Finite Group Databasebeta

← #13107164

Subgroups — #13107164

Order 1600.

968 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 235 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
1C52×C22⋊Q8C211yesmaximal, characteristic, complemented
2#11717266C212yesmaximal, complemented
3C2×#87808C211yesmaximal, characteristic
4#11717244C211yesmaximal, characteristic, complemented
5#11717217C212yesmaximal, complemented
6#11717216C212yesmaximal, complemented
7#11717216C212yesmaximal, complemented
8C22⋊C4×C5⋊D5C212yesmaximal
9#11717198C212yesmaximal, complemented
10C23×C5⋊D5C2211yescharacteristic
11C2×C52×Q8C2211yescharacteristic, complemented
12C22×C4×C52C2211yescharacteristic
13C22×#1039C2211yescharacteristic
14C2×C4×C5⋊D5C2212yes—
15C2×C4×C5⋊D5C2212yes—
16C2×#6045C2212yescomplemented
17C2×#6045C2211yescharacteristic, complemented
18C52×C4⋊C4C2211yescharacteristic, complemented
19C52×C4⋊C4C2212yescomplemented
20C52×C22⋊C4C2212yes—
21C22⋊#1039C2212yes—
22#87808C2212yes—
23#87808C2212yes—
24#87808C2212yes—
25#87808C2212yes—
26C4⋊#1039C2212yescomplemented
27C4⋊#1039C2211yescharacteristic, complemented
28#87806C2212yescomplemented
29#87806C2212yescomplemented
30#87806C2212yescomplemented
31#65968—56nomaximal
32C23×C52C2311yescharacteristic
33C22×C5⋊D5—41no—
34C22×C5⋊D5—41no—
35C22×C5⋊D5Q814yescomplemented
36C22×C5⋊D5C2312yes—
37C52×Q8—41no—
38C2×C4×C52—41no—
39C2×C4×C52C2311yescharacteristic
40C2×C4×C52C2311yescharacteristic
41C2×C4×C52C2312yes—
42C2×C4×C52C2312yes—
43C2×#1039—41no—
44C2×#1039C2312yes—
45C2×#1039C2312yes—
46C2×#1039C2312yes—
47C4×C5⋊D5—42no—
48C4×C5⋊D5—42no—
49#6045—41no—
50#6045—42no—
51C5×C22⋊Q8D516yescomplemented
52#4092—512no—
53C2×#685—56no—
54#4070—56no—
55#4043—512no—
56#4042—512no—
57#4042—512no—
58D5×C22⋊C4—512no—
59#4024—512no—
60C22×C52—21no—
61C22×C52—21no—
62C22×C52C2×Q812yes—
63C22×C52C2411yescharacteristic
64C2×C5⋊D5—22no—
65C2×C5⋊D5—22no—
66C2×C5⋊D5—44no—
67C2×C5⋊D5C2×Q814yes—
68C4×C52—22no—
69C4×C52—21no—
70C4×C52—22no—
71C4×C52—21no—
72#1039—22no—
73#1039—22no—
74#1039—22no—
75C23×D5—56no—
76C2×C5×Q8C2×D516yescomplemented
77C22×C4×C5C2×D516yes—
78C22×Dic5—56no—
79C2×C4×D5—512no—
80C2×C4×D5—512no—
81C2×Dic10—512no—
82C2×Dic10—56no—
83C5×C4⋊C4C2×D516yescomplemented
84C5×C4⋊C4C2×D5112yescomplemented
85C5×C22⋊C4C2×D5112yes—
86C22⋊Dic5—512no—
87#685—512no—
88#685—512no—
89#685—512no—
90#685—512no—
91C4⋊Dic5—512no—
92C4⋊Dic5—56no—
93#683—512no—
94#683—512no—
95#683—512no—
96#543—251noSylow
97C2×C52—22no—
98C2×C52C22×Q811yescharacteristic
99C2×C52H(5,𝔽2)11yescharacteristic
100C2×C52H(5,𝔽2)11yescharacteristic
101C5⋊D5—24no—
102C23×C5C22×D516yes—
103C22×D5—512no—
104C22×D5—524no—
105C22×D5—206no—
106C22×D5—206no—
107C5×Q8—46no—
108C2×C4×C5—46no—
109C2×C4×C5C22×D516yes—
110C2×C4×C5C22×D5112yes—
111C2×C4×C5C22×D5112yes—
112C2×C4×C5C22×D516yes—
113C2×Dic5—512no—
114C2×Dic5—512no—
115C2×Dic5—512no—
116C2×Dic5—206no—
117C4×D5—2012no—
118C4×D5—2012no—
