Finite Group Databasebeta

← #37544722

Subgroups — #37544722

Order 23168.

400 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041833C211yesmaximal, characteristic, complemented
2#124×D181C211yesmaximal, characteristic
3#26041672C211yesmaximal, characteristic, complemented
4#26041670C211yesmaximal, characteristic, complemented
5#26041640C211yesmaximal, characteristic, complemented
6#26041637C211yesmaximal, characteristic, complemented
7#26041502C211yesmaximal, characteristic, complemented
8#26041501C211yesmaximal, characteristic, complemented
9#26041498C211yesmaximal, characteristic, complemented
10#26041436C211yesmaximal, characteristic, complemented
11#26041433C211yesmaximal, characteristic, complemented
12#26041431C211yesmaximal, characteristic
13#26041428C211yesmaximal, characteristic
14#26041335C211yesmaximal, characteristic, complemented
15#464×C181C211yesmaximal, characteristic, complemented
16C2×Q8×D181C2211yescharacteristic
17C2×D4×D181C2211yescharacteristic
18C2×Dic724C2211yescharacteristic, complemented
19C2×#15668439C2211yescharacteristic, complemented
20C2×#15668439C2211yescharacteristic, complemented
21C2×#15668438C2211yescharacteristic
22C2×#15668437C2211yescharacteristic
23C2×#15668436C2211yescharacteristic
24#19153815C2211yescharacteristic, complemented
25#19153814C2211yescharacteristic, complemented
26#19153806—21no—
27#19153805C2211yescharacteristic
28#19153804C2211yescharacteristic
29#19153803C2211yescharacteristic
30#19153802—21no—
31#19153802—21no—
32#19153796C2211yescharacteristic
33#19153794—21no—
34#19153792C2211yescharacteristic
35C4×Dic362—21no—
36C4×Dic362C2211yescharacteristic, complemented
37#19153773—21no—
38C22⋊C4×D181—21no—
39C22⋊C4×D181—21no—
40C42×D181C2211yescharacteristic
41C4⋊#13089678C2211yescharacteristic, complemented
42#19153741C2211yescharacteristic, complemented
43#19153740C2211yescharacteristic, complemented
44#19153739C2211yescharacteristic, complemented
45#19153738C2211yescharacteristic
46#19153737C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153728C2211yescharacteristic
49#19153728C2211yescharacteristic
50#19153726C2211yescharacteristic, complemented
51#19153726C2211yescharacteristic, complemented
52C2×Q16×C181C2211yescharacteristic, complemented
53C2×SD16×C181C2211yescharacteristic
54#124×C181C2211yescharacteristic
55C4×Q8×C181C2211yescharacteristic, complemented
56C4⋊C8×C181C2211yescharacteristic, complemented
57#103×C181C2211yescharacteristic, complemented
58#102×C181C2211yescharacteristic
59C23×D181—21no—
60C2×Dic362—21no—
61C2×Dic362—21no—
62C2×Dic362C2311yescharacteristic
63C2×Dic362C2311yescharacteristic
64Q8×D181—41no—
65Q8×D181—41no—
66C2×D724C2311yescharacteristic
67C2×C22⋊D181—21no—
68D4×D181—41no—
69D4×D181—41no—
70C2×C4×D181—21no—
71C2×C4×D181D411yescharacteristic
72C2×C4×D181D411yescharacteristic
73C2×C4×D181C2311yescharacteristic
74Dic724—41no—
75#15668439—41no—
76#15668439—41no—
77#15668438—41no—
78#15668437—41no—
79#15668436—41no—
80C2×#13089678C2311yescharacteristic
81C2×#13089678C2311yescharacteristic
82C4⋊Dic181—21no—
83C4⋊Dic181C2311yescharacteristic
84#15668428—21no—
85#15668428—21no—
86#15668428—21no—
87#15668428—21no—
88#15668428—21no—
89C22⋊Dic181—21no—
90C22⋊Dic181—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426—21no—
97C4×Dic181—21no—
