Finite Group Databasebeta

← #37546378

Subgroups — #37546378

Order 23168.

362 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042011C211yesmaximal, characteristic, complemented
2#26041985C211yesmaximal, characteristic, complemented
3#26041635C211yesmaximal, characteristic, complemented
4#26041624C211yesmaximal, characteristic, complemented
5#26041577C211yesmaximal, characteristic, complemented
6#26041576C211yesmaximal, characteristic, complemented
7#26041572C211yesmaximal, characteristic, complemented
8#26041497C211yesmaximal, characteristic, complemented
9#26041491C211yesmaximal, characteristic, complemented
10#26041473C211yesmaximal, characteristic
11C8×Dic362C211yesmaximal, characteristic, complemented
12#26041364C211yesmaximal, characteristic, complemented
13C4×D1448C211yesmaximal, characteristic, complemented
14#26041359C211yesmaximal, characteristic, complemented
15#488×C181C211yesmaximal, characteristic, complemented
16C2×#15668436C2211yescharacteristic
17C2×D1448C2211yescharacteristic
18C2×#15668434C2211yescharacteristic
19#19153815C2211yescharacteristic
20#19153812C2211yescharacteristic
21#19153807C2211yescharacteristic
22#19153805C2211yescharacteristic
23#19153802—21no—
24#19153800—21no—
25#19153798—21no—
26#19153796—21no—
27#19153794—21no—
28#19153794—21no—
29Q8×Dic181C2211yescharacteristic
30C4×Dic362C2211yescharacteristic
31C4×D724C2211yescharacteristic
32C4×D724C2211yescharacteristic
33D4×Dic181C2211yescharacteristic
34#19153778C2211yescharacteristic
35#19153778C2211yescharacteristic
36#19153776—21no—
37C4⋊C4×D181—21no—
38#19153750C2211yescharacteristic
39#19153749C2211yescharacteristic
40#19153747C2211yescharacteristic
41C4⋊#13089678C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153745C2211yescharacteristic
44#19153740C2211yescharacteristic
45#19153738C2211yescharacteristic
46#19153738C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153736C2211yescharacteristic
49#19153736C2211yescharacteristic
50C8×Dic181C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52#125×C181C2211yescharacteristic
53#124×C181C2211yescharacteristic
54#103×C181C2211yescharacteristic
55#103×C181C2211yescharacteristic
56#102×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58C4×C8×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362D411yescharacteristic
61C2×D724C2311yescharacteristic
62C2×D724C2311yescharacteristic
63C2×C22⋊D181—21no—
64C2×C22⋊D181—21no—
65C22×Dic181—21no—
66C2×C4×D181—21no—
67C2×C4×D181—21no—
68C2×C4×D181—21no—
69C2×C4×D181—21no—
70#15668436—22no—
71D1448—22no—
72#15668434—22no—
73C2×#13089678C2311yescharacteristic
74C2×#13089678C2311yescharacteristic
75C4⋊Dic181—21no—
76C4⋊Dic181—21no—
77C4⋊Dic181D411yescharacteristic
78C4⋊Dic181C2311yescharacteristic
79C4⋊Dic181C2311yescharacteristic
80#15668428—21no—
81#15668428—21no—
82#15668428D411yescharacteristic
83#15668428D411yescharacteristic
84C22⋊Dic181—21no—
85#15668426—21no—
