Finite Group Databasebeta

← #37543071

Subgroups — #37543071

Order 23168.

724 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 133 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#26042114C211yesmaximal, characteristic, complemented
2#26042078C212yesmaximal, complemented
3#26041828C211yesmaximal, characteristic, complemented
4#26041460C212yesmaximal, complemented
5#26041456C211yesmaximal, characteristic, complemented
6#26041453C212yesmaximal, complemented
7#26041449C211yesmaximal, characteristic, complemented
8#26041447C211yesmaximal, characteristic, complemented
9#104×D181C212yesmaximal, complemented
10#26041260C211yesmaximal, characteristic, complemented
11#455×C181C211yesmaximal, characteristic, complemented
12#19153863—22no—
13#19153863C2212yes—
14#19153861—22no—
15#19153861—22no—
16#19153861C2211yescharacteristic
17C2×#15668450C2211yescharacteristic, complemented
18C2×#15668448—22no—
19C2×#15668448—22no—
20C2×#15668448C2211yescharacteristic
21#41×D181—22no—
22#41×D181C2212yes—
23#19153848—22no—
24#19153847C2212yescomplemented
25#19153844C2212yescomplemented
26#19153843—22no—
27#19153841C2212yescomplemented
28#19153839—22no—
29#19153837—22no—
30M16×D181C2212yescomplemented
31#19153814C2211yescharacteristic, complemented
32#19153812C2211yescharacteristic, complemented
33#19153804C2211yescharacteristic, complemented
34#19153789—22no—
35#19153776—22no—
36C42×D181C2211yescharacteristic, complemented
37#19153744C2212yescomplemented
38#19153743C2212yescomplemented
39#19153742C2212yescomplemented
40#19153734C2212yescomplemented
41#19153733C2211yescharacteristic, complemented
42H(5,𝔽2)×C181C2211yescharacteristic
43#137×C181C2212yescomplemented
44#124×C181C2211yescharacteristic, complemented
45#104×C181C2212yescomplemented
46#100×C181C2211yescharacteristic, complemented
47#15668450—22no—
48#15668450—22no—
49#15668450D412yescomplemented
50#15668449—24no—
51#15668449—22no—
52#15668449—22no—
53#15668449C2312yes—
54#15668448—24no—
55#15668448—24no—
56#15668448—24no—
57#15668448—22no—
58#15668448—22no—
59#15668448—24no—
60#15668448—22no—
61#15668448—22no—
62#15668448—22no—
63#15668448D412yescomplemented
64C2×Dic362—22no—
65C2×Dic362—22no—
66C2×Dic362C2311yescharacteristic
67Q8×D181—22no—
68Q8×D181D412yescomplemented
69C2×D724—21no—
70C2×D724C2311yescharacteristic
71C2×C22⋊D181—24no—
72C2×C22⋊D181—22no—
73C2×C22⋊D181—22no—
74C2×C22⋊D181—22no—
75C2×C22⋊D181D412yes—
76D4×D181—24no—
77D4×D181—22no—
78D4×D181—22no—
79D4×D181D412yescomplemented
80C22×Dic181—22no—
81C22×Dic181D412yes—
82C2×C4×D181—22no—
83C2×C4×D181—22no—
84C2×C4×D181—21no—
85C2×C4×D181—21no—
86C2×C4×D181C2311yescharacteristic
87#15668439—22no—
88#15668438—22no—
89#15668436—22no—
90#15668436—22no—
91D1448—22no—
92#15668434—22no—
93#15668433C2312yes—
94#15668432—22no—
95C8×D181—22no—
96#15668428—22no—
97#15668426—22no—
98C4×Dic181—21no—
99C4×Dic181C2311yescharacteristic
100#41×C181—22no—
101#41×C181C2312yes—
102C2×Q8×C181C2311yescharacteristic
103C2×D4×C181—22no—
104C2×D4×C181—22no—
105C2×D4×C181C2311yescharacteristic
106Q16×C181—22no—
107SD16×C181—22no—
108M16×C181C2312yes—
109C22⋊C4×C181—22no—
110C42×C181C2311yescharacteristic
