Finite Group Databasebeta

← #37545235

Subgroups — #37545235

Order 23168.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153845C211yesmaximal, characteristic, complemented
2C22×#15668437C211yesmaximal, characteristic, complemented
3#26041942C211yesmaximal, characteristic, complemented
4#26041923C211yesmaximal, characteristic
5#26041670C211yesmaximal, characteristic, complemented
6#26041524C211yesmaximal, characteristic, complemented
7#26041518C211yesmaximal, characteristic, complemented
8#26041500C211yesmaximal, characteristic, complemented
9#26041482C211yesmaximal, characteristic, complemented
10#26041442C211yesmaximal, characteristic, complemented
11#26041437C211yesmaximal, characteristic, complemented
12#26041416C211yesmaximal, characteristic, complemented
13#26041376C211yesmaximal, characteristic, complemented
14#26041347C211yesmaximal, characteristic, complemented
15#450×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C22×Dic362C2211yescharacteristic
18C2×Q8×D181C2211yescharacteristic, complemented
19#19153845—24no—
20C2×Dic724C2211yescharacteristic, complemented
21C2×Dic724C2211yescharacteristic, complemented
22C2×#15668439C2211yescharacteristic, complemented
23C2×#15668437—24no—
24C2×#15668437C2211yescharacteristic
25C2×#15668437C2211yescharacteristic
26C2×#15668437C2211yescharacteristic
27C2×#15668437C2211yescharacteristic
28C2×#15668432C2211yescharacteristic, complemented
29C22×#13089678C2211yescharacteristic
30#19153815—21no—
31#19153814C2211yescharacteristic, complemented
32#19153803C2211yescharacteristic
33#19153797—21no—
34#19153796C2211yescharacteristic, complemented
35#19153795C2211yescharacteristic
36#19153794—21no—
37#19153793—21no—
38#19153788—21no—
39#19153787C2211yescharacteristic
40#19153785—21no—
41#19153781C2211yescharacteristic, complemented
42D4×Dic181C2211yescharacteristic
43C4×C22⋊D181C2211yescharacteristic
44#19153776—21no—
45C2×C22⋊Dic181—21no—
46#19153749C2211yescharacteristic, complemented
47#19153745C2211yescharacteristic, complemented
48#19153741C2211yescharacteristic, complemented
49#19153741C2211yescharacteristic, complemented
50#19153740C2211yescharacteristic, complemented
51#19153739C2211yescharacteristic, complemented
52#19153728C2211yescharacteristic, complemented
53#19153726C2211yescharacteristic, complemented
54C22×D4×C181C2211yescharacteristic
55C2×SD16×C181C2211yescharacteristic
56C2×SD16×C181C2211yescharacteristic
57C22⋊Q8×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59#102×C181C2211yescharacteristic
60C22⋊C8×C181C2211yescharacteristic
61#15668448—22no—
62#15668448—41no—
63C2×Dic362—21no—
64C2×Dic362—22no—
65C2×Dic362—21no—
66C2×Dic362—41no—
67C2×Dic362C2311yescharacteristic
68C2×Dic362C2311yescharacteristic
69C2×Dic362C2311yescharacteristic
70Q8×D181—22no—
71Q8×D181—41no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181—21no—
74C2×C22⋊D181D411yescharacteristic
75C22×Dic181—21no—
76C22×Dic181—21no—
77C22×Dic181—21no—
78C2×C4×D181—21no—
79C2×C4×D181C2311yescharacteristic
80Dic724—22no—
81Dic724—41no—
82#15668439—22no—
83#15668437—22no—
84#15668437—22no—
85#15668437—22no—
86#15668437D414yescomplemented
87#15668432—22no—
88C2×#13089678—22no—
89C2×#13089678C2311yescharacteristic
90C2×#13089678C2311yescharacteristic
91C4⋊Dic181—21no—
92C4⋊Dic181C2311yescharacteristic
93#15668428—21no—
94#15668428—21no—
95#15668428—21no—
96#15668428—21no—
97#15668428D411yescharacteristic
98C22⋊Dic181—21no—
99C22⋊Dic181—21no—
100C22⋊Dic181—21no—
101C22⋊Dic181—21no—
102C22⋊Dic181—21no—
103C22⋊Dic181D411yescharacteristic
104#15668426—21no—
105#15668426D411yescharacteristic
106C4×Dic181—21no—
