Finite Group Databasebeta

← #37545943

Subgroups — #37545943

Order 23168.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041963C211yesmaximal, characteristic, complemented
2#26041920C211yesmaximal, characteristic
3#26041635C211yesmaximal, characteristic, complemented
4#26041617C211yesmaximal, characteristic, complemented
5#26041599C211yesmaximal, characteristic, complemented
6#26041578C211yesmaximal, characteristic, complemented
7#26041528C211yesmaximal, characteristic, complemented
8#26041503C211yesmaximal, characteristic, complemented
9#26041440C211yesmaximal, characteristic, complemented
10#26041429C211yesmaximal, characteristic, complemented
11#26041364C211yesmaximal, characteristic, complemented
12#26041355C211yesmaximal, characteristic, complemented
13#26041285C211yesmaximal, characteristic, complemented
14#26041279C211yesmaximal, characteristic, complemented
15#470×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C22×D724C2211yescharacteristic
18C2×#15668436C2211yescharacteristic
19C2×#15668436C2211yescharacteristic
20C2×#15668432C2211yescharacteristic, complemented
21#19153809—21no—
22#19153807C2211yescharacteristic, complemented
23#19153806—21no—
24#19153805C2211yescharacteristic
25#19153801—21no—
26#19153798—21no—
27#19153792C2211yescharacteristic, complemented
28#19153791—21no—
29#19153791C2211yescharacteristic
30#19153786—21no—
31Q8×Dic181C2211yescharacteristic, complemented
32#19153778C2211yescharacteristic
33C4×C22⋊D181C2211yescharacteristic
34#19153775C2211yescharacteristic
35#19153774—21no—
36C4⋊C4×D181C2211yescharacteristic, complemented
37C22⋊C4×D181—21no—
38#19153749C2211yescharacteristic, complemented
39#19153747C2211yescharacteristic, complemented
40#19153745C2211yescharacteristic, complemented
41#19153740C2211yescharacteristic, complemented
42#19153740C2211yescharacteristic, complemented
43#19153738C2211yescharacteristic
44#19153738C2211yescharacteristic
45#19153738C2211yescharacteristic
46#19153738C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153736C2211yescharacteristic
49C22⋊#13089678C2211yescharacteristic
50#19153728C2211yescharacteristic, complemented
51C8×Dic181C2211yescharacteristic, complemented
52C2×Q16×C181C2211yescharacteristic, complemented
53C2×M16×C181C2211yescharacteristic
54C22⋊Q8×C181C2211yescharacteristic
55C22⋊Q8×C181C2211yescharacteristic
56#106×C181C2211yescharacteristic, complemented
57#103×C181C2211yescharacteristic, complemented
58#103×C181C2211yescharacteristic, complemented
59C23×D181—21no—
60#15668450—22no—
61#15668450—41no—
62C2×D724—22no—
63C2×D724—21no—
64C2×D724—21no—
65C2×D724—21no—
66C2×D724—41no—
67C2×D724C2311yescharacteristic
68C2×D724C2311yescharacteristic
69C2×C22⋊D181D411yescharacteristic
70C2×C4×D181—21no—
71C2×C4×D181—21no—
72C2×C4×D181—21no—
73C2×C4×D181C2311yescharacteristic
74#15668436—22no—
75#15668436—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#15668428D411yescharacteristic
87C22⋊Dic181—21no—
88C22⋊Dic181D411yescharacteristic
89#15668426—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426D411yescharacteristic
95C4×Dic181—21no—
96C4×Dic181—21no—
97C4×Dic181C2311yescharacteristic
98C2×Q8×C181C2311yescharacteristic
