Finite Group Databasebeta

← #12003493

Subgroups — #12003493

Order 1280.

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

normal subgroups only

This group has 358 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#10601311C211yesmaximal, characteristic
5#10598203C211yesmaximal, characteristic
6C5×#533C2212yes—
7C5×#475—22no—
8C5×#470—22no—
9C5×#462—22no—
10C5×#423C2211yescharacteristic
11C2×#3954—22no—
12#65213C2212yes—
13#65213C2211yescharacteristic
14C2×C4×#168C2211yescharacteristic
15#64767—22no—
16#64767—22no—
17Q8⋊#168—24no—
18#10534—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—
36#3954—44no—
37C4⋊#168—22no—
38C4⋊#168—22no—
39C4⋊#168—22no—
40C4×#168—21no—
41C4×#168D412yes—
42#3255D511yescharacteristic
43#1848—51no—
44#1561—52no—
45#1493—52no—
46#1309—51no—
47C2×C5×Q8—22no—
48C2×C5×Q8—22no—
49C22×C4×C5C2×D411yescharacteristic
50C22×C4×C5#4112yes—
51C5×Q16—82no—
52C5×M16—42no—
53C2×C5×C8—22no—
54C5×C4⋊C4—22no—
55C5×C4⋊C4—22no—
56C5×C4⋊C4—22no—
57C5×C4⋊C4—22no—
58C5×C4⋊C4—42no—
59C5×C4⋊C4—42no—
60C5×C22⋊C4—42no—
61C42×C5—42no—
62C42×C5C2×D411yescharacteristic
63C42×C5C2×D412yes—
64C42×C5#4111yescharacteristic
65#681—42no—
66#681—82no—
67C2×#168—22no—
68C2×#168—22no—
69C2×#168—22no—
70C2×#168—41no—
71C2×#168—42no—
72#533C2×D512yes—
73#475—22no—
74#470—22no—
75#462—22no—
76C2×#108—102no—
77#423—51no—
78#423—52no—
79#423C2×D511yescharacteristic
80C2×C4×C8—51no—
81#332—102no—
82#332—102no—
83Q8⋊C8—104no—
84C23×C5#12411yescharacteristic
85C5×Q8—42no—
86C5×Q8—42no—
87C5×Q8—42no—
88C5×Q8—42no—
89C2×C4×C5—21no—
90C2×C4×C5—21no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—42no—
94C2×C4×C5—42no—
95C2×C4×C5#12011yescharacteristic
96C2×C4×C5#12111yescharacteristic
97C2×C4×C5#12112yes—
98C2×C4×C5#12312yes—
99C5×C8—42no—
100#168—24no—
101#168—81no—
102#168—82no—
103C2×Q16—22no—
104C2×M16—101no—
105C2×M16—102no—
106C2×M16C22⋊D512yes—
107C22×C8—101no—
108C4⋊Q8C22⋊D512yes—
109C4⋊Q8C22⋊D512yes—
110#125—22no—
111C22⋊Q8—22no—
112C22⋊Q8—22no—
113C4×Q8—22no—
114C4×Q8—22no—
115#117—22no—
116C2×C42C22×D511yescharacteristic
117#108—204no—
118#106—22no—
119C4⋊C8—22no—
120C4⋊C8—102no—
121C4⋊C8—102no—
122C4⋊C8—102no—
123#103—24no—
124C4×C8—52no—
125C4×C8—101no—
126C22×C5—22no—
127C22×C5—21no—
128C22×C5#39411yescharacteristic
129C4×C5—22no—
130C4×C5—21no—
131C4×C5—82no—
132C4×C5—82no—
133C4×C5#45311yescharacteristic
134C4×C5#45611yescharacteristic
135C4×C5#47112yes—
136C4×C5#47312yes—
137C2×Q8—22no—
138C2×Q8—22no—
139C22×C4#71112yes—
140C22×C4C2×C22⋊D511yescharacteristic
141Q16—82no—
142M16—42no—
143M16—202no—
144M16—402no—
145C2×C8—22no—
146C2×C8—102no—
147C2×C8—102no—
148C2×C8—102no—
149C2×C8—201no—
150C2×C8—202no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C22⋊C4—42no—
158C42—42no—
159C42D4×D511yescharacteristic
160C42#71111yescharacteristic
161C42C2×C22⋊D512yes—
162C2×C5—41no—
163C2×C5#203911yescharacteristic
164C2×C5#205212yes—
165C23#408211yescharacteristic
166Q8—42no—
167Q8—42no—
168Q8—42no—
169Q8—42no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4#408112yes—
177C2×C4#408411yescharacteristic
178C2×C4#408512yes—
179C2×C4#409911yescharacteristic
180C8—42no—
181C8—104no—
182C8—402no—
183C8—401no—
184C5#1053411yescharacteristic, complemented, Sylow
185C22—22no—
186C22—21no—
187C22#6551111yescharacteristic
188C4—22no—
189C4—21no—
190C4—82no—
191C4—82no—
192C4#6549212yes—
193C4#6549312yes—
194C4#6553011yescharacteristic
195C4#6553511yescharacteristic
196C2—41no—
197C2#1060266911yescharacteristic
198C2#1060289812yes—
subgroupG/H
centerC22#65511
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#3255C2
socleC22×C5#394
prime radicalC23×C5#124
hypercenter#3255D5
hypocenterC5#10534
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
184. C5#10534semidirect

Maximal subgroups

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

subgroupindex
C5×#32552
#106026272
#106026192
#106013112
#105982032
#105345