Finite Group Databasebeta

← #37545284

Subgroups — #37545284

Order 23168.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153845C211yesmaximal, characteristic, complemented
2C22×#15668439C211yesmaximal, characteristic, complemented
3Q8×C22⋊D181C211yesmaximal, characteristic, complemented
4#26041935C211yesmaximal, characteristic
5#26041670C211yesmaximal, characteristic, complemented
6#26041536C211yesmaximal, characteristic, complemented
7#26041506C211yesmaximal, characteristic, complemented
8#26041499C211yesmaximal, characteristic, complemented
9#26041488C211yesmaximal, characteristic, complemented
10#26041438C211yesmaximal, characteristic, complemented
11#26041437C211yesmaximal, characteristic, complemented
12#26041436C211yesmaximal, characteristic, complemented
13#26041383C211yesmaximal, characteristic, complemented
14#26041347C211yesmaximal, characteristic, complemented
15#448×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C22×Dic362C2211yescharacteristic
18C2×Q8×D181C2211yescharacteristic
19#19153845—24no—
20C2×Dic724C2211yescharacteristic
21C2×Dic724C2211yescharacteristic
22C2×#15668439—24no—
23C2×#15668439C2211yescharacteristic
24C2×#15668439C2211yescharacteristic
25C2×#15668439C2211yescharacteristic
26C2×#15668439C2211yescharacteristic
27C2×#15668437C2211yescharacteristic
28C2×#15668432C2211yescharacteristic
29C22×#13089678C2211yescharacteristic
30#19153814—21no—
31#19153814C2211yescharacteristic
32#19153803C2211yescharacteristic
33#19153802—21no—
34#19153796C2211yescharacteristic
35#19153795—21no—
36#19153795C2211yescharacteristic
37#19153793—21no—
38#19153792—21no—
39#19153792C2211yescharacteristic
40#19153788—21no—
41Q8×Dic181C2211yescharacteristic
42#19153781C2211yescharacteristic
43D4×Dic181—21no—
44C4×C22⋊D181—21no—
45C4×C22⋊D181C2211yescharacteristic
46#19153747C2211yescharacteristic
47#19153745C2211yescharacteristic
48#19153741C2211yescharacteristic
49#19153741C2211yescharacteristic
50#19153739C2211yescharacteristic
51#19153737C2211yescharacteristic
52#19153728C2211yescharacteristic
53#19153726C2211yescharacteristic
54C22×Q8×C181C2211yescharacteristic
55C2×SD16×C181C2211yescharacteristic
56C2×SD16×C181C2211yescharacteristic
57#121×C181C2211yescharacteristic
58#103×C181C2211yescharacteristic
59#103×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⋊D181D411yescharacteristic
74C22×Dic181—21no—
75C22×Dic181—21no—
76C2×C4×D181—21no—
77C2×C4×D181C2311yescharacteristic
78Dic724—22no—
79Dic724—41no—
80#15668439—22no—
81#15668439—22no—
82#15668439—22no—
83#15668439D414yescomplemented
84#15668437—22no—
85#15668432—22no—
86C2×#13089678—22no—
87C2×#13089678C2311yescharacteristic
88C2×#13089678C2311yescharacteristic
89C4⋊Dic181—21no—
90C4⋊Dic181—21no—
91C4⋊Dic181C2311yescharacteristic
92#15668428—21no—
93#15668428—21no—
94#15668428—21no—
95#15668428—21no—
96#15668428—21no—
97#15668428D411yescharacteristic
98C22⋊Dic181—21no—
99C22⋊Dic181—21no—
100C22⋊Dic181—21no—
101C22⋊Dic181D411yescharacteristic
102#15668426—21no—
103#15668426—21no—
104#15668426D411yescharacteristic
105C4×Dic181—21no—
106C4×Dic181—21no—
107C4×Dic181C2311yescharacteristic
