Finite Group Databasebeta

← #35319556

Subgroups — #35319556

Order 21248.

470 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 131 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#24909104C212yesmaximal
2#24907263C212yesmaximal, complemented
3#24905586C211yesmaximal, characteristic, complemented
4#24905584C211yesmaximal, characteristic, complemented
5C83×#2929C211yesmaximal, characteristic
6#19127685C2212yes—
7#19127675C2212yes—
8#19127656—22no—
9#19127491C414yescomplemented
10#19127488C414yescomplemented
11#19127472—22no—
12C2×#131×C83C2211yescharacteristic
13C23×C8×C83C411yescharacteristic, complemented
14#525×C83C2211yescharacteristic
15#435×C83—22no—
16C8×D4×C83—22no—
17#424×C83C411yescharacteristic, complemented
18#408×C83C2211yescharacteristic
19#407×C83C411yescharacteristic, complemented
20#405×C83C411yescharacteristic, complemented
21C2×#13050574—22no—
22#14581751—22no—
23#14581711—22no—
24#14581709C2×C414yes—
25#14581707—22no—
26#14581707C2×C414yes—
27#14581702—42no—
28C8×Dic83—42no—
29C2×#41×C83C2311yescharacteristic
30C23×C4×C83C2×C411yescharacteristic
31#131×C83—41no—
32#131×C83—41no—
33C2×M16×C83C2×C411yescharacteristic
34C2×M16×C83C2×C411yescharacteristic
35C2×M16×C83D412yes—
36C22×C8×C83—21no—
37C22×C8×C83—21no—
38C22×C8×C83—41no—
39C22×C8×C83—41no—
40C22×C8×C83C2×C411yescharacteristic
41C22×C8×C83C2×C411yescharacteristic
42C22×C8×C83D412yes—
43#123×C83C2×C412yes—
44C22⋊Q8×C83—21no—
45#121×C83—21no—
46#120×C83C2×C412yes—
47C4×D4×C83—22no—
48#117×C83C2×C411yescharacteristic
49C4⋊C8×C83—22no—
50C22⋊C8×C83—22no—
51C22⋊C8×C83—22no—
52#97×C83—21no—
53C4×C8×C83—21no—
54#13050574—24no—
55#13050574—42no—
56C2×C83⋊C8—42no—
57C4⋊Dic83—42no—
58C22⋊Dic83—24no—
59C4×Dic83—42no—
60C24×C83C4211yescharacteristic
61#41×C83—41no—
62#41×C83—41no—
63C2×Q8×C83#4111yescharacteristic
64C2×D4×C83—22no—
65C2×D4×C83C22×C412yes—
66C2×D4×C83#4111yescharacteristic
67C22×C4×C83—21no—
68C22×C4×C83—21no—
69C22×C4×C83—41no—
70C22×C4×C83—41no—
71C22×C4×C83C4211yescharacteristic
72C22×C4×C83C22×C411yescharacteristic
73C22×C4×C83C2×D412yes—
74M16×C83—41no—
75M16×C83—42no—
76M16×C83—41no—
77C2×C8×C83—21no—
78C2×C8×C83—21no—
79C2×C8×C83—22no—
80C2×C8×C83—22no—
81C2×C8×C83—21no—
82C2×C8×C83—21no—
83C2×C8×C83—42no—
84C2×C8×C83—41no—
85C2×C8×C83—41no—
86C2×C8×C83—41no—
87C2×C8×C83C22⋊C411yescharacteristic
88C2×C8×C83C22⋊C412yes—
89C2×C8×C83C22⋊C411yescharacteristic
90C4⋊C4×C83—21no—
91C4⋊C4×C83—22no—
92C22⋊C4×C83—22no—
93C22⋊C4×C83—22no—
94C22⋊C4×C83C4212yes—
95C42×C83—21no—
96C23×C83—21no—
97C23×C83—41no—
98C23×C83—42no—
99C23×C83C2×C4211yescharacteristic
100C23×C83C4×D412yes—
101Q8×C83—41no—
102D4×C83—42no—
103D4×C83—41no—
104D4×C83—44no—
105C2×C4×C83—21no—
106C2×C4×C83—22no—
107C2×C4×C83—21no—
108C2×C4×C83—22no—
109C2×C4×C83—22no—
110C2×C4×C83—21no—
111C2×C4×C83—42no—
112C2×C4×C83—41no—
113C2×C4×C83—41no—
114C2×C4×C83—41no—
115C2×C4×C83C2×C22⋊C411yescharacteristic
116C2×C4×C83#11711yescharacteristic
117C2×C4×C83C4×D412yes—
118C2×Dic83—44no—
119C8×C83—21no—
120C8×C83—22no—
121C8×C83—41no—
122C8×C83—41no—
123C8×C83#9911yescharacteristic
124C8×C83#9911yescharacteristic
125C83⋊C8—44no—
126C22×C83—22no—
127C22×C83—22no—
128C22×C83—21no—
129C22×C83—42no—
130C22×C83—41no—
131C22×C83—41no—
132C22×C83—42no—
133C22×C83—42no—
134C22×C83—41no—
135C22×C83C4×C22⋊C411yescharacteristic
136C4×C83—22no—
137C4×C83—21no—
138C4×C83—42no—
139C4×C83—41no—
140C4×C83C2×#9911yescharacteristic
141C4×C83#41011yescharacteristic
142Dic83—84no—
143#12623—831nomaximal, Sylow
144C2×C83—21no—
145C2×C83—42no—
146C2×C83—42no—
147C2×C83—41no—
148C2×C83C4×#9911yescharacteristic
149#2929D8311yescharacteristic
150#2120—832no—
151#1757—832no—
