Finite Group Databasebeta

← #37545831

Subgroups — #37545831

Order 23168.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041990C211yesmaximal, characteristic, complemented
2#26041937C211yesmaximal, characteristic
3#26041608C211yesmaximal, characteristic, complemented
4#26041596C211yesmaximal, characteristic, complemented
5#26041561C211yesmaximal, characteristic, complemented
6#26041536C211yesmaximal, characteristic, complemented
7#26041507C211yesmaximal, characteristic, complemented
8#26041498C211yesmaximal, characteristic, complemented
9#26041442C211yesmaximal, characteristic, complemented
10#26041436C211yesmaximal, characteristic, complemented
11#26041387C211yesmaximal, characteristic, complemented
12#26041352C211yesmaximal, characteristic, complemented
13#26041277C211yesmaximal, characteristic, complemented
14#103×D181C211yesmaximal, characteristic, complemented
15#489×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18C2×Dic724C2211yescharacteristic
19C2×#15668439C2211yescharacteristic
20C2×#15668437C2211yescharacteristic
21C2×#15668432C2211yescharacteristic
22C2×C8×D181C2211yescharacteristic
23#19153807—21no—
24#19153802—21no—
25#19153796C2211yescharacteristic
26#19153796C2211yescharacteristic
27#19153793—21no—
28#19153792—21no—
29#19153792—21no—
30#19153792C2211yescharacteristic
31#19153789—21no—
32#19153787C2211yescharacteristic
33C4×Dic362C2211yescharacteristic
34C4×Dic362C2211yescharacteristic
35C4×D724C2211yescharacteristic
36#19153778—21no—
37#19153776—21no—
38C4⋊C4×D181C2211yescharacteristic
39C4⋊C4×D181C2211yescharacteristic
40#19153748C2211yescharacteristic
41#19153747C2211yescharacteristic
42#19153747C2211yescharacteristic
43C4⋊#13089678C2211yescharacteristic
44#19153741C2211yescharacteristic
45#19153741C2211yescharacteristic
46#19153740C2211yescharacteristic
47#19153739C2211yescharacteristic
48#19153739C2211yescharacteristic
49#19153737C2211yescharacteristic
50#19153728C2211yescharacteristic
51#19153728C2211yescharacteristic
52C4⋊Q8×C181C2211yescharacteristic
53#124×C181C2211yescharacteristic
54#103×C181C2211yescharacteristic
55#103×C181C2211yescharacteristic
56#102×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58#97×C181C2211yescharacteristic
59#15668448—22no—
60#15668448—41no—
61C2×Dic362—21no—
62C2×Dic362C2311yescharacteristic
63C2×Dic362C2311yescharacteristic
64Q8×D181—22no—
65Q8×D181—41no—
66C2×D724D411yescharacteristic
67C2×C22⋊D181—21no—
68C2×C22⋊D181—21no—
69C22×Dic181—21no—
70C2×C4×D181—21no—
71C2×C4×D181—21no—
72C2×C4×D181—21no—
73C2×C4×D181C2311yescharacteristic
74C2×C4×D181C2311yescharacteristic
75Dic724—22no—
76#15668439—41no—
77#15668437—41no—
78#15668432—41no—
79C2×#13089678C2311yescharacteristic
80C2×#13089678C2311yescharacteristic
81C8×D181—41no—
82C4⋊Dic181—21no—
83C4⋊Dic181—21no—
84C4⋊Dic181D411yescharacteristic
85C4⋊Dic181C2311yescharacteristic
86C4⋊Dic181C2311yescharacteristic
87#15668428—21no—
88#15668428—21no—
89#15668428—21no—
90#15668428—21no—
91#15668428—21no—
92C22⋊Dic181—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426D411yescharacteristic
