Finite Group Databasebeta

← #37543906

Subgroups — #37543906

Order 23168.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041902C211yesmaximal, characteristic, complemented
2#26041754C211yesmaximal, characteristic, complemented
3#26041617C211yesmaximal, characteristic, complemented
4#26041604C211yesmaximal, characteristic, complemented
5#26041567C211yesmaximal, characteristic, complemented
6#26041557C211yesmaximal, characteristic, complemented
7#26041496C211yesmaximal, characteristic, complemented
8#26041496C211yesmaximal, characteristic, complemented
9#26041413C211yesmaximal, characteristic
10#26041388C211yesmaximal, characteristic, complemented
11#26041380C211yesmaximal, characteristic, complemented
12SD16×Dic181C211yesmaximal, characteristic, complemented
13#26041359C211yesmaximal, characteristic, complemented
14#26041337C211yesmaximal, characteristic, complemented
15#450×C181C211yesmaximal, characteristic, complemented
16C2×#15668434C2211yescharacteristic
17C2×#15668434C2211yescharacteristic
18#19153815C2211yescharacteristic, complemented
19#19153811—21no—
20#19153805C2211yescharacteristic, complemented
21#19153802—21no—
22#19153798—21no—
23#19153796—21no—
24#19153790C2211yescharacteristic
25Q8×Dic181C2211yescharacteristic, complemented
26D4×Dic181—21no—
27D4×Dic181—21no—
28D4×Dic181C2211yescharacteristic
29D4×Dic181C2211yescharacteristic
30D4×Dic181C2211yescharacteristic
31D4×Dic181C2211yescharacteristic
32#19153778C2211yescharacteristic, complemented
33#19153775C2211yescharacteristic
34#19153775C2211yescharacteristic
35C2×C22⋊Dic181—21no—
36C2×C22⋊Dic181—21no—
37#19153750C2211yescharacteristic, complemented
38#19153749C2211yescharacteristic, complemented
39#19153748C2211yescharacteristic, complemented
40#19153747C2211yescharacteristic, complemented
41#19153745C2211yescharacteristic, complemented
42#19153745C2211yescharacteristic, complemented
43#19153740C2211yescharacteristic, complemented
44#19153740C2211yescharacteristic, complemented
45#19153738C2211yescharacteristic, complemented
46#19153738C2211yescharacteristic, complemented
47#19153737C2211yescharacteristic
48#19153737C2211yescharacteristic
49C22⋊#13089678C2211yescharacteristic
50#19153726C2211yescharacteristic, complemented
51C8×Dic181C2211yescharacteristic, complemented
52C22×D4×C181C2211yescharacteristic
53C2×SD16×C181C2211yescharacteristic
54C2×SD16×C181C2211yescharacteristic
55C22⋊Q8×C181C2211yescharacteristic
56#102×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58C22⋊C8×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×D724C2311yescharacteristic
61C2×C22⋊D181—21no—
62C22×Dic181—21no—
63C22×Dic181—21no—
64C22×Dic181—21no—
65C22×Dic181—21no—
66C2×C4×D181—21no—
67#15668434—22no—
68#15668434—22no—
69C2×#13089678C2311yescharacteristic
70C2×#13089678C2311yescharacteristic
71C4⋊Dic181—21no—
72C4⋊Dic181D411yescharacteristic
73C4⋊Dic181D411yescharacteristic
74C4⋊Dic181C2311yescharacteristic
75C4⋊Dic181C2311yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77#15668428—21no—
78C22⋊Dic181—21no—
79C22⋊Dic181—21no—
80C22⋊Dic181—21no—
81C22⋊Dic181—21no—
82C22⋊Dic181—21no—
83C22⋊Dic181D411yescharacteristic
84C22⋊Dic181D411yescharacteristic
85#15668426—21no—
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89C4×Dic181—21no—
90C4×Dic181—21no—
