Finite Group Databasebeta

← #72229958

Subgroups — #72229958

Order 2880.

3036 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 3036 conjugacy classes of subgroups — too many to draw. Only its 270 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×D5×C3⋊D3C211yesmaximal, characteristic
2C2×#10624098C211yesmaximal, characteristic, complemented
3C2×C32×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4C2×(C2×AGL(1,𝔽5))⋊(C3⋊D3)C214yesmaximal, complemented
5#11752025—34nomaximal
6C23×C3⋊D15C411yescharacteristic, complemented
7C23×C5×C3⋊D3C411yescharacteristic, complemented
8C23×C32×D5C2211yescharacteristic
9C22×D5×C3⋊D3—22no—
10C22×D5×C3⋊D3—22no—
11C22×D5×C3⋊D3—28no—
12C22×D5×C3⋊D3C412yescomplemented
13C22×D5×C3⋊D3C2212yes—
14C22×C3⋊ΓL(1,𝔽16)C2212yescomplemented
15C22×C32×AGL(1,𝔽5)C2212yescomplemented
16#10624098—22no—
17C32×C22⋊AGL(1,𝔽5)—22no—
18(C2×AGL(1,𝔽5))⋊(C3⋊D3)—28no—
19#10592654—51nomaximal, Hall
20C23×D3×D5—34no—
21C2×#6763—34no—
22C2×C3×C22⋊AGL(1,𝔽5)D314yescomplemented
23C2×(C2×D3)⋊AGL(1,𝔽5)—316no—
24C23×C32×C5C2×C411yescharacteristic
25C22×C3⋊D15—22no—
26C22×C3⋊D15—24no—
27C22×C3⋊D15C2×C412yes—
28C22×C5×C3⋊D3—22no—
29C22×C5×C3⋊D3—24no—
30C22×C5×C3⋊D3C2×C412yes—
31C22×C32×D5—24no—
32C22×C32×D5C2×C411yescharacteristic
33C22×C32×D5D412yes—
34C22×C32×D5D412yes—
35C22×C32×D5C2311yescharacteristic
36C2×D5×C3⋊D3—216no—
37C2×D5×C3⋊D3—216no—
38C2×D5×C3⋊D3—24no—
39C2×D5×C3⋊D3—24no—
40C2×D5×C3⋊D3—24no—
41C2×D5×C3⋊D3—22no—
42C2×D5×C3⋊D3—24no—
43C2×D5×C3⋊D3—22no—
44C2×D5×C3⋊D3D418yes—
45C2×C3⋊ΓL(1,𝔽16)—22no—
46C2×C3⋊ΓL(1,𝔽16)—24no—
47C2×C32×AGL(1,𝔽5)—22no—
48C2×C32×AGL(1,𝔽5)—24no—
49#65800—91noHall
50C24×C3⋊D3—51no—
51C2×C32×C22⋊C4—51no—
52C2×#3757—51no—
53C2×#3752—54no—
54C23×D15—34no—
55C23×C5×D3—34no—
56C23×C3×D5C2×D314yes—
57C22×D3×D5—38no—
58C22×D3×D5—38no—
59C22×D3×D5—632no—
60C22×D3×D5—68no—
61C22×D3×D5—68no—
62C22×ΓL(1,𝔽16)—38no—
63C22×C3×AGL(1,𝔽5)C2×D318yescomplemented
64#6763—68no—
65C3×C22⋊AGL(1,𝔽5)—28no—
66(C2×D3)⋊AGL(1,𝔽5)—632no—
67#4955—154no—
68C22×C32×C5—22no—
69C22×C32×C5C22⋊C412yescomplemented
70C22×C32×C5C22×C411yescharacteristic
71C2×C3⋊D15—28no—
72C2×C3⋊D15—22no—
73C2×C3⋊D15—22no—
74C2×C3⋊D15C22⋊C414yescomplemented
75C2×C5×C3⋊D3—28no—
76C2×C5×C3⋊D3—22no—