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89#15668426—21no—
90#15668426—21no—
91C4×Dic181—21no—
92C4×Dic181C2311yescharacteristic
93C4×Dic181C2311yescharacteristic
94C2×Q8×C181C2311yescharacteristic
95C2×D4×C181C2311yescharacteristic
96C2×C8×C181C2311yescharacteristic
97C2×C8×C181C2311yescharacteristic
98C4⋊C4×C181—21no—
99C4⋊C4×C181—21no—
100C4⋊C4×C181C2311yescharacteristic
101C4⋊C4×C181C2311yescharacteristic
102C22⋊C4×C181—21no—
103C22⋊C4×C181—21no—
104C42×C181C2311yescharacteristic
105C22×D181—21no—
106C22×D181—21no—
107Dic362—22no—
108Dic362—81no—
109D724—21no—
110D724—21no—
111D724—21no—
112D724#4112yes—
113C22⋊D181—42no—
114C22⋊D181—81no—
115C2×Dic181—21no—
116C2×Dic181—21no—
117C2×Dic181—81no—
118C2×Dic181C2×D411yescharacteristic
119C2×Dic181C2×D411yescharacteristic
120C2×Dic181C2×D411yescharacteristic
121C2×Dic181C2×D411yescharacteristic
122C4×D181—42no—
123C4×D181—81no—
124C4×D181—81no—
125C4×D181—81no—
126#13089678—21no—
127#13089678—21no—
128C23×C181—21no—
129Q8×C181—21no—
130Q8×C181—21no—
131D4×C181—21no—
132D4×C181—21no—
133C2×C4×C181—21no—
134C2×C4×C181—21no—
135C2×C4×C181—21no—
136C2×C4×C181C2×D411yescharacteristic
137C2×C4×C181C2×D411yescharacteristic
138C2×C4×C181C2411yescharacteristic
139C8×C181—21no—
140C8×C181—21no—
141C22×C181—41no—
142C22×C181—41no—
143C22×C181—41no—
144C22×C181C22×D411yescharacteristic
145C2×D181—22no—
146C2×D181—41no—
147C2×D181—41no—
148C2×D181—41no—
149C2×D181—41no—
150C2×D181—41no—
151C4×C181—21no—
152C4×C181—21no—
153C4×C181—41no—
154C4×C181—41no—
155C4×C181—41no—
156C4×C181C2×#4111yescharacteristic
157C4×C1812−1+411yescharacteristic
158Dic181—21no—
159Dic181—21no—
160Dic181—21no—
161Dic181—41no—
162Dic181—41no—
163Dic181#13512yes—
164C2×C181—81no—
165C2×C181#54811yescharacteristic
166C2×C181C2×#13511yescharacteristic
167C2×C181#57611yescharacteristic
168D181—42no—
169D181—81no—
170C181#336111yescharacteristic, complemented, Sylow
171#3361—1811nomaximal, Sylow
172#559—1811no—
173#554—1811no—
174#488—1811no—
175#488D18111yescharacteristic, complemented
176#486—1811no—
177#478—1811no—
178#478—1811no—
179#477—1811no—
180#463—1811no—
181#463—1811no—
182#459—1811no—
183C8×Q8—1811no—
184C4×SD16—1811no—
185C4×D8—1811no—
186C4×D8—1811no—
187C2×SD16—1811no—
188C2×D8—1811no—
189C2×D8—1811no—
190C4⋊Q8—1811no—
191#127—1811no—
192#125—1811no—
193#125C2×D18111yescharacteristic
194#124—3621no—
195#124—1811no—
196#124C2×D18111yescharacteristic
197#123—3621no—
198#123—3621no—
199C22⋊Q8—3621no—
200C22⋊Q8—3621no—
201C22⋊Q8—3621no—
202C4×Q8—1811no—
203C4×Q8—1811no—
204C4×D4—3621no—
205C4×D4—1811no—
206C4×D4—1811no—
207C4×D4—1811no—
208C4×D4—1811no—
209C4×D4—1811no—
210C2×C4⋊C4—3621no—
211#107—1811no—
212#107—1811no—
213#106—1811no—
214C4⋊C8—1811no—
215C4⋊C8—1811no—
216C4⋊C8—1811no—
217#103—1811no—
218#103C2×D18111yescharacteristic
219#103C2×D18111yescharacteristic
220#102—1811no—
221#102—1811no—
222#102—1811no—
223#102—1811no—
224#102—1811no—
225#102C2×D18111yescharacteristic
