Finite Group Databasebeta

← #10608538

Subgroups — #10608538

Order 640.

444 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C5×#393C211yesmaximal, characteristic
2#65519C212yesmaximal, complemented
3#65238C212yesmaximal
4#65236C211yesmaximal, characteristic
5#65232C211yesmaximal, characteristic
6C22⋊C4×Dic5C212yesmaximal
7#64942C212yesmaximal, complemented
8#64941C212yesmaximal, complemented
9#64935C212yesmaximal, complemented
10C22×C5×Q8C2211yescharacteristic
11C23×Dic5C2211yescharacteristic
12C22×Dic10C2211yescharacteristic
13C2×C5×C22⋊C4C2212yes—
14C2×C5×C22⋊C4C2211yescharacteristic
15C2×C22⋊Dic5C2212yes—
16C2×C22⋊Dic5C2211yescharacteristic
17C2×C4⋊Dic5C2212yes—
18C2×#683C2212yes—
19C2×#683C2212yes—
20C2×C4×Dic5C2212yes—
21C2×C4×Dic5C2212yes—
22C5×#95C2212yes—
23C5×#95C2211yescharacteristic
24#3963C2212yes—
25#3963C2212yes—
26#3963C2212yes—
27#3963C2212yes—
28#3963C2212yes—
29#3963C2211yescharacteristic
30#3963C2212yes—
31#2607—51nomaximal, Sylow
32C24×C5C2311yescharacteristic
33C2×C5×Q8—24no—
34C2×C5×Q8—41no—
35C22×C4×C5C2311yescharacteristic
36C22×C4×C5C2311yescharacteristic
37C22×C4×C5C2312yes—
38C22×C4×C5C2312yes—
39C22×Dic5—41no—
40C22×Dic5—41no—
41C22×Dic5Q814yescomplemented
42C22×Dic5C2312yes—
43C22×Dic5C2312yes—
44C22×Dic5C2312yes—
45C22×Dic5C2312yes—
46C2×Dic10—24no—
47C2×Dic10—41no—
48C5×C22⋊C4—24no—
49C5×C22⋊C4—41no—
50C22⋊Dic5—24no—
51C22⋊Dic5—41no—
52C4⋊Dic5—24no—
53#683—24no—
54#683—42no—
55C4×Dic5—42no—
56C4×Dic5—42no—
57#399—52no—
58#393—51no—
59#393D511yescharacteristic
60#391—52no—
61#391—52no—
62#387—52no—
63#387—52no—
64#381—51no—
65C4×C22⋊C4—52no—
66C23×C5—22no—
67C23×C5—22no—
68C23×C5—21no—
69C23×C5—21no—
70C23×C5C2×Q812yes—
71C23×C5C2411yescharacteristic
72C5×Q8—44no—
73C2×C4×C5—22no—
74C2×C4×C5—22no—
75C2×C4×C5—22no—
76C2×C4×C5—22no—
77C2×C4×C5—22no—
78C2×C4×C5—22no—
79C2×C4×C5—22no—
80C2×C4×C5—21no—
81C2×C4×C5—21no—
82C2×C4×C5#4114yes—
83C2×Dic5—22no—
84C2×Dic5—22no—
85C2×Dic5—22no—
86C2×Dic5—22no—
87C2×Dic5—22no—
88C2×Dic5—22no—
89C2×Dic5—22no—
90C2×Dic5—22no—
91C2×Dic5—22no—
92C2×Dic5—22no—
93C2×Dic5—44no—
94C2×Dic5C2×Q814yes—
95C2×Dic5#4114yes—
96Dic10—44no—
97C22×Q8—51no—
98C22×Q8C2×D511yescharacteristic
99C23×C4—51no—
100C2×C4⋊C4—52no—
101C2×C4⋊C4—52no—
102C2×C4⋊C4—52no—
103C2×C22⋊C4—52no—
104C2×C22⋊C4—51no—
105C2×C22⋊C4C2×D512yes—
106C2×C22⋊C4C2×D511yescharacteristic
107C2×C42—52no—
108C2×C42—52no—
109#95—52no—
110#95—52no—
111#95—52no—
112#95—52no—
113#95—51no—
114#95—52no—
115#95—52no—
116#95C2×D511yescharacteristic
117#95C2×D512yes—
118C22×C5—24no—
119C22×C5—22no—
120C22×C5—41no—
121C22×C5—41no—
122C22×C5—42no—
123C22×C5C22×Q811yescharacteristic
124C22×C5C2×#4112yes—
125C22×C5C2×#4112yes—
126C22×C5H(5,𝔽2)11yescharacteristic
127C22×C5H(5,𝔽2)11yescharacteristic
128C4×C5—24no—
129C4×C5—41no—
130C4×C5—41no—
131C4×C5—42no—
132Dic5—24no—
133Dic5—24no—
134Dic5—42no—
135Dic5—42no—
136C24C22×D511yescharacteristic
137C2×Q8—24no—
138C2×Q8—41no—
139C2×Q8—104no—
140C2×Q8—201no—
141C22×C4—52no—
142C22×C4—52no—
143C22×C4—54no—
144C22×C4—52no—
145C22×C4—52no—
146C22×C4—201no—
147C22×C4—201no—
148C22×C4C22×D511yescharacteristic
149C22×C4C22×D511yescharacteristic
150C22×C4C22×D512yes—
151C22×C4C22×D512yes—
152C4⋊C4—104no—
153C4⋊C4—104no—
154C4⋊C4—202no—
155C22⋊C4—24no—
156C22⋊C4—41no—
157C22⋊C4—104no—
158C22⋊C4—201no—
159C42—202no—
160C42—202no—
161C2×C5—42no—
162C2×C5#53312yes—
163C2×C5#54311yescharacteristic
164C2×C5#55111yescharacteristic
165C2×C5#55111yescharacteristic
166C2×C5#55912yes—
167C23—22no—
168C23—21no—
169C23—22no—
170C23—21no—
171C23Q8×D512yes—
172C23C23×D511yescharacteristic
173Q8—44no—
174Q8—204no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—54no—
185C2×C4—54no—
186C2×C4—102no—
187C2×C4—102no—
188C2×C4—102no—
189C2×C4—102no—
190C2×C4—102no—
191C2×C4—102no—
192C2×C4—102no—
193C2×C4—102no—
194C2×C4—102no—
195C2×C4—102no—
196C2×C4—204no—
197C2×C4#71114yes—
198C5#260711yescharacteristic, complemented, Sylow
199C22—24no—
200C22—22no—
201C22—41no—
202C22—41no—
203C22—42no—
204C22C2×#71112yes—
205C22#414411yescharacteristic
206C22C2×Q8×D511yescharacteristic
207C22D5×#4112yes—
208C22#414911yescharacteristic
209C4—24no—
210C4—41no—
211C4—41no—
212C4—42no—
213C4—104no—
214C4—104no—
215C4—202no—
216C4—202no—
217C2—42no—
218C2#6592111yescharacteristic
219C2#6595612yes—
220C2#6596811yescharacteristic
221C2#6601311yescharacteristic
222C2#6601512yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#393C2
prime radicalC24×C5C23
hypercenter#393D5
hypocenterC5#2607
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #65519C2semidirect
7. #64942C2semidirect
8. #64941C2semidirect
9. #64935C2semidirect
41. C22×Dic5Q8semidirect
198. C5#2607semidirect

Maximal subgroups

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

subgroupindex
C5×#3932
#655192
#652382
#652362
#652322
C22⋊C4×Dic52
#649422
#649412
#649352
#26075