Finite Group Databasebeta

← #13101455

Subgroups — #13101455

Order 1600.

504 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C2×#87808C211yesmaximal, characteristic
2C2×C5×C4⋊Dic5C211yesmaximal, characteristic, complemented
3C2×#87780C212yesmaximal, complemented
4C2×#87779C212yesmaximal
5C5×#3963C211yesmaximal, characteristic, complemented
6C23×C5⋊D5C2211yescharacteristic
7C22×C4×C52C2211yescharacteristic
8C22×#1039C2211yescharacteristic
9C22×C5×Dic5C2212yes—
10C22×C5×Dic5C2212yes—
11#87808—22no—
12C5×C4⋊Dic5—22no—
13#87780—24no—
14#87779—24no—
15#65281—51nomaximal
16#64966—51nomaximal
17C23×C52C2311yescharacteristic
18C22×C5⋊D5—21no—
19C22×C5⋊D5—22no—
20C22×C5⋊D5—21no—
21C22×C5⋊D5—22no—
22C22×C5⋊D5Q812yes—
23C2×C4×C52—22no—
24C2×C4×C52D412yes—
25C2×#1039—21no—
26C2×#1039—22no—
27C2×C5×Dic5—22no—
28C2×C5×Dic5—22no—
29C2×C5×Dic5—22no—
30C2×C5×Dic5—22no—
31C2×C5×Dic5—22no—
32C2×C5×Dic5D414yes—
33C2×C5×C4⋊C4—51no—
34C2×#685—51no—
35C2×#685—51no—
36C2×#685—52no—
37C2×#685—52no—
38C2×#685—102no—
39C2×C4⋊Dic5—52no—
40C2×C4⋊Dic5D511yescharacteristic, complemented
41C2×#683—52no—
42C5×#95—51no—
43#3963D511yescharacteristic, complemented
44C22×C52C2×D412yes—
45C22×C52C2×D411yescharacteristic
46C22×C52C2×Q811yescharacteristic
47C22×C52#4112yes—
48C22×C52#4111yescharacteristic
49C2×C5⋊D5—22no—
50C2×C5⋊D5—24no—
51C2×C5⋊D5—42no—
52C2×C5⋊D5—41no—
53C2×C5⋊D5—41no—
54C4×C52—22no—
55#1039—41no—
56C5×Dic5—24no—
57C5×Dic5—42no—
58C23×D5—51no—
59C23×D5—51no—
60C23×D5—102no—
61C22×C4×C5—22no—
62C22×C4×C5—52no—
63C22×C4×C5—52no—
64C22×C4×C5C2×D511yescharacteristic
65C22×C4×C5C2×D511yescharacteristic
66C22×Dic5—51no—
67C22×Dic5—51no—
68C22×Dic5—102no—
69C22×Dic5C2×D512yes—
70C22×Dic5C2×D512yes—
71C5×C4⋊C4—102no—
72#685—104no—
73#685—108no—
74#685—102no—
75#685—102no—
76#685—104no—
77C4⋊Dic5—22no—
78C4⋊Dic5—104no—
79#683—104no—
80#399—251noSylow
81C2×C52C22⋊Q812yes—
82C2×C52C22⋊Q811yescharacteristic
83C2×C52#12312yes—
84C2×C52#12311yescharacteristic
85C2×C52#12711yescharacteristic
86C5⋊D5—42no—
87C23×C5—22no—
88C23×C5C22×D511yescharacteristic
89C23×C5C22×D511yescharacteristic
90C22×D5—52no—
91C22×D5—52no—
92C22×D5—104no—
93C22×D5—104no—
94C22×D5—104no—
95C22×D5—102no—
96C22×D5—101no—
97C22×D5—102no—
98C22×D5—102no—
99C22×D5—101no—
100C22×D5—102no—
101C22×D5—101no—
102C22×D5—101no—
103C22×D5—204no—
104C22×D5—204no—
105C2×C4×C5—22no—
106C2×C4×C5—22no—
107C2×C4×C5—24no—
108C2×C4×C5—44no—
109C2×C4×C5—54no—
110C2×C4×C5—102no—
111C2×C4×C5—102no—
112C2×C4×C5—102no—
113C2×C4×C5—102no—
114C2×C4×C5—102no—
115C2×C4×C5D2012yes—
116C2×C4×C5C22⋊D512yes—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5—22no—
121C2×Dic5—22no—
122C2×Dic5—104no—
123C2×Dic5—102no—
124C2×Dic5—101no—
125C2×Dic5—102no—
126C2×Dic5—101no—
127C2×Dic5—204no—
128C2×Dic5D2014yes—
129C2×C4⋊C4—51no—
