Finite Group Databasebeta

← #36364045

Subgroups — #36364045

Order 22144.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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
2C2×#19149702C211yesmaximal, characteristic, complemented
3C2×D8×D173C211yesmaximal, characteristic, complemented
4#26012059C212yesmaximal, complemented
5#26011881C212yesmaximal, complemented
6#26011877C212yesmaximal, complemented
7#26011876C211yesmaximal, characteristic, complemented
8#26011868C211yesmaximal, characteristic, complemented
9#26011866C211yesmaximal, characteristic, complemented
10#26011445C211yesmaximal, characteristic, complemented
11#495×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C2212yes—
13C2×D4×D173C2212yes—
14#19149706—28no—
15#19149702—24no—
16C2×#14583813C2212yescomplemented
17C2×#14583813C2212yescomplemented
18C2×D1384C2211yescharacteristic
19C2×D1384C2211yescharacteristic, complemented
20C2×D1384C2212yescomplemented
21C2×#14583811C2212yes—
22C2×#14583811C2212yescomplemented
23D8×D173—24no—
24C2×#14583809C2211yescharacteristic
25C2×C8×D173C2211yescharacteristic
26#19149676C2212yescomplemented
27#19149675C2212yescomplemented
28#19149668C2212yescomplemented
29#19149652—24no—
30#19149649—24no—
31#19149636C2211yescharacteristic
32#19149590C2211yescharacteristic, complemented
33#19149589C2211yescharacteristic, complemented
34C8×Dic173C2211yescharacteristic, complemented
35C2×Q16×C173C2211yescharacteristic, complemented
36C2×SD16×C173C2212yescomplemented
37C2×D8×C173C2211yescharacteristic
38#124×C173C2212yescomplemented
39C4×C8×C173C2211yescharacteristic, complemented
40C23×D173—22no—
41#14583827—24no—
42#14583827—24no—
43C2×D692—22no—
44C2×D692—24no—
45C2×D692C2312yes—
46C2×D692C2312yes—
47C2×C22⋊D173—22no—
48C2×C22⋊D173—24no—
49D4×D173—24no—
50D4×D173—24no—
51C2×C4×D173—22no—
52C2×C4×D173C2311yescharacteristic
53#14583813—24no—
54#14583813—24no—
55D1384—22no—
56D1384—22no—
57D1384—24no—
58#14583811—24no—
59#14583811—24no—
60#14583809D414yescomplemented
61C2×#13063436C2311yescharacteristic
62C2×#13063436C2311yescharacteristic
63C8×D173D414yescomplemented
64#14583805D412yes—
65#14583803—24no—
66#14583803D412yes—
67C4×Dic173C2311yescharacteristic
68C2×Q8×C173C2312yes—
69C2×D4×C173C2312yes—
70Q16×C173—22no—
71SD16×C173—24no—
72D8×C173—22no—
73C2×C8×C173C2311yescharacteristic
74C2×C8×C173C2311yescharacteristic
75C22⋊C4×C173—24no—
76C42×C173C2311yescharacteristic
77C22×D173—22no—
78C22×D173—22no—
79C22×D173—44no—
80C22×D173—84no—
81C22×D173C2×D411yescharacteristic
82D692—22no—
83D692—22no—
84D692—22no—
85D692—22no—
86D692—44no—
87D692—84no—
88C22⋊D173—44no—
89C22⋊D173—84no—
90C2×Dic173C2×D412yes—
91C2×Dic173C2×D411yescharacteristic
92C4×D173—44no—
93C4×D173C2×D412yes—
94C4×D173C2×D412yes—
95#13063436C2×D412yes—
96#13063436C2×D412yes—
97C23×C173—22no—
98Q8×C173—22no—
99Q8×C173—22no—
100D4×C173—22no—
101D4×C173—22no—
102C2×C4×C173—22no—
103C2×C4×C173C2×D412yes—
104C2×C4×C173C2411yescharacteristic
105C8×C173C2×D412yes—
106C8×C173C2×D412yes—
107C22×C173—42no—
108C22×C173—44no—
109C22×C173C22×D411yescharacteristic
110C2×D173—22no—
111C2×D173—42no—
112C2×D173—42no—
113C2×D173—44no—
114C2×D173—44no—
115C2×D173—88no—
116C2×D173#12712yes—
117C4×C173—22no—
118C4×C173—42no—
119C4×C173C22×D411yescharacteristic
120C4×C173C22×D411yescharacteristic
121Dic173—22no—
122Dic173#12712yes—
123C2×C173—82no—
124C2×C173C2×#12711yescharacteristic
125C2×C173#57612yes—
126D173—22no—
127D173—82no—
128D173—82no—
129C173#315511yescharacteristic, complemented, Sylow
130#3155—1731nomaximal, Sylow
131C2×AGL(1,ℤ/8)—1732no—
