Finite Group Databasebeta

← #35300127

Subgroups — #35300127

Order 21248.

694 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 135 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×#19127547C211yesmaximal, characteristic
3C2×#19127439C212yesmaximal
4#24905343C212yesmaximal
5C83×#1817C211yesmaximal, characteristic
6C2×#14581771C2211yescharacteristic
7C22×C4×Dic83—21no—
8#19127607C412yescomplemented
9#19127599C412yescomplemented
10#19127585—22no—
11#19127574—21no—
12#19127564—21no—
13#19127561C2211yescharacteristic
14#19127547—24no—
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—
31C2×C4×Dic83—28no—
32C2×C4×Dic83—24no—
33C83⋊#95—24no—
34C83⋊#95—22no—
35C83⋊#95—22no—
36C83⋊#95—22no—
37C83⋊#95—21no—
38C83⋊#95—22no—
39C83⋊#95D414yes—
40C23×C4×C83D411yescharacteristic
41C23×C4×C83D411yescharacteristic
42C23×C4×C83C2311yescharacteristic
43C22×C8×C83—22no—
44C2×C42×C83—22no—
45C2×C42×C83—22no—
46C2×C42×C83C2×C412yes—
47C2×C42×C83D412yes—
48C22⋊C8×C83—22no—
49C22⋊C8×C83—22no—
50C2×Dic166—22no—
51C2×Dic166—42no—
52C22×Dic83—28no—
53C22×Dic83—24no—
54C22×Dic83—21no—
55C22×Dic83—22no—
56C22×Dic83—21no—
57C22×Dic83—44no—
58C22×Dic83—42no—
59C22×Dic83—42no—
60C4⋊Dic83—22no—
61C4⋊Dic83—42no—
62Dic83⋊C4—44no—
63Dic83⋊C4—42no—
64C22⋊Dic83—41no—
65C22⋊Dic83—41no—
66C22⋊Dic83—42no—
67C4×Dic83—216no—
68C24×C83C2×D411yescharacteristic
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—24no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—22no—
75C22×C4×C83—22no—
76C22×C4×C83—42no—
77C22×C4×C83C22⋊C412yes—
78C22×C4×C83C22⋊C412yes—
79C22×C4×C83C22×C411yescharacteristic
80C22×C4×C83C2×D412yes—
81C22×C4×C83C2×D411yescharacteristic
82C22×C4×C83#4112yes—
83C2×C8×C83—44no—
84C2×C8×C83—42no—
85C42×C83—24no—
86C42×C83—24no—
87C23×C83—22no—
88C23×C83—22no—
89C23×C83C2×C22⋊C411yescharacteristic
90C23×C83#12011yescharacteristic
91C23×C83#12311yescharacteristic
92C23×C83#13512yes—
93C23×C83#13712yes—
94C2×C4×C83—24no—
95C2×C4×C83—24no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83—22no—
103C2×C4×C83—42no—
104C2×C4×C83—42no—
105C2×C4×C83—42no—
106C2×C4×C83—42no—
107C2×C4×C83#10414yes—
108C2×C4×C83C4×D412yes—
109C2×C4×C83#12112yes—
110C2×Dic83—28no—
111C2×Dic83—44no—
112C2×Dic83—48no—
113C2×Dic83—48no—
114C2×Dic83—44no—
115C2×Dic83—41no—
116C2×Dic83—41no—
117C2×Dic83—42no—
118C2×Dic83—42no—
119C2×Dic83—42no—
120C2×Dic83—42no—
121C2×Dic83—42no—
122Dic166—82no—
123C8×C83—82no—
124C22×C83—21no—
125C22×C83—21no—
126C22×C83—21no—
127C22×C83—21no—
128C22×C83—24no—
129C22×C83—24no—
130C22×C83#38611yescharacteristic
131C22×C83#41012yes—
132C22×C83#41611yescharacteristic
133C22×C83#41911yescharacteristic
134C22×C83C2×#10412yes—
135C22×C83#45212yes—
136C22×C83#48512yes—
137C4×C83—24no—
138C4×C83—22no—
139C4×C83—22no—
140C4×C83—42no—
141Dic83—48no—
142Dic83—81no—
143Dic83—81no—
144Dic83—82no—
145#9145—831nomaximal, Sylow
146C2×C83—22no—
147C2×C83—22no—
148C2×C83#150012yes—
149C2×C83#150612yes—
150C2×C83#187311yescharacteristic
151C2×C83#189611yescharacteristic
152C2×C83#189911yescharacteristic
