Finite Group Databasebeta

← #35315153

Subgroups — #35315153

Order 21248.

386 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 79 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#24913276C211yesmaximal, characteristic
2#24907543C211yesmaximal, characteristic
3#24905494C212yesmaximal, complemented
4#24905491C212yesmaximal, complemented
5C83×#1463C211yesmaximal, characteristic
6C2×#14581771C411yescharacteristic
7#19127665—21no—
8#19127640C411yescharacteristic
9#19127640C2211yescharacteristic
10#19127634—21no—
11#19127634—21no—
12#19127621C411yescharacteristic
13#19127621C2211yescharacteristic
14#19127574—21no—
15#19127564—21no—
16#19127561C411yescharacteristic
17#19127470—21no—
18#19127470C2212yes—
19C22×C4⋊C4×C83C2211yescharacteristic
20#407×C83—21no—
21#324×C83—21no—
22#324×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic
24C2×#13050574—22no—
25#14581771—41no—
26#14581771—41no—
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83C2×C411yescharacteristic
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×Dic83⋊C4—21no—
32C2×C22⋊Dic83—21no—
33C2×C22⋊Dic83—21no—
34C22⋊(C83⋊C8)—41no—
35C22⋊(C83⋊C8)—41no—
36C83⋊#95—21no—
37C83⋊#95—21no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95C2×C411yescharacteristic
43C83⋊#95C2×C411yescharacteristic
44C23×C4×C83C2×C411yescharacteristic
45C23×C4×C83D411yescharacteristic
46C23×C4×C83D411yescharacteristic
47C2×M16×C83—41no—
48C2×C4⋊C4×C83—41no—
49C2×C4⋊C4×C83—41no—
50C2×C4⋊C4×C83C2×C411yescharacteristic
51C2×C4⋊C4×C83D411yescharacteristic
52C2×C4⋊C4×C83D411yescharacteristic
53C2×C4⋊C4×C83C2311yescharacteristic
54C22⋊C8×C83—22no—
55C22⋊C8×C83—41no—
56C2×Dic166—41no—
57C2×Dic166—81no—
58C22×Dic83—21no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62#13050574—42no—
63#13050574—42no—
64C2×C83⋊C8—42no—
65C4⋊Dic83—41no—
66C4⋊Dic83—81no—
67Dic83⋊C4—42no—
68Dic83⋊C4—81no—
69Dic83⋊C4—81no—
70C22⋊Dic83—42no—
71C22⋊Dic83—41no—
72C22⋊Dic83—41no—
73C24×C83C22⋊C411yescharacteristic
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—22no—
81C22×C4×C83—41no—
82C22×C4×C83—81no—
83C22×C4×C83—81no—
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22×C411yescharacteristic
88C22×C4×C83C2×D411yescharacteristic
89C22×C4×C83C2×D411yescharacteristic
90M16×C83—82no—
91C2×C8×C83—42no—
92C4⋊C4×C83—41no—
93C4⋊C4×C83—41no—
94C4⋊C4×C83—41no—
95C4⋊C4×C83—41no—
96C23×C83—21no—
97C23×C83—21no—
98C23×C83—41no—
99C23×C83—41no—
100C23×C83#9911yescharacteristic
101C23×C83#9911yescharacteristic
102C23×C83C2×C22⋊C411yescharacteristic
103C2×C4×C83—21no—
104C2×C4×C83—21no—
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—21no—
112C2×C4×C83—42no—
113C2×C4×C83—42no—
114C2×C4×C83—42no—
115C2×C4×C83—41no—
116C2×C4×C83—41no—
117C2×C4×C83—41no—
118C2×C4×C83—41no—
119C2×C4×C83—81no—
120C2×C4×C83—81no—
121C2×C4×C83—81no—
122C2×C4×C83#13712yes—
123C2×Dic83—41no—
124C2×Dic83—41no—
125C2×Dic83—41no—
126C2×Dic83—41no—
127C2×Dic83—41no—
128C2×Dic83—41no—
129C2×Dic83—42no—
130C2×Dic83—41no—
131C2×Dic83—41no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134Dic166—82no—
135C8×C83—82no—
136C83⋊C8—82no—
137C22×C83—21no—
138C22×C83—21no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83—21no—
142C22×C83—21no—
143C22×C83—21no—
144C22×C83—21no—
145C22×C83—42no—
146C22×C83—42no—
147C22×C83C2×#9911yescharacteristic
148C22×C83#41911yescharacteristic
149C22×C83#42111yescharacteristic
150C4×C83—22no—
151C4×C83—41no—
