Finite Group Databasebeta

← #37546783

Subgroups — #37546783

Order 23168.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042012C211yesmaximal, characteristic, complemented
2#26042001C211yesmaximal, characteristic, complemented
3#26041670C211yesmaximal, characteristic, complemented
4#26041665C211yesmaximal, characteristic
5#26041663C211yesmaximal, characteristic, complemented
6#26041504C211yesmaximal, characteristic, complemented
7#26041501C211yesmaximal, characteristic, complemented
8#26041494C211yesmaximal, characteristic, complemented
9#26041493C211yesmaximal, characteristic, complemented
10#26041483C211yesmaximal, characteristic, complemented
11#26041478C211yesmaximal, characteristic, complemented
12#26041409C211yesmaximal, characteristic, complemented
13#26041390C211yesmaximal, characteristic, complemented
14#26041385C211yesmaximal, characteristic, complemented
15C4×SD16×C181C211yesmaximal, characteristic, complemented
16C2×Dic724C2211yescharacteristic
17C2×#15668439C2211yescharacteristic
18C2×#15668438C2211yescharacteristic
19C2×#15668438C2211yescharacteristic
20C2×#15668437C2211yescharacteristic
21C2×#15668436C2211yescharacteristic
22C2×D1448C2211yescharacteristic
23C2×#15668434C2211yescharacteristic
24#19153814C2211yescharacteristic
25#19153812C2211yescharacteristic
26#19153811—21no—
27#19153810—21no—
28#19153805C2211yescharacteristic
29#19153804C2211yescharacteristic
30#19153804C2211yescharacteristic
31#19153803C2211yescharacteristic
32#19153802—21no—
33#19153799—21no—
34#19153798—21no—
35#19153794—21no—
36C4×Dic362C2211yescharacteristic
37#19153781C2211yescharacteristic
38#19153781C2211yescharacteristic
39#19153778C2211yescharacteristic
40#19153778C2211yescharacteristic
41C4×C22⋊D181—21no—
42#19153773—21no—
43C4⋊#13089678C2211yescharacteristic
44#19153745C2211yescharacteristic
45#19153745C2211yescharacteristic
46#19153739C2211yescharacteristic
47#19153739C2211yescharacteristic
48#19153736C2211yescharacteristic
49#19153736C2211yescharacteristic
50#19153726C2211yescharacteristic
51#19153726C2211yescharacteristic
52C2×SD16×C181C2211yescharacteristic
53C4×Q8×C181C2211yescharacteristic
54C4×D4×C181C2211yescharacteristic
55#106×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58C4×C8×C181C2211yescharacteristic
59C2×Dic362D411yescharacteristic
60C2×Dic362C2311yescharacteristic
61C2×Dic362C2311yescharacteristic
62C2×D724C2311yescharacteristic
63C2×D724C2311yescharacteristic
64C2×C22⋊D181—21no—
65C2×C22⋊D181—21no—
66C2×C4×D181—21no—
67C2×C4×D181—21no—
68Dic724—22no—
69#15668439—41no—
70#15668438—41no—
71#15668438—41no—
72#15668437—41no—
73#15668436—41no—
74D1448—22no—
75#15668434—41no—
76C2×#13089678C2311yescharacteristic
77C2×#13089678C2311yescharacteristic
78C4⋊Dic181D411yescharacteristic
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83#15668428D411yescharacteristic
84#15668428D411yescharacteristic
85C22⋊Dic181—21no—
86C22⋊Dic181—21no—
87#15668426—21no—
88#15668426—21no—
89#15668426—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95C4×Dic181—21no—
