Finite Group Databasebeta

← #10613352

Subgroups — #10613352

Order 640.

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

normal subgroups only

This group has 350 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#66066C211yesmaximal, characteristic, complemented
2#66064C211yesmaximal, characteristic, complemented
3C5×#487C211yesmaximal, characteristic, complemented
4#65373C211yesmaximal, characteristic, complemented
5#65372C211yesmaximal, characteristic, complemented
6#65108C212yesmaximal
7#65089C212yesmaximal, complemented
8#65085C212yesmaximal, complemented
9#65077C212yesmaximal, complemented
10C4×D40C211yesmaximal, characteristic
11C8×Dic10C211yesmaximal, characteristic, complemented
12C5×C4⋊Q8C2211yescharacteristic
13C5×C4⋊Q8C2211yescharacteristic
14#4093C2212yes—
15#4092—22no—
16Q8×Dic5C2212yes—
17C2×#688C2212yes—
18C2×D40C2211yescharacteristic
19#4044—22no—
20#4043—21no—
21#4043—21no—
22#4039C2212yes—
23#4038—21no—
24D5×C4⋊C4—21no—
25#4036C2212yes—
26C4×D20C2211yescharacteristic
27C4×D20C2211yescharacteristic
28C4×Dic10C2211yescharacteristic
29C5×#103C2212yes—
30C5×#103C2212yes—
31C4×C5×C8C2211yescharacteristic
32#3967C2212yes—
33#3953C2211yescharacteristic
34#3953C2211yescharacteristic
35#3950C2211yescharacteristic
36#3946C2212yes—
37C8×Dic5C2212yes—
38#3941C2212yes—
39#3940C2212yes—
40C4⋊#168C2211yescharacteristic
41#3360—51nomaximal, Sylow
42C2×C5×Q8—21no—
43C2×C5×Q8C2312yes—
44C2×D20C2311yescharacteristic
45C2×D20C2311yescharacteristic
46C2×C4×D5—21no—
47C2×C4×D5—21no—
48C2×C4×D5—22no—
49C2×Dic10D411yescharacteristic
50C2×C5×C8C2311yescharacteristic
51C2×C5×C8C2311yescharacteristic
52C5×C4⋊C4—21no—
53C5×C4⋊C4—21no—
54C5×C4⋊C4—21no—
55C5×C4⋊C4C2312yes—
56C42×C5C2311yescharacteristic
57#688—24no—
58#685—21no—
59#685—21no—
60#685—22no—
61#685—22no—
62#685—22no—
63C4⋊Dic5—22no—
64C4⋊Dic5—22no—
65C4⋊Dic5D411yescharacteristic
66C4⋊Dic5C2311yescharacteristic
67C4⋊Dic5C2311yescharacteristic
68#683—22no—
69#683D412yes—
70C4×Dic5—22no—
71C4×Dic5C2312yes—
72C2×#168C2312yes—
73D40—22no—
74#556—51no—
75#554—51no—
76#488—52no—
77#487D511yescharacteristic, complemented
78#479—52no—
79#476—51no—
80#463—52no—
81#461—51no—
82C8×Q8—51no—
83C4×SD16—52no—
84C4×D8—51no—
85C22×D5—21no—
86C22×D5—21no—
87C5×Q8—22no—
88C5×Q8—22no—
89C5×Q8—81no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5C2×D411yescharacteristic
93C2×C4×C5C2×D411yescharacteristic
94C2×C4×C5C2411yescharacteristic
95C2×Dic5—21no—
96C2×Dic5—21no—
97C2×Dic5C2×D412yes—
98C2×Dic5C2×D412yes—
99D20—21no—
100D20—21no—
101D20—21no—
102D20#4112yes—
103C4×D5—42no—
104C4×D5—42no—
105C4×D5—82no—
106Dic10—22no—
107C5×C8—21no—
108C5×C8—21no—
109#168—22no—
110C2×SD16—52no—
111C2×D8—51no—
112C4⋊Q8C2×D511yescharacteristic
113C4⋊Q8C2×D511yescharacteristic
114#126—102no—
115#125—52no—
116#124—52no—
