Finite Group Databasebeta

← #37546499

Subgroups — #37546499

Order 23168.

340 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042004C211yesmaximal, characteristic, complemented
2#26041985C211yesmaximal, characteristic
3#26041646C211yesmaximal, characteristic, complemented
4#26041628C211yesmaximal, characteristic, complemented
5#26041588C211yesmaximal, characteristic, complemented
6#26041583C211yesmaximal, characteristic, complemented
7#26041572C211yesmaximal, characteristic, complemented
8#26041483C211yesmaximal, characteristic
9#26041483C211yesmaximal, characteristic
10#26041481C211yesmaximal, characteristic
11#26041407C211yesmaximal, characteristic, complemented
12#26041382C211yesmaximal, characteristic
13#26041381C211yesmaximal, characteristic
14#26041364C211yesmaximal, characteristic
15#491×C181C211yesmaximal, characteristic, complemented
16C2×#15668438C2211yescharacteristic
17C2×#15668436C2211yescharacteristic
18C2×#15668436C2211yescharacteristic
19#19153815C2211yescharacteristic
20#19153814C2211yescharacteristic
21#19153814C2211yescharacteristic
22#19153807—21no—
23#19153807C2211yescharacteristic
24#19153806—21no—
25#19153806—21no—
26#19153796—21no—
27#19153794—21no—
28#19153794—21no—
29Q8×Dic181C2211yescharacteristic
30Q8×Dic181C2211yescharacteristic
31C4×Dic362C2211yescharacteristic
32C4×Dic362C2211yescharacteristic
33#19153781—21no—
34C4×D724C2211yescharacteristic
35#19153778C2211yescharacteristic
36#19153778C2211yescharacteristic
37C4⋊C4×D181—21no—
38#19153750C2211yescharacteristic
39#19153749C2211yescharacteristic
40#19153747C2211yescharacteristic
41C4⋊#13089678C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153745C2211yescharacteristic
44#19153741C2211yescharacteristic
45#19153740C2211yescharacteristic
46#19153740C2211yescharacteristic
47#19153739C2211yescharacteristic
48#19153739C2211yescharacteristic
49#19153738C2211yescharacteristic
50#19153726C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C4⋊Q8×C181C2211yescharacteristic
53#125×C181C2211yescharacteristic
54#103×C181C2211yescharacteristic
55#103×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#97×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362D411yescharacteristic
61C2×Dic362C2311yescharacteristic
62C2×D724C2311yescharacteristic
63C2×C4×D181—21no—
64C2×C4×D181—21no—
65C2×C4×D181—21no—
66#15668438—22no—
67#15668436—22no—
68#15668436—22no—
69C2×#13089678C2311yescharacteristic
70C2×#13089678C2311yescharacteristic
71C4⋊Dic181—21no—
72C4⋊Dic181—21no—
73C4⋊Dic181—21no—
74C4⋊Dic181D411yescharacteristic
75C4⋊Dic181C2311yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77#15668428—21no—
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83#15668428—21no—
84#15668428D411yescharacteristic
85#15668428D411yescharacteristic
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89C4×Dic181—21no—
90C4×Dic181—21no—
91C4×Dic181—21no—
92C4×Dic181C2311yescharacteristic
93C4×Dic181C2311yescharacteristic
94C2×Q8×C181C2311yescharacteristic
95C2×Q8×C181C2311yescharacteristic
96C2×C8×C181C2311yescharacteristic
97C2×C8×C181C2311yescharacteristic
98C4⋊C4×C181—21no—
99C4⋊C4×C181—21no—
100C4⋊C4×C181—21no—
101C4⋊C4×C181—21no—
102C4⋊C4×C181C2311yescharacteristic
103C4⋊C4×C181C2311yescharacteristic
104C42×C181C2311yescharacteristic
105C22×D181—21no—
106Dic362—22no—
107Dic362—21no—
108Dic362—21no—
109Dic362—81no—
110D724—21no—
111D724#4112yes—
112C2×Dic181—21no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181C2×D411yescharacteristic
116C2×Dic181C2×D411yescharacteristic
117C2×Dic181C2×D411yescharacteristic
118C2×Dic181C2×D411yescharacteristic
119C4×D181—42no—
120C4×D181—81no—
121C4×D181—81no—
122#13089678—21no—
123#13089678—21no—
124Q8×C181—21no—
125Q8×C181—21no—
126Q8×C181—21no—
127Q8×C181—21no—
128C2×C4×C181—21no—
129C2×C4×C181—21no—
130C2×C4×C181—21no—
131C2×C4×C181—21no—
132C2×C4×C181C2×D411yescharacteristic
133C2×C4×C181C2×D411yescharacteristic
134C2×C4×C181C2411yescharacteristic
135C8×C181—21no—
136C8×C181—21no—
137C22×C181C22×D411yescharacteristic
138C2×D181—22no—
139C2×D181—41no—
140C2×D181—41no—
141C4×C181—21no—
