Finite Group Databasebeta

← #37543972

Subgroups — #37543972

Order 23168.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041887C211yesmaximal, characteristic, complemented
2#26041751C211yesmaximal, characteristic
3#26041622C211yesmaximal, characteristic, complemented
4#26041619C211yesmaximal, characteristic, complemented
5#26041582C211yesmaximal, characteristic, complemented
6#26041577C211yesmaximal, characteristic, complemented
7#26041483C211yesmaximal, characteristic, complemented
8#26041482C211yesmaximal, characteristic, complemented
9#26041431C211yesmaximal, characteristic
10#26041428C211yesmaximal, characteristic
11#26041381C211yesmaximal, characteristic, complemented
12C4×#15668436C211yesmaximal, characteristic, complemented
13C4×#15668437C211yesmaximal, characteristic, complemented
14#26041341C211yesmaximal, characteristic, complemented
15#440×C181C211yesmaximal, characteristic, complemented
16C2×D4×D181C2211yescharacteristic
17C2×#15668438C2211yescharacteristic
18C2×#15668437C2211yescharacteristic
19C2×#15668436C2211yescharacteristic
20#19153814C2211yescharacteristic, complemented
21#19153811—21no—
22#19153810—21no—
23#19153808—21no—
24#19153807C2211yescharacteristic, complemented
25#19153806—21no—
26#19153796C2211yescharacteristic
27#19153792C2211yescharacteristic
28C4×Dic362C2211yescharacteristic, complemented
29C4×D724C2211yescharacteristic
30D4×Dic181C2211yescharacteristic
31#19153778C2211yescharacteristic
32C4×C22⋊D181—21no—
33#19153776—21no—
34#19153774C2211yescharacteristic
35#19153773C2211yescharacteristic
36C2×#15668426—21no—
37C22⋊C4×D181—21no—
38#19153750C2211yescharacteristic, complemented
39#19153747C2211yescharacteristic, complemented
40#19153747C2211yescharacteristic, complemented
41#19153745C2211yescharacteristic, complemented
42#19153745C2211yescharacteristic, complemented
43#19153741C2211yescharacteristic, complemented
44#19153740C2211yescharacteristic, complemented
45#19153739C2211yescharacteristic, complemented
46#19153738C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153736C2211yescharacteristic
49#19153728C2211yescharacteristic
50#19153728C2211yescharacteristic
51C4×#13089678C2211yescharacteristic, complemented
52C2×SD16×C181C2211yescharacteristic
53C4×Q8×C181C2211yescharacteristic, complemented
54C4×D4×C181C2211yescharacteristic
55#107×C181C2211yescharacteristic, complemented
56#103×C181C2211yescharacteristic, complemented
57#102×C181C2211yescharacteristic
58#97×C181C2211yescharacteristic, complemented
59C23×D181—21no—
60C2×Dic362C2311yescharacteristic
61C2×D724C2311yescharacteristic
62C2×C22⋊D181—21no—
63D4×D181—22no—
64D4×D181—22no—
65C22×Dic181—21no—
66C2×C4×D181—21no—
67C2×C4×D181—21no—
68C2×C4×D181C2311yescharacteristic
69#15668438—22no—
70#15668437—22no—
71#15668436—22no—
72C2×#13089678C2311yescharacteristic
73C2×#13089678C2311yescharacteristic
74C4⋊Dic181—21no—
75C4⋊Dic181—21no—
76C4⋊Dic181C2311yescharacteristic
77C4⋊Dic181C2311yescharacteristic
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83#15668428D411yescharacteristic
84#15668428D411yescharacteristic
85C22⋊Dic181—21no—
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89#15668426—21no—
90#15668426—21no—
91#15668426D411yescharacteristic
92#15668426D411yescharacteristic
