Finite Group Databasebeta

← #25129422

Subgroups — #25129422

Order 11008.

1314 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1314 conjugacy classes of subgroups — too many to draw. Only its 187 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#19148239C211yesmaximal, characteristic
2#19144526C212yesmaximal, complemented
3#19135812C214yesmaximal, complemented
4#19135038C212yesmaximal, complemented
5#19134804C212yesmaximal, complemented
6#19134548C212yesmaximal, complemented
7#19134523C211yesmaximal, characteristic, complemented
8C43×#2297C211yesmaximal, characteristic, complemented
9C2×#13063370C2211yescharacteristic
10#14583548—24no—
11#14583537—24no—
12#14583473C2212yes—
13#14583458—22no—
14#14583410C2212yes—
15C2×#13063308—22no—
16#14583374C2211yescharacteristic
17#14583240—24no—
18#14583223—24no—
19#14582939C2214yescomplemented
20C2×#13063253C2214yes—
21#14582898C2214yes—
22C2×#13063244C2212yescomplemented
23C2×#13063242C2212yes—
24#14582863C2212yescomplemented
25#14582659C2214yescomplemented
26C22×#41×C43C2211yescharacteristic
27C2×#117×C43C2211yescharacteristic, complemented
28C22×C42×C43C2211yescharacteristic, complemented
29C43×#391C2212yescomplemented
30C43×#390C2212yes—
31C43×#386—22no—
32C4×C22⋊C4×C43—22no—
33#13063370—24no—
34#13063370—42no—
35C2×#10621898C2312yes—
36C2×#10621900—22no—
37C22×C22⋊D43—22no—
38C2×D4×D43—22no—
39C2×#10621877C2314yes—
40#13063323—22no—
41#13063314—22no—
42#13063308—28no—
43#13063308—22no—
44#13063308—24no—
45#13063306—24no—
46#13063304D414yes—
47#13063301—24no—
48#13063299D414yes—
49#13063297—28no—
50#13063294—22no—
51#13063293—24no—
52D4×Dic43—24no—
53C4×C22⋊D43—24no—
54#13063284C2312yes—
55C2×Dic43⋊C4—24no—
56C2×C22⋊Dic43—22no—
57C2×(C2×D43)⋊C4—24no—
58#13063258—24no—
59#13063256—24no—
60#13063253—44no—
61#13063248—24no—
62#13063245—24no—
63#13063244—42no—
64#13063242—24no—
65C22⋊(C43⋊C8)—24no—
66C2×#41×C43—24no—
67C2×#41×C43—24no—
68C2×#41×C43—42no—
69C22×Q8×C43C2311yescharacteristic
70C22×D4×C43D412yes—
71C22×D4×C43C2311yescharacteristic
72C23×C4×C43—21no—
73C23×C4×C43D412yes—
74C23×C4×C43C2311yescharacteristic
75#117×C43—24no—
76C2×C4⋊C4×C43C2312yes—
77C2×C22⋊C4×C43—21no—
78C2×C22⋊C4×C43—24no—
79C2×C42×C43—21no—
80C2×C42×C43—21no—
81C2×C42×C43—42no—
82C2×C42×C43C2312yes—
83#95×C43—24no—
84C24×C43C2×D411yescharacteristic
85C24×C43C2×D412yes—
86C23×D43—22no—
87#41×C43—416no—
88C2×Q8×C43—22no—
89C2×Q8×C43—22no—
90C2×Q8×C43—22no—
91C2×D4×C43—22no—
92C2×D4×C43—22no—
93C2×D4×C43—24no—
94C2×D4×C43—24no—
95C2×D4×C43—42no—
96C2×D4×C43#4114yes—
97C22×C4×C43—22no—
98C22×C4×C43—22no—
99C22×C4×C43—24no—
100C22×C4×C43—24no—
101C22×C4×C43—22no—
102C22×C4×C43—22no—
103C22×C4×C43—42no—
104C22×C4×C43—42no—
105C22×C4×C43—44no—
106C22×C4×C43C2×D414yes—
107C22×C4×C43C2×D412yes—
108C22×C4×C43C2×D411yescharacteristic
109C22×C4×C43C2411yescharacteristic
110C2×C22⋊D43—24no—
111C2×C22⋊D43—24no—
112C2×C22⋊D43—22no—
113C2×C22⋊D43—44no—
114C2×C22⋊D43—44no—
115C22×Dic43—22no—
116C22×Dic43—24no—
117#10621900—48no—
118D4×D43—48no—
119#10621898—24no—
120#10621898—24no—
121C2×D172C2×D412yes—
122C2×C4×D43—22no—
123C2×Dic86C2×D412yes—
124C4⋊C4×C43—24no—
125C22⋊C4×C43—48no—
