Finite Group Databasebeta

← #10609008

Subgroups — #10609008

Order 640.

408 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C5×#395C211yesmaximal, characteristic, complemented
2#65520C212yesmaximal, complemented
3#65283C212yesmaximal, complemented
4#65282C212yesmaximal, complemented
5#65277C211yesmaximal, characteristic
6#65272C211yesmaximal, characteristic, complemented
7#64964C212yesmaximal, complemented
8#64961C212yesmaximal
9#64941C212yesmaximal, complemented
10C22×C5×Q8C2211yescharacteristic
11C22×D20C2211yescharacteristic
12C2×C5×C4⋊C4C2212yes—
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×#685C2212yes—
15C2×#685C2212yes—
16C2×#685C2212yes—
17C2×#685C2212yes—
18C2×#685C2212yes—
19C2×#685C2212yes—
20C2×C4⋊Dic5C2211yescharacteristic
21C2×#683C2212yes—
22C2×#683C2212yes—
23C2×C4×Dic5C2211yescharacteristic
24C5×#95C2212yes—
25C5×#95C2211yescharacteristic
26#3963C2211yescharacteristic
27#3963C2212yes—
28#3963C2212yes—
29#3963C2212yes—
30#3963C2212yes—
31#2830—51nomaximal, Sylow
32C23×D5C2312yes—
33C2×C5×Q8—41no—
34C2×C5×Q8—42no—
35C22×C4×C5C2311yescharacteristic
36C22×C4×C5C2312yes—
37C22×C4×C5C2311yescharacteristic
38C22×C4×C5C2312yes—
39C22×C4×C5C2311yescharacteristic
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C22×Dic5C2312yes—
43C2×D20—22no—
44C2×D20—41no—
45C2×D20—41no—
46C5×C4⋊C4—41no—
47C5×C4⋊C4—42no—
48#685—42no—
49#685—42no—
50#685—42no—
51#685—42no—
52#685—42no—
53#685—42no—
54C4⋊Dic5—41no—
55#683—24no—
56#683—42no—
57C4×Dic5—41no—
58#398—51no—
59#397—52no—
60#397—52no—
61#396—52no—
62#395D511yescharacteristic, complemented
63#394—52no—
64#393—52no—
65#390—51no—
66#387—52no—
67C23×C5C2411yescharacteristic
68C22×D5—22no—
69C22×D5—22no—
70C22×D5—22no—
71C22×D5—22no—
72C22×D5—22no—
73C22×D5—22no—
74C22×D5—22no—
75C5×Q8—44no—
76C2×C4×C5—21no—
77C2×C4×C5—21no—
78C2×C4×C5—22no—
79C2×C4×C5—21no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—22no—
84C2×C4×C5—22no—
85C2×C4×C5—21no—
86C2×C4×C5—22no—
87C2×C4×C5—22no—
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—22no—
98D20—44no—
99C22×Q8C2×D511yescharacteristic
100C22×D4—51no—
101C2×C4⋊C4—52no—
102C2×C4⋊C4—52no—
103C2×C4⋊C4—51no—
104C2×C4⋊C4C2×D511yescharacteristic
105C2×C4⋊C4C2×D512yes—
106C2×C22⋊C4—52no—
107C2×C22⋊C4—52no—
108C2×C22⋊C4—52no—
109C2×C22⋊C4—52no—
110C2×C22⋊C4—52no—
111C2×C22⋊C4—52no—
112C2×C42—51no—
113#95—52no—
114#95—52no—
115#95—52no—
116#95—52no—
117#95—51no—
118#95C2×D512yes—
119#95C2×D511yescharacteristic
120C22×C5C2×#4111yescharacteristic
121C22×C5H(5,𝔽2)11yescharacteristic
122C22×C5H(5,𝔽2)12yes—
123C22×C5H(5,𝔽2)11yescharacteristic
124C22×C52−1+412yes—
125C2×D5—42no—
126C2×D5—42no—
127C2×D5—42no—
128C2×D5—42no—
129C2×D5—42no—
130C2×D5—42no—
131C2×D5—42no—
132C4×C5—22no—
133C4×C5—41no—
134C4×C5—42no—
135C4×C5—41no—
136C4×C5—42no—
137Dic5—42no—
138Dic5—42no—
139Dic5—42no—
140C24—52no—
141C2×Q8—42no—
142C2×Q8—41no—
143C2×D4—102no—
144C2×D4—201no—
145C2×D4—201no—
146C22×C4—52no—
147C22×C4—52no—
148C22×C4—52no—
149C22×C4C22×D512yes—
150C22×C4C22×D511yescharacteristic
151C22×C4C22×D512yes—
152C22×C4C22×D511yescharacteristic
153C22×C4C22×D511yescharacteristic
154C4⋊C4—41no—
155C4⋊C4—42no—
156C4⋊C4—104no—
157C4⋊C4—202no—
158C4⋊C4—201no—
159C22⋊C4—202no—
160C22⋊C4—202no—
161C22⋊C4—202no—
162C22⋊C4—202no—
163C22⋊C4—202no—
164C22⋊C4—202no—
165C42—201no—
166C2×C5#53811yescharacteristic
167C2×C5#54212yes—
168C2×C5#56011yescharacteristic
169C2×C5#56212yes—
170C2×C5#56311yescharacteristic
171D5—82no—
172C23—102no—
173C23—102no—
174C23—102no—
175C23—102no—
176C23—102no—
177C23—102no—
178C23—102no—
179C23C23×D511yescharacteristic
180Q8—44no—
181D4—204no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—102no—
195C2×C4—102no—
196C2×C4—102no—
197C2×C4—102no—
198C2×C4—102no—
199C2×C4—102no—
200C2×C4—102no—
201C2×C4—102no—
202C2×C4—102no—
203C2×C4#71312yes—
204C5#283011yescharacteristic, complemented, Sylow
205C22—202no—
206C22—202no—
207C22—202no—
208C22—202no—
209C22—202no—
210C22—202no—
211C22—202no—
212C22#414412yes—
213C22#414411yescharacteristic
214C22C2×#71311yescharacteristic
215C22#414712yes—
216C22#414911yescharacteristic
217C4—22no—
218C4—41no—
219C4—42no—
220C4—41no—
221C4—42no—
222C4—202no—
223C4—202no—
224C4—202no—
225C2—402no—
226C2#6592911yescharacteristic
227C2#6597811yescharacteristic
228C2#6598112yes—
229C2#6606712yes—
230C2#6606911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#395C2
prime radicalC22×D20C22
hypercenter#395D5
hypocenterC5#2830
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C5×#395C2semidirect
2. #65520C2semidirect
3. #65283C2semidirect
4. #65282C2semidirect
6. #65272C2semidirect
7. #64964C2semidirect
9. #64941C2semidirect
62. #395D5semidirect
204. C5#2830semidirect

Maximal subgroups

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

subgroupindex
C5×#3952
#655202
#652832
#652822
#652772
#652722
#649642
#649612
#649412
#28305