Finite Group Databasebeta

← #37546484

Subgroups — #37546484

Order 23168.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042015C211yesmaximal, characteristic
2#26042006C211yesmaximal, characteristic, complemented
3#26041637C211yesmaximal, characteristic, complemented
4#26041632C211yesmaximal, characteristic, complemented
5#26041583C211yesmaximal, characteristic, complemented
6#26041581C211yesmaximal, characteristic, complemented
7#26041567C211yesmaximal, characteristic, complemented
8#26041500C211yesmaximal, characteristic, complemented
9#26041500C211yesmaximal, characteristic, complemented
10#26041476C211yesmaximal, characteristic
11#26041407C211yesmaximal, characteristic, complemented
12#26041387C211yesmaximal, characteristic, complemented
13#26041376C211yesmaximal, characteristic, complemented
14#26041370C211yesmaximal, characteristic, complemented
15#490×C181C211yesmaximal, characteristic
16C2×Dic724C2211yescharacteristic
17C2×#15668437C2211yescharacteristic
18C2×#15668437C2211yescharacteristic
19#19153815—21no—
20#19153815C2211yescharacteristic
21#19153807C2211yescharacteristic
22#19153806—21no—
23#19153806—21no—
24#19153803—21no—
25#19153803C2211yescharacteristic
26#19153803C2211yescharacteristic
27#19153797—21no—
28#19153797—21no—
29C4×Dic362C2211yescharacteristic
30C4×Dic362C2211yescharacteristic
31C4×Dic362C2211yescharacteristic
32#19153781—21no—
33#19153781C2211yescharacteristic
34#19153781C2211yescharacteristic
35D4×Dic181—21no—
36D4×Dic181C2211yescharacteristic
37D4×Dic181C2211yescharacteristic
38#19153749C2211yescharacteristic
39#19153748C2211yescharacteristic
40#19153747C2211yescharacteristic
41C4⋊#13089678C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153745C2211yescharacteristic
44#19153741C2211yescharacteristic
45#19153741C2211yescharacteristic
46#19153739C2211yescharacteristic
47#19153739C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153737C2211yescharacteristic
50#19153726C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52#127×C181C2211yescharacteristic
53#125×C181C2211yescharacteristic
54#102×C181C2211yescharacteristic
55#102×C181C2211yescharacteristic
56#102×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58#97×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362D411yescharacteristic
61C2×Dic362C2311yescharacteristic
62C2×Dic362C2311yescharacteristic
63C22×Dic181—21no—
64C22×Dic181—21no—
65Dic724—22no—
66#15668437—22no—
67#15668437—22no—
68C2×#13089678C2311yescharacteristic
69C2×#13089678C2311yescharacteristic
70C4⋊Dic181—21no—
71C4⋊Dic181D411yescharacteristic
72C4⋊Dic181C2311yescharacteristic
73C4⋊Dic181C2311yescharacteristic
74#15668428—21no—
75#15668428—21no—
76#15668428—21no—
77#15668428—21no—
78#15668428—21no—
79#15668428—21no—
80#15668428D411yescharacteristic
81#15668428D411yescharacteristic
82C22⋊Dic181—21no—
83C22⋊Dic181—21no—
84C22⋊Dic181—21no—
85C22⋊Dic181—21no—
86C22⋊Dic181—21no—
87C22⋊Dic181—21no—
88C4×Dic181—21no—
89C4×Dic181—21no—
90C4×Dic181—21no—
91C4×Dic181—21no—
92C4×Dic181C2311yescharacteristic
93C4×Dic181C2311yescharacteristic
94C2×D4×C181—21no—
95C2×D4×C181—21no—
96C2×D4×C181C2311yescharacteristic
97C2×D4×C181C2311yescharacteristic
98C2×C8×C181C2311yescharacteristic
99C2×C8×C181C2311yescharacteristic
100C4⋊C4×C181—21no—
101C4⋊C4×C181—21no—
102C4⋊C4×C181C2311yescharacteristic
103C4⋊C4×C181C2311yescharacteristic
104C42×C181C2311yescharacteristic
105Dic362—21no—
106Dic362—21no—
107Dic362—21no—
108Dic362—41no—
109Dic362—42no—
110Dic362#4112yes—
111C2×Dic181—21no—
112C2×Dic181—21no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181—81no—
116C2×Dic181—81no—
117C2×Dic181C2×D411yescharacteristic
118C2×Dic181C2×D411yescharacteristic
119C2×Dic181C2×D411yescharacteristic
120C2×Dic181C2×D411yescharacteristic
121#13089678—21no—
122#13089678—21no—
123C23×C181—21no—
124C23×C181—21no—
125D4×C181—21no—
126D4×C181—21no—
127D4×C181—21no—
128D4×C181—21no—
129D4×C181—42no—
130D4×C181—81no—
131C2×C4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181C2×D411yescharacteristic
134C2×C4×C181C2×D411yescharacteristic
135C2×C4×C181C2411yescharacteristic
136C8×C181—21no—
137C8×C181—21no—