119Dic10—206no—
120Dic10—2012no—
121C22⋊Q8—252no—
122C22⋊Q8—251no—
123C22⋊Q8—252no—
124C22⋊Q8—252no—
125C22⋊Q8—252no—
126C22⋊Q8—252no—
127C22⋊Q8C5⋊D511yescharacteristic, complemented
128C2×C22⋊C4—251no—
129C2×C22⋊C4—252no—
130C52#54311yescharacteristic, complemented, Sylow
131C22×C5—26no—
132C22×C5—26no—
133C22×C5Q8×D5112yes—
134C22×C5C23×D516yes—
135C2×D5—524no—
136C2×D5—1012no—
137C2×D5—1012no—
138C2×D5—2024no—
139C4×C5—26no—
140C4×C5—212no—
141C4×C5—26no—
142C4×C5—212no—
143Dic5—1012no—
144Dic5—1012no—
145Dic5—1012no—
146C24—251no—
147C2×Q8—251no—
148C2×Q8—252no—
149C2×Q8C2×C5⋊D511yescharacteristic, complemented
150C22×C4—251no—
151C22×C4—252no—
152C22×C4—252no—
153C22×C4C2×C5⋊D511yescharacteristic
154C4⋊C4—252no—
155C4⋊C4—252no—
156C4⋊C4—252no—
157C4⋊C4—251no—
158C4⋊C4—252no—
159C4⋊C4C2×C5⋊D511yescharacteristic, complemented
160C4⋊C4C2×C5⋊D512yescomplemented
161C22⋊C4—252no—
162C22⋊C4—252no—
163C22⋊C4—252no—
164C22⋊C4—252no—
165C22⋊C4—252no—
166C22⋊C4C2×C5⋊D512yes—
167C2×C5—212no—
168C2×C5#414416yes—
169C2×C5C2×Q8×D516yes—
170C2×C5#414916yes—
171D5—1024no—
172C23—252no—
173C23—254no—
174C23—1001no—
175C23—1001no—
176C23C22×C5⋊D511yescharacteristic
177Q8—41no—
178Q8—1001no—
179Q8—1002no—
180C2×C4—41no—
181C2×C4—252no—
182C2×C4—252no—
183C2×C4—252no—
184C2×C4—1002no—
185C2×C4—1002no—
186C2×C4—1001no—
187C2×C4C22×C5⋊D511yescharacteristic
188C2×C4C22×C5⋊D511yescharacteristic
189C2×C4C22×C5⋊D512yes—
190C2×C4C22×C5⋊D512yes—
191C5#6596816yescomplemented
192C22—21no—
193C22—21no—
194C22—254no—
195C22—502no—
196C22—502no—
197C22—1004no—
198C22Q8×C5⋊D512yes—
199C22C23×C5⋊D511yescharacteristic
200C4—21no—
201C4—21no—
202C4—22no—
203C4—22no—
204C4—502no—
205C4—502no—
206C4—502no—
207C2—22no—
208C2—504no—
209C2#1171751711yescharacteristic
210C2C2×Q8×C5⋊D511yescharacteristic
211C2#1171752211yescharacteristic
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊Q8C2
prime radicalC23×C5⋊D5C22
hypercenterC22⋊Q8C5⋊D5
hypocenterC52#543
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C52×C22⋊Q8C2semidirect
2. #11717266C2semidirect
4. #11717244C2semidirect
5. #11717217C2semidirect
6. #11717216C2semidirect
7. #11717216C2semidirect
9. #11717198C2semidirect
11. C2×C52×Q8C22semidirect
16. C2×#6045C22semidirect
17. C2×#6045C22semidirect
18. C52×C4⋊C4C22semidirect
19. C52×C4⋊C4C22semidirect
26. C4⋊#1039C22semidirect
27. C4⋊#1039C22semidirect
28. #87806C22semidirect
29. #87806C22semidirect
30. #87806C22semidirect
35. C22×C5⋊D5Q8semidirect
51. C5×C22⋊Q8D5semidirect
76. C2×C5×Q8C2×D5semidirect
83. C5×C4⋊C4C2×D5semidirect
84. C5×C4⋊C4C2×D5semidirect
127. C22⋊Q8C5⋊D5semidirect
130. C52#543semidirect
149. C2×Q8C2×C5⋊D5semidirect
159. C4⋊C4C2×C5⋊D5semidirect
160. C4⋊C4C2×C5⋊D5semidirect
191. C5#65968semidirect

Maximal subgroups

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

subgroupindex
C52×C22⋊Q82
#117172662
C2×#878082
#117172442
#117172172
#117172162
C22⋊C4×C5⋊D52
#117171982
#659685