Finite Group Databasebeta

← #13087766

Subgroups — #13087766

Order 1440.

346 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 75 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×#1220C211yesmaximal, characteristic
2C2×#67154C211yesmaximal, characteristic, complemented
3C5×#3757C211yesmaximal, characteristic, complemented
4C3×#6719C211yesmaximal, characteristic, complemented
5#10623832C211yesmaximal, characteristic, complemented
6#10623831C211yesmaximal, characteristic, complemented
7#10623829C211yesmaximal, characteristic, complemented
8#91562—31nomaximal
9#91548—31nomaximal
10C23×C32×C5C2211yescharacteristic
11C22×C3×D15C2211yescharacteristic
12C2×C5×C3⋊Dic3C2211yescharacteristic
13C2×C5×C3⋊Dic3C2211yescharacteristic
14C3×#1220—22no—
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×Dic15C2211yescharacteristic
17C2×C3×Dic15C2211yescharacteristic
18#67154—22no—
19#63969—51nomaximal, Hall
20C2×#1220D311yescharacteristic
21C2×C3×C22⋊D5—31no—
22C2×#1200—31no—
23C2×D5×Dic3—31no—
24#6719D311yescharacteristic, complemented
25C5×#217—31no—
26C5×#217—31no—
27C5×#217—61no—
28C3×C22⋊Dic5—31no—
29Dic3⋊Dic5—31no—
30Dic3⋊Dic5—31no—
31#6669—31no—
32#6668—31no—
33#6667—31no—
34#6665—31no—
35C22×C32×C5—21no—
36C22×C32×C5—21no—
37C22×C32×C5—21no—
38C22×C32×C5C2311yescharacteristic
39C2×C3×D15—21no—
40C2×C3×D15—21no—
41C2×C3×D15D412yes—
42C5×C3⋊Dic3—21no—
43C5×C3⋊Dic3—21no—
44C3×Dic15—21no—
45C3×Dic15—21no—
46C3×Dic15—21no—
47#4029—91noHall
48C2×C3×#67—51no—
49C2×D3×Dic3—51no—
50#3757D511yescharacteristic, complemented
51C3×#217—51no—
52#3724—51no—
53Dic3⋊Dic3—51no—
54(C2×D3)⋊Dic3—51no—
55C23×C3×C5—21no—
56C23×C3×C5C2×D311yescharacteristic
57C23×C3×C5C2×D311yescharacteristic
58C22×D15C2×D311yescharacteristic
59C22×C3×D5—31no—
60#1220—22no—
61C2×Dic15C2×D311yescharacteristic
62C2×Dic15C2×D311yescharacteristic
63C2×Dic15C2×D311yescharacteristic
64C2×C5×Dic3—31no—
65C2×C5×Dic3—31no—
66C2×C5×Dic3—31no—
67C2×C5×Dic3—31no—
68C2×C5×Dic3—61no—
69C2×C5×Dic3—61no—
70C3×C22⋊D5—62no—
71C2×C3×Dic5—31no—
72C2×C3×Dic5—31no—
73C2×C3×Dic5—31no—
74#1200—62no—
75D5×Dic3—62no—
76#935—151no—
77#883—151no—
78C2×C32×C5—41no—
79C2×C32×C5C2×D411yescharacteristic
80C2×C32×C5#4111yescharacteristic
81C2×C32×C5#4111yescharacteristic
82C3×D15—22no—
83C2×C22⋊D5—31no—
84C2×C4×D5—91no—
85C5×C22⋊C4—91no—
86C22⋊Dic5—31no—
87#685—91no—
88C4⋊Dic5—91no—
89#683—91no—
90C23×C32C2×D511yescharacteristic
91C22×C3×D3—51no—
92C2×C3⋊Dic3C2×D511yescharacteristic
93C2×C3⋊Dic3C2×D511yescharacteristic
94C3×#67—102no—
95C2×C3×Dic3—51no—
96C2×C3×Dic3—51no—
97C2×C3×Dic3—51no—
98D3×Dic3—102no—
99C22×C3×C5—22no—
100C22×C3×C5—21no—
101C22×C3×C5—21no—
102C22×C3×C5—21no—
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—21no—
106C22×C3×C5—21no—
107C22×C3×C5—41no—
108C22×C3×C5—41no—
109C22×C3×C5C22×D311yescharacteristic
110C22×C3×C5C22×D311yescharacteristic
111C2×D15—21no—
112C2×D15—21no—
113C2×D15#6712yes—
114C2×C3×D5—32no—
115C2×C3×D5—61no—
116C2×C3×D5—61no—
117Dic15—21no—
118Dic15—21no—
119Dic15—21no—
120C3×Dic5—61no—
121C3×Dic5—61no—
122C3×Dic5—61no—
123C5×Dic3—61no—
124C5×Dic3—61no—
125C5×Dic3—61no—
126C5×Dic3—61no—
127C5×Dic3—121no—
128C5×Dic3—121no—
129C2×C3×D4—151no—
130C2×#67—51no—
131C22×Dic3—151no—
132C2×C4×D3—151no—
133C3×C22⋊C4—151no—
134#217—31no—
135#217—31no—
136#217—51no—
137#217—61no—
138#217—151no—
139(C2×D3)⋊C4—151no—
140C4⋊Dic3—151no—
141Dic3⋊C4—151no—
142Dic3⋊C4—151no—
143Dic3⋊C4—151no—
144C32×C5#12311yescharacteristic, complemented, Hall
145C23×C5D3211yescharacteristic
146C22×D5—31no—
147C2×C4×C5—91no—
148C2×C4×C5—91no—
149C22⋊D5—62no—
150C2×Dic5—31no—
151C2×Dic5—31no—
152C2×Dic5—31no—
153C4×D5—182no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32C22×D511yescharacteristic
