Finite Group Databasebeta

← #10624235

Subgroups — #10624235

Order 720.

232 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 51 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×(C3⋊D3)⋊D5C211yesmaximal, characteristic, complemented
3C5×#638C211yesmaximal, characteristic, complemented
4C3×#1220C211yesmaximal, characteristic, complemented
5#67157C211yesmaximal, characteristic, complemented
6#67156C211yesmaximal, characteristic, complemented
7#67154C211yesmaximal, characteristic, complemented
8#6790—31nomaximal
9D5×#67—31nomaximal
10C22×C32×C5C2211yescharacteristic, complemented
11C2×C5×C3⋊D3C2211yescharacteristic, complemented
12C2×C3×D15—22no—
13C2×C3×D15C2211yescharacteristic
14C2×C3×D15C2211yescharacteristic, complemented
15C2×C3×D15C2211yescharacteristic, complemented
16(C3⋊D3)⋊D5—22no—
17C5×C3⋊Dic3C2211yescharacteristic, complemented
18C3×Dic15C2211yescharacteristic, complemented
19D3×#67—51nomaximal, Hall
20C22×D15D311yescharacteristic, complemented
21C22×C3×D5—31no—
22C2×D3×D5—31no—
23C2×D3×D5—31no—
24#1220D311yescharacteristic, complemented
25C5×#67—31no—
26C5×#67—31no—
27C5×#67—61no—
28C3×C22⋊D5—31no—
29#1203—31no—
30#1202—31no—
31#1202—31no—
32#1201—31no—
33#1200—31no—
34D5×Dic3—31no—
35C2×C32×C5—21no—
36C2×C32×C5C2311yescharacteristic
37C5×C3⋊D3—21no—
38C3×D15—21no—
39C3×D15—21no—
40C3×D15D412yescomplemented
41D4×D5—91noHall
42C22×C3×D3—51no—
43C2×D32—51no—
44#638D511yescharacteristic, complemented
45C3×#67—51no—
46#626—51no—
47#625—51no—
48D3×Dic3—51no—
49C22×C3×C5—21no—
50C22×C3×C5C2×D311yescharacteristic, complemented
51C22×C3×C5C2×D311yescharacteristic, complemented
52C2×D15—22no—
53C2×D15C2×D311yescharacteristic
54C2×D15C2×D311yescharacteristic, complemented
55C2×D15C2×D311yescharacteristic, complemented
56C2×C5×D3—31no—
57C2×C5×D3—31no—
58C2×C5×D3—61no—
59C2×C3×D5—31no—
60C2×C3×D5—31no—
61C2×C3×D5—31no—
62C2×C3×D5—62no—
63D3×D5—62no—
64D3×D5—62no—
65Dic15C2×D311yescharacteristic, complemented
66C3×Dic5—31no—
67C5×Dic3—31no—
68C5×Dic3—31no—
69C5×Dic3—61no—
70C2×#67—151no—
71D3×D4—151no—
72C32×C5C2×D411yescharacteristic, complemented, Hall
73C22×D5—31no—
74C22×D5—91no—
75C5×D4—91no—
76C22⋊D5—31no—
77C22⋊D5—91no—
78D20—91no—
79C4×D5—91no—
80C22×C32C2×D511yescharacteristic, complemented
81C2×C3⋊D3C2×D511yescharacteristic, complemented
82C2×C3×D3—51no—
83C2×C3×D3—51no—
84C2×C3×D3—51no—
85C2×C3×D3—102no—
86D32—102no—
87C3⋊Dic3C2×D511yescharacteristic, complemented
88C3×Dic3—51no—
89C2×C3×C5—21no—
90C2×C3×C5—21no—
91C2×C3×C5—21no—
92C2×C3×C5—41no—
93C2×C3×C5C22×D311yescharacteristic
94C2×C3×C5C22×D311yescharacteristic
95D15—21no—
96D15—21no—
97D15#6712yescomplemented
98C3×D5—32no—
99C3×D5—61no—
100C3×D5—61no—
101C5×D3—61no—
102C5×D3—61no—
103C5×D3—121no—
104C23×C3—151no—
105C22×D3—51no—
106C22×D3—151no—
107C22×D3—151no—
108C3×D4—151no—
109#67—31no—
110#67—31no—
111#67—51no—
112#67—61no—
113#67—151no—
114#67—151no—
115#67—151no—
116C2×Dic3—151no—
117D12—151no—
118C4×D3—151no—
