Finite Group Databasebeta

← #36365016

Subgroups — #36365016

Order 22144.

322 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#26012104C211yesmaximal, characteristic
2#26012099C211yesmaximal, characteristic
3#26011801C211yesmaximal, characteristic, complemented
4#26011797C211yesmaximal, characteristic, complemented
5#26011795C211yesmaximal, characteristic, complemented
6#26011750C211yesmaximal, characteristic
7#26011748C211yesmaximal, characteristic
8#26011733C211yesmaximal, characteristic
9#26011731C211yesmaximal, characteristic
10#26011702C211yesmaximal, characteristic, complemented
11#26011701C211yesmaximal, characteristic, complemented
12#26011700C211yesmaximal, characteristic, complemented
13#26011700C211yesmaximal, characteristic, complemented
14#26011516C211yesmaximal, characteristic
15#479×C173C211yesmaximal, characteristic, complemented
16C2×Dic692C2211yescharacteristic
17C2×Dic692C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20C2×#14583809C2211yescharacteristic
21C2×C8×D173C2211yescharacteristic
22#19149679C2211yescharacteristic
23#19149677C2211yescharacteristic
24#19149674—21no—
25#19149674—21no—
26#19149674—21no—
27#19149673—21no—
28#19149671—21no—
29#19149670—21no—
30#19149670—21no—
31#19149670C2211yescharacteristic
32#19149669—21no—
33#19149659C2211yescharacteristic
34#19149659C2211yescharacteristic
35#19149659C2211yescharacteristic
36#19149655C2211yescharacteristic
37C4×Dic346C2211yescharacteristic
38#19149644C2211yescharacteristic
39#19149644C2211yescharacteristic
40#19149636C2211yescharacteristic
41#19149612C2211yescharacteristic
42#19149610C2211yescharacteristic
43C4⋊#13063436C2211yescharacteristic
44#19149608C2211yescharacteristic
45#19149608C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149604C2211yescharacteristic
48#19149602C2211yescharacteristic
49#19149602C2211yescharacteristic
50#19149602C2211yescharacteristic
51#19149602C2211yescharacteristic
52#125×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59C2×Dic346C2311yescharacteristic
60C2×Dic346C2311yescharacteristic
61C2×Dic346C2311yescharacteristic
62C2×C4×D173C2311yescharacteristic
63Dic692—41no—
64Dic692—41no—
65#14583816—41no—
66#14583816—41no—
67#14583809—41no—
68C2×#13063436C2311yescharacteristic
69C2×#13063436C2311yescharacteristic
70C8×D173—41no—
71C4⋊Dic173—21no—
72C4⋊Dic173—21no—
73C4⋊Dic173—21no—
74C4⋊Dic173—21no—
75C4⋊Dic173—21no—
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805D411yescharacteristic
86#14583805D411yescharacteristic
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803D411yescharacteristic
92#14583803D411yescharacteristic
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97C2×Q8×C173C2311yescharacteristic
98C2×C8×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C4⋊C4×C173—21no—
101C4⋊C4×C173—21no—
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C42×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173C2×D411yescharacteristic
109Dic346—21no—
110Dic346—21no—
111Dic346—21no—
112Dic346—21no—
113Dic346—21no—
114Dic346—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173C2×D411yescharacteristic
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C4×D173—21no—
123C4×D173—21no—
124#13063436—21no—
125#13063436—21no—
126Q8×C173—21no—
127Q8×C173—21no—
128C2×C4×C173—21no—
129C2×C4×C173—21no—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173C2×D411yescharacteristic
133C2×C4×C173C2×D411yescharacteristic
134C2×C4×C173C2411yescharacteristic
135C8×C173—21no—
136C8×C173—21no—
137C22×C173C22×D411yescharacteristic
138C2×D173—21no—
139C2×D173—21no—
140C2×D173—21no—
141C4×C173—21no—
142C4×C173—21no—
143C4×C173—41no—
144C4×C173—41no—
145C4×C173—41no—
146C4×C173—41no—
147C4×C173H(5,𝔽2)11yescharacteristic
148C4×C1732−1+411yescharacteristic
149Dic173—21no—
150Dic173—21no—
