Finite Group Databasebeta

← #37546055

Subgroups — #37546055

Order 23168.

398 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#26041984C211yesmaximal, characteristic, complemented
2#26041948C211yesmaximal, characteristic
3#26041624C211yesmaximal, characteristic, complemented
4#26041592C211yesmaximal, characteristic, complemented
5#26041591C211yesmaximal, characteristic, complemented
6#26041572C211yesmaximal, characteristic, complemented
7#26041525C211yesmaximal, characteristic, complemented
8#26041524C211yesmaximal, characteristic, complemented
9#26041478C211yesmaximal, characteristic, complemented
10#26041444C211yesmaximal, characteristic, complemented
11#26041441C211yesmaximal, characteristic, complemented
12#26041371C211yesmaximal, characteristic, complemented
13#26041355C211yesmaximal, characteristic, complemented
14#26041279C211yesmaximal, characteristic, complemented
15#417×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668439C2211yescharacteristic
18C2×#15668438C2211yescharacteristic
19C2×#15668437C2211yescharacteristic
20C2×#15668432C2211yescharacteristic
21#19153815C2211yescharacteristic
22#19153812—21no—
23#19153812C2211yescharacteristic
24#19153806—21no—
25#19153796C2211yescharacteristic
26#19153795—21no—
27#19153795C2211yescharacteristic
28#19153794—21no—
29#19153790—21no—
30#19153790C2211yescharacteristic
31#19153787—21no—
32#19153787C2211yescharacteristic
33Q8×Dic181C2211yescharacteristic
34#19153781C2211yescharacteristic
35C4×C22⋊D181—21no—
36C4×C22⋊D181—21no—
37C4×C22⋊D181C2211yescharacteristic
38C4⋊C4×D181C2211yescharacteristic
39#19153750C2211yescharacteristic
40#19153749C2211yescharacteristic
41#19153749C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153741C2211yescharacteristic
44#19153740C2211yescharacteristic
45#19153738C2211yescharacteristic
46#19153737C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153736C2211yescharacteristic
49C22⋊#13089678C2211yescharacteristic
50#19153728C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C2×#41×C181C2211yescharacteristic
53C2×M16×C181C2211yescharacteristic
54C2×C4⋊C4×C181C2211yescharacteristic
55#103×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59#15668450—22no—
60#15668450—41no—
61C2×Dic362—21no—
62C2×Dic362C2311yescharacteristic
63C2×D724—21no—
64C2×D724C2311yescharacteristic
65C2×C22⋊D181—21no—
66C2×C22⋊D181—21no—
67C2×C22⋊D181—21no—
68C2×C22⋊D181—21no—
69C2×C22⋊D181D411yescharacteristic
70C2×C4×D181—21no—
71C2×C4×D181—21no—
72C2×C4×D181C2311yescharacteristic
73#15668439—22no—
74#15668438—41no—
75#15668437—41no—
76#15668432—41no—
77C2×#13089678C2311yescharacteristic
78C2×#13089678C2311yescharacteristic
79C4⋊Dic181—21no—
80C4⋊Dic181—21no—
81C4⋊Dic181C2311yescharacteristic
82C4⋊Dic181C2311yescharacteristic
83#15668428—21no—
84#15668428—21no—
85#15668428—21no—
86#15668428—21no—
87#15668428D411yescharacteristic
88C22⋊Dic181—21no—
89C22⋊Dic181—21no—
90C22⋊Dic181D411yescharacteristic
91#15668426—21no—
92#15668426—21no—
93#15668426D411yescharacteristic
94C4×Dic181—21no—
95C4×Dic181—21no—
96C4×Dic181C2311yescharacteristic
97#41×C181—22no—
