Finite Group Databasebeta

← #37545565

Subgroups — #37545565

Order 23168.

420 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#26041925C211yesmaximal, characteristic, complemented
2#26041636C211yesmaximal, characteristic, complemented
3#26041614C211yesmaximal, characteristic, complemented
4#26041603C211yesmaximal, characteristic, complemented
5#26041576C211yesmaximal, characteristic, complemented
6#26041513C211yesmaximal, characteristic, complemented
7#26041506C211yesmaximal, characteristic, complemented
8#26041494C211yesmaximal, characteristic, complemented
9#26041422C211yesmaximal, characteristic, complemented
10#26041412C211yesmaximal, characteristic, complemented
11C4×#15668434C211yesmaximal, characteristic, complemented
12D4×#13089678C211yesmaximal, characteristic, complemented
13#26041273C211yesmaximal, characteristic, complemented
14C2×#19153736C211yesmaximal, characteristic, complemented
15#553×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C22×D724C2211yescharacteristic
18C2×#15668437C2211yescharacteristic, complemented
19C2×#15668434C2211yescharacteristic
20C2×#15668434C2211yescharacteristic
21C22×#13089678C2211yescharacteristic
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#19153749C2211yescharacteristic, complemented
33#19153747C2211yescharacteristic, complemented
34C4⋊#13089678C2211yescharacteristic, complemented
35#19153739C2211yescharacteristic, complemented
36#19153739C2211yescharacteristic, complemented
37#19153739C2211yescharacteristic, complemented
38#19153737C2211yescharacteristic, complemented
39#19153736C2211yescharacteristic
40#19153736C2211yescharacteristic
41#19153736C2211yescharacteristic
42#19153736C2211yescharacteristic
43#19153736C2211yescharacteristic
44#19153736C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic, complemented
47C4×#13089678C2211yescharacteristic, complemented
48#126×C181—21no—
49#125×C181C2211yescharacteristic, complemented
50#123×C181—21no—
51#121×C181—21no—
52#121×C181C2211yescharacteristic
53#121×C181C2211yescharacteristic, complemented
54C4×D4×C181—21no—
55C4×D4×C181C2211yescharacteristic
56C4×D4×C181C2211yescharacteristic, complemented
57#117×C181C2211yescharacteristic, complemented
58C2×C4⋊C4×C181C2211yescharacteristic
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#15668437—41no—
73#15668434—22no—
74#15668434—41no—
75C2×#13089678—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×D4×C181—21no—
89C2×D4×C181—21no—
90C2×D4×C181D411yescharacteristic
91C2×D4×C181C2311yescharacteristic
92C22×C4×C181—21no—
93C22×C4×C181—21no—
94C22×C4×C181C2311yescharacteristic
95C22×C4×C181C2311yescharacteristic
96C4⋊C4×C181—21no—
97C4⋊C4×C181—21no—
98C4⋊C4×C181—21no—
99C4⋊C4×C181D411yescharacteristic
100C4⋊C4×C181C2311yescharacteristic
101C4⋊C4×C181C2311yescharacteristic
102C4⋊C4×C181C2311yescharacteristic
103C22⋊C4×C181—21no—
104C22⋊C4×C181—21no—
105C22⋊C4×C181—21no—
106C22⋊C4×C181—21no—
107C22⋊C4×C181D411yescharacteristic
108C22⋊C4×C181D411yescharacteristic
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×C181—21no—
134C23×C181C2×D411yescharacteristic
135C23×C181C2×D411yescharacteristic
136D4×C181—22no—
137D4×C181—21no—
138D4×C181—21no—
139D4×C181—42no—
140D4×C181—81no—
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×C181—21no—
148C2×C4×C181—81no—
149C2×C4×C181—81no—
150C2×C4×C181C2×D411yescharacteristic
151C2×C4×C181C2×D411yescharacteristic
152C2×C4×C181C2×D411yescharacteristic
153C2×C4×C181C2×D411yescharacteristic
154C2×C4×C181C2411yescharacteristic
155C22×C181—21no—
156C22×C181—21no—
157C22×C181—21no—
158C22×C181—21no—
159C22×C181—21no—
160C22×C181—41no—
161C22×C181—41no—
162C22×C181—41no—
163C22×C181#13512yes—
164C22×C181C22×D411yescharacteristic
165C2×D181—22no—
166C2×D181—41no—
167C2×D181—41no—
168C2×D181—41no—
169C2×D181—41no—
170C2×D181—41no—
171C2×D181—84no—
172C4×C181—21no—
173C4×C181—21no—
174C4×C181—21no—
175C4×C181—21no—
176C4×C181—41no—
177C4×C181—41no—
178C4×C181—41no—
179C4×C181C2×#4111yescharacteristic
180C4×C181H(5,𝔽2)11yescharacteristic
181Dic181—41no—
182Dic181—41no—
183C2×C181—22no—
184C2×C181—41no—
185C2×C181—81no—
186C2×C181#54611yescharacteristic
187C2×C181C2×#13511yescharacteristic
188C2×C181#57611yescharacteristic
189D181—42no—
190D181—81no—
191C181#329711yescharacteristic, complemented, Sylow
192#3297—1811nomaximal, Sylow
193#553D18111yescharacteristic, complemented
194#546—1811no—
195#488—1811no—
196#484—1811no—
197#482—1811no—
198#478—1811no—
199#466—1811no—
200#465—1811no—
201#463—1811no—
202#449—1811no—
203#447—1811no—
204C4×D8—1811no—
205C8×D4—1811no—
