Finite Group Databasebeta

← #14282317

Subgroups — #14282317

Order 1920.

632 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C4×D5×D12C211yesmaximal, characteristic
2C4×#6758C211yesmaximal, characteristic, complemented
3C3×#65683C211yesmaximal, characteristic, complemented
4#11746490C212yesmaximal, complemented
5#11746488C212yesmaximal, complemented
6#10604407—31nomaximal, Hall
7C2×D5×D12C411yescharacteristic
8C2×C4×D3×D5—21no—
9C2×#6758—21no—
10C4×D60C411yescharacteristic
11C4×C5×D12C411yescharacteristic
12C3×C42×D5C2211yescharacteristic
13D5×(C2×D3)⋊C4—21no—
14C4×#1203—21no—
15#91418C2211yescharacteristic
16Dic5×D12C2211yescharacteristic
17D5×C4⋊Dic3C411yescharacteristic
18#91405—21no—
19#91231—21no—
20C4×C3⋊#170C2212yescomplemented
21C3×#4003—21no—
22C3×#4002—21no—
23C3×C4⋊#170C2212yescomplemented
24#70799—51nomaximal, Hall
25C4×D4×D5—31no—
26#65683D311yescharacteristic, complemented
27C4×#700—31no—
28D4⋊#170—32no—
29#64871—32no—
30C22×D3×D5—21no—
31C2×D60C2×C411yescharacteristic
32C2×C4×D15—21no—
33C2×C5×D12C2×C411yescharacteristic
34C2×C4×C5×D3—21no—
35C2×C3×C4×D5C2×C411yescharacteristic
36C2×C3×C4×D5D411yescharacteristic
37C2×C3×C4×D5D411yescharacteristic
38C2×#1203—21no—
39C2×#1200—21no—
40C2×D3×Dic5—21no—
41C2×D5×Dic3—21no—
42D5×D12—22no—
43D5×D12—22no—
44C4×D3×D5—44no—
45C2×C3⋊#170—22no—
46#6758—24no—
47C2×C3×#170—22no—
48C3×C42×C5C2×C411yescharacteristic
49#6714—21no—
50C4⋊Dic15C2×C411yescharacteristic
51C5×(C2×D3)⋊C4—21no—
52C5×C4⋊Dic3C2×C411yescharacteristic
53C3×C4×Dic5D411yescharacteristic
54C3×C4×Dic5D411yescharacteristic
55C3×C4×Dic5C2311yescharacteristic
56#6668—21no—
57#6667—21no—
58(C2×D3)⋊Dic5—21no—
59C2×C4×D12—51no—
60C3×#432—51no—
61C4×#208—51no—
62#4483—52no—
63#4483—52no—
64C2×D4×D5—31no—
65C22×C4×D5—61no—
66C2×#700—61no—
67C4×C5×D4—31no—
68D4×Dic5—31no—
69C4×C22⋊D5—61no—
70#4039—31no—
71D5×C4⋊C4—31no—
72#4027—61no—
73D5×C22⋊C4—61no—
74C4×D20—31no—
75C42×D5C2×D311yescharacteristic
76#4003—21no—
77#4002—21no—
78#4002—61no—
79C4⋊#170C2×D312yescomplemented
80C4×#170—32no—
81#1490—151noSylow
82C22×D15—21no—
83C22×C5×D3—21no—
84C22×C3×D5C22⋊C411yescharacteristic
85C2×D3×D5—24no—
86C2×D3×D5—42no—
87C2×D3×D5—42no—
88C2×C3×C4×C5C22⋊C411yescharacteristic
89C2×C3×C4×C5C22⋊C411yescharacteristic
90C2×C3×C4×C5C22×C411yescharacteristic
91C2×Dic15—21no—
92D60—21no—
93D60M1612yescomplemented
94C4×D15—42no—
95C2×C5×Dic3—21no—
96C5×D12—21no—
97C5×D12M1612yescomplemented
98C4×C5×D3—42no—
99C2×C3×Dic5C22⋊C411yescharacteristic
100C2×C3×Dic5C2×D411yescharacteristic
101C2×C3×Dic5C2×D411yescharacteristic
102C3×C4×D5—22no—
103C3×C4×D5—21no—
104C3×C4×D5—21no—
105C3×C4×D5—22no—
106#1203—24no—
107#1200—42no—
108D3×Dic5—42no—
109D5×Dic3—42no—
110C3⋊#170—24no—
111C3×#170—42no—
112C22×D12—51no—
113C22×C4×D3—101no—
114C2×C3×C42—51no—
115C2×(C2×D3)⋊C4—101no—
116C2×C4⋊Dic3—51no—
117C2×#208—101no—
118C4×D12—51no—
119C4×D12—51no—
120C4×D12—51no—
121C4×D12—101no—
122C4×D12—101no—
123C4×D12AGL(1,𝔽5)11yescharacteristic
124C3×C4⋊C8—52no—
125C3×C22⋊C8—101no—
126C3×#97—101no—
127#803—101no—
128C4×C3⋊C8—52no—
129C23×D5—61no—
130C2×C5×D4—31no—
131C22×C4×C5—61no—
132C2×C22⋊D5—61no—
133C22×Dic5—61no—
134D4×D5—62no—
135D4×D5—62no—
136C2×D20—31no—
137C2×C4×D5—61no—
138C2×C4×D5—61no—
139C2×C4×D5—61no—
140C2×C4×D5—124no—
141C2×C4×D5C4×D311yescharacteristic
142C2×C4×D5D1211yescharacteristic
