Finite Group Databasebeta

← #36363040

Subgroups — #36363040

Order 22144.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#19149706C212yesmaximal, complemented
2#127×D173C211yesmaximal, characteristic, complemented
3C4×#14583827C211yesmaximal, characteristic, complemented
4#26011691C211yesmaximal, characteristic, complemented
5#26011689C212yesmaximal, complemented
6#26011687C211yesmaximal, characteristic, complemented
7#26011685C212yesmaximal, complemented
8#26011683C211yesmaximal, characteristic, complemented
9#26011513C211yesmaximal, characteristic, complemented
10#26011494C212yesmaximal, complemented
11#460×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C2211yescharacteristic
13C2×D4×D173—21no—
14C2×D4×D173—21no—
15C2×D4×D173C2212yes—
16#19149706—28no—
17C2×#14583813C2212yescomplemented
18C2×D1384C2212yescomplemented
19C2×#14583811C2212yescomplemented
20C2×#14583809C2212yes—
21#19149676C2211yescharacteristic
22#19149675—21no—
23#19149675—21no—
24#19149675C2212yes—
25Q8×Dic173C2211yescharacteristic, complemented
26C4×D692—21no—
27C4×D692C2211yescharacteristic, complemented
28#19149641—21no—
29#19149641C2211yescharacteristic, complemented
30#19149637—21no—
31#19149637C2211yescharacteristic
32C42×D173—21no—
33C42×D173C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic, complemented
35#19149608C2212yescomplemented
36#19149601C2212yescomplemented
37#19149600C2212yescomplemented
38#19149599C2212yescomplemented
39C2×SD16×C173C2212yescomplemented
40#127×C173C2211yescharacteristic
41C4×Q8×C173C2211yescharacteristic, complemented
42C4⋊C8×C173C2211yescharacteristic, complemented
43#102×C173C2212yescomplemented
44C23×D173—22no—
45#14583827—22no—
46#14583827D414yescomplemented
47C2×D692—21no—
48C2×D692—21no—
49C2×D692—21no—
50C2×D692C2312yes—
51C2×D692C2311yescharacteristic
52C2×C22⋊D173—22no—
53C2×C22⋊D173—22no—
54C2×C22⋊D173—22no—
55D4×D173—24no—
56D4×D173—24no—
57D4×D173—44no—
58D4×D173—44no—
59C2×C4×D173—21no—
60C2×C4×D173—21no—
61C2×C4×D173—21no—
62C2×C4×D173D411yescharacteristic
63C2×C4×D173D411yescharacteristic
64C2×C4×D173C2311yescharacteristic
65#14583813—24no—
66D1384—24no—
67#14583811—24no—
68#14583809—24no—
69C2×#13063436C2312yes—
70C4⋊Dic173—21no—
71C4⋊Dic173C2311yescharacteristic
72#14583803—21no—
73#14583803—21no—
74#14583803—21no—
75C4×Dic173—21no—
76C4×Dic173—21no—
77C4×Dic173D411yescharacteristic
78C4×Dic173D411yescharacteristic
79C4×Dic173C2311yescharacteristic
80C2×Q8×C173C2311yescharacteristic
81C2×D4×C173—21no—
82C2×D4×C173—21no—
83C2×D4×C173C2312yes—
84SD16×C173—24no—
85C2×C8×C173C2312yes—
86C4⋊C4×C173—21no—
87C4⋊C4×C173C2311yescharacteristic
88C42×C173—21no—
89C42×C173C2311yescharacteristic
90C22×D173—21no—
91C22×D173—22no—
92C22×D173—44no—
93C22×D173—44no—
94C22×D173—44no—
95C22×D173C2×D411yescharacteristic
96D692—24no—
97D692—22no—
98D692—21no—
99D692—22no—
100D692—42no—
101D692—42no—
102D692C2×D412yescomplemented
