Finite Group Databasebeta

← #33008140

Subgroups — #33008140

Order 18688.

1962 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988459C212yesmaximal
2C2×#17988451C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊Dic73C211yesmaximal, characteristic
4C22×#14556498C211yesmaximal, characteristic
5#22608189C212yesmaximal, complemented
6#22608186C212yesmaximal, complemented
7#22608159C211yesmaximal, characteristic, complemented
8#22608156C211yesmaximal, characteristic, complemented
9#22608155C211yesmaximal, characteristic, complemented
10#22608153C212yesmaximal
11C73×C22⋊#95C211yesmaximal, characteristic
12C24×Dic73C2212yes—
13C22×#11924007C2212yes—
14C22×#11924007C2212yes—
15C22×C22⋊Dic73C2211yescharacteristic
16C22×C22⋊Dic73C2211yescharacteristic
17C22×C4×Dic73C2211yescharacteristic
18C22×C4×Dic73C2211yescharacteristic
19#17988459—28no—
20#17988451—24no—
21C4×C22⋊Dic73C418yescomplemented
22C2×#14556498—24no—
23C2×#14556498C2212yes—
24C2×#14556498C2212yes—
25C2×#14556498C2211yescharacteristic
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2211yescharacteristic
32C2×#14556498C2211yescharacteristic
33C2×#14556498C2211yescharacteristic
34C24×C4×C73C2211yescharacteristic
35C22×C22⋊C4×C73C2212yes—
36C2×#95×C73C2211yescharacteristic
37C2×#95×C73C2211yescharacteristic
38C2×#95×C73C2212yes—
39C23×Dic73—24no—
40C23×Dic73—24no—
41C23×Dic73—24no—
42C23×Dic73—24no—
43C23×Dic73—24no—
44C23×Dic73—42no—
45C23×Dic73—42no—
46C23×Dic73C2311yescharacteristic
47C23×Dic73C2311yescharacteristic
48C23×Dic73C2312yes—
49C23×Dic73C2312yes—
50C23×Dic73C2311yescharacteristic
51C23×Dic73C2311yescharacteristic
52C2×#11924007—24no—
53C2×#11924007—24no—
54C2×#11924007—24no—
55C2×#11924007—24no—
56C2×#11924007—42no—
57C2×#11924007—42no—
58C2×C22⋊Dic73—24no—
59C2×C22⋊Dic73C2×C414yes—
60C2×C22⋊Dic73C2×C414yes—
61C2×C22⋊Dic73D414yes—
62C2×C22⋊Dic73Q814yes—
63C2×C4×Dic73—22no—
64C2×C4×Dic73—22no—
65C2×C4×Dic73—42no—
66C2×C4×Dic73C2×C414yescomplemented
67C2×C4×Dic73C2×C414yescomplemented
68#14556498—24no—
69#14556498—24no—
70#14556498—24no—
71#14556498—24no—
72#14556498—24no—
73#14556498—22no—
74#14556498—42no—
75#14556498—42no—
76#14556498C2×C414yescomplemented
77#14556498C2×C414yescomplemented
78#14556498D414yes—
79C25×C73C2311yescharacteristic
80C23×C4×C73—24no—
81C23×C4×C73—22no—
82C23×C4×C73—22no—
83C23×C4×C73—42no—
84C23×C4×C73C2×C414yes—
85C23×C4×C73C2312yes—
86C23×C4×C73C2312yes—
87C23×C4×C73C2311yescharacteristic
88C23×C4×C73C2311yescharacteristic
89C2×C22⋊C4×C73—24no—
90C2×C22⋊C4×C73—24no—
91C2×C22⋊C4×C73—24no—
92#95×C73—22no—
93#95×C73—22no—
94#95×C73—42no—
95C22×Dic73—21no—
96C22×Dic73—21no—
97C22×Dic73—22no—
98C22×Dic73—22no—
99C22×Dic73—21no—
100C22×Dic73—21no—
101C22×Dic73—216no—