138C22×C181—41no—
139C22×C181—41no—
140C22×C181—41no—
141C22×C181—41no—
142C22×C181—41no—
143C22×C181—41no—
144C22×C181C22×D411yescharacteristic
145C4×C181—21no—
146C4×C181—21no—
147C4×C181—41no—
148C4×C181—41no—
149C4×C181C2×#4111yescharacteristic
150C4×C1812−1+411yescharacteristic
151Dic181—22no—
152Dic181—21no—
153Dic181—21no—
154Dic181—21no—
155Dic181—21no—
156Dic181—41no—
157Dic181—41no—
158Dic181—41no—
159C2×C181—81no—
160C2×C181—81no—
161C2×C181#54811yescharacteristic
162C2×C181#57511yescharacteristic
163C2×C181#57711yescharacteristic
164C181#337111yescharacteristic, complemented, Sylow
165#3371—1811nomaximal, Sylow
166#559—1811no—
167#557—1811no—
168#490D18111yescharacteristic
169#489—1811no—
170#488—1811no—
171#479—1811no—
172#478—1811no—
173#476—1811no—
174#464—1811no—
175#464—1811no—
176#460—1811no—
177#446—1811no—
178#441—1811no—
179#440—1811no—
180C4×SD16—1811no—
181C2×Q16—1811no—
182C2×SD16—1811no—
183C2×SD16—1811no—
184C4⋊Q8—3621no—
185C4⋊Q8—1811no—
186#127C2×D18111yescharacteristic
187#125—3621no—
188#125—3621no—
189#125—1811no—
190#125C2×D18111yescharacteristic
191#124—3621no—
192#124—1811no—
193#124—1811no—
194#123—3621no—
195#123—3621no—
196C4×Q8—3621no—
197C4×Q8—1811no—
198C4×Q8—1811no—
199C4×Q8—1811no—
200C4×Q8—1811no—
201C4×Q8—1811no—
202C4×D4—3621no—
203C4×D4—1811no—
204C4×D4—1811no—
205#107—1811no—
206#106—1811no—
207#106—1811no—
208C4⋊C8—1811no—
209C4⋊C8—1811no—
210C4⋊C8—1811no—
211#103—1811no—
212#103—1811no—
213#103—1811no—
214#103—1811no—
215#102—1811no—
216#102—1811no—
217#102C2×D18111yescharacteristic
218#102C2×D18111yescharacteristic
219#102C2×D18111yescharacteristic
220#102C2×D18111yescharacteristic
221#97—1811no—
222#97C2×D18111yescharacteristic
223C4×C8—1811no—
224C2×Q8—3621no—
225C2×Q8—1811no—
226C2×Q8—1811no—
227C2×Q8—1811no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4C22×D18111yescharacteristic
231C2×D4C22×D18111yescharacteristic
232C22×C4—3621no—
233C22×C4—3621no—
234Q16—3622no—
235SD16—3622no—
236SD16—3622no—
237C2×C8—1811no—
238C2×C8—1811no—
239C2×C8C22×D18111yescharacteristic
240C2×C8C22×D18111yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—21no—
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—
260C22⋊C4—3621no—
261C22⋊C4—3621no—
262C22⋊C4—3621no—
263C42—3621no—
264C42—3621no—
265C42—3621no—
266C42—3621no—
267C42—1811no—
268C42—1811no—
269C42C22×D18111yescharacteristic
270C23—21no—
271C23—21no—
272Q8—3621no—
273Q8—3621no—
274Q8—3621no—
275Q8—1812no—
276Q8—7241no—
277Q8—7242no—
278D4—21no—
279D4—21no—
280D4—21no—
281D4—21no—
282D4—42no—
283D4—81no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—3621no—
287C2×C4—3621no—
288C2×C4—3621no—
289C2×C4—3621no—
290C2×C4—14481no—
291C2×C4—14481no—
292C2×C4—1811no—
293C2×C4—1811no—
294C2×C4—1811no—
295C2×C4—1811no—
296C2×C4D4×D18111yescharacteristic
297C2×C4D4×D18111yescharacteristic
298C2×C4C23×D18111yescharacteristic
299C8—21no—
300C8—21no—
301C8—3621no—
302C8—3621no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C2×D4×D18111yescharacteristic
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—3622no—
315C4—3621no—
316C4—3621no—
317C4—3621no—
318C4—3621no—
319C4—7241no—
320C4—7241no—
321C4—7241no—
322C4#41×D18111yescharacteristic
323C4#1915386311yescharacteristic
324C2—81no—
325C2—81no—
326C2#2604193711yescharacteristic
327C2#2604207511yescharacteristic
328C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#490×C181C2
socleC22×C181C22×D4
prime radical#127×C181C22
hypercenter#490D181
hypocenterC181#3371
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
2. #26042006C2semidirect
3. #26041637C2semidirect
4. #26041632C2semidirect
5. #26041583C2semidirect
6. #26041581C2semidirect
7. #26041567C2semidirect
8. #26041500C2semidirect
9. #26041500C2semidirect
11. #26041407C2semidirect
12. #26041387C2semidirect
13. #26041376C2semidirect
14. #26041370C2semidirect
164. C181#3371semidirect

Maximal subgroups

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

subgroupindex
#260420152
#260420062
#260416372
#260416322
#260415832
#260415812
#260415672
#260415002
#260414762
#260414072
#260413872
#260413762
#260413702
#490×C1812
#3371181