Finite Group Databasebeta

← #37545753

Subgroups — #37545753

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#26041997C211yesmaximal, characteristic, complemented
2#26041917C211yesmaximal, characteristic
3#26041640C211yesmaximal, characteristic, complemented
4#26041618C211yesmaximal, characteristic, complemented
5#26041603C211yesmaximal, characteristic, complemented
6#26041567C211yesmaximal, characteristic, complemented
7#26041536C211yesmaximal, characteristic, complemented
8#26041508C211yesmaximal, characteristic, complemented
9#26041504C211yesmaximal, characteristic, complemented
10#26041441C211yesmaximal, characteristic, complemented
11#26041431C211yesmaximal, characteristic, complemented
12#26041376C211yesmaximal, characteristic, complemented
13#26041350C211yesmaximal, characteristic, complemented
14#26041282C211yesmaximal, characteristic, complemented
15#416×C181C211yesmaximal, characteristic, complemented
16C2×D4×D181C2211yescharacteristic
17C2×#15668439C2211yescharacteristic, complemented
18C2×#15668438C2211yescharacteristic
19C2×#15668437C2211yescharacteristic
20C2×#15668432C2211yescharacteristic
21#19153815C2211yescharacteristic, complemented
22#19153810—21no—
23#19153809—21no—
24#19153805C2211yescharacteristic
25#19153802—21no—
26#19153799—21no—
27#19153796C2211yescharacteristic
28#19153795C2211yescharacteristic, complemented
29#19153792C2211yescharacteristic
30#19153790C2211yescharacteristic
31#19153788—21no—
32#19153784—21no—
33C4×Dic362—21no—
34#19153781C2211yescharacteristic, complemented
35D4×Dic181C2211yescharacteristic
36C4×C22⋊D181C2211yescharacteristic
37#19153774C2211yescharacteristic
38C2×#15668426—21no—
39#19153749C2211yescharacteristic, complemented
40#19153748C2211yescharacteristic, complemented
41#19153747C2211yescharacteristic, complemented
42#19153745C2211yescharacteristic, complemented
43#19153741C2211yescharacteristic, complemented
44#19153740C2211yescharacteristic, complemented
45#19153738C2211yescharacteristic
46#19153737C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153736C2211yescharacteristic
49C22⋊#13089678C2211yescharacteristic, complemented
50#19153728C2211yescharacteristic
51#19153726C2211yescharacteristic, complemented
52C2×#41×C181C2211yescharacteristic
53C22×C8×C181C2211yescharacteristic, complemented
54#117×C181C2211yescharacteristic, complemented
55#103×C181C2211yescharacteristic, complemented
56#103×C181C2211yescharacteristic, complemented
57#102×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59C23×D181—21no—
60C2×Dic362—21no—
61C2×Dic362C2311yescharacteristic
62C2×D724C2311yescharacteristic
63C2×C22⋊D181—21no—
64C2×C22⋊D181—21no—
65C2×C22⋊D181D411yescharacteristic
66D4×D181—22no—
67D4×D181—41no—
68C22×Dic181—21no—
69C2×C4×D181C2311yescharacteristic
70#15668439—41no—
71#15668438—41no—
72#15668437—22no—
73#15668432—41no—
74C2×#13089678C2311yescharacteristic
75C2×#13089678C2311yescharacteristic
76C4⋊Dic181—21no—
77C4⋊Dic181C2311yescharacteristic
78C4⋊Dic181C2311yescharacteristic
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83#15668428D411yescharacteristic
84C22⋊Dic181—21no—
85C22⋊Dic181—21no—
86C22⋊Dic181D411yescharacteristic
87#15668426—21no—
88#15668426—21no—
