Finite Group Databasebeta

← #35315486

Subgroups — #35315486

Order 21248.

412 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 83 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#467×D83C211yesmaximal, characteristic
2#24909072C211yesmaximal, characteristic
3#24907487C211yesmaximal, characteristic
4#24905476C211yesmaximal, characteristic, complemented
5#24905475C211yesmaximal, characteristic, complemented
6#24905474C211yesmaximal, characteristic, complemented
7C83×#1348C211yesmaximal, characteristic, complemented
8C2×Q16×D83C411yescharacteristic, complemented
9C22×C8×D83C2211yescharacteristic
10C22⋊Q8×D83—21no—
11C2×#14581731—21no—
12#19127943C411yescharacteristic, complemented
13#19127942C2211yescharacteristic
14#19127941—21no—
15#19127940—21no—
16#19127939C411yescharacteristic, complemented
17#107×D83C411yescharacteristic, complemented
18C2×#14581723—21no—
19#103×D83—21no—
20#19127558—21no—
21C22⋊(C83⋊C16)C2211yescharacteristic
22(C2×D83)⋊C16—21no—
23#19127464C2211yescharacteristic
24#467×C83C2211yescharacteristic
25C22⋊C16×C83C2211yescharacteristic
26#340×C83C2211yescharacteristic
27C2×Q8×D83—21no—
28C22×C4×D83D411yescharacteristic
29C2×Dic332C2×C411yescharacteristic
30C2×#13050580—21no—
31Q16×D83—42no—
32C22×C83⋊C8D411yescharacteristic
33C2×C8×D83—42no—
34C2×C8×D83C2×C411yescharacteristic
35C2×C8×D83C2×C411yescharacteristic
36C2×C8×D83D411yescharacteristic
37C2×C8×D83D411yescharacteristic
38#14581772—21no—
39#14581771—21no—
40#14581770—41no—
41#14581769—41no—
42#14581768—21no—
43C2×C4⋊Dic83—21no—
44C4⋊C4×D83—21no—
45C4⋊C4×D83—41no—
46C2×(C2×D83)⋊C4—41no—
47C22⋊C4×D83—41no—
48#14581731—42no—
49C2×C83⋊C16—21no—
50#14581726C2×C411yescharacteristic
51#14581725—21no—
52#14581723—42no—
53#14581717—21no—
54#14581716—21no—
55#14581715—21no—
56C83⋊#95—41no—
57C2×Q16×C83C2×C411yescharacteristic
58C22×C8×C83C2311yescharacteristic
59C22⋊Q8×C83—21no—
60C2×C4⋊C4×C83—21no—
61C2×C16×C83—21no—
62#107×C83C2×C411yescharacteristic
63#103×C83—21no—
64C23×D83D811yescharacteristic
65C2×Dic166—21no—
66C2×Dic166—41no—
67Q8×D83—42no—
68Q8×D83—42no—
69C22×Dic83—41no—
70C22×Dic83D811yescharacteristic
71C2×C4×D83—41no—
72C2×C4×D83—41no—
73C2×C4×D83—41no—
74C2×C4×D83—41no—
75C2×C4×D83C22⋊C411yescharacteristic
76C2×C4×D83C22⋊C411yescharacteristic
77C2×C4×D83SD1611yescharacteristic
78C2×C4×D83SD1611yescharacteristic
79Dic332—41no—
80#13050580—42no—
81C2×C83⋊C8—22no—
82C2×C83⋊C8C22⋊C411yescharacteristic
83C2×C83⋊C8C22⋊C411yescharacteristic
84C8×D83—22no—
85C8×D83—22no—
86C8×D83—41no—
87C8×D83—41no—
88C4⋊Dic83—21no—
89C4⋊Dic83—41no—
90C4⋊Dic83—41no—
91C4⋊Dic83—41no—
92Dic83⋊C4—41no—
93Dic83⋊C4—41no—
94Dic83⋊C4—41no—
95C22⋊Dic83—41no—
96(C2×D83)⋊C4—41no—
97(C2×D83)⋊C4—41no—
98(C2×D83)⋊C4—82no—
99C83⋊C16—41no—
100C2×Q8×C83—21no—
101C22×C4×C83—41no—
102C22×C4×C83C2×D411yescharacteristic
103Q16×C83—41no—
104C2×C8×C83—41no—