102C22×Dic73—21no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—21no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—21no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—21no—
122C22×Dic73—48no—
123C22×Dic73—48no—
124C22×Dic73—48no—
125C22×Dic73—48no—
126C22×Dic73—48no—
127C22×Dic73—48no—
128C22×Dic73C22×C412yes—
129C22×Dic73C22×C412yes—
130C22×Dic73C22×C412yes—
131C22×Dic73C22×C412yes—
132C22×Dic73C2×D414yes—
133C22×Dic73C2×D412yes—
134C22×Dic73C2×D412yes—
135C22×Dic73C2×Q812yes—
136C22×Dic73C2×Q812yes—
137C22×Dic73#4114yes—
138#11924007—48no—
139#11924007—48no—
140C22⋊Dic73—28no—
141C22⋊Dic73C4⋊C4116yes—
142C4×Dic73—28no—
143C4×Dic73—44no—
144C24×C73—21no—
145C24×C73—21no—
146C24×C73—22no—
147C24×C73—22no—
148C24×C73—21no—
149C24×C73—21no—
150C24×C73C22×C412yes—
151C24×C73C2×D412yes—
152C24×C73C2×Q812yes—
153C24×C73#4114yes—
154C24×C73#4114yes—
155C24×C73C2411yescharacteristic
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—28no—
159C22×C4×C73—21no—
160C22×C4×C73—22no—
161C22×C4×C73—21no—
162C22×C4×C73—22no—
163C22×C4×C73—28no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73—22no—
170C22×C4×C73—22no—
171C22×C4×C73—22no—
172C22×C4×C73—22no—
173C22×C4×C73—22no—
174C22×C4×C73—22no—
175C22×C4×C73—22no—
176C22×C4×C73—22no—
177C22×C4×C73—22no—
178C22×C4×C73—44no—
179C22×C4×C73—48no—
180C22×C4×C73—48no—
181C22×C4×C73C22×C412yes—
182C22×C4×C73C22×C412yes—
183C22×C4×C73C2×D412yes—
184C22×C4×C73#4112yes—
185C22⋊C4×C73—48no—
186C23×C73—22no—
187C23×C73—22no—
188C23×C73—22no—
189C23×C73—22no—
190C23×C73—24no—
191C23×C73—24no—
192C23×C73—24no—
193C23×C73—24no—
194C23×C73—22no—
195C23×C73—24no—
196C23×C73—24no—
197C23×C73—22no—
198C23×C73—22no—
199C23×C73—24no—
200C23×C73—24no—
201C23×C73—24no—
202C23×C73—24no—
203C23×C73—22no—
204C23×C73C2×C4⋊C414yes—
205C23×C73#11714yes—
206C23×C73#11714yes—
207C23×C73#12418yes—
208C23×C73#12518yes—
209C23×C73C23×C411yescharacteristic
210C23×C73C22×D411yescharacteristic
211C23×C73C22×D411yescharacteristic
212C23×C73C22×Q811yescharacteristic
213C23×C73C2×#4112yes—
214C23×C73C2×#4112yes—
215C23×C73C2×#4111yescharacteristic
216C23×C73C2×#4112yes—
217C23×C73H(5,𝔽2)12yes—
218C23×C73H(5,𝔽2)11yescharacteristic
219C23×C732−1+411yescharacteristic
220C2×C4×C73—22no—
221C2×C4×C73—22no—
222C2×C4×C73—24no—
223C2×C4×C73—24no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—22no—
227C2×C4×C73—416no—
228C2×C4×C73—42no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73—42no—
232C2×C4×C73—42no—
233C2×C4×C73—42no—
234C2×C4×C73—42no—
235C2×C4×C73—42no—
236C2×C4×C73—42no—
237C2×C4×C73—42no—
238C2×C4×C73—42no—
239C2×C4×C73—42no—
240C2×C4×C73—42no—
241C2×C4×C73—42no—
242C2×C4×C73—42no—
243C2×C4×C73—42no—
244C2×C4×C73C4×D414yes—
245C2×Dic73—22no—
246C2×Dic73—22no—
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73—22no—
250C2×Dic73—24no—
251C2×Dic73—22no—
252C2×Dic73—22no—