97#15668426D411yescharacteristic
98C4×Dic181—21no—
99C4×Dic181—21no—
100C2×Q8×C181—21no—
101C2×Q8×C181C2311yescharacteristic
102C2×D4×C181C2311yescharacteristic
103C2×C8×C181C2311yescharacteristic
104C2×C8×C181C2311yescharacteristic
105C4⋊C4×C181—21no—
106C4⋊C4×C181C2311yescharacteristic
107C4⋊C4×C181C2311yescharacteristic
108C22⋊C4×C181—21no—
109C22⋊C4×C181—21no—
110C42×C181C2311yescharacteristic
111C22×D181C2×D411yescharacteristic
112C22×D181C2×D411yescharacteristic
113Dic362—21no—
114Dic362—21no—
115Dic362—21no—
116Dic362—42no—
117Dic362#4112yes—
118D724—22no—
119C22⋊D181—42no—
120C22⋊D181—42no—
121C2×Dic181—21no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181—21no—
125C2×Dic181—42no—
126C2×Dic181C2×D411yescharacteristic
127C2×Dic181C2×D411yescharacteristic
128C4×D181—21no—
129C4×D181—21no—
130C4×D181—21no—
131C4×D181—21no—
132C4×D181—42no—
133C4×D181—81no—
134C4×D181—81no—
135#13089678—21no—
136#13089678—21no—
137C23×C181—21no—
138Q8×C181—21no—
139Q8×C181—21no—
140Q8×C181—81no—
141D4×C181—21no—
142D4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181C2×D411yescharacteristic
147C2×C4×C181C2×D411yescharacteristic
148C2×C4×C181C2411yescharacteristic
149C8×C181—21no—
150C8×C181—21no—
151C22×C181—41no—
152C22×C181—41no—
153C22×C181—41no—
154C22×C181C22×D411yescharacteristic
155C2×D181—21no—
156C2×D181—21no—
157C2×D181—21no—
158C2×D181—21no—
159C2×D181—21no—
160C2×D181#13712yes—
161C4×C181—21no—
162C4×C181—21no—
163C4×C181—41no—
164C4×C181—41no—
165C4×C181—41no—
166C4×C181C2×#4111yescharacteristic
167C4×C181H(5,𝔽2)11yescharacteristic
168Dic181—21no—
169Dic181—21no—
170Dic181—22no—
171Dic181—41no—
172Dic181—41no—
173Dic181—41no—
174C2×C181—81no—
175C2×C181#54611yescharacteristic
176C2×C181C2×#13711yescharacteristic
177C2×C181#57711yescharacteristic
178D181—22no—
179D181—41no—
180C181#331611yescharacteristic, complemented, Sylow
181#3316—1811nomaximal, Sylow
182#554—1811no—
183#548—1811no—
184#489D18111yescharacteristic, complemented
185#483—1811no—
186#481—1811no—
187#475—1811no—
188#470—1811no—
189#465—1811no—
190#464—1811no—
191#452—1811no—
192#451—1811no—
193#441—1811no—
194#435—1811no—
195#417—1811no—
196C2×#103—1811no—
197C2×#41—1811no—
198C22×Q8—1811no—
199C2×Q16—1811no—
200C2×Q16—1811no—
201C2×SD16—1811no—
202C2×M16—1811no—
203C22×C8—1811no—
204C4⋊Q8C2×D18111yescharacteristic
205#125—3621no—
206#124—3621no—
207#124C2×D18111yescharacteristic
208C22⋊Q8—3621no—
209C22⋊Q8—3621no—
210C22⋊Q8—3621no—
211C22⋊Q8—1811no—
212C22⋊Q8—1811no—
213C22⋊Q8—1811no—
214#121—3621no—
215#121—1811no—
216C4×Q8—1811no—
217C4×Q8—1811no—
218C4×D4—3621no—
219C4×D4—3621no—
220C4×D4—1811no—
221C2×C4⋊C4—1811no—
222C2×C4⋊C4—1811no—
223#106—1811no—
224#106—1811no—
225#106—1811no—
226C4⋊C8—1811no—
227#103—1811no—
228#103—1811no—
229#103—1811no—
230#103—1811no—
231#103—1811no—
232#103C2×D18111yescharacteristic
233#103C2×D18111yescharacteristic
234#102—1811no—
235#102C2×D18111yescharacteristic
236#102C2×D18111yescharacteristic
237C22⋊C8—1811no—
238C22⋊C8—1811no—
