Finite Group Databasebeta

← #37545653

Subgroups — #37545653

Order 23168.

396 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042000C211yesmaximal, characteristic, complemented
2#26041948C211yesmaximal, characteristic
3#26041624C211yesmaximal, characteristic, complemented
4#26041604C211yesmaximal, characteristic, complemented
5#26041601C211yesmaximal, characteristic, complemented
6#26041562C211yesmaximal, characteristic, complemented
7#26041510C211yesmaximal, characteristic, complemented
8#26041505C211yesmaximal, characteristic, complemented
9#26041477C211yesmaximal, characteristic, complemented
10#26041440C211yesmaximal, characteristic, complemented
11Q16×Dic181C211yesmaximal, characteristic, complemented
12#26041342C211yesmaximal, characteristic, complemented
13#26041275C211yesmaximal, characteristic, complemented
14#26041274C211yesmaximal, characteristic, complemented
15#452×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668436C2211yescharacteristic
18C22×#13089678C2211yescharacteristic
19C2×C8×D181C2211yescharacteristic
20#19153815C2211yescharacteristic
21#19153812—21no—
22#19153812C2211yescharacteristic
23#19153811—21no—
24#19153809—21no—
25#19153802—21no—
26#19153792C2211yescharacteristic
27#19153790—21no—
28#19153790C2211yescharacteristic
29#19153787—21no—
30#19153787C2211yescharacteristic
31Q8×Dic181C2211yescharacteristic
32Q8×Dic181C2211yescharacteristic
33#19153781—21no—
34C4×C22⋊D181—21no—
35C4×C22⋊D181C2211yescharacteristic
36#19153775C2211yescharacteristic
37#19153774C2211yescharacteristic
38#19153750C2211yescharacteristic
39#19153748C2211yescharacteristic
40#19153747C2211yescharacteristic
41#19153745C2211yescharacteristic
42#19153740C2211yescharacteristic
43#19153740C2211yescharacteristic
44#19153740C2211yescharacteristic
45#19153738C2211yescharacteristic
46#19153738C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153736C2211yescharacteristic
50#19153728C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C2×#41×C181C2211yescharacteristic
53C2×Q16×C181C2211yescharacteristic
54C2×SD16×C181C2211yescharacteristic
55C22⋊Q8×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58C22⋊C8×C181C2211yescharacteristic
59#15668450—22no—
60#15668450—41no—
61C2×Dic362—21no—
62C2×D724—21no—
63C2×D724C2311yescharacteristic
64C2×C22⋊D181—21no—
65C2×C22⋊D181—21no—
66C2×C22⋊D181—21no—
67C2×C22⋊D181—21no—
68C2×C22⋊D181D411yescharacteristic
69C2×C4×D181—21no—
70C2×C4×D181C2311yescharacteristic
71#15668436—41no—
72C2×#13089678—41no—
73C2×#13089678C2311yescharacteristic
74C2×#13089678C2311yescharacteristic
75C8×D181—41no—
76C4⋊Dic181—21no—
77C4⋊Dic181—21no—
78C4⋊Dic181C2311yescharacteristic
79C4⋊Dic181C2311yescharacteristic
80C4⋊Dic181C2311yescharacteristic
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#15668426D411yescharacteristic
91C4×Dic181—21no—
92C4×Dic181—21no—
93C4×Dic181—21no—
94C4×Dic181—21no—
95C4×Dic181C2311yescharacteristic
96#41×C181—22no—
97#41×C181—41no—
98C2×Q8×C181C2311yescharacteristic
99C2×Q8×C181C2311yescharacteristic
100C2×D4×C181—21no—
101C2×D4×C181C2311yescharacteristic
102C22×C4×C181—21no—
103C22×C4×C181C2311yescharacteristic
104Q16×C181—22no—
105SD16×C181—41no—
106C2×C8×C181C2311yescharacteristic
107C2×C8×C181C2311yescharacteristic
108C4⋊C4×C181—21no—
109C4⋊C4×C181C2311yescharacteristic
110C22⋊C4×C181—21no—
111C22×D181—21no—
112C22×D181C2×D411yescharacteristic
113Dic362—42no—
114D724—21no—
115D724—21no—
116D724—42no—
117C22⋊D181—22no—
118C22⋊D181—42no—
119C22⋊D181—81no—
120C22⋊D181—81no—
121C22⋊D181—81no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181—21no—
125C2×Dic181C2×D411yescharacteristic
126C2×Dic181C2×D411yescharacteristic
127C2×Dic181C2×D411yescharacteristic
128C4×D181—21no—
129C4×D181—21no—
130C4×D181—42no—
131#13089678—21no—
132#13089678—21no—
133C23×C181—21no—
134C23×C181C2×D411yescharacteristic
135Q8×C181—21no—
136Q8×C181—21no—
137Q8×C181—21no—
138Q8×C181#4112yes—
139D4×C181—21no—
140D4×C181—21no—
141D4×C181—42no—
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—21no—
154C22×C181—21no—
155C22×C181—41no—
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—
165C4×C181—21no—
166C4×C181—22no—
167C4×C181—41no—
168C4×C181—41no—
169C4×C181C2×#4111yescharacteristic
170C4×C181H(5,𝔽2)11yescharacteristic
171Dic181—21no—
172Dic181—21no—
173Dic181—41no—
174Dic181—41no—
175Dic181—41no—
176Dic181#13512yes—
