Finite Group Databasebeta

← #35404149

Subgroups — #35404149

Order 21248.

1546 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1546 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#24910044C212yesmaximal
3#24907537C211yesmaximal, characteristic, complemented
4#24907536C214yesmaximal, complemented
5#24907535C212yesmaximal, complemented
6#24907194C214yesmaximal, complemented
7C83×#1819C211yesmaximal, characteristic, complemented
8C23×C22⋊D83C2211yescharacteristic
9#19128174—28no—
10C2×C4×C22⋊D83C414yescomplemented
11C2×(C22⋊D83)⋊C4C414yescomplemented
12C2×#14581748C2214yes—
13C2×#14581748C2212yes—
14C22×C22⋊Dic83C2211yescharacteristic
15C2×C22⋊C4×D83C2212yes—
16#19127601—28no—
17#19127557C2212yes—
18#19127556C2214yes—
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×C412yes—
30C23×Dic83C2311yescharacteristic
31C22×C4×D83C2×C414yes—
32C4×C22⋊D83—28no—
33C4×C22⋊D83—28no—
34(C22⋊D83)⋊C4—216no—
35#14581748—28no—
36#14581748—216no—
37C2×Dic83⋊C4—24no—
38C2×Dic83⋊C4C2×C414yes—
39C2×C22⋊Dic83—24no—
40C2×C22⋊Dic83—24no—
41C2×C22⋊Dic83C2×C412yes—
42C2×C22⋊Dic83C2314yes—
43C2×C22⋊Dic83C2312yes—
44C2×(C2×D83)⋊C4—28no—
45C2×(C2×D83)⋊C4C2×C414yes—
46C22⋊C4×D83—28no—
47C2×C4×Dic83—24no—
48C2×C4×Dic83C2×C414yes—
49C25×C83C2311yescharacteristic
50C23×C4×C83C2×C412yes—
51C2×C22⋊C4×C83—24no—
52C2×C22⋊C4×C83—24no—
53C2×C22⋊C4×C83C2×C412yes—
54C2×C22⋊C4×C83C2314yes—
55C2×C22⋊C4×C83C2312yes—
56C23×D83—24no—
57C23×D83—82no—
58C23×D83C22×C412yes—
59C23×D83C2×D414yes—
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—82no—
69C22×Dic83C22×C414yes—
70C22×Dic83C22×C412yes—
71C22×Dic83C2×D414yes—
72C22×Dic83#4118yes—
73C2×C4×D83—28no—
74C2×C4×D83—84no—
75Dic83⋊C4—28no—
76Dic83⋊C4—48no—
77C22⋊Dic83—24no—
78C22⋊Dic83—24no—
79C22⋊Dic83—28no—
80C22⋊Dic83—84no—
81(C2×D83)⋊C4—28no—
82(C2×D83)⋊C4—88no—
83C4×Dic83—28no—
84C4×Dic83—48no—
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—24no—
115C22×D83—28no—
116C22×D83—44no—
117C22×D83—48no—
118C22×D83—88no—
119C22×D83C2×C22⋊C414yes—
120C2×C4×C83—24no—
121C2×C4×C83—44no—
122C2×C4×C83—48no—
123C2×C4×C83—44no—
124C2×C4×C83—82no—
125C2×C4×C83—82no—
126C22⋊D83—216no—
127C22⋊D83—216no—
128C2×Dic83—24no—
129C2×Dic83—44no—
130C2×Dic83—48no—
131C2×Dic83—44no—
132C2×Dic83—84no—
133C2×Dic83C2×C22⋊C414yes—
134C2×Dic83#11718yes—
135C2×Dic83#12318yes—
136C2×Dic83#12318yes—
137C4×D83—88no—
138C22×C83—22no—
139C22×C83—24no—
140C22×C83—24no—
141C22×C83—22no—
142C22×C83—22no—
143C22×C83—22no—
144C22×C83—22no—
145C22×C83—44no—
146C22×C83—44no—
147C22×C83—44no—
148C22×C83—88no—
149C22×C83C22×C22⋊C411yescharacteristic
150C22×C83C2×#11712yes—
151C22×C83C2×#12314yes—
152C2×D83—28no—
153C2×D83—24no—
154C2×D83—48no—
155C2×D83—88no—
156C4×C83—44no—
157C4×C83—84no—
158Dic83—44no—
159Dic83—84no—
160Dic83#38118yes—
161#20690—831nomaximal, Sylow
162C2×C83—24no—
163C2×C83—22no—
164C2×C83—42no—
165C2×C83—42no—
166C2×C83C2×#38111yescharacteristic
167C2×C83#288312yes—
168D83—44no—
169C2×#518—831no—
170#2309—832no—
171#1819—832no—
172#1819—834no—
173#1819—831no—
174#1819D8311yescharacteristic, complemented
175#1740—834no—
176C83#2069011yescharacteristic, complemented, Sylow
177C23×D4—831no—
178#518—1668no—
179C2×C4×D4—834no—
180C2×C4×D4—834no—
181C2×#117—832no—