98C4×Dic181D411yescharacteristic
99C4×Dic181D411yescharacteristic
100C4×Dic181C2311yescharacteristic
101C2×Q8×C181C2311yescharacteristic
102C2×Q8×C181C2311yescharacteristic
103C2×D4×C181C2311yescharacteristic
104Q16×C181—41no—
105SD16×C181—41no—
106C2×C8×C181C2311yescharacteristic
107C2×C8×C181C2311yescharacteristic
108C4⋊C4×C181—21no—
109C4⋊C4×C181C2311yescharacteristic
110C22⋊C4×C181—21no—
111C22⋊C4×C181—21no—
112C42×C181—21no—
113C42×C181C2311yescharacteristic
114C22×D181—21no—
115C22×D181—42no—
116C22×D181—42no—
117C22×D181—42no—
118C22×D181C2×D411yescharacteristic
119Dic362—21no—
120Dic362—21no—
121Dic362—21no—
122Dic362—21no—
123Dic362—42no—
124Dic362—42no—
125D724—21no—
126D724—21no—
127C22⋊D181—42no—
128C2×Dic181—21no—
129C2×Dic181—21no—
130C2×Dic181—21no—
131C2×Dic181C2×D411yescharacteristic
132C2×Dic181C2×D411yescharacteristic
133C2×Dic181C2×D411yescharacteristic
134C4×D181—22no—
135C4×D181—21no—
136C4×D181—21no—
137C4×D181—41no—
138C4×D181—42no—
139#13089678—21no—
140#13089678—21no—
141C23×C181—21no—
142Q8×C181—21no—
143Q8×C181—21no—
144Q8×C181—21no—
145Q8×C181—21no—
146D4×C181—21no—
147D4×C181—21no—
148C2×C4×C181—21no—
149C2×C4×C181—21no—
150C2×C4×C181—21no—
151C2×C4×C181C2×D411yescharacteristic
152C2×C4×C181C2×D411yescharacteristic
153C2×C4×C181C2411yescharacteristic
154C8×C181—21no—
155C8×C181—21no—
156C22×C181—41no—
157C22×C181—41no—
158C22×C181—41no—
159C22×C181C22×D411yescharacteristic
160C2×D181—21no—
161C2×D181—21no—
162C2×D181—41no—
163C2×D181—41no—
164C2×D181—41no—
165C2×D181—84no—
166C2×D181#13712yes—
167C4×C181—21no—
168C4×C181—21no—
169C4×C181—41no—
170C4×C181—41no—
171C4×C181—41no—
172C4×C181H(5,𝔽2)11yescharacteristic
173C4×C181H(5,𝔽2)11yescharacteristic
174Dic181—21no—
175Dic181—21no—
176Dic181—21no—
177Dic181—41no—
178Dic181—41no—
179Dic181—41no—
180C2×C181—81no—
181C2×C181#53511yescharacteristic
182C2×C181C2×#13711yescharacteristic
183C2×C181#57511yescharacteristic
184D181—22no—
185D181—81no—
186C181#321511yescharacteristic, complemented, Sylow
187#3215—1811nomaximal, Sylow
188#533—1811no—
189C2×#124—1811no—
190#497—1811no—
191#497—1811no—
192#489—1811no—
193#489—1811no—
194#464—1811no—
195#464—1811no—
196#464—1811no—
197#464D18111yescharacteristic, complemented
198#451—1811no—
199#451—1811no—
200#450—1811no—
201#450—1811no—
202#432—1811no—
203C22×Q8—1811no—
204C22×D4—1811no—
205C2×Q16—1811no—
206C2×Q16—1811no—
207C2×Q16—1811no—
208C2×Q16C2×D18111yescharacteristic, complemented
209C2×SD16—1811no—
210C2×SD16—1811no—
211C2×SD16—1811no—
212C2×SD16C2×D18111yescharacteristic
213C4⋊Q8—1811no—
214C4⋊Q8—1811no—
215#125—3621no—
216#124—3621no—
217#124—3621no—
218#124—1811no—
219#124—1811no—
220#124—1811no—
221#124C2×D18111yescharacteristic
222C22⋊Q8—3621no—
223C22⋊Q8—1811no—
224C22⋊Q8—1811no—
225C4×Q8—3621no—
226C4×Q8—1811no—
227C4×Q8C2×D18111yescharacteristic, complemented
228#117—3621no—
229C2×C22⋊C4—3621no—
230C2×C22⋊C4—3621no—
231C2×C42—1811no—
232C4⋊C8—1811no—
233C4⋊C8C2×D18111yescharacteristic, complemented
234#103—1811no—
235#103—1811no—
236#103—1811no—
237#103C2×D18111yescharacteristic, complemented
238#102—1811no—
239#102—1811no—
240#102—1811no—
241#102C2×D18111yescharacteristic
242C22⋊C8—1811no—
243C22⋊C8—1811no—
244#97—1811no—
245#97—1811no—
246C24—3621no—
247C2×Q8—3621no—
248C2×Q8—3621no—
249C2×Q8—1811no—
250C2×Q8—1811no—
