Finite Group Databasebeta

← #37546558

Subgroups — #37546558

Order 23168.

332 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042006C211yesmaximal, characteristic, complemented
2#26042003C211yesmaximal, characteristic
3#26041640C211yesmaximal, characteristic
4#26041635C211yesmaximal, characteristic
5#26041588C211yesmaximal, characteristic, complemented
6#26041576C211yesmaximal, characteristic
7#26041572C211yesmaximal, characteristic, complemented
8#26041504C211yesmaximal, characteristic
9#26041504C211yesmaximal, characteristic
10#26041486C211yesmaximal, characteristic, complemented
11#26041407C211yesmaximal, characteristic, complemented
12#26041383C211yesmaximal, characteristic, complemented
13#26041381C211yesmaximal, characteristic
14#26041371C211yesmaximal, characteristic, complemented
15#491×C181C211yesmaximal, characteristic, complemented
16C2×#15668439C2211yescharacteristic
17C2×#15668439C2211yescharacteristic
18C2×#15668438C2211yescharacteristic
19#19153815—21no—
20#19153815C2211yescharacteristic
21#19153811—21no—
22#19153811—21no—
23#19153807C2211yescharacteristic
24#19153806—21no—
25#19153806—21no—
26#19153805C2211yescharacteristic
27#19153805C2211yescharacteristic
28#19153796—21no—
29Q8×Dic181C2211yescharacteristic
30Q8×Dic181C2211yescharacteristic
31C4×Dic362C2211yescharacteristic
32C4×Dic362C2211yescharacteristic
33#19153781—21no—
34#19153781C2211yescharacteristic
35#19153781C2211yescharacteristic
36C4×D724C2211yescharacteristic
37#19153774—21no—
38#19153750C2211yescharacteristic
39#19153749C2211yescharacteristic
40#19153747C2211yescharacteristic
41C4⋊#13089678C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153745C2211yescharacteristic
44#19153741C2211yescharacteristic
45#19153740C2211yescharacteristic
46#19153740C2211yescharacteristic
47#19153738C2211yescharacteristic
48#19153736C2211yescharacteristic
49#19153736C2211yescharacteristic
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—
64#15668439—22no—
65#15668439—22no—
66#15668438—22no—
67C2×#13089678C2311yescharacteristic
68C2×#13089678C2311yescharacteristic
69C4⋊Dic181—21no—
70C4⋊Dic181—21no—
71C4⋊Dic181—21no—
72C4⋊Dic181D411yescharacteristic
73C4⋊Dic181C2311yescharacteristic
74C4⋊Dic181C2311yescharacteristic
75#15668428—21no—
76#15668428—21no—
77#15668428—21no—
78#15668428—21no—
79#15668428—21no—
80#15668428D411yescharacteristic
81#15668428D411yescharacteristic
82#15668426—21no—
83#15668426—21no—
84#15668426—21no—
85#15668426—21no—
86#15668426—21no—
87C4×Dic181—21no—
88C4×Dic181—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—21no—
107Dic362—41no—
108Dic362—42no—
109Dic362#4112yes—
110D724—21no—
111D724—21no—
112C2×Dic181—21no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181C2×D411yescharacteristic
116C2×Dic181C2×D411yescharacteristic
117C2×Dic181C2×D411yescharacteristic
118C2×Dic181C2×D411yescharacteristic
119C4×D181—42no—
120#13089678—21no—
121#13089678—21no—
122Q8×C181—21no—
123Q8×C181—21no—
124Q8×C181—21no—
125Q8×C181—21no—
126C2×C4×C181—21no—
127C2×C4×C181—21no—
128C2×C4×C181—21no—
129C2×C4×C181—21no—
130C2×C4×C181C2×D411yescharacteristic
131C2×C4×C181C2×D411yescharacteristic
132C2×C4×C181C2411yescharacteristic
133C8×C181—21no—
134C8×C181—21no—
135C22×C181C22×D411yescharacteristic
136C2×D181—41no—
137C2×D181—41no—
138C2×D181—41no—
139C4×C181—21no—
140C4×C181—21no—
141C4×C181—41no—
142C4×C181—41no—
143C4×C181—41no—
144C4×C181—41no—
145C4×C181C2×#4111yescharacteristic
146C4×C1812−1+411yescharacteristic
147Dic181—22no—
148Dic181—21no—
149Dic181—21no—
150Dic181—21no—
151Dic181—21no—
152Dic181—41no—
153Dic181—41no—
154C2×C181#54811yescharacteristic
155C2×C181#57511yescharacteristic