142C4×C181—21no—
143C4×C181—41no—
144C4×C181—41no—
145C4×C181—41no—
146C4×C181—41no—
147C4×C181C2×#4111yescharacteristic
148C4×C1812−1+411yescharacteristic
149Dic181—21no—
150Dic181—21no—
151Dic181—21no—
152Dic181—41no—
153Dic181—41no—
154Dic181—41no—
155Dic181#13712yes—
156C2×C181#54811yescharacteristic
157C2×C181C2×#13711yescharacteristic
158C2×C181#57711yescharacteristic
159D181—42no—
160C181#337211yescharacteristic, complemented, Sylow
161#3372—1811nomaximal, Sylow
162#557—1811no—
163#554—1811no—
164#491—1811no—
165#491D18111yescharacteristic, complemented
166#487—1811no—
167#479—1811no—
168#479—1811no—
169#477—1811no—
170#461—1811no—
171#461—1811no—
172#461—1811no—
173#446—1811no—
174#440—1811no—
175#440—1811no—
176C4×SD16—1811no—
177C2×SD16—1811no—
178C2×SD16—1811no—
179C2×SD16—1811no—
180C4⋊Q8—1811no—
181C4⋊Q8—1811no—
182C4⋊Q8—1811no—
183C4⋊Q8C2×D18111yescharacteristic
184#125—3621no—
185#125—3621no—
186#125—3621no—
187#125—1811no—
188#125C2×D18111yescharacteristic
189C22⋊Q8—3621no—
190C22⋊Q8—3621no—
191C22⋊Q8—3621no—
192C4×Q8—3621no—
193C4×Q8—1811no—
194C4×Q8—1811no—
195C4×Q8—1811no—
196C4×Q8—1811no—
197C4×D4—1811no—
198C4×D4—1811no—
199C4×D4—1811no—
200C2×C4⋊C4—3621no—
201#107—1811no—
202#107—1811no—
203#106—1811no—
204C4⋊C8—1811no—
205C4⋊C8—1811no—
206C4⋊C8—1811no—
207#103—1811no—
208#103—1811no—
209#103—1811no—
210#103—1811no—
211#103—1811no—
212#103C2×D18111yescharacteristic
213#103C2×D18111yescharacteristic
214#103C2×D18111yescharacteristic
215#103C2×D18111yescharacteristic
216#102—1811no—
217#97—1811no—
218#97C2×D18111yescharacteristic
219C4×C8—1811no—
220C2×Q8—3621no—
221C2×Q8—1811no—
222C2×Q8—1811no—
223C2×Q8C22×D18111yescharacteristic
224C2×Q8C22×D18111yescharacteristic
225C2×D4—1811no—
226C22×C4—3621no—
227C22×C4—3621no—
228C22×C4—3621no—
229SD16—3622no—
230SD16—3622no—
231SD16—3622no—
232C2×C8—1811no—
233C2×C8—1811no—
234C2×C8C22×D18111yescharacteristic
235C2×C8C22×D18111yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—3621no—
241C4⋊C4—3621no—
242C4⋊C4—3621no—
243C4⋊C4—3621no—
244C4⋊C4—3621no—
245C4⋊C4—3621no—
246C4⋊C4—3621no—
247C4⋊C4—3621no—
248C4⋊C4—3621no—
249C4⋊C4—3621no—
250C4⋊C4—1811no—
251C4⋊C4—1811no—
252C4⋊C4—1811no—
253C4⋊C4—1811no—
254C4⋊C4—1811no—
255C4⋊C4C22×D18111yescharacteristic
256C4⋊C4C22×D18111yescharacteristic
257C22⋊C4—3621no—
258C22⋊C4—3621no—
259C22⋊C4—3621no—
260C42—3621no—
261C42—3621no—
262C42—3621no—
263C42—1811no—
264C42—1811no—
265C42C22×D18111yescharacteristic
266C23—3621no—
267Q8—21no—
268Q8—21no—
269Q8—21no—
270Q8—21no—
271Q8—3622no—
272Q8—3621no—
273Q8—3621no—
274Q8—14481no—
275D4—3621no—
276D4—1812no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—3621no—
282C2×C4—3621no—
283C2×C4—3621no—
284C2×C4—14481no—
285C2×C4—14481no—
286C2×C4—1811no—
287C2×C4—1811no—
288C2×C4—1811no—
289C2×C4—1811no—
290C2×C4—7242no—
291C2×C4D4×D18111yescharacteristic
292C2×C4D4×D18111yescharacteristic
293C2×C4C23×D18111yescharacteristic
294C8—21no—
295C8—21no—
296C8—3621no—
297C8—3621no—
298C22—3622no—
299C22—7241no—
300C22—7241no—
301C22C2×D4×D18111yescharacteristic
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—3621no—
309C4—3621no—
310C4—3621no—
311C4—1812no—
312C4—7241no—
313C4—7241no—
314C4—7241no—
315C4#41×D18111yescharacteristic
316C4#1915386311yescharacteristic
317C2—7242no—
318C2#2604193711yescharacteristic
319C2#137×D18111yescharacteristic
320C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#491×C181C2
socleC22×C181C22×D4
prime radicalC2×D724C23
hypercenter#491D181
hypocenterC181#3372
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #26042004C2semidirect
3. #26041646C2semidirect
4. #26041628C2semidirect
5. #26041588C2semidirect
6. #26041583C2semidirect
7. #26041572C2semidirect
11. #26041407C2semidirect
15. #491×C181C2semidirect
160. C181#3372semidirect
165. #491D181semidirect

Maximal subgroups

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

subgroupindex
#260420042
#260419852
#260416462
#260416282
#260415882
#260415832
#260415722
#260414832
#260414812
#260414072
#260413822
#260413812
#260413642
#491×C1812
#3372181