89#15668426—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426D411yescharacteristic
94C4×Dic181—21no—
95C4×Dic181—21no—
96C4×Dic181C2311yescharacteristic
97#41×C181—22no—
98#41×C181—41no—
99C2×Q8×C181C2311yescharacteristic
100C2×D4×C181—21no—
101C2×D4×C181C2311yescharacteristic
102C22×C4×C181—21no—
103C22×C4×C181C2311yescharacteristic
104C2×C8×C181—41no—
105C2×C8×C181C2311yescharacteristic
106C2×C8×C181C2311yescharacteristic
107C4⋊C4×C181C2311yescharacteristic
108C4⋊C4×C181C2311yescharacteristic
109C22⋊C4×C181—21no—
110C42×C181—21no—
111C22×D181—21no—
112C22×D181—42no—
113C22×D181—42no—
114C22×D181—81no—
115C22×D181C2×D411yescharacteristic
116Dic362—21no—
117Dic362—21no—
118Dic362—42no—
119D724—21no—
120D724—21no—
121C22⋊D181—41no—
122C22⋊D181—42no—
123C22⋊D181—81no—
124C2×Dic181—21no—
125C2×Dic181—21no—
126C2×Dic181—21no—
127C2×Dic181—81no—
128C2×Dic181C2×D411yescharacteristic
129C2×Dic181C2×D411yescharacteristic
130C2×Dic181C2×D411yescharacteristic
131C4×D181—21no—
132C4×D181—21no—
133#13089678—21no—
134#13089678—21no—
135C23×C181—21no—
136C23×C181C2×D411yescharacteristic
137Q8×C181—21no—
138Q8×C181—21no—
139D4×C181—21no—
140D4×C181—42no—
141D4×C181#4112yes—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181—42no—
148C2×C4×C181C2×D411yescharacteristic
149C2×C4×C181C2411yescharacteristic
150C8×C181—21no—
151C8×C181—21no—
152C22×C181—21no—
153C22×C181—22no—
154C22×C181—21no—
155C22×C181—21no—
156C22×C181—41no—
157C22×C181—41no—
158C22×C181C22×D411yescharacteristic
159C2×D181—21no—
160C2×D181—21no—
161C2×D181—21no—
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×C181C2×#4111yescharacteristic
171C4×C181H(5,𝔽2)11yescharacteristic
172Dic181—21no—
173Dic181—21no—
174Dic181—21no—
175Dic181—41no—
176Dic181—41no—
177Dic181—41no—
178C2×C181—42no—
179C2×C181—41no—
180C2×C181#54611yescharacteristic
181C2×C181#57511yescharacteristic
182C2×C181#57711yescharacteristic
183D181—41no—
184D181—81no—
185C181#331111yescharacteristic, complemented, Sylow
186#3311—1811nomaximal, Sylow
187#556—1811no—
188#546—1811no—
189#489—1811no—
190#485—1811no—
191#482—1811no—
192#476—1811no—
193#470—1811no—
194#465—1811no—
195#464—1811no—
196#452—1811no—
197#450—1811no—
198#440—1811no—
199#435—1811no—
200#418—1811no—
201#416D18111yescharacteristic, complemented
202C2×#41C2×D18111yescharacteristic
203C22×D4—1811no—
204C2×Q16—1811no—
205C2×SD16—1811no—
206C2×SD16—1811no—
207C2×M16—1811no—
208C22×C8C2×D18111yescharacteristic, complemented
209C4⋊Q8—1811no—
210#126—3621no—
211#126—3621no—
212#124—3621no—
213#124—1811no—
214#123—3621no—
215C22⋊Q8—1811no—
216C22⋊Q8—1811no—
217C22⋊Q8—1811no—
218#121—3621no—
219#121—1811no—
220#120—3621no—
221C4×Q8—3621no—
222C4×Q8—1811no—
223C4×D4—1811no—
224C4×D4—1811no—
225#117—1811no—
226#117C2×D18111yescharacteristic, complemented
227C2×C22⋊C4—3621no—
228#107—1811no—
229#106—1811no—
230#106—1811no—
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#102C2×D18111yescharacteristic
241#102C2×D18111yescharacteristic
242C22⋊C8—1811no—
243C22⋊C8—1811no—
244#97—1811no—
245C24—3621no—
246#41—22no—
247#41—41no—
248C2×Q8—3621no—
249C2×Q8—1811no—
250C2×Q8C22×D18111yescharacteristic
251C2×D4—21no—
252C2×D4—3622no—
253C2×D4—3621no—
254C2×D4—3621no—
