Finite Group Databasebeta

← #33096304

Subgroups — #33096304

Order 18688.

1010 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988550C211yesmaximal, characteristic
2C2×#17988549C211yesmaximal, characteristic
3#407×D73C211yesmaximal, characteristic
4#22610138C211yesmaximal, characteristic, complemented
5#22610137C212yesmaximal, complemented
6#22610136C211yesmaximal, characteristic, complemented
7#22610135C211yesmaximal, characteristic, complemented
8#22610134C212yesmaximal, complemented
9#22609972C212yesmaximal, complemented
10#22609970C212yesmaximal, complemented
11C73×#1855C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C22×#11924012C2211yescharacteristic
14C22×#11924011C2212yes—
15C2×M16×D73C2211yescharacteristic
16C2×M16×D73C2211yescharacteristic
17C2×#14556527C2212yescomplemented
18C2×#14556526C2212yescomplemented
19C2×#14556526C2212yescomplemented
20#17988551C2211yescharacteristic
21#17988550—21no—
22#17988550—21no—
23#17988550C2211yescharacteristic
24#17988550C2212yes—
25#17988550C2211yescharacteristic
26#17988550C2212yes—
27#17988549—21no—
28#17988549—21no—
29#17988549C412yes—
30#17988549C412yes—
31#17988549C412yes—
32#17988549C412yes—
33C2×C22⋊#10596831C2211yescharacteristic
34C2×#14556503—21no—
35C2×#14556503—21no—
36C2×#14556503C2212yes—
37C22⋊C8×D73—21no—
38C22⋊C8×D73—21no—
39#17988352C2212yescomplemented
40#17988351C2212yescomplemented
41#17988351C2212yescomplemented
42C22×M16×C73C2211yescharacteristic
43C2×#108×C73C2212yescomplemented
44#407×C73C2211yescharacteristic
45C2×C22⋊C8×C73C2211yescharacteristic
46#341×C73C2212yescomplemented
47C24×D73Q811yescharacteristic
48C23×Dic73Q811yescharacteristic
49C22×C4×D73—42no—
50C22×C4×D73—42no—
51C22×C4×D73C2×C411yescharacteristic
52C22×C4×D73C2×C411yescharacteristic
53C22×C4×D73C2×C411yescharacteristic
54C22×C4×D73C2×C411yescharacteristic
55C22×C4×D73D411yescharacteristic
56C22×C4×D73D411yescharacteristic
57C22×C4×D73D411yescharacteristic
58C22×C4×D73D411yescharacteristic
59C22×C4×D73Q811yescharacteristic
60C22×C4×D73Q811yescharacteristic
61C22×C4×D73C2311yescharacteristic
62C22×C4×D73C2311yescharacteristic
63C2×#11924012—21no—
64C2×#11924012—42no—
65C2×#11924012C2311yescharacteristic
66C2×#11924012C2312yes—
67C2×#11924012C2311yescharacteristic
68C2×#11924011—21no—
69C2×#11924011—21no—
70C2×#11924011—42no—
71C2×#11924011—42no—
72C2×#11924011C2×C414yes—
73C2×#11924011C2×C414yes—
74M16×D73—42no—
75M16×D73—42no—
76C22×#10596831—21no—
77C22×#10596831C2312yes—
78C2×C8×D73—21no—
79C2×C8×D73—21no—
80#14556527—42no—
81#14556526—42no—
82#14556526—42no—
83C22⋊#10596831—21no—
84C22⋊#10596831—21no—
85C22⋊#10596831C2×C412yes—
86C22⋊#10596831C2×C412yes—
87#14556503—21no—
88#14556503—21no—
89#14556503—21no—
90#14556503—21no—
91#14556503—22no—
92#14556503—22no—
93#14556503—22no—
94#14556503—22no—
95#14556503C2×C414yes—
96#14556503C2×C414yes—
97C23×C4×C73C2311yescharacteristic
98C2×M16×C73—21no—
