Finite Group Databasebeta

← #29505410

Subgroups — #29505410

Order 15296.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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×D239C211yesmaximal, characteristic, complemented
2#21439949C211yesmaximal, characteristic, complemented
3#21439943C212yesmaximal, complemented
4#21439936C211yesmaximal, characteristic, complemented
5#21439934C211yesmaximal, characteristic, complemented
6#21439932C212yesmaximal, complemented
7#21439928C212yesmaximal, complemented
8D4×Dic239C211yesmaximal, characteristic, complemented
9C4×C22⋊D239C211yesmaximal, characteristic, complemented
10C2×(C2×D239)⋊C4C212yesmaximal, complemented
11C2×#41×C239C211yesmaximal, characteristic, complemented
12C23×D239C2212yes—
13C2×D956C2211yescharacteristic
14C2×C22⋊D239C2211yescharacteristic
15C2×C22⋊D239C2212yes—
16C2×C22⋊D239C2212yescomplemented
17D4×D239—24no—
18C22×Dic239C2212yescomplemented
19C2×C4×D239C2211yescharacteristic
20C4⋊Dic239C2211yescharacteristic, complemented
21Dic239⋊C4C2211yescharacteristic, complemented
22Dic239⋊C4C2212yescomplemented
23C22⋊Dic239C2211yescharacteristic, complemented
24C22⋊Dic239C2212yescomplemented
25(C2×D239)⋊C4C2211yescharacteristic
26(C2×D239)⋊C4C2212yes—
27(C2×D239)⋊C4C2212yescomplemented
28(C2×D239)⋊C4C2212yescomplemented
29(C2×D239)⋊C4C2212yescomplemented
30C4×Dic239C2211yescharacteristic, complemented
31#41×C239—24no—
32C2×Q8×C239C2211yescharacteristic, complemented
33C2×D4×C239C2211yescharacteristic
34C2×D4×C239C2212yescomplemented
35C22×C4×C239C2212yescomplemented
36C22×C4×C239C2211yescharacteristic, complemented
37C22×D239—24no—
38C22×D239—24no—
39C22×D239—42no—
40C22×D239C2312yes—
41C22×D239C2311yescharacteristic
42D956—22no—
43C22⋊D239—24no—
44C22⋊D239—22no—
45C22⋊D239—42no—
46C2×Dic239—42no—
47C2×Dic239C2311yescharacteristic
48C2×Dic239C2311yescharacteristic
49C2×Dic239C2312yes—
50C2×Dic239C2311yescharacteristic
51C4×D239—22no—
52C23×C239C2311yescharacteristic
53C23×C239C2312yes—
54Q8×C239—22no—
55D4×C239—24no—
56D4×C239D414yescomplemented
57C2×C4×C239—24no—
58C2×C4×C239—22no—
59C2×C4×C239C2311yescharacteristic
60C2×C4×C239C2312yes—
61C2×C4×C239C2311yescharacteristic
62C22×C239—22no—
63C22×C239—22no—
64C22×C239—21no—
65C22×C239—21no—
66C22×C239—21no—
67C22×C239C2×D414yes—
68C22×C239C2411yescharacteristic
69C2×D239—21no—
70C2×D239—21no—
71C2×D239—22no—
72C2×D239—22no—
73C2×D239—22no—
74C2×D239—48no—
75C2×D239#4112yescomplemented
76C4×C239—21no—
77C4×C239—22no—
78C4×C239C2×D412yes—
79Dic239—21no—
80Dic239—21no—
81Dic239—21no—
82Dic239—22no—
83C2×C239—24no—
84C2×C239—41no—
85C2×C239C22×D411yescharacteristic
86C2×C239C2×#4111yescharacteristic
87C2×C239H(5,𝔽2)11yescharacteristic
88D239—22no—
89D239—42no—
90C239#54611yescharacteristic, complemented, Sylow
91#546—2391nomaximal, Sylow
92C2×#41D23911yescharacteristic, complemented
93C22×D4—2391no—
94#124—2391no—
95#123—2392no—
96C22⋊Q8—2391no—
97#121—2392no—
