Finite Group Databasebeta

← #33093147

Subgroups — #33093147

Order 18688.

888 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988929C211yesmaximal, characteristic
2#22612459C211yesmaximal, characteristic
3#117×Dic73C211yesmaximal, characteristic
4#22610571C211yesmaximal, characteristic, complemented
5#22610564C211yesmaximal, characteristic, complemented
6#22610508C211yesmaximal, characteristic, complemented
7#22610501C211yesmaximal, characteristic, complemented
8#22610243C211yesmaximal, characteristic
9#22610023C211yesmaximal, characteristic
10#107×Dic73C211yesmaximal, characteristic, complemented
11#106×Dic73C211yesmaximal, characteristic, complemented
12#22609750C211yesmaximal, characteristic, complemented
13#22609750C211yesmaximal, characteristic, complemented
14C22⋊C8×Dic73C211yesmaximal, characteristic
15C73×#1854C211yesmaximal, characteristic
16C2×#14556570C2211yescharacteristic
17C2×#14556556C2211yescharacteristic
18C2×#14556550C2211yescharacteristic
19C22×C4×Dic73C2211yescharacteristic
20#17988929C418yescomplemented
21C2×#14556525C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556516C2211yescharacteristic
24C2×#14556516C2211yescharacteristic
25C2×#14556514C2211yescharacteristic
26C2×#14556514C2211yescharacteristic
27C2×C22⋊#10596831C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30#17988492C2211yescharacteristic
31#17988483C2211yescharacteristic
32#17988469—21no—
33#17988459—21no—
34C4⋊C4×Dic73C2211yescharacteristic
35C4⋊C4×Dic73C2211yescharacteristic
36C4⋊C4×Dic73C2211yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39#17988438—21no—
40C22⋊C4×Dic73—21no—
41C22⋊C4×Dic73—21no—
42C22⋊C4×Dic73—21no—
43#17988433—21no—
44C42×Dic73—21no—
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×#117×C73C2211yescharacteristic
54C2×#107×C73C2211yescharacteristic
55C2×#106×C73C2211yescharacteristic
56C2×C22⋊C8×C73C2211yescharacteristic
57#380×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×Dic146C2311yescharacteristic
61C23×Dic73D411yescharacteristic
62C23×Dic73D411yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556570—22no—
66#14556570C2×C414yes—
67#14556556—22no—
68#14556556—22no—
69#14556550—22no—
70#14556550C2×C414yes—
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C4×Dic73—22no—
80C2×C4×Dic73—22no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73D411yescharacteristic
89C2×C4×Dic73D411yescharacteristic
90C2×C4×Dic73C2311yescharacteristic
91C2×C4×Dic73C2311yescharacteristic
92#14556525C2×C414yescomplemented
93#14556523C2×C414yescomplemented
94#14556516C2×C414yescomplemented
95#14556516C2×C414yescomplemented
96#14556514—22no—
97#14556514—22no—
98C22⋊#10596831—22no—
99C8×Dic73—22no—
100C8×Dic73—22no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109C23×C4×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112#117×C73—22no—
113#117×C73—22no—
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C22⋊C4×C73—21no—
119C2×C22⋊C4×C73—21no—
120C2×C42×C73—21no—
121#107×C73—22no—
122#106×C73—22no—
123C22⋊C8×C73C2×C414yes—
124#95×C73—21no—
125C2×Dic146—21no—
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146C22×C412yes—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—22no—
136C22×Dic73—22no—
137C22×Dic73—22no—
138C22×Dic73—22no—
139C22×Dic73—22no—
140C22×Dic73—41no—
141C22×Dic73C2×D411yescharacteristic
142C22×Dic73C2×D411yescharacteristic
143C22×Dic73C2×D411yescharacteristic
144C22×Dic73C2×D411yescharacteristic
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73C22×C412yes—
155C4⋊Dic73C22×C412yes—
156C4⋊Dic73C22×C412yes—
157#11924007—24no—
158#11924007—42no—
159#11924007—42no—
160C22⋊Dic73—24no—
161C22⋊Dic73—24no—
162C4×Dic73—24no—
163C4×Dic73—21no—
164C4×Dic73—21no—
165C4×Dic73—22no—
166C4×Dic73—22no—
167C4×Dic73—42no—
168C4×Dic73—42no—
169C4×Dic73—42no—
170C4×Dic73—42no—
171C4×Dic73—42no—
172C4×Dic73—42no—
173C4×Dic73#4112yes—
174C4×Dic73#4112yes—
175C24×C73C2×D411yescharacteristic