99C2×M16×C73—42no—
100C2×M16×C73C2311yescharacteristic
101C2×M16×C73C2312yes—
102C2×M16×C73C2311yescharacteristic
103C22×C8×C73—21no—
104C22×C8×C73C2312yes—
105#108×C73—42no—
106C22⋊C8×C73—21no—
107C22⋊C8×C73—21no—
108C22⋊C8×C73C2×C412yes—
109C22⋊C8×C73C2×C412yes—
110C23×D73—42no—
111C23×D73—42no—
112C23×D73—42no—
113C23×D73C4⋊C411yescharacteristic
114C23×D73C4⋊C411yescharacteristic
115C23×D73C4⋊C411yescharacteristic
116C23×D73C4⋊C411yescharacteristic
117C23×D73C2×Q811yescharacteristic
118C23×D73C2×Q811yescharacteristic
119C22×Dic73—42no—
120C22×Dic73C4⋊C411yescharacteristic
121C22×Dic73C4⋊C411yescharacteristic
122C22×Dic73C4⋊C411yescharacteristic
123C22×Dic73C4⋊C411yescharacteristic
124C22×Dic73C2×Q811yescharacteristic
125C22×Dic73C2×Q811yescharacteristic
126C2×C4×D73—22no—
127C2×C4×D73—22no—
128C2×C4×D73—22no—
129C2×C4×D73—22no—
130C2×C4×D73—42no—
131C2×C4×D73—42no—
132C2×C4×D73—42no—
133C2×C4×D73—42no—
134C2×C4×D73—41no—
135C2×C4×D73—41no—
136C2×C4×D73—41no—
137C2×C4×D73—41no—
138C2×C4×D73—41no—
139C2×C4×D73—41no—
140C2×C4×D73—41no—
141C2×C4×D73—41no—
142C2×C4×D73—42no—
143C2×C4×D73—42no—
144C2×C4×D73C4⋊C411yescharacteristic
145C2×C4×D73C4⋊C411yescharacteristic
146C2×C4×D73C4⋊C411yescharacteristic
147C2×C4×D73C4⋊C411yescharacteristic
148C2×C4×D73C4⋊C411yescharacteristic
149C2×C4×D73C4⋊C411yescharacteristic
150C2×C4×D73C4⋊C411yescharacteristic
151C2×C4×D73C4⋊C411yescharacteristic
152C2×C4×D73C22×C411yescharacteristic
153C2×C4×D73C22×C411yescharacteristic
154C2×C4×D73C22×C411yescharacteristic
155C2×C4×D73C22×C411yescharacteristic
156C2×C4×D73C2×D411yescharacteristic
157C2×C4×D73C2×D411yescharacteristic
158C2×C4×D73C2×D411yescharacteristic
159C2×C4×D73C2×D411yescharacteristic
160#11924012—22no—
161#11924012—22no—
162#11924012—22no—
163#11924012—22no—
164#11924012—42no—
165#11924011—44no—
166#11924011—44no—
167#11924011—42no—
168#11924011—42no—
169C2×#10596831—22no—
170C2×#10596831—22no—
171C2×#10596831—22no—
172C2×#10596831—21no—
173C2×#10596831—21no—
174C2×#10596831—42no—
175C2×#10596831C22×C414yes—
176C8×D73—42no—
177C8×D73—42no—
178C24×C73C2×Q811yescharacteristic
179C22×C4×C73—22no—
180C22×C4×C73—22no—
181C22×C4×C73C22×C411yescharacteristic
182C22×C4×C73C22×C411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×Q811yescharacteristic
186C22×C4×C73C2411yescharacteristic
187M16×C73—22no—
188M16×C73—22no—
189M16×C73—22no—
190M16×C73—22no—
191M16×C73—42no—
192C2×C8×C73—22no—
193C2×C8×C73—22no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—22no—
197C2×C8×C73—42no—
198C2×C8×C73C22×C414yes—
199C23×C73—22no—
200C23×C73—22no—
201C23×C73—41no—
202C23×C73C2×C4⋊C411yescharacteristic
203C23×C73C2×C4⋊C411yescharacteristic
204C23×C73C22×Q811yescharacteristic
205C22×D73—22no—
206C22×D73—22no—
207C22×D73—22no—
208C22×D73—22no—
209C22×D73—22no—
210C22×D73—22no—
211C22×D73—22no—
212C22×D73—41no—
213C22×D73—41no—
214C22×D73—41no—
215C22×D73—41no—
216C22×D73—41no—
