Finite Group Databasebeta

← #43272513

Subgroups — #43272513

Order 28736.

330 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 111 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
2C23×Dic449C211yesmaximal, characteristic, complemented
3#28385324C211yesmaximal, characteristic, complemented
4#28385321C211yesmaximal, characteristic, complemented
5#28385317C212yesmaximal, complemented
6#28385312C212yesmaximal, complemented
7#28385309C211yesmaximal, characteristic, complemented
8D4×Dic449C212yesmaximal, complemented
9#28385300C212yesmaximal, complemented
10#28385296C211yesmaximal, characteristic, complemented
11#120×C449C211yesmaximal, characteristic, complemented
12#20297735—24no—
13C2×Dic898C2211yescharacteristic, complemented
14C2×C22⋊D449C2212yes—
15C2×C22⋊D449C2211yescharacteristic
16C2×C22⋊D449C2211yescharacteristic, complemented
17C22×Dic449—24no—
18C22×Dic449C2212yes—
19C22×Dic449C2212yes—
20C22×Dic449C2211yescharacteristic
21C22×Dic449C2211yescharacteristic
22C22×Dic449C2211yescharacteristic, complemented
23C2×C4×D449C2211yescharacteristic, complemented
24C4⋊Dic449C2212yescomplemented
25#20297715C2212yescomplemented
26#20297715C2212yescomplemented
27C22⋊Dic449C2211yescharacteristic
28C22⋊Dic449C2212yes—
29C22⋊Dic449C2212yescomplemented
30#20297713C2212yescomplemented
31C4×Dic449C2212yescomplemented
32C24×C449C2211yescharacteristic
33C2×D4×C449C2212yes—
34C2×D4×C449C2211yescharacteristic
35C22⋊C4×C449C2211yescharacteristic
36C22⋊C4×C449C2212yes—
37C22×D449C2311yescharacteristic
38Dic898—22no—
39C22⋊D449—22no—
40C22⋊D449—24no—
41C22⋊D449—22no—
42C2×Dic449—22no—
43C2×Dic449—24no—
44C2×Dic449—24no—
45C2×Dic449—22no—
46C2×Dic449D414yescomplemented
47C2×Dic449C2312yes—
48C2×Dic449C2311yescharacteristic
49C2×Dic449C2312yes—
50C2×Dic449C2311yescharacteristic
51C2×Dic449C2311yescharacteristic
52C4×D449—22no—
53C23×C449—22no—
54C23×C449—24no—
55C23×C449C2311yescharacteristic
56C23×C449C2311yescharacteristic
57C23×C449C2312yes—
58D4×C449—22no—
59D4×C449—24no—
60C2×C4×C449C2311yescharacteristic
61C2×C4×C449C2312yes—
62C22×C449—28no—
63C22×C449—22no—
64C22×C449—22no—
65C22×C449—22no—
66C22×C449—21no—
67C22×C449—22no—
68C22×C449C2×D412yes—
69C22×C449#4114yescomplemented
70C22×C449C2411yescharacteristic
71C2×D449—21no—
72C2×D449—22no—
73C4×C449—21no—
74C4×C449—22no—
75Dic449—22no—
76Dic449—21no—
77Dic449—22no—
78Dic449C2×D412yes—
79Dic449C2×D412yes—
80C2×C449—22no—
81C2×C449—24no—
82C2×C449—41no—
83C2×C449C22×D411yescharacteristic
84C2×C449C2×#4112yes—
85D449—41no—
86C449#52511yescharacteristic, complemented, Sylow
87#525—4491nomaximal, Sylow
88C2×#41—4491no—
89C23×C4—4491no—
90#123—4491no—
91#123—4491no—
92C22⋊Q8—4492no—
93#121—4492no—
94#120—4491no—
95#120D44911yescharacteristic, complemented
96C4×D4—4492no—
97C4×D4—4492no—
98#117—4491no—
99C24C2×D44911yescharacteristic
100#41—8984no—
101C2×Q8—4491no—
102C2×D4—4492no—
103C2×D4—4491no—
104C2×D4—4491no—
105C2×D4C2×D44912yes—
106C2×D4C2×D44911yescharacteristic
107C22×C4—8984no—
108C22×C4—4492no—
109C22×C4—4492no—
110C22×C4—4491no—
111C22×C4—4491no—
112C22×C4—4491no—
113C22×C4—4491no—
114C4⋊C4—4492no—
115C4⋊C4—4492no—
116C4⋊C4—4492no—
117C22⋊C4—4491no—
118C22⋊C4—4492no—
119C22⋊C4—4492no—
120C22⋊C4—4492no—
121C22⋊C4C2×D44911yescharacteristic
122C22⋊C4C2×D44912yes—
123C42—4492no—
124C23—22no—
125C23—24no—
126C23—4491no—
127C23C22×D44911yescharacteristic
128C23C22×D44911yescharacteristic
129C23C22×D44912yes—
130Q8—8982no—
131D4—22no—
132D4—24no—
133D4—8982no—
134D4—8984no—
135D4—8982no—
136C2×C4—8982no—
137C2×C4—8982no—
138C2×C4—8984no—
139C2×C4—8984no—
140C2×C4—8982no—
141C2×C4—4492no—
142C2×C4—4492no—
143C2×C4—4491no—
144C2×C4—4494no—
145C2×C4—4491no—
146C2×C4—4491no—
147C2×C4C22×D44912yes—
148C2×C4C22×D44911yescharacteristic
149C22—28no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—22no—
155C22—8981no—
156C22—8982no—
157C22D4×D44912yes—
158C22#2029773514yescomplemented
159C22C23×D44911yescharacteristic
160C4—21no—
161C4—22no—
162C4—8982no—
163C4—8981no—
164C4—8982no—
165C4—4492no—
166C4—4492no—
167C2—22no—
168C2—24no—
169C2—41no—
170C2—17961no—
171C2C2×D4×D44911yescharacteristic
172C2C2×#2029773512yes—
subgroupG/H
center, FrattiniC22C23×D449
commutator, socle, cosocleC22×C449C24
Fitting#120×C449C2
hypercenter#120D449
hypocenterC449#525
LCSG ▷ C22×C449 ▷ C449
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#20297735C2semidirect
2. C23×Dic449C2semidirect
3. #28385324C2semidirect
4. #28385321C2semidirect
5. #28385317C2semidirect
6. #28385312C2semidirect
7. #28385309C2semidirect
8. D4×Dic449C2semidirect
9. #28385300C2semidirect
10. #28385296C2semidirect
11. #120×C449C2semidirect
13. C2×Dic898C22semidirect
16. C2×C22⋊D449C22semidirect
22. C22×Dic449C22semidirect
23. C2×C4×D449C22semidirect
24. C4⋊Dic449C22semidirect
25. #20297715C22semidirect
26. #20297715C22semidirect
29. C22⋊Dic449C22semidirect
30. #20297713C22semidirect
31. C4×Dic449C22semidirect
46. C2×Dic449D4semidirect
69. C22×C449#41semidirect
86. C449#525semidirect
95. #120D449semidirect
158. C22#20297735semidirect

Maximal subgroups

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

subgroupindex
C2×#202977352
C23×Dic4492
#283853242
#283853212
#283853172
#283853122
#283853092
D4×Dic4492
#283853002
#283852962
#120×C4492
#525449