Finite Group Databasebeta

← #10608080

Subgroups — #10608080

Order 640.

676 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 185 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×D4×Dic5C211yesmaximal, characteristic
2C2×#4034C211yesmaximal, characteristic, complemented
3C2×#4024C214yesmaximal, complemented
4C5×#390C211yesmaximal, characteristic
5#65510C211yesmaximal, characteristic
6#65265C211yesmaximal, characteristic, complemented
7#65238C214yesmaximal, complemented
8#65224C211yesmaximal, characteristic, complemented
9C4×#683C211yesmaximal, characteristic, complemented
10C22×C5×D4C2211yescharacteristic
11C23×Dic5C2212yes—
12C22×Dic10C2212yes—
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C5×C22⋊C4C2214yes—
15C2×C42×C5C2211yescharacteristic
16C2×C22⋊Dic5C2214yes—
17C2×C22⋊Dic5C2212yes—
18D4×Dic5—24no—
19C2×C4⋊Dic5C2214yes—
20C2×C4⋊Dic5C2211yescharacteristic
21C2×C4⋊Dic5C2211yescharacteristic
22C2×#683C2214yes—
23C2×#683C2212yes—
24C2×#683C2212yes—
25C2×C4×Dic5C2211yescharacteristic
26C2×C4×Dic5C2211yescharacteristic
27#4034—24no—
28#4024—216no—
29#3963C2212yes—
30#2542—51nomaximal, Sylow
31C24×C5C2312yes—
32C2×C5×D4—22no—
33C2×C5×D4—22no—
34C2×C5×D4Q814yescomplemented
35C22×C4×C5C2312yes—
36C22×C4×C5C2312yes—
37C22×C4×C5C2311yescharacteristic
38C22×Dic5—28no—
39C22×Dic5—24no—
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C22×Dic5C2311yescharacteristic
43C22×Dic5C2312yes—
44C22×Dic5C2311yescharacteristic
45C2×Dic10—28no—
46C2×Dic10—24no—
47C5×C4⋊C4—22no—
48C5×C22⋊C4—28no—
49C42×C5—22no—
50C22⋊Dic5—28no—
51C22⋊Dic5—24no—
52C4⋊Dic5—28no—
53C4⋊Dic5—22no—
54C4⋊Dic5—22no—
55#683—28no—
56#683—24no—
57#683D418yescomplemented
58C4×Dic5—22no—
59C4×Dic5D414yes—
60C2×C4⋊Q8—51no—
61C2×C22⋊Q8—54no—
62C2×C4×D4—51no—
63#399—54no—
64#390—51no—
65#390D511yescharacteristic
66#384—51no—
67#384—51no—
68C4×C4⋊C4—51no—
69C23×C5—24no—
70C23×C5—22no—
71C23×C5—22no—
72C23×C5—24no—
73C23×C5C2×Q814yes—
74C23×C5C2411yescharacteristic
75C5×D4—28no—
76C2×C4×C5—22no—
77C2×C4×C5—22no—
78C2×C4×C5—24no—
79C2×C4×C5—22no—
80C2×C4×C5C2×D414yes—
81C2×C4×C5C2×D412yes—
82C2×C4×C5C2×Q812yes—
83C2×C4×C5#4112yes—
84C2×Dic5—22no—
85C2×Dic5—22no—
86C2×Dic5—22no—
87C2×Dic5—22no—
88C2×Dic5—24no—
89C2×Dic5—24no—
90C2×Dic5—22no—
91C2×Dic5—48no—
92C2×Dic5C2×D412yes—
93C2×Dic5C2×D414yes—
94C2×Dic5C2×D412yes—
95C2×Dic5C2×D414yes—
96Dic10—48no—
97C22×Q8—52no—
98C22×D4C2×D511yescharacteristic
99C23×C4—52no—
100C4⋊Q8—104no—
101C22⋊Q8—1016no—
102C4×D4—104no—
103C2×C4⋊C4—52no—
104C2×C4⋊C4—51no—
105C2×C4⋊C4—54no—
106C2×C4⋊C4—54no—
107C2×C4⋊C4—51no—
108C2×C4⋊C4—52no—
109C2×C4⋊C4C2×D511yescharacteristic
110C2×C22⋊C4—52no—
111C2×C22⋊C4—54no—
