Finite Group Databasebeta

← #36366129

Subgroups — #36366129

Order 22144.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012188C211yesmaximal, characteristic
2#26012183C211yesmaximal, characteristic, complemented
3#26011892C211yesmaximal, characteristic, complemented
4#26011829C211yesmaximal, characteristic, complemented
5#26011820C211yesmaximal, characteristic, complemented
6#26011813C211yesmaximal, characteristic, complemented
7#26011807C211yesmaximal, characteristic, complemented
8#26011800C211yesmaximal, characteristic, complemented
9#26011793C211yesmaximal, characteristic, complemented
10#26011734C211yesmaximal, characteristic, complemented
11SD16×Dic173C211yesmaximal, characteristic, complemented
12#26011567C211yesmaximal, characteristic, complemented
13#26011533C211yesmaximal, characteristic, complemented
14#26011532C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C2×#14583809C2211yescharacteristic
17#19149674—21no—
18#19149672—21no—
19#19149670—21no—
20#19149670C2211yescharacteristic
21#19149670C2211yescharacteristic
22#19149663—21no—
23#19149660—21no—
24#19149655C2211yescharacteristic
25#19149650—21no—
26#19149650C2211yescharacteristic
27Q8×Dic173C2211yescharacteristic
28D4×Dic173C2211yescharacteristic
29C4×C22⋊D173C2211yescharacteristic
30#19149639—21no—
31#19149638C2211yescharacteristic
32#19149637C2211yescharacteristic
33#19149635—21no—
34C2×C4⋊Dic173C2211yescharacteristic
35C4⋊C4×D173C2211yescharacteristic
36#19149613C2211yescharacteristic
37#19149613C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149610C2211yescharacteristic
42#19149610C2211yescharacteristic
43#19149610C2211yescharacteristic
44#19149608C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C2×M16×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C2×C22⋊D173—21no—
60C2×C22⋊D173D411yescharacteristic
61C22×Dic173—21no—
62C22×Dic173—21no—
63C2×C4×D173—21no—
64C2×C4×D173C2311yescharacteristic
65#14583809—22no—
66C2×#13063436C2311yescharacteristic
67C2×#13063436C2311yescharacteristic
68C4⋊Dic173—21no—
69C4⋊Dic173—21no—
70C4⋊Dic173—21no—
71C4⋊Dic173—21no—
72C4⋊Dic173C2311yescharacteristic
73C4⋊Dic173C2311yescharacteristic
74C4⋊Dic173C2311yescharacteristic
75C4⋊Dic173C2311yescharacteristic
76#14583805—21no—
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80#14583805D411yescharacteristic
81C22⋊Dic173—21no—
82C22⋊Dic173—21no—
83C22⋊Dic173—21no—
84C22⋊Dic173D411yescharacteristic
85#14583803—21no—
86#14583803—21no—
87#14583803D411yescharacteristic
88C4×Dic173—21no—
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173C2311yescharacteristic
92C2×Q8×C173C2311yescharacteristic
93C2×D4×C173—21no—
94C2×D4×C173C2311yescharacteristic
95C22×C4×C173C2311yescharacteristic
96SD16×C173—22no—
97M16×C173—22no—
98C2×C8×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C4⋊C4×C173—21no—
101C4⋊C4×C173C2311yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C22⋊C4×C173—21no—
104C22⋊C4×C173—21no—
105C22×D173C2×D411yescharacteristic
106C22⋊D173—41no—
107C22⋊D173—42no—
108C2×Dic173—21no—
109C2×Dic173—21no—
110C2×Dic173—21no—
111C2×Dic173—21no—
112C2×Dic173—42no—
113C2×Dic173—81no—
114C2×Dic173C2×D411yescharacteristic
115C2×Dic173C2×D411yescharacteristic
116C2×Dic173C2×D411yescharacteristic
117C4×D173—21no—
118C4×D173—21no—
119C4×D173—81no—
120#13063436—21no—
121#13063436—21no—
122C23×C173—21no—
123C23×C173C2×D411yescharacteristic
124Q8×C173—21no—
125Q8×C173—21no—
126D4×C173—21no—
127D4×C173—21no—
128D4×C173—81no—
129C2×C4×C173—21no—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173C2×D411yescharacteristic
135C2×C4×C173C2411yescharacteristic
136C8×C173—21no—
137C8×C173#4112yes—
138C22×C173—21no—
139C22×C173—21no—
140C22×C173—21no—
141C22×C173—41no—
142C22×C173—41no—
143C22×C173—41no—
144C22×C173C22×D411yescharacteristic
145C2×D173—21no—
146C2×D173—21no—
147C2×D173—21no—
148C4×C173—21no—
149C4×C173—41no—
150C4×C173—41no—
151C4×C173—41no—
152C4×C173C2×#4111yescharacteristic
153C4×C1732−1+411yescharacteristic
154Dic173—21no—
155Dic173—21no—
156Dic173—21no—
157Dic173—41no—
158Dic173—41no—
159Dic173—41no—
160Dic173—41no—
