Finite Group Databasebeta

← #35293712

Subgroups — #35293712

Order 21248.

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

normal subgroups only

This group has 350 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#24915448C211yesmaximal, characteristic
2#24909025C211yesmaximal, characteristic
3#24909002C211yesmaximal, characteristic
4#24905288C211yesmaximal, characteristic
5#24905287C211yesmaximal, characteristic
6#24905286C211yesmaximal, characteristic
7C83×#2100C211yesmaximal, characteristic
8C2×#14581725—21no—
9C2×#14581715—21no—
10#19127643—21no—
11#19127632—21no—
12#19127631—21no—
13#19127607C2211yescharacteristic
14#19127605—21no—
15#19127599C2211yescharacteristic
16#19127587—21no—
17#19127582—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×Dic166D411yescharacteristic
28C22×C83⋊C8—21no—
29C22×C83⋊C8—21no—
30C2×C4⋊Dic83D411yescharacteristic
31C2×C4⋊Dic83D411yescharacteristic
32C2×C4⋊Dic83D411yescharacteristic
33C2×Dic83⋊C4—21no—
34C2×Dic83⋊C4—21no—
35C2×Dic83⋊C4—21no—
36C2×C4×Dic83—21no—
37#14581725—42no—
38#14581715—42no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95—21no—
43C83⋊#95—21no—
44C83⋊#95—21no—
45C83⋊#95—21no—
46C83⋊#95—21no—
47C22×C8×C83—21no—
48C2×C4⋊C4×C83—21no—
49C2×C4⋊C4×C83—21no—
50C2×C4⋊C4×C83—21no—
51C2×C4⋊C4×C83—21no—
52C2×C4⋊C4×C83D411yescharacteristic
53C2×C4⋊C4×C83D411yescharacteristic
54C2×C42×C83C2311yescharacteristic
55#95×C83—21no—
56C2×Dic166—21no—
57C2×Dic166—41no—
58C2×Dic166—41no—
59C2×Dic166Q1612yes—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63C22×Dic83—21no—
64C2×C83⋊C8—22no—
65C2×C83⋊C8—41no—
66C2×C83⋊C8—41no—
67C2×C83⋊C8—41no—
68C2×C83⋊C8—41no—
69C2×C83⋊C8—41no—
70C4⋊Dic83—21no—
71C4⋊Dic83—21no—
72C4⋊Dic83—41no—
73C4⋊Dic83—41no—
74Dic83⋊C4—41no—
75Dic83⋊C4—41no—
76Dic83⋊C4—81no—
77Dic83⋊C4—81no—
78C4×Dic83—81no—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—41no—
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83C2×D411yescharacteristic
85C2×C8×C83—41no—
86C2×C8×C83—41no—
87C2×C8×C83—41no—
88C4⋊C4×C83—21no—
89C4⋊C4×C83—21no—
90C4⋊C4×C83—41no—
91C4⋊C4×C83—42no—
92C4⋊C4×C83—42no—
93C4⋊C4×C83—41no—
94C4⋊C4×C83—41no—
95C4⋊C4×C83—41no—
96C4⋊C4×C83—41no—
97C42×C83—41no—
98C23×C83#12011yescharacteristic
99C2×C4×C83—21no—
100C2×C4×C83—21no—
101C2×C4×C83—21no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83—41no—
106C2×C4×C83—41no—
107C2×C4×C83—41no—
108C2×C4×C83—41no—
109C2×C4×C83—41no—
110C2×C4×C83—41no—
111C2×C4×C83—81no—
112C2×C4×C83—81no—
113C2×C4×C83—81no—
114C2×C4×C83C2×Q1611yescharacteristic
115C2×C4×C83#13511yescharacteristic
116C2×C4×C83#13511yescharacteristic
117C2×C4×C83#13711yescharacteristic
118C2×C4×C83#13711yescharacteristic
119C2×C4×C83#13711yescharacteristic
120C2×Dic83—22no—
121C2×Dic83—41no—
122C2×Dic83—41no—
123C2×Dic83—41no—
124C2×Dic83—41no—
125C2×Dic83—41no—
126C2×Dic83—41no—
127C2×Dic83—41no—
128C2×Dic83—41no—
129C2×Dic83—41no—
130C2×Dic83—41no—
131C2×Dic83—41no—
132Dic166—42no—
133Dic166—81no—
134C8×C83—81no—
135C83⋊C8—42no—
136C83⋊C8—81no—
137C22×C83#45111yescharacteristic
138C22×C83#45211yescharacteristic
139C22×C83#45211yescharacteristic
140C22×C83#45311yescharacteristic
141C22×C83#45411yescharacteristic
142C22×C83#45411yescharacteristic
143C22×C83#45811yescharacteristic
144C4×C83—21no—
145C4×C83—21no—
146C4×C83—21no—
147C4×C83—21no—
148C4×C83—21no—
149C4×C83—21no—
150C4×C83—81no—
151C4×C83—81no—
152C4×C83—161no—
153Dic83—42no—
154Dic83—81no—
155Dic83—81no—
156Dic83—81no—
157#7831—831nomaximal, Sylow
158C2×C83#161411yescharacteristic
