Finite Group Databasebeta

← #36365580

Subgroups — #36365580

Order 22144.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012176C211yesmaximal, characteristic, complemented
2D4×C22⋊D173C211yesmaximal, characteristic
3#26011854C211yesmaximal, characteristic, complemented
4#26011814C211yesmaximal, characteristic, complemented
5#26011804C211yesmaximal, characteristic, complemented
6#26011793C211yesmaximal, characteristic, complemented
7#26011736C211yesmaximal, characteristic, complemented
8#26011720C211yesmaximal, characteristic, complemented
9#26011690C211yesmaximal, characteristic, complemented
10#26011643C211yesmaximal, characteristic, complemented
11#26011642C211yesmaximal, characteristic, complemented
12#26011588C211yesmaximal, characteristic, complemented
13#26011562C211yesmaximal, characteristic, complemented
14#26011494C211yesmaximal, characteristic, complemented
15C2×#102×C173C211yesmaximal, characteristic, complemented
16C22×C22⋊D173—21no—
17C2×D4×D173C2211yescharacteristic
18C2×#14583815C2211yescharacteristic, complemented
19C2×#14583814C2211yescharacteristic
20C2×#14583814C2211yescharacteristic, complemented
21C2×#14583809C2211yescharacteristic, complemented
22#19149675C2211yescharacteristic
23#19149674—21no—
24#19149670C2211yescharacteristic, complemented
25#19149669—21no—
26#19149662—21no—
27#19149659C2211yescharacteristic, complemented
28#19149658C2211yescharacteristic, complemented
29#19149653C2211yescharacteristic
30#19149651—21no—
31#19149650C2211yescharacteristic
32#19149648—21no—
33#19149647—21no—
34#19149644C2211yescharacteristic, complemented
35D4×Dic173C2211yescharacteristic
36C4×C22⋊D173C2211yescharacteristic
37#19149638—21no—
38#19149637C2211yescharacteristic, complemented
39#19149612C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149611C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic, complemented
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic, complemented
48#19149599C2211yescharacteristic, complemented
49C22⋊#13063436C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic, complemented
51#19149589C2211yescharacteristic, complemented
52C22×D4×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic, complemented
54C2×C4⋊C4×C173C2211yescharacteristic, complemented
55#102×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C2×Dic346C2311yescharacteristic
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—44no—
68C2×C22⋊D173D411yescharacteristic
69D4×D173—22no—
70D4×D173—41no—
71C22×Dic173—21no—
72C2×C4×D173C2311yescharacteristic
73#14583815—41no—
74#14583814—22no—
75#14583814—41no—
76#14583809—41no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C4⋊Dic173—21no—
80C4⋊Dic173—21no—
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#14583803D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C24×C173—21no—
99C2×D4×C173—22no—
100C2×D4×C173—21no—
101C2×D4×C173—41no—
102C2×D4×C173C2311yescharacteristic
103C2×D4×C173C2311yescharacteristic
104C22×C4×C173—21no—
105C22×C4×C173C2311yescharacteristic
106C2×C8×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173—21no—
110C4⋊C4×C173C2311yescharacteristic
111C4⋊C4×C173C2311yescharacteristic
112C22×D173—21no—
113C22×D173—42no—
114C22×D173—42no—
115C22×D173—81no—
116C22×D173C2×D411yescharacteristic
117Dic346—21no—
118Dic346—21no—
119D692—21no—
120D692—21no—
121C22⋊D173—22no—
122C22⋊D173—42no—
123C22⋊D173—42no—
124C22⋊D173—42no—
125C22⋊D173—42no—
126C22⋊D173—81no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—42no—
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C4×D173—21no—
135C4×D173—21no—
136#13063436—21no—
137#13063436—21no—
138C23×C173—21no—
139C23×C173—21no—
140C23×C173—42no—
141C23×C173—42no—
142C23×C173—81no—
143C23×C173C2×D411yescharacteristic
144D4×C173—21no—
145D4×C173—21no—
146D4×C173—21no—
147D4×C173—42no—
148D4×C173#4112yes—
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—22no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—84no—
167C22×C173AGL(1,ℤ/8)12yes—
168C22×C173C22×D411yescharacteristic
169C2×D173—21no—
170C2×D173—21no—
171C2×D173—21no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—84no—
176C4×C173—21no—
177C4×C173—41no—
178C4×C173—41no—
179C4×C173C2×#4111yescharacteristic
180C4×C173H(5,𝔽2)11yescharacteristic
181Dic173—21no—
182Dic173—21no—
183Dic173—21no—
184Dic173—41no—
185Dic173—41no—
186Dic173—41no—
187C2×C173—22no—
188C2×C173—42no—
189C2×C173—81no—
190C2×C173#54611yescharacteristic
191C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C173#57711yescharacteristic
193D173—41no—
194D173—81no—
195C173#331011yescharacteristic, complemented, Sylow
196#3310—1731nomaximal, Sylow
197#552—1731no—
198D42—1731no—
199#490—1731no—
200#482—1731no—
201#480—1731no—
202#478—1731no—
203#468—1731no—
204#465—1731no—
205#460—1731no—
206#450—1731no—
207#450—1731no—
208#440—1731no—
209#435—1731no—