77C2×C5×C3⋊D3—22no—
78C2×C5×C3⋊D3C22⋊C414yescomplemented
79C2×C32×D5—22no—
80C2×C32×D5—24no—
81C2×C32×D5—24no—
82C2×C32×D5C22⋊C412yescomplemented
83C2×C32×D5C2×D411yescharacteristic
84C2×C32×D5C2×D414yes—
85C2×C32×D5C2×D411yescharacteristic
86D5×C3⋊D3—216no—
87D5×C3⋊D3—28no—
88D5×C3⋊D3—28no—
89C3⋊ΓL(1,𝔽16)—42no—
90C32×AGL(1,𝔽5)—42no—
91C24×D5—91no—
92C2×C22⋊AGL(1,𝔽5)—91no—
93C2×C22⋊AGL(1,𝔽5)—94no—
94C2×C22⋊AGL(1,𝔽5)C3⋊D311yescharacteristic, complemented
95C24×C32—51no—
96C23×C3⋊D3—52no—
97C23×C3⋊D3—51no—
98C23×C3⋊D3—52no—
99C23×C3⋊D3—108no—
100C23×C3⋊D3—102no—
101C23×C3⋊D3—102no—
102C23×C3⋊D3AGL(1,𝔽5)11yescharacteristic, complemented
103C22×C32×C4—52no—
104C22×C3⋊Dic3—52no—
105C32×C22⋊C4—102no—
106#3757—102no—
107#3752—108no—
108C23×C3×C5C4×D314yes—
109C22×D15—38no—
110C22×D15—616no—
111C22×D15—68no—
112C22×C5×D3—38no—
113C22×C5×D3—616no—
114C22×C5×D3—68no—
115C22×C3×D5—216no—
116C22×C3×D5C4×D314yes—
117C22×C3×D5D1218yes—
118C22×C3×D5#6718yes—
119C22×C3×D5C22×D314yes—
120C2×D3×D5—332no—
121C2×D3×D5—616no—
122C2×D3×D5—616no—
123C2×D3×D5—664no—
124C2×D3×D5—664no—
125C2×D3×D5—616no—
126C2×D3×D5—68no—
127C2×D3×D5—616no—
128C2×D3×D5—68no—
129C2×ΓL(1,𝔽16)—68no—
130C2×ΓL(1,𝔽16)—616no—
131C2×C3×AGL(1,𝔽5)—216no—
132C2×C3×AGL(1,𝔽5)—28no—
133C24×D3—154no—
134C2×C3×C22⋊C4—54no—
135C2×#217—154no—
136C2×(C2×D3)⋊C4—1516no—
137C2×C32×C5—22no—
138C2×C32×C5C2×C22⋊C412yes—
139C2×C32×C5C2×C22⋊C411yescharacteristic
140C3⋊D15—24no—
141C5×C3⋊D3—24no—
142C32×D5—22no—
143C32×D5#12012yes—
144C32×D5#12012yes—
145C24×C5—91no—
146C23×D5—92no—
147C23×D5—91no—
148C23×D5—92no—
149C23×D5—188no—
150C23×D5—182no—
151C23×D5—182no—
152C23×D5C2×C3⋊D311yescharacteristic
153C22×AGL(1,𝔽5)—92no—
154C22×AGL(1,𝔽5)C2×C3⋊D312yescomplemented
155C22⋊AGL(1,𝔽5)—22no—
156C22⋊AGL(1,𝔽5)—188no—
157C22⋊AGL(1,𝔽5)—182no—
158C23×C32—51no—
159C23×C32—52no—
160C23×C32—52no—
161C23×C32—51no—
162C23×C32—104no—
163C23×C32C2×AGL(1,𝔽5)11yescharacteristic
164C22×C3⋊D3—22no—
165C22×C3⋊D3—24no—
166C22×C3⋊D3—52no—
167C22×C3⋊D3—58no—
168C22×C3⋊D3—104no—
169C22×C3⋊D3—102no—
170C22×C3⋊D3—104no—
171C22×C3⋊D3—102no—
172C22×C3⋊D3—104no—
173C22×C3⋊D3—1016no—
174C22×C3⋊D3—1016no—
175C22×C3⋊D3—104no—
176C22×C3⋊D3—102no—
177C22×C3⋊D3—104no—
