Finite Group Databasebeta

← #13105206

Subgroups — #13105206

Order 1600.

522 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 124 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#11717201C211yesmaximal, characteristic
2C5×#4025C212yesmaximal, complemented
3#11717100C211yesmaximal, characteristic, complemented
4#11717093C212yesmaximal
5#11717075C211yesmaximal, characteristic, complemented
6C4×#6039C211yesmaximal, characteristic, complemented
7#11717018C211yesmaximal, characteristic, complemented
8#11716989C211yesmaximal, characteristic, complemented
9#11716984C211yesmaximal, characteristic, complemented
10#11716964C212yesmaximal, complemented
11#11716954C212yesmaximal, complemented
12C2×C22⋊(C5⋊D5)C2211yescharacteristic
13C22×#1039C2211yescharacteristic
14C2×C4×C5⋊D5C2211yescharacteristic
15C2×#6039C2211yescharacteristic
16C2×#6036C2212yes—
17C52×C22⋊C4C2211yescharacteristic
18#87808C2211yescharacteristic
19#87806C2211yescharacteristic
20C4×#1039C2211yescharacteristic
21C5×C22⋊Dic5C2212yes—
22C5×C22⋊Dic5C2212yes—
23C5×C4⋊Dic5C2212yes—
24C5×#683C2212yes—
25C5×#683C2212yes—
26C4×C5×Dic5C2212yes—
27#87781C2211yescharacteristic
28#87781C2211yescharacteristic
29#87781C2211yescharacteristic
30#87780C2212yes—
31#87780C2212yes—
32#87780C2212yes—
33#87779C2212yes—
34Dic52C2212yes—
35#66042—52nomaximal
36C23×C52C2311yescharacteristic
37C22×C5⋊D5C2311yescharacteristic
38C2×C4×C52C2311yescharacteristic
39C2×C4×C52C2311yescharacteristic
40C22⋊(C5⋊D5)—22no—
41C2×#1039—22no—
42C2×#1039C2311yescharacteristic
43C2×#1039C2311yescharacteristic
44C2×#1039C2311yescharacteristic
45C4×C5⋊D5—22no—
46C2×C5×Dic5C2312yes—
47C2×C5×Dic5C2312yes—
48C2×C5×Dic5C2312yes—
49C2×C5×Dic5C2312yes—
50#6039—22no—
51#6036—24no—
52C5×#126—52no—
53#4042—52no—
54#4038—52no—
55D5×C4⋊C4—52no—
56#4036—52no—
57#4035—52no—
58#4035—52no—
59#4029—52no—
60#4029—52no—
61#4027—52no—
62#4027—102no—
63#4025D512yescomplemented
64#4024—52no—
65#4023—52no—
66#4018—52no—
67C4×Dic10—52no—
68C22×C52—21no—
69C22×C52—21no—
70C22×C52—21no—
71C22×C52C2411yescharacteristic
72C2×C5⋊D5—21no—
73C2×C5⋊D5—21no—
74C2×C5⋊D5#4112yes—
75C4×C52—21no—
76C4×C52—21no—
77#1039—21no—
78#1039—21no—
79#1039#4112yes—
80C5×Dic5—22no—
81C5×Dic5—22no—
82C5×Dic5—22no—
83C5×Dic5—22no—
84C2×C22⋊D5—52no—
85C2×C22⋊D5—102no—
86C22×Dic5—52no—
87C22×Dic5—102no—
88C2×C4×D5—52no—
89C2×C4×D5—52no—
90C2×C4×D5—52no—
91C2×C4×D5—102no—
92C2×Dic10—52no—
93C5×C4⋊C4—52no—
94C5×C4⋊C4—52no—
95C5×C4⋊C4—52no—
96C5×C22⋊C4—22no—
97C5×C22⋊C4—52no—
98C5×C22⋊C4—52no—
99C5×C22⋊C4C2×D512yes—
100C42×C5—52no—
101C22⋊Dic5C2×D512yes—
102C22⋊Dic5C2×D512yes—
103#685—52no—
104#685—52no—
105#685—52no—
106#685—102no—
107C4⋊Dic5—52no—
108C4⋊Dic5—52no—
109C4⋊Dic5—52no—
110C4⋊Dic5C2×D512yes—
111#683—52no—
112#683—52no—
113#683—52no—
114#683—52no—
115#683—52no—
116#683—52no—
117#683—52no—
118#683—102no—
119#683C2×D512yes—
120#683C2×D512yes—
121C4×Dic5—52no—
122C4×Dic5—52no—
123C4×Dic5—52no—
124C4×Dic5—102no—
125C4×Dic5C2×D512yes—
126#552—251noSylow
127C2×C52—41no—
128C2×C52C2×#4111yescharacteristic
129C2×C52C2×#4111yescharacteristic
130C2×C522−1+411yescharacteristic
131C5⋊D5—22no—
132C23×C5—22no—
133C23×C5C22×D512yes—