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73C22×C412yes—
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73#4112yes—
187C22×C4×C73C2411yescharacteristic
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73C22×C412yes—
193C2×C8×C73C22×C412yes—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73#4112yes—
201C4⋊C4×C73#4112yes—
202C22⋊C4×C73—24no—
203C22⋊C4×C73—42no—
204C42×C73—24no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73C4×D412yes—
212C23×C73C22×D411yescharacteristic
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73C4×D412yes—
232C2×C4×C73C23×C411yescharacteristic
233C2×C4×C73C2×#4111yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—22no—
243C2×Dic73—22no—
244C2×Dic73—22no—
245C2×Dic73—22no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—44no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—44no—
259C2×Dic73#12312yes—
260C2×Dic73#12312yes—
261C2×Dic73#13512yes—
262C2×Dic73#13512yes—
263C2×Dic73#13712yes—
264Dic146—22no—
265Dic146—22no—
266Dic146—41no—
267Dic146—41no—
268C8×C73—22no—
269C8×C73—22no—
270#10596831—41no—
271#10596831—41no—
272C22×C73—22no—
273C22×C73—22no—
274C22×C73—22no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73C2×C4×D411yescharacteristic
280C22×C73C2×#12311yescharacteristic
281C22×C73#52511yescharacteristic
282C22×C73C2×#13511yescharacteristic
283C22×C73C2×#13511yescharacteristic
284C22×C73C2×#13711yescharacteristic
285C22×C73#57511yescharacteristic
286C4×C73—22no—
287C4×C73—42no—
288C4×C73—42no—
289C4×C73—81no—
290C4×C73—81no—
291C4×C73C4×#4111yescharacteristic
292C4×C73C4×#4111yescharacteristic
293C4×C73#52911yescharacteristic
294C4×C73#52911yescharacteristic
295Dic73—22no—
296Dic73—22no—
297Dic73—41no—
298Dic73—42no—
299Dic73—42no—
300Dic73—42no—
301Dic73—42no—
302Dic73#48514yescomplemented
303C4×#485—731nomaximal, Sylow
304C2×C73—42no—
305C2×C73C4×#12311yescharacteristic
306C2×C73C4×#13511yescharacteristic
307C2×C73C4×#13711yescharacteristic
308C2×C73C2×#48511yescharacteristic
309C2×C73#309111yescharacteristic
310C2×C73#310211yescharacteristic
311C2×C73#310711yescharacteristic
312C2×#485—731no—
313C4×C22⋊Q8—731no—
314C4×#117—731no—
315#1928—731no—
316#1927—731no—
317#1919—731no—
318#1918—731no—
319#1873—731no—
320#1854D7311yescharacteristic
321#1825—731no—
322C4×#107—731no—
323C4×#106—731no—
324C4×#103—731no—
325C4×#103—731no—
326C4×C22⋊C8—731no—
327C73C4×#48511yescharacteristic, complemented, Sylow
328C2×C22⋊Q8—731no—
329C2×C4×Q8—731no—
330C2×#117—731no—
331C2×#117C2×D7311yescharacteristic
332C22×C42—731no—
333#485—738no—
334C2×#107—731no—
335C2×#107C2×D7311yescharacteristic
336C2×#106—731no—
337C2×#106C2×D7311yescharacteristic
338C2×#103—731no—
339C2×#103—731no—
340C2×#103—731no—
341C2×#103—731no—
342C2×C22⋊C8—731no—
343C2×C22⋊C8C2×D7311yescharacteristic
344C2×C4×C8—731no—
345C2×C4×C8—731no—
346#391—731no—
347#389—731no—
348#387—1461no—
349#385—1461no—
350#380C2×D7311yescharacteristic
351C4×C4⋊C4—731no—
352C4×C4⋊C4—731no—
353C4×C4⋊C4—731no—
354C4×C4⋊C4—731no—
355C4×C4⋊C4—731no—
356C4×C4⋊C4—1461no—
357C4×C22⋊C4—1461no—
358C4×C22⋊C4—1461no—
359C4×C22⋊C4—1461no—
360#376—1461no—
361C43—1461no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340C2×D7311yescharacteristic
369#340C2×D7311yescharacteristic
370#336—731no—
371#336—731no—
372C22×Q8—731no—
373C23×C4—731no—
374C23×C4—731no—
375C23×C4C22×D7311yescharacteristic
376C22×C8—731no—
377C22×C8—731no—
378C22×C8C22×D7311yescharacteristic
379C22×C8C22×D7311yescharacteristic
380C22⋊Q8—734no—
381C22⋊Q8—1462no—
382C4×Q8—1462no—
383C4×Q8—1462no—
384#117—22no—
385#117—22no—
386#117—734no—
387#117—1462no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4C22×D7311yescharacteristic
395C2×C4⋊C4C22×D7311yescharacteristic
396C2×C4⋊C4C22×D7311yescharacteristic
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1461no—
401C2×C22⋊C4—1461no—
402C2×C42—21no—
403C2×C42—731no—
404C2×C42—731no—
405C2×C42—731no—
406C2×C42—731no—
407C2×C42—1461no—
408C2×C42—1461no—
