Finite Group Databasebeta

← #10599656

Subgroups — #10599656

Order 640.

324 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 85 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×#677C211yesmaximal, characteristic
2C2×#3946C211yesmaximal, characteristic, complemented
3#65277C211yesmaximal, characteristic
4#65263C211yesmaximal, characteristic, complemented
5C2×#3942C211yesmaximal, characteristic, complemented
6C2×#3941C211yesmaximal, characteristic
7C5×#340C211yesmaximal, characteristic, complemented
8C22×D20C2211yescharacteristic
9C22×Dic10C2211yescharacteristic
10C22×C5×C8C2211yescharacteristic
11C2×C5×C4⋊C4C2211yescharacteristic
12C2×C5×C4⋊C4C2211yescharacteristic
13C2×#685—21no—
14C2×#685—21no—
15C2×C4×Dic5C2211yescharacteristic
16C22×#168C2211yescharacteristic
17C2×#677—22no—
18C2×#677—22no—
19C2×#677—22no—
20#3963—21no—
21#3963—21no—
22#3946—22no—
23#3942—22no—
24#3941—22no—
25#2055—51nomaximal, Sylow
26C23×D5—21no—
27C22×C4×C5—21no—
28C22×C4×C5—21no—
29C22×C4×C5C2311yescharacteristic
30C22×Dic5—21no—
31C22×Dic5D411yescharacteristic
32C22×Dic5D411yescharacteristic
33C2×D20—21no—
34C2×D20—21no—
35C2×D20—21no—
36C2×D20—21no—
37C2×D20—21no—
38C2×D20D412yes—
39C2×Dic10—21no—
40C2×Dic10—21no—
41C2×Dic10—21no—
42C2×Dic10—21no—
43C2×Dic10—21no—
44C2×Dic10D412yes—
45C2×C5×C8—21no—
46C2×C5×C8—21no—
47C2×C5×C8D412yes—
48C5×C4⋊C4—21no—
49C5×C4⋊C4—21no—
50C5×C4⋊C4—21no—
51C5×C4⋊C4—21no—
52#685—42no—
53#685—42no—
54C4×Dic5—21no—
55C4×Dic5—21no—
56C2×#168—21no—
57C2×#168—21no—
58C2×#168—21no—
59#677—28no—
60C22×SD16—51no—
61C2×C4⋊C8—51no—
62C2×#103—51no—
63C2×#102—51no—
64#391—51no—
65#390—51no—
66#340D511yescharacteristic, complemented
67C23×C5C2×D411yescharacteristic
68C22×D5—22no—
69C22×D5—41no—
70C22×D5—41no—
71C22×D5—41no—
72C22×D5—41no—
73C22×D5—41no—
74C22×D5—41no—
75C2×C4×C5—41no—
76C2×C4×C5—41no—
77C2×C4×C5—41no—
78C2×C4×C5—41no—
79C2×C4×C5—41no—
80C2×C4×C5—41no—
81C2×C4×C5C2×D411yescharacteristic
82C2×C4×C5C2×D411yescharacteristic
83C2×C4×C5C2×D411yescharacteristic
84C2×C4×C5#4111yescharacteristic
85C2×C4×C5#4111yescharacteristic
86C2×C4×C5#4111yescharacteristic
87C2×Dic5—21no—
88C2×Dic5—21no—
89C2×Dic5—21no—
90C2×Dic5—22no—
91C2×Dic5—21no—
92C2×Dic5—21no—
93C2×Dic5—41no—
94C2×Dic5—41no—
95C2×Dic5SD1612yescomplemented
96D20—22no—
97D20—22no—
98D20—41no—
99D20—41no—
100Dic10—22no—
101Dic10—22no—
102Dic10—41no—
103Dic10—41no—
104C5×C8—22no—
105#168—41no—
106C22×Q8—51no—
107C22×D4—51no—
108C2×SD16—102no—
109C2×SD16—102no—
110C2×SD16—102no—
111C22×C8—51no—
112C22×C8C2×D511yescharacteristic
113C2×C4⋊C4C2×D511yescharacteristic
114C2×C4⋊C4C2×D511yescharacteristic
115C2×C22⋊C4—101no—
116C2×C22⋊C4—101no—
117C2×C42—51no—
118C4⋊C8—102no—
119#103—102no—
120#102—102no—
121#95—101no—
122#95—101no—
123C22×C5#12111yescharacteristic
124C22×C5#12111yescharacteristic
125C22×C5#12111yescharacteristic
126C22×C5C2×SD1611yescharacteristic
127C22×C5#13511yescharacteristic
128C22×C5AGL(1,ℤ/8)11yescharacteristic
129C22×C5#13711yescharacteristic
130C2×D5—42no—
131C2×D5—42no—
132C2×D5—42no—
