Finite Group Databasebeta

← #36364355

Subgroups — #36364355

Order 22144.

424 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#26011998C211yesmaximal, characteristic, complemented
3#26011830C211yesmaximal, characteristic, complemented
4#26011829C211yesmaximal, characteristic, complemented
5#26011816C211yesmaximal, characteristic, complemented
6#26011813C211yesmaximal, characteristic, complemented
7#26011729C211yesmaximal, characteristic, complemented
8#26011726C211yesmaximal, characteristic, complemented
9#26011723C211yesmaximal, characteristic, complemented
10#26011717C211yesmaximal, characteristic, complemented
11#26011652C211yesmaximal, characteristic, complemented
12#26011636C211yesmaximal, characteristic, complemented
13#26011635C211yesmaximal, characteristic, complemented
14C22⋊C8×D173C211yesmaximal, characteristic, complemented
15#452×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C22×C4×D173C2211yescharacteristic
18C2×#14583813C2211yescharacteristic, complemented
19C2×#14583811C2211yescharacteristic, complemented
20C2×C8×D173C2211yescharacteristic
21C2×C8×D173C2211yescharacteristic
22#19149664—21no—
23#19149660—21no—
24#19149655C2211yescharacteristic
25#19149655C2211yescharacteristic
26#19149655C2211yescharacteristic
27#19149654C2211yescharacteristic
28#19149653C2211yescharacteristic
29#19149650C2211yescharacteristic
30#19149650C2211yescharacteristic
31#19149649—21no—
32#19149647—21no—
33#19149641—21no—
34C4×C22⋊D173—21no—
35C4×C22⋊D173—21no—
36#19149639—21no—
37#19149638C2211yescharacteristic
38#19149637C2211yescharacteristic
39#19149613C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149611C2211yescharacteristic, complemented
42#19149610C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149603C2211yescharacteristic, complemented
45#19149603C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic, complemented
47#19149601C2211yescharacteristic, complemented
48#19149600C2211yescharacteristic, complemented
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic, complemented
55C22⋊Q8×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic
59C23×D173D411yescharacteristic
60#14583827—41no—
61#14583827—41no—
62C2×D692—21no—
63C2×D692—21no—
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C22×Dic173D411yescharacteristic
69C2×C4×D173—21no—
70C2×C4×D173—41no—
71C2×C4×D173—41no—
72C2×C4×D173D411yescharacteristic
73C2×C4×D173D411yescharacteristic
74C2×C4×D173C2311yescharacteristic
75C2×C4×D173C2311yescharacteristic
76#14583813—41no—
77#14583811—41no—
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C8×D173—41no—
81C8×D173—41no—
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84C4⋊Dic173C2311yescharacteristic
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97C4×Dic173—21no—
98C4×Dic173—21no—
99#41×C173—41no—
100#41×C173—41no—
101C2×Q8×C173C2311yescharacteristic
102C2×Q8×C173C2311yescharacteristic
103C2×D4×C173—21no—
104C2×D4×C173C2311yescharacteristic
105C22×C4×C173—21no—
106C22×C4×C173C2311yescharacteristic
107Q16×C173—41no—
108SD16×C173—41no—
109C2×C8×C173C2311yescharacteristic
110C2×C8×C173C2311yescharacteristic
111C4⋊C4×C173—21no—
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173—22no—
115C22×D173—21no—
116C22×D173—41no—
117C22×D173—41no—
118C22×D173C2×D411yescharacteristic
119C22×D173C2×D411yescharacteristic
120D692—21no—
121D692—21no—
122D692—42no—
123D692—42no—
124C22⋊D173—42no—
125C22⋊D173—42no—
126C22⋊D173—42no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—41no—
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C4×D173—21no—
134C4×D173—21no—
135C4×D173—21no—
136C4×D173—21no—
137C4×D173—41no—
138C4×D173—41no—
139C4×D173—42no—
140#13063436—21no—
141#13063436—21no—
142C23×C173—21no—
143C23×C173C2×D411yescharacteristic
144Q8×C173—21no—