132C2×#135—1731no—
133C22×D8—1731no—
134#535—1732no—
135#496—1732no—
136#496—1732no—
137#495—1731no—
138#495D17311yescharacteristic, complemented
139#493—1731no—
140#493—1731no—
141#405—1731no—
142C2×#41—1732no—
143C22×D4—1732no—
144AGL(1,ℤ/8)—3468no—
145#135—3464no—
146C2×Q16C2×D17311yescharacteristic, complemented
147C2×SD16—1732no—
148C2×SD16—1732no—
149C2×SD16C2×D17312yescomplemented
150C2×D8—3464no—
151C2×D8—1732no—
152C2×D8—1732no—
153C2×D8—1731no—
154C2×D8—1731no—
155C2×D8—1732no—
156C2×D8C2×D17311yescharacteristic
157C2×M16—1731no—
158C22×C8—1731no—
159#127—1732no—
160#127—1732no—
161#124—1732no—
162#124C2×D17312yescomplemented
163#121—3464no—
164#120—3464no—
165#117—1731no—
166#97—1731no—
167#97—1731no—
168C4×C8—1731no—
169C4×C8C2×D17311yescharacteristic, complemented
170C24—3462no—
171#41—3464no—
172#41—3464no—
173C2×Q8C22×D17312yes—
174C2×D4—3462no—
175C2×D4—3464no—
176C2×D4—3464no—
177C2×D4—3464no—
178C2×D4—3462no—
179C2×D4—3464no—
180C2×D4—1732no—
181C2×D4—1732no—
182C2×D4C22×D17312yes—
183C22×C4—3462no—
184C22×C4—1731no—
185Q16—22no—
186SD16—24no—
187SD16—3464no—
188SD16—3464no—
189D8—22no—
190D8—3464no—
191D8—3464no—
192D8—3462no—
193D8—3462no—
194D8—3464no—
195M16—1734no—
196C2×C8—1731no—
197C2×C8—1731no—
198C2×C8—1734no—
199C2×C8C22×D17311yescharacteristic
200C2×C8C22×D17311yescharacteristic
201C4⋊C4—1732no—
202C22⋊C4—24no—
203C22⋊C4—3464no—
204C22⋊C4—1732no—
205C42—1731no—
206C42C22×D17311yescharacteristic
207C23—22no—
208C23—3462no—
209C23—3462no—
210C23—13844no—
211C23—1731no—
212C23—6924no—
213Q8—22no—
214Q8—22no—
215D4—22no—
216D4—22no—
217D4—3462no—
218D4—3462no—
219D4—3462no—
220D4—3462no—
221D4—13844no—
222D4—13844no—
223D4—6924no—
224D4—6924no—
225C2×C4—22no—
226C2×C4—1732no—
227C2×C4—1731no—
228C2×C4—1732no—
229C2×C4—1732no—
230C2×C4—6924no—
231C2×C4D4×D17312yes—
232C2×C4C23×D17311yescharacteristic
233C8—1732no—
234C8—1732no—
235C8D4×D17312yes—
236C8D4×D17312yes—
237C22—42no—
238C22—44no—
239C22—3462no—
240C22—13848no—
241C22—1732no—
242C22—6922no—
243C22—6922no—
244C22—6924no—
245C22—6924no—
246C22C2×D4×D17311yescharacteristic
247C4—22no—
248C4—42no—
249C4—3462no—
250C4—1732no—
251C4C2×D4×D17311yescharacteristic
252C4C2×D4×D17311yescharacteristic
253C2—82no—
254C2—3462no—
255C2—13842no—
256C2—13842no—
257C2#127×D17311yescharacteristic
258C2#2601229312yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#495×C173C2
socleC22×C173C22×D4
hypercenter#495D173
hypocenterC173#3155
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. C2×#19149702C2semidirect
3. C2×D8×D173C2semidirect
4. #26012059C2semidirect
5. #26011881C2semidirect
6. #26011877C2semidirect
7. #26011876C2semidirect
8. #26011868C2semidirect
9. #26011866C2semidirect
10. #26011445C2semidirect
11. #495×C173C2semidirect
16. C2×#14583813C22semidirect
17. C2×#14583813C22semidirect
19. C2×D1384C22semidirect
20. C2×D1384C22semidirect
22. C2×#14583811C22semidirect
26. #19149676C22semidirect
27. #19149675C22semidirect
28. #19149668C22semidirect
32. #19149590C22semidirect
33. #19149589C22semidirect
34. C8×Dic173C22semidirect
35. C2×Q16×C173C22semidirect
36. C2×SD16×C173C22semidirect
38. #124×C173C22semidirect
39. C4×C8×C173C22semidirect
60. #14583809D4semidirect
63. C8×D173D4semidirect
129. C173#3155semidirect
138. #495D173semidirect
146. C2×Q16C2×D173semidirect
149. C2×SD16C2×D173semidirect
162. #124C2×D173semidirect
169. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
C2×#191497022
C2×D8×D1732
#260120592
#260118812
#260118772
#260118762
#260118682
#260118662
#260114452
#495×C1732
#3155173