Finite Group Databasebeta

← #13087572

Subgroups — #13087572

Order 1440.

410 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 86 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, complemented
2C2×C3×C5×#67C211yesmaximal, characteristic, complemented
3C2×#67150C211yesmaximal, characteristic, complemented
4C2×Dic3×D15C211yesmaximal, characteristic, complemented
5#10623913C211yesmaximal, characteristic, complemented
6Dic3⋊Dic15C211yesmaximal, characteristic, complemented
7(C2×D3)⋊Dic15C211yesmaximal, characteristic, complemented
8#91827—31nomaximal
9#91549—31nomaximal
10C23×C32×C5C2211yescharacteristic
11C22×C3×D15C2211yescharacteristic
12C22×C3×C5×D3C2211yescharacteristic
13C2×C3⋊Dic15C2211yescharacteristic
14C2×C3⋊Dic15C2211yescharacteristic
15C3×#1220—22no—
16C2×C3×Dic15C2211yescharacteristic
17C3×C5×#67—22no—
18C2×C3×C5×Dic3C2211yescharacteristic
19#67150—22no—
20Dic3×D15—22no—
21#63970—51nomaximal, Hall
22C2×C3×C5×D4—31no—
23C2×#1220—31no—
24C2×#1220D311yescharacteristic, complemented
25C2×C4×D15—31no—
26C2×C5×#67D311yescharacteristic, complemented
27C2×C3×C22⋊D5—31no—
28C2×#1201—31no—
29C2×D5×Dic3—31no—
30#6719—31no—
31#6719—31no—
32#6719—31no—
33#6719—61no—
34C4⋊Dic15—31no—
35Dic3⋊Dic5—31no—
36(C2×D3)⋊Dic5—31no—
37C22×C32×C5—21no—
38C22×C32×C5—21no—
39C22×C32×C5—21no—
40C22×C32×C5C2311yescharacteristic
41C2×C3×D15—21no—
42C2×C3×D15—21no—
43C2×C3×D15D412yescomplemented
44C2×C3×C5×D3—21no—
45C2×C3×C5×D3—21no—
46C2×C3×C5×D3—21no—
47C3⋊Dic15—21no—
48C3⋊Dic15—21no—
49C3×Dic15—21no—
50C3×C5×Dic3D412yescomplemented
51#4084—91noHall
52C2×C3×#67—51no—
53C2×C3×#67D511yescharacteristic, complemented
54C2×#625—51no—
55C2×D3×Dic3—51no—
56#3757—51no—
57Dic3⋊Dic3—51no—
58(C2×D3)⋊Dic3—51no—
59C23×C3×C5—21no—
60C23×C3×C5—31no—
61C23×C3×C5C2×D311yescharacteristic
62C23×C3×C5C2×D311yescharacteristic
63C22×D15C2×D311yescharacteristic
64C22×C5×D3C2×D311yescharacteristic
65C22×C3×D5—31no—
66C3×C5×D4—62no—
67C2×C3×C4×C5—31no—
68#1220—22no—
69#1220—62no—
70C2×Dic15—31no—
71C2×Dic15—31no—
72C2×Dic15—31no—
73C2×Dic15—31no—
74C2×Dic15—61no—
75C2×Dic15—61no—
76C2×Dic15C2×D311yescharacteristic
77C4×D15—62no—
78C5×#67—22no—
79C2×C5×Dic3C2×D311yescharacteristic
80C3×C22⋊D5—62no—
81C2×C3×Dic5—31no—
82#1201—62no—
83D5×Dic3—62no—
84#939—151no—
85#938—151no—
86C2×C32×C5—41no—
87C2×C32×C5C2×D411yescharacteristic
88C2×C32×C5C2×D411yescharacteristic
89C2×C32×C5#4111yescharacteristic
90C3×D15—22no—
91C3×C5×D3—41no—
92C2×C5×D4—31no—
93C2×C22⋊D5—31no—
94C2×C22⋊D5—91no—
95C2×C4×D5—91no—
96C22⋊Dic5—91no—
97C22⋊Dic5—91no—
98C4⋊Dic5—91no—
99C23×C32C2×D511yescharacteristic
100C22×C3×D3—51no—
101C22×C3×D3C2×D511yescharacteristic
102C2×C3⋊Dic3—51no—
103C2×C3⋊Dic3—51no—
104C3×#67—22no—
105C3×#67—102no—
106C2×C3×Dic3—51no—
107C2×C3×Dic3C2×D511yescharacteristic
108#625—102no—
109D3×Dic3—102no—
110C22×C3×C5—22no—
