Finite Group Databasebeta

← #14315906

Subgroups — #14315906

Order 1920.

308 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C3×#65733C211yesmaximal, characteristic, complemented
2#11751662C211yesmaximal, characteristic, complemented
3#11748265C211yesmaximal, characteristic
4#11746774C211yesmaximal, characteristic, complemented
5#11746473C211yesmaximal, characteristic, complemented
6#11746468C211yesmaximal, characteristic, complemented
7#11746453C211yesmaximal, characteristic, complemented
8#10604125—31nomaximal, Hall
9C3×#4131—21no—
10AGL(1,𝔽5)×D12—21no—
11C3×#4058C2211yescharacteristic
12D5×D24C2211yescharacteristic
13#91222C2211yescharacteristic
14#91219C2211yescharacteristic
15C3×#4008—21no—
16C3×#3995C2211yescharacteristic
17C3×#3992C2211yescharacteristic
18#91152—21no—
19#91110C411yescharacteristic, complemented
20C5×#826C411yescharacteristic, complemented
21#90943C411yescharacteristic, complemented
22#90941C411yescharacteristic, complemented
23#90934C2211yescharacteristic
24#70929—51nomaximal, Hall
25#65733D311yescharacteristic, complemented
26#65731—31no—
27#65200—31no—
28#64906—31no—
29#64903—31no—
30#64847—31no—
31#64846—31no—
32C2×D3×AGL(1,𝔽5)—41no—
33C3×#711—21no—
34D5×D12—21no—
35C4⋊ΓL(1,𝔽16)—21no—
36C3×#704—41no—
37C2×C3×#170—41no—
38C3×C4×AGL(1,𝔽5)D411yescharacteristic
39C3×#700—21no—
40C3×D5⋊C8D411yescharacteristic
41(C2×D3)⋊AGL(1,𝔽5)—41no—
42C3×C5×D8C2×C411yescharacteristic, complemented
43D120C2×C411yescharacteristic
44C5×D24C2×C411yescharacteristic
45C3×#687—21no—
46C3×Dic20C2×C411yescharacteristic, complemented
47C3×C8×D5C2311yescharacteristic
48#6654—21no—
49C3⋊#674—21no—
50C3⋊#672C2×C411yescharacteristic, complemented
51C5×C3⋊C16C2×C411yescharacteristic, complemented
52C3×#444—51no—
53#5193—51no—
54#4711—51no—
55#4566—51no—
56#4495—51no—
57#4494—51no—
58#4466—51no—
59D4×AGL(1,𝔽5)—61no—
60#4131—21no—
61#4058C2×D311yescharacteristic
62D5×D8—31no—
63#4008—21no—
64#4007—61no—
65#3995C2×D311yescharacteristic
66#3993—31no—
67#3992C2×D311yescharacteristic
68#3990—31no—
69C5×D16—31no—
70#3959—31no—
71#3958—31no—
72#3932—31no—
73#3930—31no—
74AGL(1,ℤ/16)—151noSylow
75C2×D3×D5—41no—
76C2×ΓL(1,𝔽16)—41no—
77C2×C3×AGL(1,𝔽5)—21no—
78D3×AGL(1,𝔽5)—82no—
79C3×C5×D4—21no—
80D60—21no—
81C5×D12—21no—
82C3×C22⋊D5—41no—
83C2×C3×Dic5—41no—
84C3×C4×D5C2×D411yescharacteristic
85C3×Dic10—21no—
86#1203—41no—
87C3×#170—21no—
88C3×#170—41no—
89C3×C5×C8C22×C411yescharacteristic
90C3×#168#4111yescharacteristic
91C3×#135—51no—
92C3×#131—101no—
93C2×D24—51no—
94C4×D12—101no—
95C3×#108—51no—
96C3×#104—101no—
97C3×#97—51no—
