Finite Group Databasebeta

← #36363750

Subgroups — #36363750

Order 22144.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012114C211yesmaximal, characteristic, complemented
2#26011965C211yesmaximal, characteristic, complemented
3#26011815C211yesmaximal, characteristic, complemented
4#26011780C211yesmaximal, characteristic, complemented
5#26011770C211yesmaximal, characteristic, complemented
6#26011709C211yesmaximal, characteristic, complemented
7#26011709C211yesmaximal, characteristic, complemented
8#26011641C211yesmaximal, characteristic, complemented
9#26011627C211yesmaximal, characteristic
10#26011602C211yesmaximal, characteristic, complemented
11#26011594C211yesmaximal, characteristic, complemented
12#26011576C211yesmaximal, characteristic, complemented
13#26011573C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#485×C173C211yesmaximal, characteristic, complemented
16C22×D692C2211yescharacteristic
17C2×#14583813C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C2×D1384C2211yescharacteristic
20C2×D1384C2211yescharacteristic
21#19149678C2211yescharacteristic, complemented
22#19149674—21no—
23#19149668C2211yescharacteristic, complemented
24#19149665—21no—
25#19149661—21no—
26#19149659—21no—
27#19149653C2211yescharacteristic
28Q8×Dic173C2211yescharacteristic, complemented
29C4×D692—21no—
30C4×D692—21no—
31#19149641C2211yescharacteristic
32#19149641C2211yescharacteristic
33#19149641C2211yescharacteristic
34#19149641C2211yescharacteristic
35#19149641C2211yescharacteristic, complemented
36#19149638C2211yescharacteristic
37C22⋊C4×D173—21no—
38C22⋊C4×D173—21no—
39#19149612C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149599C2211yescharacteristic
46#19149599C2211yescharacteristic
47#19149599C2211yescharacteristic, complemented
48#19149599C2211yescharacteristic, complemented
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C22⋊Q8×C173C2211yescharacteristic
53#117×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic, complemented
55#106×C173C2211yescharacteristic, complemented
56#103×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic
59C23×D173—21no—
60C2×Dic346—21no—
61C2×D692—22no—
62C2×D692—22no—
63C2×D692—21no—
64C2×D692C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C4×D173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73#14583813—22no—
74#14583813—22no—
75D1384—22no—
76D1384—22no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C4⋊Dic173—21no—
80C4⋊Dic173D411yescharacteristic
81C4⋊Dic173D411yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84C22⋊Dic173D411yescharacteristic
85C22⋊Dic173D411yescharacteristic
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96C4×Dic173C2311yescharacteristic
97C2×Q8×C173C2311yescharacteristic
98C22×C4×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C4⋊C4×C173—21no—
102C4⋊C4×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C42×C173—21no—
108C22×D173—21no—
109C22×D173—21no—
110C22×D173—24no—
111C22×D173—21no—
112C22×D173—81no—
113C22×D173—81no—
114Dic346—81no—
115D692—24no—
116D692—21no—
117D692—21no—
118D692—21no—
119D692—21no—
120D692#4112yes—
121D692#4112yes—
122C22⋊D173—81no—
123C2×Dic173—21no—
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C4×D173—81no—
129C4×D173—81no—
130C4×D173—81no—
131C4×D173—81no—
132C4×D173—81no—
133#13063436—21no—
134#13063436—21no—
135C23×C173C2×D411yescharacteristic
136Q8×C173—21no—
137Q8×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2411yescharacteristic
146C8×C173—21no—
147C8×C173—21no—
148C22×C173—21no—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173C22×D411yescharacteristic
152C2×D173—22no—
153C2×D173—22no—
154C2×D173—41no—
155C2×D173—41no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
159C2×D173—41no—
160C2×D173—41no—
161C2×D173—84no—
162C4×C173—21no—
163C4×C173—41no—
164C4×C173—41no—
165C4×C173—41no—
166C4×C173—41no—
167C4×C173C2×#4111yescharacteristic
168C4×C173C2×#4111yescharacteristic
169Dic173—21no—
170Dic173—21no—
171Dic173—21no—
172Dic173—21no—
173Dic173—41no—
174C2×C173—41no—
175C2×C173#52511yescharacteristic
176C2×C173#57611yescharacteristic
177C2×C173#57711yescharacteristic
178D173—42no—
179D173—42no—
180D173—81no—
181C173#311811yescharacteristic, complemented, Sylow
182#3118—1731nomaximal, Sylow
183#542—1731no—
184#518—1731no—
185#485D17311yescharacteristic, complemented
186#482—1731no—
187#476—1731no—
