Finite Group Databasebeta

← #14315828

Subgroups — #14315828

Order 1920.

284 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 64 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#11751690C211yesmaximal, characteristic
2#11751670C211yesmaximal, characteristic, complemented
3#11746827C211yesmaximal, characteristic
4#11746717C211yesmaximal, characteristic, complemented
5C3×ΓL(1,𝔽81)C211yesmaximal, characteristic, complemented
6#11746470C211yesmaximal, characteristic, complemented
7#11746465C211yesmaximal, characteristic
8#10604138—31nomaximal, Hall
9#91916—21no—
10ΓL(1,𝔽16)⋊Q8—21no—
11#91271C2211yescharacteristic
12D5×Dic12C2211yescharacteristic
13#91221C2211yescharacteristic
14#91219C2211yescharacteristic
15C3×#3995C2211yescharacteristic
16C3×#3993C2211yescharacteristic
17#91157—21no—
18#91153—21no—
19#91084C411yescharacteristic, complemented
20C5×Dic24C411yescharacteristic, complemented
21C3×#3930C2211yescharacteristic
22#90948C411yescharacteristic, complemented
23#90945C411yescharacteristic, complemented
24#69309—51nomaximal, Hall
25#65741—31no—
26#65739—31no—
27#65207—31no—
28#64910—31no—
29#64907—31no—
30ΓL(1,𝔽81)D311yescharacteristic, complemented
31#64845—31no—
32#6773—21no—
33D5×Dic6—21no—
34C4×ΓL(1,𝔽16)D411yescharacteristic
35D5⋊(C3⋊C8)D411yescharacteristic
36C3×C4⋊AGL(1,𝔽5)—21no—
37C3×#700—21no—
38#6740—41no—
39#6738—41no—
40Dic3⋊AGL(1,𝔽5)—41no—
41Dic3×AGL(1,𝔽5)—41no—
42Dic60C2×C411yescharacteristic
43D120C2×C411yescharacteristic
44C5×Dic12C2×C411yescharacteristic
45C5×Dic12C2×C411yescharacteristic
46C3×C8×D5C2311yescharacteristic
47#6663—21no—
48#6660—21no—
49C3⋊#674—21no—
50C3×C5×C16C2×C411yescharacteristic
51C3×#672C2×C411yescharacteristic
52#4909—51no—
53#4894—51no—
54#4853—51no—
55C3×#365—51no—
56#4512—51no—
57#4496—51no—
58#4488—51no—
59Q8×AGL(1,𝔽5)—61no—
60#4133—61no—
61#4065—31no—
62D5×Q16—31no—
63#4010—61no—
64#4009—61no—
65#3995C2×D311yescharacteristic
66#3993C2×D311yescharacteristic
67#3992—31no—
68#3990—31no—
69C5×Q32—31no—
70#3961—31no—
71#3960—31no—
72#3932—31no—
73#3930C2×D311yescharacteristic
74#2180—151noSylow
75C2×ΓL(1,𝔽16)—21no—
76C2×C3×AGL(1,𝔽5)—41no—
77D60—21no—
78Dic30—21no—
79C5×Dic6—21no—
80C5×Dic6—21no—
81C3×C4×D5C2×D411yescharacteristic
82#1204—41no—
83#1202—41no—
84#1200—41no—
85D5×Dic3—41no—
86C3⋊#170—21no—
87C3×#170—41no—
88C3×C5×C8C22×C411yescharacteristic
89C3×#168#4111yescharacteristic
90#907—101no—
91C2×Dic12—51no—
92#904—51no—
93#887—101no—
94C3×M32—51no—
95C3×#108—51no—
96C3×#106—51no—
97#825—101no—
98#815—51no—
99#812—51no—
100#810—101no—
101Dic24—51no—
102Dic24AGL(1,𝔽5)11yescharacteristic, complemented
103#800—51no—
104#800—51no—
105#713—61no—
106Q8×D5—61no—
107C4⋊AGL(1,𝔽5)—21no—
108C4⋊AGL(1,𝔽5)—121no—
109C4×AGL(1,𝔽5)—31no—
110C4×AGL(1,𝔽5)—121no—
111#700—21no—
112#700—121no—
113D5⋊C8—31no—
114D5⋊C8—121no—
115C5×Q16—31no—
116C5×Q16—31no—
117#689—61no—
118#688—61no—
119Dic20—31no—
120D40—31no—
121C8×D5C22×D311yescharacteristic
122#674—61no—
123C5×C16C4×D311yescharacteristic
124#672C4×D311yescharacteristic
125#510—151no—
126#444—151no—
127#441—151no—
128#365—51no—
129#362—151no—
130#360—151no—
131#347—151no—
132C2×D15—41no—
133C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
134ΓL(1,𝔽16)—41no—
135C3×AGL(1,𝔽5)—81no—
136C3×C4×C5C4×D411yescharacteristic
137Dic15—41no—
138C3×Dic5AGL(1,ℤ/8)11yescharacteristic
139C5×Dic3—41no—
140C5×Dic3—41no—
141#235—101no—
142C2×Dic6—101no—
143C3×M16—101no—
144C2×C3×C8—51no—