161C2×C173—41no—
162C2×C173—81no—
163C2×C173#54711yescharacteristic
164C2×C173#57511yescharacteristic
165C2×C173#57711yescharacteristic
166D173—41no—
167C173#335111yescharacteristic, complemented, Sylow
168#3351—1731nomaximal, Sylow
169#553—1731no—
170#552—1731no—
171#499—1731no—
172#485—1731no—
173#483—1731no—
174#482—1731no—
175#481—1731no—
176#479—1731no—
177#478—1731no—
178#468—1731no—
179#468D17311yescharacteristic, complemented
180C4×SD16—1731no—
181#436—1731no—
182#428—1731no—
183#428—1731no—
184C2×SD16C2×D17311yescharacteristic
185C2×M16—1731no—
186C2×M16C2×D17311yescharacteristic
187#126—3461no—
188#126—3461no—
189#125—3461no—
190#125—1731no—
191#125—1731no—
192#123—3461no—
193#123—3461no—
194C22⋊Q8—1731no—
195C22⋊Q8C2×D17311yescharacteristic
196#121—3461no—
197#121—1731no—
198#121C2×D17311yescharacteristic
199C4×Q8—1731no—
200C4×D4—3461no—
201C4×D4—1731no—
202C4×D4—1731no—
203#117—3461no—
204#117—1731no—
205#117—1731no—
206C2×C4⋊C4—1731no—
207C2×C4⋊C4—1731no—
208#107—1731no—
209#107—1731no—
210#107—1731no—
211#107C2×D17311yescharacteristic
212#106—1731no—
213#106—1731no—
214#106—1731no—
215#106—1731no—
216#106—1731no—
217C4⋊C8—1731no—
218#103—1731no—
219#103—1731no—
220#103C2×D17311yescharacteristic
221#102—1731no—
222#102—1731no—
223#102C2×D17311yescharacteristic
224C22⋊C8—1731no—
225C22⋊C8—1731no—
226C4×C8—1731no—
227C2×Q8C22×D17311yescharacteristic
228C2×D4—21no—
229C2×D4—3461no—
230C2×D4—1731no—
231C2×D4C22×D17311yescharacteristic
232C22×C4—3461no—
233C22×C4—3461no—
234C22×C4—3461no—
235C22×C4—1731no—
236C22×C4C22×D17311yescharacteristic
237SD16—22no—
238M16—22no—
239M16—3462no—
240C2×C8—1731no—
241C2×C8—1731no—
242C2×C8C22×D17311yescharacteristic
243C2×C8C22×D17311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—3461no—
246C4⋊C4—3461no—
247C4⋊C4—3461no—
248C4⋊C4—3461no—
249C4⋊C4—3461no—
250C4⋊C4—3461no—
251C4⋊C4—3461no—
252C4⋊C4—3461no—
253C4⋊C4—1731no—
254C4⋊C4—1731no—
255C4⋊C4—1731no—
256C4⋊C4—1731no—
257C4⋊C4—1731no—
258C4⋊C4C22×D17311yescharacteristic
259C4⋊C4C22×D17311yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—3461no—
263C22⋊C4—3461no—
264C22⋊C4—3461no—
265C22⋊C4—3461no—
266C22⋊C4—3461no—
267C22⋊C4—1731no—
268C22⋊C4—1731no—
269C42—3461no—
270C42—3461no—
271C42—3461no—
272C42—1731no—
273C23—21no—
274C23—1731no—
275C23D4×D17311yescharacteristic
276Q8—21no—
277Q8—21no—
278D4—21no—
279D4—21no—
280D4—81no—
281D4—6921no—
282D4—6922no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—3461no—
289C2×C4—3461no—
290C2×C4—3461no—
291C2×C4—3461no—
292C2×C4—3461no—
293C2×C4—3461no—
294C2×C4—13841no—
295C2×C4—13841no—
296C2×C4—1731no—
297C2×C4—1731no—
298C2×C4—1731no—
299C2×C4—6922no—
300C2×C4D4×D17311yescharacteristic
301C2×C4C23×D17311yescharacteristic
302C8—21no—
303C8—3461no—
304C8—3461no—
305C8#1458382512yes—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—3461no—
313C22—3461no—
314C22—3461no—
315C22C2×D4×D17311yescharacteristic
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—3461no—
321C4—3461no—
322C4—3461no—
323C4—6921no—
324C4—6921no—
325C4—6921no—
326C4—6921no—
327C4C2×#1458382511yescharacteristic
328C4#1914972611yescharacteristic
329C2—41no—
330C2—81no—
331C2—6921no—
332C2#2601214111yescharacteristic
333C2#2601229011yescharacteristic
334C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
prime radical#26012188C2
hypercenter#468D173
hypocenterC173#3351
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #26012183C2semidirect
3. #26011892C2semidirect
4. #26011829C2semidirect
5. #26011820C2semidirect
6. #26011813C2semidirect
7. #26011807C2semidirect
8. #26011800C2semidirect
9. #26011793C2semidirect
10. #26011734C2semidirect
11. SD16×Dic173C2semidirect
12. #26011567C2semidirect
13. #26011533C2semidirect
14. #26011532C2semidirect
15. #468×C173C2semidirect
167. C173#3351semidirect
179. #468D173semidirect

Maximal subgroups

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

subgroupindex
#260121882
#260121832
#260118922
#260118292
#260118202
#260118132
#260118072
#260118002
#260117932
#260117342
SD16×Dic1732
#260115672
#260115332
#260115322
#468×C1732
#3351173