Finite Group Databasebeta

← #35302103

Subgroups — #35302103

Order 21248.

554 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 107 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#24912416C211yesmaximal, characteristic
2#24906181C212yesmaximal, complemented
3#24905804C211yesmaximal, characteristic
4#24905320C212yesmaximal, complemented
5C83×#2973C211yesmaximal, characteristic
6C2×#14581792C2211yescharacteristic
7C2×#14581771—21no—
8C2×#14581771—21no—
9#19127758C2212yes—
10#19127637—21no—
11#19127637—21no—
12#19127604C412yes—
13#19127570C412yes—
14C4×C4⋊Dic83C2211yescharacteristic
15#19127448—24no—
16#19127425—21no—
17#19127425—21no—
18C2×C4×D4×C83C2211yescharacteristic
19#435×C83—24no—
20#432×C83C2212yes—
21#407×C83—21no—
22C2×C22⋊C8×C83—21no—
23C22×Dic166—21no—
24#14581792—22no—
25#14581792—22no—
26#14581771—44no—
27#14581771—44no—
28C2×C4⋊Dic83—21no—
29C2×C4⋊Dic83—21no—
30C2×C4⋊Dic83C2×C412yes—
31C2×Dic83⋊C4—22no—
32C2×Dic83⋊C4—22no—
33C2×C22⋊Dic83—22no—
34C2×C22⋊Dic83—22no—
35C2×C4×Dic83C2×C412yes—
36C4⋊(C83⋊C8)—22no—
37C22⋊(C83⋊C8)—22no—
38#14581703—22no—
39C83⋊#95—21no—
40C22×D4×C83C2×C411yescharacteristic
41C23×C4×C83D411yescharacteristic
42C23×C4×C83D411yescharacteristic
43C2×M16×C83—22no—
44C22×C8×C83—22no—
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×C83D414yes—
52C22⋊C8×C83—22no—
53C22⋊C8×C83—22no—
54C22⋊C8×C83—22no—
55#97×C83—22no—
56C2×Dic166—42no—
57C2×Dic166—42no—
58C2×Dic166—41no—
59C2×Dic166—41no—
60C22×Dic83—22no—
61C22×Dic83—22no—
62C2×C83⋊C8—42no—
63C4⋊Dic83—24no—
64C4⋊Dic83—24no—
65C4⋊Dic83—42no—
66Dic83⋊C4—44no—
67Dic83⋊C4—44no—
68C22⋊Dic83—44no—
69C22⋊Dic83—44no—
70C4×Dic83—42no—
71C24×C83C22⋊C411yescharacteristic
72C24×C83C22⋊C411yescharacteristic
73C2×D4×C83—22no—
74C2×D4×C83—22no—
75C2×D4×C83—41no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—42no—
81C22×C4×C83—42no—
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22×C411yescharacteristic
85C22×C4×C83C2×D411yescharacteristic
86C22×C4×C83C2×D411yescharacteristic
87M16×C83—82no—
88C2×C8×C83—24no—
89C2×C8×C83—82no—
90C4⋊C4×C83—22no—
91C22⋊C4×C83—22no—
92C22⋊C4×C83—22no—
93C42×C83C2×D412yes—
94C42×C83#4112yes—
95C23×C83—21no—
96C23×C83—21no—
97C23×C83—21no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83—41no—
101C23×C83—41no—
102C23×C83—41no—
103C23×C83—41no—
104C23×C83#10112yes—
105C23×C83C2×C22⋊C411yescharacteristic
106D4×C83—44no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
109C2×C4×C83—22no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—41no—
113C2×C4×C83—42no—
114C2×C4×C83—42no—
115C2×C4×C83—41no—
116C2×C4×C83—41no—
117C2×C4×C83—81no—
118C2×C4×C83—81no—
119C2×C4×C83C4×D412yes—
120C2×C4×C83#12011yescharacteristic
121C2×C4×C83#12112yes—
122C2×C4×C83#12311yescharacteristic
123C2×Dic83—24no—
124C2×Dic83—42no—
125C2×Dic83—42no—
126C2×Dic83—42no—
127C2×Dic83—42no—
128C2×Dic83—42no—
129Dic166—82no—
130Dic166—82no—
131C8×C83—44no—
132C83⋊C8—82no—
133C22×C83—21no—
134C22×C83—21no—
135C22×C83—22no—
136C22×C83—42no—
137C22×C83—42no—
138C22×C83—41no—
139C22×C83—41no—
140C22×C83—41no—
141C22×C83—41no—
142C22×C83—41no—
143C22×C83—81no—
144C22×C83—81no—
145C22×C83#38611yescharacteristic
146C22×C83C2×#10111yescharacteristic
147C22×C83#41311yescharacteristic
148C4×C83—21no—
149C4×C83—22no—
150C4×C83—22no—
151C4×C83—41no—
152C4×C83—41no—
153C4×C83#45312yes—
154Dic83—44no—
155Dic83—82no—
156#9455—831nomaximal, Sylow
