Finite Group Databasebeta

← #10604320

Subgroups — #10604320

Order 640.

408 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C22×#700C211yesmaximal, characteristic, complemented
2C2×#4039C211yesmaximal, characteristic
3C2×#4003C211yesmaximal, characteristic, complemented
4C2×#4003C212yesmaximal, complemented
5#64918C212yesmaximal, complemented
6C22×D20C411yescharacteristic
7C22×C4×D5C2211yescharacteristic
8C22×C4×D5C2211yescharacteristic
9C22×#170C2212yes—
10C22×#170C2212yes—
11C2×#700—22no—
12C2×#700—24no—
13C2×C5×C4⋊C4C411yescharacteristic
14C2×#685C411yescharacteristic
15C2×#685C411yescharacteristic
16C2×C4×Dic5C2211yescharacteristic
17#4039—22no—
18#4039—22no—
19#4003—22no—
20#4003—24no—
21#1870—51nomaximal, Sylow
22C23×D5C2×C411yescharacteristic
23C23×D5C2×C411yescharacteristic
24C22×C4×C5C2×C411yescharacteristic
25C22×C4×C5C2×C411yescharacteristic
26C22×C4×C5C2×C411yescharacteristic
27C22×Dic5C2×C411yescharacteristic
28C22×Dic5C2311yescharacteristic
29C2×D20—22no—
30C2×D20—22no—
31C2×D20—22no—
32C2×C4×D5—22no—
33C2×C4×D5—22no—
34C2×C4×D5—22no—
35C2×C4×D5—21no—
36C2×C4×D5—21no—
37C2×C4×D5—22no—
38C2×C4×D5D412yes—
39C2×C4×D5D412yes—
40C2×#170—22no—
41C2×#170—22no—
42C2×#170—22no—
43C2×#170—22no—
44C2×#170—22no—
45C2×#170D414yescomplemented
46#700—28no—
47C5×C4⋊C4—22no—
48#685—22no—
49#685—22no—
50C4×Dic5—21no—
51C4×Dic5—21no—
52C22×M16—51no—
53C2×C4×D4—51no—
54C2×C22⋊C8—52no—
55C2×C22⋊C8—51no—
56#336—52no—
57C23×C5C22×C411yescharacteristic
58C22×D5—21no—
59C22×D5—22no—
60C22×D5—22no—
61C22×D5—21no—
62C22×D5—21no—
63C22×D5—21no—
64C22×D5—22no—
65C22×D5—22no—
66C22×D5C22⋊C412yes—
67C22×D5M1612yescomplemented
68C2×C4×C5—21no—
69C2×C4×C5—22no—
70C2×C4×C5—21no—
71C2×C4×C5—21no—
72C2×C4×C5—22no—
73C2×C4×C5C22⋊C412yes—
74C2×C4×C5M1612yescomplemented
75C2×Dic5—21no—
76C2×Dic5—22no—
77C2×Dic5C2×D412yes—
78C2×Dic5C2×D411yescharacteristic
79C2×Dic5C2×D411yescharacteristic
80C2×Dic5#4112yes—
81D20—28no—
82C4×D5—22no—
83C4×D5—24no—
84C4×D5—22no—
85C4×D5—41no—
86C4×D5—42no—
87C4×D5—41no—
88#170—24no—
89#170—42no—
90C22×D4—51no—
91C23×C4—51no—
92C23×C4—51no—
93C2×M16—104no—
94C2×M16—102no—
95C22×C8—52no—
96C22×C8—52no—
97C4×D4—102no—
98C4×D4—102no—
99C2×C4⋊C4AGL(1,𝔽5)11yescharacteristic
100C2×C22⋊C4—51no—
101C2×C22⋊C4—51no—
102C2×C42—51no—
103C22⋊C8—104no—
104C22⋊C8—102no—
105C22×C5C2×C22⋊C411yescharacteristic
106C22×C5C4×D412yes—
107C22×C5C2×M1611yescharacteristic
108C22×C5C2×M1611yescharacteristic
109C22×C5#13112yes—
110C2×D5—22no—
111C2×D5—22no—
112C2×D5—22no—
113C2×D5—22no—
114C2×D5—24no—
115C2×D5—41no—
116C2×D5—41no—
