Finite Group Databasebeta

← #13087462

Subgroups — #13087462

Order 1440.

430 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 78 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×C4×C3⋊D15C211yesmaximal, characteristic, complemented
2C2×#67152C211yesmaximal, characteristic, complemented
3C2×#67151C211yesmaximal, characteristic, complemented
4C2×#67150C211yesmaximal, characteristic, complemented
5C3×#6714C211yesmaximal, characteristic, complemented
6C3×C5×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
7#10623815C211yesmaximal, characteristic, complemented
8#91773—31nomaximal
9#91461—31nomaximal
10C22×C3⋊D15C2211yescharacteristic
11C22×C3×D15C2211yescharacteristic
12C22×C3×C5×D3C2211yescharacteristic
13C2×C32×C4×C5C2211yescharacteristic
14C2×C3⋊Dic15C2211yescharacteristic
15C4×C3⋊D15—22no—
16C2×C3×Dic15C2211yescharacteristic
17C2×C3×C5×Dic3C2211yescharacteristic
18#67152—22no—
19#67151—22no—
20#67150—22no—
21#63912—51nomaximal, Hall
22C2×#1220—31no—
23C2×#1220—31no—
24C2×D60—31no—
25C2×C4×D15—31no—
26C2×C4×D15—31no—
27C2×C4×D15—61no—
28C2×#1203—31no—
29C2×#1202—31no—
30C2×#1201—31no—
31C3×C5×C22⋊C4—31no—
32#6714D311yescharacteristic, complemented
33#6712—31no—
34C5×(C2×D3)⋊C4D311yescharacteristic, complemented
35C3×#685—31no—
36#6669—31no—
37C22×C32×C5C2311yescharacteristic
38C2×C3⋊D15—21no—
39C2×C3⋊D15—21no—
40C2×C3⋊D15D412yescomplemented
41C2×C3×D15—21no—
42C2×C3×D15—21no—
43C2×C3×D15—21no—
44C2×C3×C5×D3—21no—
45C2×C3×C5×D3—21no—
46C2×C3×C5×D3—21no—
47C32×C4×C5—21no—
48C3⋊Dic15D412yescomplemented
49C3×Dic15—21no—
50C3×C5×Dic3—21no—
51#4030—91noHall
52C2×C4×C3⋊D3—51no—
53C2×#626—51no—
54C2×#626—51no—
55C2×#625—51no—
56C3×(C2×D3)⋊C4—51no—
57C3×(C2×D3)⋊C4D511yescharacteristic, complemented
58#3724—51no—
59C23×C3×C5—31no—
60C22×D15—31no—
61C22×D15—31no—
62C22×D15—61no—
63C22×D15C2×D311yescharacteristic
64C22×C5×D3C2×D311yescharacteristic
65C22×C3×D5—31no—
66C2×C3×C4×C5—21no—
67C2×C3×C4×C5—31no—
68C2×C3×C4×C5C2×D311yescharacteristic
69C2×C3×C4×C5C2×D311yescharacteristic
70#1220—62no—
71#1220—62no—
72C2×Dic15—31no—
73C2×Dic15—31no—
74C2×Dic15—61no—
75C2×Dic15C2×D311yescharacteristic
76D60—62no—
77C4×D15—62no—
78C4×D15—62no—
79C4×D15—64no—
80C2×C5×Dic3C2×D311yescharacteristic
81C2×C3×Dic5—31no—
82#1203—62no—
83#1202—62no—
84#1201—62no—
85#884—151no—
86#884—151no—
87C2×C32×C5C2×D411yescharacteristic
88C2×C32×C5C2×D411yescharacteristic
89C2×C32×C5#4111yescharacteristic
90C3⋊D15—22no—
91C3×D15—41no—
92C3×C5×D3—41no—
93C2×C22⋊D5—91no—
94C2×C22⋊D5—91no—
95C2×D20—91no—
96C2×C4×D5—91no—
97C5×C22⋊C4—31no—
98#685—31no—
99#683—91no—
100C22×C3⋊D3—51no—
101C22×C3×D3—51no—
102C22×C3×D3C2×D511yescharacteristic
103C2×C32×C4C2×D511yescharacteristic
104C2×C3⋊Dic3—51no—
105C4×C3⋊D3—102no—
106C2×C3×Dic3—51no—
107C2×C3×Dic3C2×D511yescharacteristic
108#626—102no—