96C4×Dic181—21no—
97C4×Dic181C2311yescharacteristic
98C4×Dic181C2311yescharacteristic
99C2×Q8×C181C2311yescharacteristic
100C2×D4×C181C2311yescharacteristic
101C22×C4×C181—21no—
102SD16×C181—22no—
103C2×C8×C181C2311yescharacteristic
104C2×C8×C181C2311yescharacteristic
105C4⋊C4×C181—21no—
106C4⋊C4×C181C2311yescharacteristic
107C4⋊C4×C181C2311yescharacteristic
108C22⋊C4×C181—21no—
109C42×C181—21no—
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112C22×D181—21no—
113Dic362—21no—
114Dic362—21no—
115Dic362—21no—
116Dic362—21no—
117Dic362—41no—
118D724—21no—
119D724—21no—
120D724—21no—
121D724—21no—
122C22⋊D181—42no—
123C22⋊D181—81no—
124C2×Dic181—21no—
125C2×Dic181—21no—
126C2×Dic181C2×D411yescharacteristic
127C2×Dic181C2×D411yescharacteristic
128C2×Dic181C2×D411yescharacteristic
129C2×Dic181C2×D411yescharacteristic
130C4×D181—42no—
131C4×D181—81no—
132#13089678—21no—
133#13089678—21no—
134C23×C181—21no—
135Q8×C181—21no—
136Q8×C181—21no—
137D4×C181—21no—
138D4×C181—21no—
139C2×C4×C181—21no—
140C2×C4×C181—21no—
141C2×C4×C181—21no—
142C2×C4×C181—81no—
143C2×C4×C181C2×D411yescharacteristic
144C2×C4×C181C2×D411yescharacteristic
145C2×C4×C181C2411yescharacteristic
146C8×C181—21no—
147C8×C181#4112yes—
148C22×C181—41no—
149C22×C181—41no—
150C22×C181—41no—
151C22×C181C22×D411yescharacteristic
152C2×D181—41no—
153C2×D181—41no—
154C2×D181—41no—
155C2×D181—41no—
156C2×D181—41no—
157C2×D181—41no—
158C4×C181—21no—
159C4×C181—21no—
160C4×C181—41no—
161C4×C181—41no—
162C4×C181—41no—
163C4×C181C2×#4111yescharacteristic
164C4×C181H(5,𝔽2)11yescharacteristic
165Dic181—21no—
166Dic181—21no—
167Dic181—21no—
168Dic181—21no—
169Dic181—41no—
170Dic181—41no—
171C2×C181—81no—
172C2×C181#55011yescharacteristic
173C2×C181#57511yescharacteristic
174C2×C181#57711yescharacteristic
175D181—81no—
176D181—81no—
177C181#340111yescharacteristic, complemented, Sylow
178#3401—1811nomaximal, Sylow
179#559—1811no—
180#556—1811no—
181#497—1811no—
182#496—1811no—
183#495—1811no—
184#464—1811no—
185#464—1811no—
186#463—1811no—
187#463—1811no—
188#461—1811no—
189#460—1811no—
190#446—1811no—
191#442—1811no—
192#441—1811no—
193C4×SD16D18111yescharacteristic, complemented
194C2×Q16—1811no—
195C2×Q16—1811no—
196C2×SD16—1811no—
197C2×SD16—1811no—
198C2×SD16—1811no—
199C2×SD16—1811no—
200C2×SD16C2×D18111yescharacteristic
201C2×D8—1811no—
202C2×D8—1811no—
203C4⋊Q8—1811no—
204#127—1811no—
205#126—3621no—
206#126—3621no—
207#124—3621no—
208#124—1811no—
209#124—1811no—
210#124—1811no—
211#124—1811no—
212#123—3621no—
213#123—3621no—
214C22⋊Q8—3621no—
215C4×Q8—1811no—
216C4×Q8—1811no—
217C4×Q8—1811no—
218C4×Q8C2×D18111yescharacteristic
219C4×D4—3621no—
220C4×D4—1811no—
221C4×D4—1811no—
222C4×D4C2×D18111yescharacteristic
223#117—3621no—
224#106C2×D18111yescharacteristic
225C4⋊C8—1811no—
226C4⋊C8—1811no—
227C4⋊C8—1811no—
228#103—1811no—
229#103—1811no—
230#103C2×D18111yescharacteristic
231#102—1811no—
232#102—1811no—
233#102C2×D18111yescharacteristic
234#97—1811no—
235#97—1811no—