111C22×D181—22no—
112C22×D181—22no—
113C22×D181—21no—
114C22×D181C2×D411yescharacteristic
115Dic362—24no—
116Dic362—22no—
117Dic362—22no—
118Dic362C2×D412yes—
119D724—22no—
120D724—22no—
121D724—42no—
122D724C2×D412yes—
123C22⋊D181—24no—
124C22⋊D181—24no—
125C22⋊D181—22no—
126C22⋊D181—22no—
127C22⋊D181—24no—
128C22⋊D181—22no—
129C22⋊D181—44no—
130C2×Dic181—22no—
131C2×Dic181—24no—
132C2×Dic181—21no—
133C2×Dic181—22no—
134C2×Dic181—22no—
135C2×Dic181—22no—
136C2×Dic181C2×D412yes—
137C2×Dic181C2×D411yescharacteristic
138C4×D181—24no—
139C4×D181—22no—
140C4×D181—22no—
141C4×D181—22no—
142C4×D181—42no—
143C4×D181—41no—
144C4×D181—41no—
145C4×D181C2×D412yes—
146C4×D181C2×D412yes—
147#13089678—22no—
148C23×C181—22no—
149C23×C181C2×D412yes—
150Q8×C181—22no—
151Q8×C181C2×D412yes—
152D4×C181—24no—
153D4×C181—22no—
154D4×C181—22no—
155D4×C181C2×D412yes—
156C2×C4×C181—22no—
157C2×C4×C181—22no—
158C2×C4×C181—21no—
159C2×C4×C181—21no—
160C2×C4×C181C2411yescharacteristic
161C8×C181—22no—
162C22×C181—22no—
163C22×C181—22no—
164C22×C181—42no—
165C22×C181—82no—
166C22×C181C22×D411yescharacteristic
167C2×D181—22no—
168C2×D181—41no—
169C2×D181—42no—
170C2×D181—41no—
171C2×D181—82no—
172C2×D181—82no—
173C2×D181#12012yes—
174C4×C181—22no—
175C4×C181—41no—
176C4×C181—41no—
177C4×C181C22×D412yes—
178Dic181—22no—
179Dic181—22no—
180Dic181—22no—
181Dic181#12011yescharacteristic
182Dic181#12011yescharacteristic
183C2×C181—21no—
184C2×C181—42no—
185C2×C181—42no—
186C2×C181C2×#12011yescharacteristic
187D181—22no—
188D181—42no—
189D181—81no—
190C181#301811yescharacteristic, complemented, Sylow
191#3018—1811nomaximal, Sylow
192#585—1811no—
193#575—1812no—
194#532—1811no—
195#456—1812no—
196#455—1811no—
197#455D18111yescharacteristic, complemented
198#454—1812no—
199#453—1811no—
200#453—1811no—
201C2×#104—1812no—
202#413—1811no—
2032−1+4—3622no—
2042−1+4—1812no—
205H(5,𝔽2)—3622no—
206H(5,𝔽2)—3622no—
207H(5,𝔽2)—1811no—
208H(5,𝔽2)C2×D18111yescharacteristic
209C2×#41—3622no—
210C2×#41—3622no—
211C2×#41—3622no—
212C2×#41—1812no—
213C2×#41—1811no—
214C2×#41—1811no—
215#137—3622no—
216#137—1812no—
217#137C2×D18112yescomplemented
218AGL(1,ℤ/8)—3622no—
219AGL(1,ℤ/8)—1812no—
220AGL(1,ℤ/8)—1812no—
221#135—3622no—
222#135—3622no—
223C2×M16—1812no—
224C4⋊Q8—1811no—
225#127—1811no—
226#124—1811no—
227#124C2×D18111yescharacteristic, complemented
228#121—3622no—
229C4×D4—3622no—
230C2×C42—1811no—
231#104—1812no—
232#104—1812no—
233#104—1812no—
234#104C2×D18112yescomplemented
235#101—1812no—
236#100—1811no—
237#100C2×D18111yescharacteristic, complemented
238#41—22no—
239#41—3624no—
240#41—3622no—
241#41—3622no—
242#41—3624no—
243#41—3624no—
244#41—3624no—
245#41—3622no—
246#41—3622no—
247#41—3622no—
248#41—3622no—
249#41—3624no—
250#41—3622no—
251#41—3622no—
252#41—3622no—
253#41—1812no—
254#41—1812no—
255#41—1812no—
256#41C22×D18112yes—
257C2×Q8—3622no—
258C2×Q8—3622no—
259C2×Q8—3622no—
260C2×Q8—1812no—
261C2×Q8—1811no—
262C2×Q8C22×D18111yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—3621no—
266C2×D4—3624no—
267C2×D4—3622no—
268C2×D4—3622no—
269C2×D4—3624no—
270C2×D4—3622no—
271C2×D4—3622no—
