Finite Group Databasebeta

← #36365527

Subgroups — #36365527

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#26012142C211yesmaximal, characteristic, complemented
2#26012138C211yesmaximal, characteristic, complemented
3#26011849C211yesmaximal, characteristic, complemented
4#26011823C211yesmaximal, characteristic, complemented
5#26011780C211yesmaximal, characteristic, complemented
6#26011742C211yesmaximal, characteristic, complemented
7#26011721C211yesmaximal, characteristic, complemented
8#26011705C211yesmaximal, characteristic, complemented
9#26011638C211yesmaximal, characteristic, complemented
10#26011626C211yesmaximal, characteristic, complemented
11#26011602C211yesmaximal, characteristic, complemented
12#26011565C211yesmaximal, characteristic, complemented
13#26011490C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#481×C173C211yesmaximal, characteristic, complemented
16C2×#14583826—21no—
17C2×#14583826C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19C2×#14583815C2211yescharacteristic, complemented
20C2×D1384C2211yescharacteristic
21C2×#14583811C2211yescharacteristic
22C2×#14583810C2211yescharacteristic, complemented
23C2×C8×D173C2211yescharacteristic
24#19149678C2211yescharacteristic, complemented
25#19149666C2211yescharacteristic
26#19149664—21no—
27#19149662—21no—
28#19149661—21no—
29#19149659C2211yescharacteristic
30#19149657—21no—
31#19149654C2211yescharacteristic
32#19149653C2211yescharacteristic, complemented
33#19149650—21no—
34#19149647—21no—
35#19149641C2211yescharacteristic
36#19149641C2211yescharacteristic, complemented
37C4×C22⋊D173C2211yescharacteristic
38C4⋊C4×D173C2211yescharacteristic
39C22⋊C4×D173—21no—
40#19149610C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149602C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149599C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic
51#19149589C2211yescharacteristic, complemented
52#121×C173C2211yescharacteristic
53C2×C4⋊C4×C173C2211yescharacteristic, complemented
54#106×C173C2211yescharacteristic, complemented
55#106×C173C2211yescharacteristic, complemented
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583826—22no—
61#14583826—41no—
62#14583826—44no—
63C2×Dic346—21no—
64C2×Dic346C2311yescharacteristic
65C2×D692—21no—
66C2×D692C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173D411yescharacteristic
72D4×D173—22no—
73D4×D173—41no—
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76C2×C4×D173—21no—
77C2×C4×D173—21no—
78C2×C4×D173C2311yescharacteristic
79#14583815—41no—
80D1384—22no—
81#14583811—41no—
82#14583810—41no—
83C2×#13063436C2311yescharacteristic
84C2×#13063436C2311yescharacteristic
85C8×D173—41no—
86C4⋊Dic173—21no—
87C4⋊Dic173C2311yescharacteristic
88#14583805—21no—
89#14583805—21no—
90#14583805D411yescharacteristic
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173D411yescharacteristic
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803D411yescharacteristic
100C4×Dic173C2311yescharacteristic
101C2×D4×C173—21no—
102C2×D4×C173C2311yescharacteristic
103C22×C4×C173—21no—
104C22×C4×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173—21no—
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C4⋊C4×C173C2311yescharacteristic
111C22⋊C4×C173—21no—
112C22×D173—21no—
113C22×D173—21no—
114C22×D173—42no—
115C22×D173—42no—
116C22×D173—81no—
117C22×D173C2×D411yescharacteristic
118Dic346—21no—
119Dic346—21no—
120Dic346—42no—
121D692—21no—
122D692—21no—
123D692—21no—
124D692—42no—
125D692#4112yescomplemented
126C22⋊D173—22no—
127C22⋊D173—42no—
128C22⋊D173—42no—
129C22⋊D173—42no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C2×Dic173C2×D411yescharacteristic
135C4×D173—21no—
136C4×D173—21no—
137C4×D173—42no—
138C4×D173—42no—
139C4×D173—42no—
140C4×D173—81no—
141#13063436—21no—
142#13063436—21no—
143C23×C173—21no—
144C23×C173C2×D411yescharacteristic
145D4×C173—21no—
146D4×C173—21no—
147D4×C173—81no—
148C2×C4×C173—21no—
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—21no—
160C22×C173—21no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173C22×D411yescharacteristic
165C2×D173—21no—
166C2×D173—21no—
167C2×D173—22no—
168C2×D173—41no—
169C2×D173—41no—
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×C173—41no—
179C4×C173C2×#4111yescharacteristic
180C4×C173H(5,𝔽2)11yescharacteristic
181Dic173—21no—
182Dic173—21no—
183Dic173—21no—
184Dic173—41no—
