Finite Group Databasebeta

← #13280563

Subgroups — #13280563

Order 1792.

1830 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6450C211yesmaximal, characteristic, complemented
2C2×#89719C211yesmaximal, characteristic
3C2×#89717C212yesmaximal
4C2×#89723C211yesmaximal, characteristic
5C7×#1439C211yesmaximal, characteristic
6#11729322C211yesmaximal, characteristic, complemented
7#11729301C212yesmaximal
8#11727178C211yesmaximal, characteristic, complemented
9#11727174C212yesmaximal, complemented
10#11727157C212yesmaximal, complemented
11#11727155C211yesmaximal, characteristic, complemented
12C23×Dic14C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2211yescharacteristic
15C22×C7×C22⋊C4C2212yes—
16C22×C22⋊Dic7C2211yescharacteristic
17C22×C22⋊Dic7C2212yes—
18C22×C22⋊Dic7C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
20C22×Dic7⋊C4C2212yes—
21C22×Dic7⋊C4C2212yes—
22C22×C4×Dic7C2211yescharacteristic
23C22×C4×Dic7C2212yes—
24C2×#6450—28no—
25C2×#6450—24no—
26C2×C7×#95C2211yescharacteristic
27C2×C7×#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2211yescharacteristic
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2211yescharacteristic
36C2×C7⋊#95C2212yes—
37#89723—24no—
38#89719C418yescomplemented
39#89717—28no—
40#14301—71nomaximal, Sylow
41C25×C7C2311yescharacteristic
42C23×C4×C7C2312yes—
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2312yes—
45C23×C4×C7C2311yescharacteristic
46C23×Dic7C2311yescharacteristic
47C23×Dic7C2311yescharacteristic
48C23×Dic7C2312yes—
49C23×Dic7C2311yescharacteristic
50C23×Dic7C2312yes—
51C23×Dic7C2311yescharacteristic
52C22×Dic14—22no—
53C22×Dic14—22no—
54C22×Dic14—24no—
55C22×Dic14—24no—
56C22×Dic14C2×C414yes—
57C2×C7×C4⋊C4—24no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C22⋊C4—22no—
60C2×C7×C22⋊C4—22no—
61C2×C7×C22⋊C4—24no—
62C2×C7×C22⋊C4—42no—
63C2×C7×C22⋊C4—42no—
64C2×C7×C22⋊C4C2×C414yes—
65C2×C22⋊Dic7—24no—
66C2×C22⋊Dic7—24no—
67C2×C22⋊Dic7—24no—
68C2×C22⋊Dic7—22no—
69C2×C22⋊Dic7—22no—
70C2×C22⋊Dic7—42no—
71C2×C22⋊Dic7C2×C418yes—
72C2×C4⋊Dic7—22no—
73C2×C4⋊Dic7—42no—
74C2×Dic7⋊C4—24no—
75C2×Dic7⋊C4—24no—
76C2×Dic7⋊C4—24no—
77C2×Dic7⋊C4—24no—
78C2×Dic7⋊C4—42no—
79C2×Dic7⋊C4—42no—
80C2×C4×Dic7—24no—
81C2×C4×Dic7—24no—
82C2×C4×Dic7—24no—
83C2×C4×Dic7D418yes—
84C2×C4×Dic7D414yes—
85#6450—232no—
86C7×#95—22no—
87C7×#95—42no—
88C7⋊#95—22no—
89C7⋊#95—24no—
90C7⋊#95—24no—
91C7⋊#95—22no—
92C7⋊#95—24no—
93C7⋊#95—42no—
94C7⋊#95—42no—
95C7⋊#95C2×C414yes—
96C7⋊#95C2×C418yes—
97C22×C4×Q8—71no—
98C2×#393—71no—
99C2×#388—71no—
100C2×#388—72no—
101#1446—71no—
102#1444—72no—
103#1444—71no—
104#1439—72no—
105#1439D711yescharacteristic
106#1434—71no—
107#1434—72no—
108C24×C7—22no—
109C24×C7—21no—
110C24×C7—22no—
111C24×C7—21no—
112C24×C7—22no—
113C24×C7—21no—
114C24×C7—21no—
115C24×C7—21no—
116C24×C7—21no—
117C24×C7—22no—
118C24×C7C22×C412yes—
119C24×C7C2411yescharacteristic
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—21no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—21no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7C22×C412yes—
140C22×C4×C7C22×C412yes—
