Finite Group Databasebeta

← #33007909

Subgroups — #33007909

Order 18688.

1918 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988450C211yesmaximal, characteristic
2C2×#17988445C211yesmaximal, characteristic
3C2×C22⋊C4×Dic73C211yesmaximal, characteristic
4C22×#14556498C211yesmaximal, characteristic
5#22608192C212yesmaximal, complemented
6#22608186C212yesmaximal, complemented
7#22608162C212yesmaximal, complemented
8#22608155C212yesmaximal, complemented
9#22608153C211yesmaximal, characteristic
10#22608153C211yesmaximal, characteristic
11C2×#381×C73C211yesmaximal, characteristic
12C24×Dic73C2211yescharacteristic
13C24×Dic73C2211yescharacteristic
14C22×#11924007C2212yes—
15C22×C22⋊Dic73C2211yescharacteristic
16C22×C22⋊Dic73C2211yescharacteristic
17C22×C4×Dic73C2212yes—
18#17988450—24no—
19#17988445—24no—
20C22⋊C4×Dic73—24no—
21C2×#14556498C418yescomplemented
22C2×#14556498C2212yes—
23C2×#14556498C2212yes—
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2211yescharacteristic
31C2×#14556498C2211yescharacteristic
32C2×#14556498C2211yescharacteristic
33C2×#14556498C2211yescharacteristic
34C24×C4×C73C2211yescharacteristic
35C22×C22⋊C4×C73C2211yescharacteristic
36C22×C22⋊C4×C73C2211yescharacteristic
37#381×C73—24no—
38C2×#95×C73C2212yes—
39C2×#95×C73C2212yes—
40C23×Dic73—24no—
41C23×Dic73—24no—
42C23×Dic73—22no—
43C23×Dic73—22no—
44C23×Dic73—22no—
45C23×Dic73—22no—
46C23×Dic73—42no—
47C23×Dic73—42no—
48C23×Dic73C2×C414yes—
49C23×Dic73C2×C414yes—
50C23×Dic73C2312yes—
51C23×Dic73C2311yescharacteristic
52C23×Dic73C2311yescharacteristic
53C23×Dic73C2312yes—
54C23×Dic73C2311yescharacteristic
55C23×Dic73C2311yescharacteristic
56C2×#11924007—24no—
57C2×#11924007—24no—
58C2×#11924007—42no—
59C2×C22⋊Dic73—22no—
60C2×C22⋊Dic73—22no—
61C2×C22⋊Dic73—22no—
62C2×C22⋊Dic73—22no—
63C2×C22⋊Dic73—22no—
64C2×C22⋊Dic73D414yes—
65C2×C4×Dic73—24no—
66C2×C4×Dic73—24no—
67C2×C4×Dic73—42no—
68#14556498—24no—
69#14556498—24no—
70#14556498—24no—
71#14556498—24no—
72#14556498—24no—
73#14556498—24no—
74#14556498—42no—
75#14556498—42no—
76#14556498C2×C414yes—
77#14556498C2×C414yes—
78#14556498C2×C414yes—
79#14556498C2×C414yes—
80C25×C73C2311yescharacteristic
81C23×C4×C73—22no—
82C23×C4×C73—22no—
83C23×C4×C73—22no—
84C23×C4×C73—22no—
85C23×C4×C73—22no—
86C23×C4×C73—22no—
87C23×C4×C73C2×C414yes—
88C23×C4×C73C2312yes—
89C23×C4×C73C2311yescharacteristic
90C23×C4×C73C2311yescharacteristic
91C23×C4×C73C2312yes—
92C2×C22⋊C4×C73—22no—
93C2×C22⋊C4×C73—22no—
94C2×C22⋊C4×C73—22no—
95C2×C22⋊C4×C73—22no—
96C2×C22⋊C4×C73—22no—
97C2×C22⋊C4×C73D414yes—
98#95×C73—24no—
99#95×C73—24no—
100C22×Dic73—21no—
101C22×Dic73—21no—
102C22×Dic73—21no—
103C22×Dic73—21no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—28no—
107C22×Dic73—28no—
