Finite Group Databasebeta

← #13107835

Subgroups — #13107835

Order 1600.

1784 conjugacy classes of subgroups; only the 528 normal subgroups are shown.

normal subgroups only

This group has 1784 conjugacy classes of subgroups — too many to draw. Only its 528 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×#87904C211yesmaximal, characteristic, complemented
2C5×#4147C212yesmaximal, complemented
3#11717485C214yesmaximal, complemented
4D5×#713C218yesmaximal, complemented
5#11717481C214yesmaximal, complemented
6#11717473C216yesmaximal, complemented
7#11717471C216yesmaximal, complemented
8C2×C52×Q8C2211yescharacteristic, complemented
9#87904C2214yescomplemented
10#87904C2214yescomplemented
11C2×C4⋊(C5⋊D5)C2213yescomplemented
12C2×C4×C5⋊D5C2213yescomplemented
13C5×#713C2218yescomplemented
14C5×Q8×D5C2218yescomplemented
15C5×#709C22112yescomplemented
16#87886C22112yescomplemented
17#87880C2214yescomplemented
18#87877C2216yescomplemented
19D5×D20C22124yescomplemented
20C4×D52C22112yescomplemented
21#87874C22124yescomplemented
22#87873C22112yescomplemented
23#87872C2216yescomplemented
24#87871C22112yescomplemented
25#66287—52nomaximal
26C22×C5⋊D5C2313yes—
27C2×D52C23112yes—
28C52×Q8C2314yes—
29C2×C4×C52C2313yes—
30C2×#1039C2311yescharacteristic
31C4⋊(C5⋊D5)C23112yes—
32C4×C5⋊D5C2316yes—
33C4×C5⋊D5C2316yes—
34C5×C22⋊D5C2318yes—
35C5×D20C23112yes—
36C4×C5×D5C23124yes—
37C5×Dic10C23112yes—
38#6039C2314yes—
39#6038C23124yes—
40#6037C2314yes—
41#6036C23112yes—
42D5×Dic5C2318yes—
43C5×2−1+4—52no—
44#4149—58no—
45#4149—512no—
46D5×#41—58no—
47D5×#41—512no—
48#4147—58no—
49#4147D512yescomplemented
50C2×#713—52no—
51C2×#713—58no—
52C2×#713—102no—
53C22×C52C2411yescharacteristic
54C2×C5⋊D5—43no—
55C2×C5⋊D5C2416yes—
56C2×C5×D5C2418yes—
57D52—412no—
58C4×C52C2416yes—
59#1039C2411yescharacteristic
60#1039C2411yescharacteristic
61C5×Dic5C2418yes—
62C5×#41—58no—
63C5×#41—512no—
64C2×C5×Q8—22no—
65C2×C5×Q8—58no—
66C2×C5×Q8C2×D512yescomplemented
67#713—58no—
68#713—58no—
69#713—512no—
70#713—524no—
71#713—108no—
72#713—108no—
73#713C2×D518yescomplemented
74Q8×D5—512no—
75Q8×D5C2×D518yescomplemented
76#711—58no—
77#711—512no—
78D4×D5—524no—
79D4×D5—512no—
80D4×D5—524no—
81#709—524no—
82#709—524no—
83#709C2×D5112yescomplemented
84C2×D20—56no—
85C2×D20—524no—
86C2×D20—106no—
87C2×C4×D5—56no—
88C2×C4×D5—524no—
89C2×C4×D5—106no—
90#585—251noSylow
91C2×C52—21no—
92C2×C52C2511yescharacteristic
93C5⋊D5—26no—
94C5×D5—28no—
95C22×D5—56no—
96C22×D5—524no—
97C22×D5—106no—
98C5×Q8—28no—
99C5×Q8—512no—
100C5×Q8C22×D518yes—
101C5×D4—58no—
102C5×D4—512no—
103C2×C4×C5—26no—
104C2×C4×C5—524no—
105C2×C4×C5C22×D516yes—
106C22⋊D5—58no—
107C22⋊D5—524no—
108C22⋊D5C22×D518yes—
109C2×Dic5—52no—
110C2×Dic5—58no—
111C2×Dic5—102no—
