Finite Group Databasebeta

← D3×#6705 #13087258

Subgroups — D3×#6705 #13087258

Order 1440.

292 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C4×D3×D15C211yesmaximal, characteristic
2#10623899C211yesmaximal, characteristic, complemented
3C3×#6705C211yesmaximal, characteristic, complemented
4C3×C5×D3×C8C211yesmaximal, characteristic, complemented
5#10623797C211yesmaximal, characteristic, complemented
6#10623795C211yesmaximal, characteristic, complemented
7D3×C3⋊#168C211yesmaximal, characteristic, complemented
8C2×#6705—31nomaximal
9D3×#676—31nomaximal
10C2×D3×D15C411yescharacteristic
11C4×C3⋊D15C2211yescharacteristic
12C3×C4×D15C2211yescharacteristic
13C3×C4×C5×D3C2211yescharacteristic
14#67149C411yescharacteristic
15D3×Dic15C411yescharacteristic
16Dic3×D15C411yescharacteristic
17C32×C5×C8C2211yescharacteristic, complemented
18C3⋊(C3⋊#168)C2211yescharacteristic, complemented
19C3×C3⋊#168C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21D3×ΓL(1,𝔽25)—51nomaximal, Hall
22C2×C4×D15—31no—
23C4×D3×D5—31no—
24C2×C3×C5×C8—31no—
25C2×C3⋊#168—31no—
26#6705—31no—
27#6705—31no—
28#6705—31no—
29#6705—31no—
30#6705—61no—
31#6705D311yescharacteristic, direct product factor
32C5×D3×C8D311yescharacteristic, complemented
33C3×#676—31no—
34#6651—31no—
35#6649—31no—
36D3×#168—31no—
37C2×C3⋊D15C2×C411yescharacteristic
38C2×C3×D15C2×C411yescharacteristic
39C2×C3×C5×D3C2×C411yescharacteristic
40D3×D15—22no—
41C32×C4×C5C2311yescharacteristic
42C3⋊Dic15C2×C411yescharacteristic
43C3×Dic15C2×C411yescharacteristic
44C3×C5×Dic3C2×C411yescharacteristic
45C2×#676—91noHall
46C4×D32—51no—
47#3743—51no—
48C3×ΓL(1,𝔽25)—51no—
49C3×D3×C8D511yescharacteristic, complemented
50#3712—51no—
51#3711—51no—
52D3×C3⋊C8—51no—
53C22×D15—31no—
54C2×D3×D5—31no—
55C2×C3×C4×C5—31no—
56C2×Dic15—31no—
57C4×D15—31no—
58C4×D15—31no—
59C4×D15—31no—
60C4×D15—31no—
61C4×D15—61no—
62C4×D15C2×D311yescharacteristic
63C4×C5×D3C2×D311yescharacteristic
64C3×C4×D5—31no—
65#1200—31no—
66D3×Dic5—31no—
67D5×Dic3—31no—
68C3×C5×C8—21no—
69C3×C5×C8—31no—
70C3×C5×C8C2×D311yescharacteristic, complemented
71C3×C5×C8C2×D311yescharacteristic, complemented
72C3⋊#168—31no—
73C3⋊#168—31no—
74C3⋊#168—61no—
75C3⋊#168C2×D311yescharacteristic, complemented
76C3×#168—31no—
77C5×C3⋊C8C2×D311yescharacteristic, complemented
78D3×M16—151no—
79C2×ΓL(1,𝔽25)—151no—
80C2×C32×C5C22×C411yescharacteristic
81C3⋊D15—21no—
82C3×D15—21no—
83C3×C5×D3M1612yescomplemented
84C2×C4×D5—91no—
85C2×C5×C8—31no—
86C2×#168—91no—
87#676—31no—
88#676—91no—
89#676—91no—
90#676—91no—
91C2×D32—51no—
92C4×C3⋊D3—51no—
93C3×C4×D3—51no—
94C3×C4×D3C2×D511yescharacteristic
