Finite Group Databasebeta

← #22578207

Subgroups — #22578207

Order 8768.

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#17984134C211yesmaximal, characteristic, complemented
2#17984133C211yesmaximal, characteristic, complemented
3#17984116C211yesmaximal, characteristic, complemented
4#17984115C212yesmaximal, complemented
5#17984112C212yesmaximal, complemented
6#17984109C212yesmaximal, complemented
7#17984108C211yesmaximal, characteristic, complemented
8C2×#14555495C211yesmaximal, characteristic, complemented
9D8×D137C212yesmaximal, complemented
10#17984096C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C137C211yesmaximal, characteristic, complemented
12#14555511C2212yescomplemented
13#14555510—21no—
14#14555510C2211yescharacteristic
15#14555510C2211yescharacteristic, complemented
16#14555509—21no—
17#14555509C2212yescomplemented
18C2×D548—21no—
19C2×D548C2211yescharacteristic, complemented
20C2×C22⋊D137—22no—
21D4×D137—21no—
22D4×D137—22no—
23D4×D137C2212yescomplemented
24D4×D137C2212yescomplemented
25#14555500C2211yescharacteristic, complemented
26#14555499C2212yescomplemented
27#14555498C2211yescharacteristic, complemented
28#14555497C2211yescharacteristic, complemented
29D1096C2212yescomplemented
30#14555495C2211yescharacteristic, complemented
31#14555495C2212yescomplemented
32#14555495C2212yescomplemented
33#14555493C2212yescomplemented
34C2×#11764040C2211yescharacteristic, complemented
35C8×D137C2212yescomplemented
36#41×C137C2211yescharacteristic, complemented
37C2×D4×C137C2211yescharacteristic, complemented
38SD16×C137C2212yescomplemented
39D8×C137C2212yescomplemented
40M16×C137C2211yescharacteristic, complemented
41C22×D137—21no—
42C22×D137—22no—
43C22×D137—22no—
44Dic274D411yescharacteristic
45Dic274C2311yescharacteristic
46D548—22no—
47D548—21no—
48D548D411yescharacteristic
49D548C2312yes—
50D548C2311yescharacteristic
51C22⋊D137—21no—
52C22⋊D137—22no—
53C22⋊D137—22no—
54C22⋊D137—22no—
55C22⋊D137D412yes—
56C2×Dic137—22no—
57C4×D137—21no—
58C4×D137—22no—
59C4×D137C2312yes—
60#11764040C2311yescharacteristic
61#11764040C2311yescharacteristic
62C23×C137—21no—
63Q8×C137C2311yescharacteristic
64D4×C137—21no—
65D4×C137—21no—
66D4×C137C2311yescharacteristic
67D4×C137C2312yes—
68C2×C4×C137—21no—
69C2×C4×C137C2311yescharacteristic
70C8×C137C2312yes—
71C22×C137—21no—
72C22×C137—22no—
73C22×C137—81no—
74C22×C137C2×D411yescharacteristic
75C2×D137—21no—
76C2×D137—22no—
77C2×D137—82no—
78C2×D137—82no—
79C2×D137—81no—
80C2×D137C2×D412yes—
81C4×C137—21no—
82C4×C137C2×D411yescharacteristic
83C4×C137C2411yescharacteristic
84Dic137—21no—
85Dic137C2×D412yes—
86C2×C137—21no—
87C2×C137—41no—
88C2×C137—42no—
89C2×C137C22×D411yescharacteristic
90D137—22no—
91D137—41no—
92D137—42no—
93C137#57611yescharacteristic, complemented, Sylow
94#576—1371nomaximal, Sylow
95H(5,𝔽2)—1371no—
96H(5,𝔽2)—1371no—
97AGL(1,ℤ/8)—1372no—
98AGL(1,ℤ/8)—1372no—
