Finite Group Databasebeta

← #35295396

Subgroups — #35295396

Order 21248.

510 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 121 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#24916153C211yesmaximal, characteristic
2#24907242C212yesmaximal, complemented
3#24905955C212yesmaximal
4#24905348C211yesmaximal, characteristic, complemented
5C83×#1298C211yesmaximal, characteristic, complemented
6C2×#14581800C411yescharacteristic, complemented
7C23×C83⋊C8C2211yescharacteristic
8#19128212C2211yescharacteristic
9#19127769—22no—
10(C22⋊D83)⋊C8—22no—
11#19127729C411yescharacteristic, complemented
12#19127669C411yescharacteristic, complemented
13#19127659C411yescharacteristic, complemented
14C2×#14581702—22no—
15C4×C22⋊Dic83C2212yes—
16#19127455—22no—
17#19127439C414yescomplemented
18#19127436C414yescomplemented
19#407×C83C2211yescharacteristic
20C2×C22⋊C8×C83C2212yes—
21C2×#13050590C2×C411yescharacteristic
22#14581800—42no—
23C2×#13050573—21no—
24C22×C83⋊C8—42no—
25C22×C83⋊C8C2×C411yescharacteristic
26C22×C83⋊C8C2×C411yescharacteristic
27C22×C83⋊C8D412yes—
28C22×C83⋊C8D412yes—
29C2×C8×D83—21no—
30#14581775—21no—
31#14581771C2×C412yes—
32#14581766C2×C412yes—
33#14581761—21no—
34C4×C22⋊D83—22no—
35#14581751C2×C411yescharacteristic
36C2×C22⋊Dic83—22no—
37C2×C4×Dic83—22no—
38Dic83⋊C8—22no—
39(C2×D83)⋊C8—22no—
40#14581702—21no—
41#14581702—44no—
42C8×Dic83—21no—
43C83⋊#95C2×C414yes—
44C23×C4×C83C2311yescharacteristic
45C2×M16×C83C2×C411yescharacteristic
46C2×M16×C83C2×C411yescharacteristic
47C22×C8×C83—22no—
48C22⋊C8×C83—22no—
49C22⋊C8×C83—22no—
50C22⋊C8×C83C2×C414yes—
51#13050590—41no—
52#13050590—41no—
53C2×Dic166C4211yescharacteristic
54C2×D332C4211yescharacteristic
55C2×C22⋊D83—21no—
56C2×C22⋊D83—22no—
57C22×Dic83—24no—
58C2×C4×D83—21no—
59#13050573—42no—
60C2×C83⋊C8—22no—
61C2×C83⋊C8—24no—
62C2×C83⋊C8—41no—
63C2×C83⋊C8—42no—
64C2×C83⋊C8C22⋊C411yescharacteristic
65C2×C83⋊C8C22⋊C412yes—
66C2×C83⋊C8C22⋊C411yescharacteristic
67C8×D83—42no—
68C4⋊Dic83C4212yes—
69Dic83⋊C4—22no—
70C22⋊Dic83—21no—
71C22⋊Dic83—22no—
72C22⋊Dic83—44no—
73(C2×D83)⋊C4—22no—
74C4×Dic83—21no—
75C4×Dic83—42no—
76C4×Dic83—42no—
77C24×C83#4111yescharacteristic
78C22×C4×C83—41no—
79C22×C4×C83—41no—
80C22×C4×C83C22×C411yescharacteristic
81C22×C4×C83C22×C412yes—
82C22×C4×C83C2×D412yes—
83C22×C4×C83#4111yescharacteristic
84M16×C83—41no—
85M16×C83—41no—
86C2×C8×C83—21no—
87C2×C8×C83—21no—
88C2×C8×C83—24no—
89C2×C8×C83—42no—
90C2×C8×C83—42no—
91C23×C83—24no—
92C23×C83—41no—
93C23×C83#11711yescharacteristic
94C23×C83C4×D412yes—
95C22×D83—21no—
96C2×C4×C83—22no—
97C2×C4×C83—21no—
98C2×C4×C83—22no—
99C2×C4×C83—21no—
100C2×C4×C83—42no—
101C2×C4×C83—41no—
102C2×C4×C83C2×C4211yescharacteristic
103C2×C4×C83C2×C22⋊C411yescharacteristic
104C2×C4×C83C4×D412yes—
105C22⋊D83—42no—
106C22⋊D83—44no—
107C2×Dic83—22no—
108C2×Dic83—21no—
109C2×Dic83—44no—
110C2×Dic83—44no—
111C2×Dic83—44no—
112D332—41no—
113C4×D83—41no—
114C4×D83—41no—
115Dic166—41no—
116C8×C83—44no—
117C8×C83—41no—
118C8×C83—41no—
119C83⋊C8—22no—
120C83⋊C8—21no—
121C83⋊C8#9912yes—
122C22×C83—22no—
123C22×C83—22no—
124C22×C83—21no—
125C22×C83—22no—
126C22×C83—44no—
127C22×C83C4×C22⋊C411yescharacteristic
128C22×C83#44414yes—
129C2×D83—41no—
130C2×D83—42no—
131C4×C83—22no—
132C4×C83—21no—
133C4×C83C2×#9911yescharacteristic
134C4×C83#41011yescharacteristic
135Dic83—41no—
136Dic83—42no—
137Dic83—84no—
138#8239—831nomaximal, Sylow
139C2×C83—24no—
140C2×C83—41no—
141C2×C83C4×#9911yescharacteristic
