Finite Group Databasebeta

← #35292976

Subgroups — #35292976

Order 21248.

500 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 97 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#24915812C211yesmaximal, characteristic
2#24907530C211yesmaximal, characteristic
3#24907224C211yesmaximal, characteristic
4#24905289C211yesmaximal, characteristic, complemented
5#24905288C211yesmaximal, characteristic, complemented
6#24905288C211yesmaximal, characteristic, complemented
7C83×#1272C211yesmaximal, characteristic, complemented
8C2×#14581790C411yescharacteristic
9C2×#14581772C411yescharacteristic
10C2×#14581768C411yescharacteristic
11C2×C42×D83C2211yescharacteristic
12C2×(C2×D83)⋊C8—21no—
13C2×(C2×D83)⋊C8—21no—
14C2×#14581702—21no—
15#19127607C2211yescharacteristic
16#19127598—21no—
17#19127598—21no—
18#19127570C411yescharacteristic
19#19127560—21no—
20#19127560—21no—
21#19127456C2211yescharacteristic
22#19127456C2211yescharacteristic
23#19127455—21no—
24#391×C83C2211yescharacteristic
25#336×C83C2211yescharacteristic
26#336×C83C2211yescharacteristic
27C22×Dic166C2×C411yescharacteristic
28C22×C4×D83C2×C411yescharacteristic
29C22×C4×D83D411yescharacteristic
30C22×C4×D83D411yescharacteristic
31C22×C83⋊C8—21no—
32C22×C83⋊C8—21no—
33#14581790—42no—
34#14581772—42no—
35#14581768—42no—
36C2×C4⋊Dic83—21no—
37C2×C4⋊Dic83C2×C411yescharacteristic
38C2×Dic83⋊C4—21no—
39C2×Dic83⋊C4—21no—
40C2×Dic83⋊C4—21no—
41C2×Dic83⋊C4—21no—
42C2×Dic83⋊C4—21no—
43C2×(C2×D83)⋊C4—21no—
44C2×(C2×D83)⋊C4—21no—
45C2×C4×Dic83C2×C411yescharacteristic
46C2×C4×Dic83D411yescharacteristic
47C2×C4×Dic83D411yescharacteristic
48C42×D83—42no—
49(C2×D83)⋊C8—24no—
50(C2×D83)⋊C8—42no—
51#14581702—42no—
52C83⋊#95—21no—
53C83⋊#95—21no—
54C22×Q8×C83C2×C411yescharacteristic
55C22×C8×C83—21no—
56C22×C8×C83—21no—
57C2×C4⋊C4×C83C2×C411yescharacteristic
58C2×C42×C83C2311yescharacteristic
59#95×C83—21no—
60#95×C83—21no—
61C23×D83C22⋊C411yescharacteristic
62C2×Dic166—21no—
63C2×Dic166—21no—
64C2×Dic166—41no—
65C2×Dic166—41no—
66C22×Dic83—21no—
67C22×Dic83—21no—
68C22×Dic83C22⋊C411yescharacteristic
69C22×Dic83C22⋊C411yescharacteristic
70C22×Dic83C22⋊C411yescharacteristic
71C2×C4×D83—22no—
72C2×C4×D83—21no—
73C2×C4×D83—21no—
74C2×C4×D83—21no—
75C2×C4×D83—21no—
76C2×C4×D83—21no—
77C2×C4×D83—21no—
78C2×C4×D83—22no—
79C2×C4×D83—22no—
80C2×C4×D83—41no—
81C2×C4×D83—41no—
82C2×C4×D83—41no—
83C2×C83⋊C8—22no—
84C2×C83⋊C8—41no—
85C2×C83⋊C8—41no—
86C2×C83⋊C8—41no—
87C2×C83⋊C8—41no—
88C2×C83⋊C8—41no—
89C4⋊Dic83—42no—
90C4⋊Dic83—41no—
91Dic83⋊C4—42no—
92Dic83⋊C4—42no—
93Dic83⋊C4—42no—
94Dic83⋊C4—42no—
95Dic83⋊C4—42no—
96(C2×D83)⋊C4—42no—
97(C2×D83)⋊C4—42no—
98C4×Dic83—22no—
99C4×Dic83—22no—
100C4×Dic83—41no—
101C2×Q8×C83—21no—
102C2×Q8×C83—21no—
103C2×Q8×C83—41no—
104C2×Q8×C83—41no—
105C22×C4×C83—21no—
106C22×C4×C83—21no—
107C22×C4×C83C22×C411yescharacteristic
108C22×C4×C83C2×D411yescharacteristic
109C22×C4×C83C2×D411yescharacteristic
110C2×C8×C83—22no—
111C2×C8×C83—41no—
112C2×C8×C83—41no—
113C2×C8×C83—41no—
114C2×C8×C83—41no—
115C2×C8×C83—41no—
116C4⋊C4×C83—41no—
117C42×C83—41no—
