Finite Group Databasebeta

← #36362791

Subgroups — #36362791

Order 22144.

994 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D173C211yesmaximal, characteristic, complemented
2C2×#19149706C212yesmaximal, complemented
3#26012004C211yesmaximal, characteristic, complemented
4#26011641C211yesmaximal, characteristic, complemented
5#26011640C212yesmaximal, complemented
6#26011627C211yesmaximal, characteristic, complemented
7#26011626C212yesmaximal, complemented
8#26011625C211yesmaximal, characteristic, complemented
9#26011494C212yesmaximal, complemented
10#26011453C211yesmaximal, characteristic, complemented
11#450×C173C211yesmaximal, characteristic, complemented
12C24×D173—21no—
13C2×#14583827C2211yescharacteristic, complemented
14C22×D692C2211yescharacteristic
15C22×C22⋊D173—21no—
16C2×D4×D173—28no—
17C2×D4×D173—21no—
18C2×D4×D173—21no—
19C2×D4×D173C2212yes—
20C2×D4×D173C2212yes—
21C22×C4×D173C2211yescharacteristic
22#19149706—28no—
23C2×#14583813C2212yescomplemented
24C2×D1384C2212yescomplemented
25C2×#14583811C2212yescomplemented
26C2×#14583809C2212yescomplemented
27#19149664—21no—
28#19149655C2211yescharacteristic, complemented
29#19149654C2211yescharacteristic, complemented
30#19149653C2211yescharacteristic, complemented
31#19149649—21no—
32#19149641—21no—
33#19149639—21no—
34#19149637C2211yescharacteristic, complemented
35#19149601C2212yescomplemented
36#19149600C2212yescomplemented
37#19149599C2212yescomplemented
38C22⋊#13063436C2211yescharacteristic, complemented
39#19149591C2212yescomplemented
40C22×D4×C173C2211yescharacteristic
41C2×SD16×C173C2212yescomplemented
42C22⋊Q8×C173C2211yescharacteristic, complemented
43#102×C173C2212yescomplemented
44C22⋊C8×C173C2211yescharacteristic, complemented
45C23×D173—28no—
46C23×D173—22no—
47C23×D173—22no—
48C23×D173—21no—
49C23×D173—44no—
50C23×D173—44no—
51C23×D173D411yescharacteristic
52#14583827—24no—
53C2×D692—24no—
54C2×D692—21no—
55C2×D692—21no—
56C2×D692—21no—
57C2×D692C2312yes—
58C2×D692C2311yescharacteristic, complemented
59C2×C22⋊D173—28no—
60C2×C22⋊D173—22no—
61C2×C22⋊D173—22no—
62C2×C22⋊D173—21no—
63D4×D173—28no—
64D4×D173—28no—
65D4×D173—24no—
66D4×D173—24no—
67D4×D173—24no—
68D4×D173D418yescomplemented
69C22×Dic173D411yescharacteristic
70C2×C4×D173—24no—
71C2×C4×D173—21no—
72C2×C4×D173D411yescharacteristic
73C2×C4×D173D411yescharacteristic
74C2×C4×D173C2311yescharacteristic
75C2×C4×D173C2311yescharacteristic
76#14583813—24no—
77D1384—24no—
78#14583811—24no—
79#14583809—24no—
80C2×#13063436C2312yescomplemented
81C4⋊Dic173C2311yescharacteristic, complemented
82#14583805—21no—
83#14583803—21no—
84#14583803—21no—
85#14583803—21no—
86#14583803—21no—
87C4×Dic173—21no—
88C24×C173—21no—
89C2×Q8×C173C2311yescharacteristic, complemented
90C2×D4×C173—24no—
91C2×D4×C173—21no—
92C2×D4×C173C2312yes—
93C22×C4×C173C2311yescharacteristic
94SD16×C173—24no—
95C2×C8×C173C2312yescomplemented
96C4⋊C4×C173—21no—
97C4⋊C4×C173C2311yescharacteristic, complemented
98C22⋊C4×C173—21no—
99C22×D173—22no—
100C22×D173—22no—
101C22×D173—22no—
102C22×D173—28no—
103C22×D173—24no—
104C22×D173—22no—
105C22×D173—28no—
106C22×D173—21no—
107C22×D173—432no—
108C22×D173—44no—
109C22×D173—44no—
110C22×D173—44no—
111C22×D173—42no—
112C22×D173—42no—
113C22×D173—44no—
