Finite Group Databasebeta

← #72229969

Subgroups — #72229969

Order 2880.

1404 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 1404 conjugacy classes of subgroups — too many to draw. Only its 254 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
1C23×C3⋊ΓL(1,𝔽16)C211yesmaximal, characteristic
2C2×C32×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3C2×#10623965C212yesmaximal, complemented
4D5×#3757C211yesmaximal, characteristic
5(C2×AGL(1,𝔽5))⋊(C3⋊Dic3)C212yesmaximal, complemented
6#11752029—34nomaximal
7C23×C32×D5C2211yescharacteristic
8C22×C3⋊ΓL(1,𝔽16)—22no—
9C22×C3⋊ΓL(1,𝔽16)—24no—
10C22×C3⋊ΓL(1,𝔽16)C2212yes—
11C22×C32×AGL(1,𝔽5)C2211yescharacteristic
12C22×C32×AGL(1,𝔽5)C2211yescharacteristic
13C2×D5×C3⋊Dic3C2212yes—
14C32×C22⋊AGL(1,𝔽5)—22no—
15#10623965—24no—
16#10623913C411yescharacteristic, complemented
17C5×#3757C411yescharacteristic, complemented
18#10623770C412yescomplemented
19#10592645—51nomaximal, Hall
20C23×ΓL(1,𝔽16)—34no—
21C2×C3×C22⋊AGL(1,𝔽5)D314yes—
22C2×Dic3⋊AGL(1,𝔽5)—38no—
23D5×#217—34no—
24(C2×AGL(1,𝔽5))⋊Dic3—38no—
25C23×C32×C5C2×C411yescharacteristic
26C22×C32×D5—22no—
27C22×C32×D5—22no—
28C22×C32×D5C2×C411yescharacteristic
29C22×C32×D5D412yes—
30C22×C32×D5Q812yes—
31C22×C32×D5C2311yescharacteristic
32C2×C3⋊ΓL(1,𝔽16)—28no—
33C2×C3⋊ΓL(1,𝔽16)—22no—
34C2×C3⋊ΓL(1,𝔽16)—22no—
35C2×C3⋊ΓL(1,𝔽16)D414yes—
36C2×C32×AGL(1,𝔽5)—21no—
37C2×C32×AGL(1,𝔽5)—21no—
38C2×C32×AGL(1,𝔽5)—22no—
39C2×C32×AGL(1,𝔽5)—22no—
40C2×C3⋊Dic15C2×C412yescomplemented
41C2×C5×C3⋊Dic3C2×C412yescomplemented
42D5×C3⋊Dic3—22no—
43D5×C3⋊Dic3—22no—
44#65699—91noHall
45C23×C3⋊Dic3—51no—
46C2×C32×C22⋊C4—51no—
47C2×#3757—51no—
48C2×#3750—52no—
49#63658—52no—
50C23×C3×D5C2×D314yes—
51C22×ΓL(1,𝔽16)—38no—
52C22×ΓL(1,𝔽16)—616no—
53C22×ΓL(1,𝔽16)—68no—
54C22×C3×AGL(1,𝔽5)C2×D314yes—
55C22×C3×AGL(1,𝔽5)C2×D314yes—
56C2×D5×Dic3—38no—
57C3×C22⋊AGL(1,𝔽5)—28no—
58Dic3⋊AGL(1,𝔽5)—616no—
59#6719—34no—
60C5×#217—34no—
61#6665—38no—
62#4946—154no—
63C22×C32×C5—22no—
64C22×C32×C5C4⋊C412yescomplemented
65C22×C32×C5C22×C411yescharacteristic
66C2×C32×D5—22no—
67C2×C32×D5—22no—
68C2×C32×D5—24no—
69C2×C32×D5—22no—
70C2×C32×D5C4⋊C412yescomplemented
71C2×C32×D5C2×D412yes—
72C2×C32×D5C2×D411yescharacteristic
73C2×C32×D5C2×Q811yescharacteristic
74C2×C32×D5#4112yes—
75C3⋊ΓL(1,𝔽16)—24no—
76C32×AGL(1,𝔽5)—41no—
77C32×AGL(1,𝔽5)—41no—
78C3⋊Dic15—22no—
79C5×C3⋊Dic3—22no—
80C23×AGL(1,𝔽5)—91no—