98#828—51no—
99#828—51no—
100#826—51no—
101#826AGL(1,𝔽5)11yescharacteristic, complemented
102#821—101no—
103#817—51no—
104#812—51no—
105#812—51no—
106C22×AGL(1,𝔽5)—121no—
107#711—21no—
108D4×D5—61no—
109C22⋊AGL(1,𝔽5)—121no—
110#704—41no—
111C2×#170—41no—
112C4⋊AGL(1,𝔽5)—61no—
113C4×AGL(1,𝔽5)#6711yescharacteristic
114#700—21no—
115D5⋊C8#6711yescharacteristic
116C5×D8—31no—
117C5×D8C4×D311yescharacteristic, complemented
118#687—21no—
119#686—61no—
120Dic20C4×D311yescharacteristic, complemented
121D40—31no—
122C8×D5C22×D311yescharacteristic
123#674—61no—
124C5×C16—31no—
125#672—31no—
126#509—151no—
127#444—51no—
128#442—151no—
129#365—151no—
130#361—151no—
131#360—151no—
132#347—151no—
133C22×C3×C5—41no—
134C2×D15—41no—
135C2×C5×D3—41no—
136C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
137D3×D5—81no—
138D3×D5—81no—
139ΓL(1,𝔽16)—81no—
140C3×AGL(1,𝔽5)—41no—
141C3×C4×C5C4×D411yescharacteristic
142C3×Dic5—41no—
143C3×Dic5AGL(1,ℤ/8)11yescharacteristic
144C3×#41—101no—
145C2×D12—101no—
146C2×C4×D3—201no—
147C3×Q16—51no—
148C3×SD16—101no—
149C3×D8C2×AGL(1,𝔽5)11yescharacteristic, complemented
150C3×M16—101no—
151C3×M16—201no—
152C2×C3×C8—51no—
153C2×C3×C8—51no—
154C2×C3×C8—201no—
155C3×C42—51no—
156(C2×D3)⋊C4—201no—
157C4⋊Dic3—101no—
158D24—51no—
159D24—101no—
160D24C2×AGL(1,𝔽5)11yescharacteristic
161C3⋊C16—51no—
162C3⋊C16—101no—
163C3⋊C16C2×AGL(1,𝔽5)11yescharacteristic, complemented
164C22×D5—121no—
165C2×AGL(1,𝔽5)—21no—
166C2×AGL(1,𝔽5)—121no—
167C2×AGL(1,𝔽5)—242no—
168C5×D4—21no—
169C5×D4—61no—
170C22⋊D5—41no—
171C22⋊D5—121no—
172C2×Dic5—41no—
173D20—61no—
174C4×D5C2×#6711yescharacteristic
175Dic10—21no—
176#170—21no—
177#170—41no—
178C5×C8C2×C4×D311yescharacteristic
179#168#23511yescharacteristic
180#135—51no—
181C2×D8—151no—
182#131—101no—
183C4×D4—301no—
184SD32—151no—
185SD32—151no—
186D16—31no—
187D16—151no—
188M32—151no—
189M32—151no—
190#108—51no—
191#106—151no—
192#104—101no—
193#102—301no—
194#97—51no—
195C2×C3×C5—81no—
196C2×C3×C5#44211yescharacteristic
197D15—81no—
198C3×D5—21no—
199C5×D3—81no—
200C22×D3—201no—
201C3×Q8—101no—
202C3×D4—21no—
203C3×D4—201no—
204C2×C3×C4—51no—
205C2×C3×C4—101no—
206C2×C3×C4—201no—
207C2×Dic3—201no—
208D12—21no—
209D12—101no—
210D12—201no—
211C4×D3—402no—
212C3×C8—51no—
213C3×C8—101no—
214C3×C8—201no—
215C3×C8C22×AGL(1,𝔽5)11yescharacteristic
216C22×C5—41no—
217C22×C5—121no—
218C2×D5—121no—
219C2×D5—241no—
220C2×D5—241no—
221C2×D5#94911yescharacteristic
222AGL(1,𝔽5)—41no—
223AGL(1,𝔽5)—241no—
224C4×C5C4×#6711yescharacteristic
225Dic5—41no—
226Dic5#93211yescharacteristic
