Finite Group Databasebeta

← #10609003

Subgroups — #10609003

Order 640.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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#65277C211yesmaximal, characteristic
5#65276C212yesmaximal, complemented
6#65275C211yesmaximal, characteristic, complemented
7#64957C212yesmaximal, complemented
8#64954C212yesmaximal
9#64944C212yesmaximal, complemented
10C22×C5×Q8C2211yescharacteristic
11C22×D20C2211yescharacteristic
12C22×C4×D5C2212yes—
13C2×C5×C4⋊C4C2212yes—
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×#685C2212yes—
16C2×#685C2212yes—
17C2×#685C2212yes—
18C2×#685C2212yes—
19C2×#685C2212yes—
20C2×C4⋊Dic5C2211yescharacteristic
21C2×C4⋊Dic5C2212yes—
22C2×#683C2212yes—
23C2×C4×Dic5C2211yescharacteristic
24C5×#95C2212yes—
25C5×#95C2211yescharacteristic
26#3963C2212yes—
27#3963C2212yes—
28#3963C2212yes—
29#3963C2212yes—
30#3963C2211yescharacteristic
31#2784—51nomaximal, Sylow
32C23×D5C2312yes—
33C2×C5×Q8—42no—
34C2×C5×Q8—41no—
35C22×C4×C5C2311yescharacteristic
36C22×C4×C5C2312yes—
37C22×C4×C5C2311yescharacteristic
38C22×C4×C5C2311yescharacteristic
39C22×C4×C5C2312yes—
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C22×Dic5C2312yes—
43C2×D20—41no—
44C2×D20—41no—
45C2×D20Q814yescomplemented
46C2×C4×D5—24no—
47C2×C4×D5—42no—
48C2×C4×D5—42no—
49C5×C4⋊C4—42no—
50C5×C4⋊C4—41no—
51#685—24no—
52#685—42no—
53#685—42no—
54#685—42no—
55#685—42no—
56C4⋊Dic5—42no—
57C4⋊Dic5—41no—
58#683—24no—
59C4×Dic5—41no—
60#400—51no—
61#396—52no—
62#395D511yescharacteristic, complemented
63#393—52no—
64#390—51no—
65#387—52no—
66#386—52no—
67#385—52no—
68#380—52no—
69C23×C5C2411yescharacteristic
70C22×D5—22no—
71C22×D5—22no—
72C22×D5—22no—
73C22×D5—22no—
74C22×D5—22no—
75C22×D5—22no—
76C22×D5C2×Q814yes—
77C5×Q8—44no—
78C2×C4×C5—21no—
79C2×C4×C5—22no—
80C2×C4×C5—22no—
81C2×C4×C5—21no—
82C2×C4×C5—21no—
83C2×C4×C5—22no—
84C2×C4×C5—22no—
85C2×C4×C5—22no—
86C2×C4×C5—22no—
87C2×C4×C5—22no—
88C2×C4×C5—21no—
89C2×C4×C5C2×Q812yes—
90C2×C4×C5#4114yes—
91C2×Dic5—22no—
92C2×Dic5—22no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5—22no—
96C2×Dic5—22no—
97C2×Dic5—22no—
98C2×Dic5—22no—
99C2×Dic5—22no—
100D20—44no—
101C4×D5—48no—
102C22×Q8C2×D511yescharacteristic
103C22×D4—51no—
104C23×C4—52no—
105C2×C4⋊C4—52no—
106C2×C4⋊C4—52no—
107C2×C4⋊C4—51no—
108C2×C4⋊C4C2×D512yes—
109C2×C4⋊C4C2×D511yescharacteristic
110C2×C22⋊C4—52no—
111C2×C22⋊C4—52no—
112C2×C22⋊C4—52no—
113C2×C22⋊C4—52no—
114C2×C22⋊C4—52no—
115C2×C42—51no—
116#95—51no—
117#95—52no—
118#95—52no—
119#95—52no—
120#95—52no—
121#95C2×D511yescharacteristic
122#95C2×D512yes—
123C22×C5C22×Q811yescharacteristic
124C22×C5C2×#4112yes—
125C22×C5H(5,𝔽2)12yes—
126C22×C5H(5,𝔽2)11yescharacteristic
127C22×C5H(5,𝔽2)11yescharacteristic
128C2×D5—24no—
129C2×D5—24no—
130C2×D5—24no—
131C2×D5—42no—
132C2×D5—42no—
133C2×D5—42no—
134C2×D5—42no—
135C4×C5—24no—
136C4×C5—22no—
137C4×C5—41no—
138C4×C5—41no—
139C4×C5—42no—
140Dic5—42no—
141Dic5—42no—
142Dic5—42no—
143C24—52no—
144C2×Q8—42no—
145C2×Q8—41no—
146C2×D4—54no—
147C2×D4—201no—
148C2×D4—201no—
149C22×C4—52no—
150C22×C4—52no—
151C22×C4—52no—
152C22×C4—104no—
153C22×C4—202no—
154C22×C4—202no—
155C22×C4C22×D512yes—
156C22×C4C22×D511yescharacteristic
157C22×C4C22×D511yescharacteristic
158C22×C4C22×D511yescharacteristic
159C22×C4C22×D512yes—
160C4⋊C4—42no—
161C4⋊C4—41no—
162C4⋊C4—104no—
163C4⋊C4—202no—
164C4⋊C4—201no—
165C22⋊C4—104no—
166C22⋊C4—202no—
167C22⋊C4—202no—
168C22⋊C4—202no—
169C22⋊C4—202no—
170C42—201no—
171C2×C5#53812yes—
172C2×C5#54311yescharacteristic
173C2×C5#55412yes—
174C2×C5#55511yescharacteristic
175C2×C5#56011yescharacteristic
176D5—44no—
177C23—54no—
178C23—102no—
179C23—102no—
180C23—102no—
181C23—102no—
182C23—102no—
183C23—102no—
184C23C23×D511yescharacteristic
185Q8—44no—
186D4—204no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—102no—
199C2×C4—102no—
200C2×C4—102no—
201C2×C4—102no—
202C2×C4—102no—
203C2×C4—102no—
204C2×C4—102no—
205C2×C4—102no—
206C2×C4—102no—
207C2×C4—208no—
208C2×C4Q8×D512yes—
209C2×C4#71314yes—
210C5#278411yescharacteristic, complemented, Sylow
211C22—104no—
212C22—104no—
213C22—104no—
214C22—202no—
215C22—202no—
216C22—202no—
217C22—202no—
218C22#414412yes—
219C22#414411yescharacteristic
220C22C2×Q8×D511yescharacteristic
221C22C2×#71312yes—
222C22#414911yescharacteristic
223C4—22no—
224C4—24no—
225C4—41no—
226C4—42no—
227C4—41no—
228C4—202no—
229C4—202no—
230C4—202no—
231C2—204no—
232C2#6592911yescharacteristic
233C2#6596811yescharacteristic
234C2#6597812yes—
235C2#6606412yes—
236C2#6606811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#395C2
prime radicalC22×D20C22
hypercenter#395D5
hypocenterC5#2784
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
5. #65276C2semidirect
6. #65275C2semidirect
7. #64957C2semidirect
9. #64944C2semidirect
45. C2×D20Q8semidirect
62. #395D5semidirect
210. C5#2784semidirect

Maximal subgroups

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

subgroupindex
C5×#3952
#655202
#652832
#652772
#652762
#652752
#649572
#649542
#649442
#27845