81C2×C22⋊AGL(1,𝔽5)C3⋊D311yescharacteristic
82C2×C4⋊AGL(1,𝔽5)—92no—
83D5×C22⋊C4—91no—
84(C2×AGL(1,𝔽5))⋊C4—92no—
85C24×C32—51no—
86C22×C32×C4—51no—
87C22×C32×C4—51no—
88C22×C3⋊Dic3—52no—
89C22×C3⋊Dic3—52no—
90C22×C3⋊Dic3—104no—
91C22×C3⋊Dic3—102no—
92C32×C22⋊C4—102no—
93#3757—51no—
94#3757—52no—
95#3757AGL(1,𝔽5)11yescharacteristic, complemented
96#3750—104no—
97C23×C3×C5C4×D314yes—
98C22×C3×D5—28no—
99C22×C3×D5—28no—
100C22×C3×D5Dic618yes—
101C22×C3×D5C4×D314yes—
102C22×C3×D5#6718yes—
103C22×C3×D5C22×D314yes—
104C2×ΓL(1,𝔽16)—316no—
105C2×ΓL(1,𝔽16)—632no—
106C2×ΓL(1,𝔽16)—68no—
107C2×ΓL(1,𝔽16)—68no—
108C2×C3×AGL(1,𝔽5)—28no—
109C2×C3×AGL(1,𝔽5)—28no—
110C2×C3×AGL(1,𝔽5)—24no—
111C2×C3×AGL(1,𝔽5)—24no—
112C2×Dic15—38no—
113C2×C5×Dic3—38no—
114D5×Dic3—68no—
115D5×Dic3—68no—
116C23×Dic3—154no—
117C2×C3×C22⋊C4—54no—
118C2×#217—154no—
119C2×Dic3⋊C4—158no—
120C3⋊#95—158no—
121C2×C32×C5—22no—
122C2×C32×C5C2×C4⋊C411yescharacteristic
123C2×C32×C5C4×D412yes—
124C32×D5—22no—
125C32×D5#12011yescharacteristic
126C32×D5C22⋊Q812yes—
127C32×D5#12311yescharacteristic
128C23×D5C2×C3⋊D311yescharacteristic
129C22×AGL(1,𝔽5)—92no—
130C22×AGL(1,𝔽5)—184no—
131C22×AGL(1,𝔽5)—182no—
132C22×AGL(1,𝔽5)C2×C3⋊D311yescharacteristic
133C22×AGL(1,𝔽5)C2×C3⋊D311yescharacteristic
134C2×C4×D5—92no—
135C22⋊AGL(1,𝔽5)—22no—
136C4⋊AGL(1,𝔽5)—184no—
137C5×C22⋊C4—91no—
138C22⋊Dic5—91no—
139#685—92no—
140C23×C32—51no—
141C23×C32—52no—
142C23×C32—52no—
143C23×C32—51no—
144C23×C32—102no—
145C23×C32—102no—
146C23×C32C2×AGL(1,𝔽5)11yescharacteristic
147C2×C32×C4—102no—
148C2×C32×C4—101no—
149C2×C32×C4—101no—
150C2×C32×C4—102no—
151C2×C3⋊Dic3—54no—
152C2×C3⋊Dic3—52no—
153C2×C3⋊Dic3—102no—
154C2×C3⋊Dic3—102no—
155C2×C3⋊Dic3—102no—
156C2×C3⋊Dic3—102no—
157C2×C3⋊Dic3—108no—
158C2×C3⋊Dic3C2×AGL(1,𝔽5)12yescomplemented
159#385—451noSylow
160C22×C3×C5—28no—
161C22×C3×C5Dic3⋊C418yescomplemented
162C22×C3×C5C2×C4×D314yes—
163C2×C3×D5—28no—
164C2×C3×D5—216no—
165C2×C3×D5—28no—
166C2×C3×D5—28no—
167C2×C3×D5Dic3⋊C418yescomplemented
168C2×C3×D5C2×Dic614yes—
169C2×C3×D5D3×D418yes—
170C2×C3×D5#23718yes—
171C2×C3×D5C2×#6714yes—
172ΓL(1,𝔽16)—616no—
173C3×AGL(1,𝔽5)—44no—
174C3×AGL(1,𝔽5)—44no—
175Dic15—68no—
176C5×Dic3—68no—
177C24×C3—54no—
178C22×C3×C4—54no—
179C22×C3×C4—54no—
180C22×Dic3—158no—
181C22×Dic3—158no—
182C22×Dic3—3016no—
183C22×Dic3—308no—
184C3×C22⋊C4—108no—
185#217—34no—
186#217—158no—
187#217—154no—
188Dic3⋊C4—3016no—
