Finite Group Databasebeta

← #13106100

Subgroups — #13106100

Order 1600.

550 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 128 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, complemented
2C4×C5×C22⋊D5C212yesmaximal, complemented
3#11717097C211yesmaximal, characteristic
4#11717093C211yesmaximal, characteristic, complemented
5#11717077C212yesmaximal, complemented
6#11717023C212yesmaximal, complemented
7#11717022C212yesmaximal, complemented
8#11716970C212yesmaximal, complemented
9#11716955C212yesmaximal, complemented
10C22×C4×C52C2211yescharacteristic
11C2×C22⋊(C5⋊D5)C2211yescharacteristic
12C2×C5×C22⋊D5C2212yes—
13C2×C4×C5×D5C2212yes—
14C2×#6039C2212yes—
15C2×#6038C2212yes—
16C2×#6037C2211yescharacteristic
17C2×#6036C2211yescharacteristic
18C22⋊#1039C2211yescharacteristic
19#87808C2212yes—
20#87806C2212yes—
21C5×C22⋊Dic5C2212yes—
22C5×#685C2212yes—
23C5×#683C2212yes—
24C4×C5×Dic5C2212yes—
25#87780C2212yes—
26#87779C2211yescharacteristic
27#87778C2212yes—
28#87778C2212yes—
29Dic52C2211yescharacteristic
30Dic52C2212yes—
31#65893—52nomaximal
32C23×C52C2311yescharacteristic
33C22×C5⋊D5C2311yescharacteristic
34C22×C5×D5C2312yes—
35C2×C4×C52—41no—
36C2×C4×C52C2312yes—
37C22⋊(C5⋊D5)—41no—
38C2×#1039C2312yes—
39C2×#1039C2311yescharacteristic
40C5×C22⋊D5—24no—
41C2×C5×Dic5C2312yes—
42C2×C5×Dic5C2312yes—
43C2×C5×Dic5C2312yes—
44C4×C5×D5—42no—
45#6039—24no—
46#6038—24no—
47#6037—41no—
48#6036—41no—
49C4×C5×D4—52no—
50#4086—52no—
51#4082—52no—
52#4075—52no—
53#4075—52no—
54#4075—102no—
55C4×C22⋊D5—52no—
56C4×C22⋊D5D512yescomplemented
57#4039—52no—
58#4033—52no—
59#4031—52no—
60#4031—52no—
61#4029—52no—
62#4029—52no—
63#4021—52no—
64C4×Dic10—52no—
65C22×C52—22no—
66C22×C52—21no—
67C22×C52C2411yescharacteristic
68C2×C5⋊D5—22no—
69C2×C5⋊D5—21no—
70C2×C5×D5—22no—
71C2×C5×D5—22no—
72C2×C5×D5—22no—
73C4×C52—22no—
74#1039—22no—
75#1039—21no—
76C5×Dic5—22no—
77C5×Dic5—22no—
78C5×Dic5#4114yes—
79C2×C5×D4—52no—
80C22×C4×C5—22no—
81C22×C4×C5—52no—
82C22×C4×C5C2×D512yes—
83C2×C22⋊D5—52no—
84C2×C22⋊D5—52no—
85C2×C22⋊D5—52no—
86C2×C22⋊D5—102no—
87C2×C22⋊D5C2×D512yes—
88C2×D20—52no—
89C2×C4×D5—52no—
90C2×C4×D5C2×D512yes—
91C2×Dic10—52no—
92C2×Dic10—52no—
93C5×C4⋊C4—52no—
94C5×C22⋊C4—52no—
95C5×C22⋊C4—52no—
96C42×C5—52no—
97C22⋊Dic5—52no—
98C22⋊Dic5—52no—
99C22⋊Dic5—52no—
100C22⋊Dic5—102no—
101C22⋊Dic5C2×D512yes—
102#685—52no—
103#685—52no—
104#685—52no—
105#685—52no—
106#685—52no—
107#685—104no—
108#685C2×D512yes—
109C4⋊Dic5—52no—
110#683—52no—
111#683—52no—
112#683—52no—
113#683—104no—
114#683C2×D512yes—
115C4×Dic5—52no—
116C4×Dic5—52no—
117C4×Dic5—52no—
118C4×Dic5C2×D512yes—
119#559—251noSylow
120C2×C52—41no—
121C2×C52C2×#4112yes—
122C2×C52H(5,𝔽2)11yescharacteristic
123C5⋊D5—41no—
124C5×D5—42no—
125C23×C5—22no—
126C23×C5—52no—
127C23×C5C22×D512yes—
128C22×D5—52no—
129C22×D5—102no—
130C22×D5C22×D512yes—
131C5×D4—104no—
132C2×C4×C5—24no—
133C2×C4×C5—42no—
