Finite Group Databasebeta

← #35315044

Subgroups — #35315044

Order 21248.

378 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 83 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#24912825C211yesmaximal, characteristic
2C4×#14581711C211yesmaximal, characteristic
3#24905494C212yesmaximal
4#24905491C212yesmaximal
5C83×#1467C211yesmaximal, characteristic
6C2×C4×Dic166C411yescharacteristic
7C2×#14581711—21no—
8C4×#13050574—21no—
9#19127640C411yescharacteristic
10#19127640C2211yescharacteristic
11#19127621C411yescharacteristic
12#19127621C2211yescharacteristic
13#19127608—21no—
14#19127597—21no—
15#19127585—21no—
16#19127585—21no—
17#19127567C411yescharacteristic
18#19127470C2212yes—
19C2×C4×Q8×C83C2211yescharacteristic
20#432×C83—21no—
21#324×C83—21no—
22#324×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic
24C2×#13050574—22no—
25C4×Dic166—41no—
26C4×Dic166—41no—
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83C2×C411yescharacteristic
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×Dic83⋊C4—21no—
32C2×C4×Dic83—21no—
33#14581711—24no—
34#14581703—41no—
35C4×C83⋊C8—41no—
36C83⋊#95—21no—
37C83⋊#95—21no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95—21no—
43C83⋊#95C2×C411yescharacteristic
44C83⋊#95C2×C411yescharacteristic
45C22×Q8×C83D411yescharacteristic
46C4×Q8×C83—41no—
47C4×Q8×C83—41no—
48C2×C4⋊C4×C83C2×C411yescharacteristic
49C2×C4⋊C4×C83D411yescharacteristic
50C2×C4⋊C4×C83C2311yescharacteristic
51C2×C42×C83C2×C411yescharacteristic
52C2×C42×C83D411yescharacteristic
53C2×C42×C83D411yescharacteristic
54C4⋊C8×C83—41no—
55C22⋊C8×C83—22no—
56#97×C83—41no—
57C2×Dic166—41no—
58C2×Dic166—42no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63#13050574—24no—
64#13050574—42no—
65C2×C83⋊C8—42no—
66C4⋊Dic83—42no—
67C4⋊Dic83—41no—
68Dic83⋊C4—42no—
69Dic83⋊C4—42no—
70Dic83⋊C4—41no—
71Dic83⋊C4—41no—
72C4×Dic83—42no—
73C2×Q8×C83—22no—
74C2×Q8×C83—22no—
75C2×Q8×C83—41no—
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83C22⋊C411yescharacteristic
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22×C411yescharacteristic
81C22×C4×C83C2×D411yescharacteristic
82C22×C4×C83C2×D411yescharacteristic
83C2×C8×C83—42no—
84C4⋊C4×C83—41no—
85C4⋊C4×C83—41no—
86C4⋊C4×C83—41no—
87C42×C83—21no—
88C42×C83—21no—
89C42×C83—21no—
90C42×C83—21no—
91C42×C83—21no—
92C42×C83—21no—
93C23×C83C2×C22⋊C411yescharacteristic
94Q8×C83—44no—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
97C2×C4×C83—21no—
98C2×C4×C83—21no—
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#9911yescharacteristic
110C2×C4×C83#9911yescharacteristic
111C2×C4×C83#10412yes—
112C2×C4×C83#13712yes—
113C2×Dic83—22no—
114C2×Dic83—41no—
115C2×Dic83—41no—
116C2×Dic83—41no—
117C2×Dic83—41no—
118C2×Dic83—41no—
119C2×Dic83—41no—
120C2×Dic83—42no—
121C2×Dic83—41no—
122C2×Dic83—41no—
123C2×Dic83—41no—
124Dic166—42no—
125Dic166—42no—
126C8×C83—82no—
127C83⋊C8—44no—
128C22×C83—21no—
129C22×C83—21no—
130C22×C83C2×#9911yescharacteristic
131C22×C83#41911yescharacteristic
132C22×C83C2×#10411yescharacteristic
133C4×C83—21no—
134C4×C83—21no—
135C4×C83—22no—
136C4×C83—22no—
137C4×C83—41no—
138C4×C83—41no—
139C4×C83—41no—
140C4×C83—41no—
141Dic83—42no—
142Dic83—81no—
143Dic83—81no—
144Dic83—81no—
145#11792—831nomaximal, Sylow
