Finite Group Databasebeta

← #35315360

Subgroups — #35315360

Order 21248.

336 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 81 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
1#24918028C211yesmaximal, characteristic
2#24909263C211yesmaximal, characteristic, complemented
3#24907589C211yesmaximal, characteristic, complemented
4#24905476C211yesmaximal, characteristic, complemented
5#24905474C211yesmaximal, characteristic, complemented
6#24905472C211yesmaximal, characteristic
7C83×#1347C211yesmaximal, characteristic
8#19128232—21no—
9#19128102C411yescharacteristic, complemented
10#19128084C411yescharacteristic, complemented
11#19128054—21no—
12#19127942C2211yescharacteristic
13#19127928—21no—
14#19127795C411yescharacteristic, complemented
15#19127792—21no—
16D8×Dic83C411yescharacteristic, complemented
17C2×C8×Dic83C2211yescharacteristic
18#19127570—21no—
19Dic83⋊C16—21no—
20C22⋊(C83⋊C16)C2211yescharacteristic
21#19127476—21no—
22#19127464C2211yescharacteristic
23#19127463—21no—
24#466×C83C2211yescharacteristic
25C22⋊C16×C83C2211yescharacteristic
26#340×C83C2211yescharacteristic
27C2×Dic332C2×C411yescharacteristic
28C2×#13050578—21no—
29C22×C83⋊C8D411yescharacteristic
30#14581785—41no—
31#14581783—21no—
32#14581779—21no—
33#14581773—41no—
34#14581771—21no—
35#14581757—21no—
36D4×Dic83—21no—
37D4×Dic83—41no—
38#14581748—41no—
39C2×C4⋊Dic83—21no—
40C2×Dic83⋊C4—41no—
41C2×Dic83⋊C4—41no—
42C2×C4×Dic83D411yescharacteristic
43C2×C83⋊C16—21no—
44#14581726C2×C411yescharacteristic
45#14581723—21no—
46#14581717—21no—
47#14581715—21no—
48#14581713—21no—
49C8×Dic83C2×C411yescharacteristic
50C8×Dic83C2×C411yescharacteristic
51C8×Dic83D411yescharacteristic
52C8×Dic83D411yescharacteristic
53C2×D8×C83C2×C411yescharacteristic
54C22×C8×C83C2311yescharacteristic
55#121×C83—21no—
56C2×C4⋊C4×C83—21no—
57C2×C16×C83—21no—
58#107×C83C2×C411yescharacteristic
59#102×C83—21no—
60C2×Dic166—21no—
61C22×Dic83—41no—
62C22×Dic83—41no—
63C22×Dic83D811yescharacteristic
64C22×Dic83D811yescharacteristic
65Dic332—41no—
66#13050578—42no—
67C2×C83⋊C8—41no—
68C2×C83⋊C8C22⋊C411yescharacteristic
69C2×C83⋊C8C22⋊C411yescharacteristic
70C4⋊Dic83—21no—
71C4⋊Dic83—41no—
72C4⋊Dic83—41no—
73C4⋊Dic83—41no—
74Dic83⋊C4—41no—
75Dic83⋊C4—41no—
76Dic83⋊C4—41no—
77Dic83⋊C4—82no—
78Dic83⋊C4—82no—
79C22⋊Dic83—41no—
80C22⋊Dic83—41no—
81C22⋊Dic83—41no—
82C22⋊Dic83—41no—
83C4×Dic83—41no—
84C4×Dic83C22⋊C411yescharacteristic
85C4×Dic83C22⋊C411yescharacteristic
86C4×Dic83SD1611yescharacteristic
87C4×Dic83SD1611yescharacteristic
88C83⋊C16—41no—
89C2×D4×C83—21no—
90C2×D4×C83—41no—
91C22×C4×C83—41no—
92C22×C4×C83C2×D411yescharacteristic
93D8×C83—41no—
94C2×C8×C83—41no—
95C2×C8×C83C22×C411yescharacteristic
96C2×C8×C83C2×D411yescharacteristic
97C4⋊C4×C83—21no—
98C4⋊C4×C83—41no—
99C4⋊C4×C83—41no—
100C22⋊C4×C83—41no—
101C16×C83—41no—
102C23×C83—41no—
103C23×C83C2×D811yescharacteristic
104D4×C83—41no—
105D4×C83—41no—
106D4×C83—81no—
107D4×C83—81no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—41no—
111C2×C4×C83—81no—
112C2×C4×C83—81no—
113C2×C4×C83—81no—
114C2×C4×C83C2×C22⋊C411yescharacteristic
115C2×C4×C83C2×SD1611yescharacteristic
116C2×Dic83—21no—
117C2×Dic83—21no—
118C2×Dic83—22no—
119C2×Dic83—41no—
120C2×Dic83—41no—
121C2×Dic83—82no—
122C2×Dic83—81no—
123C2×Dic83—81no—
124C2×Dic83—81no—
125C2×Dic83#10211yescharacteristic
126C2×Dic83#10211yescharacteristic
127C2×Dic83#10211yescharacteristic
128C2×Dic83#10211yescharacteristic
129Dic166—41no—
130Dic166—41no—
131C8×C83—21no—
132C8×C83—21no—
133C83⋊C8—21no—
134C83⋊C8—21no—
135C22×C83—21no—
136C22×C83—21no—
137C22×C83—81no—
138C22×C83—81no—
139C22×C83—81no—
140C22×C83C2×#10211yescharacteristic
141C22×C83#50911yescharacteristic
142C22×C83#51011yescharacteristic
143C4×C83—21no—
144C4×C83—81no—
145C4×C83—161no—
146C4×C83#41011yescharacteristic
147C4×C83#42111yescharacteristic
148Dic83—21no—
149Dic83—21no—
150Dic83—21no—
151Dic83—81no—
152Dic83—81no—
