Finite Group Databasebeta

← #13355542

Subgroups — #13355542

Order 1792.

1818 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#537C211yesmaximal, characteristic
2C22×#6511C211yesmaximal, characteristic, complemented
3#11741532C211yesmaximal, characteristic, complemented
4C2×#90004C211yesmaximal, characteristic, complemented
5C2×#90005C211yesmaximal, characteristic, complemented
6#11729451C212yesmaximal, complemented
7#11729450C212yesmaximal, complemented
8#11729436C212yesmaximal, complemented
9#11729435C212yesmaximal, complemented
10#11729312C211yesmaximal, characteristic, complemented
11C2×#89324C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C22×C7×#41C2211yescharacteristic
14C7×#537—24no—
15C7×#537C418yescomplemented
16C2×C7×C22⋊Q8—21no—
17C2×C7×C22⋊Q8C2212yes—
18C2×C7×#121—21no—
19C2×C7×#121—22no—
20C2×C7×#121C2212yes—
21C22×C7×C4⋊C4C2211yescharacteristic
22C2×#6511—24no—
23C2×#6511—24no—
24C2×#6511—24no—
25C2×#6489C2211yescharacteristic
26C22×C4⋊Dic7C2211yescharacteristic
27C22×#1136C2212yes—
28C22×#1136C2211yescharacteristic
29C23×C7⋊C8C2212yes—
30C23×C7⋊C8C2211yescharacteristic
31#90005—24no—
32#90004—24no—
33#90002—21no—
34C2×#6409C2211yescharacteristic
35C2×#6409C2212yes—
36C2×#6409C2211yescharacteristic
37#89993—21no—
38#89992—22no—
39C2×#6406C2211yescharacteristic
40C2×#6406C2212yes—
41C2×#6406C2211yescharacteristic
42#89984C2212yes—
43#89984C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#89324C2212yes—
47#89324C2212yes—
48#20976—71nomaximal, Sylow
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41—22no—
53C2×C7×#41—22no—
54C2×C7×#41—22no—
55C2×C7×#41—22no—
56C2×C7×#41—22no—
57C2×C7×#41—22no—
58C2×C7×#41C2×C414yes—
59C2×C7×#41C2×C414yes—
60C2×C7×#41D414yes—
61C22×C7×Q8D411yescharacteristic
62C22×C7×Q8C2311yescharacteristic
63C22×C7×D4—21no—
64C22×C7×D4—22no—
65C22×C7×D4D412yes—
66C22×C7×D4D411yescharacteristic
67C22×C7×D4C2311yescharacteristic
68C23×C4×C7—21no—
69C23×C4×C7—21no—
70C23×C4×C7C2312yes—
71C23×C4×C7C2311yescharacteristic
72C23×Dic7—21no—
73C7×C22⋊Q8—28no—
74C7×C22⋊Q8—24no—
75C7×C22⋊Q8C2×C418yescomplemented
76C7×#121—28no—
77C7×#121—216no—
78C7×#121—24no—
79C7×#121C2×C418yescomplemented
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4C2×C414yes—
84C2×C7×C4⋊C4C2312yes—
85C2×C7×C22⋊C4—22no—
86C2×C7×C22⋊C4—22no—
87C2×C22⋊Dic7—21no—
88C2×C22⋊Dic7—22no—
89C2×C22⋊Dic7—22no—
90#6511—232no—
91#6489—22no—
92#6489—22no—
93C2×C4⋊Dic7—22no—
94C2×C4⋊Dic7—22no—
95C2×C4⋊Dic7—21no—
96C2×C4⋊Dic7C2312yes—
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4×Dic7—21no—
100C2×#1136—24no—
101C2×#1136—22no—
102C2×#1136—22no—
103C2×#1136—24no—
104C2×#1136—24no—
105C2×#1136D414yescomplemented
106C22×C7⋊C8—24no—
107C22×C7⋊C8—22no—
108C22×C7⋊C8—22no—
109C22×C7⋊C8—24no—
110C22×C7⋊C8—24no—
