Finite Group Databasebeta

← #35204027

Subgroups — #35204027

Order 20288.

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

normal subgroups only

This group has 268 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#23778826C211yesmaximal, characteristic, complemented
2#23778825C211yesmaximal, characteristic, complemented
3#23778808C211yesmaximal, characteristic, complemented
4#23778807C212yesmaximal, complemented
5#23778804C212yesmaximal, complemented
6#23778801C212yesmaximal, complemented
7#23778800C211yesmaximal, characteristic, complemented
8C2×#19099559C211yesmaximal, characteristic, complemented
9D8×D317C212yesmaximal, complemented
10#23778788C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C317C211yesmaximal, characteristic, complemented
12#19099575C2212yescomplemented
13#19099574—21no—
14#19099574C2211yescharacteristic
15#19099574C2211yescharacteristic, complemented
16#19099573—21no—
17#19099573C2212yescomplemented
18C2×D1268—21no—
19C2×D1268C2211yescharacteristic, complemented
20C2×C22⋊D317—22no—
21D4×D317—21no—
22D4×D317—22no—
23D4×D317C2212yescomplemented
24D4×D317C2212yescomplemented
25#19099564C2211yescharacteristic, complemented
26#19099563C2212yescomplemented
27#19099562C2211yescharacteristic, complemented
28#19099561C2211yescharacteristic, complemented
29D2536C2212yescomplemented
30#19099559C2211yescharacteristic, complemented
31#19099559C2212yescomplemented
32#19099559C2212yescomplemented
33#19099557C2212yescomplemented
34C2×#14579261C2211yescharacteristic, complemented
35C8×D317C2212yescomplemented
36#41×C317C2211yescharacteristic, complemented
37C2×D4×C317C2211yescharacteristic, complemented
38SD16×C317C2212yescomplemented
39D8×C317C2212yescomplemented
40M16×C317C2211yescharacteristic, complemented
41C22×D317—21no—
42C22×D317—22no—
43C22×D317—22no—
44Dic634D411yescharacteristic
45Dic634C2311yescharacteristic
46D1268—22no—
47D1268—21no—
48D1268D411yescharacteristic
49D1268C2312yes—
50D1268C2311yescharacteristic
51C22⋊D317—21no—
52C22⋊D317—22no—
53C22⋊D317—22no—
54C22⋊D317—22no—
55C22⋊D317D412yes—
56C2×Dic317—22no—
57C4×D317—21no—
58C4×D317—22no—
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—22no—
77C2×D317—82no—
78C2×D317—82no—
79C2×D317—81no—
80C2×D317C2×D412yes—
81C4×C317—21no—
82C4×C317C2×D411yescharacteristic
83C4×C317C2411yescharacteristic
84Dic317—21no—
85Dic317C2×D412yes—
86C2×C317—21no—
87C2×C317—41no—
88C2×C317—42no—
89C2×C317C22×D411yescharacteristic
90D317—22no—
91D317—41no—
92D317—42no—
93C317#57611yescharacteristic, complemented, Sylow
94#576—3171nomaximal, Sylow
95H(5,𝔽2)—3171no—
96H(5,𝔽2)—3171no—
97AGL(1,ℤ/8)—3172no—
98AGL(1,ℤ/8)—3172no—
99AGL(1,ℤ/8)—3171no—
100AGL(1,ℤ/8)D31711yescharacteristic, complemented
101#135—3171no—
102#135—3172no—
103C2×D8—3171no—
104C2×D8—3172no—
105#131—3171no—
106#41—3171no—
107#41—3172no—
108#41—3172no—
109#41—3171no—
110#41—6341no—
111#41—6341no—
112#41C2×D31711yescharacteristic, complemented
113C2×D4—3172no—
114C2×D4—3172no—
115C2×D4—3171no—
116C2×D4—6341no—
117C2×D4—6342no—
118C2×D4—6341no—
119C2×D4—6342no—
120C2×D4C2×D31711yescharacteristic, complemented
121Q16—3171no—
122SD16—3171no—
123SD16—3171no—
124SD16—3172no—
125SD16C2×D31712yescomplemented
126D8—3172no—
127D8—3171no—
128D8—3172no—
129D8—3172no—
130D8C2×D31712yescomplemented
131M16—3172no—
132M16C2×D31711yescharacteristic, complemented
133C2×C8—3171no—
134C2×C8—3172no—
135C23—21no—
136C23—6342no—
137C23—6342no—
138C23—6341no—
139Q8—3171no—
140Q8—3171no—
141Q8C22×D31711yescharacteristic
142D4—21no—
143D4—21no—
144D4—3172no—
145D4—3171no—
146D4—3172no—
147D4—3171no—
148D4—6342no—
149D4—6342no—
150D4—6342no—
151D4—6342no—
152D4—6341no—
153D4—6341no—
154D4C22×D31712yes—
155D4C22×D31711yescharacteristic
156C2×C4—21no—
157C2×C4—3172no—
158C2×C4—6342no—
159C2×C4—6342no—
160C2×C4—6341no—
161C2×C4C22×D31711yescharacteristic
162C8—3171no—
163C8—3171no—
164C8C22×D31712yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—3172no—
169C22—6342no—
170C22—6341no—
171C22—25361no—
172C22—25362no—
173C22—25362no—
174C22D4×D31711yescharacteristic
175C4—21no—
176C4—3172no—
177C4—6341no—
178C4D4×D31711yescharacteristic
179C4C23×D31711yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—6342no—
184C2—12682no—
185C2—12681no—
186C2C2×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#576
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. #23778826C2semidirect
2. #23778825C2semidirect
3. #23778808C2semidirect
4. #23778807C2semidirect
5. #23778804C2semidirect
6. #23778801C2semidirect
7. #23778800C2semidirect
8. C2×#19099559C2semidirect
9. D8×D317C2semidirect
10. #23778788C2semidirect
11. AGL(1,ℤ/8)×C317C2semidirect
12. #19099575C22semidirect
15. #19099574C22semidirect
17. #19099573C22semidirect
19. C2×D1268C22semidirect
23. D4×D317C22semidirect
24. D4×D317C22semidirect
25. #19099564C22semidirect
26. #19099563C22semidirect
27. #19099562C22semidirect
28. #19099561C22semidirect
29. D2536C22semidirect
30. #19099559C22semidirect
31. #19099559C22semidirect
32. #19099559C22semidirect
33. #19099557C22semidirect
34. C2×#14579261C22semidirect
35. C8×D317C22semidirect
36. #41×C317C22semidirect
37. C2×D4×C317C22semidirect
38. SD16×C317C22semidirect
39. D8×C317C22semidirect
40. M16×C317C22semidirect
93. C317#576semidirect
100. AGL(1,ℤ/8)D317semidirect
112. #41C2×D317semidirect
120. C2×D4C2×D317semidirect
125. SD16C2×D317semidirect
130. D8C2×D317semidirect
132. M16C2×D317semidirect

Maximal subgroups

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

subgroupindex
#237788262
#237788252
#237788082
#237788072
#237788042
#237788012
#237788002
C2×#190995592
D8×D3172
#237787882
AGL(1,ℤ/8)×C3172
#576317