Finite Group Databasebeta

← #13086169

Subgroups — #13086169

Order 1440.

274 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 70 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
1#10623828C211yesmaximal, characteristic, complemented
2Dic3⋊Dic15C211yesmaximal, characteristic, complemented
3#10623815C211yesmaximal, characteristic, complemented
4C5×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
5#10623770C211yesmaximal, characteristic
6C3×#6665C211yesmaximal, characteristic
7C3×#6665C211yesmaximal, characteristic
8#91362—31nomaximal
9#91361—31nomaximal
10C22×C32×D5C2211yescharacteristic
11C2×C3⋊Dic15C2211yescharacteristic
12C2×C5×C3⋊Dic3C2211yescharacteristic
13C2×C3×Dic15C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17#63956—51nomaximal, Hall
18C4⋊Dic15—31no—
19#6712—31no—
20C5×C4⋊Dic3—31no—
21C5×Dic3⋊C4—31no—
22C3×#685—31no—
23C3×#685—31no—
24Dic3⋊Dic5—31no—
25#6669—31no—
26#6665—31no—
27#6665—31no—
28#6665—61no—
29#6665D311yescharacteristic
30#6665D311yescharacteristic
31Dic3×Dic5—31no—
32Dic3×Dic5—31no—
33C22×C32×C5C2311yescharacteristic
34C2×C32×D5—21no—
35C2×C32×D5—21no—
36C2×C32×D5—21no—
37C3⋊Dic15—21no—
38C5×C3⋊Dic3—21no—
39C3×Dic15—21no—
40C3×Dic15—21no—
41C3×C5×Dic3—21no—
42C3×C5×Dic3—21no—
43#4044—91noHall
44#3757—51no—
45C3×#217—51no—
46C3×#217—51no—
47#3724—51no—
48Dic3⋊Dic3—51no—
49Dic3⋊Dic3D511yescharacteristic, complemented
50Dic32—51no—
51C22×C3×D5—21no—
52C22×C3×D5C2×D311yescharacteristic
53C22×C3×D5C2×D311yescharacteristic
54C2×C3×C4×C5—31no—
55C2×C3×C4×C5—31no—
56C2×Dic15—31no—
57C2×Dic15—31no—
58C2×Dic15—61no—
59C2×Dic15C2×D311yescharacteristic
60C2×Dic15C2×D311yescharacteristic
61C2×C5×Dic3—31no—
62C2×C5×Dic3—31no—
63C2×C5×Dic3—61no—
64C2×C5×Dic3C2×D311yescharacteristic
65C2×C5×Dic3C2×D311yescharacteristic
66C2×C3×Dic5—31no—
67C2×C3×Dic5—31no—
68#879—151no—
69#879—151no—
70C2×C32×C5#4111yescharacteristic
71C2×C32×C5#4111yescharacteristic
72C2×C32×C5#4111yescharacteristic
73C32×D5—41no—
74C5×C4⋊C4—91no—
75#685—31no—
76#685—31no—
77#685—91no—
78C4⋊Dic5—91no—
79#683—91no—
80C4×Dic5—91no—
81C23×C32—51no—
82C2×C3⋊Dic3—51no—
83C2×C3⋊Dic3C2×D511yescharacteristic
84C2×C3×Dic3—51no—
85C2×C3×Dic3—51no—
86C2×C3×Dic3C2×D511yescharacteristic
87C2×C3×Dic3C2×D511yescharacteristic
88C22×C3×C5—21no—
89C22×C3×C5C22×D311yescharacteristic
90C22×C3×C5C22×D311yescharacteristic
91C2×C3×D5—22no—
92C2×C3×D5—21no—
93C2×C3×D5—21no—
94C2×C3×D5—21no—
95C2×C3×D5—21no—
96C2×C3×D5—21no—
97C2×C3×D5—21no—
98C2×C3×D5—41no—
99C2×C3×D5—41no—
100C3×C4×C5—61no—
101C3×C4×C5—61no—
102Dic15—21no—
103Dic15—21no—
104Dic15—61no—
105Dic15—61no—
106Dic15—121no—
107C3×Dic5—61no—
108C3×Dic5—61no—
109C5×Dic3—21no—
110C5×Dic3—21no—
111C5×Dic3—61no—
112C5×Dic3—61no—
113C5×Dic3—121no—
114C3×C22⋊C4—151no—
115C3×C22⋊C4—151no—
116#217—51no—
117#217—51no—
118#217—151no—
119#217—151no—
120#217—301no—
121C4⋊Dic3—31no—
122C4⋊Dic3—151no—
123Dic3⋊C4—31no—
124Dic3⋊C4—151no—
125Dic3⋊C4—151no—
126Dic3⋊C4—151no—
127C4×Dic3—151no—
128C4×Dic3—151no—
129C32×C5#12611yescharacteristic, complemented, Hall
130C22×D5D3211yescharacteristic
131C2×C4×C5—31no—
132C2×C4×C5—31no—
133C2×C4×C5—91no—
134C2×Dic5—31no—
135C2×Dic5—31no—
136C2×Dic5—91no—
137C22×C32—101no—
138C22×C32—101no—
139C22×C32—101no—
140C22×C32C22×D511yescharacteristic
141C3⋊Dic3—21no—
142C3⋊Dic3—101no—
143C3×Dic3—21no—
