Finite Group Databasebeta

← #14411577

Subgroups — #14411577

Order 1920.

1112 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91927C212yesmaximal
2C4×#6782C211yesmaximal, characteristic
3C4×#6762C211yesmaximal, characteristic, complemented
4C2×C3×#4002C211yesmaximal, characteristic, complemented
5#11751988C212yesmaximal, complemented
6#11751907C211yesmaximal, characteristic, complemented
7#11751874C212yesmaximal, complemented
8#11751873C211yesmaximal, characteristic, complemented
9#11751870C212yesmaximal, complemented
10D3×#4002C211yesmaximal, characteristic, complemented
11#11751756C211yesmaximal, characteristic, complemented
12#10617136—31nomaximal, Hall
13C2×#6782C2211yescharacteristic
14C2×#6762C2212yes—
15C22×C3×#170C2212yes—
16C2×#6740C2212yes—
17C2×#6739C2212yes—
18C2×#6738C2212yes—
19#91927C4116yescomplemented
20C2×D3×#170C2212yes—
21C4×#1220C411yescharacteristic
22C4×C5×#67C411yescharacteristic
23C2×C3×C4×Dic5C2211yescharacteristic
24#91552C411yescharacteristic
25Dic5×#67C411yescharacteristic
26#91528C2211yescharacteristic
27C4×#1204C411yescharacteristic
28C4×#1203C411yescharacteristic
29#91405C411yescharacteristic
30#91402C2211yescharacteristic
31C4×#1200C2211yescharacteristic
32#91398C2211yescharacteristic
33C4×D3×Dic5C2211yescharacteristic
34#91323C411yescharacteristic
35#91231C2211yescharacteristic
36#91231C2211yescharacteristic
37C4×C3⋊#170C2212yes—
38C3×#4002C2211yescharacteristic
39C3×#4002C2211yescharacteristic
40C3×#4002C2212yes—
41#91169C2212yes—
42#91169C2212yes—
43Dic3×#170C2212yes—
44Dic3×#170C2212yes—
45#78486—51nomaximal, Hall
46C2×#4131—32no—
47C4×#711—31no—
48C4×#704—31no—
49C4×#704—31no—
50C2×#4002—31no—
51C2×#4002D311yescharacteristic, complemented
52#65701—32no—
53#65691—32no—
54#65691—32no—
55C4×#700—31no—
56#65676—31no—
57C22×C3×C4×C5C2×C411yescharacteristic
58C2×#1220C2×C411yescharacteristic
59C2×C4×D15C2×C411yescharacteristic
60C2×C5×#67C2×C411yescharacteristic
61C2×C4×C5×D3C2×C411yescharacteristic
62C22×C3×Dic5C2×C411yescharacteristic
63C22×C3×Dic5C2311yescharacteristic
64C2×#1204C2×C411yescharacteristic
65C2×#1203C2×C411yescharacteristic
66C2×#1200C2×C411yescharacteristic
67C2×#1200C2311yescharacteristic
68#6782C2×C418yes—
69C2×D3×Dic5C2×C411yescharacteristic
70C2×D3×Dic5C2311yescharacteristic
71#6762C2×C418yescomplemented
72C2×C3⋊#170C2312yes—
73C2×C3⋊#170C2312yes—
74C2×C3×#170C2×C418yescomplemented
75C2×C3×#170C2312yes—
76C2×C3×#170C2312yes—
77#6740C2×C418yescomplemented
78#6739C2×C418yescomplemented
79#6738C2×C418yescomplemented
80D3×#170C2×C418yescomplemented
81#6719C2×C411yescharacteristic
82#6714C2×C411yescharacteristic
83#6712C2×C411yescharacteristic
84C4×Dic15C2×C411yescharacteristic
85C5×#217C2×C411yescharacteristic
86C5×(C2×D3)⋊C4C2×C411yescharacteristic
87C5×Dic3⋊C4C2×C411yescharacteristic
88C4×C5×Dic3C2×C411yescharacteristic
89C3×C4×Dic5C2×C411yescharacteristic
