Finite Group Databasebeta

← #13106041

Subgroups — #13106041

Order 1600.

470 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 120 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
1#11717249C211yesmaximal, characteristic
2C5×#4070C212yesmaximal, complemented
3#11717093C211yesmaximal, characteristic
4#11717074C212yesmaximal, complemented
5#11717071C211yesmaximal, characteristic
6#11717008C212yesmaximal, complemented
7#11716966C212yesmaximal, complemented
8#11716964C212yesmaximal, complemented
9#11716957C212yesmaximal, complemented
10C22×C4×C52C2211yescharacteristic
11C2×C22⋊(C5⋊D5)C2211yescharacteristic
12C2×C5×Dic10C2212yes—
13C2×#6039C2211yescharacteristic
14C2×#6036C2211yescharacteristic
15C22⋊#1039C2211yescharacteristic
16#87808C2212yes—
17#87806C2212yes—
18C5×C22⋊Dic5C2212yes—
19C5×C22⋊Dic5C2212yes—
20C5×C4⋊Dic5C2212yes—
21C5×#683C2212yes—
22C5×#683C2212yes—
23#87781C2212yes—
24#87780C2212yes—
25#87780C2212yes—
26#87780C2212yes—
27#87779C2211yescharacteristic
28#87779C2212yes—
29Dic52C2212yes—
30Dic52C2211yescharacteristic
31#65980—52nomaximal
32C23×C52C2311yescharacteristic
33C22×C5⋊D5C2311yescharacteristic
34C2×C4×C52—41no—
35C2×C4×C52C2312yes—
36C22⋊(C5⋊D5)—41no—
37C2×#1039C2312yes—
38C2×#1039C2311yescharacteristic
39C2×C5×Dic5C2312yes—
40C2×C5×Dic5C2312yes—
41C2×C5×Dic5C2312yes—
42C2×C5×Dic5C2312yes—
43C5×Dic10—42no—
44#6039—41no—
45#6036—41no—
46C5×C22⋊Q8—52no—
47#4092—52no—
48#4090—52no—
49#4075—52no—
50#4075—102no—
51#4070D512yescomplemented
52#4044—52no—
53#4044—52no—
54#4042—52no—
55#4036—52no—
56#4035—52no—
57#4034—52no—
58#4031—52no—
59#4029—52no—
60#4025—52no—
61#4025—52no—
62C22×C52—21no—
63C22×C52—22no—
64C22×C52C2411yescharacteristic
65C2×C5⋊D5—21no—
66C2×C5⋊D5—22no—
67C4×C52—22no—
68#1039—21no—
69#1039—22no—
70C5×Dic5—22no—
71C5×Dic5—22no—
72C5×Dic5—22no—
73C5×Dic5—22no—
74C2×C5×Q8—52no—
75C22×C4×C5—22no—
76C22×C4×C5C2×D512yes—
77C2×C22⋊D5—52no—
78C2×C22⋊D5—102no—
79C2×C4×D5—52no—
80C2×Dic10—52no—
81C2×Dic10C2×D512yes—
82C5×C4⋊C4—52no—
83C5×C4⋊C4—52no—
84C5×C4⋊C4—52no—
85C5×C22⋊C4—52no—
86C5×C22⋊C4—52no—
87C22⋊Dic5—52no—
88C22⋊Dic5—102no—
89C22⋊Dic5C2×D512yes—
90C22⋊Dic5C2×D512yes—
91#685—52no—
92#685—52no—
93#685—52no—
94#685—52no—
95#685—52no—
96#685—104no—
97C4⋊Dic5—52no—
98C4⋊Dic5—52no—
99C4⋊Dic5—52no—
100C4⋊Dic5C2×D512yes—
101#683—52no—
102#683—52no—
103#683—52no—
104#683—52no—
105#683—52no—
106#683—52no—
107#683—52no—
108#683—104no—
109#683C2×D512yes—
110#683C2×D512yes—
111C4×Dic5—52no—
112C4×Dic5—52no—
113C4×Dic5—52no—
114#563—251noSylow
115C2×C52—41no—
116C2×C52H(5,𝔽2)11yescharacteristic
117C2×C522−1+412yes—
118C5⋊D5—41no—
119C23×C5—22no—
120C23×C5C22×D512yes—
121C22×D5—52no—
122C22×D5—102no—
