Finite Group Databasebeta

← #37545751

Subgroups — #37545751

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#26041925C211yesmaximal, characteristic
2#26041641C211yesmaximal, characteristic, complemented
3#26041620C211yesmaximal, characteristic, complemented
4#26041603C211yesmaximal, characteristic, complemented
5#26041566C211yesmaximal, characteristic, complemented
6#26041535C211yesmaximal, characteristic, complemented
7#26041510C211yesmaximal, characteristic, complemented
8#26041501C211yesmaximal, characteristic, complemented
9#26041439C211yesmaximal, characteristic, complemented
10#26041429C211yesmaximal, characteristic, complemented
11#26041377C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic, complemented
13#26041283C211yesmaximal, characteristic, complemented
14#26041273C211yesmaximal, characteristic, complemented
15#556×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C22×D724C2211yescharacteristic
18C2×#15668439C2211yescharacteristic, complemented
19C2×#15668436C2211yescharacteristic
20C2×#15668436C2211yescharacteristic
21C2×#15668433C2211yescharacteristic
22C22×#13089678C2211yescharacteristic, complemented
23#19153804C2211yescharacteristic
24#19153801—21no—
25#19153795C2211yescharacteristic
26#19153790C2211yescharacteristic
27#19153789—21no—
28#19153784—21no—
29C4×D724C2211yescharacteristic
30C22⋊C4×D181—21no—
31#19153749C2211yescharacteristic, complemented
32#19153747C2211yescharacteristic, complemented
33#19153747C2211yescharacteristic, complemented
34C4⋊#13089678C2211yescharacteristic, complemented
35#19153740C2211yescharacteristic, complemented
36#19153739C2211yescharacteristic, complemented
37#19153739C2211yescharacteristic, complemented
38#19153739C2211yescharacteristic, complemented
39#19153736C2211yescharacteristic
40#19153736C2211yescharacteristic
41#19153736C2211yescharacteristic
42#19153736C2211yescharacteristic
43#19153736C2211yescharacteristic
44#19153736C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic, complemented
47#19153727C2211yescharacteristic, complemented
48C4⋊Q8×C181C2211yescharacteristic, complemented
49#126×C181—21no—
50#126×C181—21no—
51#124×C181—21no—
52C22⋊Q8×C181C2211yescharacteristic
53C22⋊Q8×C181C2211yescharacteristic, complemented
54C4×Q8×C181—21no—
55C4×Q8×C181C2211yescharacteristic, complemented
56C4×D4×C181C2211yescharacteristic
57#117×C181C2211yescharacteristic
58#117×C181C2211yescharacteristic, complemented
59C23×D181—21no—
60#15668449—22no—
61#15668449—41no—
62C2×Dic362C2311yescharacteristic
63C2×D724—22no—
64C2×D724—21no—
65C2×D724—41no—
66C2×D724C2311yescharacteristic
67C2×D724C2311yescharacteristic
68C2×C22⋊D181—21no—
69C2×C22⋊D181—21no—
70C2×C4×D181—21no—
71C2×C4×D181—21no—
72#15668439—41no—
73#15668436—22no—
74#15668436—41no—
75#15668433—41no—
76C2×#13089678—41no—
77C2×#13089678C2311yescharacteristic
78C2×#13089678C2311yescharacteristic
79C2×#13089678C2311yescharacteristic
80C2×#13089678C2311yescharacteristic
81C4⋊Dic181C2311yescharacteristic
82#15668428—21no—
83C22⋊Dic181—21no—
84#15668426—21no—
85#15668426—21no—
86#15668426—21no—
87#15668426—21no—
88C2×Q8×C181—21no—
89C2×Q8×C181C2311yescharacteristic
90C2×D4×C181D411yescharacteristic
91C22×C4×C181C2311yescharacteristic
92C22×C4×C181C2311yescharacteristic