118C23×C83C2×C22⋊C411yescharacteristic
119C22×D83—21no—
120C22×D83—21no—
121C22×D83—21no—
122C22×D83—21no—
123C22×D83—21no—
124C22×D83—21no—
125C22×D83#10012yes—
126Q8×C83—81no—
127Q8×C83—81no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83#13111yescharacteristic
141C2×C4×C83#13111yescharacteristic
142C2×C4×C83#13711yescharacteristic
143C2×C4×C83#13711yescharacteristic
144C2×C4×C83#13711yescharacteristic
145C2×C4×C83#13711yescharacteristic
146C2×Dic83—21no—
147C2×Dic83—21no—
148C2×Dic83—21no—
149C2×Dic83—21no—
150C2×Dic83—21no—
151C2×Dic83—21no—
152C2×Dic83—21no—
153C2×Dic83—41no—
154C2×Dic83—41no—
155C2×Dic83—41no—
156C2×Dic83—41no—
157C2×Dic83—41no—
158C2×Dic83—41no—
159C2×Dic83#9912yes—
160C2×Dic83#10412yes—
161C4×D83—42no—
162C4×D83—42no—
163C4×D83—42no—
164C4×D83—42no—
165C4×D83—42no—
166C4×D83—42no—
167Dic166—81no—
168Dic166—81no—
169C8×C83—42no—
170C8×C83—81no—
171C83⋊C8—42no—
172C83⋊C8—81no—
173C22×C83#40811yescharacteristic
174C22×C83C2×#9911yescharacteristic
175C22×C83C2×#10011yescharacteristic
176C22×C83#41911yescharacteristic
177C22×C83#41911yescharacteristic
178C22×C83C2×#10411yescharacteristic
179C22×C83#42111yescharacteristic
180C2×D83—22no—
181C2×D83—22no—
182C2×D83—22no—
183C2×D83—41no—
184C2×D83—41no—
185C2×D83—41no—
186C2×D83—41no—
187C4×C83—21no—
188C4×C83—21no—
189C4×C83—21no—
190C4×C83—21no—
191C4×C83—21no—
192C4×C83—21no—
193C4×C83—81no—
194C4×C83—81no—
195Dic83—22no—
196Dic83—22no—
197Dic83—41no—
198Dic83—81no—
199Dic83—81no—
200#7707—831nomaximal, Sylow
201C2×C83#147111yescharacteristic
202C2×C83#148511yescharacteristic
203C2×C83#149711yescharacteristic
204C2×C83#150811yescharacteristic
205C2×C83#151111yescharacteristic
206C2×C83#153011yescharacteristic
207C2×C83#155811yescharacteristic
208D83—42no—
209#2857—831no—
210#1817—831no—
211#1749—831no—
212#1272—831no—
213#1272—831no—
214#1272—831no—
215#1272D8311yescharacteristic, complemented
216C83#770711yescharacteristic, complemented, Sylow
217C2×C4⋊Q8—831no—
218C2×C22⋊Q8—831no—
219C2×C22⋊Q8—831no—
220C22×C42—831no—
221C2×C22⋊C8—1661no—
222C2×C22⋊C8—1661no—
223C2×#97—1661no—
224#391—831no—
225#391C2×D8311yescharacteristic
226#389—1661no—
227#389—1661no—
228#384—831no—
229#380—1661no—
230#380—1661no—
231#336—831no—
232#336—831no—
233#336—1661no—
234#336C2×D8311yescharacteristic
235#336C2×D8311yescharacteristic
236C22×Q8—831no—
237C22×Q8C4×D8311yescharacteristic
238C23×C4—831no—
239C23×C4—831no—
240C23×C4—831no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—1661no—
244C22×C8—1661no—
245C4⋊Q8—3322no—
246C22⋊Q8—3322no—
247C22⋊Q8—3322no—
248C2×C4⋊C4—831no—
249C2×C4⋊C4—1661no—
250C2×C4⋊C4—1661no—
251C2×C4⋊C4—1661no—
252C2×C4⋊C4—1661no—
253C2×C4⋊C4—1661no—
254C2×C4⋊C4—1661no—
255C2×C4⋊C4C4×D8311yescharacteristic
256C2×C22⋊C4—1661no—
257C2×C22⋊C4—1661no—
258C2×C42—3322no—
259C2×C42—831no—
260C2×C42—831no—
261C2×C42—831no—
262C2×C42C22×D8311yescharacteristic
263C22⋊C8—3322no—
264C22⋊C8—1664no—
265#97—3322no—
266#95—21no—
267#95—21no—
268#95—1661no—
269#95—1661no—
270C24—831no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—3321no—
