Finite Group Databasebeta

← #36362814

Subgroups — #36362814

Order 22144.

806 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14583827C211yesmaximal, characteristic, complemented
2C2×#19149706C212yesmaximal, complemented
3#121×D173C211yesmaximal, characteristic, complemented
4#26011639C212yesmaximal, complemented
5#26011635C212yesmaximal, complemented
6#26011630C211yesmaximal, characteristic, complemented
7#26011627C211yesmaximal, characteristic, complemented
8#26011624C211yesmaximal, characteristic, complemented
9#26011491C212yesmaximal, complemented
10#26011453C211yesmaximal, characteristic, complemented
11#448×C173C211yesmaximal, characteristic, complemented
12C2×#14583827—28no—
13C2×#14583827—21no—
14C2×#14583827—21no—
15C2×#14583827C2212yes—
16C2×#14583827C2212yescomplemented
17C22×D692—21no—
18C22×D692C2211yescharacteristic, complemented
19C2×D4×D173—21no—
20C2×D4×D173C2211yescharacteristic, complemented
21C22×C4×D173—21no—
22C22×C4×D173C2211yescharacteristic
23#19149706—28no—
24C2×#14583813C2212yescomplemented
25C2×D1384C2212yescomplemented
26C2×#14583811C2212yescomplemented
27C2×#14583809C2212yescomplemented
28#19149654C2211yescharacteristic, complemented
29#19149653C2211yescharacteristic
30#19149652—21no—
31#19149651—21no—
32#19149650C2211yescharacteristic, complemented
33C4⋊C4×D173C2211yescharacteristic, complemented
34C22⋊C4×D173—21no—
35#19149603C2212yescomplemented
36#19149601C2212yescomplemented
37#19149599C2212yescomplemented
38C22⋊#13063436C2211yescharacteristic, complemented
39#19149591C2212yescomplemented
40C22×Q8×C173C2211yescharacteristic, complemented
41C2×SD16×C173C2212yescomplemented
42#121×C173C2211yescharacteristic
43#103×C173C2212yescomplemented
44C22⋊C8×C173C2211yescharacteristic, complemented
45C23×D173—21no—
46C23×D173—21no—
47C23×D173D411yescharacteristic
48#14583827—28no—
49#14583827—28no—
50#14583827—24no—
51#14583827—24no—
52#14583827—24no—
53#14583827D418yescomplemented
54C2×D692—28no—
55C2×D692—22no—
56C2×D692—22no—
57C2×D692—24no—
58C2×D692—21no—
59C2×D692—21no—
60C2×D692C2311yescharacteristic
61C2×D692C2312yes—
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65D4×D173—24no—
66D4×D173—44no—
67C22×Dic173D411yescharacteristic
68C2×C4×D173—28no—
69C2×C4×D173—22no—
70C2×C4×D173—22no—
71C2×C4×D173—24no—
72C2×C4×D173—21no—
73C2×C4×D173D411yescharacteristic
74C2×C4×D173D411yescharacteristic
75C2×C4×D173C2311yescharacteristic
76C2×C4×D173C2311yescharacteristic
77#14583813—24no—
78D1384—24no—
79#14583811—24no—
80#14583809—24no—
81C2×#13063436C2312yes—
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84C22⋊Dic173—21no—
85#14583803—21no—
86#14583803—21no—
87C2×Q8×C173—24no—
88C2×Q8×C173—21no—
89C2×Q8×C173C2312yes—
90C2×D4×C173—21no—
91C2×D4×C173C2311yescharacteristic
92C22×C4×C173—21no—
93C22×C4×C173C2311yescharacteristic
94SD16×C173—24no—
95C2×C8×C173C2312yes—
96C4⋊C4×C173C2311yescharacteristic
97C22⋊C4×C173—21no—
98C22×D173—24no—
99C22×D173—22no—
100C22×D173—22no—
101C22×D173—22no—
102C22×D173—21no—
103C22×D173—41no—
104C22×D173—42no—
105C22×D173—42no—
106C22×D173—42no—
107C22×D173—42no—
108C22×D173—42no—
109C22×D173C2×D411yescharacteristic
110C22×D173C2×D411yescharacteristic
