Finite Group Databasebeta

← #10610973

Subgroups — #10610973

Order 640.

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

normal subgroups only

This group has 398 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
1#65855C211yesmaximal, characteristic
2#65834C211yesmaximal, characteristic, complemented
3C5×#428C211yesmaximal, characteristic, complemented
4#65420C211yesmaximal, characteristic
5#65408C211yesmaximal, characteristic, complemented
6#65287C211yesmaximal, characteristic, complemented
7C2×#3941C211yesmaximal, characteristic
8#65180C212yesmaximal, complemented
9#65179C212yesmaximal, complemented
10#65160C212yesmaximal, complemented
11#65159C212yesmaximal, complemented
12C22×D20C2211yescharacteristic
13C2×C5×M16C2211yescharacteristic
14C5×#117C2211yescharacteristic
15C2×C5×C4⋊C4C2211yescharacteristic
16#4072C2211yescharacteristic
17#4072C2211yescharacteristic
18#4070—22no—
19C22×#168C2211yescharacteristic
20#4039C2212yes—
21#4039C2212yes—
22#4036—21no—
23#4036C2212yescomplemented
24#4034—21no—
25#4034C2212yescomplemented
26D5×C22⋊C4—22no—
27C4×D20—21no—
28C4×D20—21no—
29C5×#107C2212yescomplemented
30C5×#106C2212yescomplemented
31#3953C2212yes—
32#3953C2212yes—
33#3946C2212yescomplemented
34#3946C2212yescomplemented
35#3941C2212yes—
36#3941C2212yes—
37#3940C2211yescharacteristic, complemented
38#3940C2211yescharacteristic, complemented
39#3939C2211yescharacteristic, complemented
40#3939C2211yescharacteristic, complemented
41#3081—51nomaximal, Sylow
42C23×D5—21no—
43C22×C4×C5—21no—
44C22×C4×C5C2311yescharacteristic
45C2×D20—21no—
46C2×D20—22no—
47C2×D20Q814yescomplemented
48C2×D20C2312yes—
49C2×C4×D5—22no—
50C2×C4×D5—22no—
51C2×Dic10—22no—
52C5×M16—22no—
53C2×C5×C8C2312yes—
54C5×C4⋊C4—21no—
55C5×C4⋊C4C2312yes—
56C5×C4⋊C4C2311yescharacteristic
57C5×C4⋊C4C2311yescharacteristic
58C5×C22⋊C4—21no—
59C42×C5—21no—
60C22⋊Dic5—21no—
61C22⋊Dic5D412yes—
62#685—22no—
63#685—22no—
64C4⋊Dic5—22no—
65C4⋊Dic5D411yescharacteristic
66C4⋊Dic5D411yescharacteristic
67C4⋊Dic5C2311yescharacteristic
68C4⋊Dic5C2311yescharacteristic
69#683—22no—
70#683—22no—
71C4×Dic5—21no—
72C4×Dic5C2312yes—
73C2×#168—22no—
74C2×#168C2311yescharacteristic
75C2×#168C2311yescharacteristic
76#544—51no—
77#518—51no—
78#478—52no—
79#478—52no—
80#476—52no—
81#474—52no—
82#428D511yescharacteristic, complemented
83#427—51no—
84#424—51no—
85#418—51no—
86C2×#102—51no—
87C23×C5C2×D411yescharacteristic
88C22×D5—22no—
89C22×D5—24no—
90C22×D5—81no—
91C22×D5—81no—
92C2×C4×C5—22no—
93C2×C4×C5—21no—
94C2×C4×C5—21no—
95C2×C4×C5—81no—
96C2×C4×C5C2×D411yescharacteristic
97C2×C4×C5C2×Q812yes—
98C2×C4×C5#4112yes—
99C2×C4×C5C2411yescharacteristic
100C2×Dic5—21no—
101C2×Dic5—21no—
102C2×Dic5C2×D412yes—
103C2×Dic5C2×D412yes—
104D20—24no—
105D20—22no—
106D20C2×Q814yes—
107C4×D5—82no—
108C4×D5—82no—
109Dic10—82no—
110C5×C8—22no—
111#168—21no—
112#168—21no—
113C22×D4—51no—
114C2×M16C2×D511yescharacteristic
115C22×C8—51no—
116C4⋊Q8—52no—
117C4⋊Q8—101no—
118#125—52no—
119#125—101no—
120C22⋊Q8—102no—
121C4×D4—52no—
122C4×D4—52no—
123C4×D4—101no—
124C4×D4—101no—
125#117—51no—
126#117—51no—
127#117C2×D511yescharacteristic
