Finite Group Databasebeta

← #35314692

Subgroups — #35314692

Order 21248.

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

normal subgroups only

This group has 448 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#24913284C211yesmaximal, characteristic
2#24905777C211yesmaximal, characteristic
3#24905483C214yesmaximal
4C83×#1740C211yesmaximal, characteristic
5C2×#14581792C411yescharacteristic, complemented
6#19127754—21no—
7#19127640—21no—
8#19127637—21no—
9#19127621C412yescomplemented
10#19127613C2212yes—
11#19127599—21no—
12#19127567C411yescharacteristic, complemented
13#19127561—21no—
14#19127427—21no—
15#19127425C2212yes—
16C2×#117×C83—21no—
17C2×#117×C83C2211yescharacteristic
18#342×C83—21no—
19#342×C83C2212yes—
20C22×Dic166C2×C412yes—
21#14581792—41no—
22#14581792—41no—
23C2×C4⋊Dic83—21no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83—21no—
26C2×Dic83⋊C4—22no—
27C2×C22⋊Dic83—22no—
28C4⋊(C83⋊C8)—41no—
29C22⋊(C83⋊C8)—22no—
30C4×C83⋊C8—41no—
31C83⋊#95—22no—
32C83⋊#95—21no—
33C83⋊#95—22no—
34C83⋊#95C2×C412yes—
35C23×C4×C83D411yescharacteristic
36#117×C83—24no—
37#117×C83—41no—
38#117×C83—41no—
39#117×C83—42no—
40C2×C4⋊C4×C83—41no—
41C2×C4⋊C4×C83C2×C411yescharacteristic
42C2×C4⋊C4×C83D411yescharacteristic
43C2×C22⋊C4×C83—22no—
44C2×C22⋊C4×C83D411yescharacteristic
45C2×C22⋊C4×C83C2311yescharacteristic
46C2×C42×C83—41no—
47C2×C42×C83C2×C411yescharacteristic
48C2×C42×C83D411yescharacteristic
49C2×Dic166—42no—
50C2×Dic166—82no—
51C22×Dic83—22no—
52C22×Dic83—22no—
53C2×C83⋊C8—42no—
54C4⋊Dic83—42no—
55C4⋊Dic83—41no—
56C4⋊Dic83—41no—
57C4⋊Dic83—42no—
58Dic83⋊C4—82no—
59C22⋊Dic83—82no—
60C24×C83C2×D411yescharacteristic
61C22×C4×C83—21no—
62C22×C4×C83—21no—
63C22×C4×C83—22no—
64C22×C4×C83—42no—
65C22×C4×C83C22⋊C411yescharacteristic
66C22×C4×C83C22⋊C411yescharacteristic
67C22×C4×C83C22⋊C411yescharacteristic
68C22×C4×C83C22⋊C411yescharacteristic
69C22×C4×C83SD1612yes—
70C22×C4×C83Q1612yes—
71C22×C4×C83C22×C411yescharacteristic
72C22×C4×C83C2×D411yescharacteristic
73C4⋊C4×C83—41no—
74C4⋊C4×C83—41no—
75C4⋊C4×C83—44no—
76C22⋊C4×C83—24no—
77C22⋊C4×C83—41no—
78C22⋊C4×C83—41no—
79C22⋊C4×C83—42no—
80C42×C83—21no—
81C42×C83—21no—
82C42×C83—21no—
83C42×C83—21no—
84C42×C83—44no—
85C23×C83—21no—
86C23×C83—21no—
87C23×C83—41no—
88C23×C83—41no—
89C23×C83C2×C22⋊C411yescharacteristic
90C23×C83C2×SD1611yescharacteristic
91C23×C83C2×Q1611yescharacteristic
92C2×C4×C83—21no—
93C2×C4×C83—21no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—21no—
99C2×C4×C83—21no—
100C2×C4×C83—21no—
101C2×C4×C83—21no—
102C2×C4×C83—41no—
103C2×C4×C83—41no—
104C2×C4×C83—41no—
105C2×C4×C83—41no—
106C2×C4×C83—44no—
107C2×C4×C83—82no—
108C2×C4×C83—82no—
109C2×C4×C83#9911yescharacteristic
110C2×C4×C83#9911yescharacteristic
111C2×C4×C83#10312yes—
112C2×C4×C83#10312yes—
113C2×C4×C83#10412yes—
114C2×Dic83—42no—
115C2×Dic83—42no—
116C2×Dic83—42no—
117C2×Dic83—42no—
118C2×Dic83—42no—
119C2×Dic83—42no—
120Dic166—84no—
121C83⋊C8—82no—
122C22×C83—21no—
123C22×C83—21no—
124C22×C83—21no—
125C22×C83—21no—
126C22×C83—41no—
127C22×C83—41no—
128C22×C83—41no—
129C22×C83—41no—
130C22×C83—81no—
131C22×C83C2×#9911yescharacteristic
132C22×C83C2×#10311yescharacteristic
133C22×C83C2×#10411yescharacteristic
134C4×C83—22no—
135C4×C83—41no—
136C4×C83—41no—
137C4×C83—41no—
138C4×C83—41no—
139C4×C83—84no—
140C4×C83#32812yes—
141C4×C83#32812yes—
142Dic83—82no—
143Dic83—82no—
144#11717—831nomaximal, Sylow
