Finite Group Databasebeta

← #72231004

Subgroups — #72231004

Order 2400.

428 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 67 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×#10597182C212yesmaximal, complemented
2#11925655C211yesmaximal, characteristic
3C4×C3⋊#1044C211yesmaximal, characteristic, complemented
4C3×C4⋊#1043C211yesmaximal, characteristic, complemented
5#11925355C212yesmaximal, complemented
6#11717365—31nomaximal, Hall
7C2×D15⋊D5C2212yes—
8C2×C3⋊#1044C2211yescharacteristic
9C2×C3⋊#1044C2211yescharacteristic
10C2×C3×#1043C2212yes—
11#10597182—24no—
12C5×D60C412yescomplemented
13C3×C4×C5⋊D5C2211yescharacteristic
14#10597128C412yescomplemented
15#91921—52nomaximal
16C2×D5⋊AGL(1,𝔽5)—32no—
17#87877—31no—
18C4×#1044—31no—
19C4⋊#1043D311yescharacteristic, complemented
20#87831—32no—
21C2×C5×D15C2×C414yescomplemented
22C2×C3×C5⋊D5C2311yescharacteristic
23D15⋊D5—22no—
24D15⋊D5—22no—
25C3⋊#1044—21no—
26C3⋊#1044D412yescomplemented
27C3×#1043—22no—
28C3×C4×C52C2×C411yescharacteristic, complemented
29C3×#1039C2×C411yescharacteristic, complemented
30C2×D3×AGL(1,𝔽5)—54no—
31D5×D12—52no—
32C4×ΓL(1,𝔽16)—52no—
33C3×C4⋊AGL(1,𝔽5)—52no—
34C3×C4⋊AGL(1,𝔽5)—102no—
35(C2×D3)⋊AGL(1,𝔽5)—54no—
36C2×D52—32no—
37C2×#1044—31no—
38C2×#1044—31no—
39C2×#1043C2×D312yes—
40D5⋊AGL(1,𝔽5)—64no—
41C4×C5⋊D5C2×D311yescharacteristic
42C5×D20—32no—
43#6037—32no—
44D4×AGL(1,𝔽5)—152no—
45C2×C3×C52C22×C411yescharacteristic
46C5×D15—24no—
47C3×C5⋊D5C2×D411yescharacteristic
48C3×C5⋊D5#4111yescharacteristic
49C2×D3×D5—54no—
50C2×ΓL(1,𝔽16)—52no—
51C2×ΓL(1,𝔽16)—52no—
52C2×C3×AGL(1,𝔽5)—54no—
53C2×C3×AGL(1,𝔽5)—104no—
54D3×AGL(1,𝔽5)—108no—
55D60AGL(1,𝔽5)12yescomplemented
56C5×D12—52no—
57C3×C4×D5—52no—
58C3×C4×D5—102no—
59#1203—54no—
60C2×C5⋊D5C22×D311yescharacteristic
61C2×C5×D5—34no—
62D52—62no—
63D52—62no—
64#1044—32no—
65#1044—61no—
66#1043—22no—
67C4×C52C4×D311yescharacteristic, complemented
68#1039C4×D311yescharacteristic, complemented
69#893—251noHall
70C22×AGL(1,𝔽5)—154no—
71D4×D5—152no—
72C22⋊AGL(1,𝔽5)—154no—
73C4⋊AGL(1,𝔽5)—52no—
74C4⋊AGL(1,𝔽5)—102no—
75C4×AGL(1,𝔽5)—152no—
76C3×C52C4×D411yescharacteristic, complemented, Hall
77C2×D15C2×AGL(1,𝔽5)14yescomplemented
78C2×C5×D3—54no—
79C2×C3×D5—52no—
80C2×C3×D5—102no—
81D3×D5—104no—
82D3×D5—104no—
83ΓL(1,𝔽16)—54no—
84ΓL(1,𝔽16)—102no—
85C3×AGL(1,𝔽5)—104no—
86C3×AGL(1,𝔽5)—204no—
87C3×C4×C5—22no—
88C3×C4×C5C2×AGL(1,𝔽5)12yescomplemented
89C3×Dic5—52no—
90C3×Dic5—102no—
91C2×C52C2×C4×D311yescharacteristic
92C5⋊D5D3×D411yescharacteristic
93C5⋊D5#23911yescharacteristic
94C5×D5—64no—
95C2×D12—251no—
96C2×C4×D3—252no—
97C3×C4⋊C4—251no—
98(C2×D3)⋊C4—252no—
99C4×Dic3—251no—
100C22×D5—154no—
101C2×AGL(1,𝔽5)—54no—
102C2×AGL(1,𝔽5)—104no—
103C2×AGL(1,𝔽5)—152no—
104C2×AGL(1,𝔽5)—152no—
105C2×AGL(1,𝔽5)—308no—