236C4×C8C2×D18111yescharacteristic
237C2×Q8—1811no—
238C2×Q8—1811no—
239C2×Q8—1811no—
240C2×Q8C22×D18111yescharacteristic
241C2×D4—3621no—
242C2×D4—3621no—
243C2×D4—1811no—
244C2×D4—1811no—
245C2×D4C22×D18111yescharacteristic
246C22×C4—21no—
247C22×C4—3621no—
248C22×C4—3621no—
249Q16—3622no—
250Q16—7241no—
251SD16—22no—
252SD16—7241no—
253SD16—7241no—
254SD16—7241no—
255SD16—7241no—
256D8—3622no—
257D8—7241no—
258C2×C8—1811no—
259C2×C8—1811no—
260C2×C8C22×D18111yescharacteristic
261C2×C8C22×D18111yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—3621no—
264C4⋊C4—3621no—
265C4⋊C4—3621no—
266C4⋊C4—3621no—
267C4⋊C4—1811no—
268C4⋊C4—1811no—
269C4⋊C4—1811no—
270C4⋊C4C22×D18111yescharacteristic
271C4⋊C4C22×D18111yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—3621no—
274C22⋊C4—3621no—
275C22⋊C4—3621no—
276C22⋊C4—3621no—
277C22⋊C4—3621no—
278C22⋊C4—3621no—
279C22⋊C4—3621no—
280C22⋊C4—3621no—
281C22⋊C4—3621no—
282C22⋊C4—3621no—
283C42—21no—
284C42—3621no—
285C42—3621no—
286C42—1811no—
287C42—1811no—
288C42C22×D18111yescharacteristic
289C23—21no—
290C23—3621no—
291C23—3621no—
292Q8—21no—
293Q8—21no—
294Q8—3621no—
295Q8—3621no—
296Q8—3621no—
297Q8—3621no—
298Q8—7241no—
299D4—21no—
300D4—21no—
301D4—3621no—
302D4—3621no—
303D4—3621no—
304D4—3621no—
305D4—14481no—
306D4—7242no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—3621no—
312C2×C4—3621no—
313C2×C4—14481no—
314C2×C4—1811no—
315C2×C4—1811no—
316C2×C4—1811no—
317C2×C4—1811no—
318C2×C4—7242no—
319C2×C4D4×D18111yescharacteristic
320C2×C4D4×D18111yescharacteristic
321C2×C4C23×D18111yescharacteristic
322C8—21no—
323C8—3621no—
324C8—3621no—
325C8#1566844912yes—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—7241no—
330C22—7241no—
331C22—7241no—
332C22—7241no—
333C22—7241no—
334C22—7241no—
335C22C2×D4×D18111yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—3621no—
342C4—3621no—
343C4—3621no—
344C4—3621no—
345C4—7241no—
346C4—7241no—
347C4C2×#1566844911yescharacteristic
348C4#1915386211yescharacteristic
349C2—81no—
350C2—14481no—
351C2—14481no—
352C2#2604195011yescharacteristic
353C2#2604207611yescharacteristic
354C2#2604208511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC4×SD16×C181C2
socleC22×C181C22×D4
prime radical#26041665C2
hypercenterC4×SD16D181
hypocenterC181#3401
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #26042012C2semidirect
2. #26042001C2semidirect
3. #26041670C2semidirect
5. #26041663C2semidirect
6. #26041504C2semidirect
7. #26041501C2semidirect
8. #26041494C2semidirect
9. #26041493C2semidirect
10. #26041483C2semidirect
11. #26041478C2semidirect
12. #26041409C2semidirect
13. #26041390C2semidirect
14. #26041385C2semidirect
15. C4×SD16×C181C2semidirect
177. C181#3401semidirect
193. C4×SD16D181semidirect

Maximal subgroups

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

subgroupindex
#260420122
#260420012
#260416702
#260416652
#260416632
#260415042
#260415012
#260414942
#260414932
#260414832
#260414782
#260414092
#260413902
#260413852
C4×SD16×C1812
#3401181