Finite Group Databasebeta

← #36366174

Subgroups — #36366174

Order 22144.

342 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012209C211yesmaximal, characteristic
2#26012196C211yesmaximal, characteristic
3#26011888C211yesmaximal, characteristic, complemented
4#26011834C211yesmaximal, characteristic, complemented
5#26011832C211yesmaximal, characteristic, complemented
6#26011823C211yesmaximal, characteristic, complemented
7#26011821C211yesmaximal, characteristic, complemented
8#26011785C211yesmaximal, characteristic, complemented
9#26011775C211yesmaximal, characteristic, complemented
10#26011750C211yesmaximal, characteristic, complemented
11#26011749C211yesmaximal, characteristic, complemented
12#26011585C211yesmaximal, characteristic, complemented
13#26011567C211yesmaximal, characteristic
14#26011532C211yesmaximal, characteristic, complemented
15#428×C173C211yesmaximal, characteristic, complemented
16C2×Dic692C2211yescharacteristic
17C2×#14583809C2211yescharacteristic
18#19149678C2211yescharacteristic
19#19149678C2211yescharacteristic
20#19149673—21no—
21#19149672—21no—
22#19149669—21no—
23#19149665—21no—
24#19149659C2211yescharacteristic
25#19149659C2211yescharacteristic
26#19149658—21no—
27#19149658C2211yescharacteristic
28#19149658C2211yescharacteristic
29#19149657—21no—
30C4×Dic346—21no—
31#19149644C2211yescharacteristic
32#19149644C2211yescharacteristic
33C4×C22⋊D173—21no—
34C4×C22⋊D173C2211yescharacteristic
35#19149637C2211yescharacteristic
36C4⋊C4×D173C2211yescharacteristic
37#19149612C2211yescharacteristic
38#19149611C2211yescharacteristic
39#19149610C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149610C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149602C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×M16×C173C2211yescharacteristic
53#117×C173C2211yescharacteristic
54C2×C4⋊C4×C173C2211yescharacteristic
55#107×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic
57#106×C173C2211yescharacteristic
58#106×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346—21no—
61C2×Dic346C2311yescharacteristic
62C2×Dic346C2311yescharacteristic
63C2×C22⋊D173D411yescharacteristic
64C2×C4×D173—21no—
65C2×C4×D173C2311yescharacteristic
66Dic692—22no—
67#14583809—22no—
68C2×#13063436C2311yescharacteristic
69C2×#13063436C2311yescharacteristic
70C4⋊Dic173—21no—
71C4⋊Dic173—21no—
72C4⋊Dic173C2311yescharacteristic
73C4⋊Dic173C2311yescharacteristic
74#14583805—21no—
75#14583805—21no—
76#14583805—21no—
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805D411yescharacteristic
82C22⋊Dic173—21no—
83C22⋊Dic173—21no—
84C22⋊Dic173D411yescharacteristic
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803D411yescharacteristic
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173C2311yescharacteristic
93C22×C4×C173—21no—
94C22×C4×C173C2311yescharacteristic
95M16×C173—22no—
96C2×C8×C173C2311yescharacteristic
97C2×C8×C173C2311yescharacteristic
98C4⋊C4×C173—21no—
99C4⋊C4×C173C2311yescharacteristic
100C4⋊C4×C173C2311yescharacteristic
101C4⋊C4×C173C2311yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C22⋊C4×C173—21no—
104C42×C173—21no—
105C22×D173C2×D411yescharacteristic
106Dic346—21no—
107Dic346—21no—
108Dic346—21no—
109Dic346—21no—
110Dic346—42no—
111Dic346—81no—
112C22⋊D173—22no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173C2×D411yescharacteristic
118C2×Dic173C2×D411yescharacteristic
119C2×Dic173C2×D411yescharacteristic
120C4×D173—21no—
121C4×D173—21no—
122C4×D173—81no—
123#13063436—21no—
124#13063436—21no—
125C23×C173C2×D411yescharacteristic
126C2×C4×C173—21no—
127C2×C4×C173—21no—
128C2×C4×C173—21no—
129C2×C4×C173—21no—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173—81no—
133C2×C4×C173C2×D411yescharacteristic
134C2×C4×C173C2411yescharacteristic
135C8×C173—21no—
136C8×C173#4112yes—
137C22×C173—21no—
138C22×C173—21no—
139C22×C173—21no—
140C22×C173C22×D411yescharacteristic
141C2×D173—21no—
142C2×D173—21no—
143C2×D173—21no—
144C4×C173—21no—
145C4×C173—41no—
146C4×C173—41no—
147C4×C173—41no—
148C4×C173—41no—
149C4×C173C2×#4111yescharacteristic
