Finite Group Databasebeta

← #37545873

Subgroups — #37545873

Order 23168.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041992C211yesmaximal, characteristic, complemented
2#26041917C211yesmaximal, characteristic, complemented
3#26041640C211yesmaximal, characteristic, complemented
4#26041619C211yesmaximal, characteristic, complemented
5#26041603C211yesmaximal, characteristic, complemented
6#26041557C211yesmaximal, characteristic, complemented
7#26041528C211yesmaximal, characteristic, complemented
8#26041515C211yesmaximal, characteristic, complemented
9#26041497C211yesmaximal, characteristic, complemented
10#26041424C211yesmaximal, characteristic, complemented
11#26041414C211yesmaximal, characteristic, complemented
12#26041388C211yesmaximal, characteristic, complemented
13#26041350C211yesmaximal, characteristic, complemented
14#26041282C211yesmaximal, characteristic, complemented
15#416×C181C211yesmaximal, characteristic, complemented
16C2×D4×D181C2211yescharacteristic
17C2×#15668436C2211yescharacteristic, complemented
18C2×D1448C2211yescharacteristic
19C2×#15668434C2211yescharacteristic
20C2×#15668432C2211yescharacteristic
21#19153815C2211yescharacteristic, complemented
22#19153805C2211yescharacteristic
23#19153804—21no—
24#19153802—21no—
25#19153799—21no—
26#19153792C2211yescharacteristic
27#19153791C2211yescharacteristic
28#19153790C2211yescharacteristic
29#19153790C2211yescharacteristic, complemented
30#19153789—21no—
31#19153788—21no—
32#19153784—21no—
33D4×Dic181C2211yescharacteristic
34#19153778C2211yescharacteristic, complemented
35C4×C22⋊D181C2211yescharacteristic
36#19153774C2211yescharacteristic
37#19153773—21no—
38C2×#15668426—21no—
39#19153750C2211yescharacteristic, complemented
40#19153749C2211yescharacteristic, complemented
41#19153747C2211yescharacteristic, complemented
42#19153745C2211yescharacteristic, complemented
43#19153740C2211yescharacteristic, complemented
44#19153738C2211yescharacteristic
45#19153738C2211yescharacteristic, complemented
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×D724—21no—
62C2×D724C2311yescharacteristic
63C2×D724C2311yescharacteristic
64C2×C22⋊D181—21no—
65C2×C22⋊D181—21no—
66C2×C22⋊D181—21no—
67C2×C22⋊D181D411yescharacteristic
68D4×D181—22no—
69D4×D181—41no—
70C22×Dic181—21no—
71C2×C4×D181—21no—
72C2×C4×D181C2311yescharacteristic
73#15668436—41no—
74D1448—41no—
75#15668434—22no—
76#15668432—41no—
77C2×#13089678C2311yescharacteristic
78C2×#13089678C2311yescharacteristic
79C4⋊Dic181C2311yescharacteristic
80C4⋊Dic181C2311yescharacteristic
81#15668428—21no—
82#15668428—21no—
83#15668428D411yescharacteristic
84C22⋊Dic181—21no—
85C22⋊Dic181D411yescharacteristic
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89#15668426—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426D411yescharacteristic
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—21no—
113C22×D181—42no—
114C22×D181—42no—
115C22×D181—81no—
116C22×D181C2×D411yescharacteristic
117Dic362—81no—
118D724—21no—
119D724—21no—
120D724—21no—
121D724—21no—
122D724—81no—
123C22⋊D181—41no—
124C22⋊D181—42no—
125C22⋊D181—81no—
