Finite Group Databasebeta

← #35204033

Subgroups — #35204033

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#23778825C211yesmaximal, characteristic, complemented
3#23778813C211yesmaximal, characteristic, complemented
4#23778809C212yesmaximal, complemented
5#23778804C212yesmaximal, complemented
6#23778800C211yesmaximal, characteristic, complemented
7#23778799C212yesmaximal, complemented
8C2×#19099562C211yesmaximal, characteristic, complemented
9SD16×D317C212yesmaximal, complemented
10#23778788C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C317C211yesmaximal, characteristic, complemented
12#19099574—21no—
13#19099574C2211yescharacteristic
14#19099574C2211yescharacteristic
15#19099573—21no—
16#19099573—22no—
17#19099573C2212yes—
18#19099573C2212yescomplemented
19C2×Dic634—21no—
20C2×Dic634C2211yescharacteristic, complemented
21Q8×D317C2212yescomplemented
22C2×C22⋊D317—22no—
23D4×D317—21no—
24D4×D317C2212yes—
25Dic1268C2212yescomplemented
26#19099564C2211yescharacteristic, complemented
27#19099563C2212yescomplemented
28#19099562C2212yescomplemented
29#19099562C2212yescomplemented
30#19099562C2211yescharacteristic, complemented
31#19099561C2211yescharacteristic, complemented
32#19099559C2211yescharacteristic, complemented
33#19099557C2212yescomplemented
34C2×#14579261C2211yescharacteristic, complemented
35C8×D317C2212yescomplemented
36#41×C317C2211yescharacteristic, complemented
37C2×D4×C317C2211yescharacteristic
38SD16×C317C2212yescomplemented
39D8×C317C2212yescomplemented
40M16×C317C2211yescharacteristic, complemented
41C22×D317—22no—
42Dic634—21no—
43Dic634—22no—
44Dic634D411yescharacteristic
45Dic634C2312yes—
46Dic634C2311yescharacteristic
47D1268D411yescharacteristic
48D1268C2311yescharacteristic
49C22⋊D317—21no—
50C22⋊D317—22no—
51C22⋊D317—22no—
52C22⋊D317—22no—
53C22⋊D317D412yes—
54C2×Dic317—21no—
55C2×Dic317—22no—
56C2×Dic317—22no—
57C4×D317—22no—
58C4×D317—21no—
59C4×D317C2312yes—
60#14579261C2311yescharacteristic
61#14579261C2311yescharacteristic
62C23×C317—21no—
63Q8×C317C2311yescharacteristic
64D4×C317—21no—
65D4×C317—21no—
66D4×C317C2311yescharacteristic
67D4×C317C2312yes—
68C2×C4×C317—21no—
69C2×C4×C317C2311yescharacteristic
70C8×C317C2312yes—
71C22×C317—21no—
72C22×C317—22no—
73C22×C317—81no—
74C22×C317C2×D411yescharacteristic
75C2×D317—21no—
76C2×D317—82no—
77C2×D317C2×D412yes—
78C4×C317—21no—
79C4×C317C2×D411yescharacteristic
80C4×C317C2411yescharacteristic
81Dic317—21no—
82Dic317—22no—
83Dic317C2×D412yes—
84C2×C317—21no—
85C2×C317—42no—
86C2×C317—41no—
87C2×C317C22×D411yescharacteristic
88D317—22no—
89D317—41no—
90C317#57711yescharacteristic, complemented, Sylow
91#577—3171nomaximal, Sylow
922−1+4—3171no—
93H(5,𝔽2)—3171no—
94#137—3172no—
95#137—3171no—
96AGL(1,ℤ/8)—3172no—
97AGL(1,ℤ/8)D31711yescharacteristic, complemented
98#135—3171no—
99#135—3172no—
100C2×SD16—3171no—