117C2×D5—42no—
118C2×D5—42no—
119C2×D5—42no—
120C4×C5—22no—
121C4×C5—22no—
122C4×C5—41no—
123Dic5—41no—
124Dic5#12011yescharacteristic
125Dic5#12112yes—
126Dic5#12311yescharacteristic
127C24—51no—
128C24—51no—
129C2×D4—102no—
130C2×D4—102no—
131C2×D4—102no—
132C22×C4—51no—
133C22×C4—52no—
134C22×C4—52no—
135C22×C4—51no—
136C22×C4—102no—
137C22×C4—102no—
138C22×C4—102no—
139C22×C4—101no—
140C22×C4—101no—
141C22×C4—102no—
142C22×C4C2×AGL(1,𝔽5)11yescharacteristic
143C22×C4C2×AGL(1,𝔽5)11yescharacteristic
144C22×C4C2×AGL(1,𝔽5)11yescharacteristic
145M16—108no—
146C2×C8—54no—
147C2×C8—102no—
148C2×C8—102no—
149C2×C8—102no—
150C2×C8—102no—
151C2×C8—102no—
152C4⋊C4—22no—
153C22⋊C4—102no—
154C22⋊C4—102no—
155C42—101no—
156C42—101no—
157C2×C5#38611yescharacteristic
158C2×C5#40711yescharacteristic
159C2×C5#40811yescharacteristic
160C2×C5#43512yes—
161C2×C5#43612yes—
162D5—42no—
163D5—42no—
164C23—52no—
165C23—52no—
166C23—101no—
167C23—101no—
168C23—101no—
169C23—102no—
170C23—102no—
171C23—102no—
172C23—102no—
173C23—101no—
174C23C22×AGL(1,𝔽5)11yescharacteristic
175D4—108no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—52no—
182C2×C4—52no—
183C2×C4—51no—
184C2×C4—51no—
185C2×C4—102no—
186C2×C4—104no—
187C2×C4—102no—
188C2×C4—102no—
189C2×C4—101no—
190C2×C4—201no—
191C2×C4—202no—
192C2×C4—201no—
193C2×C4#70012yescomplemented
194C2×C4C22⋊AGL(1,𝔽5)12yes—
195C8—104no—
196C8—202no—
197C5#187011yescharacteristic, complemented, Sylow
198C22—104no—
199C22—102no—
200C22—102no—
201C22—102no—
202C22—102no—
203C22—202no—
204C22—202no—
205C22—202no—
206C22—201no—
207C22—201no—
208C22C2×#70011yescharacteristic
209C22D5⋊M1611yescharacteristic
210C22D4×AGL(1,𝔽5)12yes—
211C22#413312yes—
212C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
213C4—22no—
214C4—22no—
215C4—41no—
216C4—51no—
217C4—52no—
218C4—51no—
219C4—201no—
220C2—202no—
221C2—202no—
222C2#6570412yes—
223C2#6570512yes—
224C2#6575611yescharacteristic
225C2#6577911yescharacteristic
226C2#6578711yescharacteristic
subgroupG/H
center, FrattiniC23C22×AGL(1,𝔽5)
commutator, socleC23×C5C22×C4
FittingC2×C5×C4⋊C4C4
prime radicalC22×D20C4
cosocleC22×Dic5C23
hypercenterC2×C4⋊C4AGL(1,𝔽5)
hypocenterC5#1870
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#700C2semidirect
3. C2×#4003C2semidirect
4. C2×#4003C2semidirect
5. #64918C2semidirect
45. C2×#170D4semidirect
67. C22×D5M16semidirect
74. C2×C4×C5M16semidirect
193. C2×C4#700semidirect
197. C5#1870semidirect

Maximal subgroups

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

subgroupindex
C22×#7002
C2×#40392
C2×#40032
#649182
#18705