Finite Group Databasebeta

← #10612589

Subgroups — #10612589

Order 640.

370 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 109 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#65953C212yesmaximal, complemented
2C5×#469C211yesmaximal, characteristic, complemented
3#65439C212yesmaximal, complemented
4#65424C211yesmaximal, characteristic, complemented
5#65408C211yesmaximal, characteristic, complemented
6#65327C212yesmaximal, complemented
7#65160C212yesmaximal, complemented
8#65075C212yesmaximal, complemented
9#65052C212yesmaximal, complemented
10C2×C5×D8C2211yescharacteristic
11C2×C5×M16C2211yescharacteristic
12C5×#121C2212yes—
13#4086C2212yes—
14#4085—22no—
15#4084—22no—
16D4×Dic5C2212yes—
17C2×#686C2212yes—
18#4076C2212yes—
19#4072C2211yescharacteristic
20C22×#168C2211yescharacteristic
21C2×D40C2211yescharacteristic
22#4039C2212yes—
23#4036C2212yes—
24#4032—22no—
25#4029—22no—
26C5×#106C2211yescharacteristic
27C5×#102C2212yes—
28#3964C2212yes—
29#3953C2212yes—
30#3949C2211yescharacteristic
31#3946C2212yes—
32#3946C2212yes—
33#3941C2212yes—
34#3939C2212yes—
35#3259—51nomaximal, Sylow
36C2×C5×D4—22no—
37C2×C5×D4C2312yes—
38C22×C4×C5C2311yescharacteristic
39C2×C22⋊D5—22no—
40C2×C22⋊D5—22no—
41C2×C22⋊D5—22no—
42C22×Dic5—22no—
43C2×D20C2312yes—
44C2×C4×D5—22no—
45C5×D8—41no—
46C5×M16—41no—
47C2×C5×C8C2311yescharacteristic
48C2×C5×C8C2311yescharacteristic
49C5×C4⋊C4C2312yes—
50C5×C22⋊C4—22no—
51C22⋊Dic5—22no—
52C22⋊Dic5D411yescharacteristic
53C22⋊Dic5D411yescharacteristic
54#686—42no—
55#685—22no—
56#685—22no—
57C4⋊Dic5—22no—
58C4⋊Dic5D412yes—
59C4⋊Dic5C2312yes—
60#683—22no—
61C4×Dic5C2312yes—
62C2×#168—41no—
63C2×#168C2312yes—
64D40—41no—
65#540—52no—
66#478—52no—
67#478—52no—
68#469—51no—
69#469D511yescharacteristic, complemented
70#466—52no—
71#459—52no—
72#459—52no—
73#424—51no—
74C23×C5—22no—
75C23×C5C2×D411yescharacteristic
76C22×D5—22no—
77C5×D4—22no—
78C5×D4—22no—
79C5×D4—82no—
80C2×C4×C5—21no—
81C2×C4×C5—21no—
82C2×C4×C5—22no—
83C2×C4×C5C2×D411yescharacteristic
84C2×C4×C5C2411yescharacteristic
85C22⋊D5—82no—
86C22⋊D5—82no—
87C22⋊D5—82no—
88C2×Dic5—22no—
89C2×Dic5—82no—
90C2×Dic5C2×D412yes—
91C2×Dic5C2×D412yes—
92D20—22no—
93D20—22no—
94C4×D5—82no—
95C5×C8—21no—
96C5×C8—21no—
97#168—22no—
98C2×D8—52no—
99C2×D8—51no—
100C2×D8C2×D511yescharacteristic
101C2×M16C2×D511yescharacteristic
102C22×C8—51no—
103#127—52no—
104#125—52no—
105#123—102no—
106#123—102no—
107#121—52no—
108#121—102no—
109#121—102no—
110#121C2×D512yes—
111C4×D4—52no—
112C4×D4—52no—
113#117—51no—
114#107—52no—
115#106—51no—
116#106C2×D511yescharacteristic
117C4⋊C8—52no—
118C4⋊C8—52no—
119#102—52no—
120#102—52no—
121#102—52no—
122#102C2×D512yes—
123C22×C5—21no—
124C22×C5—22no—
125C22×C5—42no—
126C22×C5—42no—
127C22×C5—42no—
128C22×C5C22×D411yescharacteristic
129C2×D5—42no—
130C2×D5—42no—
131C2×D5—42no—
132C4×C5—21no—
133C4×C5—42no—
134C4×C5H(5,𝔽2)11yescharacteristic
135C4×C52−1+411yescharacteristic
136Dic5—22no—
137Dic5—22no—
138Dic5—42no—
139C2×D4—22no—
140C2×D4—52no—
141C2×D4—102no—
142C2×D4—102no—
143C2×D4—102no—
144C2×D4C22×D512yes—
145C22×C4—102no—
146C22×C4—102no—
147C22×C4C22×D511yescharacteristic
148D8—41no—
149D8—201no—
150D8—202no—
151M16—41no—
152C2×C8—52no—
153C2×C8—201no—
154C2×C8C22×D511yescharacteristic
155C2×C8C22×D511yescharacteristic
156C4⋊C4—52no—
157C4⋊C4—52no—
158C4⋊C4—102no—
159C4⋊C4—102no—
160C4⋊C4C22×D512yes—
161C22⋊C4—22no—
162C22⋊C4—51no—
163C22⋊C4—51no—
164C22⋊C4—102no—
165C22⋊C4—102no—
166C22⋊C4—102no—
167C42—52no—
168C2×C5—41no—
169C2×C5—82no—
170C2×C5#53711yescharacteristic
171C2×C5#57612yes—
172D5—82no—
173C23—22no—
174C23—102no—
175C23D4×D511yescharacteristic
176D4—22no—
177D4—22no—
178D4—82no—
179D4—102no—
180D4—102no—
181D4—402no—
182D4—402no—
183D4—402no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—52no—
188C2×C4—52no—
189C2×C4—102no—
190C2×C4—402no—
191C2×C4—402no—
192C2×C4D4×D511yescharacteristic
193C2×C4C23×D511yescharacteristic
194C8—21no—
195C8—21no—
196C8—102no—
197C5#325911yescharacteristic, complemented, Sylow
198C22—22no—
199C22—21no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—202no—
204C22—202no—
205C22—202no—
206C22C2×D4×D511yescharacteristic
207C4—21no—
208C4—42no—
209C4—102no—
210C4—102no—
211C4—202no—
212C4#414411yescharacteristic
213C4#415011yescharacteristic
214C2—41no—
215C2—82no—
216C2—402no—
217C2#6593111yescharacteristic
218C2#6610812yes—
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#469C2
socleC22×C5C22×D4
hypercenter#469D5
hypocenterC5#3259
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #65953C2semidirect
2. C5×#469C2semidirect
3. #65439C2semidirect
4. #65424C2semidirect
5. #65408C2semidirect
6. #65327C2semidirect
7. #65160C2semidirect
8. #65075C2semidirect
9. #65052C2semidirect
69. #469D5semidirect
197. C5#3259semidirect

Maximal subgroups

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

subgroupindex
#659532
C5×#4692
#654392
#654242
#654082
#653272
#651602
#650752
#650522
#32595