227#41—101no—
228C2×D4—301no—
229C22×C4—601no—
230Q16—51no—
231SD16—101no—
232D8—31no—
233D8—151no—
234D8—301no—
235D8D3×AGL(1,𝔽5)11yescharacteristic, complemented
236M16—101no—
237M16—201no—
238C2×C8—51no—
239C2×C8—51no—
240C2×C8—201no—
241C4⋊C4—301no—
242C22⋊C4—601no—
243C42—51no—
244C16—31no—
245C16—151no—
246C16—301no—
247C3×C5AGL(1,ℤ/16)11yescharacteristic, complemented, Hall
248C22×C3—41no—
249C22×C3—51no—
250C2×D3—41no—
251C2×D3—201no—
252C2×D3—401no—
253C2×D3—401no—
254C3×C4—51no—
255C3×C4—201no—
256C3×C4—201no—
257C3×C4D4×AGL(1,𝔽5)11yescharacteristic
258Dic3—401no—
259C2×C5—81no—
260C2×C5—241no—
261C2×C5#501411yescharacteristic
262D5—21no—
263D5—241no—
264C23—601no—
265Q8—101no—
266D4—21no—
267D4—61no—
268D4—201no—
269D4—301no—
270D4—601no—
271C2×C4—51no—
272C2×C4—101no—
273C2×C4—201no—
274C2×C4—601no—
275C2×C4—1202no—
276C8—51no—
277C8—101no—
278C8—201no—
279C8C2×D3×AGL(1,𝔽5)11yescharacteristic
280C2×C3—81no—
281C2×C3—101no—
282C2×C3#6573111yescharacteristic
283D3—81no—
284D3—401no—
285C5#7092911yescharacteristic, complemented, Sylow
286C22—41no—
287C22—51no—
288C22—121no—
289C22—601no—
290C22—1201no—
291C22—1201no—
292C4—51no—
293C4—201no—
294C4—201no—
295C4—1201no—
296C4AGL(1,𝔽5)×#6711yescharacteristic
297C3#1060412511yescharacteristic, complemented, Sylow
298C2—81no—
299C2—101no—
300C2—241no—
301C2—1201no—
302C2#1175194011yescharacteristic
subgroupG/H
centerC2#11751940
commutatorC3×C5×C8C22×C4
FrattiniC8C2×D3×AGL(1,𝔽5)
FittingC3×C5×D8C2×C4
socleC2×C3×C5#442
prime radical#11748265C2
cosocleC3×C8×D5C23
hypercenterD8D3×AGL(1,𝔽5)
hypocenterC3×C5AGL(1,ℤ/16)
LCSG ▷ C3×C5×C8 ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ D8

Complements

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

NH ≅ G/Ntype
1. C3×#65733C2semidirect
2. #11751662C2semidirect
4. #11746774C2semidirect
5. #11746473C2semidirect
6. #11746468C2semidirect
7. #11746453C2semidirect
19. #91110C4semidirect
20. C5×#826C4semidirect
21. #90943C4semidirect
22. #90941C4semidirect
25. #65733D3semidirect
42. C3×C5×D8C2×C4semidirect
46. C3×Dic20C2×C4semidirect
50. C3⋊#672C2×C4semidirect
51. C5×C3⋊C16C2×C4semidirect
101. #826AGL(1,𝔽5)semidirect
117. C5×D8C4×D3semidirect
120. Dic20C4×D3semidirect
149. C3×D8C2×AGL(1,𝔽5)semidirect
163. C3⋊C16C2×AGL(1,𝔽5)semidirect
235. D8D3×AGL(1,𝔽5)semidirect
247. C3×C5AGL(1,ℤ/16)semidirect
285. C5#70929semidirect
297. C3#10604125semidirect

Maximal subgroups

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

subgroupindex
C3×#657332
#117516622
#117482652
#117467742
#117464732
#117464682
#117464532
#106041253
#709295