126C22⋊C4×C43—42no—
127C42×C43—24no—
128C42×C43—42no—
129C42×C43—42no—
130C22⋊Dic43—24no—
131C22⋊Dic43—24no—
132C22⋊Dic43—22no—
133(C2×D43)⋊C4—28no—
134(C2×D43)⋊C4—24no—
135C4⋊Dic43C2×D414yes—
136Dic43⋊C4—28no—
137Dic43⋊C4—24no—
138C4×Dic43—22no—
139#10621877—24no—
140#10621877—24no—
141C2×C43⋊C8—24no—
142C23×C43—22no—
143C23×C43—21no—
144C23×C43—21no—
145C23×C43—22no—
146C23×C43—42no—
147C23×C43—44no—
148C23×C43#12012yes—
149C23×C43#12314yes—
150C23×C43C22×D411yescharacteristic
151C22×D43—22no—
152C22×D43—44no—
153C22×D43—84no—
154Q8×C43—44no—
155D4×C43—28no—
156D4×C43—48no—
157C2×C4×C43—21no—
158C2×C4×C43—21no—
159C2×C4×C43—28no—
160C2×C4×C43—24no—
161C2×C4×C43—22no—
162C2×C4×C43—22no—
163C2×C4×C43—22no—
164C2×C4×C43—48no—
165C2×C4×C43—42no—
166C2×C4×C43—48no—
167C2×C4×C43—42no—
168C2×C4×C43—44no—
169C2×C4×C43—42no—
170C2×C4×C43#12012yes—
171C2×C4×C43C22×D412yes—
172C2×C4×C43C2×#4112yes—
173C2×C4×C43H(5,𝔽2)12yes—
174C22⋊D43—48no—
175C22⋊D43—42no—
176C22⋊D43—42no—
177C22⋊D43—84no—
178C2×Dic43—24no—
179C2×Dic43—22no—
180C2×Dic43—44no—
181C2×Dic43—84no—
182D172—24no—
183C4×D43—44no—
184Dic86—24no—
185C43⋊C8—44no—
186#24021—431nomaximal, Sylow
187C22×C43—22no—
188C22×C43—24no—
189C22×C43—22no—
190C22×C43—24no—
191C22×C43—44no—
192C22×C43—42no—
193C22×C43—42no—
194C22×C43—44no—
195C22×C43—82no—
196C22×C43C2×#12011yescharacteristic
197C22×C43C2×#12311yescharacteristic
198C22×C43#53411yescharacteristic
199C2×D43—42no—
200C2×D43—44no—
201C2×D43—164no—
202C4×C43—22no—
203C4×C43—44no—
204C4×C43—42no—
205C4×C43—82no—
206C4×C43#54614yes—
207Dic43—42no—
208Dic43—44no—
209#3485—431no—
210#3093—432no—
211#2297D4311yescharacteristic, complemented
212#2038—434no—
213#1896—432no—
214#1856—432no—
215#1792—432no—
216#1785—431no—
217C2×C43—21no—
218C2×C43—21no—
219C2×C43—44no—
220C2×C43—42no—
221C2×C43#240511yescharacteristic
222C2×C43#301812yes—
223D43—82no—
224C2×H(5,𝔽2)—431no—
225C22×#41C2×D4311yescharacteristic
226#547—864no—
227#546—864no—
228#536—432no—
229#534—862no—
230#525—432no—
231C2×#123—862no—
232#518—431no—
233C2×#117C2×D4311yescharacteristic, complemented
234C22×C42C2×D4311yescharacteristic, complemented
235#485—864no—
236#482—864no—
237#428—434no—
238C2×#104—434no—
239#417—434no—
240C2×#101—432no—
241C2×#100—432no—
242#407—432no—
243#391C2×D4312yescomplemented
244#390C2×D4312yes—
245#386—22no—
246C4×C22⋊C4—22no—
247#339—434no—
248C43#2402111yescharacteristic, complemented, Sylow
249H(5,𝔽2)—864no—
250H(5,𝔽2)—1722no—
251C2×#41—24no—
252C2×#41—24no—
253C2×#41—42no—
254C2×#41—432no—
255C2×#41—862no—
256C22×Q8C22×D4311yescharacteristic
257C22×D4—862no—
258C22×D4—862no—
259C22×D4C22⋊D4312yes—
260C22×D4C22×D4311yescharacteristic
261C23×C4—21no—
262C23×C4C22⋊D4312yes—
263C23×C4C22×D4311yescharacteristic
264C2×M16—434no—
265C4⋊Q8—862no—
266#124—862no—
267#123—868no—
268#123—864no—
269#123—862no—
270#123—864no—
271C22⋊Q8—434no—
272C22⋊Q8—864no—
273#121—434no—
274#121—868no—
275#120—864no—
276#120—862no—
277C4×D4—864no—
278C4×D4—864no—
279#117—24no—
280#117—432no—
281C2×C4⋊C4—864no—
282C2×C4⋊C4C22×D4312yes—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—864no—
286C2×C22⋊C4—862no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—42no—
