Finite Group Databasebeta

← #43272515

Subgroups — #43272515

Order 28736.

330 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 107 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×#20297735C211yesmaximal, characteristic, complemented
2C22×C4×D449C211yesmaximal, characteristic, complemented
3#28385323C211yesmaximal, characteristic, complemented
4#28385318C212yesmaximal, complemented
5#28385313C212yesmaximal, complemented
6#28385310C211yesmaximal, characteristic, complemented
7#28385308C211yesmaximal, characteristic, complemented
8#28385302C212yesmaximal, complemented
9#28385300C212yesmaximal, complemented
10#28385296C211yesmaximal, characteristic, complemented
11#123×C449C211yesmaximal, characteristic, complemented
12C23×D449C2211yescharacteristic
13#20297735—24no—
14C2×Dic898C2211yescharacteristic, complemented
15C2×D1796C2212yes—
16C2×C22⋊D449C2212yes—
17C2×C22⋊D449C2211yescharacteristic
18C2×C22⋊D449C2211yescharacteristic, complemented
19C22×Dic449C2211yescharacteristic
20C22×Dic449C2211yescharacteristic, complemented
21C2×C4×D449—24no—
22C2×C4×D449C2212yes—
23C2×C4×D449C2212yes—
24C2×C4×D449C2211yescharacteristic, complemented
25#20297715C2212yescomplemented
26#20297715C2212yescomplemented
27C22⋊Dic449C2211yescharacteristic
28#20297713C2212yes—
29#20297713C2212yes—
30#20297713C2212yescomplemented
31C4×Dic449C2212yescomplemented
32C2×D4×C449C2211yescharacteristic
33C22×C4×C449C2211yescharacteristic
34C4⋊C4×C449C2212yescomplemented
35C22⋊C4×C449C2211yescharacteristic
36C22⋊C4×C449C2212yescomplemented
37C22×D449—22no—
38C22×D449—24no—
39C22×D449C2311yescharacteristic
40C22×D449C2312yes—
41Dic898—22no—
42D1796—24no—
43C22⋊D449—22no—
44C22⋊D449—24no—
45C22⋊D449—22no—
46C2×Dic449—22no—
47C2×Dic449D414yescomplemented
48C2×Dic449C2311yescharacteristic
49C2×Dic449C2312yes—
50C2×Dic449C2311yescharacteristic
51C2×Dic449C2311yescharacteristic
52C4×D449—24no—
53C4×D449—24no—
54C4×D449—22no—
55C23×C449C2311yescharacteristic
56C23×C449C2311yescharacteristic
57D4×C449—22no—
58C2×C4×C449—22no—
59C2×C4×C449C2312yes—
60C2×C4×C449C2311yescharacteristic
61C2×C4×C449C2312yes—
62C22×C449—22no—
63C22×C449—21no—
64C22×C449—22no—
65C22×C449C2×D412yes—
66C22×C449C2411yescharacteristic
67C2×D449—28no—
68C2×D449—22no—
69C2×D449—22no—
70C2×D449—21no—
71C2×D449—22no—
72C2×D449#4114yescomplemented
73C4×C449—22no—
74C4×C449—21no—
75C4×C449—22no—
76Dic449—21no—
77Dic449—22no—
78Dic449C2×D412yes—
79Dic449C2×D412yes—
80C2×C449—22no—
81C2×C449—41no—
82C2×C449C22×D411yescharacteristic
83C2×C449C2×#4112yes—
84D449—24no—
85D449—41no—
86C449#52511yescharacteristic, complemented, Sylow
87#525—4491nomaximal, Sylow
88C2×#41—4491no—
89C23×C4—4491no—
90#123—4491no—
91#123D44911yescharacteristic, complemented
92C22⋊Q8—4492no—
93#121—4492no—
94#120—4491no—
95#120—4491no—
96C4×D4—4492no—
97C4×D4—4492no—
98#117—4491no—
99C24—4491no—
100#41—8984no—
101C2×Q8—4491no—
102C2×D4—4492no—
103C2×D4—4491no—
104C2×D4—4491no—
105C2×D4—4492no—
106C2×D4C2×D44911yescharacteristic
107C22×C4—8984no—
108C22×C4—4491no—
109C22×C4—4492no—
110C22×C4—4492no—
111C22×C4—4491no—
112C22×C4—4491no—
113C22×C4C2×D44911yescharacteristic
114C4⋊C4—4492no—
115C4⋊C4—4492no—
116C4⋊C4C2×D44912yescomplemented
117C22⋊C4—4491no—
118C22⋊C4—4492no—
119C22⋊C4—4492no—
120C22⋊C4—4492no—
121C22⋊C4C2×D44911yescharacteristic
122C22⋊C4C2×D44912yescomplemented
123C42—4492no—
124C23—8984no—
125C23—8982no—
126C23—4491no—
127C23—4492no—
128C23C22×D44911yescharacteristic
129C23C22×D44911yescharacteristic
130Q8—8982no—
131D4—22no—
132D4—8984no—
133D4—8984no—
134D4—8982no—
135D4—8982no—
136C2×C4—22no—
137C2×C4—8984no—
138C2×C4—8984no—
139C2×C4—8982no—
140C2×C4—8982no—
141C2×C4—4492no—
142C2×C4—4491no—
143C2×C4—4494no—
144C2×C4—4491no—
145C2×C4—4491no—
146C2×C4C22×D44912yes—
147C2×C4C22×D44911yescharacteristic
148C2×C4C22×D44912yes—
149C22—22no—
150C22—21no—
151C22—22no—
152C22—8988no—
153C22—8981no—
154C22—8982no—
155C22—8982no—
156C22—8982no—
157C22—4494no—
158C22D4×D44912yes—
159C22C23×D44911yescharacteristic
160C4—22no—
161C4—21no—
162C4—22no—
163C4—8981no—
164C4—8982no—
165C4—4492no—
166C4—4492no—
167C2—22no—
168C2—41no—
169C2—17961no—
170C2—8984no—
171C2C2×D4×D44911yescharacteristic
172C2#41×D44912yes—
subgroupG/H
center, FrattiniC22C23×D449
commutator, socle, cosocleC22×C449C24
Fitting#123×C449C2
hypercenter#123D449
hypocenterC449#525
LCSG ▷ C22×C449 ▷ C449
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#20297735C2semidirect
2. C22×C4×D449C2semidirect
3. #28385323C2semidirect
4. #28385318C2semidirect
5. #28385313C2semidirect
6. #28385310C2semidirect
7. #28385308C2semidirect
8. #28385302C2semidirect
9. #28385300C2semidirect
10. #28385296C2semidirect
11. #123×C449C2semidirect
14. C2×Dic898C22semidirect
18. C2×C22⋊D449C22semidirect
20. C22×Dic449C22semidirect
24. C2×C4×D449C22semidirect
25. #20297715C22semidirect
26. #20297715C22semidirect
30. #20297713C22semidirect
31. C4×Dic449C22semidirect
34. C4⋊C4×C449C22semidirect
36. C22⋊C4×C449C22semidirect
47. C2×Dic449D4semidirect
72. C2×D449#41semidirect
86. C449#525semidirect
91. #123D449semidirect
116. C4⋊C4C2×D449semidirect
122. C22⋊C4C2×D449semidirect

Maximal subgroups

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

subgroupindex
C2×#202977352
C22×C4×D4492
#283853232
#283853182
#283853132
#283853102
#283853082
#283853022
#283853002
#283852962
#123×C4492
#525449