Finite Group Databasebeta

← #37546291

Subgroups — #37546291

Order 23168.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041971C211yesmaximal, characteristic
2#26041963C211yesmaximal, characteristic
3#26041679C211yesmaximal, characteristic, complemented
4#26041618C211yesmaximal, characteristic, complemented
5#26041617C211yesmaximal, characteristic, complemented
6#26041613C211yesmaximal, characteristic, complemented
7#26041612C211yesmaximal, characteristic, complemented
8#26041588C211yesmaximal, characteristic, complemented
9#26041584C211yesmaximal, characteristic, complemented
10#26041534C211yesmaximal, characteristic, complemented
11Q16×Dic181C211yesmaximal, characteristic, complemented
12#26041355C211yesmaximal, characteristic
13#26041321C211yesmaximal, characteristic, complemented
14#26041318C211yesmaximal, characteristic, complemented
15#470×C181C211yesmaximal, characteristic, complemented
16C2×#15668432C2211yescharacteristic
17#19153811—21no—
18#19153809—21no—
19#19153807—21no—
20#19153807C2211yescharacteristic
21#19153807C2211yescharacteristic
22#19153806—21no—
23#19153800—21no—
24#19153798—21no—
25#19153792C2211yescharacteristic
26#19153792C2211yescharacteristic
27Q8×Dic181C2211yescharacteristic
28Q8×Dic181C2211yescharacteristic
29C4×C22⋊D181C2211yescharacteristic
30#19153775C2211yescharacteristic
31#19153774—21no—
32#19153774C2211yescharacteristic
33#19153772—21no—
34C2×C4⋊Dic181C2211yescharacteristic
35C4⋊C4×D181C2211yescharacteristic
36#19153750C2211yescharacteristic
37#19153750C2211yescharacteristic
38#19153749C2211yescharacteristic
39#19153749C2211yescharacteristic
40#19153749C2211yescharacteristic
41#19153748C2211yescharacteristic
42#19153748C2211yescharacteristic
43#19153747C2211yescharacteristic
44#19153745C2211yescharacteristic
45#19153740C2211yescharacteristic
46#19153740C2211yescharacteristic
47#19153740C2211yescharacteristic
48#19153740C2211yescharacteristic
49C22⋊#13089678C2211yescharacteristic
50#19153728C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C2×Q16×C181C2211yescharacteristic
53C2×M16×C181C2211yescharacteristic
54C22⋊Q8×C181C2211yescharacteristic
55C22⋊Q8×C181C2211yescharacteristic
56#106×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#103×C181C2211yescharacteristic
59C2×C22⋊D181D411yescharacteristic
60C22×Dic181—21no—
61C2×C4×D181—21no—
62C2×C4×D181C2311yescharacteristic
63#15668432—22no—
64C2×#13089678C2311yescharacteristic
65C2×#13089678C2311yescharacteristic
66C4⋊Dic181—21no—
67C4⋊Dic181—21no—
68C4⋊Dic181—21no—
69C4⋊Dic181—21no—
70C4⋊Dic181—21no—
71C4⋊Dic181C2311yescharacteristic
72C4⋊Dic181C2311yescharacteristic
73C4⋊Dic181C2311yescharacteristic
74C4⋊Dic181C2311yescharacteristic
75#15668428—21no—
76#15668428—21no—
77#15668428—21no—
78#15668428—21no—
79#15668428—21no—
80#15668428D411yescharacteristic
81C22⋊Dic181—21no—
82C22⋊Dic181D411yescharacteristic
83#15668426—21no—
84#15668426—21no—
85#15668426—21no—
86#15668426D411yescharacteristic
87C4×Dic181—21no—
88C4×Dic181—21no—
89C4×Dic181—21no—
90C4×Dic181—21no—
91C4×Dic181C2311yescharacteristic
92C2×Q8×C181C2311yescharacteristic
93C2×Q8×C181C2311yescharacteristic
94C22×C4×C181C2311yescharacteristic
95Q16×C181—22no—
96M16×C181—22no—
97C2×C8×C181C2311yescharacteristic
98C2×C8×C181C2311yescharacteristic
99C4⋊C4×C181—21no—
100C4⋊C4×C181—21no—
101C4⋊C4×C181C2311yescharacteristic
102C4⋊C4×C181C2311yescharacteristic
103C22⋊C4×C181—21no—
104C22⋊C4×C181—21no—
105C22×D181C2×D411yescharacteristic
106C22⋊D181—41no—
107C2×Dic181—21no—
108C2×Dic181—21no—
109C2×Dic181—21no—
110C2×Dic181—21no—
111C2×Dic181—42no—
112C2×Dic181C2×D411yescharacteristic
113C2×Dic181C2×D411yescharacteristic
114C2×Dic181C2×D411yescharacteristic
115C4×D181—21no—
116C4×D181—21no—
117C4×D181—42no—
118#13089678—21no—
119#13089678—21no—
120C23×C181C2×D411yescharacteristic
121Q8×C181—21no—
122Q8×C181—21no—
123Q8×C181—21no—
124Q8×C181—21no—
125C2×C4×C181—21no—
126C2×C4×C181—21no—
127C2×C4×C181—21no—
128C2×C4×C181—21no—
129C2×C4×C181—21no—
130C2×C4×C181—21no—
131C2×C4×C181C2×D411yescharacteristic
132C2×C4×C181C2411yescharacteristic
133C8×C181—21no—
134C8×C181#4112yes—
135C22×C181—21no—
136C22×C181—21no—
