Finite Group Databasebeta

← #37545806

Subgroups — #37545806

Order 23168.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041987C211yesmaximal, characteristic, complemented
2#26041923C211yesmaximal, characteristic
3#26041637C211yesmaximal, characteristic, complemented
4#26041615C211yesmaximal, characteristic, complemented
5#26041591C211yesmaximal, characteristic, complemented
6#26041557C211yesmaximal, characteristic, complemented
7#26041522C211yesmaximal, characteristic, complemented
8#26041518C211yesmaximal, characteristic, complemented
9#26041499C211yesmaximal, characteristic, complemented
10#26041438C211yesmaximal, characteristic, complemented
11#26041380C211yesmaximal, characteristic, complemented
12#26041347C211yesmaximal, characteristic, complemented
13#26041277C211yesmaximal, characteristic, complemented
14C2×#19153737C211yesmaximal, characteristic, complemented
15#450×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×#15668439C2211yescharacteristic, complemented
18C2×#15668432C2211yescharacteristic, complemented
19C22×#13089678C2211yescharacteristic
20#19153815C2211yescharacteristic, complemented
21#19153806—21no—
22#19153803C2211yescharacteristic
23#19153797—21no—
24#19153796—21no—
25#19153795C2211yescharacteristic
26#19153793—21no—
27#19153792C2211yescharacteristic, complemented
28#19153788—21no—
29#19153787C2211yescharacteristic
30#19153785—21no—
31Q8×Dic181C2211yescharacteristic, complemented
32D4×Dic181C2211yescharacteristic
33C4×C22⋊D181C2211yescharacteristic
34#19153776—21no—
35C2×C4⋊Dic181C2211yescharacteristic
36C4⋊C4×D181C2211yescharacteristic, complemented
37C2×C22⋊Dic181—21no—
38#19153750C2211yescharacteristic, complemented
39#19153750C2211yescharacteristic, complemented
40#19153749C2211yescharacteristic, complemented
41#19153745C2211yescharacteristic, complemented
42#19153741C2211yescharacteristic, complemented
43#19153741C2211yescharacteristic, complemented
44#19153740C2211yescharacteristic, complemented
45#19153739C2211yescharacteristic, complemented
46#19153737C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153737C2211yescharacteristic
50#19153728C2211yescharacteristic, complemented
51#19153726C2211yescharacteristic, complemented
52C22×D4×C181C2211yescharacteristic
53C2×SD16×C181C2211yescharacteristic
54C2×SD16×C181C2211yescharacteristic
55C22⋊Q8×C181C2211yescharacteristic
56#102×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58C22⋊C8×C181C2211yescharacteristic
59#15668448—22no—
60#15668448—41no—
61C2×Dic362—21no—
62C2×Dic362C2311yescharacteristic
63C2×C22⋊D181—21no—
64C2×C22⋊D181—21no—
65C2×C22⋊D181D411yescharacteristic
66C22×Dic181—21no—
67C22×Dic181—21no—
68C22×Dic181—21no—
69C2×C4×D181—21no—
70C2×C4×D181C2311yescharacteristic
71#15668439—41no—
72#15668432—41no—
73C2×#13089678—41no—
74C2×#13089678C2311yescharacteristic
75C2×#13089678C2311yescharacteristic
76C4⋊Dic181—21no—
77C4⋊Dic181—21no—
78C4⋊Dic181C2311yescharacteristic
79C4⋊Dic181C2311yescharacteristic
80C4⋊Dic181C2311yescharacteristic
81#15668428—21no—
82#15668428—21no—
83#15668428—21no—
84#15668428—21no—
85#15668428D411yescharacteristic
86C22⋊Dic181—21no—
87C22⋊Dic181—21no—
88C22⋊Dic181—21no—
89C22⋊Dic181—21no—
90C22⋊Dic181—21no—
91C22⋊Dic181D411yescharacteristic
92#15668426—21no—
93#15668426D411yescharacteristic
94C4×Dic181—21no—
95C4×Dic181C2311yescharacteristic
