Finite Group Databasebeta

← #1325×D83 #35313556

Subgroups — #1325×D83 #35313556

Order 21248.

598 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 101 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#384×D83C211yesmaximal, characteristic
2#324×D83C212yesmaximal
3#24905409C211yesmaximal, characteristic, complemented
4#24905408C212yesmaximal, complemented
5C83×#1325C211yesmaximal, characteristic, complemented
6C2×C4⋊C4×D83—22no—
7C2×C4⋊C4×D83C411yescharacteristic
8C2×C4⋊C4×D83C2211yescharacteristic
9C2×C42×D83C411yescharacteristic
10C22⋊C8×D83—22no—
11#19127571C411yescharacteristic
12#19127570—22no—
13#19127570C411yescharacteristic
14#19127570C2211yescharacteristic
15#19127427C2212yes—
16#19127426—22no—
17#384×C83C2211yescharacteristic
18#324×C83C2212yes—
19C22×C4×D83—21no—
20C22×C4×D83—21no—
21C22×C4×D83C2×C411yescharacteristic
22C22×C4×D83D412yes—
23C2×C8×D83—42no—
24C2×C4⋊Dic83—22no—
25C2×C4⋊Dic83C2×C411yescharacteristic
26C2×C4⋊Dic83C2×C411yescharacteristic
27C2×Dic83⋊C4—21no—
28C2×Dic83⋊C4—22no—
29C2×Dic83⋊C4—22no—
30C2×Dic83⋊C4D412yes—
31C4⋊C4×D83—41no—
32C4⋊C4×D83—41no—
33C4⋊C4×D83—41no—
34C4⋊C4×D83—41no—
35C4⋊C4×D83—48no—
36C2×C4×Dic83—21no—
37C2×C4×Dic83C2×C411yescharacteristic
38C42×D83—22no—
39C42×D83C814yescomplemented
40C22⋊(C83⋊C8)—22no—
41(C2×D83)⋊C8—42no—
42C2×C4⋊C4×C83—22no—
43C2×C4⋊C4×C83C2×C411yescharacteristic
44C2×C4⋊C4×C83C2311yescharacteristic
45C2×C42×C83C2×C411yescharacteristic
46C22⋊C8×C83—22no—
47C23×D83C22⋊C411yescharacteristic
48C22×Dic83—21no—
49C22×Dic83—21no—
50C22×Dic83C22⋊C411yescharacteristic
51C22×Dic83C22⋊C412yes—
52C2×C4×D83—24no—
53C2×C4×D83—24no—
54C2×C4×D83—24no—
55C2×C4×D83—21no—
56C2×C4×D83—21no—
57C2×C4×D83—22no—
58C2×C4×D83—22no—
59C2×C4×D83—22no—
60C2×C4×D83—44no—
61C2×C4×D83—44no—
62C2×C4×D83—41no—
63C2×C4×D83—41no—
64C2×C4×D83C2×C812yes—
65C2×C4×D83M1612yes—
66C2×C83⋊C8—42no—
67C8×D83—84no—
68C4⋊Dic83—41no—
69C4⋊Dic83—44no—
70C4⋊Dic83—41no—
71Dic83⋊C4—42no—
72Dic83⋊C4—44no—
73Dic83⋊C4—44no—
74Dic83⋊C4—42no—
75C4×Dic83—21no—
76C4×Dic83—24no—
77C4×Dic83C2×C812yescomplemented
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83C22×C411yescharacteristic
81C22×C4×C83C2×D412yes—
82C2×C8×C83—42no—
83C4⋊C4×C83—41no—
84C4⋊C4×C83—41no—
85C4⋊C4×C83—44no—
86C42×C83—21no—
87C42×C83C2×C812yescomplemented
88C23×C83C2×C22⋊C411yescharacteristic
89C22×D83—22no—
90C22×D83—22no—
91C22×D83C22⋊C812yes—
92C22×D83#9911yescharacteristic
93C22×D83#9911yescharacteristic
94C22×D83#10112yes—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—21no—
99C2×C4×C83—21no—
100C2×C4×C83—22no—
101C2×C4×C83—42no—
102C2×C4×C83—42no—
103C2×C4×C83—41no—
104C2×C4×C83C22×C811yescharacteristic
105C2×C4×C83C2×M1611yescharacteristic
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83—22no—
109C2×Dic83—21no—
110C2×Dic83—21no—
111C2×Dic83—22no—
112C2×Dic83—42no—
113C2×Dic83—42no—
114C2×Dic83—41no—
115C2×Dic83C22⋊C812yes—
116C4×D83—216no—
117C4×D83—22no—
118C4×D83—22no—
119C4×D83—22no—
120C4×D83—22no—
121C4×D83—44no—
122C4×D83—82no—
123C4×D83—82no—
124C4×D83—82no—
125C8×C83—82no—
126C83⋊C8—82no—
127C22×C83—21no—
128C22×C83—21no—
129C22×C83C2×C22⋊C811yescharacteristic
130C22×C83C2×#9911yescharacteristic
131C22×C83C2×#10111yescharacteristic
132C2×D83—21no—
133C2×D83—21no—
134C2×D83—24no—
135C2×D83—24no—
136C2×D83#32412yes—
137C2×D83#32412yes—
138C2×D83#35312yes—
139C2×D83#35612yes—
140C4×C83—24no—
141C4×C83—21no—
142C4×C83—21no—
143C4×C83—42no—
144C4×C83—81no—
145Dic83—24no—
146Dic83—22no—
147Dic83—42no—
148Dic83—81no—
149C2×#1325—831nomaximal, Sylow
150C2×C83—21no—
151C2×C83—21no—
152C2×C83C2×#32411yescharacteristic
153C2×C83C2×#35311yescharacteristic
