Finite Group Databasebeta

← #35204030

Subgroups — #35204030

Order 20288.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#23778827C211yesmaximal, characteristic, complemented
2#23778826C211yesmaximal, characteristic, complemented
3#23778812C212yesmaximal, complemented
4#23778810C211yesmaximal, characteristic, complemented
5#23778805C211yesmaximal, characteristic, complemented
6#23778804C212yesmaximal, complemented
7#23778801C212yesmaximal, complemented
8C2×#19099563C211yesmaximal, characteristic, complemented
9SD16×D317C212yesmaximal, complemented
10#23778786C211yesmaximal, characteristic, complemented
11#135×C317C211yesmaximal, characteristic, complemented
12#19099575C2212yes—
13#19099574—21no—
14#19099574—21no—
15#19099574C2211yescharacteristic
16#19099573—22no—
17#19099573C2212yescomplemented
18C2×Dic634—21no—
19C2×Dic634C2211yescharacteristic, complemented
20Q8×D317C2212yescomplemented
21C2×D1268—21no—
22C2×D1268C2211yescharacteristic
23D4×D317—22no—
24D4×D317C2212yes—
25#19099564C2212yescomplemented
26#19099563C2212yescomplemented
27#19099563C2212yescomplemented
28#19099562C2212yescomplemented
29#19099561C2212yescomplemented
30#19099559C2212yescomplemented
31#19099558C2211yescharacteristic, complemented
32#19099557C2212yescomplemented
33C8×D317C2212yescomplemented
34#41×C317C2211yescharacteristic
35#41×C317C2211yescharacteristic, complemented
36Q16×C317C2211yescharacteristic, complemented
37SD16×C317C2211yescharacteristic, complemented
38SD16×C317C2211yescharacteristic, complemented
39D8×C317C2211yescharacteristic, complemented
40C2×C8×C317C2211yescharacteristic, complemented
41C22×D317—21no—
42C22×D317—22no—
43Dic634—21no—
44Dic634—22no—
45Dic634D411yescharacteristic
46Dic634C2312yes—
47D1268—22no—
48D1268—21no—
49D1268D411yescharacteristic
50D1268C2312yes—
51C22⋊D317—22no—
52C22⋊D317—22no—
53C22⋊D317D412yes—
54C2×Dic317—21no—
55C2×Dic317—22no—
56C4×D317—22no—
57C4×D317—22no—
58C4×D317C2312yes—
59#14579261C2312yes—
60Q8×C317C2311yescharacteristic
61Q8×C317C2311yescharacteristic
62D4×C317—21no—
63D4×C317—21no—
64D4×C317C2311yescharacteristic
65D4×C317C2311yescharacteristic
66C2×C4×C317—21no—
67C2×C4×C317—21no—
68C2×C4×C317C2311yescharacteristic
69C8×C317C2311yescharacteristic
70C8×C317C2311yescharacteristic
71C22×C317—21no—
72C22×C317—21no—
73C22×C317C2×D411yescharacteristic
74C2×D317—22no—
75C2×D317—82no—
76C2×D317—81no—
77C2×D317C2×D412yes—
78C4×C317—21no—
79C4×C317—21no—
80C4×C317C2×D411yescharacteristic
81C4×C317C2411yescharacteristic
82Dic317—22no—
83Dic317C2×D412yes—
84C2×C317—21no—
85C2×C317—41no—
86C2×C317—41no—
87C2×C317C22×D411yescharacteristic
88D317—22no—
89D317—42no—
90C317#57711yescharacteristic, complemented, Sylow
91#577—3171nomaximal, Sylow
922−1+4—3171no—
93H(5,𝔽2)—3171no—
94#137—3171no—
95#137—3172no—
96AGL(1,ℤ/8)—3171no—
97AGL(1,ℤ/8)—3172no—
98#135—3172no—
99#135D31711yescharacteristic, complemented
100C2×SD16—3172no—
101C2×SD16—3171no—
102#131—3171no—
103#41—3171no—
104#41—3172no—