188#474—1731no—
189#463—1731no—
190#463—1731no—
191#450—1731no—
192#447—1731no—
193#442—1731no—
194#440—1731no—
195C4×SD16—1731no—
196C4×D8—1731no—
197#433—1731no—
198C22×D4—1731no—
199C2×SD16—1731no—
200C2×SD16—1731no—
201C2×D8—1731no—
202C2×D8—1731no—
203C4⋊Q8—1731no—
204#126—3461no—
205#124—3461no—
206#124—1731no—
207#123—3461no—
208C22⋊Q8—3461no—
209C22⋊Q8C2×D17311yescharacteristic
210#121—1731no—
211C4×Q8—1731no—
212C4×D4—3461no—
213C4×D4—3461no—
214C4×D4—1731no—
215C4×D4—1731no—
216C4×D4—1731no—
217C4×D4—1731no—
218C4×D4—1731no—
219#117—1731no—
220#117C2×D17311yescharacteristic
221C2×C22⋊C4—3461no—
222C2×C22⋊C4—3461no—
223#107—1731no—
224#107C2×D17311yescharacteristic, complemented
225#106—1731no—
226#106C2×D17311yescharacteristic, complemented
227C4⋊C8—1731no—
228C4⋊C8—1731no—
229#103C2×D17311yescharacteristic, complemented
230#103C2×D17311yescharacteristic, complemented
231#102—1731no—
232#102—1731no—
233#102—1731no—
234#102—1731no—
235#102—1731no—
236#102—1731no—
237C22⋊C8—1731no—
238C22⋊C8C2×D17311yescharacteristic
239#97—1731no—
240C4×C8—1731no—
241C24—3461no—
242C2×Q8—3461no—
243C2×Q8C22×D17311yescharacteristic
244C2×D4—3462no—
245C2×D4—3462no—
246C2×D4—3461no—
247C2×D4—3461no—
248C2×D4—1731no—
249C2×D4—1731no—
250C2×D4—1731no—
251C22×C4—3461no—
252C22×C4—3461no—
253C22×C4—3461no—
254C22×C4—3461no—
255C22×C4—3461no—
256C22×C4C22×D17311yescharacteristic
257SD16—3462no—
258SD16—3462no—
259D8—3462no—
260D8—3462no—
261C2×C8—1731no—
262C2×C8—1731no—
263C2×C8C22×D17311yescharacteristic
264C2×C8C22×D17311yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—1731no—
269C4⋊C4—1731no—
270C4⋊C4—1731no—
271C4⋊C4C22×D17311yescharacteristic
272C4⋊C4C22×D17311yescharacteristic
273C4⋊C4C22×D17311yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—1731no—
285C22⋊C4—1731no—
286C42—21no—
287C42—3461no—
288C42—1731no—
289C42—1731no—
290C23—3464no—
291C23—3461no—
292C23—3461no—
293C23—3461no—
294C23—13841no—
295C23—13841no—
296C23D4×D17311yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—13841no—
300D4—3464no—
301D4—3461no—
302D4—3461no—
303D4—3461no—
304D4—3461no—
305D4—13841no—
306D4—1732no—
307D4—1732no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—3461no—
315C2×C4—13841no—
316C2×C4—13841no—
317C2×C4—13841no—
318C2×C4—13841no—
319C2×C4—13841no—
320C2×C4—1731no—
321C2×C4—1731no—
322C2×C4—1731no—
323C2×C4—1731no—
324C2×C4D4×D17311yescharacteristic
325C2×C4C23×D17311yescharacteristic
326C8—21no—
327C8—21no—
328C8—3461no—
329C8—3461no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—3462no—
334C22—3462no—
335C22—13844no—
336C22—6921no—
337C22—6921no—
338C22—6921no—
339C22—6921no—
340C22—6921no—
341C22—6921no—
342C22—6921no—
343C22C2×D4×D17311yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—3461no—
350C4—3461no—
351C4—3461no—
352C4—3461no—
353C4—6921no—
354C4#41×D17311yescharacteristic
355C4#41×D17311yescharacteristic
356C2—41no—
357C2—13841no—
358C2—6922no—
359C2—6922no—
360C2#2601199911yescharacteristic
361C2#2601229311yescharacteristic
362C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#485×C173C2
socleC22×C173C22×D4
prime radical#26011627C2
hypercenter#485D173
hypocenterC173#3118
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #26012114C2semidirect
2. #26011965C2semidirect
3. #26011815C2semidirect
4. #26011780C2semidirect
5. #26011770C2semidirect
6. #26011709C2semidirect
7. #26011709C2semidirect
8. #26011641C2semidirect
10. #26011602C2semidirect
11. #26011594C2semidirect
12. #26011576C2semidirect
13. #26011573C2semidirect
14. #26011549C2semidirect
15. #485×C173C2semidirect
21. #19149678C22semidirect
23. #19149668C22semidirect
28. Q8×Dic173C22semidirect
35. #19149641C22semidirect
39. #19149612C22semidirect
40. #19149610C22semidirect
41. #19149608C22semidirect
42. #19149608C22semidirect
47. #19149599C22semidirect
48. #19149599C22semidirect
50. #19149589C22semidirect
51. C8×Dic173C22semidirect
54. #107×C173C22semidirect
55. #106×C173C22semidirect
56. #103×C173C22semidirect
57. #103×C173C22semidirect
181. C173#3118semidirect
185. #485D173semidirect
224. #107C2×D173semidirect
226. #106C2×D173semidirect
229. #103C2×D173semidirect
230. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121142
#260119652
#260118152
#260117802
#260117702
#260117092
#260116412
#260116272
#260116022
#260115942
#260115762
#260115732
#260115492
#485×C1732
#3118173