Finite Group Databasebeta

← #13085656

Subgroups — #13085656

Order 1440.

288 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 72 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#10624154C211yesmaximal, characteristic
2#10623818C211yesmaximal, characteristic, complemented
3#10623797C211yesmaximal, characteristic, complemented
4#10623794C211yesmaximal, characteristic, complemented
5C5×#3712C211yesmaximal, characteristic, complemented
6C3×#6649C211yesmaximal, characteristic, complemented
7C3×D5×C3⋊C8C211yesmaximal, characteristic, complemented
8#91268—31nomaximal
9#91266—31nomaximal
10C4×C3⋊D15C2211yescharacteristic
11C4×C5×C3⋊D3C2211yescharacteristic
12C32×C4×D5C2211yescharacteristic
13#67141C411yescharacteristic
14#67140C411yescharacteristic
15#67139C411yescharacteristic
16#67138C411yescharacteristic
17C3×C3⋊#168C2211yescharacteristic
18C3×C3⋊#168C2211yescharacteristic
19C3×C5×C3⋊C8C2211yescharacteristic
20C3×C5×C3⋊C8C2211yescharacteristic
21#63817—51nomaximal, Hall
22#6771—31no—
23#6771—31no—
24#6771—61no—
25#6705—31no—
26C8×D15—31no—
27C5×ΓL(1,𝔽25)—31no—
28C5×ΓL(1,𝔽25)—31no—
29C3×#676—31no—
30C3×C8×D5—31no—
31#6651—31no—
32#6649D311yescharacteristic, complemented
33#6648—31no—
34D3×#168—31no—
35D3×#168—31no—
36D5×C3⋊C8D311yescharacteristic, complemented
37C2×C3⋊D15C2×C411yescharacteristic
38C2×C5×C3⋊D3C2×C411yescharacteristic
39C2×C32×D5C2×C411yescharacteristic
40C32×C4×C5C2311yescharacteristic
41C3⋊Dic15C2×C411yescharacteristic
42C5×C3⋊Dic3C2×C411yescharacteristic
43C32×Dic5C2×C411yescharacteristic
44#4053—91noHall
45#3828—51no—
46C3×#208—51no—
47C2×C3×C3⋊C8—51no—
48#3712—51no—
49#3712D511yescharacteristic, complemented
50#3710—51no—
51#3710—51no—
52C4×D15—31no—
53C4×D15—31no—
54C4×D15—61no—
55C4×C5×D3—31no—
56C4×C5×D3—31no—
57C4×C5×D3—61no—
58C3×C4×D5—21no—
59C3×C4×D5C2×D311yescharacteristic
60C3×C4×D5C2×D311yescharacteristic
61#1204—31no—
62#1204—31no—
63#1204—61no—
64#1203—31no—
65#1203—31no—
66#1203—61no—
67#1202—31no—
68#1202—31no—
69#1202—61no—
70#1201—31no—
71#1201—31no—
72#1201—61no—
73C3×C5×C8—31no—
74C3×C5×C8—31no—
75C3⋊#168C2×D311yescharacteristic
76C3⋊#168C2×D311yescharacteristic
77C3×#168—31no—
78C3×#168—31no—
79C5×C3⋊C8C2×D311yescharacteristic
80C5×C3⋊C8C2×D311yescharacteristic
81#907—151no—
82#901—151no—
83C2×C32×C5C22×C411yescharacteristic
84C3⋊D15—21no—
85C5×C3⋊D3—21no—
86C32×D5—21no—
87#709—91no—
88C5×M16—91no—
89C2×#168—91no—
90#676—31no—
91#676—91no—
92C8×D5—31no—
93C8×D5—91no—
94C2×C32×C4—51no—
95#638—51no—
96#638—51no—
97C4⋊(C3⋊D3)—51no—
98C4×C3⋊D3—51no—
99C4×C3⋊D3C2×D511yescharacteristic
100#634—51no—
101C3×C3⋊C8—51no—
102C3×C3⋊C8—51no—
103C3×C3⋊C8C2×D511yescharacteristic
104C3×C3⋊C8C2×D511yescharacteristic
105C2×D15—31no—
106C2×D15—31no—
107C2×D15—61no—
108C2×C5×D3—31no—
109C2×C5×D3—31no—
110C2×C5×D3—61no—
111C2×C3×D5—21no—
112C2×C3×D5C4×D311yescharacteristic
113C2×C3×D5C4×D311yescharacteristic
114C3×C4×C5—21no—
115C3×C4×C5C22×D311yescharacteristic
116C3×C4×C5C22×D311yescharacteristic
117Dic15—31no—
118Dic15—31no—
119Dic15—61no—
120C3×Dic5—21no—
121C3×Dic5C4×D311yescharacteristic
122C3×Dic5C4×D311yescharacteristic
123C5×Dic3—31no—
124C5×Dic3—31no—
125C5×Dic3—61no—
126#235—151no—
127#235—151no—
128#235—301no—
129C3×M16—151no—
130C2×C3×C8—151no—
131#208—51no—
132C2×C3⋊C8—51no—
133ΓL(1,𝔽25)—31no—
134ΓL(1,𝔽25)—31no—
135ΓL(1,𝔽25)—151no—
136ΓL(1,𝔽25)—151no—
137D3×C8—151no—
138D3×C8—151no—
139D3×C8—151no—
140D3×C8—151no—
141C32×C5#13111yescharacteristic, complemented, Hall
142C2×C4×C5—91no—
