Finite Group Databasebeta

← #13088745

Subgroups — #13088745

Order 1440.

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

normal subgroups only

This group has 430 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
1C22×C3×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
2C2×ΓL(1,𝔽16)⋊D3C211yesmaximal, characteristic, complemented
3D5×#638C211yesmaximal, characteristic
4C3×#6763C211yesmaximal, characteristic, complemented
5#10623979C211yesmaximal, characteristic, complemented
6#10623978C211yesmaximal, characteristic, complemented
7ΓL(1,𝔽16)⋊Dic3C211yesmaximal, characteristic, complemented
8#67⋊AGL(1,𝔽5)—31nomaximal
9AGL(1,𝔽5)×#67—31nomaximal
10C22×C32×D5C2211yescharacteristic
11C2×D5×C3⋊D3C2211yescharacteristic
12C2×C3×ΓL(1,𝔽16)—22no—
13C2×C3×ΓL(1,𝔽16)C2211yescharacteristic
14C2×C3×ΓL(1,𝔽16)C2211yescharacteristic
15C2×C3×ΓL(1,𝔽16)C2211yescharacteristic
16C2×C3×ΓL(1,𝔽16)C2211yescharacteristic
17ΓL(1,𝔽16)⋊D3—22no—
18#67184C411yescharacteristic, complemented
19C5×#638C411yescharacteristic, complemented
20#67139C411yescharacteristic, complemented
21#67138C411yescharacteristic, complemented
22D5×C3⋊Dic3C2211yescharacteristic
23#63952—51nomaximal, Hall
24C22×ΓL(1,𝔽16)D311yescharacteristic, complemented
25C22×C3×AGL(1,𝔽5)—31no—
26C2×D3×AGL(1,𝔽5)—31no—
27C2×D3×AGL(1,𝔽5)—31no—
28D5×#67—31no—
29D5×#67—31no—
30D5×#67—61no—
31#6763D311yescharacteristic, complemented
32C3×C22⋊AGL(1,𝔽5)—31no—
33Dic3⋊AGL(1,𝔽5)—31no—
34Dic3⋊AGL(1,𝔽5)—31no—
35(C2×D3)⋊AGL(1,𝔽5)—31no—
36(C2×D3)⋊AGL(1,𝔽5)—31no—
37Dic3×AGL(1,𝔽5)—31no—
38Dic3×AGL(1,𝔽5)—31no—
39C22×C32×C5C2×C411yescharacteristic, complemented
40C2×C3⋊D15C2×C411yescharacteristic, complemented
41C2×C5×C3⋊D3C2×C411yescharacteristic, complemented
42C2×C32×D5—21no—
43C2×C32×D5—21no—
44C2×C32×D5C2×C411yescharacteristic, complemented
45C2×C32×D5C2311yescharacteristic
46D5×C3⋊D3—21no—
47D5×C3⋊D3—21no—
48C3×ΓL(1,𝔽16)—21no—
49C3×ΓL(1,𝔽16)—21no—
50C3×ΓL(1,𝔽16)—21no—
51C3×ΓL(1,𝔽16)D412yescomplemented
52C3⋊Dic15C2×C411yescharacteristic, complemented
53C5×C3⋊Dic3C2×C411yescharacteristic, complemented
54D4×AGL(1,𝔽5)—91noHall
55C2×#638—51no—
56C22×C3×Dic3—51no—
57C2×#624—51no—
58C3×#217—51no—
59#3724—51no—
60#3722—51no—
61Dic32—51no—
62C22×C3×D5—21no—
63C22×C3×D5C2×D311yescharacteristic
64C22×C3×D5C2×D311yescharacteristic
65C2×D3×D5—31no—
66C2×D3×D5—31no—
67C2×D3×D5—61no—
68C2×ΓL(1,𝔽16)—22no—
69C2×ΓL(1,𝔽16)C2×D311yescharacteristic
70C2×ΓL(1,𝔽16)C2×D311yescharacteristic
71C2×ΓL(1,𝔽16)C2×D311yescharacteristic
72C2×ΓL(1,𝔽16)C2×D311yescharacteristic
73C2×C3×AGL(1,𝔽5)—31no—
74C2×C3×AGL(1,𝔽5)—31no—
75C2×C3×AGL(1,𝔽5)—31no—
76C2×C3×AGL(1,𝔽5)—31no—
77C2×C3×AGL(1,𝔽5)—62no—
78D3×AGL(1,𝔽5)—62no—
79D3×AGL(1,𝔽5)—62no—
