Finite Group Databasebeta

← #33014441

Subgroups — #33014441

Order 18688.

1658 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988480C212yesmaximal, complemented
2#22608180C212yesmaximal, complemented
3#22608176C212yesmaximal, complemented
4#22608172C212yesmaximal
5#22608168C212yesmaximal, complemented
6#22608163C212yesmaximal, complemented
7#22608161C211yesmaximal, characteristic, complemented
8#22608158C211yesmaximal, characteristic, complemented
9C2×#390×C73C211yesmaximal, characteristic
10C22×C4⋊Dic73C2212yes—
11C22×C22⋊Dic73C2212yes—
12C22×C22⋊Dic73C2212yes—
13C22×C22⋊Dic73C2212yes—
14C22×C22⋊Dic73C2212yes—
15C22×C4×Dic73C2212yes—
16#17988480—28no—
17C2×#14556498C2212yes—
18C2×#14556498C2211yescharacteristic
19C2×#14556498C2212yes—
20C2×#14556498C2212yes—
21C2×#14556498C2212yes—
22C2×#14556498C2211yescharacteristic
23C2×#14556498C2211yescharacteristic
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C2×#14556498C2211yescharacteristic
27C23×D4×C73C2211yescharacteristic
28C22×C4⋊C4×C73C2211yescharacteristic
29C22×C22⋊C4×C73C2212yes—
30C22×C22⋊C4×C73C2212yes—
31C22×C42×C73C2211yescharacteristic
32#390×C73C418yescomplemented
33C23×Dic73C2312yes—
34C23×Dic73C2312yes—
35C23×Dic73C2312yes—
36C23×Dic73C2312yes—
37C2×C4⋊Dic73—24no—
38C2×C4⋊Dic73—24no—
39C2×C4⋊Dic73—42no—
40C2×C22⋊Dic73—24no—
41C2×C22⋊Dic73—24no—
42C2×C22⋊Dic73—24no—
43C2×C22⋊Dic73—24no—
44C2×C22⋊Dic73—42no—
45C2×C22⋊Dic73—42no—
46C2×C22⋊Dic73—42no—
47C2×C22⋊Dic73—42no—
48C2×C22⋊Dic73—42no—
49C2×C22⋊Dic73—42no—
50C2×C22⋊Dic73—42no—
51C2×C22⋊Dic73—42no—
52C2×C4×Dic73—24no—
53C2×C4×Dic73—42no—
54C2×C4×Dic73—42no—
55#14556498—24no—
56#14556498—22no—
57#14556498—24no—
58#14556498—24no—
59#14556498—42no—
60#14556498—42no—
61#14556498—42no—
62#14556498—41no—
63#14556498—41no—
64#14556498D414yescomplemented
65C25×C73C2312yes—
66C22×D4×C73—22no—
67C22×D4×C73—22no—
68C22×D4×C73—42no—
69C22×D4×C73—42no—
70C22×D4×C73C2×C414yes—
71C23×C4×C73C2311yescharacteristic
72C23×C4×C73C2312yes—
73C23×C4×C73C2311yescharacteristic
74C23×C4×C73C2311yescharacteristic
75C2×C4⋊C4×C73—42no—
76C2×C4⋊C4×C73C2×C414yes—
77C2×C22⋊C4×C73—24no—
78C2×C22⋊C4×C73—24no—
79C2×C22⋊C4×C73—42no—
80C2×C22⋊C4×C73—42no—
81C2×C22⋊C4×C73C2×C418yes—
82C2×C22⋊C4×C73C2×C418yes—
83C2×C42×C73—42no—
84C2×C42×C73C2×C414yes—
85C22×Dic73—22no—
86C22×Dic73—22no—
87C22×Dic73—22no—
88C22×Dic73—22no—
89C22×Dic73—22no—
90C22×Dic73—22no—
91C22×Dic73—22no—
92C22×Dic73—22no—
93C22×Dic73—22no—
94C22×Dic73—22no—
95C22×Dic73—22no—
96C22×Dic73—22no—
97C22×Dic73—22no—
98C22×Dic73—22no—
99C22×Dic73—22no—
100C22×Dic73—22no—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73C2×D414yes—
112C22×Dic73#4114yes—
113C4⋊Dic73—48no—
114C22⋊Dic73—48no—
115C22⋊Dic73—48no—
116C22⋊Dic73—48no—
117C22⋊Dic73—48no—
118C4×Dic73—48no—
119C24×C73—22no—
120C24×C73—22no—
121C24×C73—22no—
122C24×C73—22no—
123C24×C73—22no—
124C24×C73—22no—
125C24×C73—22no—
126C24×C73—22no—
127C24×C73—22no—
128C24×C73—22no—
129C24×C73—22no—
130C24×C73—22no—
131C24×C73—22no—
132C24×C73—22no—
133C24×C73C22×C414yes—
134C24×C73C2411yescharacteristic
