Finite Group Databasebeta

← #10614980

Subgroups — #10614980

Order 640.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×C5×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#4093C211yesmaximal, characteristic
3#65520C212yesmaximal, complemented
4#65519C211yesmaximal, characteristic, complemented
5#65283C212yesmaximal, complemented
6#65280C212yesmaximal
7#65279C212yesmaximal, complemented
8#65267C212yesmaximal, complemented
9#65229C212yesmaximal, complemented
10C22×C5×Q8C2211yescharacteristic
11C22×C5×Q8C2211yescharacteristic
12C22×D20C2211yescharacteristic
13C5×C4⋊Q8—24no—
14C2×C5×C4⋊C4C2212yes—
15C2×C5×C4⋊C4C2212yes—
16C2×C42×C5C2211yescharacteristic
17#4093—24no—
18C2×#685C2212yes—
19C2×#685C2212yes—
20C2×#685C2212yes—
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×C4⋊Dic5C2211yescharacteristic
25C2×#683C2212yes—
26C2×C4×Dic5C2211yescharacteristic
27C2×C4×Dic5C2211yescharacteristic
28#3963C2212yes—
29#3963C2212yes—
30#3963C2212yes—
31#3963C2212yes—
32#3963C2212yes—
33#2731—51nomaximal, Sylow
34C23×D5C2312yes—
35C2×C5×Q8—22no—
36C2×C5×Q8—41no—
37C2×C5×Q8—41no—
38C2×C5×Q8—41no—
39C2×C5×Q8—41no—
40C2×C5×Q8D414yes—
41C22×C4×C5C2312yes—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2312yes—
46C22×Dic5C2312yes—
47C22×Dic5C2312yes—
48C22×Dic5C2312yes—
49C2×D20—22no—
50C2×D20—41no—
51C2×D20—41no—
52C5×C4⋊C4—24no—
53C5×C4⋊C4—24no—
54C42×C5—22no—
55#685—24no—
56#685—24no—
57#685—24no—
58#685—24no—
59#685—42no—
60#685—42no—
61C4⋊Dic5—41no—
62#683—24no—
63C4×Dic5—22no—
64C4×Dic5—41no—
65C2×C4⋊Q8D511yescharacteristic, complemented
66C2×#124—51no—
67#396—52no—
68#394—52no—
69#393—52no—
70#391—51no—
71#388—52no—
72#388—52no—
73#387—52no—
74C23×C5C2411yescharacteristic
75C22×D5—22no—
76C22×D5—22no—
77C22×D5—22no—
78C22×D5—22no—
79C22×D5—22no—
80C22×D5—22no—
81C22×D5—22no—
82C5×Q8—44no—
83C5×Q8—44no—
84C2×C4×C5—22no—
85C2×C4×C5—21no—
86C2×C4×C5—21no—
87C2×C4×C5—21no—
88C2×C4×C5—21no—
89C2×C4×C5—21no—
90C2×C4×C5—21no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5C2×D412yes—
96C2×C4×C5C2×D412yes—
97C2×C4×C5C2×D412yes—
98C2×C4×C5#4114yes—
99C2×Dic5—22no—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—22no—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108D20—44no—
109C22×Q8C2×D511yescharacteristic
110C22×Q8C2×D511yescharacteristic
111C22×D4—51no—
112C4⋊Q8—24no—
113#124—104no—
114C2×C4⋊C4—52no—
115C2×C4⋊C4—51no—
116C2×C4⋊C4C2×D512yes—
117C2×C4⋊C4C2×D512yes—
118C2×C22⋊C4—52no—
119C2×C22⋊C4—52no—
120C2×C22⋊C4—52no—
121C2×C22⋊C4—52no—
122C2×C22⋊C4—52no—
123C2×C22⋊C4—52no—
124C2×C42—51no—
125C2×C42—51no—
126C2×C42C2×D511yescharacteristic
127#95—52no—
128#95—52no—
129#95—52no—
130#95—52no—
131#95—52no—
132C22×C5C22×D411yescharacteristic
133C22×C5C2×#4112yes—
134C22×C5H(5,𝔽2)11yescharacteristic
135C22×C5H(5,𝔽2)11yescharacteristic
136C22×C52−1+412yes—
137C2×D5—42no—
138C2×D5—42no—
139C2×D5—42no—
140C2×D5—42no—
141C2×D5—42no—
142C2×D5—42no—
143C2×D5—42no—
144C4×C5—22no—
145C4×C5—22no—
146C4×C5—22no—
147C4×C5—24no—
148C4×C5—42no—
149Dic5—42no—
150Dic5—42no—
151Dic5—42no—
152C24—52no—
153C2×Q8—22no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8C22⋊D514yes—
159C2×D4—102no—
160C2×D4—201no—
161C2×D4—201no—
162C22×C4—52no—
163C22×C4—52no—
164C22×C4—52no—
165C22×C4C22×D512yes—
166C22×C4C22×D512yes—
167C22×C4C22×D511yescharacteristic
168C22×C4C22×D511yescharacteristic
169C22×C4C22×D511yescharacteristic
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—104no—
173C4⋊C4—201no—
174C22⋊C4—104no—
175C22⋊C4—104no—
176C22⋊C4—104no—
177C22⋊C4—104no—
178C22⋊C4—202no—
179C22⋊C4—202no—
180C42—22no—
181C42—102no—
182C42—201no—
183C2×C5#53411yescharacteristic
184C2×C5#54212yes—
185C2×C5#54812yes—
186C2×C5#55511yescharacteristic
187C2×C5#56311yescharacteristic
188D5—82no—
189C23—102no—
190C23—102no—
191C23—102no—
192C23—102no—
193C23—102no—
194C23—102no—
195C23—102no—
196C23C23×D511yescharacteristic
197Q8—44no—
198Q8—44no—
199D4—204no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—102no—
212C2×C4—102no—
213C2×C4—102no—
214C2×C4—102no—
215C2×C4—102no—
216C2×C4—102no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4—102no—
220C2×C4#71314yes—
221C2×C4C2×C22⋊D512yes—
222C2×C4C2×C22⋊D512yes—
223C2×C4C2×C22⋊D512yes—
224C5#273111yescharacteristic, complemented, Sylow
225C22—202no—
226C22—202no—
227C22—202no—
228C22—202no—
229C22—202no—
230C22—202no—
231C22—202no—
232C22#414411yescharacteristic
233C22#414411yescharacteristic
234C22C2×#71312yes—
235C22#414712yes—
236C22C22×C22⋊D511yescharacteristic
237C4—22no—
238C4—22no—
239C4—22no—
240C4—24no—
241C4—42no—
242C4—202no—
243C4—202no—
244C4—202no—
245C2—402no—
246C2#6606712yes—
247C2#6606811yescharacteristic
248C2#6606911yescharacteristic
249C2#6613811yescharacteristic
250C2#6615112yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C4⋊Q8C2
prime radicalC22×D20C22
hypercenterC2×C4⋊Q8D5
hypocenterC5#2731
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C5×C4⋊Q8C2semidirect
3. #65520C2semidirect
4. #65519C2semidirect
5. #65283C2semidirect
7. #65279C2semidirect
8. #65267C2semidirect
9. #65229C2semidirect
65. C2×C4⋊Q8D5semidirect
224. C5#2731semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C4⋊Q82
C2×#40932
#655202
#655192
#652832
#652802
#652792
#652672
#652292
#27315