103C22⋊D173—44no—
104C22⋊D173—44no—
105C22⋊D173—44no—
106C2×Dic173—21no—
107C2×Dic173C2×D411yescharacteristic
108C2×Dic173C2×D411yescharacteristic
109C2×Dic173C2×D411yescharacteristic
110C4×D173—22no—
111C4×D173—24no—
112C4×D173—22no—
113C4×D173—42no—
114C4×D173—42no—
115C4×D173C2×D412yes—
116C4×D173#4112yes—
117#13063436—22no—
118C23×C173—22no—
119Q8×C173—21no—
120Q8×C173C2×D412yescomplemented
121D4×C173—22no—
122D4×C173—22no—
123D4×C173—42no—
124D4×C173—42no—
125C2×C4×C173—21no—
126C2×C4×C173—21no—
127C2×C4×C173C2×D411yescharacteristic
128C2×C4×C173C2×D411yescharacteristic
129C2×C4×C173C2411yescharacteristic
130C8×C173—22no—
131C22×C173—42no—
132C22×C173—42no—
133C22×C173—42no—
134C22×C173C22×D411yescharacteristic
135C2×D173—21no—
136C2×D173—21no—
137C2×D173—22no—
138C2×D173—41no—
139C2×D173—41no—
140C2×D173—42no—
141C2×D173—42no—
142C2×D173—42no—
143C2×D173—88no—
144C2×D173#12112yes—
145C4×C173—22no—
146C4×C173—21no—
147C4×C173—41no—
148C4×C173AGL(1,ℤ/8)12yescomplemented
149C4×C173C22×D411yescharacteristic
150C4×C173C2×#4111yescharacteristic
151Dic173—21no—
152Dic173—41no—
153Dic173#12112yes—
154Dic173AGL(1,ℤ/8)12yescomplemented
155C2×C173—82no—
156C2×C173C2×#12111yescharacteristic
157C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
158C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
159D173—22no—
160D173—42no—
161D173—82no—
162C173#303811yescharacteristic, complemented, Sylow
163#3038—1731nomaximal, Sylow
164C2×AGL(1,ℤ/8)—1732no—
165C2×#127—1731no—
166C4×#41—1731no—
167#460—1732no—
168#460—1731no—
169#460D17311yescharacteristic, complemented
170#459—1731no—
171#459—1731no—
172#459—1732no—
173#423—1731no—
174#418—1732no—
175C2×#41—1731no—
176C22×D4—3461no—
177C22×D4—3461no—
178C22×D4—1732no—
179AGL(1,ℤ/8)—3468no—
180C2×SD16—1732no—
181C2×SD16C2×D17312yescomplemented
182C2×D8—1732no—
183C2×D8—1732no—
184C2×M16—1732no—
185#127—3461no—
186#127—3461no—
187#127—1732no—
188#127—1731no—
189#127C2×D17311yescharacteristic
190C4×Q8—1731no—
191C4×Q8C2×D17311yescharacteristic, complemented
192C4×D4—3461no—
193C4×D4—3461no—
194C4×D4—1731no—
195C4×D4—1731no—
196#117—3461no—
197#117—1731no—
198C2×C42—3461no—
199C2×C42—1731no—
200C4⋊C8—1732no—
201C4⋊C8—1731no—
202C4⋊C8C2×D17311yescharacteristic, complemented
203#102—1732no—
204#102—1732no—
205#102—1732no—
206#102C2×D17312yescomplemented
207C24—3462no—
208#41—3462no—
209#41—1734no—
210C2×Q8C22×D17311yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—3461no—
214C2×D4—3464no—
215C2×D4—3464no—
216C2×D4—3462no—
217C2×D4—3462no—
218C2×D4—3462no—
219C2×D4—3461no—
220C2×D4—3461no—
221C2×D4—1732no—
222C2×D4—1731no—
223C2×D4—6924no—
224C2×D4—6924no—
225C2×D4C22×D17312yes—
226C22×C4—3461no—
227C22×C4—3461no—
228C22×C4—3461no—
229C22×C4—1731no—
230C22×C4—1731no—
231C22×C4—1731no—
232SD16—24no—
233SD16—3464no—
234D8—3464no—
235D8—3464no—
236M16—3464no—
237C2×C8—1732no—
238C2×C8C22×D17312yes—
239C4⋊C4—21no—