145C2×C83—21no—
146C2×C83—21no—
147C2×C83—41no—
148C2×C83—41no—
149C2×C83C2×#32811yescharacteristic
150C2×C83#212211yescharacteristic
151C2×C83#212911yescharacteristic
152#2628—831no—
153#1740D8311yescharacteristic
154#1458—831no—
155#1351—834no—
156C83#1171711yescharacteristic, complemented, Sylow
157C2×C4⋊Q8—831no—
158C2×#117—21no—
159C2×#117C2×D8311yescharacteristic
160#431—1661no—
161#400—1661no—
162#399—1661no—
163#395—832no—
164#393—832no—
165#389—1661no—
166#382—831no—
167#380—1661no—
168#342—21no—
169#342C2×D8312yes—
170#324—1661no—
171#323—832no—
172C22×Q8—832no—
173C23×C4C22⋊D8311yescharacteristic
174C4⋊Q8—3321no—
175C4⋊Q8—3321no—
176#117—24no—
177#117—41no—
178#117—41no—
179#117—42no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—1661no—
182C2×C4⋊C4—1661no—
183C2×C4⋊C4—1661no—
184C2×C4⋊C4—1662no—
185C2×C4⋊C4C4×D8311yescharacteristic
186C2×C4⋊C4D33211yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—1662no—
189C2×C22⋊C4C22⋊D8311yescharacteristic
190C2×C22⋊C4C22×D8311yescharacteristic
191C2×C42—41no—
192C2×C42C4×D8311yescharacteristic
193C2×C42D33211yescharacteristic
194C4⋊C8—3321no—
195C22⋊C8—1662no—
196C4×C8—3321no—
197#95—832no—
198#95—1662no—
199#95—1661no—
200#95—1662no—
201C24C2×C22⋊D8311yescharacteristic
202C2×Q8—3322no—
203C2×Q8—6642no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—1662no—
209C22×C4—1662no—
210C22×C4(C2×D83)⋊C411yescharacteristic
211C22×C4(C2×D83)⋊C411yescharacteristic
212C22×C4(C2×D83)⋊C411yescharacteristic
213C22×C4(C2×D83)⋊C411yescharacteristic
214C22×C4#1305057812yes—
215C22×C4#1305058012yes—
216C22×C4C2×C4×D8311yescharacteristic
217C22×C4C2×D33211yescharacteristic
218C2×C8—3322no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—44no—
222C4⋊C4—3322no—
223C4⋊C4—3321no—
224C4⋊C4—3321no—
225C4⋊C4—3322no—
226C4⋊C4—6642no—
227C22⋊C4—24no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C22⋊C4—6642no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—44no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23C2×(C2×D83)⋊C411yescharacteristic
242C23C2×#1305057811yescharacteristic
243C23C2×#1305058011yescharacteristic
244Q8—6644no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—44no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—3322no—
263C2×C4—3322no—
264C2×C4—3322no—
265C2×C4—3322no—
266C2×C4—3322no—
267C2×C4—3322no—
268C2×C4#1458170611yescharacteristic
269C2×C4#1458170611yescharacteristic
270C2×C4#1458171512yes—
271C2×C4#1458171512yes—
272C2×C4#1458172012yes—
273C8—6642no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22C2×#1458170611yescharacteristic
284C22C2×#1458171511yescharacteristic
285C22C2×#1458172011yescharacteristic
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—84no—
292C4—6642no—
293C4—6642no—
294C4#1912743712yes—
295C4#1912743712yes—
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2C2×#1912743711yescharacteristic
301C2#2490911211yescharacteristic
302C2#2490913711yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC83×#1740C2
socleC22×C83C2×#99
prime radicalC24×C83C2×D4
cosocleC2×C22⋊C4×C83C23
hypercenter#1740D83
hypocenterC83#11717
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
5. C2×#14581792C4semidirect
9. #19127621C4semidirect
12. #19127567C4semidirect
156. C83#11717semidirect

Maximal subgroups

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

subgroupindex
#249132842
#249057772
#249054832
C83×#17402
#1171783