Finite Group Databasebeta

← #35315348

Subgroups — #35315348

Order 21248.

334 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 89 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#24918044C211yesmaximal, characteristic
2#24909259C211yesmaximal, characteristic
3#24907589C211yesmaximal, characteristic
4#24905476C211yesmaximal, characteristic
5#24905475C211yesmaximal, characteristic
6#24905474C211yesmaximal, characteristic
7C83×#1348C211yesmaximal, characteristic
8#19128253—21no—
9#19128105C411yescharacteristic, complemented
10#19128079C411yescharacteristic, complemented
11#19128050—21no—
12#19127942C2211yescharacteristic
13#19127940—21no—
14Q16×Dic83C411yescharacteristic, complemented
15#19127795C411yescharacteristic, complemented
16#19127793—21no—
17C2×C8×Dic83C2211yescharacteristic
18#19127570—21no—
19Dic83⋊C16—21no—
20C22⋊(C83⋊C16)C2211yescharacteristic
21C16×Dic83—21no—
22#19127464C2211yescharacteristic
23#19127463—21no—
24#467×C83C2211yescharacteristic
25C22⋊C16×C83C2211yescharacteristic
26#340×C83C2211yescharacteristic
27C2×Dic332C2×C411yescharacteristic
28C2×#13050580—21no—
29C22×C83⋊C8D411yescharacteristic
30#14581791—21no—
31#14581790—21no—
32#14581782—41no—
33#14581771—21no—
34#14581769—41no—
35Q8×Dic83—21no—
36#14581757—21no—
37#14581757—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#14581725—21no—
46#14581717—21no—
47#14581716—21no—
48#14581715—21no—
49C8×Dic83C2×C411yescharacteristic
50C8×Dic83C2×C411yescharacteristic
51C8×Dic83D411yescharacteristic
52C8×Dic83D411yescharacteristic
53C2×Q16×C83C2×C411yescharacteristic
54C22×C8×C83C2311yescharacteristic
55C22⋊Q8×C83—21no—
56C2×C4⋊C4×C83—21no—
57C2×C16×C83—21no—
58#107×C83C2×C411yescharacteristic
59#103×C83—21no—
60C2×Dic166—21no—
61C2×Dic166—41no—
62C22×Dic83—41no—
63C22×Dic83D811yescharacteristic
64C22×Dic83D811yescharacteristic
65Dic332—41no—
66#13050580—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—41no—
78Dic83⋊C4—41no—
79Dic83⋊C4—82no—
80Dic83⋊C4—82no—
81C22⋊Dic83—41no—
82C4×Dic83—41no—
83C4×Dic83—41no—
84C4×Dic83C22⋊C411yescharacteristic
85C4×Dic83C22⋊C411yescharacteristic
86C4×Dic83SD1611yescharacteristic
87C4×Dic83SD1611yescharacteristic
88C83⋊C16—41no—
89C2×Q8×C83—21no—
90C22×C4×C83—41no—
91C22×C4×C83C2×D411yescharacteristic
92Q16×C83—41no—
93C2×C8×C83—41no—
94C2×C8×C83C22×C411yescharacteristic
95C2×C8×C83C2×D411yescharacteristic
96C4⋊C4×C83—21no—
97C4⋊C4×C83—41no—
98C4⋊C4×C83—41no—
99C4⋊C4×C83—41no—
100C22⋊C4×C83—41no—
101C16×C83—41no—
102C23×C83C2×D811yescharacteristic
103Q8×C83—41no—
104Q8×C83—41no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—41no—
108C2×C4×C83—41no—
109C2×C4×C83—81no—
110C2×C4×C83—81no—
111C2×C4×C83—81no—
112C2×C4×C83C2×C22⋊C411yescharacteristic
113C2×C4×C83C2×SD1611yescharacteristic
114C2×Dic83—22no—
115C2×Dic83—41no—
116C2×Dic83—41no—
117C2×Dic83—81no—
118C2×Dic83—81no—
119C2×Dic83—81no—
120C2×Dic83#10211yescharacteristic
121C2×Dic83#10211yescharacteristic
122C2×Dic83#10211yescharacteristic
123C2×Dic83#10211yescharacteristic
124C2×Dic83SD3212yes—
125C2×Dic83Q3212yes—
126Dic166—41no—
127Dic166—41no—
128Dic166—82no—
129C8×C83—21no—
130C8×C83—21no—
131C83⋊C8—21no—
132C83⋊C8—21no—
133C22×C83—21no—
134C22×C83—21no—
135C22×C83C2×#10211yescharacteristic
136C22×C83C2×SD3211yescharacteristic
137C22×C83C2×Q3211yescharacteristic
138C4×C83—21no—
139C4×C83—81no—
140C4×C83—81no—
141C4×C83—161no—
142C4×C83#41011yescharacteristic
143C4×C83#42111yescharacteristic
144Dic83—21no—
145Dic83—81no—
146Dic83—81no—
147Dic83—161no—
148Dic83#35812yes—
149Dic83#35812yes—
150Dic83#36012yes—
151#11857—831nomaximal, Sylow
