Finite Group Databasebeta

← #37544816

Subgroups — #37544816

Order 23168.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041903C211yesmaximal, characteristic, complemented
2#26041847C211yesmaximal, characteristic, complemented
3#26041668C211yesmaximal, characteristic, complemented
4#26041661C211yesmaximal, characteristic, complemented
5#26041627C211yesmaximal, characteristic, complemented
6#26041504C211yesmaximal, characteristic, complemented
7#26041497C211yesmaximal, characteristic, complemented
8#26041478C211yesmaximal, characteristic, complemented
9#26041473C211yesmaximal, characteristic, complemented
10#26041444C211yesmaximal, characteristic, complemented
11#26041431C211yesmaximal, characteristic, complemented
12#26041427C211yesmaximal, characteristic, complemented
13#26041414C211yesmaximal, characteristic, complemented
14#26041341C211yesmaximal, characteristic, complemented
15#489×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×D4×D181C2211yescharacteristic
18C2×#15668439C2211yescharacteristic, complemented
19C2×#15668438C2211yescharacteristic
20C2×#15668438C2211yescharacteristic, complemented
21C2×#15668437C2211yescharacteristic
22C2×#15668436C2211yescharacteristic, complemented
23C2×D1448C2211yescharacteristic
24C2×D1448C2211yescharacteristic, complemented
25C2×#15668434C2211yescharacteristic
26#19153813C2211yescharacteristic
27#19153812C2211yescharacteristic
28#19153811—21no—
29#19153805—21no—
30#19153805C2211yescharacteristic
31#19153804C2211yescharacteristic, complemented
32#19153801—21no—
33#19153796C2211yescharacteristic
34#19153792—21no—
35#19153791C2211yescharacteristic
36#19153789—21no—
37#19153789—21no—
38#19153786—21no—
39#19153786—21no—
40#19153781C2211yescharacteristic, complemented
41#19153778C2211yescharacteristic, complemented
42#19153773C2211yescharacteristic
43#19153745C2211yescharacteristic, complemented
44#19153745C2211yescharacteristic, complemented
45#19153736C2211yescharacteristic
46#19153736C2211yescharacteristic
47#19153736C2211yescharacteristic, complemented
48#19153736C2211yescharacteristic, complemented
49#19153728C2211yescharacteristic
50#19153728C2211yescharacteristic
51C4×#13089678C2211yescharacteristic, complemented
52C4⋊Q8×C181C2211yescharacteristic, complemented
53#124×C181C2211yescharacteristic
54#103×C181C2211yescharacteristic, complemented
55#103×C181C2211yescharacteristic, complemented
56#102×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58#97×C181C2211yescharacteristic, complemented
59C23×D181—21no—
60#15668450—41no—
61#15668450—41no—
62C2×Dic362C2311yescharacteristic
63C2×D724—21no—
64C2×D724—21no—
65C2×D724—21no—
66C2×D724—21no—
67C2×D724C2311yescharacteristic
68C2×D724C2311yescharacteristic
69C2×D724C2311yescharacteristic
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73D4×D181—41no—
74D4×D181—41no—
75C2×C4×D181—21no—
76C2×C4×D181—21no—
77C2×C4×D181C2311yescharacteristic
78#15668439—41no—
79#15668438—41no—
80#15668438—41no—
81#15668437—41no—
82#15668436—41no—
83D1448—41no—
84D1448—41no—
85#15668434—41no—
86C2×#13089678C2311yescharacteristic
87C2×#13089678C2311yescharacteristic
88C4⋊Dic181—21no—
89#15668428—21no—
90#15668428D411yescharacteristic
91#15668428D411yescharacteristic
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426—21no—
97#15668426—21no—
98#15668426—21no—
99#15668426D411yescharacteristic
100#15668426D411yescharacteristic
101C4×Dic181—21no—
