Finite Group Databasebeta

← #72230437

Subgroups — #72230437

Order 2880.

564 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 98 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#13088723C211yesmaximal, characteristic
2C3⋊D3×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3D5×#3712C211yesmaximal, characteristic
4#13085455C211yesmaximal, characteristic, complemented
5#13085454C211yesmaximal, characteristic, complemented
6C3×#91160C211yesmaximal, characteristic, complemented
7C3×#91159C211yesmaximal, characteristic, complemented
8#11751654—31nomaximal
9#11751653—31nomaximal
10C2×C3⋊D3×AGL(1,𝔽5)—21no—
11C4×D5×C3⋊D3C2211yescharacteristic
12C4⋊(C3⋊ΓL(1,𝔽16))C2211yescharacteristic
13C3×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
14C3×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
15C32×C4⋊AGL(1,𝔽5)C2211yescharacteristic
16#10623978—21no—
17ΓL(1,𝔽16)⋊Dic3—21no—
18#10623965—21no—
19#10623820C411yescharacteristic, complemented
20#10623797C411yescharacteristic, complemented
21#10623797C411yescharacteristic, complemented
22C5×#3712C411yescharacteristic, complemented
23C3×D5×C3⋊C8C2211yescharacteristic
24C3×D5×C3⋊C8C2211yescharacteristic
25#10590583—51nomaximal, Hall
26D3×C4⋊AGL(1,𝔽5)—31no—
27D3×C4⋊AGL(1,𝔽5)—31no—
28D3×C4⋊AGL(1,𝔽5)—61no—
29#91907—31no—
30#91907—31no—
31D5×ΓL(1,𝔽25)—31no—
32D5×ΓL(1,𝔽25)—31no—
33#91223—31no—
34#91222—31no—
35C3×#3994—31no—
36C3×#3993—31no—
37#91160—31no—
38#91160D311yescharacteristic, complemented
39#91159—31no—
40#91159D311yescharacteristic, complemented
41C2×D5×C3⋊D3D411yescharacteristic
42C2×C3⋊ΓL(1,𝔽16)—21no—
43C2×C3×ΓL(1,𝔽16)—21no—
44C2×C3×ΓL(1,𝔽16)—21no—
45C2×C32×AGL(1,𝔽5)—21no—
46C3⋊D3×AGL(1,𝔽5)—42no—
47C4×C3⋊D15C2×C411yescharacteristic
48C4×C5×C3⋊D3C2×C411yescharacteristic
49C32×C4×D5C2311yescharacteristic
50#67137Q811yescharacteristic
51C3⋊D3×Dic5Q811yescharacteristic
52D5×C3⋊Dic3D411yescharacteristic
53C3×C3⋊#168C2×C411yescharacteristic, complemented
54C3×C3⋊#168C2×C411yescharacteristic, complemented
55C3×C5×C3⋊C8C2×C411yescharacteristic, complemented
56C3×C5×C3⋊C8C2×C411yescharacteristic, complemented
57#65727—91noHall
58C4⋊C4×C3⋊D3—51no—
59#63849—51no—
60C2×#3712—51no—
61C3×#808—51no—
62C3×#807—51no—
63#63575—51no—
64#63573—51no—
65C2×D3×AGL(1,𝔽5)—61no—