141C22×C4×C7C2×D414yes—
142C22×C4×C7C2×D412yes—
143C22×C4×C7#4112yes—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—21no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—22no—
163C22×Dic7—21no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—21no—
167C22×Dic7—22no—
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—21no—
171C22×Dic7C22×C414yes—
172C22×Dic7C22×C414yes—
173C22×Dic7C2×D412yes—
174C22×Dic7C2×D414yes—
175C22×Dic7C2×D414yes—
176C22×Dic7C2×D412yes—
177C22×Dic7#4114yes—
178C2×Dic14—28no—
179C2×Dic14—216no—
180C2×Dic14—28no—
181C2×Dic14—28no—
182C2×Dic14—216no—
183C7×C4⋊C4—28no—
184C7×C22⋊C4—28no—
185C7×C22⋊C4—48no—
186C22⋊Dic7—216no—
187C22⋊Dic7—44no—
188C22⋊Dic7—44no—
189C4⋊Dic7—44no—
190Dic7⋊C4—216no—
191Dic7⋊C4—48no—
192C4×Dic7—216no—
193C4×Dic7—28no—
194C23×Q8—71no—
195C2×C4×Q8—144no—
196C2×C4×Q8—148no—
197C22×C4⋊C4—72no—
198C22×C4⋊C4—72no—
199C22×C4⋊C4—71no—
200C22×C4⋊C4C2×D711yescharacteristic
201C22×C22⋊C4—72no—
202C22×C22⋊C4—71no—
203C22×C22⋊C4—71no—
204C22×C22⋊C4C2×D711yescharacteristic
205C22×C22⋊C4C2×D712yes—
206C22×C42—72no—
207C22×C42—71no—
208#393—78no—
209#388—148no—
210#388—144no—
211C2×#95—72no—
212C2×#95—71no—
213C2×#95—72no—
214C2×#95—71no—
215C2×#95—72no—
216C2×#95—72no—
217C2×#95—71no—
218C2×#95—72no—
219C2×#95—72no—
220C2×#95C2×D711yescharacteristic
221C2×#95C2×D712yes—
222C23×C7—24no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—24no—
227C23×C7—41no—
228C23×C7—42no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—42no—
232C23×C7—41no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—41no—
236C23×C7—42no—
237C23×C7—41no—
238C23×C7—42no—
239C23×C7—41no—
240C23×C7—41no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7C23×C411yescharacteristic
246C23×C7C22×D411yescharacteristic
247C23×C7C22×D412yes—
248C23×C7C2×#4111yescharacteristic
249C23×C7C2×#4112yes—
250C23×C7H(5,𝔽2)12yes—
251C23×C7H(5,𝔽2)11yescharacteristic
252C23×C72−1+411yescharacteristic
253C23×C72−1+411yescharacteristic
254C23×C72−1+411yescharacteristic
255C23×C72−1+412yes—
256C2×C4×C7—24no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—24no—
260C2×C4×C7—22no—
261C2×C4×C7—24no—
262C2×C4×C7—24no—
263C2×C4×C7—24no—
264C2×C4×C7—24no—
265C2×C4×C7—41no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—41no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7C4×D414yes—
280C2×Dic7—24no—
281C2×Dic7—24no—
282C2×Dic7—24no—
283C2×Dic7—24no—
284C2×Dic7—24no—
285C2×Dic7—24no—
286C2×Dic7—22no—
287C2×Dic7—24no—
288C2×Dic7—24no—
289C2×Dic7—22no—
290C2×Dic7—24no—
291C2×Dic7—24no—
292C2×Dic7—24no—
293C2×Dic7—22no—
294C2×Dic7—22no—
295C2×Dic7—22no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7—42no—
299C2×Dic7—42no—
300C2×Dic7—42no—
301C2×Dic7—42no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—42no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—42no—
309C2×Dic7—41no—
310C2×Dic7—42no—
311C2×Dic7—42no—
312C2×Dic7C4×D418yes—
313C2×Dic7#12014yes—
314Dic14—232no—
315C25C22×D711yescharacteristic
316C22×Q8—74no—
317C22×Q8—144no—
318C22×Q8—144no—
319C22×Q8—142no—
320C22×Q8—142no—
321C23×C4—72no—
322C23×C4—71no—