90C3×C4×Dic5C2×C411yescharacteristic
91C3×C4×Dic5C2311yescharacteristic
92C3×C4×Dic5C2311yescharacteristic
93Dic3⋊Dic5C2×C411yescharacteristic
94#6669C2×C411yescharacteristic
95#6668C2×C411yescharacteristic
96#6668C2×C411yescharacteristic
97#6667C2×C411yescharacteristic
98(C2×D3)⋊Dic5C2×C411yescharacteristic
99Dic3×Dic5C2×C411yescharacteristic
100Dic3×Dic5C2×C411yescharacteristic
101Dic3×Dic5C2311yescharacteristic
102Dic3×Dic5C2311yescharacteristic
103C2×#901—52no—
104C4×#235—51no—
105C2×C3×#97—51no—
106#5101—52no—
107C4×#208—51no—
108#4708—52no—
109#4708—52no—
110#4707—51no—
111#4707—51no—
112D3×#97—51no—
113D3×#97—51no—
114C2×#711—31no—
115C2×#704—32no—
116C2×#704—32no—
117C22×#170—32no—
118C22×#170C2×D312yes—
119#4131—316no—
120C2×#700—32no—
121C2×D5⋊C8—32no—
122C4×C5×D4—31no—
123D4×Dic5—31no—
124C4×C22⋊D5—31no—
125C4×C22⋊D5—31no—
126C2×C4×Dic5—31no—
127C2×C4×Dic5C2×D311yescharacteristic
128#4038—31no—
129#4033—31no—
130#4027—31no—
131#4027—31no—
132#4023—31no—
133#4023—31no—
134C42×D5—31no—
135C4×Dic10—31no—
136#4002—32no—
137#4002—32no—
138#4002—31no—
139#4002—31no—
140#4002C2×D311yescharacteristic
141#4002C2×D311yescharacteristic
142#4002C2×D312yes—
143C4×#170—32no—
144C4×#170—32no—
145C4×#170—32no—
146#2875—151noSylow
147C23×C3×C5C22×C411yescharacteristic
148C22×D15C22×C411yescharacteristic
149C22×C5×D3C22×C411yescharacteristic
150C2×C3×C4×C5—41no—
151C2×C3×C4×C5C22×C411yescharacteristic
152C2×C3×C4×C5C22×C411yescharacteristic
153#1220C4214yes—
154C2×Dic15C22×C411yescharacteristic
155C2×Dic15C22×C411yescharacteristic
156C2×Dic15C22×C411yescharacteristic
157C4×D15—41no—
158C5×#67C4214yes—
159C2×C5×Dic3C22×C411yescharacteristic
160C2×C5×Dic3C22×C411yescharacteristic
161C2×C5×Dic3C22×C411yescharacteristic
162C4×C5×D3—41no—
163C2×C3×Dic5—41no—
164C2×C3×Dic5C22×C411yescharacteristic
165C2×C3×Dic5C22×C411yescharacteristic
166C2×C3×Dic5C22×C411yescharacteristic
167C2×C3×Dic5C22×C414yes—
168C2×C3×Dic5C2411yescharacteristic
169#1204C4214yes—
170#1203C4214yes—
171#1200—41no—
172#1200C22×C414yes—
173D3×Dic5—41no—
174D3×Dic5C22×C414yes—
175C3⋊#170C22×C414yes—
176C3⋊#170C22×C414yes—
177C3×#170C22×C414yes—
178C3×#170C22×C414yes—
179C2×#235—51no—
180C22×C3×C8—52no—
181C2×C3×C42—51no—
182C4×#67—51no—
183C4×#67—51no—
184C4×#67—51no—
185C4×#67AGL(1,𝔽5)11yescharacteristic
186#926—51no—
187C2×#208—52no—
188#901—516no—
189C2×ΓL(1,𝔽25)—52no—
190C2×ΓL(1,𝔽25)—52no—
191C2×D3×C8—52no—
192C2×D3×C8—52no—
193C4×D12—51no—
194C4×D12—51no—
195#872—51no—
196#872—51no—
197C42×D3—51no—
198C42×D3—51no—
199C4×Dic6—51no—
200C4×Dic6—51no—
201C3×#97—51no—
202C3×#97—51no—
203C3×#97—52no—
204#815—52no—
205#815—52no—
206C8×Dic3—52no—
207C8×Dic3—52no—
208#803—51no—
209#803—51no—
210C4×C3⋊C8—52no—
211C2×C5×D4—31no—
212C22×C4×C5—31no—
213C22×C4×C5C4×D311yescharacteristic
214C2×C22⋊D5—31no—
215C2×C22⋊D5—31no—
216C22×Dic5—31no—
217C22×Dic5—31no—