189C32×C5#38511yescharacteristic, complemented, Hall
190C23×C5C4×C3⋊D311yescharacteristic
191C22×D5—22no—
192C22×D5—22no—
193C22×D5#63412yes—
194C22×D5C4×C3⋊D311yescharacteristic
195C22×D5#63812yes—
196C22×D5C22×C3⋊D311yescharacteristic
197C2×AGL(1,𝔽5)—22no—
198C2×AGL(1,𝔽5)—22no—
199C2×AGL(1,𝔽5)—21no—
200C2×AGL(1,𝔽5)—21no—
201C2×AGL(1,𝔽5)—94no—
202C2×AGL(1,𝔽5)—188no—
203C2×AGL(1,𝔽5)—182no—
204C2×AGL(1,𝔽5)—182no—
205C2×C4×C5—92no—
206C2×Dic5—92no—
207C4×D5—182no—
208C4×D5—182no—
209C22×C32—22no—
210C22×C32—51no—
211C22×C32—51no—
212C22×C32—52no—
213C22×C32—52no—
214C22×C32—52no—
215C22×C32—102no—
216C22×C32—104no—
217C22×C32—102no—
218C22×C32—102no—
219C22×C32C4⋊AGL(1,𝔽5)12yescomplemented
220C22×C32C22×AGL(1,𝔽5)11yescharacteristic
221C32×C4—201no—
222C32×C4—201no—
223C3⋊Dic3—22no—
224C3⋊Dic3—104no—
225C3⋊Dic3—102no—
226C23×C4—451no—
227C2×C4⋊C4—452no—
228C2×C22⋊C4—51no—
229C2×C22⋊C4—451no—
230#95—452no—
231C2×C3×C5—28no—
232C2×C3×C5#67⋊C418yes—
233C2×C3×C5C2×Dic3⋊C414yes—
234C3×D5—28no—
235C3×D5#87818yes—
236C3×D5#93514yes—
237C3×D5#93714yes—
238C23×C3—54no—
239C23×C3—58no—
240C23×C3—58no—
241C23×C3—54no—
242C23×C3—108no—
243C23×C3—108no—
244C23×C3D3×AGL(1,𝔽5)14yes—
245C2×C3×C4—108no—
246C2×C3×C4—104no—
247C2×C3×C4—104no—
248C2×C3×C4—108no—
249C2×Dic3—38no—
250C2×Dic3—1516no—
251C2×Dic3—158no—
252C2×Dic3—308no—
253C2×Dic3—308no—
254C2×Dic3—308no—
255C2×Dic3—308no—
256C2×Dic3—3032no—
257C22×C5—22no—
258C22×C5#375012yescomplemented
259C22×C5C2×C4×C3⋊D311yescharacteristic
260C2×D5—22no—
261C2×D5—24no—
262C2×D5—22no—
263C2×D5—22no—
264C2×D5#375012yescomplemented
265C2×D5C2×#63411yescharacteristic
266C2×D5D4×C3⋊D312yes—
267C2×D5#383012yes—
268C2×D5C2×#63811yescharacteristic
269AGL(1,𝔽5)—41no—
270AGL(1,𝔽5)—41no—
271AGL(1,𝔽5)—184no—
272C4×C5—182no—
273Dic5—182no—
274C2×C32—22no—
275C2×C32—51no—
276C2×C32—52no—
277C2×C32—51no—
278C2×C32—102no—
279C2×C32C2×C4⋊AGL(1,𝔽5)11yescharacteristic
280C2×C32D4×AGL(1,𝔽5)12yes—
281C24—51no—
282C22×C4—51no—
283C22×C4—51no—
284C22×C4—452no—
285C22×C4—452no—
286C22×C4—904no—
287C22×C4—902no—
288C4⋊C4—904no—
289C22⋊C4—91no—
290C22⋊C4—102no—
291C22⋊C4—452no—
292C22⋊C4—451no—
293C3×C5#494614yescomplemented
294C22×C3—28no—
295C22×C3—54no—
296C22×C3—54no—
297C22×C3—58no—
298C22×C3—58no—
299C22×C3—58no—
300C22×C3—108no—
301C22×C3—108no—
302C22×C3—1016no—
303C22×C3—108no—
304C22×C3Dic3⋊AGL(1,𝔽5)18yescomplemented
