Finite Group Databasebeta

← #37544988

Subgroups — #37544988

Order 23168.

308 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041906C211yesmaximal, characteristic
2#26041887C211yesmaximal, characteristic
3#26041687C211yesmaximal, characteristic, complemented
4#26041680C211yesmaximal, characteristic, complemented
5#26041646C211yesmaximal, characteristic, complemented
6#26041628C211yesmaximal, characteristic, complemented
7#26041622C211yesmaximal, characteristic
8#26041619C211yesmaximal, characteristic
9#26041616C211yesmaximal, characteristic
10#26041608C211yesmaximal, characteristic
11#26041587C211yesmaximal, characteristic, complemented
12#26041586C211yesmaximal, characteristic, complemented
13#26041561C211yesmaximal, characteristic, complemented
14#26041339C211yesmaximal, characteristic
15#477×C181C211yesmaximal, characteristic, complemented
16#19153815C2211yescharacteristic
17#19153814—21no—
18#19153814C2211yescharacteristic
19#19153811—21no—
20#19153810—21no—
21#19153808—21no—
22#19153808C2211yescharacteristic
23#19153807C2211yescharacteristic
24#19153807C2211yescharacteristic
25#19153806—21no—
26#19153806—21no—
27#19153796C2211yescharacteristic
28#19153794—21no—
29#19153792—21no—
30#19153792C2211yescharacteristic
31C4×Dic362C2211yescharacteristic
32#19153774C2211yescharacteristic
33#19153773C2211yescharacteristic
34C4⋊C4×D181C2211yescharacteristic
35#19153750C2211yescharacteristic
36#19153749C2211yescharacteristic
37#19153748C2211yescharacteristic
38#19153747C2211yescharacteristic
39#19153747C2211yescharacteristic
40#19153747C2211yescharacteristic
41C4⋊#13089678C2211yescharacteristic
42#19153741C2211yescharacteristic
43#19153741C2211yescharacteristic
44#19153740C2211yescharacteristic
45#19153740C2211yescharacteristic
46#19153739C2211yescharacteristic
47#19153739C2211yescharacteristic
48#19153728C2211yescharacteristic
49#19153728C2211yescharacteristic
50#19153726C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C4⋊Q8×C181C2211yescharacteristic
53C4×Q8×C181C2211yescharacteristic
54#107×C181C2211yescharacteristic
55#107×C181C2211yescharacteristic
56C4⋊C8×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#103×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362C2311yescharacteristic
61C2×C4×D181—21no—
62C2×C4×D181C2311yescharacteristic
63C2×#13089678C2311yescharacteristic
64C2×#13089678C2311yescharacteristic
65C4⋊Dic181—21no—
66C4⋊Dic181—21no—
67C4⋊Dic181—21no—
68C4⋊Dic181C2311yescharacteristic
69C4⋊Dic181C2311yescharacteristic
70C4⋊Dic181C2311yescharacteristic
71#15668428—21no—
72#15668428—21no—
73#15668428—21no—
74#15668428—21no—
75#15668428—21no—
76#15668428—21no—
77#15668428—21no—
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428D411yescharacteristic
82#15668428D411yescharacteristic
83#15668426—21no—
84#15668426—21no—
85#15668426—21no—
86#15668426D411yescharacteristic
87#15668426D411yescharacteristic
88C4×Dic181—21no—
89C4×Dic181—21no—
90C4×Dic181C2311yescharacteristic
91C2×Q8×C181—21no—
92C2×Q8×C181C2311yescharacteristic
93C2×C8×C181C2311yescharacteristic
94C2×C8×C181C2311yescharacteristic
95C4⋊C4×C181—21no—
96C4⋊C4×C181—21no—
97C4⋊C4×C181C2311yescharacteristic
98C4⋊C4×C181C2311yescharacteristic
99C4⋊C4×C181C2311yescharacteristic
100C42×C181—21no—
101C42×C181C2311yescharacteristic
102C22×D181C2×D411yescharacteristic
103Dic362—21no—
104Dic362—21no—
105Dic362—81no—
106C2×Dic181—21no—
107C2×Dic181—21no—
108C2×Dic181—21no—
109C2×Dic181—21no—
110C2×Dic181C2×D411yescharacteristic
111C2×Dic181C2×D411yescharacteristic
112C2×Dic181C2×D411yescharacteristic
113C4×D181—21no—
114C4×D181—21no—
115C4×D181—81no—
116#13089678—21no—
117#13089678—21no—
118Q8×C181—21no—
119Q8×C181—21no—
120Q8×C181—81no—
121C2×C4×C181—21no—
122C2×C4×C181—21no—
123C2×C4×C181—21no—
124C2×C4×C181—21no—
125C2×C4×C181C2×D411yescharacteristic
126C2×C4×C181C2×D411yescharacteristic
127C2×C4×C181C2411yescharacteristic
128C8×C181—21no—
129C8×C181—21no—
130C22×C181C22×D411yescharacteristic
131C2×D181—21no—
132C2×D181—21no—
133C2×D181—21no—
134C4×C181—21no—
135C4×C181—21no—