210#418—1731no—
211C2×#102D17311yescharacteristic, complemented
212C22×D4—3461no—
213C22×D4—1731no—
214C22×D4C2×D17311yescharacteristic
215C2×SD16—1731no—
216C2×SD16—1731no—
217C2×SD16—1731no—
218C2×M16—1731no—
219C22×C8C2×D17311yescharacteristic, complemented
220#127—1731no—
221#126—3461no—
222#125—3461no—
223#125—1731no—
224#123—3461no—
225C22⋊Q8—1731no—
226C22⋊Q8—1731no—
227#121—3461no—
228#121—1731no—
229#121—1731no—
230#120—3461no—
231#120—3461no—
232C4×Q8—1731no—
233C4×D4—1731no—
234C4×D4—1731no—
235#117—3461no—
236#117—1731no—
237C2×C4⋊C4C2×D17311yescharacteristic, complemented
238#107—1731no—
239#107—1731no—
240#106—1731no—
241C4⋊C8—1731no—
242#103—1731no—
243#102—1731no—
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102—1731no—
248#102C2×D17311yescharacteristic
249#102C2×D17311yescharacteristic
250#102C2×D17311yescharacteristic, complemented
251#102C2×D17311yescharacteristic, complemented
252C22⋊C8—1731no—
253C22⋊C8—1731no—
254#97—1731no—
255C24—21no—
256C24—3461no—
257C2×Q8—1731no—
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—3462no—
262C2×D4—3461no—
263C2×D4—3461no—
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—3461no—
267C2×D4—1731no—
268C2×D4—1731no—
269C2×D4—6921no—
270C2×D4—6924no—
271C2×D4C22×D17311yescharacteristic
272C2×D4C22×D17311yescharacteristic
273C22×C4—21no—
274C22×C4—3461no—
275C22×C4—1731no—
276C22×C4C22×D17311yescharacteristic
277SD16—3462no—
278SD16—6921no—
279SD16—6921no—
280M16—6921no—
281C2×C8—41no—
282C2×C8—1731no—
283C2×C8—1731no—
284C2×C8C22×D17311yescharacteristic
285C2×C8C22×D17311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—3461no—
288C4⋊C4—3461no—
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—3461no—
292C4⋊C4—1731no—
293C4⋊C4—1731no—
294C4⋊C4—1731no—
295C4⋊C4C22×D17311yescharacteristic
296C4⋊C4C22×D17311yescharacteristic
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4—3461no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—1731no—
304C22⋊C4—1731no—
305C42—3461no—
306C42—3461no—
307C42—1731no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—3461no—
314C23—13841no—
315C23—1731no—
316C23—6922no—
317C23—6922no—
318C23D4×D17311yescharacteristic
319Q8—3461no—
320Q8—3461no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—3462no—
326D4—3461no—
327D4—3461no—
328D4—13841no—
329D4—6922no—
330D4—6922no—
331D4—6922no—
332D4—6922no—
333D4#1458382612yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—3461no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—1731no—
345C2×C4—1731no—
346C2×C4—1731no—
347C2×C4—6922no—
348C2×C4D4×D17311yescharacteristic
349C2×C4C23×D17311yescharacteristic
350C8—21no—
351C8—21no—
352C8—3461no—
353C8—3461no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—84no—
363C22—3461no—
364C22—3461no—
365C22—3461no—
366C22—13844no—
367C22—6921no—
368C22—6921no—
369C22—6921no—
370C22#1914970312yes—
371C22C2×D4×D17311yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—3461no—
376C4—3461no—
377C4—3461no—
378C4—6921no—
379C4—6921no—
380C4—6921no—
381C4C2×#1458382611yescharacteristic
382C4#1914972411yescharacteristic
383C2—22no—
384C2—42no—
385C2—81no—
386C2—13841no—
387C2—6921no—
388C2#2601213111yescharacteristic
389C2C2×#1914970311yescharacteristic
390C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#102×C173C2
socleC22×C173C22×D4
prime radicalD4×C22⋊D173C2
hypercenterC2×#102D173
hypocenterC173#3310
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #26012176C2semidirect
3. #26011854C2semidirect
4. #26011814C2semidirect
5. #26011804C2semidirect
6. #26011793C2semidirect
7. #26011736C2semidirect
8. #26011720C2semidirect
9. #26011690C2semidirect
10. #26011643C2semidirect
11. #26011642C2semidirect
12. #26011588C2semidirect
13. #26011562C2semidirect
14. #26011494C2semidirect
15. C2×#102×C173C2semidirect
18. C2×#14583815C22semidirect
20. C2×#14583814C22semidirect
21. C2×#14583809C22semidirect
24. #19149670C22semidirect
27. #19149659C22semidirect
28. #19149658C22semidirect
34. #19149644C22semidirect
38. #19149637C22semidirect
39. #19149612C22semidirect
40. #19149612C22semidirect
41. #19149611C22semidirect
42. #19149608C22semidirect
43. #19149604C22semidirect
44. #19149601C22semidirect
46. #19149600C22semidirect
47. #19149599C22semidirect
48. #19149599C22semidirect
49. C22⋊#13063436C22semidirect
50. #19149591C22semidirect
51. #19149589C22semidirect
53. C22×C8×C173C22semidirect
54. C2×C4⋊C4×C173C22semidirect
57. #102×C173C22semidirect
58. #102×C173C22semidirect
195. C173#3310semidirect
211. C2×#102D173semidirect
219. C22×C8C2×D173semidirect
237. C2×C4⋊C4C2×D173semidirect
250. #102C2×D173semidirect
251. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121762
D4×C22⋊D1732
#260118542
#260118142
#260118042
#260117932
#260117362
#260117202
#260116902
#260116432
#260116422
#260115882
#260115622
#260114942
C2×#102×C1732
#3310173