Finite Group Databasebeta

← #36365517

Subgroups — #36365517

Order 22144.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012141C211yesmaximal, characteristic, complemented
2#26012129C211yesmaximal, characteristic, complemented
3#26011852C211yesmaximal, characteristic, complemented
4#26011820C211yesmaximal, characteristic, complemented
5#26011810C211yesmaximal, characteristic, complemented
6#26011779C211yesmaximal, characteristic, complemented
7#26011742C211yesmaximal, characteristic, complemented
8#26011717C211yesmaximal, characteristic, complemented
9#26011708C211yesmaximal, characteristic, complemented
10#26011623C211yesmaximal, characteristic, complemented
11#26011603C211yesmaximal, characteristic, complemented
12#26011565C211yesmaximal, characteristic, complemented
13#26011492C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic, complemented
16C2×#14583825—21no—
17C2×D4×D173C2211yescharacteristic
18C2×#14583811C2211yescharacteristic
19C2×#14583810C2211yescharacteristic, complemented
20C2×C8×D173C2211yescharacteristic
21#19149678C2211yescharacteristic, complemented
22#19149668C2211yescharacteristic
23#19149662—21no—
24#19149661—21no—
25#19149660—21no—
26#19149656—21no—
27#19149655C2211yescharacteristic
28#19149653C2211yescharacteristic
29#19149651—21no—
30#19149650C2211yescharacteristic
31#19149647—21no—
32D4×Dic173C2211yescharacteristic
33D4×Dic173C2211yescharacteristic, complemented
34C4×C22⋊D173C2211yescharacteristic
35C2×C4⋊Dic173C2211yescharacteristic, complemented
36C4⋊C4×D173C2211yescharacteristic
37C2×#14583803—21no—
38#19149611C2211yescharacteristic, complemented
39#19149611C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149603C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic, complemented
49#19149599C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149589C2211yescharacteristic, complemented
52C2×#41×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic, complemented
54C2×D8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic, complemented
56#103×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583825—44no—
61C2×Dic346—21no—
62C2×D692C2311yescharacteristic
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173D411yescharacteristic
67D4×D173—22no—
68D4×D173—41no—
69C22×Dic173—21no—
70C22×Dic173—21no—
71C22×Dic173—21no—
72C2×C4×D173—21no—
73C2×C4×D173C2311yescharacteristic
74#14583811—41no—
75#14583810—41no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C8×D173—41no—
79C4⋊Dic173—21no—
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805D411yescharacteristic
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173D411yescharacteristic
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803D411yescharacteristic
96C4×Dic173C2311yescharacteristic
97#41×C173—22no—
98#41×C173—41no—
99C2×Q8×C173C2311yescharacteristic
100C2×D4×C173—21no—
101C2×D4×C173—21no—
102C2×D4×C173C2311yescharacteristic
103C2×D4×C173C2311yescharacteristic
104C22×C4×C173—21no—
105C22×C4×C173C2311yescharacteristic
106SD16×C173—41no—
107D8×C173—22no—
108C2×C8×C173C2311yescharacteristic
109C2×C8×C173C2311yescharacteristic
110C4⋊C4×C173C2311yescharacteristic
111C22⋊C4×C173—21no—
112C22×D173—21no—
113C22×D173—42no—
114C22×D173—42no—
115C22×D173—81no—
116C22×D173C2×D411yescharacteristic
117Dic346—42no—
118D692—21no—
119D692—21no—
120C22⋊D173—22no—
121C22⋊D173—42no—
122C22⋊D173—42no—
123C22⋊D173—81no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—42no—
128C2×Dic173—42no—
129C2×Dic173—81no—
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C4×D173—21no—
134C4×D173—21no—
135C4×D173—42no—
136#13063436—21no—
137#13063436—21no—
138C23×C173—21no—
139C23×C173—21no—
140C23×C173C2×D411yescharacteristic
141Q8×C173—21no—
142Q8×C173—21no—
143D4×C173—21no—
144D4×C173—21no—
145D4×C173—21no—
146D4×C173—42no—
147D4×C173—42no—
148D4×C173#4112yescomplemented
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—42no—
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2411yescharacteristic
156C8×C173—21no—
157C8×C173—21no—
158C22×C173—21no—
159C22×C173—22no—
160C22×C173—21no—
161C22×C173—21no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173C22×D411yescharacteristic
168C2×D173—21no—
169C2×D173—21no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173—41no—
173C2×D173—84no—
174C2×D173AGL(1,ℤ/8)12yes—
175C4×C173—21no—
176C4×C173—41no—
177C4×C173—41no—
178C4×C173C2×#4111yescharacteristic
179C4×C173H(5,𝔽2)11yescharacteristic
180Dic173—21no—
181Dic173—21no—
182Dic173—21no—
183Dic173—41no—
184Dic173—41no—
185Dic173—41no—
186C2×C173—42no—
187C2×C173—41no—
188C2×C173—81no—
189C2×C173#54611yescharacteristic
190C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C173#57711yescharacteristic
192D173—22no—
193D173—81no—
194C173#330611yescharacteristic, complemented, Sylow
195#3306—1731nomaximal, Sylow
