Finite Group Databasebeta

← #35313677

Subgroups — #35313677

Order 21248.

460 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 89 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#24910002C211yesmaximal, characteristic
2#24905785C212yesmaximal, complemented
3#24905407C211yesmaximal, characteristic, complemented
4#24905394C212yesmaximal
5C83×#1324C211yesmaximal, characteristic, complemented
6C2×#14581780C411yescharacteristic
7C2×#14581757C2211yescharacteristic
8C2×#14581754C411yescharacteristic
9#19127751—22no—
10#19127606C411yescharacteristic
11#19127602C2211yescharacteristic
12#19127590—22no—
13#19127548—22no—
14C42×Dic83C411yescharacteristic
15#19127427C2212yes—
16#19127426—22no—
17#382×C83C2211yescharacteristic
18#324×C83C2212yes—
19C22×Dic166C2×C411yescharacteristic
20C22×D332C2×C411yescharacteristic
21C22×C4×D83—21no—
22#14581780—22no—
23#14581780C814yescomplemented
24#14581757—41no—
25#14581757—41no—
26#14581754—41no—
27#14581754—41no—
28C2×Dic83⋊C4D412yes—
29C2×(C2×D83)⋊C4—22no—
30C2×(C2×D83)⋊C4—21no—
31C2×(C2×D83)⋊C4—22no—
32C2×C4×Dic83—21no—
33C2×C4×Dic83—21no—
34C2×C4×Dic83C2×C411yescharacteristic
35C2×C4×Dic83C2×C411yescharacteristic
36C2×C4×Dic83D412yes—
37Dic83⋊C8—42no—
38C22⋊(C83⋊C8)—22no—
39C8×Dic83—42no—
40C83⋊#95—22no—
41C2×C4⋊C4×C83C2×C411yescharacteristic
42C2×C4⋊C4×C83C2311yescharacteristic
43C2×C42×C83C2×C411yescharacteristic
44C22⋊C8×C83—22no—
45#95×C83—22no—
46C23×D83—21no—
47C2×Dic166—41no—
48C2×Dic166—41no—
49C2×Dic166C2×C812yes—
50C2×Dic166M1612yes—
51C2×D332—21no—
52C2×D332—41no—
53C2×D332—41no—
54C2×D332C2×C812yes—
55C22×Dic83—21no—
56C22×Dic83C22⋊C412yes—
57C22×Dic83C22⋊C411yescharacteristic
58C22×Dic83C22⋊C411yescharacteristic
59C2×C4×D83—44no—
60C2×C4×D83—41no—
61C2×C4×D83—41no—
62C2×C83⋊C8—42no—
63Dic83⋊C4—42no—
64(C2×D83)⋊C4—28no—
65(C2×D83)⋊C4—44no—
66(C2×D83)⋊C4—42no—
67C4×Dic83—22no—
68C4×Dic83—22no—
69C4×Dic83—21no—
70C4×Dic83—21no—
71C4×Dic83—42no—
72C4×Dic83—42no—
73C4×Dic83—41no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83C22×C411yescharacteristic
77C22×C4×C83C2×D412yes—
78C2×C8×C83—42no—
79C4⋊C4×C83—41no—
80C4⋊C4×C83—41no—
81C42×C83—21no—
82C42×C83C2×C812yes—
83C23×C83C2×C22⋊C411yescharacteristic
84C22×D83—22no—
85C22×D83—42no—
86C22×D83—42no—
87C22×D83—41no—
88C22×D83—41no—
89C2×C4×C83—21no—
90C2×C4×C83—21no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—41no—
94C2×C4×C83—41no—
95C2×C4×C83—42no—
96C2×C4×C83—42no—
97C2×C4×C83—41no—
98C2×C4×C83C22×C811yescharacteristic
99C2×C4×C83C2×M1611yescharacteristic
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—21no—
105C2×Dic83—41no—
106C2×Dic83—41no—
107C2×Dic83—41no—
108C2×Dic83C22⋊C814yes—
109C2×Dic83#9911yescharacteristic
110C2×Dic83#9911yescharacteristic
111D332—42no—
112D332—42no—
113C4×D83—82no—
114C4×D83—82no—
115Dic166—42no—
116Dic166—42no—
117C8×C83—82no—
118C83⋊C8—82no—
119C22×C83—21no—
120C22×C83—21no—
121C22×C83C2×C22⋊C811yescharacteristic
122C22×C83C2×#9911yescharacteristic
123C22×C83#41311yescharacteristic
124C2×D83—44no—
125C2×D83—42no—
126C2×D83—82no—
127C2×D83—81no—
128C2×D83—81no—
129C4×C83—21no—
130C4×C83—21no—
131C4×C83—42no—
132C4×C83—42no—
133C4×C83—81no—
134Dic83—21no—
135Dic83—24no—
136Dic83—42no—
137Dic83—41no—
138Dic83#35512yes—
139#11476—831nomaximal, Sylow
140C2×C83—21no—
141C2×C83—21no—
142C2×C83#145911yescharacteristic
