Finite Group Databasebeta

← #33143200

Subgroups — #33143200

Order 18688.

958 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988896C211yesmaximal, characteristic, complemented
2#22613638C211yesmaximal, characteristic, complemented
3#22612772C211yesmaximal, characteristic
4#22611457C211yesmaximal, characteristic, complemented
5#22611317C211yesmaximal, characteristic, complemented
6#22611301C211yesmaximal, characteristic, complemented
7#22611075C211yesmaximal, characteristic, complemented
8#22610187C211yesmaximal, characteristic, complemented
9#22610183C211yesmaximal, characteristic, complemented
10#22610130C211yesmaximal, characteristic, complemented
11#22610109C211yesmaximal, characteristic, complemented
12#22609960C211yesmaximal, characteristic, complemented
13#22609934C211yesmaximal, characteristic, complemented
14#22609871C211yesmaximal, characteristic
15C2×#416×C73C211yesmaximal, characteristic
16C2×#14556582C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×Q8×Dic73C2211yescharacteristic
19#17988896—24no—
20C2×#14556525C2211yescharacteristic
21C2×#14556525C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×#14556520C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556512C2211yescharacteristic
29C2×#14556512C2211yescharacteristic
30C2×C22⋊#10596831C2211yescharacteristic
31#17988523—21no—
32#17988510—21no—
33#17988490—21no—
34#17988490—21no—
35#17988490C2211yescharacteristic
36#17988490C2211yescharacteristic
37#17988486C2211yescharacteristic
38#17988485—21no—
39#17988483C2211yescharacteristic
40#17988471—21no—
41#17988456C2211yescharacteristic
42#17988447—21no—
43#17988447C2211yescharacteristic
44#17988447C2211yescharacteristic
45C4×C22⋊Dic73—21no—
46C4×C22⋊Dic73C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C22×#41×C73C2211yescharacteristic
54C23×C8×C73C2211yescharacteristic
55C2×#117×C73C2211yescharacteristic
56#416×C73—24no—
57C2×#103×C73C2211yescharacteristic
58C2×#103×C73C2211yescharacteristic
59C2×#102×C73C2211yescharacteristic
60C2×#102×C73C2211yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63#14556582—22no—
64#14556582—41no—
65Q8×Dic73—22no—
66Q8×Dic73—22no—
67Q8×Dic73—41no—
68Q8×Dic73—41no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C22⋊Dic73—21no—
85C2×C22⋊Dic73—21no—
86C2×C22⋊Dic73—21no—
87C2×C22⋊Dic73D411yescharacteristic
88C2×C22⋊Dic73D411yescharacteristic
89C2×C4×Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73C2311yescharacteristic
92C2×C4×Dic73C2311yescharacteristic
93#14556525—22no—
94#14556525—41no—
95#14556524—41no—
96#14556522—22no—
97#14556520—22no—
98#14556520—41no—
99#14556515—22no—
100#14556515—22no—
101#14556512—41no—
102#14556512—41no—
103C22⋊#10596831—22no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498D411yescharacteristic
109#14556498D411yescharacteristic
110C2×#41×C73—22no—
111C2×#41×C73—22no—
112C2×#41×C73—22no—
113C2×#41×C73—41no—
114C2×#41×C73—41no—
115C2×#41×C73Q814yes—
