Finite Group Databasebeta

← #25963360

Subgroups — #25963360

Order 11008.

4854 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 4854 conjugacy classes of subgroups — too many to draw. Only its 807 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#19148235C213yesmaximal, complemented
2#19144001C214yesmaximal, complemented
3#19143992C218yesmaximal, complemented
4#19143987C214yesmaximal, complemented
5C2×#14583089C218yesmaximal, complemented
6#19143056C213yesmaximal, complemented
7C43×H(7,𝔽2)C211yesmaximal, characteristic, complemented
8C22×#10621900C2211yescharacteristic, complemented
9#14583548—212no—
10#14583543—212no—
11#14583470C2216yescomplemented
12#14583458C22112yescomplemented
13#14583406C2216yescomplemented
14C2×#13063306C22112yescomplemented
15#14583374C2213yescomplemented
16#14583089C22116yescomplemented
17#14583089C22132yescomplemented
18#14583086C22116yescomplemented
19#14582873C22112yescomplemented
20C2×#13063240C22124yescomplemented
21C43×#585—212no—
22C43×#585C2214yes—
23C2×H(5,𝔽2)×C43—212no—
24C2×H(5,𝔽2)×C43C2218yes—
25C2×H(5,𝔽2)×C43C2213yes—
26C2×#10621900—212no—
27C22×Dic86C2311yescharacteristic, complemented
28C22×C22⋊D43C2312yescomplemented
29C23×Dic43C2312yescomplemented
30C22×C4×D43C2311yescharacteristic, complemented
31#13063323—26no—
32#13063312—26no—
33#13063308—212no—
34#13063306—224no—
35#13063306C23112yescomplemented
36#13063306C23124yescomplemented
37#13063304—212no—
38#13063301—26no—
39#13063297—224no—
40#13063296—26no—
41#13063294—212no—
42#13063293C23112yescomplemented
43D4×Dic43—212no—
44C4×C22⋊D43—212no—
45#13063284C2316yescomplemented
46C2×Dic43⋊C4—212no—
47C2×C22⋊Dic43—212no—
48C2×C22⋊Dic43C23112yescomplemented
49#13063240C23148yescomplemented
502−1+4×C43—216no—
512−1+4×C43—212no—
52H(5,𝔽2)×C43—248no—
53H(5,𝔽2)×C43—212no—
54H(5,𝔽2)×C43—248no—
55H(5,𝔽2)×C43—224no—
56H(5,𝔽2)×C43C23116yes—
57C2×#41×C43—248no—
58C2×#41×C43—224no—
59C2×#41×C43—224no—
60C2×#41×C43D4112yescomplemented
61C2×#41×C43C2316yes—
62C22×D4×C43—216no—
63C22×D4×C43—224no—
64C22×D4×C43D4112yescomplemented
65C22×D4×C43C23112yes—
66C22×D4×C43C2311yescharacteristic
67C24×C43—28no—
68C24×C43C2×D4112yes—
69C24×C43C2412yes—
70C23×D43C2411yescharacteristic
71#41×C43—296no—
72#41×C43—248no—
73#41×C43—216no—
74#41×C43—248no—
75#41×C43—248no—
76#41×C43—224no—
77C2×Q8×C43—224no—
78C2×Q8×C43—28no—
79C2×Q8×C43C2×D416yes—
80C2×D4×C43—224no—
81C2×D4×C43—296no—
82C2×D4×C43—224no—
83C2×D4×C43—248no—
84C2×D4×C43—248no—
85C2×D4×C43—248no—
86C2×D4×C43C2×D416yes—
87C2×D4×C43C2×D4124yes—
88C2×D4×C43C2×D4112yes—
89C2×D4×C43C24112yes—
90C22×C4×C43—216no—
91C22×C4×C43—224no—
92C22×C4×C43C2×D4112yes—
93C22×C4×C43C2×D4112yes—
94C22×C4×C43C2411yescharacteristic
95C2×C22⋊D43—212no—
96C2×C22⋊D43—212no—
97C22×Dic43—212no—
98C22×Dic43—212no—
99C22×Dic43C2411yescharacteristic
