Finite Group Databasebeta

← D3×#1220 #10624217

Subgroups — D3×#1220 #10624217

Order 720.

232 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 58 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×C5×D3C211yesmaximal, characteristic, complemented
2C2×D3×D15C211yesmaximal, characteristic, complemented
3#67184C211yesmaximal, characteristic, complemented
4C3×#1220C211yesmaximal, characteristic, complemented
5#67152C211yesmaximal, characteristic, complemented
6#67150C211yesmaximal, characteristic, complemented
7D3×Dic15C211yesmaximal, characteristic, complemented
8C2×#1220—31nomaximal
9D3×C22⋊D5—31nomaximal
10C22×C32×C5C2211yescharacteristic, complemented
11C2×C3⋊D15C2211yescharacteristic, complemented
12C2×C3×D15C2211yescharacteristic, complemented
13C2×C3×C5×D3—22no—
14C2×C3×C5×D3C2211yescharacteristic
15C2×C3×C5×D3C2211yescharacteristic, complemented
16D3×D15—22no—
17C3⋊Dic15C2211yescharacteristic, complemented
18C3×Dic15C2211yescharacteristic, complemented
19D3×#67—51nomaximal, Hall
20C23×C3×C5—31no—
21C22×D15—31no—
22C22×C5×D3D311yescharacteristic, complemented
23C2×D3×D5—31no—
24#1220—31no—
25#1220—31no—
26#1220—31no—
27#1220—31no—
28#1220—61no—
29#1220D311yescharacteristic, direct product factor
30C2×Dic15—31no—
31C3×C22⋊D5—31no—
32#1203—31no—
33#1201—31no—
34D3×Dic5—31no—
35C2×C32×C5—21no—
36C2×C32×C5C2311yescharacteristic
37C3⋊D15—21no—
38C3×D15—21no—
39C3×C5×D3—21no—
40C3×C5×D3D412yescomplemented
41C2×C22⋊D5—91noHall
42C22×C3×D3D511yescharacteristic, complemented
43C2×D32—51no—
44#638—51no—
45C3×#67—51no—
46#626—51no—
47#625—51no—
48D3×Dic3—51no—
49C22×C3×C5—21no—
50C22×C3×C5—31no—
51C22×C3×C5—31no—
52C22×C3×C5—62no—
53C22×C3×C5C2×D311yescharacteristic, complemented
54C22×C3×C5C2×D311yescharacteristic, complemented
55C2×D15—31no—
56C2×D15—31no—
57C2×D15—61no—
58C2×D15—62no—
59C2×D15C2×D311yescharacteristic, complemented
60C2×C5×D3—22no—
61C2×C5×D3C2×D311yescharacteristic
62C2×C5×D3C2×D311yescharacteristic, complemented
63C2×C3×D5—31no—
64D3×D5—62no—
65Dic15—31no—
66Dic15—31no—
67Dic15—61no—
68Dic15C2×D311yescharacteristic, complemented
69C3×Dic5—31no—
70C2×#67—151no—
71D3×D4—151no—
72C32×C5C2×D411yescharacteristic, complemented, Hall
73C23×C5—31no—
74C22×D5—91no—
75C22⋊D5—31no—
76C22⋊D5—91no—
77C22⋊D5—91no—
78C22⋊D5—91no—
79C2×Dic5—91no—
80C22×C32C2×D511yescharacteristic, complemented
81C2×C3⋊D3—51no—
82C2×C3×D3—22no—
83C2×C3×D3—51no—
84C2×C3×D3C2×D511yescharacteristic
85C2×C3×D3C2×D511yescharacteristic, complemented
86D32—102no—
87C3⋊Dic3—51no—
88C3×Dic3—51no—
89C2×C3×C5—21no—
90C2×C3×C5—21no—
91C2×C3×C5—21no—
92C2×C3×C5—32no—
93C2×C3×C5—41no—
94C2×C3×C5—61no—
95C2×C3×C5C22×D311yescharacteristic
96C2×C3×C5C22×D311yescharacteristic
97D15—21no—
98D15—61no—
99D15—61no—
100D15—121no—
101C3×D5—61no—
102C5×D3—21no—
103C5×D3#6712yescomplemented
104C23×C3—31no—
105C22×D3—151no—
106C22×D3—151no—
107C22×D3D1511yescharacteristic, complemented
108C3×D4—151no—
109#67—51no—
110#67—151no—
111#67—151no—
112#67—151no—
113#67—151no—
114#67—151no—
115#67—301no—
116C2×Dic3—151no—
117D12—151no—
118C4×D3—151no—
119C22×C5—31no—
120C22×C5—31no—
121C22×C5—62no—
