Finite Group Databasebeta

← #36363699

Subgroups — #36363699

Order 22144.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012023C211yesmaximal, characteristic, complemented
2C4×D4×D173C211yesmaximal, characteristic, complemented
3#26011823C211yesmaximal, characteristic, complemented
4#26011803C211yesmaximal, characteristic, complemented
5#26011790C211yesmaximal, characteristic, complemented
6#26011789C211yesmaximal, characteristic, complemented
7#26011705C211yesmaximal, characteristic, complemented
8#26011704C211yesmaximal, characteristic, complemented
9#26011643C211yesmaximal, characteristic, complemented
10#26011623C211yesmaximal, characteristic
11#26011601C211yesmaximal, characteristic, complemented
12#26011593C211yesmaximal, characteristic, complemented
13#26011591C211yesmaximal, characteristic, complemented
14#26011546C211yesmaximal, characteristic, complemented
15#442×C173C211yesmaximal, characteristic, complemented
16C2×D4×D173C2211yescharacteristic
17C22×C4×D173—21no—
18C2×#14583815C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21#19149672—21no—
22#19149670C2211yescharacteristic, complemented
23#19149666C2211yescharacteristic, complemented
24#19149661—21no—
25#19149659C2211yescharacteristic
26#19149650C2211yescharacteristic
27C4×Dic346C2211yescharacteristic, complemented
28C4×D692C2211yescharacteristic
29D4×Dic173C2211yescharacteristic
30#19149641C2211yescharacteristic
31C4×C22⋊D173—21no—
32C4×C22⋊D173—21no—
33#19149639—21no—
34#19149638—21no—
35C4⋊C4×D173C2211yescharacteristic
36C22⋊C4×D173—21no—
37C42×D173C2211yescharacteristic
38#19149613C2211yescharacteristic, complemented
39#19149612C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic, complemented
48#19149599C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149590C2211yescharacteristic, complemented
52C2×D8×C173C2211yescharacteristic
53C4×D4×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic, complemented
55#106×C173C2211yescharacteristic, complemented
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic, complemented
58#97×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C2×Dic346C2311yescharacteristic
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64D4×D173—22no—
65D4×D173—22no—
66C22×Dic173—21no—
67C2×C4×D173—21no—
68C2×C4×D173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—44no—
71C2×C4×D173D411yescharacteristic
72C2×C4×D173D411yescharacteristic
73C2×C4×D173C2311yescharacteristic
74#14583815—22no—
75#14583814—22no—
76#14583811—22no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C4⋊Dic173—21no—
80C4⋊Dic173—21no—
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93C4×Dic173—21no—
94C4×Dic173D411yescharacteristic
95C4×Dic173D411yescharacteristic
96C4×Dic173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C2×D4×C173C2311yescharacteristic
99C22×C4×C173—21no—
100C22×C4×C173—21no—
101D8×C173—22no—
102C2×C8×C173C2311yescharacteristic
103C2×C8×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C42×C173C2311yescharacteristic
109C22×D173—24no—
110C22×D173—21no—
111C22×D173—42no—
112C22×D173—42no—
113C22×D173C2×D411yescharacteristic
114Dic346—21no—
115Dic346—21no—
116D692—21no—
117D692#4112yes—
118C22⋊D173—24no—
119C22⋊D173—42no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—42no—
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C4×D173—22no—
128C4×D173—21no—
129C4×D173—21no—
130C4×D173—41no—
131C4×D173—42no—
132C4×D173—42no—
133C4×D173—42no—
134#13063436—21no—
135#13063436—21no—
136C23×C173—21no—
137C23×C173—21no—
138D4×C173—21no—
139D4×C173—21no—
140D4×C173—21no—
141D4×C173#4112yes—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—42no—
145C2×C4×C173—42no—
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2411yescharacteristic
149C8×C173—21no—
150C8×C173—21no—
151C22×C173—22no—
152C22×C173—41no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173C22×D411yescharacteristic
158C2×D173—21no—
159C2×D173—21no—
160C2×D173—22no—
161C2×D173—48no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173AGL(1,ℤ/8)12yes—
165C4×C173—21no—
166C4×C173—21no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173C2×#4111yescharacteristic
170C4×C173C2×#4111yescharacteristic
171Dic173—21no—
172Dic173—21no—
173Dic173—21no—
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177C2×C173—42no—
178C2×C173—81no—
179C2×C173#52511yescharacteristic
180C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C173#57511yescharacteristic
182D173—22no—
183D173—42no—
184C173#311411yescharacteristic, complemented, Sylow
185#3114—1731nomaximal, Sylow
