Finite Group Databasebeta

← #37545622

Subgroups — #37545622

Order 23168.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041982C211yesmaximal, characteristic, complemented
2#26041937C211yesmaximal, characteristic, complemented
3#26041618C211yesmaximal, characteristic, complemented
4#26041596C211yesmaximal, characteristic, complemented
5#26041583C211yesmaximal, characteristic, complemented
6#26041514C211yesmaximal, characteristic, complemented
7#26041511C211yesmaximal, characteristic, complemented
8#26041491C211yesmaximal, characteristic, complemented
9#26041425C211yesmaximal, characteristic, complemented
10#26041419C211yesmaximal, characteristic, complemented
11C4×#15668438C211yesmaximal, characteristic, complemented
12C8×D724C211yesmaximal, characteristic, complemented
13#26041272C211yesmaximal, characteristic, complemented
14#103×D181C211yesmaximal, characteristic, complemented
15#488×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18C2×#15668438C2211yescharacteristic
19C2×#15668436C2211yescharacteristic
20C2×#15668434C2211yescharacteristic
21C2×C8×D181C2211yescharacteristic
22C2×C8×D181C2211yescharacteristic
23#19153811—21no—
24#19153802—21no—
25#19153801—21no—
26#19153793—21no—
27#19153792C2211yescharacteristic
28#19153791—21no—
29#19153791C2211yescharacteristic
30#19153791C2211yescharacteristic
31#19153789—21no—
32#19153787C2211yescharacteristic
33C4×Dic362C2211yescharacteristic
34C4×D724C2211yescharacteristic
35C4×D724C2211yescharacteristic
36#19153778—21no—
37#19153776—21no—
38#19153774C2211yescharacteristic
39C4⋊C4×D181C2211yescharacteristic
40#19153749C2211yescharacteristic
41#19153748C2211yescharacteristic
42#19153747C2211yescharacteristic
43C4⋊#13089678C2211yescharacteristic
44#19153741C2211yescharacteristic
45#19153740C2211yescharacteristic
46#19153739C2211yescharacteristic
47#19153739C2211yescharacteristic
48#19153738C2211yescharacteristic
49#19153737C2211yescharacteristic
50#19153728C2211yescharacteristic
51#19153728C2211yescharacteristic
52#125×C181C2211yescharacteristic
53#124×C181C2211yescharacteristic
54#103×C181C2211yescharacteristic
55#103×C181C2211yescharacteristic
56#102×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58C4×C8×C181C2211yescharacteristic
59#15668448—22no—
60#15668448—41no—
61C2×Dic362—21no—
62C2×Dic362C2311yescharacteristic
63Q8×D181—22no—
64Q8×D181—41no—
65C2×D724—21no—
66C2×D724—21no—
67C2×D724D411yescharacteristic
68C2×D724C2311yescharacteristic
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C22×Dic181—21no—
72C2×C4×D181—21no—
73C2×C4×D181—21no—
74C2×C4×D181—21no—
75C2×C4×D181C2311yescharacteristic
76C2×C4×D181C2311yescharacteristic
77#15668438—22no—
78#15668436—41no—
79#15668434—41no—
80C2×#13089678C2311yescharacteristic
81C2×#13089678C2311yescharacteristic
82C8×D181—41no—
83C8×D181—41no—
84C4⋊Dic181D411yescharacteristic
85C4⋊Dic181C2311yescharacteristic
86C4⋊Dic181C2311yescharacteristic
87#15668428—21no—
88#15668428—21no—
89#15668428—21no—
90C22⋊Dic181—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426D411yescharacteristic
97#15668426D411yescharacteristic
