Finite Group Databasebeta

← #36364429

Subgroups — #36364429

Order 22144.

410 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#26012064C211yesmaximal, characteristic, complemented
2#26012057C211yesmaximal, characteristic
3#26011835C211yesmaximal, characteristic, complemented
4#26011830C211yesmaximal, characteristic, complemented
5#26011817C211yesmaximal, characteristic, complemented
6#26011814C211yesmaximal, characteristic, complemented
7#26011739C211yesmaximal, characteristic, complemented
8#26011735C211yesmaximal, characteristic, complemented
9#26011720C211yesmaximal, characteristic, complemented
10#26011643C211yesmaximal, characteristic, complemented
11#26011640C211yesmaximal, characteristic, complemented
12#26011631C211yesmaximal, characteristic, complemented
13#26011628C211yesmaximal, characteristic, complemented
14#26011457C211yesmaximal, characteristic, complemented
15#465×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic, complemented
18C2×D4×D173C2211yescharacteristic
19C2×#14583815C2211yescharacteristic
20C2×#14583814C2211yescharacteristic
21C2×#14583813C2211yescharacteristic
22C2×#14583810C2211yescharacteristic, complemented
23#19149678—21no—
24#19149675—21no—
25#19149665—21no—
26#19149665—21no—
27#19149664—21no—
28#19149661—21no—
29#19149659C2211yescharacteristic
30#19149658C2211yescharacteristic, complemented
31#19149655C2211yescharacteristic
32#19149654C2211yescharacteristic
33#19149653C2211yescharacteristic
34#19149650C2211yescharacteristic
35#19149649—21no—
36#19149647—21no—
37#19149637C2211yescharacteristic
38#19149637C2211yescharacteristic, complemented
39#19149612C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149611C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic, complemented
52C2×SD16×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic, complemented
54C22⋊Q8×C173C2211yescharacteristic, complemented
55#121×C173C2211yescharacteristic
56#106×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic
59C23×D173—21no—
60#14583826—41no—
61#14583826—41no—
62C2×Dic346—21no—
63C2×Dic346D411yescharacteristic
64C2×Dic346C2311yescharacteristic
65Q8×D173—41no—
66Q8×D173—41no—
67C2×D692—21no—
68C2×D692D411yescharacteristic
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173D411yescharacteristic
74C2×C22⋊D173D411yescharacteristic
75D4×D173—41no—
76D4×D173—41no—
77C2×C4×D173—21no—
78C2×C4×D173C2311yescharacteristic
79C2×C4×D173C2311yescharacteristic
80#14583815—41no—
81#14583814—41no—
82#14583813—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—
106C2×C8×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—42no—
116C22×D173—81no—
117C22×D173—81no—
118C22×D173C2×D411yescharacteristic
119C22×D173C2×D411yescharacteristic
120Dic346—21no—
121Dic346—21no—
122Dic346—41no—
123Dic346—42no—
124D692—21no—
125D692—21no—
126D692—41no—
127D692—81no—
128C22⋊D173—41no—
129C22⋊D173—41no—
130C22⋊D173—42no—
131C22⋊D173—81no—
132C22⋊D173—81no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173C2×D411yescharacteristic
137C2×Dic173C2×D411yescharacteristic
138C4×D173—21no—
139C4×D173—21no—
140C4×D173—21no—
141C4×D173—21no—
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—81no—
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—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—84no—
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#57711yescharacteristic
193C2×C173#57711yescharacteristic
194D173—41no—
195D173—41no—
196D173—81no—
197C173#320011yescharacteristic, complemented, Sylow
198#3200—1731nomaximal, Sylow
199#536—1731no—
200#535—1731no—
201#485—1731no—
202#485—1731no—