159C2×C83#161511yescharacteristic
160C2×C83#161511yescharacteristic
161C2×C83#163411yescharacteristic
162C2×C83#163611yescharacteristic
163C2×C83#163611yescharacteristic
164C2×C83#165711yescharacteristic
165#2809—831no—
166#2100—831no—
167#2100D8311yescharacteristic
168#2095—831no—
169#1272—831no—
170#1271—831no—
171#1271—831no—
172C83#783111yescharacteristic, complemented, Sylow
173C2×#107—1661no—
174C2×#103—1661no—
175#400—1661no—
176#398—1661no—
177#398—1661no—
178#391—831no—
179#391—1661no—
180#389—831no—
181#389C2×D8311yescharacteristic
182#387—1661no—
183#387—1661no—
184#384C2×D8311yescharacteristic
185#340—21no—
186#340—21no—
187#340—1661no—
188#340—1661no—
189#336—831no—
190#336—831no—
191#336C2×D8311yescharacteristic
192C22×Q8—831no—
193C22×C8—21no—
194C22×C8—1661no—
195C22×C8—1661no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—831no—
201C2×C4⋊C4—831no—
202C2×C4⋊C4—831no—
203C2×C4⋊C4—1661no—
204C2×C4⋊C4—1661no—
205C2×C4⋊C4—1661no—
206C2×C4⋊C4C22⋊D8311yescharacteristic
207C2×C4⋊C4C22⋊D8311yescharacteristic
208C2×C42—1661no—
209C2×C42C22×D8311yescharacteristic
210#107—3322no—
211#103—3322no—
212#95—21no—
213#95—1661no—
214#95—1661no—
215#95—1661no—
216#95—1661no—
217#95—1661no—
218#95—1661no—
219#95—1661no—
220#95—1661no—
221C2×Q8—3321no—
222C2×Q8—3321no—
223C2×Q8—832no—
224C2×Q8—1661no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—41no—
228C22×C4—1661no—
229C22×C4—1661no—
230C22×C4—1661no—
231C22×C4—1661no—
232C22×C4D4×D8311yescharacteristic
233C22×C4D4×D8311yescharacteristic
234C22×C4C2×C22⋊D8311yescharacteristic
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—3321no—
239C2×C8—3321no—
240C2×C8—3321no—
241C2×C8—3321no—
242C2×C8—3321no—
243C2×C8—1662no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—3321no—
254C4⋊C4—3321no—
255C4⋊C4—3321no—
256C4⋊C4—3321no—
257C4⋊C4—6641no—
258C4⋊C4—6641no—
259C4⋊C4—1661no—
260C4⋊C4—1661no—
261C42—41no—
262C42—6641no—
263C23#1458176011yescharacteristic
264Q8—3322no—
265Q8—6641no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—3321no—
282C2×C4—3321no—
283C2×C4—3321no—
284C2×C4—3321no—
285C2×C4—3321no—
286C2×C4—3321no—
287C2×C4—3321no—
288C2×C4—3321no—
289C2×C4—3321no—
290C2×C4—3321no—
291C2×C4—3321no—
292C2×C4—1662no—
293C2×C4Q16×D8311yescharacteristic
294C2×C4#1458181211yescharacteristic
295C2×C4#1458181311yescharacteristic
296C2×C4#1458182111yescharacteristic
297C2×C4#1458182311yescharacteristic
298C2×C4#1458182411yescharacteristic
299C8—81no—
300C8—3322no—
301C8—6641no—
302C22#1912785511yescharacteristic
303C22#1912786011yescharacteristic
304C22#1912786111yescharacteristic
305C22#1912786711yescharacteristic
306C22#1912787211yescharacteristic
307C22#1912787311yescharacteristic
308C22#1912789011yescharacteristic
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—81no—
316C4—81no—
317C4—161no—
318C4—3322no—
319C4—6641no—
320C4—6641no—
321C4—6641no—
322C2#2490648611yescharacteristic
323C2#2490649111yescharacteristic
324C2#2490649211yescharacteristic
325C2#2490660711yescharacteristic
326C2#2490661811yescharacteristic
327C2#2490661911yescharacteristic
328C2#2490676011yescharacteristic
subgroupG/H
centerC23#14581760
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#2100C2
socle, prime radicalC23×C83#120
hypercenter#2100D83
hypocenterC83#7831
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
172. C83#7831semidirect

Maximal subgroups

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

subgroupindex
#249154482
#249090252
#249090022
#249052882
#249052872
#249052862
C83×#21002
#783183