Finite Group Databasebeta

← #67×D15 #10624216

Subgroups — #67×D15 #10624216

Order 720.

232 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 54 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×D15C211yesmaximal, characteristic, complemented
2C2×D3×D15C211yesmaximal, characteristic, complemented
3#67184C211yesmaximal, characteristic, complemented
4C3×C5×#67C211yesmaximal, characteristic, complemented
5#67151C211yesmaximal, characteristic, complemented
6#67150C211yesmaximal, characteristic, complemented
7Dic3×D15C211yesmaximal, characteristic, complemented
8D4×D15—31nomaximal
9D5×#67—31nomaximal
10C22×C32×C5C2211yescharacteristic, complemented
11C2×C3⋊D15C2211yescharacteristic, complemented
12C2×C3×D15—22no—
13C2×C3×D15C2211yescharacteristic
14C2×C3×D15C2211yescharacteristic, complemented
15C2×C3×C5×D3C2211yescharacteristic, complemented
16D3×D15—22no—
17C3⋊Dic15C2211yescharacteristic, complemented
18C3×C5×Dic3C2211yescharacteristic, complemented
19D3×#67—51nomaximal, Hall
20C22×D15—31no—
21C22×D15D311yescharacteristic, complemented
22C22×C3×D5—31no—
23C2×D3×D5—31no—
24C3×C5×D4—31no—
25#1220—31no—
26#1220—31no—
27#1220—31no—
28#1220—61no—
29D60—31no—
30C4×D15—31no—
31C5×#67D311yescharacteristic, complemented
32#1202—31no—
33#1201—31no—
34D5×Dic3—31no—
35C2×C32×C5—21no—
36C2×C32×C5C2311yescharacteristic
37C3⋊D15—21no—
38C3×D15—21no—
39C3×D15D412yescomplemented
40C3×C5×D3—21no—
41D4×D5—91noHall
42C22×C3×D3—51no—
43C2×D32—51no—
44#638—51no—
45C3×#67D511yescharacteristic, complemented
46#626—51no—
47#625—51no—
48D3×Dic3—51no—
49C22×C3×C5—21no—
50C22×C3×C5—31no—
51C22×C3×C5C2×D311yescharacteristic, complemented
52C22×C3×C5C2×D311yescharacteristic, complemented
53C2×D15—22no—
54C2×D15—31no—
55C2×D15—31no—
56C2×D15—61no—
57C2×D15—62no—
58C2×D15C2×D311yescharacteristic
59C2×D15C2×D311yescharacteristic, complemented
60C2×C5×D3C2×D311yescharacteristic, complemented
61C2×C3×D5—31no—
62C2×C3×D5—31no—
63C2×C3×D5—62no—
64D3×D5—62no—
65C3×C4×C5—31no—
66Dic15—31no—
67Dic15—31no—
68Dic15—61no—
69C5×Dic3C2×D311yescharacteristic, complemented
70C2×#67—151no—
71D3×D4—151no—
72C32×C5C2×D411yescharacteristic, complemented, Hall
73C22×D5—31no—
74C22×D5—91no—
75C5×D4—31no—
76C22⋊D5—91no—
77C22⋊D5—91no—
78D20—91no—
79C4×D5—91no—
80C22×C32C2×D511yescharacteristic, complemented
81C2×C3⋊D3—51no—
82C2×C3×D3—51no—
83C2×C3×D3—51no—
84C2×C3×D3—102no—
85C2×C3×D3C2×D511yescharacteristic, complemented
86D32—102no—
87C3⋊Dic3—51no—
88C3×Dic3C2×D511yescharacteristic, complemented
89C2×C3×C5—21no—
90C2×C3×C5—21no—
91C2×C3×C5—21no—
92C2×C3×C5—41no—
93C2×C3×C5—61no—
94C2×C3×C5C22×D311yescharacteristic
95C2×C3×C5C22×D311yescharacteristic
96D15—21no—
97D15—61no—
98D15—61no—
99D15—121no—
100D15#6712yesdirect product factor
101C3×D5—32no—
102C3×D5—61no—
103C5×D3—21no—
104C23×C3—151no—
105C22×D3—51no—
106C22×D3—151no—
107C22×D3—151no—
108C3×D4—31no—
109#67—151no—
110#67—151no—
111#67—151no—
112#67—151no—
113#67—151no—
114#67—301no—
115#67D1511yescharacteristic, direct product factor
