Finite Group Databasebeta

← #35302154

Subgroups — #35302154

Order 21248.

554 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 107 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#24910066C211yesmaximal, characteristic, complemented
2#24907543C211yesmaximal, characteristic, complemented
3#24905992C212yesmaximal
4#24905316C212yesmaximal, complemented
5C83×#2504C211yesmaximal, characteristic
6#19127665C2212yes—
7#19127587—24no—
8#19127564C2212yes—
9#19127561—21no—
10C4×C22⋊Dic83—21no—
11#19127470—22no—
12#19127435—24no—
13C2×#121×C83C2211yescharacteristic
14C22×C4⋊C4×C83C2211yescharacteristic
15#397×C83—22no—
16#390×C83C412yescomplemented
17#386×C83—21no—
18#386×C83—21no—
19#382×C83C412yescomplemented
20C4×C22⋊C4×C83C2211yescharacteristic
21C2×#13050574—22no—
22C2×C4⋊Dic83—22no—
23C2×C22⋊Dic83—22no—
24C2×C4×Dic83—22no—
25C22⋊(C83⋊C8)—22no—
26C22⋊(C83⋊C8)—22no—
27C83⋊#95—22no—
28C83⋊#95—22no—
29C83⋊#95—22no—
30C83⋊#95D414yes—
31C22×D4×C83—21no—
32C22×D4×C83C2×C411yescharacteristic
33C23×C4×C83D411yescharacteristic
34C23×C4×C83D411yescharacteristic
35C23×C4×C83C2311yescharacteristic
36#121×C83—22no—
37#121×C83—22no—
38C2×C4⋊C4×C83—22no—
39C2×C4⋊C4×C83—22no—
40C2×C4⋊C4×C83C2×C411yescharacteristic
41C2×C4⋊C4×C83C2×C412yes—
42C2×C4⋊C4×C83D412yes—
43C2×C22⋊C4×C83—21no—
44C2×C22⋊C4×C83—21no—
45C2×C22⋊C4×C83—22no—
46C2×C22⋊C4×C83—22no—
47C2×C42×C83C2×C412yes—
48#95×C83—22no—
49#95×C83—22no—
50#95×C83—21no—
51C22×Dic83—24no—
52#13050574—44no—
53C2×C83⋊C8—42no—
54C4⋊Dic83—82no—
55C22⋊Dic83—44no—
56C4×Dic83—82no—
57C24×C83—21no—
58C24×C83C2×D411yescharacteristic
59C2×D4×C83—22no—
60C2×D4×C83—42no—
61C2×D4×C83—44no—
62C2×D4×C83—41no—
63C2×D4×C83—41no—
64C22×C4×C83—21no—
65C22×C4×C83—21no—
66C22×C4×C83—21no—
67C22×C4×C83—21no—
68C22×C4×C83—21no—
69C22×C4×C83—22no—
70C22×C4×C83—42no—
71C22×C4×C83—81no—
72C22×C4×C83—81no—
73C22×C4×C83C22⋊C412yes—
74C22×C4×C83C22⋊C412yes—
75C22×C4×C83C22×C411yescharacteristic
76C22×C4×C83C2×D412yes—
77C22×C4×C83C2×D411yescharacteristic
78C22×C4×C83#4112yes—
79C4⋊C4×C83—24no—
80C4⋊C4×C83—24no—
81C4⋊C4×C83—22no—
82C22⋊C4×C83—42no—
83C22⋊C4×C83—42no—
84C22⋊C4×C83—82no—
85C22⋊C4×C83—82no—
86C42×C83—42no—
87C23×C83—22no—
88C23×C83—22no—
89C23×C83—41no—
90C23×C83—42no—
91C23×C83—42no—
92C23×C83—41no—
93C23×C83—42no—
94C23×C83C2×C22⋊C411yescharacteristic
95C23×C83#12011yescharacteristic
96C23×C83#12311yescharacteristic
97D4×C83—82no—
98D4×C83—82no—
99D4×C83—82no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83—22no—
103C2×C4×C83—22no—
104C2×C4×C83—42no—
105C2×C4×C83—42no—
106C2×C4×C83—42no—
107C2×C4×C83—42no—
108C2×C4×C83—42no—
109C2×C4×C83—42no—
110C2×C4×C83—42no—
111C2×C4×C83—42no—
112C2×C4×C83—42no—
113C2×C4×C83—41no—
114C2×C4×C83—42no—
115C2×C4×C83—81no—
116C2×C4×C83—81no—
117C2×C4×C83C4×D412yes—
118C2×C4×C83#12112yes—
119C2×Dic83—44no—
120C2×Dic83—44no—
121C2×Dic83—44no—
122C83⋊C8—82no—
123C22×C83—21no—
124C22×C83—21no—
125C22×C83—21no—
126C22×C83—21no—
127C22×C83—21no—
128C22×C83—21no—
129C22×C83—21no—
130C22×C83—44no—
131C22×C83—84no—
132C22×C83—81no—
133C22×C83—81no—
134C22×C83—81no—
135C22×C83#38611yescharacteristic
136C22×C83#41012yes—
137C22×C83#45412yes—
138C4×C83—22no—
139C4×C83—22no—
140C4×C83—42no—
141C4×C83—42no—
142C4×C83—82no—
143C4×C83—81no—
144Dic83—84no—
145#9465—831nomaximal, Sylow
146C2×C83—22no—
