Finite Group Databasebeta

← #36365183

Subgroups — #36365183

Order 22144.

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

normal subgroups only

This group has 556 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×#19149704C212yesmaximal, complemented
2D4×D692C211yesmaximal, characteristic, complemented
3#26011867C211yesmaximal, characteristic, complemented
4#26011742C212yesmaximal, complemented
5#26011691C211yesmaximal, characteristic, complemented
6#26011683C211yesmaximal, characteristic, complemented
7#26011636C212yesmaximal, complemented
8#26011624C212yesmaximal, complemented
9C4×#14583813C211yesmaximal, characteristic, complemented
10#26011569C211yesmaximal, characteristic, complemented
11#550×C173C211yesmaximal, characteristic, complemented
12C22×D692C2212yes—
13C2×D4×D173—21no—
14#19149704—28no—
15C2×#14583813C2211yescharacteristic
16C2×#14583813C2212yescomplemented
17C2×#14583811C2212yes—
18C2×#14583811C2212yescomplemented
19C2×#14583811C2212yescomplemented
20C2×#14583810C2212yescomplemented
21#19149676C2211yescharacteristic
22#19149654—21no—
23#19149653C2212yes—
24#19149649—22no—
25C4×D692C2211yescharacteristic
26#19149610C2211yescharacteristic, complemented
27C4⋊#13063436C2211yescharacteristic, complemented
28#19149603C2211yescharacteristic, complemented
29#19149600C2212yescomplemented
30#19149599C2211yescharacteristic
31#19149599C2212yescomplemented
32C22⋊#13063436C2212yescomplemented
33C4×#13063436C2211yescharacteristic, complemented
34C2×#41×C173C2212yescomplemented
35#127×C173—21no—
36#127×C173C2211yescharacteristic, complemented
37#121×C173—22no—
38#121×C173C2212yescomplemented
39C4×Q8×C173C2211yescharacteristic, complemented
40C4×D4×C173—21no—
41C4×D4×C173C2211yescharacteristic
42C23×D173—22no—
43C2×D692—24no—
44C2×D692—22no—
45C2×D692—21no—
46C2×D692—21no—
47C2×D692—42no—
48C2×D692C2312yes—
49C2×D692C2311yescharacteristic
50C2×C22⋊D173—22no—
51D4×D173—44no—
52C2×C4×D173—21no—
53#14583813—24no—
54#14583813D414yescomplemented
55#14583811—24no—
56#14583811—24no—
57#14583811—42no—
58#14583810—24no—
59C2×#13063436C2311yescharacteristic
60C2×#13063436C2312yes—
61C2×#13063436C2311yescharacteristic
62C4⋊Dic173C2311yescharacteristic
63#14583803—21no—
64#14583803—22no—
65#41×C173—24no—
66#41×C173—42no—
67C2×Q8×C173C2311yescharacteristic
68C2×D4×C173—22no—
69C2×D4×C173—22no—
70C2×D4×C173—21no—
71C2×D4×C173—21no—
72C2×D4×C173D411yescharacteristic
73C2×D4×C173C2312yes—
74C22×C4×C173—22no—
75C22×C4×C173C2312yes—
76C4⋊C4×C173—21no—
77C4⋊C4×C173D411yescharacteristic
78C4⋊C4×C173C2311yescharacteristic
79C22⋊C4×C173—22no—
80C22⋊C4×C173D412yes—
81C42×C173—21no—
82C42×C173C2311yescharacteristic
83C22×D173—21no—
84C22×D173—22no—
85C22×D173—44no—
86C22×D173—44no—
87C22×D173—82no—
88D692—22no—
89D692—21no—
90D692—22no—
91D692—44no—
92D692—42no—
93D692—81no—
94D692C2×D412yes—
95C22⋊D173—44no—
96C2×Dic173—21no—
97C4×D173—42no—
98#13063436—22no—
99#13063436—21no—
100#13063436C2×D412yescomplemented
101C23×C173—22no—
102C23×C173C2×D412yes—
103Q8×C173—21no—
104Q8×C173C2×D412yescomplemented
105D4×C173—22no—
106D4×C173—22no—
107D4×C173—22no—
108D4×C173—44no—
109D4×C173—42no—
110D4×C173—81no—
111D4×C173—82no—
112C2×C4×C173—21no—
113C2×C4×C173—22no—
114C2×C4×C173—22no—
115C2×C4×C173—21no—
116C2×C4×C173—44no—
117C2×C4×C173C2×D411yescharacteristic
118C2×C4×C173C2×D412yes—
119C2×C4×C173C2×D411yescharacteristic
120C2×C4×C173C2411yescharacteristic
121C22×C173—22no—
122C22×C173—22no—
123C22×C173—22no—
124C22×C173—42no—
125C22×C173—42no—
126C22×C173—42no—
127C22×C173C22×D411yescharacteristic
128C2×D173—22no—
129C2×D173—41no—
130C2×D173—41no—
131C2×D173—42no—
132C2×D173—42no—
133C2×D173—42no—
134C2×D173—88no—
135C4×C173—22no—
136C4×C173—22no—
137C4×C173—21no—
138C4×C173—41no—
139C4×C173AGL(1,ℤ/8)12yescomplemented
140C4×C173C22×D411yescharacteristic
141C4×C173H(5,𝔽2)11yescharacteristic
142Dic173—41no—
143C2×C173—42no—
144C2×C173—82no—
145C2×C173D4211yescharacteristic
146C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C173#57611yescharacteristic
148D173—42no—
149D173—82no—
150C173#327411yescharacteristic, complemented, Sylow
151#3274—1731nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1732no—
