Finite Group Databasebeta

← #13087699

Subgroups — #13087699

Order 1440.

402 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 79 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×D60C211yesmaximal, characteristic, complemented
2C2×#67157C211yesmaximal, characteristic, complemented
3C2×#67157C211yesmaximal, characteristic, complemented
4C2×#67154C211yesmaximal, characteristic, complemented
5C5×#3750C211yesmaximal, characteristic, complemented
6C3×#6714C211yesmaximal, characteristic, complemented
7#10623829C211yesmaximal, characteristic, complemented
8#91459—31nomaximal
9#91409—31nomaximal
10C22×C3×D15C2211yescharacteristic
11C22×C3×D15C2211yescharacteristic
12C22×C3×D15C2211yescharacteristic
13C2×C32×C4×C5C2211yescharacteristic
14C2×C5×C3⋊Dic3C2211yescharacteristic
15C2×C5×C3⋊Dic3C2211yescharacteristic
16C2×C3×Dic15C2211yescharacteristic
17C3×D60—22no—
18#67157—22no—
19#67157—22no—
20#67154—22no—
21#63885—51nomaximal, Hall
22C2×D60D311yescharacteristic, complemented
23C2×C3×D20—31no—
24C2×#1202—31no—
25C2×#1202—31no—
26C2×#1202—31no—
27C2×#1202—31no—
28C2×#1200—31no—
29C2×D5×Dic3—31no—
30#6714D311yescharacteristic, complemented
31C5×Dic3⋊C4—31no—
32C5×Dic3⋊C4—31no—
33C5×Dic3⋊C4—61no—
34C3×#685—31no—
35#6667—31no—
36#6665—31no—
37C22×C32×C5C2311yescharacteristic
38C2×C3×D15—21no—
39C2×C3×D15—21no—
40C2×C3×D15—21no—
41C2×C3×D15—21no—
42C2×C3×D15—21no—
43C2×C3×D15—21no—
44C2×C3×D15—21no—
45C2×C3×D15—21no—
46C2×C3×D15D412yescomplemented
47C32×C4×C5—21no—
48C5×C3⋊Dic3—21no—
49C5×C3⋊Dic3D412yescomplemented
50C3×Dic15—21no—
51#4041—91noHall
52C2×C3×D12—51no—
53C2×#625—51no—
54C2×#625—51no—
55C2×D3×Dic3—51no—
56#3750D511yescharacteristic, complemented
57C3×(C2×D3)⋊C4—51no—
58(C2×D3)⋊Dic3—51no—
59C22×D15C2×D311yescharacteristic
60C22×D15C2×D311yescharacteristic
61C22×D15C2×D311yescharacteristic
62C22×C3×D5—31no—
63C22×C3×D5—31no—
64C22×C3×D5—31no—
65C2×C3×C4×C5—21no—
66C2×C3×C4×C5C2×D311yescharacteristic
67C2×C3×C4×C5C2×D311yescharacteristic
68C2×Dic15C2×D311yescharacteristic
69D60—22no—
70C2×C5×Dic3—31no—
71C2×C5×Dic3—31no—
72C2×C5×Dic3—31no—
73C2×C5×Dic3—31no—
74C2×C5×Dic3—61no—
75C2×C5×Dic3—61no—
76C2×C3×Dic5—31no—
77C3×D20—62no—
78#1202—62no—
79#1202—62no—
80#1202—62no—
81#1202—62no—
82#1200—62no—
83D5×Dic3—62no—
84#939—151no—
85#884—151no—
86C2×C32×C5C2×D411yescharacteristic
87C2×C32×C5C2×D411yescharacteristic
88C2×C32×C5#4111yescharacteristic
89C3×D15—22no—
90C3×D15—41no—
91C3×D15—41no—
92C2×D20—31no—
93C2×D20—91no—
94C2×D20—91no—
95C2×C4×D5—91no—
96C5×C4⋊C4—91no—
97#685—31no—
98#685—91no—
99C22×C3×D3—51no—
100C22×C3×D3—51no—
101C22×C3×D3—51no—
102C2×C32×C4C2×D511yescharacteristic
103C2×C3⋊Dic3C2×D511yescharacteristic
104C2×C3⋊Dic3C2×D511yescharacteristic
105C2×C3×Dic3—51no—
106C3×D12—102no—