101C2×SD16—3172no—
102#131—3171no—
103#41—3171no—
104#41—3172no—
105#41—3172no—
106#41—3171no—
107#41—6341no—
108#41—6342no—
109#41—6341no—
110#41C2×D31711yescharacteristic, complemented
111C2×Q8—3172no—
112C2×Q8—3171no—
113C2×Q8—6341no—
114C2×D4—3172no—
115C2×D4—6341no—
116C2×D4—6342no—
117C2×D4C2×D31711yescharacteristic
118Q16—3171no—
119Q16—3172no—
120SD16—3171no—
121SD16—3172no—
122SD16—3172no—
123SD16—3171no—
124SD16—3172no—
125SD16C2×D31712yescomplemented
126D8—3171no—
127D8C2×D31712yescomplemented
128M16—3172no—
129M16C2×D31711yescharacteristic, complemented
130C2×C8—3171no—
131C2×C8—3172no—
132C23—21no—
133C23—6342no—
134Q8—3171no—
135Q8—3172no—
136Q8—3171no—
137Q8—6342no—
138Q8—6341no—
139Q8C22×D31711yescharacteristic
140D4—21no—
141D4—21no—
142D4—3172no—
143D4—3171no—
144D4—3171no—
145D4—6342no—
146D4—6342no—
147D4—6342no—
148D4—6341no—
149D4C22×D31711yescharacteristic
150D4C22×D31712yes—
151C2×C4—21no—
152C2×C4—3172no—
153C2×C4—6342no—
154C2×C4—6342no—
155C2×C4—6342no—
156C2×C4—6341no—
157C2×C4—6341no—
158C2×C4C22×D31711yescharacteristic
159C8—3171no—
160C8—3171no—
161C8C22×D31712yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—3172no—
166C22—6341no—
167C22—25362no—
168C22D4×D31711yescharacteristic
169C4—21no—
170C4—3172no—
171C4—6341no—
172C4—6342no—
173C4D4×D31711yescharacteristic
174C4C23×D31711yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—6342no—
179C2—12681no—
180C2C2×D4×D31711yescharacteristic
subgroupG/H
centerC2C2×D4×D317
commutator, cosocleC4×C317C24
FrattiniC4C23×D317
FittingAGL(1,ℤ/8)×C317C2
socleC2×C317C22×D4
hypercenterAGL(1,ℤ/8)D317
hypocenterC317#577
LCSG ▷ C4×C317 ▷ C2×C317 ▷ C317
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #23778827C2semidirect
2. #23778825C2semidirect
3. #23778813C2semidirect
4. #23778809C2semidirect
5. #23778804C2semidirect
6. #23778800C2semidirect
7. #23778799C2semidirect
8. C2×#19099562C2semidirect
9. SD16×D317C2semidirect
10. #23778788C2semidirect
11. AGL(1,ℤ/8)×C317C2semidirect
18. #19099573C22semidirect
20. C2×Dic634C22semidirect
21. Q8×D317C22semidirect
25. Dic1268C22semidirect
26. #19099564C22semidirect
27. #19099563C22semidirect
28. #19099562C22semidirect
29. #19099562C22semidirect
30. #19099562C22semidirect
31. #19099561C22semidirect
32. #19099559C22semidirect
33. #19099557C22semidirect
34. C2×#14579261C22semidirect
35. C8×D317C22semidirect
36. #41×C317C22semidirect
38. SD16×C317C22semidirect
39. D8×C317C22semidirect
40. M16×C317C22semidirect
90. C317#577semidirect
97. AGL(1,ℤ/8)D317semidirect
110. #41C2×D317semidirect
125. SD16C2×D317semidirect
127. D8C2×D317semidirect
129. M16C2×D317semidirect

Maximal subgroups

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

subgroupindex
#237788272
#237788252
#237788132
#237788092
#237788042
#237788002
#237787992
C2×#190995622
SD16×D3172
#237787882
AGL(1,ℤ/8)×C3172
#577317