Finite Group Databasebeta

← #20299397

Subgroups — #20299397

Order 7232.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 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
1C2×#14268648C211yesmaximal, characteristic, complemented
2#41×D113C211yesmaximal, characteristic, complemented
3#16813654C212yesmaximal, complemented
4#16813653C211yesmaximal, characteristic, complemented
5#16813650C211yesmaximal, characteristic, complemented
6#16813649C212yesmaximal, complemented
7#16813647C211yesmaximal, characteristic, complemented
8#16813645C212yesmaximal, complemented
9#16813643C212yesmaximal, complemented
10M16×D113C211yesmaximal, characteristic, complemented
11#137×C113C211yesmaximal, characteristic, complemented
12#14268648—21no—
13#14268648—21no—
14#14268648C2212yes—
15#14268648C2212yescomplemented
16#14268648C2211yescharacteristic, complemented
17#14268647—21no—
18#14268647C2211yescharacteristic
19#14268646—21no—
20#14268646C2211yescharacteristic, complemented
21Q8×D113C2211yescharacteristic, complemented
22C2×D452—21no—
23C2×D452C2211yescharacteristic, complemented
24D4×D113—21no—
25D4×D113C2211yescharacteristic, complemented
26C2×C4×D113—21no—
27C2×C4×D113—21no—
28C2×C4×D113C2211yescharacteristic
29#14268637C2212yescomplemented
30#14268636C2212yescomplemented
31#14268634C2212yescomplemented
32#14268634C2212yescomplemented
33D904C2212yescomplemented
34#14268632C2212yescomplemented
35#14268631C2211yescharacteristic, complemented
36#14268630C2212yescomplemented
37C8×D113C2212yescomplemented
38#41×C113C2211yescharacteristic
39C2×Q8×C113C2211yescharacteristic, complemented
40Q16×C113C2212yescomplemented
41SD16×C113C2212yescomplemented
42M16×C113C2211yescharacteristic, complemented
43C22×D113—21no—
44C22×D113—21no—
45C22×D113D411yescharacteristic
46Q8×C113—21no—
47Q8×C113C2312yes—
48Q8×C113C2311yescharacteristic
49D4×C113—21no—
50D4×C113C2311yescharacteristic
51C2×C4×C113—21no—
52C2×C4×C113—21no—
53C2×C4×C113C2311yescharacteristic
54C22⋊D113—21no—
55C22⋊D113—21no—
56C22⋊D113—21no—
57C2×Dic113—21no—
58C2×Dic113D411yescharacteristic
59D452—21no—
60D452—22no—
61D452—22no—
62D452—21no—
63D452C2312yes—
64D452C2311yescharacteristic
65C4×D113—21no—
66C4×D113—21no—
67C4×D113—21no—
68C4×D113—22no—
69C4×D113—22no—
70C4×D113D411yescharacteristic
71C4×D113D411yescharacteristic
72C4×D113C2311yescharacteristic
73C4×D113C2311yescharacteristic
74Dic226—21no—
75Dic226C2311yescharacteristic
76C8×C113C2312yes—
77#11744751C2312yes—
78C22×C113—21no—
79C22×C113C2×D411yescharacteristic
80C2×D113—22no—
81C2×D113—21no—
82C2×D113—41no—
83C2×D113—42no—
84C2×D113—42no—
85C2×D113C2×D411yescharacteristic
86C2×D113C2×D411yescharacteristic
87C4×C113—21no—
88C4×C113—22no—
89C4×C113C2×D411yescharacteristic
90C4×C113C2411yescharacteristic
91Dic113—21no—
92Dic113C2×D411yescharacteristic
93Dic113C2×D411yescharacteristic
94C2×C113—21no—
95C2×C113—41no—
96C2×C113C22×D411yescharacteristic
97D113—21no—
98D113—21no—
99D113—42no—
100D113—41no—
101C113#57511yescharacteristic, complemented, Sylow