98#121—2391no—
99#120—2392no—
100C4×D4—2391no—
101C4×D4—2391no—
102C2×C22⋊C4—2392no—
103C24—2392no—
104#41—24no—
105C2×Q8C2×D23911yescharacteristic, complemented
106C2×D4—4784no—
107C2×D4—2392no—
108C2×D4—2391no—
109C2×D4—2392no—
110C2×D4—2391no—
111C2×D4C2×D23911yescharacteristic
112C2×D4C2×D23912yescomplemented
113C22×C4—2392no—
114C22×C4—2391no—
115C22×C4C2×D23912yescomplemented
116C22×C4C2×D23911yescharacteristic, complemented
117C4⋊C4—2391no—
118C4⋊C4—2391no—
119C4⋊C4—2392no—
120C22⋊C4—2392no—
121C22⋊C4—2391no—
122C22⋊C4—2391no—
123C22⋊C4—2392no—
124C22⋊C4—2392no—
125C22⋊C4—2392no—
126C22⋊C4—2392no—
127C42—2391no—
128C23—9562no—
129C23—4784no—
130C23—4784no—
131C23—2392no—
132C23—2391no—
133C23C22×D23911yescharacteristic
134C23C22×D23912yes—
135Q8—22no—
136D4—24no—
137D4—9562no—
138D4—4782no—
139D4—4784no—
140D4—4782no—
141D4C22⋊D23914yescomplemented
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—9562no—
145C2×C4—4782no—
146C2×C4—2391no—
147C2×C4—2391no—
148C2×C4—2392no—
149C2×C4—2391no—
150C2×C4C22×D23911yescharacteristic
151C2×C4C22×D23912yes—
152C2×C4C22×D23911yescharacteristic
153C22—22no—
154C22—22no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—9568no—
159C22—4782no—
160C22—4782no—
161C22—4782no—
162C22—4781no—
163C22—4781no—
164C22—2392no—
165C22C2×C22⋊D23914yes—
166C22C23×D23911yescharacteristic
167C4—21no—
168C4—22no—
169C4—4781no—
170C4—4781no—
171C4—4782no—
172C4—4781no—
173C4C2×C22⋊D23912yes—
174C2—24no—
175C2—41no—
176C2—9562no—
177C2—4782no—
178C2C22×C22⋊D23911yescharacteristic
179C2#41×D23911yescharacteristic
180C2#2144000611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D239
commutator, socle, cosocleC22×C239C24
FittingC2×#41×C239C2
hypercenterC2×#41D239
hypocenterC239#546
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×D4×D239C2semidirect
2. #21439949C2semidirect
3. #21439943C2semidirect
4. #21439936C2semidirect
5. #21439934C2semidirect
6. #21439932C2semidirect
7. #21439928C2semidirect
8. D4×Dic239C2semidirect
9. C4×C22⋊D239C2semidirect
10. C2×(C2×D239)⋊C4C2semidirect
11. C2×#41×C239C2semidirect
16. C2×C22⋊D239C22semidirect
18. C22×Dic239C22semidirect
20. C4⋊Dic239C22semidirect
21. Dic239⋊C4C22semidirect
22. Dic239⋊C4C22semidirect
23. C22⋊Dic239C22semidirect
24. C22⋊Dic239C22semidirect
27. (C2×D239)⋊C4C22semidirect
28. (C2×D239)⋊C4C22semidirect
29. (C2×D239)⋊C4C22semidirect
30. C4×Dic239C22semidirect
32. C2×Q8×C239C22semidirect
34. C2×D4×C239C22semidirect
35. C22×C4×C239C22semidirect
36. C22×C4×C239C22semidirect
56. D4×C239D4semidirect
75. C2×D239#41semidirect
90. C239#546semidirect
92. C2×#41D239semidirect
105. C2×Q8C2×D239semidirect
112. C2×D4C2×D239semidirect
115. C22×C4C2×D239semidirect
116. C22×C4C2×D239semidirect
141. D4C22⋊D239semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D2392
#214399492
#214399432
#214399362
#214399342
#214399322
#214399282
D4×Dic2392
C4×C22⋊D2392
C2×(C2×D239)⋊C42
C2×#41×C2392
#546239