Finite Group Databasebeta

← #36363865

Subgroups — #36363865

Order 22144.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012104C211yesmaximal, characteristic
2#26011967C211yesmaximal, characteristic
3#26011835C211yesmaximal, characteristic
4#26011834C211yesmaximal, characteristic
5#26011795C211yesmaximal, characteristic, complemented
6#26011702C211yesmaximal, characteristic, complemented
7#26011701C211yesmaximal, characteristic, complemented
8#26011648C211yesmaximal, characteristic
9#26011648C211yesmaximal, characteristic
10#26011597C211yesmaximal, characteristic, complemented
11#26011595C211yesmaximal, characteristic, complemented
12#26011585C211yesmaximal, characteristic, complemented
13#26011583C211yesmaximal, characteristic, complemented
14#26011551C211yesmaximal, characteristic
15#479×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×Dic692C2211yescharacteristic
18C2×Dic692C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20C2×#14583816C2211yescharacteristic
21#19149679C2211yescharacteristic
22#19149674—21no—
23#19149674—21no—
24#19149670—21no—
25#19149670—21no—
26#19149659C2211yescharacteristic
27#19149659C2211yescharacteristic
28Q8×Dic173C2211yescharacteristic
29C4×Dic346—21no—
30C4×Dic346C2211yescharacteristic
31#19149644—21no—
32#19149644C2211yescharacteristic
33#19149644C2211yescharacteristic
34#19149644C2211yescharacteristic
35#19149637—21no—
36#19149637C2211yescharacteristic
37#19149636—21no—
38#19149636C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149610C2211yescharacteristic
41C4⋊#13063436C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149602C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52#125×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61C2×Dic346C2311yescharacteristic
62C2×Dic346C2311yescharacteristic
63Q8×D173—22no—
64Q8×D173—22no—
65C2×C4×D173—21no—
66C2×C4×D173C2311yescharacteristic
67Dic692—22no—
68Dic692—22no—
69#14583816—22no—
70#14583816—22no—
71C2×#13063436C2311yescharacteristic
72C2×#13063436C2311yescharacteristic
73C4⋊Dic173—21no—
74C4⋊Dic173—21no—
75C4⋊Dic173—21no—
76C4⋊Dic173—21no—
77C4⋊Dic173—21no—
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805D411yescharacteristic
84#14583805D411yescharacteristic
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803D411yescharacteristic
89#14583803D411yescharacteristic
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
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—24no—
110Dic346—21no—
111Dic346—21no—
112Dic346—21no—
113Dic346—21no—
114Dic346—21no—
115Dic346#4112yes—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C4×D173—24no—
124C4×D173—21no—
125C4×D173—21no—
126#13063436—21no—
127#13063436—21no—
128Q8×C173—21no—
129Q8×C173#4112yes—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173C2×D411yescharacteristic
135C2×C4×C173C2×D411yescharacteristic
136C2×C4×C173C2411yescharacteristic
137C8×C173—21no—
138C8×C173—21no—
139C22×C173C22×D411yescharacteristic
140C2×D173—21no—
141C2×D173—21no—
142C2×D173—21no—
143C4×C173—22no—
144C4×C173—21no—
145C4×C173—21no—
146C4×C173—41no—
147C4×C173—41no—
148C4×C173—41no—
149C4×C173C2×#4111yescharacteristic
150C4×C173C2×#4111yescharacteristic
151Dic173—22no—
152Dic173—21no—
153Dic173—21no—
154Dic173—21no—
155Dic173—41no—
156Dic173—41no—
157Dic173—41no—
158C2×C173#52511yescharacteristic
159C2×C173#57811yescharacteristic