111C22×C7⋊C8D414yescomplemented
112C22×C7⋊C8C2311yescharacteristic
113C22×C7⋊C8C2312yes—
114C22×C7⋊C8C2311yescharacteristic
115#6409—24no—
116#6409—22no—
117#6409—22no—
118#6406—24no—
119#6406—22no—
120#6406—22no—
121C7⋊#95—22no—
122C7⋊#95—22no—
123C22×#131—71no—
124C2×#537D711yescharacteristic
125C2×#417—71no—
126C2×#416—71no—
127#2311—71no—
128#1895—72no—
129#1894—72no—
130#1893—72no—
131#1892—72no—
132#1736—71no—
133C2×#340—71no—
134C24×C7—21no—
135C24×C7C2×D412yes—
136C24×C7C2×D411yescharacteristic
137C7×#41—28no—
138C7×#41—28no—
139C7×#41—28no—
140C7×#41—28no—
141C7×#41—28no—
142C7×#41—28no—
143C7×#41—44no—
144C7×#41—44no—
145C2×C7×Q8—22no—
146C2×C7×Q8—22no—
147C2×C7×Q8—21no—
148C2×C7×Q8—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8C22⋊C414yes—
153C2×C7×Q8C22×C412yes—
154C2×C7×D4—24no—
155C2×C7×D4—24no—
156C2×C7×D4—24no—
157C2×C7×D4—28no—
158C2×C7×D4—22no—
159C2×C7×D4—22no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C2×C7×D4—21no—
165C2×C7×D4—42no—
166C2×C7×D4—42no—
167C2×C7×D4—82no—
168C2×C7×D4—82no—
169C2×C7×D4C22⋊C418yes—
170C2×C7×D4C22⋊C414yes—
171C2×C7×D4C22×C412yes—
172C22×C4×C7—22no—
173C22×C4×C7—22no—
174C22×C4×C7—22no—
175C22×C4×C7—22no—
176C22×C4×C7—22no—
177C22×C4×C7—21no—
178C22×C4×C7—24no—
179C22×C4×C7—24no—
180C22×C4×C7—42no—
181C22×C4×C7—42no—
182C22×C4×C7—81no—
183C22×C4×C7—81no—
184C22×C4×C7C22×C414yes—
185C22×C4×C7C22×C412yes—
186C22×C4×C7C2×D412yes—
187C22×C4×C7C2×D414yes—
188C22×C4×C7C2×D411yescharacteristic
189C22×C4×C7C2×D412yes—
190C22×C4×C7C2×D412yes—
191C22×C4×C7C2×D412yes—
192C22×C4×C7#4112yes—
193C22×C4×C7#4112yes—
194C22×C4×C7C2411yescharacteristic
195C22×Dic7—21no—
196C22×Dic7—21no—
197C22×Dic7—22no—
198C22×Dic7—42no—
199C22×Dic7—81no—
200C22×Dic7—81no—
201C7×C4⋊C4—24no—
202C7×C4⋊C4—28no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4C22×C414yes—
205C7×C22⋊C4—28no—
206C7×C22⋊C4—28no—
207C22⋊Dic7—42no—
208C22⋊Dic7—82no—
209C22⋊Dic7—82no—
210C4⋊Dic7—22no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—22no—
214C4⋊Dic7—21no—
215C4⋊Dic7—21no—
216C4⋊Dic7—42no—
217C4×Dic7—42no—
218#1136—216no—
219#1136—28no—
220C2×C7⋊C8—216no—
221C2×C7⋊C8—28no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—22no—
226C2×C7⋊C8—22no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8C2×D412yes—
229C2×C7⋊C8C2×D414yes—
230C2×C7⋊C8C2×D412yes—
231C22×#41C2×D711yescharacteristic
232C22×#41C2×D711yescharacteristic
233C2×#131—144no—
234C2×#131—144no—
235C2×#131—144no—
236C22×M16—71no—
237C22×M16—72no—
238C23×C8—71no—
239C23×C8—72no—
240#537—24no—
241#537Dic718yescomplemented
242C2×C22⋊Q8—21no—
243C2×C22⋊Q8C2×D712yes—
244C2×#121—21no—
245C2×#121—22no—
246C2×#121C2×D712yes—
247C2×#117—71no—
248C22×C4⋊C4—71no—
249C22×C4⋊C4C2×D711yescharacteristic
250#417—144no—
251#416—144no—