240C4⋊C4—3461no—
241C4⋊C4—1731no—
242C4⋊C4C22×D17311yescharacteristic
243C22⋊C4—3461no—
244C22⋊C4—3461no—
245C22⋊C4—3461no—
246C42—21no—
247C42—3461no—
248C42—3461no—
249C42—1731no—
250C42—1731no—
251C42—1731no—
252C42C22×D17311yescharacteristic
253C23—22no—
254C23—3461no—
255C23—3462no—
256C23—1731no—
257C23—6924no—
258C23—6924no—
259C23—6924no—
260Q8—21no—
261Q8D4×D17312yescomplemented
262D4—22no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—3464no—
267D4—3462no—
268D4—3461no—
269D4—3462no—
270D4—1732no—
271D4—6922no—
272D4—6924no—
273D4—6924no—
274D4—6924no—
275D4—6922no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—3464no—
279C2×C4—3462no—
280C2×C4—3462no—
281C2×C4—3461no—
282C2×C4—1731no—
283C2×C4—1731no—
284C2×C4—1731no—
285C2×C4—1732no—
286C2×C4—1732no—
287C2×C4—6922no—
288C2×C4—6922no—
289C2×C4D4×D17311yescharacteristic
290C2×C4D4×D17311yescharacteristic
291C2×C4C23×D17311yescharacteristic
292C8—22no—
293C8—3462no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—3461no—
298C22—3461no—
299C22—3462no—
300C22—13848no—
301C22—1732no—
302C22—6922no—
303C22—6922no—
304C22—6922no—
305C22—6921no—
306C22—6921no—
307C22C2×D4×D17311yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—3461no—
312C4—1732no—
313C4—1732no—
314C4—6921no—
315C4#1914970612yescomplemented
316C4C2×D4×D17311yescharacteristic
317C4#41×D17311yescharacteristic
318C2—82no—
319C2—3462no—
320C2—13842no—
321C2—6922no—
322C2#121×D17311yescharacteristic
323C2AGL(1,ℤ/8)×D17311yescharacteristic
324C2C2×#1914970611yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#460×C173C2
socleC22×C173C22×D4
hypercenter#460D173
hypocenterC173#3038
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. #127×D173C2semidirect
3. C4×#14583827C2semidirect
4. #26011691C2semidirect
5. #26011689C2semidirect
6. #26011687C2semidirect
7. #26011685C2semidirect
8. #26011683C2semidirect
9. #26011513C2semidirect
10. #26011494C2semidirect
11. #460×C173C2semidirect
17. C2×#14583813C22semidirect
18. C2×D1384C22semidirect
19. C2×#14583811C22semidirect
25. Q8×Dic173C22semidirect
27. C4×D692C22semidirect
29. #19149641C22semidirect
34. C4⋊#13063436C22semidirect
35. #19149608C22semidirect
36. #19149601C22semidirect
37. #19149600C22semidirect
38. #19149599C22semidirect
39. C2×SD16×C173C22semidirect
41. C4×Q8×C173C22semidirect
42. C4⋊C8×C173C22semidirect
43. #102×C173C22semidirect
46. #14583827D4semidirect
102. D692C2×D4semidirect
120. Q8×C173C2×D4semidirect
148. C4×C173AGL(1,ℤ/8)semidirect
154. Dic173AGL(1,ℤ/8)semidirect
162. C173#3038semidirect
169. #460D173semidirect
181. C2×SD16C2×D173semidirect
191. C4×Q8C2×D173semidirect
202. C4⋊C8C2×D173semidirect
206. #102C2×D173semidirect
261. Q8D4×D173semidirect
315. C4#19149706semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
#127×D1732
C4×#145838272
#260116912
#260116892
#260116872
#260116852
#260116832
#260115132
#260114942
#460×C1732
#3038173