109#626—102no—
110#625—102no—
111C22×C3×C5—21no—
112C22×C3×C5—61no—
113C22×C3×C5—61no—
114C22×C3×C5—61no—
115C22×C3×C5C22×D311yescharacteristic
116C22×C3×C5C22×D311yescharacteristic
117C2×D15—21no—
118C2×D15—21no—
119C2×D15—21no—
120C2×D15—32no—
121C2×D15—32no—
122C2×D15—62no—
123C2×D15—62no—
124C2×D15—62no—
125C2×D15—61no—
126C2×D15—61no—
127C2×D15—61no—
128C2×D15—61no—
129C2×C5×D3—21no—
130C2×C5×D3—21no—
131C2×C5×D3—21no—
132C2×C3×D5—61no—
133C2×C3×D5—61no—
134C2×C3×D5—61no—
135C3×C4×C5—22no—
136C3×C4×C5—21no—
137C3×C4×C5—21no—
138C3×C4×C5—61no—
139Dic15—21no—
140Dic15—32no—
141Dic15—32no—
142Dic15—62no—
143C3×Dic5—61no—
144C5×Dic3—21no—
145C2×#67—151no—
146C2×#67—151no—
147C2×#67—151no—
148C2×#67—151no—
149C2×D12—151no—
150C2×D12—151no—
151C2×C4×D3—151no—
152C2×C4×D3—151no—
153C2×C4×D3—301no—
154C3×C22⋊C4—31no—
155C3×C22⋊C4—151no—
156(C2×D3)⋊C4—51no—
157(C2×D3)⋊C4D1511yescharacteristic, complemented
158Dic3⋊C4—151no—
159Dic3⋊C4—151no—
160C32×C5#12111yescharacteristic, complemented, Hall
161C23×C5—31no—
162C22×D5—31no—
163C22×D5—91no—
164C2×C4×C5—31no—
165C2×C4×C5D3211yescharacteristic
166C22⋊D5—182no—
167C22⋊D5—182no—
168C2×Dic5—31no—
169C2×Dic5—91no—
170D20—182no—
171C4×D5—182no—
172C22×C32C22×D511yescharacteristic
173C2×C3⋊D3—52no—
174C2×C3⋊D3—101no—
175C2×C3⋊D3—101no—
176C2×C3×D3—21no—
177C2×C3×D3—21no—
178C2×C3×D3—21no—
179C2×C3×D3—101no—
180C2×C3×D3—101no—
181C2×C3×D3—101no—
182C32×C4—21no—
183C3⋊Dic3—52no—
184C3×Dic3—21no—
185C3×Dic3—101no—
186#121—451noSylow
187C2×C3×C5—21no—
188C2×C3×C5—21no—
189C2×C3×C5—21no—
190C2×C3×C5—121no—
191C2×C3×C5#23511yescharacteristic
192C2×C3×C5#23511yescharacteristic
193C2×C3×C5D3×D411yescharacteristic
194C2×C3×C5D3×D411yescharacteristic
195C2×C3×C5D3×D411yescharacteristic
196C2×C3×C5D3×D411yescharacteristic
197D15—41no—
198D15—64no—
199D15—62no—
200D15—62no—
201C3×D5—121no—
202C5×D3—41no—
203C23×C3—31no—
204C23×C3—151no—
205C22×D3—51no—
206C22×D3—151no—
207C22×D3—151no—
208C22×D3—301no—
209C22×D3C2×D1511yescharacteristic
210C2×C3×C4—21no—
211C2×C3×C4—31no—
212C2×C3×C4—151no—
213C2×C3×C4D3×D511yescharacteristic
214C2×C3×C4C2×D1511yescharacteristic
215#67—302no—
216#67—302no—
217#67—302no—
218#67—302no—
219C2×Dic3—51no—
220C2×Dic3—151no—
221C2×Dic3—151no—
222C2×Dic3—301no—
223C2×Dic3C2×D1511yescharacteristic
224D12—302no—
225D12—302no—
226C4×D3—302no—
227C4×D3—302no—
228C4×D3—304no—
229C22×C5—61no—
230C22×C5—61no—
231C22×C5—61no—
232C22×C5C2×D3211yescharacteristic
233C2×D5—61no—
234C2×D5—61no—
235C2×D5—61no—
236C2×D5—92no—
237C2×D5—181no—
238C2×D5—181no—
239C4×C5—21no—
240C4×C5—61no—
241Dic5—61no—
242Dic5—92no—
243C2×C32#70911yescharacteristic
244C2×C32D4×D511yescharacteristic
245C2×C32D4×D511yescharacteristic
246C3⋊D3—102no—
247C3×D3—41no—
248C3×D3—201no—
249C2×D4—451no—
250C2×D4—451no—
251C2×D4—451no—
252C22×C4—451no—
