Finite Group Databasebeta

← #10614949

Subgroups — #10614949

Order 640.

362 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 125 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×C5×C4⋊Q8C211yesmaximal, characteristic, complemented
2#65813C211yesmaximal, characteristic
3#65383C212yesmaximal, complemented
4#65378C212yesmaximal, complemented
5#65376C212yesmaximal, complemented
6#65369C212yesmaximal, complemented
7#65259C212yesmaximal, complemented
8#65257C212yesmaximal
9#65215C211yesmaximal, characteristic, complemented
10C22×C5×Q8—21no—
11C5×C4⋊Q8—21no—
12C5×C4⋊Q8—21no—
13C5×C4⋊Q8C2212yes—
14C5×C4⋊Q8C2212yes—
15C2×C5×C4⋊C4—21no—
16C2×C5×C4⋊C4C2212yes—
17C2×C42×C5C2211yescharacteristic
18#4076C2211yescharacteristic
19#4075—21no—
20C4×C22⋊D5—21no—
21#4070C2211yescharacteristic
22#4022—21no—
23#4021C2211yescharacteristic
24C4×D20C2211yescharacteristic
25#4018—21no—
26#4016C2211yescharacteristic
27C4×Dic10C2211yescharacteristic
28C22⋊#168C2212yes—
29#3942C2214yes—
30#3941C2214yes—
31#3940C2214yes—
32#3939C2214yes—
33C4⋊#168C2212yes—
34#3935C2212yes—
35#3239—51nomaximal, Sylow
36C2×C5×Q8—22no—
37C2×C5×Q8—22no—
38C2×C5×Q8—44no—
39C22×C4×C5—22no—
40C22×C4×C5D411yescharacteristic
41C22×C4×C5D411yescharacteristic
42C22×C4×C5C2311yescharacteristic
43C2×C22⋊D5—21no—
44C2×D20C2311yescharacteristic
45C2×C4×D5—21no—
46C2×Dic10C2311yescharacteristic
47C5×C4⋊C4—22no—
48C5×C4⋊C4—22no—
49C5×C4⋊C4—22no—
50C5×C4⋊C4C2314yes—
51C42×C5D411yescharacteristic
52C42×C5D411yescharacteristic
53C42×C5C2311yescharacteristic
54C42×C5C2311yescharacteristic
55C22⋊Dic5—21no—
56#685—21no—
57#685—21no—
58#685—21no—
59#685—21no—
60C4⋊Dic5C2311yescharacteristic
61C4⋊Dic5C2311yescharacteristic
62#683—21no—
63#683—21no—
64#683—21no—
65#683—21no—
66C4×Dic5—21no—
67C2×#168C2314yes—
68C2×C4⋊Q8D511yescharacteristic, complemented
69#529—51no—
70#501—52no—
71#489—52no—
72#484—52no—
73#481—52no—
74#475—52no—
75#474—52no—
76#432—51no—
77C23×C5C2×D411yescharacteristic
78C22×D5—21no—
79C5×Q8—44no—
80C5×Q8—44no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—22no—
84C2×C4×C5—24no—
85C2×C4×C5—44no—
86C2×C4×C5C2×D411yescharacteristic
87C2×C4×C5C2×D411yescharacteristic
88C2×C4×C5C2×D411yescharacteristic
89C2×C4×C5C2×D411yescharacteristic
90C2×C4×C5C2×D411yescharacteristic
91C2×C4×C5C2411yescharacteristic
92C22⋊D5—42no—
93C2×Dic5—21no—
94C2×Dic5—21no—
95C2×Dic5—21no—
96D20—22no—
97C4×D5—42no—
98Dic10—22no—
99#168—24no—
100C22×Q8—21no—
101C4⋊Q8—21no—
102C4⋊Q8—21no—
103C4⋊Q8C2×D512yes—
104C4⋊Q8C2×D512yes—
105#126—101no—
106#125—51no—
107#124—51no—
108#123—101no—
109C22⋊Q8—51no—
110#121—51no—
111C4×Q8—51no—
112C4×D4—51no—
113C4×D4—101no—
114#117—101no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4C2×D512yes—
117C2×C42C2×D511yescharacteristic
118#107—54no—
119#106—54no—
120C4⋊C8—52no—
121#103—54no—
122#102—54no—
123C22⋊C8—52no—
124#97—52no—
125C22×C5—21no—
126C22×C5—21no—
127C22×C5#13712yes—
128C22×C5C22×D411yescharacteristic
129C2×D5—41no—
130C2×D5—41no—
131C2×D5—41no—
132C4×C5—21no—
133C4×C5—21no—
134C4×C5—21no—
135C4×C5—21no—
136C4×C5—44no—
137C4×C5AGL(1,ℤ/8)12yes—
138C4×C52−1+412yes—
139Dic5—41no—
140Dic5—41no—
141Dic5—41no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—44no—
145C2×Q8—51no—
146C2×D4—51no—
147C2×D4—101no—
148C22×C4—22no—
149C22×C4—101no—
150C22×C4C22⋊D511yescharacteristic
151C22×C4C22⋊D511yescharacteristic
152C22×C4C22×D511yescharacteristic
153C2×C8—54no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—51no—
158C4⋊C4—51no—
159C4⋊C4—101no—
160C4⋊C4—101no—
161C4⋊C4—101no—
162C4⋊C4—101no—
163C4⋊C4C22×D514yes—
164C22⋊C4—101no—
165C22⋊C4—101no—
166C22⋊C4—101no—
167C22⋊C4—101no—
168C22⋊C4—101no—
169C42—101no—
170C42C22⋊D511yescharacteristic
171C42C22⋊D511yescharacteristic
172C42C22×D511yescharacteristic
173C42C22×D511yescharacteristic
174C2×C5—22no—
175C2×C5#53611yescharacteristic
176C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
177C2×C5C2×#13711yescharacteristic
178D5—81no—
179C23—101no—
180C23C2×C22⋊D511yescharacteristic
181Q8—44no—
182Q8—44no—
183Q8—102no—
184D4—102no—
185D4—202no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—44no—
191C2×C4—101no—
192C2×C4—101no—
193C2×C4—101no—
194C2×C4—202no—
195C2×C4C2×C22⋊D511yescharacteristic
196C2×C4C2×C22⋊D511yescharacteristic
197C2×C4C2×C22⋊D511yescharacteristic
198C2×C4C2×C22⋊D511yescharacteristic
199C2×C4C2×C22⋊D511yescharacteristic
200C2×C4C23×D511yescharacteristic
201C8—104no—
202C5#323911yescharacteristic, complemented, Sylow
203C22—21no—
204C22—21no—
205C22—201no—
206C22—201no—
207C22—201no—
208C22#408812yes—
209C22C22×C22⋊D511yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—44no—
215C4—201no—
216C4—201no—
217C4—201no—
218C4#407812yes—
219C4#414712yes—
220C2—22no—
221C2—401no—
222C2C2×#407811yescharacteristic
223C2C2×#408811yescharacteristic
224C2#6615011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC2×C5×C4⋊Q8C2
socleC22×C5C22×D4
prime radical#4076C22
hypercenterC2×C4⋊Q8D5
hypocenterC5#3239
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C5×C4⋊Q8C2semidirect
3. #65383C2semidirect
4. #65378C2semidirect
5. #65376C2semidirect
6. #65369C2semidirect
7. #65259C2semidirect
9. #65215C2semidirect
68. C2×C4⋊Q8D5semidirect
202. C5#3239semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C4⋊Q82
#658132
#653832
#653782
#653762
#653692
#652592
#652572
#652152
#32395