253C2×Dic73—24no—
254C2×Dic73—22no—
255C2×Dic73—22no—
256C2×Dic73—22no—
257C2×Dic73—22no—
258C2×Dic73—22no—
259C2×Dic73—22no—
260C2×Dic73—22no—
261C2×Dic73—24no—
262C2×Dic73—24no—
263C2×Dic73—24no—
264C2×Dic73—24no—
265C2×Dic73—24no—
266C2×Dic73—24no—
267C2×Dic73—432no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73—42no—
273C2×Dic73—42no—
274C2×Dic73—42no—
275C2×Dic73—42no—
276C2×Dic73—42no—
277C2×Dic73—42no—
278C2×Dic73—42no—
279C2×Dic73—42no—
280C2×Dic73C2×C4⋊C414yes—
281C2×Dic73C2×C4⋊C414yes—
282C22×C73—24no—
283C22×C73—24no—
284C22×C73—24no—
285C22×C73—24no—
286C22×C73—24no—
287C22×C73—28no—
288C22×C73—28no—
289C22×C73—28no—
290C22×C73—28no—
291C22×C73—24no—
292C22×C73#38218yescomplemented
293C22×C73C22×C4⋊C411yescharacteristic
294C22×C73C2×#11711yescharacteristic
295C22×C73C2×#11711yescharacteristic
296C22×C73C2×C4×D411yescharacteristic
297C22×C73C4×#4111yescharacteristic
298C22×C73#51811yescharacteristic
299C22×C73#52011yescharacteristic
300C22×C73#52512yes—
301C22×C73C2×#12412yes—
302C22×C73C2×#12512yes—
303C22×C73#52912yes—
304C22×C73#53812yes—
305C22×C73#53912yes—
306C22×C73D4211yescharacteristic
307C22×C73#54612yes—
308C22×C73#54612yes—
309C22×C73#54711yescharacteristic
310C22×C73D4×Q811yescharacteristic
311C22×C73#55112yes—
312C22×C73#55112yes—
313C22×C73#55212yes—
314C22×C73#55412yes—
315C22×C73#55411yescharacteristic
316C4×C73—24no—
317C4×C73—42no—
318C4×C73—82no—
319C4×C73—82no—
320Dic73—24no—
321Dic73—24no—
322Dic73—42no—
323Dic73—44no—
324Dic73—44no—
325Dic73—42no—
326#13896—731nomaximal, Sylow
327C2×C73—28no—
328C2×C73C2×#38211yescharacteristic
329C2×C73#229011yescharacteristic
330C2×C73#231811yescharacteristic
331C2×C73#234211yescharacteristic
332C2×C73#234311yescharacteristic
333C2×C73D4×C4⋊C411yescharacteristic
334C2×C73#234911yescharacteristic
335C2×C73#247212yes—
336C2×C73#247312yes—
337C2×C73#248812yes—
338C2×C73#250512yes—
339C2×#385—732no—
340C2×#382—731no—
341C2×C4×C22⋊C4—731no—
342C22×#95—731no—
343#1442—732no—
344#1442—732no—
345C4⋊#95—731no—
346#1434—731no—
347#1434—731no—
348C22⋊#95—732no—
349C22⋊#95D7311yescharacteristic
350C73#1389611yescharacteristic, complemented, Sylow
351C24×C4—732no—
352C24×C4C2×D7311yescharacteristic
353C22×C4⋊C4—732no—
354C22×C4⋊C4—732no—
355C22×C22⋊C4—731no—
356C22×C22⋊C4—731no—
357C22×C22⋊C4C2×D7312yes—
358C22×C42—731no—
359C22×C42—731no—
360#385—1468no—
361#382—1464no—
362C4×C22⋊C4—738no—
363C2×#95—732no—
364C2×#95—732no—
365C2×#95—731no—
366C2×#95—731no—
367C2×#95—731no—
368C2×#95—732no—
369C2×#95—732no—
370C2×#95—731no—
371C2×#95—732no—
372C2×#95—732no—
373C2×#95—732no—
374C2×#95—1464no—
375C2×#95C2×D7311yescharacteristic
376C2×#95C2×D7311yescharacteristic
377C2×#95C2×D7312yes—
378C25C22×D7311yescharacteristic
379C23×C4—24no—
380C23×C4—22no—
381C23×C4—22no—