157C2×C83—21no—
158C2×C83—21no—
159C2×C83—42no—
160C2×C83—81no—
161C2×C83#188711yescharacteristic
162C2×C83#190111yescharacteristic
163C2×C83#190911yescharacteristic
164#2973D8311yescharacteristic
165#2542—831no—
166#1556—832no—
167#1466—831no—
168#1285—832no—
169C83#945511yescharacteristic, complemented, Sylow
170C2×C4⋊Q8—831no—
171C2×C22⋊Q8—1661no—
172C2×C22⋊Q8—1661no—
173C2×C4×D4C2×D8311yescharacteristic
174#435—24no—
175#432—832no—
176#432C2×D8312yes—
177#407—21no—
178C2×C22⋊C8—21no—
179#399—1661no—
180#399—1661no—
181#390—832no—
182#384—832no—
183C4×C4⋊C4—831no—
184#332—1664no—
185#323—1661no—
186#323—1661no—
187C22×Q8—1661no—
188C22×D4C4×D8311yescharacteristic
189C23×C4C22⋊D8311yescharacteristic
190C23×C4C22⋊D8311yescharacteristic
191C2×M16—22no—
192C22×C8—22no—
193C4⋊Q8—1662no—
194C4⋊Q8—1662no—
195C22⋊Q8—3324no—
196C22⋊Q8—3324no—
197C4×D4—22no—
198C4×D4—22no—
199C2×C4⋊C4—832no—
200C2×C4⋊C4—1662no—
201C2×C4⋊C4—1662no—
202C2×C4⋊C4—1661no—
203C2×C4⋊C4—1661no—
204C2×C4⋊C4C4×D8311yescharacteristic
205C2×C22⋊C4—1662no—
206C2×C22⋊C4—1662no—
207C2×C22⋊C4D33211yescharacteristic
208C2×C22⋊C4D33211yescharacteristic
209C2×C42—832no—
210C2×C42C22×D8311yescharacteristic
211C4⋊C8—1662no—
212C4⋊C8C22⋊D8314yes—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—1662no—
217#97—22no—
218#97—1662no—
219#95—1661no—
220C24(C2×D83)⋊C411yescharacteristic
221C24(C2×D83)⋊C411yescharacteristic
222C2×Q8—3322no—
223C2×Q8—3322no—
224C2×Q8—3321no—
225C2×Q8—3321no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—1662no—
236C22×C4—1662no—
237C22×C4(C2×D83)⋊C411yescharacteristic
238C22×C4(C2×D83)⋊C411yescharacteristic
239C22×C4C2×C4×D8311yescharacteristic
240C22×C4C2×C22⋊D8311yescharacteristic
241C22×C4C2×D33211yescharacteristic
242M16—82no—
243C2×C8—24no—
244C2×C8—82no—
245C2×C8—3322no—
246C4⋊C4—22no—
247C4⋊C4—3324no—
248C4⋊C4—3324no—
249C4⋊C4—3322no—
250C4⋊C4—1664no—
251C4⋊C4—1664no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—3324no—
255C22⋊C4—3324no—
256C42—3322no—
257C42C2×C22⋊D8312yes—
258C42#1305059012yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23#1458171012yes—
269C23C2×(C2×D83)⋊C411yescharacteristic
270Q8—6642no—
271Q8—6642no—
272D4—44no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—3322no—
286C2×C4—3322no—
287C2×C4—3322no—
288C2×C4—3322no—
289C2×C4—3322no—
290C2×C4—1664no—
291C2×C4C4×C22⋊D8312yes—
292C2×C4#1458176111yescharacteristic
293C2×C4#1458176612yes—
294C2×C4#1458177511yescharacteristic
295C8—44no—
296C8—6642no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—81no—
309C22#1912758011yescharacteristic
310C22C2×#1458171011yescharacteristic
311C22#1912768311yescharacteristic
312C4—21no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—3324no—
318C4—6642no—
319C4#1912786912yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—81no—
324C2#2490786111yescharacteristic
325C2#2490793111yescharacteristic
326C2#2490798511yescharacteristic
subgroupG/H
centerC22#19127580
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#2973C2
socleC22×C83#386
prime radicalC22×D4×C83C2×C4
cosocleC2×C42×C83C23
hypercenter#2973D83
hypocenterC83#9455
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #24906181C2semidirect
4. #24905320C2semidirect
169. C83#9455semidirect

Maximal subgroups

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

subgroupindex
#249124162
#249061812
#249058042
#249053202
C83×#29732
#945583