143C2×C83#212611yescharacteristic
144C2×C83C2×#35511yescharacteristic
145D83—82no—
146C4×#124—831no—
147#1461—832no—
148#1324—831no—
149#1324D8311yescharacteristic, complemented
150#1318—832no—
151C83#1147611yescharacteristic, complemented, Sylow
152C2×#124—831no—
153C2×C4×Q8—831no—
154C2×C4×D4—831no—
155#431—1662no—
156#391—831no—
157#390—831no—
158#388—1662no—
159#382C2×D8311yescharacteristic
160C4×C22⋊C4—1662no—
161C43—831no—
162#324—832no—
163#324—1662no—
164#324C2×D8312yes—
165C22×Q8—831no—
166C22×D4—831no—
167C23×C4—1661no—
168#124—834no—
169#124—1662no—
170C4×Q8—3321no—
171C4×Q8—3321no—
172C4×D4—3321no—
173C4×D4—3321no—
174C2×C4⋊C4—832no—
175C2×C4⋊C4C4×D8311yescharacteristic
176C2×C4⋊C4C22×D8311yescharacteristic
177C2×C22⋊C4—1661no—
178C2×C22⋊C4—1662no—
179C2×C22⋊C4—1662no—
180C2×C42—831no—
181C2×C42—831no—
182C2×C42—832no—
183C2×C42—1661no—
184C2×C42—1661no—
185C2×C42C4×D8311yescharacteristic
186C4⋊C8—3322no—
187C22⋊C8—22no—
188C22⋊C8—1662no—
189C4×C8—3322no—
190#95—22no—
191#95—1662no—
192C24—1661no—
193C2×Q8—3321no—
194C2×Q8—3321no—
195C2×Q8—832no—
196C2×Q8—832no—
197C2×D4—3321no—
198C2×D4—3321no—
199C2×D4—832no—
200C2×D4—1661no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—3321no—
204C22×C4—3321no—
205C22×C4—3324no—
206C22×C4—831no—
207C22×C4—832no—
208C22×C4—831no—
209C22×C4—1661no—
210C22×C4C2×C4×D8311yescharacteristic
211C22×C4D4×D8312yes—
212C2×C8—42no—
213C2×C8—3322no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—3322no—
217C22⋊C4—3322no—
218C22⋊C4—3324no—
219C22⋊C4—1668no—
220C42—21no—
221C42—3321no—
222C42—3322no—
223C42—3322no—
224C42—1662no—
225C42—1662no—
226C42—1661no—
227C42—1661no—
228C42C8×D8312yes—
229C23—3322no—
230C23—3322no—
231C23—3321no—
232C23—3321no—
233C23—1662no—
234C23C22⋊C4×D8311yescharacteristic
235Q8—3322no—
236Q8—3322no—
237D4—3322no—
238D4—3322no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—3321no—
249C2×C4—3321no—
250C2×C4—3321no—
251C2×C4—831no—
252C2×C4—834no—
253C2×C4—831no—
254C2×C4—6642no—
255C2×C4—6642no—
256C2×C4—1662no—
257C2×C4—1661no—
258C2×C4—1662no—
259C2×C4—1662no—
260C2×C4—1662no—
261C2×C4C2×C8×D8311yescharacteristic
262C2×C4M16×D8311yescharacteristic
263C8—82no—
264C8—6642no—
265C22—21no—
266C22—21no—
267C22—3324no—
268C22—3322no—
269C22—6642no—
270C22—6641no—
271C22—6641no—
272C22C22⋊C8×D8311yescharacteristic
273C22#99×D8311yescharacteristic
274C22#1912768411yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—81no—
280C4—3322no—
281C4—3321no—
282C4—832no—
283C4—1664no—
284C4—1661no—
285C2—21no—
286C2—21no—
287C2—6642no—
288C2#2490578011yescharacteristic
289C2#2490912711yescharacteristic
290C2#355×D8311yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC2×C4×C83C22×C8
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1324C2
socleC22×C83C2×#99
prime radicalC22×D332C2×C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1324D83
hypocenterC83#11476
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1324

Complements

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

NH ≅ G/Ntype
2. #24905785C2semidirect
3. #24905407C2semidirect
5. C83×#1324C2semidirect
23. #14581780C8semidirect
149. #1324D83semidirect
151. C83#11476semidirect

Maximal subgroups

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

subgroupindex
#249100022
#249057852
#249054072
#249053942
C83×#13242
#1147683