Finite Group Databasebeta

← #36364401

Subgroups — #36364401

Order 22144.

418 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012004C211yesmaximal, characteristic, complemented
2#26012001C211yesmaximal, characteristic, complemented
3#26011834C211yesmaximal, characteristic, complemented
4#26011831C211yesmaximal, characteristic, complemented
5#26011818C211yesmaximal, characteristic, complemented
6#26011813C211yesmaximal, characteristic, complemented
7#26011747C211yesmaximal, characteristic, complemented
8#26011742C211yesmaximal, characteristic, complemented
9#26011737C211yesmaximal, characteristic, complemented
10#26011656C211yesmaximal, characteristic, complemented
11#26011650C211yesmaximal, characteristic, complemented
12#26011637C211yesmaximal, characteristic, complemented
13#26011635C211yesmaximal, characteristic, complemented
14#26011454C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19C2×#14583816C2211yescharacteristic, complemented
20C2×#14583815C2211yescharacteristic, complemented
21C2×#14583811C2211yescharacteristic, complemented
22C2×#14583810C2211yescharacteristic
23#19149664—21no—
24#19149661—21no—
25#19149659C2211yescharacteristic
26#19149658C2211yescharacteristic
27#19149655C2211yescharacteristic
28#19149654C2211yescharacteristic
29#19149653C2211yescharacteristic
30#19149650C2211yescharacteristic
31#19149649—21no—
32#19149647—21no—
33D4×Dic173—21no—
34#19149641—21no—
35#19149639—21no—
36#19149639—21no—
37#19149637C2211yescharacteristic
38#19149637C2211yescharacteristic
39#19149613C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41#19149610C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic, complemented
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic, complemented
53C2×M16×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic, complemented
59C23×D173D411yescharacteristic
60#14583827—41no—
61#14583827—41no—
62#14583825—41no—
63#14583825—41no—
64C2×Dic346C2311yescharacteristic
65C2×D692—21no—
66C2×D692—21no—
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C22×Dic173—21no—
72C22×Dic173D411yescharacteristic
73C2×C4×D173—21no—
74C2×C4×D173—41no—
75C2×C4×D173—41no—
76C2×C4×D173D411yescharacteristic
77C2×C4×D173D411yescharacteristic
78C2×C4×D173C2311yescharacteristic
79C2×C4×D173C2311yescharacteristic
80#14583816—41no—
81#14583815—41no—
82#14583811—41no—
83#14583810—41no—
84C2×#13063436C2311yescharacteristic
85C2×#13063436C2311yescharacteristic
86C4⋊Dic173—21no—
87C4⋊Dic173C2311yescharacteristic
88C4⋊Dic173C2311yescharacteristic
89#14583805—21no—
90#14583805—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99C4×Dic173—21no—
100C4×Dic173—21no—
101C2×Q8×C173C2311yescharacteristic
102C2×D4×C173—21no—
103C2×D4×C173C2311yescharacteristic
104C22×C4×C173C2311yescharacteristic
105SD16×C173—41no—
106M16×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173—21no—
110C4⋊C4×C173C2311yescharacteristic
111C4⋊C4×C173C2311yescharacteristic
112C22⋊C4×C173—21no—
113C22⋊C4×C173—21no—
114C22×D173—21no—
115C22×D173—41no—
116C22×D173—41no—
117C22×D173—41no—
118C22×D173C2×D411yescharacteristic
119C22×D173C2×D411yescharacteristic
120Dic346—21no—
121Dic346—21no—
122D692—21no—
123D692—21no—
124D692—42no—
125D692—42no—
126C22⋊D173—42no—
127C22⋊D173—42no—
128C22⋊D173—42no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—41no—
133C2×Dic173—42no—
134C2×Dic173C2×D411yescharacteristic
135C2×Dic173C2×D411yescharacteristic
136C4×D173—21no—
137C4×D173—21no—
138C4×D173—21no—
139C4×D173—21no—
140C4×D173—41no—
141C4×D173—41no—
142C4×D173—42no—
143#13063436—21no—
144#13063436—21no—
145C23×C173—21no—
146C23×C173C2×D411yescharacteristic
147Q8×C173—21no—
148Q8×C173—21no—
149D4×C173—21no—
150D4×C173—21no—
151D4×C173—42no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173C2×D411yescharacteristic
158C2×C4×C173C2411yescharacteristic
159C8×C173—21no—
160C8×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—21no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173C22×D411yescharacteristic
168C2×D173—21no—
169C2×D173—21no—
170C2×D173—21no—
171C2×D173—21no—
172C2×D173—21no—
173C2×D173—21no—
174C2×D173—44no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—41no—
178C4×C173—21no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173—41no—
182C4×C173H(5,𝔽2)11yescharacteristic
183C4×C173H(5,𝔽2)11yescharacteristic
184Dic173—21no—
185Dic173—21no—
186Dic173—41no—
187Dic173—41no—
188Dic173—41no—
189C2×C173—41no—
190C2×C173—81no—
191C2×C173#53411yescharacteristic
192C2×C173#57511yescharacteristic
193C2×C173#57511yescharacteristic
194D173—41no—
195D173—41no—
196D173—81no—
197C173#319711yescharacteristic, complemented, Sylow
198#3197—1731nomaximal, Sylow
