Finite Group Databasebeta

← C3×#11717365 #72230950

Subgroups — C3×#11717365 #72230950

Order 2400.

428 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 94 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×C3×D5⋊AGL(1,𝔽5)C212yesmaximal, complemented
2C3×#87877C211yesmaximal, characteristic
3C3×C4×#1044C211yesmaximal, characteristic, complemented
4C3×C4⋊#1043C211yesmaximal, characteristic, complemented
5C3×#87831C212yesmaximal, complemented
6#11717365C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×D52C2212yes—
8C2×C3×#1044C2211yescharacteristic
9C2×C3×#1044C2211yescharacteristic
10C2×C3×#1043C2212yes—
11C3×D5⋊AGL(1,𝔽5)—24no—
12C3×C4×C5⋊D5C2211yescharacteristic
13C3×C5×D20C412yescomplemented
14C3×#6037C412yescomplemented
15C3×D4×AGL(1,𝔽5)—52nomaximal
16C2×D5⋊AGL(1,𝔽5)C2×C312yescomplemented
17#87877C2×C311yescharacteristic
18C4×#1044C2×C311yescharacteristic, complemented
19C4⋊#1043C2×C311yescharacteristic, complemented
20#87831C2×C312yescomplemented
21C2×C3×C5⋊D5C2311yescharacteristic
22C2×C3×C5×D5C2×C414yescomplemented
23C3×D52—22no—
24C3×D52—22no—
25C3×#1044—21no—
26C3×#1044D412yescomplemented
27C3×#1043—22no—
28C3×C4×C52C2×C411yescharacteristic, complemented
29C3×#1039C2×C411yescharacteristic, complemented
30C22×C3×AGL(1,𝔽5)—54no—
31C3×D4×D5—52no—
32C3×C22⋊AGL(1,𝔽5)—54no—
33C3×C4⋊AGL(1,𝔽5)—52no—
34C3×C4⋊AGL(1,𝔽5)—102no—
35C3×C4×AGL(1,𝔽5)—52no—
36C2×D52C22×C312yes—
37C2×#1044C22×C311yescharacteristic
38C2×#1044C22×C311yescharacteristic
39C2×#1043C22×C312yes—
40D5⋊AGL(1,𝔽5)—24no—
41C4×C5⋊D5C22×C311yescharacteristic
42C5×D20C3×C412yescomplemented
43#6037C3×C412yescomplemented
44D4×AGL(1,𝔽5)—52no—
45C2×C3×C52C22×C411yescharacteristic
46C3×C5⋊D5C2×D411yescharacteristic
47C3×C5⋊D5#4111yescharacteristic
48C3×C5×D5—24no—
49C22×C3×D5—54no—
50C2×C3×AGL(1,𝔽5)—52no—
51C2×C3×AGL(1,𝔽5)—54no—
52C2×C3×AGL(1,𝔽5)—52no—
53C2×C3×AGL(1,𝔽5)—108no—
54C2×C3×AGL(1,𝔽5)—104no—
55C3×C5×D4—52no—
56C3×C22⋊D5—54no—
57C3×D20AGL(1,𝔽5)12yescomplemented
58C3×C4×D5—52no—
59C3×C4×D5—102no—
60C2×C5⋊D5C23×C311yescharacteristic
61C2×C5×D5C2×C3×C414yescomplemented
62D52—22no—
63D52—22no—
64#1044—21no—
65#1044C3×D412yescomplemented
66#1043—22no—
67C4×C52C2×C3×C411yescharacteristic, complemented
68#1039C2×C3×C411yescharacteristic, complemented
69C3×C4×D4—251noHall
70C22×AGL(1,𝔽5)—54no—
71D4×D5—52no—
72C22⋊AGL(1,𝔽5)—54no—
73C4⋊AGL(1,𝔽5)—52no—
74C4⋊AGL(1,𝔽5)—102no—
75C4×AGL(1,𝔽5)—52no—
76C3×C52C4×D411yescharacteristic, complemented, Hall
77C22×C3×C5—54no—
78C2×C3×D5—52no—
79C2×C3×D5—102no—
80C2×C3×D5—104no—
81C2×C3×D5—104no—
82C2×C3×D5C2×AGL(1,𝔽5)14yescomplemented
83C3×AGL(1,𝔽5)—54no—
84C3×AGL(1,𝔽5)—104no—
85C3×AGL(1,𝔽5)—102no—
86C3×AGL(1,𝔽5)—204no—
87C3×C4×C5—22no—
88C3×C4×C5C2×AGL(1,𝔽5)12yescomplemented
89C3×Dic5—52no—
90C3×Dic5—102no—
91C2×C52C22×C3×C411yescharacteristic
92C5⋊D5C2×C3×D411yescharacteristic
93C5⋊D5C3×#4111yescharacteristic
94C5×D5—24no—
95C2×C3×D4—251no—
96C22×C3×C4—252no—
97C3×C4⋊C4—251no—
98C3×C22⋊C4—252no—
99C3×C42—251no—
100C22×D5—54no—
101C2×AGL(1,𝔽5)—52no—
102C2×AGL(1,𝔽5)—54no—
103C2×AGL(1,𝔽5)—52no—
104C2×AGL(1,𝔽5)—104no—
105C2×AGL(1,𝔽5)—108no—
106C5×D4—52no—
107C22⋊D5—54no—
108D20C3×AGL(1,𝔽5)12yescomplemented
