Finite Group Databasebeta

← #35313586

Subgroups — #35313586

Order 21248.

434 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 115 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#24910083C211yesmaximal, characteristic
2#24905777C211yesmaximal, characteristic
3#24905412C214yesmaximal
4C83×#1458C211yesmaximal, characteristic
5C2×#14581751C411yescharacteristic
6C22×C4⋊Dic83C411yescharacteristic
7#19127754—21no—
8#19127599—21no—
9#19127599—21no—
10#19127573C412yes—
11#19127573C2212yes—
12#19127570—21no—
13#19127566—21no—
14#19127427—21no—
15#19127425C2212yes—
16C2×#117×C83C2211yescharacteristic
17#431×C83—21no—
18#324×C83—21no—
19#323×C83C2212yes—
20C23×Dic83C2×C412yes—
21#14581751—22no—
22#14581751C814yescomplemented
23C2×C4⋊Dic83—21no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83—41no—
27C2×C4⋊Dic83—41no—
28C2×Dic83⋊C4—22no—
29C2×Dic83⋊C4—22no—
30C2×C22⋊Dic83C2×C412yes—
31C2×C4×Dic83—21no—
32C4⋊(C83⋊C8)—41no—
33C22⋊(C83⋊C8)—22no—
34C4×C83⋊C8—41no—
35C83⋊#95—22no—
36C83⋊#95—22no—
37C23×C4×C83C2×C411yescharacteristic
38#117×C83—41no—
39#117×C83—41no—
40C2×C4⋊C4×C83D411yescharacteristic
41C2×C4⋊C4×C83D411yescharacteristic
42C2×C22⋊C4×C83C2×C411yescharacteristic
43C2×C22⋊C4×C83C2311yescharacteristic
44C2×C42×C83D411yescharacteristic
45C2×C42×C83D411yescharacteristic
46C4⋊C8×C83—41no—
47C22⋊C8×C83—22no—
48C4×C8×C83—41no—
49C22×Dic83—22no—
50C22×Dic83—22no—
51C22×Dic83—42no—
52C22×Dic83—82no—
53C2×C83⋊C8—42no—
54C4⋊Dic83—24no—
55C4⋊Dic83—42no—
56C4⋊Dic83—41no—
57C4⋊Dic83—41no—
58Dic83⋊C4—82no—
59Dic83⋊C4—82no—
60C22⋊Dic83—22no—
61C22⋊Dic83C2×C814yes—
62C4×Dic83—24no—
63C24×C83C22×C411yescharacteristic
64C22×C4×C83—21no—
65C22×C4×C83—21no—
66C22×C4×C83—22no—
67C22×C4×C83C22⋊C411yescharacteristic
68C22×C4×C83C22⋊C411yescharacteristic
69C22×C4×C83C22⋊C411yescharacteristic
70C22×C4×C83C22⋊C411yescharacteristic
71C22×C4×C83C2×C812yes—
72C22×C4×C83M1612yes—
73C22×C4×C83C2×D411yescharacteristic
74C22×C4×C83C2×D411yescharacteristic
75C2×C8×C83—42no—
76C4⋊C4×C83—41no—
77C4⋊C4×C83—41no—
78C22⋊C4×C83—41no—
79C22⋊C4×C83—41no—
80C42×C83—21no—
81C42×C83—21no—
82C42×C83—21no—
83C42×C83—21no—
84C23×C83—21no—
85C23×C83—21no—
86C23×C83—41no—
87C23×C83—41no—
88C23×C83C2×C22⋊C411yescharacteristic
89C23×C83C22×C811yescharacteristic
90C23×C83C2×M1611yescharacteristic
91C2×C4×C83—21no—
92C2×C4×C83—21no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—21no—
98C2×C4×C83—21no—
99C2×C4×C83—21no—
100C2×C4×C83—21no—
101C2×C4×C83—41no—
102C2×C4×C83—41no—
103C2×C4×C83—41no—
104C2×C4×C83—41no—
105C2×C4×C83C22⋊C812yes—
106C2×C4×C83C22⋊C812yes—
107C2×C4×C83#9911yescharacteristic
108C2×C4×C83#9911yescharacteristic
109C2×C4×C83#10112yes—
110C2×Dic83—24no—
111C2×Dic83—42no—
112C2×Dic83—42no—
113C2×Dic83—42no—
114C2×Dic83—42no—
115C2×Dic83—42no—
116C2×Dic83—84no—
117C8×C83—82no—
118C83⋊C8—82no—
119C22×C83—21no—
120C22×C83—21no—
121C22×C83—21no—
122C22×C83—21no—
123C22×C83—41no—
124C22×C83—41no—
125C22×C83—42no—
126C22×C83—41no—
127C22×C83—41no—
128C22×C83C2×C22⋊C811yescharacteristic
129C22×C83C2×#9911yescharacteristic
130C22×C83C2×#10111yescharacteristic
131C4×C83—22no—
132C4×C83—41no—
133C4×C83—41no—
134C4×C83—41no—
135C4×C83—41no—
136C4×C83#32412yes—
137C4×C83#32412yes—
138Dic83—44no—
139Dic83—82no—
140#11454—831nomaximal, Sylow
141C2×C83—21no—
