Finite Group Databasebeta

← #37544687

Subgroups — #37544687

Order 23168.

410 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041814C211yesmaximal, characteristic, complemented
2#124×D181C211yesmaximal, characteristic, complemented
3#26041663C211yesmaximal, characteristic, complemented
4#26041661C211yesmaximal, characteristic, complemented
5#26041636C211yesmaximal, characteristic, complemented
6#26041504C211yesmaximal, characteristic, complemented
7#26041500C211yesmaximal, characteristic, complemented
8#26041497C211yesmaximal, characteristic, complemented
9#26041493C211yesmaximal, characteristic, complemented
10#26041436C211yesmaximal, characteristic, complemented
11#26041431C211yesmaximal, characteristic, complemented
12#26041419C211yesmaximal, characteristic, complemented
13#26041414C211yesmaximal, characteristic
14#26041330C211yesmaximal, characteristic, complemented
15#488×C181C211yesmaximal, characteristic, complemented
16C2×Q8×D181C2211yescharacteristic
17C2×D4×D181C2211yescharacteristic
18C2×Dic724C2211yescharacteristic, complemented
19C2×#15668439C2211yescharacteristic, complemented
20C2×#15668438C2211yescharacteristic
21C2×#15668438C2211yescharacteristic, complemented
22C2×#15668437C2211yescharacteristic
23C2×#15668436C2211yescharacteristic, complemented
24C2×D1448C2211yescharacteristic
25C2×#15668434C2211yescharacteristic
26#19153811—21no—
27#19153805C2211yescharacteristic
28#19153804C2211yescharacteristic
29#19153804C2211yescharacteristic, complemented
30#19153803C2211yescharacteristic
31#19153802—21no—
32#19153802—21no—
33#19153801—21no—
34#19153796C2211yescharacteristic
35#19153791C2211yescharacteristic
36#19153781C2211yescharacteristic, complemented
37#19153778—21no—
38#19153778C2211yescharacteristic, complemented
39#19153774—21no—
40C22⋊C4×D181—21no—
41C22⋊C4×D181—21no—
42C42×D181C2211yescharacteristic
43#19153745C2211yescharacteristic, complemented
44#19153745C2211yescharacteristic, complemented
45#19153739C2211yescharacteristic, complemented
46#19153739C2211yescharacteristic, complemented
47#19153736C2211yescharacteristic
48#19153736C2211yescharacteristic
49#19153728C2211yescharacteristic
50#19153728C2211yescharacteristic
51C4×#13089678C2211yescharacteristic, complemented
52#125×C181C2211yescharacteristic, complemented
53#124×C181C2211yescharacteristic
54#103×C181C2211yescharacteristic, complemented
55#103×C181C2211yescharacteristic, complemented
56#102×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58C4×C8×C181C2211yescharacteristic, complemented
59C23×D181—21no—
60C2×Dic362—21no—
61C2×Dic362C2311yescharacteristic
62C2×Dic362C2311yescharacteristic
63Q8×D181—41no—
64Q8×D181—41no—
65C2×D724—21no—
66C2×D724C2311yescharacteristic
67C2×D724C2311yescharacteristic
68C2×C22⋊D181—21no—
69D4×D181—41no—
70D4×D181—41no—
71C2×C4×D181—21no—
72C2×C4×D181—21no—
73C2×C4×D181D411yescharacteristic
74C2×C4×D181D411yescharacteristic
75C2×C4×D181C2311yescharacteristic
76Dic724—41no—
77#15668439—41no—
78#15668438—41no—
79#15668438—41no—
80#15668437—41no—
81#15668436—41no—
82D1448—41no—
83#15668434—41no—
84C2×#13089678C2311yescharacteristic
85C2×#13089678C2311yescharacteristic
86C4⋊Dic181—21no—
87#15668428—21no—
88C22⋊Dic181—21no—
89C22⋊Dic181—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426—21no—
97#15668426—21no—
98#15668426—21no—
99C4×Dic181—21no—
100C4×Dic181D411yescharacteristic
101C4×Dic181D411yescharacteristic
