Finite Group Databasebeta

← #33012707

Subgroups — #33012707

Order 18688.

1566 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988449C211yesmaximal, characteristic
2C2×#17988444C211yesmaximal, characteristic
3C2×#17988434C211yesmaximal, characteristic
4#22608196C211yesmaximal, characteristic, complemented
5#22608194C211yesmaximal, characteristic, complemented
6#22608187C211yesmaximal, characteristic, complemented
7#22608177C212yesmaximal
8#22608175C212yesmaximal
9#22608165C212yesmaximal, complemented
10#22608162C212yesmaximal, complemented
11C73×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C22×C4⋊Dic73C2212yes—
14C22×C4⋊Dic73C2211yescharacteristic
15C22×#11924007C2212yes—
16C22×#11924005C2211yescharacteristic
17C22×#11924005C2211yescharacteristic
18C22×#11924005C2212yes—
19C22×#11924005C2212yes—
20C22×C4×Dic73C2211yescharacteristic
21C22×C4×Dic73C2211yescharacteristic
22#17988449C418yescomplemented
23#17988444—24no—
24#17988434—24no—
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2211yescharacteristic
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2211yescharacteristic
33C22×C4⋊C4×C73C2211yescharacteristic
34C2×#95×C73C2211yescharacteristic
35C2×#95×C73C2212yes—
36C2×#95×C73C2212yes—
37C2×#95×C73C2211yescharacteristic
38C24×D73C2311yescharacteristic
39C23×Dic73C2312yes—
40C23×Dic73C2312yes—
41C23×Dic73C2311yescharacteristic
42C23×Dic73C2311yescharacteristic
43C23×Dic73C2311yescharacteristic
44C22×C4×D73—42no—
45C22×C4×D73—42no—
46C22×C4×D73C2×C414yes—
47C22×C4×D73D414yes—
48C22×C4×D73Q814yes—
49C2×C4⋊Dic73—24no—
50C2×C4⋊Dic73—22no—
51C2×C4⋊Dic73—42no—
52C2×C4⋊Dic73—42no—
53C2×C4⋊Dic73—42no—
54C2×#11924007—24no—
55C2×#11924007—42no—
56C2×#11924007—42no—
57C2×#11924005—22no—
58C2×#11924005—22no—
59C2×#11924005—22no—
60C2×#11924005—22no—
61C2×#11924005—24no—
62C2×#11924005—22no—
63C2×#11924005—22no—
64C2×#11924005—42no—
65C2×#11924005—42no—
66C2×#11924005—42no—
67C2×#11924005—42no—
68C2×#11924005C2×C418yes—
69C2×C4×Dic73—22no—
70C2×C4×Dic73—22no—
71C2×C4×Dic73—42no—
72C2×C4×Dic73—42no—
73#14556498—24no—
74#14556498—22no—
75#14556498—24no—
76#14556498—42no—
77#14556498—42no—
78#14556498—42no—
79#14556498C2×C418yes—
80#14556498C2×C414yes—
81C23×C4×C73C2312yes—
82C23×C4×C73C2312yes—
83C23×C4×C73C2311yescharacteristic
84C23×C4×C73C2311yescharacteristic
85C23×C4×C73C2311yescharacteristic
86C2×C4⋊C4×C73—22no—
87C2×C4⋊C4×C73—42no—
88#95×C73—22no—
89#95×C73—24no—
90#95×C73—42no—
91#95×C73C2×C414yes—
92C23×D73—21no—
93C23×D73—21no—
94C23×D73—21no—
95C23×D73—21no—
96C23×D73—22no—
97C23×D73—22no—
98C23×D73—22no—
99C23×D73—22no—
100C23×D73C22×C412yes—
101C23×D73C2×D412yes—
102C23×D73C2×Q812yes—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—21no—
106C22×Dic73—21no—
107C22×Dic73—22no—
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—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73C22×C414yes—
127C22×Dic73C22×C412yes—
128C22×Dic73C2×D412yes—
129C22×Dic73C2×Q812yes—
130C22×Dic73#4112yes—
131C22×Dic73#4112yes—
132C2×C4×D73—48no—
133C2×C4×D73—44no—
134C2×C4×D73—44no—
135C2×C4×D73—48no—
136C2×C4×D73C4⋊C4116yescomplemented
137C4⋊Dic73—48no—
138C4⋊Dic73—44no—
139#11924007—48no—