151Dic173—21no—
152Dic173—41no—
153Dic173—41no—
154Dic173—41no—
155Dic173—41no—
156C2×C173#53711yescharacteristic
157C2×C173#57811yescharacteristic
158C2×C173#57811yescharacteristic
159D173—41no—
160C173#326111yescharacteristic, complemented, Sylow
161#3261—1731nomaximal, Sylow
162#541—1731no—
163#541—1731no—
164#479—1731no—
165#479—1731no—
166#479—1731no—
167#479D17311yescharacteristic, complemented
168#470—1731no—
169#470—1731no—
170#467—1731no—
171#467—1731no—
172#462—1731no—
173#462—1731no—
174#462—1731no—
175#462—1731no—
176#424—1731no—
177C2×Q16—1731no—
178C2×Q16—1731no—
179C2×Q16—1731no—
180C2×Q16—1731no—
181C2×M16—1731no—
182C22×C8—1731no—
183C4⋊Q8—1731no—
184C4⋊Q8—1731no—
185#126—3461no—
186#126—3461no—
187#126—3461no—
188#126—3461no—
189#125—3461no—
190#125—3461no—
191#125—3461no—
192#125—3461no—
193#125—1731no—
194#125C2×D17311yescharacteristic
195C22⋊Q8—1731no—
196C22⋊Q8—1731no—
197C22⋊Q8—1731no—
198C22⋊Q8—1731no—
199C4×Q8—1731no—
200C4×Q8—1731no—
201C4×Q8—1731no—
202C4×Q8C2×D17311yescharacteristic
203#117—1731no—
204#107—1731no—
205#107C2×D17311yescharacteristic
206#106—1731no—
207#106C2×D17311yescharacteristic
208C4⋊C8—1731no—
209C4⋊C8—1731no—
210C4⋊C8—1731no—
211C4⋊C8C2×D17311yescharacteristic
212#103—1731no—
213#103—1731no—
214#103—1731no—
215#103—1731no—
216#103—1731no—
217#103—1731no—
218#103C2×D17311yescharacteristic
219#103C2×D17311yescharacteristic
220C2×Q8—1731no—
221C2×Q8—1731no—
222C2×Q8—1731no—
223C2×Q8C22×D17311yescharacteristic
224C22×C4—1731no—
225Q16—6921no—
226Q16—6921no—
227Q16—6921no—
228Q16—6921no—
229M16—6921no—
230C2×C8—1731no—
231C2×C8—1731no—
232C2×C8—6921no—
233C2×C8C22×D17311yescharacteristic
234C2×C8C22×D17311yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
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—3461no—
248C4⋊C4—3461no—
249C4⋊C4—3461no—
250C4⋊C4—3461no—
251C4⋊C4—1731no—
252C4⋊C4—1731no—
253C4⋊C4—1731no—
254C4⋊C4C22×D17311yescharacteristic
255C4⋊C4C22×D17311yescharacteristic
256C4⋊C4C22×D17311yescharacteristic
257C22⋊C4—3461no—
258C22⋊C4—3461no—
259C22⋊C4—3461no—
260C22⋊C4—3461no—
261C22⋊C4—1731no—
262C22⋊C4—1731no—
263C42—21no—
264C42—3461no—
265C42—3461no—
266C42—3461no—
267C42—1731no—
268C42C22×D17311yescharacteristic
269C23—1731no—
270Q8—21no—
271Q8—21no—
272Q8—3461no—
273Q8—3461no—
274Q8—3461no—
275Q8—3461no—
276Q8—3461no—
277Q8—3461no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—3461no—
283C2×C4—3461no—
284C2×C4—3461no—
285C2×C4—3461no—
286C2×C4—3461no—
287C2×C4—3461no—
288C2×C4—1731no—
289C2×C4—1731no—
290C2×C4—1731no—
291C2×C4D4×D17311yescharacteristic
292C2×C4D4×D17311yescharacteristic
293C2×C4C23×D17311yescharacteristic
294C8—21no—
295C8—21no—
296C8—3461no—
297C8—3461no—
298C22—3461no—
299C22—3461no—
300C22—3461no—
301C22C2×D4×D17311yescharacteristic
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—3461no—
309C4—3461no—
310C4—3461no—
311C4—6921no—
312C4—6921no—
313C4—6921no—
314C4—6921no—
315C4#1914972511yescharacteristic
316C4#1914972711yescharacteristic
317C2—6921no—
318C2#2601207111yescharacteristic
319C2#2601230511yescharacteristic
320C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#479×C173C2
socleC22×C173C22×D4
prime radicalC22×D173C2×D4
hypercenter#479D173
hypocenterC173#3261
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
3. #26011801C2semidirect
4. #26011797C2semidirect
5. #26011795C2semidirect
10. #26011702C2semidirect
11. #26011701C2semidirect
12. #26011700C2semidirect
13. #26011700C2semidirect
15. #479×C173C2semidirect
160. C173#3261semidirect
167. #479D173semidirect

Maximal subgroups

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

subgroupindex
#260121042
#260120992
#260118012
#260117972
#260117952
#260117502
#260117482
#260117332
#260117312
#260117022
#260117012
#260117002
#260115162
#479×C1732
#3261173