Finite Group Databasebeta

← #13101641

Subgroups — #13101641

Order 1600.

372 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 64 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×#4144C211yesmaximal, characteristic
2C5×#4081C211yesmaximal, characteristic, complemented
3#11717079C212yesmaximal, complemented
4#11716701C211yesmaximal, characteristic, complemented
5#11716621C212yesmaximal, complemented
6C2×C52×D4C2211yescharacteristic
7C2×C5×C22⋊D5—21no—
8C2×C5×C22⋊D5C2212yes—
9C22×C5×Dic5C2211yescharacteristic, complemented
10C5×#711—22no—
11C5×D4×D5—22no—
12C5×#709—21no—
13C22⋊#1039C2212yescomplemented
14C5×C22⋊Dic5—21no—
15C5×C22⋊Dic5C2211yescharacteristic, complemented
16C5×#683—21no—
17#87781—22no—
18#87778—22no—
19#65532—51nomaximal
20#65342—51nomaximal
21C23×C52D411yescharacteristic
22C23×C52C2311yescharacteristic
23C22×C5×D5—21no—
24C22×C5×D5D412yes—
25C52×D4—22no—
26C2×C4×C52D411yescharacteristic
27C2×#1039—22no—
28C5×C22⋊D5—22no—
29C5×C22⋊D5—22no—
30C5×C22⋊D5—22no—
31C2×C5×Dic5—21no—
32C2×C5×Dic5—22no—
33C2×C5×Dic5—21no—
34C2×C5×Dic5—21no—
35C2×C5×Dic5D412yes—
36C5×D20—21no—
37C4×C5×D5—22no—
38C5×Dic10—21no—
39C5×H(5,𝔽2)—51no—
40#4144D511yescharacteristic
41C5×#123—51no—
42#4081—52no—
43#4081D511yescharacteristic, complemented
44#4075—52no—
45#3966—51no—
46#3966—51no—
47#3966—52no—
48#3966—102no—
49#3938—52no—
50C22×C52—21no—
51C22×C52—41no—
52C22×C52—41no—
53C22×C52C2×D412yes—
54C22×C52C2×D411yescharacteristic
55C2×C5×D5—22no—
56C2×C5×D5—42no—
57C2×C5×D5—81no—
58C4×C52—21no—
59#1039—42no—
60C5×Dic5—22no—
61C5×Dic5—41no—
62C5×Dic5—41no—
63C5×#41—102no—
64C5×#41—101no—
65C2×C5×D4—22no—
66C2×C5×D4—52no—
67C2×C5×D4—102no—
68C2×C5×D4—101no—
69C2×C5×D4C2×D511yescharacteristic
70C2×C5×D4C2×D511yescharacteristic
71C22×C4×C5—51no—
72C2×C22⋊D5—21no—
73C2×C22⋊D5C2×D512yes—
74C22×Dic5C2×D511yescharacteristic, complemented
75#711—22no—
76D4×D5—22no—
77#709—21no—
78C5×C4⋊C4—101no—
79C5×C22⋊C4—51no—
80C5×C22⋊C4—101no—
81C22⋊Dic5—21no—
82C22⋊Dic5—52no—
83C22⋊Dic5—52no—
84C22⋊Dic5—104no—
85C22⋊Dic5—102no—
86C22⋊Dic5C2×D511yescharacteristic, complemented
87#685—102no—
88#683—21no—
89#683—102no—
90#683—102no—
91#458—251noSylow
92C2×C52—21no—
93C2×C52—22no—
94C2×C52—41no—
95C2×C52#12011yescharacteristic
96C5×D5—42no—
97C23×C5—22no—
98C23×C5—22no—
99C23×C5—52no—
100C23×C5—101no—
101C23×C5C22⋊D511yescharacteristic
102C23×C5C22⋊D511yescharacteristic
103C23×C5C22×D511yescharacteristic
104C23×C5C22×D511yescharacteristic
105C22×D5—21no—
106C22×D5C22⋊D512yes—
107C5×Q8—101no—
108C5×D4—22no—
109C5×D4—22no—
110C5×D4—44no—
111C5×D4—101no—
112C5×D4—102no—
113C5×D4—102no—
114C5×D4—102no—
115C2×C4×C5—22no—
116C2×C4×C5—52no—
117C2×C4×C5—102no—
118C2×C4×C5—101no—
119C2×C4×C5—102no—
120C2×C4×C5—101no—
121C2×C4×C5—101no—
122C2×C4×C5C22⋊D511yescharacteristic
123C2×C4×C5C22⋊D511yescharacteristic