106C5×D4—152no—
107C22⋊D5—154no—
108D20—32no—
109C4×D5—52no—
110C4×D5—102no—
111C4×D4—751noSylow
112C2×C3×C5—22no—
113C2×C3×C5C22×AGL(1,𝔽5)12yes—
114D15—24no—
115C3×D5—52no—
116C3×D5—52no—
117C3×D5—102no—
118C3×D5—102no—
119C5×D3—104no—
120C52#89311yescharacteristic, complemented, Sylow
121C22×D3—252no—
122C2×C3×C4—251no—
123C2×C3×C4—252no—
124C2×Dic3—251no—
125C2×Dic3—251no—
126D12—52no—
127D12—252no—
128C4×D3—504no—
129C22×C5—154no—
130C2×D5—34no—
131C2×D5—52no—
132C2×D5—102no—
133C2×D5—304no—
134C2×D5—304no—
135AGL(1,𝔽5)—104no—
136AGL(1,𝔽5)—154no—
137AGL(1,𝔽5)—204no—
138AGL(1,𝔽5)—302no—
139C4×C5—22no—
140C4×C5D3×AGL(1,𝔽5)12yescomplemented
141Dic5—52no—
142Dic5—102no—
143C2×D4—751no—
144C22×C4—752no—
145C4⋊C4—251no—
146C22⋊C4—752no—
147C42—751no—
148C3×C5—22no—
149C3×C5D4×AGL(1,𝔽5)12yescomplemented
150C22×C3—251no—
151C2×D3—54no—
152C2×D3—502no—
153C2×D3—502no—
154C3×C4—251no—
155C3×C4—502no—
156C3×C4D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
157Dic3—252no—
158Dic3—501no—
159C2×C5—22no—
160C2×C5—304no—
161C2×C5C2×D3×AGL(1,𝔽5)12yes—
162D5—52no—
163D5—52no—
164D5—64no—
165D5—102no—
166D5—102no—
167C23—752no—
168D4—152no—
169D4—752no—
170C2×C4—251no—
171C2×C4—252no—
172C2×C4—751no—
173C2×C4—751no—
174C2×C4—1504no—
175C2×C3—251no—
176C2×C3—251no—
177C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
178D3—104no—
179C5—22no—
180C5#9192112yescomplemented
181C22—154no—
182C22—251no—
183C22—1502no—
184C22—1502no—
185C4—251no—
186C4—502no—
187C4—752no—
188C4—1501no—
189C4#1059718211yescharacteristic, complemented
190C3#1171736511yescharacteristic, complemented, Sylow
191C2—251no—
192C2—251no—
193C2—304no—
194C2C2×#1059718211yescharacteristic
subgroupG/H
center, FrattiniC2C2×#10597182
commutator, socleC2×C3×C52C22×C4
FittingC3×C4×C52C2×C4
prime radical#11925655C2
cosocleC2×C3×C5⋊D5C23
hypercenterC4#10597182
hypocenterC3×C52C4×D4
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×#10597182C2semidirect
3. C4×C3⋊#1044C2semidirect
4. C3×C4⋊#1043C2semidirect
5. #11925355C2semidirect
12. C5×D60C4semidirect
14. #10597128C4semidirect
19. C4⋊#1043D3semidirect
21. C2×C5×D15C2×C4semidirect
26. C3⋊#1044D4semidirect
28. C3×C4×C52C2×C4semidirect
29. C3×#1039C2×C4semidirect
55. D60AGL(1,𝔽5)semidirect
67. C4×C52C4×D3semidirect
68. #1039C4×D3semidirect
76. C3×C52C4×D4semidirect
77. C2×D15C2×AGL(1,𝔽5)semidirect
88. C3×C4×C5C2×AGL(1,𝔽5)semidirect
120. C52#893semidirect
140. C4×C5D3×AGL(1,𝔽5)semidirect
149. C3×C5D4×AGL(1,𝔽5)semidirect
156. C3×C4D5⋊AGL(1,𝔽5)semidirect
180. C5#91921semidirect
189. C4#10597182semidirect
190. C3#11717365semidirect

Maximal subgroups

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

subgroupindex
C2×#105971822
#119256552
C4×C3⋊#10442
C3×C4⋊#10432
#119253552
#117173653
#919215