105C2×C8×C83C22×C411yescharacteristic
106C2×C8×C83C2×D411yescharacteristic
107C4⋊C4×C83—21no—
108C4⋊C4×C83—41no—
109C4⋊C4×C83—41no—
110C4⋊C4×C83—41no—
111C22⋊C4×C83—41no—
112C16×C83—41no—
113C23×C83C2×D811yescharacteristic
114C22×D83—21no—
115C22×D83—21no—
116C22×D83—22no—
117C22×D83—41no—
118C22×D83#10211yescharacteristic
119C22×D83#10211yescharacteristic
120Q8×C83—41no—
121Q8×C83—41no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—81no—
127C2×C4×C83—81no—
128C2×C4×C83—81no—
129C2×C4×C83C2×C22⋊C411yescharacteristic
130C2×C4×C83C2×SD1611yescharacteristic
131C2×Dic83—41no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134C2×Dic83—81no—
135C2×Dic83—81no—
136C2×Dic83—81no—
137C2×Dic83#10211yescharacteristic
138C2×Dic83#10211yescharacteristic
139C4×D83—21no—
140C4×D83—21no—
141C4×D83—21no—
142C4×D83—21no—
143C4×D83—41no—
144C4×D83—41no—
145C4×D83—82no—
146C4×D83—82no—
147Dic166—41no—
148Dic166—41no—
149Dic166—82no—
150C8×C83—21no—
151C8×C83—21no—
152C83⋊C8—21no—
153C83⋊C8#9912yes—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83C2×#10211yescharacteristic
157C22×C83#51011yescharacteristic
158C22×C83#51011yescharacteristic
159C2×D83—21no—
160C2×D83—21no—
161C2×D83—21no—
162C2×D83—21no—
163C2×D83—21no—
164C2×D83—42no—
165C2×D83—42no—
166C2×D83#36012yes—
167C4×C83—21no—
168C4×C83—81no—
169C4×C83—81no—
170C4×C83—161no—
171C4×C83C2×#9911yescharacteristic
172C4×C83#42111yescharacteristic
173Dic83—21no—
174Dic83—21no—
175Dic83—81no—
176Dic83—81no—
177Dic83—161no—
178#11889—831nomaximal, Sylow
179C2×C83—21no—
180C2×C83—21no—
181C2×C83#151911yescharacteristic
182C2×C83#214111yescharacteristic
183C2×C83C2×#36011yescharacteristic
184D83—22no—
185D83—41no—
186C2×#467—831no—
187#2112—831no—
188#1807—831no—
189#1348—831no—
190#1348—831no—
191#1348—831no—
192#1348D8311yescharacteristic, complemented
193C83#1188911yescharacteristic, complemented, Sylow
194C22×Q16—831no—
195C23×C8—831no—
196C2×C22⋊Q8—1661no—
197C2×M32—1661no—
198#467—831no—
199#467—831no—
200#467—831no—
201#467—1661no—
202#467—1661no—
203#467C2×D8311yescharacteristic
204C2×#107—831no—
205C2×#106—1661no—
206C2×#103—1661no—
207#380—1661no—
208C22⋊C16—831no—
209C22⋊C16—1661no—
210C22⋊C16C2×D8311yescharacteristic
211#340—831no—
212#340C2×D8311yescharacteristic
213C22×Q8—1661no—
214C23×C4—831no—
215C2×Q16—3322no—
216C2×Q16—831no—
217C2×Q16—1661no—
218C2×Q16C4×D8311yescharacteristic
219C22×C8—3322no—
220C22×C8—831no—
221C22×C8—831no—
222C22×C8—831no—
223C22×C8—831no—
224C22×C8—831no—
225C22×C8C22×D8311yescharacteristic
226C22⋊Q8—21no—
227C22⋊Q8—3321no—
228C22⋊Q8—3321no—
229C22⋊Q8—1661no—
230C22⋊Q8—1661no—
231C22⋊Q8—1661no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—3321no—
234C2×C4⋊C4—1661no—
235C2×C4⋊C4—1661no—
236C2×C22⋊C4—3321no—
237C2×C22⋊C4—3321no—
238M32—3322no—
239C2×C16—21no—
240C2×C16—1661no—
241#107—831no—
242#107—1661no—
