Finite Group Databasebeta

← #14282869

Subgroups — #14282869

Order 1920.

608 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 102 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#11754527C211yesmaximal, characteristic
2C4×#6763C211yesmaximal, characteristic, complemented
3C2×C3×#4003C211yesmaximal, characteristic, complemented
4#11746500C212yesmaximal, complemented
5#11746499C212yesmaximal, complemented
6#10604498—31nomaximal, Hall
7C22×C3×C4×D5C2211yescharacteristic
8C2×#6773C411yescharacteristic, complemented
9C2×#6763—21no—
10C2×C4×ΓL(1,𝔽16)—21no—
11C22×C3×#170—21no—
12#91819C411yescharacteristic, complemented
13C5×#924C411yescharacteristic, complemented
14#91564—21no—
15#91552—21no—
16#91450C2211yescharacteristic
17#91444—21no—
18#91359C2211yescharacteristic
19#91358C411yescharacteristic, complemented
20#91348—21no—
21#91235C2212yes—
22C3×#4003—21no—
23C3×#4003C2212yes—
24#69942—51nomaximal, Hall
25#65961—31no—
26C4×C22⋊AGL(1,𝔽5)—31no—
27C2×#4003D311yescharacteristic, complemented
28#64926—32no—
29#64868—32no—
30C23×C3×D5D411yescharacteristic
31C22×ΓL(1,𝔽16)—22no—
32C22×C3×C4×C5C2×C411yescharacteristic
33C2×#1220—21no—
34C2×D60C2×C411yescharacteristic
35C2×C5×Dic6C2×C411yescharacteristic
36C22×C3×Dic5D411yescharacteristic
37C2×C3×C4×D5—42no—
38C2×C3×C4×D5C2×C411yescharacteristic
39C2×C3×C4×D5D411yescharacteristic
40C2×C3×C4×D5D411yescharacteristic
41C2×C3×C4×D5C2311yescharacteristic
42C2×#1202—21no—
43C2×#1202—21no—
44C2×#1200—21no—
45#6773—42no—
46#6763—24no—
47C4×ΓL(1,𝔽16)—42no—
48C2×C3×#170—22no—
49C2×C3×#170—42no—
50#6714—21no—
51C4⋊Dic15C2×C411yescharacteristic
52C5×#217—21no—
53C5×C4⋊Dic3C2×C411yescharacteristic
54C5×Dic3⋊C4—21no—
55#6669—21no—
56#6667—21no—
57#6665—21no—
58#6665—21no—
59Dic3×Dic5—21no—
60#5791—51no—
61C2×C3×C22⋊C8—51no—
62C4×#217—51no—
63#4470—52no—
64#4467—52no—
65C2×#713—31no—
66C22×C4×D5C2×D311yescharacteristic
67C2×C22⋊AGL(1,𝔽5)—61no—
68C22×#170—21no—
69C2×C4×AGL(1,𝔽5)—61no—
70C5×C22⋊Q8—31no—
71#4092—31no—
72#4076—31no—
73#4041—31no—
74#4040—61no—
75#4039—61no—
76#4038—31no—
77#4028—61no—
78#4027—61no—
79(C2×AGL(1,𝔽5))⋊C4—32no—
80#4003—21no—
81#4003C2×D312yes—
82#1501—151noSylow
83C23×C3×C5C22⋊C411yescharacteristic
84C22×D15—21no—
85C22×C3×D5—22no—
86C22×C3×D5—22no—
87C22×C3×D5—41no—
88C22×C3×D5C22⋊C411yescharacteristic
89C22×C3×D5C2×D411yescharacteristic
90C2×ΓL(1,𝔽16)—24no—
91C2×ΓL(1,𝔽16)—42no—
92C2×ΓL(1,𝔽16)—42no—
93C2×C3×C4×C5—41no—
94C2×C3×C4×C5C22⋊C411yescharacteristic
95C2×C3×C4×C5C22×C411yescharacteristic
96#1220—42no—
97C2×Dic15—21no—
98D60—41no—
99C2×C5×Dic3—21no—
100C2×C5×Dic3—21no—
101C5×Dic6—41no—
102C2×C3×Dic5—22no—
103C2×C3×Dic5C22⋊C411yescharacteristic
104C2×C3×Dic5C2×D411yescharacteristic
105C3×C4×D5—22no—
106C3×C4×D5—22no—
107C3×C4×D5—41no—
108C3×C4×D5—41no—
109#1202—42no—
110#1202—42no—
111#1200—42no—
112C3×#170—42no—
113C23×C3×C4—51no—
114C2×#235—51no—
115C22×C3×C8—101no—
116#953—101no—
117C2×#217—101no—
118#930—51no—
119#930—51no—
120#929—101no—
121C4×#67—101no—
122C4×#67—101no—
123#926—51no—
124#924—51no—
125#924AGL(1,𝔽5)11yescharacteristic, complemented
126C2×C4×Dic3—101no—
127C3×C22⋊C8—52no—
128C3×C22⋊C8—101no—
129C3⋊#95—52no—
130C23×D5D1211yescharacteristic
131C22×AGL(1,𝔽5)—62no—
132C2×C5×Q8—31no—
133C22×C4×C5C4×D311yescharacteristic
134C2×C22⋊D5—61no—
135C22×Dic5#6711yescharacteristic
136#713—122no—
137C2×D20—31no—
138C2×D20—61no—
139C2×D20—61no—
140C2×C4×D5—42no—
141C2×C4×D5—61no—
