Finite Group Databasebeta

← #14302872

Subgroups — #14302872

Order 1920.

688 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 110 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×#66260C211yesmaximal, characteristic, complemented
2#11755272C211yesmaximal, characteristic
3#11752444C211yesmaximal, characteristic, complemented
4#11751707C212yesmaximal, complemented
5#11746520C212yesmaximal, complemented
6#10604571—31nomaximal, Hall
7C2×C3×#713C2211yescharacteristic
8C2×D5×D12C2211yescharacteristic
9#91989C2212yes—
10C3×#4133—24no—
11C3×D5⋊M16—22no—
12C3×D5⋊M16C2212yes—
13#91836C411yescharacteristic, complemented
14C5×#947C411yescharacteristic, complemented
15#91474—22no—
16#91463C411yescharacteristic, complemented
17#91458C411yescharacteristic, complemented
18#91448—22no—
19#91357C2211yescharacteristic
20#91228—22no—
21#91152—24no—
22#72025—51nomaximal, Hall
23#66260D311yescharacteristic, complemented
24#66008—31no—
25#65785—31no—
26#65767—32no—
27#64863—32no—
28C22×D3×D5—21no—
29C2×C3×C5×Q8C2×C411yescharacteristic
30C2×D60C2×C411yescharacteristic
31C2×C5×D12C2×C411yescharacteristic
32C3×#713—22no—
33C3×#713—22no—
34C2×C3×D20C2×C411yescharacteristic
35C2×C3×D20D412yes—
36C2×C3×C4×D5D412yes—
37C2×C3×C4×D5C2311yescharacteristic
38C2×#1203—21no—
39C2×#1202—22no—
40C2×#1201—22no—
41D5×D12—22no—
42D5×D12—22no—
43#6763—22no—
44C4⋊ΓL(1,𝔽16)—22no—
45C4×ΓL(1,𝔽16)—22no—
46C3×#704—22no—
47C3×#700—24no—
48C3×#700—22no—
49C3×D5⋊C8—24no—
50C3×D5⋊C8D414yescomplemented
51#6714—22no—
52C4×Dic15C2×C411yescharacteristic
53C5×(C2×D3)⋊C4—22no—
54C4×C5×Dic3C2×C411yescharacteristic
55#6668—21no—
56#6665—21no—
57C3×#568—51no—
58#5830—51no—
59#5240—51no—
60#5131—52no—
61#4557—52no—
62C2×#713C2×D311yescharacteristic
63C2×D4×D5—31no—
64#4133—24no—
65#4130—32no—
66D5⋊M16—22no—
67D5⋊M16C2×D312yes—
68C5×#124—31no—
69#4093—31no—
70#4086—31no—
71#4030—62no—
72#4028—62no—
73#4020—31no—
74#4018—31no—
75#4007—64no—
76#3999—62no—
77#1910—151noSylow
78C22×D15—21no—
79C22×C5×D3—21no—
80C22×C3×D5C22⋊C412yes—
81C22×C3×D5C2×D411yescharacteristic
82C2×D3×D5—42no—
83C2×D3×D5—82no—
84C2×ΓL(1,𝔽16)—42no—
85C3×C5×Q8—22no—
86C2×C3×C4×C5C22⋊C412yes—
87C2×C3×C4×C5C22×C411yescharacteristic
88C2×Dic15—21no—
89D60—22no—
90C2×C5×Dic3—21no—
91C5×D12—22no—
92C2×C3×Dic5C2×D411yescharacteristic
93C3×D20—24no—
94C3×D20—22no—
95C3×C4×D5—22no—
96C3×C4×D5—22no—
97C3×C4×D5C2×D412yes—
98C3×C4×D5#4112yes—
99#1203—41no—
100#1203—41no—
101#1202—82no—
102#1201—82no—
103C3×#170—24no—
104C2×C3×#41—51no—
105C22×D12—51no—
106C3×#131—104no—
107C2×C3×M16—52no—
108C2×C3×M16—102no—
109#947—51no—
110#947AGL(1,𝔽5)11yescharacteristic, complemented
111#940—51no—
112#940—51no—
113#937—102no—
114#930—102no—
115#926—51no—
116#926—52no—
117#834—102no—
118#821—104no—
119C23×D5—61no—
120C2×C5×Q8C4×D311yescharacteristic
121C2×C5×D4—31no—
122C2×C22⋊D5—62no—
123C2×C22⋊D5—61no—
124#713—22no—
125#713—22no—
126D4×D5—62no—
127D4×D5—62no—
128C2×D20—31no—