91C4×Dic181C2311yescharacteristic
92C4×Dic181C2311yescharacteristic
93C24×C181—21no—
94C2×Q8×C181C2311yescharacteristic
95C2×D4×C181—22no—
96C2×D4×C181—22no—
97C2×D4×C181—21no—
98C2×D4×C181C2311yescharacteristic
99C2×D4×C181C2311yescharacteristic
100C22×C4×C181C2311yescharacteristic
101SD16×C181—22no—
102SD16×C181—22no—
103C2×C8×C181C2311yescharacteristic
104C2×C8×C181C2311yescharacteristic
105C4⋊C4×C181—21no—
106C4⋊C4×C181C2311yescharacteristic
107C22⋊C4×C181—21no—
108C22×D181—21no—
109Dic362—81no—
110D724—21no—
111D724—21no—
112C22⋊D181—81no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181—21no—
116C2×Dic181—81no—
117C2×Dic181—81no—
118C2×Dic181—81no—
119C2×Dic181—81no—
120C2×Dic181C2×D411yescharacteristic
121C2×Dic181C2×D411yescharacteristic
122C2×Dic181C2×D411yescharacteristic
123C2×Dic181C2×D411yescharacteristic
124C4×D181—81no—
125#13089678—21no—
126#13089678—21no—
127C23×C181—24no—
128C23×C181—21no—
129C23×C181—21no—
130C23×C181—81no—
131C23×C181—81no—
132C23×C181C2×D411yescharacteristic
133Q8×C181—21no—
134Q8×C181—21no—
135D4×C181—24no—
136D4×C181—21no—
137D4×C181—21no—
138D4×C181#4112yes—
139D4×C181#4112yes—
140C2×C4×C181—21no—
141C2×C4×C181—21no—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181C2×D411yescharacteristic
145C2×C4×C181C2411yescharacteristic
146C8×C181—21no—
147C8×C181—21no—
148C22×C181—21no—
149C22×C181—22no—
150C22×C181—22no—
151C22×C181—21no—
152C22×C181—21no—
153C22×C181—41no—
154C22×C181—41no—
155C22×C181—41no—
156C22×C181—41no—
157C22×C181—84no—
158C22×C181C22×D411yescharacteristic
159C2×D181—41no—
160C2×D181—41no—
161C2×D181—41no—
162C4×C181—21no—
163C4×C181—41no—
164C4×C181—41no—
165C4×C181C2×#4111yescharacteristic
166C4×C181C2×#4111yescharacteristic
167Dic181—21no—
168Dic181—21no—
169Dic181—21no—
170Dic181—21no—
171Dic181—41no—
172Dic181—41no—
173Dic181—41no—
174C2×C181—41no—
175C2×C181—42no—
176C2×C181—42no—
177C2×C181#52511yescharacteristic
178C2×C181#57611yescharacteristic
179C2×C181#57711yescharacteristic
180D181—81no—
181C181#311811yescharacteristic, complemented, Sylow
182#3118—1811nomaximal, Sylow
183#542—1811no—
184#518—1811no—
185#485—1811no—
186#482—1811no—
187#476—1811no—
188#474—1811no—
189#463—1811no—
190#463—1811no—
191#450D18111yescharacteristic, complemented
192#447—1811no—
193#442—1811no—
194#440—1811no—
195C4×SD16—1811no—
196C4×D8—1811no—
197#433—1811no—
198C22×D4C2×D18111yescharacteristic
199C2×SD16C2×D18111yescharacteristic
200C2×SD16C2×D18111yescharacteristic
201C2×D8—1811no—
202C2×D8—1811no—
203C4⋊Q8—1811no—
204#126—3621no—
205#124—3621no—
206#124—1811no—
207#123—3621no—
208C22⋊Q8—3621no—
209C22⋊Q8C2×D18111yescharacteristic
210#121—1811no—
211C4×Q8—1811no—
212C4×D4—3621no—
213C4×D4—3621no—
214C4×D4—1811no—
215C4×D4—1811no—
216C4×D4—1811no—
217C4×D4—1811no—
218C4×D4—1811no—
219#117—1811no—
220#117—1811no—
221C2×C22⋊C4—3621no—
222C2×C22⋊C4—3621no—
223#107—1811no—
224#107—1811no—
225#106—1811no—
226#106—1811no—
227C4⋊C8—1811no—
228C4⋊C8—1811no—
229#103—1811no—
230#103—1811no—
231#102—1811no—
232#102—1811no—
233#102—1811no—
234#102—1811no—
235#102C2×D18111yescharacteristic
236#102C2×D18111yescharacteristic
237C22⋊C8—1811no—
238C22⋊C8C2×D18111yescharacteristic
239#97—1811no—
240C4×C8—1811no—
241C24—21no—
242C2×Q8—3621no—