116C22×Q8×C73C2311yescharacteristic
117C22×D4×C73—21no—
118C22×D4×C73C2311yescharacteristic
119C23×C4×C73—21no—
120C23×C4×C73C2311yescharacteristic
121C22×C8×C73—22no—
122C22×C8×C73—22no—
123C22×C8×C73—41no—
124C22×C8×C73C2311yescharacteristic
125C22×C8×C73C2311yescharacteristic
126#117×C73—22no—
127#117×C73—41no—
128C2×C4⋊C4×C73C2311yescharacteristic
129C2×C4⋊C4×C73C2311yescharacteristic
130C2×C22⋊C4×C73—21no—
131C2×C42×C73—21no—
132#103×C73—22no—
133#103×C73D414yes—
134#102×C73—22no—
135#102×C73—22no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73C2×D411yescharacteristic
141C22×Dic73C2×D411yescharacteristic
142C22×Dic73C2×D411yescharacteristic
143C22×Dic73C2×D411yescharacteristic
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—42no—
159C4⋊Dic73—42no—
160C4⋊Dic73—42no—
161C4⋊Dic73—42no—
162#11924007—42no—
163#11924007—42no—
164#11924007—81no—
165#11924007—81no—
166C22⋊Dic73—22no—
167C22⋊Dic73—41no—
168C22⋊Dic73—42no—
169C22⋊Dic73—42no—
170C22⋊Dic73—81no—
171C22⋊Dic73—81no—
172C22⋊Dic73—81no—
173C22⋊Dic73—81no—
174C4×Dic73—21no—
175C4×Dic73—21no—
176C4×Dic73—21no—
177C4×Dic73—21no—
178C4×Dic73—42no—
179C4×Dic73—42no—
180C24×C73—21no—
181C24×C73C2×D411yescharacteristic
182#41×C73—28no—
183#41×C73—44no—
184#41×C73—44no—
185#41×C73—44no—
186C2×Q8×C73—21no—
187C2×Q8×C73—21no—
188C2×Q8×C73—21no—
189C2×Q8×C73C2×D412yes—
190C2×Q8×C73C2×Q812yes—
191C2×Q8×C73#4112yes—
192C2×D4×C73—24no—
193C2×D4×C73—21no—
194C2×D4×C73—21no—
195C2×D4×C73—21no—
196C2×D4×C73—21no—
197C2×D4×C73—21no—
198C2×D4×C73—42no—
199C2×D4×C73—42no—
200C2×D4×C73C2×Q812yes—
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73—21no—
204C22×C4×C73—21no—
205C22×C4×C73—21no—
206C22×C4×C73—21no—
207C22×C4×C73—24no—
208C22×C4×C73—21no—
209C22×C4×C73—42no—
210C22×C4×C73—42no—
211C22×C4×C73C2×D411yescharacteristic
212C22×C4×C73C2×Q812yes—
213C22×C4×C73#4112yes—
214C22×C4×C73C2411yescharacteristic
215C2×C8×C73—21no—
216C2×C8×C73—21no—
217C2×C8×C73—21no—
218C2×C8×C73—21no—
219C2×C8×C73—21no—
220C2×C8×C73—44no—
221C2×C8×C73C2×D412yes—
222C4⋊C4×C73—21no—
223C4⋊C4×C73—21no—
224C4⋊C4×C73—21no—
225C4⋊C4×C73C2×D412yes—
226C22⋊C4×C73—42no—
227C42×C73—42no—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—21no—
231C23×C73—21no—
232C23×C73—21no—
233C23×C73—21no—
234C23×C73—22no—
235C23×C73—41no—
236C23×C73—41no—
237C23×C73—41no—
238C23×C73—41no—
239C23×C73—41no—
240C23×C73—41no—
241C23×C73C22⋊Q812yes—
242C23×C73C22×D411yescharacteristic
243Q8×C73—22no—
244Q8×C73—22no—
245Q8×C73—22no—
246Q8×C73C22⋊Q814yes—
247D4×C73—22no—
248D4×C73—22no—
249D4×C73—48no—
250D4×C73—41no—
251D4×C73—41no—
252C2×C4×C73—22no—
253C2×C4×C73—21no—
254C2×C4×C73—22no—
255C2×C4×C73—21no—
256C2×C4×C73—22no—
257C2×C4×C73—22no—
258C2×C4×C73—22no—
259C2×C4×C73—22no—
260C2×C4×C73—22no—
261C2×C4×C73—22no—
262C2×C4×C73—48no—
263C2×C4×C73—41no—
264C2×C4×C73—41no—
265C2×C4×C73—41no—
266C2×C4×C73—41no—
267C2×C4×C73—41no—