122C22×C5D3211yescharacteristic, complemented
123C2×D5—31no—
124C2×D5—91no—
125C2×D5—182no—
126Dic5—31no—
127Dic5—91no—
128C2×C32—21no—
129C2×C32C22×D511yescharacteristic
130C3⋊D3—101no—
131C3×D3—21no—
132C3×D3—101no—
133C3×D3C22⋊D512yescomplemented
134C2×D4—451noSylow
135C3×C5—21no—
136C3×C5D3×D411yescharacteristic, complemented
137C3×C5C2×#6711yescharacteristic, complemented
138C22×C3—21no—
139C22×C3—31no—
140C22×C3—31no—
141C22×C3—62no—
142C22×C3—151no—
143C22×C3D3×D511yescharacteristic, complemented
144C22×C3C2×D1511yescharacteristic, complemented
145C2×D3—22no—
146C2×D3—51no—
147C2×D3—151no—
148C2×D3—151no—
149C2×D3—301no—
150C2×D3—302no—
151C2×D3—302no—
152C2×D3C2×D1511yescharacteristic
153C2×D3C2×D1511yescharacteristic, complemented
154C3×C4—151no—
155Dic3—51no—
156Dic3—151no—
157Dic3—151no—
158Dic3—301no—
159C2×C5—21no—
160C2×C5—32no—
161C2×C5—61no—
162C2×C5C2×D3211yescharacteristic
163D5—61no—
164D5—181no—
165C32C2×C22⋊D511yescharacteristic, complemented, Sylow
166C23—31no—
167C23—451no—
168D4—151no—
169D4—451no—
170D4—451no—
171D4—451no—
172C2×C4—451no—
173C2×C3—21no—
174C2×C3—21no—
175C2×C3—21no—
176C2×C3—32no—
177C2×C3—41no—
178C2×C3—61no—
179C2×C3—301no—
180C2×C3C2×D3×D511yescharacteristic
181C2×C3C22×D1511yescharacteristic
182D3—21no—
183D3—101no—
184D3—301no—
185D3—301no—
186D3—601no—
187D3#122012yesdirect product factor
188C5D3×#6711yescharacteristic, complemented, Sylow
189C22—31no—
190C22—31no—
191C22—62no—
192C22—151no—
193C22—451no—
194C22—902no—
195C22D3×D1511yescharacteristic, complemented
196C4—151no—
197C4—451no—
198C3—21no—
199C3D3×C22⋊D511yescharacteristic, complemented
200C3C2×#122011yescharacteristic, complemented
201C2—21no—
202C2—32no—
203C2—61no—
204C2—301no—
205C2—901no—
206C2C2×D3×D1511yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×D15
commutator, socle, cosocleC2×C32×C5C23
FittingC22×C32×C5C22
hypercenterC22D3×D15
hypocenterC32×C5C2×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×C5×D3C2semidirect
2. C2×D3×D15C2semidirect
3. #67184C2semidirect
4. C3×#1220C2semidirect
5. #67152C2semidirect
6. #67150C2semidirect
7. D3×Dic15C2semidirect
10. C22×C32×C5C22semidirect
11. C2×C3⋊D15C22semidirect
12. C2×C3×D15C22semidirect
15. C2×C3×C5×D3C22semidirect
17. C3⋊Dic15C22semidirect
18. C3×Dic15C22semidirect
22. C22×C5×D3D3semidirect
29. #1220D3direct (N × H)
40. C3×C5×D3D4semidirect
42. C22×C3×D3D5semidirect
53. C22×C3×C5C2×D3semidirect
54. C22×C3×C5C2×D3semidirect
59. C2×D15C2×D3semidirect
62. C2×C5×D3C2×D3semidirect
68. Dic15C2×D3semidirect
72. C32×C5C2×D4semidirect
80. C22×C32C2×D5semidirect
85. C2×C3×D3C2×D5semidirect
103. C5×D3#67semidirect
107. C22×D3D15semidirect
122. C22×C5D32semidirect
133. C3×D3C22⋊D5semidirect
136. C3×C5D3×D4semidirect
137. C3×C5C2×#67semidirect
143. C22×C3D3×D5semidirect
144. C22×C3C2×D15semidirect
153. C2×D3C2×D15semidirect
165. C32C2×C22⋊D5semidirect
187. D3#1220direct (N × H)
188. C5D3×#67semidirect
195. C22D3×D15semidirect
199. C3D3×C22⋊D5semidirect
200. C3C2×#1220semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C5×D32
C2×D3×D152
#671842
C3×#12202
#671522
#671502
D3×Dic152
C2×#12203
D3×C22⋊D53
D3×#675