Finite Group Databasebeta

← #37546780

Subgroups — #37546780

Order 23168.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042015C211yesmaximal, characteristic, complemented
2#26042001C211yesmaximal, characteristic, complemented
3#26041670C211yesmaximal, characteristic, complemented
4#26041666C211yesmaximal, characteristic
5#26041660C211yesmaximal, characteristic, complemented
6#26041502C211yesmaximal, characteristic, complemented
7#26041500C211yesmaximal, characteristic, complemented
8#26041496C211yesmaximal, characteristic, complemented
9#26041493C211yesmaximal, characteristic, complemented
10#26041482C211yesmaximal, characteristic, complemented
11#26041408C211yesmaximal, characteristic, complemented
12#26041388C211yesmaximal, characteristic, complemented
13#26041384C211yesmaximal, characteristic, complemented
14#26041370C211yesmaximal, characteristic, complemented
15#460×C181C211yesmaximal, characteristic, complemented
16C2×Dic724C2211yescharacteristic
17C2×#15668439C2211yescharacteristic
18C2×#15668438C2211yescharacteristic
19C2×#15668437C2211yescharacteristic
20C2×#15668437C2211yescharacteristic
21C2×#15668434C2211yescharacteristic
22C2×#15668434C2211yescharacteristic
23#19153814C2211yescharacteristic
24#19153811—21no—
25#19153810—21no—
26#19153805C2211yescharacteristic
27#19153804C2211yescharacteristic
28#19153803—21no—
29#19153803C2211yescharacteristic
30#19153803C2211yescharacteristic
31#19153797—21no—
32#19153797—21no—
33#19153794—21no—
34C4×Dic362C2211yescharacteristic
35C4×Dic362C2211yescharacteristic
36#19153781C2211yescharacteristic
37D4×Dic181—21no—
38D4×Dic181C2211yescharacteristic
39D4×Dic181C2211yescharacteristic
40#19153778C2211yescharacteristic
41#19153773—21no—
42#19153747C2211yescharacteristic
43#19153745C2211yescharacteristic
44#19153745C2211yescharacteristic
45#19153741C2211yescharacteristic
46#19153740C2211yescharacteristic
47#19153739C2211yescharacteristic
48#19153738C2211yescharacteristic
49#19153727C2211yescharacteristic
50#19153726C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C2×SD16×C181C2211yescharacteristic
53C2×SD16×C181C2211yescharacteristic
54#127×C181C2211yescharacteristic
55C4×Q8×C181C2211yescharacteristic
56C4⋊C8×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59C2×Dic362D411yescharacteristic
60C2×Dic362C2311yescharacteristic
61C2×Dic362C2311yescharacteristic
62C2×D724C2311yescharacteristic
63C22×Dic181—21no—
64C22×Dic181—21no—
65C2×C4×D181—21no—
66Dic724—41no—
67#15668439—22no—
68#15668438—41no—
69#15668437—22no—
70#15668437—41no—
71#15668434—22no—
72#15668434—41no—
73C2×#13089678C2311yescharacteristic
74C2×#13089678C2311yescharacteristic
75C4⋊Dic181D411yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77#15668428—21no—
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428D411yescharacteristic
82#15668428D411yescharacteristic
83C22⋊Dic181—21no—
84C22⋊Dic181—21no—
85C22⋊Dic181—21no—
86C22⋊Dic181—21no—
87C22⋊Dic181—21no—
88C22⋊Dic181—21no—
89#15668426—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94C4×Dic181—21no—