137C22×C181—21no—
138C22×C181C22×D411yescharacteristic
139C2×D181—21no—
140C2×D181—21no—
141C2×D181—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—41no—
156C2×C181—41no—
157C2×C181#54711yescharacteristic
158C2×C181#57511yescharacteristic
159C2×C181#57811yescharacteristic
160D181—41no—
161C181#335211yescharacteristic, complemented, Sylow
162#3352—1811nomaximal, Sylow
163#552—1811no—
164#552—1811no—
165#499—1811no—
166#485—1811no—
167#485—1811no—
168#484—1811no—
169#484—1811no—
170#479—1811no—
171#479—1811no—
172#470—1811no—
173#470D18111yescharacteristic, complemented
174C4×Q16—1811no—
175#436—1811no—
176#428—1811no—
177#428—1811no—
178C2×Q16C2×D18111yescharacteristic
179C2×M16—1811no—
180C2×M16C2×D18111yescharacteristic
181#126—3621no—
182#126—3621no—
183#125—3621no—
184#125—3621no—
185#125—1811no—
186#125—1811no—
187#123—3621no—
188#123—3621no—
189C22⋊Q8—1811no—
190C22⋊Q8—1811no—
191C22⋊Q8C2×D18111yescharacteristic
192C22⋊Q8C2×D18111yescharacteristic
193C4×Q8—1811no—
194C4×Q8—1811no—
195C4×D4—1811no—
196#117—3621no—
197#117—3621no—
198#117—1811no—
199#117—1811no—
200C2×C4⋊C4—1811no—
201C2×C4⋊C4—1811no—
202#107—1811no—
203#107—1811no—
204#107—1811no—
205#107—1811no—
206#107—1811no—
207#106—1811no—
208#106—1811no—
209#106—1811no—
210#106C2×D18111yescharacteristic
211C4⋊C8—1811no—
212#103—1811no—
213#103—1811no—
214#103—1811no—
215#103—1811no—
216#103C2×D18111yescharacteristic
217#103C2×D18111yescharacteristic
218C22⋊C8—1811no—
219C22⋊C8—1811no—
220C4×C8—1811no—
221C2×Q8C22×D18111yescharacteristic
222C2×Q8C22×D18111yescharacteristic
223C2×D4—1811no—
224C22×C4—3621no—
225C22×C4—3621no—
226C22×C4—1811no—
227C22×C4C22×D18111yescharacteristic
228Q16—22no—
229M16—22no—
230M16—3622no—
231C2×C8—1811no—
232C2×C8—1811no—
233C2×C8C22×D18111yescharacteristic
234C2×C8C22×D18111yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—3621no—
238C4⋊C4—3621no—
239C4⋊C4—3621no—
240C4⋊C4—3621no—
241C4⋊C4—3621no—
242C4⋊C4—3621no—
243C4⋊C4—3621no—
244C4⋊C4—3621no—
245C4⋊C4—3621no—
246C4⋊C4—3621no—
247C4⋊C4—1811no—
248C4⋊C4—1811no—
249C4⋊C4—1811no—
250C4⋊C4—1811no—
251C4⋊C4—1811no—
252C4⋊C4C22×D18111yescharacteristic
253C4⋊C4C22×D18111yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—3621no—
257C22⋊C4—3621no—
258C22⋊C4—3621no—
259C22⋊C4—3621no—
260C22⋊C4—1811no—
261C22⋊C4—1811no—
262C42—3621no—
263C42—3621no—
264C42—3621no—
265C42—3621no—
266C42—1811no—
267C23—1811no—
268C23D4×D18111yescharacteristic
269Q8—21no—
270Q8—21no—
271Q8—21no—
272Q8—21no—
273D4—7241no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—3621no—
281C2×C4—3621no—
282C2×C4—3621no—
283C2×C4—3621no—
284C2×C4—3621no—
285C2×C4—3621no—
286C2×C4—1811no—
287C2×C4—1811no—
288C2×C4—1811no—
289C2×C4—7242no—
290C2×C4—7242no—
291C2×C4D4×D18111yescharacteristic
292C2×C4C23×D18111yescharacteristic
293C8—21no—
294C8—3621no—
295C8—3621no—
296C8#1566844812yes—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—3621no—
301C22—3621no—
302C22—3621no—
303C22C2×D4×D18111yescharacteristic
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—3621no—
310C4—3621no—
311C4—3621no—
312C4—7241no—
313C4—7241no—
314C4—7241no—
315C4—7241no—
316C4C2×#1566844811yescharacteristic
317C4#1915386311yescharacteristic
318C2—41no—
319C2—7241no—
320C2#2604192911yescharacteristic
321C2#2604207811yescharacteristic
322C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#470×C181C2
socleC22×C181C22×D4
prime radicalC2×C22⋊D181D4
hypercenter#470D181
hypocenterC181#3352
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
3. #26041679C2semidirect
4. #26041618C2semidirect
5. #26041617C2semidirect
6. #26041613C2semidirect
7. #26041612C2semidirect
8. #26041588C2semidirect
9. #26041584C2semidirect
10. #26041534C2semidirect
11. Q16×Dic181C2semidirect
13. #26041321C2semidirect
14. #26041318C2semidirect
15. #470×C181C2semidirect
161. C181#3352semidirect
173. #470D181semidirect

Maximal subgroups

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

subgroupindex
#260419712
#260419632
#260416792
#260416182
#260416172
#260416132
#260416122
#260415882
#260415842
#260415342
Q16×Dic1812
#260413552
#260413212
#260413182
#470×C1812
#3352181