255C2×D4—1811no—
256C2×D4—1811no—
257C2×D4—7241no—
258C2×D4C22×D18111yescharacteristic
259C22×C4—21no—
260C22×C4—3621no—
261C22×C4—1811no—
262C22×C4C22×D18111yescharacteristic
263Q16—7241no—
264SD16—3622no—
265SD16—7241no—
266M16—7241no—
267C2×C8—41no—
268C2×C8—1811no—
269C2×C8—1811no—
270C2×C8C22×D18111yescharacteristic
271C2×C8C22×D18111yescharacteristic
272C4⋊C4—3621no—
273C4⋊C4—3621no—
274C4⋊C4—3621no—
275C4⋊C4—3621no—
276C4⋊C4—3621no—
277C4⋊C4—1811no—
278C4⋊C4—1811no—
279C4⋊C4—1811no—
280C4⋊C4C22×D18111yescharacteristic
281C4⋊C4C22×D18111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—3621no—
284C22⋊C4—3621no—
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—21no—
294C42—3621no—
295C42—3621no—
296C42—1811no—
297C23—21no—
298C23—3621no—
299C23—14481no—
300C23—1811no—
301C23—7242no—
302C23—7242no—
303C23D4×D18111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3621no—
307Q8—3621no—
308Q8—7242no—
309D4—21no—
310D4—42no—
311D4—3621no—
312D4—3621no—
313D4—14481no—
314D4—7241no—
315D4—7242no—
316D4#1566844912yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—3621no—
324C2×C4—3621no—
325C2×C4—3621no—
326C2×C4—3621no—
327C2×C4—3621no—
328C2×C4—14481no—
329C2×C4—1811no—
330C2×C4—1811no—
331C2×C4—1811no—
332C2×C4D4×D18111yescharacteristic
333C2×C4C23×D18111yescharacteristic
334C8—21no—
335C8—21no—
336C8—3621no—
337C8—3621no—
338C22—21no—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—3621no—
345C22—3621no—
346C22—3621no—
347C22—14484no—
348C22—7241no—
349C22—7241no—
350C22—7241no—
351C22C2×D4×D18111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—3621no—
357C4—3621no—
358C4—3621no—
359C4—7241no—
360C4—7241no—
361C4—7241no—
362C4C2×#1566844911yescharacteristic
363C4#1915386111yescharacteristic
364C2—41no—
365C2—42no—
366C2—14481no—
367C2—7241no—
368C2#2604191911yescharacteristic
369C2#2604207611yescharacteristic
370C2#2604208611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#416×C181C2
socleC22×C181C22×D4
prime radical#26041917C2
hypercenter#416D181
hypocenterC181#3311
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #26041997C2semidirect
3. #26041640C2semidirect
4. #26041618C2semidirect
5. #26041603C2semidirect
6. #26041567C2semidirect
7. #26041536C2semidirect
8. #26041508C2semidirect
9. #26041504C2semidirect
10. #26041441C2semidirect
11. #26041431C2semidirect
12. #26041376C2semidirect
13. #26041350C2semidirect
14. #26041282C2semidirect
15. #416×C181C2semidirect
17. C2×#15668439C22semidirect
21. #19153815C22semidirect
28. #19153795C22semidirect
34. #19153781C22semidirect
39. #19153749C22semidirect
40. #19153748C22semidirect
41. #19153747C22semidirect
42. #19153745C22semidirect
43. #19153741C22semidirect
44. #19153740C22semidirect
49. C22⋊#13089678C22semidirect
51. #19153726C22semidirect
53. C22×C8×C181C22semidirect
54. #117×C181C22semidirect
55. #103×C181C22semidirect
56. #103×C181C22semidirect
185. C181#3311semidirect
201. #416D181semidirect
208. C22×C8C2×D181semidirect
226. #117C2×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
#260419972
#260419172
#260416402
#260416182
#260416032
#260415672
#260415362
#260415082
#260415042
#260414412
#260414312
#260413762
#260413502
#260412822
#416×C1812
#3311181