Finite Group Databasebeta

← #35300167

Subgroups — #35300167

Order 21248.

586 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 127 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#24915802C211yesmaximal, characteristic
2C2×#19127569C211yesmaximal, characteristic
3C2×#19127439C212yesmaximal
4#24905343C212yesmaximal
5C83×#1817C211yesmaximal, characteristic
6C2×#14581771C2211yescharacteristic
7#19127607C412yescomplemented
8#19127599C412yescomplemented
9#19127585—22no—
10#19127574—21no—
11#19127569—24no—
12#19127564—21no—
13#19127561C2211yescharacteristic
14C2×C83⋊#95—21no—
15C2×C83⋊#95—21no—
16C2×C83⋊#95C2212yes—
17#19127439—24no—
18C22×C42×C83C2211yescharacteristic
19C2×C22⋊C8×C83C2212yes—
20#336×C83—22no—
21C22×Dic166C2×C411yescharacteristic
22C23×Dic83—22no—
23#14581771—22no—
24#14581771—22no—
25C2×C4⋊Dic83C2×C411yescharacteristic
26C2×C4⋊Dic83C2×C412yes—
27C2×Dic83⋊C4—22no—
28C2×Dic83⋊C4—21no—
29C2×C22⋊Dic83—21no—
30C2×C22⋊Dic83—21no—
31C83⋊#95—24no—
32C83⋊#95—24no—
33C83⋊#95—22no—
34C83⋊#95—22no—
35C83⋊#95—22no—
36C83⋊#95—21no—
37C83⋊#95—22no—
38C83⋊#95D414yes—
39C23×C4×C83D411yescharacteristic
40C23×C4×C83D411yescharacteristic
41C23×C4×C83C2311yescharacteristic
42C22×C8×C83—22no—
43C2×C42×C83—22no—
44C2×C42×C83—22no—
45C2×C42×C83C2×C412yes—
46C2×C42×C83D412yes—
47C22⋊C8×C83—22no—
48C22⋊C8×C83—22no—
49C2×Dic166—22no—
50C2×Dic166—42no—
51C22×Dic83—21no—
52C22×Dic83—22no—
53C22×Dic83—21no—
54C22×Dic83—24no—
55C22×Dic83—44no—
56C22×Dic83—44no—
57C22×Dic83—42no—
58C22×Dic83—42no—
59C4⋊Dic83—22no—
60C4⋊Dic83—42no—
61Dic83⋊C4—44no—
62Dic83⋊C4—42no—
63C22⋊Dic83—41no—
64C22⋊Dic83—41no—
65C22⋊Dic83—42no—
66C24×C83C2×D411yescharacteristic
67C22×C4×C83—21no—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—22no—
72C22×C4×C83—22no—
73C22×C4×C83—42no—
74C22×C4×C83—42no—
75C22×C4×C83C22⋊C412yes—
76C22×C4×C83C22⋊C412yes—
77C22×C4×C83C22×C411yescharacteristic
78C22×C4×C83C2×D412yes—
79C22×C4×C83C2×D411yescharacteristic
80C22×C4×C83#4112yes—
81C2×C8×C83—44no—
82C2×C8×C83—42no—
83C42×C83—24no—
84C42×C83—24no—
85C23×C83—22no—
86C23×C83—22no—
87C23×C83C2×C22⋊C411yescharacteristic
88C23×C83#12011yescharacteristic
89C23×C83#12311yescharacteristic
90C23×C83#13512yes—
91C23×C83#13712yes—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—42no—
101C2×C4×C83—42no—
102C2×C4×C83—42no—
103C2×C4×C83—42no—
104C2×C4×C83—42no—
105C2×C4×C83—42no—
106C2×C4×C83C4×D412yes—
107C2×C4×C83#12112yes—
108C2×Dic83—44no—
109C2×Dic83—44no—
110C2×Dic83—41no—
111C2×Dic83—41no—
112C2×Dic83—42no—
113C2×Dic83—42no—
114C2×Dic83—42no—
115C2×Dic83—42no—
116C2×Dic83—42no—
117C2×Dic83—44no—
118C2×Dic83—84no—
119C2×Dic83—84no—
120Dic166—82no—
121C8×C83—82no—
122C22×C83—21no—
123C22×C83—21no—
124C22×C83—21no—
125C22×C83—21no—
126C22×C83—24no—
127C22×C83—24no—
128C22×C83#38611yescharacteristic
129C22×C83#41012yes—
130C22×C83#41611yescharacteristic
131C22×C83#41911yescharacteristic
132C22×C83#42112yes—
133C22×C83#45212yes—
134C22×C83#48512yes—
135C4×C83—22no—
136C4×C83—22no—
137C4×C83—42no—
138C4×C83—42no—
139Dic83—84no—
140Dic83—81no—
141Dic83—81no—
142Dic83—82no—
143#9153—831nomaximal, Sylow
144C2×C83—22no—
145C2×C83—22no—
146C2×C83#151512yes—
147C2×C83#152112yes—
148C2×C83#187311yescharacteristic