382C23×C4—42no—
383C23×C4—2922no—
384C23×C4—2922no—
385C23×C4—731no—
386C23×C4—731no—
387C23×C4—731no—
388C23×C4—731no—
389C23×C4—732no—
390C23×C4—732no—
391C23×C4—1464no—
392C23×C4—1464no—
393C23×C4—1464no—
394C23×C4—1464no—
395C23×C4—1464no—
396C23×C4C4×D7314yes—
397C23×C4C22×D7312yes—
398C23×C4C22×D7312yes—
399C23×C4C22×D7311yescharacteristic
400C23×C4C22×D7311yescharacteristic
401C2×C4⋊C4—2922no—
402C2×C4⋊C4—2922no—
403C2×C4⋊C4—1464no—
404C2×C4⋊C4—1464no—
405C2×C4⋊C4—1464no—
406C2×C4⋊C4—1464no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—734no—
411C2×C22⋊C4—734no—
412C2×C22⋊C4—734no—
413C2×C22⋊C4—734no—
414C2×C22⋊C4—1464no—
415C2×C42—2922no—
416C2×C42—734no—
417C2×C42—734no—
418C2×C42—1462no—
419C2×C42—1462no—
420#95—22no—
421#95—22no—
422#95—42no—
423#95—2922no—
424#95—2922no—
425#95—734no—
426#95—734no—
427#95—734no—
428#95—1464no—
429#95—1464no—
430#95—1464no—
431#95—1464no—
432#95—1464no—
433#95—1462no—
434C24—21no—
435C24—22no—
436C24—21no—
437C24—22no—
438C24—21no—
439C24—21no—
440C24C2×C4×D7312yes—
441C24D4×D7312yes—
442C24Q8×D7312yes—
443C24#1192402714yes—
444C24#1192402814yes—
445C24C23×D7311yescharacteristic
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—28no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—2928no—
472C22×C4—2928no—
473C22×C4—2928no—
474C22×C4—2928no—
475C22×C4—2928no—
476C22×C4—2928no—
477C22×C4—732no—
478C22×C4—732no—
479C22×C4—732no—
480C22×C4—732no—
481C22×C4—732no—
482C22×C4—732no—
483C22×C4—732no—
484C22×C4—732no—
485C22×C4—734no—
486C22×C4—734no—
487C22×C4—14616no—
488C22×C4—1461no—
489C22×C4—1462no—
490C22×C4—1462no—
491C22×C4—1462no—
492C22×C4—1462no—
493C22×C4—1462no—
494C22×C4—1461no—
495C22×C4—1462no—
496C22×C4—1462no—
497C22×C4—1462no—
498C22×C4—1462no—
499C22×C4—1462no—
500C22×C4—1462no—
501C22×C4—1462no—
502C22×C4—1462no—
503C22×C4—1462no—
504C22×C4—1461no—
505C22×C4—1462no—
506C22×C4—1462no—
507C22×C4—1461no—
508C22×C4—1461no—
509C22×C4—1462no—
510C22×C4—1461no—
511C22×C4—1462no—
512C22×C4—1461no—
513C22×C4—1461no—
514C22×C4C2×C4×D7312yes—
515C22×C4C2×C4×D7312yes—
516C22×C4D4×D7312yes—
517C22×C4#1192402712yes—
518C4⋊C4—2928no—
519C4⋊C4—2928no—
520C22⋊C4—48no—
521C22⋊C4—7316no—
522C22⋊C4—1468no—
523C42—2924no—
524C42—1468no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23C4⋊C4×D7314yes—
544C23#1455654714yes—
545C23#1455654814yes—
546C23#1455657718yes—
547C23#1455658118yes—
548C23C22×C4×D7311yescharacteristic
549C23C2×D4×D7311yescharacteristic
550C23C2×D4×D7311yescharacteristic
551C23C2×Q8×D7311yescharacteristic
552C23#41×D7312yes—
553C23C2×#1192402712yes—
554C23C2×#1192402711yescharacteristic
555C23C2×#1192402812yes—
556C23#1455663612yes—
557C23#1455663711yescharacteristic
558C23#1455663811yescharacteristic
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—416no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—29232no—