80#1220—31no—
81#1220—31no—
82#1220—61no—
83C5×#67—31no—
84C5×#67—31no—
85C5×#67—61no—
86#1202—31no—
87#1202—31no—
88#1202—61no—
89#1201—31no—
90#1201—31no—
91#1201—61no—
92D5×Dic3—31no—
93D5×Dic3—31no—
94D5×Dic3—61no—
95C4×#67—151no—
96#67⋊C4—151no—
97C2×C32×C5—21no—
98C2×C32×C5C22×C411yescharacteristic
99C3⋊D15—21no—
100C5×C3⋊D3—21no—
101C32×D5—21no—
102C32×D5C2×D411yescharacteristic
103C32×D5#4111yescharacteristic
104C22×AGL(1,𝔽5)—31no—
105C22×AGL(1,𝔽5)—91no—
106D4×D5—91no—
107C22⋊AGL(1,𝔽5)—31no—
108C22⋊AGL(1,𝔽5)—91no—
109C4⋊AGL(1,𝔽5)—91no—
110C4×AGL(1,𝔽5)—91no—
111C23×C32—51no—
112C22×C3⋊D3—51no—
113#638—51no—
114#638—51no—
115#638—51no—
116#638AGL(1,𝔽5)11yescharacteristic, complemented
117C2×C3⋊Dic3—51no—
118C2×C3×Dic3—51no—
119C2×C3×Dic3—51no—
120C2×C3×Dic3—51no—
121C2×C3×Dic3—51no—
122C2×C3×Dic3—102no—
123#624—102no—
124C22×C3×C5—21no—
125C22×C3×C5C4×D311yescharacteristic, complemented
126C22×C3×C5C4×D311yescharacteristic, complemented
127C2×D15—31no—
128C2×D15—31no—
129C2×D15—61no—
130C2×C5×D3—31no—
131C2×C5×D3—31no—
132C2×C5×D3—61no—
133C2×C3×D5—21no—
134C2×C3×D5—21no—
135C2×C3×D5—21no—
136C2×C3×D5—21no—
137C2×C3×D5—21no—
138C2×C3×D5—21no—
139C2×C3×D5—41no—
140C2×C3×D5—41no—
141C2×C3×D5C4×D311yescharacteristic, complemented
142C2×C3×D5C4×D311yescharacteristic, complemented
143C2×C3×D5C22×D311yescharacteristic
144C2×C3×D5C22×D311yescharacteristic
145D3×D5—61no—
146D3×D5—61no—
147D3×D5—61no—
148D3×D5—61no—
149D3×D5—121no—
150D3×D5—121no—
151ΓL(1,𝔽16)—21no—
152ΓL(1,𝔽16)—21no—
153ΓL(1,𝔽16)—21no—
154ΓL(1,𝔽16)#6712yescomplemented
155C3×AGL(1,𝔽5)—32no—
156C3×AGL(1,𝔽5)—61no—
157C3×AGL(1,𝔽5)—61no—
158C3×AGL(1,𝔽5)—61no—
159Dic15—31no—
160Dic15—31no—
161Dic15—61no—
162C5×Dic3—31no—
163C5×Dic3—31no—
164C5×Dic3—61no—
165C22×C3×C4—151no—
166C2×#67—151no—
167C2×#67—151no—
168C2×#67—301no—
169C22×Dic3—51no—
170C2×C4×D3—151no—
171C2×C4×D3—151no—
172C3×C22⋊C4—151no—
173#217—51no—
174(C2×D3)⋊C4—151no—
175(C2×D3)⋊C4—151no—
176Dic3⋊C4—151no—
177Dic3⋊C4—151no—
178C4×Dic3—151no—
179C4×Dic3—151no—
180C32×C5C4×D411yescharacteristic, complemented, Hall
181C22×D5—91no—
182C22×D5D3211yescharacteristic
183C2×AGL(1,𝔽5)—31no—
184C2×AGL(1,𝔽5)—31no—
185C2×AGL(1,𝔽5)—31no—
186C2×AGL(1,𝔽5)—31no—
187C2×AGL(1,𝔽5)—62no—
188C2×AGL(1,𝔽5)—182no—
189C5×D4—91no—
190C22⋊D5—91no—
191C22⋊D5—91no—
192D20—91no—
193C4×D5—91no—
194C22×C32—51no—
195C22×C32—51no—
196C22×C32—101no—
197C22×C32—101no—
198C22×C32C2×AGL(1,𝔽5)11yescharacteristic, complemented
199C2×C3⋊D3—51no—
200C2×C3⋊D3—101no—
201C2×C3⋊D3—101no—
202C2×C3⋊D3C2×AGL(1,𝔽5)11yescharacteristic, complemented
203C3⋊Dic3—51no—
204C3⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic, complemented
205C3×Dic3—52no—