111D692—28no—
112D692—24no—
113D692—28no—
114D692—22no—
115D692—22no—
116D692—42no—
117D692C2×D414yescomplemented
118C22⋊D173—42no—
119C22⋊D173—42no—
120C22⋊D173—42no—
121C2×Dic173—22no—
122C2×Dic173—21no—
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C4×D173—28no—
126C4×D173—22no—
127C4×D173—22no—
128C4×D173—28no—
129C4×D173—22no—
130C4×D173—42no—
131C4×D173C2×D414yes—
132#13063436—22no—
133C23×C173—21no—
134C23×C173C2×D411yescharacteristic
135Q8×C173—24no—
136Q8×C173—22no—
137Q8×C173C2×D414yescomplemented
138D4×C173—22no—
139D4×C173—42no—
140C2×C4×C173—24no—
141C2×C4×C173—22no—
142C2×C4×C173—22no—
143C2×C4×C173—21no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2411yescharacteristic
146C8×C173—22no—
147C22×C173—21no—
148C22×C173—21no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173AGL(1,ℤ/8)12yescomplemented
153C22×C173C22×D411yescharacteristic
154C2×D173—21no—
155C2×D173—21no—
156C2×D173—21no—
157C2×D173—24no—
158C2×D173—21no—
159C2×D173—21no—
160C2×D173—44no—
161C2×D173—48no—
162C2×D173—42no—
163C2×D173—42no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—84no—
168C2×D173#12012yes—
169C4×C173—24no—
170C4×C173—21no—
171C4×C173—41no—
172C4×C173C22×D412yes—
173Dic173—21no—
174Dic173—41no—
175Dic173#12012yes—
176C2×C173—22no—
177C2×C173—81no—
178C2×C173C2×#12011yescharacteristic
179C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C173#57511yescharacteristic
181D173—22no—
182D173—41no—
183D173—44no—
184D173—81no—
185C173#300611yescharacteristic, complemented, Sylow
186#3006—1731nomaximal, Sylow
187C22×#41—1731no—
188C2×AGL(1,ℤ/8)—1732no—
189C2×#121—1731no—
190#449—1732no—
191#449—1732no—
192#448—1731no—
193#448D17311yescharacteristic, complemented
194#447—1731no—
195#447—1731no—
196#417—1732no—
197#407—1731no—
198C2×#41—3468no—
199C2×#41—3461no—
200C2×#41—3461no—
201C2×#41—1732no—
202C2×#41—1732no—
203C22×Q8C2×D17311yescharacteristic, complemented
204C22×D4—3461no—
205C22×D4—3461no—
206C22×D4—1731no—
207C22×D4—1731no—
208C23×C4—3461no—
209C23×C4—1731no—
210AGL(1,ℤ/8)—3468no—
211C2×SD16—1732no—
212C2×SD16C2×D17312yescomplemented
213C2×D8—1732no—
214C2×D8—1732no—
215C2×M16—1732no—
216#121—3461no—
217#121—3461no—
218#121—1731no—
219#121—1731no—
220#121—1731no—
221#121C2×D17311yescharacteristic
222C2×C4⋊C4—1731no—
223C2×C22⋊C4—3461no—
224#103—1732no—
225#103C2×D17312yescomplemented
226#102—1732no—
227#102—1732no—
228C22⋊C8—1731no—
229C22⋊C8—1732no—
230C22⋊C8C2×D17311yescharacteristic, complemented
231C24—3461no—
232C24—3461no—
233C24—1731no—
234#41—3468no—
235#41—3468no—
236#41—3464no—
237#41—3464no—
238#41—3464no—
239#41—1738no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8C22×D17312yes—
243C2×D4—21no—
244C2×D4—3468no—
245C2×D4—3462no—
246C2×D4—3462no—
247C2×D4—3464no—
248C2×D4—3461no—
249C2×D4—3461no—
250C2×D4—3461no—
251C2×D4—3464no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—1731no—
255C2×D4—1732no—
256C2×D4—6924no—
257C2×D4C22×D17311yescharacteristic
258C22×C4—21no—
259C22×C4—3468no—
260C22×C4—3462no—
261C22×C4—3462no—
262C22×C4—3464no—
263C22×C4—3461no—
264C22×C4—1731no—
265C22×C4—1731no—
266C22×C4—1731no—
267C22×C4—1731no—
268C22×C4—1731no—