100C22×Dic43C2412yes—
101#10621900—416no—
102C2×C4×D43—26no—
103C2×Dic86—26no—
104C22⋊Dic43—212no—
105C22⋊Dic43—224no—
106C22⋊Dic43—212no—
107C22⋊Dic43C24112yes—
108(C2×D43)⋊C4—212no—
109C4⋊Dic43—26no—
110Dic43⋊C4—212no—
111Dic43⋊C4—224no—
112C4×Dic43—26no—
113C23×C43—216no—
114C23×C43—224no—
115C23×C43—82no—
116C23×C43—812no—
117C23×C43—168no—
118C23×C43C22×D4112yes—
119C23×C43C22×D4112yes—
120C23×C43C2511yescharacteristic
121C22×D43—23no—
122C22×D43—81no—
123Q8×C43—216no—
124Q8×C43—212no—
125D4×C43—248no—
126D4×C43—248no—
127D4×C43—224no—
128D4×C43—212no—
129D4×C43#120116yes—
130C2×C4×C43—248no—
131C2×C4×C43—224no—
132C2×C4×C43—224no—
133C2×C4×C43C22×D4112yes—
134C2×C4×C43C22×D416yes—
135C22⋊D43—48no—
136C22⋊D43—812no—
137C2×Dic43—212no—
138C2×Dic43—26no—
139C2×Dic43—23no—
140C2×Dic43—48no—
141C2×Dic43—812no—
142C4×D43—44no—
143Dic86—44no—
144#60401—431nomaximal, Sylow
145C22×C43—212no—
146C22×C43—41no—
147C22×C43—412no—
148C22×C43—412no—
149C22×C43—824no—
150C22×C43—816no—
151C22×C43C2×#12018yes—
152C22×C43C23×D413yes—
153C2×D43—46no—
154C2×D43—41no—
155C4×C43—212no—
156C4×C43C2×#12014yes—
157Dic43—412no—
158Dic43—42no—
159Dic43—41no—
160H(7,𝔽2)D4311yescharacteristic, complemented
161#3485—433no—
162#3028—434no—
163#3026—438no—
164#3025—434no—
165C2×#458—438no—
166#2883—433no—
167C2×C43—23no—
168C2×C43—28no—
169C2×C43—412no—
170C2×C43C22×#12011yescharacteristic
171D43—81no—
172#585—212no—
173#585C2×D4314yes—
174C2×H(5,𝔽2)—212no—
175C2×H(5,𝔽2)C2×D4318yes—
176C2×H(5,𝔽2)C2×D4313yes—
177C22×#41—431no—
178#547—8612no—
179#546—8612no—
180#536—436no—
181#534—4312no—
182#525—436no—
183C2×#123—4312no—
184#518—433no—
185#458—4316no—
186#458—4332no—
187B(4,𝔽2)—4316no—
188#410—4312no—
189C2×#99—4324no—
190C43#6040111yescharacteristic, complemented, Sylow
1912−1+4—216no—
1922−1+4—212no—
193H(5,𝔽2)—248no—
194H(5,𝔽2)—212no—
195H(5,𝔽2)—248no—
196H(5,𝔽2)—224no—
197H(5,𝔽2)C22×D43116yes—
198C2×#41—248no—
199C2×#41—224no—
200C2×#41—224no—
201C2×#41—8612no—
202C2×#41C22⋊D43112yescomplemented
203C2×#41C22×D4316yes—
204C22×Q8—431no—
205C22×D4—216no—
206C22×D4—224no—
207C22×D4—432no—
208C22×D4C22⋊D43112yescomplemented
209C22×D4C22×D43112yes—
210C22×D4C22×D4311yescharacteristic
211C23×C4—432no—
212C23×C4—431no—
213C4⋊Q8—866no—
214#124—866no—
215#123—4312no—
216#123—4324no—
217#123—8624no—
218#123—8612no—
219C22⋊Q8—866no—
220C22⋊Q8—8612no—
221#121—866no—
222#121—8624no—
223#120—4312no—
224#120—8612no—
225C4×D4—8612no—
226C4×D4—8612no—
227#117—436no—
228C2×C4⋊C4—8612no—
229C2×C22⋊C4—4312no—
230C2×C22⋊C4—8612no—
231#99—4348no—
232C24—28no—
233C24—431no—
234C24C2×C22⋊D43112yes—
235C24C23×D4312yes—
236#41—296no—
237#41—248no—
238#41—216no—
239#41—248no—
240#41—248no—
241#41—224no—
242#41—17216no—
243C2×Q8—224no—
244C2×Q8—28no—
245C2×Q8—866no—
246C2×Q8C2×C22⋊D4316yes—
247C2×D4—296no—
248C2×D4—224no—
