Finite Group Databasebeta

← #13106430

Subgroups — #13106430

Order 1600.

486 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 134 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
2C5×#4090C212yesmaximal, complemented
3#11717008C212yesmaximal
4#11717001C211yesmaximal, characteristic
5#11716986C212yesmaximal, complemented
6#11716984C211yesmaximal, characteristic
7#11716963C212yesmaximal, complemented
8#11716957C212yesmaximal, complemented
9#11716954C212yesmaximal
10C2×C52×Q8C2211yescharacteristic
11C2×C4×C5⋊D5C2211yescharacteristic
12C2×C5×Dic10C2212yes—
13C2×#6039C2211yescharacteristic
14C2×#6036C2211yescharacteristic
15C2×#6036C2211yescharacteristic
16#87808C2212yes—
17C4⋊#1039C2211yescharacteristic
18#87806C2212yes—
19C5×#683C2212yes—
20C5×#683C2212yes—
21C5×#683C2212yes—
22C5×#683C2212yes—
23C4×C5×Dic5C2212yes—
24#87781C2212yes—
25#87781C2211yescharacteristic
26#87780C2212yes—
27#87780C2212yes—
28#87780C2212yes—
29#87779C2212yes—
30Dic52C2212yes—
31#66063—52nomaximal
32C22×C5⋊D5C2311yescharacteristic
33C52×Q8—41no—
34C2×C4×C52C2312yes—
35C2×C4×C52C2311yescharacteristic
36C2×#1039C2312yes—
37C2×#1039C2311yescharacteristic
38C4×C5⋊D5—41no—
39C2×C5×Dic5C2312yes—
40C2×C5×Dic5C2312yes—
41C2×C5×Dic5C2312yes—
42C2×C5×Dic5C2312yes—
43C5×Dic10—42no—
44#6039—41no—
45#6036—41no—
46#6036Q814yescomplemented
47C5×C4⋊Q8—52no—
48#4092—52no—
49#4092—52no—
50#4092—102no—
51#4090—52no—
52#4090D512yescomplemented
53#4042—52no—
54#4042—52no—
55#4038—52no—
56D5×C4⋊C4—52no—
57#4036—52no—
58#4035—52no—
59#4035—52no—
60#4034—52no—
61#4018—52no—
62#4016—52no—
63C22×C52C2411yescharacteristic
64C2×C5⋊D5—21no—
65C2×C5⋊D5—21no—
66C2×C5⋊D5C2×Q812yes—
67C4×C52—21no—
68C4×C52—22no—
69#1039—21no—
70#1039—22no—
71C5×Dic5—22no—
72C5×Dic5—22no—
73C5×Dic5—22no—
74C5×Dic5C2×Q814yes—
75C2×C5×Q8—22no—
76C2×C5×Q8—52no—
77C2×C5×Q8C2×D512yes—
78C2×C4×D5—52no—
79C2×C4×D5—52no—
80C2×C4×D5—52no—
81C2×C4×D5—102no—
82C2×Dic10—52no—
83C2×Dic10C2×D512yes—
84C5×C4⋊C4—52no—
85C5×C4⋊C4—52no—
86C5×C4⋊C4—52no—
87C5×C4⋊C4—52no—
88C42×C5—52no—
89#685—52no—
90#685—52no—
91#685—52no—
92#685—52no—
93#685—104no—
94C4⋊Dic5—52no—
95C4⋊Dic5—52no—
96C4⋊Dic5—52no—
97C4⋊Dic5—52no—
98C4⋊Dic5—102no—
99#683—52no—
100#683—52no—
101#683—52no—
102#683—52no—
103#683—52no—
104#683—52no—
105#683—52no—
106#683—52no—
107#683—104no—
108#683C2×D512yes—
109#683C2×D512yes—
110#683C2×D512yes—
111#683C2×D512yes—
112C4×Dic5—52no—
113C4×Dic5—52no—
114C4×Dic5C2×D512yes—
115#544—251noSylow
116C2×C52C22×Q811yescharacteristic
117C2×C52H(5,𝔽2)11yescharacteristic
118C2×C522−1+411yescharacteristic
119C5⋊D5—22no—
120C22×D5—52no—
121C22×D5—102no—
122C5×Q8—42no—
123C5×Q8—44no—