186#529—1731no—
187C2×C4×D4—1731no—
188#483—1731no—
189#480—1731no—
190#478—1731no—
191#478—1731no—
192#463—1731no—
193#463—1731no—
194#450—1731no—
195#447—1731no—
196#442—1731no—
197#442D17311yescharacteristic, complemented
198#440—1731no—
199#440—1731no—
200#432—1731no—
201C22×D4—1731no—
202C23×C4—3461no—
203C2×SD16—1731no—
204C2×SD16—1731no—
205C2×D8—1731no—
206C2×D8C2×D17311yescharacteristic
207#126—3461no—
208#125—1731no—
209#124—1731no—
210#123—3461no—
211C22⋊Q8—1731no—
212#121—1731no—
213C4×Q8—1731no—
214C4×D4—3461no—
215C4×D4—3461no—
216C4×D4—3461no—
217C4×D4—1731no—
218C4×D4—1731no—
219C4×D4—1731no—
220C4×D4C2×D17311yescharacteristic
221C4×D4C2×D17311yescharacteristic, complemented
222#117—3461no—
223C2×C4⋊C4—1731no—
224C2×C22⋊C4—3461no—
225C2×C42—1731no—
226#107—1731no—
227#107—1731no—
228#106—1731no—
229#106C2×D17311yescharacteristic, complemented
230C4⋊C8—1731no—
231C4⋊C8—1731no—
232#103—1731no—
233#103—1731no—
234#102—1731no—
235#102—1731no—
236#102—1731no—
237#102—1731no—
238#102C2×D17311yescharacteristic
239#102C2×D17311yescharacteristic, complemented
240C22⋊C8—1731no—
241C22⋊C8—1731no—
242#97—1731no—
243#97C2×D17311yescharacteristic, complemented
244C24—3461no—
245C2×Q8—1731no—
246C2×D4—3462no—
247C2×D4—3462no—
248C2×D4—3461no—
249C2×D4—3461no—
250C2×D4—1731no—
251C2×D4C22×D17311yescharacteristic
252C2×D4C22×D17311yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—3461no—
256C22×C4—3461no—
257C22×C4—3461no—
258C22×C4—3461no—
259C22×C4—1731no—
260C22×C4—1731no—
261C22×C4—1731no—
262C22×C4—6924no—
263SD16—3462no—
264SD16—3462no—
265D8—22no—
266D8—3462no—
267C2×C8—1731no—
268C2×C8—1731no—
269C2×C8C22×D17311yescharacteristic
270C2×C8C22×D17311yescharacteristic
271C4⋊C4—3461no—
272C4⋊C4—3461no—
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—1731no—
277C4⋊C4—1731no—
278C4⋊C4C22×D17311yescharacteristic
279C4⋊C4C22×D17311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C42—3461no—
290C42—1731no—
291C42—1731no—
292C42—1731no—
293C42C22×D17311yescharacteristic
294C23—21no—
295C23—21no—
296C23—3464no—
297C23—3461no—
298C23—1731no—
299C23—6922no—
300C23—6922no—
301Q8—3461no—
302Q8—3461no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—3464no—
307D4—3461no—
308D4—1732no—
309D4—6922no—
310D4#1458382612yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—3462no—
316C2×C4—3461no—
317C2×C4—3461no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—1731no—
322C2×C4—1731no—
323C2×C4—1731no—
324C2×C4—6921no—
325C2×C4—6922no—
326C2×C4—6922no—
327C2×C4—6922no—
328C2×C4—6922no—
329C2×C4D4×D17311yescharacteristic
330C2×C4D4×D17311yescharacteristic
331C2×C4C23×D17311yescharacteristic
332C8—21no—
333C8—21no—
334C8—3461no—
335C8—3461no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—3461no—
343C22—3461no—
344C22—3462no—
345C22—1732no—
346C22—6928no—
347C22—6921no—
348C22—6921no—
349C22C2×D4×D17311yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—3461no—
355C4—3461no—
356C4—3461no—
357C4—6921no—
358C4—6921no—
359C4—6921no—
360C4#41×D17311yescharacteristic
361C4C2×#1458382611yescharacteristic
362C2—42no—
363C2—81no—
364C2—3462no—
365C2—6922no—
366C2#2601200011yescharacteristic
367C2AGL(1,ℤ/8)×D17311yescharacteristic
368C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#442×C173C2
socleC22×C173C22×D4
prime radical#26011623C2
hypercenter#442D173
hypocenterC173#3114
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #26012023C2semidirect
2. C4×D4×D173C2semidirect
3. #26011823C2semidirect
4. #26011803C2semidirect
5. #26011790C2semidirect
6. #26011789C2semidirect
7. #26011705C2semidirect
8. #26011704C2semidirect
9. #26011643C2semidirect
11. #26011601C2semidirect
12. #26011593C2semidirect
13. #26011591C2semidirect
14. #26011546C2semidirect
15. #442×C173C2semidirect
22. #19149670C22semidirect
23. #19149666C22semidirect
27. C4×Dic346C22semidirect
38. #19149613C22semidirect
39. #19149612C22semidirect
40. #19149610C22semidirect
41. #19149608C22semidirect
42. #19149608C22semidirect
43. #19149604C22semidirect
44. #19149602C22semidirect
47. #19149600C22semidirect
51. #19149590C22semidirect
54. C4×D4×C173C22semidirect
55. #106×C173C22semidirect
57. #102×C173C22semidirect
58. #97×C173C22semidirect
184. C173#3114semidirect
197. #442D173semidirect
221. C4×D4C2×D173semidirect
229. #106C2×D173semidirect
239. #102C2×D173semidirect
243. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120232
C4×D4×D1732
#260118232
#260118032
#260117902
#260117892
#260117052
#260117042
#260116432
#260116232
#260116012
#260115932
#260115912
#260115462
#442×C1732
#3114173