102#575—1131nomaximal, Sylow
103C2×#41—1131no—
104C2×#41—1131no—
105#137—1131no—
106#137—1132no—
107#137D11311yescharacteristic, complemented
108AGL(1,ℤ/8)—1131no—
109AGL(1,ℤ/8)—1132no—
110AGL(1,ℤ/8)—1131no—
111#135—1132no—
112#135—1132no—
113C2×M16—1131no—
114#41—1131no—
115#41—1132no—
116#41—1132no—
117#41—1131no—
118#41—1131no—
119#41—2261no—
120#41—2261no—
121#41—2261no—
122#41—2261no—
123#41C2×D11311yescharacteristic
124C2×Q8—1131no—
125C2×Q8C2×D11311yescharacteristic, complemented
126C2×D4—1131no—
127C2×D4—1131no—
128C2×D4—2261no—
129C2×D4—2261no—
130C22×C4—1131no—
131C22×C4—2261no—
132C22×C4—2261no—
133Q16—1132no—
134Q16C2×D11312yescomplemented
135SD16—1132no—
136SD16—1132no—
137SD16—1132no—
138SD16C2×D11312yescomplemented
139D8—1132no—
140D8—1132no—
141M16—1131no—
142M16—1132no—
143M16C2×D11311yescharacteristic, complemented
144C2×C8—1132no—
145C23—1131no—
146C23—2261no—
147C23—2261no—
148Q8—21no—
149Q8—1131no—
150Q8—2261no—
151Q8C22×D11311yescharacteristic
152Q8C22×D11312yes—
153D4—21no—
154D4—1132no—
155D4—1131no—
156D4—2261no—
157D4—2262no—
158D4—2262no—
159D4—2261no—
160D4—2261no—
161D4—2261no—
162D4—2261no—
163D4C22×D11311yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—1131no—
167C2×C4—1131no—
168C2×C4—1131no—
169C2×C4—1131no—
170C2×C4—1131no—
171C2×C4—2261no—
172C2×C4—2262no—
173C2×C4—2262no—
174C2×C4—2261no—
175C2×C4—2261no—
176C2×C4—2261no—
177C2×C4C22×D11311yescharacteristic
178C8—1132no—
179C8C22×D11312yes—
180C22—21no—
181C22—1131no—
182C22—1131no—
183C22—4522no—
184C22—4521no—
185C22—4522no—
186C22—2262no—
187C22—2261no—
188C22D4×D11311yescharacteristic
189C4—21no—
190C4—22no—
191C4—1131no—
192C4—1131no—
193C4—2261no—
194C4D4×D11311yescharacteristic
195C4C23×D11311yescharacteristic
196C2—21no—
197C2—41no—
198C2—4522no—
199C2—4521no—
200C2—2261no—
201C2—2261no—
202C2C2×D4×D11311yescharacteristic
subgroupG/H
centerC2C2×D4×D113
commutatorC4×C113C24
FrattiniC4C23×D113
Fitting#137×C113C2
socleC2×C113C22×D4
hypercenter#137D113
hypocenterC113#575
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. C2×#14268648C2semidirect
2. #41×D113C2semidirect
3. #16813654C2semidirect
4. #16813653C2semidirect
5. #16813650C2semidirect
6. #16813649C2semidirect
7. #16813647C2semidirect
8. #16813645C2semidirect
9. #16813643C2semidirect
10. M16×D113C2semidirect
11. #137×C113C2semidirect
15. #14268648C22semidirect
16. #14268648C22semidirect
20. #14268646C22semidirect
21. Q8×D113C22semidirect
23. C2×D452C22semidirect
25. D4×D113C22semidirect
29. #14268637C22semidirect
30. #14268636C22semidirect
31. #14268634C22semidirect
32. #14268634C22semidirect
33. D904C22semidirect
34. #14268632C22semidirect
35. #14268631C22semidirect
36. #14268630C22semidirect
37. C8×D113C22semidirect
39. C2×Q8×C113C22semidirect
40. Q16×C113C22semidirect
41. SD16×C113C22semidirect
42. M16×C113C22semidirect
101. C113#575semidirect
107. #137D113semidirect
125. C2×Q8C2×D113semidirect
134. Q16C2×D113semidirect
138. SD16C2×D113semidirect
143. M16C2×D113semidirect