105#41—3172no—
106#41—6341no—
107#41—6341no—
108#41—6342no—
109#41C2×D31711yescharacteristic
110#41C2×D31711yescharacteristic, complemented
111C2×Q8—3172no—
112C2×Q8—3171no—
113C2×Q8—6341no—
114C2×D4—3172no—
115C2×D4—3171no—
116C2×D4—6341no—
117C2×D4—6342no—
118Q16—3172no—
119Q16C2×D31711yescharacteristic, complemented
120SD16—3172no—
121SD16—3172no—
122SD16—3172no—
123SD16—3172no—
124SD16C2×D31711yescharacteristic, complemented
125SD16C2×D31711yescharacteristic, complemented
126D8—3172no—
127D8C2×D31711yescharacteristic, complemented
128M16—3171no—
129M16—3172no—
130C2×C8—3172no—
131C2×C8C2×D31711yescharacteristic, complemented
132C23—6342no—
133C23—6341no—
134Q8—3171no—
135Q8—3172no—
136Q8—6342no—
137Q8—6341no—
138Q8C22×D31711yescharacteristic
139Q8C22×D31711yescharacteristic
140D4—21no—
141D4—21no—
142D4—3172no—
143D4—3171no—
144D4—3172no—
145D4—6342no—
146D4—6342no—
147D4—6342no—
148D4—6341no—
149D4C22×D31711yescharacteristic
150D4C22×D31711yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—3172no—
154C2×C4—6342no—
155C2×C4—6342no—
156C2×C4—6342no—
157C2×C4—6341no—
158C2×C4C22×D31711yescharacteristic
159C8—3172no—
160C8C22×D31711yescharacteristic
161C8C22×D31711yescharacteristic
162C22—21no—
163C22—21no—
164C22—3172no—
165C22—6342no—
166C22—25361no—
167C22—25362no—
168C22D4×D31711yescharacteristic
169C4—21no—
170C4—21no—
171C4—3172no—
172C4—6342no—
173C4D4×D31711yescharacteristic
174C4C23×D31711yescharacteristic
175C2—21no—
176C2—41no—
177C2—41no—
178C2—6342no—
179C2—12682no—
180C2C2×D4×D31711yescharacteristic
subgroupG/H
centerC2C2×D4×D317
commutator, cosocleC4×C317C24
FrattiniC4C23×D317
Fitting#135×C317C2
socleC2×C317C22×D4
hypercenter#135D317
hypocenterC317#577
LCSG ▷ C4×C317 ▷ C2×C317 ▷ C317
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #23778827C2semidirect
2. #23778826C2semidirect
3. #23778812C2semidirect
4. #23778810C2semidirect
5. #23778805C2semidirect
6. #23778804C2semidirect
7. #23778801C2semidirect
8. C2×#19099563C2semidirect
9. SD16×D317C2semidirect
10. #23778786C2semidirect
11. #135×C317C2semidirect
17. #19099573C22semidirect
19. C2×Dic634C22semidirect
20. Q8×D317C22semidirect
25. #19099564C22semidirect
26. #19099563C22semidirect
27. #19099563C22semidirect
28. #19099562C22semidirect
29. #19099561C22semidirect
30. #19099559C22semidirect
31. #19099558C22semidirect
32. #19099557C22semidirect
33. C8×D317C22semidirect
35. #41×C317C22semidirect
36. Q16×C317C22semidirect
37. SD16×C317C22semidirect
38. SD16×C317C22semidirect
39. D8×C317C22semidirect
40. C2×C8×C317C22semidirect
90. C317#577semidirect
99. #135D317semidirect
110. #41C2×D317semidirect
119. Q16C2×D317semidirect
124. SD16C2×D317semidirect
125. SD16C2×D317semidirect
127. D8C2×D317semidirect
131. C2×C8C2×D317semidirect

Maximal subgroups

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

subgroupindex
#237788272
#237788262
#237788122
#237788102
#237788052
#237788042
#237788012
C2×#190995632
SD16×D3172
#237787862
#135×C3172
#577317