Finite Group Databasebeta

← C3×#11716899 #72259822

Subgroups — C3×#11716899 #72259822

Order 2400.

412 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 88 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×C3×#1044C212yesmaximal
2C2×C3×C4×C5⋊D5C211yesmaximal, characteristic
3#11716899C311yesmaximal, characteristic, direct product factor, Hall
4C22×C3×C5⋊D5C2211yescharacteristic
5C2×C3×#1044—22no—
6C2×C3×#1044—22no—
7C2×C3×#1044C414yescomplemented
8C2×C3×C4×C52C411yescharacteristic, complemented
9C2×C3×#1039C411yescharacteristic, complemented
10C3×C4×C5⋊D5—22no—
11C3×(C2×AGL(1,𝔽5))⋊C4—52nomaximal
12C22×#1044C2×C312yes—
13C2×C4×C5⋊D5C2×C311yescharacteristic
14C22×C3×C52C2×C411yescharacteristic
15C2×C3×C5⋊D5C2×C412yes—
16C2×C3×C5⋊D5D411yescharacteristic
17C2×C3×C5⋊D5D412yes—
18C2×C3×C5⋊D5Q811yescharacteristic
19C3×#1044—24no—
20C3×C4×C52—21no—
21C3×#1039—21no—
22C22×C3×AGL(1,𝔽5)—54no—
23C2×C3×C4×D5—52no—
24C2×C3×C4×D5—102no—
25C22×C5⋊D5C22×C311yescharacteristic
26C2×#1044—22no—
27C2×#1044—22no—
28C2×#1044C3×C414yescomplemented
29C2×C4×C52C3×C411yescharacteristic, complemented
30C2×#1039C3×C411yescharacteristic, complemented
31C4×C5⋊D5—22no—
32(C2×AGL(1,𝔽5))⋊C4—52no—
33C2×C3×C52C4211yescharacteristic
34C2×C3×C52C22⋊C411yescharacteristic
35C2×C3×C52C4⋊C411yescharacteristic
36C3×C5⋊D5C22⋊C412yes—
37C3×C5⋊D5C4⋊C412yes—
38C22×C3×D5—52no—
39C22×C3×D5—102no—
40C2×C3×AGL(1,𝔽5)—58no—
41C2×C3×AGL(1,𝔽5)—104no—
42C2×C3×AGL(1,𝔽5)—104no—
43C2×C3×C4×C5—22no—
44C2×C3×C4×C5AGL(1,𝔽5)12yescomplemented
45C2×C3×Dic5—52no—
46C2×C3×Dic5—102no—
47C3×C4×D5—104no—
48C3×C4×D5—108no—
49C22×C52C2×C3×C411yescharacteristic
50C2×C5⋊D5C2×C3×C412yes—
51C2×C5⋊D5C3×D411yescharacteristic
52C2×C5⋊D5C3×D412yes—
53C2×C5⋊D5C3×Q811yescharacteristic
54#1044—24no—
55C4×C52—21no—
56#1039—21no—
57C3×#95—251noHall
58C22×AGL(1,𝔽5)—54no—
59C2×C4×D5—52no—
60C2×C4×D5—102no—
61C3×C52#9511yescharacteristic, complemented, Hall
62C22×C3×C5—22no—
63C22×C3×C5C2×AGL(1,𝔽5)12yes—
64C2×C3×D5—54no—
65C2×C3×D5—52no—
66C2×C3×D5—52no—
67C2×C3×D5—54no—
68C2×C3×D5—104no—
69C2×C3×D5—104no—
70C2×C3×D5—102no—
71C2×C3×D5—102no—
72C3×AGL(1,𝔽5)—108no—
73C3×C4×C5—24no—
74C3×C4×C5—22no—
75C3×Dic5—104no—
76C3×Dic5—102no—
77C2×C52C3×C4211yescharacteristic
78C2×C52C3×C22⋊C411yescharacteristic
79C2×C52C3×C4⋊C411yescharacteristic
80C5⋊D5C3×C22⋊C412yes—
81C5⋊D5C3×C4⋊C412yes—
82C22×C3×C4—252no—
83C22×C3×C4—251no—
84C22×D5—52no—
85C22×D5—102no—
86C2×AGL(1,𝔽5)—58no—
87C2×AGL(1,𝔽5)—104no—
88C2×AGL(1,𝔽5)—104no—
89C2×C4×C5—22no—
90C2×C4×C5C3×AGL(1,𝔽5)12yescomplemented
91C2×Dic5—52no—
92C2×Dic5—102no—
93C4×D5—104no—
94C4×D5—108no—
95#95—251noSylow
