Finite Group Databasebeta

← #25963348

Subgroups — #25963348

Order 11008.

5128 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 5128 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#19148226C213yesmaximal, complemented
2#19143997C214yesmaximal, complemented
3#19143992C218yesmaximal, complemented
4#19143987C214yesmaximal, complemented
5C2×#14583086C218yesmaximal, complemented
6#19143056C213yesmaximal, complemented
7C43×H(7,𝔽2)C211yesmaximal, characteristic, complemented
8C22×D4×D43C2211yescharacteristic, complemented
9#14583537—212no—
10D4×C22⋊D43—212no—
11#14583463C2216yescomplemented
12#14583458C22112yescomplemented
13#14583406C2216yescomplemented
14C2×#13063293C22112yescomplemented
15#14583374C2213yescomplemented
16#14583089C22116yescomplemented
17#14583086C22116yescomplemented
18#14583086C22132yescomplemented
19#14582873C22112yescomplemented
20C2×#13063240C22124yescomplemented
21C43×#585—212no—
22C43×#585C2214yescomplemented
23C2×H(5,𝔽2)×C43—212no—
24C2×H(5,𝔽2)×C43C2213yes—
25C2×H(5,𝔽2)×C43C2218yescomplemented
26C24×D43C2312yescomplemented
27C22×D172C2311yescharacteristic, complemented
28C22×C22⋊D43—212no—
29C22×C22⋊D43C2312yescomplemented
30C2×D4×D43—212no—
31C22×C4×D43C2311yescharacteristic, complemented
32#13063321—26no—
33#13063314—26no—
34#13063308—212no—
35#13063306C23112yescomplemented
36#13063301—26no—
37#13063299—212no—
38#13063297—224no—
39#13063296—26no—
40#13063294—212no—
41#13063293—224no—
42#13063293C23112yescomplemented
43#13063293C23124yescomplemented
44D4×Dic43—212no—
45C4×C22⋊D43—212no—
46#13063284C2316yescomplemented
47C2×C22⋊Dic43C23112yescomplemented
48C2×(C2×D43)⋊C4—212no—
49#13063240C23148yescomplemented
502−1+4×C43—216no—
512−1+4×C43—212no—
52H(5,𝔽2)×C43—248no—
53H(5,𝔽2)×C43—224no—
54H(5,𝔽2)×C43—212no—
55H(5,𝔽2)×C43—248no—
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×D4112yescomplemented
69C24×C43C2412yes—
70C23×D43—212no—
71C23×D43—84no—
72C23×D43C2411yescharacteristic
73C23×D43C2412yes—
74#41×C43—296no—
75#41×C43—248no—
76#41×C43—216no—
77#41×C43—248no—
78#41×C43—248no—
79#41×C43—224no—
80C2×Q8×C43—224no—
81C2×Q8×C43—28no—
82C2×Q8×C43C2×D416yescomplemented
83C2×D4×C43—224no—
84C2×D4×C43—296no—
85C2×D4×C43—224no—
86C2×D4×C43—248no—
87C2×D4×C43—248no—
88C2×D4×C43—248no—
89C2×D4×C43C2×D4112yes—
90C2×D4×C43C2×D416yescomplemented
91C2×D4×C43C2×D4124yescomplemented
92C2×D4×C43C24112yes—
93C22×C4×C43—216no—
94C22×C4×C43—224no—
95C22×C4×C43C2×D4112yescomplemented
96C22×C4×C43C2×D4112yescomplemented
97C22×C4×C43C2411yescharacteristic
98C2×C22⋊D43—212no—
99C2×C22⋊D43—212no—
100C2×C22⋊D43—224no—
101C2×C22⋊D43—824no—
102C22×Dic43—212no—
103C22×Dic43C2411yescharacteristic
104D4×D43—416no—
105C2×D172—26no—
106C2×C4×D43—26no—
107C22⋊Dic43—212no—
108C22⋊Dic43—212no—
109C22⋊Dic43C24112yes—
110(C2×D43)⋊C4—212no—
111(C2×D43)⋊C4—224no—
112C4⋊Dic43—26no—
113Dic43⋊C4—212no—
114C4×Dic43—26no—
115C23×C43—216no—
116C23×C43—224no—
117C23×C43—82no—
118C23×C43—812no—
119C23×C43—168no—
120C23×C43C22×D4112yes—
121C23×C43C22×D4112yes—
