Finite Group Databasebeta

← #35404144

Subgroups — #35404144

Order 21248.

1602 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1602 conjugacy classes of subgroups — too many to draw. Only its 275 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
1C2×#19128174C211yesmaximal, characteristic
2#24910039C212yesmaximal
3#24907539C214yesmaximal, complemented
4#24907537C211yesmaximal, characteristic, complemented
5#24907535C212yesmaximal, complemented
6#342×D83C214yesmaximal, complemented
7C83×#1819C211yesmaximal, characteristic, complemented
8C23×C22⋊D83C2211yescharacteristic
9#19128174—28no—
10C2×C4×C22⋊D83C414yescomplemented
11C2×(C22⋊D83)⋊C4C414yescomplemented
12C2×#14581748C2212yes—
13C22×C22⋊Dic83C2211yescharacteristic
14C2×C22⋊C4×D83C2214yes—
15C2×C22⋊C4×D83C2212yes—
16#19127577—28no—
17#19127562C2214yes—
18#19127557C2212yes—
19#19127468C2214yes—
20#19127467C2218yes—
21C22×C22⋊C4×C83C2211yescharacteristic
22#379×C83C2212yes—
23#342×C83C2214yes—
24C24×D83C2311yescharacteristic
25C22×C22⋊D83—28no—
26C22×C22⋊D83C2×C412yes—
27C22×C22⋊D83C2×C412yes—
28C22×C22⋊D83D418yes—
29C23×Dic83C2×C412yescomplemented
30C23×Dic83C2311yescharacteristic
31C22×C4×D83—24no—
32C22×C4×D83C2×C414yes—
33C4×C22⋊D83—28no—
34C4×C22⋊D83—28no—
35(C22⋊D83)⋊C4—216no—
36#14581748—28no—
37C2×Dic83⋊C4C2×C414yescomplemented
38C2×C22⋊Dic83—24no—
39C2×C22⋊Dic83—24no—
40C2×C22⋊Dic83C2×C412yescomplemented
41C2×C22⋊Dic83C2314yes—
42C2×C22⋊Dic83C2312yes—
43C2×(C2×D83)⋊C4—24no—
44C2×(C2×D83)⋊C4C2×C414yes—
45C22⋊C4×D83—28no—
46C22⋊C4×D83—216no—
47C2×C4×Dic83C2×C414yescomplemented
48C83⋊#95—28no—
49C25×C83C2311yescharacteristic
50C23×C4×C83C2×C412yescomplemented
51C2×C22⋊C4×C83—24no—
52C2×C22⋊C4×C83—24no—
53C2×C22⋊C4×C83C2×C412yescomplemented
54C2×C22⋊C4×C83C2314yes—
55C2×C22⋊C4×C83C2312yes—
56C23×D83—82no—
57C23×D83C22×C412yes—
58C23×D83C2×D414yes—
59C23×D83#4118yes—
60C2×C22⋊D83—28no—
61C2×C22⋊D83—28no—
62C2×C22⋊D83—216no—
63C2×C22⋊D83—216no—
64C2×C22⋊D83C22⋊C418yes—
65C2×C22⋊D83C22⋊C418yes—
66C22×Dic83—24no—
67C22×Dic83—24no—
68C22×Dic83—24no—
69C22×Dic83—82no—
70C22×Dic83C22×C412yes—
71C22×Dic83C22×C414yes—
72C22×Dic83C2×D414yes—
73C2×C4×D83—28no—
74C2×C4×D83—48no—
75C2×C4×D83—88no—
76C2×C4×D83—84no—
77Dic83⋊C4—28no—
78C22⋊Dic83—24no—
79C22⋊Dic83—24no—
80C22⋊Dic83—28no—
81C22⋊Dic83—84no—
82(C2×D83)⋊C4—28no—
83(C2×D83)⋊C4—48no—
84C4×Dic83—28no—
85C24×C83—24no—
86C24×C83—82no—
87C24×C83C22×C411yescharacteristic
88C24×C83C22×C411yescharacteristic
89C24×C83C2×D414yes—
90C24×C83C2411yescharacteristic
91C22×C4×C83—22no—
92C22×C4×C83—22no—
93C22×C4×C83—24no—
94C22×C4×C83—82no—
95C22×C4×C83C22×C414yes—
96C22⋊C4×C83—24no—
97C22⋊C4×C83—24no—
98C22⋊C4×C83—28no—