98#41×C181—41no—
99C2×Q8×C181C2311yescharacteristic
100C2×D4×C181—21no—
101C2×D4×C181C2311yescharacteristic
102C22×C4×C181—21no—
103C22×C4×C181—21no—
104C22×C4×C181C2311yescharacteristic
105M16×C181—41no—
106C2×C8×C181C2311yescharacteristic
107C2×C8×C181C2311yescharacteristic
108C4⋊C4×C181—21no—
109C4⋊C4×C181C2311yescharacteristic
110C4⋊C4×C181C2311yescharacteristic
111C22×D181—21no—
112C22×D181C2×D411yescharacteristic
113Dic362—21no—
114Dic362—21no—
115Dic362—81no—
116D724—21no—
117D724—21no—
118D724—42no—
119C22⋊D181—22no—
120C22⋊D181—42no—
121C22⋊D181—81no—
122C22⋊D181—81no—
123C22⋊D181—81no—
124C2×Dic181—21no—
125C2×Dic181—21no—
126C2×Dic181—21no—
127C2×Dic181C2×D411yescharacteristic
128C2×Dic181C2×D411yescharacteristic
129C2×Dic181C2×D411yescharacteristic
130C4×D181—21no—
131C4×D181—21no—
132C4×D181—42no—
133C4×D181—81no—
134#13089678—21no—
135#13089678—21no—
136C23×C181—21no—
137C23×C181C2×D411yescharacteristic
138Q8×C181—21no—
139Q8×C181#4112yes—
140D4×C181—21no—
141D4×C181—21no—
142D4×C181—42no—
143C2×C4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181—21no—
148C2×C4×C181—42no—
149C2×C4×C181—81no—
150C2×C4×C181C2×D411yescharacteristic
151C2×C4×C181C2411yescharacteristic
152C8×C181—21no—
153C8×C181—21no—
154C22×C181—21no—
155C22×C181—21no—
156C22×C181—21no—
157C22×C181—41no—
158C22×C181—41no—
159C22×C181—41no—
160C22×C181C22×D411yescharacteristic
161C2×D181—21no—
162C2×D181—21no—
163C2×D181—21no—
164C2×D181—41no—
165C2×D181—41no—
166C2×D181—41no—
167C4×C181—21no—
168C4×C181—22no—
169C4×C181—41no—
170C4×C181—41no—
171C4×C181C2×#4111yescharacteristic
172C4×C181H(5,𝔽2)11yescharacteristic
173Dic181—21no—
174Dic181—21no—
175Dic181—41no—
176Dic181—41no—
177Dic181—41no—
178Dic181#13712yes—
179C2×C181—41no—
180C2×C181—81no—
181C2×C181#54611yescharacteristic
182C2×C181C2×#13711yescharacteristic
183C2×C181#57611yescharacteristic
184D181—41no—
185D181—81no—
186C181#333311yescharacteristic, complemented, Sylow
187#3333—1811nomaximal, Sylow
188#554—1811no—
189#550—1811no—
190#486—1811no—
191#480—1811no—
192#480—1811no—
193#477—1811no—
194#468—1811no—
195#468—1811no—
196#460—1811no—
197#452—1811no—
198#452—1811no—
199C4×Q16—1811no—
200#436—1811no—
201#417—1811no—
202#417D18111yescharacteristic, complemented
203C2×#41—1811no—
204C2×#41C2×D18111yescharacteristic
205C2×Q16—1811no—
206C2×SD16—1811no—
207C2×SD16—1811no—
208C2×M16—1811no—
209C2×M16C2×D18111yescharacteristic
210C4⋊Q8—1811no—
211#127—3621no—
212#127—1811no—
213#125—3621no—
214C22⋊Q8—3621no—
215C22⋊Q8—3621no—
216C22⋊Q8—1811no—
217C22⋊Q8—1811no—
218#121—3621no—
219#121—3621no—
220#121—1811no—
221#121—1811no—
222C4×Q8—1811no—
223C4×Q8—1811no—
224C4×D4—3621no—
225C4×D4—3621no—
226C4×D4—1811no—
227C2×C4⋊C4—1811no—
228C2×C4⋊C4C2×D18111yescharacteristic
229#107—1811no—
230#107—1811no—
231#107—1811no—
232C4⋊C8—1811no—
233#103—1811no—
234#103—1811no—
235#103C2×D18111yescharacteristic
236#103C2×D18111yescharacteristic
237#102—1811no—
238#102—1811no—
239#102—1811no—
240#102—1811no—
241#102C2×D18111yescharacteristic
242#102C2×D18111yescharacteristic
243C22⋊C8—1811no—