119C22×C5—91no—
120C22×C5D3211yescharacteristic, complemented
121C2×D5—31no—
122C2×D5—31no—
123C2×D5—31no—
124C2×D5—62no—
125C2×D5—182no—
126C4×C5—91no—
127Dic5—31no—
128C2×C32—21no—
129C2×C32C22×D511yescharacteristic
130C3⋊D3—21no—
131C3×D3—52no—
132C3×D3—101no—
133C3×D3—101no—
134C2×D4—451noSylow
135C3×C5—21no—
136C3×C5D3×D411yescharacteristic, complemented
137C3×C5C2×#6711yescharacteristic, complemented
138C22×C3—21no—
139C22×C3—151no—
140C22×C3—151no—
141C22×C3—151no—
142C22×C3—302no—
143C22×C3D3×D511yescharacteristic, complemented
144C22×C3D3×D511yescharacteristic, complemented
145C2×D3—31no—
146C2×D3—31no—
147C2×D3—51no—
148C2×D3—51no—
149C2×D3—51no—
150C2×D3—61no—
151C2×D3—102no—
152C2×D3—302no—
153C2×D3—302no—
154C3×C4—151no—
155Dic3—31no—
156Dic3—31no—
157Dic3—51no—
158Dic3—61no—
159C2×C5—21no—
160C2×C5—181no—
161C2×C5C2×D3211yescharacteristic
162D5—32no—
163D5—61no—
164D5—61no—
165C32D4×D511yescharacteristic, complemented, Sylow
166C23—151no—
167C23—451no—
168D4—91no—
169D4—151no—
170D4—451no—
171D4—451no—
172C2×C4—451no—
173C2×C3—21no—
174C2×C3—21no—
175C2×C3—21no—
176C2×C3—41no—
177C2×C3—152no—
178C2×C3—301no—
179C2×C3—301no—
180C2×C3C2×D3×D511yescharacteristic
181C2×C3C2×D3×D511yescharacteristic
182D3—52no—
183D3—61no—
184D3—61no—
185D3—101no—
186D3—101no—
187D3—121no—
188C5D3×#6711yescharacteristic, complemented, Sylow
189C22—91no—
190C22—151no—
191C22—151no—
192C22—151no—
193C22—302no—
194C22—902no—
195C22(C3⋊D3)⋊D511yescharacteristic, complemented
196C4—91no—
197C4—151no—
198C3—21no—
199C3D5×#6711yescharacteristic, complemented
200C3#679011yescharacteristic, complemented
201C2—21no—
202C2—152no—
203C2—181no—
204C2—301no—
205C2—301no—
206C2C2×(C3⋊D3)⋊D511yescharacteristic
subgroupG/H
center, FrattiniC2C2×(C3⋊D3)⋊D5
commutator, socle, cosocleC2×C32×C5C23
FittingC22×C32×C5C22
hypercenterC22(C3⋊D3)⋊D5
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×(C3⋊D3)⋊D5C2semidirect
3. C5×#638C2semidirect
4. C3×#1220C2semidirect
5. #67157C2semidirect
6. #67156C2semidirect
7. #67154C2semidirect
10. C22×C32×C5C22semidirect
11. C2×C5×C3⋊D3C22semidirect
14. C2×C3×D15C22semidirect
15. C2×C3×D15C22semidirect
17. C5×C3⋊Dic3C22semidirect
18. C3×Dic15C22semidirect
20. C22×D15D3semidirect
24. #1220D3semidirect
40. C3×D15D4semidirect
44. #638D5semidirect
50. C22×C3×C5C2×D3semidirect
51. C22×C3×C5C2×D3semidirect
54. C2×D15C2×D3semidirect
55. C2×D15C2×D3semidirect
65. Dic15C2×D3semidirect
72. C32×C5C2×D4semidirect
80. C22×C32C2×D5semidirect
81. C2×C3⋊D3C2×D5semidirect
87. C3⋊Dic3C2×D5semidirect
97. D15#67semidirect
120. C22×C5D32semidirect
136. C3×C5D3×D4semidirect
137. C3×C5C2×#67semidirect
143. C22×C3D3×D5semidirect
144. C22×C3D3×D5semidirect
165. C32D4×D5semidirect
188. C5D3×#67semidirect
195. C22(C3⋊D3)⋊D5semidirect
199. C3D5×#67semidirect
200. C3#6790semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D152
C2×(C3⋊D3)⋊D52
C5×#6382
C3×#12202
#671572
#671562
#671542
#67903
D5×#673
D3×#675