Finite Group Databasebeta

← #35203695

Subgroups — #35203695

Order 20288.

338 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 151 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×D4×D317C211yesmaximal, characteristic
2C4×D1268C211yesmaximal, characteristic, complemented
3D4×Dic317C211yesmaximal, characteristic, complemented
4#23778742C211yesmaximal, characteristic, complemented
5C4×C22⋊D317C212yesmaximal, complemented
6#23778740C212yesmaximal, complemented
7#23778738C211yesmaximal, characteristic, complemented
8#23778737C211yesmaximal, characteristic, complemented
9C2×#19099551C212yesmaximal, complemented
10C22⋊C4×D317C212yesmaximal, complemented
11C4×D4×C317C211yesmaximal, characteristic, complemented
12C23×D317C2212yes—
13C2×D1268C2211yescharacteristic
14C2×C22⋊D317C2212yes—
15D4×D317C418yescomplemented
16C22×Dic317C2212yescomplemented
17C2×C4×D317C2211yescharacteristic
18C2×C4×D317C2212yescomplemented
19C2×C4×D317C2212yescomplemented
20C4⋊Dic317C2211yescharacteristic, complemented
21#19099553C2211yescharacteristic, complemented
22#19099553C2211yescharacteristic, complemented
23C22⋊Dic317C2212yescomplemented
24#19099551C2211yescharacteristic
25#19099551C2212yes—
26#19099551C2211yescharacteristic
27#19099551C2212yescomplemented
28#19099551C2212yescomplemented
29C4×Dic317C2211yescharacteristic, complemented
30C4×Dic317C2212yescomplemented
31C2×D4×C317C2211yescharacteristic
32C22×C4×C317C2212yescomplemented
33C4⋊C4×C317C2211yescharacteristic, complemented
34C22⋊C4×C317C2212yescomplemented
35C42×C317C2211yescharacteristic, complemented
36C22×D317—42no—
37C22×D317—42no—
38C22×D317C2×C418yescomplemented
39C22×D317C2311yescharacteristic
40C22×D317C2312yes—
41D1268C2×C414yescomplemented
42C22⋊D317C2×C418yescomplemented
43C2×Dic317—42no—
44C2×Dic317C2311yescharacteristic
45C2×Dic317C2312yes—
46C2×Dic317C2311yescharacteristic
47C2×Dic317C2311yescharacteristic
48C4×D317—42no—
49C4×D317—42no—
50C4×D317C2×C414yescomplemented
51C23×C317C2312yes—
52D4×C317C2×C414yescomplemented
53C2×C4×C317—42no—
54C2×C4×C317C2311yescharacteristic
55C2×C4×C317C2311yescharacteristic
56C2×C4×C317C2312yes—
57C2×C4×C317C2311yescharacteristic
58C22×C317—22no—
59C22×C317—22no—
60C22×C317C22×C414yes—
61C22×C317C2411yescharacteristic
62C2×D317—22no—
63C2×D317—22no—
64C2×D317—21no—
65C2×D317—21no—
66C2×D317—48no—
67C2×D317C22×C412yes—
68C2×D317C22×C414yes—
69C4×C317—21no—
70C4×C317—22no—
71C4×C317—21no—
72C4×C317C22×C412yes—
73Dic317—21no—
74Dic317—22no—
75Dic317—21no—
76Dic317C22×C412yes—
77C2×C317—24no—
78C2×C317C23×C411yescharacteristic
79C2×C317H(5,𝔽2)11yescharacteristic
80C2×C317H(5,𝔽2)11yescharacteristic
81D317—22no—
82D317—24no—
83C317#51811yescharacteristic, complemented, Sylow
84#518—3171nomaximal, Sylow
85C22×D4—3171no—
86C4×D4—3171no—
87C4×D4—3172no—
88C4×D4—3171no—
89C4×D4—3172no—
90C4×D4—3171no—
91C4×D4D31711yescharacteristic, complemented
92#117—3171no—
