Finite Group Databasebeta

← #37545003

Subgroups — #37545003

Order 23168.

320 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041909C211yesmaximal, characteristic, complemented
2#26041894C211yesmaximal, characteristic
3#26041686C211yesmaximal, characteristic, complemented
4#26041683C211yesmaximal, characteristic, complemented
5#26041637C211yesmaximal, characteristic, complemented
6#26041632C211yesmaximal, characteristic, complemented
7#26041623C211yesmaximal, characteristic, complemented
8#26041616C211yesmaximal, characteristic, complemented
9#26041601C211yesmaximal, characteristic
10#26041591C211yesmaximal, characteristic
11#26041583C211yesmaximal, characteristic, complemented
12#26041575C211yesmaximal, characteristic, complemented
13#26041559C211yesmaximal, characteristic, complemented
14#26041339C211yesmaximal, characteristic, complemented
15#478×C181C211yesmaximal, characteristic, complemented
16#19153816C2211yescharacteristic
17#19153815—21no—
18#19153815C2211yescharacteristic
19#19153815C2211yescharacteristic
20#19153809—21no—
21#19153807C2211yescharacteristic
22#19153806—21no—
23#19153803C2211yescharacteristic
24#19153802—21no—
25#19153799—21no—
26#19153796—21no—
27#19153796C2211yescharacteristic
28#19153795—21no—
29#19153794—21no—
30#19153787C2211yescharacteristic
31C4×Dic362C2211yescharacteristic
32#19153774C2211yescharacteristic
33#19153773C2211yescharacteristic
34C4⋊C4×D181C2211yescharacteristic
35#19153750C2211yescharacteristic
36#19153750C2211yescharacteristic
37#19153749C2211yescharacteristic
38#19153749C2211yescharacteristic
39#19153749C2211yescharacteristic
40#19153747C2211yescharacteristic
41C4⋊#13089678C2211yescharacteristic
42#19153741C2211yescharacteristic
43#19153741C2211yescharacteristic
44#19153739C2211yescharacteristic
45#19153739C2211yescharacteristic
46#19153737C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153728C2211yescharacteristic
49#19153728C2211yescharacteristic
50#19153726C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52#125×C181C2211yescharacteristic
53C4×D4×C181C2211yescharacteristic
54#107×C181C2211yescharacteristic
55#106×C181C2211yescharacteristic
56C4⋊C8×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362—21no—
61C2×Dic362—21no—
62C2×Dic362C2311yescharacteristic
63C2×C22⋊D181—21no—
64C2×C4×D181—21no—
65C2×C4×D181C2311yescharacteristic
66C2×#13089678C2311yescharacteristic
67C2×#13089678C2311yescharacteristic
68C4⋊Dic181—21no—
69C4⋊Dic181—21no—
70C4⋊Dic181—21no—
71C4⋊Dic181C2311yescharacteristic
72C4⋊Dic181C2311yescharacteristic
73C4⋊Dic181C2311yescharacteristic
74#15668428—21no—
75#15668428—21no—
76#15668428—21no—
77#15668428—21no—
78#15668428—21no—
79#15668428D411yescharacteristic
80#15668428D411yescharacteristic
81C22⋊Dic181—21no—
82C22⋊Dic181—21no—
83C22⋊Dic181—21no—
84#15668426—21no—
85#15668426—21no—
86#15668426D411yescharacteristic
87#15668426D411yescharacteristic
88C4×Dic181—21no—
89C4×Dic181—21no—
90C4×Dic181C2311yescharacteristic
91C2×D4×C181C2311yescharacteristic
92C22×C4×C181—21no—
93C2×C8×C181C2311yescharacteristic
94C2×C8×C181C2311yescharacteristic
95C4⋊C4×C181—21no—
96C4⋊C4×C181—21no—
97C4⋊C4×C181C2311yescharacteristic
98C4⋊C4×C181C2311yescharacteristic
99C4⋊C4×C181C2311yescharacteristic
100C22⋊C4×C181—21no—
101C42×C181C2311yescharacteristic
102C22×D181C2×D411yescharacteristic
103Dic362—21no—
104Dic362—21no—
105Dic362—81no—
106Dic362—81no—
107Dic362—81no—
108C22⋊D181—81no—
109C2×Dic181—21no—
110C2×Dic181—21no—
111C2×Dic181—21no—
112C2×Dic181—21no—
113C2×Dic181C2×D411yescharacteristic
114C2×Dic181C2×D411yescharacteristic
115C2×Dic181C2×D411yescharacteristic
116C4×D181—21no—
117C4×D181—21no—
118C4×D181—81no—
119#13089678—21no—
120#13089678—21no—
121C23×C181—21no—
122D4×C181—21no—
123D4×C181—21no—
124C2×C4×C181—21no—
125C2×C4×C181—21no—
126C2×C4×C181—21no—
127C2×C4×C181—81no—
128C2×C4×C181C2×D411yescharacteristic
129C2×C4×C181C2×D411yescharacteristic
130C2×C4×C181C2411yescharacteristic
131C8×C181—21no—
132C8×C181—21no—
133C22×C181—41no—
134C22×C181—41no—
135C22×C181—41no—
136C22×C181C22×D411yescharacteristic
137C2×D181—21no—
138C2×D181—21no—
139C2×D181—21no—
140C4×C181—21no—
141C4×C181—21no—
142C4×C181—41no—
143C4×C181—41no—
