Finite Group Databasebeta

← #35293703

Subgroups — #35293703

Order 21248.

394 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 87 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#24914494C211yesmaximal, characteristic
2#24909025C211yesmaximal, characteristic
3#24909002C211yesmaximal, characteristic
4#24905288C211yesmaximal, characteristic
5#24905286C211yesmaximal, characteristic
6#24905285C211yesmaximal, characteristic
7C83×#2100C211yesmaximal, characteristic
8C2×#14581791—21no—
9C2×#14581725—21no—
10C2×#14581715—21no—
11#19127634—21no—
12#19127622—21no—
13#19127621—21no—
14#19127607C2211yescharacteristic
15#19127599C2211yescharacteristic
16#19127587—21no—
17#19127584—21no—
18#19127463—21no—
19#19127463—21no—
20#19127456C2211yescharacteristic
21#19127456C2211yescharacteristic
22#389×C83C2211yescharacteristic
23#384×C83C2211yescharacteristic
24#340×C83—21no—
25#340×C83—21no—
26#336×C83C2211yescharacteristic
27C22×Dic166—21no—
28C22×Dic166D411yescharacteristic
29C22×C83⋊C8—21no—
30C22×C83⋊C8—21no—
31#14581791—42no—
32#14581791—42no—
33C2×C4⋊Dic83—21no—
34C2×C4⋊Dic83D411yescharacteristic
35C2×C4⋊Dic83D411yescharacteristic
36C2×C4⋊Dic83D411yescharacteristic
37C2×Dic83⋊C4—21no—
38C2×Dic83⋊C4—21no—
39C2×Dic83⋊C4—21no—
40C2×C4×Dic83—21no—
41#14581725—42no—
42#14581715—42no—
43C83⋊#95—21no—
44C83⋊#95—21no—
45C83⋊#95—21no—
46C83⋊#95—21no—
47C83⋊#95—21no—
48C83⋊#95—21no—
49C22×C8×C83—21no—
50C2×C4⋊C4×C83—21no—
51C2×C4⋊C4×C83—21no—
52C2×C4⋊C4×C83—21no—
53C2×C4⋊C4×C83—21no—
54C2×C4⋊C4×C83D411yescharacteristic
55C2×C4⋊C4×C83D411yescharacteristic
56C2×C42×C83C2311yescharacteristic
57#95×C83—21no—
58C2×Dic166—21no—
59C2×Dic166—41no—
60C2×Dic166—41no—
61C2×Dic166—42no—
62C2×Dic166—81no—
63C2×Dic166—81no—
64C2×Dic166SD1612yes—
65C22×Dic83—21no—
66C22×Dic83—21no—
67C22×Dic83—21no—
68C22×Dic83—21no—
69C2×C83⋊C8—22no—
70C2×C83⋊C8—41no—
71C2×C83⋊C8—41no—
72C2×C83⋊C8—41no—
73C2×C83⋊C8—41no—
74C2×C83⋊C8—41no—
75C4⋊Dic83—21no—
76C4⋊Dic83—42no—
77C4⋊Dic83—41no—
78C4⋊Dic83—41no—
79C4⋊Dic83Q1612yes—
80Dic83⋊C4—42no—
81Dic83⋊C4—41no—
82Dic83⋊C4—41no—
83Dic83⋊C4—81no—
84C4×Dic83—42no—
85C22×C4×C83—21no—
86C22×C4×C83—21no—
87C22×C4×C83—41no—
88C22×C4×C83C2×D411yescharacteristic
89C22×C4×C83C2×D411yescharacteristic
90C22×C4×C83C2×D411yescharacteristic
91C2×C8×C83—41no—
92C2×C8×C83—41no—
93C2×C8×C83—41no—
94C4⋊C4×C83—21no—
95C4⋊C4×C83—41no—
96C4⋊C4×C83—42no—
97C4⋊C4×C83—42no—
98C4⋊C4×C83—41no—
99C4⋊C4×C83—41no—
100C4⋊C4×C83—41no—
101C4⋊C4×C83—41no—
102C4⋊C4×C83Q1612yes—
103C42×C83—41no—
104C23×C83#12011yescharacteristic
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—22no—
112C2×C4×C83—41no—
113C2×C4×C83—41no—
114C2×C4×C83—41no—
115C2×C4×C83—41no—
116C2×C4×C83—41no—
117C2×C4×C83—81no—
118C2×C4×C83—81no—
119C2×C4×C83—81no—
120C2×C4×C83C2×SD1611yescharacteristic
121C2×C4×C83C2×Q1611yescharacteristic
122C2×C4×C83C2×Q1611yescharacteristic
123C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C83#13711yescharacteristic
125C2×C4×C83#13711yescharacteristic
126C2×Dic83—22no—
127C2×Dic83—22no—
128C2×Dic83—41no—
129C2×Dic83—41no—
130C2×Dic83—41no—
131C2×Dic83—41no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134C2×Dic83—41no—
135C2×Dic83—41no—
136C2×Dic83—41no—
137C2×Dic83—41no—
138Dic166—42no—
139Dic166—81no—
140Dic166—84no—
141C8×C83—81no—
142C83⋊C8—42no—
143C83⋊C8—81no—
144C22×C83#45011yescharacteristic
145C22×C83#45111yescharacteristic
146C22×C83#45111yescharacteristic
147C22×C83#45311yescharacteristic
148C22×C83#45411yescharacteristic
149C22×C83#45411yescharacteristic
150C22×C83#45811yescharacteristic
151C4×C83—21no—
152C4×C83—21no—
153C4×C83—21no—
154C4×C83—21no—
155C4×C83—21no—
156C4×C83—21no—
157C4×C83—42no—
158C4×C83—81no—
159C4×C83—161no—
160Dic83—42no—
161Dic83—42no—
162Dic83—81no—
163Dic83—81no—
164#7829—831nomaximal, Sylow