96C24×C181—21no—
97C2×Q8×C181C2311yescharacteristic
98C2×D4×C181—22no—
99C2×D4×C181—21no—
100C2×D4×C181—41no—
101C2×D4×C181C2311yescharacteristic
102C2×D4×C181C2311yescharacteristic
103C22×C4×C181C2311yescharacteristic
104SD16×C181—22no—
105SD16×C181—41no—
106C2×C8×C181C2311yescharacteristic
107C2×C8×C181C2311yescharacteristic
108C4⋊C4×C181—21no—
109C4⋊C4×C181C2311yescharacteristic
110C22⋊C4×C181—21no—
111C22×D181C2×D411yescharacteristic
112Dic362—21no—
113Dic362—21no—
114Dic362—81no—
115C22⋊D181—22no—
116C22⋊D181—42no—
117C22⋊D181—81no—
118C2×Dic181—21no—
119C2×Dic181—21no—
120C2×Dic181—21no—
121C2×Dic181—21no—
122C2×Dic181—42no—
123C2×Dic181—81no—
124C2×Dic181—81no—
125C2×Dic181C2×D411yescharacteristic
126C2×Dic181C2×D411yescharacteristic
127C2×Dic181C2×D411yescharacteristic
128C4×D181—21no—
129C4×D181—21no—
130C4×D181—81no—
131#13089678—21no—
132#13089678—21no—
133C23×C181—21no—
134C23×C181—21no—
135C23×C181—42no—
136C23×C181—42no—
137C23×C181—81no—
138C23×C181C2×D411yescharacteristic
139Q8×C181—21no—
140Q8×C181—21no—
141D4×C181—21no—
142D4×C181—21no—
143D4×C181—21no—
144D4×C181—42no—
145D4×C181#4112yes—
146C2×C4×C181—21no—
147C2×C4×C181—21no—
148C2×C4×C181—21no—
149C2×C4×C181—21no—
150C2×C4×C181C2×D411yescharacteristic
151C2×C4×C181C2411yescharacteristic
152C8×C181—21no—
153C8×C181—21no—
154C22×C181—21no—
155C22×C181—21no—
156C22×C181—22no—
157C22×C181—41no—
158C22×C181—41no—
159C22×C181—41no—
160C22×C181—41no—
161C22×C181—41no—
162C22×C181—84no—
163C22×C181#13712yes—
164C22×C181C22×D411yescharacteristic
165C2×D181—21no—
166C2×D181—21no—
167C2×D181—21no—
168C4×C181—21no—
169C4×C181—41no—
170C4×C181—41no—
171C4×C181C2×#4111yescharacteristic
172C4×C181H(5,𝔽2)11yescharacteristic
173Dic181—21no—
174Dic181—21no—
175Dic181—21no—
176Dic181—41no—
177Dic181—41no—
178Dic181—41no—
179Dic181—41no—
180C2×C181—22no—
181C2×C181—42no—
182C2×C181—81no—
183C2×C181#54611yescharacteristic
184C2×C181C2×#13711yescharacteristic
185C2×C181#57611yescharacteristic
186D181—41no—
187C181#331411yescharacteristic, complemented, Sylow
188#3314—1811nomaximal, Sylow
189#554—1811no—
190#546—1811no—
191#489—1811no—
192#484—1811no—
193#480—1811no—
194#474—1811no—
195#468—1811no—
196#467—1811no—
197#464—1811no—
198#452—1811no—
199#450D18111yescharacteristic, complemented
200#440—1811no—
201#435—1811no—
202#417—1811no—
203C2×#102—1811no—
204C2×#41—1811no—
205C22×D4C2×D18111yescharacteristic
206C2×Q16—1811no—
207C2×SD16C2×D18111yescharacteristic
208C2×SD16C2×D18111yescharacteristic
209C2×M16—1811no—
210C22×C8—1811no—
211C4⋊Q8—1811no—
212#125—3621no—
213#124—1811no—
214#123—3621no—
215C22⋊Q8—3621no—
216C22⋊Q8—3621no—
217C22⋊Q8—1811no—
218C22⋊Q8—1811no—
219C22⋊Q8C2×D18111yescharacteristic
220#121—3621no—
221#121—1811no—
222#120—3621no—
223C4×Q8—1811no—
224C4×D4—3621no—
225C4×D4—1811no—
226C4×D4—1811no—
227C2×C4⋊C4—1811no—
228C2×C4⋊C4—1811no—
229C2×C22⋊C4—3621no—
230#107—1811no—
231#107—1811no—
232#107—1811no—
233C4⋊C8—1811no—
234#103—1811no—
235#103—1811no—
236#103—1811no—
237#103—1811no—
238#102—1811no—
239#102—1811no—
240#102—1811no—
241#102—1811no—
242#102C2×D18111yescharacteristic
243#102C2×D18111yescharacteristic
244C22⋊C8—1811no—
245C22⋊C8C2×D18111yescharacteristic
246#97—1811no—
247C24—21no—
248#41—3622no—
249#41—7241no—
250C2×Q8—3621no—
251C2×Q8—1811no—
252C2×Q8C22×D18111yescharacteristic