66C2×D3×AGL(1,𝔽5)—61no—
67C2×D3×AGL(1,𝔽5)—61no—
68C2×D3×AGL(1,𝔽5)—61no—
69C4×D3×D5—31no—
70C4×D3×D5—31no—
71C4×D3×D5—61no—
72C4⋊ΓL(1,𝔽16)—31no—
73C4⋊ΓL(1,𝔽16)—31no—
74C4⋊ΓL(1,𝔽16)—61no—
75C4⋊ΓL(1,𝔽16)C2×D311yescharacteristic
76C4⋊ΓL(1,𝔽16)C2×D311yescharacteristic
77C3×C4⋊AGL(1,𝔽5)—21no—
78C3×C4⋊AGL(1,𝔽5)—31no—
79C3×C4⋊AGL(1,𝔽5)—31no—
80C3×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
81C3×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
82Dic3⋊AGL(1,𝔽5)—61no—
83Dic3⋊AGL(1,𝔽5)—61no—
84Dic3⋊AGL(1,𝔽5)—61no—
85Dic3⋊AGL(1,𝔽5)—61no—
86(C2×D3)⋊AGL(1,𝔽5)—61no—
87(C2×D3)⋊AGL(1,𝔽5)—61no—
88Dic3×AGL(1,𝔽5)—61no—
89Dic3×AGL(1,𝔽5)—61no—
90#6705—31no—
91#6705—31no—
92C5×ΓL(1,𝔽25)—31no—
93C5×ΓL(1,𝔽25)—31no—
94C3×C8×D5—31no—
95C3×C8×D5—31no—
96#6651—31no—
97#6651—31no—
98#6650—31no—
99#6650—31no—
100D5×C3⋊C8C2×D311yescharacteristic
101D5×C3⋊C8C2×D311yescharacteristic
102#4881—151no—
103#4861—151no—
104C2×C3⋊D15C4⋊C411yescharacteristic
105C2×C5×C3⋊D3C4⋊C411yescharacteristic
106C2×C32×D5C2×D411yescharacteristic
107D5×C3⋊D3—21no—
108D5×C3⋊D3—21no—
109C3⋊ΓL(1,𝔽16)—41no—
110C3×ΓL(1,𝔽16)—41no—
111C3×ΓL(1,𝔽16)—41no—
112C32×AGL(1,𝔽5)—41no—
113C32×C4×C5C22×C411yescharacteristic
114C3⋊Dic15C4⋊C411yescharacteristic
115C5×C3⋊Dic3C4⋊C411yescharacteristic
116C32×Dic5C2×Q811yescharacteristic
117#4130—91no—
118C2×C4⋊AGL(1,𝔽5)—91no—
119D5×M16—91no—
120#3994—31no—
121#3994—91no—
122#3993—31no—
123#3993—91no—
124C2×C4×C3⋊D3—51no—
125C2×C4×C3⋊D3—101no—
126C32×C4⋊C4—51no—
127C4⋊(C3⋊Dic3)—51no—
128#3750—101no—
129C3×C4⋊Dic3—51no—
130C3×C4⋊Dic3—51no—
131C2×C3×C3⋊C8—51no—
132C2×C3×C3⋊C8—51no—
133#3722—101no—
134Dic32—101no—
135#3712—51no—
136#3712—51no—
137#3712—51no—
138#3712AGL(1,𝔽5)11yescharacteristic, complemented
139C2×D3×D5—31no—
140C2×D3×D5—31no—
141C2×D3×D5—61no—
142C2×ΓL(1,𝔽16)—21no—
143C2×ΓL(1,𝔽16)—21no—
144C2×ΓL(1,𝔽16)—61no—
145C2×ΓL(1,𝔽16)—61no—
146C2×ΓL(1,𝔽16)—61no—
147C2×ΓL(1,𝔽16)—61no—
148C2×C3×AGL(1,𝔽5)—21no—
149C2×C3×AGL(1,𝔽5)—21no—
150C2×C3×AGL(1,𝔽5)—21no—
151C2×C3×AGL(1,𝔽5)—21no—
152C2×C3×AGL(1,𝔽5)—61no—
153C2×C3×AGL(1,𝔽5)—61no—
154D3×AGL(1,𝔽5)—122no—
155D3×AGL(1,𝔽5)—122no—
156D3×AGL(1,𝔽5)—122no—
157D3×AGL(1,𝔽5)—122no—
158C4×D15—31no—
159C4×D15—31no—
160C4×D15—61no—
161C4×C5×D3—31no—
162C4×C5×D3—31no—
163C4×C5×D3—61no—
164C3×C4×D5—21no—