133C2×D5—81no—
134C2×D5—81no—
135C2×D5—81no—
136C2×D5—81no—
137C4×C5—81no—
138C4×C5—81no—
139C4×C5#12011yescharacteristic
140C4×C5#12311yescharacteristic
141C4×C5#12311yescharacteristic
142C4×C5#12411yescharacteristic
143Dic5—22no—
144Dic5—42no—
145Dic5—41no—
146C24—101no—
147C2×Q8—52no—
148C2×Q8—101no—
149C2×Q8—101no—
150C2×Q8—101no—
151C2×Q8—101no—
152C2×Q8—101no—
153C2×D4—52no—
154C2×D4—101no—
155C2×D4—101no—
156C2×D4—101no—
157C2×D4—101no—
158C2×D4—101no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—51no—
162C22×C4—51no—
163C22×C4—101no—
164C22×C4C22×D511yescharacteristic
165SD16—108no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—101no—
169C2×C8—101no—
170C2×C8—101no—
171C2×C8D2012yes—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C22⋊C4—202no—
177C22⋊C4—202no—
178C42—101no—
179C42—101no—
180C2×C5#39411yescharacteristic
181C2×C5#46011yescharacteristic
182C2×C5#46111yescharacteristic
183C2×C5#46311yescharacteristic
184C2×C5#46411yescharacteristic
185C2×C5#46511yescharacteristic
186C2×C5#46811yescharacteristic
187D5—82no—
188C23—102no—
189C23—201no—
190C23—201no—
191C23—201no—
192C23—201no—
193C23—201no—
194C23—201no—
195C23D4×D511yescharacteristic
196Q8—102no—
197Q8—102no—
198Q8—201no—
199Q8—201no—
200D4—102no—
201D4—102no—
202D4—201no—
203D4—201no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—52no—
211C2×C4—101no—
212C2×C4—101no—
213C2×C4—102no—
214C2×C4—101no—
215C2×C4—101no—
216C2×C4—101no—
217C2×C4—201no—
218C2×C4—201no—
219C2×C4C2×D2011yescharacteristic
220C2×C4#70911yescharacteristic
221C2×C4#70911yescharacteristic
222C2×C4D4×D511yescharacteristic
223C2×C4D4×D511yescharacteristic
224C2×C4#71311yescharacteristic
225C8—22no—
226C8—201no—
227C5#205511yescharacteristic, complemented, Sylow
228C22—202no—
229C22—202no—
230C22—202no—
231C22—401no—
232C22—401no—
233C22—401no—
234C22—401no—
235C22#403011yescharacteristic
236C22#403011yescharacteristic
237C22#404111yescharacteristic
238C22#405711yescharacteristic
239C22D5×SD1611yescharacteristic
240C22#406211yescharacteristic
241C22#406411yescharacteristic
242C4—81no—
243C4—81no—
244C4—102no—
245C4—201no—
246C4—202no—
247C4#402111yescharacteristic
248C4#402811yescharacteristic
249C4#404011yescharacteristic
250C4#404011yescharacteristic
251C2—402no—
252C2#6496111yescharacteristic
253C2#6505111yescharacteristic
254C2#6507611yescharacteristic
255C2#6508711yescharacteristic
256C2#6510711yescharacteristic
257C2#6515311yescharacteristic
258C2#6517511yescharacteristic
subgroupG/H
centerC23D4×D5
commutator, cosocleC22×C4×C5C23
FrattiniC22×C4C22×D5
FittingC5×#340C2
socleC23×C5C2×D4
prime radicalC22×D20C22
hypercenter#340D5
hypocenterC5#2055
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#3946C2semidirect
4. #65263C2semidirect
5. C2×#3942C2semidirect
7. C5×#340C2semidirect
66. #340D5semidirect
95. C2×Dic5SD16semidirect
227. C5#2055semidirect

Maximal subgroups

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

subgroupindex
C22×#6772
C2×#39462
#652772
#652632
C2×#39422
C2×#39412
C5×#3402
#20555