Finite Group Databasebeta

← #35413370

Subgroups — #35413370

Order 21248.

1294 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1294 conjugacy classes of subgroups — too many to draw. Only its 235 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×#19128277C211yesmaximal, characteristic
2#24910048C212yesmaximal, complemented
3#24907920C218yesmaximal, complemented
4#24907540C211yesmaximal, characteristic, complemented
5#24907380C212yesmaximal
6C83×#1820C211yesmaximal, characteristic, complemented
7C22×#13050590C2211yescharacteristic
8C23×Dic166C2211yescharacteristic
9#19128277—28no—
10C2×#14581792C412yescomplemented
11C2×#14581780C412yescomplemented
12C2×#14581776C2214yes—
13C2×#14581772C414yescomplemented
14C2×#14581748C2214yes—
15C2×#14581706C2218yes—
16#19127605—28no—
17#19127564C2212yes—
18#19127562C2214yes—
19#19127468C2214yes—
20C2×#117×C83C2211yescharacteristic
21#381×C83C2212yes—
22#342×C83C2214yes—
23C2×#13050590—24no—
24C2×#13050590—44no—
25C22×Dic166—44no—
26C22×Dic166C2×C411yescharacteristic
27C22×Dic166C2×C412yes—
28C22×Dic166C2×C412yes—
29C22×Dic166D418yes—
30C22×D332C2×C411yescharacteristic
31C22×C22⋊D83C2312yes—
32C23×Dic83C2312yes—
33C22×C4×D83C2×C412yes—
34#14581792—28no—
35#14581780—28no—
36#14581776—28no—
37#14581776—28no—
38#14581772—216no—
39#14581748—48no—
40C2×C4⋊Dic83C2×C418yes—
41C2×Dic83⋊C4—24no—
42C2×C22⋊Dic83C2314yes—
43C2×(C2×D83)⋊C4C2×C418yes—
44C2×C4×Dic83—24no—
45#14581706—48no—
46C83⋊#95—28no—
47C83⋊#95—24no—
48C23×C4×C83C2311yescharacteristic
49#117×C83—24no—
50C2×C4⋊C4×C83C2×C412yes—
51C2×C22⋊C4×C83C2314yes—
52C2×C22⋊C4×C83C2312yes—
53C2×C42×C83C2×C412yes—
54#95×C83—24no—
55C23×D83C22×C412yes—
56#13050590—416no—
57C2×Dic166—22no—
58C2×Dic166—44no—
59C2×Dic166—44no—
60C2×Dic166—48no—
61C2×Dic166—48no—
62C2×Dic166—42no—
63C2×Dic166C22⋊C418yes—
64C2×Dic166C22⋊C418yes—
65C2×D332—22no—
66C2×D332—42no—
67C2×C22⋊D83—24no—
68C2×C22⋊D83—28no—
69C22×Dic83—28no—
70C22×Dic83—24no—
71C22×Dic83C22×C412yes—
72C22×Dic83C22×C414yes—
73C22×Dic83C2×D414yes—
74C22×Dic83C2×D414yes—
75C2×C4×D83—24no—
76C2×C4×D83—44no—
77C4⋊Dic83—216no—
78Dic83⋊C4—48no—
79C22⋊Dic83—28no—
80(C2×D83)⋊C4—216no—
81C4×Dic83—48no—
82C24×C83C2411yescharacteristic
83C22×C4×C83—24no—
84C22×C4×C83—24no—
85C22×C4×C83C22×C411yescharacteristic
86C22×C4×C83C22×C411yescharacteristic
87C22×C4×C83C22×C414yes—
88C22×C4×C83C2×D414yes—
89C4⋊C4×C83—24no—
90C22⋊C4×C83—24no—
91C22⋊C4×C83—28no—
92C42×C83—24no—
93C23×C83—42no—
94C23×C83C23×C411yescharacteristic
95C23×C83C22×D412yes—
96C23×C83H(5,𝔽2)14yes—
97C22×D83—22no—
98C22×D83—24no—
99C22×D83—44no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83—28no—
103C2×C4×C83—24no—
104C2×C4×C83—44no—
105C2×C4×C83—48no—
106C2×C4×C83—44no—
107C2×C4×C83C2×C22⋊C412yes—
108C2×C4×C83C2×C22⋊C412yes—
109C22⋊D83—48no—
110C2×Dic83—28no—
111C2×Dic83—24no—
112C2×Dic83—24no—
113C2×Dic83—28no—
114C2×Dic83—22no—
115C2×Dic83—24no—
116C2×Dic83—48no—
117C2×Dic83—44no—
118C2×Dic83C2×C22⋊C418yes—
119D332—44no—
120C4×D83—48no—
121Dic166—48no—
122Dic166—48no—
123Dic166—44no—
124C22×C83—24no—
125C22×C83—22no—
126C22×C83—44no—
127C22×C83C22×C22⋊C411yescharacteristic
128C22×C83#51812yes—
129C22×C83#53414yes—
130C2×D83—42no—
131C2×D83—44no—
132C2×D83—84no—
133C4×C83—22no—
134C4×C83—22no—
135C4×C83—44no—
136C4×C83—84no—
137Dic83—28no—
138Dic83—42no—
139Dic83—84no—
140#21751—831nomaximal, Sylow
141C2×C83—24no—
142C2×C83—22no—