149C2×C83#189711yescharacteristic
150C2×C83#189911yescharacteristic
151#2855—831no—
152C2×#383—831no—
153#1817D8311yescharacteristic
154C2×#328—832no—
155#1294—832no—
156C83#915311yescharacteristic, complemented, Sylow
157C2×C22⋊Q8—831no—
158C22×C42C2×D8311yescharacteristic
159C2×C22⋊C8C2×D8312yes—
160#391—832no—
161#389—832no—
162#387—1662no—
163#385—1661no—
164#383—1664no—
165#381—1661no—
166#380—831no—
167C2×#95—832no—
168C2×#95—1661no—
169C2×#95—1661no—
170#336—22no—
171#328—1664no—
172C22×Q8—831no—
173C23×C4—1662no—
174C23×C4D33211yescharacteristic
175C23×C4D33211yescharacteristic
176C23×C4C22×D8311yescharacteristic
177C22×C8—22no—
178C22⋊Q8—1662no—
179C22⋊Q8—1662no—
180C2×C4⋊C4—832no—
181C2×C4⋊C4—831no—
182C2×C4⋊C4—1662no—
183C2×C4⋊C4—1661no—
184C2×C22⋊C4—1661no—
185C2×C22⋊C4—1661no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42C4×D8312yes—
189C2×C42C22⋊D8312yes—
190C22⋊C8—22no—
191C22⋊C8—22no—
192#95—834no—
193#95—1664no—
194#95—1664no—
195#95—1662no—
196#95—1662no—
197#95—1662no—
198#95—1662no—
199#95—1661no—
200C24C2×D33211yescharacteristic
201C2×Q8—3322no—
202C2×Q8—1662no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—3324no—
212C22×C4—3324no—
213C22×C4—3322no—
214C22×C4—3322no—
215C22×C4—1664no—
216C22×C4—1662no—
217C22×C4—1661no—
218C22×C4—1661no—
219C22×C4(C2×D83)⋊C412yes—
220C22×C4(C2×D83)⋊C412yes—
221C22×C4C2×C4×D8311yescharacteristic
222C22×C4D4×D8312yes—
223C22×C4C2×C22⋊D8311yescharacteristic
224C22×C4#1305058912yes—
225C2×C8—44no—
226C2×C8—42no—
227C4⋊C4—3322no—
228C4⋊C4—3324no—
229C4⋊C4—3322no—
230C4⋊C4—1662no—
231C22⋊C4—3321no—
232C22⋊C4—3321no—
233C22⋊C4—3322no—
234C42—24no—
235C42—24no—
236C23—22no—
237C23—22no—
238C23C2×(C2×D83)⋊C411yescharacteristic
239C23#1458176211yescharacteristic
240C23#1458177611yescharacteristic
241C23#1458181412yes—
242C23#1458182512yes—
243Q8—6642no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—3324no—
259C2×C4—3324no—
260C2×C4—3321no—
261C2×C4—3321no—
262C2×C4—3324no—
263C2×C4—3322no—
264C2×C4—3322no—
265C2×C4—3322no—
266C2×C4—3322no—
267C2×C4—3322no—
268C2×C4—6644no—
269C2×C4—6644no—
270C2×C4(C22⋊D83)⋊C412yes—
271C2×C4#1458176412yes—
272C8—82no—
273C22—24no—
274C22—24no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22#1912757811yescharacteristic
280C22#1912767412yes—
281C22#1912769811yescharacteristic
282C22#1912771111yescharacteristic
283C22#1912771712yes—
284C22#1912786512yes—
285C22#1912804312yes—
286C4—22no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—6644no—
291C4—6642no—
292C4—6641no—
293C4—6641no—
294C2—22no—
295C2—22no—
296C2#2490601512yes—
297C2#2490604112yes—
298C2#2490777311yescharacteristic
299C2#2490790911yescharacteristic
300C2#2490791911yescharacteristic
subgroupG/H
centerC23C2×(C2×D83)⋊C4
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#1817C2
socleC23×C83C2×C22⋊C4
prime radicalC24×C83C2×D4
cosocleC23×C4×C83C23
hypercenter#1817D83
hypocenterC83#9153
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
7. #19127607C4semidirect
8. #19127599C4semidirect
156. C83#9153semidirect

Maximal subgroups

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

subgroupindex
#249158022
C2×#191275692
C2×#191274392
#249053432
C83×#18172
#915383