323C23×C4—71no—
324C23×C4—72no—
325C23×C4—71no—
326C23×C4—71no—
327C23×C4C22×D712yes—
328C23×C4C22×D711yescharacteristic
329C23×C4C22×D712yes—
330C23×C4C22×D711yescharacteristic
331C4×Q8—1432no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—144no—
335C2×C4⋊C4—144no—
336C2×C4⋊C4—144no—
337C2×C4⋊C4—142no—
338C2×C4⋊C4—144no—
339C2×C4⋊C4—282no—
340C2×C4⋊C4—282no—
341C2×C4⋊C4—282no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—42no—
346C2×C22⋊C4—42no—
347C2×C22⋊C4—78no—
348C2×C22⋊C4—144no—
349C2×C22⋊C4—144no—
350C2×C22⋊C4—144no—
351C2×C22⋊C4—142no—
352C2×C22⋊C4—142no—
353C2×C22⋊C4—282no—
354C2×C22⋊C4C4×D714yes—
355C2×C42—78no—
356C2×C42—74no—
357C2×C42—144no—
358C2×C42—144no—
359C2×C42—144no—
360#95—22no—
361#95—42no—
362#95—78no—
363#95—74no—
364#95—144no—
365#95—142no—
366#95—144no—
367#95—144no—
368#95—142no—
369#95—282no—
370#95—282no—
371C22×C7—42no—
372C22×C7—44no—
373C22×C7—44no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7—81no—
377C22×C7—81no—
378C22×C7—82no—
379C22×C7—81no—
380C22×C7—81no—
381C22×C7—82no—
382C22×C7C2×C4×D411yescharacteristic
383C22×C7C2×C4×D412yes—
384C22×C7#51911yescharacteristic
385C22×C7#52011yescharacteristic
386C22×C7#52012yes—
387C22×C7C2×#12011yescharacteristic
388C22×C7#52512yes—
389C22×C7#52511yescharacteristic
390C22×C7#53511yescharacteristic
391C22×C7#53512yes—
392C22×C7#53611yescharacteristic
393C22×C7#53612yes—
394C22×C7#53612yes—
395C22×C7#53611yescharacteristic
396C22×C7#53712yes—
397C22×C7#53711yescharacteristic
398C22×C7#54112yes—
399C22×C7#54111yescharacteristic
400C22×C7#54211yescharacteristic
401C22×C7#54212yes—
402C22×C7#54212yes—
403C22×C7#54212yes—
404C22×C7#54212yes—
405C4×C7—24no—
406C4×C7—42no—
407C4×C7—44no—
408C4×C7—82no—
409Dic7—24no—
410Dic7—28no—
411Dic7—44no—
412Dic7—42no—
413Dic7—81no—
414Dic7—81no—
415C24—21no—
416C24—22no—
417C24—21no—
418C24—21no—
419C24—22no—
420C24—21no—
421C24—22no—
422C24—21no—
423C24—21no—
424C24—22no—
425C24C2×C4×D712yes—
426C24C23×D711yescharacteristic
427C2×Q8—1416no—
428C2×Q8—148no—
429C2×Q8—1416no—
430C2×Q8—148no—
431C2×Q8—148no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
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—74no—
452C22×C4—72no—
453C22×C4—74no—
454C22×C4—74no—
455C22×C4—74no—
456C22×C4—74no—
457C22×C4—72no—
458C22×C4—142no—
459C22×C4—142no—
460C22×C4—142no—
461C22×C4—141no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4—141no—
465C22×C4—142no—
466C22×C4—142no—
467C22×C4—141no—
468C22×C4—142no—
469C22×C4—142no—
470C22×C4—142no—
471C22×C4—142no—
472C22×C4—142no—
473C22×C4—142no—
474C22×C4—141no—
475C22×C4—141no—
476C22×C4—141no—
477C22×C4—142no—
478C22×C4—142no—
479C22×C4—142no—
480C22×C4—141no—
481C22×C4—141no—
482C22×C4—142no—
483C22×C4—141no—
484C22×C4—141no—
485C22×C4C2×C4×D712yes—
486C22×C4C2×C4×D712yes—
487C22×C4D4×D714yes—
488C22×C4D4×D712yes—
489C22×C4#115912yes—
490C4⋊C4—28no—
491C4⋊C4—1416no—
492C4⋊C4—288no—
493C4⋊C4—284no—
494C22⋊C4—28no—
495C22⋊C4—48no—
496C22⋊C4—1416no—
497C22⋊C4—284no—
498C22⋊C4—284no—
499C42—148no—
500C42—1416no—
501C2×C7—82no—
502C2×C7C4×#12011yescharacteristic