95#624—51no—
96D3×Dic3—51no—
97D3×Dic3—51no—
98C32×C8C2×D511yescharacteristic, complemented
99C3⋊(C3⋊C8)—51no—
100C3×C3⋊C8—51no—
101C3×C3⋊C8C2×D511yescharacteristic, complemented
102C22×C3×C5—31no—
103C2×D15—31no—
104C2×D15—31no—
105C2×D15—61no—
106C2×D15—62no—
107C2×D15C4×D311yescharacteristic
108C2×C5×D3C4×D311yescharacteristic
109C2×C3×D5—31no—
110D3×D5—62no—
111C3×C4×C5—21no—
112C3×C4×C5—31no—
113C3×C4×C5C22×D311yescharacteristic
114C3×C4×C5C22×D311yescharacteristic
115Dic15—31no—
116Dic15—31no—
117Dic15—61no—
118Dic15C4×D311yescharacteristic
119C3×Dic5—31no—
120C5×Dic3C4×D311yescharacteristic
121C2×C4×D3—151no—
122C2×C4×D3—151no—
123C3×M16—151no—
124C2×C3×C8—31no—
125#208—151no—
126C2×C3⋊C8—151no—
127ΓL(1,𝔽25)—51no—
128ΓL(1,𝔽25)—151no—
129ΓL(1,𝔽25)—151no—
130ΓL(1,𝔽25)—151no—
131ΓL(1,𝔽25)—151no—
132ΓL(1,𝔽25)—151no—
133ΓL(1,𝔽25)—301no—
134D3×C8—151no—
135D3×C8D1511yescharacteristic, complemented
136C32×C5C2×M1611yescharacteristic, complemented, Hall
137C22×D5—91no—
138C2×C4×C5—31no—
139C2×Dic5—91no—
140C4×D5—31no—
141C4×D5—91no—
142C4×D5—91no—
143C4×D5—91no—
144C5×C8—31no—
145C5×C8D3211yescharacteristic, complemented
146#168—31no—
147#168—91no—
148C2×C3⋊D3—51no—
149C2×C3×D3—51no—
150C2×C3×D3C4×D511yescharacteristic
151D32—102no—
152C32×C4C22×D511yescharacteristic
153C3⋊Dic3—51no—
154C3×Dic3—51no—
155C3×Dic3C4×D511yescharacteristic
156C2×M16—451noSylow
157C2×C3×C5—21no—
158C2×C3×C5—32no—
159C2×C3×C5C2×C4×D311yescharacteristic
160C2×C3×C5C2×C4×D311yescharacteristic
161D15—21no—
162D15—61no—
163D15—61no—
164D15—121no—
165C3×D5—61no—
166C5×D3ΓL(1,𝔽25)12yescomplemented
167C22×D3—151no—
168C22×D3—151no—
169C2×C3×C4—31no—
170C2×C3×C4—151no—
171C2×Dic3—151no—
172C2×Dic3—151no—
173C4×D3—51no—
174C4×D3—151no—
175C4×D3—151no—
176C4×D3—151no—
177C4×D3—151no—
178C4×D3—151no—
179C4×D3—151no—
180C4×D3—301no—
181C4×D3C2×D1511yescharacteristic
182C3×C8—21no—
183C3×C8—31no—
184C3×C8—151no—
185C3×C8D3×D511yescharacteristic, complemented
186C3×C8C2×D1511yescharacteristic, complemented
187C3⋊C8—51no—
188C3⋊C8—151no—
189C3⋊C8—151no—
190C3⋊C8—301no—
191C3⋊C8C2×D1511yescharacteristic, complemented
192C22×C5—31no—
193C2×D5—31no—
194C2×D5—91no—
195C2×D5—182no—
196C4×C5—31no—
197C4×C5C2×D3211yescharacteristic
198Dic5—31no—
199Dic5—91no—
200C2×C32C2×C4×D511yescharacteristic
201C3⋊D3—101no—
202C3×D3—101no—
203C3×D3#67612yescomplemented
204C22×C4—451no—
205M16—151no—
206M16—451no—
207M16—451no—
208M16—451no—
209C2×C8—31no—
210C2×C8—451no—
211C3×C5—21no—
212C3×C5C2×ΓL(1,𝔽25)11yescharacteristic, complemented
213C3×C5D3×M1611yescharacteristic, complemented
214C22×C3—31no—
215C22×C3—151no—
216C2×D3—51no—