165C3×C4×D5C22×D311yescharacteristic
166C3×C4×D5C22×D311yescharacteristic
167#1200—31no—
168#1200—31no—
169#1200—61no—
170D3×Dic5—31no—
171D3×Dic5—31no—
172D3×Dic5—61no—
173D5×Dic3—31no—
174D5×Dic3—31no—
175D5×Dic3—61no—
176C3×C5×C8—31no—
177C3×C5×C8—31no—
178C3⋊#168C4×D311yescharacteristic, complemented
179C3⋊#168C4×D311yescharacteristic, complemented
180C3×#168—31no—
181C3×#168—31no—
182C5×C3⋊C8C4×D311yescharacteristic, complemented
183C5×C3⋊C8C4×D311yescharacteristic, complemented
184C2×ΓL(1,𝔽25)—151no—
185C2×ΓL(1,𝔽25)—151no—
186#892—151no—
187#892—151no—
188D3×C4⋊C4—151no—
189D3×C4⋊C4—151no—
190D3×C4⋊C4—301no—
191C3×#107—151no—
192C3×#106—151no—
193#818—151no—
194#817—151no—
195#808—51no—
196#808—151no—
197#807—51no—
198#807—151no—
199C2×C32×C5C2×C4⋊C411yescharacteristic
200C3⋊D15—21no—
201C5×C3⋊D3—21no—
202C32×D5AGL(1,ℤ/8)11yescharacteristic
203C32×D5#13711yescharacteristic
204C22×AGL(1,𝔽5)—181no—
205C2×C4×D5—91no—
206C22⋊AGL(1,𝔽5)—181no—
207C4⋊AGL(1,𝔽5)—31no—
208C4⋊AGL(1,𝔽5)—31no—
209C4⋊AGL(1,𝔽5)—91no—
210C4⋊AGL(1,𝔽5)—181no—
211C4⋊AGL(1,𝔽5)D3211yescharacteristic
212C4×AGL(1,𝔽5)—181no—
213C5×M16—91no—
214#681—91no—
215#676—91no—
216#676—91no—
217C8×D5—31no—
218C8×D5—31no—
219C22×C3⋊D3—51no—
220C2×C32×C4—51no—
221C2×C32×C4—101no—
222C2×C3⋊Dic3—51no—
223C2×C3⋊Dic3—101no—
224C4×C3⋊D3—51no—
225C4×C3⋊D3—51no—
226C4×C3⋊D3—51no—
227C4×C3⋊D3—202no—
228C4×C3⋊D3C2×AGL(1,𝔽5)11yescharacteristic
229C2×C3×Dic3—101no—
230C2×C3×Dic3—101no—
231C3×C3⋊C8—51no—
232C3×C3⋊C8—51no—
233C3×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic, complemented
234C3×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic, complemented
235#428—451noSylow
236C2×D15—31no—
237C2×D15—31no—
238C2×D15—61no—
239C2×C5×D3—31no—
240C2×C5×D3—31no—
241C2×C5×D3—61no—
242C2×C3×D5—21no—
243C2×C3×D5D3×D411yescharacteristic
244C2×C3×D5D3×D411yescharacteristic
245D3×D5—61no—
246D3×D5—61no—
247D3×D5—61no—
248D3×D5—61no—
249D3×D5—61no—
250D3×D5—61no—
251D3×D5—61no—
252D3×D5—61no—
253ΓL(1,𝔽16)—41no—
254ΓL(1,𝔽16)—41no—
255ΓL(1,𝔽16)—121no—
256ΓL(1,𝔽16)—121no—
257ΓL(1,𝔽16)—121no—
258ΓL(1,𝔽16)—121no—
259C3×AGL(1,𝔽5)—41no—
260C3×AGL(1,𝔽5)—41no—
261C3×AGL(1,𝔽5)—41no—
262C3×AGL(1,𝔽5)—41no—
263C3×AGL(1,𝔽5)—121no—
264C3×AGL(1,𝔽5)—121no—
265C3×C4×C5—21no—
266C3×C4×C5C2×C4×D311yescharacteristic
