Finite Group Databasebeta

← #13106489

Subgroups — #13106489

Order 1600.

510 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 122 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#11717266C211yesmaximal, characteristic, complemented
2C5×#4092C212yesmaximal, complemented
3#11717030C211yesmaximal, characteristic
4#11717023C212yesmaximal, complemented
5#11717022C212yesmaximal, complemented
6#11717001C211yesmaximal, characteristic, complemented
7#11716977C212yesmaximal, complemented
8#11716972C212yesmaximal, complemented
9#11716965C212yesmaximal, complemented
10C2×C52×Q8C2211yescharacteristic
11C2×C4×C5⋊D5C2211yescharacteristic
12C2×C4×C5×D5C2212yes—
13C2×#6038C2212yes—
14C2×#6037C2211yescharacteristic
15C2×#6036C2211yescharacteristic
16C2×#6036C2211yescharacteristic
17#87808C2212yes—
18C4⋊#1039C2211yescharacteristic
19#87806C2212yes—
20C5×#685C2212yes—
21C5×#685C2212yes—
22C5×C4⋊Dic5C2212yes—
23C5×#683C2212yes—
24C5×#683C2212yes—
25#87781C2211yescharacteristic
26#87781C2212yes—
27#87780C2212yes—
28#87778C2212yes—
29#87778C2212yes—
30Dic52C2212yes—
31#65972—52nomaximal
32C22×C5⋊D5C2311yescharacteristic
33C22×C5×D5C2312yes—
34C52×Q8—41no—
35C2×C4×C52C2312yes—
36C2×C4×C52C2311yescharacteristic
37C2×#1039C2312yes—
38C2×#1039C2311yescharacteristic
39C4×C5⋊D5—41no—
40C2×C5×Dic5C2312yes—
41C2×C5×Dic5C2312yes—
42C2×C5×Dic5C2312yes—
43C4×C5×D5—42no—
44#6038—24no—
45#6037—41no—
46#6036—22no—
47#6036—41no—
48C5×C22⋊Q8—52no—
49#4092—52no—
50#4092—102no—
51#4092D512yescomplemented
52#4075—52no—
53C4×C22⋊D5—52no—
54#4044—52no—
55#4040—52no—
56#4039—52no—
57D5×C4⋊C4—52no—
58#4035—52no—
59#4035—52no—
60#4030—52no—
61#4029—52no—
62#4029—52no—
63#4025—52no—
64C22×C52C2411yescharacteristic
65C2×C5⋊D5—21no—
66C2×C5⋊D5—21no—
67C2×C5⋊D5#4112yes—
68C2×C5×D5—22no—
69C2×C5×D5—22no—
70C2×C5×D5—22no—
71C4×C52—21no—
72C4×C52—22no—
73#1039—21no—
74#1039—22no—
75C5×Dic5—22no—
76C5×Dic5—22no—
77C5×Dic5—22no—
78C2×C5×Q8—22no—
79C2×C5×Q8C2×D512yes—
80C22×C4×C5—52no—
81C2×C22⋊D5—52no—
82C2×C22⋊D5—52no—
83C2×D20—52no—
84C2×C4×D5—52no—
85C2×C4×D5—52no—
86C2×C4×D5—52no—
87C2×C4×D5—102no—
88C2×C4×D5C2×D512yes—
89C5×C4⋊C4—52no—
90C5×C4⋊C4—52no—
91C5×C4⋊C4—52no—
92C5×C22⋊C4—52no—
93C5×C22⋊C4—52no—
94C22⋊Dic5—52no—
95C22⋊Dic5—52no—
96#685—52no—
97#685—52no—
98#685—52no—
99#685—52no—
100#685—104no—
101#685C2×D512yes—
102#685C2×D512yes—
103C4⋊Dic5—52no—
104C4⋊Dic5—52no—
105C4⋊Dic5—102no—
106C4⋊Dic5C2×D512yes—
107#683—52no—
108#683—52no—
109#683—52no—
110#683—52no—
111#683—52no—
112#683—52no—
113#683—104no—
114#683C2×D512yes—
115#683C2×D512yes—
116C4×Dic5—52no—
117C4×Dic5—52no—
118#539—251noSylow
119C2×C52C2×#4111yescharacteristic
120C2×C52H(5,𝔽2)11yescharacteristic
121C2×C522−1+411yescharacteristic
122C5⋊D5—22no—
123C5×D5—42no—
124C23×C5—52no—
125C22×D5—52no—
126C22×D5—102no—
127C22×D5C22×D512yes—
128C5×Q8—42no—
129C5×Q8—44no—
130C2×C4×C5—22no—
