Finite Group Databasebeta

← #10614205

Subgroups — #10614205

Order 640.

576 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 169 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×C4×C5×Q8C211yesmaximal, characteristic, complemented
2C2×#4093C211yesmaximal, characteristic
3C2×#4090C211yesmaximal, characteristic, complemented
4C2×#4021C211yesmaximal, characteristic
5#65283C214yesmaximal, complemented
6#65277C211yesmaximal, characteristic
7#65263C211yesmaximal, characteristic, complemented
8#65229C214yesmaximal, complemented
9C4×C4⋊Dic5C211yesmaximal, characteristic, complemented
10C22×C5×Q8C2211yescharacteristic
11C22×D20C2211yescharacteristic
12C22×Dic10C2211yescharacteristic
13C4×C5×Q8—24no—
14C2×C5×C4⋊C4C2212yes—
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C42×C5C2212yes—
17C2×C42×C5C2211yescharacteristic
18#4093—24no—
19#4090—24no—
20C2×#685C2212yes—
21C2×#685C2214yes—
22C2×#685C2214yes—
23C2×#685C2212yes—
24C2×C4⋊Dic5C2212yes—
25C2×#683C2214yes—
26C2×C4×Dic5C2211yescharacteristic
27C2×C4×Dic5C2211yescharacteristic
28#4021—24no—
29#3963C2214yes—
30#3963C2212yes—
31#2545—51nomaximal, Sylow
32C23×D5C2312yes—
33C2×C5×Q8—22no—
34C2×C5×Q8—22no—
35C2×C5×Q8D414yes—
36C22×C4×C5C2312yes—
37C22×C4×C5C2311yescharacteristic
38C22×C4×C5C2311yescharacteristic
39C22×C4×C5C2312yes—
40C22×C4×C5C2311yescharacteristic
41C22×Dic5C2312yes—
42C22×Dic5C2312yes—
43C22×Dic5C2312yes—
44C2×D20—22no—
45C2×D20—22no—
46C2×D20—22no—
47C2×Dic10—22no—
48C2×Dic10—22no—
49C2×Dic10—22no—
50C5×C4⋊C4—24no—
51C5×C4⋊C4—22no—
52C42×C5—24no—
53C42×C5D414yes—
54#685—24no—
55#685—24no—
56#685—28no—
57#685—28no—
58C4⋊Dic5—24no—
59#683—28no—
60C4×Dic5—22no—
61C4×Dic5—22no—
62C2×C4⋊Q8—51no—
63C2×#124—51no—
64C2×#124—51no—
65C2×C4×Q8D511yescharacteristic, complemented
66#396—54no—
67#391—51no—
68#390—51no—
69#388—54no—
70C4×C4⋊C4—51no—
71C23×C5C2411yescharacteristic
72C22×D5—22no—
73C22×D5—22no—
74C22×D5—24no—
75C22×D5—22no—
76C22×D5—24no—
77C5×Q8—28no—
78C2×C4×C5—22no—
79C2×C4×C5—22no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—24no—
84C2×C4×C5C2×D414yes—
85C2×C4×C5C2×D412yes—
86C2×C4×C5C2×D412yes—
87C2×C4×C5C2×D412yes—
88C2×C4×C5C2×D412yes—
89C2×C4×C5#4112yes—
90C2×Dic5—22no—
91C2×Dic5—22no—
92C2×Dic5—22no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5—22no—
96C2×Dic5—22no—
97C2×Dic5—22no—
98C2×Dic5#4114yes—
99D20—44no—
100Dic10—44no—
101C22×Q8—51no—
102C22×Q8C2×D511yescharacteristic
103C22×D4—51no—
104C4⋊Q8—104no—
105#124—104no—
106#124—104no—
107C4×Q8—24no—
108C2×C4⋊C4—54no—
109C2×C4⋊C4—52no—
110C2×C4⋊C4C2×D511yescharacteristic
111C2×C4⋊C4C2×D512yes—
112C2×C22⋊C4—52no—
113C2×C22⋊C4—54no—
