Finite Group Databasebeta

← #35302091

Subgroups — #35302091

Order 21248.

498 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 103 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#24916492C211yesmaximal, characteristic
2#24912636C211yesmaximal, characteristic
3#24906181C212yesmaximal, complemented
4#24905318C212yesmaximal, complemented
5C83×#1466C211yesmaximal, characteristic
6C2×#14581792C2211yescharacteristic
7#19127773—24no—
8#19127758C2212yes—
9#19127665—21no—
10C2×C22⋊(C83⋊C8)—21no—
11#19127613—21no—
12#19127613—21no—
13#19127604C412yes—
14#19127566C412yes—
15#19127561—21no—
16#19127561—21no—
17C4×C4⋊Dic83C2211yescharacteristic
18#19127448—24no—
19C2×C4×D4×C83C2211yescharacteristic
20#432×C83C2212yes—
21#323×C83—21no—
22#323×C83—21no—
23C22×Dic166—21no—
24C2×#13050574—22no—
25C22×C83⋊C8—22no—
26#14581792—22no—
27#14581792—22no—
28C2×C4⋊Dic83—21no—
29C2×C4⋊Dic83—21no—
30C2×C4⋊Dic83C2×C412yes—
31C2×C22⋊Dic83—22no—
32C2×C22⋊Dic83—22no—
33C2×C4×Dic83C2×C412yes—
34C4⋊(C83⋊C8)D414yes—
35C22⋊(C83⋊C8)—22no—
36C22⋊(C83⋊C8)—22no—
37C22⋊(C83⋊C8)—22no—
38#14581703—22no—
39C83⋊#95—22no—
40C83⋊#95—22no—
41C83⋊#95—21no—
42C22×D4×C83C2×C411yescharacteristic
43C23×C4×C83D411yescharacteristic
44C23×C4×C83D411yescharacteristic
45C4×D4×C83—22no—
46C4×D4×C83—22no—
47C2×C4⋊C4×C83C2×C411yescharacteristic
48C2×C22⋊C4×C83D411yescharacteristic
49C2×C22⋊C4×C83D411yescharacteristic
50C2×C42×C83C2311yescharacteristic
51C4⋊C8×C83—22no—
52C22⋊C8×C83—22no—
53#97×C83—22no—
54C2×Dic166—42no—
55C2×Dic166—82no—
56C22×Dic83—22no—
57C22×Dic83—22no—
58#13050574—82no—
59C2×C83⋊C8—24no—
60C2×C83⋊C8—82no—
61C4⋊Dic83—24no—
62C4⋊Dic83—42no—
63C4⋊Dic83—41no—
64C4⋊Dic83—41no—
65C22⋊Dic83—82no—
66C22⋊Dic83—82no—
67C4×Dic83—42no—
68C24×C83C22⋊C411yescharacteristic
69C24×C83C22⋊C411yescharacteristic
70C2×D4×C83—22no—
71C2×D4×C83—22no—
72C2×D4×C83—41no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—42no—
78C22×C4×C83—42no—
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22×C411yescharacteristic
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C2×C8×C83—42no—
85C4⋊C4×C83—22no—
86C22⋊C4×C83—22no—
87C22⋊C4×C83—22no—
88C42×C83C2×D412yes—
89C42×C83#4112yes—
90C23×C83—21no—
91C23×C83—21no—
92C23×C83—21no—
93C23×C83—21no—
94C23×C83—21no—
95C23×C83—41no—
96C23×C83—41no—
97C23×C83—41no—
98C23×C83—41no—
99C23×C83#10112yes—
100C23×C83C2×C22⋊C411yescharacteristic
101D4×C83—44no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—22no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—41no—
108C2×C4×C83—42no—
109C2×C4×C83—42no—
110C2×C4×C83—41no—
111C2×C4×C83—41no—
112C2×C4×C83—81no—
113C2×C4×C83—81no—
114C2×C4×C83C4×D412yes—
115C2×C4×C83#12011yescharacteristic
116C2×C4×C83#12112yes—
117C2×C4×C83#12311yescharacteristic
118C2×Dic83—42no—
119C2×Dic83—42no—
120C2×Dic83—42no—
121C2×Dic83—42no—
122C2×Dic83—42no—
123C2×Dic83—42no—
124Dic166—84no—
125C8×C83—82no—
126C83⋊C8—44no—
127C22×C83—21no—
128C22×C83—21no—
129C22×C83—22no—
130C22×C83—42no—
131C22×C83—42no—
132C22×C83—41no—
133C22×C83—41no—
134C22×C83—41no—
135C22×C83—41no—
136C22×C83—41no—
137C22×C83—81no—
138C22×C83—81no—
139C22×C83#38611yescharacteristic
140C22×C83C2×#10111yescharacteristic
141C22×C83#41311yescharacteristic
142C4×C83—21no—
143C4×C83—22no—
144C4×C83—22no—
145C4×C83—41no—
146C4×C83—41no—
147C4×C83#45512yes—
148Dic83—82no—
