Finite Group Databasebeta

← #35313693

Subgroups — #35313693

Order 21248.

440 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 87 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#24910171C211yesmaximal, characteristic
2#24905785C212yesmaximal, complemented
3#24905409C211yesmaximal, characteristic, complemented
4#24905394C212yesmaximal
5C83×#1325C211yesmaximal, characteristic, complemented
6C2×#14581780C411yescharacteristic
7C2×#14581757C2211yescharacteristic
8C2×#14581754C411yescharacteristic
9#19127751—22no—
10#19127606C411yescharacteristic
11#19127602C2211yescharacteristic
12#19127591—22no—
13#19127572—22no—
14#19127547C411yescharacteristic
15#19127427C2212yes—
16#19127426—22no—
17#384×C83C2211yescharacteristic
18#324×C83C2212yes—
19C22×Dic166C2×C411yescharacteristic
20C22×D332C2×C411yescharacteristic
21C22×C4×D83—21no—
22#14581780—22no—
23#14581780C814yescomplemented
24#14581757—42no—
25#14581754—42no—
26C2×Dic83⋊C4D412yes—
27C2×(C2×D83)⋊C4—22no—
28C2×(C2×D83)⋊C4—21no—
29C2×(C2×D83)⋊C4—22no—
30C2×C4×Dic83C2×C411yescharacteristic
31C2×C4×Dic83C2×C411yescharacteristic
32C2×C4×Dic83D412yes—
33Dic83⋊C8—42no—
34C22⋊(C83⋊C8)—22no—
35C8×Dic83—42no—
36C83⋊#95—21no—
37C83⋊#95—22no—
38C83⋊#95—21no—
39C2×C4⋊C4×C83—22no—
40C2×C4⋊C4×C83C2×C411yescharacteristic
41C2×C4⋊C4×C83C2311yescharacteristic
42C2×C42×C83C2×C411yescharacteristic
43C22⋊C8×C83—22no—
44C23×D83—21no—
45C2×Dic166—41no—
46C2×Dic166—41no—
47C2×Dic166C2×C812yes—
48C2×Dic166M1612yes—
49C2×D332—21no—
50C2×D332—41no—
51C2×D332—41no—
52C2×D332C2×C812yes—
53C22×Dic83—21no—
54C22×Dic83C22⋊C412yes—
55C22×Dic83C22⋊C411yescharacteristic
56C22×Dic83C22⋊C411yescharacteristic
57C2×C4×D83—42no—
58C2×C4×D83—82no—
59C2×C83⋊C8—42no—
60Dic83⋊C4—42no—
61(C2×D83)⋊C4—28no—
62(C2×D83)⋊C4—42no—
63(C2×D83)⋊C4—82no—
64C4×Dic83—22no—
65C4×Dic83—22no—
66C4×Dic83—22no—
67C4×Dic83—41no—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83C22×C411yescharacteristic
71C22×C4×C83C2×D412yes—
72C2×C8×C83—42no—
73C4⋊C4×C83—41no—
74C4⋊C4×C83—41no—
75C4⋊C4×C83—82no—
76C42×C83—21no—
77C42×C83C2×C812yes—
78C23×C83C2×C22⋊C411yescharacteristic
79C22×D83—22no—
80C22×D83—42no—
81C22×D83—42no—
82C22×D83—41no—
83C22×D83—41no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—41no—
89C2×C4×C83—41no—
90C2×C4×C83—42no—
91C2×C4×C83—42no—
92C2×C4×C83—41no—
93C2×C4×C83C22×C811yescharacteristic
94C2×C4×C83C2×M1611yescharacteristic
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—21no—
98C2×Dic83—41no—
99C2×Dic83—41no—
100C2×Dic83—41no—
101C2×Dic83—41no—
102C2×Dic83—41no—
103C2×Dic83C22⋊C814yes—
104C2×Dic83#9912yes—
105D332—42no—
106D332—42no—
107C4×D83—84no—
108Dic166—42no—
109Dic166—42no—
110C8×C83—82no—
111C83⋊C8—82no—
112C22×C83—21no—
113C22×C83—21no—
114C22×C83C2×C22⋊C811yescharacteristic
115C22×C83C2×#9911yescharacteristic
116C22×C83#41311yescharacteristic
117C2×D83—44no—
118C2×D83—42no—
119C2×D83—82no—
120C2×D83—81no—
121C2×D83—81no—
122C4×C83—21no—
123C4×C83—21no—
124C4×C83—42no—
125C4×C83—42no—
126C4×C83—81no—
127Dic83—22no—
128Dic83—24no—
129Dic83—41no—
130Dic83—81no—
131#11480—831nomaximal, Sylow
132C2×C83—21no—
133C2×C83—21no—
134C2×C83#145911yescharacteristic
