Finite Group Databasebeta

← #123×D257 #30657237

Subgroups — #123×D257 #30657237

Order 16448.

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×D257C211yesmaximal, characteristic
2C22×C4×D257C211yesmaximal, characteristic, complemented
3#22551577C212yesmaximal, complemented
4#22551576C211yesmaximal, characteristic, complemented
5#22551575C211yesmaximal, characteristic, complemented
6#22551574C212yesmaximal, complemented
7#22551573C211yesmaximal, characteristic, complemented
8C4⋊C4×D257C212yesmaximal, complemented
9C22⋊C4×D257C212yesmaximal, complemented
10C22⋊C4×D257C211yesmaximal, characteristic, complemented
11#123×C257C211yesmaximal, characteristic, complemented
12C23×D257C2211yescharacteristic
13C23×D257C2211yescharacteristic
14C2×D1028C2211yescharacteristic
15C2×C22⋊D257C2211yescharacteristic
16C2×C22⋊D257C2211yescharacteristic
17D4×D257—24no—
18C22×Dic257C2211yescharacteristic, complemented
19C2×C4×D257—24no—
20C2×C4×D257C2212yes—
21C2×C4×D257C2211yescharacteristic
22C2×C4×D257C2212yes—
23C2×C4×D257C2212yescomplemented
24C4⋊Dic257C2212yescomplemented
25#16861597C2212yescomplemented
26#16861597C2212yescomplemented
27C22⋊Dic257C2212yescomplemented
28C22⋊Dic257C2211yescharacteristic, complemented
29#16861595C2212yescomplemented
30#16861595C2212yescomplemented
31#16861595C2212yescomplemented
32C2×D4×C257C2211yescharacteristic
33C22×C4×C257C2211yescharacteristic, complemented
34C4⋊C4×C257C2212yescomplemented
35C22⋊C4×C257C2212yescomplemented
36C22⋊C4×C257C2211yescharacteristic, complemented
37C22×D257—24no—
38C22×D257—22no—
39C22×D257—24no—
40C22×D257D414yes—
41C22×D257C2311yescharacteristic
42C22×D257C2311yescharacteristic
43C22×D257C2311yescharacteristic
44D1028—22no—
45C22⋊D257—22no—
46C22⋊D257—22no—
47C2×Dic257—22no—
48C2×Dic257C2312yes—
49C2×Dic257C2312yes—
50C2×Dic257C2311yescharacteristic
51C4×D257—24no—
52C4×D257—24no—
53C4×D257—22no—
54C4×D257—24no—
55C23×C257C2311yescharacteristic
56C23×C257C2311yescharacteristic
57D4×C257—22no—
58C2×C4×C257—22no—
59C2×C4×C257C2312yes—
60C2×C4×C257C2312yes—
61C2×C4×C257C2311yescharacteristic
62C22×C257—22no—
63C22×C257—21no—
64C22×C257—22no—
65C22×C257C2×D412yes—
66C22×C257C2411yescharacteristic
67C2×D257—22no—
68C2×D257—28no—
69C2×D257—21no—
70C2×D257—22no—
71C2×D257—44no—
72C2×D257C2×D412yes—
73C2×D257C2×D412yes—
74C2×D257#4114yes—
75C4×C257—22no—
76C4×C257—21no—
77C4×C257—22no—
78Dic257—22no—
79Dic257—21no—
80Dic257—22no—
81C2×C257—22no—
82C2×C257—41no—
83C2×C257C22×D411yescharacteristic
84C2×C257C2×#4112yes—
85D257—22no—
86D257—41no—
87D257#12314yesdirect product factor
88C257C2×#12311yescharacteristic, complemented, Sylow
89C2×#123—2571nomaximal, Sylow
90C22×D4—2571no—
91C23×C4—2571no—
92#123—2571no—
93#123—2571no—
94#123—2572no—
95#123—2572no—
96#123—2571no—