226#102C2×D18111yescharacteristic
227C4×C8—1811no—
228C4×C8—1811no—
229C4×C8C2×D18111yescharacteristic
230C2×Q8—3621no—
231C2×Q8—1811no—
232C2×Q8C22×D18111yescharacteristic
233C2×D4—3621no—
234C2×D4—3621no—
235C2×D4—1811no—
236C2×D4—1811no—
237C2×D4C22×D18111yescharacteristic
238C22×C4—3621no—
239C22×C4—3621no—
240C22×C4—3621no—
241C22×C4—3621no—
242C22×C4—3621no—
243SD16—3622no—
244D8—3622no—
245D8—3622no—
246C2×C8—1811no—
247C2×C8—1811no—
248C2×C8C22×D18111yescharacteristic
249C2×C8C22×D18111yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—3621no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—3621no—
256C4⋊C4—1811no—
257C4⋊C4—1811no—
258C4⋊C4—1811no—
259C4⋊C4—1811no—
260C4⋊C4—1811no—
261C4⋊C4C22×D18111yescharacteristic
262C4⋊C4C22×D18111yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—3621no—
266C22⋊C4—3621no—
267C22⋊C4—3621no—
268C22⋊C4—3621no—
269C22⋊C4—3621no—
270C22⋊C4—3621no—
271C22⋊C4—3621no—
272C42—3621no—
273C42—1811no—
274C42—1811no—
275C42C22×D18111yescharacteristic
276C23—21no—
277C23—3621no—
278C23—3621no—
279Q8—21no—
280Q8—21no—
281Q8—3622no—
282Q8—14481no—
283D4—21no—
284D4—21no—
285D4—3621no—
286D4—3621no—
287D4—3621no—
288D4—14481no—
289D4—1812no—
290D4—7242no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—3621no—
295C2×C4—3621no—
296C2×C4—14481no—
297C2×C4—14481no—
298C2×C4—14481no—
299C2×C4—14481no—
300C2×C4—1811no—
301C2×C4—1811no—
302C2×C4—1811no—
303C2×C4—1811no—
304C2×C4—7242no—
305C2×C4D4×D18111yescharacteristic
306C2×C4D4×D18111yescharacteristic
307C2×C4C23×D18111yescharacteristic
308C8—21no—
309C8—21no—
310C8—3621no—
311C8—3621no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—3622no—
316C22—7241no—
317C22—7241no—
318C22—7241no—
319C22—7241no—
320C22—7241no—
321C22C2×D4×D18111yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—3621no—
328C4—3621no—
329C4—3621no—
330C4—1812no—
331C4—7241no—
332C4—7241no—
333C4#41×D18111yescharacteristic
334C4#1915386311yescharacteristic
335C2—81no—
336C2—14481no—
337C2—7242no—
338C2#2604193711yescharacteristic
339C2#135×D18111yescharacteristic
340C2#2604208111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#488×C181C2
socleC22×C181C22×D4
prime radical#26041473C2
hypercenter#488D181
hypocenterC181#3361
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #26042011C2semidirect
2. #26041985C2semidirect
3. #26041635C2semidirect
4. #26041624C2semidirect
5. #26041577C2semidirect
6. #26041576C2semidirect
7. #26041572C2semidirect
8. #26041497C2semidirect
9. #26041491C2semidirect
11. C8×Dic362C2semidirect
12. #26041364C2semidirect
13. C4×D1448C2semidirect
14. #26041359C2semidirect
15. #488×C181C2semidirect
170. C181#3361semidirect
175. #488D181semidirect

Maximal subgroups

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

subgroupindex
#260420112
#260419852
#260416352
#260416242
#260415772
#260415762
#260415722
#260414972
#260414912
#260414732
C8×Dic3622
#260413642
C4×D14482
#260413592
#488×C1812
#3361181