114C2×C22⋊C4—52no—
115C2×C22⋊C4—54no—
116C2×C42—51no—
117C2×C42—51no—
118C2×C42C2×D512yes—
119C2×C42C2×D511yescharacteristic
120#95—52no—
121#95—54no—
122C22×C5C22×D411yescharacteristic
123C22×C5C22×D411yescharacteristic
124C22×C5C2×#4111yescharacteristic
125C22×C5C2×#4112yes—
126C22×C52−1+412yes—
127C2×D5—42no—
128C2×D5—42no—
129C2×D5—42no—
130C2×D5—44no—
131C2×D5—44no—
132C4×C5—24no—
133C4×C5—22no—
134C4×C5—22no—
135C4×C5—42no—
136C4×C5#12114yes—
137Dic5—24no—
138Dic5—42no—
139Dic5—42no—
140C24—52no—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—102no—
144C2×Q8—102no—
145C2×Q8—102no—
146C2×Q8D2014yes—
147C2×D4—102no—
148C2×D4—102no—
149C2×D4—102no—
150C22×C4—52no—
151C22×C4—52no—
152C22×C4—52no—
153C22×C4C22×D512yes—
154C22×C4C22×D512yes—
155C22×C4C22×D511yescharacteristic
156C22×C4C22×D511yescharacteristic
157C22×C4C22×D511yescharacteristic
158C4⋊C4—24no—
159C4⋊C4—22no—
160C4⋊C4—104no—
161C4⋊C4—108no—
162C22⋊C4—108no—
163C22⋊C4—108no—
164C22⋊C4—104no—
165C22⋊C4—104no—
166C42—24no—
167C42—102no—
168C42—102no—
169C42C22⋊D514yes—
170C2×C5C2×#12111yescharacteristic
171C2×C5#53211yescharacteristic
172C2×C5#53611yescharacteristic
173C2×C5#54812yes—
174C2×C5#55612yes—
175D5—82no—
176C23—102no—
177C23—102no—
178C23—104no—
179C23—104no—
180C23—102no—
181C23C23×D511yescharacteristic
182Q8—28no—
183Q8—204no—
184D4—204no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—54no—
192C2×C4—102no—
193C2×C4—102no—
194C2×C4—102no—
195C2×C4—102no—
196C2×C4—102no—
197C2×C4—102no—
198C2×C4—102no—
199C2×C4—102no—
200C2×C4C2×D2014yes—
201C2×C4C2×D2012yes—
202C2×C4#70912yes—
203C2×C4C2×C22⋊D512yes—
204C2×C4C2×C22⋊D512yes—
205C2×C4C2×C22⋊D512yes—
206C5#254511yescharacteristic, complemented, Sylow
207C22—202no—
208C22—202no—
209C22—204no—
210C22—204no—
211C22—202no—
212C22C22×D2011yescharacteristic
213C22C2×#70911yescharacteristic
214C22#414712yes—
215C22D5×#4112yes—
216C22C22×C22⋊D511yescharacteristic
217C4—24no—
218C4—22no—
219C4—22no—
220C4—42no—
221C4—104no—
222C4—202no—
223C4—202no—
224C4#407614yes—
225C2—402no—
226C2#6590212yes—
227C2#6591012yes—
228C2C2×#407611yescharacteristic
229C2#6615011yescharacteristic
230C2#6616611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C4×C5×Q8C2
prime radicalC22×D20C22
hypercenterC2×C4×Q8D5
hypocenterC5#2545
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C4×C5×Q8C2semidirect
3. C2×#4090C2semidirect
5. #65283C2semidirect
7. #65263C2semidirect
8. #65229C2semidirect
9. C4×C4⋊Dic5C2semidirect
65. C2×C4×Q8D5semidirect
206. C5#2545semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×Q82
C2×#40932
C2×#40902
C2×#40212
#652832
#652772
#652632
#652292
C4×C4⋊Dic52
#25455