Finite Group Databasebeta

← #35314930

Subgroups — #35314930

Order 21248.

420 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 91 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#24907202C211yesmaximal, characteristic
2#24905797C211yesmaximal, characteristic
3#24905490C212yesmaximal, complemented
4#24905469C212yesmaximal, complemented
5C83×#2611C211yesmaximal, characteristic
6C22×#13050574—21no—
7#19127758—21no—
8#19127470—21no—
9#19127470C2212yes—
10#19127427C2212yes—
11#19127425—21no—
12C2×#125×C83C411yescharacteristic
13C2×#117×C83C2211yescharacteristic
14#400×C83C2211yescharacteristic
15#399×C83C411yescharacteristic
16#398×C83C2211yescharacteristic
17#397×C83C411yescharacteristic
18#389×C83—21no—
19#388×C83—21no—
20#387×C83—21no—
21#385×C83—21no—
22#376×C83C411yescharacteristic
23C2×#13050574—24no—
24C2×#13050574—22no—
25C2×#13050574—41no—
26C2×#13050574—42no—
27C22×C83⋊C8—41no—
28C4⋊(C83⋊C8)—41no—
29C22⋊(C83⋊C8)—22no—
30#14581703—41no—
31C23×C4×C83D411yescharacteristic
32#125×C83—41no—
33#125×C83—41no—
34#117×C83—41no—
35#117×C83—41no—
36C2×C4⋊C4×C83—21no—
37C2×C4⋊C4×C83—21no—
38C2×C4⋊C4×C83—21no—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83C2×C411yescharacteristic
41C2×C4⋊C4×C83C2×C411yescharacteristic
42C2×C4⋊C4×C83D411yescharacteristic
43C2×C4⋊C4×C83C2311yescharacteristic
44C2×C22⋊C4×C83—21no—
45C2×C22⋊C4×C83—21no—
46C2×C22⋊C4×C83C2×C411yescharacteristic
47C2×C22⋊C4×C83D411yescharacteristic
48C2×C42×C83—21no—
49C2×C42×C83C2×C411yescharacteristic
50C2×C42×C83D411yescharacteristic
51#95×C83—21no—
52#95×C83—21no—
53#95×C83—21no—
54#95×C83—21no—
55#95×C83—21no—
56#95×C83C2×C411yescharacteristic
57#95×C83C2×C411yescharacteristic
58#13050574—24no—
59#13050574—44no—
60#13050574—42no—
61C2×C83⋊C8—44no—
62C2×C83⋊C8—42no—
63C2×C83⋊C8—42no—
64C24×C83C22⋊C411yescharacteristic
65C22×C4×C83—22no—
66C22×C4×C83—21no—
67C22×C4×C83—21no—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83C22⋊C411yescharacteristic
75C22×C4×C83C22⋊C411yescharacteristic
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83C22×C411yescharacteristic
78C22×C4×C83C2×D411yescharacteristic
79C22×C4×C83C2×D411yescharacteristic
80C4⋊C4×C83—41no—
81C4⋊C4×C83—41no—
82C4⋊C4×C83—41no—
83C4⋊C4×C83—41no—
84C4⋊C4×C83—42no—
85C4⋊C4×C83—42no—
86C4⋊C4×C83—41no—
87C4⋊C4×C83—41no—
88C4⋊C4×C83—41no—
89C4⋊C4×C83—41no—
90C22⋊C4×C83—41no—
91C22⋊C4×C83—41no—
92C22⋊C4×C83—42no—
93C22⋊C4×C83—42no—
94C42×C83—21no—
95C42×C83—21no—
96C42×C83—21no—
97C42×C83—21no—
98C42×C83—42no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—41no—
102C23×C83—41no—
103C23×C83#10412yes—
104C23×C83C2×C22⋊C411yescharacteristic
105C2×C4×C83—22no—
106C2×C4×C83—22no—
107C2×C4×C83—22no—
108C2×C4×C83—22no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—22no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—41no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—41no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—41no—
130C2×C4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—41no—
133C2×C4×C83#9911yescharacteristic
134C2×C4×C83#9911yescharacteristic
135C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C83#13711yescharacteristic
137C83⋊C8—44no—
138C83⋊C8—82no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83—22no—
142C22×C83—42no—
143C22×C83—41no—
144C22×C83—41no—
145C22×C83—81no—
146C22×C83C2×#9911yescharacteristic