252C2×#103—71no—
253C2×#103—71no—
254C2×#103—72no—
255C2×#102—71no—
256C2×#102—71no—
257C2×#102—72no—
258#391—141no—
259#390—141no—
260#386—142no—
261#380—72no—
262#380—72no—
263#340—72no—
264#340—72no—
265#340—72no—
266#340—72no—
267C23×C7—21no—
268C23×C7—21no—
269C23×C7—21no—
270C23×C7—21no—
271C23×C7—22no—
272C23×C7—22no—
273C23×C7—22no—
274C23×C7—22no—
275C23×C7—22no—
276C23×C7—22no—
277C23×C7—22no—
278C23×C7—41no—
279C23×C7—41no—
280C23×C7—41no—
281C23×C7—41no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7C2×C22⋊C414yes—
285C23×C7C2×C22⋊C412yes—
286C23×C7#12112yes—
287C23×C7#12112yes—
288C23×C7C22×D411yescharacteristic
289C7×Q8—28no—
290C7×Q8—22no—
291C7×Q8—22no—
292C7×Q8—41no—
293C7×Q8—41no—
294C7×D4—216no—
295C7×D4—22no—
296C7×D4—22no—
297C7×D4—28no—
298C7×D4—48no—
299C7×D4—41no—
300C7×D4—41no—
301C7×D4—88no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—24no—
305C2×C4×C7—24no—
306C2×C4×C7—24no—
307C2×C4×C7—24no—
308C2×C4×C7—24no—
309C2×C4×C7—22no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—42no—
313C2×C4×C7—42no—
314C2×C4×C7—48no—
315C2×C4×C7—84no—
316C2×C4×C7C2×C22⋊C414yes—
317C2×C4×C7C2×C22⋊C412yes—
318C2×C4×C7C4×D414yes—
319C2×C4×C7C4×D414yes—
320C2×C4×C7#12014yes—
321C2×C4×C7#12112yes—
322C2×C4×C7#12112yes—
323C2×C4×C7#12314yes—
324C2×C4×C7C23×C411yescharacteristic
325C2×C4×C7C22×D411yescharacteristic
326C2×C4×C7C22×D411yescharacteristic
327C2×C4×C7C22×D411yescharacteristic
328C2×C4×C7C2×#4111yescharacteristic
329C2×C4×C7C2×#4111yescharacteristic
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—42no—
333C2×Dic7—41no—
334C2×Dic7—42no—
335C2×Dic7—41no—
336C2×Dic7—42no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—84no—
340C7⋊C8—22no—
341C7⋊C8—24no—
342C7⋊C8—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41—22no—
350C2×#41—22no—
351C2×#41—22no—
352C2×#41C2×Dic714yes—
353C2×#41C2×Dic714yes—
354C2×#41C22⋊D714yes—
355C22×Q8C22⋊D711yescharacteristic
356C22×Q8C22×D711yescharacteristic
357C22×D4—21no—
358C22×D4—22no—
359C22×D4C22⋊D712yes—
360C22×D4C22⋊D711yescharacteristic
361C22×D4C22×D711yescharacteristic
362C23×C4—21no—
363C23×C4—21no—
364C23×C4—141no—
365C23×C4C22×D712yes—
366C23×C4C22×D711yescharacteristic
367#131—1432no—
368C2×M16—74no—
369C2×M16—144no—
370C2×M16—144no—
371C2×M16—142no—
372C2×M16—144no—
373C2×M16—142no—
374C22×C8—74no—
375C22×C8—71no—
376C22×C8—71no—
377C22×C8—72no—
378C22×C8—144no—
379C22×C8—144no—
380C22×C8—142no—
381C22×C8—144no—
382C22×C8—142no—
383C22⋊Q8—28no—
384C22⋊Q8—24no—
385C22⋊Q8C2×Dic718yescomplemented
386#121—28no—
387#121—216no—
388#121—24no—
389#121C2×Dic718yescomplemented
390#117—142no—
391#117—142no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—72no—
396C2×C4⋊C4—71no—
397C2×C4⋊C4—71no—
398C2×C4⋊C4—142no—
399C2×C4⋊C4—142no—
