Finite Group Databasebeta

← #35299084

Subgroups — #35299084

Order 21248.

542 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C4×#14581781C211yesmaximal, characteristic
2#24905853C212yesmaximal
3#24905787C211yesmaximal, characteristic, complemented
4#24905295C212yesmaximal, complemented
5C83×#2983C211yesmaximal, characteristic, complemented
6C2×#14581781C411yescharacteristic
7C2×C4×D332C2211yescharacteristic
8#19127754C2212yes—
9#19127608C411yescharacteristic
10#19127602C411yescharacteristic
11#19127591—22no—
12#19127566C411yescharacteristic
13C4×(C2×D83)⋊C4—22no—
14C42×Dic83C2211yescharacteristic
15#19127428—24no—
16#19127427—22no—
17C2×C4×Q8×C83C2211yescharacteristic
18#446×C83—24no—
19#432×C83—22no—
20#431×C83C2212yes—
21C22×D332C2×C411yescharacteristic
22C22×C4×D83—21no—
23#14581781—22no—
24#14581781—22no—
25C4×D332—22no—
26C4×D332—22no—
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×(C2×D83)⋊C4—22no—
29C2×(C2×D83)⋊C4—22no—
30C2×(C2×D83)⋊C4—21no—
31C2×C4×Dic83—21no—
32C2×C4×Dic83—21no—
33C2×C4×Dic83C2×C411yescharacteristic
34C2×C4×Dic83C2×C411yescharacteristic
35C4⋊(C83⋊C8)—22no—
36C22⋊(C83⋊C8)—22no—
37C4×C83⋊C8—22no—
38C83⋊#95—22no—
39C83⋊#95—22no—
40C22×Q8×C83C2×C411yescharacteristic
41C4×Q8×C83—22no—
42C4×Q8×C83—22no—
43C2×C4⋊C4×C83C2×C411yescharacteristic
44C2×C4⋊C4×C83D412yes—
45C2×C42×C83D412yes—
46C2×C42×C83C2311yescharacteristic
47C4⋊C8×C83—24no—
48C4⋊C8×C83D414yes—
49C22⋊C8×C83—22no—
50#97×C83—24no—
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—44no—
61C2×C4×D83—41no—
62C2×C4×D83—41no—
63C2×C83⋊C8—42no—
64C4⋊Dic83—22no—
65(C2×D83)⋊C4—28no—
66(C2×D83)⋊C4—44no—
67(C2×D83)⋊C4—42no—
68C4×Dic83—21no—
69C4×Dic83—22no—
70C4×Dic83—42no—
71C4×Dic83—42no—
72C4×Dic83M1612yes—
73C2×Q8×C83—22no—
74C2×Q8×C83—21no—
75C2×Q8×C83—21no—
76C2×Q8×C83—22no—
77C22×C4×C83C22⋊C412yes—
78C22×C4×C83C22⋊C412yes—
79C22×C4×C83C22×C411yescharacteristic
80C22×C4×C83C2×D411yescharacteristic
81C22×C4×C83C2×D411yescharacteristic
82C2×C8×C83—24no—
83C4⋊C4×C83—22no—
84C4⋊C4×C83—24no—
85C42×C83—22no—
86C42×C83—22no—
87C42×C83C2×D412yes—
88C42×C83#4112yes—
89C23×C83C2×C22⋊C411yescharacteristic
90C22×D83—22no—
91C22×D83—42no—
92C22×D83—42no—
93C22×D83—41no—
94C22×D83—41no—
95Q8×C83—42no—
96Q8×C83—42no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83—41no—
103C2×C4×C83—42no—
104C2×C4×C83C4×D412yes—
105C2×C4×C83#12011yescharacteristic
106C2×C4×C83#12112yes—
107C2×C4×C83#12311yescharacteristic
108C2×C4×C83C2×M1611yescharacteristic
109C2×C4×C83C2×M1611yescharacteristic
110C2×C4×C83#13112yes—
111C2×Dic83—22no—
112C2×Dic83—22no—
113C2×Dic83—41no—
114C2×Dic83—41no—
115C2×Dic83—41no—
116C2×Dic83—41no—
117C2×Dic83—42no—
118C2×Dic83—41no—
119D332—24no—
120D332—42no—
121C4×D83—82no—
122C4×D83—82no—
123C8×C83—44no—
124C83⋊C8—82no—
125C22×C83—21no—
126C22×C83—21no—
127C22×C83#38611yescharacteristic
128C22×C83#40711yescharacteristic
129C22×C83#40811yescharacteristic
130C2×D83—44no—
131C2×D83—42no—
132C2×D83—82no—
133C2×D83—81no—
134C2×D83—81no—
135C4×C83—22no—
136C4×C83—21no—
137C4×C83—42no—
138C4×C83—42no—
139C4×C83#43512yes—
140C4×C83#43612yes—
141C4×C83#45412yes—
142Dic83—42no—
143Dic83—42no—
144Dic83—81no—
145#8964—831nomaximal, Sylow
146C2×C83—21no—
147C2×C83—21no—
148C2×C83#187011yescharacteristic
149C2×C83#188411yescharacteristic
150C2×C83#188811yescharacteristic
151D83—82no—
152#2983D8311yescharacteristic, complemented