95C4×Dic181—21no—
96C4×Dic181C2311yescharacteristic
97C4×Dic181C2311yescharacteristic
98C2×Q8×C181C2311yescharacteristic
99C2×D4×C181—21no—
100C2×D4×C181—21no—
101C2×D4×C181C2311yescharacteristic
102C2×D4×C181C2311yescharacteristic
103SD16×C181—41no—
104SD16×C181—41no—
105C2×C8×C181C2311yescharacteristic
106C2×C8×C181C2311yescharacteristic
107C4⋊C4×C181—21no—
108C4⋊C4×C181C2311yescharacteristic
109C42×C181—21no—
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112Dic362—21no—
113Dic362—21no—
114Dic362—21no—
115Dic362—21no—
116Dic362—41no—
117D724—21no—
118D724—21no—
119C2×Dic181—21no—
120C2×Dic181—21no—
121C2×Dic181—21no—
122C2×Dic181—81no—
123C2×Dic181—81no—
124C2×Dic181C2×D411yescharacteristic
125C2×Dic181C2×D411yescharacteristic
126C2×Dic181C2×D411yescharacteristic
127C2×Dic181C2×D411yescharacteristic
128C4×D181—42no—
129#13089678—21no—
130#13089678#4112yes—
131C23×C181—21no—
132C23×C181—21no—
133Q8×C181—21no—
134Q8×C181—21no—
135D4×C181—21no—
136D4×C181—21no—
137D4×C181—21no—
138D4×C181—21no—
139D4×C181—42no—
140D4×C181—81no—
141C2×C4×C181—21no—
142C2×C4×C181—21no—
143C2×C4×C181C2×D411yescharacteristic
144C2×C4×C181C2×D411yescharacteristic
145C2×C4×C181C2411yescharacteristic
146C8×C181—21no—
147C8×C181—21no—
148C22×C181—41no—
149C22×C181—41no—
150C22×C181—41no—
151C22×C181—41no—
152C22×C181—41no—
153C22×C181—41no—
154C22×C181C22×D411yescharacteristic
155C2×D181—41no—
156C2×D181—41no—
157C2×D181—41no—
158C4×C181—21no—
159C4×C181—21no—
160C4×C181—41no—
161C4×C181—41no—
162C4×C181C2×#4111yescharacteristic
163C4×C181H(5,𝔽2)11yescharacteristic
164Dic181—21no—
165Dic181—21no—
166Dic181—21no—
167Dic181—21no—
168Dic181—41no—
169Dic181—41no—
170Dic181—41no—
171C2×C181—81no—
172C2×C181—81no—
173C2×C181#55011yescharacteristic
174C2×C181#57511yescharacteristic
175C2×C181#57711yescharacteristic
176D181—81no—
177C181#340111yescharacteristic, complemented, Sylow
178#3401—1811nomaximal, Sylow
179#559—1811no—
180#556—1811no—
181#497—1811no—
182#496—1811no—
183#495—1811no—
184#464—1811no—
185#464—1811no—
186#463—1811no—
187#463—1811no—
188#461—1811no—
189#460D18111yescharacteristic, complemented
190#446—1811no—
191#442—1811no—
192#441—1811no—
193C4×SD16—1811no—
194C2×Q16—1811no—
195C2×Q16—1811no—
196C2×SD16—1811no—
197C2×SD16—1811no—
198C2×SD16—1811no—
199C2×SD16C2×D18111yescharacteristic
200C2×SD16C2×D18111yescharacteristic
201C2×D8—1811no—
202C2×D8—1811no—
203C4⋊Q8—1811no—
204#127C2×D18111yescharacteristic
205#126—3621no—
206#126—3621no—
207#124—3621no—
208#124—1811no—
209#124—1811no—
210#124—1811no—
211#124—1811no—
212#123—3621no—
213#123—3621no—
214C22⋊Q8—3621no—
215C4×Q8—1811no—
216C4×Q8—1811no—
217C4×Q8—1811no—
218C4×Q8C2×D18111yescharacteristic
219C4×D4—3621no—
220C4×D4—1811no—
221C4×D4—1811no—
222C4×D4—1811no—
223#117—3621no—
224#106—1811no—
225C4⋊C8—1811no—
226C4⋊C8—1811no—
227C4⋊C8C2×D18111yescharacteristic
228#103—1811no—
229#103—1811no—
230#103—1811no—
231#102—1811no—
232#102C2×D18111yescharacteristic