177C2×C181—41no—
178C2×C181—81no—
179C2×C181#54611yescharacteristic
180C2×C181C2×#13511yescharacteristic
181C2×C181#57711yescharacteristic
182D181—41no—
183D181—81no—
184C181#330411yescharacteristic, complemented, Sylow
185#3304—1811nomaximal, Sylow
186#556—1811no—
187#550—1811no—
188#486—1811no—
189#482—1811no—
190#482—1811no—
191#475—1811no—
192#465—1811no—
193#465—1811no—
194#460—1811no—
195#452—1811no—
196#452D18111yescharacteristic, complemented
197C4×Q16—1811no—
198C8×D4—1811no—
199#416—1811no—
200#416—1811no—
201C2×#41—1811no—
202C2×#41C2×D18111yescharacteristic
203C2×Q16C2×D18111yescharacteristic
204C2×SD16—1811no—
205C2×SD16C2×D18111yescharacteristic
206C22×C8—1811no—
207C22×C8—1811no—
208C4⋊Q8—1811no—
209#127—3621no—
210#127—1811no—
211#126—3621no—
212#126—3621no—
213#124—3621no—
214C22⋊Q8—1811no—
215C22⋊Q8C2×D18111yescharacteristic
216#121—3621no—
217#121—3621no—
218#121—1811no—
219#121—1811no—
220C4×Q8—3621no—
221C4×Q8—1811no—
222C4×Q8—1811no—
223C4×D4—3621no—
224C4×D4—1811no—
225#117—1811no—
226#117—1811no—
227#107—1811no—
228#106—1811no—
229#106—1811no—
230C4⋊C8—1811no—
231#103—1811no—
232#103—1811no—
233#103—1811no—
234#103C2×D18111yescharacteristic
235#102—1811no—
236#102—1811no—
237#102—1811no—
238#102—1811no—
239#102—1811no—
240#102C2×D18111yescharacteristic
241C22⋊C8—1811no—
242C22⋊C8C2×D18111yescharacteristic
243C4×C8—1811no—
244#41—22no—
245#41—41no—
246#41—3622no—
247#41—7241no—
248C2×Q8—3621no—
249C2×Q8C22×D18111yescharacteristic
250C2×Q8C22×D18111yescharacteristic
251C2×D4—21no—
252C2×D4—3621no—
253C2×D4—3621no—
254C2×D4—3621no—
255C2×D4—3621no—
256C2×D4—3621no—
257C2×D4—1811no—
258C2×D4—1811no—
259C2×D4C22×D18111yescharacteristic
260C22×C4—21no—
261C22×C4—3621no—
262C22×C4—1811no—
263C22×C4C22×D18111yescharacteristic
264Q16—22no—
265SD16—41no—
266SD16—7241no—
267C2×C8—1811no—
268C2×C8—1811no—
269C2×C8—7241no—
270C2×C8—7241no—
271C2×C8C22×D18111yescharacteristic
272C2×C8C22×D18111yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—3621no—
275C4⋊C4—3621no—
276C4⋊C4—3621no—
277C4⋊C4—3621no—
278C4⋊C4—1811no—
279C4⋊C4—1811no—
280C4⋊C4—1811no—
281C4⋊C4—1811no—
282C4⋊C4C22×D18111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—3621no—
285C22⋊C4—3621no—
286C22⋊C4—3621no—
287C22⋊C4—3621no—
288C22⋊C4—3621no—
289C22⋊C4—1811no—
290C22⋊C4—1811no—
291C42—3621no—
292C42—3621no—
293C42—3621no—
294C42—3621no—
295C42—1811no—
296C23—21no—
297C23—3621no—
298C23—1811no—
299C23D4×D18111yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—7242no—
304Q8#1566844812yes—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—3622no—
309D4—3621no—
310D4—3621no—
311D4—14481no—
312D4—14481no—
313D4—14481no—
314D4—7242no—
315D4—7242no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—3621no—
323C2×C4—3621no—
324C2×C4—3621no—
325C2×C4—3621no—
326C2×C4—3621no—
327C2×C4—1811no—
328C2×C4—1811no—
329C2×C4—1811no—
330C2×C4—7242no—
331C2×C4D4×D18111yescharacteristic
332C2×C4C23×D18111yescharacteristic
333C8—21no—
334C8—21no—
335C8—3621no—
336C8—3621no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—3621no—
344C22—3621no—
345C22—3621no—
346C22—7241no—
347C22—7241no—
348C22—7241no—
349C22C2×D4×D18111yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—3621no—
355C4—3621no—
356C4—1812no—
357C4—7241no—
358C4—7241no—
359C4—7241no—
360C4C2×#1566844811yescharacteristic
361C4#1915386111yescharacteristic
362C2—41no—
363C2—81no—
364C2—14481no—
365C2—7241no—
366C2#2604191611yescharacteristic
367C2#135×D18111yescharacteristic
368C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#452×C181C2
socleC22×C181C22×D4
prime radical#26041948C2
hypercenter#452D181
hypocenterC181#3304
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #26042000C2semidirect
3. #26041624C2semidirect
4. #26041604C2semidirect
5. #26041601C2semidirect
6. #26041562C2semidirect
7. #26041510C2semidirect
8. #26041505C2semidirect
9. #26041477C2semidirect
10. #26041440C2semidirect
11. Q16×Dic181C2semidirect
12. #26041342C2semidirect
13. #26041275C2semidirect
14. #26041274C2semidirect
15. #452×C181C2semidirect
184. C181#3304semidirect
196. #452D181semidirect

Maximal subgroups

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

subgroupindex
#260420002
#260419482
#260416242
#260416042
#260416012
#260415622
#260415102
#260415052
#260414772
#260414402
Q16×Dic1812
#260413422
#260412752
#260412742
#452×C1812
#3304181