107#625—102no—
108#625—102no—
109D3×Dic3—102no—
110C22×C3×C5—21no—
111C22×C3×C5C22×D311yescharacteristic
112C22×C3×C5C22×D311yescharacteristic
113C2×D15—21no—
114C2×D15—21no—
115C2×D15—21no—
116C2×D15—21no—
117C2×D15—21no—
118C2×D15—21no—
119C2×D15—21no—
120C2×D15—21no—
121C2×D15#6712yescomplemented
122C2×C3×D5—32no—
123C2×C3×D5—61no—
124C2×C3×D5—61no—
125C2×C3×D5—61no—
126C2×C3×D5—61no—
127C2×C3×D5—61no—
128C2×C3×D5—61no—
129C2×C3×D5—61no—
130C2×C3×D5—61no—
131C3×C4×C5—21no—
132C3×C4×C5—21no—
133C3×C4×C5—41no—
134Dic15—21no—
135C3×Dic5—61no—
136C5×Dic3—32no—
137C5×Dic3—32no—
138C5×Dic3—62no—
139C5×Dic3—61no—
140C5×Dic3—61no—
141C5×Dic3—121no—
142C2×C3×D4—151no—
143C2×#67—151no—
144C2×#67—151no—
145C2×#67—151no—
146C2×#67—151no—
147C22×Dic3—151no—
148C2×D12—51no—
149C2×C4×D3—151no—
150C3×C22⋊C4—151no—
151#217—151no—
152(C2×D3)⋊C4—51no—
153(C2×D3)⋊C4—151no—
154Dic3⋊C4—31no—
155Dic3⋊C4—31no—
156Dic3⋊C4—61no—
157C32×C5#12111yescharacteristic, complemented, Hall
158C22×D5—31no—
159C22×D5—31no—
160C22×D5—31no—
161C2×C4×C5—91no—
162C2×C4×C5—91no—
163C2×C4×C5D3211yescharacteristic
164C2×Dic5—31no—
165D20—62no—
166D20—182no—
167D20—182no—
168C4×D5—182no—
169C22×C32C22×D511yescharacteristic
170C2×C3×D3—52no—
171C2×C3×D3—101no—
172C2×C3×D3—101no—
173C2×C3×D3—101no—
174C2×C3×D3—101no—
175C2×C3×D3—101no—
176C2×C3×D3—101no—
177C2×C3×D3—101no—
178C2×C3×D3—101no—
179C32×C4—21no—
180C3⋊Dic3—21no—
181C3⋊Dic3D2012yescomplemented
182C3×Dic3—101no—
183#121—451noSylow
184C2×C3×C5—21no—
185C2×C3×C5—21no—
186C2×C3×C5—21no—
187C2×C3×C5#23511yescharacteristic
188C2×C3×C5D3×D411yescharacteristic
189C2×C3×C5D3×D411yescharacteristic
190C2×C3×C5D3×D411yescharacteristic
191C2×C3×C5#23711yescharacteristic
192C2×C3×C5C2×#6711yescharacteristic
193D15—22no—
194D15—41no—
195D15—41no—
196C3×D5—62no—
197C3×D5—121no—
198C3×D5—121no—
199C23×C3—151no—
200C23×C3—151no—
201C23×C3—151no—
202C22×D3—51no—
203C22×D3—51no—
204C22×D3—51no—
205C3×D4—302no—
206C2×C3×C4—21no—
207C2×C3×C4—151no—
208C2×C3×C4D3×D511yescharacteristic
209C2×C3×C4D3×D511yescharacteristic
210#67—302no—
211#67—302no—
212#67—302no—
213#67—302no—
214C2×Dic3—31no—
215C2×Dic3—31no—
216C2×Dic3—31no—
217C2×Dic3—31no—
218C2×Dic3—51no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—302no—
222D12—102no—
223C4×D3—302no—
224C22×C5C2×D3211yescharacteristic
225C2×D5—32no—
226C2×D5—61no—
227C2×D5—61no—
228C2×D5—61no—
229C2×D5—61no—
230C2×D5—61no—
231C2×D5—61no—
232C2×D5—61no—
233C2×D5—61no—
234C4×C5—21no—
235C4×C5—92no—
236C4×C5—181no—
237Dic5—61no—
238C2×C32C2×D2011yescharacteristic
239C2×C32D4×D511yescharacteristic
240C2×C32#71311yescharacteristic
241C3×D3—102no—
242C3×D3—201no—
243C3×D3—201no—
244C2×D4—151no—
245C2×D4—451no—
246C2×D4—451no—
247C22×C4—451no—
248C4⋊C4—91no—