239#97C2×D18111yescharacteristic
240#41—3622no—
241#41—7241no—
242C2×Q8—21no—
243C2×Q8—3621no—
244C2×Q8—3622no—
245C2×Q8—1811no—
246C2×Q8—1811no—
247C2×Q8—7241no—
248C2×Q8C22×D18111yescharacteristic
249C2×D4—3621no—
250C2×D4—3621no—
251C2×D4—1811no—
252C2×D4C22×D18111yescharacteristic
253C22×C4—3621no—
254C22×C4—3621no—
255C22×C4—3621no—
256C22×C4—3621no—
257C22×C4—1811no—
258C22×C4—1811no—
259Q16—3622no—
260Q16—7241no—
261SD16—7241no—
262M16—7241no—
263C2×C8—1811no—
264C2×C8—1811no—
265C2×C8—7241no—
266C2×C8C22×D18111yescharacteristic
267C2×C8C22×D18111yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—3621no—
270C4⋊C4—3621no—
271C4⋊C4—3621no—
272C4⋊C4—3621no—
273C4⋊C4—3621no—
274C4⋊C4—3621no—
275C4⋊C4—3621no—
276C4⋊C4—1811no—
277C4⋊C4—1811no—
278C4⋊C4—1811no—
279C4⋊C4C22×D18111yescharacteristic
280C4⋊C4C22×D18111yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—3621no—
284C22⋊C4—3621no—
285C22⋊C4—3621no—
286C22⋊C4—3621no—
287C22⋊C4—1811no—
288C22⋊C4—1811no—
289C42—3621no—
290C42—3621no—
291C42C22×D18111yescharacteristic
292C23—21no—
293C23—1811no—
294C23—1811no—
295Q8—21no—
296Q8—21no—
297Q8—81no—
298Q8—3621no—
299Q8—3621no—
300Q8—3621no—
301Q8—1812no—
302Q8—7242no—
303D4—21no—
304D4—21no—
305D4—3622no—
306D4—7242no—
307D4—7242no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—3621no—
312C2×C4—3621no—
313C2×C4—3621no—
314C2×C4—3621no—
315C2×C4—3621no—
316C2×C4—3621no—
317C2×C4—3621no—
318C2×C4—3621no—
319C2×C4—14481no—
320C2×C4—14481no—
321C2×C4—1811no—
322C2×C4—1811no—
323C2×C4—7242no—
324C2×C4—7242no—
325C2×C4D4×D18111yescharacteristic
326C2×C4D4×D18111yescharacteristic
327C2×C4C23×D18111yescharacteristic
328C8—21no—
329C8—21no—
330C8—3621no—
331C8—3621no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—3621no—
336C22—3621no—
337C22—3621no—
338C22—3621no—
339C22—3621no—
340C22—1812no—
341C22C2×D4×D18111yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—3621no—
348C4—3621no—
349C4—3622no—
350C4—7241no—
351C4—7241no—
352C4—7241no—
353C4#41×D18111yescharacteristic
354C4#1915386211yescharacteristic
355C2—81no—
356C2—3622no—
357C2—7241no—
358C2#2604192211yescharacteristic
359C2#137×D18111yescharacteristic
360C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#489×C181C2
socleC22×C181C22×D4
prime radical#26041937C2
hypercenter#489D181
hypocenterC181#3316
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. #26041990C2semidirect
3. #26041608C2semidirect
4. #26041596C2semidirect
5. #26041561C2semidirect
6. #26041536C2semidirect
7. #26041507C2semidirect
8. #26041498C2semidirect
9. #26041442C2semidirect
10. #26041436C2semidirect
11. #26041387C2semidirect
12. #26041352C2semidirect
13. #26041277C2semidirect
14. #103×D181C2semidirect
15. #489×C181C2semidirect
180. C181#3316semidirect
184. #489D181semidirect

Maximal subgroups

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

subgroupindex
#260419902
#260419372
#260416082
#260415962
#260415612
#260415362
#260415072
#260414982
#260414422
#260414362
#260413872
#260413522
#260412772
#103×D1812
#489×C1812
#3316181