114C22×D173C2×D411yescharacteristic
115C22×D173C2×D411yescharacteristic
116D692—24no—
117D692—22no—
118D692—22no—
119D692—42no—
120D692—42no—
121D692C2×D414yes—
122C22⋊D173—28no—
123C22⋊D173—28no—
124C22⋊D173—42no—
125C2×Dic173—22no—
126C2×Dic173—21no—
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—22no—
130C4×D173—22no—
131C4×D173—22no—
132C4×D173—42no—
133C4×D173C2×D414yes—
134#13063436—22no—
135C23×C173—24no—
136C23×C173—22no—
137C23×C173—42no—
138C23×C173—42no—
139C23×C173C2×D411yescharacteristic
140Q8×C173—22no—
141D4×C173—24no—
142D4×C173—22no—
143D4×C173C2×D414yes—
144C2×C4×C173—22no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2411yescharacteristic
149C8×C173—22no—
150C22×C173—21no—
151C22×C173—21no—
152C22×C173—24no—
153C22×C173—48no—
154C22×C173—42no—
155C22×C173—42no—
156C22×C173AGL(1,ℤ/8)12yescomplemented
157C22×C173C22×D411yescharacteristic
158C2×D173—28no—
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—21no—
162C2×D173—21no—
163C2×D173—24no—
164C2×D173—416no—
165C2×D173—416no—
166C2×D173—48no—
167C2×D173—42no—
168C2×D173—42no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173#12012yes—
173C2×D173AGL(1,ℤ/8)12yescomplemented
174C4×C173—21no—
175C4×C173—41no—
176C4×C173—41no—
177C4×C173C22×D412yes—
178Dic173—21no—
179Dic173—41no—
180Dic173#12012yes—
181C2×C173—22no—
182C2×C173—44no—
183C2×C173C2×#12011yescharacteristic
184C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
186D173—22no—
187D173—22no—
188D173—44no—
189D173—81no—
190C173#300311yescharacteristic, complemented, Sylow
191#3003—1731nomaximal, Sylow
192C23×D4—1731no—
193C2×AGL(1,ℤ/8)—1732no—
194#525—1731no—
195#450—1732no—
196#450—1731no—
197#450D17311yescharacteristic, complemented
198#447—1731no—
199#447—1732no—
200#447—1731no—
201#418—1732no—
202#407—1731no—
203C25—3461no—
204C2×#41—1731no—
205C22×D4—3468no—
206C22×D4—3461no—
207C22×D4—3461no—
208C22×D4—3461no—
209C22×D4—1732no—
210C22×D4—1732no—
211C22×D4—1731no—
212C22×D4C2×D17311yescharacteristic
213C23×C4—1731no—
214AGL(1,ℤ/8)—3468no—
215C2×SD16—1732no—
216C2×SD16C2×D17312yescomplemented
217C2×D8—1732no—
218C2×D8—1732no—
219C2×M16—1732no—
220#123—3461no—
221C22⋊Q8—1731no—
222C22⋊Q8C2×D17311yescharacteristic, complemented
223#121—1731no—
224#121—1731no—
225#120—3461no—
226C4×D4—3461no—
227C4×D4—3461no—
228#117—1731no—
229#102—1732no—
230#102—1732no—
231#102—1732no—
232#102C2×D17312yescomplemented
233C22⋊C8—1731no—
234C22⋊C8—1732no—
235C22⋊C8C2×D17311yescharacteristic, complemented
236C24—21no—
237C24—3468no—
238C24—3462no—
239C24—3462no—
240C24—3461no—
241C24—1731no—
242C24—6924no—
243C24—6924no—
244#41—3464no—
245C2×Q8C22×D17311yescharacteristic, complemented
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—3464no—
249C2×D4—3461no—
250C2×D4—3461no—
251C2×D4—3468no—
252C2×D4—3468no—
253C2×D4—3464no—
254C2×D4—3468no—
255C2×D4—3464no—
256C2×D4—3461no—
257C2×D4—3464no—
258C2×D4—3462no—
259C2×D4—3462no—
260C2×D4—3461no—
261C2×D4—1738no—
262C2×D4—1731no—
263C2×D4—1732no—
264C2×D4C22×D17312yes—
265C22×C4—3464no—
266C22×C4—3461no—
267C22×C4—1731no—
268C22×C4—1731no—
269C22×C4—1731no—
270C22×C4—1731no—
271C22×C4—1731no—
272C22×C4C22×D17311yescharacteristic
273SD16—24no—
274SD16—3464no—
275D8—3464no—
276D8—3464no—
277M16—3464no—
278C2×C8—1732no—