218C22×Dic5C4×D311yescharacteristic
219C22×Dic5C22×D311yescharacteristic
220#711—38no—
221C2×C4×D5—31no—
222C2×C4×D5—31no—
223C2×C4×D5—31no—
224C2×Dic10—31no—
225#704—38no—
226#704—38no—
227C2×#170—32no—
228C2×#170—32no—
229C2×#170—38no—
230C2×#170C4×D318yescomplemented
231C2×#170C22×D312yes—
232C2×#170C22×D312yes—
233#700—38no—
234D5⋊C8—38no—
235C5×C4⋊C4—31no—
236C5×C22⋊C4—31no—
237C5×C22⋊C4—31no—
238C42×C5—31no—
239C22⋊Dic5—31no—
240C22⋊Dic5—31no—
241#685—31no—
242#685—31no—
243C4⋊Dic5—31no—
244#683—31no—
245#683—31no—
246#683—31no—
247#683—31no—
248C4×Dic5—31no—
249C4×Dic5—31no—
250C4×Dic5—31no—
251C4×Dic5—31no—
252C4×Dic5—31no—
253C4×Dic5C4×D311yescharacteristic
254C4×Dic5C4×D311yescharacteristic
255C4×Dic5C22×D311yescharacteristic
256C4×Dic5C22×D311yescharacteristic
257C2×#131—152no—
258C4×#41—151no—
259#405—152no—
260#405—152no—
261#405—152no—
262C4×M16—151no—
263C4×M16—151no—
264C4×M16—151no—
265C2×#97—51no—
266C2×#97—151no—
267C2×#97—151no—
268C22×C3×C5—21no—
269C22×C3×C5—21no—
270C22×C3×C5C2×C4212yes—
271C22×C3×C5C23×C411yescharacteristic
272C2×D15—21no—
273C2×D15—21no—
274C2×D15C2×C4212yes—
275C2×C5×D3—21no—
276C2×C5×D3—21no—
277C2×C5×D3C2×C4212yes—
278C3×C4×C5—21no—
279C3×C4×C5—21no—
280Dic15—21no—
281Dic15—21no—
282Dic15C2×C4212yes—
283C3×Dic5—21no—
284C3×Dic5—21no—
285C3×Dic5C2×C4212yes—
286C3×Dic5C23×C412yes—
287C5×Dic3—21no—
288C5×Dic3—21no—
289C5×Dic3C2×C4212yes—
290C22×C3×C4—51no—
291C22×C3×C4—51no—
292C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
293C2×#67—51no—
294C2×#67C2×AGL(1,𝔽5)11yescharacteristic
295#235—58no—
296C2×D12—51no—
297C2×C4×D3—51no—
298C2×C4×D3—51no—
299C2×C4×D3—51no—
300C2×C4×D3—51no—
301C2×C4×D3—51no—
302C2×C4×D3C2×AGL(1,𝔽5)11yescharacteristic
303C2×Dic6—51no—
304C2×C3×C8—52no—
305C2×C3×C8—52no—
306C2×C3×C8—58no—
307C3×C42—51no—
308C3×C42—51no—
309C3×C42—51no—
310C3×C42—51no—
311#217—51no—
312#217C2×AGL(1,𝔽5)11yescharacteristic
313(C2×D3)⋊C4—51no—
314(C2×D3)⋊C4—51no—
315(C2×D3)⋊C4—51no—
316(C2×D3)⋊C4C2×AGL(1,𝔽5)11yescharacteristic
317C4⋊Dic3—51no—
318C4⋊Dic3—51no—
319Dic3⋊C4—51no—
320Dic3⋊C4—51no—
321Dic3⋊C4—51no—
322Dic3⋊C4C2×AGL(1,𝔽5)11yescharacteristic
323C4×Dic3—51no—
324C4×Dic3—51no—
325C4×Dic3—51no—
326C4×Dic3—51no—
327C4×Dic3—51no—
328C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
329#208—58no—
330C2×C3⋊C8—52no—
331C2×C3⋊C8—52no—
332ΓL(1,𝔽25)—58no—
333ΓL(1,𝔽25)—58no—
334D3×C8—58no—
335D3×C8—58no—
336C23×C5—31no—
337C23×C5C2×C4×D311yescharacteristic
338C22×D5—31no—
339C5×D4—34no—
340C2×C4×C5—31no—
341C2×C4×C5—31no—
342C2×C4×C5—31no—
343C2×C4×C5—41no—
344C2×C4×C5—121no—
345C2×C4×C5C2×C4×D311yescharacteristic
346C2×C4×C5C2×C4×D311yescharacteristic
347C22⋊D5—34no—
348C22⋊D5—34no—
349C2×Dic5—31no—
350C2×Dic5—34no—
351C2×Dic5—31no—
352C2×Dic5—31no—
353C2×Dic5—41no—
354C2×Dic5—121no—
355C2×Dic5C2×C4×D311yescharacteristic
356C2×Dic5C2×C4×D311yescharacteristic
