Finite Group Databasebeta

← #27213953

Subgroups — #27213953

Order 13504.

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#20270024C211yesmaximal, characteristic, complemented
2#20270023C211yesmaximal, characteristic, complemented
3#20270009C212yesmaximal, complemented
4#20270007C211yesmaximal, characteristic, complemented
5#20270002C211yesmaximal, characteristic, complemented
6#20270001C212yesmaximal, complemented
7#20269998C212yesmaximal, complemented
8C2×#16809951C211yesmaximal, characteristic, complemented
9SD16×D211C212yesmaximal, complemented
10#20269983C211yesmaximal, characteristic, complemented
11#135×C211C211yesmaximal, characteristic, complemented
12#16809963C2212yes—
13#16809962—21no—
14#16809962—21no—
15#16809962C2211yescharacteristic
16#16809961—22no—
17#16809961C2212yescomplemented
18C2×Dic422—21no—
19C2×Dic422C2211yescharacteristic, complemented
20Q8×D211C2212yescomplemented
21C2×D844—21no—
22C2×D844C2211yescharacteristic
23D4×D211—22no—
24D4×D211C2212yes—
25#16809952C2212yescomplemented
26#16809951C2212yescomplemented
27#16809951C2212yescomplemented
28#16809950C2212yescomplemented
29#16809949C2212yescomplemented
30#16809947C2212yescomplemented
31#16809946C2211yescharacteristic, complemented
32#16809945C2212yescomplemented
33C8×D211C2212yescomplemented
34#41×C211C2211yescharacteristic
35#41×C211C2211yescharacteristic, complemented
36Q16×C211C2211yescharacteristic, complemented
37SD16×C211C2211yescharacteristic, complemented
38SD16×C211C2211yescharacteristic, complemented
39D8×C211C2211yescharacteristic, complemented
40C2×C8×C211C2211yescharacteristic, complemented
41C22×D211—21no—
42C22×D211—22no—
43Dic422—21no—
44Dic422—22no—
45Dic422D411yescharacteristic
46Dic422C2312yes—
47D844—22no—
48D844—21no—
49D844D411yescharacteristic
50D844C2312yes—
51C22⋊D211—22no—
52C22⋊D211—22no—
53C22⋊D211D412yes—
54C2×Dic211—21no—
55C2×Dic211—22no—
56C4×D211—22no—
57C4×D211—22no—
58C4×D211C2312yes—
59C211⋊C8C2312yes—
60Q8×C211C2311yescharacteristic
61Q8×C211C2311yescharacteristic
62D4×C211—21no—
63D4×C211—21no—
64D4×C211C2311yescharacteristic
65D4×C211C2311yescharacteristic
66C2×C4×C211—21no—
67C2×C4×C211—21no—
68C2×C4×C211C2311yescharacteristic
69C8×C211C2311yescharacteristic
70C8×C211C2311yescharacteristic
71C22×C211—21no—
72C22×C211—21no—
73C22×C211C2×D411yescharacteristic
74C2×D211—22no—
75C2×D211—82no—
76C2×D211—81no—
77C2×D211C2×D412yes—
78C4×C211—21no—
79C4×C211—21no—
80C4×C211C2×D411yescharacteristic
81C4×C211C2411yescharacteristic
82Dic211—22no—
83Dic211C2×D412yes—
84C2×C211—21no—
85C2×C211—41no—
86C2×C211—41no—
87C2×C211C22×D411yescharacteristic
88D211—22no—
89D211—42no—
90C211#57711yescharacteristic, complemented, Sylow
91#577—2111nomaximal, Sylow
922−1+4—2111no—
93H(5,𝔽2)—2111no—
94#137—2111no—
95#137—2112no—
96AGL(1,ℤ/8)—2111no—
97AGL(1,ℤ/8)—2112no—
98#135—2112no—
99#135D21111yescharacteristic, complemented
100C2×SD16—2112no—
101C2×SD16—2111no—
102#131—2111no—
103#41—4221no—