178C22×C3⋊D3C2×AGL(1,𝔽5)12yes—
179C2×C32×C4—104no—
180C2×C32×C4—102no—
181C2×C3⋊Dic3—104no—
182C2×C3⋊Dic3—102no—
183#379—451noSylow
184C22×C3×C5—28no—
185C22×C3×C5(C2×D3)⋊C418yescomplemented
186C22×C3×C5C2×C4×D314yes—
187C2×D15—316no—
188C2×D15—632no—
189C2×D15—68no—
190C2×D15—68no—
191C2×C5×D3—316no—
192C2×C5×D3—632no—
193C2×C5×D3—68no—
194C2×C5×D3—68no—
195C2×C3×D5—28no—
196C2×C3×D5—216no—
197C2×C3×D5—216no—
198C2×C3×D5(C2×D3)⋊C418yescomplemented
199C2×C3×D5C2×D1214yes—
200C2×C3×D5D3×D4116yes—
201C2×C3×D5C2×#6714yes—
202D3×D5—664no—
203D3×D5—632no—
204D3×D5—632no—
205ΓL(1,𝔽16)—128no—
206C3×AGL(1,𝔽5)—48no—
207C24×C3—54no—
208C23×D3—34no—
209C23×D3—158no—
210C23×D3—158no—
211C23×D3—154no—
212C23×D3—3032no—
213C23×D3—308no—
214C23×D3—308no—
215C22×C3×C4—58no—
216C22×Dic3—158no—
217C3×C22⋊C4—108no—
218#217—308no—
219(C2×D3)⋊C4—3032no—
220C32×C5#37911yescharacteristic, complemented, Hall
221C23×C5—92no—
222C23×C5—184no—
223C23×C5—182no—
224C23×C5C4×C3⋊D311yescharacteristic
225C22×D5—24no—
226C22×D5—92no—
227C22×D5—98no—
228C22×D5—184no—
229C22×D5—184no—
230C22×D5—1816no—
231C22×D5—1816no—
232C22×D5—184no—
233C22×D5—182no—
234C22×D5—184no—
235C22×D5—182no—
236C22×D5—184no—
237C22×D5—182no—
238C22×D5C4×C3⋊D311yescharacteristic
239C22×D5C4⋊(C3⋊D3)12yes—
240C22×D5#63812yes—
241C22×D5C22×C3⋊D311yescharacteristic
242C2×AGL(1,𝔽5)—24no—
243C2×AGL(1,𝔽5)—22no—
244C2×AGL(1,𝔽5)—182no—
245C2×AGL(1,𝔽5)—184no—
246C22×C32—22no—
247C22×C32—51no—
248C22×C32—54no—
249C22×C32—51no—
250C22×C32—52no—
251C22×C32—104no—
252C22×C32—104no—
253C22×C32—102no—
254C22×C32C22⋊AGL(1,𝔽5)12yescomplemented
255C22×C32C22×AGL(1,𝔽5)11yescharacteristic
256C2×C3⋊D3—28no—
257C2×C3⋊D3—22no—
258C2×C3⋊D3—22no—
259C2×C3⋊D3—54no—
260C2×C3⋊D3—108no—
261C2×C3⋊D3—108no—
262C2×C3⋊D3—1016no—
263C2×C3⋊D3—108no—
264C2×C3⋊D3—102no—
265C2×C3⋊D3—102no—
266C2×C3⋊D3C22⋊AGL(1,𝔽5)14yescomplemented
267C32×C4—202no—
268C3⋊Dic3—202no—
269C25—451no—
270C2×C22⋊C4—51no—
271C2×C22⋊C4—454no—
272C2×C22⋊C4—451no—
273C2×C3×C5—28no—
274C2×C3×C5D3×C22⋊C418yes—
275C2×C3×C5C2×(C2×D3)⋊C414yes—
276D15—616no—
277C3×D5—28no—
278C3×D5#88218yes—
279C3×D5#93718yes—
280C5×D3—616no—
281C23×C3—54no—
282C23×C3—58no—
283C23×C3—58no—
284C23×C3—54no—
285C23×C3—1016no—