153C4×#124—831no—
154#1478—832no—
155#1461—831no—
156#1276—832no—
157C83#896411yescharacteristic, complemented, Sylow
158C2×#124—831no—
159C2×C4×Q8C2×D8311yescharacteristic
160C2×C4×D4—831no—
161#446—24no—
162#432—22no—
163#431—832no—
164#431C2×D8312yes—
165#391—831no—
166#390—831no—
167#388—1662no—
168#382—831no—
169C4×C22⋊C4—1662no—
170C43—831no—
171D4⋊C8—1664no—
172#324—1662no—
173C22×Q8C4×D8311yescharacteristic
174C22×D4—831no—
175C23×C4—1661no—
176#124—1662no—
177#124—1662no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×D4—1662no—
181C4×D4—1662no—
182C2×C4⋊C4—831no—
183C2×C4⋊C4C4×D8311yescharacteristic
184C2×C4⋊C4D33212yes—
185C2×C22⋊C4—1662no—
186C2×C22⋊C4—1661no—
187C2×C22⋊C4—1662no—
188C2×C42—831no—
189C2×C42—831no—
190C2×C42—1661no—
191C2×C42—1661no—
192C2×C42C22⋊D8312yes—
193C2×C42C22×D8311yescharacteristic
194C4⋊C8—24no—
195C4⋊C8—1662no—
196C4⋊C8C22⋊D8314yes—
197C22⋊C8—22no—
198C22⋊C8—1662no—
199#97—24no—
200C4×C8—22no—
201C4×C8—1662no—
202#95—1662no—
203#95—1662no—
204C24—1661no—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×D4—832no—
210C2×D4—1661no—
211C2×D4—1662no—
212C2×D4—1662no—
213C22×C4—3324no—
214C22×C4—3321no—
215C22×C4—3321no—
216C22×C4—1661no—
217C22×C4—1661no—
218C22×C4—1661no—
219C22×C4(C2×D83)⋊C412yes—
220C22×C4(C2×D83)⋊C412yes—
221C22×C4C2×C4×D8311yescharacteristic
222C22×C4C2×C22⋊D8311yescharacteristic
223C22×C4C2×D33211yescharacteristic
224C2×C8—24no—
225C2×C8—3322no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—1662no—
229C22⋊C4—3324no—
230C22⋊C4—3322no—
231C22⋊C4—1668no—
232C42—22no—
233C42—22no—
234C42—3322no—
235C42—3322no—
236C42—832no—
237C42—1662no—
238C42—1661no—
239C42C2×C22⋊D8312yes—
240C42#1305059012yes—
241C23—3322no—
242C23—3322no—
243C23—3321no—
244C23—3321no—
245C23—1662no—
246C23C2×(C2×D83)⋊C411yescharacteristic
247Q8—42no—
248Q8—42no—
249D4—3322no—
250D4—1664no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—3321no—
259C2×C4—3321no—
260C2×C4—3321no—
261C2×C4—3321no—
262C2×C4—3322no—
263C2×C4—3321no—
264C2×C4—6642no—
265C2×C4—6642no—
266C2×C4—1662no—
267C2×C4—1662no—
268C2×C4C4×C22⋊D8312yes—
269C2×C4#1458176111yescharacteristic
270C2×C4#1458176612yes—
271C2×C4#1458177511yescharacteristic
272C2×C4M16×D8311yescharacteristic
273C2×C4M16×D8311yescharacteristic
274C2×C4#1458180012yes—
275C8—44no—
276C8—6642no—
277C22—21no—
278C22—21no—
279C22—3324no—
280C22—3322no—
281C22—6642no—
282C22—6641no—
283C22—6641no—
284C22#1912758011yescharacteristic
285C22#1912766411yescharacteristic
286C22#1912766911yescharacteristic
287C4—22no—
288C4—21no—
289C4—42no—
290C4—42no—
291C4—3322no—
292C4—3322no—
293C4—6641no—
294C4#1912777012yes—
295C4#1912777712yes—
296C4#1912787412yes—
297C2—21no—
298C2—21no—
299C2—6642no—
300C2#2490776011yescharacteristic
301C2#2490784611yescharacteristic
302C2#2490786611yescharacteristic
subgroupG/H
centerC2×C4#14581761
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#2983C2
socleC22×C83#386
prime radicalC22×D332C2×C4
cosocleC2×C42×C83C23
hypercenter#2983D83
hypocenterC83#8964
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
3. #24905787C2semidirect
4. #24905295C2semidirect
5. C83×#2983C2semidirect
152. #2983D83semidirect
157. C83#8964semidirect

Maximal subgroups

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

subgroupindex
C4×#145817812
#249058532
#249057872
#249052952
C83×#29832
#896483