Finite Group Databasebeta

← #123×D449 #43272507

Subgroups — #123×D449 #43272507

Order 28736.

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

normal subgroups only

This group has 342 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×D4×D449C211yesmaximal, characteristic
2C22×C4×D449C211yesmaximal, characteristic, complemented
3#28385325C212yesmaximal, complemented
4#28385324C211yesmaximal, characteristic, complemented
5#28385323C211yesmaximal, characteristic, complemented
6#28385322C212yesmaximal, complemented
7#28385321C211yesmaximal, characteristic, complemented
8C4⋊C4×D449C212yesmaximal, complemented
9C22⋊C4×D449C211yesmaximal, characteristic, complemented
10C22⋊C4×D449C212yesmaximal, complemented
11#123×C449C211yesmaximal, characteristic, complemented
12C23×D449C2211yescharacteristic
13C23×D449C2211yescharacteristic
14C2×D1796C2211yescharacteristic
15C2×C22⋊D449C2211yescharacteristic
16C2×C22⋊D449C2211yescharacteristic
17D4×D449—24no—
18C22×Dic449C2211yescharacteristic, complemented
19C2×C4×D449—24no—
20C2×C4×D449C2212yes—
21C2×C4×D449C2212yes—
22C2×C4×D449C2211yescharacteristic
23C2×C4×D449C2212yescomplemented
24C4⋊Dic449C2212yescomplemented
25#20297715C2212yescomplemented
26#20297715C2212yescomplemented
27C22⋊Dic449C2212yescomplemented
28C22⋊Dic449C2211yescharacteristic, complemented
29#20297713C2212yescomplemented
30#20297713C2212yescomplemented
31#20297713C2212yescomplemented
32C2×D4×C449C2211yescharacteristic
33C22×C4×C449C2211yescharacteristic, complemented
34C4⋊C4×C449C2212yescomplemented
35C22⋊C4×C449C2212yescomplemented
36C22⋊C4×C449C2211yescharacteristic, complemented
37C22×D449—24no—
38C22×D449—22no—
39C22×D449—24no—
40C22×D449D414yes—
41C22×D449C2311yescharacteristic
42C22×D449C2311yescharacteristic
43C22×D449C2311yescharacteristic
44D1796—22no—
45C22⋊D449—22no—
46C22⋊D449—22no—
47C2×Dic449—22no—
48C2×Dic449C2312yes—
49C2×Dic449C2311yescharacteristic
50C2×Dic449C2312yes—
51C4×D449—24no—
52C4×D449—24no—
53C4×D449—22no—
54C4×D449—24no—
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—22no—
68C2×D449—28no—
69C2×D449—21no—
70C2×D449—22no—
71C2×D449—44no—
72C2×D449C2×D412yes—
73C2×D449C2×D412yes—
74C2×D449#4114yes—
75C4×C449—22no—
76C4×C449—21no—
77C4×C449—22no—
78Dic449—22no—
79Dic449—21no—
80Dic449—22no—
81C2×C449—22no—
82C2×C449—41no—
83C2×C449C22×D411yescharacteristic
84C2×C449C2×#4112yes—
85D449—22no—
86D449—41no—
87D449#12314yesdirect product factor
88C449C2×#12311yescharacteristic, complemented, Sylow
89C2×#123—4491nomaximal, Sylow
90C22×D4—4491no—
91C23×C4—4491no—
92#123—4491no—
93#123—4491no—
94#123—4492no—
95#123—4492no—
96#123—4491no—
97#123D44911yescharacteristic, direct product factor
98C2×C4⋊C4—4492no—
99C2×C22⋊C4—4491no—
100C2×C22⋊C4—4492no—
101C24—4491no—
102C24—4491no—
103C2×D4—8984no—
104C2×D4—4491no—
105C2×D4—4491no—
106C2×D4—4491no—
107C2×D4C2×D44911yescharacteristic
108C22×C4—8984no—
109C22×C4—4491no—
110C22×C4—4492no—
111C22×C4—4492no—
112C22×C4—4491no—
113C22×C4—4492no—
114C22×C4C2×D44911yescharacteristic, complemented
115C4⋊C4—4492no—
116C4⋊C4—4492no—
117C4⋊C4—4492no—
118C4⋊C4C2×D44912yescomplemented
119C22⋊C4—4491no—
120C22⋊C4—4492no—
121C22⋊C4—4492no—
122C22⋊C4—4492no—
123C22⋊C4—4492no—
124C22⋊C4C2×D44911yescharacteristic, complemented
125C22⋊C4C2×D44912yescomplemented
126C23—8982no—
127C23—8984no—
128C23—8984no—
129C23—4491no—
130C23—4494no—
131C23—4491no—
132C23—4491no—
133C23C22×D44911yescharacteristic
134C23C22×D44911yescharacteristic
135D4—22no—
136D4—8982no—
137D4—8982no—
138D4—8982no—
139C2×C4—22no—
140C2×C4—8984no—
141C2×C4—8984no—
142C2×C4—8984no—
143C2×C4—8982no—
144C2×C4—8982no—
145C2×C4—4492no—
146C2×C4—4492no—
147C2×C4—4491no—
148C2×C4C22×D44912yes—
149C2×C4C22×D44912yes—
150C2×C4C22×D44911yescharacteristic
151C22—22no—
152C22—21no—
153C22—22no—
154C22—17964no—
155C22—8988no—
156C22—8981no—
157C22—8982no—
158C22—8982no—
159C22—4492no—
160C22—4492no—
161C22—4494no—
162C22D4×D44912yes—
163C22C23×D44911yescharacteristic
164C4—22no—
165C4—21no—
166C4—22no—
167C4—8982no—
168C4—8981no—
169C4—8982no—
170C2—22no—
171C2—41no—
172C2—17961no—
173C2—8982no—
174C2—4494no—
175C2C2×D4×D44911yescharacteristic
176C2#41×D44912yes—
subgroupG/H
center, FrattiniC22C23×D449
commutator, socle, cosocleC22×C449C24
Fitting#123×C449C2
prime radicalC2×D4×D449C2
hypercenter#123D449
hypocenterC449C2×#123
LCSG ▷ C22×C449 ▷ C449
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×D449C2semidirect
3. #28385325C2semidirect
4. #28385324C2semidirect
5. #28385323C2semidirect
6. #28385322C2semidirect
7. #28385321C2semidirect
8. C4⋊C4×D449C2semidirect
9. C22⋊C4×D449C2semidirect
10. C22⋊C4×D449C2semidirect
11. #123×C449C2semidirect
18. C22×Dic449C22semidirect
23. C2×C4×D449C22semidirect
24. C4⋊Dic449C22semidirect
25. #20297715C22semidirect
26. #20297715C22semidirect
27. C22⋊Dic449C22semidirect
28. C22⋊Dic449C22semidirect
29. #20297713C22semidirect
30. #20297713C22semidirect
31. #20297713C22semidirect
33. C22×C4×C449C22semidirect
34. C4⋊C4×C449C22semidirect
35. C22⋊C4×C449C22semidirect
36. C22⋊C4×C449C22semidirect
87. D449#123direct (N × H)
88. C449C2×#123semidirect
97. #123D449direct (N × H)
114. C22×C4C2×D449semidirect
118. C4⋊C4C2×D449semidirect
124. C22⋊C4C2×D449semidirect
125. C22⋊C4C2×D449semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D4492
C22×C4×D4492
#283853252
#283853242
#283853232
#283853222
#283853212
C4⋊C4×D4492
C22⋊C4×D4492
#123×C4492
C2×#123449