203#482—1731no—
204#482—1731no—
205#468—1731no—
206#468—1731no—
207#465—1731no—
208#465D17311yescharacteristic, complemented
209#450—1731no—
210#450—1731no—
211#448—1731no—
212#448—1731no—
213#408—1731no—
214C2×#41—1731no—
215C22×Q8—1731no—
216C22×D4—1731no—
217C2×SD16—1731no—
218C2×SD16—1731no—
219C2×SD16—1731no—
220C2×SD16C2×D17311yescharacteristic
221C2×M16—1731no—
222C22×C8C2×D17311yescharacteristic, complemented
223C4⋊Q8—3461no—
224#127—3461no—
225#124—3461no—
226#124—3461no—
227#123—3461no—
228#123—3461no—
229C22⋊Q8—1731no—
230C22⋊Q8—1731no—
231C22⋊Q8—1731no—
232C22⋊Q8C2×D17311yescharacteristic, complemented
233#121—1731no—
234#121—1731no—
235#121—1731no—
236#121C2×D17311yescharacteristic
237#120—3461no—
238#120—3461no—
239#117—1731no—
240#117—1731no—
241#107—1731no—
242#107—1731no—
243#106—1731no—
244#106C2×D17311yescharacteristic, complemented
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
253C22⋊C8—1731no—
254C22⋊C8—1731no—
255C22⋊C8—1731no—
256C22⋊C8—1731no—
257C24—3461no—
258#41—6921no—
259#41—6921no—
260C2×Q8—3461no—
261C2×Q8—1731no—
262C2×Q8—1731no—
263C2×Q8—6921no—
264C2×Q8—6921no—
265C2×Q8C22×D17311yescharacteristic
266C2×D4—21no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—1731no—
272C2×D4—1731no—
273C2×D4—1731no—
274C2×D4—1731no—
275C2×D4—6921no—
276C2×D4—6921no—
277C2×D4C22×D17311yescharacteristic
278C22×C4—3461no—
279C22×C4—1731no—
280C22×C4—1731no—
281C22×C4C22×D17311yescharacteristic
282SD16—41no—
283SD16—6921no—
284SD16—6921no—
285SD16—6921no—
286M16—6921no—
287C2×C8—41no—
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—13841no—
315C23—13841no—
316C23—1731no—
317C23—1731no—
318C23—6922no—
319C23D4×D17311yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—3461no—
323Q8—3461no—
324Q8—6921no—
325Q8—6922no—
326D4—21no—
327D4—21no—
328D4—81no—
329D4—3461no—
330D4—3461no—
331D4—13841no—
332D4—13841no—
333D4—13841no—
334D4—6921no—
335D4—6921no—
336D4—6921no—
337D4—6922no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—3461no—
347C2×C4—3461no—
348C2×C4—3461no—
349C2×C4—3461no—
350C2×C4—1731no—
351C2×C4—1731no—
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—13844no—
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#2601229711yescharacteristic
394C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#465×C173C2
socleC22×C173C22×D4
prime radical#26012057C2
hypercenter#465D173
hypocenterC173#3200
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #26012064C2semidirect
3. #26011835C2semidirect
4. #26011830C2semidirect
5. #26011817C2semidirect
6. #26011814C2semidirect
7. #26011739C2semidirect
8. #26011735C2semidirect
9. #26011720C2semidirect
10. #26011643C2semidirect
11. #26011640C2semidirect
12. #26011631C2semidirect
13. #26011628C2semidirect
14. #26011457C2semidirect
15. #465×C173C2semidirect
17. C2×Q8×D173C22semidirect
22. C2×#14583810C22semidirect
30. #19149658C22semidirect
38. #19149637C22semidirect
39. #19149612C22semidirect
40. #19149612C22semidirect
41. #19149611C22semidirect
42. #19149604C22semidirect
43. #19149603C22semidirect
44. #19149602C22semidirect
50. #19149591C22semidirect
51. #19149591C22semidirect
53. C22×C8×C173C22semidirect
54. C22⋊Q8×C173C22semidirect
56. #106×C173C22semidirect
57. #103×C173C22semidirect
197. C173#3200semidirect
208. #465D173semidirect
222. C22×C8C2×D173semidirect
232. C22⋊Q8C2×D173semidirect
244. #106C2×D173semidirect
248. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120642
#260120572
#260118352
#260118302
#260118172
#260118142
#260117392
#260117352
#260117202
#260116432
#260116402
#260116312
#260116282
#260114572
#465×C1732
#3200173