276C2×Q8—3321no—
277C2×Q8—1661no—
278C2×Q8—1661no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—3321no—
282C22×C4—3321no—
283C22×C4—3321no—
284C22×C4—831no—
285C22×C4—831no—
286C22×C4—831no—
287C22×C4—1662no—
288C22×C4—1662no—
289C22×C4—1662no—
290C22×C4—1661no—
291C22×C4—1661no—
292C22×C4—1661no—
293C22×C4—1661no—
294C22×C4—1661no—
295C22×C4—1661no—
296C22×C4—1661no—
297C22×C4—1661no—
298C22×C4C2×C4×D8311yescharacteristic
299C22×C4D4×D8311yescharacteristic
300C22×C4D4×D8311yescharacteristic
301C2×C8—22no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—3321no—
308C2×C8—3321no—
309C2×C8—3321no—
310C2×C8—3321no—
311C2×C8—3321no—
312C2×C8—1662no—
313C4⋊C4—41no—
314C4⋊C4—3321no—
315C4⋊C4—3322no—
316C4⋊C4—3322no—
317C4⋊C4—3322no—
318C4⋊C4—3322no—
319C4⋊C4—3322no—
320C4⋊C4—3322no—
321C22⋊C4—3322no—
322C22⋊C4—3322no—
323C42—41no—
324C42—3321no—
325C42—1662no—
326C42—1662no—
327C23—832no—
328C23—1661no—
329C23—1661no—
330C23—1661no—
331C23—1661no—
332C23—1661no—
333C23—1661no—
334C23C22⋊C4×D8311yescharacteristic
335Q8—81no—
336Q8—81no—
337Q8—6641no—
338Q8—6641no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—3322no—
352C2×C4—3322no—
353C2×C4—3322no—
354C2×C4—3322no—
355C2×C4—3322no—
356C2×C4—3322no—
357C2×C4—3321no—
358C2×C4—3321no—
359C2×C4—3321no—
360C2×C4—3321no—
361C2×C4—3321no—
362C2×C4—3321no—
363C2×C4—832no—
364C2×C4—832no—
365C2×C4—1661no—
366C2×C4—1661no—
367C2×C4—1661no—
368C2×C4—1661no—
369C2×C4—1661no—
370C2×C4—1661no—
371C2×C4—1661no—
372C2×C4#1458179811yescharacteristic
373C2×C4#1458180011yescharacteristic
374C2×C4#1458182111yescharacteristic
375C2×C4#1458182111yescharacteristic
376C2×C4#1458182411yescharacteristic
377C2×C4#1458182411yescharacteristic
378C8—42no—
379C8—81no—
380C8—3322no—
381C8—6641no—
382C22—3321no—
383C22—3321no—
384C22—3321no—
385C22—3321no—
386C22—1662no—
387C22—1662no—
388C22—1662no—
389C22#1912766611yescharacteristic
390C22#99×D8311yescharacteristic
391C22#100×D8311yescharacteristic
392C22#1912770811yescharacteristic
393C22#1912770811yescharacteristic
394C22#104×D8311yescharacteristic
395C22#1912771611yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—81no—
403C4—81no—
404C4—3321no—
405C4—6641no—
406C4—6641no—
407C4—1662no—
408C4—1662no—
409C2—3322no—
410C2#2490582111yescharacteristic
411C2#2490588211yescharacteristic
412C2#2490593411yescharacteristic
413C2#2490598111yescharacteristic
414C2#2490599411yescharacteristic
415C2#2490607211yescharacteristic
416C2#2490618711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D83
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1272C2
socleC23×C83C2×C22⋊C4
prime radicalC23×D83C22⋊C4
cosocleC2×C42×C83C23
hypercenter#1272D83
hypocenterC83#7707
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
4. #24905289C2semidirect
5. #24905288C2semidirect
6. #24905288C2semidirect
7. C83×#1272C2semidirect
215. #1272D83semidirect
216. C83#7707semidirect

Maximal subgroups

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

subgroupindex
#249158122
#249075302
#249072242
#249052892
#249052882
C83×#12722
#770783