143C2×C4×D5#6711yescharacteristic
144C2×#170—22no—
145C2×#170—62no—
146#700—64no—
147C5×C4⋊C4—31no—
148C5×C22⋊C4—61no—
149C42×C5C4×D311yescharacteristic
150C22⋊Dic5—61no—
151#685—61no—
152#685—61no—
153C4⋊Dic5—31no—
154#683—61no—
155C4×Dic5D1211yescharacteristic
156C4×Dic5#6711yescharacteristic
157C4×Dic5C22×D311yescharacteristic
158C2×C4×D4—151no—
159#432—51no—
160C4×M16—151no—
161D4⋊C8—152no—
162D4⋊C8—152no—
163C22×C3×C5C2×C22⋊C411yescharacteristic
164C2×D15—22no—
165C2×D15—41no—
166C2×D15—41no—
167C2×C5×D3—22no—
168C2×C5×D3—41no—
169C2×C5×D3—41no—
170C2×C3×D5—21no—
171C2×C3×D5—21no—
172C2×C3×D5#10412yes—
173D3×D5—48no—
174C3×C4×C5—21no—
175C3×C4×C5—21no—
176C3×C4×C5C2×M1611yescharacteristic
177C3×C4×C5C2×M1611yescharacteristic
178Dic15—41no—
179C3×Dic5—21no—
180C3×Dic5—21no—
181C3×Dic5AGL(1,ℤ/8)12yes—
182C5×Dic3—41no—
183C23×D3—101no—
184C22×C3×C4—51no—
185C22×C3×C4—51no—
186C22×C3×C4—51no—
187C22×Dic3—101no—
188C2×D12—51no—
189C2×D12—102no—
190C2×D12—101no—
191C2×D12—102no—
192C2×D12C2×AGL(1,𝔽5)11yescharacteristic
193C2×C4×D3—21no—
194C2×C4×D3—101no—
195C2×C4×D3—101no—
196C2×C4×D3—101no—
197C2×C4×D3—204no—
198C2×C3×C8—102no—
199C3×C42—51no—
200C3×C42—51no—
201C3×C42—51no—
202C3×C42C2×AGL(1,𝔽5)11yescharacteristic
203(C2×D3)⋊C4—21no—
204(C2×D3)⋊C4—101no—
205(C2×D3)⋊C4—101no—
206(C2×D3)⋊C4—101no—
207C4⋊Dic3—51no—
208C4⋊Dic3—101no—
209C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
210#208—104no—
211C2×C3⋊C8—102no—
212C23×C5—61no—
213C22×D5—64no—
214C22×D5—61no—
215C22×D5—122no—
216C22×D5—122no—
217C22×D5(C2×D3)⋊C411yescharacteristic
218C5×D4—32no—
219C5×D4—61no—
220C2×C4×C5—61no—
221C2×C4×C5—122no—
222C2×C4×C5(C2×D3)⋊C411yescharacteristic
223C2×C4×C5(C2×D3)⋊C411yescharacteristic
224C2×C4×C5C2×C4×D311yescharacteristic
225C22⋊D5—64no—
226C2×Dic5—61no—
227C2×Dic5—122no—
228C2×Dic5(C2×D3)⋊C411yescharacteristic
229C2×Dic5C2×D1211yescharacteristic
230C2×Dic5C2×#6711yescharacteristic
231D20—32no—
232D20—61no—
233C4×D5—22no—
234C4×D5—21no—
235C4×D5—21no—
236C4×D5—22no—
237C4×D5—122no—
238C4×D5—122no—
239C4×D5—122no—
240#170—42no—
241#170—64no—
242C22×D4—151no—
243C23×C4—301no—
244C2×M16—301no—
245C4×D4—31no—
246C4×D4—151no—
247C4×D4—151no—
248C4×D4—151no—
249C4×D4—301no—
250C4×D4—301no—
251C2×C4⋊C4—151no—
252C2×C22⋊C4—301no—
253C2×C42—51no—
254C4⋊C8—52no—
255C22⋊C8—101no—
256#97—101no—
257#97—301no—
258C4×C8—152no—
259C2×C3×C5#40711yescharacteristic
260C2×C3×C5#41811yescharacteristic
261C2×C3×C5C2×#10411yescharacteristic
262D15—42no—
263C3×D5—22no—
264C5×D3—42no—
265C23×C3—51no—
266C22×D3—21no—
267C22×D3—104no—
268C22×D3—101no—
269C22×D3—202no—
270C22×D3—202no—
271C2×C3×C4—51no—
272C2×C3×C4—51no—
273C2×C3×C4—51no—
274C2×C3×C4—102no—
275C2×C3×C4—102no—
276C2×C3×C4—101no—
277C2×C3×C4—101no—
278C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
279C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
280C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
281C2×Dic3—21no—
282C2×Dic3—101no—
283C2×Dic3—202no—
284D12—21no—
285D12—52no—
286D12—104no—
287D12—101no—
288D12#70012yescomplemented
289C4×D3—42no—
290C4×D3—202no—
291C4×D3—202no—
292C4×D3—202no—
293C3×C8—202no—
294C3⋊C8—104no—
295C22×C5—62no—
296C22×C5—121no—
297C22×C5—121no—
298C22×C5C2×(C2×D3)⋊C411yescharacteristic
299C2×D5—21no—
300C2×D5—21no—
301C2×D5—62no—
302C2×D5—128no—
303C2×D5—121no—