206#416—1811no—
207C2×#102—1811no—
208C2×#41—1811no—
209C22×D4—1811no—
210C2×SD16—1811no—
211C2×D8—1811no—
212C2×D8—1811no—
213C22×C8—1811no—
214C22×C8—1811no—
215#126—21no—
216#125C2×D18111yescharacteristic, complemented
217#124—1811no—
218#123—21no—
219#123—3621no—
220C22⋊Q8—1811no—
221#121—21no—
222#121—3621no—
223#121—1811no—
224#121C2×D18111yescharacteristic
225#121C2×D18111yescharacteristic, complemented
226#120—3621no—
227C4×D4—21no—
228C4×D4—1811no—
229C4×D4C2×D18111yescharacteristic
230C4×D4C2×D18111yescharacteristic, complemented
231#117C2×D18111yescharacteristic, complemented
232C2×C4⋊C4C2×D18111yescharacteristic
233C2×C22⋊C4—3621no—
234#107—1811no—
235#107—1811no—
236#106—1811no—
237C4⋊C8—1811no—
238#103—1811no—
239#103—1811no—
240#103—1811no—
241#102—1811no—
242#102—1811no—
243#102—1811no—
244#102—1811no—
245#102—1811no—
246#102—1811no—
247#102—1811no—
248C22⋊C8—1811no—
249C22⋊C8—1811no—
250C4×C8—1811no—
251C24—3621no—
252#41—3622no—
253#41—7241no—
254C2×Q8—1811no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—3622no—
258C2×D4—3621no—
259C2×D4—3621no—
260C2×D4—3621no—
261C2×D4—1811no—
262C2×D4—1811no—
263C2×D4—7241no—
264C2×D4C22⋊D18111yescharacteristic
265C2×D4C22×D18111yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—3621no—
269C22×C4—3621no—
270C22×C4C22×D18111yescharacteristic
271C22×C4C22×D18111yescharacteristic
272SD16—7241no—
273D8—3622no—
274D8—7241no—
275C2×C8—1811no—
276C2×C8—1811no—
277C2×C8—1811no—
278C2×C8—1811no—
279C2×C8—7241no—
280C2×C8—7241no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—3621no—
285C4⋊C4—1811no—
286C4⋊C4C22⋊D18111yescharacteristic
287C4⋊C4C22×D18111yescharacteristic
288C4⋊C4C22×D18111yescharacteristic
289C4⋊C4C22×D18111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—3621no—
295C22⋊C4—3621no—
296C22⋊C4—3621no—
297C22⋊C4—3621no—
298C22⋊C4—3621no—
299C22⋊C4C22⋊D18111yescharacteristic
300C22⋊C4C22⋊D18111yescharacteristic
301C42—21no—
302C42C22×D18111yescharacteristic
303C23—21no—
304C23—3621no—
305C23—3621no—
306C23—14481no—
307C23—7242no—
308C23—7242no—
309C23C2×C22⋊D18111yescharacteristic
310C23C2×C22⋊D18111yescharacteristic
311Q8—3621no—
312Q8—3621no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—3621no—
319D4—3621no—
320D4—3621no—
321D4—14481no—
322D4—1812no—
323D4—7242no—
324D4—7242no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—3621no—
335C2×C4—3621no—
336C2×C4—14481no—
337C2×C4—7242no—
338C2×C4C2×C22⋊D18111yescharacteristic
339C2×C4C2×C22⋊D18111yescharacteristic
340C2×C4C2×C22⋊D18111yescharacteristic
341C2×C4C2×C22⋊D18111yescharacteristic
342C2×C4C23×D18111yescharacteristic
343C8—3621no—
344C8—3621no—
345C8—3621no—
346C8—3621no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—3622no—
356C22—14484no—
357C22—7241no—
358C22—7241no—
359C22—7241no—
360C22—7241no—
361C22—7241no—
362C22#1915383612yes—
363C22C22×C22⋊D18111yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—7241no—
372C4—7241no—
373C4#41×D18111yescharacteristic
374C4#1915386211yescharacteristic
375C2—22no—
376C2—41no—
377C2—81no—
378C2—14481no—
379C2—7242no—
380C2#2604191711yescharacteristic
381C2C2×#1915383611yescharacteristic
382C2#2604208211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#553×C181C2
socleC22×C181C22×D4
hypercenter#553D181
hypocenterC181#3297
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #26041925C2semidirect
2. #26041636C2semidirect
3. #26041614C2semidirect
4. #26041603C2semidirect
5. #26041576C2semidirect
6. #26041513C2semidirect
7. #26041506C2semidirect
8. #26041494C2semidirect
9. #26041422C2semidirect
10. #26041412C2semidirect
11. C4×#15668434C2semidirect
12. D4×#13089678C2semidirect
13. #26041273C2semidirect
14. C2×#19153736C2semidirect
15. #553×C181C2semidirect
18. C2×#15668437C22semidirect
22. C22×#13089678C22semidirect
31. #19153749C22semidirect
32. #19153749C22semidirect
33. #19153747C22semidirect
34. C4⋊#13089678C22semidirect
35. #19153739C22semidirect
36. #19153739C22semidirect
37. #19153739C22semidirect
38. #19153737C22semidirect
46. C22⋊#13089678C22semidirect
47. C4×#13089678C22semidirect
49. #125×C181C22semidirect
53. #121×C181C22semidirect
56. C4×D4×C181C22semidirect
57. #117×C181C22semidirect
191. C181#3297semidirect
193. #553D181semidirect
216. #125C2×D181semidirect
225. #121C2×D181semidirect
230. C4×D4C2×D181semidirect
231. #117C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260419252
#260416362
#260416142
#260416032
#260415762
#260415132
#260415062
#260414942
#260414222
#260414122
C4×#156684342
D4×#130896782
#260412732
C2×#191537362
#553×C1812
#3297181