199#525—1731no—
200#525—1731no—
201#485—1731no—
202#485—1731no—
203#482—1731no—
204#482—1731no—
205#470—1731no—
206#469—1731no—
207#468—1731no—
208#468D17311yescharacteristic, complemented
209#452—1731no—
210#452—1731no—
211#449—1731no—
212#449—1731no—
213#407—1731no—
214C2×#41—1731no—
215C2×#41—1731no—
216C23×C4—1731no—
217C2×Q16—1731no—
218C2×SD16—1731no—
219C2×SD16C2×D17311yescharacteristic, complemented
220C2×D8—1731no—
221C2×M16—1731no—
222C2×M16C2×D17311yescharacteristic
223#123—3461no—
224#123—3461no—
225C22⋊Q8—1731no—
226C22⋊Q8—1731no—
227C22⋊Q8—1731no—
228C22⋊Q8C2×D17311yescharacteristic
229#121—1731no—
230#121—1731no—
231#121—1731no—
232#121C2×D17311yescharacteristic
233#120—3461no—
234#120—3461no—
235C4×D4—3461no—
236C4×D4—3461no—
237C4×D4—3461no—
238C4×D4—3461no—
239#117—1731no—
240#117—1731no—
241#107—1731no—
242#107C2×D17311yescharacteristic, complemented
243#106—1731no—
244#106—1731no—
245#103—1731no—
246#103—1731no—
247#103—1731no—
248#103C2×D17311yescharacteristic, complemented
249#102—1731no—
250#102—1731no—
251#102—1731no—
252#102C2×D17311yescharacteristic, complemented
253C22⋊C8—1731no—
254C22⋊C8—1731no—
255C22⋊C8—1731no—
256C22⋊C8—1731no—
257C24—1731no—
258#41—6921no—
259#41—6921no—
260#41—6921no—
261#41—6921no—
262C2×Q8—1731no—
263C2×Q8C22×D17311yescharacteristic
264C2×D4—21no—
265C2×D4—3461no—
266C2×D4—3461no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—1731no—
271C2×D4C22×D17311yescharacteristic
272C22×C4—3461no—
273C22×C4—3461no—
274C22×C4—1731no—
275C22×C4—1731no—
276C22×C4—1731no—
277C22×C4—1731no—
278C22×C4—1731no—
279C22×C4—6921no—
280C22×C4—6921no—
281C22×C4C22×D17311yescharacteristic
282Q16—6921no—
283SD16—41no—
284SD16—6921no—
285D8—6921no—
286M16—41no—
287M16—6921no—
288C2×C8—1731no—
289C2×C8—1731no—
290C2×C8C22×D17311yescharacteristic
291C2×C8C22×D17311yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—3461no—
294C4⋊C4—3461no—
295C4⋊C4—3461no—
296C4⋊C4—1731no—
297C4⋊C4—1731no—
298C4⋊C4C22×D17311yescharacteristic
299C4⋊C4C22×D17311yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C42—3461no—
311C42—3461no—
312C23—21no—
313C23—3461no—
314C23—1731no—
315C23—1731no—
316C23—6921no—
317C23—6921no—
318C23—6921no—
319C23D4×D17311yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—3461no—
323Q8—3461no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—3461no—
328D4—3461no—
329D4—6922no—
330D4—6922no—
331D4—6922no—
332D4—6922no—
333D4—6922no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—3461no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—1731no—
347C2×C4—1731no—
348C2×C4—6921no—
349C2×C4—6921no—
350C2×C4—6922no—
351C2×C4—6921no—
352C2×C4—6922no—
353C2×C4D4×D17311yescharacteristic
354C2×C4C23×D17311yescharacteristic
355C8—21no—
356C8—21no—
357C8—3461no—
358C8—3461no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—3461no—
366C22—3461no—
367C22—3461no—
368C22—3461no—
369C22—3461no—
370C22—3461no—
371C22—6924no—
372C22—6921no—
373C22—6921no—
374C22—6921no—
375C22C2×D4×D17311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—3461no—
381C4—3461no—
382C4—6921no—
383C4—6921no—
384C4—6921no—
385C4#1914972411yescharacteristic
386C4#1914972511yescharacteristic
387C2—41no—
388C2—81no—
389C2—13841no—
390C2—6921no—
391C2—6921no—
392C2#2601204911yescharacteristic
393C2#2601228711yescharacteristic
394C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
hypercenter#468D173
hypocenterC173#3197
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
1. #26012004C2semidirect
2. #26012001C2semidirect
3. #26011834C2semidirect
4. #26011831C2semidirect
5. #26011818C2semidirect
6. #26011813C2semidirect
7. #26011747C2semidirect
8. #26011742C2semidirect
9. #26011737C2semidirect
10. #26011656C2semidirect
11. #26011650C2semidirect
12. #26011637C2semidirect
13. #26011635C2semidirect
14. #26011454C2semidirect
15. #468×C173C2semidirect
19. C2×#14583816C22semidirect
20. C2×#14583815C22semidirect
21. C2×#14583811C22semidirect
39. #19149613C22semidirect
40. #19149610C22semidirect
41. #19149610C22semidirect
42. #19149604C22semidirect
43. #19149603C22semidirect
44. #19149602C22semidirect
45. #19149601C22semidirect
46. #19149600C22semidirect
47. #19149599C22semidirect
52. C2×SD16×C173C22semidirect
56. #107×C173C22semidirect
57. #103×C173C22semidirect
58. #102×C173C22semidirect
197. C173#3197semidirect
208. #468D173semidirect
219. C2×SD16C2×D173semidirect
242. #107C2×D173semidirect
248. #103C2×D173semidirect
252. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120042
#260120012
#260118342
#260118312
#260118182
#260118132
#260117472
#260117422
#260117372
#260116562
#260116502
#260116372
#260116352
#260114542
#468×C1732
#3197173