146C2×C83—21no—
147C2×C83—21no—
148C2×C83#150811yescharacteristic
149C2×C83#212011yescharacteristic
150C2×C83#213311yescharacteristic
151#2581—831no—
152C4×#101—831no—
153#1467D8311yescharacteristic
154#1354—832no—
155#1353—832no—
156C83#1179211yescharacteristic, complemented, Sylow
157C2×C4×Q8—831no—
158C2×C4×Q8C2×D8311yescharacteristic
159#432—21no—
160C2×#101—1661no—
161C4×M16—1661no—
162#400—831no—
163#400—831no—
164#395—831no—
165#395—831no—
166#391—1661no—
167#389—1661no—
168#387—1661no—
169#387—1661no—
170#382—831no—
171#343—832no—
172#324—21no—
173#324C2×D8312yes—
174C22×Q8—831no—
175C22×Q8D33211yescharacteristic
176C2×M16—1662no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×Q8—3321no—
180C4×Q8—3321no—
181C2×C4⋊C4—831no—
182C2×C4⋊C4—1661no—
183C2×C4⋊C4—1661no—
184C2×C4⋊C4—1661no—
185C2×C4⋊C4—1661no—
186C2×C4⋊C4C4×D8311yescharacteristic
187C2×C4⋊C4C22⋊D8311yescharacteristic
188C2×C4⋊C4C22×D8311yescharacteristic
189C2×C42—1661no—
190C2×C42C4×D8311yescharacteristic
191C2×C42D33211yescharacteristic
192C2×C42C22⋊D8311yescharacteristic
193C4⋊C8—41no—
194#101—1664no—
195C22⋊C8—22no—
196#97—41no—
197#97—3321no—
198C4×C8—3321no—
199#95—831no—
200#95—831no—
201#95—1661no—
202#95—1661no—
203#95—1661no—
204#95—1661no—
205#95—1661no—
206#95—1661no—
207#95—1661no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—41no—
211C2×Q8—3321no—
212C2×Q8—3322no—
213C22×C4—1661no—
214C22×C4—1661no—
215C22×C4—1661no—
216C22×C4—1661no—
217C22×C4(C2×D83)⋊C411yescharacteristic
218C22×C4(C2×D83)⋊C411yescharacteristic
219C22×C4(C2×D83)⋊C411yescharacteristic
220C22×C4(C2×D83)⋊C411yescharacteristic
221C22×C4C2×C4×D8311yescharacteristic
222C22×C4C2×C22⋊D8311yescharacteristic
223C22×C4C2×D33211yescharacteristic
224M16—3322no—
225M16—1664no—
226C2×C8—42no—
227C2×C8—3322no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—3321no—
232C4⋊C4—3322no—
233C4⋊C4—3322no—
234C4⋊C4—3321no—
235C4⋊C4—3321no—
236C4⋊C4—3322no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—3322no—
244C23C2×(C2×D83)⋊C411yescharacteristic
245Q8—44no—
246Q8—3322no—
247Q8—3322no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—3321no—
263C2×C4—3321no—
264C2×C4—3321no—
265C2×C4—3321no—
266C2×C4—3321no—
267C2×C4—3321no—
268C2×C4—3322no—
269C2×C4—3321no—
270C2×C4—3321no—
271C2×C4—3321no—
272C2×C4—1662no—
273C2×C4#1458170611yescharacteristic
274C2×C4#1458170611yescharacteristic
275C2×C4#1458171812yes—
276C2×C4#1458182512yes—
277C8—82no—
278C8—3324no—
279C22—21no—
280C22—21no—
281C22C2×#1458170611yescharacteristic
282C22#1912771111yescharacteristic
283C22C2×#1458171811yescharacteristic
284C4—22no—
285C4—22no—
286C4—21no—
287C4—21no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—3322no—
293C4—6641no—
294C4—6641no—
295C4—6641no—
296C2—21no—
297C2—21no—
298C2#2490598411yescharacteristic
299C2#2490910211yescharacteristic
300C2#2490915011yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1467C2
socleC22×C83C2×#99
prime radicalC23×C83C2×C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1467D83
hypocenterC83#11792
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
156. C83#11792semidirect

Maximal subgroups

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

subgroupindex
#249128252
C4×#145817112
#249054942
#249054912
C83×#14672
#1179283