206C3×Dic3—101no—
207C3×Dic3—101no—
208C3×Dic3—101no—
209C4×D4—451noSylow
210C2×C3×C5—21no—
211C2×C3×C5—21no—
212C2×C3×C5—21no—
213C2×C3×C5—41no—
214C2×C3×C5C2×C4×D311yescharacteristic
215C2×C3×C5C2×C4×D311yescharacteristic
216D15—61no—
217D15—61no—
218D15—121no—
219C3×D5—21no—
220C3×D5—21no—
221C3×D5—21no—
222C3×D5—21no—
223C3×D5—41no—
224C3×D5#23511yescharacteristic
225C3×D5D3×D411yescharacteristic
226C3×D5#23711yescharacteristic
227C3×D5C2×#6711yescharacteristic
228C5×D3—61no—
229C5×D3—61no—
230C5×D3—121no—
231C23×C3—51no—
232C23×C3—51no—
233C23×C3—101no—
234C22×D3—151no—
235C22×D3—151no—
236C22×D3—301no—
237C2×C3×C4—151no—
238C2×C3×C4—151no—
239C2×C3×C4—151no—
240C2×C3×C4—151no—
241C2×C3×C4—302no—
242#67—31no—
243#67—31no—
244#67—61no—
245#67—151no—
246#67—151no—
247#67—151no—
248#67—151no—
249#67—151no—
250#67—151no—
251#67—301no—
252#67—301no—
253#67—301no—
254C2×Dic3—51no—
255C2×Dic3—51no—
256C2×Dic3—51no—
257C2×Dic3—51no—
258C2×Dic3—102no—
259C2×Dic3—151no—
260C2×Dic3—151no—
261C2×Dic3—301no—
262C4×D3—302no—
263C4×D3—302no—
264C22×C5—91no—
265C22×C5#62411yescharacteristic, complemented
266C2×D5—21no—
267C2×D5—21no—
268C2×D5—91no—
269C2×D5—181no—
270C2×D5—181no—
271C2×D5#62411yescharacteristic, complemented
272C2×D5C2×D3211yescharacteristic
273AGL(1,𝔽5)—32no—
274AGL(1,𝔽5)—61no—
275AGL(1,𝔽5)—61no—
276AGL(1,𝔽5)—61no—
277C4×C5—91no—
278Dic5—91no—
279C2×C32—21no—
280C2×C32—51no—
281C2×C32—51no—
282C2×C32—101no—
283C2×C32C22×AGL(1,𝔽5)11yescharacteristic
284C3⋊D3—21no—
285C3⋊D3—101no—
286C2×D4—451no—
287C22×C4—151no—
288C22×C4—451no—
289C4⋊C4—451no—
290C22⋊C4—151no—
291C22⋊C4—451no—
292C42—451no—
293C3×C5—21no—
294C3×C5#67⋊C411yescharacteristic, complemented
295C3×C5C4×#6711yescharacteristic, complemented
296C22×C3—21no—
297C22×C3—51no—
298C22×C3—51no—
299C22×C3—51no—
300C22×C3—51no—
301C22×C3—101no—
302C22×C3—101no—
303C22×C3—101no—
304C22×C3—101no—
305C22×C3—101no—
306C22×C3—101no—
307C22×C3—201no—
308C22×C3—201no—
309C22×C3D3×AGL(1,𝔽5)11yescharacteristic, complemented
310C22×C3D3×AGL(1,𝔽5)11yescharacteristic, complemented
311C2×D3—31no—
312C2×D3—31no—
313C2×D3—61no—
314C2×D3—151no—
315C2×D3—151no—
316C2×D3—301no—
317C2×D3—301no—
318C2×D3—301no—
319C2×D3—301no—
320C2×D3—301no—
321C2×D3—601no—
322C2×D3—601no—
323C3×C4—152no—
324C3×C4—301no—
325C3×C4—301no—
326C3×C4—301no—
327Dic3—31no—
328Dic3—31no—
329Dic3—52no—
330Dic3—61no—
331Dic3—101no—
332Dic3—101no—
333Dic3—101no—
334Dic3—151no—
335Dic3—151no—
336Dic3—301no—
337C2×C5—21no—
338C2×C5—181no—
339C2×C5C2×#62411yescharacteristic
340D5—21no—
341D5—181no—
342D5#380411yescharacteristic
343D5D3×#6711yescharacteristic
344C32D4×AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
