Finite Group Databasebeta

← #13101514

Subgroups — #13101514

Order 1600.

366 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 80 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
1C22×C5×Dic10C211yesmaximal, characteristic, complemented
2C5×#4076C211yesmaximal, characteristic
3C2×#87772C212yesmaximal, complemented
4C22⋊#6029C211yesmaximal, characteristic, complemented
5#11716606C212yesmaximal, complemented
6C22×C4×C52C2211yescharacteristic
7C2×C5×C22⋊D5—21no—
8C22×C5×Dic5—21no—
9C2×C5×D20C2211yescharacteristic
10C2×C5×Dic10—21no—
11C2×C5×Dic10—24no—
12C2×C5×Dic10C2212yes—
13C2×#6029C2212yes—
14C5×#685—21no—
15C5×C4⋊Dic5C2211yescharacteristic
16#87772—24no—
17#65516—51nomaximal
18#65329—51nomaximal
19C23×C52D411yescharacteristic
20C22×C5×D5—21no—
21C2×C4×C52—22no—
22C2×C4×C52D411yescharacteristic
23C2×C4×C52C2311yescharacteristic
24C5×C22⋊D5—42no—
25C2×C5×Dic5—24no—
26C2×C5×Dic5—22no—
27C2×C5×Dic5—21no—
28C5×D20—22no—
29C5×Dic10—24no—
30C5×Dic10—22no—
31C5×Dic10D414yescomplemented
32#6029—22no—
33C22×C5×Q8—51no—
34C22×Dic10D511yescharacteristic, complemented
35C5×#121—51no—
36C2×#688—52no—
37C2×#687—52no—
38#4076D511yescharacteristic
39#3967—52no—
40C22⋊#168—51no—
41C22⋊#168—51no—
42C22⋊#168—102no—
43#3942—52no—
44C22×C52—21no—
45C22×C52—21no—
46C22×C52SD1612yescomplemented
47C22×C52C2×D411yescharacteristic
48C2×C5×D5—41no—
49C2×C5×D5—41no—
50C2×C5×D5—41no—
51C4×C52—21no—
52C4×C52C2×D412yes—
53C5×Dic5—24no—
54C5×Dic5—41no—
55C2×C5×Q8—52no—
56C2×C5×Q8—104no—
57C2×C5×Q8—101no—
58C2×C5×D4—51no—
59C2×C5×D4—101no—
60C22×C4×C5—22no—
61C22×C4×C5—101no—
62C22×C4×C5C2×D511yescharacteristic
63C22×C4×C5C2×D511yescharacteristic
64C2×C22⋊D5—21no—
65C22×Dic5—21no—
66C2×D20C2×D511yescharacteristic
67C2×Dic10—21no—
68C2×Dic10—24no—
69C2×Dic10C2×D512yes—
70C5×C4⋊C4—51no—
71C5×C22⋊C4—101no—
72#688—104no—
73#687—104no—
74#685—21no—
75C4⋊Dic5C2×D511yescharacteristic
76C2×#168—52no—
77C2×#168—52no—
78C2×#168—104no—
79#448—251noSylow
80C2×C52—22no—
81C2×C52#12011yescharacteristic
82C2×C52C2×SD1611yescharacteristic
83C2×C52#13711yescharacteristic
84C5×D5—81no—
85C23×C5—22no—
86C23×C5—101no—
87C23×C5C22⋊D511yescharacteristic
88C23×C5C22⋊D511yescharacteristic
89C22×D5—21no—
90C5×Q8—54no—
91C5×Q8—104no—
92C5×Q8—102no—
93C5×D4—102no—
94C5×D4—202no—
95C2×C4×C5—22no—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—44no—
100C2×C4×C5—104no—
101C2×C4×C5—102no—
102C2×C4×C5—101no—
103C2×C4×C5C22⋊D511yescharacteristic
104C2×C4×C5C22⋊D511yescharacteristic
105C2×C4×C5C22×D511yescharacteristic
106C2×C4×C5C22×D511yescharacteristic
107C22⋊D5—42no—
108C2×Dic5—24no—
109C2×Dic5—22no—
110C2×Dic5—21no—
111D20—22no—
112Dic10—24no—
113Dic10—22no—