123C5×Q8—202no—
124C2×C4×C5—24no—
125C2×C4×C5—42no—
126C2×C4×C5—44no—
127C2×C4×C5—52no—
128C2×C4×C5—52no—
129C2×C4×C5—52no—
130C2×C4×C5—52no—
131C2×C4×C5C22×D512yes—
132C2×C4×C5C22×D512yes—
133C22⋊D5—202no—
134C22⋊D5—204no—
135C2×Dic5—52no—
136C2×Dic5—52no—
137C2×Dic5—52no—
138C2×Dic5—102no—
139C2×Dic5—104no—
140C2×Dic5C22×D512yes—
141C2×Dic5C22×D512yes—
142C2×Dic5C22×D512yes—
143C2×Dic5C22×D512yes—
144C4×D5—202no—
145Dic10—42no—
146Dic10—202no—
147C4⋊Q8—252no—
148#126—252no—
149#126—252no—
150#125—252no—
151#124—251no—
152#123—251no—
153#123—251no—
154C22⋊Q8—52no—
155C22⋊Q8—252no—
156C52#56311yescharacteristic, complemented, Sylow
157C22×C5—24no—
158C22×C5—22no—
159C22×C5—22no—
160C22×C5—22no—
161C22×C5—22no—
162C22×C5—44no—
163C22×C5C23×D512yes—
164C2×D5—102no—
165C2×D5—102no—
166C2×D5—102no—
167C2×D5—202no—
168C2×D5—204no—
169C4×C5—22no—
170C4×C5—22no—
171C4×C5—44no—
172C4×C5—102no—
173C4×C5—102no—
174C4×C5—102no—
175C4×C5—102no—
176Dic5—22no—
177Dic5—22no—
178Dic5—22no—
179Dic5—22no—
180Dic5—102no—
181Dic5—102no—
182Dic5—102no—
183Dic5—204no—
184Dic5—202no—
185C2×Q8—52no—
186C2×Q8—251no—
187C2×D4—251no—
188C22×C4—251no—
189C22×C4D5211yescharacteristic
190C4⋊C4—52no—
191C4⋊C4—52no—
192C4⋊C4—52no—
193C4⋊C4—252no—
194C4⋊C4—252no—
195C4⋊C4—252no—
196C4⋊C4—252no—
197C4⋊C4—252no—
198C22⋊C4—52no—
199C22⋊C4—52no—
200C22⋊C4—252no—
201C22⋊C4—251no—
202C22⋊C4—251no—
203C22⋊C4—252no—
204C42—251no—
205C42—252no—
206C2×C5—24no—
207C2×C5—22no—
208C2×C5—44no—
209C2×C5—42no—
210C2×C5#414412yes—
211C2×C5#414712yes—
212C2×C5#415012yes—
213D5—202no—
214D5—402no—
215C23—251no—
216C23C2×D5211yescharacteristic
217Q8—202no—
218Q8—1001no—
219D4—1001no—
220C2×C4—41no—
221C2×C4—52no—
222C2×C4—52no—
223C2×C4—52no—
224C2×C4—52no—
225C2×C4—252no—
226C2×C4—251no—
227C2×C4—1001no—
228C2×C4C2×D5212yes—
229C5—22no—
230C5#6598012yescomplemented
231C22—22no—
232C22—21no—
233C22—502no—
234C22—501no—
235C22C22×D5211yescharacteristic
236C4—22no—
237C4—102no—
238C4—102no—
239C4—102no—
240C4—102no—
241C4—502no—
242C4—501no—
243C2—41no—
244C2—1001no—
245C2#1171746212yes—
246C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC22×C4×C52C22
prime radicalC2×C22⋊(C5⋊D5)C22
hypercenterC22×C4D52
hypocenterC52#563
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C5×#4070C2semidirect
4. #11717074C2semidirect
6. #11717008C2semidirect
7. #11716966C2semidirect
8. #11716964C2semidirect
9. #11716957C2semidirect
51. #4070D5semidirect
156. C52#563semidirect
230. C5#65980semidirect

Maximal subgroups

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

subgroupindex
#117172492
C5×#40702
#117170932
#117170742
#117170712
#117170082
#117169662
#117169642
#117169572
#659805