152C4×C83—41no—
153C4×C83—41no—
154C4×C83—41no—
155C4×C83—41no—
156Dic83—81no—
157Dic83—81no—
158Dic83—81no—
159Dic83—81no—
160#11817—831nomaximal, Sylow
161C2×C83—21no—
162C2×C83—21no—
163C2×C83—41no—
164C2×C83—41no—
165C2×C83#151111yescharacteristic
166C2×C83#212111yescharacteristic
167C2×C83#213311yescharacteristic
168#2627—831no—
169#1821—831no—
170#1463D8311yescharacteristic
171#1354—832no—
172#1353—832no—
173C83#1181711yescharacteristic, complemented, Sylow
174C2×C22⋊Q8—831no—
175C22×C4⋊C4C2×D8311yescharacteristic
176#407—21no—
177#407—1661no—
178#400—831no—
179#400—831no—
180#398—1661no—
181#398—1661no—
182#395—831no—
183#395—831no—
184#385—1661no—
185#381—1661no—
186#380—831no—
187#343—832no—
188#343—1661no—
189#324—21no—
190#324C2×D8312yes—
191C22×Q8—831no—
192C23×C4C4×D8311yescharacteristic
193C23×C4D33211yescharacteristic
194C23×C4C22⋊D8311yescharacteristic
195C2×M16—41no—
196C2×M16—1662no—
197C22⋊Q8—3321no—
198C22⋊Q8—3321no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—831no—
202C2×C4⋊C4—1661no—
203C2×C4⋊C4—1661no—
204C2×C4⋊C4—1661no—
205C2×C4⋊C4—1661no—
206C2×C4⋊C4C4×D8311yescharacteristic
207C2×C4⋊C4D33211yescharacteristic
208C2×C4⋊C4C22⋊D8311yescharacteristic
209C2×C4⋊C4C22×D8311yescharacteristic
210C2×C22⋊C4—1661no—
211C2×C22⋊C4—1661no—
212C22⋊C8—22no—
213C22⋊C8—41no—
214C22⋊C8—3321no—
215C22⋊C8—3321no—
216#95—831no—
217#95—831no—
218#95—1661no—
219#95—1661no—
220#95—1661no—
221#95—1661no—
222#95—1661no—
223#95—1661no—
224C24(C2×D83)⋊C411yescharacteristic
225C2×Q8—3321no—
226C2×Q8—6641no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—1661no—
238C22×C4—1661no—
239C22×C4—1661no—
240C22×C4—1661no—
241C22×C4(C2×D83)⋊C411yescharacteristic
242C22×C4(C2×D83)⋊C411yescharacteristic
243C22×C4(C2×D83)⋊C411yescharacteristic
244C22×C4C2×C4×D8311yescharacteristic
245C22×C4C2×C22⋊D8311yescharacteristic
246C22×C4C2×D33211yescharacteristic
247M16—82no—
248M16—3322no—
249M16—3322no—
250C2×C8—42no—
251C2×C8—3322no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—3321no—
257C4⋊C4—3322no—
258C4⋊C4—6641no—
259C4⋊C4—6641no—
260C4⋊C4—6641no—
261C22⋊C4—3322no—
262C22⋊C4—3321no—
263C22⋊C4—3321no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23#1458170611yescharacteristic
269C23#1458170611yescharacteristic
270C23C2×(C2×D83)⋊C411yescharacteristic
271Q8—6642no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—3321no—
292C2×C4—3321no—
293C2×C4—3321no—
294C2×C4—3321no—
295C2×C4—3321no—
296C2×C4—3321no—
297C2×C4—3322no—
298C2×C4—3321no—
299C2×C4—3321no—
300C2×C4—3321no—
301C2×C4—3321no—
302C2×C4#1458182512yes—
303C8—82no—
304C8—6642no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×#1458170611yescharacteristic
316C22#1912771111yescharacteristic
317C22#1912771711yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—6641no—
325C4—6641no—
326C4—6641no—
327C4—6641no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2#2490599711yescharacteristic
333C2#2490910711yescharacteristic
334C2#2490915011yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1463C2
socleC22×C83C2×#99
prime radicalC24×C83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1463D83
hypocenterC83#11817
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #24905494C2semidirect
4. #24905491C2semidirect
173. C83#11817semidirect

Maximal subgroups

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

subgroupindex
#249132762
#249075432
#249054942
#249054912
C83×#14632
#1181783