112D20—524no—
113D20—524no—
114D20—1024no—
115D20C22×D5112yes—
116C4×D5—58no—
117C4×D5—512no—
118C4×D5—512no—
119C4×D5—524no—
120C4×D5—1012no—
121C4×D5—1012no—
122C4×D5C22×D5124yes—
123Dic10—58no—
124Dic10C22×D5112yes—
1252−1+4—52no—
1262−1+4—254no—
127H(5,𝔽2)—254no—
128H(5,𝔽2)—256no—
129C2×#41—258no—
130C2×#41—256no—
131C2×#41—251no—
132C52#58511yescharacteristic, complemented, Sylow
133C22×C5—22no—
134C22×C5—58no—
135C22×C5C23×D512yes—
136C2×D5—512no—
137C2×D5—1012no—
138C2×D5—2024no—
139C2×D5—206no—
140C2×D5—2012no—
141C2×D5C23×D518yes—
142C4×C5—212no—
143C4×C5—58no—
144C4×C5C23×D5112yes—
145Dic5—52no—
146Dic5—52no—
147Dic5—102no—
148Dic5—102no—
149Dic5C23×D518yes—
150#41—58no—
151#41—512no—
152#41—2512no—
153#41—2524no—
154#41—254no—
155#41—2512no—
156#41—254no—
157#41—254no—
158C2×Q8—58no—
159C2×Q8—256no—
160C2×Q8D5211yescharacteristic, complemented
161C2×D4—2524no—
162C2×D4—2512no—
163C2×D4—256no—
164C2×D4—253no—
165C22×C4—2512no—
166C22×C4—253no—
167C2×C5—24no—
168C2×C5—22no—
169C2×C5—22no—
170C2×C5—108no—
171C2×C5C24×D512yes—
172D5—28no—
173D5—1012no—
174D5—2012no—
175C23—2512no—
176C23—253no—
177Q8—512no—
178Q8—254no—
179Q8C2×D5214yes—
180D4—58no—
181D4—512no—
182D4—2524no—
183D4—254no—
184D4—2512no—
185C2×C4—524no—
186C2×C4—2512no—
187C2×C4—258no—
188C2×C4—251no—
189C2×C4—256no—
190C2×C4—256no—
191C2×C4C2×D5213yes—
192C5—22no—
193C5#6628712yescomplemented
194C22—58no—
195C22—256no—
196C22—1003no—
197C22—10012no—
198C22C22×D5211yescharacteristic
199C4—58no—
200C4—251no—
201C4—251no—
202C4C22×D5216yes—
203C2—21no—
204C2—108no—
205C2—506no—
206C2C23×D5211yescharacteristic
subgroupG/H
center, FrattiniC2C23×D52
commutator, socle, cosocleC2×C52C25
FittingC2×C52×Q8C22
hypercenterC2×Q8D52
hypocenterC52#585
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#87904C2semidirect
2. C5×#4147C2semidirect
3. #11717485C2semidirect
4. D5×#713C2semidirect
5. #11717481C2semidirect
6. #11717473C2semidirect
7. #11717471C2semidirect
8. C2×C52×Q8C22semidirect
9. #87904C22semidirect
10. #87904C22semidirect
11. C2×C4⋊(C5⋊D5)C22semidirect
12. C2×C4×C5⋊D5C22semidirect
13. C5×#713C22semidirect
14. C5×Q8×D5C22semidirect
15. C5×#709C22semidirect
16. #87886C22semidirect
17. #87880C22semidirect
18. #87877C22semidirect
19. D5×D20C22semidirect
20. C4×D52C22semidirect
21. #87874C22semidirect
22. #87873C22semidirect
23. #87872C22semidirect
24. #87871C22semidirect
49. #4147D5semidirect
66. C2×C5×Q8C2×D5semidirect
73. #713C2×D5semidirect
75. Q8×D5C2×D5semidirect
83. #709C2×D5semidirect
132. C52#585semidirect
160. C2×Q8D52semidirect
193. C5#66287semidirect

Maximal subgroups

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

subgroupindex
C2×#879042
C5×#41472
#117174852
D5×#7132
#117174812
#117174732
#117174712
#662875