108C22×Dic73—21no—
109C22×Dic73—21no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—21no—
117C22×Dic73—21no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—28no—
123C22×Dic73—28no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73—22no—
128C22×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73—22no—
131C22×Dic73—44no—
132C22×Dic73—44no—
133C22×Dic73—48no—
134C22×Dic73—48no—
135C22×Dic73—48no—
136C22×Dic73—48no—
137C22×Dic73C22×C412yes—
138C22×Dic73C22×C412yes—
139C22×Dic73C22×C412yes—
140C22×Dic73C22×C412yes—
141C22×Dic73C2×D414yes—
142C22×Dic73#4114yes—
143C22×Dic73#4114yes—
144#11924007—48no—
145C22⋊Dic73—28no—
146C22⋊Dic73—44no—
147C4×Dic73—48no—
148C24×C73—22no—
149C24×C73—22no—
150C24×C73—22no—
151C24×C73—22no—
152C24×C73C22×C412yes—
153C24×C73C2×D412yes—
154C24×C73C2×D412yes—
155C24×C73#4112yes—
156C24×C73#4112yes—
157C24×C73#4112yes—
158C24×C73#4112yes—
159C24×C73C2411yescharacteristic
160C22×C4×C73—28no—
161C22×C4×C73—28no—
162C22×C4×C73—28no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—22no—
178C22×C4×C73—22no—
179C22×C4×C73—22no—
180C22×C4×C73—22no—
181C22×C4×C73—22no—
182C22×C4×C73—22no—
183C22×C4×C73—22no—
184C22×C4×C73—22no—
185C22×C4×C73—22no—
186C22×C4×C73—22no—
187C22×C4×C73—22no—
188C22×C4×C73—44no—
189C22×C4×C73—44no—
190C22×C4×C73—44no—
191C22×C4×C73—44no—
192C22×C4×C73C22×C412yes—
193C22×C4×C73C22×C412yes—
194C22×C4×C73C2×D414yes—
195C22⋊C4×C73—28no—
196C22⋊C4×C73—44no—
197C23×C73—22no—
198C23×C73—22no—
199C23×C73—22no—
200C23×C73—22no—
201C23×C73—24no—
202C23×C73—24no—
203C23×C73—24no—
204C23×C73—24no—
205C23×C73—24no—
206C23×C73—24no—
207C23×C73—24no—
208C23×C73—24no—
209C23×C73—24no—
210C23×C73—24no—
211C23×C73—24no—
212C23×C73—24no—
213C23×C73#11714yes—
214C23×C73C4×D414yes—
215C23×C73C4×D414yes—
216C23×C73#12114yes—
217C23×C73#12314yes—
218C23×C73#12414yes—
219C23×C73#12614yes—
220C23×C73C23×C411yescharacteristic
221C23×C73C22×D411yescharacteristic
222C23×C73C22×D411yescharacteristic
223C23×C73C2×#4112yes—
224C23×C73C2×#4112yes—
225C23×C73C2×#4111yescharacteristic
226C23×C73C2×#4111yescharacteristic
227C23×C73C2×#4111yescharacteristic
228C23×C73C2×#4111yescharacteristic
229C23×C73H(5,𝔽2)12yes—
230C23×C732−1+412yes—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—22no—
234C2×C4×C73—22no—
235C2×C4×C73—22no—
236C2×C4×C73—22no—
237C2×C4×C73—24no—
238C2×C4×C73—24no—
239C2×C4×C73—24no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—42no—
249C2×C4×C73—42no—
250C2×C4×C73—42no—
251C2×C4×C73—42no—
252C2×C4×C73—416no—
253C2×C4×C73—42no—
254C2×C4×C73—42no—
255C2×C4×C73—42no—
256C2×C4×C73—42no—
257C2×C4×C73—42no—
258C2×C4×C73—42no—
259C2×C4×C73—42no—
260C2×Dic73—22no—
261C2×Dic73—22no—
262C2×Dic73—22no—
263C2×Dic73—22no—
264C2×Dic73—24no—