99AGL(1,ℤ/8)—1371no—
100AGL(1,ℤ/8)D13711yescharacteristic, complemented
101#135—1371no—
102#135—1372no—
103C2×D8—1371no—
104C2×D8—1372no—
105#131—1371no—
106#41—2741no—
107#41—2741no—
108#41—1372no—
109#41—1372no—
110#41—1371no—
111#41—1371no—
112#41C2×D13711yescharacteristic, complemented
113C2×D4—2741no—
114C2×D4—2742no—
115C2×D4—2741no—
116C2×D4—2742no—
117C2×D4—1372no—
118C2×D4—1372no—
119C2×D4—1371no—
120C2×D4C2×D13711yescharacteristic, complemented
121Q16—1371no—
122SD16—1371no—
123SD16—1371no—
124SD16—1372no—
125SD16C2×D13712yescomplemented
126D8—1372no—
127D8—1371no—
128D8—1372no—
129D8—1372no—
130D8C2×D13712yescomplemented
131M16—1372no—
132M16C2×D13711yescharacteristic, complemented
133C2×C8—1371no—
134C2×C8—1372no—
135C23—21no—
136C23—2742no—
137C23—2742no—
138C23—2741no—
139Q8—1371no—
140Q8—1371no—
141Q8C22×D13711yescharacteristic
142D4—21no—
143D4—21no—
144D4—2742no—
145D4—2742no—
146D4—2741no—
147D4—2742no—
148D4—2742no—
149D4—2741no—
150D4—1372no—
151D4—1371no—
152D4—1372no—
153D4—1371no—
154D4C22×D13712yes—
155D4C22×D13711yescharacteristic
156C2×C4—21no—
157C2×C4—2742no—
158C2×C4—2742no—
159C2×C4—2741no—
160C2×C4—1372no—
161C2×C4C22×D13711yescharacteristic
162C8—1371no—
163C8—1371no—
164C8C22×D13712yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—2742no—
169C22—2741no—
170C22—10961no—
171C22—10962no—
172C22—10962no—
173C22—1372no—
174C22D4×D13711yescharacteristic
175C4—21no—
176C4—2741no—
177C4—1372no—
178C4D4×D13711yescharacteristic
179C4C23×D13711yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—2742no—
184C2—5482no—
185C2—5481no—
186C2C2×D4×D13711yescharacteristic
subgroupG/H
centerC2C2×D4×D137
commutatorC4×C137C24
FrattiniC4C23×D137
FittingAGL(1,ℤ/8)×C137C2
socleC2×C137C22×D4
hypercenterAGL(1,ℤ/8)D137
hypocenterC137#576
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #17984134C2semidirect
2. #17984133C2semidirect
3. #17984116C2semidirect
4. #17984115C2semidirect
5. #17984112C2semidirect
6. #17984109C2semidirect
7. #17984108C2semidirect
8. C2×#14555495C2semidirect
9. D8×D137C2semidirect
10. #17984096C2semidirect
11. AGL(1,ℤ/8)×C137C2semidirect
12. #14555511C22semidirect
15. #14555510C22semidirect
17. #14555509C22semidirect
19. C2×D548C22semidirect
23. D4×D137C22semidirect
24. D4×D137C22semidirect
25. #14555500C22semidirect
26. #14555499C22semidirect
27. #14555498C22semidirect
28. #14555497C22semidirect
29. D1096C22semidirect
30. #14555495C22semidirect
31. #14555495C22semidirect
32. #14555495C22semidirect
33. #14555493C22semidirect
34. C2×#11764040C22semidirect
35. C8×D137C22semidirect
36. #41×C137C22semidirect
37. C2×D4×C137C22semidirect
38. SD16×C137C22semidirect
39. D8×C137C22semidirect
40. M16×C137C22semidirect
93. C137#576semidirect
100. AGL(1,ℤ/8)D137semidirect
112. #41C2×D137semidirect
120. C2×D4C2×D137semidirect
125. SD16C2×D137semidirect
130. D8C2×D137semidirect
132. M16C2×D137semidirect