269C22×C4C22×D17311yescharacteristic
270SD16—24no—
271SD16—3464no—
272D8—3464no—
273D8—3464no—
274M16—3464no—
275C2×C8—1732no—
276C2×C8C22×D17312yes—
277C4⋊C4—3461no—
278C4⋊C4—1731no—
279C4⋊C4C22×D17311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C23—21no—
285C23—3464no—
286C23—3462no—
287C23—3462no—
288C23—3462no—
289C23—3461no—
290C23—1731no—
291C23—1731no—
292C23—6921no—
293C23—6922no—
294C23—6922no—
295C23—6922no—
296C23—6922no—
297C23—6922no—
298C23D4×D17311yescharacteristic
299Q8—24no—
300Q8—22no—
301Q8D4×D17314yescomplemented
302D4—22no—
303D4—42no—
304D4—3468no—
305D4—3464no—
306D4—3468no—
307D4—3462no—
308D4—3462no—
309D4—1734no—
310D4—6922no—
311D4—6922no—
312D4—6922no—
313D4—6922no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—3468no—
319C2×C4—3462no—
320C2×C4—3468no—
321C2×C4—3462no—
322C2×C4—3462no—
323C2×C4—3462no—
324C2×C4—3461no—
325C2×C4—1731no—
326C2×C4—1731no—
327C2×C4—1734no—
328C2×C4—6922no—
329C2×C4D4×D17311yescharacteristic
330C2×C4C23×D17311yescharacteristic
331C8—22no—
332C8—3462no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—3461no—
339C22—3461no—
340C22—3461no—
341C22—3464no—
342C22—3461no—
343C22—3461no—
344C22—13844no—
345C22—1732no—
346C22—6924no—
347C22—6928no—
348C22—6921no—
349C22—6921no—
350C22—6921no—
351C22—6922no—
352C22—6922no—
353C22#1914970612yescomplemented
354C22C2×D4×D17311yescharacteristic
355C4—24no—
356C4—21no—
357C4—41no—
358C4—3461no—
359C4—1732no—
360C4—6921no—
361C4C2×D4×D17312yes—
362C2—22no—
363C2—81no—
364C2—3462no—
365C2—13841no—
366C2—6921no—
367C2—6924no—
368C2#120×D17311yescharacteristic
369C2C2×#1914970611yescharacteristic
370C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#448×C173C2
socleC22×C173C22×D4
hypercenter#448D173
hypocenterC173#3006
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#14583827C2semidirect
2. C2×#19149706C2semidirect
3. #121×D173C2semidirect
4. #26011639C2semidirect
5. #26011635C2semidirect
6. #26011630C2semidirect
7. #26011627C2semidirect
8. #26011624C2semidirect
9. #26011491C2semidirect
10. #26011453C2semidirect
11. #448×C173C2semidirect
16. C2×#14583827C22semidirect
18. C22×D692C22semidirect
20. C2×D4×D173C22semidirect
24. C2×#14583813C22semidirect
25. C2×D1384C22semidirect
26. C2×#14583811C22semidirect
27. C2×#14583809C22semidirect
28. #19149654C22semidirect
32. #19149650C22semidirect
33. C4⋊C4×D173C22semidirect
35. #19149603C22semidirect
36. #19149601C22semidirect
37. #19149599C22semidirect
38. C22⋊#13063436C22semidirect
39. #19149591C22semidirect
40. C22×Q8×C173C22semidirect
41. C2×SD16×C173C22semidirect
43. #103×C173C22semidirect
44. C22⋊C8×C173C22semidirect
53. #14583827D4semidirect
117. D692C2×D4semidirect
137. Q8×C173C2×D4semidirect
152. C22×C173AGL(1,ℤ/8)semidirect
185. C173#3006semidirect
193. #448D173semidirect
203. C22×Q8C2×D173semidirect
212. C2×SD16C2×D173semidirect
225. #103C2×D173semidirect
230. C22⋊C8C2×D173semidirect
301. Q8D4×D173semidirect
353. C22#19149706semidirect

Maximal subgroups

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

subgroupindex
C22×#145838272
C2×#191497062
#121×D1732
#260116392
#260116352
#260116302
#260116272
#260116242
#260114912
#260114532
#448×C1732
#3006173