268C2×C4×C73C22⋊Q812yes—
269C2×C4×C73C22×D411yescharacteristic
270C2×C4×C73C22×Q811yescharacteristic
271C2×C4×C73C2×#4111yescharacteristic
272C2×C4×C73C2×#4111yescharacteristic
273C2×C4×C73H(5,𝔽2)11yescharacteristic
274C2×C4×C73H(5,𝔽2)11yescharacteristic
275C2×Dic73—21no—
276C2×Dic73—21no—
277C2×Dic73—21no—
278C2×Dic73—21no—
279C2×Dic73—21no—
280C2×Dic73—21no—
281C2×Dic73—21no—
282C2×Dic73—21no—
283C2×Dic73—21no—
284C2×Dic73—21no—
285C2×Dic73—21no—
286C2×Dic73—41no—
287C2×Dic73—41no—
288C2×Dic73—41no—
289C2×Dic73—41no—
290C2×Dic73—41no—
291C2×Dic73—41no—
292C2×Dic73—41no—
293C2×Dic73—41no—
294C2×Dic73—41no—
295C2×Dic73—41no—
296C2×Dic73—41no—
297C2×Dic73—41no—
298C2×Dic73#13512yes—
299C8×C73—22no—
300C8×C73—41no—
301#10596831—41no—
302#10596831—41no—
303C22×C73—22no—
304C22×C73—22no—
305C22×C73—22no—
306C22×C73—41no—
307C22×C73—41no—
308C22×C73—41no—
309C22×C73—41no—
310C22×C73—42no—
311C22×C73—42no—
312C22×C73—42no—
313C22×C73—81no—
314C22×C73—81no—
315C22×C73—81no—
316C22×C73—81no—
317C22×C73C2×C22⋊Q811yescharacteristic
318C22×C73D4211yescharacteristic
319C22×C73#54611yescharacteristic
320C22×C73C2×#13511yescharacteristic
321C22×C73#57511yescharacteristic
322C22×C73#57611yescharacteristic
323C22×C73#57811yescharacteristic
324C4×C73—22no—
325C4×C73—24no—
326C4×C73—42no—
327C4×C73—81no—
328C4×C73C2×C22⋊Q811yescharacteristic
329C4×C73#54311yescharacteristic
330C4×C73#54611yescharacteristic
331C4×C73#55111yescharacteristic
332Dic73—22no—
333Dic73—41no—
334Dic73—41no—
335Dic73—41no—
336Dic73—81no—
337Dic73—81no—
338Dic73—81no—
339Dic73—81no—
340#25961—731nomaximal, Sylow
341C2×C73—42no—
342C2×C73—82no—
343C2×C73#243711yescharacteristic
344C2×C73#307611yescharacteristic
345C2×C73#308311yescharacteristic
346C2×C73#326911yescharacteristic
347C2×C73#328711yescharacteristic
348C2×C73#330011yescharacteristic
349C2×C73#330911yescharacteristic
350C2×#479—731no—
351C2×#416D7311yescharacteristic
352#2441—731no—
353#2341—731no—
354#2093—731no—
355#2067—731no—
356#2065—731no—
357#2025—731no—
358#1863—731no—
359#1863—731no—
360#1854—731no—
361#1850—731no—
362#1808—731no—
363#1803—731no—
364#1788—731no—
365C73#2596111yescharacteristic, complemented, Sylow
366C22×#41C2×D7311yescharacteristic
367C23×C8C2×D7311yescharacteristic
368C2×#125—731no—
369C2×C4×Q8—731no—
370C2×C4×Q8—731no—
371C2×#117C2×D7311yescharacteristic
372#479—1464no—
373C2×#107—731no—
374C2×#107—731no—
375C2×#107—731no—
376C2×#106—731no—
377C2×C4⋊C8—731no—
378C2×C4⋊C8—731no—
379#416—24no—
380C2×#103—731no—
381C2×#103—731no—
382C2×#103C2×D7311yescharacteristic
383C2×#103C2×D7311yescharacteristic
384C2×#102—731no—
385C2×#102—731no—
386C2×#102C2×D7311yescharacteristic
387C2×#102C2×D7311yescharacteristic
388C2×C22⋊C8—731no—
389#399—1461no—
390#396—1461no—
391#390—731no—
392#390—731no—
393#390—1461no—
394#390—1461no—
395#389—731no—
396#389—731no—
397#389—1461no—
398#387—1461no—
399#384—731no—
400#380—731no—
401#380—731no—