290C2×C42C22×D4312yes—
291#107—864no—
292#106—864no—
293#104—1724no—
294#103—864no—
295#102—864no—
296#101—1722no—
297#100—864no—
298C22⋊C8—864no—
299#95—24no—
300C24—862no—
301C24C2×C22⋊D4311yescharacteristic
302C24C2×C22⋊D4312yes—
303#41—416no—
304#41—864no—
305#41—864no—
306#41—1728no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—432no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—42no—
316C2×D4—432no—
317C2×D4—864no—
318C2×D4—864no—
319C2×D4—862no—
320C2×D4—1728no—
321C2×D4—1724no—
322C2×D4—1724no—
323C2×D4#1062189814yes—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—864no—
334C22×C4—862no—
335C22×C4—862no—
336C22×C4D4×D4314yes—
337C22×C4C2×C22⋊D4312yes—
338C22×C4C2×C22⋊D4311yescharacteristic
339C22×C4C23×D4311yescharacteristic
340M16—864no—
341M16—864no—
342C2×C8—864no—
343C4⋊C4—24no—
344C4⋊C4—434no—
345C4⋊C4—868no—
346C4⋊C4—864no—
347C22⋊C4—48no—
348C22⋊C4—42no—
349C22⋊C4—868no—
350C22⋊C4—864no—
351C22⋊C4—864no—
352C22⋊C4—862no—
353C22⋊C4—864no—
354C42—24no—
355C42—42no—
356C42—42no—
357C42—862no—
358C23—22no—
359C23—21no—
360C23—21no—
361C23—22no—
362C23—42no—
363C23—44no—
364C23—862no—
365C23—3444no—
366C23—1724no—
367C23#1306329312yes—
368C23#1306330814yes—
369C23C22×C22⋊D4311yescharacteristic
370Q8—44no—
371Q8—864no—
372D4—28no—
373D4—48no—
374D4—864no—
375D4—3444no—
376D4—1728no—
377D4—1722no—
378D4—1722no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—28no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—48no—
387C2×C4—42no—
388C2×C4—48no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—864no—
393C2×C4—862no—
394C2×C4—3444no—
395C2×C4—1724no—
396C2×C4—1724no—
397C2×C4#1306329312yes—
398C2×C4C2×D4×D4312yes—
399C2×C4C2×#1062189812yes—
400C2×C4#1306337012yes—
401C8—1724no—
402C22—22no—
403C22—24no—
404C22—22no—
405C22—24no—
406C22—44no—
407C22—42no—
408C22—42no—
409C22—44no—
410C22—82no—
411C22—1724no—
412C22—1722no—
413C22—6884no—
414C22C2×#1306329311yescharacteristic
415C22C2×#1306330811yescharacteristic
416C22#1458345811yescharacteristic
417C4—22no—
418C4—44no—
419C4—42no—
420C4—82no—
421C4—1724no—
422C4—1722no—
423C4#1458353914yes—
424C2—21no—
425C2—21no—
426C2—42no—
427C2—44no—
428C2—3442no—
429C2#1913792211yescharacteristic
430C2#1914394312yes—
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2297C2
socleC22×C43C2×#120
prime radical#19148239C2
hypercenter#2297D43
hypocenterC43#24021
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. #19144526C2semidirect
3. #19135812C2semidirect
4. #19135038C2semidirect
5. #19134804C2semidirect
6. #19134548C2semidirect
7. #19134523C2semidirect
8. C43×#2297C2semidirect
19. #14582939C22semidirect
22. C2×#13063244C22semidirect
24. #14582863C22semidirect
25. #14582659C22semidirect
27. C2×#117×C43C22semidirect
28. C22×C42×C43C22semidirect
29. C43×#391C22semidirect
211. #2297D43semidirect
233. C2×#117C2×D43semidirect
234. C22×C42C2×D43semidirect
243. #391C2×D43semidirect
248. C43#24021semidirect

Maximal subgroups

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

subgroupindex
#191482392
#191445262
#191358122
#191350382
#191348042
#191345482
#191345232
C43×#22972
#2402143