279C2×C8C22×D17312yescomplemented
280C4⋊C4—21no—
281C4⋊C4—3461no—
282C4⋊C4—1731no—
283C4⋊C4C22×D17311yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C42—3461no—
290C23—24no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—3462no—
295C23—3462no—
296C23—3464no—
297C23—3468no—
298C23—3468no—
299C23—3462no—
300C23—3462no—
301C23—3461no—
302C23—1731no—
303C23—1731no—
304C23—6922no—
305C23—69232no—
306C23—6924no—
307C23—6924no—
308C23—6924no—
309C23—6924no—
310C23—6922no—
311C23D4×D17311yescharacteristic
312Q8—22no—
313D4—24no—
314D4—22no—
315D4—3464no—
316D4—3468no—
317D4—3468no—
318D4—3462no—
319D4—3462no—
320D4—1734no—
321D4—6922no—
322D4—6922no—
323D4—6922no—
324D4D4×D17314yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—3462no—
329C2×C4—3462no—
330C2×C4—3462no—
331C2×C4—3462no—
332C2×C4—3461no—
333C2×C4—1731no—
334C2×C4—1731no—
335C2×C4—1734no—
336C2×C4—6922no—
337C2×C4D4×D17311yescharacteristic
338C2×C4C23×D17311yescharacteristic
339C8—22no—
340C8—3462no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—3468no—
348C22—3461no—
349C22—3461no—
350C22—3461no—
351C22—3461no—
352C22—3464no—
353C22—1732no—
354C22—1732no—
355C22—6928no—
356C22—69216no—
357C22—69216no—
358C22—6921no—
359C22—6921no—
360C22—6921no—
361C22—6922no—
362C22—6922no—
363C22#1914970612yescomplemented
364C22C2×D4×D17311yescharacteristic
365C4—21no—
366C4—41no—
367C4—41no—
368C4—3461no—
369C4—1732no—
370C4—6921no—
371C4C2×D4×D17312yes—
372C2—22no—
373C2—44no—
374C2—3462no—
375C2—3462no—
376C2—13841no—
377C2—6924no—
378C2#120×D17311yescharacteristic
379C2AGL(1,ℤ/8)×D17311yescharacteristic
380C2C2×#1914970611yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
hypercenter#450D173
hypocenterC173#3003
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×D4×D173C2semidirect
2. C2×#19149706C2semidirect
3. #26012004C2semidirect
4. #26011641C2semidirect
5. #26011640C2semidirect
6. #26011627C2semidirect
7. #26011626C2semidirect
8. #26011625C2semidirect
9. #26011494C2semidirect
10. #26011453C2semidirect
11. #450×C173C2semidirect
13. C2×#14583827C22semidirect
23. C2×#14583813C22semidirect
24. C2×D1384C22semidirect
25. C2×#14583811C22semidirect
26. C2×#14583809C22semidirect
28. #19149655C22semidirect
29. #19149654C22semidirect
30. #19149653C22semidirect
34. #19149637C22semidirect
35. #19149601C22semidirect
36. #19149600C22semidirect
37. #19149599C22semidirect
38. C22⋊#13063436C22semidirect
39. #19149591C22semidirect
41. C2×SD16×C173C22semidirect
42. C22⋊Q8×C173C22semidirect
43. #102×C173C22semidirect
44. C22⋊C8×C173C22semidirect
58. C2×D692C23semidirect
68. D4×D173D4semidirect
80. C2×#13063436C23semidirect
81. C4⋊Dic173C23semidirect
89. C2×Q8×C173C23semidirect
95. C2×C8×C173C23semidirect
97. C4⋊C4×C173C23semidirect
156. C22×C173AGL(1,ℤ/8)semidirect
173. C2×D173AGL(1,ℤ/8)semidirect
190. C173#3003semidirect
197. #450D173semidirect
216. C2×SD16C2×D173semidirect
222. C22⋊Q8C2×D173semidirect
232. #102C2×D173semidirect
235. C22⋊C8C2×D173semidirect
245. C2×Q8C22×D173semidirect
279. C2×C8C22×D173semidirect
283. C4⋊C4C22×D173semidirect
363. C22#19149706semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1732
C2×#191497062
#260120042
#260116412
#260116402
#260116272
#260116262
#260116252
#260114942
#260114532
#450×C1732
#3003173