402#380—1461no—
403C4×C22⋊C4—731no—
404C4×C22⋊C4—1461no—
405#340—731no—
406#340—731no—
407#340—731no—
408#340—731no—
409#336—731no—
410#336—731no—
411C2×#41—22no—
412C2×#41—22no—
413C2×#41—22no—
414C2×#41—41no—
415C2×#41—41no—
416C2×#41Dic14614yes—
417C22×Q8C22×D7311yescharacteristic
418C22×D4—21no—
419C22×D4C22×D7311yescharacteristic
420C23×C4—21no—
421C23×C4C22×D7311yescharacteristic
422C22×C8—22no—
423C22×C8—22no—
424C22×C8—41no—
425C22×C8—731no—
426C22×C8—731no—
427C22×C8C22×D7311yescharacteristic
428C22×C8C22×D7311yescharacteristic
429#125—2921no—
430#125—1462no—
431C4×Q8—2921no—
432C4×Q8—2921no—
433C4×Q8—1462no—
434C4×Q8—1462no—
435#117—22no—
436#117—41no—
437C2×C4⋊C4—731no—
438C2×C4⋊C4—731no—
439C2×C4⋊C4—731no—
440C2×C4⋊C4—731no—
441C2×C4⋊C4—1461no—
442C2×C4⋊C4—1461no—
443C2×C4⋊C4—1461no—
444C2×C4⋊C4—1461no—
445C2×C4⋊C4—1461no—
446C2×C4⋊C4—1461no—
447C2×C4⋊C4—1461no—
448C2×C4⋊C4—1461no—
449C2×C4⋊C4C22×D7311yescharacteristic
450C2×C4⋊C4C22×D7311yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—731no—
453C2×C22⋊C4—731no—
454C2×C22⋊C4—1461no—
455C2×C22⋊C4—1461no—
456C2×C22⋊C4—1461no—
457C2×C22⋊C4—1461no—
458C2×C22⋊C4—1461no—
459C2×C22⋊C4—1461no—
460C2×C42—21no—
461C2×C42—731no—
462C2×C42—731no—
463C2×C42—1461no—
464C2×C42—1461no—
465#107—2921no—
466#107—2921no—
467#107—1462no—
468#106—1462no—
469C4⋊C8—2921no—
470C4⋊C8—1462no—
471#103—22no—
472#103—1462no—
473#103—1462no—
474#103C22⋊D7314yes—
475#102—22no—
476#102—22no—
477#102—2921no—
478#102—2921no—
479C22⋊C8—1462no—
480#95—731no—
481#95—731no—
482#95—1461no—
483#95—1461no—
484#95—1461no—
485#95—1461no—
486C24—21no—
487C24D4×D7311yescharacteristic
488#41—28no—
489#41—44no—
490#41—44no—
491#41—44no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8C2×C22⋊D7312yes—
496C2×Q8C2×Dic14612yes—
497C2×Q8#1192402812yes—
498C2×D4—24no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4—42no—
505C2×D4—42no—
506C2×D4C2×Dic14612yes—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—24no—
514C22×C4—21no—
515C22×C4—42no—
516C22×C4—42no—
517C22×C4—731no—
518C22×C4—731no—
519C22×C4—731no—
520C22×C4—731no—
521C22×C4—1461no—
522C22×C4—1461no—
523C22×C4—1461no—
524C22×C4—1461no—
525C22×C4D4×D7311yescharacteristic
526C22×C4C2×Dic14612yes—
527C22×C4#1192402712yes—
528C22×C4C23×D7311yescharacteristic
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—44no—
535C2×C8—1461no—
536C2×C8—1461no—
537C2×C8—1461no—
538C2×C8—1461no—
539C2×C8—1461no—
540C2×C8—1461no—
541C2×C8C2×C22⋊D7312yes—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—2922no—
546C4⋊C4—2922no—
547C4⋊C4—2922no—
548C4⋊C4—2922no—
549C4⋊C4—2922no—
550C4⋊C4—2922no—
551C4⋊C4—5841no—
552C4⋊C4—5841no—
553C4⋊C4—1461no—
554C4⋊C4—1461no—
555C4⋊C4—1461no—
556C4⋊C4—1461no—
557C4⋊C4—1461no—
558C4⋊C4—1461no—
559C4⋊C4—1461no—
560C4⋊C4—1461no—
561C4⋊C4C2×C22⋊D7312yes—
562C22⋊C4—42no—
563C22⋊C4—2922no—
564C22⋊C4—2922no—
565C22⋊C4—2921no—
566C22⋊C4—5841no—