196#547—1731no—
197#546—1731no—
198#489—1731no—
199#483—1731no—
200#481—1731no—
201#476—1731no—
202#469—1731no—
203#465—1731no—
204#463—1731no—
205#449D17311yescharacteristic, complemented
206#447—1731no—
207#442—1731no—
208#435—1731no—
209#417—1731no—
210C2×#102—1731no—
211C2×#41—3461no—
212C2×#41C2×D17311yescharacteristic
213C22×D4—1731no—
214C2×SD16C2×D17311yescharacteristic, complemented
215C2×D8—1731no—
216C2×D8C2×D17311yescharacteristic
217C2×M16—1731no—
218C22×C8—1731no—
219C4⋊Q8—1731no—
220#124—1731no—
221#123—3461no—
222#123—3461no—
223#123—3461no—
224C22⋊Q8—3461no—
225C22⋊Q8—1731no—
226#121—3461no—
227#121—1731no—
228#121—1731no—
229#121C2×D17311yescharacteristic, complemented
230#120—3461no—
231C4×D4—1731no—
232C4×D4—1731no—
233C4×D4—1731no—
234C2×C4⋊C4—1731no—
235C2×C4⋊C4—1731no—
236C2×C22⋊C4—3461no—
237#106—1731no—
238#106—1731no—
239#106—1731no—
240C4⋊C8—1731no—
241#103—1731no—
242#103—1731no—
243#103C2×D17311yescharacteristic, complemented
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102—1731no—
250#102C2×D17311yescharacteristic
251C22⋊C8—1731no—
252C22⋊C8C2×D17311yescharacteristic, complemented
253#97—1731no—
254C24—3461no—
255#41—22no—
256#41—41no—
257#41—6924no—
258C2×Q8—3461no—
259C2×Q8C22×D17311yescharacteristic
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—3462no—
263C2×D4—3461no—
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—1731no—
267C2×D4—1731no—
268C2×D4—6921no—
269C2×D4C22×D17311yescharacteristic
270C2×D4C22×D17311yescharacteristic
271C22×C4—21no—
272C22×C4—3461no—
273C22×C4—3461no—
274C22×C4—3461no—
275C22×C4—3461no—
276C22×C4—1731no—
277C22×C4C22×D17311yescharacteristic
278SD16—41no—
279D8—22no—
280D8—6921no—
281M16—6921no—
282C2×C8—1731no—
283C2×C8—1731no—
284C2×C8—6921no—
285C2×C8C22×D17311yescharacteristic
286C2×C8C22×D17311yescharacteristic
287C4⋊C4—3461no—
288C4⋊C4—3461no—
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—1731no—
292C4⋊C4—1731no—
293C4⋊C4—1731no—
294C4⋊C4—1731no—
295C4⋊C4C22×D17311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4—3461no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—1731no—
305C22⋊C4—1731no—
306C42—1731no—
307C23—21no—
308C23—21no—
309C23—3461no—
310C23—13841no—
311C23—1731no—
312C23—6922no—
313C23—6922no—
314C23D4×D17311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—6922no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—3462no—
324D4—3461no—
325D4—3461no—
326D4—13841no—
327D4—6922no—
328D4—6922no—
329D4#1458382512yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—3461no—
336C2×C4—3461no—
337C2×C4—3461no—
338C2×C4—3461no—
339C2×C4—3461no—
340C2×C4—13841no—
341C2×C4—1731no—
342C2×C4—1731no—
343C2×C4—1731no—
344C2×C4—6922no—
345C2×C4—6922no—
346C2×C4—6922no—
347C2×C4D4×D17311yescharacteristic
348C2×C4C23×D17311yescharacteristic
349C8—21no—
350C8—21no—
351C8—3461no—
352C8—3461no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—3461no—
363C22—3461no—
364C22—13844no—
365C22—1732no—
366C22—6921no—
367C22—6921no—
368C22—6921no—
369C22C2×D4×D17311yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—3461no—
374C4—3461no—
375C4—3461no—
376C4—6921no—
377C4—6921no—
378C4—6921no—
379C4C2×#1458382511yescharacteristic
380C4#1914972411yescharacteristic
381C2—41no—
382C2—42no—
383C2—81no—
384C2—3462no—
385C2—13841no—
386C2#2601212811yescharacteristic
387C2AGL(1,ℤ/8)×D17311yescharacteristic
388C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3306
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #26012141C2semidirect
2. #26012129C2semidirect
3. #26011852C2semidirect
4. #26011820C2semidirect
5. #26011810C2semidirect
6. #26011779C2semidirect
7. #26011742C2semidirect
8. #26011717C2semidirect
9. #26011708C2semidirect
10. #26011623C2semidirect
11. #26011603C2semidirect
12. #26011565C2semidirect
13. #26011492C2semidirect
14. #102×D173C2semidirect
15. #449×C173C2semidirect
19. C2×#14583810C22semidirect
21. #19149678C22semidirect
33. D4×Dic173C22semidirect
35. C2×C4⋊Dic173C22semidirect
38. #19149611C22semidirect
39. #19149611C22semidirect
40. #19149610C22semidirect
41. #19149608C22semidirect
42. #19149603C22semidirect
43. #19149603C22semidirect
48. #19149600C22semidirect
51. #19149589C22semidirect
53. C2×SD16×C173C22semidirect
55. #121×C173C22semidirect
56. #103×C173C22semidirect
58. C22⋊C8×C173C22semidirect
148. D4×C173#41semidirect
194. C173#3306semidirect
205. #449D173semidirect
214. C2×SD16C2×D173semidirect
229. #121C2×D173semidirect
243. #103C2×D173semidirect
252. C22⋊C8C2×D173semidirect
329. D4#14583825semidirect

Maximal subgroups

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

subgroupindex
#260121412
#260121292
#260118522
#260118202
#260118102
#260117792
#260117422
#260117172
#260117082
#260116232
#260116032
#260115652
#260114922
#102×D1732
#449×C1732
#3306173