Finite Group Databasebeta

← #10604205

Subgroups — #10604205

Order 640.

280 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 62 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×#4032C211yesmaximal, characteristic
2#65798C211yesmaximal, characteristic, complemented
3C2×C22⋊#170C211yesmaximal, characteristic, complemented
4#64915C212yesmaximal, complemented
5#64913C212yesmaximal, complemented
6C22×C22⋊D5C411yescharacteristic
7C23×Dic5C2211yescharacteristic
8C2×#704—21no—
9C22×#170—21no—
10C2×C5×C22⋊C4C411yescharacteristic
11C2×#685C411yescharacteristic
12C2×#685C2211yescharacteristic
13C2×C4⋊Dic5C411yescharacteristic
14C2×C4⋊Dic5C2211yescharacteristic
15#4032—21no—
16#4032—21no—
17#4032—21no—
18#4032—21no—
19C22⋊#170—21no—
20C22⋊#170—21no—
21C22⋊#170C2212yes—
22C22⋊#170C2212yes—
23#1468—51nomaximal, Sylow
24C24×C5C2×C411yescharacteristic
25C23×D5C2×C411yescharacteristic
26C22×C4×C5C2×C411yescharacteristic
27C22×C4×C5C2×C411yescharacteristic
28C2×C22⋊D5—21no—
29C2×C22⋊D5—21no—
30C2×C22⋊D5—42no—
31C22×Dic5—42no—
32C22×Dic5C2×C411yescharacteristic
33C22×Dic5C2×C411yescharacteristic
34C22×Dic5D411yescharacteristic
35C22×Dic5D411yescharacteristic
36C22×Dic5D411yescharacteristic
37C22×Dic5D411yescharacteristic
38C22×Dic5C2311yescharacteristic
39#704—42no—
40C2×#170—22no—
41C2×#170—22no—
42C2×#170—42no—
43C5×C22⋊C4—21no—
44C5×C22⋊C4—21no—
45#685—21no—
46#685—21no—
47#685—21no—
48#685—21no—
49C4⋊Dic5—21no—
50C4⋊Dic5—21no—
51C4⋊Dic5—21no—
52C4⋊Dic5—21no—
53C2×#123—51no—
54#407—51no—
55C2×C22⋊C8—51no—
56#324—52no—
57#323—52no—
58C23×C5—22no—
59C23×C5—22no—
60C23×C5—41no—
61C23×C5C22⋊C411yescharacteristic
62C23×C5C22⋊C411yescharacteristic
63C23×C5C22×C411yescharacteristic
64C22×D5—21no—
65C22×D5—22no—
66C22×D5—41no—
67C22×D5—41no—
68C2×C4×C5—21no—
69C2×C4×C5—21no—
70C2×C4×C5—22no—
71C2×C4×C5—41no—
72C22⋊D5—42no—
73C22⋊D5—42no—
74C2×Dic5—22no—
75C2×Dic5—22no—
76C2×Dic5—21no—
77C2×Dic5—41no—
78C2×Dic5—41no—
79C2×Dic5—41no—
80C2×Dic5—41no—
81C2×Dic5—41no—
82C2×Dic5C22⋊C411yescharacteristic
83C2×Dic5C22⋊C411yescharacteristic
84C2×Dic5C2×D411yescharacteristic
85C2×Dic5C2×D411yescharacteristic
86#170—42no—
87#170—42no—
88C22×D4—51no—
89C23×C4—51no—
90C2×M16—101no—
91C22×C8—101no—
92#123—101no—
93#123—101no—
94#123—101no—
95#123—101no—
96C2×C4⋊C4—51no—
97C2×C4⋊C4—51no—
98C2×C22⋊C4—51no—
99C2×C22⋊C4—51no—
100C2×C22⋊C4AGL(1,𝔽5)11yescharacteristic
101C22⋊C8—52no—
102C22⋊C8—52no—
103C22⋊C8—101no—
104C22⋊C8—101no—
105C22×C5—21no—
106C22×C5—21no—
107C22×C5—21no—
108C22×C5—21no—
109C22×C5—21no—
110C22×C5—21no—
111C22×C5—21no—
112C22×C5—44no—
113C22×C5#9912yes—
114C22×C5C2×C22⋊C411yescharacteristic
115C22×C5#13112yes—
116C2×D5—41no—
117C2×D5—41no—
118C2×D5—41no—
119C2×D5—82no—
120C4×C5—41no—
121C4×C5—41no—
122Dic5—21no—
123Dic5—21no—
124Dic5—21no—
125Dic5—21no—
126Dic5—41no—
127C24—51no—
128C24C2×AGL(1,𝔽5)11yescharacteristic
129C2×D4—101no—
130C2×D4—101no—
131C2×D4—202no—
132C22×C4—51no—
133C22×C4—51no—
134C22×C4—51no—
135C22×C4—51no—
136C22×C4—51no—
137C22×C4—51no—
138C22×C4—51no—
139C22×C4—202no—
140C22×C4C2×AGL(1,𝔽5)11yescharacteristic
141C22×C4C2×AGL(1,𝔽5)11yescharacteristic
142M16—202no—
143C2×C8—102no—
144C2×C8—102no—
145C2×C8—202no—
146C4⋊C4—101no—
147C4⋊C4—101no—
148C4⋊C4—101no—
149C4⋊C4—101no—
150C22⋊C4—21no—
151C22⋊C4—21no—
152C22⋊C4—101no—
153C22⋊C4—101no—
154C22⋊C4—101no—
155C22⋊C4—101no—
156C2×C5—22no—
157C2×C5—22no—
158C2×C5—41no—
159C2×C5#40811yescharacteristic
160C2×C5C2×#9911yescharacteristic
161C2×C5#41311yescharacteristic
162D5—81no—
163C23—22no—
164C23—22no—
165C23—41no—
166C23—101no—
167C23—102no—
168C23—201no—
169C23—201no—
170C23C22⋊AGL(1,𝔽5)11yescharacteristic
171C23C22⋊AGL(1,𝔽5)11yescharacteristic
172C23C22×AGL(1,𝔽5)11yescharacteristic
173D4—202no—
174D4—202no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—51no—
180C2×C4—51no—
181C2×C4—51no—
182C2×C4—51no—
183C2×C4—102no—
184C2×C4—102no—
185C2×C4—101no—
186C2×C4—201no—
187C2×C4—201no—
188C2×C4—201no—
189C2×C4—201no—
190C2×C4—201no—
191C8—202no—
192C8—202no—
193C5#146811yescharacteristic, complemented, Sylow
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—201no—
203C22—201no—
204C22—201no—
205C22—402no—
206C22#399912yes—
207C22#413112yes—
208C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
209C4—41no—
210C4—41no—
211C4—101no—
212C4—101no—
213C4—101no—
214C4—101no—
215C4—201no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—401no—
220C2C2×#399911yescharacteristic
221C2#6577511yescharacteristic
222C2#6577711yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C22×C4
FrattiniC23C22×AGL(1,𝔽5)
FittingC2×C5×C22⋊C4C4
socleC22×C5C2×C22⋊C4
prime radicalC22×C22⋊D5C4
cosocleC22×Dic5C23
hypercenterC2×C22⋊C4AGL(1,𝔽5)
hypocenterC5#1468
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #65798C2semidirect
3. C2×C22⋊#170C2semidirect
4. #64915C2semidirect
5. #64913C2semidirect
193. C5#1468semidirect

Maximal subgroups

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

subgroupindex
C2×#40322
#657982
C2×C22⋊#1702
#649152
#649132
#14685