357C2×Dic5C2×C4×D311yescharacteristic
358C2×Dic5C2×C4×D314yes—
359C2×Dic5C23×D311yescharacteristic
360C4×D5—34no—
361C4×D5—121no—
362C4×D5—121no—
363Dic10—34no—
364#170—34no—
365#170—34no—
366#170C2×C4×D314yes—
367#170C2×C4×D314yes—
368C2×#41—151no—
369#131—1516no—
370C2×M16—152no—
371C2×M16—152no—
372C2×M16—152no—
373C22×C8—52no—
374C22×C8—152no—
375C22×C8—152no—
376C4×Q8—151no—
377C4×Q8—151no—
378C4×D4—31no—
379C4×D4—151no—
380C4×D4—151no—
381C4×D4—151no—
382C4×D4—151no—
383C4×D4—151no—
384#117—151no—
385#117—151no—
386#117—151no—
387C2×C42—51no—
388C2×C42—151no—
389C2×C42—151no—
390#97—51no—
391#97—51no—
392#97—52no—
393#97—151no—
394#97—151no—
395#97—152no—
396#97—152no—
397C4×C8—152no—
398C4×C8—152no—
399C4×C8—152no—
400C2×C3×C5—22no—
401C2×C3×C5C22×C4211yescharacteristic
402C2×C3×C5#56811yescharacteristic
403C2×C3×C5#56811yescharacteristic
404D15—22no—
405C5×D3—22no—
406C23×C3C22×AGL(1,𝔽5)11yescharacteristic
407C22×D3—51no—
408C22×D3C22×AGL(1,𝔽5)11yescharacteristic
409C2×C3×C4—41no—
410C2×C3×C4—54no—
411C2×C3×C4—51no—
412C2×C3×C4—51no—
413C2×C3×C4—51no—
414C2×C3×C4—51no—
415C2×C3×C4—201no—
416C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
417C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
418#67—54no—
419#67C4×AGL(1,𝔽5)14yes—
420C2×Dic3—51no—
421C2×Dic3—51no—
422C2×Dic3—51no—
423C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
424C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
425C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
426D12—54no—
427C4×D3—41no—
428C4×D3—54no—
429C4×D3—54no—
430C4×D3—201no—
431C4×D3—201no—
432C4×D3—201no—
433Dic6—54no—
434C3×C8—54no—
435C3×C8—54no—
436C3⋊C8—54no—
437C3⋊C8—54no—
438C22×C5—21no—
439C22×C5—21no—
440C22×C5—32no—
441C22×C5—61no—
442C22×C5—61no—
443C22×C5C42×D312yes—
444C22×C5C22×C4×D311yescharacteristic
445C2×D5—32no—
446C2×D5—61no—
447C2×D5—61no—
448C4×C5—21no—
449C4×C5—21no—
450C4×C5—32no—
451C4×C5—61no—
452C4×C5—61no—
453Dic5—21no—
454Dic5—21no—
455Dic5—32no—
456Dic5—61no—
457Dic5—61no—
458Dic5C42×D312yes—
459Dic5C22×C4×D312yes—
460#41—158no—
461C2×Q8—151no—
462C2×D4—31no—
463C2×D4—151no—
464C2×D4—151no—
465C22×C4—31no—
466C22×C4—51no—
467C22×C4—51no—
468C22×C4—151no—
469C22×C4—151no—
470C22×C4—151no—
471C22×C4—151no—
472C22×C4—151no—
473C22×C4D3×AGL(1,𝔽5)11yescharacteristic
474M16—158no—
475M16—158no—
476M16—158no—
477C2×C8—52no—
478C2×C8—52no—
479C2×C8—58no—
480C2×C8—158no—
481C2×C8—158no—
482C2×C8—152no—
483C2×C8—152no—
484C4⋊C4—31no—
485C4⋊C4—151no—
486C4⋊C4—151no—
487C4⋊C4—151no—
488C4⋊C4—151no—
489C4⋊C4—151no—
490C22⋊C4—31no—
491C22⋊C4—31no—
492C22⋊C4—151no—
493C22⋊C4—151no—
494C22⋊C4—151no—
495C22⋊C4—151no—
496C42—31no—
497C42—51no—
498C42—51no—
499C42—51no—
500C42—51no—
501C42—151no—
502C42—151no—
503C42—151no—
504C42—151no—
505C42—151no—
506C3×C5#287511yescharacteristic, complemented, Hall
507C22×C3—21no—
508C22×C3—21no—