144C4×C181—41no—
145C4×C1812−1+411yescharacteristic
146C4×C1812−1+411yescharacteristic
147Dic181—21no—
148Dic181—21no—
149Dic181—21no—
150Dic181—41no—
151Dic181—41no—
152Dic181—41no—
153Dic181—41no—
154C2×C181—81no—
155C2×C181#53611yescharacteristic
156C2×C181#57611yescharacteristic
157C2×C181#57811yescharacteristic
158D181—41no—
159C181#324311yescharacteristic, complemented, Sylow
160#3243—1811nomaximal, Sylow
161#544—1811no—
162#542—1811no—
163#501—1811no—
164#500—1811no—
165#489—1811no—
166#488—1811no—
167#485—1811no—
168#484—1811no—
169#482—1811no—
170#480—1811no—
171#479—1811no—
172#478D18111yescharacteristic, complemented
173#477—1811no—
174#474—1811no—
175#433—1811no—
176C4⋊Q8—3621no—
177C4⋊Q8—1811no—
178C4⋊Q8—1811no—
179C4⋊Q8—1811no—
180#126—3621no—
181#125—3621no—
182#125—1811no—
183#125C2×D18111yescharacteristic
184#124—3621no—
185#124—1811no—
186#123—3621no—
187C22⋊Q8—3621no—
188C22⋊Q8—3621no—
189C22⋊Q8—3621no—
190C22⋊Q8—1811no—
191#121—1811no—
192C4×Q8—1811no—
193C4×D4C2×D18111yescharacteristic
194#117—1811no—
195#117—1811no—
196C2×C4⋊C4—1811no—
197#107—1811no—
198#107—1811no—
199#107—1811no—
200#107—1811no—
201#107—1811no—
202#107C2×D18111yescharacteristic
203#106—1811no—
204#106C2×D18111yescharacteristic
205C4⋊C8—1811no—
206C4⋊C8C2×D18111yescharacteristic
207#103—1811no—
208#103—1811no—
209#103—1811no—
210#103—1811no—
211#102—1811no—
212#102—1811no—
213#102C2×D18111yescharacteristic
214#102C2×D18111yescharacteristic
215C22⋊C8—1811no—
216C22⋊C8—1811no—
217#97—1811no—
218C4×C8—1811no—
219C2×Q8—3621no—
220C2×Q8—3621no—
221C2×Q8—3621no—
222C2×Q8—1811no—
223C2×D4—3621no—
224C2×D4C22×D18111yescharacteristic
225C22×C4—21no—
226C22×C4—3621no—
227C22×C4—1811no—
228C2×C8—1811no—
229C2×C8—1811no—
230C2×C8C22×D18111yescharacteristic
231C2×C8C22×D18111yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—3621no—
235C4⋊C4—3621no—
236C4⋊C4—3621no—
237C4⋊C4—3621no—
238C4⋊C4—3621no—
239C4⋊C4—3621no—
240C4⋊C4—3621no—
241C4⋊C4—3621no—
242C4⋊C4—1811no—
243C4⋊C4—1811no—
244C4⋊C4—1811no—
245C4⋊C4—1811no—
246C4⋊C4—1811no—
247C4⋊C4C22×D18111yescharacteristic
248C4⋊C4C22×D18111yescharacteristic
249C4⋊C4C22×D18111yescharacteristic
250C22⋊C4—21no—
251C22⋊C4—3621no—
252C22⋊C4—3621no—
253C22⋊C4—3621no—
254C22⋊C4—3621no—
255C22⋊C4—3621no—
256C22⋊C4—1811no—
257C22⋊C4—1811no—
258C42—3621no—
259C42—3621no—
260C42—1811no—
261C42C22×D18111yescharacteristic
262C23—21no—
263C23—1811no—
264Q8—3621no—
265Q8—3621no—
266Q8—14481no—
267Q8—14481no—
268Q8—14481no—
269D4—21no—
270D4—21no—
271D4—14481no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—81no—
276C2×C4—3621no—
277C2×C4—3621no—
278C2×C4—3621no—
279C2×C4—3621no—
280C2×C4—3621no—
281C2×C4—3621no—
282C2×C4—14481no—
283C2×C4—1811no—
284C2×C4—1811no—
285C2×C4—1811no—
286C2×C4D4×D18111yescharacteristic
287C2×C4D4×D18111yescharacteristic
288C2×C4C23×D18111yescharacteristic
289C8—21no—
290C8—21no—
291C8—3621no—
292C8—3621no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—3621no—
297C22—3621no—
298C22—3621no—
299C22C2×D4×D18111yescharacteristic
300C4—21no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—3621no—
306C4—3621no—
307C4—3621no—
308C4—7241no—
309C4—7241no—
310C4—7241no—
311C4—7241no—
312C4#1915386311yescharacteristic
313C4#1915386411yescharacteristic
314C2—81no—
315C2—7241no—
316C2#2604185211yescharacteristic
317C2#2604208311yescharacteristic
318C2#2604209311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#478×C181C2
socleC22×C181C22×D4
prime radical#19153787C22
hypercenter#478D181
hypocenterC181#3243
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #26041909C2semidirect
3. #26041686C2semidirect
4. #26041683C2semidirect
5. #26041637C2semidirect
6. #26041632C2semidirect
7. #26041623C2semidirect
8. #26041616C2semidirect
11. #26041583C2semidirect
12. #26041575C2semidirect
13. #26041559C2semidirect
14. #26041339C2semidirect
15. #478×C181C2semidirect
159. C181#3243semidirect
172. #478D181semidirect

Maximal subgroups

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

subgroupindex
#260419092
#260418942
#260416862
#260416832
#260416372
#260416322
#260416232
#260416162
#260416012
#260415912
#260415832
#260415752
#260415592
#260413392
#478×C1812
#3243181