182C2×#117—834no—
183C22×C22⋊C4—831no—
184C22×C22⋊C4—832no—
185C22×C22⋊C4C2×D8311yescharacteristic
186#390—1668no—
187#379—832no—
188#379—834no—
189#379C2×D8312yes—
190#342—838no—
191#342—834no—
192#342C2×D8314yes—
193C25—831no—
194C25C22×D8311yescharacteristic
195C22×D4—832no—
196C22×D4—832no—
197C22×D4—838no—
198C22×D4—1668no—
199C23×C4—831no—
200C23×C4—832no—
201C23×C4—834no—
202C23×C4C4×D8312yes—
203C4×D4—16616no—
204C4×D4—1668no—
205C4×D4—1668no—
206#117—16616no—
207#117—1668no—
208C2×C4⋊C4—834no—
209C2×C4⋊C4—1664no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—832no—
213C2×C22⋊C4—832no—
214C2×C22⋊C4—834no—
215C2×C22⋊C4—834no—
216C2×C22⋊C4—1668no—
217C2×C22⋊C4—1668no—
218C2×C22⋊C4—1664no—
219C2×C22⋊C4—1664no—
220C2×C22⋊C4C4×D8312yes—
221C2×C22⋊C4C22×D8314yes—
222C2×C22⋊C4C22×D8312yes—
223C2×C42—834no—
224C2×C42—1664no—
225C24—24no—
226C24—82no—
227C24—832no—
228C24—834no—
229C24—6642no—
230C24—1664no—
231C24C2×C4×D8311yescharacteristic
232C24C2×C4×D8311yescharacteristic
233C24D4×D8314yes—
234C24C23×D8311yescharacteristic
235C2×D4—838no—
236C2×D4—838no—
237C2×D4—1668no—
238C2×D4—1668no—
239C2×D4—16616no—
240C2×D4—16616no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—82no—
245C22×C4—834no—
246C22×C4—832no—
247C22×C4—834no—
248C22×C4—838no—
249C22×C4—6644no—
250C22×C4—6642no—
251C22×C4—1668no—
252C22×C4—1664no—
253C22×C4—1664no—
254C22×C4C2×C4×D8314yes—
255C4⋊C4—3328no—
256C4⋊C4—1668no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—28no—
260C22⋊C4—84no—
261C22⋊C4—6648no—
262C22⋊C4—6644no—
263C22⋊C4—1668no—
264C22⋊C4—1668no—
265C22⋊C4—1664no—
266C22⋊C4—1664no—
267C42—3328no—
268C42—1668no—
269C23—22no—
270C23—22no—
271C23—28no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—44no—
276C23—44no—
277C23—88no—
278C23—3324no—
279C23—3328no—
280C23—834no—
281C23—6648no—
282C23—1664no—
283C23—1668no—
284C23C22⋊C4×D8312yes—
285C23C22⋊C4×D8312yes—
286C23C22×C4×D8311yescharacteristic
287C23C2×D4×D8312yes—
288C23#41×D8314yes—
289D4—16616no—
290D4—16616no—
291C2×C4—24no—
292C2×C4—44no—
293C2×C4—48no—
294C2×C4—44no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—3324no—
298C2×C4—3328no—
299C2×C4—3324no—
300C2×C4—834no—
301C2×C4—838no—
302C2×C4—838no—
303C2×C4—838no—
304C2×C4—6648no—
305C2×C4—6644no—
306C2×C4—1664no—
307C22—22no—
308C22—24no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—24no—
314C22—44no—
315C22—44no—
316C22—44no—
317C22—88no—
318C22—3328no—
319C22—6648no—
320C22—1664no—
321C22—1668no—
322C22C2×C22⋊C4×D8311yescharacteristic
323C22#117×D8312yes—
324C22#123×D8314yes—
325C4—44no—
326C4—84no—
327C4—3324no—
328C4—838no—
329C4—6644no—
330C2—24no—
331C2—22no—
332C2—42no—
333C2—42no—
334C2—3324no—
335C2#381×D8311yescharacteristic
336C2#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#20690
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. #24907537C2semidirect
4. #24907536C2semidirect
5. #24907535C2semidirect
6. #24907194C2semidirect
7. C83×#1819C2semidirect
10. C2×C4×C22⋊D83C4semidirect
11. C2×(C22⋊D83)⋊C4C4semidirect
174. #1819D83semidirect
176. C83#20690semidirect

Maximal subgroups

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

subgroupindex
C2×#191281742
#249100442
#249075372
#249075362
#249075352
#249071942
C83×#18192
#2069083