140#11924005—28no—
141#11924005—28no—
142#11924005—48no—
143#11924005—48no—
144C4×Dic73—44no—
145C4×Dic73—44no—
146C24×C73C2411yescharacteristic
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73—21no—
150C22×C4×C73—21no—
151C22×C4×C73—22no—
152C22×C4×C73—21no—
153C22×C4×C73—21no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
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×C73C22×C414yes—
171C22×C4×C73C22×C412yes—
172C22×C4×C73C2×D412yes—
173C22×C4×C73C2×Q812yes—
174C22×C4×C73#4112yes—
175C22×C4×C73#4112yes—
176C4⋊C4×C73—44no—
177C23×C73C23×C411yescharacteristic
178C23×C73C22×D411yescharacteristic
179C23×C73C22×Q811yescharacteristic
180C23×C73C2×#4111yescharacteristic
181C23×C73C2×#4111yescharacteristic
182C23×C73C2×#4111yescharacteristic
183C23×C73C2×#4111yescharacteristic
184C23×C73H(5,𝔽2)12yes—
185C23×C73H(5,𝔽2)12yes—
186C23×C73H(5,𝔽2)12yes—
187C23×C732−1+412yes—
188C22×D73—22no—
189C22×D73—22no—
190C22×D73—24no—
191C22×D73—24no—
192C22×D73—22no—
193C22×D73—22no—
194C22×D73—22no—
195C22×D73—22no—
196C22×D73—22no—
197C22×D73—22no—
198C22×D73—22no—
199C22×D73—22no—
200C22×D73—24no—
201C22×D73—24no—
202C22×D73—42no—
203C22×D73—42no—
204C22×D73—42no—
205C22×D73—42no—
206C22×D73—42no—
207C22×D73—42no—
208C22×D73—42no—
209C22×D73—42no—
210C22×D73C2×C4⋊C414yes—
211C2×C4×C73—24no—
212C2×C4×C73—22no—
213C2×C4×C73—22no—
214C2×C4×C73—24no—
215C2×C4×C73—24no—
216C2×C4×C73—24no—
217C2×C4×C73—24no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—24no—
221C2×C4×C73—24no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—42no—
227C2×C4×C73—42no—
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×C73C2×C4⋊C414yes—
240C2×Dic73—24no—
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—24no—
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—24no—
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73—24no—
250C2×Dic73—24no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—42no—
256C2×Dic73—42no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—42no—
268C2×Dic73C2×C4⋊C414yes—
269C4×D73—416no—
270C22×C73C22×C4⋊C411yescharacteristic
271C22×C73C4×#4111yescharacteristic
272C22×C73C4×#4111yescharacteristic
273C22×C73#51812yes—
274C22×C73#52012yes—
275C22×C73#52511yescharacteristic
276C22×C73#52511yescharacteristic
277C22×C73#53311yescharacteristic
278C22×C73#53311yescharacteristic
279C22×C73#53411yescharacteristic
280C22×C73#53411yescharacteristic
281C22×C73#53511yescharacteristic
282C22×C73#53611yescharacteristic
283C22×C73#53812yes—
284C22×C73#53812yes—
285C22×C73#53912yes—
286C22×C73#53912yes—
287C22×C73#54012yes—
288C22×C73#54012yes—
289C22×C73#54212yes—
290C22×C73#54212yes—
291C22×C73#54312yes—
292C22×C73#54412yes—
293C2×D73—24no—
294C2×D73—24no—
295C2×D73—24no—
296C2×D73—44no—
297C2×D73—44no—
298C2×D73—42no—
299C2×D73—42no—
300C2×D73—42no—
301C2×D73—42no—
302C2×D73—44no—
303C2×D73—44no—
304C4×C73—24no—
305C4×C73—44no—
306C4×C73—42no—
307C4×C73—42no—
308C4×C73—82no—
309Dic73—24no—
310Dic73—44no—
311Dic73—42no—
312Dic73—42no—
313Dic73—82no—
314#14295—731nomaximal, Sylow
315C2×C73#229611yescharacteristic
316C2×C73#231511yescharacteristic
317C2×C73#231711yescharacteristic