153#550D17311yescharacteristic, complemented
154D42—1731no—
155#493—1731no—
156#469—1732no—
157#460—1731no—
158#459—1731no—
159#449—1732no—
160#447—1732no—
161C4×SD16—1731no—
162#436—1731no—
163C2×#41C2×D17312yescomplemented
164C22×D4—3461no—
165C22×D4—1732no—
166AGL(1,ℤ/8)—3468no—
167C2×SD16—1732no—
168C2×SD16—1731no—
169C2×D8—1732no—
170C2×D8—1732no—
171C2×D8—1732no—
172C2×M16—1732no—
173#127—21no—
174#127—1731no—
175#127C2×D17311yescharacteristic, complemented
176#121—22no—
177#121—3461no—
178#121—1732no—
179#121C2×D17312yescomplemented
180#120—3462no—
181C4×Q8C2×D17311yescharacteristic, complemented
182C4×D4—21no—
183C4×D4—1731no—
184C4×D4C2×D17311yescharacteristic
185#106—1731no—
186C4⋊C8—1731no—
187#103—1731no—
188#102—1732no—
189#102—1731no—
190#102—1732no—
191C22⋊C8—1732no—
192C4×C8—1731no—
193C24—3462no—
194#41—24no—
195#41—42no—
196C2×Q8C22×D17311yescharacteristic
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—3464no—
202C2×D4—3462no—
203C2×D4—3461no—
204C2×D4—3462no—
205C2×D4—3461no—
206C2×D4—1732no—
207C2×D4—1731no—
208C2×D4—6922no—
209C2×D4—6924no—
210C2×D4C22⋊D17311yescharacteristic
211C2×D4C22×D17312yes—
212C22×C4—22no—
213C22×C4—3461no—
214C22×C4C22×D17312yes—
215SD16—3464no—
216SD16—1734no—
217D8—3464no—
218D8—3464no—
219D8—6922no—
220M16—3464no—
221C2×C8—1731no—
222C2×C8—1732no—
223C2×C8—1731no—
224C4⋊C4—21no—
225C4⋊C4—1731no—
226C4⋊C4C22⋊D17311yescharacteristic
227C4⋊C4C22×D17311yescharacteristic
228C22⋊C4—22no—
229C22⋊C4—3461no—
230C22⋊C4—3462no—
231C22⋊C4C22⋊D17312yes—
232C42—21no—
233C42C22×D17311yescharacteristic
234C23—22no—
235C23—3461no—
236C23—3462no—
237C23—13842no—
238C23—6924no—
239C23—6924no—
240C23C2×C22⋊D17312yes—
241Q8—21no—
242Q8D4×D17312yescomplemented
243D4—22no—
244D4—22no—
245D4—22no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—82no—
250D4—3462no—
251D4—3461no—
252D4—3462no—
253D4—13841no—
254D4—1732no—
255D4—6924no—
256D4—6924no—
257D4—6922no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—3461no—
264C2×C4—6922no—
265C2×C4C2×C22⋊D17311yescharacteristic
266C2×C4C2×C22⋊D17312yes—
267C2×C4C2×C22⋊D17311yescharacteristic
268C2×C4C23×D17311yescharacteristic
269C8—3462no—
270C8—3461no—
271C8—1732no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—3462no—
279C22—13848no—
280C22—6922no—
281C22—6922no—
282C22—6922no—
283C22—6921no—
284C22—6921no—
285C22C22×C22⋊D17311yescharacteristic
286C4—21no—
287C4—22no—
288C4—22no—
289C4—41no—
290C4—6921no—
291C4#1914970412yescomplemented
292C4C2×D4×D17311yescharacteristic
293C4#1914972411yescharacteristic
294C2—42no—
295C2—82no—
296C2—13842no—
297C2—6922no—
298C2D4×C22⋊D17311yescharacteristic
299C2C2×#1914970411yescharacteristic
300C2#2601229411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#550×C173C2
socleC22×C173C22×D4
hypercenter#550D173
hypocenterC173#3274
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#19149704C2semidirect
2. D4×D692C2semidirect
3. #26011867C2semidirect
4. #26011742C2semidirect
5. #26011691C2semidirect
6. #26011683C2semidirect
7. #26011636C2semidirect
8. #26011624C2semidirect
9. C4×#14583813C2semidirect
10. #26011569C2semidirect
11. #550×C173C2semidirect
16. C2×#14583813C22semidirect
18. C2×#14583811C22semidirect
19. C2×#14583811C22semidirect
20. C2×#14583810C22semidirect
26. #19149610C22semidirect
27. C4⋊#13063436C22semidirect
28. #19149603C22semidirect
29. #19149600C22semidirect
31. #19149599C22semidirect
32. C22⋊#13063436C22semidirect
33. C4×#13063436C22semidirect
34. C2×#41×C173C22semidirect
36. #127×C173C22semidirect
38. #121×C173C22semidirect
39. C4×Q8×C173C22semidirect
54. #14583813D4semidirect
100. #13063436C2×D4semidirect
104. Q8×C173C2×D4semidirect
139. C4×C173AGL(1,ℤ/8)semidirect
150. C173#3274semidirect
153. #550D173semidirect
163. C2×#41C2×D173semidirect
175. #127C2×D173semidirect
179. #121C2×D173semidirect
181. C4×Q8C2×D173semidirect
242. Q8D4×D173semidirect
291. C4#19149704semidirect

Maximal subgroups

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

subgroupindex
C2×#191497042
D4×D6922
#260118672
#260117422
#260116912
#260116832
#260116362
#260116242
C4×#145838132
#260115692
#550×C1732
#3274173