124C5×Q8—202no—
125C2×C4×C5—22no—
126C2×C4×C5—24no—
127C2×C4×C5—52no—
128C2×C4×C5—52no—
129C2×C4×C5—52no—
130C2×C4×C5—52no—
131C2×C4×C5C22×D512yes—
132C2×C4×C5C22×D512yes—
133C2×C4×C5C22×D512yes—
134C2×Dic5—52no—
135C2×Dic5—52no—
136C2×Dic5—52no—
137C2×Dic5—102no—
138C2×Dic5—104no—
139C2×Dic5C22×D512yes—
140C2×Dic5C22×D512yes—
141C2×Dic5C22×D512yes—
142C2×Dic5C22×D512yes—
143C4×D5—58no—
144C4×D5—202no—
145C4×D5—202no—
146C4×D5—204no—
147Dic10—42no—
148Dic10—202no—
149C4⋊Q8—52no—
150C4⋊Q8—252no—
151#125—252no—
152#125—252no—
153C22⋊Q8—251no—
154C22⋊Q8—252no—
155C22⋊Q8—251no—
156#117—252no—
157C2×C4⋊C4—251no—
158C52#54411yescharacteristic, complemented, Sylow
159C22×C5—22no—
160C22×C5C23×D512yes—
161C2×D5—54no—
162C2×D5—104no—
163C2×D5—102no—
164C2×D5—102no—
165C2×D5—202no—
166C2×D5—202no—
167C4×C5—24no—
168C4×C5—22no—
169C4×C5—22no—
170C4×C5—22no—
171C4×C5—44no—
172C4×C5—54no—
173C4×C5—102no—
174C4×C5—102no—
175C4×C5—102no—
176Dic5—22no—
177Dic5—22no—
178Dic5—22no—
179Dic5—102no—
180Dic5—102no—
181Dic5—102no—
182Dic5—202no—
183Dic5—204no—
184Dic5Q8×D514yes—
185C2×Q8—52no—
186C2×Q8—251no—
187C2×Q8D5211yescharacteristic
188C22×C4—251no—
189C22×C4—251no—
190C22×C4—251no—
191C4⋊C4—52no—
192C4⋊C4—52no—
193C4⋊C4—52no—
194C4⋊C4—52no—
195C4⋊C4—252no—
196C4⋊C4—252no—
197C4⋊C4—251no—
198C4⋊C4—252no—
199C4⋊C4—252no—
200C4⋊C4—251no—
201C4⋊C4—252no—
202C22⋊C4—252no—
203C22⋊C4—252no—
204C42—52no—
205C42—252no—
206C2×C5—22no—
207C2×C5—22no—
208C2×C5—22no—
209C2×C5#414412yes—
210C2×C5C2×Q8×D512yes—
211C2×C5#414712yes—
212D5—104no—
213D5—204no—
214C23—251no—
215Q8—41no—
216Q8—202no—
217Q8—1001no—
218C2×C4—52no—
219C2×C4—52no—
220C2×C4—52no—
221C2×C4—52no—
222C2×C4—254no—
223C2×C4—252no—
224C2×C4—251no—
225C2×C4—1001no—
226C2×C4—1001no—
227C2×C4C2×D5212yes—
228C2×C4C2×D5211yescharacteristic
229C5—22no—
230C5#6606312yescomplemented
231C22—252no—
232C22—501no—
233C22—501no—
234C22C22×D5211yescharacteristic
235C4—21no—
236C4—22no—
237C4—54no—
238C4—102no—
239C4—102no—
240C4—102no—
241C4—502no—
242C4—501no—
243C2—502no—
244C2#1171748111yescharacteristic
245C2Q8×D5211yescharacteristic
246C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×Q8C22
prime radicalC22×C5⋊D5C23
hypercenterC2×Q8D52
hypocenterC52#544
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C5×#4090C2semidirect
5. #11716986C2semidirect
7. #11716963C2semidirect
8. #11716957C2semidirect
46. #6036Q8semidirect
52. #4090D5semidirect
158. C52#544semidirect
230. C5#66063semidirect

Maximal subgroups

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

subgroupindex
#117172662
C5×#40902
#117170082
#117170012
#117169862
#117169842
#117169632
#117169572
#117169542
#660635