217C22×D73—41no—
218C22×D73—41no—
219C22×D73—41no—
220C22×D73—41no—
221C22×D73—41no—
222C22×D73—41no—
223C22×D73—416no—
224C22×D73C2×C4⋊C411yescharacteristic
225C22×D73C2×C4⋊C411yescharacteristic
226C22×D73C2×C4⋊C411yescharacteristic
227C22×D73C2×C4⋊C411yescharacteristic
228C2×C4×C73—22no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73C2×C4⋊C411yescharacteristic
236C2×C4×C73C2×C4⋊C411yescharacteristic
237C2×C4×C73C23×C411yescharacteristic
238C2×C4×C73C22×D411yescharacteristic
239C2×C4×C73H(5,𝔽2)12yes—
240C2×C4×C73H(5,𝔽2)12yes—
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73C2×C4⋊C411yescharacteristic
248C2×Dic73C2×C4⋊C411yescharacteristic
249C2×Dic73C2×C4⋊C411yescharacteristic
250C2×Dic73C2×C4⋊C411yescharacteristic
251C4×D73—22no—
252C4×D73—22no—
253C4×D73—22no—
254C4×D73—22no—
255C4×D73—21no—
256C4×D73—21no—
257C4×D73—21no—
258C4×D73—21no—
259C4×D73—21no—
260C4×D73—21no—
261C4×D73—21no—
262C4×D73—21no—
263C4×D73—41no—
264C4×D73—41no—
265C4×D73—41no—
266C4×D73—41no—
267C4×D73—41no—
268C4×D73—41no—
269C4×D73—41no—
270C4×D73—41no—
271C8×C73—24no—
272C8×C73—42no—
273C8×C73—41no—
274C8×C73—41no—
275#10596831—24no—
276#10596831—41no—
277#10596831—41no—
278#10596831—42no—
279C22×C73—21no—
280C22×C73—21no—
281C22×C73—21no—
282C22×C73—21no—
283C22×C73—21no—
284C22×C73—21no—
285C22×C73—21no—
286C22×C73—44no—
287C22×C73#43012yes—
288C22×C73C22×C4⋊C411yescharacteristic
289C22×C73#54312yes—
290C2×D73—21no—
291C2×D73—21no—
292C2×D73—21no—
293C2×D73—21no—
294C2×D73—21no—
295C2×D73—21no—
296C2×D73—21no—
297C2×D73—21no—
298C2×D73—21no—
299C2×D73—21no—
300C2×D73—21no—
301C2×D73—44no—
302C2×D73—44no—
303C2×D73—44no—
304C2×D73—44no—
305C2×D73—44no—
306C2×D73—44no—
307C2×D73#43012yes—
308C4×C73—21no—
309C4×C73—21no—
310C4×C73#51812yes—
311C4×C73#53411yescharacteristic
312C4×C73#53411yescharacteristic
313Dic73—21no—
314Dic73—21no—
315Dic73—21no—
316Dic73—21no—
317#21949—731nomaximal, Sylow
318C2×C73—22no—
319C2×C73—22no—
320C2×C73—41no—
321C2×C73#231511yescharacteristic
322C2×C73C2×#43011yescharacteristic
323C2×C73C2×#43011yescharacteristic
324D73—22no—
325D73—41no—
326D73—41no—
327D73—41no—
328C2×#407—731no—
329C2×#407—731no—
330C2×#407—731no—
331#1855—732no—
332#1855—731no—
333#1855—731no—
334#1855—732no—
335#1855—731no—
336#1855D7311yescharacteristic, complemented
337#1811—732no—
338#1811—732no—
339C73#2194911yescharacteristic, complemented, Sylow
340C24×C4—731no—
341C22×M16—732no—
342C22×M16—731no—
343C22×M16—731no—
344C22×M16—731no—
345C22×M16C2×D7311yescharacteristic
346C2×#108—732no—
347C2×#108—732no—
348C2×#108—732no—
349C2×#108C2×D7312yescomplemented
350#407—732no—
351#407—732no—
352#407—732no—
353#407—732no—
354#407—731no—
355#407—731no—
356#407—731no—
357#407—732no—
358#407—732no—
359#407—1461no—
360#407—1461no—
361#407—1461no—
362#407—1461no—
363#407C2×D7311yescharacteristic
364C2×C22⋊C8—731no—
365C2×C22⋊C8—732no—
366C2×C22⋊C8—1461no—