130C2×C4⋊C4—252no—
131C2×C22⋊C4—252no—
132C2×C22⋊C4—251no—
133#95—51no—
134C52#39911yescharacteristic, complemented, Sylow
135C22×C5—22no—
136C22×C5—22no—
137C22×C5—24no—
138C22×C5—24no—
139C22×C5—22no—
140C22×C5C2×D2012yes—
141C22×C5C2×D2011yescharacteristic
142C22×C5#70912yes—
143C22×C5D4×D512yes—
144C22×C5#71111yescharacteristic
145C22×C5#71112yes—
146C22×C5Q8×D511yescharacteristic
147C22×C5Q8×D511yescharacteristic
148C22×C5#71311yescharacteristic
149C22×C5C2×C22⋊D511yescharacteristic
150C2×D5—108no—
151C2×D5—104no—
152C2×D5—102no—
153C2×D5—104no—
154C2×D5—102no—
155C2×D5—208no—
156C2×D5—208no—
157C2×D5—204no—
158C2×D5—204no—
159C2×D5—201no—
160C2×D5—202no—
161C2×D5—201no—
162C2×D5—202no—
163C2×D5—201no—
164C2×D5—201no—
165C4×C5—22no—
166C4×C5—22no—
167C4×C5—44no—
168C4×C5—104no—
169C4×C5—202no—
170Dic5—24no—
171Dic5—42no—
172Dic5—204no—
173Dic5—201no—
174Dic5—201no—
175C24—251no—
176C22×C4—52no—
177C22×C4—52no—
178C22×C4—251no—
179C22×C4D5211yescharacteristic
180C4⋊C4—102no—
181C4⋊C4—504no—
182C22⋊C4—504no—
183C22⋊C4—502no—
184C2×C5—22no—
185C2×C5—24no—
186C2×C5—24no—
187C2×C5—22no—
188C2×C5—22no—
189C2×C5#402011yescharacteristic
190C2×C5#403212yes—
191C2×C5#403211yescharacteristic
192C2×C5#404012yes—
193C2×C5#404212yes—
194C2×C5#404311yescharacteristic
195C2×C5#404312yes—
196C2×C5#407511yescharacteristic
197C2×C5#408611yescharacteristic
198C2×C5#409211yescharacteristic
199D5—208no—
200D5—202no—
201D5—202no—
202C23—252no—
203C23—501no—
204C23—502no—
205C23—502no—
206C23—501no—
207C23C2×D5211yescharacteristic
208C2×C4—22no—
209C2×C4—54no—
210C2×C4—102no—
211C2×C4—102no—
212C2×C4—102no—
213C2×C4—102no—
214C2×C4—102no—
215C2×C4—502no—
216C2×C4—501no—
217C2×C4#603812yes—
218C5—22no—
219C5#6496611yescharacteristic, complemented
220C5#6528111yescharacteristic, complemented
221C22—502no—
222C22—504no—
223C22—1002no—
224C22—1001no—
225C22—1001no—
226C22#8787111yescharacteristic
227C22#8787211yescharacteristic
228C22D5×D2012yes—
229C22#8787912yes—
230C22C2×#603811yescharacteristic
231C4—22no—
232C4—104no—
233C4—202no—
234C4—1001no—
235C2—1002no—
236C2#1171699011yescharacteristic
237C2#1171700912yes—
238C2#1171701112yes—
239C2#1171702911yescharacteristic
240C2#1171708411yescharacteristic
subgroupG/H
center, FrattiniC23C2×D52
commutator, socle, cosocleC23×C52C23
FittingC22×C4×C52C22
prime radicalC23×C5⋊D5C22
hypercenterC22×C4D52
hypocenterC52#399
LCSG ▷ C23×C52 ▷ C52
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C5×C4⋊Dic5C2semidirect
3. C2×#87780C2semidirect
5. C5×#3963C2semidirect
40. C2×C4⋊Dic5D5semidirect
43. #3963D5semidirect
134. C52#399semidirect
219. C5#64966semidirect
220. C5#65281semidirect

Maximal subgroups

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

subgroupindex
C2×#878082
C2×C5×C4⋊Dic52
C2×#877802
C2×#877792
C5×#39632
#652815
#649665