Finite Group Databasebeta

← #36366131

Subgroups — #36366131

Order 22144.

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#26012183C211yesmaximal, characteristic
2#26012180C211yesmaximal, characteristic
3#26011892C211yesmaximal, characteristic, complemented
4#26011834C211yesmaximal, characteristic, complemented
5#26011827C211yesmaximal, characteristic, complemented
6#26011825C211yesmaximal, characteristic, complemented
7#26011801C211yesmaximal, characteristic, complemented
8#26011800C211yesmaximal, characteristic, complemented
9#26011748C211yesmaximal, characteristic, complemented
10#26011747C211yesmaximal, characteristic, complemented
11#26011583C211yesmaximal, characteristic, complemented
12#26011566C211yesmaximal, characteristic
13#26011534C211yesmaximal, characteristic, complemented
14#26011530C211yesmaximal, characteristic, complemented
15#485×C173C211yesmaximal, characteristic, complemented
16C2×#14583816C2211yescharacteristic
17C2×#14583810C2211yescharacteristic
18C2×#14583809C2211yescharacteristic
19#19149674—21no—
20#19149672—21no—
21#19149671—21no—
22#19149670—21no—
23#19149670C2211yescharacteristic
24#19149670C2211yescharacteristic
25#19149663—21no—
26#19149661—21no—
27#19149659C2211yescharacteristic
28#19149658C2211yescharacteristic
29#19149655C2211yescharacteristic
30Q8×Dic173C2211yescharacteristic
31#19149644C2211yescharacteristic
32C4×C22⋊D173C2211yescharacteristic
33#19149637C2211yescharacteristic
34#19149636—21no—
35#19149635—21no—
36C2×C4⋊Dic173C2211yescharacteristic
37C4⋊C4×D173C2211yescharacteristic
38#19149613C2211yescharacteristic
39#19149613C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149612C2211yescharacteristic
42#19149610C2211yescharacteristic
43#19149610C2211yescharacteristic
44#19149610C2211yescharacteristic
45#19149608C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149604C2211yescharacteristic
48#19149603C2211yescharacteristic
49#19149602C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C22⋊Q8×C173C2211yescharacteristic
53#117×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59C2×Dic346C2311yescharacteristic
60C2×C22⋊D173D411yescharacteristic
61C22×Dic173—21no—
62C2×C4×D173—21no—
63C2×C4×D173C2311yescharacteristic
64#14583816—22no—
65#14583810—22no—
66#14583809—22no—
67C2×#13063436C2311yescharacteristic
68C2×#13063436C2311yescharacteristic
69C4⋊Dic173—21no—
70C4⋊Dic173—21no—
71C4⋊Dic173—21no—
72C4⋊Dic173—21no—
73C4⋊Dic173—21no—
74C4⋊Dic173C2311yescharacteristic
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805D411yescharacteristic
84C22⋊Dic173—21no—
85C22⋊Dic173D411yescharacteristic
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803D411yescharacteristic
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173C2311yescharacteristic
94C2×Q8×C173C2311yescharacteristic
95C22×C4×C173C2311yescharacteristic
96C2×C8×C173C2311yescharacteristic
97C2×C8×C173C2311yescharacteristic
98C4⋊C4×C173—21no—
99C4⋊C4×C173C2311yescharacteristic
100C4⋊C4×C173C2311yescharacteristic
101C4⋊C4×C173C2311yescharacteristic
102C22⋊C4×C173—21no—
103C22⋊C4×C173—21no—
104C42×C173—21no—
105C22×D173C2×D411yescharacteristic
106Dic346—21no—
107Dic346—21no—
108C22⋊D173—41no—
109C2×Dic173—21no—
110C2×Dic173—21no—
111C2×Dic173—21no—
112C2×Dic173—21no—
113C2×Dic173—42no—
114C2×Dic173C2×D411yescharacteristic
115C2×Dic173C2×D411yescharacteristic
116C2×Dic173C2×D411yescharacteristic
117C4×D173—21no—
118C4×D173—21no—
119C4×D173—42no—
120#13063436—21no—
121#13063436#4112yes—
122C23×C173C2×D411yescharacteristic
123Q8×C173—21no—
124Q8×C173—21no—
125C2×C4×C173—21no—
126C2×C4×C173—21no—
127C2×C4×C173—21no—
128C2×C4×C173—21no—
129C2×C4×C173—21no—
130C2×C4×C173—21no—
131C2×C4×C173C2×D411yescharacteristic
132C2×C4×C173C2411yescharacteristic
133C8×C173—21no—
134C8×C173—21no—
135C22×C173—21no—
136C22×C173—21no—
137C22×C173—21no—
138C22×C173C22×D411yescharacteristic
139C2×D173—21no—
140C2×D173—21no—
141C2×D173—21no—
142C4×C173—21no—
143C4×C173—41no—
144C4×C173—41no—
145C4×C173—41no—
146C4×C173—41no—
147C4×C173C2×#4111yescharacteristic
148C4×C1732−1+411yescharacteristic
149Dic173—21no—
