Finite Group Databasebeta

← #25069375

Subgroups — #25069375

Order 11008.

1250 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1250 conjugacy classes of subgroups — too many to draw. Only its 167 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#19148359C211yesmaximal, characteristic
2#19144717C212yesmaximal, complemented
3#19133580C214yesmaximal, complemented
4#19133546C214yesmaximal, complemented
5#19133102C212yesmaximal, complemented
6#19132936C211yesmaximal, characteristic, complemented
7C43×#2398C211yesmaximal, characteristic, complemented
8C2×#13063370—21no—
9#14583553—22no—
10#14583544—24no—
11D4×D172—22no—
12#14583480—21no—
13#14583461C2214yes—
14#14583449C2212yes—
15C2×#13063299—22no—
16#14583212C2214yescomplemented
17#14583194—22no—
18#14583099—22no—
19#14583038C2214yescomplemented
20#14582997—24no—
21#14582956C2212yescomplemented
22#14582632C2218yescomplemented
23#14582623C2214yescomplemented
24C2×C4×Q8×C43C2211yescharacteristic, complemented
25C2×C4×D4×C43C2211yescharacteristic
26C43×#400—21no—
27C43×#397C2212yescomplemented
28C43×#394—21no—
29C43×#393C2212yescomplemented
30C4×C22⋊C4×C43—22no—
31C43×#376C2211yescharacteristic, complemented
32#13063370—48no—
33C2×#10621898—22no—
34C2×#10621900—24no—
35C22×D172—22no—
36C22×C22⋊D43—24no—
37C2×D4×D43—24no—
38C2×#10621884—24no—
39#13063325D412yes—
40#13063322D412yes—
41#13063313—22no—
42#13063309—28no—
43#13063305—24no—
44#13063300—24no—
45#13063299—24no—
46#13063299—24no—
47#13063299—22no—
48#13063299—24no—
49#13063299C2314yes—
50#13063295—28no—
51C4×D172—22no—
52C4×D172D414yes—
53C2×C4⋊Dic43—22no—
54C2×(C2×D43)⋊C4—24no—
55#13063258—24no—
56C4⋊(C43⋊C8)—22no—
57#13063249—24no—
58#13063245—24no—
59C22⋊(C43⋊C8)C2314yes—
60#13063236—22no—
61C22×Q8×C43C2311yescharacteristic
62C22×D4×C43D411yescharacteristic
63C23×C4×C43D412yes—
64C4×Q8×C43—22no—
65C4×Q8×C43—22no—
66C4×D4×C43—22no—
67C4×D4×C43—22no—
68C2×C4⋊C4×C43—21no—
69C2×C4⋊C4×C43D411yescharacteristic
70C2×C4⋊C4×C43C2311yescharacteristic
71C2×C22⋊C4×C43—22no—
72C2×C22⋊C4×C43—22no—
73C2×C22⋊C4×C43C2312yes—
74C2×C42×C43—21no—
75C2×C42×C43—21no—
76C2×C42×C43C2311yescharacteristic
77#95×C43—21no—
78#95×C43—22no—
79#95×C43—22no—
80#95×C43C2312yes—
81C24×C43C2×D412yes—
82C23×D43—24no—
83C2×Q8×C43—22no—
84C2×Q8×C43—42no—
85C2×D4×C43—22no—
86C2×D4×C43—24no—
87C22×C4×C43—22no—
88C22×C4×C43—22no—
89C22×C4×C43—21no—
90C22×C4×C43—22no—
91C22×C4×C43—21no—
92C22×C4×C43—44no—
93C22×C4×C43C2×D411yescharacteristic
94C22×C4×C43C2×D411yescharacteristic
95C22×C4×C43C2×D412yes—
96C22×C4×C43C2411yescharacteristic
97C2×C22⋊D43—28no—
98C2×C22⋊D43—24no—
99C2×C22⋊D43—48no—
100C2×C22⋊D43—48no—
101C22×Dic43—24no—
102#10621900—416no—
103D4×D43—416no—
104#10621898—48no—
105C2×D172—22no—
106C2×D172—22no—
107C2×D172—24no—
108C2×D172—84no—
109C2×D172C2×D414yes—
110C2×C4×D43—24no—
111C2×Dic86—22no—
112C4⋊C4×C43—22no—
113C4⋊C4×C43—22no—
114C4⋊C4×C43—42no—
115C22⋊C4×C43—24no—
116C22⋊C4×C43—44no—
117C22⋊C4×C43—44no—
118C42×C43—42no—
119C42×C43—41no—
120C42×C43—41no—
121C42×C43C2×D412yes—