152#1392—831no—
153#1391—831no—
154C83#1262311yescharacteristic, complemented, Sylow
155C2×#131C2×D8311yescharacteristic
156C23×C8Dic8311yescharacteristic, complemented
157#525C2×D8311yescharacteristic
158#435—22no—
159C8×D4—22no—
160#424Dic8311yescharacteristic, complemented
161#413—832no—
162#410—832no—
163#408C2×D8311yescharacteristic
164#407Dic8311yescharacteristic, complemented
165#405—1662no—
166#405Dic8311yescharacteristic, complemented
167#352—834no—
168AΣL(1,𝔽16)—834no—
169#344—1662no—
170C2×#41C22×D8311yescharacteristic
171C23×C4C2×Dic8311yescharacteristic
172#131—41no—
173#131—41no—
174C2×M16—1662no—
175C2×M16C2×Dic8311yescharacteristic
176C2×M16C2×Dic8311yescharacteristic
177C2×M16C22⋊D8312yes—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—41no—
181C22×C8—41no—
182C22×C8C2×Dic8311yescharacteristic
183C22×C8C2×Dic8311yescharacteristic
184C22×C8C22⋊D8312yes—
185#123C4×D8312yes—
186C22⋊Q8—21no—
187#121—21no—
188#120C4×D8312yes—
189C4×D4—22no—
190#117—1662no—
191#117C2×Dic8311yescharacteristic
192C4⋊C8—22no—
193#101—1662no—
194#100—834no—
195#99—834no—
196#99—1662no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199#97—21no—
200#97—3322no—
201C4×C8—21no—
202C4×C8—3322no—
203C24C4×Dic8311yescharacteristic
204#41—41no—
205#41—41no—
206C2×Q8#1305059011yescharacteristic
207C2×D4—22no—
208C2×D4C2×C4×D8312yes—
209C2×D4#1305059011yescharacteristic
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4C4×Dic8311yescharacteristic
215C22×C4C22×Dic8311yescharacteristic
216C22×C4C2×C22⋊D8312yes—
217M16—41no—
218M16—42no—
219M16—41no—
220M16—3322no—
221M16—1664no—
222C2×C8—22no—
223C2×C8—21no—
224C2×C8—22no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—42no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—3322no—
233C2×C8C22⋊Dic8311yescharacteristic
234C2×C8C22⋊Dic8312yes—
235C2×C8C22⋊Dic8311yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—22no—
238C4⋊C4—3322no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—1664no—
242C22⋊C4C4×Dic8312yes—
243C42—21no—
244C42—3322no—
245C23—21no—
246C23—41no—
247C23—42no—
248C23C2×C4×Dic8311yescharacteristic
249C23C4×C22⋊D8312yes—
250Q8—41no—
251D4—42no—
252D4—41no—
253D4—44no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—3324no—
265C2×C4C2×C22⋊Dic8311yescharacteristic
266C2×C4#1458175111yescharacteristic
267C2×C4C4×C22⋊D8312yes—
268C8—21no—
269C8—22no—
270C8—41no—
271C8—41no—
272C8—3324no—
273C8#1458170711yescharacteristic
274C8#1458170711yescharacteristic
275C22—22no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—41no—
284C22C4×C22⋊Dic8311yescharacteristic
285C4—22no—
286C4—21no—
287C4—42no—
288C4—41no—
289C4—6644no—
290C4C2×#1458170711yescharacteristic
291C4#1912767511yescharacteristic
292C2—21no—
293C2—42no—
294C2—42no—
295C2—41no—
296C2C4×#1458170711yescharacteristic
subgroupG/H
centerC8#14581707
commutatorC23×C83C2×C42
FrattiniC2×#41C22×D83
FittingC83×#2929C2
socleC2×C83C4×#99
prime radical#525×C83C22
cosocleC2×#41×C83C23
hypercenter#2929D83
hypocenterC83#12623
LCSG ▷ C23×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ #2929

Complements

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

NH ≅ G/Ntype
2. #24907263C2semidirect
3. #24905586C2semidirect
4. #24905584C2semidirect
9. #19127491C4semidirect
10. #19127488C4semidirect
13. C23×C8×C83C4semidirect
17. #424×C83C4semidirect
19. #407×C83C4semidirect
20. #405×C83C4semidirect
154. C83#12623semidirect
156. C23×C8Dic83semidirect
160. #424Dic83semidirect
164. #407Dic83semidirect
166. #405Dic83semidirect

Maximal subgroups

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

subgroupindex
#249091042
#249072632
#249055862
#249055842
C83×#29292
#1262383