111C22×C3×C5—21no—
112C22×C3×C5—21no—
113C22×C3×C5—21no—
114C22×C3×C5—21no—
115C22×C3×C5—21no—
116C22×C3×C5—21no—
117C22×C3×C5—21no—
118C22×C3×C5—41no—
119C22×C3×C5—41no—
120C22×C3×C5—61no—
121C22×C3×C5—61no—
122C22×C3×C5—61no—
123C22×C3×C5C22×D311yescharacteristic
124C22×C3×C5C22×D311yescharacteristic
125C2×D15—21no—
126C2×D15—21no—
127C2×D15#6712yescomplemented
128C2×C5×D3—21no—
129C2×C5×D3—21no—
130C2×C5×D3—21no—
131C2×C3×D5—32no—
132C2×C3×D5—61no—
133C2×C3×D5—61no—
134C3×C4×C5—32no—
135Dic15—21no—
136Dic15—61no—
137Dic15—61no—
138Dic15—61no—
139Dic15—61no—
140Dic15—121no—
141Dic15—121no—
142C3×Dic5—61no—
143C5×Dic3#6712yescomplemented
144C2×C3×D4—31no—
145C2×C3×D4—151no—
146C2×#67—51no—
147C2×#67—151no—
148C2×#67—151no—
149C2×#67D1511yescharacteristic, complemented
150C22×Dic3—151no—
151C2×C4×D3—151no—
152#217—151no—
153#217—151no—
154#217—151no—
155#217—301no—
156(C2×D3)⋊C4—151no—
157C4⋊Dic3—151no—
158Dic3⋊C4—151no—
159C32×C5#12111yescharacteristic, complemented, Hall
160C23×C5—31no—
161C23×C5D3211yescharacteristic
162C22×D5—31no—
163C5×D4—62no—
164C2×C4×C5—31no—
165C22⋊D5—62no—
166C22⋊D5—182no—
167C2×Dic5—31no—
168C2×Dic5—91no—
169C2×Dic5—91no—
170C4×D5—182no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32C22×D511yescharacteristic
175C2×C3×D3—21no—
176C2×C3×D3—21no—
177C2×C3×D3—21no—
178C2×C3×D3—52no—
179C2×C3×D3—101no—
180C2×C3×D3—101no—
181C3⋊Dic3—101no—
182C3⋊Dic3—101no—
183C3×Dic3—101no—
184C3×Dic3C22⋊D512yescomplemented
185#121—451noSylow
186C2×C3×C5—21no—
187C2×C3×C5—21no—
188C2×C3×C5—21no—
189C2×C3×C5—42no—
190C2×C3×C5—41no—
191C2×C3×C5—41no—
192C2×C3×C5—121no—
193C2×C3×C5D3×D411yescharacteristic
194C2×C3×C5D3×D411yescharacteristic
195C2×C3×C5#23711yescharacteristic
196C2×C3×C5#23711yescharacteristic
197C2×C3×C5C2×#6711yescharacteristic
198C2×C3×C5C2×#6711yescharacteristic
199D15—22no—
200C3×D5—62no—
201C5×D3—41no—
202C23×C3—21no—
203C23×C3—31no—
204C23×C3—151no—
205C23×C3D3×D511yescharacteristic
206C23×C3C2×D1511yescharacteristic
207C22×D3—51no—
208C22×D3C2×D1511yescharacteristic
209C3×D4—62no—
210C3×D4—302no—
211C2×C3×C4—31no—
212C2×C3×C4—151no—
213#67—22no—
214#67—102no—
215#67—302no—
216#67—302no—
217C2×Dic3—51no—
218C2×Dic3—151no—
219C2×Dic3—151no—
220C2×Dic3—151no—
221C2×Dic3—151no—
222C2×Dic3—301no—
223C2×Dic3—301no—
224C2×Dic3—302no—
225C2×Dic3C2×D1511yescharacteristic
226C4×D3—302no—
227C22×C5—21no—
228C22×C5—21no—
229C22×C5—21no—
230C22×C5—61no—
231C22×C5—61no—
232C22×C5—61no—
233C22×C5C2×D3211yescharacteristic
234C2×D5—32no—
235C2×D5—61no—
236C2×D5—61no—
237C4×C5—32no—
238Dic5—61no—
239Dic5—181no—
240Dic5—181no—
241C2×C32—41no—
242C2×C32D4×D511yescharacteristic
243C2×C32#71111yescharacteristic
244C2×C32C2×C22⋊D511yescharacteristic
245C3×D3—41no—
246C3×D3—102no—
247C2×D4—31no—
248C2×D4—151no—
249C2×D4—451no—
250C22×C4—451no—
251C4⋊C4—451no—
