Finite Group Databasebeta

← #20299407

Subgroups — #20299407

Order 7232.

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#16813670C211yesmaximal, characteristic, complemented
2#16813668C211yesmaximal, characteristic, complemented
3#16813654C212yesmaximal, complemented
4#16813650C211yesmaximal, characteristic, complemented
5#16813649C212yesmaximal, complemented
6#16813645C212yesmaximal, complemented
7#16813642C211yesmaximal, characteristic, complemented
8C2×#14268634C211yesmaximal, characteristic, complemented
9SD16×D113C212yesmaximal, complemented
10#16813630C211yesmaximal, characteristic, complemented
11#137×C113C211yesmaximal, characteristic, complemented
12#14268648—21no—
13#14268648C2212yes—
14#14268648C2212yescomplemented
15#14268647—22no—
16#14268647—21no—
17#14268647C2211yescharacteristic
18#14268647C2211yescharacteristic, complemented
19Q8×D113—21no—
20Q8×D113C2212yescomplemented
21C2×D452—21no—
22C2×D452C2211yescharacteristic
23D4×D113—22no—
24D4×D113C2212yes—
25#14268637C2211yescharacteristic, complemented
26#14268636C2212yescomplemented
27#14268635C2211yescharacteristic, complemented
28#14268634C2212yescomplemented
29#14268634C2212yescomplemented
30#14268634C2211yescharacteristic, complemented
31D904C2212yescomplemented
32#14268632C2211yescharacteristic, complemented
33#14268630C2212yescomplemented
34C2×#11744751C2211yescharacteristic, complemented
35C8×D113C2212yescomplemented
36#41×C113C2211yescharacteristic
37C2×Q8×C113C2211yescharacteristic, complemented
38Q16×C113C2212yescomplemented
39SD16×C113C2212yescomplemented
40M16×C113C2211yescharacteristic, complemented
41C22×D113—21no—
42C22×D113—22no—
43Q8×C113—21no—
44Q8×C113C2312yes—
45Q8×C113C2311yescharacteristic
46D4×C113—21no—
47D4×C113C2311yescharacteristic
48C2×C4×C113—21no—
49C2×C4×C113—21no—
50C2×C4×C113C2311yescharacteristic
51C22⋊D113—21no—
52C22⋊D113—22no—
53C22⋊D113D412yes—
54D452—22no—
55D452—22no—
56D452—21no—
57D452D411yescharacteristic
58D452C2311yescharacteristic
59D452C2312yes—
60C4×D113—22no—
61C4×D113—21no—
62C4×D113—22no—
63C4×D113—22no—
64C4×D113C2312yes—
65Dic226—22no—
66Dic226D411yescharacteristic
67Dic226C2311yescharacteristic
68C8×C113C2312yes—
69#11744751C2311yescharacteristic
70#11744751C2311yescharacteristic
71C22×C113—21no—
72C22×C113C2×D411yescharacteristic
73C2×D113—21no—
74C2×D113—22no—
75C2×D113—82no—
76C2×D113—81no—
77C2×D113C2×D412yes—
78C4×C113—22no—
79C4×C113—21no—
80C4×C113C2×D411yescharacteristic
81C4×C113C2411yescharacteristic
82Dic113—21no—
83Dic113C2×D412yes—
84C2×C113—21no—
85C2×C113—41no—
86C2×C113C22×D411yescharacteristic
87D113—22no—
88D113—41no—
89D113—42no—
90C113#57711yescharacteristic, complemented, Sylow
91#577—1131nomaximal, Sylow
922−1+4—1131no—
93H(5,𝔽2)—1131no—
94#137—1132no—
95#137D11311yescharacteristic, complemented
96AGL(1,ℤ/8)—1132no—
97AGL(1,ℤ/8)—1131no—
98#135—1131no—
99#135—1132no—
100C2×SD16—1131no—
101C2×SD16—1132no—
102#131—1131no—
103#41—1131no—
104#41—1132no—
105#41—1132no—
106#41—1131no—
107#41—2262no—
108#41—2261no—
109#41—2261no—
110#41C2×D11311yescharacteristic
111C2×Q8—1132no—
112C2×Q8—2261no—
113C2×Q8C2×D11311yescharacteristic, complemented
114C2×D4—1132no—
115C2×D4—1131no—
116C2×D4—2261no—
117C2×D4—2262no—
118Q16—1131no—
119Q16C2×D11312yescomplemented
120SD16—1132no—
121SD16—1132no—
122SD16—1131no—
123SD16—1131no—
124SD16—1132no—
125SD16C2×D11312yescomplemented
126D8—1131no—
127D8—1132no—
128M16—1132no—
129M16C2×D11311yescharacteristic, complemented
130C2×C8—1131no—
131C2×C8—1132no—
132C23—2262no—
133C23—2261no—
134Q8—21no—
135Q8—1131no—
136Q8—1131no—
137Q8—2262no—
138Q8C22×D11312yes—
139Q8C22×D11311yescharacteristic
140D4—21no—
141D4—1132no—
142D4—1131no—
143D4—1131no—
144D4—1132no—
145D4—2262no—
146D4—2262no—
147D4—2262no—
148D4—2261no—
149D4—2261no—
150D4C22×D11311yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—1132no—
154C2×C4—2262no—
155C2×C4—2262no—
156C2×C4—2262no—
157C2×C4—2261no—
158C2×C4C22×D11311yescharacteristic
159C8—1131no—
160C8—1131no—
161C8C22×D11312yes—
162C22—21no—
163C22—1132no—
164C22—9041no—
165C22—9042no—
166C22—2261no—
167C22—2262no—
168C22D4×D11311yescharacteristic
169C4—21no—
170C4—22no—
171C4—1132no—
172C4—2261no—
173C4D4×D11311yescharacteristic
174C4C23×D11311yescharacteristic
175C2—21no—
176C2—41no—
177C2—4522no—
178C2—4521no—
179C2—2262no—
180C2C2×D4×D11311yescharacteristic
subgroupG/H
centerC2C2×D4×D113
commutatorC4×C113C24
FrattiniC4C23×D113
Fitting#137×C113C2
socleC2×C113C22×D4
hypercenter#137D113
hypocenterC113#577
LCSG ▷ C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #16813670C2semidirect
2. #16813668C2semidirect
3. #16813654C2semidirect
4. #16813650C2semidirect
5. #16813649C2semidirect
6. #16813645C2semidirect
7. #16813642C2semidirect
8. C2×#14268634C2semidirect
9. SD16×D113C2semidirect
10. #16813630C2semidirect
11. #137×C113C2semidirect
14. #14268648C22semidirect
18. #14268647C22semidirect
20. Q8×D113C22semidirect
25. #14268637C22semidirect
26. #14268636C22semidirect
27. #14268635C22semidirect
28. #14268634C22semidirect
29. #14268634C22semidirect
30. #14268634C22semidirect
31. D904C22semidirect
32. #14268632C22semidirect
33. #14268630C22semidirect
34. C2×#11744751C22semidirect
35. C8×D113C22semidirect
37. C2×Q8×C113C22semidirect
38. Q16×C113C22semidirect
39. SD16×C113C22semidirect
40. M16×C113C22semidirect
90. C113#577semidirect
95. #137D113semidirect
113. C2×Q8C2×D113semidirect
119. Q16C2×D113semidirect
125. SD16C2×D113semidirect
129. M16C2×D113semidirect