584C2×C4—2922no—
585C2×C4—2922no—
586C2×C4—2922no—
587C2×C4—2922no—
588C2×C4—2922no—
589C2×C4—2922no—
590C2×C4—2922no—
591C2×C4—2922no—
592C2×C4—2922no—
593C2×C4—2922no—
594C2×C4—2922no—
595C2×C4—2922no—
596C2×C4—734no—
597C2×C4—734no—
598C2×C4—1464no—
599C2×C4—1464no—
600C2×C4—1464no—
601C2×C4—1464no—
602C2×C4—1462no—
603C2×C4—1462no—
604C2×C4—1462no—
605C2×C4—1462no—
606C2×C4—1462no—
607C2×C4—1462no—
608C2×C4—1464no—
609C2×C4—1464no—
610C2×C4—1462no—
611C2×C4—1464no—
612C2×C4—1462no—
613C2×C4—1462no—
614C2×C4—1464no—
615C2×C4—1462no—
616C2×C4—1462no—
617C2×C4—1462no—
618C2×C4—1462no—
619C2×C4—1462no—
620C2×C4#1455655114yes—
621C22—28no—
622C22—24no—
623C22—24no—
624C22—28no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—28no—
630C22—28no—
631C22#1798845118yescomplemented
632C22C2×C4⋊C4×D7311yescharacteristic
633C22C2×#1455654711yescharacteristic
634C22C2×#1455654811yescharacteristic
635C22C2×#1455655111yescharacteristic
636C22#1798905511yescharacteristic
637C22#1798906011yescharacteristic
638C22#1798906711yescharacteristic
639C22#1798909612yes—
640C22C2×#1455657712yes—
641C22C2×#1455658112yes—
642C22#1798911712yes—
643C22#1798917012yes—
644C22#1798917912yes—
645C22#1798922111yescharacteristic
646C22#1798922412yes—
647C22#1798922612yes—
648C22#1798923711yescharacteristic
649C22#1798925011yescharacteristic
650C22#1798926012yes—
651C22#1798926812yes—
652C22#1798927412yes—
653C22#1798929112yes—
654C22#1798929511yescharacteristic
655C4—24no—
656C4—42no—
657C4—82no—
658C4—82no—
659C4—2924no—
660C4—2924no—
661C4—2922no—
662C4—2922no—
663C4—1464no—
664C4—1464no—
665C2—28no—
666C2C2×#1798845111yescharacteristic
667C2#2261237011yescharacteristic
668C2#2261255811yescharacteristic
669C2#2261278211yescharacteristic
670C2#2261279311yescharacteristic
671C2#2261282711yescharacteristic
672C2#2261283511yescharacteristic
673C2#2261405912yes—
674C2#2261407412yes—
675C2#2261425912yes—
676C2#2261443412yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×C22⋊#95C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC22⋊#95D73
hypocenterC73#13896
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#17988451C2semidirect
5. #22608189C2semidirect
6. #22608186C2semidirect
7. #22608159C2semidirect
8. #22608156C2semidirect
9. #22608155C2semidirect
21. C4×C22⋊Dic73C4semidirect
66. C2×C4×Dic73C2×C4semidirect
67. C2×C4×Dic73C2×C4semidirect
76. #14556498C2×C4semidirect
77. #14556498C2×C4semidirect
292. C22×C73#382semidirect
350. C73#13896semidirect
631. C22#17988451semidirect

Maximal subgroups

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

subgroupindex
C2×#179884592
C2×#179884512
C2×C4×C22⋊Dic732
C22×#145564982
#226081892
#226081862
#226081592
#226081562
#226081552
#226081532
C73×C22⋊#952
#1389673