152C2×C83—21no—
153C2×C83—21no—
154C2×C83#151711yescharacteristic
155C2×C83C2×#35811yescharacteristic
156C2×C83C2×#36011yescharacteristic
157#3170—831no—
158#2162—831no—
159#1834—831no—
160#1348—831no—
161#1348—831no—
162#1348—831no—
163#1348D8311yescharacteristic
164C83#1185711yescharacteristic, complemented, Sylow
165#533—1661no—
166#500—831no—
167#494—831no—
168#487—1661no—
169#467—831no—
170#467—1661no—
171#467C2×D8311yescharacteristic
172C4×Q16—831no—
173C4×Q16—831no—
174C4×Q16—1661no—
175C2×C4×C8—831no—
176#384—1661no—
177C4⋊C16—1661no—
178C22⋊C16—831no—
179C22⋊C16C2×D8311yescharacteristic
180C4×C16—1661no—
181#340—831no—
182#340—1661no—
183#340C2×D8311yescharacteristic
184C2×Q16—831no—
185C2×Q16—1661no—
186C2×Q16C4×D8311yescharacteristic
187C22×C8—831no—
188C22×C8C22×D8311yescharacteristic
189C4⋊Q8—1661no—
190C4⋊Q8—1661no—
191#125—3321no—
192C22⋊Q8—21no—
193C22⋊Q8—3321no—
194C22⋊Q8—1661no—
195C4×Q8—3321no—
196C4×Q8—1661no—
197C4×Q8—1661no—
198#117—3321no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—3321no—
201C2×C4⋊C4—3321no—
202C2×C4⋊C4—1661no—
203C2×C42—831no—
204C2×C16—21no—
205C2×C16—1661no—
206#107—831no—
207#107—1661no—
208#107C4×D8311yescharacteristic
209#103—21no—
210#103—1661no—
211#103—1661no—
212#103—1661no—
213C4×C8—831no—
214C4×C8—831no—
215C4×C8—831no—
216C4×C8—831no—
217C2×Q8—21no—
218C2×Q8—3321no—
219C2×Q8—1661no—
220C22×C4—41no—
221C22×C4—3321no—
222C22×C4—831no—
223C22×C4—831no—
224C22×C4D4×D8311yescharacteristic
225Q16—41no—
226Q16—3321no—
227Q16—3322no—
228C2×C8—41no—
229C2×C8—3321no—
230C2×C8—831no—
231C2×C8—831no—
232C2×C8C2×C4×D8311yescharacteristic
233C2×C8D4×D8311yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—3321no—
239C4⋊C4—3321no—
240C4⋊C4—3321no—
241C4⋊C4—3321no—
242C4⋊C4—3321no—
243C4⋊C4—3321no—
244C4⋊C4—3321no—
245C4⋊C4—3321no—
246C4⋊C4—6642no—
247C4⋊C4—6642no—
248C4⋊C4—1661no—
249C22⋊C4—41no—
250C22⋊C4—3321no—
251C42—3321no—
252C42—3321no—
253C42—831no—
254C42—831no—
255C42—831no—
256C42—831no—
257C16—41no—
258C16—3321no—
259C23D8×D8311yescharacteristic
260Q8—41no—
261Q8—41no—
262Q8—3321no—
263Q8—3321no—
264Q8—6642no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—3321no—
273C2×C4—3321no—
274C2×C4—832no—
275C2×C4—832no—
276C2×C4—831no—
277C2×C4—831no—
278C2×C4—831no—
279C2×C4—831no—
280C2×C4—6641no—
281C2×C4—6641no—
282C2×C4—6641no—
283C2×C4—1662no—
284C2×C4C22⋊C4×D8311yescharacteristic
285C2×C4SD16×D8311yescharacteristic
286C8—21no—
287C8—21no—
288C8—1661no—
289C8—1661no—
290C22—21no—
291C22—21no—
292C22#102×D8311yescharacteristic
293C22SD32×D8311yescharacteristic
294C22Q32×D8311yescharacteristic
295C4—21no—
296C4—81no—
297C4—81no—
298C4—161no—
299C4—13281no—
300C4—832no—
301C4—832no—
302C4—832no—
303C4—6641no—
304C4—6641no—
305C4—1661no—
306C4#1912767311yescharacteristic
307C4#1912771611yescharacteristic
308C2—21no—
309C2—21no—
310C2#2490602111yescharacteristic
311C2#358×D8311yescharacteristic
312C2#360×D8311yescharacteristic
subgroupG/H
centerC22#102×D83
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC83×#1348C2
socleC22×C83C2×#102
prime radicalC23×C83C2×D8
cosocleC22×C8×C83C23
hypercenter#1348D83
hypocenterC83#11857
LCSG ▷ C2×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1348

Complements

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

NH ≅ G/Ntype
9. #19128105C4semidirect
10. #19128079C4semidirect
14. Q16×Dic83C4semidirect
15. #19127795C4semidirect
164. C83#11857semidirect

Maximal subgroups

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

subgroupindex
#249180442
#249092592
#249075892
#249054762
#249054752
#249054742
C83×#13482
#1185783