160C2×C173#57811yescharacteristic
161D173—41no—
162C173#312711yescharacteristic, complemented, Sylow
163#3127—1731nomaximal, Sylow
164#541—1731no—
165#519—1731no—
166#485—1731no—
167#485—1731no—
168#479—1731no—
169#479D17311yescharacteristic, complemented
170#462—1731no—
171#462—1731no—
172#451—1731no—
173#451—1731no—
174#441—1731no—
175#441—1731no—
176C4×Q16—1731no—
177C4×Q16—1731no—
178#433—1731no—
179C22×Q8—1731no—
180C2×Q16—1731no—
181C2×Q16—1731no—
182C2×Q16—1731no—
183C2×Q16—1731no—
184C4⋊Q8—1731no—
185#126—3461no—
186#126—3461no—
187#125—3461no—
188#125—3461no—
189#125C2×D17311yescharacteristic
190C22⋊Q8—1731no—
191C22⋊Q8—1731no—
192C4×Q8—3461no—
193C4×Q8—3461no—
194C4×Q8—1731no—
195C4×Q8—1731no—
196C4×Q8—1731no—
197C4×Q8—1731no—
198C4×Q8—1731no—
199C4×Q8C2×D17311yescharacteristic
200#117—3461no—
201#117—3461no—
202#117—1731no—
203#117—1731no—
204#107—1731no—
205#107C2×D17311yescharacteristic
206#106—1731no—
207#106C2×D17311yescharacteristic
208C4⋊C8—1731no—
209C4⋊C8C2×D17311yescharacteristic
210#103—1731no—
211#103—1731no—
212#103—1731no—
213#103—1731no—
214#103—1731no—
215#103—1731no—
216#103C2×D17311yescharacteristic
217#103C2×D17311yescharacteristic
218C22⋊C8—1731no—
219C22⋊C8—1731no—
220#97—1731no—
221C4×C8—1731no—
222C2×Q8—3461no—
223C2×Q8—3462no—
224C2×Q8—3462no—
225C2×Q8—1731no—
226C2×Q8—1731no—
227C2×Q8—1731no—
228C2×Q8C22×D17311yescharacteristic
229C22×C4—3461no—
230C22×C4—1731no—
231Q16—3462no—
232Q16—3462no—
233Q16—3462no—
234Q16—3462no—
235C2×C8—1731no—
236C2×C8—1731no—
237C2×C8C22×D17311yescharacteristic
238C2×C8C22×D17311yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
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—1731no—
261C22⋊C4—1731no—
262C42—21no—
263C42—3461no—
264C42—3461no—
265C42—3461no—
266C42—3461no—
267C42—3461no—
268C42—3461no—
269C42—1731no—
270C42C22×D17311yescharacteristic
271C23—1731no—
272Q8—21no—
273Q8—3464no—
274Q8—3461no—
275Q8—3461no—
276Q8—3461no—
277Q8—3461no—
278Q8—3461no—
279Q8—1732no—
280Q8#1458382712yes—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—3464no—
286C2×C4—3461no—
287C2×C4—3461no—
288C2×C4—3461no—
289C2×C4—3461no—
290C2×C4—3461no—
291C2×C4—3461no—
292C2×C4—1731no—
293C2×C4—1731no—
294C2×C4—1731no—
295C2×C4D4×D17311yescharacteristic
296C2×C4D4×D17311yescharacteristic
297C2×C4C23×D17311yescharacteristic
298C8—21no—
299C8—21no—
300C8—3461no—
301C8—3461no—
302C22—3461no—
303C22—3461no—
304C22—3461no—
305C22C2×D4×D17311yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—3462no—
313C4—3461no—
314C4—3461no—
315C4—3461no—
316C4—6921no—
317C4—6921no—
318C4—6921no—
319C4#41×D17311yescharacteristic
320C4C2×#1458382711yescharacteristic
321C2—6921no—
322C2#2601200411yescharacteristic
323C2#2601230511yescharacteristic
324C2#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#3127
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
5. #26011795C2semidirect
6. #26011702C2semidirect
7. #26011701C2semidirect
10. #26011597C2semidirect
11. #26011595C2semidirect
12. #26011585C2semidirect
13. #26011583C2semidirect
15. #479×C173C2semidirect
162. C173#3127semidirect
169. #479D173semidirect

Maximal subgroups

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

subgroupindex
#260121042
#260119672
#260118352
#260118342
#260117952
#260117022
#260117012
#260116482
#260115972
#260115952
#260115852
#260115832
#260115512
#479×C1732
#3127173