97#123D25711yescharacteristic, direct product factor
98C2×C4⋊C4—2572no—
99C2×C22⋊C4—2571no—
100C2×C22⋊C4—2572no—
101C24—2571no—
102C24—2571no—
103C2×D4—2571no—
104C2×D4—2571no—
105C2×D4—2571no—
106C2×D4—5144no—
107C2×D4C2×D25711yescharacteristic
108C22×C4—2571no—
109C22×C4—2572no—
110C22×C4—2572no—
111C22×C4—2571no—
112C22×C4—2572no—
113C22×C4—5144no—
114C22×C4C2×D25711yescharacteristic, complemented
115C4⋊C4—2572no—
116C4⋊C4—2572no—
117C4⋊C4—2572no—
118C4⋊C4C2×D25712yescomplemented
119C22⋊C4—2571no—
120C22⋊C4—2572no—
121C22⋊C4—2572no—
122C22⋊C4—2572no—
123C22⋊C4—2572no—
124C22⋊C4C2×D25711yescharacteristic, complemented
125C22⋊C4C2×D25712yescomplemented
126C23—2571no—
127C23—2574no—
128C23—2571no—
129C23—2571no—
130C23—5144no—
131C23—5142no—
132C23—5144no—
133C23C22×D25711yescharacteristic
134C23C22×D25711yescharacteristic
135D4—22no—
136D4—5142no—
137D4—5142no—
138D4—5142no—
139C2×C4—2572no—
140C2×C4—2572no—
141C2×C4—2571no—
142C2×C4—22no—
143C2×C4—5144no—
144C2×C4—5144no—
145C2×C4—5142no—
146C2×C4—5144no—
147C2×C4—5142no—
148C2×C4C22×D25712yes—
149C2×C4C22×D25712yes—
150C2×C4C22×D25711yescharacteristic
151C22—2572no—
152C22—2574no—
153C22—2572no—
154C22—22no—
155C22—21no—
156C22—22no—
157C22—5148no—
158C22—5142no—
159C22—5141no—
160C22—5142no—
161C22—10284no—
162C22D4×D25712yes—
163C22C23×D25711yescharacteristic
164C4—22no—
165C4—21no—
166C4—22no—
167C4—5142no—
168C4—5141no—
169C4—5142no—
170C2—2574no—
171C2—22no—
172C2—5142no—
173C2—41no—
174C2—10281no—
175C2C2×D4×D25711yescharacteristic
176C2#41×D25712yes—
subgroupG/H
center, FrattiniC22C23×D257
commutator, socle, cosocleC22×C257C24
Fitting#123×C257C2
prime radicalC2×D4×D257C2
hypercenter#123D257
hypocenterC257C2×#123
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×D257C2semidirect
3. #22551577C2semidirect
4. #22551576C2semidirect
5. #22551575C2semidirect
6. #22551574C2semidirect
7. #22551573C2semidirect
8. C4⋊C4×D257C2semidirect
9. C22⋊C4×D257C2semidirect
10. C22⋊C4×D257C2semidirect
11. #123×C257C2semidirect
18. C22×Dic257C22semidirect
23. C2×C4×D257C22semidirect
24. C4⋊Dic257C22semidirect
25. #16861597C22semidirect
26. #16861597C22semidirect
27. C22⋊Dic257C22semidirect
28. C22⋊Dic257C22semidirect
29. #16861595C22semidirect
30. #16861595C22semidirect
31. #16861595C22semidirect
33. C22×C4×C257C22semidirect
34. C4⋊C4×C257C22semidirect
35. C22⋊C4×C257C22semidirect
36. C22⋊C4×C257C22semidirect
87. D257#123direct (N × H)
88. C257C2×#123semidirect
97. #123D257direct (N × H)
114. C22×C4C2×D257semidirect
118. C4⋊C4C2×D257semidirect
124. C22⋊C4C2×D257semidirect
125. C22⋊C4C2×D257semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D2572
C22×C4×D2572
#225515772
#225515762
#225515752
#225515742
#225515732
C4⋊C4×D2572
C22⋊C4×D2572
#123×C2572
C2×#123257