126C22⋊D181—81no—
127C2×Dic181—21no—
128C2×Dic181—21no—
129C2×Dic181—81no—
130C2×Dic181C2×D411yescharacteristic
131C2×Dic181C2×D411yescharacteristic
132C2×Dic181C2×D411yescharacteristic
133C4×D181—21no—
134C4×D181—21no—
135C4×D181—42no—
136#13089678—21no—
137#13089678—21no—
138C23×C181—21no—
139C23×C181C2×D411yescharacteristic
140Q8×C181—21no—
141Q8×C181—21no—
142D4×C181—21no—
143D4×C181—42no—
144D4×C181#4112yes—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181—21no—
148C2×C4×C181—21no—
149C2×C4×C181—21no—
150C2×C4×C181—42no—
151C2×C4×C181C2×D411yescharacteristic
152C2×C4×C181C2411yescharacteristic
153C8×C181—21no—
154C8×C181—21no—
155C22×C181—21no—
156C22×C181—22no—
157C22×C181—21no—
158C22×C181—21no—
159C22×C181—41no—
160C22×C181—41no—
161C22×C181C22×D411yescharacteristic
162C2×D181—21no—
163C2×D181—21no—
164C2×D181—21no—
165C2×D181—41no—
166C2×D181—41no—
167C2×D181—41no—
168C2×D181—41no—
169C2×D181—41no—
170C2×D181—41no—
171C2×D181—84no—
172C4×C181—21no—
173C4×C181—41no—
174C4×C181—41no—
175C4×C181—41no—
176C4×C181C2×#4111yescharacteristic
177C4×C181H(5,𝔽2)11yescharacteristic
178Dic181—21no—
179Dic181—21no—
180Dic181—21no—
181Dic181—41no—
182Dic181—41no—
183C2×C181—42no—
184C2×C181—41no—
185C2×C181#54611yescharacteristic
186C2×C181#57511yescharacteristic
187C2×C181#57611yescharacteristic
188D181—41no—
189D181—81no—
190D181—81no—
191C181#331911yescharacteristic, complemented, Sylow
192#3319—1811nomaximal, Sylow
193#555—1811no—
194#546—1811no—
195#489—1811no—
196#485—1811no—
197#482—1811no—
198#474—1811no—
199#468—1811no—
200#466—1811no—
201#463—1811no—
202#449—1811no—
203#447—1811no—
204#442—1811no—
205#435—1811no—
206#418—1811no—
207#416D18111yescharacteristic, complemented
208C2×#41C2×D18111yescharacteristic
209C22×D4—1811no—
210C2×SD16—1811no—
211C2×D8—1811no—
212C2×D8—1811no—
213C2×M16—1811no—
214C22×C8C2×D18111yescharacteristic, complemented
215C4⋊Q8—1811no—
216#124—3621no—
217#124—3621no—
218#124—1811no—
219#123—3621no—
220C22⋊Q8—1811no—
221#121—3621no—
222#121—3621no—
223#121—1811no—
224#121—1811no—
225#121—1811no—
226#120—3621no—
227C4×D4—1811no—
228C4×D4—1811no—
229C4×D4—1811no—
230#117—3621no—
231#117—1811no—
232#117C2×D18111yescharacteristic, complemented
233C2×C22⋊C4—3621no—
234#107—1811no—
235#107—1811no—
236#106—1811no—
237C4⋊C8—1811no—
238#103—1811no—
239#103C2×D18111yescharacteristic, complemented
240#103C2×D18111yescharacteristic, complemented
241#102—1811no—
242#102—1811no—
243#102—1811no—
244#102—1811no—
245#102—1811no—
246#102C2×D18111yescharacteristic
247#102C2×D18111yescharacteristic
248C22⋊C8—1811no—
249C22⋊C8—1811no—
250#97—1811no—
251C24—3621no—
252#41—22no—
253#41—41no—
254C2×Q8—3621no—
255C2×Q8C22×D18111yescharacteristic
256C2×D4—21no—
257C2×D4—3621no—
258C2×D4—3622no—
259C2×D4—3621no—
260C2×D4—3621no—
261C2×D4—3621no—
262C2×D4—1811no—
263C2×D4—1811no—
264C2×D4—1811no—
265C2×D4—7241no—
266C2×D4C22×D18111yescharacteristic
267C22×C4—21no—
268C22×C4—3621no—
269C22×C4—3621no—
270C22×C4—1811no—
271C22×C4C22×D18111yescharacteristic
272SD16—7241no—
273D8—3622no—
274D8—7241no—
275M16—7241no—
276C2×C8—41no—
277C2×C8—1811no—
