Finite Group Databasebeta

← #13086197

Subgroups — #13086197

Order 1440.

298 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 90 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×#67153C211yesmaximal, characteristic
2C2×C5×#627C211yesmaximal, characteristic
3#10623831C211yesmaximal, characteristic
4#10623814C211yesmaximal, characteristic
5#10623773C211yesmaximal, characteristic
6C3×#6668C211yesmaximal, characteristic
7C3×Dic3×Dic5C211yesmaximal, characteristic
8#91376—31nomaximal
9#91329—31nomaximal
10C2×C3⋊Dic15C2211yescharacteristic
11C2×C5×C3⋊Dic3C2211yescharacteristic
12C2×C3×Dic15C2211yescharacteristic
13C2×C3×Dic15C2211yescharacteristic
14C2×C3×C5×Dic3C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C32×Dic5C2211yescharacteristic
17#67153—22no—
18C5×#627—22no—
19#63918—51nomaximal, Hall
20C2×Dic30—31no—
21C2×C5×Dic6—31no—
22C2×C5×Dic6—31no—
23C2×#1204—31no—
24#6712—31no—
25C3×#683—31no—
26C3×C4×Dic5—31no—
27#6669—31no—
28#6668—31no—
29#6668—31no—
30#6668—31no—
31#6668—31no—
32#6668—61no—
33#6668D311yescharacteristic
34Dic3×Dic5D311yescharacteristic
35C22×C32×C5C2311yescharacteristic
36C3⋊Dic15—21no—
37C5×C3⋊Dic3—21no—
38C3×Dic15—21no—
39C3×Dic15Q812yescomplemented
40C3×C5×Dic3—21no—
41C3×C5×Dic3D412yes—
42C32×Dic5Q812yescomplemented
43#4090—91noHall
44C2×#627—51no—
45C2×#627D511yescharacteristic
46C4⋊(C3⋊Dic3)—51no—
47C3×C4⋊Dic3—51no—
48C3×C4×Dic3—51no—
49Dic3⋊Dic3—51no—
50Dic3⋊Dic3—51no—
51C2×C3×C4×C5—31no—
52C2×C3×C4×C5—31no—
53C2×Dic15—31no—
54C2×Dic15—31no—
55C2×Dic15—61no—
56C2×Dic15C2×D311yescharacteristic
57C2×Dic15C2×D311yescharacteristic
58Dic30—62no—
59C2×C5×Dic3—31no—
60C2×C5×Dic3—31no—
61C2×C5×Dic3—61no—
62C2×C5×Dic3C2×D311yescharacteristic
63C2×C5×Dic3C2×D311yescharacteristic
64C5×Dic6—62no—
65C5×Dic6—62no—
66C2×C3×Dic5—21no—
67C2×C3×Dic5—31no—
68C2×C3×Dic5—31no—
69C2×C3×Dic5C2×D311yescharacteristic
70C2×C3×Dic5C2×D311yescharacteristic
71#1204—62no—
72#888—151no—
73#869—151no—
74C2×C32×C5C2×D411yescharacteristic
75C2×C32×C5C2×Q811yescharacteristic
76C2×C32×C5C2×Q811yescharacteristic
77C2×C5×Q8—91no—
78C2×Dic10—91no—
79#683—31no—
80#683—91no—
81#683—91no—
82#683—91no—
83C4×Dic5—31no—
84C2×C32×C4—51no—
85C2×C3⋊Dic3—51no—
86C2×C3⋊Dic3C2×D511yescharacteristic
87C2×C3×Dic3—51no—
88C2×C3×Dic3—51no—
89C2×C3×Dic3C2×D511yescharacteristic
90C2×C3×Dic3C2×D511yescharacteristic
91#627—22no—
92#627—102no—
93C22×C3×C5—21no—
94C22×C3×C5C22×D311yescharacteristic
95C22×C3×C5C22×D311yescharacteristic
96C3×C4×C5—32no—
97C3×C4×C5—61no—
98Dic15—21no—
99Dic15—61no—
100Dic15—61no—
101Dic15—121no—
102Dic15Dic612yescomplemented
103C3×Dic5—22no—
104C3×Dic5—32no—
105C3×Dic5—61no—
106C3×Dic5Dic612yescomplemented
107C3×Dic5Dic612yescomplemented
108C5×Dic3—21no—
109C5×Dic3—61no—
110C5×Dic3—61no—
111C5×Dic3—121no—
112C5×Dic3D1212yes—
113C2×Dic6—31no—
114C2×Dic6—31no—
115C2×Dic6—151no—
116C2×Dic6—151no—
117C3×C4⋊C4—151no—
118C3×C42—151no—
119C4⋊Dic3—51no—
120C4⋊Dic3—151no—
121C4⋊Dic3—151no—
122C4⋊Dic3—151no—
123C4⋊Dic3—151no—
124C4⋊Dic3—301no—
125Dic3⋊C4—151no—
126Dic3⋊C4—151no—
127C4×Dic3—51no—
128C32×C5C4⋊Q811yescharacteristic, complemented, Hall
129C5×Q8—182no—
130C2×C4×C5—31no—
131C2×C4×C5—31no—
132C2×C4×C5—91no—
133C2×Dic5—31no—
134C2×Dic5—31no—
135C2×Dic5—91no—
136C2×Dic5D3211yescharacteristic
137Dic10—182no—