150C4×C1732−1+411yescharacteristic
151Dic173—21no—
152Dic173—21no—
153Dic173—41no—
154Dic173—41no—
155Dic173—41no—
156Dic173—41no—
157Dic173#13712yes—
158C2×C173—41no—
159C2×C173#54711yescharacteristic
160C2×C173C2×#13711yescharacteristic
161C2×C173#57711yescharacteristic
162D173—41no—
163C173#335611yescharacteristic, complemented, Sylow
164#3356—1731nomaximal, Sylow
165#556—1731no—
166#554—1731no—
167#498—1731no—
168#485—1731no—
169#485—1731no—
170#483—1731no—
171#483—1731no—
172#477—1731no—
173#475—1731no—
174#470—1731no—
175#470—1731no—
176C4×Q16—1731no—
177#436—1731no—
178#428—1731no—
179#428D17311yescharacteristic, complemented
180C2×Q16—1731no—
181C2×M16—1731no—
182C2×M16C2×D17311yescharacteristic
183C4⋊Q8—1731no—
184C4⋊Q8—1731no—
185#126—3461no—
186#126—3461no—
187#125—3461no—
188#124—3461no—
189C22⋊Q8—3461no—
190C22⋊Q8—3461no—
191C22⋊Q8—1731no—
192C22⋊Q8—1731no—
193C22⋊Q8—1731no—
194C22⋊Q8—1731no—
195C4×Q8—3461no—
196C4×Q8—1731no—
197C4×Q8—1731no—
198C4×D4—3461no—
199C4×D4—1731no—
200#117—1731no—
201#117C2×D17311yescharacteristic
202C2×C4⋊C4—1731no—
203C2×C4⋊C4C2×D17311yescharacteristic
204#107—1731no—
205#107C2×D17311yescharacteristic
206#107C2×D17311yescharacteristic
207#106—1731no—
208#106—1731no—
209#106—1731no—
210#106—1731no—
211#106C2×D17311yescharacteristic
212#106C2×D17311yescharacteristic
213C4⋊C8—1731no—
214#103—1731no—
215#103—1731no—
216#103—1731no—
217#103—1731no—
218#103—1731no—
219#103—1731no—
220C22⋊C8—1731no—
221C22⋊C8—1731no—
222C4×C8—1731no—
223C2×Q8—3461no—
224C2×Q8—3461no—
225C2×Q8—1731no—
226C2×Q8—1731no—
227C2×D4—1731no—
228C22×C4—21no—
229C22×C4—3461no—
230C22×C4—1731no—
231C22×C4C22×D17311yescharacteristic
232Q16—3462no—
233M16—22no—
234M16—3462no—
235C2×C8—1731no—
236C2×C8—1731no—
237C2×C8C22×D17311yescharacteristic
238C2×C8C22×D17311yescharacteristic
239C4⋊C4—21no—
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—1731no—
250C4⋊C4—1731no—
251C4⋊C4—1731no—
252C4⋊C4C22×D17311yescharacteristic
253C4⋊C4C22×D17311yescharacteristic
254C4⋊C4C22×D17311yescharacteristic
255C4⋊C4C22×D17311yescharacteristic
256C22⋊C4—21no—
257C22⋊C4—3461no—
258C22⋊C4—3461no—
259C22⋊C4—3461no—
260C22⋊C4—3461no—
261C22⋊C4—3461no—
262C22⋊C4—1731no—
263C22⋊C4—1731no—
264C42—21no—
265C42—3461no—
266C42—3461no—
267C42—3461no—
268C42—1731no—
269C23—1731no—
270C23D4×D17311yescharacteristic
271Q8—3461no—
272Q8—3461no—
273Q8—3461no—
274Q8—3461no—
275Q8—13841no—
276Q8—6922no—
277D4—3462no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—3461no—
286C2×C4—3461no—
287C2×C4—3461no—
288C2×C4—3461no—
289C2×C4—3461no—
290C2×C4—3461no—
291C2×C4—13841no—
292C2×C4—1731no—
293C2×C4—1731no—
294C2×C4—1731no—
295C2×C4D4×D17311yescharacteristic
296C2×C4C23×D17311yescharacteristic
297C8—21no—
298C8—3461no—
299C8—3461no—
300C8#1458382612yes—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—3461no—
305C22—3461no—
306C22—3461no—
307C22C2×D4×D17311yescharacteristic
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—3461no—
314C4—3461no—
315C4—1732no—
316C4—6921no—
317C4—6921no—
318C4—6921no—
319C4—6921no—
320C4C2×#1458382611yescharacteristic
321C4#1914972711yescharacteristic
322C2—41no—
323C2—6921no—
324C2#2601214211yescharacteristic
325C2#137×D17311yescharacteristic
326C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#428×C173C2
socleC22×C173C22×D4
prime radicalC2×C22⋊D173D4
hypercenter#428D173
hypocenterC173#3356
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #26011888C2semidirect
4. #26011834C2semidirect
5. #26011832C2semidirect
6. #26011823C2semidirect
7. #26011821C2semidirect
8. #26011785C2semidirect
9. #26011775C2semidirect
10. #26011750C2semidirect
11. #26011749C2semidirect
12. #26011585C2semidirect
14. #26011532C2semidirect
15. #428×C173C2semidirect
163. C173#3356semidirect
179. #428D173semidirect

Maximal subgroups

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

subgroupindex
#260122092
#260121962
#260118882
#260118342
#260118322
#260118232
#260118212
#260117852
#260117752
#260117502
#260117492
#260115852
#260115672
#260115322
#428×C1732
#3356173