112C2×C22⋊C4C2×D514yes—
113C2×C42—51no—
114C2×C42—51no—
115C2×C42C2×D511yescharacteristic
116#95—52no—
117C22×C5—24no—
118C22×C5—28no—
119C22×C5—42no—
120C22×C5—42no—
121C22×C5—44no—
122C22×C5C22×D411yescharacteristic
123C22×C5C22×D412yes—
124C22×C5C22×Q811yescharacteristic
125C22×C5C2×#4111yescharacteristic
126C22×C52−1+412yes—
127C4×C5—24no—
128C4×C5—42no—
129C4×C5C22⋊Q814yescomplemented
130Dic5—22no—
131Dic5—24no—
132Dic5—42no—
133Dic5—42no—
134Dic5#12714yes—
135C24C22×D512yes—
136C2×Q8—108no—
137C2×Q8—104no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4Dic1014yescomplemented
141C22×C4—51no—
142C22×C4—51no—
143C22×C4—52no—
144C22×C4—52no—
145C22×C4—52no—
146C22×C4—104no—
147C22×C4—108no—
148C22×C4C22×D512yes—
149C22×C4C22×D512yes—
150C22×C4C22×D511yescharacteristic
151C4⋊C4—22no—
152C4⋊C4—58no—
153C4⋊C4—108no—
154C4⋊C4—104no—
155C4⋊C4—102no—
156C4⋊C4—108no—
157C4⋊C4—102no—
158C22⋊C4—28no—
159C22⋊C4—104no—
160C22⋊C4—108no—
161C42—22no—
162C42—54no—
163C42—102no—
164C2×C5—44no—
165C2×C5C2×C22⋊Q811yescharacteristic
166C2×C5C2×#12711yescharacteristic
167C2×C5#53611yescharacteristic
168C2×C5#54712yes—
169C2×C5D4×Q812yes—
170C23—24no—
171C23—22no—
172C23—22no—
173C23—24no—
174C23C2×Dic1014yes—
175C23C23×D511yescharacteristic
176Q8—208no—
177D4—28no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—52no—
183C2×C4—54no—
184C2×C4—52no—
185C2×C4—54no—
186C2×C4—102no—
187C2×C4—102no—
188C2×C4—102no—
189C2×C4—102no—
190C2×C4—104no—
191C2×C4—104no—
192C2×C4—102no—
193C2×C4—208no—
194C2×C4C2×Dic1012yes—
195C2×C4D4×D514yes—
196C2×C4D4×D512yes—
197C2×C4#71112yes—
198C5#254211yescharacteristic, complemented, Sylow
199C22—24no—
200C22—28no—
201C22—42no—
202C22—44no—
203C22—42no—
204C22C22×Dic1011yescharacteristic
205C22C2×D4×D511yescharacteristic
206C22C2×D4×D512yes—
207C22C2×#71111yescharacteristic
208C22#415012yes—
209C4—24no—
210C4—42no—
211C4—54no—
212C4—104no—
213C4—102no—
214C4—202no—
215C4—202no—
216C4#402414yescomplemented
217C2—44no—
218C2C2×#402411yescharacteristic
219C2D4×Dic1012yes—
220C2#6594512yes—
221C2#6600911yescharacteristic
222C2D5×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#390C2
prime radicalC22×C5×D4C22
hypercenter#390D5
hypocenterC5#2542
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#4034C2semidirect
3. C2×#4024C2semidirect
6. #65265C2semidirect
7. #65238C2semidirect
8. #65224C2semidirect
9. C4×#683C2semidirect
34. C2×C5×D4Q8semidirect
57. #683D4semidirect
129. C4×C5C22⋊Q8semidirect
140. C2×D4Dic10semidirect
198. C5#2542semidirect
216. C4#4024semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic52
C2×#40342
C2×#40242
C5×#3902
#655102
#652652
#652382
#652242
C4×#6832
#25425