345C23—51no—
346C23—451no—
347D4—91no—
348D4—451no—
349D4—451no—
350D4—451no—
351C2×C4—151no—
352C2×C4—151no—
353C2×C4—151no—
354C2×C4—151no—
355C2×C4—302no—
356C2×C4—451no—
357C2×C4—902no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—41no—
362C2×C3—51no—
363C2×C3—51no—
364C2×C3—51no—
365C2×C3—51no—
366C2×C3—101no—
367C2×C3—101no—
368C2×C3—101no—
369C2×C3—101no—
370C2×C3—201no—
371C2×C3C2×D3×AGL(1,𝔽5)11yescharacteristic
372C2×C3C2×D3×AGL(1,𝔽5)11yescharacteristic
373D3—61no—
374D3—61no—
375D3—121no—
376D3—301no—
377D3—301no—
378D3—601no—
379C5#6395211yescharacteristic, complemented, Sylow
380C22—51no—
381C22—51no—
382C22—91no—
383C22—101no—
384C22—101no—
385C22—451no—
386C22—901no—
387C22—901no—
388C22ΓL(1,𝔽16)⋊D311yescharacteristic, complemented
389C4—91no—
390C4—152no—
391C4—301no—
392C4—301no—
393C4—301no—
394C4—451no—
395C3—21no—
396C3AGL(1,𝔽5)×#6711yescharacteristic, complemented
397C3#67⋊AGL(1,𝔽5)11yescharacteristic, complemented
398C2—21no—
399C2—51no—
400C2—51no—
401C2—101no—
402C2—181no—
403C2—901no—
404C2C2×ΓL(1,𝔽16)⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C2×ΓL(1,𝔽16)⋊D3
commutator, socleC2×C32×C5C22×C4
FittingC22×C32×C5C2×C4
prime radicalD5×#638C2
cosocleC2×C32×D5C23
hypercenterC22ΓL(1,𝔽16)⋊D3
hypocenterC32×C5C4×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×ΓL(1,𝔽16)C2semidirect
2. C2×ΓL(1,𝔽16)⋊D3C2semidirect
4. C3×#6763C2semidirect
5. #10623979C2semidirect
6. #10623978C2semidirect
7. ΓL(1,𝔽16)⋊Dic3C2semidirect
18. #67184C4semidirect
19. C5×#638C4semidirect
20. #67139C4semidirect
21. #67138C4semidirect
24. C22×ΓL(1,𝔽16)D3semidirect
31. #6763D3semidirect
39. C22×C32×C5C2×C4semidirect
40. C2×C3⋊D15C2×C4semidirect
41. C2×C5×C3⋊D3C2×C4semidirect
44. C2×C32×D5C2×C4semidirect
51. C3×ΓL(1,𝔽16)D4semidirect
52. C3⋊Dic15C2×C4semidirect
53. C5×C3⋊Dic3C2×C4semidirect
116. #638AGL(1,𝔽5)semidirect
125. C22×C3×C5C4×D3semidirect
126. C22×C3×C5C4×D3semidirect
141. C2×C3×D5C4×D3semidirect
142. C2×C3×D5C4×D3semidirect
154. ΓL(1,𝔽16)#67semidirect
180. C32×C5C4×D4semidirect
198. C22×C32C2×AGL(1,𝔽5)semidirect
202. C2×C3⋊D3C2×AGL(1,𝔽5)semidirect
204. C3⋊Dic3C2×AGL(1,𝔽5)semidirect
265. C22×C5#624semidirect
271. C2×D5#624semidirect
294. C3×C5#67⋊C4semidirect
295. C3×C5C4×#67semidirect
309. C22×C3D3×AGL(1,𝔽5)semidirect
310. C22×C3D3×AGL(1,𝔽5)semidirect
344. C32D4×AGL(1,𝔽5)semidirect
379. C5#63952semidirect
388. C22ΓL(1,𝔽16)⋊D3semidirect
396. C3AGL(1,𝔽5)×#67semidirect
397. C3#67⋊AGL(1,𝔽5)semidirect

Maximal subgroups

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

subgroupindex
C22×C3×ΓL(1,𝔽16)2
C2×ΓL(1,𝔽16)⋊D32
D5×#6382
C3×#67632
#106239792
#106239782
ΓL(1,𝔽16)⋊Dic32
#67⋊AGL(1,𝔽5)3
AGL(1,𝔽5)×#673
#639525