134C2×C4×C5—44no—
135C2×C4×C5—52no—
136C2×C4×C5—52no—
137C2×C4×C5—52no—
138C2×C4×C5—202no—
139C2×C4×C5C22×D512yes—
140C2×C4×C5C22×D512yes—
141C22⋊D5—24no—
142C22⋊D5—104no—
143C22⋊D5—202no—
144C22⋊D5—202no—
145C22⋊D5—204no—
146C2×Dic5—52no—
147C2×Dic5—52no—
148C2×Dic5—52no—
149C2×Dic5—102no—
150C2×Dic5—104no—
151C2×Dic5C22×D512yes—
152C2×Dic5C22×D512yes—
153C2×Dic5C22×D512yes—
154D20—104no—
155C4×D5—42no—
156C4×D5—202no—
157Dic10—104no—
158Dic10—104no—
159#127—251no—
160#124—252no—
161#124—252no—
162#123—251no—
163#123—252no—
164#123—251no—
165C4×Q8—252no—
166C4×D4—52no—
167C4×D4—252no—
168C52#55911yescharacteristic, complemented, Sylow
169C22×C5—24no—
170C22×C5—22no—
171C22×C5—22no—
172C22×C5—22no—
173C22×C5—22no—
174C22×C5—44no—
175C22×C5—102no—
176C22×C5—102no—
177C22×C5—102no—
178C22×C5C23×D512yes—
179C2×D5—22no—
180C2×D5—22no—
181C2×D5—22no—
182C2×D5—102no—
183C2×D5—102no—
184C2×D5—102no—
185C2×D5—202no—
186C2×D5—204no—
187C4×C5—22no—
188C4×C5—22no—
189C4×C5—44no—
190C4×C5—54no—
191C4×C5—102no—
192C4×C5—102no—
193Dic5—22no—
194Dic5—22no—
195Dic5—102no—
196Dic5—102no—
197Dic5—102no—
198Dic5—204no—
199Dic5—202no—
200Dic5#70914yes—
201C2×Q8—252no—
202C2×D4—52no—
203C2×D4—252no—
204C2×D4—251no—
205C2×D4—251no—
206C22×C4—52no—
207C22×C4—251no—
208C22×C4D5211yescharacteristic
209C4⋊C4—52no—
210C4⋊C4—252no—
211C4⋊C4—252no—
212C22⋊C4—52no—
213C22⋊C4—52no—
214C22⋊C4—251no—
215C22⋊C4—252no—
216C22⋊C4—252no—
217C22⋊C4—251no—
218C22⋊C4—252no—
219C42—52no—
220C42—252no—
221C42—251no—
222C2×C5—24no—
223C2×C5—22no—
224C2×C5—44no—
225C2×C5—42no—
226C2×C5—202no—
227C2×C5C2×#70912yes—
228C2×C5#414412yes—
229C2×C5D5×#4112yes—
230D5—42no—
231D5—202no—
232D5—402no—
233C23—52no—
234C23—251no—
235C23C2×D5211yescharacteristic
236Q8—504no—
237D4—104no—
238D4—504no—
239D4—1001no—
240D4—1001no—
241C2×C4—41no—
242C2×C4—52no—
243C2×C4—52no—
244C2×C4—52no—
245C2×C4—202no—
246C2×C4—252no—
247C2×C4—251no—
248C2×C4—1001no—
249C2×C4C2×D5212yes—
250C5—22no—
251C5#6589312yescomplemented
252C22—22no—
253C22—21no—
254C22—102no—
255C22—102no—
256C22—102no—
257C22—502no—
258C22—501no—
259C22C22×D5211yescharacteristic
260C4—22no—
261C4—54no—
262C4—102no—
263C4—102no—
264C4—502no—
265C4—501no—
266C2—41no—
267C2—202no—
268C2—1001no—
269C2D5×#70912yes—
270C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC22×C4×C52C22
prime radical#11717097C2
hypercenterC22×C4D52
hypocenterC52#559
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11717249C2semidirect
2. C4×C5×C22⋊D5C2semidirect
4. #11717093C2semidirect
5. #11717077C2semidirect
6. #11717023C2semidirect
7. #11717022C2semidirect
8. #11716970C2semidirect
9. #11716955C2semidirect
56. C4×C22⋊D5D5semidirect
168. C52#559semidirect
251. C5#65893semidirect

Maximal subgroups

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

subgroupindex
#117172492
C4×C5×C22⋊D52
#117170972
#117170932
#117170772
#117170232
#117170222
#117169702
#117169552
#658935