318C2×C73#233512yes—
319C2×C73#233512yes—
320C2×C73#263511yescharacteristic
321C2×C73#263511yescharacteristic
322C2×C73#263811yescharacteristic
323C2×C73#263811yescharacteristic
324C2×C73#266512yes—
325C2×C73#266712yes—
326D73—44no—
327C2×#381—731no—
328C2×#380—731no—
329C2×C4×C22⋊C4—731no—
330#1444—731no—
331#1444—731no—
332#1442—731no—
333#1442D7311yescharacteristic, complemented
334#1439—732no—
335#1439—732no—
336#1436—732no—
337#1436—732no—
338C73#1429511yescharacteristic, complemented, Sylow
339C24×C4—731no—
340C22×C4⋊C4—732no—
341C22×C4⋊C4—732no—
342C22×C4⋊C4—731no—
343C22×C4⋊C4C2×D7311yescharacteristic
344C22×C22⋊C4—732no—
345C22×C22⋊C4—732no—
346C22×C22⋊C4—731no—
347C22×C22⋊C4—731no—
348C22×C42—731no—
349C22×C42—731no—
350#381—738no—
351#380—1464no—
352C4×C22⋊C4—1464no—
353C2×#95—732no—
354C2×#95—732no—
355C2×#95—732no—
356C2×#95—732no—
357C2×#95—731no—
358C2×#95—732no—
359C2×#95—732no—
360C2×#95—731no—
361C2×#95C2×D7312yes—
362C2×#95C2×D7311yescharacteristic
363C2×#95C2×D7312yes—
364C2×#95C2×D7311yescharacteristic
365C25—731no—
366C23×C4—2922no—
367C23×C4—2922no—
368C23×C4—734no—
369C23×C4—734no—
370C23×C4—734no—
371C23×C4—731no—
372C23×C4—731no—
373C23×C4—731no—
374C23×C4—732no—
375C23×C4—732no—
376C23×C4C22×D7312yes—
377C23×C4C22×D7312yes—
378C23×C4C22×D7311yescharacteristic
379C23×C4C22×D7311yescharacteristic
380C23×C4C22×D7311yescharacteristic
381C2×C4⋊C4—22no—
382C2×C4⋊C4—42no—
383C2×C4⋊C4—2922no—
384C2×C4⋊C4—2922no—
385C2×C4⋊C4—2922no—
386C2×C4⋊C4—2922no—
387C2×C4⋊C4—2922no—
388C2×C4⋊C4—1464no—
389C2×C4⋊C4—1464no—
390C2×C4⋊C4—1462no—
391C2×C22⋊C4—2922no—
392C2×C22⋊C4—2922no—
393C2×C22⋊C4—2922no—
394C2×C22⋊C4—2922no—
395C2×C22⋊C4—738no—
396C2×C22⋊C4—1464no—
397C2×C22⋊C4—1462no—
398C2×C22⋊C4—1462no—
399C2×C22⋊C4—1462no—
400C2×C22⋊C4—1462no—
401C2×C22⋊C4—1462no—
402C2×C22⋊C4—1462no—
403C2×C42—2922no—
404C2×C42—2922no—
405C2×C42—1462no—
406C2×C42—1462no—
407#95—22no—
408#95—24no—
409#95—42no—
410#95—2922no—
411#95—2922no—
412#95—2922no—
413#95—738no—
414#95—734no—
415#95—1464no—
416#95—1462no—
417#95—1464no—
418#95C4×D7314yes—
419C24—732no—
420C24—732no—
421C24—732no—
422C24—1461no—
423C24—1461no—
424C24—1461no—
425C24—1461no—
426C24—1462no—
427C24—1462no—
428C24—1462no—
429C24—1462no—
430C24C23×D7311yescharacteristic
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—2924no—
455C22×C4—2924no—
456C22×C4—2928no—
457C22×C4—2928no—
458C22×C4—7316no—
459C22×C4—732no—
460C22×C4—732no—
461C22×C4—732no—
462C22×C4—734no—
463C22×C4—732no—
464C22×C4—732no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4—1462no—
470C22×C4—1462no—
471C22×C4—1462no—
472C22×C4—1461no—
473C22×C4—1461no—
474C22×C4—1462no—
475C22×C4—1461no—
476C22×C4—1462no—
477C22×C4—1461no—
478C22×C4—1462no—
479C22×C4—1462no—
480C22×C4—1462no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4C2×C4×D7314yes—
489C22×C4C2×C4×D7312yes—
490C22×C4D4×D7312yes—
491C22×C4Q8×D7312yes—
492C22×C4#1192402712yes—
493C22×C4#1192402912yes—
494C4⋊C4—44no—
495C4⋊C4—2924no—
496C4⋊C4—2928no—
497C4⋊C4—2928no—
498C22⋊C4—2928no—
499C22⋊C4—2928no—
500C22⋊C4—1468no—
501C22⋊C4—1468no—
502C42—2924no—
503C42—2924no—
504C23—2922no—
505C23—2922no—
506C23—2922no—
