Finite Group Databasebeta

← #10614517

Subgroups — #10614517

Order 640.

528 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 157 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×C22⋊Q8C211yesmaximal, characteristic
2C2×#4090C211yesmaximal, characteristic, complemented
3C2×#4072C211yesmaximal, characteristic
4C2×#4070C211yesmaximal, characteristic
5#65519C211yesmaximal, characteristic, complemented
6#65273C214yesmaximal, complemented
7#65266C211yesmaximal, characteristic, complemented
8#65263C211yesmaximal, characteristic, complemented
9#65241C214yesmaximal
10C22×C5×Q8C2211yescharacteristic
11C23×C4×C5C2211yescharacteristic
12C22×Dic10C2211yescharacteristic
13C5×C22⋊Q8—24no—
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C4⋊C4C2212yes—
16C2×C5×C22⋊C4C2212yes—
17C2×C22⋊Dic5C2212yes—
18C2×C22⋊Dic5C2212yes—
19#4090—24no—
20#4072—24no—
21C2×C4⋊Dic5C2211yescharacteristic
22C2×C4⋊Dic5C2212yes—
23#4070—24no—
24C2×#683C2212yes—
25C2×#683C2214yes—
26C2×C4×Dic5C2211yescharacteristic
27C2×C4×Dic5C2211yescharacteristic
28#3963C2214yes—
29#3963C2214yes—
30#3963C2214yes—
31#2621—51nomaximal, Sylow
32C24×C5C2311yescharacteristic
33C2×C5×Q8—22no—
34C2×C5×Q8—22no—
35C2×C5×Q8—22no—
36C22×C4×C5—22no—
37C22×C4×C5—22no—
38C22×C4×C5D414yes—
39C22×C4×C5C2312yes—
40C22×C4×C5C2312yes—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2311yescharacteristic
43C22×Dic5C2312yes—
44C22×Dic5C2312yes—
45C22×Dic5C2312yes—
46C22×Dic5C2312yes—
47C2×Dic10—22no—
48C2×Dic10—22no—
49C2×Dic10—22no—
50C5×C4⋊C4—22no—
51C5×C4⋊C4—24no—
52C5×C22⋊C4—24no—
53C22⋊Dic5—24no—
54C22⋊Dic5—24no—
55C4⋊Dic5—22no—
56C4⋊Dic5—24no—
57#683—24no—
58#683—28no—
59C4×Dic5—22no—
60C4×Dic5D414yes—
61C2×C4⋊Q8—51no—
62C2×C22⋊Q8—51no—
63C2×C22⋊Q8D511yescharacteristic
64C2×#117—51no—
65#398—54no—
66#397—54no—
67#391—51no—
68#391—51no—
69#382—51no—
70C23×C5—21no—
71C23×C5—21no—
72C23×C5—22no—
73C23×C5—22no—
74C23×C5C2×D412yes—
75C23×C5C2411yescharacteristic
76C5×Q8—44no—
77C2×C4×C5—21no—
78C2×C4×C5—21no—
79C2×C4×C5—28no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—22no—
84C2×C4×C5—24no—
85C2×C4×C5C2×D412yes—
86C2×C4×C5C2×D412yes—
87C2×C4×C5C2×D412yes—
88C2×C4×C5#4112yes—
89C2×Dic5—22no—
90C2×Dic5—22no—
91C2×Dic5—22no—
92C2×Dic5—22no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5—22no—
96C2×Dic5—22no—
97C2×Dic5—24no—
98C2×Dic5—22no—
99C2×Dic5C2×D414yes—
100Dic10—44no—
101C22×Q8—51no—
102C22×Q8C2×D511yescharacteristic
103C23×C4C2×D511yescharacteristic
104C4⋊Q8—104no—
105C22⋊Q8—24no—
106C22⋊Q8—104no—
107#117—104no—
108C2×C4⋊C4—52no—
109C2×C4⋊C4—51no—
110C2×C4⋊C4—52no—
111C2×C4⋊C4—54no—
112C2×C4⋊C4C2×D511yescharacteristic
113C2×C4⋊C4C2×D512yes—
114C2×C22⋊C4—52no—
115C2×C22⋊C4—52no—
116C2×C22⋊C4C2×D512yes—
117C2×C42—51no—
118C2×C42—51no—
119#95—54no—
120#95—54no—
121#95—54no—
122C22×C5—22no—
123C22×C5—22no—
124C22×C5—22no—
125C22×C5—42no—
126C22×C5—42no—
127C22×C5C22×D411yescharacteristic
128C22×C5C22×D411yescharacteristic
129C22×C5C2×#4111yescharacteristic
130C22×C52−1+412yes—
131C22×C52−1+412yes—
132C4×C5—22no—
133C4×C5—42no—
134C4×C5—42no—
135C4×C5#12114yes—
136Dic5—24no—
137Dic5—42no—
138Dic5—42no—
139Dic5—42no—
140C24C22×D511yescharacteristic
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—102no—
145C2×Q8—102no—
146C2×Q8—102no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—52no—
150C22×C4—52no—
151C22×C4—52no—
152C22×C4—52no—
153C22×C4C22⋊D514yes—
154C22×C4C22×D512yes—
155C22×C4C22×D512yes—
156C22×C4C22×D511yescharacteristic
157C22×C4C22×D511yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—24no—
160C4⋊C4—104no—
161C4⋊C4—108no—
162C4⋊C4—104no—
163C4⋊C4—102no—
164C22⋊C4—24no—
165C22⋊C4—104no—
166C22⋊C4—104no—
167C42—54no—
168C42—102no—
169C2×C5—42no—
170C2×C5C2×#12111yescharacteristic
171C2×C5#53611yescharacteristic
172C2×C5#53611yescharacteristic
173C2×C5#53612yes—
174C2×C5#54112yes—
175C23—21no—
176C23—21no—
177C23—22no—
178C23—22no—
179C23C2×C22⋊D512yes—
180C23C23×D511yescharacteristic
181Q8—44no—
182Q8—204no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—28no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—54no—
192C2×C4—102no—
193C2×C4—102no—
194C2×C4—102no—
195C2×C4—102no—
196C2×C4—102no—
197C2×C4—104no—
198C2×C4—102no—
199C2×C4—102no—
200C2×C4—102no—
201C2×C4—102no—
202C2×C4D4×D512yes—
203C2×C4#71112yes—
204C2×C4C2×C22⋊D512yes—
205C2×C4C2×C22⋊D512yes—
206C5#262111yescharacteristic, complemented, Sylow
207C22—22no—
208C22—22no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22C2×D4×D511yescharacteristic
213C22C2×#71111yescharacteristic
214C22#414712yes—
215C22#415012yes—
216C22C22×C22⋊D511yescharacteristic
217C4—22no—
218C4—42no—
219C4—42no—
220C4—104no—
221C4—202no—
222C4—202no—
223C4—202no—
224C4#408514yes—
225C2—42no—
226C2#6595912yes—
227C2#6596212yes—
228C2C2×#408511yescharacteristic
229C2#6615011yescharacteristic
230C2#6615911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC24×C5C23
hypercenterC2×C22⋊Q8D5
hypocenterC5#2621
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#4090C2semidirect
5. #65519C2semidirect
6. #65273C2semidirect
7. #65266C2semidirect
8. #65263C2semidirect
206. C5#2621semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C22⋊Q82
C2×#40902
C2×#40722
C2×#40702
#655192
#652732
#652662
#652632
#652412
#26215