Finite Group Databasebeta

← #10612722

Subgroups — #10612722

Order 640.

386 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#66060C211yesmaximal, characteristic, complemented
2#65897C211yesmaximal, characteristic, complemented
3C5×#474C211yesmaximal, characteristic, complemented
4#65368C211yesmaximal, characteristic, complemented
5C4×#686C211yesmaximal, characteristic, complemented
6#65174C212yesmaximal, complemented
7#65170C212yesmaximal, complemented
8#65131C211yesmaximal, characteristic, complemented
9#65127C211yesmaximal, characteristic, complemented
10#65071C212yesmaximal, complemented
11#65066C212yesmaximal, complemented
12C2×#713—21no—
13C5×C4⋊Q8C2211yescharacteristic
14C4×C5×D4C2211yescharacteristic
15#4092—21no—
16#4084C2212yes—
17C2×#686C2211yescharacteristic
18#4076—21no—
19#4072—21no—
20C2×#676C2212yes—
21#4041C2212yes—
22#4040—22no—
23#4038C2212yes—
24D5×C4⋊C4C2212yes—
25#4031—22no—
26C4×D20C2211yescharacteristic
27C4×D20C2211yescharacteristic
28#4015C2211yescharacteristic
29C5×#107C2212yes—
30C5×C4⋊C8C2211yescharacteristic
31C5×#102C2212yes—
32#3964C2211yescharacteristic
33#3953C2212yes—
34#3952C2212yes—
35#3949C2212yes—
36#3941C2211yescharacteristic
37#3940C2212yes—
38#3940C2212yes—
39#3939C2211yescharacteristic
40C4×#168C2211yescharacteristic
41#3354—51nomaximal, Sylow
42C2×C5×Q8—21no—
43C2×C5×D4C2311yescharacteristic
44C22×C4×C5—21no—
45C2×C22⋊D5—22no—
46#713—44no—
47C2×D20—22no—
48C2×D20D411yescharacteristic
49C2×D20C2311yescharacteristic
50C2×C4×D5—22no—
51C2×C4×D5—21no—
52C2×C4×D5C2312yes—
53C2×Dic10—21no—
54C2×C5×C8C2312yes—
55C5×C4⋊C4—21no—
56C5×C4⋊C4C2311yescharacteristic
57C5×C4⋊C4C2312yes—
58C5×C22⋊C4—21no—
59C42×C5C2311yescharacteristic
60C22⋊Dic5—22no—
61#686—22no—
62#685—22no—
63#685—21no—
64#685—22no—
65#685D412yes—
66C4⋊Dic5—21no—
67C4⋊Dic5D411yescharacteristic
68C4⋊Dic5C2311yescharacteristic
69C4⋊Dic5C2312yes—
70#683—22no—
71C4×Dic5—22no—
72C2×#168C2311yescharacteristic
73C2×#168C2311yescharacteristic
74#676—24no—
75#555—51no—
76#547—51no—
77#498—51no—
78#482—52no—
79#481—52no—
80#476—51no—
81#474D511yescharacteristic, complemented
82#469—52no—
83C4×D8—51no—
84#436—51no—
85#428—52no—
86C23×C5—21no—
87C22×D5—21no—
88C22×D5C2×D412yes—
89C5×Q8—42no—
90C5×D4—21no—
91C5×D4—21no—
92C2×C4×C5—21no—
93C2×C4×C5—22no—
94C2×C4×C5—42no—
95C2×C4×C5C2×D411yescharacteristic
96C2×C4×C5C2×D411yescharacteristic
97C2×C4×C5C2411yescharacteristic
98C22⋊D5—82no—
99C2×Dic5—22no—
100C2×Dic5—21no—
101C2×Dic5C2×D412yes—
102D20—22no—
103D20—21no—
104D20—21no—
105D20—44no—
106C4×D5—22no—
107C4×D5—22no—
108C4×D5—44no—
109C4×D5—42no—
110Dic10—81no—
111C5×C8—22no—
112#168—21no—
113#168#4112yescomplemented
114C2×#41—101no—
115C2×D8—51no—
116C2×M16—52no—
117C4⋊Q8—51no—
118C4⋊Q8C2×D511yescharacteristic
119#124—102no—
120#123—102no—
121C22⋊Q8—101no—
122#121—52no—
123#121—52no—
124#121—101no—
125C4×D4—51no—
126C4×D4—51no—
127C4×D4C2×D511yescharacteristic
128#117—52no—