108C2×Q8×C181—21no—
109C2×Q8×C181—22no—
110C2×Q8×C181—41no—
111C2×Q8×C181C2311yescharacteristic
112C2×Q8×C181C2311yescharacteristic
113C2×D4×C181—21no—
114C2×D4×C181C2311yescharacteristic
115C22×C4×C181—21no—
116C22×C4×C181C2311yescharacteristic
117SD16×C181—22no—
118SD16×C181—41no—
119C2×C8×C181C2311yescharacteristic
120C2×C8×C181C2311yescharacteristic
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—
134C2×Dic181—21no—
135C2×Dic181—21no—
136C2×Dic181—21no—
137C2×Dic181—21no—
138C2×Dic181—42no—
139C2×Dic181—42no—
140C2×Dic181C2×D411yescharacteristic
141C2×Dic181C2×D411yescharacteristic
142C2×Dic181C2×D411yescharacteristic
143C4×D181—21no—
144C4×D181—21no—
145C4×D181—42no—
146#13089678—21no—
147#13089678C2×D412yes—
148C23×C181—21no—
149C23×C181C2×D411yescharacteristic
150Q8×C181—21no—
151Q8×C181—21no—
152Q8×C181—21no—
153Q8×C181—42no—
154Q8×C181C2×D412yes—
155D4×C181—21no—
156D4×C181—21no—
157D4×C181—42no—
158C2×C4×C181—21no—
159C2×C4×C181—21no—
160C2×C4×C181—21no—
161C2×C4×C181—21no—
162C2×C4×C181—21no—
163C2×C4×C181—42no—
164C2×C4×C181C2×D411yescharacteristic
165C2×C4×C181C2411yescharacteristic
166C8×C181—21no—
167C8×C181—21no—
168C22×C181—21no—
169C22×C181—21no—
170C22×C181—41no—
171C22×C181—41no—
172C22×C181—41no—
173C22×C181#13712yescomplemented
174C22×C181C22×D411yescharacteristic
175C2×D181—21no—
176C2×D181—21no—
177C2×D181—21no—
178C4×C181—21no—
179C4×C181—22no—
180C4×C181—41no—
181C4×C181—41no—
182C4×C181C22×D411yescharacteristic
183C4×C181H(5,𝔽2)11yescharacteristic
184Dic181—21no—
185Dic181—22no—
186Dic181—21no—
187Dic181—21no—
188Dic181—41no—
189Dic181—41no—
190Dic181—41no—
191C2×C181—22no—
192C2×C181—81no—
193C2×C181D4211yescharacteristic
194C2×C181C2×#13711yescharacteristic
195C2×C181#57811yescharacteristic
196D181—41no—
197C181#327011yescharacteristic, complemented, Sylow
198#3270—1811nomaximal, Sylow
199C2×#137—1811no—
200C22×Q16—1811no—
201D4×Q8—1811no—
202#548—1811no—
203#497—1811no—
204#470—1811no—
205#465—1811no—
206#464—1811no—
207#462—1811no—
208#452—1811no—
209#451—1811no—
210#451—1811no—
211#448D18111yescharacteristic, complemented
212#441—1811no—
213#435—1811no—
214C2×#41—1811no—
215C22×Q8—1811no—
216C22×Q8—1811no—
217C22×Q8C2×D18111yescharacteristic
218#137—3624no—
219C2×Q16—3624no—
220C2×Q16—1811no—
221C2×Q16—1811no—
222C2×Q16—1811no—
223C2×Q16—1811no—
224C2×Q16—1811no—
225C2×Q16—1811no—
226C2×SD16—1811no—
227C2×SD16C2×D18111yescharacteristic
228C2×SD16C2×D18111yescharacteristic
229C2×M16—1811no—
230C22×C8—1811no—
231C4⋊Q8—3621no—
232C4⋊Q8—1811no—
233#124—3621no—
234#124—1811no—
235C22⋊Q8—3621no—
236C22⋊Q8—3621no—
237C22⋊Q8—3621no—
238C22⋊Q8—1811no—
239C22⋊Q8—1811no—
240C22⋊Q8—1811no—
241#121—3621no—
242#121C2×D18111yescharacteristic
243C4×Q8—1811no—
244C4×Q8—1811no—
245C4×D4—3621no—
246C4×D4—3621no—
247C4×D4—1811no—
248#106—1811no—
249C4⋊C8—1811no—
250#103—1811no—
251#103—1811no—
252#103—1811no—
253#103C2×D18111yescharacteristic
254#103C2×D18111yescharacteristic
255#102—1811no—
256C22⋊C8—1811no—
257C22⋊C8C2×D18111yescharacteristic
258#97—1811no—
259#41—3622no—
260#41—7241no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—3621no—