400C2×C4⋊C4—141no—
401C2×C4⋊C4C2×Dic714yes—
402C2×C4⋊C4C22×D712yes—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—141no—
406C2×C22⋊C4—142no—
407C2×C22⋊C4—142no—
408C2×C42—141no—
409#103—142no—
410#103—142no—
411#103—144no—
412#102—142no—
413#102—142no—
414#102—144no—
415#95—142no—
416#95—142no—
417C22×C7—22no—
418C22×C7—22no—
419C22×C7—22no—
420C22×C7—22no—
421C22×C7—22no—
422C22×C7—22no—
423C22×C7—24no—
424C22×C7—24no—
425C22×C7—24no—
426C22×C7—42no—
427C22×C7—42no—
428C22×C7—42no—
429C22×C7—42no—
430C22×C7—42no—
431C22×C7—42no—
432C22×C7—42no—
433C22×C7—81no—
434C22×C7—81no—
435C22×C7—81no—
436C22×C7—81no—
437C22×C7#38614yes—
438C22×C7C22×C22⋊C411yescharacteristic
439C22×C7C2×#12111yescharacteristic
440C22×C7C2×#12111yescharacteristic
441C22×C7#57611yescharacteristic
442C22×C7#57711yescharacteristic
443C22×C7#57711yescharacteristic
444C22×C7#57811yescharacteristic
445C4×C7—24no—
446C4×C7—44no—
447C4×C7—42no—
448C4×C7#38614yes—
449C4×C7C2×C4×D411yescharacteristic
450C4×C7C2×C4×D411yescharacteristic
451C4×C7C2×#12011yescharacteristic
452C4×C7C2×#12311yescharacteristic
453Dic7—81no—
454Dic7—81no—
455Dic7—82no—
456C24—21no—
457C24C2×C22⋊D712yes—
458C24C2×C22⋊D711yescharacteristic
459#41—28no—
460#41—28no—
461#41—28no—
462#41—28no—
463#41—28no—
464#41—28no—
465#41—44no—
466#41—44no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8C22⋊Dic714yes—
475C2×Q8C22×Dic712yes—
476C2×D4—24no—
477C2×D4—24no—
478C2×D4—24no—
479C2×D4—28no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—82no—
490C2×D4—82no—
491C2×D4C22⋊Dic718yes—
492C2×D4C22⋊Dic714yes—
493C2×D4C22×Dic712yes—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—81no—
505C22×C4—81no—
506C22×C4—141no—
507C22×C4—141no—
508C22×C4—142no—
509C22×C4—282no—
510C22×C4—561no—
511C22×C4—561no—
512C22×C4D4×D712yes—
513C22×C4D4×D712yes—
514C22×C4#115912yes—
515C22×C4#115912yes—
516C22×C4C22×Dic714yes—
517C22×C4C22×Dic712yes—
518C22×C4C2×C22⋊D712yes—
519C22×C4C2×C22⋊D714yes—
520C22×C4C2×C22⋊D712yes—
521C22×C4C2×C22⋊D711yescharacteristic
522C22×C4C23×D711yescharacteristic
523M16—1416no—
524M16—148no—
525C2×C8—72no—
526C2×C8—72no—
527C2×C8—74no—
528C2×C8—1416no—
529C2×C8—148no—
530C2×C8—141no—
531C2×C8—142no—
532C2×C8—141no—
533C2×C8—141no—
534C2×C8—141no—
535C2×C8—142no—
536C4⋊C4—24no—
537C4⋊C4—28no—
538C4⋊C4—22no—
539C4⋊C4—142no—
540C4⋊C4—141no—
541C4⋊C4—141no—
542C4⋊C4—142no—
543C4⋊C4—141no—
544C4⋊C4—141no—
545C4⋊C4—282no—
546C4⋊C4C22×Dic714yes—
547C22⋊C4—28no—
548C22⋊C4—28no—
549C22⋊C4—282no—
550C22⋊C4—562no—
551C22⋊C4—562no—
552C42—282no—
553C2×C7—24no—
554C2×C7—44no—
555C2×C7—82no—
556C2×C7C2×#38611yescharacteristic
557C2×C7#289711yescharacteristic
558C2×C7#289811yescharacteristic
559C2×C7#306011yescharacteristic
560C2×C7#306111yescharacteristic