305C22×C3C2×D3×AGL(1,𝔽5)14yes—
306C3×C4—204no—
307C3×C4—204no—
308Dic3—68no—
309Dic3—3016no—
310Dic3—308no—
311C2×C5—22no—
312C2×C5#6409012yes—
313C2×C5C2×#375011yescharacteristic
314D5—22no—
315D5#6408712yes—
316D5#6414411yescharacteristic
317D5#6414611yescharacteristic
318C32#6569911yescharacteristic, complemented, Sylow
319C23—51no—
320C23—51no—
321C23—52no—
322C23—52no—
323C23—102no—
324C23—102no—
325C23C3⋊D3×AGL(1,𝔽5)11yescharacteristic
326C2×C4—92no—
327C2×C4—101no—
328C2×C4—101no—
329C2×C4—102no—
330C2×C4—102no—
331C2×C4—454no—
332C2×C4—452no—
333C2×C4—902no—
334C2×C4—902no—
335C2×C4—902no—
336C2×C4—902no—
337C2×C4—908no—
338C2×C3—28no—
339C2×C3—54no—
340C2×C3—54no—
341C2×C3—58no—
342C2×C3—108no—
343C2×C3C2×Dic3⋊AGL(1,𝔽5)14yes—
344C2×C3#67⋊AGL(1,𝔽5)18yes—
345C5#1059264511yescharacteristic, complemented, Sylow
346C22—22no—
347C22—51no—
348C22—51no—
349C22—52no—
350C22—52no—
351C22—52no—
352C22—102no—
353C22—102no—
354C22—104no—
355C22—102no—
356C22#1062396512yescomplemented
357C22C2×C3⋊D3×AGL(1,𝔽5)11yescharacteristic
358C4—182no—
359C4—201no—
360C4—201no—
361C4—904no—
362C4—902no—
363C3#1175202914yescomplemented
364C2—22no—
365C2—51no—
366C2—51no—
367C2—52no—
368C2—102no—
369C2C2×#1062396511yescharacteristic
370C2#1308869012yes—
subgroupG/H
center, FrattiniC22C2×C3⋊D3×AGL(1,𝔽5)
commutator, socleC22×C32×C5C22×C4
FittingC23×C32×C5C2×C4
prime radicalC23×C32×D5C22
cosocleC22×C32×D5C23
hypercenterC23C3⋊D3×AGL(1,𝔽5)
hypocenterC32×C5#385
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
3. C2×#10623965C2semidirect
5. (C2×AGL(1,𝔽5))⋊(C3⋊Dic3)C2semidirect
16. #10623913C4semidirect
17. C5×#3757C4semidirect
18. #10623770C4semidirect
40. C2×C3⋊Dic15C2×C4semidirect
41. C2×C5×C3⋊Dic3C2×C4semidirect
64. C22×C32×C5C4⋊C4semidirect
70. C2×C32×D5C4⋊C4semidirect
95. #3757AGL(1,𝔽5)semidirect
158. C2×C3⋊Dic3C2×AGL(1,𝔽5)semidirect
161. C22×C3×C5Dic3⋊C4semidirect
167. C2×C3×D5Dic3⋊C4semidirect
189. C32×C5#385semidirect
219. C22×C32C4⋊AGL(1,𝔽5)semidirect
258. C22×C5#3750semidirect
264. C2×D5#3750semidirect
293. C3×C5#4946semidirect
304. C22×C3Dic3⋊AGL(1,𝔽5)semidirect
318. C32#65699semidirect
345. C5#10592645semidirect
356. C22#10623965semidirect
363. C3#11752029semidirect

Maximal subgroups

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

subgroupindex
C23×C3⋊ΓL(1,𝔽16)2
C2×C32×C22⋊AGL(1,𝔽5)2
C2×#106239652
D5×#37572
(C2×AGL(1,𝔽5))⋊(C3⋊Dic3)2
#117520293
#105926455