142C2×C83—21no—
143C2×C83—41no—
144C2×C83—41no—
145C2×C83C2×#32411yescharacteristic
146C2×C83#212211yescharacteristic
147C2×C83#212311yescharacteristic
148#2317—831no—
149#1458—831no—
150#1458D8311yescharacteristic
151#1326—834no—
152C83#1145411yescharacteristic, complemented, Sylow
153C2×#117—831no—
154C2×#117C2×D8311yescharacteristic
155C22×C4⋊C4—831no—
156#431—21no—
157#431—1661no—
158#389—1661no—
159#389—1661no—
160#385—832no—
161#385—832no—
162#384—1661no—
163#382—1661no—
164#324—21no—
165#324—1661no—
166#323—832no—
167#323C2×D8312yes—
168C23×C4—832no—
169C23×C4C4×D8311yescharacteristic
170#117—41no—
171#117—41no—
172#117—834no—
173#117—1662no—
174C2×C4⋊C4—3321no—
175C2×C4⋊C4—3321no—
176C2×C4⋊C4—1661no—
177C2×C4⋊C4—1661no—
178C2×C4⋊C4—1662no—
179C2×C4⋊C4—1662no—
180C2×C4⋊C4—1661no—
181C2×C4⋊C4D33211yescharacteristic
182C2×C4⋊C4C22⋊D8311yescharacteristic
183C2×C22⋊C4—832no—
184C2×C22⋊C4C4×D8311yescharacteristic
185C2×C22⋊C4C22×D8311yescharacteristic
186C2×C42—1661no—
187C2×C42D33211yescharacteristic
188C2×C42C22⋊D8311yescharacteristic
189C4⋊C8—41no—
190C4⋊C8—3321no—
191C22⋊C8—22no—
192C22⋊C8—1662no—
193C4×C8—41no—
194C4×C8—3321no—
195#95—1662no—
196#95—1662no—
197C24C2×C4×D8311yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—3322no—
202C22×C4—6642no—
203C22×C4—1662no—
204C22×C4—1662no—
205C22×C4(C2×D83)⋊C411yescharacteristic
206C22×C4(C2×D83)⋊C411yescharacteristic
207C22×C4(C2×D83)⋊C411yescharacteristic
208C22×C4(C2×D83)⋊C411yescharacteristic
209C22×C4C8×D8312yes—
210C22×C4#1305057312yes—
211C22×C4C2×C22⋊D8311yescharacteristic
212C22×C4C2×D33211yescharacteristic
213C2×C8—42no—
214C2×C8—3322no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—3322no—
218C4⋊C4—3321no—
219C4⋊C4—3321no—
220C4⋊C4—6642no—
221C4⋊C4—6642no—
222C4⋊C4—1664no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—834no—
226C22⋊C4—1662no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—1664no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23C2×(C2×D83)⋊C411yescharacteristic
237C23C2×C8×D8311yescharacteristic
238C23C2×#1305057311yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—3322no—
254C2×C4—3322no—
255C2×C4—3322no—
256C2×C4—3322no—
257C2×C4—3322no—
258C2×C4—6644no—
259C2×C4—1664no—
260C2×C4(C2×D83)⋊C812yes—
261C2×C4(C2×D83)⋊C812yes—
262C2×C4#1458170611yescharacteristic
263C2×C4#1458170611yescharacteristic
264C2×C4#1458171012yes—
265C8—82no—
266C8—6642no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22C2×(C2×D83)⋊C811yescharacteristic
277C22C2×#1458170611yescharacteristic
278C22C2×#1458171011yescharacteristic
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—3324no—
285C4—6642no—
286C4#1912742612yes—
287C4#1912742612yes—
288C2—21no—
289C2—21no—
290C2—41no—
291C2—41no—
292C2C2×#1912742611yescharacteristic
293C2#2490911411yescharacteristic
294C2#2490911611yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC23×C83C22×C8
FrattiniC2×C22⋊C4C22×D83
FittingC83×#1458C2
socleC22×C83C2×#99
prime radicalC24×C83C22×C4
cosocleC2×C22⋊C4×C83C23
hypercenter#1458D83
hypocenterC83#11454
LCSG ▷ C23×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
22. #14581751C8semidirect
152. C83#11454semidirect

Maximal subgroups

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

subgroupindex
#249100832
#249057772
#249054122
C83×#14582
#1145483