102C4×Dic181C2311yescharacteristic
103C2×Q8×C181C2311yescharacteristic
104C2×D4×C181C2311yescharacteristic
105C2×C8×C181C2311yescharacteristic
106C2×C8×C181C2311yescharacteristic
107C4⋊C4×C181—21no—
108C4⋊C4×C181—21no—
109C4⋊C4×C181C2311yescharacteristic
110C4⋊C4×C181C2311yescharacteristic
111C22⋊C4×C181—21no—
112C22⋊C4×C181—21no—
113C42×C181C2311yescharacteristic
114C22×D181—21no—
115C22×D181—21no—
116C22×D181—42no—
117C22×D181—42no—
118C22×D181—42no—
119C22×D181C2×D411yescharacteristic
120Dic362—21no—
121Dic362—21no—
122Dic362—21no—
123Dic362—21no—
124Dic362—42no—
125D724—21no—
126D724—21no—
127D724—21no—
128D724—21no—
129D724—42no—
130C22⋊D181—42no—
131C2×Dic181—21no—
132C2×Dic181—21no—
133C2×Dic181C2×D411yescharacteristic
134C2×Dic181C2×D411yescharacteristic
135C2×Dic181C2×D411yescharacteristic
136C4×D181—22no—
137C4×D181—21no—
138C4×D181—21no—
139C4×D181—41no—
140C4×D181—42no—
141C4×D181—42no—
142#13089678—21no—
143#13089678—21no—
144C23×C181—21no—
145Q8×C181—21no—
146Q8×C181—21no—
147D4×C181—21no—
148D4×C181—21no—
149C2×C4×C181—21no—
150C2×C4×C181—21no—
151C2×C4×C181—21no—
152C2×C4×C181C2×D411yescharacteristic
153C2×C4×C181C2×D411yescharacteristic
154C2×C4×C181C2411yescharacteristic
155C8×C181—21no—
156C8×C181—21no—
157C22×C181—41no—
158C22×C181—41no—
159C22×C181—41no—
160C22×C181C22×D411yescharacteristic
161C2×D181—21no—
162C2×D181—21no—
163C2×D181—41no—
164C2×D181—41no—
165C2×D181—41no—
166C2×D181—41no—
167C2×D181—41no—
168C2×D181—41no—
169C2×D181—84no—
170C2×D181#13512yes—
171C4×C181—21no—
172C4×C181—21no—
173C4×C181—41no—
174C4×C181—41no—
175C4×C181—41no—
176C4×C181H(5,𝔽2)11yescharacteristic
177C4×C181H(5,𝔽2)11yescharacteristic
178Dic181—21no—
179Dic181—21no—
180Dic181—21no—
181Dic181—41no—
182Dic181—41no—
183C2×C181—81no—
184C2×C181#53511yescharacteristic
185C2×C181C2×#13511yescharacteristic
186C2×C181#57511yescharacteristic
187D181—22no—
188D181—81no—
189D181—81no—
190C181#321111yescharacteristic, complemented, Sylow
191#3211—1811nomaximal, Sylow
192#529—1811no—
193C2×#124—1811no—
194#495—1811no—
195#495—1811no—
196#488—1811no—
197#488D18111yescharacteristic, complemented
198#464—1811no—
199#464—1811no—
200#463—1811no—
201#463—1811no—
202#451—1811no—
203#450—1811no—
204#448—1811no—
205#447—1811no—
206#431—1811no—
207C22×Q8—1811no—
208C22×D4—1811no—
209C2×Q16—1811no—
210C2×Q16—1811no—
211C2×SD16—1811no—
212C2×SD16—1811no—
213C2×SD16—1811no—
214C2×SD16—1811no—
215C2×D8—1811no—
216C2×D8—1811no—
217#126—3621no—
218#125C2×D18111yescharacteristic, complemented
219#124—3621no—
220#124—3621no—
221#124—1811no—
222#124—1811no—
223#124—1811no—
224#124—1811no—
225#124C2×D18111yescharacteristic
226#123—3621no—
227C22⋊Q8—1811no—
228#121—1811no—
229C4×Q8—1811no—
230C4×D4—3621no—
231C4×D4—1811no—
232#117—3621no—
233C2×C22⋊C4—3621no—
234C2×C22⋊C4—3621no—
235C2×C42—1811no—
236C4⋊C8—1811no—
237C4⋊C8—1811no—
238#103—1811no—
239#103—1811no—
240#103C2×D18111yescharacteristic, complemented
241#103C2×D18111yescharacteristic, complemented
242#102—1811no—
243#102—1811no—
244#102C2×D18111yescharacteristic
245#102C2×D18111yescharacteristic
246C22⋊C8—1811no—
247C22⋊C8—1811no—
248C4×C8—1811no—
249C4×C8C2×D18111yescharacteristic, complemented
250C24—3621no—
251C2×Q8—3621no—
252C2×Q8—1811no—
253C2×Q8—1811no—
254C2×Q8—7241no—
255C2×Q8—7241no—
256C2×Q8C22×D18111yescharacteristic
257C2×D4—3621no—
258C2×D4—3621no—
259C2×D4—1811no—
260C2×D4—1811no—