107C4×Dic181C2311yescharacteristic
108C24×C181—21no—
109C2×Q8×C181C2311yescharacteristic
110C2×D4×C181—22no—
111C2×D4×C181—21no—
112C2×D4×C181—41no—
113C2×D4×C181C2311yescharacteristic
114C2×D4×C181C2311yescharacteristic
115C22×C4×C181C2311yescharacteristic
116SD16×C181—22no—
117SD16×C181—41no—
118C2×C8×C181C2311yescharacteristic
119C2×C8×C181C2311yescharacteristic
120C4⋊C4×C181—21no—
121C4⋊C4×C181C2311yescharacteristic
122C22⋊C4×C181—21no—
123C22×D181C2×D411yescharacteristic
124Dic362—21no—
125Dic362—21no—
126Dic362—21no—
127Dic362—21no—
128Dic362—21no—
129Dic362—42no—
130Dic362—42no—
131Dic362C2×D412yes—
132C22⋊D181—22no—
133C22⋊D181—42no—
134C22⋊D181—81no—
135C2×Dic181—21no—
136C2×Dic181—21no—
137C2×Dic181—21no—
138C2×Dic181—21no—
139C2×Dic181—42no—
140C2×Dic181—42no—
141C2×Dic181—81no—
142C2×Dic181C2×D411yescharacteristic
143C2×Dic181C2×D411yescharacteristic
144C2×Dic181C2×D411yescharacteristic
145C4×D181—21no—
146C4×D181—21no—
147C4×D181—42no—
148#13089678—21no—
149#13089678C2×D412yes—
150C23×C181—21no—
151C23×C181—21no—
152C23×C181—42no—
153C23×C181—42no—
154C23×C181—81no—
155C23×C181C2×D411yescharacteristic
156Q8×C181—21no—
157Q8×C181—21no—
158D4×C181—21no—
159D4×C181—21no—
160D4×C181—21no—
161D4×C181—42no—
162D4×C181C2×D412yes—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—21no—
166C2×C4×C181—21no—
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2411yescharacteristic
169C8×C181—21no—
170C8×C181—21no—
171C22×C181—21no—
172C22×C181—21no—
173C22×C181—22no—
174C22×C181—41no—
175C22×C181—41no—
176C22×C181—41no—
177C22×C181—41no—
178C22×C181—41no—
179C22×C181—84no—
180C22×C181#13712yescomplemented
181C22×C181C22×D411yescharacteristic
182C2×D181—21no—
183C2×D181—21no—
184C2×D181—21no—
185C4×C181—21no—
186C4×C181—41no—
187C4×C181—41no—
188C4×C181C22×D411yescharacteristic
189C4×C181H(5,𝔽2)11yescharacteristic
190Dic181—21no—
191Dic181—22no—
192Dic181—21no—
193Dic181—21no—
194Dic181—41no—
195Dic181—41no—
196Dic181—41no—
197C2×C181—22no—
198C2×C181—42no—
199C2×C181—81no—
200C2×C181D4211yescharacteristic
201C2×C181C2×#13711yescharacteristic
202C2×C181#57711yescharacteristic
203D181—41no—
204C181#326611yescharacteristic, complemented, Sylow
205#3266—1811nomaximal, Sylow
206C2×#137—1811no—
207C22×SD16—1811no—
208D4×Q8—1811no—
209#546—1811no—
210#497—1811no—
211#468—1811no—
212#467—1811no—
213#464—1811no—
214#461—1811no—
215#452—1811no—
216#451—1811no—
217#450D18111yescharacteristic, complemented
218#448—1811no—
219#440—1811no—
220#435—1811no—
221C2×#41—1811no—
222C22×Q8—1811no—
223C22×Q8—1811no—
224C22×D4C2×D18111yescharacteristic
225#137—3624no—
226C2×Q16—1811no—
227C2×Q16—1811no—
228C2×Q16—1811no—
229C2×SD16—3624no—
230C2×SD16—1811no—
231C2×SD16—1811no—
232C2×SD16—1811no—
233C2×SD16—1811no—
234C2×SD16C2×D18111yescharacteristic
235C2×SD16C2×D18111yescharacteristic
236C2×M16—1811no—
237C22×C8—1811no—
238C4⋊Q8—3621no—
239C4⋊Q8—1811no—
240#124—1811no—
241#123—3621no—
242C22⋊Q8—3621no—
243C22⋊Q8—3621no—
244C22⋊Q8—1811no—
245C22⋊Q8—1811no—
246C22⋊Q8C2×D18111yescharacteristic
247#121—3621no—
248#121—1811no—
249#120—3621no—
250C4×Q8—1811no—
251C4×D4—3621no—
252C4×D4—1811no—
253C4×D4—1811no—
254C2×C22⋊C4—3621no—
255#107—1811no—
256C4⋊C8—1811no—
257#103—1811no—
258#103—1811no—
259#103—1811no—
260#103—1811no—
261#102C2×D18111yescharacteristic
262#102C2×D18111yescharacteristic
263C22⋊C8—1811no—
264C22⋊C8C2×D18111yescharacteristic
265#97—1811no—
266C24—21no—
267#41—3622no—
268#41—7241no—
269C2×Q8—3621no—
270C2×Q8—3622no—
271C2×Q8—3621no—
272C2×Q8—3622no—
273C2×Q8—1811no—
274C2×Q8—1811no—
275C2×Q8—1811no—
276C2×Q8—7241no—
277C2×Q8—7241no—