99C2×Q8×C181C2311yescharacteristic
100C22×C4×C181C2311yescharacteristic
101Q16×C181—41no—
102M16×C181—41no—
103C2×C8×C181C2311yescharacteristic
104C2×C8×C181C2311yescharacteristic
105C4⋊C4×C181—21no—
106C4⋊C4×C181—21no—
107C4⋊C4×C181C2311yescharacteristic
108C4⋊C4×C181C2311yescharacteristic
109C22⋊C4×C181—21no—
110C22⋊C4×C181—21no—
111C22×D181—21no—
112C22×D181—21no—
113C22×D181—42no—
114C22×D181—42no—
115C22×D181—81no—
116C22×D181C2×D411yescharacteristic
117D724—21no—
118D724—21no—
119D724—21no—
120D724—42no—
121D724—42no—
122D724—81no—
123D724#4112yes—
124C22⋊D181—41no—
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—42no—
134C4×D181—81no—
135#13089678—21no—
136#13089678—21no—
137C23×C181C2×D411yescharacteristic
138Q8×C181—21no—
139Q8×C181—21no—
140Q8×C181—21no—
141Q8×C181—21no—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181—21no—
148C2×C4×C181C2×D411yescharacteristic
149C2×C4×C181C2411yescharacteristic
150C8×C181—21no—
151C8×C181—21no—
152C22×C181—21no—
153C22×C181—21no—
154C22×C181—21no—
155C22×C181C22×D411yescharacteristic
156C2×D181—21no—
157C2×D181—22no—
158C2×D181—21no—
159C2×D181—21no—
160C2×D181—41no—
161C2×D181—41no—
162C2×D181—41no—
163C2×D181—41no—
164C2×D181—41no—
165C2×D181—84no—
166C4×C181—21no—
167C4×C181—41no—
168C4×C181—41no—
169C4×C181—41no—
170C4×C181—41no—
171C4×C181C2×#4111yescharacteristic
172C4×C181H(5,𝔽2)11yescharacteristic
173Dic181—21no—
174Dic181—21no—
175Dic181—21no—
176Dic181—41no—
177Dic181—41no—
178C2×C181—41no—
179C2×C181#54611yescharacteristic
180C2×C181#57511yescharacteristic
181C2×C181#57711yescharacteristic
182D181—41no—
183D181—42no—
184D181—81no—
185C181#332311yescharacteristic, complemented, Sylow
186#3323—1811nomaximal, Sylow
187#552—1811no—
188#546—1811no—
189#488—1811no—
190#485—1811no—
191#481—1811no—
192#478—1811no—
193#470D18111yescharacteristic, complemented
194#468—1811no—
195#464—1811no—
196#452—1811no—
197#450—1811no—
198C4×SD16—1811no—
199#436—1811no—
200#418—1811no—
201#417—1811no—
202C2×#41—1811no—
203C22×D4—1811no—
204C2×Q16C2×D18111yescharacteristic, complemented
205C2×SD16—1811no—
206C2×SD16—1811no—
207C2×M16—1811no—
208C2×M16C2×D18111yescharacteristic
209#126—3621no—
210#125—3621no—
211#125—1811no—
212#124—1811no—
213#123—3621no—
214#123—3621no—
215C22⋊Q8—1811no—
216C22⋊Q8C2×D18111yescharacteristic
217C22⋊Q8C2×D18111yescharacteristic
218#121—3621no—
219#121—1811no—
220#120—3621no—
221C4×Q8—1811no—
222C4×D4—1811no—
223C4×D4—1811no—
224#117—3621no—
225#117—1811no—
226C2×C4⋊C4—1811no—
227C2×C22⋊C4—3621no—
228#107—1811no—
229#106—1811no—
230#106C2×D18111yescharacteristic, complemented
231C4⋊C8—1811no—
232#103—1811no—
233#103—1811no—
234#103C2×D18111yescharacteristic, complemented
235#103C2×D18111yescharacteristic, complemented
236#102—1811no—
237#102—1811no—
238#102—1811no—
239#102—1811no—
240#102—1811no—
241#102—1811no—
242C22⋊C8—1811no—
243C22⋊C8—1811no—
244C4×C8—1811no—
245C24—3621no—
246#41—3622no—
247#41—7241no—
248C2×Q8C22×D18111yescharacteristic
249C2×Q8C22×D18111yescharacteristic
250C2×D4—3622no—
251C2×D4—3621no—
252C2×D4—3621no—
253C2×D4—3621no—
254C2×D4—1811no—
255C2×D4—1811no—
256C2×D4—1811no—
257C2×D4—7241no—
258C22×C4—3621no—
259C22×C4—3621no—
260C22×C4—3621no—