145C3×C4⋊C4—101no—
146Dic3⋊C4—201no—
147C4×Dic3—51no—
148C4×Dic3—201no—
149C2×C3⋊C8—51no—
150Dic12—51no—
151Dic12—101no—
152Dic12C2×AGL(1,𝔽5)11yescharacteristic
153Dic12C2×AGL(1,𝔽5)11yescharacteristic
154D24—51no—
155#204—101no—
156ΓL(1,𝔽25)—201no—
157D3×C8—201no—
158C3×C16—51no—
159C3×C16C2×AGL(1,𝔽5)11yescharacteristic
160C3⋊C16—101no—
161C2×AGL(1,𝔽5)—41no—
162C2×AGL(1,𝔽5)—61no—
163C5×Q8—61no—
164C5×Q8—61no—
165D20—61no—
166D20—121no—
167C4×D5—121no—
168C4×D5—121no—
169C4×D5D3×D411yescharacteristic
170Dic10—61no—
171Dic10—121no—
172#170—41no—
173#170—61no—
174C5×C8C2×C4×D311yescharacteristic
175#168#23911yescharacteristic
176#135—151no—
177C2×Q16—151no—
178#131—301no—
179C4×Q8—301no—
180Q32—31no—
181Q32—151no—
182SD32—151no—
183SD32—151no—
184M32—51no—
185M32—151no—
186#108—51no—
187#106—51no—
188#104—301no—
189#103—301no—
190#97—151no—
191C2×C3×C5#44211yescharacteristic
192D15—81no—
193C3×D5—21no—
194C2×C3×C4—51no—
195C2×C3×C4—201no—
196#67—201no—
197C2×Dic3—101no—
198C2×Dic3—201no—
199D12—101no—
200C4×D3—201no—
201Dic6—21no—
202Dic6—21no—
203Dic6—101no—
204Dic6—201no—
205C3×C8—51no—
206C3×C8—201no—
207C3×C8C22×AGL(1,𝔽5)11yescharacteristic
208C3⋊C8—101no—
209C2×D5—121no—
210C2×D5#91411yescharacteristic
211AGL(1,𝔽5)—81no—
212AGL(1,𝔽5)—121no—
213C4×C5—121no—
214C4×C5—121no—
215C4×C5#89311yescharacteristic
216Dic5—121no—
217Dic5#91411yescharacteristic
218#41—301no—
219C2×Q8—301no—
220Q16—31no—
221Q16—31no—
222Q16—151no—
223Q16—301no—
224SD16—301no—
225D8—151no—
226M16—101no—
227M16—601no—
228C2×C8—51no—
229C2×C8—151no—
230C2×C8—601no—
231C4⋊C4—101no—
232C4⋊C4—601no—
233C42—151no—
234C42—601no—
235C16—51no—
236C16—301no—
237C16D3×AGL(1,𝔽5)11yescharacteristic
238C3×C5#218011yescharacteristic, complemented, Hall
239C22×C3—51no—
240C2×D3—201no—
241C3×C4—51no—
242C3×C4—401no—
243C3×C4D4×AGL(1,𝔽5)11yescharacteristic
244Dic3—41no—
245Dic3—41no—
246Dic3—201no—
247Dic3—201no—
248C2×C5#486911yescharacteristic
249D5—21no—
250D5—241no—
251Q8—61no—
252Q8—61no—
253Q8—301no—
254Q8—601no—
255D4—301no—
256D4—601no—
257C2×C4—51no—
258C2×C4—201no—
259C2×C4—301no—
260C2×C4—601no—
261C2×C4—601no—
262C8—51no—
263C8—201no—
264C8—301no—
265C8C2×D3×AGL(1,𝔽5)11yescharacteristic
266C2×C3—101no—
267C2×C3#6573111yescharacteristic
268D3—401no—
269C5#6930911yescharacteristic, complemented, Sylow
270C22—51no—
271C22—601no—
272C4—51no—
273C4—121no—
274C4—121no—
275C4—401no—
276C4—601no—
277C4—601no—
278C4#9192111yescharacteristic
279C3#1060413811yescharacteristic, complemented, Sylow
280C2—101no—
281C2—1201no—
282C2#1175166511yescharacteristic
subgroupG/H
centerC2#11751665
commutatorC3×C5×C8C22×C4
FrattiniC8C2×D3×AGL(1,𝔽5)
FittingC3×C5×C16C2×C4
socleC2×C3×C5#442
prime radical#91271C22
cosocleC3×C8×D5C23
hypercenterC16D3×AGL(1,𝔽5)
hypocenterC3×C5#2180
LCSG ▷ C3×C5×C8 ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16

Complements

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

NH ≅ G/Ntype
2. #11751670C2semidirect
4. #11746717C2semidirect
5. C3×ΓL(1,𝔽81)C2semidirect
6. #11746470C2semidirect
19. #91084C4semidirect
20. C5×Dic24C4semidirect
22. #90948C4semidirect
23. #90945C4semidirect
30. ΓL(1,𝔽81)D3semidirect
102. Dic24AGL(1,𝔽5)semidirect
238. C3×C5#2180semidirect
269. C5#69309semidirect
279. C3#10604138semidirect

Maximal subgroups

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

subgroupindex
#117516902
#117516702
#117468272
#117467172
C3×ΓL(1,𝔽81)2
#117464702
#117464652
#106041383
#693095