Finite Group Databasebeta

← #35299932

Subgroups — #35299932

Order 21248.

522 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 115 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#24916548C211yesmaximal, characteristic, complemented
2#24910170C211yesmaximal, characteristic
3#24905853C212yesmaximal
4#24905291C212yesmaximal, complemented
5C83×#1461C211yesmaximal, characteristic, complemented
6C2×#14581781C411yescharacteristic
7C2×C4×D332C2211yescharacteristic
8#19127832—24no—
9#19127758—22no—
10#19127754C2212yes—
11#19127608C411yescharacteristic
12#19127602C411yescharacteristic
13#19127592—22no—
14#19127572—22no—
15#19127570C411yescharacteristic
16#19127547C2211yescharacteristic
17#19127428—24no—
18C2×C4×Q8×C83C2211yescharacteristic
19#431×C83C2212yes—
20#324×C83—22no—
21C22×D332C2×C411yescharacteristic
22C22×C4×D83—21no—
23#14581781—22no—
24#14581781—22no—
25C4×D332—22no—
26C4×D332—22no—
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×Dic83⋊C4—22no—
29C2×(C2×D83)⋊C4—22no—
30C2×(C2×D83)⋊C4—22no—
31C2×(C2×D83)⋊C4—21no—
32C2×C4×Dic83C2×C411yescharacteristic
33C2×C4×Dic83C2×C411yescharacteristic
34C4⋊(C83⋊C8)—24no—
35C4⋊(C83⋊C8)—22no—
36C22⋊(C83⋊C8)—22no—
37#14581703—24no—
38C4×C83⋊C8D414yes—
39C83⋊#95—22no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C22×Q8×C83C2×C411yescharacteristic
43C4×Q8×C83—22no—
44C4×Q8×C83—22no—
45C2×C4⋊C4×C83C2×C411yescharacteristic
46C2×C4⋊C4×C83D412yes—
47C2×C42×C83D412yes—
48C2×C42×C83C2311yescharacteristic
49C4⋊C8×C83—22no—
50C22⋊C8×C83—22no—
51C4×C8×C83—22no—
52C23×D83—21no—
53C2×D332—22no—
54C2×D332—22no—
55C2×D332—21no—
56C2×D332M1612yes—
57C22×Dic83—21no—
58C22×Dic83—21no—
59C22×Dic83—21no—
60C2×C4×D83—42no—
61C2×C4×D83—82no—
62C2×C83⋊C8—24no—
63C4⋊Dic83—22no—
64Dic83⋊C4—82no—
65(C2×D83)⋊C4—28no—
66(C2×D83)⋊C4—41no—
67(C2×D83)⋊C4—41no—
68(C2×D83)⋊C4—82no—
69C4×Dic83—21no—
70C4×Dic83—22no—
71C4×Dic83M1612yes—
72C2×Q8×C83—22no—
73C2×Q8×C83—21no—
74C2×Q8×C83—21no—
75C2×Q8×C83—22no—
76C22×C4×C83C22⋊C412yes—
77C22×C4×C83C22⋊C412yes—
78C22×C4×C83C22×C411yescharacteristic
79C22×C4×C83C2×D411yescharacteristic
80C22×C4×C83C2×D411yescharacteristic
81C2×C8×C83—42no—
82C4⋊C4×C83—22no—
83C4⋊C4×C83—24no—
84C42×C83—22no—
85C42×C83—22no—
86C42×C83C2×D412yes—
87C42×C83#4112yes—
88C23×C83C2×C22⋊C411yescharacteristic
89C22×D83—22no—
90C22×D83—42no—
91C22×D83—42no—
92C22×D83—41no—
93C22×D83—41no—
94Q8×C83—42no—
95Q8×C83—42no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—41no—
101C2×C4×C83—41no—
102C2×C4×C83—42no—
103C2×C4×C83C4×D412yes—
104C2×C4×C83#12011yescharacteristic
105C2×C4×C83#12112yes—
106C2×C4×C83#12311yescharacteristic
107C2×C4×C83C2×M1611yescharacteristic
108C2×C4×C83C2×M1611yescharacteristic
109C2×C4×C83#13112yes—
110C2×Dic83—22no—
111C2×Dic83—41no—
112C2×Dic83—41no—
113C2×Dic83—41no—
114C2×Dic83—42no—
115C2×Dic83—41no—
116C2×Dic83—41no—
117C2×Dic83—41no—
118D332—24no—
119D332—42no—
120C4×D83—84no—
121C8×C83—82no—
122C83⋊C8—44no—
123C22×C83—21no—
124C22×C83—21no—
125C22×C83#38611yescharacteristic
126C22×C83#40711yescharacteristic
127C22×C83#40811yescharacteristic
128C2×D83—44no—
129C2×D83—42no—
130C2×D83—82no—
131C2×D83—81no—
132C2×D83—81no—
133C4×C83—22no—
134C4×C83—21no—
135C4×C83—21no—
136C4×C83—42no—
137C4×C83—42no—
138C4×C83#43512yes—
139C4×C83#43612yes—
140Dic83—42no—
141Dic83—81no—
142Dic83—81no—
143#9106—831nomaximal, Sylow
144C2×C83—21no—
145C2×C83—21no—
146C2×C83#187011yescharacteristic
147C2×C83#191011yescharacteristic