135C2×D4×C73—28no—
136C2×D4×C73—44no—
137C2×D4×C73—44no—
138C2×D4×C73—48no—
139C2×D4×C73—48no—
140C22×C4×C73—22no—
141C22×C4×C73—22no—
142C22×C4×C73—22no—
143C22×C4×C73—22no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73—22no—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—21no—
153C22×C4×C73—21no—
154C22×C4×C73C22×C412yes—
155C22×C4×C73C22×C414yes—
156C22×C4×C73C22×C412yes—
157C22×C4×C73C22×C412yes—
158C22×C4×C73C2×D412yes—
159C22×C4×C73#4112yes—
160C22×C4×C73#4112yes—
161C22×C4×C73#4112yes—
162C4⋊C4×C73—44no—
163C22⋊C4×C73—216no—
164C22⋊C4×C73—48no—
165C42×C73—44no—
166C23×C73—24no—
167C23×C73—24no—
168C23×C73—24no—
169C23×C73—24no—
170C23×C73—24no—
171C23×C73—24no—
172C23×C73—24no—
173C23×C73—42no—
174C23×C73—42no—
175C23×C73—42no—
176C23×C73—42no—
177C23×C73—42no—
178C23×C73—42no—
179C23×C73—42no—
180C23×C73—42no—
181C23×C73—42no—
182C23×C73—42no—
183C23×C73—42no—
184C23×C73—42no—
185C23×C73—42no—
186C23×C73—42no—
187C23×C73—42no—
188C23×C73—42no—
189C23×C73—42no—
190C23×C73—42no—
191C23×C73—42no—
192C23×C73—42no—
193C23×C73—42no—
194C23×C73—42no—
195C23×C73—42no—
196C23×C73—42no—
197C23×C73—42no—
198C23×C73—42no—
199C23×C73—42no—
200C23×C73—42no—
201C23×C73C23×C411yescharacteristic
202C23×C73C22×D411yescharacteristic
203C23×C73C2×#4112yes—
204C23×C73C2×#4111yescharacteristic
205C23×C73C2×#4111yescharacteristic
206C23×C73C2×#4111yescharacteristic
207C23×C73H(5,𝔽2)12yes—
208C23×C73H(5,𝔽2)12yes—
209C23×C73H(5,𝔽2)12yes—
210C23×C73H(5,𝔽2)11yescharacteristic
211C23×C732−1+411yescharacteristic
212D4×C73—416no—
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—24no—
218C2×C4×C73—24no—
219C2×C4×C73—24no—
220C2×C4×C73—24no—
221C2×C4×C73—24no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—24no—
225C2×C4×C73—24no—
226C2×C4×C73—22no—
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#11714yes—
234C2×C4×C73C4×D414yes—
235C2×Dic73—24no—
236C2×Dic73—24no—
237C2×Dic73—24no—
238C2×Dic73—24no—
239C2×Dic73—24no—
240C2×Dic73—24no—
241C2×Dic73—42no—
242C2×Dic73—42no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—42no—
249C2×Dic73—42no—
250C2×Dic73—42no—
251C2×Dic73—42no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—42no—
255C2×Dic73—42no—
256C2×Dic73—42no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C22×C73—44no—
264C22×C73—44no—
265C22×C73—44no—
266C22×C73—44no—
267C22×C73—44no—
268C22×C73—44no—
269C22×C73—44no—
270C22×C73—82no—
271C22×C73—82no—
272C22×C73—82no—
273C22×C73—82no—
274C22×C73—82no—
275C22×C73—82no—
276C22×C73—82no—
277C22×C73—82no—
278C22×C73C2×#11711yescharacteristic
279C22×C73C2×C4×D411yescharacteristic
280C22×C73C4×#4111yescharacteristic
281C22×C73#51811yescharacteristic
282C22×C73#51811yescharacteristic
283C22×C73#51811yescharacteristic
284C22×C73#52011yescharacteristic
285C22×C73#53512yes—
286C22×C73#53812yes—
287C22×C73#53812yes—
288C22×C73#53812yes—
289C22×C73#54012yes—
290C22×C73#54212yes—
291C22×C73#54612yes—
292C22×C73#54711yescharacteristic
293C22×C73#55011yescharacteristic
294C22×C73#55312yes—
295C22×C73#55312yes—
296C22×C73#55511yescharacteristic
297C22×C73#55611yescharacteristic