93C4⋊C4×C181—21no—
94C4⋊C4×C181—21no—
95C4⋊C4×C181—21no—
96C4⋊C4×C181—21no—
97C4⋊C4×C181D411yescharacteristic
98C4⋊C4×C181C2311yescharacteristic
99C4⋊C4×C181C2311yescharacteristic
100C4⋊C4×C181C2311yescharacteristic
101C22⋊C4×C181—21no—
102C22⋊C4×C181—21no—
103C22⋊C4×C181—21no—
104C22⋊C4×C181—21no—
105C22⋊C4×C181D411yescharacteristic
106C22⋊C4×C181D411yescharacteristic
107C42×C181—21no—
108C42×C181—21no—
109C42×C181—21no—
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112C22×D181—21no—
113C22×D181—42no—
114C22×D181—42no—
115C22×D181—81no—
116Dic362—21no—
117Dic362—21no—
118D724—21no—
119D724—21no—
120D724—21no—
121D724—42no—
122D724#4112yes—
123C22⋊D181—42no—
124C22⋊D181—81no—
125C2×Dic181—21no—
126C2×Dic181—21no—
127C4×D181—42no—
128C4×D181—81no—
129#13089678—21no—
130#13089678—21no—
131#13089678—21no—
132#13089678—21no—
133C23×C181C2×D411yescharacteristic
134C23×C181C2×D411yescharacteristic
135Q8×C181—21no—
136Q8×C181—21no—
137Q8×C181—42no—
138D4×C181—41no—
139C2×C4×C181—21no—
140C2×C4×C181—21no—
141C2×C4×C181—21no—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181C2×D411yescharacteristic
148C2×C4×C181C2×D411yescharacteristic
149C2×C4×C181C2×D411yescharacteristic
150C2×C4×C181C2×D411yescharacteristic
151C2×C4×C181C2411yescharacteristic
152C22×C181—21no—
153C22×C181—21no—
154C22×C181—21no—
155C22×C181—21no—
156C22×C181—21no—
157C22×C181—21no—
158C22×C181C22×D411yescharacteristic
159C2×D181—22no—
160C2×D181—41no—
161C2×D181—41no—
162C2×D181—41no—
163C2×D181—41no—
164C2×D181—41no—
165C2×D181—84no—
166C4×C181—21no—
167C4×C181—21no—
168C4×C181—21no—
169C4×C181—21no—
170C4×C181—41no—
171C4×C181—41no—
172C4×C181—41no—
173C4×C181—41no—
174C4×C181C2×#4111yescharacteristic
175C4×C181H(5,𝔽2)11yescharacteristic
176Dic181—41no—
177Dic181—41no—
178C2×C181—41no—
179C2×C181—41no—
180C2×C181#54611yescharacteristic
181C2×C181#57511yescharacteristic
182C2×C181#57711yescharacteristic
183D181—42no—
184D181—81no—
185C181#331111yescharacteristic, complemented, Sylow
186#3311—1811nomaximal, Sylow
187#556D18111yescharacteristic, complemented
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#416—1811no—
202C2×#41—1811no—
203C22×D4—1811no—
204C2×Q16—1811no—
205C2×SD16—1811no—
206C2×SD16—1811no—
207C2×M16—1811no—
208C22×C8—1811no—
209C4⋊Q8C2×D18111yescharacteristic, complemented
210#126—21no—
211#126—21no—
212#124—21no—
213#124—1811no—
214#123—3621no—
215C22⋊Q8—1811no—
216C22⋊Q8C2×D18111yescharacteristic
217C22⋊Q8C2×D18111yescharacteristic, complemented
218#121—3621no—
219#121—1811no—
220#120—3621no—
221C4×Q8—21no—
222C4×Q8C2×D18111yescharacteristic, complemented
223C4×D4—1811no—
224C4×D4C2×D18111yescharacteristic
225#117C2×D18111yescharacteristic
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#103—1811no—
235#103—1811no—
236#102—1811no—
237#102—1811no—
238#102—1811no—
239#102—1811no—
240#102—1811no—
241#102—1811no—
242C22⋊C8—1811no—
243C22⋊C8—1811no—
244#97—1811no—
245C24—3621no—
246#41—3622no—
247#41—7241no—
248C2×Q8—21no—
249C2×Q8—1811no—
250C2×Q8C22×D18111yescharacteristic
251C2×D4—3622no—
252C2×D4—3621no—
253C2×D4—3621no—
254C2×D4—3621no—
255C2×D4—1811no—
256C2×D4—1811no—
257C2×D4—7241no—
258C2×D4C22⋊D18111yescharacteristic
259C22×C4—3621no—
260C22×C4—3621no—