99C22⋊C4×C83—84no—
100C23×C83—22no—
101C23×C83—22no—
102C23×C83—28no—
103C23×C83—42no—
104C23×C83—42no—
105C23×C83—42no—
106C23×C83—44no—
107C23×C83—44no—
108C23×C83—88no—
109C23×C83C2×C22⋊C412yes—
110C23×C83C2×C22⋊C412yes—
111C23×C83C23×C411yescharacteristic
112C23×C83C22×D412yes—
113C23×C83C2×#4114yes—
114C22×D83—44no—
115C22×D83—48no—
116C22×D83—88no—
117C22×D83C2×C22⋊C414yes—
118C22×D83#11718yes—
119C22×D83#12318yes—
120C22×D83#12318yes—
121C2×C4×C83—24no—
122C2×C4×C83—44no—
123C2×C4×C83—48no—
124C2×C4×C83—44no—
125C2×C4×C83—82no—
126C2×C4×C83—82no—
127C22⋊D83—216no—
128C22⋊D83—216no—
129C2×Dic83—24no—
130C2×Dic83—28no—
131C2×Dic83—24no—
132C2×Dic83—44no—
133C2×Dic83—48no—
134C2×Dic83—44no—
135C2×Dic83—84no—
136C2×Dic83C2×C22⋊C414yes—
137C4×D83—88no—
138C4×D83—816no—
139C22×C83—22no—
140C22×C83—24no—
141C22×C83—24no—
142C22×C83—22no—
143C22×C83—22no—
144C22×C83—22no—
145C22×C83—22no—
146C22×C83—44no—
147C22×C83—44no—
148C22×C83—44no—
149C22×C83—88no—
150C22×C83C22×C22⋊C411yescharacteristic
151C22×C83C2×#11712yes—
152C22×C83C2×#12314yes—
153C2×D83—28no—
154C2×D83—48no—
155C2×D83—48no—
156C2×D83—48no—
157C2×D83#38118yes—
158C4×C83—44no—
159C4×C83—84no—
160Dic83—24no—
161Dic83—44no—
162Dic83—84no—
163#20689—831nomaximal, Sylow
164C2×C83—24no—
165C2×C83—22no—
166C2×C83—42no—
167C2×C83—42no—
168C2×C83C2×#38111yescharacteristic
169C2×C83#288312yes—
170D83—28no—
171C2×#518—831no—
172#2308—832no—
173#1820—834no—
174#1819—832no—
175#1819—831no—
176#1819D8311yescharacteristic, complemented
177C2×#342—834no—
178C83#2068911yescharacteristic, complemented, Sylow
179C23×D4—831no—
180#518—1668no—
181C2×C4×D4—834no—
182C2×C4×D4—834no—
183C2×#117—832no—
184C22×C22⋊C4—831no—
185C22×C22⋊C4—832no—
186C22×C22⋊C4—834no—
187C22×C22⋊C4C2×D8311yescharacteristic
188#386—1668no—
189#381—834no—
190#379—832no—
191#379C2×D8312yes—
192#342—838no—
193#342—834no—
194#342C2×D8314yes—
195C25—831no—
196C25C22×D8311yescharacteristic
197C22×D4—832no—
198C22×D4—832no—
199C22×D4—838no—
200C22×D4—1668no—
201C23×C4—831no—
202C23×C4—832no—
203C23×C4—834no—
204C23×C4—1664no—
205C23×C4C4×D8312yescomplemented
206C4×D4—16616no—
207C4×D4—1668no—
208C4×D4—1668no—
209#117—1668no—
210C2×C4⋊C4—834no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—832no—
214C2×C22⋊C4—832no—
215C2×C22⋊C4—834no—
216C2×C22⋊C4—834no—
217C2×C22⋊C4—16616no—
218C2×C22⋊C4—1668no—
219C2×C22⋊C4—1664no—
220C2×C22⋊C4—1664no—
221C2×C22⋊C4—1664no—
222C2×C22⋊C4C4×D8312yescomplemented
223C2×C22⋊C4C22×D8314yes—
224C2×C22⋊C4C22×D8312yes—
225C2×C42—834no—
226#95—1668no—
227C24—24no—
228C24—82no—
229C24—832no—
230C24—834no—
231C24—838no—
232C24—6642no—
233C24C2×C4×D8311yescharacteristic