104#41—4221no—
105#41—4222no—
106#41—2112no—
107#41—2112no—
108#41—2111no—
109#41C2×D21111yescharacteristic
110#41C2×D21111yescharacteristic, complemented
111C2×Q8—4221no—
112C2×Q8—2112no—
113C2×Q8—2111no—
114C2×D4—4221no—
115C2×D4—4222no—
116C2×D4—2112no—
117C2×D4—2111no—
118Q16—2112no—
119Q16C2×D21111yescharacteristic, complemented
120SD16—2112no—
121SD16—2112no—
122SD16—2112no—
123SD16—2112no—
124SD16C2×D21111yescharacteristic, complemented
125SD16C2×D21111yescharacteristic, complemented
126D8—2112no—
127D8C2×D21111yescharacteristic, complemented
128M16—2111no—
129M16—2112no—
130C2×C8—2112no—
131C2×C8C2×D21111yescharacteristic, complemented
132C23—4222no—
133C23—4221no—
134Q8—4222no—
135Q8—4221no—
136Q8—2112no—
137Q8—2111no—
138Q8C22×D21111yescharacteristic
139Q8C22×D21111yescharacteristic
140D4—21no—
141D4—21no—
142D4—4222no—
143D4—4222no—
144D4—4222no—
145D4—4221no—
146D4—2112no—
147D4—2111no—
148D4—2112no—
149D4C22×D21111yescharacteristic
150D4C22×D21111yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—4222no—
154C2×C4—4222no—
155C2×C4—4221no—
156C2×C4—4222no—
157C2×C4—2112no—
158C2×C4C22×D21111yescharacteristic
159C8—2112no—
160C8C22×D21111yescharacteristic
161C8C22×D21111yescharacteristic
162C22—21no—
163C22—21no—
164C22—16881no—
165C22—16882no—
166C22—4222no—
167C22—2112no—
168C22D4×D21111yescharacteristic
169C4—21no—
170C4—21no—
171C4—4222no—
172C4—2112no—
173C4D4×D21111yescharacteristic
174C4C23×D21111yescharacteristic
175C2—21no—
176C2—41no—
177C2—41no—
178C2—8442no—
179C2—4222no—
180C2C2×D4×D21111yescharacteristic
subgroupG/H
centerC2C2×D4×D211
commutator, cosocleC4×C211C24
FrattiniC4C23×D211
Fitting#135×C211C2
socleC2×C211C22×D4
hypercenter#135D211
hypocenterC211#577
LCSG ▷ C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #20270024C2semidirect
2. #20270023C2semidirect
3. #20270009C2semidirect
4. #20270007C2semidirect
5. #20270002C2semidirect
6. #20270001C2semidirect
7. #20269998C2semidirect
8. C2×#16809951C2semidirect
9. SD16×D211C2semidirect
10. #20269983C2semidirect
11. #135×C211C2semidirect
17. #16809961C22semidirect
19. C2×Dic422C22semidirect
20. Q8×D211C22semidirect
25. #16809952C22semidirect
26. #16809951C22semidirect
27. #16809951C22semidirect
28. #16809950C22semidirect
29. #16809949C22semidirect
30. #16809947C22semidirect
31. #16809946C22semidirect
32. #16809945C22semidirect
33. C8×D211C22semidirect
35. #41×C211C22semidirect
36. Q16×C211C22semidirect
37. SD16×C211C22semidirect
38. SD16×C211C22semidirect
39. D8×C211C22semidirect
40. C2×C8×C211C22semidirect
90. C211#577semidirect
99. #135D211semidirect
110. #41C2×D211semidirect
119. Q16C2×D211semidirect
124. SD16C2×D211semidirect
125. SD16C2×D211semidirect
127. D8C2×D211semidirect
131. C2×C8C2×D211semidirect

Maximal subgroups

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

subgroupindex
#202700242
#202700232
#202700092
#202700072
#202700022
#202700012
#202699982
C2×#168099512
SD16×D2112
#202699832
#135×C2112
#577211