Finite Group Databasebeta

← #27164827

Subgroups — #27164827

Order 12736.

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#20264003C211yesmaximal, characteristic, complemented
2#20264001C211yesmaximal, characteristic, complemented
3#20263987C212yesmaximal, complemented
4#20263983C211yesmaximal, characteristic, complemented
5#20263982C212yesmaximal, complemented
6#20263978C212yesmaximal, complemented
7#20263975C211yesmaximal, characteristic, complemented
8C2×#15691873C211yesmaximal, characteristic, complemented
9SD16×D199C212yesmaximal, complemented
10#20263963C211yesmaximal, characteristic, complemented
11#137×C199C211yesmaximal, characteristic, complemented
12#15691887—21no—
13#15691887C2212yes—
14#15691887C2212yescomplemented
15#15691886—21no—
16#15691886—22no—
17#15691886C2211yescharacteristic
18#15691886C2211yescharacteristic, complemented
19Q8×D199—21no—
20Q8×D199C2212yescomplemented
21C2×D796—21no—
22C2×D796C2211yescharacteristic
23D4×D199—22no—
24D4×D199C2212yes—
25#15691876C2211yescharacteristic, complemented
26#15691875C2212yescomplemented
27#15691874C2211yescharacteristic, complemented
28#15691873C2212yescomplemented
29#15691873C2212yescomplemented
30#15691873C2211yescharacteristic, complemented
31D1592C2212yescomplemented
32#15691871C2211yescharacteristic, complemented
33#15691869C2212yescomplemented
34C2×C199⋊C8C2211yescharacteristic, complemented
35C8×D199C2212yescomplemented
36#41×C199C2211yescharacteristic
37C2×Q8×C199C2211yescharacteristic, complemented
38Q16×C199C2212yescomplemented
39SD16×C199C2212yescomplemented
40M16×C199C2211yescharacteristic, complemented
41C22×D199—21no—
42C22×D199—22no—
43Dic398—22no—
44Dic398D411yescharacteristic
45Dic398C2311yescharacteristic
46D796—22no—
47D796—22no—
48D796—21no—
49D796D411yescharacteristic
50D796C2311yescharacteristic
51D796C2312yes—
52C22⋊D199—21no—
53C22⋊D199—22no—
54C22⋊D199D412yes—
55C4×D199—22no—
56C4×D199—21no—
57C4×D199—22no—
58C4×D199—22no—
59C4×D199C2312yes—
60C199⋊C8C2311yescharacteristic
61C199⋊C8C2311yescharacteristic
62Q8×C199—21no—
63Q8×C199C2312yes—
64Q8×C199C2311yescharacteristic
65D4×C199—21no—
66D4×C199C2311yescharacteristic
67C2×C4×C199—21no—
68C2×C4×C199—21no—
69C2×C4×C199C2311yescharacteristic
70C8×C199C2312yes—
71C22×C199—21no—
72C22×C199C2×D411yescharacteristic
73C2×D199—21no—
74C2×D199—22no—
75C2×D199—82no—
76C2×D199—81no—
77C2×D199C2×D412yes—
78C4×C199—21no—
79C4×C199—22no—
80C4×C199C2×D411yescharacteristic
81C4×C199C2411yescharacteristic
82Dic199—21no—
83Dic199C2×D412yes—
84C2×C199—21no—
85C2×C199—41no—
86C2×C199C22×D411yescharacteristic
87D199—22no—
88D199—41no—
89D199—42no—
90C199#57711yescharacteristic, complemented, Sylow
91#577—1991nomaximal, Sylow
922−1+4—1991no—
93H(5,𝔽2)—1991no—
94#137—1992no—
95#137D19911yescharacteristic, complemented
96AGL(1,ℤ/8)—1992no—
97AGL(1,ℤ/8)—1991no—
98#135—1991no—
99#135—1992no—
100C2×SD16—1991no—
101C2×SD16—1992no—
102#131—1991no—
103#41—3981no—
104#41—3982no—
105#41—3981no—
106#41—1992no—
107#41—1992no—
108#41—1991no—
109#41—1991no—
110#41C2×D19911yescharacteristic
111C2×Q8—3981no—
112C2×Q8—1992no—
113C2×Q8C2×D19911yescharacteristic, complemented
114C2×D4—3981no—
115C2×D4—3982no—
116C2×D4—1992no—
117C2×D4—1991no—
118Q16—1991no—
119Q16C2×D19912yescomplemented
120SD16—1992no—
121SD16—1992no—
122SD16—1991no—
123SD16—1991no—
124SD16—1992no—
125SD16C2×D19912yescomplemented
126D8—1991no—
127D8—1992no—
128M16—1992no—
129M16C2×D19911yescharacteristic, complemented
130C2×C8—1991no—
131C2×C8—1992no—
132C23—3982no—
133C23—3981no—
134Q8—21no—
135Q8—3982no—
136Q8—1991no—
137Q8—1991no—
138Q8C22×D19912yes—
139Q8C22×D19911yescharacteristic
140D4—21no—
141D4—3982no—
142D4—3982no—
143D4—3981no—
144D4—3982no—
145D4—3981no—
146D4—1992no—
147D4—1991no—
148D4—1991no—
149D4—1992no—
150D4C22×D19911yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—3982no—
154C2×C4—3982no—
155C2×C4—3982no—
156C2×C4—3981no—
157C2×C4—1992no—
158C2×C4C22×D19911yescharacteristic
159C8—1991no—
160C8—1991no—
161C8C22×D19912yes—
162C22—21no—
163C22—15921no—
164C22—15922no—
165C22—3981no—
166C22—3982no—
167C22—1992no—
168C22D4×D19911yescharacteristic
169C4—21no—
170C4—22no—
171C4—3981no—
172C4—1992no—
173C4D4×D19911yescharacteristic
174C4C23×D19911yescharacteristic
175C2—21no—
176C2—41no—
177C2—7962no—
178C2—7961no—
179C2—3982no—
180C2C2×D4×D19911yescharacteristic
subgroupG/H
centerC2C2×D4×D199
commutator, cosocleC4×C199C24
FrattiniC4C23×D199
Fitting#137×C199C2
socleC2×C199C22×D4
hypercenter#137D199
hypocenterC199#577
LCSG ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #20264003C2semidirect
2. #20264001C2semidirect
3. #20263987C2semidirect
4. #20263983C2semidirect
5. #20263982C2semidirect
6. #20263978C2semidirect
7. #20263975C2semidirect
8. C2×#15691873C2semidirect
9. SD16×D199C2semidirect
10. #20263963C2semidirect
11. #137×C199C2semidirect
14. #15691887C22semidirect
18. #15691886C22semidirect
20. Q8×D199C22semidirect
25. #15691876C22semidirect
26. #15691875C22semidirect
27. #15691874C22semidirect
28. #15691873C22semidirect
29. #15691873C22semidirect
30. #15691873C22semidirect
31. D1592C22semidirect
32. #15691871C22semidirect
33. #15691869C22semidirect
34. C2×C199⋊C8C22semidirect
35. C8×D199C22semidirect
37. C2×Q8×C199C22semidirect
38. Q16×C199C22semidirect
39. SD16×C199C22semidirect
40. M16×C199C22semidirect
90. C199#577semidirect
95. #137D199semidirect
113. C2×Q8C2×D199semidirect
119. Q16C2×D199semidirect
125. SD16C2×D199semidirect
129. M16C2×D199semidirect

Maximal subgroups

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

subgroupindex
#202640032
#202640012
#202639872
#202639832
#202639822
#202639782
#202639752
C2×#156918732
SD16×D1992
#202639632
#137×C1992
#577199