93#117—3171no—
94C2×C22⋊C4—3172no—
95C2×C22⋊C4—3172no—
96C24—3172no—
97C2×D4—3172no—
98C2×D4—3178no—
99C2×D4—3171no—
100C2×D4C2×D31711yescharacteristic
101C22×C4—3171no—
102C22×C4—3172no—
103C22×C4—3172no—
104C22×C4—3172no—
105C22×C4C2×D31712yescomplemented
106C4⋊C4—3171no—
107C4⋊C4—3171no—
108C4⋊C4—3171no—
109C4⋊C4C2×D31711yescharacteristic, complemented
110C22⋊C4—3172no—
111C22⋊C4—3172no—
112C22⋊C4—3171no—
113C22⋊C4—3171no—
114C22⋊C4—3172no—
115C22⋊C4—3172no—
116C22⋊C4C2×D31712yescomplemented
117C42—3172no—
118C42—3171no—
119C42C2×D31711yescharacteristic, complemented
120C23—3171no—
121C23—3172no—
122C23—3178no—
123C23—12682no—
124C23—12682no—
125C23C22×D31712yes—
126D4—3178no—
127D4—3174no—
128D4C4×D31714yescomplemented
129C2×C4—42no—
130C2×C4—3171no—
131C2×C4—3174no—
132C2×C4—3171no—
133C2×C4—3172no—
134C2×C4—3171no—
135C2×C4—12682no—
136C2×C4—12682no—
137C2×C4—12682no—
138C2×C4C22×D31712yes—
139C2×C4C22×D31711yescharacteristic
140C2×C4C22×D31711yescharacteristic
141C2×C4C22×D31711yescharacteristic
142C22—22no—
143C22—22no—
144C22—3172no—
145C22—3174no—
146C22—6342no—
147C22—6341no—
148C22—6342no—
149C22—6341no—
150C22—12688no—
151C22C2×C4×D31714yes—
152C22C23×D31711yescharacteristic
153C4—21no—
154C4—22no—
155C4—21no—
156C4—3172no—
157C4—6341no—
158C4—6342no—
159C4—6341no—
160C4C2×C4×D31712yes—
161C2—24no—
162C2—6342no—
163C2—6344no—
164C2C22×C4×D31711yescharacteristic
165C2#2377882511yescharacteristic
166C2#2377882611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D317
commutatorC2×C317C23×C4
FittingC4×D4×C317C2
socle, cosocleC22×C317C24
prime radicalC2×D4×D317C2
hypercenterC4×D4D317
hypocenterC317#518
LCSG ▷ C2×C317 ▷ C317
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×D1268C2semidirect
3. D4×Dic317C2semidirect
4. #23778742C2semidirect
5. C4×C22⋊D317C2semidirect
6. #23778740C2semidirect
7. #23778738C2semidirect
8. #23778737C2semidirect
9. C2×#19099551C2semidirect
10. C22⋊C4×D317C2semidirect
11. C4×D4×C317C2semidirect
15. D4×D317C4semidirect
16. C22×Dic317C22semidirect
18. C2×C4×D317C22semidirect
19. C2×C4×D317C22semidirect
20. C4⋊Dic317C22semidirect
21. #19099553C22semidirect
22. #19099553C22semidirect
23. C22⋊Dic317C22semidirect
27. #19099551C22semidirect
28. #19099551C22semidirect
29. C4×Dic317C22semidirect
30. C4×Dic317C22semidirect
32. C22×C4×C317C22semidirect
33. C4⋊C4×C317C22semidirect
34. C22⋊C4×C317C22semidirect
35. C42×C317C22semidirect
38. C22×D317C2×C4semidirect
41. D1268C2×C4semidirect
42. C22⋊D317C2×C4semidirect
50. C4×D317C2×C4semidirect
52. D4×C317C2×C4semidirect
83. C317#518semidirect
91. C4×D4D317semidirect
105. C22×C4C2×D317semidirect
109. C4⋊C4C2×D317semidirect
116. C22⋊C4C2×D317semidirect
119. C42C2×D317semidirect
128. D4C4×D317semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D3172
C4×D12682
D4×Dic3172
#237787422
C4×C22⋊D3172
#237787402
#237787382
#237787372
C2×#190995512
C22⋊C4×D3172
C4×D4×C3172
#518317