Finite Group Databasebeta

← #22578210

Subgroups — #22578210

Order 8768.

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#17984135C211yesmaximal, characteristic, complemented
2#17984134C211yesmaximal, characteristic, complemented
3#17984120C212yesmaximal, complemented
4#17984118C211yesmaximal, characteristic, complemented
5#17984113C211yesmaximal, characteristic, complemented
6#17984112C212yesmaximal, complemented
7#17984109C212yesmaximal, complemented
8C2×#14555499C211yesmaximal, characteristic, complemented
9SD16×D137C212yesmaximal, complemented
10#17984094C211yesmaximal, characteristic, complemented
11#135×C137C211yesmaximal, characteristic, complemented
12#14555511C2212yes—
13#14555510—21no—
14#14555510—21no—
15#14555510C2211yescharacteristic
16#14555509—22no—
17#14555509C2212yescomplemented
18C2×Dic274—21no—
19C2×Dic274C2211yescharacteristic, complemented
20Q8×D137C2212yescomplemented
21C2×D548—21no—
22C2×D548C2211yescharacteristic
23D4×D137—22no—
24D4×D137C2212yes—
25#14555500C2212yescomplemented
26#14555499C2212yescomplemented
27#14555499C2212yescomplemented
28#14555498C2212yescomplemented
29#14555497C2212yescomplemented
30#14555495C2212yescomplemented
31#14555494C2211yescharacteristic, complemented
32#14555493C2212yescomplemented
33C8×D137C2212yescomplemented
34#41×C137C2211yescharacteristic
35#41×C137C2211yescharacteristic, complemented
36Q16×C137C2211yescharacteristic, complemented
37SD16×C137C2211yescharacteristic, complemented
38SD16×C137C2211yescharacteristic, complemented
39D8×C137C2211yescharacteristic, complemented
40C2×C8×C137C2211yescharacteristic, complemented
41C22×D137—21no—
42C22×D137—22no—
43Dic274—21no—
44Dic274—22no—
45Dic274D411yescharacteristic
46Dic274C2312yes—
47D548—22no—
48D548—21no—
49D548D411yescharacteristic
50D548C2312yes—
51C22⋊D137—22no—
52C22⋊D137—22no—
53C22⋊D137D412yes—
54C2×Dic137—21no—
55C2×Dic137—22no—
56C4×D137—22no—
57C4×D137—22no—
58C4×D137C2312yes—
59#11764040C2312yes—
60Q8×C137C2311yescharacteristic
61Q8×C137C2311yescharacteristic
62D4×C137—21no—
63D4×C137—21no—
64D4×C137C2311yescharacteristic
65D4×C137C2311yescharacteristic
66C2×C4×C137—21no—
67C2×C4×C137—21no—
68C2×C4×C137C2311yescharacteristic
69C8×C137C2311yescharacteristic
70C8×C137C2311yescharacteristic
71C22×C137—21no—
72C22×C137—21no—
73C22×C137C2×D411yescharacteristic
74C2×D137—22no—
75C2×D137—82no—
76C2×D137—81no—
77C2×D137C2×D412yes—
78C4×C137—21no—
79C4×C137—21no—
80C4×C137C2×D411yescharacteristic
81C4×C137C2411yescharacteristic
82Dic137—22no—
83Dic137C2×D412yes—
84C2×C137—21no—
85C2×C137—41no—
86C2×C137—41no—
87C2×C137C22×D411yescharacteristic
88D137—22no—
89D137—42no—
90C137#57711yescharacteristic, complemented, Sylow
91#577—1371nomaximal, Sylow
922−1+4—1371no—
93H(5,𝔽2)—1371no—
94#137—1371no—
95#137—1372no—
96AGL(1,ℤ/8)—1371no—
97AGL(1,ℤ/8)—1372no—
98#135—1372no—
99#135D13711yescharacteristic, complemented
100C2×SD16—1372no—
101C2×SD16—1371no—
102#131—1371no—
103#41—2741no—
104#41—2741no—
105#41—2742no—
106#41—1372no—
107#41—1372no—
108#41—1371no—
109#41C2×D13711yescharacteristic
110#41C2×D13711yescharacteristic, complemented
111C2×Q8—2741no—
112C2×Q8—1372no—
113C2×Q8—1371no—
114C2×D4—2741no—
115C2×D4—2742no—
116C2×D4—1372no—
117C2×D4—1371no—
118Q16—1372no—
119Q16C2×D13711yescharacteristic, complemented
120SD16—1372no—
121SD16—1372no—
122SD16—1372no—
123SD16—1372no—
124SD16C2×D13711yescharacteristic, complemented
125SD16C2×D13711yescharacteristic, complemented
126D8—1372no—
127D8C2×D13711yescharacteristic, complemented
128M16—1371no—
129M16—1372no—
130C2×C8—1372no—
131C2×C8C2×D13711yescharacteristic, complemented
132C23—2742no—
133C23—2741no—
134Q8—2742no—
135Q8—2741no—
136Q8—1372no—
137Q8—1371no—
138Q8C22×D13711yescharacteristic
139Q8C22×D13711yescharacteristic
140D4—21no—
141D4—21no—
142D4—2742no—
143D4—2742no—
144D4—2742no—
145D4—2741no—
146D4—1372no—
147D4—1371no—
148D4—1372no—
149D4C22×D13711yescharacteristic
150D4C22×D13711yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—2742no—
154C2×C4—2742no—
155C2×C4—2741no—
156C2×C4—2742no—
157C2×C4—1372no—
158C2×C4C22×D13711yescharacteristic
159C8—1372no—
160C8C22×D13711yescharacteristic
161C8C22×D13711yescharacteristic
162C22—21no—
163C22—21no—
164C22—2742no—
165C22—10961no—
166C22—10962no—
167C22—1372no—
168C22D4×D13711yescharacteristic
169C4—21no—
170C4—21no—
171C4—2742no—
172C4—1372no—
173C4D4×D13711yescharacteristic
174C4C23×D13711yescharacteristic
175C2—21no—
176C2—41no—
177C2—41no—
178C2—2742no—
179C2—5482no—
180C2C2×D4×D13711yescharacteristic
subgroupG/H
centerC2C2×D4×D137
commutatorC4×C137C24
FrattiniC4C23×D137
Fitting#135×C137C2
socleC2×C137C22×D4
hypercenter#135D137
hypocenterC137#577
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #17984135C2semidirect
2. #17984134C2semidirect
3. #17984120C2semidirect
4. #17984118C2semidirect
5. #17984113C2semidirect
6. #17984112C2semidirect
7. #17984109C2semidirect
8. C2×#14555499C2semidirect
9. SD16×D137C2semidirect
10. #17984094C2semidirect
11. #135×C137C2semidirect
17. #14555509C22semidirect
19. C2×Dic274C22semidirect
20. Q8×D137C22semidirect
25. #14555500C22semidirect
26. #14555499C22semidirect
27. #14555499C22semidirect
28. #14555498C22semidirect
29. #14555497C22semidirect
30. #14555495C22semidirect
31. #14555494C22semidirect
32. #14555493C22semidirect
33. C8×D137C22semidirect
35. #41×C137C22semidirect
36. Q16×C137C22semidirect
37. SD16×C137C22semidirect
38. SD16×C137C22semidirect
39. D8×C137C22semidirect
40. C2×C8×C137C22semidirect
90. C137#577semidirect
99. #135D137semidirect
110. #41C2×D137semidirect
119. Q16C2×D137semidirect
124. SD16C2×D137semidirect
125. SD16C2×D137semidirect
127. D8C2×D137semidirect
131. C2×C8C2×D137semidirect