148C2×C83#191211yescharacteristic
149D83—82no—
150#2983—831no—
151#2326—831no—
152#1478—832no—
153#1461D8311yescharacteristic, complemented
154#1274—832no—
155C83#910611yescharacteristic, complemented, Sylow
156C2×#124—831no—
157C2×C4×Q8C2×D8311yescharacteristic
158C2×C4×D4—831no—
159#446—1664no—
160#432—1662no—
161#431—832no—
162#431C2×D8312yes—
163#391—831no—
164#390—831no—
165#388—1662no—
166#385—1662no—
167#384—831no—
168#376—831no—
169D4⋊C8—1664no—
170#324—22no—
171C22×Q8C4×D8311yescharacteristic
172C22×D4—831no—
173C23×C4—1661no—
174#124—1662no—
175#124—1662no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—1662no—
179C4×D4—1662no—
180C2×C4⋊C4—831no—
181C2×C4⋊C4—1662no—
182C2×C4⋊C4C4×D8311yescharacteristic
183C2×C4⋊C4C22⋊D8312yes—
184C2×C22⋊C4—1661no—
185C2×C22⋊C4—1662no—
186C2×C22⋊C4—1662no—
187C2×C42—831no—
188C2×C42—831no—
189C2×C42D33212yes—
190C2×C42C22×D8311yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—1664no—
193C4⋊C8—1662no—
194C22⋊C8—22no—
195C22⋊C8—1662no—
196#97—1664no—
197C4×C8—22no—
198C4×C8—834no—
199#95—1662no—
200#95—1661no—
201#95—1661no—
202C24—1661no—
203C2×Q8—21no—
204C2×Q8—21no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×D4—832no—
208C2×D4—1661no—
209C2×D4—1662no—
210C2×D4—1662no—
211C22×C4—3322no—
212C22×C4—6642no—
213C22×C4—1661no—
214C22×C4—1661no—
215C22×C4—1661no—
216C22×C4(C2×D83)⋊C412yes—
217C22×C4(C2×D83)⋊C412yes—
218C22×C4C2×C4×D8311yescharacteristic
219C22×C4C2×C22⋊D8311yescharacteristic
220C22×C4C2×D33211yescharacteristic
221C2×C8—42no—
222C2×C8—1664no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—6642no—
226C4⋊C4—1662no—
227C22⋊C4—3321no—
228C22⋊C4—3321no—
229C22⋊C4—6642no—
230C22⋊C4—1668no—
231C42—22no—
232C42—22no—
233C42—832no—
234C42—1662no—
235C42—1661no—
236C42D4×D8312yes—
237C42#1305058912yes—
238C23—3322no—
239C23—3322no—
240C23—3321no—
241C23—3321no—
242C23—1662no—
243C23C2×(C2×D83)⋊C411yescharacteristic
244Q8—42no—
245Q8—42no—
246D4—3322no—
247D4—1664no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—3321no—
256C2×C4—3321no—
257C2×C4—3321no—
258C2×C4—3322no—
259C2×C4—3321no—
260C2×C4—3321no—
261C2×C4—3321no—
262C2×C4—6644no—
263C2×C4—1662no—
264C2×C4(C22⋊D83)⋊C412yes—
265C2×C4#1458176211yescharacteristic
266C2×C4#1458176412yes—
267C2×C4#1458177611yescharacteristic
268C2×C4M16×D8311yescharacteristic
269C2×C4M16×D8311yescharacteristic
270C2×C4#1458179812yes—
271C8—82no—
272C8—3324no—
273C22—21no—
274C22—21no—
275C22—3324no—
276C22—3322no—
277C22—6642no—
278C22—6641no—
279C22—6641no—
280C22#1912757811yescharacteristic
281C22#1912766411yescharacteristic
282C22#1912766711yescharacteristic
283C4—22no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—3322no—
289C4—6641no—
290C4—6641no—
291C4#1912777512yes—
292C4#1912777612yes—
293C2—21no—
294C2—21no—
295C2—6642no—
296C2#2490775811yescharacteristic
297C2#2490798811yescharacteristic
298C2#2490799811yescharacteristic
subgroupG/H
centerC22#19127578
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1461C2
socleC22×C83#386
prime radicalC22×D332C2×C4
cosocleC2×C42×C83C23
hypercenter#1461D83
hypocenterC83#9106
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
1. #24916548C2semidirect
4. #24905291C2semidirect
5. C83×#1461C2semidirect
153. #1461D83semidirect
155. C83#9106semidirect

Maximal subgroups

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

subgroupindex
#249165482
#249101702
#249058532
#249052912
C83×#14612
#910683