243#107C4×D8311yescharacteristic
244#106—3322no—
245#103—21no—
246#103—1661no—
247#103—1661no—
248#103—1661no—
249#95—3321no—
250C24—831no—
251C2×Q8—21no—
252C2×Q8—3321no—
253C2×Q8—3322no—
254C2×Q8—3322no—
255C2×Q8—1661no—
256C22×C4—41no—
257C22×C4—3321no—
258C22×C4—3321no—
259C22×C4—3321no—
260C22×C4—3321no—
261C22×C4—3321no—
262C22×C4—831no—
263C22×C4—831no—
264C22×C4—831no—
265C22×C4—831no—
266C22×C4—831no—
267C22×C4D4×D8311yescharacteristic
268Q16—41no—
269Q16—3321no—
270Q16—3322no—
271C2×C8—41no—
272C2×C8—3321no—
273C2×C8—3321no—
274C2×C8—831no—
275C2×C8—831no—
276C2×C8—1662no—
277C2×C8—1662no—
278C2×C8—1662no—
279C2×C8C2×C4×D8311yescharacteristic
280C2×C8D4×D8311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—3321no—
286C4⋊C4—3321no—
287C4⋊C4—3321no—
288C4⋊C4—3321no—
289C4⋊C4—3321no—
290C4⋊C4—3321no—
291C4⋊C4—1661no—
292C22⋊C4—41no—
293C22⋊C4—3321no—
294C22⋊C4—3321no—
295C22⋊C4—3321no—
296C22⋊C4—6642no—
297C16—41no—
298C16—3321no—
299C23—3321no—
300C23—831no—
301C23—831no—
302C23—1662no—
303C23—1661no—
304C23—1661no—
305C23D8×D8311yescharacteristic
306Q8—41no—
307Q8—41no—
308Q8—3321no—
309Q8—3321no—
310Q8—6642no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—3321no—
319C2×C4—3321no—
320C2×C4—3321no—
321C2×C4—3321no—
322C2×C4—3321no—
323C2×C4—831no—
324C2×C4—831no—
325C2×C4—6642no—
326C2×C4—6642no—
327C2×C4—6641no—
328C2×C4—6641no—
329C2×C4—6641no—
330C2×C4—1661no—
331C2×C4—1661no—
332C2×C4—1661no—
333C2×C4—1661no—
334C2×C4C22⋊C4×D8311yescharacteristic
335C2×C4SD16×D8311yescharacteristic
336C8—21no—
337C8—21no—
338C8—832no—
339C8—1661no—
340C22—21no—
341C22—21no—
342C22—3322no—
343C22—3322no—
344C22—832no—
345C22—1661no—
346C22—1661no—
347C22—1661no—
348C22—1661no—
349C22—1661no—
350C22#102×D8311yescharacteristic
351C22#1912814011yescharacteristic
352C22#1912814311yescharacteristic
353C4—21no—
354C4—81no—
355C4—81no—
356C4—161no—
357C4—13281no—
358C4—6641no—
359C4—6641no—
360C4—1661no—
361C4—1661no—
362C4#99×D8311yescharacteristic
363C4#1912771611yescharacteristic
364C2—21no—
365C2—21no—
366C2—3321no—
367C2—1662no—
368C2#2490602911yescharacteristic
369C2#2490918111yescharacteristic
370C2#360×D8311yescharacteristic
subgroupG/H
centerC22#102×D83
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC83×#1348C2
socleC22×C83C2×#102
prime radicalC23×D83D8
cosocleC22×C8×C83C23
hypercenter#1348D83
hypocenterC83#11889
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
4. #24905476C2semidirect
5. #24905475C2semidirect
6. #24905474C2semidirect
7. C83×#1348C2semidirect
8. C2×Q16×D83C4semidirect
12. #19127943C4semidirect
16. #19127939C4semidirect
17. #107×D83C4semidirect
192. #1348D83semidirect
193. C83#11889semidirect

Maximal subgroups

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

subgroupindex
#467×D832
#249090722
#249074872
#249054762
#249054752
#249054742
C83×#13482
#1188983