Finite Group Databasebeta

← #13102914

Subgroups — #13102914

Order 1600.

608 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 127 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×#87900C211yesmaximal, characteristic
2C52×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#87812C211yesmaximal, characteristic, complemented
4C2×#87811C211yesmaximal, characteristic, complemented
5C22⋊#6029C211yesmaximal, characteristic, complemented
6#11716677C211yesmaximal, characteristic, complemented
7#11716676C211yesmaximal, characteristic, complemented
8C2×C52×Q8C2211yescharacteristic
9C22×C4×C52C2211yescharacteristic
10C2×C22⋊(C5⋊D5)—21no—
11#87900—22no—
12#87900—22no—
13C2×C4⋊(C5⋊D5)C2211yescharacteristic
14C2×C4×C5⋊D5—21no—
15C2×#6045C2211yescharacteristic
16C52×C4⋊C4—21no—
17C52×C4⋊C4C2211yescharacteristic
18C52×C22⋊C4—21no—
19#87812—22no—
20#87811—22no—
21C2×#6029C2211yescharacteristic
22C2×#6029C2211yescharacteristic
23#65336—56nomaximal
24C23×C52D411yescharacteristic
25C22×C5⋊D5—21no—
26C52×Q8—21no—
27C52×Q8—21no—
28C2×C4×C52—21no—
29C2×C4×C52—21no—
30C2×C4×C52—21no—
31C2×C4×C52—21no—
32C2×C4×C52D411yescharacteristic
33C2×C4×C52C2311yescharacteristic
34C22⋊(C5⋊D5)—24no—
35C2×#1039—21no—
36C4⋊(C5⋊D5)—21no—
37C4⋊(C5⋊D5)D412yes—
38C4×C5⋊D5—24no—
39#6045—21no—
40#6045D412yes—
41#6029—21no—
42#6029—21no—
43C2×#709—56no—
44C5×C22⋊Q8D516yescomplemented
45C2×#689—56no—
46C2×#688—56no—
47C22⋊#168—56no—
48#3942—56no—
49#3941—56no—
50C22×C52—21no—
51C22×C52—21no—
52C22×C52—21no—
53C22×C52C2×D411yescharacteristic
54C2×C5⋊D5—22no—
55C2×C5⋊D5—41no—
56C2×C5⋊D5—41no—
57C4×C52—21no—
58C4×C52—41no—
59C4×C52—41no—
60C4×C52C2×D411yescharacteristic
61C4×C52C2×D411yescharacteristic
62#1039—22no—
63C2×C5×Q8C2×D516yes—
64C22×C4×C5C2×D516yes—
65C2×C22⋊D5—106no—
66#709—1012no—
67#709—1012no—
68C2×D20—56no—
69C2×C4×D5—106no—
70C2×Dic10—56no—
71C5×C4⋊C4—26no—
72C5×C4⋊C4C2×D516yes—
73C5×C22⋊C4—26no—
74#689—1012no—
75#688—1012no—
76C2×#168—56no—
77C2×#168—56no—
78#452—251noSylow
79C2×C52—41no—
80C2×C52#12011yescharacteristic
81C2×C52#13511yescharacteristic
82C2×C52#13711yescharacteristic
83C5⋊D5—42no—
84C23×C5C22⋊D516yes—
85C22×D5—106no—
86C5×Q8—26no—
87C5×Q8—26no—
88C2×C4×C5—26no—
89C2×C4×C5—26no—
90C2×C4×C5—26no—
91C2×C4×C5—26no—
92C2×C4×C5C22⋊D516yes—
93C2×C4×C5C22×D516yes—
94C22⋊D5—1024no—
95C2×Dic5—106no—
96D20—512no—
97D20—106no—
98C4×D5—1024no—
99Dic10—512no—
100Dic10—106no—
101#168—106no—
102#168—106no—
103C2×#41—251no—
104C2×Q16—251no—
105C2×SD16—251no—
106C22⋊Q8C5⋊D511yescharacteristic, complemented
107#103—251no—
108#102—251no—
109C22⋊C8—251no—
110C52#45211yescharacteristic, complemented, Sylow
111C22×C5—26no—
112C22×C5—26no—
113C22×C5—26no—
114C22×C5C2×C22⋊D516yes—
115C2×D5—1012no—
116C2×D5—206no—
117C2×D5—206no—
118C4×C5—26no—
119C4×C5—46no—
120C4×C5—46no—
121C4×C5D4×D516yes—
122C4×C5D4×D516yes—
123Dic5—1012no—
124#41—502no—
125#41—502no—
126C2×Q8—251no—
127C2×Q8C2×C5⋊D511yescharacteristic
128C2×D4—251no—
129C2×D4—501no—
130C22×C4—501no—
131C22×C4C2×C5⋊D511yescharacteristic
132Q16—502no—
133SD16—502no—
134C2×C8—251no—
135C2×C8—251no—
136C4⋊C4—21no—
137C4⋊C4C2×C5⋊D511yescharacteristic
138C22⋊C4—21no—
139C2×C5—46no—
140C2×C5#408316yes—
141C2×C5#408816yes—
142C2×C5#409616yes—
143D5—2012no—
144C23—501no—
145C23C22⋊(C5⋊D5)11yescharacteristic
146Q8—21no—
147Q8—21no—
148Q8—252no—
149Q8—501no—
150D4—252no—
151D4—504no—
152D4—501no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—504no—
158C2×C4—501no—
159C2×C4C22⋊(C5⋊D5)11yescharacteristic
160C2×C4C22×C5⋊D511yescharacteristic
161C8—501no—
162C8—501no—
163C5#6533616yescomplemented
164C22—21no—
165C22—21no—
166C22—21no—
167C22—502no—
168C22—1001no—
169C22—1001no—
170C22C2×C22⋊(C5⋊D5)11yescharacteristic
171C4—21no—
172C4—41no—
173C4—41no—
174C4—502no—
175C4D4×C5⋊D511yescharacteristic
176C4D4×C5⋊D511yescharacteristic
177C2—41no—
178C2—1002no—
179C2#1171725711yescharacteristic
180C2#1171726211yescharacteristic
181C2#1171727011yescharacteristic
subgroupG/H
centerC22C2×C22⋊(C5⋊D5)
commutator, cosocleC2×C4×C52C23
FrattiniC2×C4C22×C5⋊D5
FittingC52×C22⋊Q8C2
socleC22×C52C2×D4
prime radicalC2×#87900C2
hypercenterC22⋊Q8C5⋊D5
hypocenterC52#452
LCSG ▷ C2×C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C52×C22⋊Q8C2semidirect
3. C2×#87812C2semidirect
4. C2×#87811C2semidirect
5. C22⋊#6029C2semidirect
6. #11716677C2semidirect
7. #11716676C2semidirect
44. C5×C22⋊Q8D5semidirect
106. C22⋊Q8C5⋊D5semidirect
110. C52#452semidirect
163. C5#65336semidirect

Maximal subgroups

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

subgroupindex
C2×#879002
C52×C22⋊Q82
C2×#878122
C2×#878112
C22⋊#60292
#117166772
#117166762
#653365