Finite Group Databasebeta

← #13106485

Subgroups — #13106485

Order 1600.

494 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 130 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
1Q8×#1039C211yesmaximal, characteristic, complemented
2C5×#4092C212yesmaximal, complemented
3#11717037C211yesmaximal, characteristic, complemented
4#11717033C212yesmaximal
5C4×#6037C211yesmaximal, characteristic
6#11716977C214yesmaximal, complemented
7#11716959C212yesmaximal, complemented
8#11716956C212yesmaximal, complemented
9C2×C52×Q8C2211yescharacteristic
10C2×C4×C5×D5C2212yes—
11C2×#6039C2212yes—
12C2×#6037C2211yescharacteristic
13C4⋊#1039C2211yescharacteristic
14C4⋊#1039C2212yes—
15C4×#1039C2212yes—
16C4×#1039C2211yescharacteristic
17C5×#685C2214yes—
18C5×C4⋊Dic5C2212yes—
19C5×#683C2214yes—
20#87781C2211yescharacteristic
21#87781C2212yes—
22#87778C2214yes—
23#87778C2212yes—
24Dic52C2214yes—
25#65975—52nomaximal
26C22×C5×D5C2312yes—
27C52×Q8—22no—
28C2×C4×C52C2312yes—
29C2×C4×C52C2311yescharacteristic
30C2×#1039C2312yes—
31C2×#1039C2311yescharacteristic
32C2×#1039C2311yescharacteristic
33C2×C5×Dic5C2314yes—
34C2×C5×Dic5C2312yes—
35C4×C5×D5—24no—
36#6039—24no—
37#6037—22no—
38C5×C22⋊Q8—52no—
39#4092D512yescomplemented
40Q8×Dic5—52no—
41Q8×Dic5—102no—
42C4×C22⋊D5—52no—
43#4072—52no—
44#4044—54no—
45#4038—52no—
46#4034—52no—
47#4033—54no—
48#4031—54no—
49#4025—54no—
50C22×C52C2411yescharacteristic
51C2×C5×D5—22no—
52C2×C5×D5—22no—
53C2×C5×D5—22no—
54C4×C52—22no—
55C4×C52#4112yes—
56#1039—22no—
57#1039—21no—
58#1039#4112yes—
59C5×Dic5—24no—
60C5×Dic5—22no—
61C2×C5×Q8—22no—
62C2×C5×Q8C2×D512yes—
63C22×C4×C5—52no—
64C2×C22⋊D5—52no—
65C2×C4×D5C2×D512yes—
66C2×Dic10—54no—
67C5×C4⋊C4—54no—
68C5×C4⋊C4—52no—
69C5×C22⋊C4—54no—
70C22⋊Dic5—52no—
71C22⋊Dic5—54no—
72#685—54no—
73#685—52no—
74#685C2×D514yes—
75C4⋊Dic5—54no—
76C4⋊Dic5—52no—
77C4⋊Dic5—102no—
78C4⋊Dic5—104no—
79C4⋊Dic5C2×D512yes—
80#683—54no—
81#683—52no—
82#683C2×D514yes—
83C4×Dic5—54no—
84C4×Dic5—52no—
85C4×Dic5—54no—
86C4×Dic5—54no—
87C4×Dic5—102no—
88C4×Dic5—104no—
89#556—251noSylow
90C2×C52C2×#4111yescharacteristic
91C2×C52C2×#4111yescharacteristic
92C2×C522−1+411yescharacteristic
93C5×D5—42no—
94C23×C5—52no—
95C22×D5C22×D512yes—
96C5×Q8—28no—
97C5×Q8—24no—
98C2×C4×C5—24no—
99C2×C4×C5—22no—
100C2×C4×C5—52no—
101C2×C4×C5—54no—
102C2×C4×C5—104no—
103C2×C4×C5C22×D514yes—
104C2×C4×C5C22×D512yes—
105C22⋊D5—104no—
106C2×Dic5—52no—
107C2×Dic5—54no—
108C2×Dic5—52no—
109C2×Dic5—102no—
110C2×Dic5—104no—
111C2×Dic5—102no—
112C2×Dic5C22×D512yes—
113C2×Dic5C22×D514yes—
114C4×D5—24no—
115Dic10—108no—
116C4⋊Q8—251no—
117#126—254no—
118#124—252no—
119C22⋊Q8—52no—
120C4×Q8—252no—
121C4×Q8—251no—
122C4×D4—251no—
123#117—252no—
124C52#55611yescharacteristic, complemented, Sylow
125C22×C5—22no—
126C22×C5—102no—
127C22×C5—102no—
128C22×C5—102no—
129C22×C5C23×D512yes—
130C2×D5—22no—
131C2×D5—22no—
132C2×D5—22no—
133C4×C5—28no—
134C4×C5—24no—
135C4×C5—24no—
136C4×C5—102no—
137C4×C5—104no—
138C4×C5#71114yes—
139Dic5—22no—
140Dic5—24no—
141Dic5—54no—
142Dic5—104no—
143Dic5—104no—
144Dic5—102no—
145Dic5—204no—
146Dic5—202no—
147C2×Q8—252no—
148C2×Q8D5211yescharacteristic
149C2×D4—251no—
150C22×C4—52no—
151C4⋊C4—52no—
152C4⋊C4—54no—
153C4⋊C4—251no—
154C4⋊C4—252no—
155C4⋊C4—251no—
156C4⋊C4—252no—
157C22⋊C4—54no—
158C22⋊C4—254no—
159C22⋊C4—252no—
160C42—254no—
161C42—251no—
162C42—252no—
163C2×C5—22no—
164C2×C5—22no—
165C2×C5—22no—
166C2×C5—202no—
167C2×C5C2×#71112yes—
168C2×C5#414712yes—
169C2×C5D5×#4112yes—
170D5—42no—
171C23—52no—
172Q8—22no—
173Q8—504no—
174D4—502no—
175C2×C4—52no—
176C2×C4—54no—
177C2×C4—104no—
178C2×C4—251no—
179C2×C4—252no—
180C2×C4—251no—
181C2×C4C2×D5212yes—
182C2×C4C2×D5211yescharacteristic
183C5—22no—
184C5#6597512yescomplemented
185C22—102no—
186C22—102no—
187C22—102no—
188C22C22×D5211yescharacteristic
189C4—22no—
190C4—104no—
191C4—102no—
192C4—252no—
193C4—502no—
194C4—501no—
195C4#8788012yes—
196C2—202no—
197C2#1171748111yescharacteristic
198C2#1171748411yescharacteristic
199C2C2×#8788011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×Q8C22
prime radicalC2×#6037C22
hypercenterC2×Q8D52
hypocenterC52#556
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×#1039C2semidirect
2. C5×#4092C2semidirect
3. #11717037C2semidirect
6. #11716977C2semidirect
7. #11716959C2semidirect
8. #11716956C2semidirect
39. #4092D5semidirect
124. C52#556semidirect
184. C5#65975semidirect

Maximal subgroups

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

subgroupindex
Q8×#10392
C5×#40922
#117170372
#117170332
C4×#60372
#117169772
#117169592
#117169562
#659755