98C4×Dic181—21no—
99C4×Dic181—21no—
100C2×Q8×C181C2311yescharacteristic
101C2×D4×C181C2311yescharacteristic
102C2×C8×C181C2311yescharacteristic
103C2×C8×C181C2311yescharacteristic
104C4⋊C4×C181—21no—
105C4⋊C4×C181—21no—
106C4⋊C4×C181C2311yescharacteristic
107C4⋊C4×C181C2311yescharacteristic
108C22⋊C4×C181—21no—
109C22⋊C4×C181—21no—
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112C22×D181C2×D411yescharacteristic
113C22×D181C2×D411yescharacteristic
114Dic362—21no—
115Dic362—42no—
116Dic362#4112yes—
117D724—22no—
118D724—21no—
119D724—21no—
120D724—42no—
121D724—81no—
122C22⋊D181—42no—
123C22⋊D181—42no—
124C2×Dic181—21no—
125C2×Dic181—21no—
126C2×Dic181—21no—
127C2×Dic181—42no—
128C2×Dic181C2×D411yescharacteristic
129C2×Dic181C2×D411yescharacteristic
130C4×D181—21no—
131C4×D181—21no—
132C4×D181—21no—
133C4×D181—21no—
134C4×D181—42no—
135C4×D181—81no—
136C4×D181—81no—
137#13089678—21no—
138#13089678—21no—
139C23×C181—21no—
140Q8×C181—21no—
141Q8×C181—21no—
142D4×C181—21no—
143D4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181C2×D411yescharacteristic
148C2×C4×C181C2×D411yescharacteristic
149C2×C4×C181C2411yescharacteristic
150C8×C181—21no—
151C8×C181—21no—
152C22×C181—41no—
153C22×C181—41no—
154C22×C181—41no—
155C22×C181C22×D411yescharacteristic
156C2×D181—21no—
157C2×D181—21no—
158C2×D181—21no—
159C2×D181—21no—
160C2×D181—21no—
161C2×D181—41no—
162C2×D181—41no—
163C2×D181—41no—
164C2×D181#13512yes—
165C4×C181—21no—
166C4×C181—21no—
167C4×C181—41no—
168C4×C181—41no—
169C4×C181—41no—
170C4×C181C2×#4111yescharacteristic
171C4×C181H(5,𝔽2)11yescharacteristic
172Dic181—21no—
173Dic181—21no—
174Dic181—22no—
175Dic181—41no—
176Dic181—41no—
177C2×C181—81no—
178C2×C181#54611yescharacteristic
179C2×C181C2×#13511yescharacteristic
180C2×C181#57711yescharacteristic
181D181—22no—
182D181—41no—
183D181—81no—
184C181#330111yescharacteristic, complemented, Sylow
185#3301—1811nomaximal, Sylow
186#553—1811no—
187#548—1811no—
188#488D18111yescharacteristic, complemented
189#485—1811no—
190#481—1811no—
191#479—1811no—
192#466—1811no—
193#465—1811no—
194#463—1811no—
195#449—1811no—
196#448—1811no—
197C4×SD16—1811no—
198C8×D4—1811no—
199#416—1811no—
200C2×#103—1811no—
201C2×#41—1811no—
202C22×Q8—1811no—
203C2×SD16—1811no—
204C2×SD16—1811no—
205C2×D8—1811no—
206C22×C8—1811no—
207C22×C8—1811no—
208#126—3621no—
209#125C2×D18111yescharacteristic
210#124—3621no—
211#124C2×D18111yescharacteristic
212#123—3621no—
213C22⋊Q8—3621no—
214C22⋊Q8—1811no—
215#121—3621no—
216#121—3621no—
217#121—1811no—
218#121—1811no—
219#121—1811no—
220C4×Q8—1811no—
221C4×D4—3621no—
222C4×D4—3621no—
223C4×D4—1811no—
224C4×D4—1811no—
225#117—1811no—
226C2×C4⋊C4—1811no—
227#107—1811no—
228#106—1811no—
229#106—1811no—
230C4⋊C8—1811no—
231#103—1811no—
232#103—1811no—
233#103—1811no—
234#103—1811no—
235#103C2×D18111yescharacteristic
236#103C2×D18111yescharacteristic
237#102—1811no—
238#102—1811no—
239#102C2×D18111yescharacteristic
240#102C2×D18111yescharacteristic
241C22⋊C8—1811no—
242C22⋊C8—1811no—
243C4×C8C2×D18111yescharacteristic
244#41—3622no—
