Finite Group Databasebeta

← #13285150

Subgroups — #13285150

Order 1792.

1770 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6442C211yesmaximal, characteristic
2C2×C42×Dic7C211yesmaximal, characteristic, complemented
3C2×#89423C211yesmaximal, characteristic, complemented
4C7×#1437C211yesmaximal, characteristic
5#11729299C211yesmaximal, characteristic
6#11729299C212yesmaximal
7#11729299C212yesmaximal
8#11729290C211yesmaximal, characteristic, complemented
9#11727161C212yesmaximal, complemented
10#11727160C212yesmaximal, complemented
11#11727159C211yesmaximal, characteristic, complemented
12C22×C7×C22⋊C4C2212yes—
13C22×C7×C22⋊C4C2211yescharacteristic
14C22×C42×C7C2211yescharacteristic
15C22×C22⋊Dic7C2212yes—
16C22×C22⋊Dic7C2211yescharacteristic
17C22×C22⋊Dic7C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C22×Dic7⋊C4C2211yescharacteristic
20C22×Dic7⋊C4C2211yescharacteristic
21C22×C4×Dic7C2211yescharacteristic
22C22×C4×Dic7C2212yes—
23C22×C4×Dic7C2212yes—
24C22×C4×Dic7C2211yescharacteristic
25C2×#6442—24no—
26C2×#6442—24no—
27C2×#6442C418yescomplemented
28C2×C7×#95C2211yescharacteristic
29C2×C7×#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2211yescharacteristic
38C42×Dic7—24no—
39#89423—24no—
40#14711—71nomaximal, Sylow
41C25×C7C2311yescharacteristic
42C23×C4×C7C2312yes—
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2312yes—
46C23×Dic7C2312yes—
47C23×Dic7C2311yescharacteristic
48C23×Dic7C2311yescharacteristic
49C23×Dic7C2311yescharacteristic
50C23×Dic7C2311yescharacteristic
51C23×Dic7C2312yes—
52C2×C7×C22⋊C4—22no—
53C2×C7×C22⋊C4—22no—
54C2×C7×C22⋊C4—24no—
55C2×C7×C22⋊C4—24no—
56C2×C7×C22⋊C4—24no—
57C2×C7×C22⋊C4C2×C414yes—
58C2×C42×C7—22no—
59C2×C42×C7—24no—
60C2×C22⋊Dic7—22no—
61C2×C22⋊Dic7—22no—
62C2×C22⋊Dic7—22no—
63C2×C22⋊Dic7—22no—
64C2×C22⋊Dic7—24no—
65C2×C22⋊Dic7—24no—
66C2×C22⋊Dic7—24no—
67C2×C22⋊Dic7C2×C414yes—
68C2×C22⋊Dic7C2×C414yes—
69C2×C4⋊Dic7—22no—
70C2×C4⋊Dic7—22no—
71C2×C4⋊Dic7C2×C414yes—
72C2×Dic7⋊C4—22no—
73C2×Dic7⋊C4—22no—
74C2×Dic7⋊C4—22no—
75C2×Dic7⋊C4—22no—
76C2×Dic7⋊C4C2×C414yes—
77C2×Dic7⋊C4C2×C414yes—
78C2×C4×Dic7—22no—
79C2×C4×Dic7—24no—
80C2×C4×Dic7—24no—
81C2×C4×Dic7—24no—
82C2×C4×Dic7—24no—
83C2×C4×Dic7—24no—
84C2×C4×Dic7—24no—
85C2×C4×Dic7—24no—
86C2×C4×Dic7C2×C414yes—
87C2×C4×Dic7D414yes—
88C2×C4×Dic7D414yes—
89#6442—232no—
90C7×#95—22no—
91C7×#95—24no—
92C7⋊#95—22no—
93C7⋊#95—24no—
94C7⋊#95—24no—
95C7⋊#95—24no—
96C7⋊#95—24no—
97C7⋊#95—24no—
98C7⋊#95—24no—
99C7⋊#95—24no—
100C22×#126—71no—
101C2×#383—71no—
102C2×C43—71no—
103#1437—72no—
104#1437—72no—
105#1437—71no—
106#1437D711yescharacteristic
107#1436—71no—
108#1436—71no—
109#1436—72no—
110#1436—72no—
111C24×C7—21no—
112C24×C7—21no—
113C24×C7—22no—
114C24×C7—22no—
115C24×C7—22no—
116C24×C7—22no—
117C24×C7—22no—
118C24×C7—22no—
119C24×C7C22×C412yes—
120C24×C7C2411yescharacteristic
