Finite Group Databasebeta

← #43272806

Subgroups — #43272806

Order 28736.

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×#20297737C211yesmaximal, characteristic, complemented
2#41×D449C211yesmaximal, characteristic, complemented
3#28385372C212yesmaximal, complemented
4#28385371C211yesmaximal, characteristic, complemented
5#28385368C211yesmaximal, characteristic, complemented
6#28385367C212yesmaximal, complemented
7#28385365C211yesmaximal, characteristic, complemented
8#28385363C212yesmaximal, complemented
9#28385361C212yesmaximal, complemented
10M16×D449C211yesmaximal, characteristic, complemented
11#137×C449C211yesmaximal, characteristic, complemented
12#20297737—21no—
13#20297737—21no—
14#20297737C2212yes—
15#20297737C2212yescomplemented
16#20297737C2211yescharacteristic, complemented
17#20297736—21no—
18#20297736C2211yescharacteristic
19#20297735—21no—
20#20297735C2211yescharacteristic, complemented
21Q8×D449C2211yescharacteristic, complemented
22C2×D1796—21no—
23C2×D1796C2211yescharacteristic, complemented
24D4×D449—21no—
25D4×D449C2211yescharacteristic, complemented
26C2×C4×D449—21no—
27C2×C4×D449—21no—
28C2×C4×D449C2211yescharacteristic
29#20297726C2212yescomplemented
30#20297725C2212yescomplemented
31#20297723C2212yescomplemented
32#20297723C2212yescomplemented
33D3592C2212yescomplemented
34#20297721C2212yescomplemented
35#20297720C2211yescharacteristic, complemented
36#20297719C2212yescomplemented
37C8×D449C2212yescomplemented
38#41×C449C2211yescharacteristic
39C2×Q8×C449C2211yescharacteristic, complemented
40Q16×C449C2212yescomplemented
41SD16×C449C2212yescomplemented
42M16×C449C2211yescharacteristic, complemented
43C22×D449—21no—
44C22×D449—21no—
45C22×D449D411yescharacteristic
46Dic898—21no—
47Dic898C2311yescharacteristic
48D1796—21no—
49D1796—22no—
50D1796—22no—
51D1796—21no—
52D1796C2311yescharacteristic
53D1796C2312yes—
54C22⋊D449—21no—
55C22⋊D449—21no—
56C22⋊D449—21no—
57C2×Dic449—21no—
58C2×Dic449D411yescharacteristic
59C4×D449—21no—
60C4×D449—21no—
61C4×D449—21no—
62C4×D449—22no—
63C4×D449—22no—
64C4×D449D411yescharacteristic
65C4×D449D411yescharacteristic
66C4×D449C2311yescharacteristic
67C4×D449C2311yescharacteristic
68#16813311C2312yes—
69Q8×C449—21no—
70Q8×C449C2311yescharacteristic
71Q8×C449C2312yes—
72D4×C449—21no—
73D4×C449C2311yescharacteristic
74C2×C4×C449—21no—
75C2×C4×C449—21no—
76C2×C4×C449C2311yescharacteristic
77C8×C449C2312yes—
78C22×C449—21no—
79C22×C449C2×D411yescharacteristic
80C2×D449—21no—
81C2×D449—22no—
82C2×D449—41no—
83C2×D449—42no—
84C2×D449—42no—
85C2×D449C2×D411yescharacteristic
86C2×D449C2×D411yescharacteristic
87C4×C449—22no—
88C4×C449—21no—
89C4×C449C2×D411yescharacteristic
90C4×C449C2411yescharacteristic
91Dic449—21no—
92Dic449C2×D411yescharacteristic
93Dic449C2×D411yescharacteristic
94C2×C449—21no—
95C2×C449—41no—
96C2×C449C22×D411yescharacteristic
97D449—21no—
98D449—21no—
99D449—41no—
100D449—42no—
101C449#57511yescharacteristic, complemented, Sylow
102#575—4491nomaximal, Sylow
103C2×#41—4491no—
104C2×#41—4491no—
105#137—4491no—
106#137—4492no—
107#137D44911yescharacteristic, complemented