142C2×C83#176512yes—
143D83—81no—
144#2929—831no—
145#1752—832no—
146#1501—832no—
147#1298D8311yescharacteristic, complemented
148#1297—831no—
149C83#823911yescharacteristic, complemented, Sylow
150C2×#131—831no—
151C23×C8—831no—
152#525—831no—
153#435—1662no—
154C8×D4—1662no—
155#424—831no—
156#408—831no—
157#407C2×D8311yescharacteristic
158C2×C22⋊C8C2×D8312yes—
159#405—831no—
160C2×#97—1662no—
161C4×C22⋊C4—832no—
162#336—1662no—
163#328—834no—
164#327—834no—
165C2×#41—831no—
166C23×C4C22×D8311yescharacteristic
167#131—3322no—
168C2×M16—1661no—
169C2×M16C4×D8311yescharacteristic
170C2×M16C4×D8311yescharacteristic
171C22×C8—22no—
172C22×C8—3322no—
173C22×C8—831no—
174C22×C8—832no—
175C22×C8—831no—
176C22×C8—832no—
177C22×C8—1661no—
178#123—1661no—
179C22⋊Q8—832no—
180#121—832no—
181#120—1661no—
182C4×D4—1662no—
183#117—831no—
184C2×C22⋊C4—1662no—
185C2×C42—1662no—
186C4⋊C8—1662no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—1662no—
190C22⋊C8C4×D8314yes—
191#97—3324no—
192#97—1661no—
193C4×C8—1661no—
194#95—834no—
195C24#1305059111yescharacteristic
196#41—3321no—
197#41—3321no—
198C2×Q8—831no—
199C2×D4—831no—
200C2×D4—1661no—
201C2×D4—1662no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—1664no—
205C22×C4—1661no—
206C22×C4C2×C4×D8311yescharacteristic
207C22×C4C2×C4×D8312yes—
208C22×C4D4×D8312yes—
209C22×C4#1305058911yescharacteristic
210M16—41no—
211M16—41no—
212M16—3322no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—24no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—3322no—
219C2×C8—3322no—
220C2×C8—3321no—
221C2×C8—831no—
222C2×C8—831no—
223C2×C8—832no—
224C2×C8—1664no—
225C2×C8—1662no—
226C4⋊C4—832no—
227C4⋊C4—1662no—
228C22⋊C4—3324no—
229C22⋊C4—1661no—
230C22⋊C4—1662no—
231C22⋊C4—1662no—
232C42—3322no—
233C42—3322no—
234C42—1661no—
235C23—24no—
236C23—41no—
237C23—1661no—
238C23#1458175011yescharacteristic
239C23#1458175412yes—
240Q8—3321no—
241D4—3321no—
242D4—3322no—
243D4—3324no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—3321no—
251C2×C4—3321no—
252C2×C4—3324no—
253C2×C4—3324no—
254C2×C4—3324no—
255C2×C4—1662no—
256C2×C4—1661no—
257C2×C4C42×D8311yescharacteristic
258C2×C4C22⋊C4×D8311yescharacteristic
259C2×C4(C22⋊D83)⋊C412yes—
260C8—44no—
261C8—41no—
262C8—41no—
263C8—832no—
264C8—1661no—
265C8—1662no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—21no—
270C22—44no—
271C22—3322no—
272C22—3321no—
273C22#1912754811yescharacteristic
274C22#1912782214yes—
275C4—22no—
276C4—21no—
277C4—3322no—
278C4—3321no—
279C4—6644no—
280C4#99×D8311yescharacteristic
281C4#1912767311yescharacteristic
282C2—24no—
283C2—41no—
284C2—6641no—
285C2#2490725011yescharacteristic
286C2#2490730612yes—
subgroupG/H
centerC2×C4C22⋊C4×D83
commutatorC2×C4×C83C2×C42
FrattiniC23×C4C22×D83
FittingC83×#1298C2
socleC22×C83C4×C22⋊C4
prime radical#19128212C22
cosocleC23×C4×C83C23
hypercenter#1298D83
hypocenterC83#8239
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1298

Complements

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

NH ≅ G/Ntype
2. #24907242C2semidirect
4. #24905348C2semidirect
5. C83×#1298C2semidirect
6. C2×#14581800C4semidirect
11. #19127729C4semidirect
12. #19127669C4semidirect
13. #19127659C4semidirect
17. #19127439C4semidirect
18. #19127436C4semidirect
147. #1298D83semidirect
149. C83#8239semidirect

Maximal subgroups

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

subgroupindex
#249161532
#249072422
#249059552
#249053482
C83×#12982
#823983