149Dic83—82no—
150#9453—831nomaximal, Sylow
151C2×C83—21no—
152C2×C83—21no—
153C2×C83—42no—
154C2×C83—81no—
155C2×C83#188511yescharacteristic
156C2×C83#190111yescharacteristic
157C2×C83#191011yescharacteristic
158#2973—831no—
159#2563—831no—
160#1556—832no—
161#1466D8311yescharacteristic
162#1285—832no—
163C83#945311yescharacteristic, complemented, Sylow
164C2×C4⋊Q8—831no—
165C2×C4×D4C2×D8311yescharacteristic
166#435—1664no—
167#432—832no—
168#432C2×D8312yes—
169#407—1661no—
170C2×C22⋊C8—1661no—
171#393—1661no—
172#393—1661no—
173#390—832no—
174#382—832no—
175#380—1661no—
176#380—1661no—
177C4×C4⋊C4—831no—
178#332—1664no—
179#323—21no—
180#323—21no—
181C22×Q8—1661no—
182C22×D4C4×D8311yescharacteristic
183C23×C4D33211yescharacteristic
184C23×C4D33211yescharacteristic
185C2×M16—1662no—
186C22×C8—1662no—
187C4⋊Q8—1662no—
188C4⋊Q8—1662no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—832no—
192C2×C4⋊C4—1661no—
193C2×C4⋊C4—1661no—
194C2×C4⋊C4C4×D8311yescharacteristic
195C2×C22⋊C4—1662no—
196C2×C22⋊C4—1662no—
197C2×C22⋊C4C22⋊D8311yescharacteristic
198C2×C22⋊C4C22⋊D8311yescharacteristic
199C2×C42—832no—
200C2×C42C22×D8311yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—834no—
203C22⋊C8—22no—
204C22⋊C8—1662no—
205C22⋊C8—1662no—
206C22⋊C8—1662no—
207#97—22no—
208#97—1662no—
209#95—1662no—
210#95—1662no—
211#95—1661no—
212C24(C2×D83)⋊C411yescharacteristic
213C24(C2×D83)⋊C411yescharacteristic
214C2×Q8—3322no—
215C2×Q8—6642no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—1662no—
226C22×C4—1662no—
227C22×C4(C2×D83)⋊C411yescharacteristic
228C22×C4(C2×D83)⋊C411yescharacteristic
229C22×C4C2×C4×D8311yescharacteristic
230C22×C4C2×C22⋊D8311yescharacteristic
231C22×C4C2×D33211yescharacteristic
232M16—6642no—
233C2×C8—42no—
234C2×C8—6642no—
235C2×C8—1664no—
236C4⋊C4—22no—
237C4⋊C4—3322no—
238C4⋊C4—3321no—
239C4⋊C4—3321no—
240C4⋊C4—1664no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—6642no—
244C22⋊C4—6642no—
245C42—3322no—
246C42D4×D8312yes—
247C42#1305058912yes—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23#1458171012yes—
258C23C2×(C2×D83)⋊C411yescharacteristic
259Q8—6644no—
260D4—44no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—3322no—
274C2×C4—3322no—
275C2×C4—3322no—
276C2×C4—3322no—
277C2×C4—3322no—
278C2×C4—3322no—
279C2×C4(C22⋊D83)⋊C412yes—
280C2×C4#1458176211yescharacteristic
281C2×C4#1458176412yes—
282C2×C4#1458177611yescharacteristic
283C8—82no—
284C8—3324no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1912757811yescharacteristic
298C22C2×#1458171011yescharacteristic
299C22#1912768311yescharacteristic
300C4—21no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—6642no—
306C4—6642no—
307C4#1912788112yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2#2490784911yescharacteristic
313C2#2490792911yescharacteristic
314C2#2490798811yescharacteristic
subgroupG/H
centerC22#19127578
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1466C2
socleC22×C83#386
prime radicalC22×D4×C83C2×C4
cosocleC2×C42×C83C23
hypercenter#1466D83
hypocenterC83#9453
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #24906181C2semidirect
4. #24905318C2semidirect
163. C83#9453semidirect

Maximal subgroups

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

subgroupindex
#249164922
#249126362
#249061812
#249053182
C83×#14662
#945383