251C2×Q8—7241no—
252C2×Q8—7241no—
253C2×Q8C22×D18111yescharacteristic
254C2×Q8C22×D18111yescharacteristic
255C2×D4—3621no—
256C2×D4—1811no—
257C2×D4—7241no—
258C2×D4—7241no—
259C2×D4C22×D18111yescharacteristic
260C22×C4—3621no—
261C22×C4—1811no—
262C22×C4—1811no—
263C22×C4—1811no—
264Q16—41no—
265Q16—7241no—
266Q16—7241no—
267Q16—7241no—
268SD16—41no—
269SD16—7241no—
270SD16—7241no—
271SD16—7241no—
272C2×C8—1811no—
273C2×C8—1811no—
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⋊C4C22×D18111yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—3621no—
288C22⋊C4—3621no—
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4—3621no—
292C22⋊C4—3621no—
293C22⋊C4—3621no—
294C22⋊C4—3621no—
295C42—21no—
296C42—3621no—
297C42—1811no—
298C42—1811no—
299C42—1811no—
300C42C22×D18111yescharacteristic
301C23—21no—
302C23—3621no—
303C23—1811no—
304C23—7242no—
305C23—7242no—
306C23—7242no—
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—3621no—
312Q8—3621no—
313Q8—3621no—
314Q8—3621no—
315Q8—7242no—
316Q8—7242no—
317D4—21no—
318D4—21no—
319D4—3621no—
320D4—3621no—
321D4—7242no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—3622no—
326C2×C4—3621no—
327C2×C4—3621no—
328C2×C4—3621no—
329C2×C4—3621no—
330C2×C4—3621no—
331C2×C4—1811no—
332C2×C4—1811no—
333C2×C4—1811no—
334C2×C4—7241no—
335C2×C4—7242no—
336C2×C4D4×D18111yescharacteristic
337C2×C4D4×D18111yescharacteristic
338C2×C4C23×D18111yescharacteristic
339C8—21no—
340C8—21no—
341C8—3621no—
342C8—3621no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—3621no—
347C22—3621no—
348C22—14484no—
349C22—1812no—
350C22—7241no—
351C22—7241no—
352C22—7241no—
353C22C2×D4×D18111yescharacteristic
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—3621no—
360C4—3621no—
361C4—3621no—
362C4—7241no—
363C4—7241no—
364C4—7241no—
365C4#1915386111yescharacteristic
366C4#1915386211yescharacteristic
367C2—81no—
368C2—3622no—
369C2—14481no—
370C2#2604184511yescharacteristic
371C2#137×D18111yescharacteristic
372C2#2604207611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#464×C181C2
socleC22×C181C22×D4
prime radicalC2×D4×D181C22
hypercenter#464D181
hypocenterC181#3215
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #26041833C2semidirect
3. #26041672C2semidirect
4. #26041670C2semidirect
5. #26041640C2semidirect
6. #26041637C2semidirect
7. #26041502C2semidirect
8. #26041501C2semidirect
9. #26041498C2semidirect
10. #26041436C2semidirect
11. #26041433C2semidirect
14. #26041335C2semidirect
15. #464×C181C2semidirect
18. C2×Dic724C22semidirect
19. C2×#15668439C22semidirect
20. C2×#15668439C22semidirect
24. #19153815C22semidirect
25. #19153814C22semidirect
36. C4×Dic362C22semidirect
41. C4⋊#13089678C22semidirect
42. #19153741C22semidirect
43. #19153740C22semidirect
44. #19153739C22semidirect
50. #19153726C22semidirect
51. #19153726C22semidirect
52. C2×Q16×C181C22semidirect
55. C4×Q8×C181C22semidirect
56. C4⋊C8×C181C22semidirect
57. #103×C181C22semidirect
186. C181#3215semidirect
197. #464D181semidirect
208. C2×Q16C2×D181semidirect
227. C4×Q8C2×D181semidirect
233. C4⋊C8C2×D181semidirect
237. #103C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418332
#124×D1812
#260416722
#260416702
#260416402
#260416372
#260415022
#260415012
#260414982
#260414362
#260414332
#260414312
#260414282
#260413352
#464×C1812
#3215181