278C2×C8—1811no—
279C2×C8C22×D18111yescharacteristic
280C2×C8C22×D18111yescharacteristic
281C4⋊C4—3621no—
282C4⋊C4—3621no—
283C4⋊C4—1811no—
284C4⋊C4—1811no—
285C4⋊C4—1811no—
286C4⋊C4C22×D18111yescharacteristic
287C4⋊C4C22×D18111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4—3621no—
292C22⋊C4—3621no—
293C22⋊C4—3621no—
294C22⋊C4—3621no—
295C22⋊C4—3621no—
296C22⋊C4—3621no—
297C22⋊C4—3621no—
298C22⋊C4—1811no—
299C22⋊C4—1811no—
300C42—21no—
301C42—3621no—
302C42—1811no—
303C23—21no—
304C23—3621no—
305C23—3621no—
306C23—14481no—
307C23—1811no—
308C23—7242no—
309C23—7242no—
310C23D4×D18111yescharacteristic
311Q8—21no—
312Q8—21no—
313Q8—14481no—
314D4—21no—
315D4—42no—
316D4—3621no—
317D4—3621no—
318D4—3621no—
319D4—3621no—
320D4—14481no—
321D4—14481no—
322D4—14481no—
323D4—7241no—
324D4—7242no—
325D4#1566844912yes—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—3621no—
333C2×C4—3621no—
334C2×C4—3621no—
335C2×C4—3621no—
336C2×C4—14481no—
337C2×C4—1811no—
338C2×C4—1811no—
339C2×C4—1811no—
340C2×C4—7242no—
341C2×C4D4×D18111yescharacteristic
342C2×C4C23×D18111yescharacteristic
343C8—21no—
344C8—21no—
345C8—3621no—
346C8—3621no—
347C22—21no—
348C22—22no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—3621no—
354C22—3621no—
355C22—3621no—
356C22—14484no—
357C22—7241no—
358C22—7241no—
359C22—7241no—
360C22—7241no—
361C22—7241no—
362C22—7241no—
363C22C2×D4×D18111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—3621no—
369C4—3621no—
370C4—3621no—
371C4—7241no—
372C4—7241no—
373C4C2×#1566844911yescharacteristic
374C4#1915386111yescharacteristic
375C2—41no—
376C2—42no—
377C2—14481no—
378C2—14481no—
379C2—7241no—
380C2#2604191911yescharacteristic
381C2#2604207611yescharacteristic
382C2#2604208111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#416×C181C2
socleC22×C181C22×D4
hypercenter#416D181
hypocenterC181#3319
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. #26041992C2semidirect
2. #26041917C2semidirect
3. #26041640C2semidirect
4. #26041619C2semidirect
5. #26041603C2semidirect
6. #26041557C2semidirect
7. #26041528C2semidirect
8. #26041515C2semidirect
9. #26041497C2semidirect
10. #26041424C2semidirect
11. #26041414C2semidirect
12. #26041388C2semidirect
13. #26041350C2semidirect
14. #26041282C2semidirect
15. #416×C181C2semidirect
17. C2×#15668436C22semidirect
21. #19153815C22semidirect
29. #19153790C22semidirect
34. #19153778C22semidirect
39. #19153750C22semidirect
40. #19153749C22semidirect
41. #19153747C22semidirect
42. #19153745C22semidirect
43. #19153740C22semidirect
45. #19153738C22semidirect
49. C22⋊#13089678C22semidirect
51. #19153726C22semidirect
53. C22×C8×C181C22semidirect
54. #117×C181C22semidirect
55. #103×C181C22semidirect
56. #103×C181C22semidirect
191. C181#3319semidirect
207. #416D181semidirect
214. C22×C8C2×D181semidirect
232. #117C2×D181semidirect
239. #103C2×D181semidirect
240. #103C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260419922
#260419172
#260416402
#260416192
#260416032
#260415572
#260415282
#260415152
#260414972
#260414242
#260414142
#260413882
#260413502
#260412822
#416×C1812
#3319181