93C4×Dic181—21no—
94C4×Dic181—21no—
95C4×Dic181C2311yescharacteristic
96C2×Q8×C181C2311yescharacteristic
97C2×D4×C181C2311yescharacteristic
98C22×C4×C181—21no—
99SD16×C181—22no—
100C2×C8×C181C2311yescharacteristic
101C2×C8×C181C2311yescharacteristic
102C4⋊C4×C181—21no—
103C4⋊C4×C181C2311yescharacteristic
104C4⋊C4×C181C2311yescharacteristic
105C22⋊C4×C181—21no—
106C42×C181—21no—
107C42×C181C2311yescharacteristic
108C22×D181—21no—
109C22×D181—24no—
110C22×D181—81no—
111C22×D181—81no—
112C22×D181C2×D411yescharacteristic
113Dic362—21no—
114Dic362—21no—
115D724—21no—
116D724#4112yes—
117C22⋊D181—24no—
118C2×Dic181—21no—
119C2×Dic181—21no—
120C2×Dic181—21no—
121C2×Dic181—81no—
122C2×Dic181C2×D411yescharacteristic
123C2×Dic181C2×D411yescharacteristic
124C2×Dic181C2×D411yescharacteristic
125C4×D181—21no—
126C4×D181—21no—
127C4×D181—81no—
128C4×D181—81no—
129#13089678—21no—
130#13089678—21no—
131C23×C181—21no—
132Q8×C181—21no—
133Q8×C181—21no—
134D4×C181—21no—
135D4×C181#4112yes—
136C2×C4×C181—21no—
137C2×C4×C181—21no—
138C2×C4×C181—21no—
139C2×C4×C181—81no—
140C2×C4×C181C2×D411yescharacteristic
141C2×C4×C181C2×D411yescharacteristic
142C2×C4×C181C2411yescharacteristic
143C8×C181—21no—
144C8×C181—21no—
145C22×C181—22no—
146C22×C181—41no—
147C22×C181—41no—
148C22×C181C22×D411yescharacteristic
149C2×D181—21no—
150C2×D181—22no—
151C2×D181—21no—
152C2×D181—21no—
153C2×D181—41no—
154C2×D181—41no—
155C2×D181—84no—
156C4×C181—21no—
157C4×C181—21no—
158C4×C181—41no—
159C4×C181—41no—
160C4×C181—41no—
161C4×C181C2×#4111yescharacteristic
162C4×C181C2×#4111yescharacteristic
163Dic181—21no—
164Dic181—21no—
165Dic181—21no—
166Dic181—41no—
167Dic181—41no—
168Dic181—41no—
169C2×C181—42no—
170C2×C181#52511yescharacteristic
171C2×C181#57711yescharacteristic
172C2×C181#57711yescharacteristic
173D181—41no—
174D181—42no—
175C181#312311yescharacteristic, complemented, Sylow
176#3123—1811nomaximal, Sylow
177#541—1811no—
178#518—1811no—
179#485—1811no—
180#485—1811no—
181#478—1811no—
182#478—1811no—
183#461—1811no—
184#461—1811no—
185#450—1811no—
186#450—1811no—
187#440—1811no—
188#440D18111yescharacteristic, complemented
189C4×SD16—1811no—
190C4×SD16—1811no—
191#433—1811no—
192C22×D4—1811no—
193C2×SD16—1811no—
194C2×SD16—1811no—
195C2×SD16—1811no—
196C2×SD16C2×D18111yescharacteristic
197C4⋊Q8—1811no—
198#126—3621no—
199#126—3621no—
200#125—3621no—
201#125—3621no—
202#125—1811no—
203C22⋊Q8—1811no—
204C22⋊Q8—1811no—
205C4×Q8—1811no—
206C4×Q8C2×D18111yescharacteristic, complemented
207C4×D4—3621no—
208C4×D4—3621no—
209C4×D4—1811no—
210C4×D4—1811no—
211C4×D4—1811no—
212C4×D4C2×D18111yescharacteristic
213#117—1811no—
214#117—1811no—
215C2×C22⋊C4—3621no—
216C2×C22⋊C4—3621no—
217#107—1811no—
218#107C2×D18111yescharacteristic, complemented
219#106—1811no—
220#106—1811no—
221C4⋊C8—1811no—
222C4⋊C8—1811no—
223#103—1811no—
224#103—1811no—
225#103—1811no—
226#103C2×D18111yescharacteristic, complemented
227#102—1811no—
228#102—1811no—
229#102—1811no—
230#102C2×D18111yescharacteristic
231C22⋊C8—1811no—
232C22⋊C8—1811no—
233#97C2×D18111yescharacteristic, complemented
234C4×C8—1811no—
235C24—3621no—
236C2×Q8—1811no—
237C2×Q8C22×D18111yescharacteristic
238C2×D4—3622no—
239C2×D4—3622no—
240C2×D4—3621no—
241C2×D4—1811no—
