Finite Group Databasebeta

← #10608250

Subgroups — #10608250

Order 640.

544 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 169 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×#4092C211yesmaximal, characteristic
2C2×D5×C4⋊C4C211yesmaximal, characteristic
3C5×#391C211yesmaximal, characteristic, complemented
4#65519C211yesmaximal, characteristic, complemented
5#65265C211yesmaximal, characteristic, complemented
6C4×#685C211yesmaximal, characteristic
7#65224C211yesmaximal, characteristic, complemented
8#64957C214yesmaximal
9#64946C214yesmaximal, complemented
10C22×C5×Q8C2211yescharacteristic
11C22×C4×D5C2211yescharacteristic
12C22×C4×D5C2212yes—
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C42×C5C2211yescharacteristic
15#4092—24no—
16C2×#685C2212yes—
17C2×#685C2212yes—
18C2×C4⋊Dic5C2211yescharacteristic
19C2×C4⋊Dic5C2211yescharacteristic
20C2×C4⋊Dic5C2214yes—
21C2×#683C2212yes—
22C2×#683C2212yes—
23C2×#683C2214yes—
24C2×C4×Dic5C2211yescharacteristic
25D5×C4⋊C4—24no—
26C5×#95C2214yes—
27#3963C2212yes—
28#3963C2214yes—
29#2502—51nomaximal, Sylow
30C23×D5C2311yescharacteristic
31C2×C5×Q8—22no—
32C2×C5×Q8—22no—
33C2×C5×Q8—22no—
34C22×C4×C5C2312yes—
35C22×C4×C5C2312yes—
36C22×C4×C5C2312yes—
37C22×C4×C5C2311yescharacteristic
38C22×Dic5C2312yes—
39C22×Dic5C2312yes—
40C22×Dic5C2311yescharacteristic
41C22×Dic5C2312yes—
42C2×C4×D5—22no—
43C2×C4×D5—22no—
44C2×C4×D5—24no—
45C2×C4×D5—44no—
46C2×C4×D5D414yes—
47C5×C4⋊C4—22no—
48C42×C5—22no—
49#685—24no—
50#685Q818yescomplemented
51C4⋊Dic5—22no—
52C4⋊Dic5—22no—
53C4⋊Dic5—44no—
54#683—24no—
55#683—24no—
56#683—44no—
57C4×Dic5—22no—
58C2×C22⋊Q8—51no—
59C22×C4⋊C4—51no—
60#398—54no—
61#391—51no—
62#391D511yescharacteristic, complemented
63#385—54no—
64#384—51no—
65#384—51no—
66C4×C22⋊C4—51no—
67C23×C5C2411yescharacteristic
68C22×D5—21no—
69C22×D5—21no—
70C22×D5—22no—
71C22×D5C2×D412yes—
72C22×D5C2×Q814yes—
73C5×Q8—44no—
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—24no—
81C2×C4×C5C2×D412yes—
82C2×C4×C5C2×Q814yes—
83C2×C4×C5#4112yes—
84C2×C4×C5#4112yes—
85C2×Dic5—22no—
86C2×Dic5—21no—
87C2×Dic5—21no—
88C2×Dic5—22no—
89C2×Dic5—22no—
90C2×Dic5—22no—
91C2×Dic5—24no—
92C2×Dic5—24no—
93C2×Dic5C2×D412yes—
94C2×Dic5C2×Q814yes—
95C4×D5—28no—
96C4×D5—48no—
97C22×Q8C2×D511yescharacteristic
98C23×C4—51no—
99C23×C4—52no—
100C22⋊Q8—104no—
101C2×C4⋊C4—51no—
102C2×C4⋊C4—52no—
103C2×C4⋊C4—51no—
104C2×C4⋊C4—52no—
105C2×C4⋊C4—54no—
106C2×C4⋊C4—54no—
107C2×C4⋊C4—104no—
108C2×C4⋊C4C2×D511yescharacteristic
109C2×C22⋊C4—52no—
110C2×C22⋊C4—52no—
111C2×C42—51no—
112C2×C42C2×D511yescharacteristic
113#95—52no—
114#95—54no—
115#95C2×D514yes—
116C22×C5C22×D411yescharacteristic
117C22×C5C22×Q812yes—
118C22×C5C2×#4111yescharacteristic
119C22×C5C2×#4111yescharacteristic
120C22×C5H(5,𝔽2)12yes—
121C2×D5—22no—
122C2×D5—22no—
123C2×D5—24no—
124C2×D5—24no—
125C2×D5C4⋊Q814yes—
126C4×C5—24no—
127C4×C5—42no—
128C4×C5—42no—
129C4×C5#12314yes—
130Dic5—22no—
131Dic5—24no—
132Dic5—42no—
133Dic5—42no—
134C24—51no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8—22no—
138C22×C4—52no—
139C22×C4—51no—
140C22×C4—54no—
141C22×C4—52no—
142C22×C4—52no—
143C22×C4—104no—
144C22×C4—102no—
145C22×C4—102no—
146C22×C4—204no—
147C22×C4C22×D512yes—
148C22×C4C22×D512yes—
149C22×C4C22×D512yes—
150C22×C4C22×D511yescharacteristic
151C4⋊C4—22no—
152C4⋊C4—104no—
153C4⋊C4—104no—
154C4⋊C4—102no—
155C4⋊C4—102no—
156C4⋊C4—204no—
157C4⋊C4—204no—
158C22⋊C4—58no—
159C22⋊C4—104no—
160C42—22no—
161C42—102no—
162C2×C5C2×#12311yescharacteristic
163C2×C5C2×C4⋊Q811yescharacteristic
164C2×C5#53511yescharacteristic
165C2×C5#55412yes—
166C2×C5#55412yes—
167D5—24no—
168C23—52no—
169C23—54no—
170C23—101no—
171C23—101no—
172C23—102no—
173C23C23×D511yescharacteristic
174Q8—44no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—52no—
183C2×C4—54no—
184C2×C4—102no—
185C2×C4—101no—
186C2×C4—101no—
187C2×C4—102no—
188C2×C4—108no—
189C2×C4—102no—
190C2×C4—102no—
191C2×C4—104no—
192C2×C4—104no—
193C2×C4—208no—
194C2×C4#70912yes—
195C2×C4D4×D512yes—
196C2×C4#71112yes—
197C2×C4Q8×D514yes—
198C5#250211yescharacteristic, complemented, Sylow
199C22—54no—
200C22—102no—
201C22—104no—
202C22—104no—
203C22—102no—
204C22C2×#70911yescharacteristic
205C22C2×D4×D511yescharacteristic
206C22C2×#71111yescharacteristic
207C22C2×Q8×D512yes—
208C22#414912yes—
209C4—24no—
210C4—42no—
211C4—42no—
212C4—102no—
213C4—104no—
214C4—202no—
215C4—202no—
216C4#402914yes—
217C2—104no—
218C2C2×#402911yescharacteristic
219C2#6591312yes—
220C2#6596912yes—
221C2#6600811yescharacteristic
222C2D5×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#391C2
prime radicalC23×D5C23
hypercenter#391D5
hypocenterC5#2502
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
3. C5×#391C2semidirect
4. #65519C2semidirect
5. #65265C2semidirect
7. #65224C2semidirect
9. #64946C2semidirect
50. #685Q8semidirect
62. #391D5semidirect
198. C5#2502semidirect

Maximal subgroups

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

subgroupindex
C2×#40922
C2×D5×C4⋊C42
C5×#3912
#655192
#652652
C4×#6852
#652242
#649572
#649462
#25025