265C2×Dic73—24no—
266C2×Dic73—24no—
267C2×Dic73—24no—
268C2×Dic73—22no—
269C2×Dic73—22no—
270C2×Dic73—22no—
271C2×Dic73—22no—
272C2×Dic73—22no—
273C2×Dic73—22no—
274C2×Dic73—24no—
275C2×Dic73—24no—
276C2×Dic73—24no—
277C2×Dic73—416no—
278C2×Dic73—416no—
279C2×Dic73—42no—
280C2×Dic73—42no—
281C2×Dic73—42no—
282C2×Dic73—42no—
283C2×Dic73—42no—
284C2×Dic73—42no—
285C2×Dic73—42no—
286C2×Dic73—42no—
287C2×Dic73—42no—
288C2×Dic73—42no—
289C2×Dic73—42no—
290C2×Dic73—42no—
291C2×Dic73—42no—
292C2×Dic73—42no—
293C2×Dic73—42no—
294C2×Dic73#11714yes—
295C2×Dic73#11714yes—
296C22×C73—24no—
297C22×C73—24no—
298C22×C73—24no—
299C22×C73—24no—
300C22×C73—24no—
301C22×C73—28no—
302C22×C73—28no—
303C22×C73—28no—
304C22×C73—28no—
305C22×C73—24no—
306C22×C73#38818yes—
307C22×C73C2×#11711yescharacteristic
308C22×C73C2×#11711yescharacteristic
309C22×C73C2×#11711yescharacteristic
310C22×C73C2×C4×D411yescharacteristic
311C22×C73C2×C4×D411yescharacteristic
312C22×C73#51811yescharacteristic
313C22×C73#51911yescharacteristic
314C22×C73C2×#12111yescharacteristic
315C22×C73C2×#12311yescharacteristic
316C22×C73C2×#12411yescharacteristic
317C22×C73C2×#12611yescharacteristic
318C22×C73#52912yes—
319C22×C73#52912yes—
320C22×C73#53912yes—
321C22×C73#53912yes—
322C22×C73#54612yes—
323C22×C73#54612yes—
324C22×C73#54712yes—
325C22×C73#54712yes—
326C22×C73#55112yes—
327C22×C73#55112yes—
328C22×C73#55212yes—
329C22×C73#55212yes—
330C4×C73—44no—
331C4×C73—42no—
332C4×C73—42no—
333C4×C73—82no—
334Dic73—24no—
335Dic73—24no—
336Dic73—42no—
337Dic73—42no—
338Dic73—44no—
339Dic73—82no—
340#13874—731nomaximal, Sylow
341C2×C73—28no—
342C2×C73#228311yescharacteristic
343C2×C73C2×#38811yescharacteristic
344C2×C73#231311yescharacteristic
345C2×C73#234211yescharacteristic
346C2×C73#234211yescharacteristic
347C2×C73#234411yescharacteristic
348C2×C73#234411yescharacteristic
349C2×C73#241912yes—
350C2×C73#242512yes—
351C2×C73#248812yes—
352C2×C73#249312yes—
353C2×#381—731no—
354C2×#381D7311yescharacteristic
355C2×#380—731no—
356C2×C4×C22⋊C4—731no—
357C22×#95—731no—
358#1443—732no—
359#1442—732no—
360#1436—732no—
361#1434—732no—
362C22⋊#95—731no—
363C22⋊#95—731no—
364C73#1387411yescharacteristic, complemented, Sylow
365C24×C4—731no—
366C24×C4—731no—
367C24×C4C2×D7311yescharacteristic
368C22×C4⋊C4—732no—
369C22×C22⋊C4—731no—
370C22×C22⋊C4—731no—
371C22×C22⋊C4C2×D7311yescharacteristic
372C22×C22⋊C4C2×D7311yescharacteristic
373C22×C42—732no—
374#381—24no—
375#381—1464no—
376#380—1464no—
377C4×C22⋊C4—1464no—
378C2×#95—732no—
379C2×#95—732no—
380C2×#95—738no—
381C2×#95—731no—
382C2×#95—731no—
383C2×#95—731no—
384C2×#95—731no—
385C2×#95—732no—
386C2×#95—732no—
387C2×#95—732no—
388C2×#95—732no—
389C2×#95—732no—
390C2×#95—732no—
391C2×#95C2×D7312yes—