108AGL(1,ℤ/8)—4491no—
109AGL(1,ℤ/8)—4492no—
110AGL(1,ℤ/8)—4491no—
111#135—4492no—
112#135—4492no—
113C2×M16—4491no—
114#41—8981no—
115#41—8981no—
116#41—8981no—
117#41—8981no—
118#41—4492no—
119#41—4491no—
120#41—4492no—
121#41—4491no—
122#41—4491no—
123#41C2×D44911yescharacteristic
124C2×Q8—4491no—
125C2×Q8C2×D44911yescharacteristic, complemented
126C2×D4—8981no—
127C2×D4—8981no—
128C2×D4—4491no—
129C2×D4—4491no—
130C22×C4—8981no—
131C22×C4—8981no—
132C22×C4—4491no—
133Q16—4492no—
134Q16C2×D44912yescomplemented
135SD16—4492no—
136SD16—4492no—
137SD16—4492no—
138SD16C2×D44912yescomplemented
139D8—4492no—
140D8—4492no—
141M16—4491no—
142M16—4492no—
143M16C2×D44911yescharacteristic, complemented
144C2×C8—4492no—
145C23—8981no—
146C23—8981no—
147C23—4491no—
148Q8—21no—
149Q8—8981no—
150Q8—4491no—
151Q8C22×D44911yescharacteristic
152Q8C22×D44912yes—
153D4—21no—
154D4—8981no—
155D4—8982no—
156D4—8982no—
157D4—8981no—
158D4—8981no—
159D4—8981no—
160D4—8981no—
161D4—4491no—
162D4—4492no—
163D4C22×D44911yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—8981no—
167C2×C4—8981no—
168C2×C4—8982no—
169C2×C4—8982no—
170C2×C4—8981no—
171C2×C4—8981no—
172C2×C4—4491no—
173C2×C4—4491no—
174C2×C4—4491no—
175C2×C4—4491no—
176C2×C4—4491no—
177C2×C4C22×D44911yescharacteristic
178C8—4492no—
179C8C22×D44912yes—
180C22—21no—
181C22—17962no—
182C22—17961no—
183C22—17962no—
184C22—8982no—
185C22—8981no—
186C22—4491no—
187C22—4491no—
188C22D4×D44911yescharacteristic
189C4—21no—
190C4—22no—
191C4—8981no—
192C4—4491no—
193C4—4491no—
194C4D4×D44911yescharacteristic
195C4C23×D44911yescharacteristic
196C2—21no—
197C2—41no—
198C2—17962no—
199C2—17961no—
200C2—8981no—
201C2—8981no—
202C2C2×D4×D44911yescharacteristic
subgroupG/H
centerC2C2×D4×D449
commutator, cosocleC4×C449C24
FrattiniC4C23×D449
Fitting#137×C449C2
socleC2×C449C22×D4
hypercenter#137D449
hypocenterC449#575
LCSG ▷ C4×C449 ▷ C2×C449 ▷ C449
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#20297737C2semidirect
2. #41×D449C2semidirect
3. #28385372C2semidirect
4. #28385371C2semidirect
5. #28385368C2semidirect
6. #28385367C2semidirect
7. #28385365C2semidirect
8. #28385363C2semidirect
9. #28385361C2semidirect
10. M16×D449C2semidirect
11. #137×C449C2semidirect
15. #20297737C22semidirect
16. #20297737C22semidirect
20. #20297735C22semidirect
21. Q8×D449C22semidirect
23. C2×D1796C22semidirect
25. D4×D449C22semidirect
29. #20297726C22semidirect
30. #20297725C22semidirect
31. #20297723C22semidirect
32. #20297723C22semidirect
33. D3592C22semidirect
34. #20297721C22semidirect
35. #20297720C22semidirect
36. #20297719C22semidirect
37. C8×D449C22semidirect
39. C2×Q8×C449C22semidirect
40. Q16×C449C22semidirect
41. SD16×C449C22semidirect
42. M16×C449C22semidirect
101. C449#575semidirect
107. #137D449semidirect
125. C2×Q8C2×D449semidirect
134. Q16C2×D449semidirect
138. SD16C2×D449semidirect
143. M16C2×D449semidirect

Maximal subgroups

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

subgroupindex
C2×#202977372
#41×D4492
#283853722
#283853712
#283853682
#283853672
#283853652
#283853632
#283853612
M16×D4492
#137×C4492
#575449