253C4⋊C4—451no—
254C22⋊C4—31no—
255C22⋊C4—151no—
256C3×C5—21no—
257C3×C5#88411yescharacteristic, complemented
258C3×C5#88411yescharacteristic, complemented
259C22×C3—21no—
260C22×C3—61no—
261C22×C3—61no—
262C22×C3—61no—
263C22×C3—301no—
264C22×C3—301no—
265C22×C3—301no—
266C22×C3C2×D3×D511yescharacteristic
267C22×C3C22×D1511yescharacteristic
268C2×D3—21no—
269C2×D3—21no—
270C2×D3—21no—
271C2×D3—101no—
272C2×D3—101no—
273C2×D3—101no—
274C2×D3—152no—
275C2×D3—152no—
276C2×D3—302no—
277C2×D3—302no—
278C2×D3—302no—
279C2×D3—301no—
280C2×D3—301no—
281C2×D3—301no—
282C2×D3—301no—
283C3×C4—22no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—61no—
287C3×C4—301no—
288Dic3—21no—
289Dic3—101no—
290Dic3—152no—
291Dic3—152no—
292Dic3—302no—
293C2×C5—121no—
294C2×C5#379811yescharacteristic
295C2×C5#380211yescharacteristic
296C2×C5#381111yescharacteristic
297D5—121no—
298D5—182no—
299C32#403011yescharacteristic, complemented, Sylow
300C23—31no—
301C23—151no—
302C23—451no—
303D4—902no—
304D4—902no—
305D4—902no—
306C2×C4—31no—
307C2×C4—151no—
308C2×C4—451no—
309C2×C4—902no—
310C2×C4D3×D1511yescharacteristic
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—121no—
315C2×C3—601no—
316C2×C3#677111yescharacteristic
317C2×C3#677711yescharacteristic
318C2×C3#679011yescharacteristic
319C2×C3#681711yescharacteristic
320C2×C3D4×D1511yescharacteristic
321C2×C3D4×D1511yescharacteristic
322D3—41no—
323D3—201no—
324D3—304no—
325D3—302no—
326D3—302no—
327C5#6391211yescharacteristic, complemented, Sylow
328C22—61no—
329C22—61no—
330C22—61no—
331C22—301no—
332C22—301no—
333C22—301no—
334C22—452no—
335C22—901no—
336C22—901no—
337C22C2×D3×D1511yescharacteristic
338C4—21no—
339C4—61no—
340C4—301no—
341C4—452no—
342C3—21no—
343C3#9146111yescharacteristic, complemented
344C3#9177311yescharacteristic, complemented
345C2—121no—
346C2—601no—
347C2—902no—
348C2#1062419911yescharacteristic
349C2#1062420511yescharacteristic
350C2#1062421811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×D15
commutator, socle, cosocleC22×C32×C5C23
FittingC2×C32×C4×C5C22
hypercenterC2×C4D3×D15
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×C4×C3⋊D15C2semidirect
2. C2×#67152C2semidirect
3. C2×#67151C2semidirect
4. C2×#67150C2semidirect
5. C3×#6714C2semidirect
6. C3×C5×(C2×D3)⋊C4C2semidirect
7. #10623815C2semidirect
32. #6714D3semidirect
34. C5×(C2×D3)⋊C4D3semidirect
40. C2×C3⋊D15D4semidirect
48. C3⋊Dic15D4semidirect
57. C3×(C2×D3)⋊C4D5semidirect
157. (C2×D3)⋊C4D15semidirect
160. C32×C5#121semidirect
257. C3×C5#884semidirect
258. C3×C5#884semidirect
299. C32#4030semidirect
327. C5#63912semidirect
343. C3#91461semidirect
344. C3#91773semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊D152
C2×#671522
C2×#671512
C2×#671502
C3×#67142
C3×C5×(C2×D3)⋊C42
#106238152
#917733
#914613
#639125