261C22×C4—1811no—
262C22×C4C22×D18111yescharacteristic
263Q16—41no—
264SD16—3622no—
265SD16—7241no—
266M16—41no—
267M16—7241no—
268C2×C8—1811no—
269C2×C8—1811no—
270C2×C8C22×D18111yescharacteristic
271C2×C8C22×D18111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—3621no—
275C4⋊C4—3621no—
276C4⋊C4—3621no—
277C4⋊C4—3621no—
278C4⋊C4—3621no—
279C4⋊C4—1811no—
280C4⋊C4—1811no—
281C4⋊C4—1811no—
282C4⋊C4C22×D18111yescharacteristic
283C4⋊C4C22×D18111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—3621no—
287C22⋊C4—3621no—
288C22⋊C4—3621no—
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4—3621no—
292C22⋊C4—1811no—
293C22⋊C4—1811no—
294C42—3621no—
295C42—3621no—
296C42—1811no—
297C23—3621no—
298C23—3621no—
299C23—14481no—
300C23—1811no—
301C23—7242no—
302C23—7242no—
303C23D4×D18111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—3621no—
309D4—3621no—
310D4—3621no—
311D4—14481no—
312D4—1812no—
313D4—7242no—
314D4—7242no—
315D4—7241no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—3621no—
323C2×C4—3621no—
324C2×C4—3621no—
325C2×C4—3621no—
326C2×C4—14481no—
327C2×C4—1811no—
328C2×C4—1811no—
329C2×C4—1811no—
330C2×C4—7242no—
331C2×C4—7242no—
332C2×C4D4×D18111yescharacteristic
333C2×C4C23×D18111yescharacteristic
334C8—21no—
335C8—21no—
336C8—3621no—
337C8—3621no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—3621no—
342C22—3622no—
343C22—3621no—
344C22—3621no—
345C22—14484no—
346C22—7241no—
347C22—7241no—
348C22—7241no—
349C22—7241no—
350C22—7241no—
351C22C2×D4×D18111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3621no—
358C4—3621no—
359C4—3621no—
360C4—7241no—
361C4—7241no—
362C4#41×D18111yescharacteristic
363C4#1915386111yescharacteristic
364C2—41no—
365C2—14481no—
366C2—7241no—
367C2—7242no—
368C2#2604191811yescharacteristic
369C2#2604207811yescharacteristic
370C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#470×C181C2
socleC22×C181C22×D4
prime radical#26041920C2
hypercenter#470D181
hypocenterC181#3323
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #26041963C2semidirect
3. #26041635C2semidirect
4. #26041617C2semidirect
5. #26041599C2semidirect
6. #26041578C2semidirect
7. #26041528C2semidirect
8. #26041503C2semidirect
9. #26041440C2semidirect
10. #26041429C2semidirect
11. #26041364C2semidirect
12. #26041355C2semidirect
13. #26041285C2semidirect
14. #26041279C2semidirect
15. #470×C181C2semidirect
20. C2×#15668432C22semidirect
22. #19153807C22semidirect
27. #19153792C22semidirect
31. Q8×Dic181C22semidirect
36. C4⋊C4×D181C22semidirect
38. #19153749C22semidirect
39. #19153747C22semidirect
40. #19153745C22semidirect
41. #19153740C22semidirect
42. #19153740C22semidirect
50. #19153728C22semidirect
51. C8×Dic181C22semidirect
52. C2×Q16×C181C22semidirect
56. #106×C181C22semidirect
57. #103×C181C22semidirect
58. #103×C181C22semidirect
185. C181#3323semidirect
193. #470D181semidirect
204. C2×Q16C2×D181semidirect
230. #106C2×D181semidirect
234. #103C2×D181semidirect
235. #103C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260419632
#260419202
#260416352
#260416172
#260415992
#260415782
#260415282
#260415032
#260414402
#260414292
#260413642
#260413552
#260412852
#260412792
#470×C1812
#3323181