138C22×C32C22×D511yescharacteristic
139C32×C4—52no—
140C3⋊Dic3—21no—
141C3⋊Dic3—101no—
142C3×Dic3—21no—
143C3×Dic3—52no—
144C3×Dic3—101no—
145C3×Dic3C22⋊D512yes—
146C4⋊Q8—451noSylow
147C2×C3×C5—21no—
148C2×C3×C5—21no—
149C2×C3×C5—21no—
150C2×C3×C5C2×Dic611yescharacteristic
151C2×C3×C5C2×Dic611yescharacteristic
152C2×C3×C5C2×Dic611yescharacteristic
153C2×C3×C5C2×D1211yescharacteristic
154C2×C3×C5D3×D411yescharacteristic
155C2×C3×C5D3×Q811yescharacteristic
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—51no—
159C2×C3×C4—51no—
160C2×C3×C4—101no—
161C2×C3×C4—151no—
162C2×C3×C4—151no—
163C2×Dic3—31no—
164C2×Dic3—31no—
165C2×Dic3—51no—
166C2×Dic3—51no—
167C2×Dic3—61no—
168C2×Dic3—151no—
169C2×Dic3—151no—
170C2×Dic3—301no—
171C2×Dic3D3×D511yescharacteristic
172C2×Dic3D3×D511yescharacteristic
173Dic6—62no—
174Dic6—62no—
175Dic6—302no—
176Dic6—302no—
177C22×C5C2×D3211yescharacteristic
178C4×C5—32no—
179C4×C5—61no—
180C4×C5—181no—
181Dic5—32no—
182Dic5—61no—
183Dic5—181no—
184Dic5#62712yescomplemented
185C2×C32Q8×D511yescharacteristic
186C2×C32Q8×D511yescharacteristic
187C2×C32C2×C22⋊D511yescharacteristic
188C2×Q8—91no—
189C2×Q8—451no—
190C4⋊C4—151no—
191C4⋊C4—451no—
192C4⋊C4—451no—
193C4⋊C4—451no—
194C42—151no—
195C3×C5—21no—
196C3×C5#86911yescharacteristic, complemented
197C3×C5#88811yescharacteristic, complemented
198C22×C3—21no—
199C22×C3C2×D3×D511yescharacteristic
200C22×C3C2×D3×D511yescharacteristic
201C3×C4—32no—
202C3×C4—52no—
203C3×C4—52no—
204C3×C4—61no—
205C3×C4—102no—
206C3×C4—152no—
207C3×C4—301no—
208Dic3—21no—
209Dic3—52no—
210Dic3—61no—
211Dic3—61no—
212Dic3—101no—
213Dic3—121no—
214Dic3—301no—
215Dic3—301no—
216Dic3—601no—
217Dic3#120312yes—
218C2×C5D3×Dic611yescharacteristic
219C2×C5D3×D1211yescharacteristic
220C2×C5C2×#62711yescharacteristic
221C32#409011yescharacteristic, complemented, Sylow
222Q8—182no—
223Q8—902no—
224C2×C4—31no—
225C2×C4—31no—
226C2×C4—51no—
227C2×C4—91no—
228C2×C4—151no—
229C2×C4—151no—
230C2×C4—451no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—21no—
234C2×C3D5×Dic611yescharacteristic
235C2×C3D5×Dic611yescharacteristic
236C2×C3D5×Dic611yescharacteristic
237C2×C3D15⋊Q811yescharacteristic
238C2×C3C2×#120311yescharacteristic
239C2×C3D3×C22⋊D511yescharacteristic
240C5#6391811yescharacteristic, complemented, Sylow
241C22D32×D511yescharacteristic
242C4—32no—
243C4—52no—
244C4—61no—
245C4—152no—
246C4—181no—
247C4—301no—
248C4—901no—
249C3—21no—
250C3#9132911yescharacteristic, complemented
251C3#9137611yescharacteristic, complemented
252C2D3×#120311yescharacteristic
253C2D5×#62711yescharacteristic
254C2#1062403811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×D5
commutator, Fitting, socle, prime radical, cosocleC22×C32×C5C23
hypocenterC32×C5C4⋊Q8
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
39. C3×Dic15Q8semidirect
42. C32×Dic5Q8semidirect
102. Dic15Dic6semidirect
106. C3×Dic5Dic6semidirect
107. C3×Dic5Dic6semidirect
128. C32×C5C4⋊Q8semidirect
184. Dic5#627semidirect
196. C3×C5#869semidirect
197. C3×C5#888semidirect
221. C32#4090semidirect
240. C5#63918semidirect
250. C3#91329semidirect
251. C3#91376semidirect

Maximal subgroups

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

subgroupindex
C2×#671532
C2×C5×#6272
#106238312
#106238142
#106237732
C3×#66682
C3×Dic3×Dic52
#913763
#913293
#639185