147C22×C83#41711yescharacteristic
148C22×C83C2×#10411yescharacteristic
149C4×C83—21no—
150C4×C83—21no—
151C4×C83—21no—
152C4×C83—21no—
153C4×C83—42no—
154C4×C83—41no—
155C4×C83—41no—
156C4×C83—41no—
157C4×C83—41no—
158C4×C83—81no—
159C4×C83—81no—
160C4×C83—81no—
161#11767—831nomaximal, Sylow
162C2×C83—21no—
163C2×C83—21no—
164C2×C83—42no—
165C2×C83#150411yescharacteristic
166C2×C83#212311yescharacteristic
167C2×C83#213211yescharacteristic
168#2611D8311yescharacteristic
169#1742—831no—
170#1464—831no—
171#1353—832no—
172#1346—832no—
173C83#1176711yescharacteristic, complemented, Sylow
174C22×M16—1661no—
175C2×#125Dic8311yescharacteristic
176C2×#117C2×D8311yescharacteristic
177#432—1661no—
178#400C2×D8311yescharacteristic
179#399Dic8311yescharacteristic
180#398C2×D8311yescharacteristic
181#397Dic8311yescharacteristic
182#389—21no—
183#388—21no—
184#387—21no—
185#385—21no—
186#376Dic8311yescharacteristic
187#343—832no—
188#343—1661no—
189#324—832no—
190#323—1661no—
191C23×C4C22⋊D8311yescharacteristic
192C2×M16—3321no—
193C2×M16—3322no—
194C2×M16—1664no—
195C2×M16—1662no—
196C22×C8—3321no—
197#125—41no—
198#125—41no—
199#117—41no—
200#117—41no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C2×Dic8311yescharacteristic
206C2×C4⋊C4C2×Dic8311yescharacteristic
207C2×C4⋊C4C22⋊D8311yescharacteristic
208C2×C4⋊C4C22×D8311yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4C2×Dic8311yescharacteristic
212C2×C22⋊C4C22⋊D8311yescharacteristic
213C2×C42—21no—
214C2×C42C2×Dic8311yescharacteristic
215C2×C42C22⋊D8311yescharacteristic
216C4⋊C8—3321no—
217C22⋊C8—1662no—
218#97—3321no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95C2×Dic8311yescharacteristic
225#95C2×Dic8311yescharacteristic
226C24C22⋊Dic8311yescharacteristic
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4C22⋊Dic8311yescharacteristic
237C22×C4C22⋊Dic8311yescharacteristic
238C22×C4C22⋊Dic8311yescharacteristic
239C22×C4C22×Dic8311yescharacteristic
240C22×C4C2×C22⋊D8311yescharacteristic
241C22×C4C2×C22⋊D8311yescharacteristic
242M16—3324no—
243M16—3322no—
244M16—1664no—
245C2×C8—3324no—
246C2×C8—3322no—
247C2×C8—3322no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—42no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23#1458171912yes—
272C23C2×C22⋊Dic8311yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4#1458170711yescharacteristic
302C2×C4#1458170711yescharacteristic
303C2×C4#1458181711yescharacteristic
304C2×C4#1458182211yescharacteristic
305C8—3324no—
306C8—6642no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—81no—
314C22C2×#1458170711yescharacteristic
315C22#1912770211yescharacteristic
316C22C2×#1458171911yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—42no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—81no—
327C4—81no—
328C4—81no—
329C2—21no—
330C2—21no—
331C2—42no—
332C2#2490596711yescharacteristic
333C2#2490911711yescharacteristic
334C2#2490914811yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#2611C2
socleC22×C83C2×#99
prime radicalC24×C83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#2611D83
hypocenterC83#11767
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
3. #24905490C2semidirect
4. #24905469C2semidirect
173. C83#11767semidirect

Maximal subgroups

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

subgroupindex
#249072022
#249057972
#249054902
#249054692
C83×#26112
#1176783