409C2×C42—1461no—
410C2×C42—1461no—
411C2×C42—1461no—
412C2×C42—1461no—
413C2×C42—1462no—
414C2×C42—1462no—
415C2×C42—1461no—
416#107—22no—
417#107—734no—
418#106—22no—
419#106—734no—
420#103—734no—
421#103—734no—
422#103—1462no—
423#103—1462no—
424C22⋊C8—1462no—
425C22⋊C8C4×D7314yes—
426C4×C8—1462no—
427C4×C8—1462no—
428#95—21no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437C24D4×D7311yescharacteristic
438C2×Q8—732no—
439C2×Q8—1461no—
440C2×Q8—1461no—
441C2×Q8—1461no—
442C2×Q8—1461no—
443C2×Q8—1461no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—2921no—
453C22×C4—731no—
454C22×C4—731no—
455C22×C4—731no—
456C22×C4—731no—
457C22×C4—1462no—
458C22×C4—1462no—
459C22×C4—1462no—
460C22×C4—1462no—
461C22×C4—1462no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4C2×C4×D7312yes—
467C22×C4D4×D7311yescharacteristic
468C22×C4#1192402912yes—
469C22×C4C23×D7311yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8C2×C4×D7312yes—
481C2×C8C2×C4×D7312yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—2922no—
489C4⋊C4—2922no—
490C4⋊C4—732no—
491C4⋊C4—732no—
492C4⋊C4—732no—
493C4⋊C4—1464no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4#1192402712yes—
498C4⋊C4#1192402712yes—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—1464no—
502C22⋊C4—1464no—
503C42—24no—
504C42—2922no—
505C42—2922no—
506C42—2922no—
507C42—2922no—
508C42—2922no—
509C42—2922no—
510C42—732no—
511C42—732no—
512C42—1462no—
513C42—1462no—
514C42—1464no—
515C42—1461no—
516C42—1461no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#1455655312yes—
524C23C2×D4×D7311yescharacteristic
525Q8—2921no—
526Q8—2921no—
527Q8—1462no—
528Q8—1462no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2924no—
556C2×C4—2924no—
557C2×C4—732no—
558C2×C4—732no—
559C2×C4—732no—
560C2×C4—732no—
561C2×C4—732no—
562C2×C4—1461no—
563C2×C4—1462no—
564C2×C4—1462no—
565C2×C4—1462no—
566C2×C4—1462no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4#1455655312yes—
574C2×C4C22×C4×D7311yescharacteristic
575C2×C4#41×D7311yescharacteristic
576C2×C4#41×D7311yescharacteristic
577C2×C4C2×#1192402711yescharacteristic
578C2×C4C2×#1192402711yescharacteristic
579C2×C4C2×#1192402911yescharacteristic
580C8—22no—
581C8—22no—
582C8—2921no—
583C8—2921no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×#1455655311yescharacteristic
592C22#123×D7311yescharacteristic
593C22#1798909311yescharacteristic
594C22#135×D7311yescharacteristic
595C22#135×D7311yescharacteristic
596C22#137×D7311yescharacteristic
597C22#1798938311yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—81no—
602C4—81no—
603C4—2922no—
604C4—2922no—
605C4—2922no—
606C4—2921no—
607C4—2922no—
608C4—734no—
609C4—1462no—
610C4—1462no—
611C4C4×#1192402711yescharacteristic
612C4C4×#1192402711yescharacteristic
613C4#1798912311yescharacteristic
614C4#1798912311yescharacteristic
615C2—42no—
616C2#2261245011yescharacteristic
617C2#2261870211yescharacteristic
618C2#2261874911yescharacteristic
619C2#485×D7311yescharacteristic
620C2#2261984311yescharacteristic
621C2#2261994211yescharacteristic
622C2#2261999511yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1854C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#1854D73
hypocenterC73C4×#485
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
4. #22610571C2semidirect
5. #22610564C2semidirect
6. #22610508C2semidirect
7. #22610501C2semidirect
10. #107×Dic73C2semidirect
11. #106×Dic73C2semidirect
12. #22609750C2semidirect
13. #22609750C2semidirect
20. #17988929C4semidirect
92. #14556525C2×C4semidirect
93. #14556523C2×C4semidirect
94. #14556516C2×C4semidirect
95. #14556516C2×C4semidirect
302. Dic73#485semidirect
327. C73C4×#485semidirect

Maximal subgroups

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

subgroupindex
C2×#179889292
#226124592
#117×Dic732
#226105712
#226105642
#226105082
#226105012
#226102432
#226100232
#107×Dic732
#106×Dic732
#226097502
C22⋊C8×Dic732
C73×#18542
C4×#48573