509C22×C3C2×C4×AGL(1,𝔽5)12yes—
510C22×C3C23×AGL(1,𝔽5)11yescharacteristic
511C2×D3—21no—
512C2×D3—21no—
513C2×D3—52no—
514C2×D3—101no—
515C2×D3—101no—
516C2×D3C2×C4×AGL(1,𝔽5)12yes—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—52no—
520C3×C4—52no—
521C3×C4—101no—
522C3×C4—101no—
523Dic3—21no—
524Dic3—21no—
525Dic3—52no—
526Dic3—101no—
527Dic3—101no—
528Dic3C2×C4×AGL(1,𝔽5)12yes—
529C2×C5—22no—
530C2×C5—62no—
531C2×C5C2×C42×D311yescharacteristic
532C2×C5#574511yescharacteristic
533C2×C5#574511yescharacteristic
534D5—62no—
535C23—31no—
536C23—151no—
537C23C2×D3×AGL(1,𝔽5)11yescharacteristic
538Q8—154no—
539D4—34no—
540D4—154no—
541D4—154no—
542C2×C4—31no—
543C2×C4—31no—
544C2×C4—31no—
545C2×C4—41no—
546C2×C4—54no—
547C2×C4—51no—
548C2×C4—51no—
549C2×C4—51no—
550C2×C4—51no—
551C2×C4—121no—
552C2×C4—154no—
553C2×C4—154no—
554C2×C4—151no—
555C2×C4—151no—
556C2×C4—151no—
557C2×C4—201no—
558C2×C4—601no—
559C2×C4—601no—
560C2×C4—601no—
561C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
562C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
563C8—54no—
564C8—54no—
565C8—154no—
566C8—154no—
567C2×C3—22no—
568C2×C3C22×C4×AGL(1,𝔽5)11yescharacteristic
569C2×C3#6625611yescharacteristic
570C2×C3#6626411yescharacteristic
571D3—22no—
572D3—102no—
573C5#7848611yescharacteristic, complemented, Sylow
574C22—21no—
575C22—21no—
576C22—32no—
577C22—61no—
578C22—61no—
579C22—152no—
580C22—301no—
581C22—301no—
582C22C4×D3×AGL(1,𝔽5)12yes—
583C22C22×D3×AGL(1,𝔽5)11yescharacteristic
584C4—21no—
585C4—21no—
586C4—32no—
587C4—52no—
588C4—52no—
589C4—61no—
590C4—61no—
591C4—101no—
592C4—101no—
593C4—152no—
594C4—301no—
595C4—301no—
596C3#1061713611yescharacteristic, complemented, Sylow
597C2—22no—
598C2—62no—
599C2—302no—
600C2#1175682111yescharacteristic
601C2C2×C4×D3×AGL(1,𝔽5)11yescharacteristic
602C2#1175686311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutatorC2×C3×C5C22×C42
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C23×C4
prime radicalC2×#6782C22
cosocleC2×C3×Dic5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2875
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C4×#6762C2semidirect
4. C2×C3×#4002C2semidirect
5. #11751988C2semidirect
6. #11751907C2semidirect
7. #11751874C2semidirect
8. #11751873C2semidirect
9. #11751870C2semidirect
10. D3×#4002C2semidirect
11. #11751756C2semidirect
19. #91927C4semidirect
51. C2×#4002D3semidirect
71. #6762C2×C4semidirect
74. C2×C3×#170C2×C4semidirect
77. #6740C2×C4semidirect
78. #6739C2×C4semidirect
79. #6738C2×C4semidirect
80. D3×#170C2×C4semidirect
230. C2×#170C4×D3semidirect
506. C3×C5#2875semidirect
573. C5#78486semidirect
596. C3#10617136semidirect

Maximal subgroups

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

subgroupindex
C2×#919272
C4×#67822
C4×#67622
C2×C3×#40022
#117519882
#117519072
#117518742
#117518732
#117518702
D3×#40022
#117517562
#106171363
#784865