Finite Group Databasebeta

← #12305723

Subgroups — #12305723

Order 1280.

1778 conjugacy classes of subgroups; only the 539 normal subgroups are shown.

normal subgroups only

This group has 1778 conjugacy classes of subgroups — too many to draw. Only its 539 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
1C5×#2840C211yesmaximal, characteristic
2#10608592C2112yesmaximal, complemented
3#10608509C2112yesmaximal, complemented
4#10605576C213yesmaximal
5#10605548C213yesmaximal
6C24×Dic5C2211yescharacteristic
7C2×Q8×Dic5C2214yescomplemented
8C2×#4023C2216yes—
9C5×#393C22112yes—
10C5×#379C2213yes—
11#65536C2213yes—
12#65518C22112yescomplemented
13#65238C22112yes—
14#65237C22124yes—
15#65232C2216yes—
16#65231C22112yes—
17C22⋊C4×Dic5C22112yes—
18#64948C22112yescomplemented
19#64946C22124yescomplemented
20#64938C22112yescomplemented
21#36674—51nomaximal, Sylow
22C25×C5C2311yescharacteristic
23C22×C5×Q8C2314yes—
24C23×Dic5—44no—
25C23×Dic5Q818yescomplemented
26C23×Dic5C2313yes—
27C23×Dic5C2313yes—
28C2×C5×C22⋊C4C2316yes—
29C2×C5×C22⋊C4C23112yes—
30C2×C22⋊Dic5C2316yes—
31C2×C22⋊Dic5C23112yes—
32Q8×Dic5—48no—
33C2×C4⋊Dic5C23112yes—
34C2×#683C23112yes—
35C2×#683C23124yes—
36C2×C4×Dic5C23112yes—
37#4023—412no—
38C5×#95C23112yes—
39#3963C23112yes—
40#3963C23124yes—
41#2842—53no—
42#2841—512no—
43#2840D511yescharacteristic
44#2396—512no—
45#2280—53no—
46C24×C5—42no—
47C24×C5—44no—
48C24×C5C2×Q814yes—
49C24×C5C2413yes—
50C2×C5×Q8—48no—
51C2×C5×Q8—44no—
52C22×C4×C5C24112yes—
53C22×Dic5—212no—
54C22×Dic5—416no—
55C22×Dic5—46no—
56C22×Dic5C2×Q8112yes—
57C22×Dic5C2×Q8112yes—
58C22×Dic5C24112yes—
59C22×Dic5C2413yes—
60C22×Dic5C2411yescharacteristic
61C5×C22⋊C4—412no—
62C5×C22⋊C4—46no—
63C22⋊Dic5—412no—
64C22⋊Dic5—46no—
65C4⋊Dic5—412no—
66#683—412no—
67#683—424no—
68C4×Dic5—224no—
69C24×C4—51no—
70C2×C4×Q8—54no—
71C2×#117—56no—
72#400—512no—
73#399—512no—
74#398—524no—
75#396—524no—
76#395—512no—
77#393C2×D5112yes—
78#381—56no—
79#380—512no—
80#379—53no—
81#379C2×D513yes—
82C4×C4⋊C4—512no—
83C4×C22⋊C4—512no—
84C23×C5—26no—
85C23×C5—212no—
86C23×C5—44no—
87C23×C5—48no—
88C23×C5—416no—
89C23×C5C22×Q816yes—
90C23×C5C2511yescharacteristic
91C5×Q8—416no—
92C2×C4×C5—212no—
93C2×C4×C5—224no—
94C2×Dic5—26no—
95C2×Dic5—224no—
96C2×Dic5—212no—
97C2×Dic5—224no—
98C2×Dic5—412no—
99C2×Dic5C22×Q816yes—
100C2×Dic5C22×Q812yes—
101C2×Dic5H(5,𝔽2)14yes—
102C25C22×D511yescharacteristic
103C22×Q8C22×D514yes—
104C23×C4—53no—
105C23×C4—58no—
106C23×C4—53no—
107C23×C4—204no—
108C4×Q8—208no—
109#117—2012no—
110C2×C4⋊C4—512no—
111C2×C4⋊C4—512no—
112C2×C4⋊C4—524no—
113C2×C22⋊C4—56no—
114C2×C22⋊C4—512no—
115C2×C22⋊C4C22×D516yes—
116C2×C22⋊C4C22×D5112yes—
117C2×C42—512no—
118#95—524no—
119#95—512no—
120#95C22×D5112yes—
121C22×C5—26no—
122C22×C5—212no—
123C22×C5—416no—
124C22×C5—412no—
125C22×C5C23×Q811yescharacteristic
126C22×C5C2×H(5,𝔽2)12yes—
127C22×C5#58514yes—
128C4×C5—412no—
129Dic5—26no—
130Dic5—412no—
131Dic5#54314yes—
132C24—42no—
133C24—44no—
134C24Q8×D514yes—
135C24C23×D513yes—
136C2×Q8—44no—
137C2×Q8—48no—
138C22×C4—51no—
139C22×C4—512no—
140C22×C4—53no—
141C22×C4—512no—
142C22×C4—512no—
143C22×C4—1012no—
144C22×C4—2016no—
145C22×C4—206no—
146C22×C4C23×D5112yes—
147C4⋊C4—2012no—
148C4⋊C4—2012no—
149C4⋊C4—2024no—
150C22⋊C4—46no—
151C22⋊C4—412no—
152C22⋊C4—206no—
153C22⋊C4—2012no—
154C42—1024no—
155C2×C5—46no—
156C2×C5C2×#54311yescharacteristic
157C2×C5#345812yes—
158C2×C5#352914yes—
159C23—26no—
160C23—212no—
161C23—44no—
162C23—48no—
163C23—416no—
164C23C2×Q8×D516yes—
165C23C24×D511yescharacteristic
166Q8—416no—
167C2×C4—212no—
168C2×C4—224no—
169C2×C4—52no—
170C2×C4—54no—
171C2×C4—56no—
172C2×C4—1024no—
173C2×C4—106no—
174C2×C4—1012no—
175C2×C4—1024no—
176C2×C4—2012no—
177C5#3667411yescharacteristic, complemented, Sylow
178C22—212no—
179C22—26no—
180C22—416no—
181C22—412no—
182C22C22×Q8×D511yescharacteristic
183C22D5×H(5,𝔽2)12yes—
184C22#6628514yes—
185C4—412no—
186C4—54no—
187C4—106no—
188C4—2012no—
189C2—46no—
190C2#1061847212yes—
191C2D5×#54311yescharacteristic
192C2#1061898614yes—
subgroupG/H
center, FrattiniC23C24×D5
commutator, socle, cosocleC23×C5C25
FittingC5×#2840C2
prime radicalC25×C5C23
hypercenter#2840D5
hypocenterC5#36674
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #2840

Complements

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

NH ≅ G/Ntype
2. #10608592C2semidirect
3. #10608509C2semidirect
7. C2×Q8×Dic5C22semidirect
12. #65518C22semidirect
18. #64948C22semidirect
19. #64946C22semidirect
20. #64938C22semidirect
25. C23×Dic5Q8semidirect
177. C5#36674semidirect

Maximal subgroups

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

subgroupindex
C5×#28402
#106085922
#106085092
#106055762
#106055482
#366745