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4—3621no—
257C2×D4—3621no—
258C2×D4—1811no—
259C2×D4C22×D18111yescharacteristic
260C2×D4C22×D18111yescharacteristic
261C22×C4—3621no—
262C22×C4—3621no—
263C22×C4—3621no—
264C22×C4—3621no—
265C22×C4—1811no—
266C22×C4C22×D18111yescharacteristic
267Q16—7241no—
268SD16—22no—
269SD16—41no—
270M16—7241no—
271C2×C8—1811no—
272C2×C8—1811no—
273C2×C8—7241no—
274C2×C8C22×D18111yescharacteristic
275C2×C8C22×D18111yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—3621no—
278C4⋊C4—3621no—
279C4⋊C4—3621no—
280C4⋊C4—3621no—
281C4⋊C4—3621no—
282C4⋊C4—3621no—
283C4⋊C4—1811no—
284C4⋊C4—1811no—
285C4⋊C4—1811no—
286C4⋊C4—1811no—
287C4⋊C4C22×D18111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4—3621no—
292C22⋊C4—3621no—
293C22⋊C4—3621no—
294C22⋊C4—3621no—
295C22⋊C4—1811no—
296C22⋊C4—1811no—
297C42—3621no—
298C42—1811no—
299C23—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23—1811no—
305C23D4×D18111yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—3621no—
309Q8—3621no—
310Q8—14481no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—3622no—
316D4—14481no—
317D4—7242no—
318D4#1566844812yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—3621no—
324C2×C4—3621no—
325C2×C4—3621no—
326C2×C4—3621no—
327C2×C4—3621no—
328C2×C4—3621no—
329C2×C4—14481no—
330C2×C4—14481no—
331C2×C4—14481no—
332C2×C4—1811no—
333C2×C4—1811no—
334C2×C4—1811no—
335C2×C4—7242no—
336C2×C4D4×D18111yescharacteristic
337C2×C4C23×D18111yescharacteristic
338C8—21no—
339C8—21no—
340C8—3621no—
341C8—3621no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22—3621no—
352C22—3621no—
353C22—3621no—
354C22#1915384512yes—
355C22C2×D4×D18111yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—3621no—
360C4—3621no—
361C4—3621no—
362C4—7241no—
363C4—7241no—
364C4—7241no—
365C4—7241no—
366C4C2×#1566844811yescharacteristic
367C4#1915386111yescharacteristic
368C2—22no—
369C2—42no—
370C2—81no—
371C2—7241no—
372C2#2604191611yescharacteristic
373C2C2×#1915384511yescharacteristic
374C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#450×C181C2
socleC22×C181C22×D4
prime radical#26041923C2
hypercenter#450D181
hypocenterC181#3314
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #26041987C2semidirect
3. #26041637C2semidirect
4. #26041615C2semidirect
5. #26041591C2semidirect
6. #26041557C2semidirect
7. #26041522C2semidirect
8. #26041518C2semidirect
9. #26041499C2semidirect
10. #26041438C2semidirect
11. #26041380C2semidirect
12. #26041347C2semidirect
13. #26041277C2semidirect
14. C2×#19153737C2semidirect
15. #450×C181C2semidirect
17. C2×#15668439C22semidirect
18. C2×#15668432C22semidirect
20. #19153815C22semidirect
27. #19153792C22semidirect
31. Q8×Dic181C22semidirect
36. C4⋊C4×D181C22semidirect
38. #19153750C22semidirect
39. #19153750C22semidirect
40. #19153749C22semidirect
41. #19153745C22semidirect
42. #19153741C22semidirect
43. #19153741C22semidirect
44. #19153740C22semidirect
45. #19153739C22semidirect
50. #19153728C22semidirect
51. #19153726C22semidirect
187. C181#3314semidirect
199. #450D181semidirect

Maximal subgroups

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

subgroupindex
#260419872
#260419232
#260416372
#260416152
#260415912
#260415572
#260415222
#260415182
#260414992
#260414382
#260413802
#260413472
#260412772
C2×#191537372
#450×C1812
#3314181