252C22⋊C4—451no—
253C22⋊C4—451no—
254C3×C5—21no—
255C3×C5#93811yescharacteristic, complemented
256C3×C5#93911yescharacteristic, complemented
257C22×C3—22no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—61no—
268C22×C3—61no—
269C22×C3—61no—
270C22×C3—152no—
271C22×C3—301no—
272C22×C3—301no—
273C22×C3C2×D3×D511yescharacteristic
274C22×C3C22×D1511yescharacteristic
275C2×D3—21no—
276C2×D3—21no—
277C2×D3—21no—
278C2×D3—52no—
279C2×D3—101no—
280C2×D3—101no—
281C3×C4—32no—
282C3×C4—301no—
283Dic3—101no—
284Dic3—301no—
285Dic3—301no—
286Dic3—301no—
287Dic3—301no—
288Dic3—601no—
289Dic3—601no—
290Dic3#122012yescomplemented
291C2×C5—41no—
292C2×C5—121no—
293C2×C5#380511yescharacteristic
294C2×C5D3×#6711yescharacteristic
295C2×C5D3×#6711yescharacteristic
296D5—62no—
297C32#408411yescharacteristic, complemented, Sylow
298C23—31no—
299C23—151no—
300C23D3×D1511yescharacteristic
301D4—62no—
302D4—302no—
303D4—902no—
304C2×C4—31no—
305C2×C4—151no—
306C2×C4—451no—
307C2×C4—451no—
308C2×C4—902no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—42no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—121no—
316C2×C3—302no—
317C2×C3#678011yescharacteristic
318C2×C3D5×#6711yescharacteristic
319C2×C3D3×C22⋊D511yescharacteristic
320C2×C3D4×D1511yescharacteristic
321C2×C3#681911yescharacteristic
322C2×C3C2×#122011yescharacteristic
323D3—41no—
324D3—102no—
325C5#6397011yescharacteristic, complemented, Sylow
326C22—21no—
327C22—21no—
328C22—21no—
329C22—61no—
330C22—61no—
331C22—61no—
332C22—152no—
333C22—301no—
334C22—301no—
335C22C2×D3×D1511yescharacteristic
336C4—32no—
337C4—301no—
338C4—901no—
339C4—901no—
340C3—21no—
341C3#9154911yescharacteristic, complemented
342C3#9182711yescharacteristic, complemented
343C2—41no—
344C2—121no—
345C2—302no—
346C2#1062420811yescharacteristic
347C2#67×D1511yescharacteristic
348C2D3×#122011yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×D15
commutator, socle, cosocleC22×C32×C5C23
FittingC23×C32×C5C22
hypercenterC23D3×D15
hypocenterC32×C5#121
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1220C2semidirect
2. C2×C3×C5×#67C2semidirect
3. C2×#67150C2semidirect
4. C2×Dic3×D15C2semidirect
5. #10623913C2semidirect
6. Dic3⋊Dic15C2semidirect
7. (C2×D3)⋊Dic15C2semidirect
24. C2×#1220D3semidirect
26. C2×C5×#67D3semidirect
43. C2×C3×D15D4semidirect
50. C3×C5×Dic3D4semidirect
53. C2×C3×#67D5semidirect
127. C2×D15#67semidirect
143. C5×Dic3#67semidirect
149. C2×#67D15semidirect
159. C32×C5#121semidirect
184. C3×Dic3C22⋊D5semidirect
255. C3×C5#938semidirect
256. C3×C5#939semidirect
290. Dic3#1220semidirect
297. C32#4084semidirect
325. C5#63970semidirect
341. C3#91549semidirect
342. C3#91827semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#12202
C2×C3×C5×#672
C2×#671502
C2×Dic3×D152
#106239132
Dic3⋊Dic152
(C2×D3)⋊Dic152
#918273
#915493
#639705