503C2×C7#232311yescharacteristic
504C2×C7#232312yes—
505C2×C7#232612yes—
506C2×C7#232611yescharacteristic
507C2×C7#261411yescharacteristic
508C2×C7#261511yescharacteristic
509C2×C7#263212yes—
510C2×C7#263211yescharacteristic
511C2×C7#263711yescharacteristic
512C2×C7#263712yes—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23C22×C4×D711yescharacteristic
537C23C2×D4×D711yescharacteristic
538C23C2×D4×D712yes—
539C23C2×#115911yescharacteristic
540C23#654812yes—
541C23#654811yescharacteristic
542C23#655111yescharacteristic
543C23D7×#4112yes—
544C23#655411yescharacteristic
545C23#655412yes—
546C23#655411yescharacteristic
547Q8—1432no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—78no—
572C2×C4—74no—
573C2×C4—144no—
574C2×C4—144no—
575C2×C4—144no—
576C2×C4—144no—
577C2×C4—144no—
578C2×C4—142no—
579C2×C4—144no—
580C2×C4—144no—
581C2×C4—144no—
582C2×C4—142no—
583C2×C4—142no—
584C2×C4—144no—
585C2×C4—144no—
586C2×C4—144no—
587C2×C4—142no—
588C2×C4—142no—
589C2×C4—282no—
590C2×C4—282no—
591C2×C4—282no—
592C2×C4—282no—
593C2×C4—282no—
594C2×C4—282no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—282no—
599C2×C4—282no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—282no—
603C2×C4—281no—
604C2×C4—282no—
605C2×C4(C22⋊D7)⋊C414yes—
606C7#1430111yescharacteristic, complemented, Sylow
607C22—42no—
608C22—44no—
609C22—42no—
610C22—44no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—82no—
615C22—81no—
616C22—81no—
617C22—82no—
618C22C2×(C22⋊D7)⋊C411yescharacteristic
619C22#9019511yescharacteristic
620C22#9020711yescharacteristic
621C22C4×D4×D712yes—
622C22#9023712yes—
623C22D7×#12011yescharacteristic
624C22#9028312yes—
625C22#9028812yes—
626C22#9029411yescharacteristic
627C22#9029412yes—
628C22#9029712yes—
629C22#9029911yescharacteristic
630C22#9031611yescharacteristic
631C22#9031911yescharacteristic
632C22#9031912yes—
633C22#9033112yes—
634C22#9034711yescharacteristic
635C22#9035312yes—
636C22#9035312yes—
637C22#9036612yes—
638C22#9036611yescharacteristic
639C22#9039212yes—
640C22#9040611yescharacteristic
641C4—24no—
642C4—44no—
643C4—42no—
644C4—82no—
645C4—144no—
646C4—148no—
647C4—284no—
648C4—282no—
649C4—561no—
650C4—561no—
651C2—82no—
652C2#1173131611yescharacteristic
653C2#1173183112yes—
654C2#1173210411yescharacteristic
655C2#1173290412yes—
656C2#1173374211yescharacteristic
657C2#1173414211yescharacteristic
658C2#1173431511yescharacteristic
659C2#1173442212yes—
660C2#1173478112yes—
661C2#1173498911yescharacteristic
662C2#1173549511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1439C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenter#1439D7
hypocenterC7#14301
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C22×#6450C2semidirect
6. #11729322C2semidirect
8. #11727178C2semidirect
9. #11727174C2semidirect
10. #11727157C2semidirect
11. #11727155C2semidirect
38. #89719C4semidirect
606. C7#14301semidirect

Maximal subgroups

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

subgroupindex
C22×#64502
C2×#897192
C2×#897172
C2×#897232
C7×#14392
#117293222
#117293012
#117271782
#117271742
#117271572
#117271552
#143017