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
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×C7C22×C412yes—
139C22×C4×C7C22×C412yes—
140C22×C4×C7C2×D412yes—
141C22×C4×C7C2×D412yes—
142C22×C4×C7#4114yes—
143C22×C4×C7#4114yes—
144C22×C4×C7#4114yes—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7C22×C412yes—
169C22×Dic7C22×C412yes—
170C22×Dic7C22×C412yes—
171C22×Dic7C22×C412yes—
172C22×Dic7C2×D412yes—
173C22×Dic7C2×D412yes—
174C22×Dic7C2×D412yes—
175C22×Dic7C2×D412yes—
176C22×Dic7#4114yes—
177C22×Dic7#4114yes—
178C22×Dic7#4114yes—
179C7×C22⋊C4—28no—
180C7×C22⋊C4—48no—
181C42×C7—28no—
182C22⋊Dic7—28no—
183C22⋊Dic7—28no—
184C22⋊Dic7—48no—
185C4⋊Dic7—28no—
186Dic7⋊C4—28no—
187Dic7⋊C4—28no—
188C4×Dic7—216no—
189C4×Dic7—216no—
190C4×Dic7—28no—
191C4×Dic7C22⋊C4116yescomplemented
192C2×#126—78no—
193C2×#126—144no—
194C2×#126—144no—
195C22×C4⋊C4—71no—
196C22×C4⋊C4—71no—
197C22×C4⋊C4—71no—
198C22×C22⋊C4—71no—
199C22×C22⋊C4—71no—
200C22×C22⋊C4—72no—
201C22×C22⋊C4C2×D712yes—
202C22×C22⋊C4C2×D711yescharacteristic
203C22×C42—71no—
204C22×C42—72no—
205C22×C42—72no—
206C22×C42—71no—
207C22×C42C2×D711yescharacteristic
208#383—144no—
209C2×#95—72no—
210C2×#95—72no—
211C2×#95—72no—
212C2×#95—72no—
213C2×#95—72no—
214C2×#95—72no—
215C2×#95—72no—
216C2×#95—71no—
217C2×#95C2×D712yes—
218C2×#95C2×D711yescharacteristic
219C43—144no—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—22no—
223C23×C7—24no—
224C23×C7—24no—
225C23×C7—42no—
226C23×C7—41no—
227C23×C7—42no—
228C23×C7—41no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—41no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—41no—
241C23×C7C23×C411yescharacteristic
242C23×C7C22×D411yescharacteristic
243C23×C7C22×D411yescharacteristic
244C23×C7C2×#4112yes—
245C23×C7C2×#4112yes—
246C23×C7C2×#4112yes—
247C23×C7C2×#4112yes—
248C23×C7C2×#4112yes—
249C23×C7C2×#4112yes—
250C2×C4×C7—24no—
251C2×C4×C7—22no—
252C2×C4×C7—24no—
253C2×C4×C7—24no—
254C2×C4×C7—22no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—22no—
261C2×C4×C7—24no—
262C2×C4×C7—42no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7C2×C22⋊C414yes—
273C2×C4×C7#12618yescomplemented
274C2×Dic7—22no—
275C2×Dic7—22no—
276C2×Dic7—24no—
277C2×Dic7—24no—
278C2×Dic7—24no—
279C2×Dic7—24no—
280C2×Dic7—24no—
281C2×Dic7—24no—
282C2×Dic7—24no—
283C2×Dic7—24no—
284C2×Dic7—24no—
285C2×Dic7—22no—
286C2×Dic7—24no—
287C2×Dic7—22no—
288C2×Dic7—22no—
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—22no—
292C2×Dic7—24no—
293C2×Dic7—42no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7—42no—
299C2×Dic7—42no—
300C2×Dic7—42no—
301C2×Dic7—42no—
302C2×Dic7—42no—
303C2×Dic7—42no—
304C2×Dic7—42no—
305C2×Dic7C2×C22⋊C414yes—
306C2×Dic7C2×C22⋊C414yes—
307C2×Dic7#12618yescomplemented
308C25C22×D711yescharacteristic
309C23×C4—71no—
310C23×C4—71no—
311C23×C4—72no—
312C23×C4—71no—
313C23×C4—71no—
314C23×C4—72no—