267C3×C4×C5C2×C4×D311yescharacteristic
268Dic15—31no—
269Dic15—31no—
270Dic15—61no—
271C3×Dic5—21no—
272C3×Dic5D3×Q811yescharacteristic
273C3×Dic5D3×Q811yescharacteristic
274C5×Dic3—31no—
275C5×Dic3—31no—
276C5×Dic3—61no—
277C2×C4×D3—151no—
278C2×C4×D3—151no—
279C2×C4×D3—301no—
280C2×C4×D3—301no—
281C2×C4×D3—301no—
282C2×C4×D3—301no—
283C2×C4×D3—301no—
284C2×C3×C8—151no—
285C2×C3×C8—151no—
286C3×C4⋊C4—51no—
287C3×C4⋊C4—51no—
288C3×C4⋊C4—101no—
289C3×C4⋊C4—151no—
290C3×C4⋊C4—151no—
291(C2×D3)⋊C4—301no—
292(C2×D3)⋊C4—301no—
293C4⋊Dic3—51no—
294C4⋊Dic3—51no—
295C4⋊Dic3—151no—
296C4⋊Dic3—151no—
297C4⋊Dic3—301no—
298Dic3⋊C4—301no—
299Dic3⋊C4—301no—
300Dic3⋊C4—301no—
301Dic3⋊C4—301no—
302C4×Dic3—301no—
303C4×Dic3—301no—
304C2×C3⋊C8—51no—
305C2×C3⋊C8—51no—
306ΓL(1,𝔽25)—31no—
307ΓL(1,𝔽25)—31no—
308ΓL(1,𝔽25)—151no—
309ΓL(1,𝔽25)—151no—
310ΓL(1,𝔽25)—151no—
311ΓL(1,𝔽25)—151no—
312ΓL(1,𝔽25)—151no—
313ΓL(1,𝔽25)—151no—
314C32×C5#42811yescharacteristic, complemented, Hall
315C22×D5—91no—
316C2×AGL(1,𝔽5)—21no—
317C2×AGL(1,𝔽5)—61no—
318C2×AGL(1,𝔽5)—61no—
319C2×AGL(1,𝔽5)—181no—
320C2×AGL(1,𝔽5)—362no—
321C2×C4×C5—91no—
322C2×Dic5—91no—
323C4×D5—91no—
324C4×D5—91no—
325C4×D5—91no—
326C4×D5C2×D3211yescharacteristic
327C5×C8—31no—
328C5×C8—31no—
329#168—31no—
330#168—31no—
331C22×C32—51no—
332C2×C3⋊D3—51no—
333C2×C3⋊D3—101no—
334C2×C3⋊D3—101no—
335C2×C3⋊D3C4⋊AGL(1,𝔽5)11yescharacteristic
336C32×C4—51no—
337C32×C4—201no—
338C32×C4C22×AGL(1,𝔽5)11yescharacteristic
339C3⋊Dic3—51no—
340C3⋊Dic3—201no—
341C3⋊Dic3C4⋊AGL(1,𝔽5)11yescharacteristic
342C3×Dic3—201no—
343C3×Dic3—201no—
344C2×M16—451no—
345#117—451no—
346C2×C4⋊C4—451no—
347#107—151no—
348#107—451no—
349#106—151no—
350#106—451no—
351C2×C3×C5—21no—
352C2×C3×C5D3×C4⋊C411yescharacteristic
353C2×C3×C5D3×C4⋊C411yescharacteristic
354D15—61no—
355D15—61no—
356D15—61no—
357D15—61no—
358C3×D5—21no—
359C3×D5—21no—
360C3×D5#91111yescharacteristic
361C3×D5#91411yescharacteristic
362C3×D5#91511yescharacteristic
363C3×D5#91811yescharacteristic
364C5×D3—61no—
365C5×D3—61no—
366C5×D3—61no—
367C5×D3—61no—
368C22×D3—151no—
369C22×D3—151no—
370C22×D3—301no—
371C2×C3×C4—51no—
372C2×C3×C4—51no—
373C2×C3×C4—101no—
374C2×C3×C4—101no—
375C2×C3×C4—101no—
376C2×C3×C4—101no—
377C2×C3×C4—101no—
378C2×C3×C4—301no—