128C2×C4⋊C4C2×D511yescharacteristic
129C2×C22⋊C4—102no—
130#107—51no—
131#107—51no—
132#107C2×D512yescomplemented
133#106—51no—
134#106—51no—
135#106C2×D512yescomplemented
136C4⋊C8—52no—
137C4⋊C8—52no—
138#102—52no—
139#102—52no—
140#102—52no—
141#102—52no—
142C22×C5—21no—
143C22×C5—21no—
144C22×C5C22⋊Q812yes—
145C22×C5C22×D411yescharacteristic
146C2×D5—24no—
147C2×D5—42no—
148C2×D5—42no—
149C2×D5—84no—
150C4×C5—42no—
151C4×C5—41no—
152C4×C5—41no—
153C4×C5C22⋊Q812yes—
154C4×C5C22×Q811yescharacteristic
155C4×C5C2×#4111yescharacteristic
156Dic5—22no—
157Dic5—22no—
158Dic5—41no—
159Dic5—41no—
160C24—101no—
161C2×Q8—102no—
162C2×D4—54no—
163C2×D4—52no—
164C2×D4—102no—
165C2×D4—101no—
166C22×C4—21no—
167C22×C4—102no—
168C22×C4—102no—
169C22×C4C22×D511yescharacteristic
170M16—22no—
171C2×C8—51no—
172C2×C8—51no—
173C2×C8—102no—
174C2×C8C22×D512yes—
175C4⋊C4—21no—
176C4⋊C4—51no—
177C4⋊C4—51no—
178C4⋊C4—51no—
179C4⋊C4—51no—
180C4⋊C4—102no—
181C4⋊C4—102no—
182C4⋊C4—102no—
183C4⋊C4C22×D512yes—
184C4⋊C4C22×D511yescharacteristic
185C4⋊C4C22×D511yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—52no—
188C22⋊C4—102no—
189C22⋊C4—102no—
190C22⋊C4—101no—
191C42—21no—
192C42—52no—
193C42—101no—
194C2×C5—22no—
195C2×C5C2×C22⋊Q811yescharacteristic
196C2×C5#57611yescharacteristic
197C2×C5#57711yescharacteristic
198D5—44no—
199C23—102no—
200C23—104no—
201C23—401no—
202C23—401no—
203C23D4×D511yescharacteristic
204Q8—402no—
205D4—54no—
206D4—102no—
207D4—104no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—52no—
212C2×C4—52no—
213C2×C4—81no—
214C2×C4—101no—
215C2×C4—101no—
216C2×C4—402no—
217C2×C4—402no—
218C2×C4#70912yes—
219C2×C4D4×D511yescharacteristic
220C2×C4Q8×D512yes—
221C2×C4C23×D511yescharacteristic
222C8—22no—
223C8—101no—
224C8—101no—
225C5#308111yescharacteristic, complemented, Sylow
226C22—21no—
227C22—21no—
228C22—104no—
229C22—202no—
230C22—202no—
231C22—404no—
232C22#404212yes—
233C22C2×D4×D511yescharacteristic
234C4—42no—
235C4—41no—
236C4—41no—
237C4—102no—
238C4—102no—
239C4—201no—
240C4—201no—
241C4#404212yes—
242C4C2×#70911yescharacteristic
243C4C2×Q8×D511yescharacteristic
244C2—22no—
245C2—204no—
246C2C2×#404211yescharacteristic
247C2#6610811yescharacteristic
248C2#6611211yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#428C2
socleC22×C5C22×D4
prime radicalC22×D20C22
hypercenter#428D5
hypocenterC5#3081
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #65834C2semidirect
3. C5×#428C2semidirect
5. #65408C2semidirect
6. #65287C2semidirect
8. #65180C2semidirect
9. #65179C2semidirect
10. #65160C2semidirect
11. #65159C2semidirect
23. #4036C22semidirect
25. #4034C22semidirect
29. C5×#107C22semidirect
30. C5×#106C22semidirect
33. #3946C22semidirect
34. #3946C22semidirect
37. #3940C22semidirect
38. #3940C22semidirect
39. #3939C22semidirect
40. #3939C22semidirect
47. C2×D20Q8semidirect
82. #428D5semidirect
132. #107C2×D5semidirect
135. #106C2×D5semidirect
225. C5#3081semidirect

Maximal subgroups

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

subgroupindex
#658552
#658342
C5×#4282
#654202
#654082
#652872
C2×#39412
#651802
#651792
#651602
#651592
#30815