135C2×C83#212811yescharacteristic
136C2×C83#213311yescharacteristic
137D83—82no—
138#2326—831no—
139#1461—832no—
140#1325—831no—
141#1325D8311yescharacteristic, complemented
142#1318—832no—
143C83#1148011yescharacteristic, complemented, Sylow
144C2×#124—831no—
145C2×C4×Q8—831no—
146C2×C4×D4—831no—
147#431—1662no—
148#391—831no—
149#390—831no—
150#388—1662no—
151#385—1662no—
152#384C2×D8311yescharacteristic
153#376—831no—
154#324—832no—
155#324—1662no—
156#324C2×D8312yes—
157C22×Q8—831no—
158C22×D4—831no—
159C23×C4—1661no—
160#124—834no—
161#124—1662no—
162C4×Q8—3322no—
163C4×D4—3322no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—832no—
166C2×C4⋊C4C4×D8311yescharacteristic
167C2×C4⋊C4C22×D8311yescharacteristic
168C2×C22⋊C4—1661no—
169C2×C22⋊C4—1662no—
170C2×C22⋊C4—1662no—
171C2×C42—831no—
172C2×C42—831no—
173C2×C42—832no—
174C2×C42C4×D8311yescharacteristic
175C4⋊C8—3322no—
176C22⋊C8—22no—
177C22⋊C8—1662no—
178C4×C8—3322no—
179#95—1661no—
180#95—1661no—
181#95—1662no—
182C24—1661no—
183C2×Q8—3321no—
184C2×Q8—3321no—
185C2×Q8—832no—
186C2×Q8—832no—
187C2×D4—3321no—
188C2×D4—3321no—
189C2×D4—832no—
190C2×D4—1661no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—3322no—
194C22×C4—831no—
195C22×C4—831no—
196C22×C4—832no—
197C22×C4—6642no—
198C22×C4—1661no—
199C22×C4C2×C4×D8311yescharacteristic
200C22×C4D4×D8312yes—
201C2×C8—42no—
202C2×C8—3322no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—82no—
206C4⋊C4—3322no—
207C22⋊C4—3322no—
208C22⋊C4—6642no—
209C22⋊C4—1668no—
210C42—21no—
211C42—3321no—
212C42—1662no—
213C42—1662no—
214C42—1662no—
215C42C8×D8312yes—
216C23—3322no—
217C23—3322no—
218C23—3321no—
219C23—3321no—
220C23—1662no—
221C23C22⋊C4×D8311yescharacteristic
222Q8—3322no—
223Q8—3322no—
224D4—3322no—
225D4—3322no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—3321no—
236C2×C4—3321no—
237C2×C4—3321no—
238C2×C4—3321no—
239C2×C4—3321no—
240C2×C4—834no—
241C2×C4—832no—
242C2×C4—6644no—
243C2×C4—1661no—
244C2×C4—1662no—
245C2×C4—1662no—
246C2×C4C2×C8×D8311yescharacteristic
247C2×C4M16×D8311yescharacteristic
248C8—82no—
249C8—6642no—
250C22—21no—
251C22—21no—
252C22—3324no—
253C22—3322no—
254C22—6642no—
255C22—6641no—
256C22—6641no—
257C22C22⋊C8×D8311yescharacteristic
258C22#99×D8311yescharacteristic
259C22#1912768411yescharacteristic
260C4—21no—
261C4—21no—
262C4—42no—
263C4—42no—
264C4—81no—
265C4—3321no—
266C4—6641no—
267C4—1664no—
268C4—1662no—
269C2—21no—
270C2—21no—
271C2—6642no—
272C2#2490578011yescharacteristic
273C2#2490913311yescharacteristic
274C2#2490915111yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC2×C4×C83C22×C8
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1325C2
socleC22×C83C2×#99
prime radicalC22×D332C2×C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1325D83
hypocenterC83#11480
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
2. #24905785C2semidirect
3. #24905409C2semidirect
5. C83×#1325C2semidirect
23. #14581780C8semidirect
141. #1325D83semidirect
143. C83#11480semidirect

Maximal subgroups

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

subgroupindex
#249101712
#249057852
#249054092
#249053942
C83×#13252
#1148083