249C22⋊C4—151no—
250C22⋊C4—451no—
251C3×C5—21no—
252C3×C5#88411yescharacteristic, complemented
253C3×C5#93911yescharacteristic, complemented
254C22×C3—21no—
255C22×C3—152no—
256C22×C3—301no—
257C22×C3—301no—
258C22×C3—301no—
259C22×C3—301no—
260C22×C3—301no—
261C22×C3—301no—
262C22×C3—301no—
263C22×C3—301no—
264C22×C3C2×D3×D511yescharacteristic
265C22×C3C2×D3×D511yescharacteristic
266C2×D3—52no—
267C2×D3—101no—
268C2×D3—101no—
269C2×D3—101no—
270C2×D3—101no—
271C2×D3—101no—
272C2×D3—101no—
273C2×D3—101no—
274C2×D3—101no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—41no—
278C3×C4—301no—
279Dic3—32no—
280Dic3—32no—
281Dic3—62no—
282Dic3—61no—
283Dic3—61no—
284Dic3—101no—
285Dic3—121no—
286C2×C5#379511yescharacteristic
287C2×C5#380211yescharacteristic
288C2×C5D3×#6711yescharacteristic
289D5—62no—
290D5—121no—
291D5—121no—
292C32#404111yescharacteristic, complemented, Sylow
293C23—151no—
294C23—151no—
295C23—151no—
296D4—302no—
297D4—902no—
298D4—902no—
299C2×C4—91no—
300C2×C4—91no—
301C2×C4—151no—
302C2×C4—902no—
303C2×C4(C3⋊D3)⋊D511yescharacteristic
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—302no—
308C2×C3—601no—
309C2×C3—601no—
310C2×C3#676611yescharacteristic
311C2×C3#677311yescharacteristic
312C2×C3D3×D2011yescharacteristic
313C2×C3D3×D2011yescharacteristic
314C2×C3D5×#6711yescharacteristic
315C2×C3#679011yescharacteristic
316D3—102no—
317D3—201no—
318D3—201no—
319C5#6388511yescharacteristic, complemented, Sylow
320C22—152no—
321C22—301no—
322C22—301no—
323C22—301no—
324C22—301no—
325C22—301no—
326C22—301no—
327C22—301no—
328C22—301no—
329C22C2×(C3⋊D3)⋊D511yescharacteristic
330C4—21no—
331C4—92no—
332C4—181no—
333C4—301no—
334C3—21no—
335C3#9140911yescharacteristic, complemented
336C3#9145911yescharacteristic, complemented
337C2—302no—
338C2—601no—
339C2—601no—
340C2#1062422111yescharacteristic
341C2#1062422711yescharacteristic
342C2#1062423511yescharacteristic
subgroupG/H
center, FrattiniC22C2×(C3⋊D3)⋊D5
commutator, socle, cosocleC22×C32×C5C23
FittingC2×C32×C4×C5C22
hypercenterC2×C4(C3⋊D3)⋊D5
hypocenterC32×C5#121
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D60C2semidirect
2. C2×#67157C2semidirect
3. C2×#67157C2semidirect
4. C2×#67154C2semidirect
5. C5×#3750C2semidirect
6. C3×#6714C2semidirect
7. #10623829C2semidirect
22. C2×D60D3semidirect
30. #6714D3semidirect
46. C2×C3×D15D4semidirect
49. C5×C3⋊Dic3D4semidirect
56. #3750D5semidirect
121. C2×D15#67semidirect
157. C32×C5#121semidirect
181. C3⋊Dic3D20semidirect
252. C3×C5#884semidirect
253. C3×C5#939semidirect
292. C32#4041semidirect
319. C5#63885semidirect
335. C3#91409semidirect
336. C3#91459semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D602
C2×#671572
C2×#671542
C5×#37502
C3×#67142
#106238292
#914593
#914093
#638855