142C2×C4×D5C4×D311yescharacteristic
143C2×C4×D5D1211yescharacteristic
144C2×C4×D5#6711yescharacteristic
145C2×C4×D5C22×D311yescharacteristic
146C22⋊AGL(1,𝔽5)—64no—
147C2×#170—22no—
148C2×#170—42no—
149C4×AGL(1,𝔽5)—122no—
150C5×C4⋊C4—31no—
151C5×C4⋊C4—61no—
152C5×C22⋊C4—61no—
153#685—61no—
154#685—61no—
155#685—61no—
156#685—61no—
157C4⋊Dic5—31no—
158#683—61no—
159C4×Dic5—61no—
160#525—151no—
161C2×C22⋊C8—51no—
162C4×C22⋊C4—151no—
163#328—152no—
164#327—152no—
165C22×C3×C5—21no—
166C22×C3×C5—21no—
167C22×C3×C5#10412yes—
168C22×C3×C5C2×C22⋊C411yescharacteristic
169C2×D15—41no—
170C2×D15—41no—
171C2×D15—41no—
172C2×C3×D5—21no—
173C2×C3×D5—21no—
174C2×C3×D5—21no—
175C2×C3×D5—21no—
176C2×C3×D5—21no—
177C2×C3×D5—44no—
178C2×C3×D5#13512yes—
179ΓL(1,𝔽16)—44no—
180C3×C4×C5—21no—
181C3×C4×C5—21no—
182Dic15—41no—
183C3×Dic5—21no—
184C3×Dic5#9912yes—
185C5×Dic3—41no—
186C5×Dic3—41no—
187C24×C3—51no—
188C22×C3×C4—51no—
189C22×C3×C4—51no—
190C22×C3×C4—51no—
191C22×C3×C4—51no—
192C22×C3×C4—51no—
193C22×C3×C4—202no—
194C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
195C2×#67—101no—
196C2×#67—101no—
197C2×#67—101no—
198C22×Dic3—102no—
199#235—202no—
200C2×D12—51no—
201C2×C4×D3—101no—
202C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
203C2×C3×C8—102no—
204C2×C3×C8—202no—
205#217—21no—
206#217—101no—
207#217—101no—
208#217—104no—
209(C2×D3)⋊C4—101no—
210(C2×D3)⋊C4—101no—
211C4⋊Dic3—51no—
212C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
213Dic3⋊C4—21no—
214Dic3⋊C4—101no—
215C4×Dic3—101no—
216C4×Dic3—202no—
217C23×C5(C2×D3)⋊C411yescharacteristic
218C22×D5—22no—
219C22×D5—22no—
220C22×D5—41no—
221C22×D5—61no—
222C22×D5(C2×D3)⋊C411yescharacteristic
223C22×D5C2×D1211yescharacteristic
224C2×AGL(1,𝔽5)—64no—
225C2×AGL(1,𝔽5)—122no—
226C2×AGL(1,𝔽5)—122no—
227C5×Q8—121no—
228C2×C4×C5—41no—
229C2×C4×C5—61no—
230C2×C4×C5—61no—
231C2×C4×C5(C2×D3)⋊C411yescharacteristic
232C2×C4×C5C2×C4×D311yescharacteristic
233C22⋊D5—122no—
234C2×Dic5—22no—
235C2×Dic5—61no—
236C2×Dic5(C2×D3)⋊C411yescharacteristic
237C2×Dic5C2×#6711yescharacteristic
238D20—121no—
239D20—122no—
240D20—122no—
241C4×D5—22no—
242C4×D5—22no—
243C4×D5—41no—
244C4×D5—41no—
245C4×D5—122no—
246#170—42no—
247C2×#41—151no—
248C23×C4—51no—
249C22×C8—101no—
250#123—301no—
251C22⋊Q8—31no—
252C22⋊Q8—151no—
253#121—151no—
254#121—151no—
255#120—301no—
256C4×D4—301no—
257C4×D4—301no—
258#117—151no—
259C2×C22⋊C4—301no—
260C2×C42—301no—
261C22⋊C8—52no—
262C22⋊C8—101no—
263#95—152no—
264C2×C3×C5—22no—
265C2×C3×C5C2×#9911yescharacteristic
266C2×C3×C5#41611yescharacteristic
267C2×C3×C5C2×#10411yescharacteristic
268D15—81no—
269C3×D5—22no—
270C3×D5—41no—
271C23×C3—51no—
272C23×C3—51no—
273C23×C3—102no—
274C23×C3—102no—
275C23×C3—201no—
276C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
277C22×D3—101no—
278C2×C3×C4—41no—
279C2×C3×C4—51no—
280C2×C3×C4—51no—
281C2×C3×C4—102no—
282C2×C3×C4—102no—
283C2×C3×C4—102no—
284C2×C3×C4—201no—
285C2×C3×C4—201no—
286C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
287C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
288#67—202no—
289#67—202no—
290#67—202no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—104no—
294C2×Dic3—101no—
295C2×Dic3—202no—
296C2×Dic3—202no—
297D12—201no—
298C4×D3—202no—
299Dic6—41no—
300C3×C8—202no—
301C22×C5—21no—
302C22×C5—21no—
303C22×C5#82512yes—
304C22×C5C2×(C2×D3)⋊C411yescharacteristic
305C2×D5—21no—
306C2×D5—21no—