116C2×Dic3—151no—
117D12—151no—
118C4×D3—151no—
119C22×C5—31no—
120C22×C5D3211yescharacteristic, complemented
121C2×D5—31no—
122C2×D5—31no—
123C2×D5—62no—
124C2×D5—91no—
125C2×D5—182no—
126C4×C5—31no—
127Dic5—91no—
128C2×C32—21no—
129C2×C32C22×D511yescharacteristic
130C3⋊D3—101no—
131C3×D3—21no—
132C3×D3—52no—
133C3×D3—101no—
134C2×D4—451noSylow
135C3×C5—21no—
136C3×C5D3×D411yescharacteristic, complemented
137C3×C5C2×#6711yescharacteristic, complemented
138C22×C3—21no—
139C22×C3—31no—
140C22×C3—151no—
141C22×C3—151no—
142C22×C3—302no—
143C22×C3D3×D511yescharacteristic, complemented
144C22×C3C2×D1511yescharacteristic, complemented
145C2×D3—51no—
146C2×D3—51no—
147C2×D3—102no—
148C2×D3—151no—
149C2×D3—151no—
150C2×D3—301no—
151C2×D3—302no—
152C2×D3—302no—
153C2×D3C2×D1511yescharacteristic, complemented
154C3×C4—31no—
155Dic3—151no—
156Dic3—151no—
157Dic3—301no—
158Dic3C2×D1511yescharacteristic, complemented
159C2×C5—21no—
160C2×C5—61no—
161C2×C5C2×D3211yescharacteristic
162D5—32no—
163D5—61no—
164D5—181no—
165C32D4×D511yescharacteristic, complemented, Sylow
166C23—151no—
167C23—451no—
168D4—31no—
169D4—451no—
170D4—451no—
171D4—451no—
172C2×C4—451no—
173C2×C3—21no—
174C2×C3—21no—
175C2×C3—21no—
176C2×C3—41no—
177C2×C3—61no—
178C2×C3—152no—
179C2×C3—301no—
180C2×C3C2×D3×D511yescharacteristic
181C2×C3C22×D1511yescharacteristic
182D3—21no—
183D3—52no—
184D3—101no—
185D3—301no—
186D3—301no—
187D3—601no—
188C5D3×#6711yescharacteristic, complemented, Sylow
189C22—31no—
190C22—151no—
191C22—151no—
192C22—302no—
193C22—451no—
194C22—902no—
195C22D3×D1511yescharacteristic, complemented
196C4—31no—
197C4—451no—
198C3—21no—
199C3D5×#6711yescharacteristic, complemented
200C3D4×D1511yescharacteristic, complemented
201C2—21no—
202C2—61no—
203C2—152no—
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×D15C2semidirect
2. C2×D3×D15C2semidirect
3. #67184C2semidirect
4. C3×C5×#67C2semidirect
5. #67151C2semidirect
6. #67150C2semidirect
7. Dic3×D15C2semidirect
10. C22×C32×C5C22semidirect
11. C2×C3⋊D15C22semidirect
14. C2×C3×D15C22semidirect
15. C2×C3×C5×D3C22semidirect
17. C3⋊Dic15C22semidirect
18. C3×C5×Dic3C22semidirect
21. C22×D15D3semidirect
31. C5×#67D3semidirect
39. C3×D15D4semidirect
45. C3×#67D5semidirect
51. C22×C3×C5C2×D3semidirect
52. C22×C3×C5C2×D3semidirect
59. C2×D15C2×D3semidirect
60. C2×C5×D3C2×D3semidirect
69. C5×Dic3C2×D3semidirect
72. C32×C5C2×D4semidirect
80. C22×C32C2×D5semidirect
85. C2×C3×D3C2×D5semidirect
88. C3×Dic3C2×D5semidirect
100. D15#67direct (N × H)
115. #67D15direct (N × H)
120. C22×C5D32semidirect
136. C3×C5D3×D4semidirect
137. C3×C5C2×#67semidirect
143. C22×C3D3×D5semidirect
144. C22×C3C2×D15semidirect
153. C2×D3C2×D15semidirect
158. Dic3C2×D15semidirect
165. C32D4×D5semidirect
188. C5D3×#67semidirect
195. C22D3×D15semidirect
199. C3D5×#67semidirect
200. C3D4×D15semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D152
C2×D3×D152
#671842
C3×C5×#672
#671512
#671502
Dic3×D152
D4×D153
D5×#673
D3×#675