561C2×C7#306211yescharacteristic
562C2×C7#306311yescharacteristic
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23#650212yes—
581C23#650212yes—
582C23C2×C22⋊Dic714yes—
583C23C2×C22⋊Dic712yes—
584C23C22×C22⋊D711yescharacteristic
585Q8—28no—
586Q8—22no—
587Q8—22no—
588Q8—41no—
589Q8—41no—
590D4—216no—
591D4—22no—
592D4—22no—
593D4—28no—
594D4—48no—
595D4—41no—
596D4—41no—
597D4—88no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—48no—
611C2×C4—84no—
612C2×C4—281no—
613C2×C4—281no—
614C2×C4—282no—
615C2×C4—281no—
616C2×C4—282no—
617C2×C4—281no—
618C2×C4—281no—
619C2×C4—282no—
620C2×C4—281no—
621C2×C4—564no—
622C2×C4D4×Dic714yes—
623C2×C4D4×Dic714yes—
624C2×C4#649814yes—
625C2×C4#650014yes—
626C2×C4#650212yes—
627C2×C4#650212yes—
628C2×C4C2×C22⋊Dic714yes—
629C2×C4C2×C22⋊Dic712yes—
630C2×C4C2×D4×D711yescharacteristic
631C2×C4C2×D4×D711yescharacteristic
632C2×C4C2×#115911yescharacteristic
633C2×C4C2×#115911yescharacteristic
634C2×C4C23×Dic711yescharacteristic
635C2×C4C22×C22⋊D711yescharacteristic
636C8—142no—
637C8—142no—
638C8—144no—
639C7#2097611yescharacteristic, complemented, Sylow
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22#8999214yes—
661C22C2×#650211yescharacteristic
662C22C2×#650211yescharacteristic
663C22#9052611yescharacteristic
664C22#9052711yescharacteristic
665C22#9052811yescharacteristic
666C22#9052911yescharacteristic
667C22C22×C22⋊Dic711yescharacteristic
668C4—24no—
669C4—42no—
670C4—44no—
671C4—561no—
672C4—561no—
673C4—562no—
674C4#8999214yes—
675C4C2×D4×Dic711yescharacteristic
676C4C2×D4×Dic711yescharacteristic
677C4C2×#649811yescharacteristic
678C4C2×#650011yescharacteristic
679C2—24no—
680C2—44no—
681C2—82no—
682C2#1174086511yescharacteristic
683C2#1174086611yescharacteristic
684C2#1174086711yescharacteristic
685C2#1174086811yescharacteristic
686C2C2×#8999211yescharacteristic
687C2#1174154411yescharacteristic
688C2#1174155611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#537C2
socleC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenterC2×#537D7
hypocenterC7#20976
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
2. C22×#6511C2semidirect
3. #11741532C2semidirect
4. C2×#90004C2semidirect
5. C2×#90005C2semidirect
6. #11729451C2semidirect
7. #11729450C2semidirect
8. #11729436C2semidirect
9. #11729435C2semidirect
10. #11729312C2semidirect
11. C2×#89324C2semidirect
15. C7×#537C4semidirect
75. C7×C22⋊Q8C2×C4semidirect
79. C7×#121C2×C4semidirect
105. C2×#1136D4semidirect
111. C22×C7⋊C8D4semidirect
241. #537Dic7semidirect
385. C22⋊Q8C2×Dic7semidirect
389. #121C2×Dic7semidirect
639. C7#20976semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5372
C22×#65112
#117415322
C2×#900042
C2×#900052
#117294512
#117294502
#117294362
#117294352
#117293122
C2×#893242
#209767