Finite Group Databasebeta

← #13105101

Subgroups — #13105101

Order 1600.

768 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 144 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×#87877C211yesmaximal, characteristic
2C2×#87872C211yesmaximal, characteristic, complemented
3C42×C5⋊D5C211yesmaximal, characteristic, complemented
4C5×#4021C212yesmaximal, complemented
5#11717039C214yesmaximal, complemented
6#11717033C214yesmaximal, complemented
7#11716981C212yesmaximal, complemented
8C22×D52C2212yes—
9C2×C4×C5⋊D5C2212yes—
10C2×C4×C5⋊D5C2211yescharacteristic
11C2×C5×D20C2212yes—
12C2×C5×Dic10C2212yescomplemented
13C2×#6039C2212yescomplemented
14C2×#6037C2212yes—
15C2×#6036C2212yes—
16#87877—24no—
17#87872—24no—
18C42×C52C2211yescharacteristic, complemented
19C4×#1039C2211yescharacteristic, complemented
20C4×#1039C2212yescomplemented
21C5×#685C2218yescomplemented
22#87778C2218yescomplemented
23D5×#124—52nomaximal
24C22×C5⋊D5C2311yescharacteristic
25C22×C5×D5C2314yes—
26C2×D52—24no—
27C2×D52—28no—
28C2×C4×C52C2312yes—
29C2×C4×C52C2311yescharacteristic
30C2×#1039C2312yes—
31C2×#1039C2311yescharacteristic
32C4×C5⋊D5—24no—
33C4×C5⋊D5D414yes—
34C2×C5×Dic5C2314yes—
35C5×D20—24no—
36C5×Dic10—24no—
37#6039—24no—
38#6037—24no—
39#6036—24no—
40C2×Q8×D5—52no—
41C2×D4×D5—52no—
42C5×#124—52no—
43#4093—52no—
44#4082—52no—
45#4031—58no—
46D5×C22⋊C4—58no—
47#4021D512yescomplemented
48C42×D5—52no—
49C42×D5—102no—
50C22×C52C2411yescharacteristic
51C2×C5⋊D5C2×D412yes—
52C2×C5⋊D5#4114yes—
53C2×C5×D5—24no—
54C2×C5×D5—28no—
55D52—48no—
56C4×C52—22no—
57C4×C52C2×D412yes—
58#1039—22no—
59#1039C2×D412yes—
60C5×Dic5—24no—
61C23×D5—54no—
62C2×C5×Q8—52no—
63C2×C5×D4—52no—
64C2×C22⋊D5—54no—
65Q8×D5—108no—
66D4×D5—108no—
67C2×D20C2×D512yes—
68C2×C4×D5—52no—
69C2×C4×D5—54no—
70C2×C4×D5—54no—
71C2×C4×D5—102no—
72C2×C4×D5—104no—
73C2×Dic10—54no—
74C2×Dic10C2×D512yescomplemented
75C5×C22⋊C4—58no—
76C42×C5—22no—
77C42×C5C2×D512yescomplemented
78C22⋊Dic5—58no—
79#685—58no—
80#685C2×D518yescomplemented
81C4×Dic5—54no—
82C4×Dic5—52no—
83C4×Dic5—102no—
84C4×Dic5—104no—
85C2×#124—251noSylow
86C2×C52C22×D411yescharacteristic
87C2×C52C2×#4112yes—
88C5⋊D5#12414yescomplemented
89C5×D5—44no—
90C23×C5—54no—
91C22×D5—52no—
92C22×D5—102no—
93C22×D5—108no—
94C22×D5—1016no—
95C22×D5C22×D514yes—
96C5×Q8—104no—
97C5×D4—104no—
98C2×C4×C5—24no—
99C2×C4×C5—22no—
100C2×C4×C5—54no—
101C2×C4×C5C22×D514yes—
102C2×C4×C5C22×D512yes—
103C22⋊D5—108no—
104C2×Dic5—52no—
105C2×Dic5—54no—
106C2×Dic5—102no—
107C2×Dic5—104no—
108C2×Dic5C22×D514yes—
109D20—24no—
110C4×D5—58no—
111C4×D5—108no—
112C4×D5—108no—
113C4×D5—108no—
114C4×D5—1016no—
115Dic10—24no—
116Dic10—108no—
117C22×Q8—251no—
118C22×D4—251no—
119#124—52no—
120#124—254no—
121#124—252no—
122C2×C22⋊C4—254no—
123C2×C42—251no—
124C52C2×#12411yescharacteristic, complemented, Sylow
125C22×C5—22no—
126C22×C5—104no—
127C22×C5—108no—
128C22×C5C23×D512yes—
129C2×D5—24no—
130C2×D5—28no—
131C2×D5—54no—
132C2×D5—58no—
133C2×D5—104no—
134C2×D5—108no—
135C2×D5—2016no—
136C4×C5—28no—
137C4×C5—24no—
138C4×C5—24no—
139C4×C5—104no—
140C4×C5D4×D514yes—
141Dic5—24no—
142Dic5—54no—
143Dic5—104no—
144Dic5—108no—
145Dic5—104no—
146C24—252no—
147C2×Q8—52no—
148C2×Q8—252no—
149C2×Q8—504no—
150C2×D4—52no—
151C2×D4—252no—
152C2×D4—504no—
153C22×C4—252no—
154C22×C4—251no—
155C22×C4—252no—
156C22⋊C4—58no—
157C22⋊C4—258no—
158C42—252no—
159C42—251no—
160C42D5211yescharacteristic, complemented
161C2×C5—22no—
162C2×C5—24no—
163C2×C5—204no—
164C2×C5C2×D4×D512yes—
165C2×C5D5×#4114yes—
166D5—44no—
167D5—58no—
168D5—108no—
169C23—54no—
170C23—251no—
171C23—504no—
172C23—508no—
173Q8—104no—
174Q8—504no—
175D4—104no—
176D4—504no—
177C2×C4—54no—
178C2×C4—251no—
179C2×C4—254no—
180C2×C4—252no—
181C2×C4—504no—
182C2×C4—504no—
183C2×C4C2×D5212yes—
184C2×C4C2×D5211yescharacteristic
185C5—22no—
186C5D5×#12412yescomplemented
187C22—104no—
188C22—108no—
189C22—252no—
190C22—254no—
191C22—1008no—
192C22C22×D5211yescharacteristic
193C4—22no—
194C4—104no—
195C4—252no—
196C4—502no—
197C4#8787712yes—
198C2—204no—
199C2—254no—
200C2C2×#8787711yescharacteristic
201C2#1171747112yes—
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC42×C52C22
prime radicalC2×#87877C2
hypercenterC42D52
hypocenterC52C2×#124
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×#87872C2semidirect
3. C42×C5⋊D5C2semidirect
4. C5×#4021C2semidirect
5. #11717039C2semidirect
6. #11717033C2semidirect
7. #11716981C2semidirect
12. C2×C5×Dic10C22semidirect
13. C2×#6039C22semidirect
18. C42×C52C22semidirect
19. C4×#1039C22semidirect
20. C4×#1039C22semidirect
21. C5×#685C22semidirect
22. #87778C22semidirect
47. #4021D5semidirect
74. C2×Dic10C2×D5semidirect
77. C42×C5C2×D5semidirect
80. #685C2×D5semidirect
88. C5⋊D5#124semidirect
124. C52C2×#124semidirect
160. C42D52semidirect
186. C5D5×#124semidirect

Maximal subgroups

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

subgroupindex
C2×#878772
C2×#878722
C42×C5⋊D52
C5×#40212
#117170392
#117170332
#117169812
D5×#1245