136C4×C181—41no—
137C4×C181—41no—
138C4×C181—41no—
139C4×C181—41no—
140C4×C1812−1+411yescharacteristic
141C4×C1812−1+411yescharacteristic
142Dic181—21no—
143Dic181—21no—
144Dic181—21no—
145Dic181—41no—
146Dic181—41no—
147Dic181—41no—
148Dic181—41no—
149C2×C181#53611yescharacteristic
150C2×C181#57711yescharacteristic
151C2×C181#57811yescharacteristic
152D181—41no—
153C181#324211yescharacteristic, complemented, Sylow
154#3242—1811nomaximal, Sylow
155#544—1811no—
156#541—1811no—
157#501—1811no—
158#499—1811no—
159#491—1811no—
160#487—1811no—
161#485—1811no—
162#485—1811no—
163#484—1811no—
164#483—1811no—
165#479—1811no—
166#479—1811no—
167#477D18111yescharacteristic, complemented
168#475—1811no—
169#433—1811no—
170C4⋊Q8—3621no—
171C4⋊Q8—1811no—
172C4⋊Q8—1811no—
173C4⋊Q8C2×D18111yescharacteristic
174#126—3621no—
175#126—3621no—
176#125—3621no—
177#125—3621no—
178#125—3621no—
179#125—1811no—
180#125—1811no—
181#125—1811no—
182C22⋊Q8—3621no—
183C22⋊Q8—3621no—
184C22⋊Q8—1811no—
185C22⋊Q8—1811no—
186C4×Q8—1811no—
187C4×Q8C2×D18111yescharacteristic
188#117—1811no—
189#117—1811no—
190C2×C4⋊C4—1811no—
191#107—1811no—
192#107—1811no—
193#107C2×D18111yescharacteristic
194#107C2×D18111yescharacteristic
195#106—1811no—
196#106—1811no—
197#106—1811no—
198#106—1811no—
199C4⋊C8—1811no—
200C4⋊C8C2×D18111yescharacteristic
201#103—1811no—
202#103—1811no—
203#103—1811no—
204#103—1811no—
205#103—1811no—
206#103—1811no—
207#103C2×D18111yescharacteristic
208#103C2×D18111yescharacteristic
209C22⋊C8—1811no—
210C22⋊C8—1811no—
211#97—1811no—
212C4×C8—1811no—
213C2×Q8—21no—
214C2×Q8—3621no—
215C2×Q8—1811no—
216C2×Q8C22×D18111yescharacteristic
217C22×C4—3621no—
218C22×C4—1811no—
219C2×C8—1811no—
220C2×C8—1811no—
221C2×C8C22×D18111yescharacteristic
222C2×C8C22×D18111yescharacteristic
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—3621no—
226C4⋊C4—3621no—
227C4⋊C4—3621no—
228C4⋊C4—3621no—
229C4⋊C4—3621no—
230C4⋊C4—3621no—
231C4⋊C4—3621no—
232C4⋊C4—3621no—
233C4⋊C4—3621no—
234C4⋊C4—3621no—
235C4⋊C4—3621no—
236C4⋊C4—3621no—
237C4⋊C4—3621no—
238C4⋊C4—1811no—
239C4⋊C4—1811no—
240C4⋊C4—1811no—
241C4⋊C4—1811no—
242C4⋊C4—1811no—
243C4⋊C4C22×D18111yescharacteristic
244C4⋊C4C22×D18111yescharacteristic
245C4⋊C4C22×D18111yescharacteristic
246C22⋊C4—3621no—
247C22⋊C4—3621no—
248C22⋊C4—3621no—
249C22⋊C4—1811no—
250C22⋊C4—1811no—
251C42—21no—
252C42—3621no—
253C42—3621no—
254C42—1811no—
255C42C22×D18111yescharacteristic
256C23—1811no—
257Q8—21no—
258Q8—21no—
259Q8—81no—
260Q8—3621no—
261Q8—3621no—
262Q8—14481no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—3621no—
268C2×C4—3621no—
269C2×C4—3621no—
270C2×C4—3621no—
271C2×C4—3621no—
272C2×C4—3621no—
273C2×C4—14481no—
274C2×C4—1811no—
275C2×C4—1811no—
276C2×C4—1811no—
277C2×C4D4×D18111yescharacteristic
278C2×C4D4×D18111yescharacteristic
279C2×C4C23×D18111yescharacteristic
280C8—21no—
281C8—21no—
282C8—3621no—
283C8—3621no—
284C22—3621no—
285C22—3621no—
286C22—3621no—
287C22C2×D4×D18111yescharacteristic
288C4—21no—
289C4—21no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—3621no—
295C4—3621no—
296C4—3621no—
297C4—7241no—
298C4—7241no—
299C4—7241no—
300C4—7241no—
301C4#1915386311yescharacteristic
302C4#1915386411yescharacteristic
303C2—7241no—
304C2#2604185211yescharacteristic
305C2#2604208911yescharacteristic
306C2#2604209211yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#477×C181C2
socleC22×C181C22×D4
prime radicalC22×D181C2×D4
hypercenter#477D181
hypocenterC181#3242
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
3. #26041687C2semidirect
4. #26041680C2semidirect
5. #26041646C2semidirect
6. #26041628C2semidirect
11. #26041587C2semidirect
12. #26041586C2semidirect
13. #26041561C2semidirect
15. #477×C181C2semidirect
153. C181#3242semidirect
167. #477D181semidirect

Maximal subgroups

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

subgroupindex
#260419062
#260418872
#260416872
#260416802
#260416462
#260416282
#260416222
#260416192
#260416162
#260416082
#260415872
#260415862
#260415612
#260413392
#477×C1812
#3242181