185Dic173—41no—
186C2×C173—41no—
187C2×C173—81no—
188C2×C173#54611yescharacteristic
189C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C173#57711yescharacteristic
191D173—22no—
192D173—42no—
193D173—81no—
194C173#330611yescharacteristic, complemented, Sylow
195#3306—1731nomaximal, Sylow
196#547—1731no—
197#546—1731no—
198#489—1731no—
199#483—1731no—
200#481D17311yescharacteristic, complemented
201#476—1731no—
202#469—1731no—
203#465—1731no—
204#463—1731no—
205#449—1731no—
206#447—1731no—
207#442—1731no—
208#435—1731no—
209#417—1731no—
210C2×#102—1731no—
211C2×#41—3461no—
212C2×#41—1731no—
213C22×D4—1731no—
214C2×SD16—1731no—
215C2×D8—1731no—
216C2×D8—1731no—
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
230#120—3461no—
231C4×D4—1731no—
232C4×D4—1731no—
233C4×D4—1731no—
234C2×C4⋊C4—1731no—
235C2×C4⋊C4C2×D17311yescharacteristic, complemented
236C2×C22⋊C4—3461no—
237#106—1731no—
238#106C2×D17311yescharacteristic, complemented
239#106C2×D17311yescharacteristic, complemented
240C4⋊C8—1731no—
241#103—1731no—
242#103—1731no—
243#103—1731no—
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102C2×D17311yescharacteristic
250#102C2×D17311yescharacteristic
251C22⋊C8—1731no—
252C22⋊C8C2×D17311yescharacteristic, complemented
253#97—1731no—
254C24—3461no—
255#41—3462no—
256#41—6921no—
257#41—6924no—
258C2×Q8—3461no—
259C2×Q8—1731no—
260C2×D4—21no—
261C2×D4—3461no—
262C2×D4—3462no—
263C2×D4—3461no—
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—1731no—
267C2×D4—1731no—
268C2×D4—1731no—
269C2×D4—6921no—
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—6921no—
279D8—3462no—
280D8—6921no—
281M16—6921no—
282C2×C8—1731no—
283C2×C8—1731no—
284C2×C8—6921no—
285C2×C8C22×D17311yescharacteristic
286C2×C8C22×D17311yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—3461no—
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—1731no—
292C4⋊C4—1731no—
293C4⋊C4C22×D17311yescharacteristic
294C4⋊C4C22×D17311yescharacteristic
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—3461no—
309C23—3461no—
310C23—13841no—
311C23—1731no—
312C23—6922no—
313C23—6922no—
314C23D4×D17311yescharacteristic
315Q8—3461no—
316Q8—3461no—
317Q8—6922no—
318D4—21no—
319D4—21no—
320D4—81no—
321D4—3462no—
322D4—3461no—
323D4—3461no—
324D4—3461no—
325D4—1732no—
326D4—6922no—
327D4—6922no—
328D4—6922no—
329D4—6922no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
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—41no—
357C22—41no—
358C22—41no—
359C22—3461no—
360C22—3461no—
361C22—3462no—
362C22—13844no—
363C22—1732no—
364C22—6921no—
365C22—6921no—
366C22—6921no—
367C22—6921no—
368C22—6921no—
369C22C2×D4×D17311yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—3461no—
375C4—3461no—
376C4—3461no—
377C4—6921no—
378C4—6921no—
379C4#41×D17311yescharacteristic
380C4#1914972511yescharacteristic
381C2—41no—
382C2—81no—
383C2—3462no—
384C2—13841no—
385C2—6922no—
386C2#2601213311yescharacteristic
387C2AGL(1,ℤ/8)×D17311yescharacteristic
388C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#481×C173C2
socleC22×C173C22×D4
hypercenter#481D173
hypocenterC173#3306
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #26012142C2semidirect
2. #26012138C2semidirect
3. #26011849C2semidirect
4. #26011823C2semidirect
5. #26011780C2semidirect
6. #26011742C2semidirect
7. #26011721C2semidirect
8. #26011705C2semidirect
9. #26011638C2semidirect
10. #26011626C2semidirect
11. #26011602C2semidirect
12. #26011565C2semidirect
13. #26011490C2semidirect
14. #102×D173C2semidirect
15. #481×C173C2semidirect
19. C2×#14583815C22semidirect
22. C2×#14583810C22semidirect
24. #19149678C22semidirect
32. #19149653C22semidirect
36. #19149641C22semidirect
40. #19149610C22semidirect
41. #19149608C22semidirect
42. #19149604C22semidirect
43. #19149602C22semidirect
44. #19149602C22semidirect
49. #19149599C22semidirect
51. #19149589C22semidirect
53. C2×C4⋊C4×C173C22semidirect
54. #106×C173C22semidirect
55. #106×C173C22semidirect
58. C22⋊C8×C173C22semidirect
125. D692#41semidirect
194. C173#3306semidirect
200. #481D173semidirect
235. C2×C4⋊C4C2×D173semidirect
238. #106C2×D173semidirect
239. #106C2×D173semidirect
252. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121422
#260121382
#260118492
#260118232
#260117802
#260117422
#260117212
#260117052
#260116382
#260116262
#260116022
#260115652
#260114902
#102×D1732
#481×C1732
#3306173