129C2×D20—62no—
130C2×D20C4×D311yescharacteristic
131C2×D20D1212yes—
132C2×C4×D5#6712yes—
133C2×C4×D5C22×D311yescharacteristic
134C22⋊AGL(1,𝔽5)—62no—
135#704—22no—
136C4⋊AGL(1,𝔽5)—62no—
137C4×AGL(1,𝔽5)—62no—
138#700—24no—
139#700—22no—
140D5⋊C8—24no—
141D5⋊C8#6714yescomplemented
142C5×C22⋊C4—62no—
143C42×C5—31no—
144#685—61no—
145#685—62no—
146#683—61no—
147C4×Dic5—31no—
148#568—51no—
149#535—151no—
150#418—152no—
151#410—151no—
152H(3,ℤ/4)—152no—
153C22×C3×C5C2×C22⋊C411yescharacteristic
154C2×D15—42no—
155C2×D15—41no—
156C2×C5×D3—42no—
157C2×C5×D3—41no—
158C2×C3×D5—22no—
159C2×C3×D5—41no—
160C2×C3×D5—42no—
161C2×C3×D5#12112yes—
162D3×D5—84no—
163ΓL(1,𝔽16)—82no—
164C3×C4×C5—22no—
165C3×C4×C5C4×D412yes—
166Dic15—41no—
167C3×Dic5#12011yescharacteristic
168C3×Dic5#12311yescharacteristic
169C5×Dic3—41no—
170C23×D3—101no—
171C3×#41—102no—
172C3×#41—102no—
173C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
174C2×C3×D4—52no—
175C2×C3×D4—51no—
176C22×C3×C4—51no—
177C22×C3×C4—52no—
178C2×#67—102no—
179C2×#67—102no—
180C2×D12—51no—
181C2×D12—102no—
182C2×D12—102no—
183C2×D12—101no—
184C2×D12C2×AGL(1,𝔽5)11yescharacteristic
185C3×M16—102no—
186C3×M16—102no—
187C3×M16—104no—
188C2×C3×C8—54no—
189C2×C3×C8—104no—
190#217—101no—
191#217—102no—
192(C2×D3)⋊C4—22no—
193(C2×D3)⋊C4—102no—
194C4⋊Dic3—101no—
195C4⋊Dic3—102no—
196C4×Dic3—51no—
197C4×Dic3—102no—
198C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
199C23×C5—61no—
200C22×D5—61no—
201C22×D5—122no—
202C22×D5—242no—
203C22×D5(C2×D3)⋊C412yes—
204C22×D5C2×D1211yescharacteristic
205C2×AGL(1,𝔽5)—122no—
206C5×Q8—22no—
207C5×D4—62no—
208C2×C4×C5—61no—
209C2×C4×C5(C2×D3)⋊C412yes—
210C2×C4×C5C2×C4×D311yescharacteristic
211C22⋊D5—121no—
212C22⋊D5—121no—
213C22⋊D5—242no—
214C2×Dic5—61no—
215C2×Dic5C2×#6711yescharacteristic
216D20—24no—
217D20—22no—
218D20—62no—
219D20—242no—
220C4×D5—22no—
221C4×D5—22no—
222C4×D5#23512yes—
223C4×D5C2×#6712yes—
224#170—24no—
225C2×#41—51no—
226C22×D4—151no—
227#131—104no—
228C2×M16—52no—
229C2×M16—102no—
230#127—151no—
231#127—151no—
232#124—31no—
233#124—151no—
234#121—302no—
235#120—302no—
236#117—151no—
237#117—152no—
238#102—304no—
239#99—302no—
240C2×C3×C5—21no—
241C2×C3×C5#38611yescharacteristic
242D15—81no—
243C3×D5—22no—
244C3×D5—42no—
245C5×D3—81no—
246C23×C3—51no—
247C23×C3—52no—
248C22×D3—21no—
249C22×D3—101no—
250C22×D3—202no—
251C22×D3—402no—
252C3×Q8—22no—
253C3×D4—104no—
254C3×D4—102no—
255C2×C3×C4—51no—
256C2×C3×C4—52no—
257C2×C3×C4—52no—
258C2×C3×C4—102no—
259C2×C3×C4—102no—
260C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
261C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
262#67—402no—
263#67—402no—
264C2×Dic3—21no—
265C2×Dic3—101no—
266C2×Dic3—202no—
267D12—22no—
268D12—102no—
269D12—201no—
270D12—201no—
271C3×C8—104no—
272C22×C5—122no—
273C22×C5—121no—
274C22×C5C2×(C2×D3)⋊C411yescharacteristic
275C2×D5—22no—
276C2×D5—41no—
277C2×D5—42no—
278C2×D5—122no—
279C2×D5—121no—
280C2×D5—244no—