144C3×Dic3—21no—
145C3×Dic3—101no—
146C3×Dic3—101no—
147#126—451noSylow
148C2×C3×C5—21no—
149C2×C3×C5—21no—
150C2×C3×C5—21no—
151C2×C3×C5#23511yescharacteristic
152C2×C3×C5#23511yescharacteristic
153C2×C3×C5#23711yescharacteristic
154C2×C3×C5#23711yescharacteristic
155C2×C3×C5#23711yescharacteristic
156C2×C3×C5#23711yescharacteristic
157C3×D5—42no—
158C3×D5—41no—
159C3×D5—41no—
160C23×C3—51no—
161C23×C3—51no—
162C23×C3—101no—
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—151no—
166C2×C3×C4—151no—
167C2×Dic3—31no—
168C2×Dic3—31no—
169C2×Dic3—51no—
170C2×Dic3—51no—
171C2×Dic3—61no—
172C2×Dic3—151no—
173C2×Dic3—151no—
174C2×Dic3—301no—
175C2×Dic3D3×D511yescharacteristic
176C2×Dic3D3×D511yescharacteristic
177C22×C5C2×D3211yescharacteristic
178C2×D5—21no—
179C2×D5—21no—
180C2×D5—21no—
181C4×C5—61no—
182C4×C5—61no—
183C4×C5—181no—
184Dic5—61no—
185Dic5—61no—
186Dic5—181no—
187C2×C32—201no—
188C2×C32#70911yescharacteristic
189C2×C32#71111yescharacteristic
190C2×C32#71311yescharacteristic
191C4⋊C4—91no—
192C4⋊C4—451no—
193C4⋊C4—451no—
194C22⋊C4—151no—
195C22⋊C4—151no—
196C22⋊C4—451no—
197C42—451no—
198C3×C5—21no—
199C3×C5#87911yescharacteristic, complemented
200C3×C5#87911yescharacteristic, complemented
201C22×C3—21no—
202C22×C3—102no—
203C22×C3—101no—
204C22×C3—101no—
205C22×C3—101no—
206C22×C3—101no—
207C22×C3—101no—
208C22×C3—101no—
209C22×C3—201no—
210C22×C3—201no—
211C22×C3C2×D3×D511yescharacteristic
212C22×C3C2×D3×D511yescharacteristic
213C3×C4—61no—
214C3×C4—61no—
215C3×C4—301no—
216C3×C4—301no—
217Dic3—21no—
218Dic3—21no—
219Dic3—61no—
220Dic3—61no—
221Dic3—101no—
222Dic3—101no—
223Dic3—121no—
224Dic3—301no—
225Dic3—301no—
226Dic3—601no—
227C2×C5#380411yescharacteristic
228C2×C5#380411yescharacteristic
229C2×C5#380511yescharacteristic
230D5—41no—
231C32#404411yescharacteristic, complemented, Sylow
232C23—51no—
233C2×C4—31no—
234C2×C4—31no—
235C2×C4—91no—
236C2×C4—151no—
237C2×C4—151no—
238C2×C4—451no—
239C2×C3—21no—
240C2×C3—21no—
241C2×C3—21no—
242C2×C3—202no—
243C2×C3—201no—
244C2×C3—201no—
245C2×C3#676511yescharacteristic
246C2×C3#676611yescharacteristic
247C2×C3#677211yescharacteristic
248C2×C3#677311yescharacteristic
249C2×C3#677911yescharacteristic
250C2×C3#678011yescharacteristic
251C5#6395611yescharacteristic, complemented, Sylow
252C22—101no—
253C22—101no—
254C22—101no—
255C22D32×D511yescharacteristic
256C4—61no—
257C4—61no—
258C4—181no—
259C4—301no—
260C4—301no—
261C4—901no—
262C3—21no—
263C3#9136111yescharacteristic, complemented
264C3#9136211yescharacteristic, complemented
265C2—201no—
266C2#1062404311yescharacteristic
267C2#1062404411yescharacteristic
268C2#1062404511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×D5
commutator, Fitting, socle, cosocleC22×C32×C5C23
prime radicalC22×C32×D5C22
hypocenterC32×C5#126
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #10623828C2semidirect
2. Dic3⋊Dic15C2semidirect
3. #10623815C2semidirect
4. C5×Dic3⋊Dic3C2semidirect
49. Dic3⋊Dic3D5semidirect
129. C32×C5#126semidirect
199. C3×C5#879semidirect
200. C3×C5#879semidirect
231. C32#4044semidirect
251. C5#63956semidirect
263. C3#91361semidirect
264. C3#91362semidirect

Maximal subgroups

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

subgroupindex
#106238282
Dic3⋊Dic152
#106238152
C5×Dic3⋊Dic32
#106237702
C3×#66652
#913623
#913613
#639565