143C22⋊D5—91no—
144C22⋊D5—91no—
145D20—91no—
146C4×D5—91no—
147C4×D5D3211yescharacteristic
148Dic10—91no—
149C5×C8—31no—
150C5×C8—31no—
151#168—31no—
152#168—31no—
153C22×C32—51no—
154C2×C3⋊D3—51no—
155C2×C3⋊D3C4×D511yescharacteristic
156C32×C4—51no—
157C32×C4C22×D511yescharacteristic
158C3⋊Dic3—51no—
159C3⋊Dic3C4×D511yescharacteristic
160#131—451noSylow
161C2×C3×C5—21no—
162C2×C3×C5C2×C4×D311yescharacteristic
163C2×C3×C5C2×C4×D311yescharacteristic
164D15—61no—
165D15—61no—
166D15—121no—
167C3×D5—21no—
168C3×D5—21no—
169C3×D5—41no—
170C5×D3—61no—
171C5×D3—61no—
172C5×D3—121no—
173C2×C3×C4—51no—
174C2×C3×C4—51no—
175C2×C3×C4—101no—
176#67—151no—
177#67—151no—
178#67—151no—
179#67—151no—
180#67—301no—
181#67—301no—
182D12—151no—
183D12—151no—
184D12—301no—
185C4×D3—31no—
186C4×D3—31no—
187C4×D3—61no—
188C4×D3—151no—
189C4×D3—151no—
190C4×D3—301no—
191Dic6—151no—
192Dic6—151no—
193Dic6—301no—
194C3×C8—31no—
195C3×C8—31no—
196C3×C8—151no—
197C3×C8—151no—
198C3⋊C8—51no—
199C3⋊C8—51no—
200C3⋊C8D3×D511yescharacteristic
201C3⋊C8D3×D511yescharacteristic
202C22×C5—91no—
203C2×D5—91no—
204C2×D5#62411yescharacteristic
205C4×C5—91no—
206C4×C5C2×D3211yescharacteristic
207Dic5—91no—
208Dic5#62411yescharacteristic
209C2×C32—101no—
210C2×C32C2×C4×D511yescharacteristic
211C3⋊D3—21no—
212C3⋊D3—101no—
213#41—451no—
214M16—91no—
215M16—151no—
216M16—451no—
217C2×C8—151no—
218C2×C8—451no—
219C2×C8—451no—
220C3×C5—21no—
221C3×C5#90111yescharacteristic, complemented
222C3×C5#90711yescharacteristic, complemented
223C22×C3—51no—
224C22×C3—51no—
225C22×C3—101no—
226C2×D3—31no—
227C2×D3—31no—
228C2×D3—61no—
229C2×D3—151no—
230C2×D3—151no—
231C2×D3—301no—
232C3×C4—21no—
233C3×C4—51no—
234C3×C4—51no—
235C3×C4—101no—
236C3×C4C2×D3×D511yescharacteristic
237C3×C4C2×D3×D511yescharacteristic
238Dic3—31no—
239Dic3—31no—
240Dic3—61no—
241Dic3—151no—
242Dic3—151no—
243Dic3—301no—
244C2×C5—181no—
245C2×C5C2×#62411yescharacteristic
246D5—21no—
247D5—181no—
248C32#405311yescharacteristic, complemented, Sylow
249Q8—451no—
250D4—451no—
251D4—451no—
252D4—451no—
253C2×C4—51no—
254C2×C4—91no—
255C2×C4—451no—
256C8—31no—
257C8—31no—
258C8—151no—
259C8—151no—
260C2×C3—21no—
261C2×C3—101no—
262C2×C3—101no—
263C2×C3—201no—
264C2×C3C4×D3×D511yescharacteristic
265C2×C3C4×D3×D511yescharacteristic
266D3—61no—
267D3—61no—
268D3—121no—
269D3—301no—
270D3—301no—
271D3—601no—
272C5#6381711yescharacteristic, complemented, Sylow
273C22—51no—
274C22—91no—
275C22—451no—
276C4—51no—
277C4—91no—
278C4—451no—
279C4D32×D511yescharacteristic
280C3—21no—
281C3#9126611yescharacteristic, complemented
282C3#9126811yescharacteristic, complemented
283C2—101no—
284C2—181no—
285C2—901no—
286C2D5×#62411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D32×D5
commutator, socleC2×C32×C5C22×C4
Fitting, cosocleC32×C4×C5C23
prime radical#10624154C2
hypocenterC32×C5#131
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #10623818C2semidirect
3. #10623797C2semidirect
4. #10623794C2semidirect
5. C5×#3712C2semidirect
6. C3×#6649C2semidirect
7. C3×D5×C3⋊C8C2semidirect
32. #6649D3semidirect
36. D5×C3⋊C8D3semidirect
49. #3712D5semidirect
141. C32×C5#131semidirect
221. C3×C5#901semidirect
222. C3×C5#907semidirect
248. C32#4053semidirect
272. C5#63817semidirect
281. C3#91266semidirect
282. C3#91268semidirect

Maximal subgroups

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

subgroupindex
#106241542
#106238182
#106237972
#106237942
C5×#37122
C3×#66492
C3×D5×C3⋊C82
#912683
#912663
#638175