367C2×C22⋊C8—1461no—
368C2×C22⋊C8—1461no—
369C2×C22⋊C8—1461no—
370C2×C22⋊C8C2×D7311yescharacteristic
371#341—732no—
372#341—732no—
373#341—732no—
374#341C2×D7312yescomplemented
375C25—731no—
376C23×C4—2922no—
377C23×C4—2922no—
378C23×C4—731no—
379C23×C4—731no—
380C23×C4—731no—
381C23×C4—731no—
382C23×C4—731no—
383C23×C4—731no—
384C23×C4—731no—
385C23×C4—731no—
386C23×C4—731no—
387C23×C4—731no—
388C23×C4—731no—
389C23×C4—731no—
390C23×C4—731no—
391C23×C4C22×D7311yescharacteristic
392C2×M16—21no—
393C2×M16—42no—
394C2×M16—2922no—
395C2×M16—2922no—
396C2×M16—2922no—
397C2×M16—2922no—
398C2×M16—2922no—
399C2×M16—734no—
400C2×M16—734no—
401C2×M16—731no—
402C2×M16—732no—
403C2×M16—731no—
404C2×M16—1461no—
405C2×M16—1461no—
406C2×M16—1461no—
407C2×M16C22×D7311yescharacteristic
408C2×M16C22×D7311yescharacteristic
409C2×M16C22×D7312yes—
410C22×C8—21no—
411C22×C8—732no—
412C22×C8—1461no—
413C22×C8—1461no—
414C22×C8—1461no—
415C22×C8C22×D7312yes—
416#108—42no—
417#108—2922no—
418#108—2922no—
419#108—2922no—
420C22⋊C8—21no—
421C22⋊C8—21no—
422C22⋊C8—732no—
423C22⋊C8—732no—
424C22⋊C8—734no—
425C22⋊C8—734no—
426C22⋊C8—1461no—
427C22⋊C8—1461no—
428C22⋊C8—1461no—
429C22⋊C8—1461no—
430C22⋊C8—1462no—
431C22⋊C8—1462no—
432C22⋊C8—1462no—
433C22⋊C8—1462no—
434C22⋊C8—1461no—
435C22⋊C8—1461no—
436C22⋊C8C4×D7312yes—
437C22⋊C8C4×D7312yes—
438C24—2922no—
439C24—2922no—
440C24—2922no—
441C24—731no—
442C24—731no—
443C24—731no—
444C24—731no—
445C24—731no—
446C24—731no—
447C24Q8×D7311yescharacteristic
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—2922no—
451C22×C4—2922no—
452C22×C4—2921no—
453C22×C4—2921no—
454C22×C4—2921no—
455C22×C4—2921no—
456C22×C4—2921no—
457C22×C4—2921no—
458C22×C4—2921no—
459C22×C4—2921no—
460C22×C4—2922no—
461C22×C4—2922no—
462C22×C4—2922no—
463C22×C4—2922no—
464C22×C4—2922no—
465C22×C4—731no—
466C22×C4—731no—
467C22×C4—731no—
468C22×C4—731no—
469C22×C4—731no—
470C22×C4—731no—
471C22×C4—731no—
472C22×C4—731no—
473C22×C4—731no—
474C22×C4—731no—
475C22×C4—731no—
476C22×C4—731no—
477C22×C4—731no—
478C22×C4—731no—
479C22×C4—731no—
480C22×C4—731no—
481C22×C4—731no—
482C22×C4—731no—
483C22×C4—731no—
484C22×C4—731no—
485C22×C4—731no—
486C22×C4—731no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4—1462no—
491C22×C4C2×C4×D7311yescharacteristic
492C22×C4C2×C4×D7311yescharacteristic
493C22×C4D4×D7311yescharacteristic
494C22×C4D4×D7311yescharacteristic
495C22×C4Q8×D7311yescharacteristic
496C22×C4C23×D7311yescharacteristic
497M16—22no—
498M16—22no—
499M16—22no—
500M16—22no—
501M16—42no—
502M16—2924no—
503M16—2924no—
504M16—2922no—
505M16—2922no—
506M16—2922no—
507M16—1462no—
508M16—1462no—
509M16—1462no—
510M16—1462no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—22no—
516C2×C8—42no—
517C2×C8—2922no—
518C2×C8—2922no—
519C2×C8—2922no—
520C2×C8—734no—
521C2×C8—1461no—
522C2×C8—1462no—
523C2×C8—1461no—
524C2×C8—1462no—
525C2×C8—1462no—
526C2×C8C2×C4×D7314yes—
