Finite Group Databasebeta

← #12003498

Subgroups — #12003498

Order 1280.

342 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3255C211yesmaximal, characteristic
2#10602627C212yesmaximal
3#10602619C212yesmaximal
4#10601309C211yesmaximal, characteristic
5#10598203C211yesmaximal, characteristic
6C5×#533C2212yes—
7C5×#475—22no—
8C5×#470—22no—
9C5×#462—22no—
10C5×#423C2211yescharacteristic
11#65213C2212yes—
12#65213C2211yescharacteristic
13C2×C4×#168C2211yescharacteristic
14#64778—22no—
15#64767—22no—
16#64767—22no—
17Q8⋊#168—24no—
18#10535—51nomaximal, Sylow
19C2×C5×Q16—22no—
20C2×C5×M16D412yes—
21C5×C4⋊Q8D412yes—
22C5×C4⋊Q8D412yes—
23C5×#125—22no—
24C5×C22⋊Q8—22no—
25C5×C22⋊Q8—22no—
26C4×C5×Q8—22no—
27C4×C5×Q8—22no—
28C5×#117—22no—
29C2×C42×C5C2311yescharacteristic
30C2×#681—21no—
31C2×#681—22no—
32C22×#168—21no—
33C5×#106—22no—
34C5×C4⋊C8—22no—
35C5×#103—24no—
36C4⋊#168—22no—
37C4⋊#168—22no—
38C4⋊#168—22no—
39C4×#168—21no—
40C4×#168D412yes—
41#3255D511yescharacteristic
42#1837—51no—
43#1561—52no—
44#1493—52no—
45#1309—51no—
46C2×C5×Q8—22no—
47C2×C5×Q8—22no—
48C22×C4×C5C2×D411yescharacteristic
49C22×C4×C5#4112yes—
50C5×Q16—82no—
51C5×M16—42no—
52C2×C5×C8—22no—
53C5×C4⋊C4—22no—
54C5×C4⋊C4—22no—
55C5×C4⋊C4—22no—
56C5×C4⋊C4—22no—
57C5×C4⋊C4—42no—
58C5×C4⋊C4—42no—
59C5×C22⋊C4—42no—
60C42×C5—42no—
61C42×C5C2×D411yescharacteristic
62C42×C5C2×D412yes—
63C42×C5#4111yescharacteristic
64#681—42no—
65#681—82no—
66C2×#168—22no—
67C2×#168—41no—
68C2×#168—41no—
69C2×#168—41no—
70C2×#168—42no—
71#533C2×D512yes—
72#475—22no—
73#470—22no—
74#462—22no—
75#423—51no—
76#423—52no—
77#423C2×D511yescharacteristic
78C2×C4×C8—51no—
79#341—102no—
80#332—102no—
81#332—102no—
82Q8⋊C8—104no—
83C23×C5#12411yescharacteristic
84C5×Q8—42no—
85C5×Q8—42no—
86C5×Q8—42no—
87C5×Q8—42no—
88C2×C4×C5—21no—
89C2×C4×C5—21no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—42no—
93C2×C4×C5—42no—
94C2×C4×C5#12011yescharacteristic
95C2×C4×C5#12111yescharacteristic
96C2×C4×C5#12112yes—
97C2×C4×C5#12312yes—
98C5×C8—42no—
99#168—42no—
100#168—81no—
101#168—82no—
102C2×Q16—22no—
103C2×M16—101no—
104C2×M16—102no—
105C2×M16C22⋊D512yes—
106C22×C8—101no—
107C4⋊Q8C22⋊D512yes—
108C4⋊Q8C22⋊D512yes—
109#125—22no—
110C22⋊Q8—22no—
111C22⋊Q8—22no—
112C4×Q8—22no—
113C4×Q8—22no—
114#117—22no—
115C2×C42C22×D511yescharacteristic
116#106—22no—
117C4⋊C8—22no—
118C4⋊C8—102no—
119C4⋊C8—102no—
120C4⋊C8—102no—
121#103—24no—
122C4×C8—52no—
123C4×C8—101no—
124C22×C5—22no—
125C22×C5—21no—
126C22×C5#39411yescharacteristic
127C4×C5—22no—
128C4×C5—21no—
129C4×C5—82no—
130C4×C5—82no—
131C4×C5#45411yescharacteristic
132C4×C5#45511yescharacteristic
133C4×C5#47112yes—
134C4×C5#47312yes—
135C2×Q8—22no—
136C2×Q8—22no—
137C22×C4#71112yes—
138C22×C4C2×C22⋊D511yescharacteristic
139Q16—82no—
140M16—42no—
141M16—202no—
142M16—402no—
143C2×C8—22no—
144C2×C8—102no—
145C2×C8—201no—
146C2×C8—201no—
147C2×C8—201no—
148C2×C8—202no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C22⋊C4—42no—
156C42—42no—
157C42D4×D511yescharacteristic
158C42#71111yescharacteristic
159C42C2×C22⋊D512yes—
160C2×C5—41no—
161C2×C5#204011yescharacteristic
162C2×C5#205212yes—
163C23#408211yescharacteristic
164Q8—42no—
165Q8—42no—
166Q8—42no—
167Q8—42no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4#408112yes—
175C2×C4#408411yescharacteristic
176C2×C4#408512yes—
177C2×C4#409911yescharacteristic
178C8—42no—
179C8—202no—
180C8—402no—
181C8—401no—
182C5#1053511yescharacteristic, complemented, Sylow
183C22—22no—
184C22—21no—
185C22#6551111yescharacteristic
186C4—22no—
187C4—21no—
188C4—82no—
189C4—82no—
190C4#6549212yes—
191C4#6549312yes—
192C4#6553111yescharacteristic
193C4#6553411yescharacteristic
194C2—41no—
195C2#1060267011yescharacteristic
196C2#1060289812yes—
subgroupG/H
centerC22#65511
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#3255C2
socleC22×C5#394
prime radicalC23×C5#124
hypercenter#3255D5
hypocenterC5#10535
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3255

Complements

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

NH ≅ G/Ntype
182. C5#10535semidirect

Maximal subgroups

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

subgroupindex
C5×#32552
#106026272
#106026192
#106013092
#105982032
#105355