143C2×C83#230811yescharacteristic
144C2×C83#288412yes—
145D83—82no—
146C2×#536—831no—
147#2310—832no—
148#1899—838no—
149#1820—831no—
150#1820D8311yescharacteristic, complemented
151#1782—832no—
152C83#2175111yescharacteristic, complemented, Sylow
153C22×#41—831no—
154C23×Q8—831no—
155#536—1668no—
156C2×C4⋊Q8—832no—
157C2×#124—832no—
158C2×#123—834no—
159C2×C22⋊Q8—834no—
160C2×#117—834no—
161C2×#117C2×D8311yescharacteristic
162C2×#99—838no—
163#391—1668no—
164#381—832no—
165#381—834no—
166#381C2×D8312yes—
167#342—834no—
168#342C2×D8314yes—
169C2×#41—3324no—
170C2×#41—1664no—
171C22×Q8—3324no—
172C22×Q8—832no—
173C22×Q8—832no—
174C22×Q8—831no—
175C22×Q8—838no—
176C22×D4—832no—
177C22×D4—831no—
178C23×C4—832no—
179C23×C4—832no—
180C23×C4C22×D8311yescharacteristic
181C4⋊Q8—1668no—
182#124—1668no—
183#123—1668no—
184#123—1668no—
185C22⋊Q8—16616no—
186#117—24no—
187#117—3328no—
188C2×C4⋊C4—838no—
189C2×C4⋊C4—1664no—
190C2×C4⋊C4C4×D8312yes—
191C2×C22⋊C4—838no—
192C2×C22⋊C4—834no—
193C2×C22⋊C4C22×D8314yes—
194C2×C22⋊C4C22×D8312yes—
195C2×C42—1664no—
196C2×C42C4×D8312yes—
197#99—3328no—
198#95—24no—
199#95—1668no—
200#95—1664no—
201C24—832no—
202C24C23×D8311yescharacteristic
203#41—33216no—
204C2×Q8—3324no—
205C2×Q8—3324no—
206C2×Q8—3328no—
207C2×Q8—3328no—
208C2×Q8—3322no—
209C2×Q8—838no—
210C2×Q8—838no—
211C2×Q8—1662no—
212C2×D4—3322no—
213C2×D4—1664no—
214C2×D4—1668no—
215C2×D4—1662no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—3324no—
219C22×C4—834no—
220C22×C4—834no—
221C22×C4—834no—
222C22×C4—832no—
223C22×C4—1664no—
224C22×C4—1664no—
225C22×C4—1668no—
226C22×C4C2×C4×D8311yescharacteristic
227C22×C4C2×C4×D8311yescharacteristic
228C22×C4C2×C4×D8314yes—
229C22×C4D4×D8314yes—
230C4⋊C4—24no—
231C4⋊C4—3328no—
232C4⋊C4—16616no—
233C22⋊C4—24no—
234C22⋊C4—28no—
235C22⋊C4—16616no—
236C22⋊C4—1668no—
237C42—24no—
238C42—3328no—
239C23—42no—
240C23—3324no—
241C23—1662no—
242C23—1664no—
243C23C22×C4×D8311yescharacteristic
244C23C2×D4×D8312yes—
245C23#1458183814yes—
246Q8—3324no—
247Q8—3328no—
248Q8—3328no—
249D4—3324no—
250D4—3328no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—28no—
254C2×C4—24no—
255C2×C4—44no—
256C2×C4—48no—
257C2×C4—44no—
258C2×C4—3328no—
259C2×C4—3328no—
260C2×C4—3324no—
261C2×C4—838no—
262C2×C4—1662no—
263C2×C4—1664no—
264C2×C4—1668no—
265C2×C4—1664no—
266C2×C4—1664no—
267C2×C4—1668no—
268C2×C4C22⋊C4×D8312yes—
269C2×C4C22⋊C4×D8312yes—
270C22—24no—
271C22—22no—
272C22—44no—
273C22—3324no—
274C22—3322no—
275C22—6644no—
276C22C2×C22⋊C4×D8311yescharacteristic
277C22#1912817512yes—
278C22#1912826214yes—
279C4—22no—
280C4—22no—
281C4—44no—
282C4—84no—
283C4—3322no—
284C4—6644no—
285C4—1668no—
286C2—24no—
287C2—22no—
288C2—6642no—
289C2#2491004011yescharacteristic
290C2#2491592512yes—
subgroupG/H
centerC22C2×C22⋊C4×D83
commutatorC23×C83C23×C4
FrattiniC24C23×D83
FittingC83×#1820C2
socleC22×C83C22×C22⋊C4
prime radicalC22×#13050590C22
cosocleC24×C83C24
hypercenter#1820D83
hypocenterC83#21751
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
2. #24910048C2semidirect
3. #24907920C2semidirect
4. #24907540C2semidirect
6. C83×#1820C2semidirect
10. C2×#14581792C4semidirect
11. C2×#14581780C4semidirect
13. C2×#14581772C4semidirect
150. #1820D83semidirect
152. C83#21751semidirect

Maximal subgroups

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

subgroupindex
C2×#191282772
#249100482
#249079202
#249075402
#249073802
C83×#18202
#2175183