233#102C2×D18111yescharacteristic
234#97—1811no—
235#97—1811no—
236C4×C8—1811no—
237C2×Q8—1811no—
238C2×Q8—1811no—
239C2×Q8—1811no—
240C2×Q8C22×D18111yescharacteristic
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—1811no—
244C2×D4C22×D18111yescharacteristic
245C2×D4C22×D18111yescharacteristic
246C22×C4—3621no—
247C22×C4—3621no—
248C22×C4—3621no—
249Q16—3622no—
250Q16—7241no—
251SD16—41no—
252SD16—41no—
253SD16—3622no—
254SD16—7241no—
255SD16—7241no—
256D8—3622no—
257D8—7241no—
258C2×C8—1811no—
259C2×C8—1811no—
260C2×C8C22×D18111yescharacteristic
261C2×C8C22×D18111yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—3621no—
264C4⋊C4—3621no—
265C4⋊C4—3621no—
266C4⋊C4—3621no—
267C4⋊C4—1811no—
268C4⋊C4—1811no—
269C4⋊C4—1811no—
270C4⋊C4—1811no—
271C4⋊C4C22×D18111yescharacteristic
272C22⋊C4—3621no—
273C22⋊C4—3621no—
274C22⋊C4—3621no—
275C22⋊C4—3621no—
276C22⋊C4—3621no—
277C22⋊C4—3621no—
278C22⋊C4—3621no—
279C22⋊C4—3621no—
280C22⋊C4—3621no—
281C22⋊C4—3621no—
282C22⋊C4—3621no—
283C42—21no—
284C42—3621no—
285C42—3621no—
286C42—1811no—
287C42—1811no—
288C42C22×D18111yescharacteristic
289C23—21no—
290C23—21no—
291C23—3621no—
292Q8—21no—
293Q8—21no—
294Q8—3621no—
295Q8—3621no—
296Q8—3621no—
297Q8—3621no—
298Q8—7241no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—42no—
304D4—81no—
305D4—3621no—
306D4—3621no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3621no—
310C2×C4—3621no—
311C2×C4—3621no—
312C2×C4—14481no—
313C2×C4—14481no—
314C2×C4—1811no—
315C2×C4—1811no—
316C2×C4—1811no—
317C2×C4—1811no—
318C2×C4—7242no—
319C2×C4D4×D18111yescharacteristic
320C2×C4D4×D18111yescharacteristic
321C2×C4C23×D18111yescharacteristic
322C8—21no—
323C8—21no—
324C8—3621no—
325C8—1812no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—7241no—
333C22—7241no—
334C22—7241no—
335C22C2×D4×D18111yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—3621no—
341C4—3621no—
342C4—3621no—
343C4—3621no—
344C4—7241no—
345C4—7241no—
346C4—7241no—
347C4#41×D18111yescharacteristic
348C4#1915386111yescharacteristic
349C2—81no—
350C2—81no—
351C2—14481no—
352C2#2604194711yescharacteristic
353C2#2604207511yescharacteristic
354C2#2604208511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#460×C181C2
socleC22×C181C22×D4
prime radical#26041666C2
hypercenter#460D181
hypocenterC181#3401
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #26042015C2semidirect
2. #26042001C2semidirect
3. #26041670C2semidirect
5. #26041660C2semidirect
6. #26041502C2semidirect
7. #26041500C2semidirect
8. #26041496C2semidirect
9. #26041493C2semidirect
10. #26041482C2semidirect
11. #26041408C2semidirect
12. #26041388C2semidirect
13. #26041384C2semidirect
14. #26041370C2semidirect
15. #460×C181C2semidirect
177. C181#3401semidirect
189. #460D181semidirect

Maximal subgroups

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

subgroupindex
#260420152
#260420012
#260416702
#260416662
#260416602
#260415022
#260415002
#260414962
#260414932
#260414822
#260414082
#260413882
#260413842
#260413702
#460×C1812
#3401181