527C23—22no—
528C23—22no—
529C23—41no—
530C23—2921no—
531C23—2921no—
532C23—2921no—
533C23—2921no—
534C23—2921no—
535C23—2921no—
536C23—2921no—
537C23—2921no—
538C23—2921no—
539C23—2921no—
540C23—2921no—
541C23—29216no—
542C23—731no—
543C23—731no—
544C23—731no—
545C23—731no—
546C23—1462no—
547C23—1462no—
548C23—1462no—
549C23—1462no—
550C23—1462no—
551C23—1462no—
552C23—1462no—
553C23C4⋊C4×D7311yescharacteristic
554C23C4⋊C4×D7311yescharacteristic
555C23C2×Q8×D7311yescharacteristic
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—731no—
576C2×C4—731no—
577C2×C4—731no—
578C2×C4—731no—
579C2×C4—1462no—
580C2×C4—1462no—
581C2×C4—1461no—
582C2×C4—1461no—
583C2×C4—1461no—
584C2×C4—1461no—
585C2×C4—1461no—
586C2×C4—1461no—
587C2×C4—1461no—
588C2×C4—1461no—
589C2×C4—1462no—
590C2×C4—1462no—
591C2×C4—1462no—
592C2×C4—1462no—
593C2×C4C4⋊C4×D7311yescharacteristic
594C2×C4C4⋊C4×D7311yescharacteristic
595C2×C4C22×C4×D7311yescharacteristic
596C2×C4C2×D4×D7311yescharacteristic
597C2×C4#1455663612yes—
598C2×C4#1455663712yes—
599C8—24no—
600C8—42no—
601C8—41no—
602C8—41no—
603C8—2921no—
604C8—2921no—
605C8—2922no—
606C8—1464no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—44no—
615C22—2924no—
616C22—2924no—
617C22—2924no—
618C22—2924no—
619C22—2924no—
620C22—2924no—
621C22—732no—
622C22—1461no—
623C22—1461no—
624C22—1461no—
625C22—1461no—
626C22—1461no—
627C22—1461no—
628C22—1461no—
629C22—1461no—
630C22—1461no—
631C22—1461no—
632C22—1461no—
633C22#1798863412yes—
634C22C2×C4⋊C4×D7311yescharacteristic
635C22#1798921212yes—
636C4—21no—
637C4—21no—
638C4—1461no—
639C4—1461no—
640C4—1461no—
641C4—1461no—
642C4#1798905912yes—
643C4#1798914711yescharacteristic
644C4#1798914811yescharacteristic
645C2—22no—
646C2—22no—
647C2—41no—
648C2—2921no—
649C2—2921no—
650C2—2921no—
651C2—1462no—
652C2#2261254211yescharacteristic
653C2#430×D7311yescharacteristic
654C2C2×#1798863411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1855C2
socleC22×C73C22×C4⋊C4
prime radicalC24×D73Q8
cosocleC22×C4×C73C24
hypercenter#1855D73
hypocenterC73#21949
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1855

Complements

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

NH ≅ G/Ntype
4. #22610138C2semidirect
5. #22610137C2semidirect
6. #22610136C2semidirect
7. #22610135C2semidirect
8. #22610134C2semidirect
9. #22609972C2semidirect
10. #22609970C2semidirect
11. C73×#1855C2semidirect
17. C2×#14556527C22semidirect
18. C2×#14556526C22semidirect
19. C2×#14556526C22semidirect
39. #17988352C22semidirect
40. #17988351C22semidirect
41. #17988351C22semidirect
43. C2×#108×C73C22semidirect
46. #341×C73C22semidirect
336. #1855D73semidirect
339. C73#21949semidirect
349. C2×#108C2×D73semidirect
374. #341C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179885502
C2×#179885492
#407×D732
#226101382
#226101372
#226101362
#226101352
#226101342
#226099722
#226099702
C73×#18552
#2194973