102C4×Dic181C2311yescharacteristic
103C2×Q8×C181—21no—
104C2×Q8×C181C2311yescharacteristic
105C2×D4×C181C2311yescharacteristic
106C2×C8×C181C2311yescharacteristic
107C2×C8×C181C2311yescharacteristic
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—21no—
117C22×D181—42no—
118C22×D181—81no—
119C22×D181—81no—
120C22×D181C2×D411yescharacteristic
121Dic362—21no—
122Dic362—21no—
123D724—21no—
124D724—21no—
125D724—21no—
126D724—21no—
127D724—21no—
128D724—21no—
129D724—42no—
130D724—81no—
131D724—81no—
132D724—81no—
133C22⋊D181—42no—
134C22⋊D181—81no—
135C22⋊D181—81no—
136C2×Dic181—21no—
137C2×Dic181C2×D411yescharacteristic
138C2×Dic181C2×D411yescharacteristic
139C2×Dic181C2×D411yescharacteristic
140C4×D181—21no—
141C4×D181—21no—
142C4×D181—42no—
143C4×D181—81no—
144#13089678—21no—
145#13089678—21no—
146C23×C181—21no—
147Q8×C181—21no—
148Q8×C181—21no—
149Q8×C181—81no—
150D4×C181—21no—
151D4×C181—21no—
152C2×C4×C181—21no—
153C2×C4×C181—21no—
154C2×C4×C181—21no—
155C2×C4×C181C2×D411yescharacteristic
156C2×C4×C181C2×D411yescharacteristic
157C2×C4×C181C2411yescharacteristic
158C8×C181—21no—
159C8×C181—21no—
160C22×C181—41no—
161C22×C181—41no—
162C22×C181—41no—
163C22×C181C22×D411yescharacteristic
164C2×D181—21no—
165C2×D181—21no—
166C2×D181—21no—
167C2×D181—41no—
168C2×D181—41no—
169C2×D181—41no—
170C2×D181—41no—
171C2×D181—41no—
172C2×D181—41no—
173C2×D181—41no—
174C2×D181—41no—
175C2×D181—41no—
176C2×D181—84no—
177C4×C181—21no—
178C4×C181—21no—
179C4×C181—41no—
180C4×C181—41no—
181C4×C181—41no—
182C4×C181H(5,𝔽2)11yescharacteristic
183C4×C181H(5,𝔽2)11yescharacteristic
184Dic181—21no—
185Dic181—21no—
186Dic181—21no—
187Dic181—41no—
188C2×C181—81no—
189C2×C181#53511yescharacteristic
190C2×C181#57611yescharacteristic
191C2×C181#57711yescharacteristic
192D181—41no—
193D181—81no—
194D181—81no—
195D181—81no—
196C181#322411yescharacteristic, complemented, Sylow
197#3224—1811nomaximal, Sylow
198#542—1811no—
199#535—1811no—
200#496—1811no—
201#495—1811no—
202#489D18111yescharacteristic, complemented
203#486—1811no—
204#464—1811no—
205#463—1811no—
206#460—1811no—
207#459—1811no—
208#452—1811no—
209#450—1811no—
210#449—1811no—
211#447—1811no—
212#433—1811no—
213C2×#41—1811no—
214C22×D4—1811no—
215C2×Q16—1811no—
216C2×SD16—1811no—
217C2×SD16—1811no—
218C2×SD16—1811no—
219C2×SD16—1811no—
220C2×D8—1811no—
221C2×D8—1811no—
222C2×D8—1811no—
223C4⋊Q8C2×D18111yescharacteristic, complemented
224#127—1811no—
225#127—1811no—
226#126—3621no—
227#124—3621no—
228#124—1811no—
229#124—1811no—
230#124C2×D18111yescharacteristic
231#123—3621no—
232C22⋊Q8—3621no—
233C22⋊Q8—1811no—
234#121—3621no—
235#121—3621no—
236#121—1811no—
237#120—3621no—
238#120—3621no—
239C4×Q8—1811no—
240C4×D4—1811no—
241#117—1811no—
242C4⋊C8—1811no—
243C4⋊C8—1811no—
244#103C2×D18111yescharacteristic, complemented
245#103C2×D18111yescharacteristic, complemented
246#102—1811no—
247#102—1811no—
248#102—1811no—
249#102—1811no—
250#102C2×D18111yescharacteristic
251#102C2×D18111yescharacteristic
252C22⋊C8—1811no—
253C22⋊C8—1811no—
254#97C2×D18111yescharacteristic, complemented
255C4×C8—1811no—
256C24—3621no—
257#41—7241no—
258#41—7241no—
259C2×Q8—21no—
260C2×Q8—1811no—
261C2×Q8C22×D18111yescharacteristic
262C2×D4—3621no—
263C2×D4—3621no—
264C2×D4—3621no—
265C2×D4—3621no—
266C2×D4—3621no—
267C2×D4—3621no—
268C2×D4—3621no—
269C2×D4—1811no—
270C2×D4—1811no—
271C2×D4—1811no—
272C2×D4—7241no—
273C2×D4—7241no—
274C2×D4C22×D18111yescharacteristic