298C22×C73#55912yes—
299C22×C73#56012yes—
300C22×C73#56212yes—
301C4×C73—24no—
302C4×C73—24no—
303C4×C73—44no—
304C4×C73—42no—
305Dic73—44no—
306Dic73—44no—
307Dic73—82no—
308Dic73—82no—
309#14444—731nomaximal, Sylow
310C2×C73—84no—
311C2×C73#231211yescharacteristic
312C2×C73#232811yescharacteristic
313C2×C73#233711yescharacteristic
314C2×C73#235411yescharacteristic
315C2×C73#235711yescharacteristic
316C2×C73#236411yescharacteristic
317C2×C73#237711yescharacteristic
318C2×C73#271012yes—
319C2×C73#271412yes—
320C2×C73#280012yes—
321C2×C73#280312yes—
322C2×#390D7311yescharacteristic
323C2×#388—732no—
324#1440—732no—
325#1439—732no—
326#1438—732no—
327#1437—732no—
328#1436—732no—
329C4⋊#95—731no—
330#1434—731no—
331C73#1444411yescharacteristic, complemented, Sylow
332C23×D4C2×D7311yescharacteristic
333C22×C4⋊C4—732no—
334C22×C4⋊C4C2×D7311yescharacteristic
335C22×C22⋊C4—732no—
336C22×C22⋊C4—732no—
337C22×C22⋊C4—732no—
338C22×C22⋊C4—732no—
339C22×C22⋊C4C2×D7312yes—
340C22×C22⋊C4C2×D7312yes—
341C22×C42—732no—
342C22×C42C2×D7311yescharacteristic
343#390Dic7318yescomplemented
344#388—1468no—
345C2×#95—732no—
346C2×#95—731no—
347C2×#95—731no—
348C2×#95—732no—
349C2×#95—732no—
350C2×#95—731no—
351C2×#95—732no—
352C2×#95—731no—
353C2×#95—732no—
354C2×#95—732no—
355C25C22×D7312yes—
356C22×D4—22no—
357C22×D4—22no—
358C22×D4—42no—
359C22×D4—42no—
360C22×D4C2×Dic7314yes—
361C23×C4—732no—
362C23×C4—732no—
363C23×C4—732no—
364C23×C4—732no—
365C23×C4C22×D7312yes—
366C23×C4C22×D7311yescharacteristic
367C23×C4C22×D7311yescharacteristic
368C23×C4C22×D7311yescharacteristic
369C2×C4⋊C4—42no—
370C2×C4⋊C4—2922no—
371C2×C4⋊C4—1464no—
372C2×C4⋊C4—1464no—
373C2×C4⋊C4C2×Dic7314yes—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—42no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—2922no—
379C2×C22⋊C4—2922no—
380C2×C22⋊C4—2922no—
381C2×C22⋊C4—2922no—
382C2×C22⋊C4—2922no—
383C2×C22⋊C4—2922no—
384C2×C22⋊C4—2922no—
385C2×C22⋊C4—2922no—
386C2×C22⋊C4—1464no—
387C2×C22⋊C4—1464no—
388C2×C22⋊C4—1464no—
389C2×C22⋊C4—1464no—
390C2×C22⋊C4C2×Dic7318yes—
391C2×C22⋊C4C2×Dic7318yes—
392C2×C42—42no—
393C2×C42—2922no—
394C2×C42—2922no—
395C2×C42—1464no—
396C2×C42C2×Dic7314yes—
397#95—2922no—
398#95—2922no—
399#95—2922no—
400#95—2921no—
401#95—2921no—
402#95—734no—
403#95—1464no—
404#95—1462no—
405#95—1464no—
406#95—1464no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24C22×Dic7314yes—
422C24C23×D7311yescharacteristic
423C2×D4—28no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—48no—
427C2×D4—48no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—734no—
443C22×C4—734no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4—1462no—
448C22×C4—1462no—
449C22×C4—1462no—
450C22×C4—1462no—
451C22×C4—1462no—
452C22×C4—1462no—
453C22×C4—1462no—
454C22×C4—1462no—
455C22×C4—1462no—
456C22×C4—1462no—
457C22×C4—1462no—
458C22×C4—1462no—
459C22×C4—1462no—
460C22×C4—1462no—
461C22×C4—1462no—
462C22×C4—1462no—
463C22×C4—1462no—
464C22×C4—1462no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4—1462no—
470C22×C4C22×Dic7312yes—