243C2×Q8C22×D18111yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—3621no—
248C2×D4—1811no—
249C2×D4C22×D18111yescharacteristic
250C2×D4C22×D18111yescharacteristic
251C22×C4—3621no—
252C22×C4—3621no—
253C22×C4—3621no—
254C22×C4—3621no—
255C22×C4—3621no—
256C22×C4C22×D18111yescharacteristic
257SD16—22no—
258SD16—22no—
259D8—3622no—
260D8—3622no—
261C2×C8—1811no—
262C2×C8—1811no—
263C2×C8C22×D18111yescharacteristic
264C2×C8C22×D18111yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—3621no—
267C4⋊C4—3621no—
268C4⋊C4—1811no—
269C4⋊C4—1811no—
270C4⋊C4—1811no—
271C4⋊C4—1811no—
272C4⋊C4—1811no—
273C4⋊C4C22×D18111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—3621no—
276C22⋊C4—3621no—
277C22⋊C4—3621no—
278C22⋊C4—3621no—
279C22⋊C4—3621no—
280C22⋊C4—3621no—
281C22⋊C4—3621no—
282C22⋊C4—3621no—
283C22⋊C4—3621no—
284C22⋊C4—1811no—
285C22⋊C4—1811no—
286C42—3621no—
287C42—3621no—
288C42—1811no—
289C42—1811no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—3621no—
296C23D4×D18111yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—14481no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—3621no—
304D4—3621no—
305D4—14481no—
306D4#1566844812yes—
307D4#1566844812yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—3621no—
313C2×C4—3621no—
314C2×C4—3621no—
315C2×C4—14481no—
316C2×C4—14481no—
317C2×C4—14481no—
318C2×C4—14481no—
319C2×C4—14481no—
320C2×C4—1811no—
321C2×C4—1811no—
322C2×C4—1811no—
323C2×C4—1811no—
324C2×C4D4×D18111yescharacteristic
325C2×C4C23×D18111yescharacteristic
326C8—21no—
327C8—21no—
328C8—3621no—
329C8—3621no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—7241no—
341C22—7241no—
342C22—7241no—
343C22C2×D4×D18111yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—3621no—
348C4—3621no—
349C4—3621no—
350C4—3621no—
351C4—7241no—
352C4—7241no—
353C4—7241no—
354C4C2×#1566844811yescharacteristic
355C4C2×#1566844811yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—14481no—
360C2#2604178511yescharacteristic
361C2#2604208311yescharacteristic
362C2#2604208511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#450×C181C2
socleC22×C181C22×D4
prime radical#26041413C2
hypercenter#450D181
hypocenterC181#3118
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #26041902C2semidirect
2. #26041754C2semidirect
3. #26041617C2semidirect
4. #26041604C2semidirect
5. #26041567C2semidirect
6. #26041557C2semidirect
7. #26041496C2semidirect
8. #26041496C2semidirect
10. #26041388C2semidirect
11. #26041380C2semidirect
12. SD16×Dic181C2semidirect
13. #26041359C2semidirect
14. #26041337C2semidirect
15. #450×C181C2semidirect
18. #19153815C22semidirect
20. #19153805C22semidirect
25. Q8×Dic181C22semidirect
32. #19153778C22semidirect
37. #19153750C22semidirect
38. #19153749C22semidirect
39. #19153748C22semidirect
40. #19153747C22semidirect
41. #19153745C22semidirect
42. #19153745C22semidirect
43. #19153740C22semidirect
44. #19153740C22semidirect
45. #19153738C22semidirect
46. #19153738C22semidirect
50. #19153726C22semidirect
51. C8×Dic181C22semidirect
181. C181#3118semidirect
191. #450D181semidirect

Maximal subgroups

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

subgroupindex
#260419022
#260417542
#260416172
#260416042
#260415672
#260415572
#260414962
#260414132
#260413882
#260413802
SD16×Dic1812
#260413592
#260413372
#450×C1812
#3118181