261C22×C4C22×D18111yescharacteristic
262C22×C4C22×D18111yescharacteristic
263Q16—7241no—
264SD16—3622no—
265SD16—7241no—
266M16—7241no—
267C2×C8—1811no—
268C2×C8—1811no—
269C2×C8—1811no—
270C2×C8—1811no—
271C2×C8—7241no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—3621no—
277C4⋊C4—1811no—
278C4⋊C4C22⋊D18111yescharacteristic
279C4⋊C4C22×D18111yescharacteristic
280C4⋊C4C22×D18111yescharacteristic
281C4⋊C4C22×D18111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—3621no—
287C22⋊C4—3621no—
288C22⋊C4—3621no—
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4C22⋊D18111yescharacteristic
292C22⋊C4C22⋊D18111yescharacteristic
293C42—21no—
294C42—21no—
295C42—21no—
296C42C22×D18111yescharacteristic
297C23—3621no—
298C23—3621no—
299C23—14481no—
300C23—7242no—
301C23—7242no—
302C23C2×C22⋊D18111yescharacteristic
303C23C2×C22⋊D18111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—42no—
307Q8—3621no—
308Q8—3621no—
309D4—41no—
310D4—3621no—
311D4—3621no—
312D4—3621no—
313D4—14481no—
314D4—1812no—
315D4—7242no—
316D4—7242no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—3621no—
326C2×C4—3621no—
327C2×C4—14481no—
328C2×C4—7242no—
329C2×C4C2×C22⋊D18111yescharacteristic
330C2×C4C2×C22⋊D18111yescharacteristic
331C2×C4C2×C22⋊D18111yescharacteristic
332C2×C4C2×C22⋊D18111yescharacteristic
333C2×C4C23×D18111yescharacteristic
334C8—3621no—
335C8—3621no—
336C8—3621no—
337C8—3621no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—3622no—
345C22—14484no—
346C22—7241no—
347C22—7241no—
348C22—7241no—
349C22—7241no—
350C22—7241no—
351C22C22×C22⋊D18111yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—7241no—
361C4—7241no—
362C4#41×D18111yescharacteristic
363C4#1915386211yescharacteristic
364C2—41no—
365C2—41no—
366C2—14481no—
367C2—7242no—
368C2#2604191711yescharacteristic
369C2#2604207711yescharacteristic
370C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#556×C181C2
socleC22×C181C22×D4
prime radical#26041925C2
hypercenter#556D181
hypocenterC181#3311
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #26041641C2semidirect
3. #26041620C2semidirect
4. #26041603C2semidirect
5. #26041566C2semidirect
6. #26041535C2semidirect
7. #26041510C2semidirect
8. #26041501C2semidirect
9. #26041439C2semidirect
10. #26041429C2semidirect
11. #26041377C2semidirect
12. #26041351C2semidirect
13. #26041283C2semidirect
14. #26041273C2semidirect
15. #556×C181C2semidirect
18. C2×#15668439C22semidirect
22. C22×#13089678C22semidirect
31. #19153749C22semidirect
32. #19153747C22semidirect
33. #19153747C22semidirect
34. C4⋊#13089678C22semidirect
35. #19153740C22semidirect
36. #19153739C22semidirect
37. #19153739C22semidirect
38. #19153739C22semidirect
46. C22⋊#13089678C22semidirect
47. #19153727C22semidirect
48. C4⋊Q8×C181C22semidirect
53. C22⋊Q8×C181C22semidirect
55. C4×Q8×C181C22semidirect
58. #117×C181C22semidirect
185. C181#3311semidirect
187. #556D181semidirect
209. C4⋊Q8C2×D181semidirect
217. C22⋊Q8C2×D181semidirect
222. C4×Q8C2×D181semidirect
226. #117C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260419252
#260416412
#260416202
#260416032
#260415662
#260415352
#260415102
#260415012
#260414392
#260414292
#260413772
#260413512
#260412832
#260412732
#556×C1812
#3311181