242C2×D4C22×D18111yescharacteristic
243C22×C4—21no—
244C22×C4—3621no—
245C22×C4—3621no—
246C22×C4—3621no—
247C22×C4—1811no—
248SD16—22no—
249SD16—3622no—
250SD16—3622no—
251SD16—3622no—
252C2×C8—1811no—
253C2×C8—1811no—
254C2×C8C22×D18111yescharacteristic
255C2×C8C22×D18111yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—3621no—
258C4⋊C4—3621no—
259C4⋊C4—3621no—
260C4⋊C4—3621no—
261C4⋊C4—3621no—
262C4⋊C4—3621no—
263C4⋊C4—3621no—
264C4⋊C4—1811no—
265C4⋊C4—1811no—
266C4⋊C4—1811no—
267C4⋊C4—1811no—
268C4⋊C4C22×D18111yescharacteristic
269C4⋊C4C22×D18111yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—3621no—
272C22⋊C4—3621no—
273C22⋊C4—3621no—
274C22⋊C4—3621no—
275C22⋊C4—3621no—
276C22⋊C4—3621no—
277C22⋊C4—1811no—
278C22⋊C4—1811no—
279C42—21no—
280C42—3621no—
281C42—3621no—
282C42—1811no—
283C42C22×D18111yescharacteristic
284C23—21no—
285C23—3624no—
286C23—3621no—
287C23—14481no—
288C23—14481no—
289C23—1811no—
290Q8—21no—
291Q8—21no—
292Q8—3621no—
293Q8—3621no—
294D4—21no—
295D4—3624no—
296D4—3621no—
297D4—1812no—
298D4#1566844912yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—3621no—
304C2×C4—3621no—
305C2×C4—3621no—
306C2×C4—3621no—
307C2×C4—3621no—
308C2×C4—14481no—
309C2×C4—14481no—
310C2×C4—14481no—
311C2×C4—1811no—
312C2×C4—1811no—
313C2×C4—1811no—
314C2×C4D4×D18111yescharacteristic
315C2×C4D4×D18111yescharacteristic
316C2×C4C23×D18111yescharacteristic
317C8—21no—
318C8—21no—
319C8—3621no—
320C8—3621no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—3621no—
325C22—3622no—
326C22—3621no—
327C22—3621no—
328C22—14484no—
329C22—7241no—
330C22—7241no—
331C22C2×D4×D18111yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—3621no—
338C4—3621no—
339C4—3621no—
340C4—7241no—
341C4—7241no—
342C4—7241no—
343C4#41×D18111yescharacteristic
344C4C2×#1566844911yescharacteristic
345C2—42no—
346C2—7241no—
347C2—7242no—
348C2#2604178811yescharacteristic
349C2#2604208811yescharacteristic
350C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#440×C181C2
socleC22×C181C22×D4
prime radicalC2×D4×D181C22
hypercenter#440D181
hypocenterC181#3123
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #26041887C2semidirect
3. #26041622C2semidirect
4. #26041619C2semidirect
5. #26041582C2semidirect
6. #26041577C2semidirect
7. #26041483C2semidirect
8. #26041482C2semidirect
11. #26041381C2semidirect
12. C4×#15668436C2semidirect
13. C4×#15668437C2semidirect
14. #26041341C2semidirect
15. #440×C181C2semidirect
20. #19153814C22semidirect
24. #19153807C22semidirect
28. C4×Dic362C22semidirect
38. #19153750C22semidirect
39. #19153747C22semidirect
40. #19153747C22semidirect
41. #19153745C22semidirect
42. #19153745C22semidirect
43. #19153741C22semidirect
44. #19153740C22semidirect
45. #19153739C22semidirect
51. C4×#13089678C22semidirect
53. C4×Q8×C181C22semidirect
55. #107×C181C22semidirect
56. #103×C181C22semidirect
58. #97×C181C22semidirect
175. C181#3123semidirect
188. #440D181semidirect
206. C4×Q8C2×D181semidirect
218. #107C2×D181semidirect
226. #103C2×D181semidirect
233. #97C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418872
#260417512
#260416222
#260416192
#260415822
#260415772
#260414832
#260414822
#260414312
#260414282
#260413812
C4×#156684362
C4×#156684372
#260413412
#440×C1812
#3123181