278C2×Q8C22×D18111yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—3621no—
283C2×D4—3621no—
284C2×D4—1811no—
285C2×D4C22×D18111yescharacteristic
286C2×D4C22×D18111yescharacteristic
287C22×C4—3621no—
288C22×C4—3621no—
289C22×C4—3621no—
290C22×C4—3621no—
291C22×C4—1811no—
292C22×C4C22×D18111yescharacteristic
293Q16—3622no—
294Q16—3622no—
295Q16—7241no—
296SD16—22no—
297SD16—41no—
298SD16—3622no—
299SD16—3622no—
300SD16—3622no—
301SD16—1814no—
302M16—3622no—
303C2×C8—3622no—
304C2×C8—1811no—
305C2×C8—1811no—
306C2×C8C22×D18111yescharacteristic
307C2×C8C22×D18111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—3621no—
310C4⋊C4—3621no—
311C4⋊C4—3621no—
312C4⋊C4—3621no—
313C4⋊C4—3621no—
314C4⋊C4—1811no—
315C4⋊C4—1811no—
316C4⋊C4C22×D18111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—3621no—
319C22⋊C4—3621no—
320C22⋊C4—3621no—
321C22⋊C4—3621no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4—1811no—
325C22⋊C4—1811no—
326C42—3621no—
327C42—1811no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1811no—
334C23D4×D18111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—3621no—
338Q8—3621no—
339Q8—3621no—
340Q8—3621no—
341Q8—3621no—
342Q8—1812no—
343Q8—7242no—
344Q8—7242no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—3622no—
350D4—14481no—
351D4—7242no—
352D4D4×D18112yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—3621no—
358C2×C4—3621no—
359C2×C4—3621no—
360C2×C4—3621no—
361C2×C4—3621no—
362C2×C4—3621no—
363C2×C4—14481no—
364C2×C4—1811no—
365C2×C4—1811no—
366C2×C4—1811no—
367C2×C4—7242no—
368C2×C4—7242no—
369C2×C4—7242no—
370C2×C4D4×D18111yescharacteristic
371C2×C4C23×D18111yescharacteristic
372C8—21no—
373C8—21no—
374C8—3621no—
375C8—1812no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—3621no—
386C22—3621no—
387C22—3621no—
388C22#1915384512yescomplemented
389C22C2×D4×D18111yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—3621no—
394C4—3622no—
395C4—3621no—
396C4—3621no—
397C4—7241no—
398C4—7241no—
399C4—7241no—
400C4C2×D4×D18111yescharacteristic
401C4#1915386111yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—7241no—
406C2#2604191111yescharacteristic
407C2C2×#1915384511yescharacteristic
408C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#450×C181C2
socleC22×C181C22×D4
prime radical#26041923C2
hypercenter#450D181
hypocenterC181#3266
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. C2×#19153845C2semidirect
2. C22×#15668437C2semidirect
3. #26041942C2semidirect
5. #26041670C2semidirect
6. #26041524C2semidirect
7. #26041518C2semidirect
8. #26041500C2semidirect
9. #26041482C2semidirect
10. #26041442C2semidirect
11. #26041437C2semidirect
12. #26041416C2semidirect
13. #26041376C2semidirect
14. #26041347C2semidirect
15. #450×C181C2semidirect
18. C2×Q8×D181C22semidirect
20. C2×Dic724C22semidirect
21. C2×Dic724C22semidirect
22. C2×#15668439C22semidirect
28. C2×#15668432C22semidirect
31. #19153814C22semidirect
34. #19153796C22semidirect
41. #19153781C22semidirect
46. #19153749C22semidirect
47. #19153745C22semidirect
48. #19153741C22semidirect
49. #19153741C22semidirect
50. #19153740C22semidirect
51. #19153739C22semidirect
52. #19153728C22semidirect
53. #19153726C22semidirect
86. #15668437D4semidirect
180. C22×C181#137semidirect
204. C181#3266semidirect
217. #450D181semidirect
388. C22#19153845semidirect

Maximal subgroups

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

subgroupindex
C2×#191538452
C22×#156684372
#260419422
#260419232
#260416702
#260415242
#260415182
#260415002
#260414822
#260414422
#260414372
#260414162
#260413762
#260413472
#450×C1812
#3266181