507C23—2922no—
508C23—2922no—
509C23—2922no—
510C23—2922no—
511C23—2922no—
512C23—734no—
513C23—1462no—
514C23—1462no—
515C23—1462no—
516C23—1462no—
517C23—1462no—
518C23—1462no—
519C23—1464no—
520C23—1464no—
521C23—1462no—
522C23—1462no—
523C23—1464no—
524C23—1464no—
525C23—1462no—
526C23—1462no—
527C23C22×C4×D7311yescharacteristic
528C23C2×D4×D7311yescharacteristic
529C23C2×Q8×D7311yescharacteristic
530C23#41×D7311yescharacteristic
531C23#41×D7311yescharacteristic
532C23C2×#1192402711yescharacteristic
533C23C2×#1192402911yescharacteristic
534C23#1455663612yes—
535C23#1455663712yes—
536C23#1455663712yes—
537C23#1455663812yes—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—29216no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2922no—
572C2×C4—2922no—
573C2×C4—2922no—
574C2×C4—2922no—
575C2×C4—2922no—
576C2×C4—2922no—
577C2×C4—2922no—
578C2×C4—2922no—
579C2×C4—2922no—
580C2×C4—2922no—
581C2×C4—2922no—
582C2×C4—2922no—
583C2×C4—2922no—
584C2×C4—734no—
585C2×C4—1464no—
586C2×C4—1464no—
587C2×C4—1464no—
588C2×C4—1464no—
589C2×C4—1464no—
590C2×C4—1462no—
591C2×C4—1462no—
592C2×C4—1462no—
593C2×C4—1462no—
594C2×C4—1464no—
595C2×C4—1464no—
596C2×C4C4⋊C4×D7314yes—
597C22—2922no—
598C22—2922no—
599C22—2924no—
600C22—2922no—
601C22—2922no—
602C22—2924no—
603C22—2924no—
604C22—2924no—
605C22—1464no—
606C22—1464no—
607C22—1464no—
608C22C2×C4⋊C4×D7311yescharacteristic
609C22C4×#1192402711yescharacteristic
610C22C4×#1192402911yescharacteristic
611C22#1798905912yes—
612C22#1798906712yes—
613C22#1798909711yescharacteristic
614C22#1798910011yescharacteristic
615C22#1798914211yescharacteristic
616C22#1798914311yescharacteristic
617C22#1798914711yescharacteristic
618C22#1798914811yescharacteristic
619C22#1798915511yescharacteristic
620C22#1798915711yescharacteristic
621C22#1798917312yes—
622C22#1798917612yes—
623C22#1798918112yes—
624C22#1798918612yes—
625C22#1798919012yes—
626C22#1798919312yes—
627C22#1798920112yes—
628C22#1798920312yes—
629C22#1798921212yes—
630C22#1798921712yes—
631C4—24no—
632C4—42no—
633C4—42no—
634C4—44no—
635C4—82no—
636C4—2922no—
637C4—2922no—
638C4—2924no—
639C4—5842no—
640C4—1464no—
641C2—2924no—
642C2#2261241211yescharacteristic
643C2#2261254211yescharacteristic
644C2#2261255611yescharacteristic
645C2#2261272012yes—
646C2#2261272812yes—
647C2#2261582311yescharacteristic
648C2#2261582411yescharacteristic
649C2#2261585311yescharacteristic
650C2#2261585411yescharacteristic
651C2#2261610812yes—
652C2#2261613012yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1442C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenter#1442D73
hypocenterC73#14295
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
4. #22608196C2semidirect
5. #22608194C2semidirect
6. #22608187C2semidirect
9. #22608165C2semidirect
10. #22608162C2semidirect
11. C73×#1442C2semidirect
22. #17988449C4semidirect
136. C2×C4×D73C4⋊C4semidirect
333. #1442D73semidirect
338. C73#14295semidirect

Maximal subgroups

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

subgroupindex
C2×#179884492
C2×#179884442
C2×#179884342
#226081962
#226081942
#226081872
#226081772
#226081752
#226081652
#226081622
C73×#14422
#1429573