315C23×C4C22×D712yes—
316C23×C4C22×D711yescharacteristic
317C23×C4C22×D711yescharacteristic
318C23×C4C22×D712yes—
319#126—1432no—
320C2×C4⋊C4—74no—
321C2×C4⋊C4—74no—
322C2×C4⋊C4—74no—
323C2×C4⋊C4—142no—
324C2×C4⋊C4—142no—
325C2×C4⋊C4—142no—
326C2×C4⋊C4—142no—
327C2×C4⋊C4—142no—
328C2×C4⋊C4—142no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—74no—
335C2×C22⋊C4—74no—
336C2×C22⋊C4—142no—
337C2×C22⋊C4—142no—
338C2×C22⋊C4—142no—
339C2×C22⋊C4—142no—
340C2×C22⋊C4—144no—
341C2×C22⋊C4—144no—
342C2×C22⋊C4—144no—
343C2×C22⋊C4C4×D714yes—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42—74no—
347C2×C42—74no—
348C2×C42—74no—
349C2×C42—144no—
350C2×C42—144no—
351C2×C42—144no—
352C2×C42—144no—
353C2×C42—144no—
354C2×C42—144no—
355C2×C42—142no—
356C2×C42—144no—
357#95—22no—
358#95—24no—
359#95—144no—
360#95—144no—
361#95—144no—
362#95—144no—
363#95—144no—
364#95—144no—
365#95—142no—
366#95—144no—
367C22×C7—44no—
368C22×C7—44no—
369C22×C7—42no—
370C22×C7—42no—
371C22×C7—42no—
372C22×C7—82no—
373C22×C7—82no—
374C22×C7—82no—
375C22×C7—82no—
376C22×C7C22×C22⋊C411yescharacteristic
377C22×C7C4×#4111yescharacteristic
378C22×C7C4×#4111yescharacteristic
379C22×C7C4×#4112yes—
380C22×C7C4×#4112yes—
381C22×C7#52512yes—
382C22×C7#52511yescharacteristic
383C22×C7#52512yes—
384C22×C7#52511yescharacteristic
385C22×C7C2×#12612yes—
386C22×C7C2×#12612yes—
387C22×C7#52912yes—
388C22×C7#52912yes—
389C22×C7#52912yes—
390C22×C7#52912yes—
391C22×C7#52912yes—
392C22×C7#52912yes—
393C22×C7#53211yescharacteristic
394C22×C7#53212yes—
395C22×C7#53212yes—
396C22×C7#53211yescharacteristic
397C4×C7—28no—
398C4×C7—24no—
399C4×C7—42no—
400C4×C7—82no—
401Dic7—28no—
402Dic7—24no—
403Dic7—24no—
404Dic7—42no—
405Dic7—42no—
406Dic7—82no—
407C24—22no—
408C24—21no—
409C24—21no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24C2×C4×D712yes—
416C24C23×D711yescharacteristic
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—72no—
435C22×C4—72no—
436C22×C4—72no—
437C22×C4—72no—
438C22×C4—72no—
439C22×C4—72no—
440C22×C4—74no—
441C22×C4—74no—
442C22×C4—74no—
443C22×C4—72no—
444C22×C4—72no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—142no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—142no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—141no—
457C22×C4—141no—
458C22×C4—142no—
459C22×C4—142no—
460C22×C4—142no—
461C22×C4—142no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4—142no—
465C22×C4—142no—
466C22×C4—142no—
467C22×C4—142no—
468C22×C4C2×C4×D712yes—
469C22×C4C2×C4×D712yes—
470C22×C4#115714yes—
471C22×C4D4×D712yes—
472C22×C4D4×D712yes—
473C22×C4#115914yes—
474C22×C4#115914yes—
475C4⋊C4—148no—
476C4⋊C4—148no—
477C4⋊C4—148no—
478C22⋊C4—28no—
479C22⋊C4—48no—
480C22⋊C4—148no—
481C22⋊C4—148no—
482C22⋊C4—288no—
483C42—28no—
484C42—716no—
485C42—1416no—
486C42—1416no—
487C42—148no—
488C2×C7—82no—