261C2×D4—7241no—
262C2×D4—7241no—
263C2×D4C22×D18111yescharacteristic
264C22×C4—3621no—
265C22×C4—3621no—
266C22×C4—1811no—
267C22×C4—1811no—
268C22×C4—1811no—
269Q16—7241no—
270Q16—7241no—
271SD16—7241no—
272SD16—7241no—
273SD16—7241no—
274SD16—7241no—
275D8—7241no—
276D8—7241no—
277C2×C8—1811no—
278C2×C8—1811no—
279C2×C8C22×D18111yescharacteristic
280C2×C8C22×D18111yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—3621no—
284C4⋊C4—3621no—
285C4⋊C4C22×D18111yescharacteristic
286C4⋊C4C22×D18111yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4—3621no—
292C22⋊C4—3621no—
293C22⋊C4—3621no—
294C22⋊C4—3621no—
295C22⋊C4—3621no—
296C22⋊C4—3621no—
297C22⋊C4—3621no—
298C22⋊C4—3621no—
299C22⋊C4—3621no—
300C42—3621no—
301C42—1811no—
302C42—1811no—
303C42—1811no—
304C42C22×D18111yescharacteristic
305C23—21no—
306C23—3621no—
307C23—3621no—
308C23—1811no—
309C23—7242no—
310C23—7242no—
311C23—7242no—
312Q8—21no—
313Q8—21no—
314Q8—3621no—
315Q8—3621no—
316Q8—3621no—
317Q8—3621no—
318Q8—7242no—
319D4—21no—
320D4—21no—
321D4—3621no—
322D4—3621no—
323D4—3621no—
324D4—3621no—
325D4—7242no—
326D4—7242no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—3622no—
331C2×C4—3621no—
332C2×C4—3621no—
333C2×C4—3621no—
334C2×C4—3621no—
335C2×C4—1811no—
336C2×C4—1811no—
337C2×C4—1811no—
338C2×C4—7241no—
339C2×C4—7242no—
340C2×C4—7242no—
341C2×C4D4×D18111yescharacteristic
342C2×C4D4×D18111yescharacteristic
343C2×C4C23×D18111yescharacteristic
344C8—21no—
345C8—21no—
346C8—3621no—
347C8—3621no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—3621no—
352C22—3621no—
353C22—14484no—
354C22—1812no—
355C22—7241no—
356C22—7241no—
357C22—7241no—
358C22—7241no—
359C22—7241no—
360C22—7241no—
361C22C2×D4×D18111yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—3621no—
368C4—3621no—
369C4—3621no—
370C4—7241no—
371C4—7241no—
372C4#1915386211yescharacteristic
373C4#1915386211yescharacteristic
374C2—81no—
375C2—3622no—
376C2—14481no—
377C2—14481no—
378C2#2604184711yescharacteristic
379C2#135×D18111yescharacteristic
380C2#2604207611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#488×C181C2
socleC22×C181C22×D4
prime radical#26041414C2
hypercenter#488D181
hypocenterC181#3211
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #26041814C2semidirect
2. #124×D181C2semidirect
3. #26041663C2semidirect
4. #26041661C2semidirect
5. #26041636C2semidirect
6. #26041504C2semidirect
7. #26041500C2semidirect
8. #26041497C2semidirect
9. #26041493C2semidirect
10. #26041436C2semidirect
11. #26041431C2semidirect
12. #26041419C2semidirect
14. #26041330C2semidirect
15. #488×C181C2semidirect
18. C2×Dic724C22semidirect
19. C2×#15668439C22semidirect
21. C2×#15668438C22semidirect
23. C2×#15668436C22semidirect
29. #19153804C22semidirect
36. #19153781C22semidirect
38. #19153778C22semidirect
43. #19153745C22semidirect
44. #19153745C22semidirect
45. #19153739C22semidirect
46. #19153739C22semidirect
51. C4×#13089678C22semidirect
52. #125×C181C22semidirect
54. #103×C181C22semidirect
55. #103×C181C22semidirect
58. C4×C8×C181C22semidirect
190. C181#3211semidirect
197. #488D181semidirect
218. #125C2×D181semidirect
240. #103C2×D181semidirect
241. #103C2×D181semidirect
249. C4×C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418142
#124×D1812
#260416632
#260416612
#260416362
#260415042
#260415002
#260414972
#260414932
#260414362
#260414312
#260414192
#260414142
#260413302
#488×C1812
#3211181