286C23×C3D3×AGL(1,𝔽5)14yes—
287C22×D3—38no—
288C22×D3—616no—
289C22×D3—68no—
290C22×D3—1532no—
291C22×D3—158no—
292C22×D3—3016no—
293C22×D3—3016no—
294C22×D3—308no—
295C22×D3—308no—
296C22×D3—3016no—
297C22×D3—3016no—
298C22×D3—3064no—
299C22×D3—3064no—
300C22×D3—3016no—
301C22×D3—308no—
302C2×C3×C4—1016no—
303C2×C3×C4—108no—
304C2×Dic3—3016no—
305C2×Dic3—308no—
306C22×C5—22no—
307C22×C5—94no—
308C22×C5—188no—
309C22×C5—182no—
310C22×C5—182no—
311C22×C5#375212yescomplemented
312C22×C5C2×C4×C3⋊D311yescharacteristic
313C2×D5—22no—
314C2×D5—24no—
315C2×D5—24no—
316C2×D5—94no—
317C2×D5—1816no—
318C2×D5—188no—
319C2×D5—182no—
320C2×D5—182no—
321C2×D5—188no—
322C2×D5—188no—
323C2×D5#375212yescomplemented
324C2×D5C2×C4⋊(C3⋊D3)11yescharacteristic
325C2×D5D4×C3⋊D314yes—
326C2×D5C2×#63811yescharacteristic
327AGL(1,𝔽5)—42no—
328AGL(1,𝔽5)—362no—
329C2×C32—22no—
330C2×C32—52no—
331C2×C32—52no—
332C2×C32—102no—
333C2×C32C2×C22⋊AGL(1,𝔽5)11yescharacteristic
334C2×C32C2×C22⋊AGL(1,𝔽5)12yes—
335C3⋊D3—24no—
336C3⋊D3—104no—
337C24—51no—
338C24—91no—
339C24—452no—
340C24—452no—
341C24—451no—
342C24—902no—
343C24—902no—
344C24—908no—
345C22×C4—52no—
346C22×C4—452no—
347C22⋊C4—102no—
348C22⋊C4—908no—
349C22⋊C4—902no—
350C3×C5#495514yescomplemented
351C22×C3—28no—
352C22×C3—54no—
353C22×C3—54no—
354C22×C3—516no—
355C22×C3—58no—
356C22×C3—1016no—
357C22×C3—1016no—
358C22×C3—108no—
359C22×C3(C2×D3)⋊AGL(1,𝔽5)18yescomplemented
360C22×C3C2×D3×AGL(1,𝔽5)14yes—
361C2×D3—316no—
362C2×D3—632no—
363C2×D3—68no—
364C2×D3—68no—
365C2×D3—1516no—
366C2×D3—3032no—
367C2×D3—3032no—
368C2×D3—3064no—
369C2×D3—3032no—
370C2×D3—308no—
371C2×D3—308no—
372C3×C4—208no—
373Dic3—608no—
374C2×C5—22no—
375C2×C5—184no—
376C2×C5C22⋊C4×C3⋊D312yes—
377C2×C5C2×#375211yescharacteristic
378D5—22no—
379D5—184no—
380D5#6409112yes—
381D5#6414612yes—
382C32#6580011yescharacteristic, complemented, Sylow
383C23—51no—
384C23—51no—
385C23—52no—
386C23—52no—
387C23—92no—
388C23—104no—
389C23—184no—
390C23—182no—
391C23—458no—
392C23—452no—
393C23—904no—
394C23—902no—
395C23—902no—
396C23—904no—
397C23—904no—
398C23—904no—
399C23—904no—
400C23—9016no—
401C23—9016no—
402C23—902no—
403C23C3⋊D3×AGL(1,𝔽5)11yescharacteristic
404C2×C4—102no—
405C2×C4—104no—