471C22×C4C22×Dic7314yes—
472C22×C4C22×Dic7312yes—
473C22×C4C22×Dic7312yes—
474C22×C4D4×D7312yes—
475C22×C4#1192402712yes—
476C22×C4#1192402812yes—
477C22×C4#1192402812yes—
478C4⋊C4—44no—
479C4⋊C4—2928no—
480C22⋊C4—216no—
481C22⋊C4—48no—
482C22⋊C4—2928no—
483C22⋊C4—2928no—
484C22⋊C4—2928no—
485C22⋊C4—2928no—
486C42—44no—
487C42—2928no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23C23×Dic7311yescharacteristic
524C23C2×D4×D7311yescharacteristic
525C23#41×D7312yes—
526C23C2×#1192402711yescharacteristic
527C23C2×#1192402811yescharacteristic
528C23C2×#1192402811yescharacteristic
529C23#1455663612yes—
530C23#1455663612yes—
531C23#1455663612yes—
532C23#1455663711yescharacteristic
533C23#1455663811yescharacteristic
534D4—416no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—2922no—
556C2×C4—2922no—
557C2×C4—2922no—
558C2×C4—2922no—
559C2×C4—2922no—
560C2×C4—2922no—
561C2×C4—2922no—
562C2×C4—2922no—
563C2×C4—2922no—
564C2×C4—2922no—
565C2×C4—2922no—
566C2×C4—2922no—
567C2×C4—2922no—
568C2×C4—2922no—
569C2×C4—2922no—
570C2×C4—2922no—
571C2×C4—2922no—
572C2×C4—2922no—
573C2×C4—2922no—
574C2×C4—2922no—
575C2×C4—2922no—
576C2×C4—2922no—
577C2×C4—1464no—
578C2×C4—1464no—
579C2×C4—1464no—
580C2×C4—1464no—
581C2×C4—1464no—
582C2×C4—1464no—
583C2×C4#1455655014yes—
584C2×C4D4×Dic7314yes—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22C2×#1455655011yescharacteristic
601C22C2×D4×Dic7311yescharacteristic
602C22#41×Dic7311yescharacteristic
603C22#1798906211yescharacteristic
604C22#1798906211yescharacteristic
605C22#1798906211yescharacteristic
606C22#1798907011yescharacteristic
607C22#1798915012yes—
608C22#1798917112yes—
609C22#1798917512yes—
610C22#1798917512yes—
611C22#1798919112yes—
612C22#1798920112yes—
613C22#1798922612yes—
614C22#1798924211yescharacteristic
615C22#1798925811yescharacteristic
616C22#1798928012yes—
617C22#1798928712yes—
618C22#1798930211yescharacteristic
619C22#1798930711yescharacteristic
620C22#1798932012yes—
621C22#1798932512yes—
622C22#1798933212yes—
623C4—24no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—2924no—
628C4—2924no—
629C4—5842no—
630C4—5842no—
631C2—84no—
632C2#2261252911yescharacteristic
633C2#2261265711yescharacteristic
634C2#2261274611yescharacteristic
635C2#2261288011yescharacteristic
636C2#2261290411yescharacteristic
637C2#2261298311yescharacteristic
638C2#2261308811yescharacteristic
639C2#2261667012yes—
640C2#2261672412yes—
641C2#2261782812yes—
642C2#2261786512yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×#390×C73C2
socle, cosocleC24×C73C24
prime radicalC23×D4×C73C22
hypercenterC2×#390D73
hypocenterC73#14444
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C2×#17988480C2semidirect
2. #22608180C2semidirect
3. #22608176C2semidirect
5. #22608168C2semidirect
6. #22608163C2semidirect
7. #22608161C2semidirect
8. #22608158C2semidirect
32. #390×C73C4semidirect
64. #14556498D4semidirect
331. C73#14444semidirect
343. #390Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179884802
#226081802
#226081762
#226081722
#226081682
#226081632
#226081612
#226081582
C2×#390×C732
#1444473