275C22×C4—3621no—
276C22×C4—3621no—
277C22×C4—1811no—
278Q16—7241no—
279SD16—7241no—
280SD16—7241no—
281SD16—7241no—
282SD16—7241no—
283D8—7241no—
284D8—7241no—
285D8—7241no—
286C2×C8—1811no—
287C2×C8—1811no—
288C2×C8C22×D18111yescharacteristic
289C2×C8C22×D18111yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—3621no—
292C4⋊C4—3621no—
293C4⋊C4—1811no—
294C4⋊C4—1811no—
295C4⋊C4C22×D18111yescharacteristic
296C4⋊C4C22×D18111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—3621no—
300C22⋊C4—3621no—
301C22⋊C4—3621no—
302C22⋊C4—3621no—
303C22⋊C4—3621no—
304C22⋊C4—3621no—
305C22⋊C4—3621no—
306C22⋊C4—1811no—
307C22⋊C4—1811no—
308C42—3621no—
309C42—1811no—
310C42C22×D18111yescharacteristic
311C23—21no—
312C23—3621no—
313C23—3621no—
314C23—3621no—
315C23—14481no—
316C23—14481no—
317C23—1811no—
318C23—7242no—
319Q8—21no—
320Q8—21no—
321Q8—81no—
322Q8—3621no—
323Q8—3621no—
324D4—21no—
325D4—21no—
326D4—3621no—
327D4—3621no—
328D4—3621no—
329D4—3621no—
330D4—3621no—
331D4—3621no—
332D4—14481no—
333D4—14481no—
334D4—14481no—
335D4—14481no—
336D4—14481no—
337D4—7242no—
338D4—7242no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—3621no—
343C2×C4—3621no—
344C2×C4—3621no—
345C2×C4—14481no—
346C2×C4—1811no—
347C2×C4—1811no—
348C2×C4—1811no—
349C2×C4—7242no—
350C2×C4D4×D18111yescharacteristic
351C2×C4D4×D18111yescharacteristic
352C2×C4C23×D18111yescharacteristic
353C8—21no—
354C8—21no—
355C8—3621no—
356C8—3621no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—3621no—
361C22—3621no—
362C22—3621no—
363C22—14484no—
364C22—7241no—
365C22—7241no—
366C22—7241no—
367C22—7241no—
368C22—7241no—
369C22—7241no—
370C22—7241no—
371C22—7241no—
372C22—7241no—
373C22C2×D4×D18111yescharacteristic
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—3621no—
380C4—3621no—
381C4—3621no—
382C4—7241no—
383C4#1915386211yescharacteristic
384C4#1915386211yescharacteristic
385C2—81no—
386C2—14481no—
387C2—14481no—
388C2—14481no—
389C2—7241no—
390C2#2604184711yescharacteristic
391C2#2604208311yescharacteristic
392C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#489×C181C2
socleC22×C181C22×D4
hypercenter#489D181
hypocenterC181#3224
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #26041903C2semidirect
2. #26041847C2semidirect
3. #26041668C2semidirect
4. #26041661C2semidirect
5. #26041627C2semidirect
6. #26041504C2semidirect
7. #26041497C2semidirect
8. #26041478C2semidirect
9. #26041473C2semidirect
10. #26041444C2semidirect
11. #26041431C2semidirect
12. #26041427C2semidirect
13. #26041414C2semidirect
14. #26041341C2semidirect
15. #489×C181C2semidirect
18. C2×#15668439C22semidirect
20. C2×#15668438C22semidirect
22. C2×#15668436C22semidirect
24. C2×D1448C22semidirect
31. #19153804C22semidirect
40. #19153781C22semidirect
41. #19153778C22semidirect
43. #19153745C22semidirect
44. #19153745C22semidirect
47. #19153736C22semidirect
48. #19153736C22semidirect
51. C4×#13089678C22semidirect
52. C4⋊Q8×C181C22semidirect
54. #103×C181C22semidirect
55. #103×C181C22semidirect
58. #97×C181C22semidirect
196. C181#3224semidirect
202. #489D181semidirect
223. C4⋊Q8C2×D181semidirect
244. #103C2×D181semidirect
245. #103C2×D181semidirect
254. #97C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260419032
#260418472
#260416682
#260416612
#260416272
#260415042
#260414972
#260414782
#260414732
#260414442
#260414312
#260414272
#260414142
#260413412
#489×C1812
#3224181