281C2×D5#93012yes—
282AGL(1,𝔽5)—242no—
283C4×C5—22no—
284C4×C5—121no—
285C4×C5C4×#6712yes—
286Dic5—121no—
287Dic5#92911yescharacteristic
288Dic5#95311yescharacteristic
289C24—301no—
290#41—102no—
291#41—102no—
292C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
293C2×D4—31no—
294C2×D4—52no—
295C2×D4—51no—
296C2×D4—151no—
297C2×D4—302no—
298C2×D4—302no—
299C2×D4—301no—
300C2×D4—302no—
301C2×D4—302no—
302C22×C4—51no—
303C22×C4—52no—
304M16—102no—
305M16—102no—
306M16—104no—
307C2×C8—54no—
308C2×C8—104no—
309C4⋊C4—301no—
310C4⋊C4—302no—
311C22⋊C4—62no—
312C22⋊C4—301no—
313C22⋊C4—302no—
314C22⋊C4—302no—
315C42—31no—
316C42—151no—
317C42—302no—
318C3×C5#191011yescharacteristic, complemented, Hall
319C22×C3—52no—
320C22×C3—102no—
321C22×C3—201no—
322C22×C3—202no—
323C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
324C2×D3—42no—
325C2×D3—41no—
326C2×D3—202no—
327C2×D3—201no—
328C2×D3—404no—
329C3×C4—22no—
330C3×C4—51no—
331C3×C4—51no—
332C3×C4D4×AGL(1,𝔽5)12yes—
333Dic3—41no—
334Dic3—201no—
335Dic3—402no—
336C2×C5—21no—
337C2×C5—241no—
338C2×C5#521311yescharacteristic
339D5—22no—
340D5—42no—
341D5—241no—
342C23—51no—
343C23—52no—
344C23—61no—
345C23—301no—
346C23—602no—
347C23—1202no—
348Q8—22no—
349D4—62no—
350D4—104no—
351D4—102no—
352D4—302no—
353D4—601no—
354D4—601no—
355D4—1202no—
356D4—1202no—
357C2×C4—51no—
358C2×C4—52no—
359C2×C4—52no—
360C2×C4—61no—
361C2×C4—102no—
362C2×C4—102no—
363C2×C4—301no—
364C2×C4—602no—
365C2×C4(C2×D3)⋊AGL(1,𝔽5)12yes—
366C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
367C8—104no—
368C2×C3—21no—
369C2×C3—102no—
370C2×C3—202no—
371C2×C3#6577911yescharacteristic
372D3—81no—
373D3—401no—
374C5#7202511yescharacteristic, complemented, Sylow
375C22—52no—
376C22—102no—
377C22—122no—
378C22—121no—
379C22—201no—
380C22—202no—
381C22—602no—
382C22—601no—
383C22—1204no—
384C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
385C4—22no—
386C4—51no—
387C4—51no—
388C4—121no—
389C4—601no—
390C4—1202no—
391C4AGL(1,𝔽5)×#6712yes—
392C3#1060457111yescharacteristic, complemented, Sylow
393C2—21no—
394C2—102no—
395C2—202no—
396C2—241no—
397C2—1201no—
398C2#1175202211yescharacteristic
subgroupG/H
centerC2#11752022
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC2×C3×C5#386
prime radical#11755272C2
cosocleC2×C3×C4×D5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1910
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#66260C2semidirect
3. #11752444C2semidirect
4. #11751707C2semidirect
5. #11746520C2semidirect
13. #91836C4semidirect
14. C5×#947C4semidirect
16. #91463C4semidirect
17. #91458C4semidirect
23. #66260D3semidirect
50. C3×D5⋊C8D4semidirect
110. #947AGL(1,𝔽5)semidirect
141. D5⋊C8#67semidirect
318. C3×C5#1910semidirect
374. C5#72025semidirect
392. C3#10604571semidirect

Maximal subgroups

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

subgroupindex
C3×#662602
#117552722
#117524442
#117517072
#117465202
#106045713
#720255