272C2×D4—3622no—
273C2×D4—1812no—
274C2×D4—1812no—
275C2×D4—1811no—
276C2×D4C22×D18111yescharacteristic
277C22×C4—3622no—
278C22×C4—3622no—
279C22×C4—3621no—
280C22×C4—3621no—
281C22×C4—3622no—
282C22×C4—1812no—
283C22×C4—1811no—
284Q16—22no—
285Q16—3622no—
286SD16—22no—
287SD16—3622no—
288SD16—3622no—
289SD16—3622no—
290D8—3622no—
291D8—3622no—
292M16—3622no—
293M16—1812no—
294M16C22×D18112yes—
295C2×C8—3622no—
296C4⋊C4—3622no—
297C22⋊C4—22no—
298C22⋊C4—3622no—
299C42—3621no—
300C42—1811no—
301C42C22×D18111yescharacteristic
302C23—22no—
303C23—3622no—
304C23—3622no—
305C23—3621no—
306C23—1811no—
307C23D4×D18112yes—
308Q8—22no—
309Q8—3624no—
310Q8—3622no—
311Q8—3622no—
312Q8—1812no—
313Q8D4×D18112yes—
314D4—24no—
315D4—22no—
316D4—22no—
317D4—3622no—
318D4—3624no—
319D4—3624no—
320D4—3622no—
321D4—3624no—
322D4—3622no—
323D4—3622no—
324D4—3622no—
325D4—1812no—
326D4—7242no—
327D4—7244no—
328D4D4×D18112yes—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—3624no—
334C2×C4—3622no—
335C2×C4—3622no—
336C2×C4—3622no—
337C2×C4—3624no—
338C2×C4—3621no—
339C2×C4—3622no—
340C2×C4—3622no—
341C2×C4—3622no—
342C2×C4—3622no—
343C2×C4—1812no—
344C2×C4—1811no—
345C2×C4—1812no—
346C2×C4—1812no—
347C2×C4—7242no—
348C2×C4—7241no—
349C2×C4—7241no—
350C2×C4C23×D18111yescharacteristic
351C8—22no—
352C8—3622no—
353C22—22no—
354C22—22no—
355C22—42no—
356C22—82no—
357C22—3622no—
358C22—14482no—
359C22—14482no—
360C22—1812no—
361C22—7241no—
362C22—7242no—
363C22—7241no—
364C22C2×D4×D18111yescharacteristic
365C4—22no—
366C4—41no—
367C4—41no—
368C4—3622no—
369C4—3622no—
370C4—3622no—
371C4—1811no—
372C4—1811no—
373C4C2×D4×D18112yes—
374C2—21no—
375C2—42no—
376C2—42no—
377C2—3622no—
378C2—14481no—
379C2—7242no—
380C2#120×D18111yescharacteristic
subgroupG/H
centerC2#120×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#455×C181C2
socleC2×C181C2×#120
hypercenter#455D181
hypocenterC181#3018
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #26042114C2semidirect
2. #26042078C2semidirect
3. #26041828C2semidirect
4. #26041460C2semidirect
5. #26041456C2semidirect
6. #26041453C2semidirect
7. #26041449C2semidirect
8. #26041447C2semidirect
9. #104×D181C2semidirect
10. #26041260C2semidirect
11. #455×C181C2semidirect
17. C2×#15668450C22semidirect
24. #19153847C22semidirect
25. #19153844C22semidirect
27. #19153841C22semidirect
30. M16×D181C22semidirect
31. #19153814C22semidirect
32. #19153812C22semidirect
33. #19153804C22semidirect
36. C42×D181C22semidirect
37. #19153744C22semidirect
38. #19153743C22semidirect
39. #19153742C22semidirect
40. #19153734C22semidirect
41. #19153733C22semidirect
43. #137×C181C22semidirect
44. #124×C181C22semidirect
45. #104×C181C22semidirect
46. #100×C181C22semidirect
49. #15668450D4semidirect
63. #15668448D4semidirect
68. Q8×D181D4semidirect
79. D4×D181D4semidirect
190. C181#3018semidirect
197. #455D181semidirect
217. #137C2×D181semidirect
227. #124C2×D181semidirect
234. #104C2×D181semidirect
237. #100C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260421142
#260420782
#260418282
#260414602
#260414562
#260414532
#260414492
#260414472
#104×D1812
#260412602
#455×C1812
#3018181