122C42×C43C2×D412yes—
123#10621884—48no—
124(C2×D43)⋊C4—28no—
125(C2×D43)⋊C4—24no—
126(C2×D43)⋊C4—24no—
127C4⋊Dic43—24no—
128C4⋊Dic43—22no—
129C4⋊Dic43C2×D414yes—
130C2×C43⋊C8—24no—
131C23×C43—22no—
132C23×C43—22no—
133C23×C43—42no—
134C23×C43—42no—
135C23×C43AGL(1,ℤ/8)14yes—
136C23×C43C22×D411yescharacteristic
137C22×D43—24no—
138C22×D43—48no—
139C22×D43—84no—
140C22×D43—84no—
141Q8×C43—24no—
142Q8×C43—42no—
143D4×C43—28no—
144C2×C4×C43—22no—
145C2×C4×C43—22no—
146C2×C4×C43—22no—
147C2×C4×C43—22no—
148C2×C4×C43—41no—
149C2×C4×C43—41no—
150C2×C4×C43—41no—
151C2×C4×C43—44no—
152C2×C4×C43—42no—
153C2×C4×C43—42no—
154C2×C4×C43—42no—
155C2×C4×C43—41no—
156C2×C4×C43—42no—
157C2×C4×C43—42no—
158C2×C4×C43—82no—
159C2×C4×C43#12011yescharacteristic
160C2×C4×C43#12012yes—
161C2×C4×C43#12011yescharacteristic
162C2×C4×C43C22×D412yes—
163C22⋊D43—416no—
164C22⋊D43—48no—
165C2×Dic43—24no—
166C2×Dic43—48no—
167D172—44no—
168D172—44no—
169D172—44no—
170D172—84no—
171C4×D43—48no—
172Dic86—44no—
173C43⋊C8—44no—
174#18179—431nomaximal, Sylow
175C22×C43—21no—
176C22×C43—21no—
177C22×C43—24no—
178C22×C43—44no—
179C22×C43—42no—
180C22×C43—42no—
181C22×C43—82no—
182C22×C43C2×#12011yescharacteristic
183C22×C43C2×AGL(1,ℤ/8)12yes—
184C2×D43—44no—
185C2×D43—44no—
186C2×D43—44no—
187C2×D43—168no—
188C4×C43—22no—
189C4×C43—21no—
190C4×C43—22no—
191C4×C43—42no—
192C4×C43—42no—
193C4×C43—82no—
194C4×C43—81no—
195C4×C43#45512yes—
196Dic43—44no—
197#3496—431no—
198#3112—432no—
199#2398D4311yescharacteristic, complemented
200#1607—434no—
201#1601—434no—
202#1505—432no—
203#1466—431no—
204C2×C43—22no—
205C2×C43—44no—
206C2×C43#300311yescharacteristic
207C2×C43C2×#45511yescharacteristic
208C2×C43#302511yescharacteristic
209D43—84no—
210C2×H(5,𝔽2)—861no—
211#547—862no—
212#546—864no—
213D42—862no—
214#537—861no—
215#534—434no—
216#532—432no—
217C2×#121—862no—
218C2×C4×Q8C2×D4311yescharacteristic, complemented
219C2×C4×D4C2×D4311yescharacteristic
220#480—434no—
221#477—862no—
222#460—862no—
223#448—434no—
224#440—864no—
225#432—432no—
226#400—21no—
227#397C2×D4312yescomplemented
228#394—21no—
229#393C2×D4312yescomplemented
230C4×C22⋊C4—22no—
231#376C2×D4311yescharacteristic, complemented
232#327—438no—
233#323—434no—
234C43#1817911yescharacteristic, complemented, Sylow
235H(5,𝔽2)—1728no—
236C2×#41—864no—
237C2×#41—862no—
238C22×Q8C22×D4311yescharacteristic
239C22×D4—862no—
240C22×D4—864no—
241C22×D4—864no—
242C22×D4C22⋊D4311yescharacteristic
243C23×C4C22⋊D4312yes—
244C2×SD16—864no—
245C4⋊Q8—432no—
246#127—432no—
247#124—862no—
248#123—868no—
249C22⋊Q8—864no—
250#121—434no—
251#121—864no—
252#121—864no—
253#121—864no—
254#121—862no—
255#121—864no—
256#120—868no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—434no—
262C4×D4—862no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—862no—
265C2×C4⋊C4C22⋊D4311yescharacteristic
266C2×C4⋊C4C22×D4311yescharacteristic
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—864no—
270C2×C22⋊C4C22×D4312yes—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42C22×D4311yescharacteristic