158C2×C3×D3—52no—
159C2×C3×D3—101no—
160C2×C3×D3—101no—
161C3⋊Dic3—21no—
162C3⋊Dic3—21no—
163C3×Dic3—101no—
164C3×Dic3—101no—
165C3×Dic3—101no—
166#123—451noSylow
167C2×C3×C5—21no—
168C2×C3×C5—21no—
169C2×C3×C5—21no—
170C2×C3×C5—42no—
171C2×C3×C5—41no—
172C2×C3×C5—41no—
173C2×C3×C5#23511yescharacteristic
174C2×C3×C5D3×D411yescharacteristic
175C2×C3×C5#23711yescharacteristic
176C2×C3×C5#23711yescharacteristic
177C2×C3×C5#23711yescharacteristic
178C2×C3×C5C2×#6711yescharacteristic
179D15—22no—
180C3×D5—62no—
181C23×C3—21no—
182C23×C3—151no—
183C23×C3D3×D511yescharacteristic
184C23×C3D3×D511yescharacteristic
185C22×D3—51no—
186C3×D4—302no—
187C2×C3×C4—151no—
188C2×C3×C4—151no—
189C2×C3×C4—151no—
190#67—102no—
191C2×Dic3—31no—
192C2×Dic3—31no—
193C2×Dic3—31no—
194C2×Dic3—31no—
195C2×Dic3—51no—
196C2×Dic3—51no—
197C2×Dic3—51no—
198C2×Dic3—61no—
199C2×Dic3—61no—
200C2×Dic3—302no—
201C4×D3—302no—
202C22×C5—21no—
203C22×C5—21no—
204C22×C5—21no—
205C22×C5C2×D3211yescharacteristic
206C2×D5—32no—
207C2×D5—61no—
208C2×D5—61no—
209C4×C5—181no—
210C4×C5—181no—
211Dic5—61no—
212Dic5—61no—
213Dic5—61no—
214C2×C32—41no—
215C2×C32#70911yescharacteristic
216C2×C32D4×D511yescharacteristic
217C2×C32#71111yescharacteristic
218C3×D3—102no—
219C2×D4—151no—
220C22×C4—451no—
221C4⋊C4—451no—
222C4⋊C4—451no—
223C22⋊C4—91no—
224C22⋊C4—151no—
225C22⋊C4—451no—
226C3×C5—21no—
227C3×C5#88311yescharacteristic, complemented
228C3×C5#93511yescharacteristic, complemented
229C22×C3—22no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—152no—
240C22×C3—301no—
241C22×C3—301no—
242C22×C3C2×D3×D511yescharacteristic
243C22×C3C2×D3×D511yescharacteristic
244C2×D3—52no—
245C2×D3—101no—
246C2×D3—101no—
247C3×C4—301no—
248C3×C4—301no—
249C3×C4—301no—
250Dic3—61no—
251Dic3—61no—
252Dic3—61no—
253Dic3—61no—
254Dic3—101no—
255Dic3—101no—
256Dic3—101no—
257Dic3—121no—
258Dic3—121no—
259C2×C5—41no—
260C2×C5#380411yescharacteristic
261C2×C5#380511yescharacteristic
262C2×C5D3×#6711yescharacteristic
263D5—62no—
264C32#402911yescharacteristic, complemented, Sylow
265C23—151no—
266C23(C3⋊D3)⋊D511yescharacteristic
267D4—302no—
268C2×C4—91no—
269C2×C4—91no—
270C2×C4—151no—
271C2×C4—151no—
272C2×C4—151no—
273C2×C4—902no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—42no—
278C2×C3—41no—
279C2×C3—41no—
280C2×C3—302no—
281C2×C3#677911yescharacteristic
282C2×C3#677911yescharacteristic
283C2×C3#678011yescharacteristic
284C2×C3#678211yescharacteristic
285C2×C3D5×#6711yescharacteristic
286C2×C3#679011yescharacteristic
287D3—102no—
288C5#6396911yescharacteristic, complemented, Sylow
289C22—21no—
290C22—21no—
291C22—21no—
292C22—152no—
293C22—301no—
294C22—301no—
295C22C2×(C3⋊D3)⋊D511yescharacteristic
296C4—181no—
297C4—181no—
298C4—301no—
299C4—301no—
300C4—301no—
301C3—21no—
302C3#9154811yescharacteristic, complemented
303C3#9156211yescharacteristic, complemented
304C2—41no—
305C2—302no—
306C2#1062422911yescharacteristic
307C2#1062423011yescharacteristic
308C2#1062423511yescharacteristic
subgroupG/H
center, FrattiniC22C2×(C3⋊D3)⋊D5
commutator, socle, cosocleC22×C32×C5C23
FittingC23×C32×C5C22
prime radicalC2×C3×#1220C2
hypercenterC23(C3⋊D3)⋊D5
hypocenterC32×C5#123
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#67154C2semidirect
3. C5×#3757C2semidirect
4. C3×#6719C2semidirect
5. #10623832C2semidirect
6. #10623831C2semidirect
7. #10623829C2semidirect
24. #6719D3semidirect
50. #3757D5semidirect
144. C32×C5#123semidirect
227. C3×C5#883semidirect
228. C3×C5#935semidirect
264. C32#4029semidirect
288. C5#63969semidirect
302. C3#91548semidirect
303. C3#91562semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#12202
C2×#671542
C5×#37572
C3×#67192
#106238322
#106238312
#106238292
#915623
#915483
#639695