392C2×#95C2×D7312yes—
393C25C22×D7311yescharacteristic
394C23×C4—22no—
395C23×C4—22no—
396C23×C4—22no—
397C23×C4—22no—
398C23×C4—22no—
399C23×C4—22no—
400C23×C4—2922no—
401C23×C4—2922no—
402C23×C4—734no—
403C23×C4—734no—
404C23×C4—731no—
405C23×C4—731no—
406C23×C4—732no—
407C23×C4—731no—
408C23×C4—731no—
409C23×C4—732no—
410C23×C4—1464no—
411C23×C4—1464no—
412C23×C4—1462no—
413C23×C4—1462no—
414C23×C4—1462no—
415C23×C4—1462no—
416C23×C4C4×D7314yes—
417C23×C4C22×D7312yes—
418C23×C4C22×D7311yescharacteristic
419C23×C4C22×D7311yescharacteristic
420C23×C4C22×D7312yes—
421C2×C4⋊C4—2922no—
422C2×C4⋊C4—1464no—
423C2×C4⋊C4—1464no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—734no—
430C2×C22⋊C4—1462no—
431C2×C22⋊C4—1462no—
432C2×C22⋊C4—1462no—
433C2×C22⋊C4—1462no—
434C2×C22⋊C4—1462no—
435C2×C22⋊C4C22⋊D7314yes—
436C2×C42—2922no—
437C2×C42—1464no—
438C2×C42—1464no—
439#95—24no—
440#95—24no—
441#95—2922no—
442#95—2922no—
443#95—734no—
444#95—734no—
445#95—734no—
446#95—734no—
447#95—1464no—
448#95—1464no—
449#95—1464no—
450#95—1464no—
451#95—1464no—
452#95—1464no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24—22no—
457C24C2×C4×D7312yes—
458C24D4×D7312yes—
459C24C2×C22⋊D7312yes—
460C24#1192402712yes—
461C24#1192402712yes—
462C24#1192402712yes—
463C24#1192402812yes—
464C24C23×D7311yescharacteristic
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—2928no—
498C22×C4—2928no—
499C22×C4—2928no—
500C22×C4—2928no—
501C22×C4—2924no—
502C22×C4—2924no—
503C22×C4—732no—
504C22×C4—732no—
505C22×C4—734no—
506C22×C4—734no—
507C22×C4—732no—
508C22×C4—732no—
509C22×C4—734no—
510C22×C4—1462no—
511C22×C4—1462no—
512C22×C4—1462no—
513C22×C4—1462no—
514C22×C4—1462no—
515C22×C4—1462no—
516C22×C4—1462no—
517C22×C4—1468no—
518C22×C4—1468no—
519C22×C4—1468no—
520C22×C4—1468no—
521C22×C4—1461no—
522C22×C4—1461no—
523C22×C4—1462no—
524C22×C4—1462no—
525C22×C4—1462no—
526C22×C4—1462no—
527C22×C4—1462no—
528C22×C4—1462no—
529C22×C4—1461no—
530C22×C4—1461no—
531C22×C4—1462no—
532C22×C4—1462no—
533C22×C4—1462no—
534C22×C4—1462no—
535C22×C4—1461no—
536C22×C4—1461no—
537C22×C4—1461no—
538C22×C4—1461no—
539C22×C4—1462no—
540C22×C4—1462no—
541C22×C4C2×C4×D7312yes—
542C22×C4C2×C4×D7312yes—
543C22×C4C2×C22⋊D7314yes—
544C4⋊C4—2928no—
545C22⋊C4—28no—
546C22⋊C4—44no—
547C22⋊C4—2924no—
548C22⋊C4—1468no—
549C42—2928no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—24no—
565C23—24no—
566C23#1455654714yes—
567C23#1455655114yes—
568C23C4×C22⋊D7314yes—
569C23#1455656214yes—
570C23#1455657214yes—
571C23#1455657714yes—
572C23#1455658414yes—
573C23C22×C4×D7311yescharacteristic
574C23C2×D4×D7311yescharacteristic
575C23C22×C22⋊D7311yescharacteristic
576C23#41×D7312yes—
577C23#41×D7312yes—
578C23C2×#1192402711yescharacteristic