379C2×C3×C4—301no—
380C2×Dic3—101no—
381C2×Dic3—101no—
382C2×Dic3—151no—
383C2×Dic3—151no—
384C2×Dic3—301no—
385C2×Dic3—301no—
386C2×Dic3—301no—
387C2×Dic3—301no—
388C2×Dic3—301no—
389C4×D3—31no—
390C4×D3—31no—
391C4×D3—61no—
392C4×D3—151no—
393C4×D3—151no—
394C4×D3—151no—
395C4×D3—151no—
396C4×D3—151no—
397C4×D3—151no—
398C4×D3—301no—
399C4×D3—301no—
400C4×D3—301no—
401C4×D3—602no—
402C4×D3—602no—
403C4×D3—602no—
404C4×D3—602no—
405C3×C8—31no—
406C3×C8—31no—
407C3×C8—151no—
408C3×C8—151no—
409C3⋊C8—51no—
410C3⋊C8—51no—
411C3⋊C8D3×AGL(1,𝔽5)11yescharacteristic, complemented
412C3⋊C8D3×AGL(1,𝔽5)11yescharacteristic, complemented
413C22×C5—91no—
414C2×D5—91no—
415C2×D5—181no—
416C2×D5—181no—
417C2×D5#381111yescharacteristic
418AGL(1,𝔽5)—41no—
419AGL(1,𝔽5)—121no—
420AGL(1,𝔽5)—121no—
421AGL(1,𝔽5)—361no—
422C4×C5—91no—
423C4×C5C4×D3211yescharacteristic
424Dic5—91no—
425Dic5#379711yescharacteristic
426C2×C32—51no—
427C2×C32—51no—
428C2×C32C2×C4⋊AGL(1,𝔽5)11yescharacteristic
429C3⋊D3—21no—
430C3⋊D3—101no—
431C22×C4—451no—
432C22×C4—901no—
433M16—91no—
434M16—451no—
435M16—451no—
436M16—451no—
437C2×C8—151no—
438C2×C8—151no—
439C4⋊C4—51no—
440C4⋊C4—151no—
441C4⋊C4—151no—
442C4⋊C4—451no—
443C4⋊C4—901no—
444C22⋊C4—901no—
445C42—901no—
446C3×C5—21no—
447C3×C5#486111yescharacteristic, complemented
448C3×C5#488111yescharacteristic, complemented
449C22×C3—51no—
450C22×C3—51no—
451C22×C3—101no—
452C2×D3—31no—
453C2×D3—31no—
454C2×D3—61no—
455C2×D3—151no—
456C2×D3—151no—
457C2×D3—301no—
458C2×D3—301no—
459C2×D3—301no—
460C2×D3—301no—
461C2×D3—301no—
462C2×D3—301no—
463C2×D3—301no—
464C2×D3—301no—
465C2×D3—301no—
466C3×C4—21no—
467C3×C4—51no—
468C3×C4—51no—
469C3×C4—101no—
470C3×C4—201no—
471C3×C4—201no—
472C3×C4—201no—
473C3×C4—201no—
474C3×C4—601no—
475C3×C4—601no—
476C3×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
477C3×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
478Dic3—31no—
479Dic3—31no—
480Dic3—61no—
481Dic3—151no—
482Dic3—151no—
483Dic3—201no—
484Dic3—201no—
485Dic3—301no—
486Dic3—601no—
487Dic3—601no—
488Dic3—601no—
489Dic3—601no—
490C2×C5—181no—
491C2×C5#6388311yescharacteristic
492D5—181no—
493D5#6393711yescharacteristic
494D5#6395111yescharacteristic
495C32#6572711yescharacteristic, complemented, Sylow