114Dic10C22⋊D514yescomplemented
115#168—102no—
116#168—102no—
117#168—204no—
118C22×Q8—51no—
119C2×SD16—252no—
120#121—51no—
121#103—252no—
122C22⋊C8—251no—
123C52#44811yescharacteristic, complemented, Sylow
124C22×C5—21no—
125C22×C5—21no—
126C22×C5—21no—
127C22×C5—21no—
128C22×C5—22no—
129C22×C5—24no—
130C22×C5—42no—
131C22×C5—42no—
132C22×C5—201no—
133C22×C5—201no—
134C22×C5—201no—
135C22×C5#68712yescomplemented
136C22×C5#68812yescomplemented
137C22×C5C2×C22⋊D511yescharacteristic
138C22×C5C2×C22⋊D511yescharacteristic
139C2×D5—41no—
140C2×D5—41no—
141C2×D5—41no—
142C4×C5—21no—
143C4×C5—21no—
144C4×C5—24no—
145C4×C5—42no—
146C4×C5—104no—
147C4×C5—201no—
148C4×C5D4×D512yes—
149C4×C5C2×C22⋊D512yes—
150Dic5—24no—
151Dic5—41no—
152C2×Q8—52no—
153C2×Q8—104no—
154C2×Q8—101no—
155C2×D4—51no—
156C2×D4—101no—
157C22×C4—101no—
158C22×C4D5211yescharacteristic
159SD16—504no—
160C2×C8—252no—
161C4⋊C4—51no—
162C22⋊C4—101no—
163C2×C5—22no—
164C2×C5—22no—
165C2×C5—22no—
166C2×C5—22no—
167C2×C5—22no—
168C2×C5—44no—
169C2×C5—401no—
170C2×C5C2×#68711yescharacteristic
171C2×C5#408311yescharacteristic
172C2×C5C2×#68811yescharacteristic
173C2×C5#408811yescharacteristic
174C2×C5#409711yescharacteristic
175C2×C5#409911yescharacteristic
176D5—81no—
177C23—101no—
178C23#603711yescharacteristic
179Q8—54no—
180Q8—104no—
181Q8—102no—
182D4—102no—
183D4—202no—
184C2×C4—22no—
185C2×C4—104no—
186C2×C4—102no—
187C2×C4—101no—
188C2×C4#603711yescharacteristic
189C2×C4C2×D5211yescharacteristic
190C8—502no—
191C5—22no—
192C5#6532911yescharacteristic, complemented
193C5#6551611yescharacteristic, complemented
194C22—21no—
195C22—21no—
196C22—201no—
197C22—201no—
198C22—201no—
199C22#8777212yescomplemented
200C22C2×#603711yescharacteristic
201C4—21no—
202C4—104no—
203C4—201no—
204C4D5×C22⋊D512yes—
205C2—22no—
206C2—401no—
207C2C2×#8777211yescharacteristic
208C2#1171694511yescharacteristic
209C2#1171709411yescharacteristic
subgroupG/H
centerC22C2×#6037
commutator, cosocleC2×C4×C52C23
FrattiniC2×C4C2×D52
FittingC22×C4×C52C22
socleC22×C52C2×D4
prime radicalC5×#4076C2
hypercenterC22×C4D52
hypocenterC52#448
LCSG ▷ C2×C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C5×Dic10C2semidirect
3. C2×#87772C2semidirect
4. C22⋊#6029C2semidirect
5. #11716606C2semidirect
31. C5×Dic10D4semidirect
34. C22×Dic10D5semidirect
46. C22×C52SD16semidirect
114. Dic10C22⋊D5semidirect
123. C52#448semidirect
135. C22×C5#687semidirect
136. C22×C5#688semidirect
192. C5#65329semidirect
193. C5#65516semidirect
199. C22#87772semidirect

Maximal subgroups

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

subgroupindex
C22×C5×Dic102
C5×#40762
C2×#877722
C22⋊#60292
#117166062
#655165
#653295