Finite Group Databasebeta

← #13105113

Subgroups — #13105113

Order 1600.

564 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 120 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
1#11717196C211yesmaximal, characteristic, complemented
2C5×#4022C211yesmaximal, characteristic, complemented
3C5×#4020C211yesmaximal, characteristic, complemented
4#11717040C213yesmaximal, complemented
5#11717005C213yesmaximal, complemented
6#11717004C213yesmaximal, complemented
7#11717002C213yesmaximal, complemented
8C22×D52C2211yescharacteristic
9C2×C5×D20C2213yes—
10C2×C5×D20C2213yescomplemented
11C2×#6038C2213yes—
12C2×#6037C2213yes—
13C2×D5×Dic5C2213yescomplemented
14C42×C52C2211yescharacteristic, complemented
15#87808C2213yes—
16#87806C2213yescomplemented
17C5×#685C2213yes—
18C5×#683C2213yescomplemented
19#87778C2213yes—
20#87778C2213yescomplemented
21#66054—51nomaximal
22#66045—51nomaximal
23C22×C5⋊D5C2311yescharacteristic
24C22×C5×D5C2313yes—
25C22×C5×D5C2311yescharacteristic
26C22×C5×D5C2311yescharacteristic
27C2×D52—43no—
28C2×C4×C52C2313yes—
29C2×#1039C2313yes—
30C2×C5×Dic5C2313yes—
31C5×D20—43no—
32C5×D20—43no—
33#6038—43no—
34#6037—43no—
35D5×Dic5—43no—
36C5×#127—51no—
37C5×#126—51no—
38#4085—53no—
39#4085—53no—
40#4083—53no—
41#4081—53no—
42#4041—53no—
43#4040—53no—
44#4030—53no—
45#4028—53no—
46#4022—51no—
47#4022—51no—
48#4022—102no—
49#4022D511yescharacteristic, complemented
50#4020D511yescharacteristic, complemented
51C22×C52C2411yescharacteristic
52C2×C5⋊D5—23no—
53C2×C5×D5—23no—
54C2×C5×D5—23no—
55C2×C5×D5—23no—
56C2×C5×D5—23no—
57C2×C5×D5—23no—
58D52—44no—
59C4×C52—23no—
60#1039—23no—
61C5×Dic5—23no—
62C23×D5—51no—
63C23×D5—51no—
64C2×C5×D4—53no—
65C2×C5×D4—53no—
66C2×C22⋊D5—53no—
67C2×C22⋊D5—53no—
68C2×C22⋊D5—53no—
69C22×Dic5—53no—
70C2×D20—53no—
71C2×D20C2×D513yes—
72C2×D20C2×D513yescomplemented
73C2×C4×D5—53no—
74C5×C4⋊C4—53no—
75C5×C22⋊C4—53no—
76C42×C5—22no—
77C42×C5C2×D511yescharacteristic, complemented
78C42×C5C2×D511yescharacteristic, complemented
79C22⋊Dic5—53no—
80C22⋊Dic5—53no—
81#685—53no—
82#685—53no—
83#685—53no—
84#685—53no—
85#685—106no—
86#685C2×D513yes—
87#683—53no—
88#683—53no—
89#683—106no—
90#683C2×D513yescomplemented
91#560—251noSylow
92C2×C52H(5,𝔽2)13yes—
93C5⋊D5—41no—
94C5×D5—41no—
95C5×D5—41no—
96C5×D5—43no—
97C23×C5—51no—
98C23×C5—53no—
99C23×C5—51no—
100C22×D5—51no—
101C22×D5—51no—
102C22×D5—102no—
103C22×D5—203no—
104C22×D5—203no—
105C22×D5C22×D511yescharacteristic
106C22×D5C22×D513yes—
107C22×D5C22×D511yescharacteristic
108C5×D4—203no—
109C5×D4—203no—
110C2×C4×C5—26no—
111C2×C4×C5—53no—
112C2×C4×C5C22×D513yes—
113C2×C4×C5C22×D513yes—
114C22⋊D5—203no—