234C24C2×C4×D8311yescharacteristic
235C24D4×D8314yes—
236C24C23×D8311yescharacteristic
237C2×D4—838no—
238C2×D4—838no—
239C2×D4—1668no—
240C2×D4—1668no—
241C2×D4—16616no—
242C2×D4—16616no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—82no—
247C22×C4—3328no—
248C22×C4—834no—
249C22×C4—832no—
250C22×C4—834no—
251C22×C4—6648no—
252C22×C4—6644no—
253C22×C4—6642no—
254C22×C4—1668no—
255C22×C4—1664no—
256C22×C4—1664no—
257C22×C4—1664no—
258C22×C4C2×C4×D8314yes—
259C4⋊C4—1668no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—28no—
263C22⋊C4—84no—
264C22⋊C4—3328no—
265C22⋊C4—6644no—
266C22⋊C4—1668no—
267C22⋊C4—1668no—
268C22⋊C4—1664no—
269C22⋊C4—1664no—
270C42—1668no—
271C23—22no—
272C23—22no—
273C23—28no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—44no—
278C23—44no—
279C23—88no—
280C23—3324no—
281C23—3328no—
282C23—838no—
283C23—838no—
284C23—838no—
285C23—834no—
286C23—6648no—
287C23C22⋊C4×D8312yes—
288C23C22⋊C4×D8312yes—
289C23C22×C4×D8311yescharacteristic
290C23C2×D4×D8312yes—
291C23#41×D8314yes—
292D4—16616no—
293D4—16616no—
294C2×C4—24no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—44no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—3324no—
301C2×C4—3328no—
302C2×C4—3324no—
303C2×C4—834no—
304C2×C4—6648no—
305C2×C4—66416no—
306C2×C4—6644no—
307C2×C4—1664no—
308C2×C4—1664no—
309C2×C4—1668no—
310C22—22no—
311C22—24no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—24no—
317C22—44no—
318C22—44no—
319C22—44no—
320C22—88no—
321C22—3328no—
322C22—3328no—
323C22—3328no—
324C22—838no—
325C22—1668no—
326C22C2×C22⋊C4×D8311yescharacteristic
327C22#117×D8312yes—
328C22#123×D8314yes—
329C4—44no—
330C4—84no—
331C4—3324no—
332C4—6644no—
333C4—1664no—
334C2—24no—
335C2—22no—
336C2—42no—
337C2—42no—
338C2—1668no—
339C2#381×D8311yescharacteristic
340C2#2491592112yes—
subgroupG/H
centerC22C2×C22⋊C4×D83
commutatorC23×C83C23×C4
FrattiniC24C23×D83
FittingC83×#1819C2
socleC22×C83C22×C22⋊C4
prime radicalC23×C22⋊D83C22
cosocleC24×C83C24
hypercenter#1819D83
hypocenterC83#20689
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #24907539C2semidirect
4. #24907537C2semidirect
5. #24907535C2semidirect
6. #342×D83C2semidirect
7. C83×#1819C2semidirect
10. C2×C4×C22⋊D83C4semidirect
11. C2×(C22⋊D83)⋊C4C4semidirect
29. C23×Dic83C2×C4semidirect
37. C2×Dic83⋊C4C2×C4semidirect
40. C2×C22⋊Dic83C2×C4semidirect
47. C2×C4×Dic83C2×C4semidirect
50. C23×C4×C83C2×C4semidirect
53. C2×C22⋊C4×C83C2×C4semidirect
176. #1819D83semidirect
178. C83#20689semidirect
205. C23×C4C4×D83semidirect
222. C2×C22⋊C4C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191281742
#249100392
#249075392
#249075372
#249075352
#342×D832
C83×#18192
#2068983