Finite Group Databasebeta

← #10614945

Subgroups — #10614945

Order 640.

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

normal subgroups only

This group has 386 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#65517C212yesmaximal, complemented
4#65516C212yesmaximal
5#65379C212yesmaximal, complemented
6#65375C212yesmaximal, complemented
7#65372C212yesmaximal, complemented
8#65371C212yesmaximal, complemented
9#65215C211yesmaximal, characteristic, complemented
10C22×C5×Q8C2212yes—
11C5×C4⋊Q8—21no—
12C5×C4⋊Q8—21no—
13C5×C4⋊Q8C2212yes—
14C5×C4⋊Q8C2212yes—
15C2×C5×C4⋊C4—21no—
16C2×C5×C4⋊C4—21no—
17C2×C42×C5C2211yescharacteristic
18C2×#689C2214yes—
19C2×#688C2214yes—
20#4076C2211yescharacteristic
21#4075—21no—
22C4×C22⋊D5—21no—
23#4070C2211yescharacteristic
24#4022—21no—
25#4021C2211yescharacteristic
26C4×D20C2211yescharacteristic
27#4018—21no—
28#4016C2211yescharacteristic
29C4×Dic10C2211yescharacteristic
30#3967C2214yes—
31#3967C2214yes—
32C22⋊#168C2212yes—
33C4⋊#168C2212yes—
34#3935C2212yes—
35#3218—51nomaximal, Sylow
36C2×C5×Q8—22no—
37C2×C5×Q8—44no—
38C2×C5×Q8C2314yes—
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⋊C4—22no—
51C42×C5D411yescharacteristic
52C42×C5D411yescharacteristic
53C42×C5C2311yescharacteristic
54C42×C5C2311yescharacteristic
55C22⋊Dic5—21no—
56#689—44no—
57#688—44no—
58#685—21no—
59#685—21no—
60#685—21no—
61#685—21no—
62C4⋊Dic5C2311yescharacteristic
63C4⋊Dic5C2311yescharacteristic
64#683—21no—
65#683—21no—
66#683—21no—
67#683—21no—
68C4×Dic5—21no—
69C2×#168C2314yes—
70C2×C4⋊Q8D511yescharacteristic, complemented
71#529—51no—
72#497—52no—
73#491—52no—
74#462—52no—
75#461—52no—
76#451—52no—
77#448—52no—
78#432—51no—
79C23×C5C2×D411yescharacteristic
80C22×D5—21no—
81C5×Q8—24no—
82C5×Q8—24no—
83C5×Q8—44no—
84C2×C4×C5—22no—
85C2×C4×C5—22no—
86C2×C4×C5—22no—
87C2×C4×C5—24no—
88C2×C4×C5—44no—
89C2×C4×C5C2×D411yescharacteristic
90C2×C4×C5C2×D411yescharacteristic
91C2×C4×C5C2×D411yescharacteristic
92C2×C4×C5C2×D411yescharacteristic
93C2×C4×C5C2×D411yescharacteristic
94C2×C4×C5C2411yescharacteristic
95C22⋊D5—42no—
96C2×Dic5—21no—
97C2×Dic5—21no—
98C2×Dic5—21no—
99D20—22no—
100C4×D5—42no—
101Dic10—22no—
102#168—24no—
103C22×Q8C2×D512yes—
104C2×Q16—54no—
105C2×SD16—54no—
106C4⋊Q8—21no—
107C4⋊Q8—21no—
108C4⋊Q8C2×D512yes—
109C4⋊Q8C2×D512yes—
110#126—101no—
111#125—51no—
112#124—51no—
113#123—101no—
114C22⋊Q8—51no—
115#121—51no—
116C4×Q8—51no—
117C4×D4—51no—
118C4×D4—101no—
119#117—101no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C42C2×D511yescharacteristic
123C4⋊C8—52no—
124#103—54no—
125#103—54no—
126C22⋊C8—52no—
127#97—52no—
128C22×C5—21no—
129C22×C5—21no—
130C22×C5#13712yes—
131C22×C5C22×D411yescharacteristic
132C2×D5—41no—
133C2×D5—41no—
134C2×D5—41no—
135C4×C5—21no—
136C4×C5—21no—
137C4×C5—21no—
138C4×C5—21no—
139C4×C5—44no—
140C4×C5#13712yes—
141C4×C5H(5,𝔽2)12yes—
142Dic5—41no—
143Dic5—41no—
144Dic5—41no—
145C2×Q8—22no—
146C2×Q8—44no—
147C2×Q8—51no—
148C2×Q8C22×D514yes—
149C2×D4—51no—
150C2×D4—101no—
151C22×C4—22no—
152C22×C4—101no—
153C22×C4C22⋊D511yescharacteristic
154C22×C4C22⋊D511yescharacteristic
155C22×C4C22×D511yescharacteristic
156Q16—204no—
157SD16—204no—
158C2×C8—54no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—51no—
164C4⋊C4—51no—
165C4⋊C4—101no—
166C4⋊C4—101no—
167C4⋊C4—101no—
168C4⋊C4—101no—
169C22⋊C4—101no—
170C22⋊C4—101no—
171C22⋊C4—101no—
172C22⋊C4—101no—
173C22⋊C4—101no—
174C42—101no—
175C42C22⋊D511yescharacteristic
176C42C22⋊D511yescharacteristic
177C42C22×D511yescharacteristic
178C42C22×D511yescharacteristic
179C2×C5—22no—
180C2×C5#53511yescharacteristic
181C2×C5C2×#13711yescharacteristic
182C2×C5C2×#13711yescharacteristic
183D5—81no—
184C23—101no—
185C23C2×C22⋊D511yescharacteristic
186Q8—24no—
187Q8—24no—
188Q8—44no—
189Q8—102no—
190D4—102no—
191D4—202no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—44no—
197C2×C4—101no—
198C2×C4—101no—
199C2×C4—101no—
200C2×C4—202no—
201C2×C4C2×C22⋊D511yescharacteristic
202C2×C4C2×C22⋊D511yescharacteristic
203C2×C4C2×C22⋊D511yescharacteristic
204C2×C4C2×C22⋊D511yescharacteristic
205C2×C4C2×C22⋊D511yescharacteristic
206C2×C4C23×D511yescharacteristic
207C8—104no—
208C5#321811yescharacteristic, complemented, Sylow
209C22—21no—
210C22—21no—
211C22—201no—
212C22—201no—
213C22—201no—
214C22#408812yes—
215C22C22×C22⋊D511yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—44no—
221C4—201no—
222C4—201no—
223C4—201no—
224C4#408812yes—
225C4#414412yes—
226C2—22no—
227C2—401no—
228C2#6613911yescharacteristic
229C2C2×#408811yescharacteristic
230C2C2×#408811yescharacteristic
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#3218
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. #65517C2semidirect
5. #65379C2semidirect
6. #65375C2semidirect
7. #65372C2semidirect
8. #65371C2semidirect
9. #65215C2semidirect
70. C2×C4⋊Q8D5semidirect
208. C5#3218semidirect

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
#655172
#655162
#653792
#653752
#653722
#653712
#652152
#32185