304C2×D5—121no—
305C2×D5#82512yes—
306C4×C5—21no—
307C4×C5—21no—
308C4×C5—121no—
309C4×C5D3×M1611yescharacteristic
310C4×C5D3×M1611yescharacteristic
311Dic5—21no—
312Dic5—21no—
313Dic5—121no—
314Dic5#94912yes—
315C24—301no—
316C2×D4—31no—
317C2×D4—151no—
318C2×D4—301no—
319C2×D4—302no—
320C2×D4—302no—
321C22×C4—51no—
322C22×C4—51no—
323C22×C4—51no—
324C22×C4—61no—
325C22×C4—301no—
326C22×C4—301no—
327C22×C4—301no—
328C22×C4—301no—
329C22×C4—604no—
330M16—304no—
331C2×C8—102no—
332C2×C8—302no—
333C4⋊C4—31no—
334C4⋊C4—151no—
335C4⋊C4—301no—
336C22⋊C4—61no—
337C22⋊C4—301no—
338C22⋊C4—301no—
339C22⋊C4—301no—
340C42—51no—
341C42—51no—
342C42—51no—
343C42D3×AGL(1,𝔽5)11yescharacteristic
344C3×C5#149011yescharacteristic, complemented, Hall
345C22×C3—52no—
346C22×C3—101no—
347C22×C3—101no—
348C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
349C2×D3—22no—
350C2×D3—41no—
351C2×D3—41no—
352C2×D3—102no—
353C2×D3—208no—
354C2×D3—201no—
355C2×D3—201no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—52no—
359C3×C4—101no—
360C3×C4—101no—
361C3×C4C2×#70011yescharacteristic
362C3×C4D5⋊M1611yescharacteristic
363Dic3—41no—
364Dic3—201no—
365C2×C5—122no—
366C2×C5#500111yescharacteristic
367C2×C5#512611yescharacteristic
368C2×C5C2×#82511yescharacteristic
369D5—22no—
370D5—122no—
371C23—51no—
372C23—61no—
373C23—304no—
374C23—301no—
375C23—602no—
376C23—602no—
377D4—32no—
378D4—61no—
379D4—152no—
380D4—304no—
381D4—301no—
382C2×C4—51no—
383C2×C4—51no—
384C2×C4—51no—
385C2×C4—61no—
386C2×C4—102no—
387C2×C4—102no—
388C2×C4—101no—
389C2×C4—101no—
390C2×C4—122no—
391C2×C4—301no—
392C2×C4—602no—
393C2×C4—602no—
394C2×C4—602no—
395C2×C4—602no—
396C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
397C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
398C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
399C8—202no—
400C8—304no—
401C2×C3—102no—
402C2×C3#6575611yescharacteristic
403C2×C3#6576711yescharacteristic
404C2×C3D5⋊#10411yescharacteristic
405D3—42no—
406D3—202no—
407C5#7079911yescharacteristic, complemented, Sylow
408C22—52no—
409C22—62no—
410C22—101no—
411C22—101no—
412C22—121no—
413C22—121no—
414C22—302no—
415C22—608no—
416C22—601no—
417C22—601no—
418C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
419C4—21no—
420C4—21no—
421C4—52no—
422C4—101no—
423C4—101no—
424C4—121no—
425C4—601no—
426C4D3×#70011yescharacteristic
427C4#9191511yescharacteristic
428C3#1060440711yescharacteristic, complemented, Sylow
429C2—102no—
430C2—122no—
431C2—602no—
432C2#1175170611yescharacteristic
433C2D5⋊#82511yescharacteristic
434C2#1175180611yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C42×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×D5×D12C4
cosocleC3×C4×Dic5C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#1490
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×#6758C2semidirect
3. C3×#65683C2semidirect
4. #11746490C2semidirect
5. #11746488C2semidirect
20. C4×C3⋊#170C22semidirect
23. C3×C4⋊#170C22semidirect
26. #65683D3semidirect
79. C4⋊#170C2×D3semidirect
93. D60M16semidirect
97. C5×D12M16semidirect
288. D12#700semidirect
344. C3×C5#1490semidirect
407. C5#70799semidirect
428. C3#10604407semidirect

Maximal subgroups

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

subgroupindex
C4×D5×D122
C4×#67582
C3×#656832
#117464902
#117464882
#106044073
#707995