154C2×C83C2×#35611yescharacteristic
155D83—22no—
156D83#132514yesdirect product factor
157C2×#384—831no—
158C2×#324—832no—
159#1325—831no—
160#1325—832no—
161#1325D8311yescharacteristic, direct product factor
162C83C2×#132511yescharacteristic, complemented, Sylow
163C22×C4⋊C4—831no—
164C22×C4⋊C4—831no—
165C22×C4⋊C4—1662no—
166C22×C42—831no—
167C2×C22⋊C8—1662no—
168#384—831no—
169#384—831no—
170#384—831no—
171#384—1662no—
172#384C2×D8311yescharacteristic
173#324—832no—
174#324—1662no—
175#324C2×D8312yes—
176C23×C4—832no—
177C23×C4—831no—
178C23×C4—1661no—
179C23×C4—1661no—
180C22×C8—3322no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—3328no—
183C2×C4⋊C4—3321no—
184C2×C4⋊C4—3321no—
185C2×C4⋊C4—3321no—
186C2×C4⋊C4—3321no—
187C2×C4⋊C4—831no—
188C2×C4⋊C4—832no—
189C2×C4⋊C4—831no—
190C2×C4⋊C4—1661no—
191C2×C4⋊C4—1662no—
192C2×C4⋊C4—1662no—
193C2×C4⋊C4—1662no—
194C2×C4⋊C4C4×D8311yescharacteristic
195C2×C4⋊C4C22×D8311yescharacteristic
196C2×C42—831no—
197C2×C42—834no—
198C2×C42—1662no—
199C2×C42—1661no—
200C2×C42C4×D8311yescharacteristic
201C22⋊C8—22no—
202C22⋊C8—3322no—
203C22⋊C8—1662no—
204C24—831no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—3324no—
208C22×C4—3321no—
209C22×C4—3321no—
210C22×C4—3324no—
211C22×C4—832no—
212C22×C4—831no—
213C22×C4—832no—
214C22×C4—832no—
215C22×C4—1662no—
216C22×C4—1662no—
217C22×C4—1664no—
218C22×C4—1661no—
219C22×C4—1661no—
220C22×C4—1662no—
221C22×C4—1664no—
222C22×C4—1664no—
223C22×C4—1661no—
224C22×C4—1661no—
225C22×C4C2×C4×D8311yescharacteristic
226C22×C4D4×D8312yes—
227C2×C8—42no—
228C2×C8—3322no—
229C2×C8—6644no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—44no—
233C4⋊C4—3321no—
234C4⋊C4—3324no—
235C4⋊C4—3322no—
236C4⋊C4—3321no—
237C4⋊C4—3324no—
238C4⋊C4—3324no—
239C4⋊C4—3322no—
240C42—21no—
241C42—832no—
242C42—1664no—
243C42—1661no—
244C42C8×D8312yescomplemented
245C23—831no—
246C23—832no—
247C23—831no—
248C23—832no—
249C23—1662no—
250C23—1662no—
251C23C22⋊C4×D8311yescharacteristic
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—3324no—
262C2×C4—3322no—
263C2×C4—3322no—
264C2×C4—3321no—
265C2×C4—832no—
266C2×C4—6642no—
267C2×C4—6642no—
268C2×C4—6642no—
269C2×C4—1662no—
270C2×C4—1662no—
271C2×C4—1662no—
272C2×C4—1662no—
273C2×C4—16616no—
274C2×C4—1662no—
275C2×C4—1662no—
276C2×C4—1662no—
277C2×C4—1662no—
278C2×C4—1661no—
279C2×C4—1661no—
280C2×C4C2×C8×D8311yescharacteristic
281C2×C4M16×D8311yescharacteristic
282C8—82no—
283C8—6642no—
284C22—21no—
285C22—21no—
286C22—832no—
287C22—832no—
288C22—832no—
289C22—832no—
290C22—1664no—
291C22—1664no—
292C22—1661no—
293C22—1661no—
294C22C22⋊C8×D8311yescharacteristic
295C22#99×D8311yescharacteristic
296C22#101×D8311yescharacteristic
297C4—24no—
298C4—21no—
299C4—21no—
300C4—42no—
301C4—81no—
302C4—3322no—
303C4—6641no—
304C4—1664no—
305C4—1662no—
306C2—21no—
307C2—21no—
308C2—834no—
309C2—1662no—
310C2#324×D8311yescharacteristic
311C2#353×D8311yescharacteristic
312C2#356×D8311yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC2×C4×C83C22×C8
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1325C2
socleC22×C83C2×#99
prime radicalC23×D83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1325D83
hypocenterC83C2×#1325
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
3. #24905409C2semidirect
4. #24905408C2semidirect
5. C83×#1325C2semidirect
39. C42×D83C8semidirect
77. C4×Dic83C2×C8semidirect
87. C42×C83C2×C8semidirect
156. D83#1325direct (N × H)
161. #1325D83direct (N × H)
162. C83C2×#1325semidirect
244. C42C8×D83semidirect

Maximal subgroups

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

subgroupindex
#384×D832
#324×D832
#249054092
#249054082
C83×#13252
C2×#132583