147C2×C83—22no—
148C2×C83—41no—
149C2×C83—161no—
150C2×C83#188811yescharacteristic
151C2×C83#189911yescharacteristic
152C2×C83#191111yescharacteristic
153#2504D8311yescharacteristic
154#2313—831no—
155#1821—831no—
156#1510—832no—
157#1284—832no—
158C83#946511yescharacteristic, complemented, Sylow
159C2×#121C2×D8311yescharacteristic
160C22×C4⋊C4C2×D8311yescharacteristic
161#407—832no—
162#397—22no—
163#390Dic8312yescomplemented
164#387—1664no—
165#386—21no—
166#386—21no—
167#382Dic8312yescomplemented
168#381—832no—
169#380—1661no—
170C4×C22⋊C4—1661no—
171C4×C22⋊C4C2×D8311yescharacteristic
172#343—1662no—
173#327—1664no—
174C22×D4—21no—
175C22×D4C2×Dic8311yescharacteristic
176C23×C4C22⋊D8311yescharacteristic
177C23×C4C22⋊D8311yescharacteristic
178C23×C4C22×D8311yescharacteristic
179C2×M16—1662no—
180#121—22no—
181#121—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—1662no—
185C2×C4⋊C4C2×Dic8311yescharacteristic
186C2×C4⋊C4C2×Dic8312yes—
187C2×C4⋊C4C22⋊D8312yes—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—1662no—
193C2×C42—1662no—
194C2×C42C2×Dic8312yes—
195C22⋊C8—1662no—
196C22⋊C8—1662no—
197#95—22no—
198#95—22no—
199#95—21no—
200#95—834no—
201#95—1662no—
202#95—1662no—
203#95—1662no—
204C24—21no—
205C24C2×C22⋊D8311yescharacteristic
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—44no—
209C2×D4—41no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—1664no—
221C22×C4C22⋊Dic8312yes—
222C22×C4C22⋊Dic8312yes—
223C22×C4C22×Dic8311yescharacteristic
224C22×C4D4×D8312yes—
225C22×C4C2×C22⋊D8311yescharacteristic
226C22×C4#1305058912yes—
227M16—3324no—
228C2×C8—3322no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—6642no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C22⋊C4—3324no—
238C42—42no—
239C42—6642no—
240C23—22no—
241C23—22no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—42no—
247C23C2×C22⋊Dic8311yescharacteristic
248C23#1458176011yescharacteristic
249C23#1458177311yescharacteristic
250D4—82no—
251D4—82no—
252D4—82no—
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—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—3324no—
271C2×C4—3324no—
272C2×C4—3324no—
273C2×C4D4×Dic8312yes—
274C2×C4#1458176412yes—
275C8—6642no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—84no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22#1912758111yescharacteristic
289C22#1912767512yes—
290C22#1912787312yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—82no—
296C4—81no—
297C4—6644no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—161no—
302C2#2490786711yescharacteristic
303C2#2490792211yescharacteristic
304C2#2490799611yescharacteristic
subgroupG/H
centerC22#19127581
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2504C2
socleC22×C83#386
prime radicalC2×#121×C83C22
cosocleC23×C4×C83C23
hypercenter#2504D83
hypocenterC83#9465
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2504

Complements

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

NH ≅ G/Ntype
1. #24910066C2semidirect
2. #24907543C2semidirect
4. #24905316C2semidirect
16. #390×C83C4semidirect
19. #382×C83C4semidirect
158. C83#9465semidirect
163. #390Dic83semidirect
167. #382Dic83semidirect

Maximal subgroups

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

subgroupindex
#249100662
#249075432
#249059922
#249053162
C83×#25042
#946583