145Q8×C173—21no—
146Q8×C173—21no—
147Q8×C173—21no—
148D4×C173—21no—
149D4×C173—21no—
150D4×C173—42no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—42no—
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—44no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173#13512yes—
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×C173C2×#13511yescharacteristic
193C2×C173#57511yescharacteristic
194D173—22no—
195D173—41no—
196D173—81no—
197C173#319111yescharacteristic, complemented, Sylow
198#3191—1731nomaximal, Sylow
199#525—1731no—
200#525—1731no—
201#485—1731no—
202#485—1731no—
203#482—1731no—
204#482—1731no—
205#467—1731no—
206#466—1731no—
207#465—1731no—
208#465—1731no—
209#452—1731no—
210#452D17311yescharacteristic, complemented
211#449—1731no—
212#449—1731no—
213C2×C22⋊C8—1731no—
214C2×#41—1731no—
215C2×#41C2×D17311yescharacteristic
216C23×C4—1731no—
217C2×Q16C2×D17311yescharacteristic, complemented
218C2×SD16—1731no—
219C2×SD16C2×D17311yescharacteristic, complemented
220C2×D8—1731no—
221C22×C8—1731no—
222C22×C8—1731no—
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#121—1731no—
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#107—1731no—
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⋊C8C2×D17311yescharacteristic
257C24—1731no—
258#41—41no—
259#41—41no—
260#41—6921no—
261#41—6921no—
262C2×Q8C22×D17311yescharacteristic
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—21no—
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—41no—
283SD16—41no—
284SD16—6921no—
285D8—6921no—
286C2×C8—1731no—
287C2×C8—1731no—
288C2×C8—6921no—
289C2×C8—6921no—
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⋊C4—1731no—
299C4⋊C4C22×D17311yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—3461no—
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—3462no—
314C23—3461no—
315C23—1731no—
316C23—1731no—
317C23—6921no—
318C23—6921no—
319C23D4×D17311yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
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—42no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—3461no—
347C2×C4—1731no—
348C2×C4—1731no—
349C2×C4—6921no—
350C2×C4—6921no—
351C2×C4—6922no—
352C2×C4—6921no—
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—1732no—
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#1914972411yescharacteristic
387C2—41no—
388C2—81no—
389C2—3462no—
390C2—13841no—
391C2—6921no—
392C2#2601205111yescharacteristic
393C2#135×D17311yescharacteristic
394C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#452×C173C2
socleC22×C173C22×D4
hypercenter#452D173
hypocenterC173#3191
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #26012004C2semidirect
2. #26011998C2semidirect
3. #26011830C2semidirect
4. #26011829C2semidirect
5. #26011816C2semidirect
6. #26011813C2semidirect
7. #26011729C2semidirect
8. #26011726C2semidirect
9. #26011723C2semidirect
10. #26011717C2semidirect
11. #26011652C2semidirect
12. #26011636C2semidirect
13. #26011635C2semidirect
14. C22⋊C8×D173C2semidirect
15. #452×C173C2semidirect
18. C2×#14583813C22semidirect
19. C2×#14583811C22semidirect
39. #19149613C22semidirect
40. #19149612C22semidirect
41. #19149611C22semidirect
42. #19149610C22semidirect
43. #19149603C22semidirect
44. #19149603C22semidirect
45. #19149603C22semidirect
46. #19149601C22semidirect
47. #19149601C22semidirect
48. #19149600C22semidirect
53. C2×Q16×C173C22semidirect
54. C2×SD16×C173C22semidirect
56. #103×C173C22semidirect
57. #102×C173C22semidirect
197. C173#3191semidirect
210. #452D173semidirect
217. C2×Q16C2×D173semidirect
219. C2×SD16C2×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
#260119982
#260118302
#260118292
#260118162
#260118132
#260117292
#260117262
#260117232
#260117172
#260116522
#260116362
#260116352
C22⋊C8×D1732
#452×C1732
#3191173