150Dic173—21no—
151Dic173—21no—
152Dic173—41no—
153Dic173—41no—
154Dic173—41no—
155Dic173—41no—
156C2×C173—41no—
157C2×C173#54711yescharacteristic
158C2×C173#57511yescharacteristic
159C2×C173#57811yescharacteristic
160D173—41no—
161C173#335211yescharacteristic, complemented, Sylow
162#3352—1731nomaximal, Sylow
163#552—1731no—
164#552—1731no—
165#499—1731no—
166#485—1731no—
167#485D17311yescharacteristic, complemented
168#484—1731no—
169#484—1731no—
170#479—1731no—
171#479—1731no—
172#470—1731no—
173#470—1731no—
174C4×Q16—1731no—
175#436—1731no—
176#428—1731no—
177#428—1731no—
178C2×Q16—1731no—
179C2×M16—1731no—
180C2×M16—1731no—
181#126—3461no—
182#126—3461no—
183#125—3461no—
184#125—3461no—
185#125—1731no—
186#125—1731no—
187#123—3461no—
188#123—3461no—
189C22⋊Q8—1731no—
190C22⋊Q8—1731no—
191C22⋊Q8—1731no—
192C22⋊Q8C2×D17311yescharacteristic
193C4×Q8—1731no—
194C4×Q8—1731no—
195C4×D4—1731no—
196#117—3461no—
197#117—3461no—
198#117—1731no—
199#117C2×D17311yescharacteristic
200C2×C4⋊C4—1731no—
201C2×C4⋊C4—1731no—
202#107—1731no—
203#107—1731no—
204#107—1731no—
205#107—1731no—
206#107C2×D17311yescharacteristic
207#106—1731no—
208#106—1731no—
209#106—1731no—
210#106C2×D17311yescharacteristic
211C4⋊C8—1731no—
212#103—1731no—
213#103—1731no—
214#103—1731no—
215#103—1731no—
216#103C2×D17311yescharacteristic
217#103C2×D17311yescharacteristic
218C22⋊C8—1731no—
219C22⋊C8C2×D17311yescharacteristic
220C4×C8—1731no—
221C2×Q8—1731no—
222C2×Q8C22×D17311yescharacteristic
223C2×D4—1731no—
224C22×C4—3461no—
225C22×C4—3461no—
226C22×C4—1731no—
227C22×C4C22×D17311yescharacteristic
228Q16—3462no—
229M16—3462no—
230M16—3462no—
231C2×C8—1731no—
232C2×C8—1731no—
233C2×C8C22×D17311yescharacteristic
234C2×C8C22×D17311yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—3461no—
237C4⋊C4—3461no—
238C4⋊C4—3461no—
239C4⋊C4—3461no—
240C4⋊C4—3461no—
241C4⋊C4—3461no—
242C4⋊C4—3461no—
243C4⋊C4—3461no—
244C4⋊C4—3461no—
245C4⋊C4—3461no—
246C4⋊C4—3461no—
247C4⋊C4—1731no—
248C4⋊C4—1731no—
249C4⋊C4—1731no—
250C4⋊C4—1731no—
251C4⋊C4C22×D17311yescharacteristic
252C4⋊C4C22×D17311yescharacteristic
253C4⋊C4C22×D17311yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—3461no—
257C22⋊C4—3461no—
258C22⋊C4—3461no—
259C22⋊C4—3461no—
260C22⋊C4—1731no—
261C22⋊C4—1731no—
262C42—21no—
263C42—3461no—
264C42—3461no—
265C42—3461no—
266C42—1731no—
267C23—1731no—
268C23D4×D17311yescharacteristic
269Q8—21no—
270Q8—21no—
271Q8—3461no—
272Q8—3461no—
273D4—6921no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—3461no—
281C2×C4—3461no—
282C2×C4—3461no—
283C2×C4—3461no—
284C2×C4—3461no—
285C2×C4—3461no—
286C2×C4—1731no—
287C2×C4—1731no—
288C2×C4—1731no—
289C2×C4—6922no—
290C2×C4—6922no—
291C2×C4D4×D17311yescharacteristic
292C2×C4C23×D17311yescharacteristic
293C8—21no—
294C8—21no—
295C8—3461no—
296C8—1732no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—3461no—
301C22—3461no—
302C22—3461no—
303C22C2×D4×D17311yescharacteristic
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—3461no—
310C4—3461no—
311C4—3461no—
312C4—6921no—
313C4—6921no—
314C4—6921no—
315C4—6921no—
316C4#41×D17311yescharacteristic
317C4#1914972611yescharacteristic
318C2—41no—
319C2—6921no—
320C2#2601213911yescharacteristic
321C2#2601229011yescharacteristic
322C2#2601230511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#485×C173C2
socleC22×C173C22×D4
prime radicalC2×C22⋊D173D4
hypercenter#485D173
hypocenterC173#3352
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
3. #26011892C2semidirect
4. #26011834C2semidirect
5. #26011827C2semidirect
6. #26011825C2semidirect
7. #26011801C2semidirect
8. #26011800C2semidirect
9. #26011748C2semidirect
10. #26011747C2semidirect
11. #26011583C2semidirect
13. #26011534C2semidirect
14. #26011530C2semidirect
15. #485×C173C2semidirect
161. C173#3352semidirect
167. #485D173semidirect

Maximal subgroups

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

subgroupindex
#260121832
#260121802
#260118922
#260118342
#260118272
#260118252
#260118012
#260118002
#260117482
#260117472
#260115832
#260115662
#260115342
#260115302
#485×C1732
#3352173