244C22⋊C8—1811no—
245C4×C8—1811no—
246#41—22no—
247#41—41no—
248#41—3622no—
249#41—7241no—
250C2×Q8—3621no—
251C2×Q8—1811no—
252C2×Q8C22×D18111yescharacteristic
253C2×D4—21no—
254C2×D4—3621no—
255C2×D4—3621no—
256C2×D4—3621no—
257C2×D4—3621no—
258C2×D4—3621no—
259C2×D4—1811no—
260C2×D4—1811no—
261C2×D4C22×D18111yescharacteristic
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—3621no—
265C22×C4—3621no—
266C22×C4—1811no—
267C22×C4C22×D18111yescharacteristic
268Q16—3622no—
269SD16—7241no—
270SD16—7241no—
271M16—41no—
272M16—7241no—
273C2×C8—1811no—
274C2×C8—1811no—
275C2×C8C22×D18111yescharacteristic
276C2×C8C22×D18111yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—3621no—
279C4⋊C4—3621no—
280C4⋊C4—3621no—
281C4⋊C4—3621no—
282C4⋊C4—3621no—
283C4⋊C4—3621no—
284C4⋊C4—1811no—
285C4⋊C4—1811no—
286C4⋊C4—1811no—
287C4⋊C4C22×D18111yescharacteristic
288C4⋊C4C22×D18111yescharacteristic
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4—3621no—
292C22⋊C4—3621no—
293C22⋊C4—1811no—
294C22⋊C4—1811no—
295C42—3621no—
296C42—3621no—
297C42—1811no—
298C23—21no—
299C23—3621no—
300C23—1811no—
301C23D4×D18111yescharacteristic
302Q8—21no—
303Q8—3621no—
304Q8—3621no—
305Q8—14481no—
306Q8#1566844912yes—
307D4—21no—
308D4—21no—
309D4—42no—
310D4—3622no—
311D4—3621no—
312D4—3621no—
313D4—14481no—
314D4—14481no—
315D4—14481no—
316D4—7242no—
317D4—7242no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—81no—
325C2×C4—3621no—
326C2×C4—3621no—
327C2×C4—3621no—
328C2×C4—3621no—
329C2×C4—3621no—
330C2×C4—14481no—
331C2×C4—1811no—
332C2×C4—1811no—
333C2×C4—1811no—
334C2×C4—7242no—
335C2×C4D4×D18111yescharacteristic
336C2×C4C23×D18111yescharacteristic
337C8—21no—
338C8—21no—
339C8—3621no—
340C8—3621no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—3621no—
348C22—3621no—
349C22—3621no—
350C22—7241no—
351C22—7241no—
352C22—7241no—
353C22C2×D4×D18111yescharacteristic
354C4—21no—
355C4—22no—
356C4—41no—
357C4—41no—
358C4—3621no—
359C4—3621no—
360C4—1812no—
361C4—7241no—
362C4—7241no—
363C4—7241no—
364C4C2×#1566844911yescharacteristic
365C4#1915386111yescharacteristic
366C2—41no—
367C2—81no—
368C2—14481no—
369C2—7241no—
370C2#2604191911yescharacteristic
371C2#137×D18111yescharacteristic
372C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#417×C181C2
socleC22×C181C22×D4
prime radical#26041948C2
hypercenter#417D181
hypocenterC181#3333
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #26041984C2semidirect
3. #26041624C2semidirect
4. #26041592C2semidirect
5. #26041591C2semidirect
6. #26041572C2semidirect
7. #26041525C2semidirect
8. #26041524C2semidirect
9. #26041478C2semidirect
10. #26041444C2semidirect
11. #26041441C2semidirect
12. #26041371C2semidirect
13. #26041355C2semidirect
14. #26041279C2semidirect
15. #417×C181C2semidirect
186. C181#3333semidirect
202. #417D181semidirect

Maximal subgroups

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

subgroupindex
#260419842
#260419482
#260416242
#260415922
#260415912
#260415722
#260415252
#260415242
#260414782
#260414442
#260414412
#260413712
#260413552
#260412792
#417×C1812
#3333181