165C2×C83#161311yescharacteristic
166C2×C83#161711yescharacteristic
167C2×C83#161711yescharacteristic
168C2×C83#163311yescharacteristic
169C2×C83#163711yescharacteristic
170C2×C83#163711yescharacteristic
171C2×C83#165711yescharacteristic
172#2734—831no—
173#2100—831no—
174#2100D8311yescharacteristic
175#2095—831no—
176#1272—831no—
177#1271—831no—
178#1271—831no—
179C83#782911yescharacteristic, complemented, Sylow
180C2×C4⋊Q8—1661no—
181C2×#107—1661no—
182C2×#103—1661no—
183#398—1661no—
184#395—1661no—
185#395—1661no—
186#391—831no—
187#389—831no—
188#389C2×D8311yescharacteristic
189#387—1661no—
190#387—1661no—
191#384C2×D8311yescharacteristic
192#340—21no—
193#340—21no—
194#340—1661no—
195#340—1661no—
196#336—831no—
197#336—831no—
198#336C2×D8311yescharacteristic
199C22×Q8—831no—
200C22×Q8—1661no—
201C22×C8—21no—
202C22×C8—1661no—
203C22×C8—1661no—
204C4⋊Q8—3322no—
205C4⋊Q8—3322no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—831no—
211C2×C4⋊C4—831no—
212C2×C4⋊C4—831no—
213C2×C4⋊C4—1661no—
214C2×C4⋊C4—1661no—
215C2×C4⋊C4—1661no—
216C2×C4⋊C4—1661no—
217C2×C4⋊C4C22⋊D8311yescharacteristic
218C2×C4⋊C4C22⋊D8311yescharacteristic
219C2×C42—1661no—
220C2×C42C22×D8311yescharacteristic
221#107—3322no—
222#103—3322no—
223#95—21no—
224#95—1661no—
225#95—1661no—
226#95—1661no—
227#95—1661no—
228#95—1661no—
229#95—1661no—
230C2×Q8—3321no—
231C2×Q8—3321no—
232C2×Q8—3322no—
233C2×Q8—832no—
234C2×Q8—6641no—
235C2×Q8—6641no—
236C2×Q8—1661no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—41no—
240C22×C4—1661no—
241C22×C4—1661no—
242C22×C4—1661no—
243C22×C4—1661no—
244C22×C4D4×D8311yescharacteristic
245C22×C4D4×D8311yescharacteristic
246C22×C4C2×C22⋊D8311yescharacteristic
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—3321no—
251C2×C8—3321no—
252C2×C8—3321no—
253C2×C8—3321no—
254C2×C8—3321no—
255C2×C8—1662no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—3321no—
265C4⋊C4—3321no—
266C4⋊C4—3322no—
267C4⋊C4—3322no—
268C4⋊C4—3321no—
269C4⋊C4—3321no—
270C4⋊C4—832no—
271C4⋊C4—6641no—
272C4⋊C4—1661no—
273C4⋊C4#1305058012yes—
274C42—41no—
275C42—3322no—
276C23#1458176011yescharacteristic
277Q8—3322no—
278Q8—6641no—
279Q8—6644no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—3321no—
296C2×C4—3321no—
297C2×C4—3321no—
298C2×C4—3321no—
299C2×C4—3321no—
300C2×C4—3321no—
301C2×C4—3321no—
302C2×C4—3321no—
303C2×C4—3321no—
304C2×C4—3321no—
305C2×C4—1662no—
306C2×C4—1662no—
307C2×C4SD16×D8311yescharacteristic
308C2×C4Q16×D8311yescharacteristic
309C2×C4C2×#1305058011yescharacteristic
310C2×C4#1458181911yescharacteristic
311C2×C4#1458182211yescharacteristic
312C2×C4#1458182411yescharacteristic
313C8—81no—
314C8—3322no—
315C8—6641no—
316C22#1912785011yescharacteristic
317C22#1912785511yescharacteristic
318C22#1912785611yescharacteristic
319C22#1912786711yescharacteristic
320C22#1912787211yescharacteristic
321C22#1912787311yescharacteristic
322C22#1912789011yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—42no—
330C4—81no—
331C4—161no—
332C4—3322no—
333C4—3322no—
334C4—6641no—
335C4—6641no—
336C2#2490648111yescharacteristic
337C2#2490650311yescharacteristic
338C2#2490650411yescharacteristic
339C2#2490660211yescharacteristic
340C2#2490662511yescharacteristic
341C2#2490662711yescharacteristic
342C2#2490676011yescharacteristic
subgroupG/H
centerC23#14581760
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#2100C2
socle, prime radicalC23×C83#120
hypercenter#2100D83
hypocenterC83#7829
LCSG ▷ C2×C42×C83 ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2100

Complements

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

NH ≅ G/Ntype
179. C83#7829semidirect

Maximal subgroups

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

subgroupindex
#249144942
#249090252
#249090022
#249052882
#249052862
#249052852
C83×#21002
#782983