122C23×C43C2511yescharacteristic
123C22×D43—23no—
124C22×D43—26no—
125C22×D43—42no—
126C22×D43—424no—
127C22×D43—48no—
128C22×D43—82no—
129C22×D43—812no—
130Q8×C43—216no—
131Q8×C43—212no—
132D4×C43—248no—
133D4×C43—248no—
134D4×C43—224no—
135D4×C43—212no—
136D4×C43#120116yes—
137C2×C4×C43—248no—
138C2×C4×C43—224no—
139C2×C4×C43—224no—
140C2×C4×C43C22×D4112yes—
141C2×C4×C43C22×D416yes—
142C22⋊D43—424no—
143C22⋊D43—48no—
144C22⋊D43—824no—
145C2×Dic43—212no—
146C2×Dic43—23no—
147C2×Dic43—812no—
148D172—44no—
149C4×D43—44no—
150#60399—431nomaximal, Sylow
151C22×C43—212no—
152C22×C43—41no—
153C22×C43—412no—
154C22×C43—412no—
155C22×C43—824no—
156C22×C43—816no—
157C22×C43C2×#12018yes—
158C22×C43C23×D413yes—
159C2×D43—26no—
160C2×D43—43no—
161C2×D43—41no—
162C2×D43—412no—
163C2×D43—42no—
164C2×D43—816no—
165C4×C43—212no—
166C4×C43C2×#12014yes—
167Dic43—412no—
168Dic43—41no—
169H(7,𝔽2)D4311yescharacteristic, complemented
170#3484—433no—
171#3027—434no—
172#3026—438no—
173#3025—434no—
174C2×B(4,𝔽2)—438no—
175#2883—433no—
176C2×C43—23no—
177C2×C43—28no—
178C2×C43—412no—
179C2×C43C22×#12011yescharacteristic
180D43—42no—
181D43—82no—
182#585—212no—
183#585C2×D4314yescomplemented
184C2×H(5,𝔽2)—212no—
185C2×H(5,𝔽2)C2×D4313yes—
186C2×H(5,𝔽2)C2×D4318yescomplemented
187C23×D4—431no—
188#546—8612no—
189D42—8612no—
190#535—436no—
191#534—4312no—
192#525—436no—
193C2×#120—4312no—
194#518—433no—
195#458—4316no—
196B(4,𝔽2)—4316no—
197B(4,𝔽2)—4332no—
198#410—4312no—
199C2×#99—4324no—
200C43#6039911yescharacteristic, complemented, Sylow
201C25—432no—
2022−1+4—216no—
2032−1+4—212no—
204H(5,𝔽2)—248no—
205H(5,𝔽2)—224no—
206H(5,𝔽2)—212no—
207H(5,𝔽2)—248no—
208H(5,𝔽2)C22×D43116yes—
209C2×#41—248no—
210C2×#41—224no—
211C2×#41—224no—
212C2×#41C22⋊D43112yescomplemented
213C2×#41C22×D4316yes—
214C22×D4—216no—
215C22×D4—224no—
216C22×D4—432no—
217C22×D4—431no—
218C22×D4—8612no—
219C22×D4—8612no—
220C22×D4C22⋊D43112yescomplemented
221C22×D4C22×D43112yes—
222C22×D4C22×D4311yescharacteristic
223C23×C4—431no—
224#127—866no—
225#124—866no—
226#123—4312no—
227#123—8612no—
228C22⋊Q8—866no—
229#121—866no—
230#121—8624no—
231#121—8612no—
232#120—4312no—
233#120—4324no—
234#120—8612no—
235#120—8624no—
236C4×D4—8612no—
237C4×D4—8612no—
238#117—436no—
239C2×C22⋊C4—4312no—
240C2×C22⋊C4—8612no—
241#99—4348no—
242C24—28no—
243C24—432no—
244C24—431no—
245C24—8612no—
246C24—3444no—
247C24C2×C22⋊D43112yescomplemented
248C24C23×D4312yes—
249#41—296no—
250#41—248no—
251#41—216no—
252#41—248no—
253#41—248no—
254#41—224no—
255C2×Q8—224no—
256C2×Q8—28no—
257C2×Q8C2×C22⋊D4316yescomplemented
258C2×D4—296no—
259C2×D4—224no—
260C2×D4—224no—
261C2×D4—248no—
262C2×D4—248no—
263C2×D4—248no—
264C2×D4—8624no—
265C2×D4—8612no—
266C2×D4—8612no—
267C2×D4—866no—
268C2×D4—34424no—
269C2×D4—17216no—
270C2×D4C2×C22⋊D43112yes—
271C2×D4C2×C22⋊D43124yescomplemented
272C2×D4C2×C22⋊D4316yescomplemented
273C2×D4C23×D43112yes—
274C22×C4—224no—
275C22×C4—216no—
276C22×C4—431no—
277C22×C4—8612no—
278C22×C4—866no—