124C22⋊D5—22no—
125C22⋊D5—22no—
126C22⋊D5—22no—
127C2×Dic5—21no—
128C2×Dic5—22no—
129C2×Dic5—21no—
130C2×Dic5—21no—
131C2×Dic5—102no—
132C2×Dic5—102no—
133C2×Dic5—204no—
134C2×Dic5C22⋊D512yes—
135D20—21no—
136C4×D5—22no—
137Dic10—21no—
138H(5,𝔽2)—51no—
139#123—51no—
140#123—252no—
141#99—252no—
142#99—251no—
143C52#45811yescharacteristic, complemented, Sylow
144C22×C5—21no—
145C22×C5—21no—
146C22×C5—22no—
147C22×C5—24no—
148C22×C5—42no—
149C22×C5—41no—
150C22×C5—41no—
151C22×C5—41no—
152C22×C5—41no—
153C22×C5—82no—
154C22×C5—82no—
155C22×C5—102no—
156C22×C5—202no—
157C22×C5—401no—
158C22×C5D4×D512yes—
159C22×C5C2×C22⋊D511yescharacteristic
160C22×C5C2×C22⋊D511yescharacteristic
161C22×C5C2×C22⋊D512yes—
162C2×D5—22no—
163C2×D5—42no—
164C2×D5—81no—
165C4×C5—21no—
166C4×C5—21no—
167C4×C5—42no—
168C4×C5—102no—
169C4×C5—201no—
170C4×C5—201no—
171Dic5—22no—
172Dic5—41no—
173Dic5—41no—
174Dic5—202no—
175Dic5—202no—
176Dic5—404no—
177#41—102no—
178#41—101no—
179C2×D4—52no—
180C2×D4—102no—
181C2×D4—101no—
182C2×D4D5211yescharacteristic
183C22×C4—51no—
184C4⋊C4—101no—
185C4⋊C4—502no—
186C22⋊C4—51no—
187C22⋊C4—101no—
188C22⋊C4—252no—
189C22⋊C4—502no—
190C2×C5—21no—
191C2×C5—22no—
192C2×C5—22no—
193C2×C5—21no—
194C2×C5—22no—
195C2×C5—44no—
196C2×C5—42no—
197C2×C5—41no—
198C2×C5—41no—
199C2×C5—82no—
200C2×C5—202no—
201C2×C5#408311yescharacteristic
202C2×C5#409911yescharacteristic
203D5—42no—
204C23—52no—
205C23—101no—
206C23#603711yescharacteristic
207C23C2×D5211yescharacteristic
208Q8—101no—
209D4—22no—
210D4—101no—
211D4—102no—
212D4—102no—
213D4—102no—
214C2×C4—52no—
215C2×C4—102no—
216C2×C4—101no—
217C2×C4—101no—
218C2×C4—101no—
219C2×C4—102no—
220C2×C4—502no—
221C2×C4#603711yescharacteristic
222C5—22no—
223C5#6534211yescharacteristic, complemented
224C5#6553211yescharacteristic, complemented
225C22—21no—
226C22—41no—
227C22—41no—
228C22—102no—
229C22—202no—
230C22—401no—
231C22C2×#603711yescharacteristic
232C22D5×C22⋊D512yes—
233C4—21no—
234C4—102no—
235C4—201no—
236C4—201no—
237C4—1002no—
238C2—21no—
239C2—22no—
240C2—41no—
241C2—202no—
242C2#1171709411yescharacteristic
subgroupG/H
centerC2#11717094
commutator, cosocleC23×C52C23
FrattiniC23C2×D52
FittingC2×C52×D4C22
socleC2×C52#120
prime radicalC5×#4144C2
hypercenterC2×D4D52
hypocenterC52#458
LCSG ▷ C23×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C5×#4081C2semidirect
3. #11717079C2semidirect
4. #11716701C2semidirect
5. #11716621C2semidirect
9. C22×C5×Dic5C22semidirect
13. C22⋊#1039C22semidirect
15. C5×C22⋊Dic5C22semidirect
43. #4081D5semidirect
74. C22×Dic5C2×D5semidirect
86. C22⋊Dic5C2×D5semidirect
143. C52#458semidirect
223. C5#65342semidirect
224. C5#65532semidirect

Maximal subgroups

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

subgroupindex
C5×#41442
C5×#40812
#117170792
#117167012
#117166212
#655325
#653425