129#117—101no—
130C2×C4⋊C4—52no—
131#107—51no—
132#107C2×D512yes—
133#106—52no—
134#106—52no—
135#106—52no—
136C4⋊C8C2×D511yescharacteristic
137#102—51no—
138#102—52no—
139#102—51no—
140#102C2×D512yes—
141C22⋊C8—52no—
142C4×C8—51no—
143C22×C5—41no—
144C22×C5—41no—
145C22×C5—41no—
146C22×C5C22×D411yescharacteristic
147C2×D5—22no—
148C2×D5—22no—
149C2×D5—22no—
150C2×D5—41no—
151C2×D5—41no—
152C2×D5—41no—
153C4×C5—21no—
154C4×C5—42no—
155C4×C5—41no—
156C4×C5AGL(1,ℤ/8)12yes—
157C4×C5C2×#4111yescharacteristic
158C4×C52−1+411yescharacteristic
159Dic5—22no—
160Dic5—41no—
161Dic5—42no—
162#41—204no—
163C2×Q8—21no—
164C2×Q8—101no—
165C2×D4—51no—
166C2×D4—51no—
167C2×D4—102no—
168C2×D4—102no—
169C2×D4C22×D511yescharacteristic
170C22×C4—21no—
171C22×C4—52no—
172C22×C4—102no—
173C22×C4—101no—
174D8—102no—
175M16—104no—
176C2×C8—51no—
177C2×C8—51no—
178C2×C8C22×D512yes—
179C4⋊C4—21no—
180C4⋊C4—51no—
181C4⋊C4—52no—
182C4⋊C4—51no—
183C4⋊C4—102no—
184C4⋊C4—101no—
185C4⋊C4C22×D511yescharacteristic
186C4⋊C4C22×D512yes—
187C22⋊C4—21no—
188C22⋊C4—52no—
189C22⋊C4—102no—
190C22⋊C4—102no—
191C22⋊C4—102no—
192C22⋊C4—101no—
193C42—102no—
194C42C22×D511yescharacteristic
195C2×C5—81no—
196C2×C5#54711yescharacteristic
197C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
198C2×C5#57711yescharacteristic
199D5—42no—
200D5—81no—
201C23—21no—
202C23—52no—
203C23—101no—
204Q8—42no—
205Q8—401no—
206D4—21no—
207D4—21no—
208D4—101no—
209D4—101no—
210D4—102no—
211D4—204no—
212D4—402no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—52no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4—102no—
220C2×C4—101no—
221C2×C4—204no—
222C2×C4—202no—
223C2×C4D4×D511yescharacteristic
224C2×C4D4×D511yescharacteristic
225C2×C4C23×D511yescharacteristic
226C8—22no—
227C8—52no—
228C8—101no—
229C5#335411yescharacteristic, complemented, Sylow
230C22—41no—
231C22—41no—
232C22—41no—
233C22—102no—
234C22—102no—
235C22—102no—
236C22—201no—
237C22—201no—
238C22—201no—
239C22C2×D4×D511yescharacteristic
240C4—21no—
241C4—41no—
242C4—42no—
243C4—102no—
244C4—201no—
245C4—202no—
246C4#405712yes—
247C4#414711yescharacteristic
248C4D5×#4111yescharacteristic
249C2—81no—
250C2—202no—
251C2—401no—
252C2#6596511yescharacteristic
253C2C2×#405711yescharacteristic
254C2#6611211yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#474C2
socleC22×C5C22×D4
hypercenter#474D5
hypocenterC5#3354
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #66060C2semidirect
2. #65897C2semidirect
3. C5×#474C2semidirect
4. #65368C2semidirect
5. C4×#686C2semidirect
6. #65174C2semidirect
7. #65170C2semidirect
8. #65131C2semidirect
9. #65127C2semidirect
10. #65071C2semidirect
11. #65066C2semidirect
81. #474D5semidirect
113. #168#41semidirect
229. C5#3354semidirect

Maximal subgroups

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

subgroupindex
#660602
#658972
C5×#4742
#653682
C4×#6862
#651742
#651702
#651312
#651272
#650712
#650662
#33545