307C2×D5—21no—
308C2×D5—21no—
309C2×D5—21no—
310C2×D5—44no—
311C2×D5—121no—
312C2×D5—121no—
313C2×D5—121no—
314C2×D5#90412yes—
315AGL(1,𝔽5)—124no—
316C4×C5—21no—
317C4×C5—21no—
318C4×C5—121no—
319C4×C5—121no—
320Dic5—21no—
321Dic5—121no—
322Dic5#80612yes—
323C24—51no—
324#41—602no—
325C2×Q8—31no—
326C2×D4—151no—
327C2×D4—301no—
328C2×D4—301no—
329C2×D4—301no—
330C22×C4—51no—
331C22×C4—51no—
332C22×C4—51no—
333C22×C4—51no—
334C22×C4—51no—
335C22×C4—202no—
336C22×C4—301no—
337C22×C4—302no—
338C22×C4D3×AGL(1,𝔽5)11yescharacteristic
339C2×C8—102no—
340C2×C8—202no—
341C4⋊C4—31no—
342C4⋊C4—61no—
343C4⋊C4—151no—
344C4⋊C4—301no—
345C22⋊C4—61no—
346C22⋊C4—301no—
347C22⋊C4—301no—
348C22⋊C4—301no—
349C22⋊C4—304no—
350C22⋊C4—301no—
351C42—301no—
352C42—602no—
353C3×C5#150111yescharacteristic, complemented, Hall
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—52no—
357C22×C3—101no—
358C22×C3—101no—
359C22×C3—101no—
360C22×C3—101no—
361C22×C3—101no—
362C22×C3—204no—
363C22×C3#401012yes—
364C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
365C2×D3—201no—
366C2×D3—201no—
367C2×D3—201no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—52no—
371C3×C4—101no—
372Dic3—41no—
373Dic3—41no—
374Dic3—204no—
375Dic3—201no—
376C2×C5—22no—
377C2×C5C2×#80611yescharacteristic
378C2×C5#511311yescharacteristic
379C2×C5C2×#82511yescharacteristic
380D5—22no—
381D5—41no—
382D5—241no—
383C23—51no—
384C23—51no—
385C23—102no—
386C23—102no—
387C23—201no—
388C23—301no—
389C23(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
390Q8—121no—
391D4—601no—
392D4—602no—
393D4—602no—
394D4—602no—
395C2×C4—41no—
396C2×C4—51no—
397C2×C4—51no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—102no—
401C2×C4—102no—
402C2×C4—102no—
403C2×C4—201no—
404C2×C4—201no—
405C2×C4—304no—
406C2×C4—301no—
407C2×C4—602no—
408C2×C4—602no—
409C2×C4—602no—
410C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
411C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
412C8—202no—
413C2×C3—22no—
414C2×C3—102no—
415C2×C3—201no—
416C2×C3D5⋊#9911yescharacteristic
417C2×C3C2×#401011yescharacteristic
418C2×C3#6579411yescharacteristic
419D3—401no—
420C5#6994211yescharacteristic, complemented, Sylow
421C22—21no—
422C22—21no—
423C22—52no—
424C22—101no—
425C22—101no—
426C22—101no—
427C22—101no—
428C22—101no—
429C22—204no—
430C22—601no—
431C22—601no—
432C22—601no—
433C22#9115712yes—
434C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
435C4—21no—
436C4—21no—
437C4—52no—
438C4—101no—
439C4—121no—
440C4—121no—
441C4—604no—
442C4—601no—
443C3#1060449811yescharacteristic, complemented, Sylow
444C2—22no—
445C2—102no—
446C2—201no—
447C2—1201no—
448C2#1175170411yescharacteristic
449C2C2×#9115711yescharacteristic
450C2D5⋊#80611yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radical#11754527C2
cosocleC2×C3×C4×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1501
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C4×#6763C2semidirect
3. C2×C3×#4003C2semidirect
4. #11746500C2semidirect
5. #11746499C2semidirect
8. C2×#6773C4semidirect
12. #91819C4semidirect
13. C5×#924C4semidirect
19. #91358C4semidirect
27. C2×#4003D3semidirect
125. #924AGL(1,𝔽5)semidirect
353. C3×C5#1501semidirect
420. C5#69942semidirect
443. C3#10604498semidirect

Maximal subgroups

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

subgroupindex
#117545272
C4×#67632
C2×C3×#40032
#117465002
#117464992
#106044983
#699425