279C22×C4C2×C22⋊D43112yescomplemented
280C22×C4C2×C22⋊D43112yescomplemented
281C22×C4C23×D4311yescharacteristic
282C4⋊C4—866no—
283C4⋊C4—8612no—
284C22⋊C4—4312no—
285C22⋊C4—8612no—
286C22⋊C4—8612no—
287C22⋊C4—8612no—
288C22⋊C4—8624no—
289C42—866no—
290C23—216no—
291C23—224no—
292C23—82no—
293C23—812no—
294C23—168no—
295C23—863no—
296C23—866no—
297C23—34412no—
298C23—3442no—
299C23—1722no—
300C23—17224no—
301C23—1728no—
302C23C22×C22⋊D43112yes—
303C23C22×C22⋊D43112yes—
304C23C24×D4311yescharacteristic
305Q8—216no—
306Q8—212no—
307D4—248no—
308D4—248no—
309D4—224no—
310D4—212no—
311D4—34424no—
312D4—17224no—
313D4—1728no—
314D4—1724no—
315D4#13063294116yes—
316C2×C4—248no—
317C2×C4—224no—
318C2×C4—224no—
319C2×C4—8612no—
320C2×C4—863no—
321C2×C4—34412no—
322C2×C4—1724no—
323C2×C4C22×C22⋊D43112yes—
324C2×C4C22×C22⋊D4316yes—
325C22—212no—
326C22—412no—
327C22—41no—
328C22—412no—
329C22—824no—
330C22—816no—
331C22—866no—
332C22—34416no—
333C22—17212no—
334C22—1723no—
335C22—1722no—
336C22—1721no—
337C22C2×#1306329418yes—
338C22C23×C22⋊D4313yes—
339C4—212no—
340C4—17212no—
341C4—1721no—
342C4C2×#1306329414yes—
343C2—23no—
344C2—28no—
345C2—412no—
346C2—3442no—
347C2—1722no—
348C2C22×#1306329411yescharacteristic
subgroupG/H
centerC2C22×#13063294
commutator, cosocleC23×C43C25
FrattiniC23C24×D43
FittingC43×H(7,𝔽2)C2
socleC2×C43C22×#120
hypercenterH(7,𝔽2)D43
hypocenterC43#60399
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. #19148226C2semidirect
2. #19143997C2semidirect
3. #19143992C2semidirect
4. #19143987C2semidirect
5. C2×#14583086C2semidirect
6. #19143056C2semidirect
7. C43×H(7,𝔽2)C2semidirect
8. C22×D4×D43C22semidirect
11. #14583463C22semidirect
12. #14583458C22semidirect
13. #14583406C22semidirect
14. C2×#13063293C22semidirect
15. #14583374C22semidirect
16. #14583089C22semidirect
17. #14583086C22semidirect
18. #14583086C22semidirect
19. #14582873C22semidirect
20. C2×#13063240C22semidirect
22. C43×#585C22semidirect
25. C2×H(5,𝔽2)×C43C22semidirect
26. C24×D43C23semidirect
27. C22×D172C23semidirect
29. C22×C22⋊D43C23semidirect
31. C22×C4×D43C23semidirect
35. #13063306C23semidirect
42. #13063293C23semidirect
43. #13063293C23semidirect
46. #13063284C23semidirect
47. C2×C22⋊Dic43C23semidirect
49. #13063240C23semidirect
60. C2×#41×C43D4semidirect
64. C22×D4×C43D4semidirect
68. C24×C43C2×D4semidirect
82. C2×Q8×C43C2×D4semidirect
90. C2×D4×C43C2×D4semidirect
91. C2×D4×C43C2×D4semidirect
95. C22×C4×C43C2×D4semidirect
96. C22×C4×C43C2×D4semidirect
169. H(7,𝔽2)D43semidirect
183. #585C2×D43semidirect
186. C2×H(5,𝔽2)C2×D43semidirect
200. C43#60399semidirect
212. C2×#41C22⋊D43semidirect
220. C22×D4C22⋊D43semidirect
247. C24C2×C22⋊D43semidirect
257. C2×Q8C2×C22⋊D43semidirect
271. C2×D4C2×C22⋊D43semidirect
272. C2×D4C2×C22⋊D43semidirect
279. C22×C4C2×C22⋊D43semidirect
280. C22×C4C2×C22⋊D43semidirect

Maximal subgroups

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

subgroupindex
#191482262
#191439972
#191439922
#191439872
C2×#145830862
#191430562
C43×H(7,𝔽2)2
#6039943