117C22⋊Q8—101no—
118C22⋊Q8—101no—
119C22⋊Q8—102no—
120C4×Q8—51no—
121C4×Q8—52no—
122C4×D4—52no—
123C4×D4—51no—
124C4×D4—51no—
125#117—101no—
126C2×C4⋊C4—101no—
127#107—51no—
128#106—52no—
129C4⋊C8—52no—
130C4⋊C8—51no—
131#103—52no—
132#103C2×D512yes—
133#103C2×D512yes—
134#102—51no—
135#102—51no—
136#102—52no—
137C4×C8—52no—
138C4×C8C2×D511yescharacteristic
139C22×C5C22×D411yescharacteristic
140C2×D5—22no—
141C2×D5—41no—
142C2×D5—41no—
143C2×D5—41no—
144C2×D5—41no—
145C2×D5—41no—
146C4×C5—21no—
147C4×C5—42no—
148C4×C5—42no—
149C4×C5#13512yes—
150C4×C5C2×#4111yescharacteristic
151C4×C52−1+411yescharacteristic
152Dic5—22no—
153Dic5—22no—
154Dic5—41no—
155Dic5—41no—
156C2×Q8—21no—
157C2×Q8—51no—
158C2×Q8C22×D512yes—
159C2×D4—51no—
160C2×D4—51no—
161C22×C4—102no—
162C22×C4—101no—
163C22×C4—101no—
164SD16—104no—
165D8—102no—
166C2×C8—52no—
167C2×C8C22×D511yescharacteristic
168C2×C8C22×D511yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—51no—
173C4⋊C4—52no—
174C4⋊C4—51no—
175C4⋊C4—51no—
176C4⋊C4—102no—
177C4⋊C4—102no—
178C4⋊C4—102no—
179C4⋊C4C22×D512yes—
180C22⋊C4—102no—
181C22⋊C4—102no—
182C22⋊C4—102no—
183C22⋊C4—101no—
184C22⋊C4—101no—
185C42—52no—
186C42—102no—
187C42C22×D511yescharacteristic
188C2×C5#54811yescharacteristic
189C2×C5C2×#13511yescharacteristic
190C2×C5#57711yescharacteristic
191D5—42no—
192D5—81no—
193C23—101no—
194C23—101no—
195Q8—22no—
196Q8—22no—
197Q8—81no—
198Q8—102no—
199D4—52no—
200D4—101no—
201D4—101no—
202D4—101no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—52no—
206C2×C4—52no—
207C2×C4—101no—
208C2×C4—101no—
209C2×C4—202no—
210C2×C4—202no—
211C2×C4—402no—
212C2×C4D4×D511yescharacteristic
213C2×C4D4×D511yescharacteristic
214C2×C4C23×D511yescharacteristic
215C8—21no—
216C8—21no—
217C8—102no—
218C5#336011yescharacteristic, complemented, Sylow
219C22—102no—
220C22—201no—
221C22—201no—
222C22—201no—
223C22—201no—
224C22—201no—
225C22C2×D4×D511yescharacteristic
226C4—21no—
227C4—42no—
228C4—42no—
229C4—102no—
230C4—102no—
231C4—201no—
232C4—201no—
233C4#406512yes—
234C4D5×#4111yescharacteristic
235C4#415011yescharacteristic
236C2—202no—
237C2—401no—
238C2#6601111yescharacteristic
239C2#6609611yescharacteristic
240C2C2×#406511yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#487C2
socleC22×C5C22×D4
prime radicalC2×D40C22
hypercenter#487D5
hypocenterC5#3360
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #66066C2semidirect
2. #66064C2semidirect
3. C5×#487C2semidirect
4. #65373C2semidirect
5. #65372C2semidirect
7. #65089C2semidirect
8. #65085C2semidirect
9. #65077C2semidirect
11. C8×Dic10C2semidirect
77. #487D5semidirect
218. C5#3360semidirect

Maximal subgroups

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

subgroupindex
#660662
#660642
C5×#4872
#653732
#653722
#651082
#650892
#650852
#650772
C4×D402
C8×Dic102
#33605