217C2×D3—151no—
218C2×D3—151no—
219C2×D3—301no—
220C2×D3—302no—
221C2×D3—302no—
222C2×D3C4×D1511yescharacteristic
223C3×C4—21no—
224C3×C4—31no—
225C3×C4—151no—
226C3×C4C2×D3×D511yescharacteristic
227C3×C4C22×D1511yescharacteristic
228Dic3—51no—
229Dic3—151no—
230Dic3—151no—
231Dic3—301no—
232Dic3C4×D1511yescharacteristic
233C2×C5—32no—
234C2×C5C4×D3211yescharacteristic
235D5—61no—
236D5—181no—
237C32C2×#67611yescharacteristic, complemented, Sylow
238C23—451no—
239C2×C4—31no—
240C2×C4—151no—
241C2×C4—451no—
242C2×C4—451no—
243C2×C4—451no—
244C2×C4—451no—
245C8—31no—
246C8—151no—
247C8—451no—
248C8D3×D1511yescharacteristic, complemented
249C2×C3—21no—
250C2×C3—32no—
251C2×C3—301no—
252C2×C3C4×D3×D511yescharacteristic
253C2×C3C2×C4×D1511yescharacteristic
254D3—101no—
255D3—301no—
256D3—301no—
257D3—601no—
258D3#670512yesdirect product factor
259C5D3×ΓL(1,𝔽25)11yescharacteristic, complemented, Sylow
260C22—31no—
261C22—151no—
262C22—451no—
263C22—902no—
264C4—31no—
265C4—151no—
266C4—451no—
267C4C2×D3×D1511yescharacteristic
268C3—21no—
269C3D3×#67611yescharacteristic, complemented
270C3C2×#670511yescharacteristic, complemented
271C2—32no—
272C2—301no—
273C2—901no—
274C2C4×D3×D1511yescharacteristic
subgroupG/H
center, FrattiniC4C2×D3×D15
commutator, socleC2×C32×C5C22×C4
FittingC32×C5×C8C22
prime radicalC2×D3×D15C4
cosocleC32×C4×C5C23
hypercenterC8D3×D15
hypocenterC32×C5C2×M16
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #10623899C2semidirect
3. C3×#6705C2semidirect
4. C3×C5×D3×C8C2semidirect
5. #10623797C2semidirect
6. #10623795C2semidirect
7. D3×C3⋊#168C2semidirect
17. C32×C5×C8C22semidirect
18. C3⋊(C3⋊#168)C22semidirect
19. C3×C3⋊#168C22semidirect
20. C3×C5×C3⋊C8C22semidirect
31. #6705D3direct (N × H)
32. C5×D3×C8D3semidirect
49. C3×D3×C8D5semidirect
70. C3×C5×C8C2×D3semidirect
71. C3×C5×C8C2×D3semidirect
75. C3⋊#168C2×D3semidirect
77. C5×C3⋊C8C2×D3semidirect
83. C3×C5×D3M16semidirect
98. C32×C8C2×D5semidirect
101. C3×C3⋊C8C2×D5semidirect
135. D3×C8D15semidirect
136. C32×C5C2×M16semidirect
145. C5×C8D32semidirect
166. C5×D3ΓL(1,𝔽25)semidirect
185. C3×C8D3×D5semidirect
186. C3×C8C2×D15semidirect
191. C3⋊C8C2×D15semidirect
203. C3×D3#676semidirect
212. C3×C5C2×ΓL(1,𝔽25)semidirect
213. C3×C5D3×M16semidirect
237. C32C2×#676semidirect
248. C8D3×D15semidirect
258. D3#6705direct (N × H)
259. C5D3×ΓL(1,𝔽25)semidirect
269. C3D3×#676semidirect
270. C3C2×#6705semidirect

Maximal subgroups

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

subgroupindex
C4×D3×D152
#106238992
C3×#67052
C3×C5×D3×C82
#106237972
#106237952
D3×C3⋊#1682
C2×#67053
D3×#6763
D3×ΓL(1,𝔽25)5