245#41—7241no—
246C2×Q8—3621no—
247C2×Q8—3622no—
248C2×Q8—1811no—
249C2×Q8—7241no—
250C2×Q8C22×D18111yescharacteristic
251C2×D4—3621no—
252C2×D4—3621no—
253C2×D4—3621no—
254C2×D4—3621no—
255C2×D4—1811no—
256C2×D4—1811no—
257C2×D4C22×D18111yescharacteristic
258C22×C4—3621no—
259C22×C4—3621no—
260C22×C4—3621no—
261C22×C4—3621no—
262C22×C4—1811no—
263C22×C4—1811no—
264SD16—3622no—
265SD16—7241no—
266D8—7241no—
267C2×C8—1811no—
268C2×C8—1811no—
269C2×C8—7241no—
270C2×C8—7241no—
271C2×C8C22×D18111yescharacteristic
272C2×C8C22×D18111yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—3621no—
276C4⋊C4—3621no—
277C4⋊C4—3621no—
278C4⋊C4—1811no—
279C4⋊C4—1811no—
280C4⋊C4—1811no—
281C4⋊C4C22×D18111yescharacteristic
282C4⋊C4C22×D18111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—3621no—
286C22⋊C4—3621no—
287C22⋊C4—3621no—
288C22⋊C4—3621no—
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4—1811no—
292C22⋊C4—1811no—
293C42—3621no—
294C42—3621no—
295C42C22×D18111yescharacteristic
296C23—21no—
297C23—3621no—
298C23—1811no—
299C23—1811no—
300Q8—21no—
301Q8—21no—
302Q8—3621no—
303Q8—1812no—
304Q8—7242no—
305D4—21no—
306D4—21no—
307D4—3622no—
308D4—3621no—
309D4—3621no—
310D4—14481no—
311D4—7242no—
312D4—7242no—
313D4—7242no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—3621no—
318C2×C4—3621no—
319C2×C4—3621no—
320C2×C4—3621no—
321C2×C4—3621no—
322C2×C4—3621no—
323C2×C4—3621no—
324C2×C4—14481no—
325C2×C4—14481no—
326C2×C4—1811no—
327C2×C4—1811no—
328C2×C4—7242no—
329C2×C4—7242no—
330C2×C4D4×D18111yescharacteristic
331C2×C4D4×D18111yescharacteristic
332C2×C4C23×D18111yescharacteristic
333C8—21no—
334C8—21no—
335C8—3621no—
336C8—3621no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—3621no—
341C22—3621no—
342C22—3621no—
343C22—3621no—
344C22—3621no—
345C22—1812no—
346C22—7241no—
347C22—7241no—
348C22—7241no—
349C22C2×D4×D18111yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—3621no—
356C4—3621no—
357C4—3622no—
358C4—7241no—
359C4—7241no—
360C4#41×D18111yescharacteristic
361C4#1915386211yescharacteristic
362C2—81no—
363C2—3622no—
364C2—14481no—
365C2—7241no—
366C2#2604192211yescharacteristic
367C2#135×D18111yescharacteristic
368C2#2604208611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#488×C181C2
socleC22×C181C22×D4
hypercenter#488D181
hypocenterC181#3301
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. #26041982C2semidirect
2. #26041937C2semidirect
3. #26041618C2semidirect
4. #26041596C2semidirect
5. #26041583C2semidirect
6. #26041514C2semidirect
7. #26041511C2semidirect
8. #26041491C2semidirect
9. #26041425C2semidirect
10. #26041419C2semidirect
11. C4×#15668438C2semidirect
12. C8×D724C2semidirect
13. #26041272C2semidirect
14. #103×D181C2semidirect
15. #488×C181C2semidirect
184. C181#3301semidirect
188. #488D181semidirect

Maximal subgroups

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

subgroupindex
#260419822
#260419372
#260416182
#260415962
#260415832
#260415142
#260415112
#260414912
#260414252
#260414192
C4×#156684382
C8×D7242
#260412722
#103×D1812
#488×C1812
#3301181