96C2×C3×C5—22no—
97C2×C3×C5—22no—
98C2×C3×C5—22no—
99C2×C3×C5C4×AGL(1,𝔽5)12yes—
100C2×C3×C5C4⋊AGL(1,𝔽5)12yes—
101C2×C3×C5C22⋊AGL(1,𝔽5)12yes—
102C3×D5—54no—
103C3×D5—54no—
104C3×D5—104no—
105C3×D5—104no—
106C52C3×#9511yescharacteristic, complemented, Sylow
107C23×C3—251no—
108C2×C3×C4—254no—
109C2×C3×C4—251no—
110C2×C3×C4—502no—
111C2×C3×C4—502no—
112C2×C3×C4—502no—
113C2×C3×C4#104411yescharacteristic, complemented
114C22×C5—22no—
115C22×C5C2×C3×AGL(1,𝔽5)12yes—
116C2×D5—54no—
117C2×D5—52no—
118C2×D5—52no—
119C2×D5—54no—
120C2×D5—104no—
121C2×D5—104no—
122C2×D5—102no—
123C2×D5—102no—
124AGL(1,𝔽5)—108no—
125C4×C5—24no—
126C4×C5—22no—
127Dic5—104no—
128Dic5—102no—
129C22×C4—252no—
130C22×C4—251no—
131C3×C5—22no—
132C3×C5(C2×AGL(1,𝔽5))⋊C412yescomplemented
133C22×C3—252no—
134C22×C3—251no—
135C22×C3—251no—
136C22×C3—252no—
137C22×C3C2×#104411yescharacteristic
138C3×C4—21no—
139C3×C4—504no—
140C3×C4—501no—
141C2×C5—22no—
142C2×C5—22no—
143C2×C5—22no—
144C2×C5C3×C4×AGL(1,𝔽5)12yes—
145C2×C5C3×C4⋊AGL(1,𝔽5)12yes—
146C2×C5C3×C22⋊AGL(1,𝔽5)12yes—
147D5—54no—
148D5—54no—
149D5—104no—
150D5—104no—
151C23—251no—
152C2×C4—254no—
153C2×C4—251no—
154C2×C4—502no—
155C2×C4—502no—
156C2×C4—502no—
157C2×C4C3×#104411yescharacteristic, complemented
158C2×C3—252no—
159C2×C3—252no—
160C2×C3C4×#104411yescharacteristic
161C2×C3C4⋊#104411yescharacteristic
162C2×C3C22⋊#104411yescharacteristic
163C5—22no—
164C5C3×(C2×AGL(1,𝔽5))⋊C412yescomplemented
165C22—252no—
166C22—251no—
167C22—251no—
168C22—252no—
169C22C2×C3×#104411yescharacteristic
170C4—21no—
171C4—504no—
172C4—501no—
173C3#1171689911yescharacteristic, direct product factor, Sylow
174C2—252no—
175C2—252no—
176C2C3×C4×#104411yescharacteristic
177C2C3×C4⋊#104411yescharacteristic
178C2C3×C22⋊#104411yescharacteristic
subgroupG/H
centerC22×C3C2×#1044
commutatorC2×C52C3×C42
FrattiniC22C2×C3×#1044
FittingC2×C3×C4×C52C4
socleC22×C3×C52C2×C4
prime radicalC22×C3×C5⋊D5C22
cosocleC22×C5⋊D5C22×C3
hypercenterC2×C3×C4#1044
hypocenterC52C3×#95
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. #11716899C3direct (N × H)
7. C2×C3×#1044C4semidirect
8. C2×C3×C4×C52C4semidirect
9. C2×C3×#1039C4semidirect
28. C2×#1044C3×C4semidirect
29. C2×C4×C52C3×C4semidirect
30. C2×#1039C3×C4semidirect
44. C2×C3×C4×C5AGL(1,𝔽5)semidirect
61. C3×C52#95semidirect
90. C2×C4×C5C3×AGL(1,𝔽5)semidirect
106. C52C3×#95semidirect
113. C2×C3×C4#1044semidirect
132. C3×C5(C2×AGL(1,𝔽5))⋊C4semidirect
157. C2×C4C3×#1044semidirect
164. C5C3×(C2×AGL(1,𝔽5))⋊C4semidirect
173. C3#11716899direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#10442
C2×C3×C4×C5⋊D52
#117168993
C3×(C2×AGL(1,𝔽5))⋊C45