567C22⋊C4—5841no—
568C22⋊C4—5841no—
569C22⋊C4—5841no—
570C22⋊C4—1462no—
571C42—42no—
572C42—2922no—
573C42—2922no—
574C42—1461no—
575C42—1461no—
576C42—1461no—
577C42—1461no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—22no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23#1455656812yes—
592C23C2×D4×D7311yescharacteristic
593Q8—22no—
594Q8—22no—
595Q8—22no—
596Q8#1455657014yes—
597D4—22no—
598D4—22no—
599D4—48no—
600D4—41no—
601D4—41no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—48no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—2921no—
619C2×C4—2921no—
620C2×C4—2921no—
621C2×C4—2921no—
622C2×C4—2921no—
623C2×C4—2921no—
624C2×C4—2921no—
625C2×C4—2921no—
626C2×C4—2921no—
627C2×C4—2921no—
628C2×C4—2921no—
629C2×C4—2921no—
630C2×C4—732no—
631C2×C4—1461no—
632C2×C4—1461no—
633C2×C4—1461no—
634C2×C4—1461no—
635C2×C4—1461no—
636C2×C4—1461no—
637C2×C4—1461no—
638C2×C4—1461no—
639C2×C4—1461no—
640C2×C4—1461no—
641C2×C4—1461no—
642C2×C4#1455656812yes—
643C2×C4C22×C22⋊D7311yescharacteristic
644C2×C4C22×Dic14611yescharacteristic
645C2×C4C2×#1192402711yescharacteristic
646C2×C4C2×#1192402811yescharacteristic
647C2×C4#1455663611yescharacteristic
648C2×C4#1455663611yescharacteristic
649C8—22no—
650C8—41no—
651C8—2921no—
652C8—2921no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22C2×#1455656811yescharacteristic
668C22D4×C22⋊D7311yescharacteristic
669C22#1798922711yescharacteristic
670C22#135×D7311yescharacteristic
671C22#1798938411yescharacteristic
672C22#1798938911yescharacteristic
673C22#1798940111yescharacteristic
674C4—22no—
675C4—24no—
676C4—42no—
677C4—81no—
678C4—2921no—
679C4—2921no—
680C4—2921no—
681C4—5841no—
682C4—5841no—
683C4—5841no—
684C4—5841no—
685C4—1462no—
686C4C2×#1455657011yescharacteristic
687C4#1798921311yescharacteristic
688C4#1798923111yescharacteristic
689C4#1798926311yescharacteristic
690C2—42no—
691C2—82no—
692C2#2261359211yescharacteristic
693C2#2261971511yescharacteristic
694C2#2261978011yescharacteristic
695C2#2262148611yescharacteristic
696C2#2262168511yescharacteristic
697C2#2262181911yescharacteristic
698C2#2262193311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#416×C73C2
socleC23×C73C22×D4
prime radicalC22×#41×C73C22
hypercenterC2×#416D73
hypocenterC73#25961
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
1. C2×#17988896C2semidirect
2. #22613638C2semidirect
4. #22611457C2semidirect
5. #22611317C2semidirect
6. #22611301C2semidirect
7. #22611075C2semidirect
8. #22610187C2semidirect
9. #22610183C2semidirect
10. #22610130C2semidirect
11. #22610109C2semidirect
12. #22609960C2semidirect
13. #22609934C2semidirect
365. C73#25961semidirect

Maximal subgroups

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

subgroupindex
C2×#179888962
#226136382
#226127722
#226114572
#226113172
#226113012
#226110752
#226101872
#226101832
#226101302
#226101092
#226099602
#226099342
#226098712
C2×#416×C732
#2596173