579C23C2×#1192402711yescharacteristic
580C23C2×#1192402711yescharacteristic
581C23C2×#1192402811yescharacteristic
582C23#1455663712yes—
583C23#1455663812yes—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—42no—
594C2×C4—416no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—2922no—
614C2×C4—2922no—
615C2×C4—2922no—
616C2×C4—2922no—
617C2×C4—2922no—
618C2×C4—2922no—
619C2×C4—2922no—
620C2×C4—2922no—
621C2×C4—29216no—
622C2×C4—29216no—
623C2×C4—2922no—
624C2×C4—2922no—
625C2×C4—2922no—
626C2×C4—2922no—
627C2×C4—2922no—
628C2×C4—2922no—
629C2×C4—2922no—
630C2×C4—734no—
631C2×C4—734no—
632C2×C4—1464no—
633C2×C4—1464no—
634C2×C4—1462no—
635C2×C4—1462no—
636C2×C4—1462no—
637C2×C4—1462no—
638C2×C4—1462no—
639C2×C4—1462no—
640C2×C4—1464no—
641C2×C4—1464no—
642C2×C4—1464no—
643C2×C4—1464no—
644C2×C4—1464no—
645C2×C4—1462no—
646C2×C4—1462no—
647C2×C4—1462no—
648C2×C4—1462no—
649C22—24no—
650C22—28no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—28no—
657C22—28no—
658C22—28no—
659C22#1798847518yes—
660C22#117×D7311yescharacteristic
661C22#117×D7311yescharacteristic
662C22C2×#1455654711yescharacteristic
663C22C2×#1455655111yescharacteristic
664C22C2×C4×C22⋊D7311yescharacteristic
665C22#1798906111yescharacteristic
666C22#1798906411yescharacteristic
667C22C2×#1455656211yescharacteristic
668C22C2×#1455657211yescharacteristic
669C22C2×#1455657711yescharacteristic
670C22C2×#1455658411yescharacteristic
671C22#1798911412yes—
672C22#1798911412yes—
673C22#1798918112yes—
674C22#1798918112yes—
675C22#1798922512yes—
676C22#1798922912yes—
677C22#1798923512yes—
678C22#1798923612yes—
679C22#1798926012yes—
680C22#1798926112yes—
681C22#1798927012yes—
682C22#1798927412yes—
683C4—42no—
684C4—42no—
685C4—44no—
686C4—82no—
687C4—2922no—
688C4—2922no—
689C4—2924no—
690C4—5842no—
691C4—1464no—
692C4—1464no—
693C2—28no—
694C2#2261232211yescharacteristic
695C2C2×#1798847511yescharacteristic
696C2#2261253211yescharacteristic
697C2#2261277311yescharacteristic
698C2#2261278111yescharacteristic
699C2#2261279511yescharacteristic
700C2#2261280111yescharacteristic
701C2#2261340412yes—
702C2#2261347812yes—
703C2#2261425312yes—
704C2#2261431212yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×#381×C73C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC2×#381D73
hypocenterC73#13874
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
5. #22608192C2semidirect
6. #22608186C2semidirect
7. #22608162C2semidirect
8. #22608155C2semidirect
21. C2×#14556498C4semidirect
364. C73#13874semidirect

Maximal subgroups

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

subgroupindex
C2×#179884502
C2×#179884452
C2×C22⋊C4×Dic732
C22×#145564982
#226081922
#226081862
#226081622
#226081552
#226081532
C2×#381×C732
#1387473