylow
291C22—61no—
292C22—61no—
293C22—61no—
294C22—452no—
295C22—901no—
296C22—901no—
297C22D32×D511yescharacteristic
298C4—61no—
299C4—101no—
300C4—181no—
301C4—301no—
302C4—301no—
303C3—21no—
304C3#9143811yescharacteristic, complemented
305C3#9156211yescharacteristic, complemented
306C2—121no—
307C2—902no—
308C2#1062403011yescharacteristic
309C2#1062404811yescharacteristic
310C2#1062405511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×D5
commutator, Fitting, socle, cosocleC22×C32×C5C23
prime radicalC2×#67152C2
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×#67137C2semidirect
3. #10623831C2semidirect
4. #10623813C2semidirect
5. C5×(C2×D3)⋊Dic3C2semidirect
6. C3×Dic3⋊Dic5C2semidirect
7. C3×(C2×D3)⋊Dic5C2semidirect
30. Dic3⋊Dic5D3semidirect
34. (C2×D3)⋊Dic5D3semidirect
54. (C2×D3)⋊Dic3D5semidirect
145. C32×C5#123semidirect
229. C3×C5#883semidirect
230. C3×C5#894semidirect
266. C32#4029semidirect
290. C5#63853semidirect
304. C3#91438semidirect
305. C3#91562semidirect

Maximal subgroups

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

subgroupindex
C2×#671522
C2×#671372
#106238312
#106238132
C5×(C2×D3)⋊Dic32
C3×Dic3⋊Dic52
C3×(C2×D3)⋊Dic52
#915623
#914383
#638535