134C22×D5—52no—
135C22×D5—102no—
136C2×C4×C5—22no—
137C2×C4×C5—22no—
138C2×C4×C5—52no—
139C2×C4×C5—52no—
140C2×C4×C5—52no—
141C2×C4×C5—52no—
142C2×C4×C5C22×D512yes—
143C2×C4×C5C22×D512yes—
144C22⋊D5—104no—
145C22⋊D5—108no—
146C2×Dic5—52no—
147C2×Dic5—52no—
148C2×Dic5—52no—
149C2×Dic5—102no—
150C2×Dic5—102no—
151C2×Dic5—102no—
152C2×Dic5—104no—
153C2×Dic5—204no—
154C2×Dic5C22×D512yes—
155C2×Dic5C22×D512yes—
156C2×Dic5C22×D512yes—
157C2×Dic5C22×D512yes—
158C4×D5—104no—
159C4×D5—104no—
160C4×D5—104no—
161C4×D5—204no—
162Dic10—104no—
163#126—52no—
164#125—251no—
165#125—252no—
166#123—252no—
167C22⋊Q8—251no—
168C22⋊Q8—251no—
169C4×Q8—251no—
170C4×D4—251no—
171#117—252no—
172#117—251no—
173C2×C4⋊C4—251no—
174C52#55211yescharacteristic, complemented, Sylow
175C22×C5—24no—
176C22×C5—22no—
177C22×C5—22no—
178C22×C5—22no—
179C22×C5—22no—
180C22×C5—42no—
181C22×C5—42no—
182C22×C5C23×D512yes—
183C2×D5—54no—
184C2×D5—104no—
185C2×D5—102no—
186C2×D5—102no—
187C2×D5—202no—
188C2×D5—202no—
189C4×C5—22no—
190C4×C5—22no—
191C4×C5—42no—
192C4×C5—42no—
193C4×C5—102no—
194C4×C5—102no—
195C4×C5—102no—
196C4×C5—102no—
197Dic5—22no—
198Dic5—22no—
199Dic5—22no—
200Dic5—22no—
201Dic5—54no—
202Dic5—104no—
203Dic5—104no—
204Dic5—102no—
205Dic5—102no—
206Dic5—202no—
207C2×Q8—251no—
208C2×D4—251no—
209C22×C4—252no—
210C22×C4—251no—
211C22×C4—251no—
212C4⋊C4—52no—
213C4⋊C4—52no—
214C4⋊C4—52no—
215C4⋊C4—251no—
216C4⋊C4—251no—
217C4⋊C4—251no—
218C4⋊C4—252no—
219C4⋊C4—252no—
220C4⋊C4—252no—
221C4⋊C4—251no—
222C22⋊C4—52no—
223C22⋊C4—52no—
224C22⋊C4—252no—
225C22⋊C4—251no—
226C22⋊C4D5211yescharacteristic
227C42—52no—
228C42—252no—
229C42—251no—
230C2×C5—22no—
231C2×C5—22no—
232C2×C5—22no—
233C2×C5—44no—
234C2×C5—42no—
235C2×C5D5×#4112yes—
236C2×C5D5×#4112yes—
237C2×C5#415012yes—
238D5—104no—
239D5—204no—
240C23—251no—
241C23C2×D5211yescharacteristic
242Q8—502no—
243D4—502no—
244C2×C4—52no—
245C2×C4—52no—
246C2×C4—52no—
247C2×C4—52no—
248C2×C4—251no—
249C2×C4—251no—
250C2×C4—251no—
251C2×C4—502no—
252C2×C4—504no—
253C2×C4—502no—
254C2×C4C2×D5211yescharacteristic
255C2×C4C2×D5211yescharacteristic
256C5—22no—
257C5#6604212yescomplemented
258C22—21no—
259C22—21no—
260C22—21no—
261C22—252no—
262C22—501no—
263C22—501no—
264C22C22×D5211yescharacteristic
265C4—21no—
266C4—21no—
267C4—102no—
268C4—102no—
269C4—102no—
270C4—102no—
271C4—252no—
272C4—501no—
273C4—501no—
274C2—41no—
275C2—502no—
276C2#1171747111yescharacteristic
277C2#1171747411yescharacteristic
278C2#1171747711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
prime radicalC2×C22⋊(C5⋊D5)C22
hypercenterC22⋊C4D52
hypocenterC52#552
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#4025C2semidirect
3. #11717100C2semidirect
5. #11717075C2semidirect
6. C4×#6039C2semidirect
7. #11717018C2semidirect
8. #11716989C2semidirect
9. #11716984C2semidirect
10. #11716964C2semidirect
11. #11716954C2semidirect
63. #4025D5semidirect
174. C52#552semidirect
257. C5#66042semidirect

Maximal subgroups

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

subgroupindex
#117172012
C5×#40252
#117171002
#117170932
#117170752
C4×#60392
#117170182
#117169892
#117169842
#117169642
#117169542
#660425