109C4×D5—52no—
110C4×D5—102no—
111C4×D4—251noSylow
112C2×C3×C5—22no—
113C2×C3×C5—104no—
114C2×C3×C5C22×AGL(1,𝔽5)12yes—
115C3×D5—24no—
116C3×D5—52no—
117C3×D5—52no—
118C3×D5—102no—
119C3×D5—102no—
120C52C3×C4×D411yescharacteristic, complemented, Sylow
121C23×C3—252no—
122C3×D4—52no—
123C3×D4—252no—
124C2×C3×C4—251no—
125C2×C3×C4—251no—
126C2×C3×C4—252no—
127C2×C3×C4—251no—
128C2×C3×C4—504no—
129C22×C5—54no—
130C2×D5—52no—
131C2×D5—102no—
132C2×D5—104no—
133C2×D5—104no—
134C2×D5C2×C3×AGL(1,𝔽5)14yescomplemented
135AGL(1,𝔽5)—54no—
136AGL(1,𝔽5)—104no—
137AGL(1,𝔽5)—102no—
138AGL(1,𝔽5)—204no—
139C4×C5—22no—
140C4×C5C2×C3×AGL(1,𝔽5)12yescomplemented
141Dic5—52no—
142Dic5—102no—
143C2×D4—251no—
144C22×C4—252no—
145C4⋊C4—251no—
146C22⋊C4—252no—
147C42—251no—
148C3×C5—22no—
149C3×C5D4×AGL(1,𝔽5)12yescomplemented
150C22×C3—54no—
151C22×C3—251no—
152C22×C3—502no—
153C22×C3—502no—
154C3×C4—252no—
155C3×C4—251no—
156C3×C4—502no—
157C3×C4—501no—
158C3×C4D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
159C2×C5—22no—
160C2×C5—104no—
161C2×C5C22×C3×AGL(1,𝔽5)12yes—
162D5—24no—
163D5—52no—
164D5—52no—
165D5—102no—
166D5—102no—
167C23—252no—
168D4—52no—
169D4—252no—
170C2×C4—251no—
171C2×C4—251no—
172C2×C4—251no—
173C2×C4—252no—
174C2×C4—504no—
175C2×C3—104no—
176C2×C3—251no—
177C2×C3—251no—
178C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
179C5—22no—
180C5C3×D4×AGL(1,𝔽5)12yescomplemented
181C22—54no—
182C22—251no—
183C22—502no—
184C22—502no—
185C4—252no—
186C4—251no—
187C4—502no—
188C4—501no—
189C4C3×D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
190C3#1171736511yescharacteristic, direct product factor, Sylow
191C2—104no—
192C2—251no—
193C2—251no—
194C2C2×C3×D5⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2×C3C2×D5⋊AGL(1,𝔽5)
commutatorC2×C52C22×C3×C4
FrattiniC2C2×C3×D5⋊AGL(1,𝔽5)
FittingC3×C4×C52C2×C4
socleC2×C3×C52C22×C4
prime radicalC3×#87877C2
cosocleC2×C5⋊D5C23×C3
hypercenterC3×C4D5⋊AGL(1,𝔽5)
hypocenterC52C3×C4×D4
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D5⋊AGL(1,𝔽5)C2semidirect
3. C3×C4×#1044C2semidirect
4. C3×C4⋊#1043C2semidirect
5. C3×#87831C2semidirect
6. #11717365C3direct (N × H)
13. C3×C5×D20C4semidirect
14. C3×#6037C4semidirect
16. C2×D5⋊AGL(1,𝔽5)C2×C3semidirect
18. C4×#1044C2×C3semidirect
19. C4⋊#1043C2×C3semidirect
20. #87831C2×C3semidirect
22. C2×C3×C5×D5C2×C4semidirect
26. C3×#1044D4semidirect
28. C3×C4×C52C2×C4semidirect
29. C3×#1039C2×C4semidirect
42. C5×D20C3×C4semidirect
43. #6037C3×C4semidirect
57. C3×D20AGL(1,𝔽5)semidirect
61. C2×C5×D5C2×C3×C4semidirect
65. #1044C3×D4semidirect
67. C4×C52C2×C3×C4semidirect
68. #1039C2×C3×C4semidirect
76. C3×C52C4×D4semidirect
82. C2×C3×D5C2×AGL(1,𝔽5)semidirect
88. C3×C4×C5C2×AGL(1,𝔽5)semidirect
108. D20C3×AGL(1,𝔽5)semidirect
120. C52C3×C4×D4semidirect
134. C2×D5C2×C3×AGL(1,𝔽5)semidirect
140. C4×C5C2×C3×AGL(1,𝔽5)semidirect
149. C3×C5D4×AGL(1,𝔽5)semidirect
158. C3×C4D5⋊AGL(1,𝔽5)semidirect
180. C5C3×D4×AGL(1,𝔽5)semidirect
189. C4C3×D5⋊AGL(1,𝔽5)semidirect
190. C3#11717365direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D5⋊AGL(1,𝔽5)2
C3×#878772
C3×C4×#10442
C3×C4⋊#10432
C3×#878312
#117173653
C3×D4×AGL(1,𝔽5)5