496C23—451no—
497C2×C4—51no—
498C2×C4—91no—
499C2×C4—101no—
500C2×C4—301no—
501C2×C4—301no—
502C2×C4—451no—
503C2×C4—451no—
504C2×C4—451no—
505C2×C4—451no—
506C2×C4—901no—
507C2×C4—1802no—
508C8—31no—
509C8—31no—
510C8—151no—
511C8—151no—
512C2×C3—21no—
513C2×C3—51no—
514C2×C3—51no—
515C2×C3—51no—
516C2×C3—51no—
517C2×C3—101no—
518C2×C3—101no—
519C2×C3D3×C4⋊AGL(1,𝔽5)11yescharacteristic
520C2×C3D3×C4⋊AGL(1,𝔽5)11yescharacteristic
521D3—61no—
522D3—61no—
523D3—61no—
524D3—61no—
525D3—301no—
526D3—301no—
527D3—301no—
528D3—301no—
529C5#1059058311yescharacteristic, complemented, Sylow
530C22—51no—
531C22—91no—
532C22—451no—
533C22—901no—
534C22—901no—
535C4—51no—
536C4—91no—
537C4—201no—
538C4—451no—
539C4—601no—
540C4—601no—
541C4—1801no—
542C4D32×AGL(1,𝔽5)11yescharacteristic
543C3—21no—
544C3#1175165311yescharacteristic, complemented
545C3#1175165411yescharacteristic, complemented
546C2—51no—
547C2—51no—
548C2—181no—
549C2—901no—
550C2#1308840711yescharacteristic
subgroupG/H
centerC2#13088407
commutator, FittingC32×C4×C5C22×C4
Frattini, hypercenterC4D32×AGL(1,𝔽5)
socleC2×C32×C5C2×C4⋊C4
prime radicalC2×D5×C3⋊D3D4
cosocleC32×C4×D5C23
hypocenterC32×C5#428
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
4. #13085455C2semidirect
5. #13085454C2semidirect
6. C3×#91160C2semidirect
7. C3×#91159C2semidirect
19. #10623820C4semidirect
20. #10623797C4semidirect
21. #10623797C4semidirect
22. C5×#3712C4semidirect
38. #91160D3semidirect
40. #91159D3semidirect
53. C3×C3⋊#168C2×C4semidirect
54. C3×C3⋊#168C2×C4semidirect
55. C3×C5×C3⋊C8C2×C4semidirect
56. C3×C5×C3⋊C8C2×C4semidirect
138. #3712AGL(1,𝔽5)semidirect
178. C3⋊#168C4×D3semidirect
179. C3⋊#168C4×D3semidirect
182. C5×C3⋊C8C4×D3semidirect
183. C5×C3⋊C8C4×D3semidirect
233. C3×C3⋊C8C2×AGL(1,𝔽5)semidirect
234. C3×C3⋊C8C2×AGL(1,𝔽5)semidirect
314. C32×C5#428semidirect
411. C3⋊C8D3×AGL(1,𝔽5)semidirect
412. C3⋊C8D3×AGL(1,𝔽5)semidirect
447. C3×C5#4861semidirect
448. C3×C5#4881semidirect
495. C32#65727semidirect
529. C5#10590583semidirect
544. C3#11751653semidirect
545. C3#11751654semidirect

Maximal subgroups

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

subgroupindex
#130887232
C3⋊D3×C4⋊AGL(1,𝔽5)2
D5×#37122
#130854552
#130854542
C3×#911602
C3×#911592
#117516543
#117516533
#105905835