131C2×C4×C5—24no—
132C2×C4×C5—52no—
133C2×C4×C5—52no—
134C2×C4×C5—52no—
135C2×C4×C5—202no—
136C2×C4×C5C22×D512yes—
137C2×C4×C5C22×D512yes—
138C2×C4×C5C22×D512yes—
139C22⋊D5—104no—
140C22⋊D5—202no—
141C2×Dic5—52no—
142C2×Dic5—52no—
143C2×Dic5—52no—
144C2×Dic5—102no—
145C2×Dic5—104no—
146C2×Dic5C22×D512yes—
147C2×Dic5C22×D512yes—
148C2×Dic5C22×D512yes—
149D20—104no—
150C4×D5—42no—
151C4×D5—104no—
152C4×D5—202no—
153C4×D5—202no—
154C4×D5—204no—
155#126—252no—
156#125—252no—
157#123—252no—
158#123—252no—
159C22⋊Q8—52no—
160C22⋊Q8—251no—
161#121—251no—
162C4×D4—252no—
163C2×C4⋊C4—251no—
164C52#53911yescharacteristic, complemented, Sylow
165C22×C5—22no—
166C22×C5—102no—
167C22×C5—102no—
168C22×C5—102no—
169C22×C5C23×D512yes—
170C2×D5—22no—
171C2×D5—22no—
172C2×D5—22no—
173C2×D5—54no—
174C2×D5—104no—
175C2×D5—102no—
176C2×D5—102no—
177C2×D5—202no—
178C2×D5—202no—
179C4×C5—24no—
180C4×C5—22no—
181C4×C5—22no—
182C4×C5—22no—
183C4×C5—44no—
184C4×C5—102no—
185C4×C5—102no—
186C4×C5—102no—
187Dic5—22no—
188Dic5—22no—
189Dic5—22no—
190Dic5—102no—
191Dic5—102no—
192Dic5—102no—
193Dic5—202no—
194Dic5—204no—
195C2×Q8D5211yescharacteristic
196C2×D4—252no—
197C2×D4—251no—
198C22×C4—52no—
199C22×C4—251no—
200C22×C4—251no—
201C22×C4—251no—
202C4⋊C4—52no—
203C4⋊C4—52no—
204C4⋊C4—52no—
205C4⋊C4—252no—
206C4⋊C4—251no—
207C4⋊C4—252no—
208C4⋊C4—252no—
209C4⋊C4—251no—
210C22⋊C4—52no—
211C22⋊C4—52no—
212C22⋊C4—252no—
213C22⋊C4—252no—
214C22⋊C4—252no—
215C42—252no—
216C2×C5—22no—
217C2×C5—22no—
218C2×C5—22no—
219C2×C5—202no—
220C2×C5#414412yes—
221C2×C5#414712yes—
222C2×C5D5×#4112yes—
223D5—42no—
224D5—104no—
225D5—204no—
226C23—52no—
227C23—251no—
228Q8—41no—
229D4—504no—
230D4—1001no—
231C2×C4—52no—
232C2×C4—52no—
233C2×C4—52no—
234C2×C4—202no—
235C2×C4—252no—
236C2×C4—251no—
237C2×C4—502no—
238C2×C4—1001no—
239C2×C4—1001no—
240C2×C4C2×D5211yescharacteristic
241C2×C4C2×D5212yes—
242C5—22no—
243C5#6597212yescomplemented
244C22—102no—
245C22—102no—
246C22—102no—
247C22—252no—
248C22—501no—
249C22—501no—
250C22C22×D5211yescharacteristic
251C4—22no—
252C4—21no—
253C4—102no—
254C4—102no—
255C4—102no—
256C4—502no—
257C4—501no—
258C2—202no—
259C2—502no—
260C2#1171748111yescharacteristic
261C2#1171748411yescharacteristic
262C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×Q8C22
prime radical#11717030C2
hypercenterC2×Q8D52
hypocenterC52#539
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11717266C2semidirect
2. C5×#4092C2semidirect
4. #11717023C2semidirect
5. #11717022C2semidirect
6. #11717001C2semidirect
7. #11716977C2semidirect
8. #11716972C2semidirect
9. #11716965C2semidirect
51. #4092D5semidirect
164. C52#539semidirect
243. C5#65972semidirect

Maximal subgroups

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

subgroupindex
#117172662
C5×#40922
#117170302
#117170232
#117170222
#117170012
#117169772
#117169722
#117169652
#659725