115C22⋊D5—203no—
116C22⋊D5—203no—
117C2×Dic5—53no—
118C2×Dic5—53no—
119C2×Dic5—106no—
120C2×Dic5—203no—
121C2×Dic5C22×D513yes—
122D20—43no—
123D20—43no—
124D20—203no—
125C4×D5—203no—
126#127—51no—
127#126—51no—
128#126—251no—
129#123—253no—
130#121—253no—
131#121—253no—
132#120—253no—
133C52#56011yescharacteristic, complemented, Sylow
134C22×C5—22no—
135C22×C5—103no—
136C22×C5—103no—
137C22×C5—103no—
138C22×C5—103no—
139C22×C5—103no—
140C22×C5C23×D511yescharacteristic
141C22×C5C23×D511yescharacteristic
142C2×D5—23no—
143C2×D5—23no—
144C2×D5—23no—
145C2×D5—23no—
146C2×D5—23no—
147C2×D5—103no—
148C2×D5—103no—
149C2×D5—206no—
150C2×D5—204no—
151C2×D5—204no—
152C4×C5—23no—
153C4×C5—23no—
154C4×C5—46no—
155C4×C5—103no—
156Dic5—23no—
157Dic5—103no—
158Dic5—103no—
159Dic5—206no—
160C24—251no—
161C2×D4—53no—
162C2×D4—53no—
163C2×D4—253no—
164C2×D4—253no—
165C22×C4—253no—
166C4⋊C4—53no—
167C4⋊C4—253no—
168C22⋊C4—53no—
169C22⋊C4—253no—
170C22⋊C4—253no—
171C22⋊C4—253no—
172C42D5211yescharacteristic, complemented
173C2×C5—26no—
174C2×C5—201no—
175C2×C5—201no—
176C2×C5—203no—
177C2×C5#414413yes—
178C2×C5#414913yes—
179D5—41no—
180D5—43no—
181D5—41no—
182D5—201no—
183D5—201no—
184D5—402no—
185C23—51no—
186C23—51no—
187C23—53no—
188C23—251no—
189C23—1003no—
190D4—203no—
191D4—203no—
192D4—1003no—
193D4—1003no—
194C2×C4—53no—
195C2×C4—253no—
196C2×C4—1003no—
197C2×C4C2×D5213yes—
198C5—22no—
199C5#6604511yescharacteristic, complemented
200C5#6605411yescharacteristic, complemented
201C22—103no—
202C22—103no—
203C22—103no—
204C22—103no—
205C22—103no—
206C22—503no—
207C22—1004no—
208C22C22×D5211yescharacteristic
209C4—23no—
210C4—103no—
211C4—503no—
212C2—201no—
213C2—201no—
214C2—203no—
215C2—1001no—
216C2#1171747213yes—
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC42×C52C22
hypercenterC42D52
hypocenterC52#560
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11717196C2semidirect
2. C5×#4022C2semidirect
3. C5×#4020C2semidirect
4. #11717040C2semidirect
5. #11717005C2semidirect
6. #11717004C2semidirect
7. #11717002C2semidirect
10. C2×C5×D20C22semidirect
13. C2×D5×Dic5C22semidirect
14. C42×C52C22semidirect
16. #87806C22semidirect
18. C5×#683C22semidirect
20. #87778C22semidirect
49. #4022D5semidirect
50. #4020D5semidirect
72. C2×D20C2×D5semidirect
77. C42×C5C2×D5semidirect
78. C42×C5C2×D5semidirect
90. #683C2×D5semidirect
133. C52#560semidirect
172. C42D52semidirect
199. C5#66045semidirect
200. C5#66054semidirect

Maximal subgroups

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

subgroupindex
#117171962
C5×#40222
C5×#40202
#117170402
#117170052
#117170042
#117170022
#660545
#660455