489C2×C7#229711yescharacteristic
490C2×C7#229711yescharacteristic
491C2×C7#229711yescharacteristic
492C2×C7C4×#12612yes—
493C2×C7C4×#12612yes—
494C2×C7#285212yes—
495C2×C7#285212yes—
496C2×C7#286012yes—
497C2×C7#286012yes—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23C22×C4×D711yescharacteristic
520C23C2×#115712yes—
521C23C2×D4×D711yescharacteristic
522C23C2×D4×D711yescharacteristic
523C23C2×#115912yes—
524C23C2×#115912yes—
525C23D7×#4112yes—
526C23D7×#4112yes—
527C23D7×#4112yes—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—78no—
551C2×C4—74no—
552C2×C4—74no—
553C2×C4—144no—
554C2×C4—142no—
555C2×C4—142no—
556C2×C4—142no—
557C2×C4—144no—
558C2×C4—144no—
559C2×C4—144no—
560C2×C4—144no—
561C2×C4—144no—
562C2×C4—144no—
563C2×C4—144no—
564C2×C4—144no—
565C2×C4—144no—
566C2×C4—142no—
567C2×C4—142no—
568C2×C4—142no—
569C2×C4—142no—
570C2×C4—142no—
571C2×C4—144no—
572C2×C4—282no—
573C2×C4—282no—
574C2×C4—281no—
575C2×C4—282no—
576C2×C4—281no—
577C2×C4—282no—
578C2×C4—282no—
579C2×C4—282no—
580C2×C4—282no—
581C2×C4—282no—
582C2×C4—282no—
583C2×C4—282no—
584C2×C4#644218yescomplemented
585C2×C4D7×C22⋊C414yes—
586C7#1471111yescharacteristic, complemented, Sylow
587C22—42no—
588C22—42no—
589C22—44no—
590C22—44no—
591C22—42no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22C4×#115711yescharacteristic
597C22C2×#644212yes—
598C22C2×D7×C22⋊C411yescharacteristic
599C22#9021311yescharacteristic
600C22#9021912yes—
601C22C4×#115912yes—
602C22C4×#115912yes—
603C22#9022912yes—
604C22#9023812yes—
605C22#9023912yes—
606C22#9024812yes—
607C22#9027711yescharacteristic
608C22#9028712yes—
609C22#9029112yes—
610C22#9029712yes—
611C22#9034012yes—
612C22#9034711yescharacteristic
613C22#9035911yescharacteristic
614C22#9037112yes—
615C22D7×#12612yes—
616C22#9040711yescharacteristic
617C4—28no—
618C4—24no—
619C4—42no—
620C4—82no—
621C4—144no—
622C4—148no—
623C4—144no—
624C4—282no—
625C4—282no—
626C4—562no—
627C2—82no—
628C2C4×#644212yes—
629C2#1173100811yescharacteristic
630C2#1173210311yescharacteristic
631C2#1173231512yes—
632C2#1173290012yes—
633C2#1173326512yes—
634C2#1173340312yes—
635C2#1173373511yescharacteristic
636C2#1173387012yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1437C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenter#1437D7
hypocenterC7#14711
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×C42×Dic7C2semidirect
3. C2×#89423C2semidirect
8. #11729290C2semidirect
9. #11727161C2semidirect
10. #11727160C2semidirect
11. #11727159C2semidirect
27. C2×#6442C4semidirect
191. C4×Dic7C22⋊C4semidirect
273. C2×C4×C7#126semidirect
307. C2×Dic7#126semidirect
584. C2×C4#6442semidirect
586. C7#14711semidirect

Maximal subgroups

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

subgroupindex
C22×#64422
C2×C42×Dic72
C2×#894232
C7×#14372
#117292992
#117292902
#117271612
#117271602
#117271592
#147117