153Dic83—161no—
154Dic83#36012yes—
155#11860—831nomaximal, Sylow
156C2×C83—21no—
157C2×C83—21no—
158C2×C83—161no—
159C2×C83#151711yescharacteristic
160C2×C83#213911yescharacteristic
161C2×C83C2×#36011yescharacteristic
162#3168—831no—
163#2163—831no—
164#1834—831no—
165#1348—831no—
166#1348—831no—
167#1347—831no—
168#1347D8311yescharacteristic
169C83#1186011yescharacteristic, complemented, Sylow
170#529—1661no—
171#499—831no—
172#495—831no—
173#488—1661no—
174#467—831no—
175#466C2×D8311yescharacteristic
176#465—1661no—
177C4×Q16—831no—
178C4×SD16—1661no—
179C4×D8—831no—
180C2×C4×C8—831no—
181#384—1661no—
182C4⋊C16—1661no—
183C22⋊C16—831no—
184C22⋊C16C2×D8311yescharacteristic
185#346—1661no—
186#340—831no—
187#340—1661no—
188#340C2×D8311yescharacteristic
189C2×Q16—831no—
190C2×SD16—1661no—
191C2×D8C4×D8311yescharacteristic
192C22×C8—831no—
193C22×C8C22×D8311yescharacteristic
194#126—3321no—
195#125—1661no—
196#124—1661no—
197#123—3321no—
198C22⋊Q8—1661no—
199#121—21no—
200C4×Q8—1661no—
201C4×D4—3321no—
202C4×D4—1661no—
203#117—3321no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—3321no—
206C2×C4⋊C4—3321no—
207C2×C4⋊C4—1661no—
208C2×C42—831no—
209C2×C16—21no—
210C2×C16—1661no—
211#107—831no—
212#107C4×D8311yescharacteristic
213#106—1661no—
214#103—1661no—
215#103—1661no—
216#102—21no—
217#102—1661no—
218C4×C8—831no—
219C4×C8—831no—
220C4×C8—831no—
221C4×C8—831no—
222C2×Q8—1661no—
223C2×D4—21no—
224C2×D4—41no—
225C22×C4—41no—
226C22×C4—3321no—
227C22×C4—3321no—
228C22×C4—831no—
229C22×C4—831no—
230C22×C4D4×D8311yescharacteristic
231Q16—3321no—
232SD16—3322no—
233D8—41no—
234C2×C8—41no—
235C2×C8—3321no—
236C2×C8—831no—
237C2×C8—831no—
238C2×C8C2×C4×D8311yescharacteristic
239C2×C8D4×D8311yescharacteristic
240C4⋊C4—21no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—3321no—
244C4⋊C4—3321no—
245C4⋊C4—3321no—
246C4⋊C4—3321no—
247C4⋊C4—3321no—
248C4⋊C4—3321no—
249C4⋊C4—6642no—
250C4⋊C4—6642no—
251C4⋊C4—1661no—
252C22⋊C4—41no—
253C22⋊C4—3321no—
254C22⋊C4—3321no—
255C22⋊C4—3321no—
256C22⋊C4—3321no—
257C42—3321no—
258C42—831no—
259C42—831no—
260C42—831no—
261C42—831no—
262C16—41no—
263C16—3321no—
264C23—41no—
265C23D8×D8311yescharacteristic
266Q8—3321no—
267Q8—3321no—
268D4—41no—
269D4—41no—
270D4—81no—
271D4—81no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—3321no—
279C2×C4—3321no—
280C2×C4—831no—
281C2×C4—831no—
282C2×C4—831no—
283C2×C4—831no—
284C2×C4—6641no—
285C2×C4—6641no—
286C2×C4—6641no—
287C2×C4—6642no—
288C2×C4—1662no—
289C2×C4—1661no—
290C2×C4—1661no—
291C2×C4C22⋊C4×D8311yescharacteristic
292C2×C4SD16×D8311yescharacteristic
293C8—21no—
294C8—21no—
295C8—1661no—
296C8—1661no—
297C22—21no—
298C22—21no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22#102×D8311yescharacteristic
303C22#1912813511yescharacteristic
304C22#1912814011yescharacteristic
305C4—21no—
306C4—81no—
307C4—161no—
308C4—13281no—
309C4—832no—
310C4—6641no—
311C4—6641no—
312C4—1661no—
313C4—1661no—
314C4—1661no—
315C4#1912767311yescharacteristic
316C4#1912771611yescharacteristic
317C2—21no—
318C2—21no—
319C2—161no—
320C2#2490602111yescharacteristic
321C2#2490917211yescharacteristic
322C2#360×D8311yescharacteristic
subgroupG/H
centerC22#102×D83
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC83×#1347C2
socleC22×C83C2×#102
prime radical#466×C83C22
cosocleC22×C8×C83C23
hypercenter#1347D83
hypocenterC83#11860
LCSG ▷ C2×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1347

Complements

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

NH ≅ G/Ntype
2. #24909263C2semidirect
3. #24907589C2semidirect
4. #24905476C2semidirect
5. #24905474C2semidirect
9. #19128102C4semidirect
10. #19128084C4semidirect
14. #19127795C4semidirect
16. D8×Dic83C4semidirect
169. C83#11860semidirect

Maximal subgroups

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

subgroupindex
#249180282
#249092632
#249075892
#249054762
#249054742
#249054722
C83×#13472
#1186083