406C2×C4—902no—
407C2×C4—904no—
408C2×C3—28no—
409C2×C3—58no—
410C2×C3—58no—
411C2×C3—108no—
412C2×C3C2×(C2×D3)⋊AGL(1,𝔽5)14yes—
413C2×C3D3×C22⋊AGL(1,𝔽5)18yes—
414D3—616no—
415D3—3016no—
416C5#1059265411yescharacteristic, complemented, Sylow
417C22—22no—
418C22—51no—
419C22—51no—
420C22—54no—
421C22—52no—
422C22—94no—
423C22—104no—
424C22—104no—
425C22—102no—
426C22—188no—
427C22—182no—
428C22—182no—
429C22—454no—
430C22—908no—
431C22—908no—
432C22—908no—
433C22—9016no—
434C22—902no—
435C22—902no—
436C22(C2×AGL(1,𝔽5))⋊(C3⋊D3)12yescomplemented
437C22C2×C3⋊D3×AGL(1,𝔽5)11yescharacteristic
438C4—202no—
439C4—1802no—
440C3#1175202514yescomplemented
441C2—22no—
442C2—52no—
443C2—52no—
444C2—102no—
445C2—184no—
446C2—904no—
447C2C2×(C2×AGL(1,𝔽5))⋊(C3⋊D3)11yescharacteristic
448C2C3⋊D3×C22⋊AGL(1,𝔽5)12yes—
subgroupG/H
center, FrattiniC22C2×C3⋊D3×AGL(1,𝔽5)
commutator, socleC22×C32×C5C22×C4
FittingC23×C32×C5C2×C4
prime radicalC23×D5×C3⋊D3C2
cosocleC22×C32×D5C23
hypercenterC23C3⋊D3×AGL(1,𝔽5)
hypocenterC32×C5#379
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#10624098C2semidirect
3. C2×C32×C22⋊AGL(1,𝔽5)C2semidirect
4. C2×(C2×AGL(1,𝔽5))⋊(C3⋊D3)C2semidirect
6. C23×C3⋊D15C4semidirect
7. C23×C5×C3⋊D3C4semidirect
12. C22×D5×C3⋊D3C4semidirect
14. C22×C3⋊ΓL(1,𝔽16)C22semidirect
15. C22×C32×AGL(1,𝔽5)C22semidirect
22. C2×C3×C22⋊AGL(1,𝔽5)D3semidirect
63. C22×C3×AGL(1,𝔽5)C2×D3semidirect
69. C22×C32×C5C22⋊C4semidirect
74. C2×C3⋊D15C22⋊C4semidirect
78. C2×C5×C3⋊D3C22⋊C4semidirect
82. C2×C32×D5C22⋊C4semidirect
94. C2×C22⋊AGL(1,𝔽5)C3⋊D3semidirect
102. C23×C3⋊D3AGL(1,𝔽5)semidirect
154. C22×AGL(1,𝔽5)C2×C3⋊D3semidirect
185. C22×C3×C5(C2×D3)⋊C4semidirect
198. C2×C3×D5(C2×D3)⋊C4semidirect
220. C32×C5#379semidirect
254. C22×C32C22⋊AGL(1,𝔽5)semidirect
266. C2×C3⋊D3C22⋊AGL(1,𝔽5)semidirect
311. C22×C5#3752semidirect
323. C2×D5#3752semidirect
350. C3×C5#4955semidirect
359. C22×C3(C2×D3)⋊AGL(1,𝔽5)semidirect
382. C32#65800semidirect
416. C5#10592654semidirect
436. C22(C2×AGL(1,𝔽5))⋊(C3⋊D3)semidirect
440. C3#11752025semidirect

Maximal subgroups

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

subgroupindex
C23×D5×C3⋊D32
C2×#106240982
C2×C32×C22⋊AGL(1,𝔽5)2
C2×(C2×AGL(1,𝔽5))⋊(C3⋊D3)2
#117520253
#105926545