156C2×C181#57811yescharacteristic
157D181—81no—
158C181#337711yescharacteristic, complemented, Sylow
159#3377—1811nomaximal, Sylow
160#557—1811no—
161#556—1811no—
162#491D18111yescharacteristic, complemented
163#489—1811no—
164#488—1811no—
165#479—1811no—
166#478—1811no—
167#477—1811no—
168#464—1811no—
169#464—1811no—
170#462—1811no—
171#446—1811no—
172#441—1811no—
173#440—1811no—
174C4×Q16—1811no—
175C2×Q16—1811no—
176C2×Q16—1811no—
177C2×SD16—1811no—
178C4⋊Q8—3621no—
179C4⋊Q8—1811no—
180C4⋊Q8C2×D18111yescharacteristic
181#126—3621no—
182#126—3621no—
183#125—3621no—
184#125—3621no—
185#125—1811no—
186#125C2×D18111yescharacteristic
187#124—1811no—
188#124—1811no—
189C22⋊Q8—3621no—
190C4×Q8—3621no—
191C4×Q8—1811no—
192C4×Q8—1811no—
193C4×Q8—1811no—
194C4×Q8—1811no—
195C4×Q8—1811no—
196C4×Q8—1811no—
197C4×D4—1811no—
198#117—3621no—
199#107—1811no—
200#107—1811no—
201#106—1811no—
202C4⋊C8—1811no—
203C4⋊C8—1811no—
204C4⋊C8—1811no—
205#103—1811no—
206#103—1811no—
207#103—1811no—
208#103C2×D18111yescharacteristic
209#103C2×D18111yescharacteristic
210#103C2×D18111yescharacteristic
211#103C2×D18111yescharacteristic
212#102—1811no—
213#102—1811no—
214#102—1811no—
215#97—1811no—
216#97C2×D18111yescharacteristic
217C4×C8—1811no—
218C2×Q8—3621no—
219C2×Q8—1811no—
220C2×Q8—1811no—
221C2×Q8C22×D18111yescharacteristic
222C2×Q8C22×D18111yescharacteristic
223C2×D4—1811no—
224C22×C4—3621no—
225Q16—3622no—
226Q16—3622no—
227SD16—3622no—
228C2×C8—1811no—
229C2×C8—1811no—
230C2×C8C22×D18111yescharacteristic
231C2×C8C22×D18111yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—3621no—
237C4⋊C4—3621no—
238C4⋊C4—3621no—
239C4⋊C4—3621no—
240C4⋊C4—3621no—
241C4⋊C4—3621no—
242C4⋊C4—3621no—
243C4⋊C4—3621no—
244C4⋊C4—1811no—
245C4⋊C4—1811no—
246C4⋊C4—1811no—
247C4⋊C4—1811no—
248C4⋊C4—1811no—
249C4⋊C4C22×D18111yescharacteristic
250C4⋊C4C22×D18111yescharacteristic
251C22⋊C4—3621no—
252C22⋊C4—3621no—
253C22⋊C4—3621no—
254C22⋊C4—3621no—
255C22⋊C4—3621no—
256C42—3621no—
257C42—3621no—
258C42—3621no—
259C42—3621no—
260C42—3621no—
261C42—1811no—
262C42—1811no—
263C42C22×D18111yescharacteristic
264C23—3621no—
265Q8—21no—
266Q8—21no—
267Q8—21no—
268Q8—21no—
269Q8—3621no—
270Q8—1812no—
271Q8—7241no—
272Q8—7242no—
273D4—3621no—
274D4—3621no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—3621no—
280C2×C4—3621no—
281C2×C4—3621no—
282C2×C4—1811no—
283C2×C4—1811no—
284C2×C4—1811no—
285C2×C4—1811no—
286C2×C4—7242no—
287C2×C4D4×D18111yescharacteristic
288C2×C4D4×D18111yescharacteristic
289C2×C4C23×D18111yescharacteristic
290C8—21no—
291C8—21no—
292C8—3621no—
293C8—3621no—
294C22—7241no—
295C22—7241no—
296C22—7241no—
297C22C2×D4×D18111yescharacteristic
298C4—21no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—3622no—
305C4—3621no—
306C4—3621no—
307C4—3621no—
308C4—3621no—
309C4—7241no—
310C4—7241no—
311C4#41×D18111yescharacteristic
312C4#1915386311yescharacteristic
313C2—14481no—
314C2#2604193711yescharacteristic
315C2#2604207811yescharacteristic
316C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#491×C181C2
socleC22×C181C22×D4
prime radicalC2×D724C23
hypercenter#491D181
hypocenterC181#3377
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. #26042006C2semidirect
5. #26041588C2semidirect
7. #26041572C2semidirect
10. #26041486C2semidirect
11. #26041407C2semidirect
12. #26041383C2semidirect
14. #26041371C2semidirect
15. #491×C181C2semidirect
158. C181#3377semidirect
162. #491D181semidirect

Maximal subgroups

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

subgroupindex
#260420062
#260420032
#260416402
#260416352
#260415882
#260415762
#260415722
#260415042
#260414862
#260414072
#260413832
#260413812
#260413712
#491×C1812
#3377181