265C2×Q8—3622no—
266C2×Q8—3621no—
267C2×Q8—3622no—
268C2×Q8—1811no—
269C2×Q8—1811no—
270C2×Q8—1811no—
271C2×Q8—7241no—
272C2×Q8—7241no—
273C2×Q8C22×D18111yescharacteristic
274C2×Q8C22×D18111yescharacteristic
275C2×D4—21no—
276C2×D4—3621no—
277C2×D4—1811no—
278C2×D4C22×D18111yescharacteristic
279C22×C4—21no—
280C22×C4—3621no—
281C22×C4—3621no—
282C22×C4—3621no—
283C22×C4—1811no—
284C22×C4C22×D18111yescharacteristic
285Q16—3622no—
286Q16—3622no—
287Q16—3622no—
288Q16—3622no—
289Q16—1814no—
290Q16—7241no—
291SD16—22no—
292SD16—41no—
293SD16—3622no—
294M16—3622no—
295C2×C8—3622no—
296C2×C8—1811no—
297C2×C8—1811no—
298C2×C8C22×D18111yescharacteristic
299C2×C8C22×D18111yescharacteristic
300C4⋊C4—3621no—
301C4⋊C4—3621no—
302C4⋊C4—3621no—
303C4⋊C4—3621no—
304C4⋊C4—3621no—
305C4⋊C4—3621no—
306C4⋊C4—3621no—
307C4⋊C4—1811no—
308C4⋊C4—1811no—
309C4⋊C4C22×D18111yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—3621no—
312C22⋊C4—3621no—
313C22⋊C4—3621no—
314C22⋊C4—3621no—
315C22⋊C4—3621no—
316C22⋊C4—1811no—
317C22⋊C4—1811no—
318C42—3621no—
319C42—3621no—
320C42—1811no—
321C23—21no—
322C23—1811no—
323C23D4×D18111yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—42no—
328Q8—3621no—
329Q8—3621no—
330Q8—3621no—
331Q8—3621no—
332Q8—3621no—
333Q8—1812no—
334Q8—7242no—
335Q8—7242no—
336Q8D4×D18112yes—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—3622no—
341D4—7242no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—3621no—
349C2×C4—3621no—
350C2×C4—3621no—
351C2×C4—3621no—
352C2×C4—3621no—
353C2×C4—3621no—
354C2×C4—1811no—
355C2×C4—1811no—
356C2×C4—1811no—
357C2×C4—7242no—
358C2×C4—7242no—
359C2×C4—7242no—
360C2×C4D4×D18111yescharacteristic
361C2×C4C23×D18111yescharacteristic
362C8—21no—
363C8—21no—
364C8—3621no—
365C8—1812no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—3621no—
372C22—3621no—
373C22—3621no—
374C22#1915384512yescomplemented
375C22C2×D4×D18111yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—3621no—
381C4—3622no—
382C4—3621no—
383C4—3621no—
384C4—7241no—
385C4—7241no—
386C4—7241no—
387C4C2×D4×D18111yescharacteristic
388C4#1915386111yescharacteristic
389C2—22no—
390C2—81no—
391C2—7241no—
392C2#2604191111yescharacteristic
393C2C2×#1915384511yescharacteristic
394C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#448×C181C2
socleC22×C181C22×D4
prime radical#26041935C2
hypercenter#448D181
hypocenterC181#3270
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#19153845C2semidirect
2. C22×#15668439C2semidirect
3. Q8×C22⋊D181C2semidirect
5. #26041670C2semidirect
6. #26041536C2semidirect
7. #26041506C2semidirect
8. #26041499C2semidirect
9. #26041488C2semidirect
10. #26041438C2semidirect
11. #26041437C2semidirect
12. #26041436C2semidirect
13. #26041383C2semidirect
14. #26041347C2semidirect
15. #448×C181C2semidirect
83. #15668439D4semidirect
173. C22×C181#137semidirect
197. C181#3270semidirect
211. #448D181semidirect
374. 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×#156684392
Q8×C22⋊D1812
#260419352
#260416702
#260415362
#260415062
#260414992
#260414882
#260414382
#260414372
#260414362
#260413832
#260413472
#448×C1812
#3270181