249C2×D4—224no—
250C2×D4—248no—
251C2×D4—248no—
252C2×D4—248no—
253C2×D4—8612no—
254C2×D4—8612no—
255C2×D4C2×C22⋊D43124yes—
256C2×D4C2×C22⋊D4316yes—
257C2×D4C2×C22⋊D43112yes—
258C2×D4C23×D43112yes—
259C22×C4—224no—
260C22×C4—216no—
261C22×C4—431no—
262C22×C4—432no—
263C22×C4—8612no—
264C22×C4—8612no—
265C22×C4—866no—
266C22×C4C2×C22⋊D43112yes—
267C22×C4C2×C22⋊D43112yes—
268C22×C4C23×D4311yescharacteristic
269C4⋊C4—8612no—
270C4⋊C4—8624no—
271C4⋊C4—866no—
272C22⋊C4—4312no—
273C22⋊C4—8612no—
274C22⋊C4—8612no—
275C22⋊C4—8624no—
276C22⋊C4—8612no—
277C42—866no—
278C23—216no—
279C23—224no—
280C23—82no—
281C23—812no—
282C23—168no—
283C23—863no—
284C23—3441no—
285C23C22×C22⋊D43112yes—
286C23C22×C22⋊D43112yes—
287C23C24×D4311yescharacteristic
288Q8—216no—
289Q8—212no—
290Q8—1724no—
291D4—248no—
292D4—248no—
293D4—224no—
294D4—212no—
295D4—34412no—
296D4—1728no—
297D4#13063294116yes—
298C2×C4—248no—
299C2×C4—224no—
300C2×C4—224no—
301C2×C4—8612no—
302C2×C4—863no—
303C2×C4—866no—
304C2×C4—34412no—
305C2×C4—1728no—
306C2×C4—1724no—
307C2×C4C22×C22⋊D43112yes—
308C2×C4C22×C22⋊D4316yes—
309C22—212no—
310C22—412no—
311C22—41no—
312C22—412no—
313C22—824no—
314C22—816no—
315C22—1726no—
316C22—1721no—
317C22C2×#1306329418yes—
318C22C23×C22⋊D4313yes—
319C4—212no—
320C4—17212no—
321C4—1721no—
322C4—1722no—
323C4C2×#1306329414yes—
324C2—23no—
325C2—28no—
326C2—412no—
327C2—3441no—
328C2C22×#1306329411yescharacteristic
subgroupG/H
centerC2C22×#13063294
commutator, cosocleC23×C43C25
FrattiniC23C24×D43
FittingC43×H(7,𝔽2)C2
socleC2×C43C22×#120
hypercenterH(7,𝔽2)D43
hypocenterC43#60401
LCSG ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C22×D4 ◁ H(7,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #19148235C2semidirect
2. #19144001C2semidirect
3. #19143992C2semidirect
4. #19143987C2semidirect
5. C2×#14583089C2semidirect
6. #19143056C2semidirect
7. C43×H(7,𝔽2)C2semidirect
8. C22×#10621900C22semidirect
11. #14583470C22semidirect
12. #14583458C22semidirect
13. #14583406C22semidirect
14. C2×#13063306C22semidirect
15. #14583374C22semidirect
16. #14583089C22semidirect
17. #14583089C22semidirect
18. #14583086C22semidirect
19. #14582873C22semidirect
20. C2×#13063240C22semidirect
27. C22×Dic86C23semidirect
28. C22×C22⋊D43C23semidirect
29. C23×Dic43C23semidirect
30. C22×C4×D43C23semidirect
35. #13063306C23semidirect
36. #13063306C23semidirect
42. #13063293C23semidirect
45. #13063284C23semidirect
48. C2×C22⋊Dic43C23semidirect
49. #13063240C23semidirect
60. C2×#41×C43D4semidirect
64. C22×D4×C43D4semidirect
160. H(7,𝔽2)D43semidirect
190. C43#60401semidirect
202. C2×#41C22⋊D43semidirect
208. C22×D4C22⋊D43semidirect

Maximal subgroups

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

subgroupindex
#191482352
#191440012
#191439922
#191439872
C2×#145830892
#191430562
C43×H(7,𝔽2)2
#6040143