274#107—864no—
275C4⋊C8—862no—
276#103—864no—
277#102—864no—
278C22⋊C8—434no—
279#97—862no—
280#95—21no—
281#95—22no—
282#95—22no—
283#95C22×D4312yes—
284C24—864no—
285C24C2×C22⋊D4312yes—
286#41—1728no—
287#41—17216no—
288C2×Q8—22no—
289C2×Q8—42no—
290C2×Q8—862no—
291C2×D4—24no—
292C2×D4—22no—
293C2×D4—434no—
294C2×D4—864no—
295C2×D4—868no—
296C2×D4—864no—
297C2×D4—862no—
298C2×D4—862no—
299C2×D4—3444no—
300C2×D4—17216no—
301C2×D4—1728no—
302C2×D4—1728no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—44no—
309C22×C4—864no—
310C22×C4—864no—
311C22×C4C2×C22⋊D4311yescharacteristic
312C22×C4C2×C22⋊D4311yescharacteristic
313C22×C4C2×C22⋊D4312yes—
314C22×C4C23×D4311yescharacteristic
315SD16—1728no—
316C2×C8—864no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—434no—
321C4⋊C4—864no—
322C4⋊C4—862no—
323C22⋊C4—24no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—868no—
327C22⋊C4—864no—
328C22⋊C4—864no—
329C42—42no—
330C42—41no—
331C42—41no—
332C42D4×D4312yes—
333C42D4×D4312yes—
334C23—22no—
335C23—22no—
336C23—42no—
337C23—42no—
338C23—864no—
339C23—3444no—
340C23—3444no—
341C23—1728no—
342C23#1306335014yes—
343C23C22×C22⋊D4311yescharacteristic
344Q8—24no—
345Q8—42no—
346Q8—1724no—
347D4—28no—
348D4—3444no—
349D4—1724no—
350D4—1724no—
351D4—17216no—
352D4—1728no—
353D4—1724no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—864no—
370C2×C4—1728no—
371C2×C4—1728no—
372C2×C4#1306329311yescharacteristic
373C2×C4#1306329312yes—
374C2×C4#1306329311yescharacteristic
375C2×C4C2×D4×D4312yes—
376C8—1724no—
377C22—21no—
378C22—21no—
379C22—24no—
380C22—44no—
381C22—42no—
382C22—42no—
383C22—82no—
384C22—1724no—
385C22—1724no—
386C22—1724no—
387C22—6888no—
388C22C2×#1306329311yescharacteristic
389C22C2×#1306335012yes—
390C4—22no—
391C4—22no—
392C4—21no—
393C4—42no—
394C4—42no—
395C4—82no—
396C4—81no—
397C4—1724no—
398C4#1458307612yes—
399C2—22no—
400C2—44no—
401C2—3444no—
402C2#1914382111yescharacteristic
403C2C2×#1458307611yescharacteristic
404C2#1914398611yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2398C2
socleC22×C43C2×#120
prime radical#19148359C2
hypercenter#2398D43
hypocenterC43#18179
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2398

Complements

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

NH ≅ G/Ntype
2. #19144717C2semidirect
3. #19133580C2semidirect
4. #19133546C2semidirect
5. #19133102C2semidirect
6. #19132936C2semidirect
7. C43×#2398C2semidirect
16. #14583212C22semidirect
19. #14583038C22semidirect
21. #14582956C22semidirect
22. #14582632C22semidirect
23. #14582623C22semidirect
24. C2×C4×Q8×C43C22semidirect
27. C43×#397C22semidirect
29. C43×#393C22semidirect
31. C43×#376C22semidirect
199. #2398D43semidirect
218. C2×C4×Q8C2×D43semidirect
227. #397C2×D43semidirect
229. #393C2×D43semidirect
231. #376C2×D43semidirect
234. C43#18179semidirect

Maximal subgroups

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

subgroupindex
#191483592
#191447172
#191335802
#191335462
#191331022
#191329362
C43×#23982
#1817943