153#2855—831no—
154C2×#376—831no—
155#1817D8311yescharacteristic
156C2×#328—832no—
157#1294—832no—
158C83#914511yescharacteristic, complemented, Sylow
159C2×C22⋊Q8—831no—
160C22×C42—1661no—
161C22×C42C2×D8311yescharacteristic
162C2×C22⋊C8C2×D8312yes—
163#391—832no—
164#389—832no—
165#387—1662no—
166#385—1661no—
167#381—1661no—
168#380—831no—
169#376—1664no—
170C2×#95—832no—
171C2×#95—1661no—
172#336—22no—
173#328—1664no—
174C22×Q8—831no—
175C23×C4—1662no—
176C23×C4D33211yescharacteristic
177C23×C4D33211yescharacteristic
178C23×C4C22×D8311yescharacteristic
179C22×C8—22no—
180C22⋊Q8—1662no—
181C22⋊Q8—1662no—
182C2×C4⋊C4—832no—
183C2×C4⋊C4—831no—
184C2×C4⋊C4—1662no—
185C2×C4⋊C4—1661no—
186C2×C22⋊C4—1661no—
187C2×C22⋊C4—1661no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—1664no—
191C2×C42—1668no—
192C2×C42C4×D8312yes—
193C2×C42C22⋊D8312yes—
194C22⋊C8—22no—
195C22⋊C8—22no—
196#95—834no—
197#95—1664no—
198#95—1662no—
199#95—1662no—
200#95—1662no—
201#95—1662no—
202#95—1661no—
203C24C2×D33211yescharacteristic
204C2×Q8—3322no—
205C2×Q8—1662no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—3324no—
215C22×C4—3322no—
216C22×C4—3322no—
217C22×C4—1662no—
218C22×C4—1661no—
219C22×C4—1661no—
220C22×C4—1668no—
221C22×C4—1664no—
222C22×C4(C2×D83)⋊C412yes—
223C22×C4(C2×D83)⋊C412yes—
224C22×C4C2×C4×D8311yescharacteristic
225C22×C4D4×D8312yes—
226C22×C4C2×C22⋊D8311yescharacteristic
227C22×C4#1305058912yes—
228C2×C8—44no—
229C2×C8—42no—
230C4⋊C4—3322no—
231C4⋊C4—3322no—
232C4⋊C4—3324no—
233C4⋊C4—1662no—
234C22⋊C4—3321no—
235C22⋊C4—3321no—
236C22⋊C4—3322no—
237C42—24no—
238C42—24no—
239C42—16616no—
240C23—22no—
241C23—22no—
242C23C2×(C2×D83)⋊C411yescharacteristic
243C23#1458176211yescharacteristic
244C23#1458177611yescharacteristic
245C23#1458181412yes—
246C23#1458182512yes—
247Q8—6642no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—3324no—
262C2×C4—3328no—
263C2×C4—3328no—
264C2×C4—3324no—
265C2×C4—3321no—
266C2×C4—3321no—
267C2×C4—3322no—
268C2×C4—3322no—
269C2×C4—3322no—
270C2×C4—3322no—
271C2×C4—3322no—
272C2×C4—1668no—
273C2×C4#1458171814yes—
274C2×C4(C22⋊D83)⋊C412yes—
275C2×C4#1458176412yes—
276C8—82no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—24no—
282C22—24no—
283C22#1912757811yescharacteristic
284C22#1912767412yes—
285C22#1912769811yescharacteristic
286C22#1912771111yescharacteristic
287C22C2×#1458171812yes—
288C22#1912786512yes—
289C22#1912804312yes—
290C4—24no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—3328no—
295C4—6641no—
296C4—6641no—
297C4—6642no—
298C2—22no—
299C2—22no—
300C2#2490595012yes—
301C2#2490597612yes—
302C2#2490777311yescharacteristic
303C2#2490790411yescharacteristic
304C2#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#9145
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
8. #19127607C4semidirect
9. #19127599C4semidirect
158. C83#9145semidirect

Maximal subgroups

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

subgroupindex
#249158022
C2×#191275472
C2×#191274392
#249053432
C83×#18172
#914583