Finite Group Databasebeta

← #13380661

Subgroups — #13380661

Order 1792.

2014 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×C7×#537C211yesmaximal, characteristic
2C2×#90516C211yesmaximal, characteristic, complemented
3C22×#6514C211yesmaximal, characteristic, complemented
4C2×#90006C211yesmaximal, characteristic, complemented
5C2×#90005C211yesmaximal, characteristic, complemented
6C22×#6384C211yesmaximal, characteristic, complemented
7#11729454C214yesmaximal, complemented
8#11729439C214yesmaximal, complemented
9#11729312C211yesmaximal, characteristic, complemented
10C22×C7×#41C2211yescharacteristic
11C22×C7×#41C2211yescharacteristic
12C23×Dic14C2211yescharacteristic
13C7×#537—24no—
14C7×#537—24no—
15C2×C7×C22⋊Q8—21no—
16C2×C7×C22⋊Q8C2212yes—
17C2×C7×#121—21no—
18C2×C7×#121—22no—
19C2×C7×#121C2212yes—
20C22×C7×C4⋊C4C2211yescharacteristic
21#90516—24no—
22#90516—24no—
23C2×#6514—24no—
24C2×#6514—24no—
25C2×#6514—24no—
26C2×#6507—21no—
27C22×#1144C2212yes—
28C2×#6499—21no—
29C2×#6498—22no—
30C22×#1142C2212yes—
31C2×#6489C2211yescharacteristic
32C2×#6487C2214yes—
33C22×#1136C2211yescharacteristic
34C23×C7⋊C8C2211yescharacteristic
35#90006—24no—
36#90005—24no—
37C2×#6409C2212yes—
38C2×#6406C2212yes—
39C2×C22⋊(C7⋊C8)C2214yes—
40C2×#6384—24no—
41C2×#6384C2214yes—
42#89324C2214yes—
43#23394—71nomaximal, Sylow
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C2×C7×#41—22no—
48C2×C7×#41—22no—
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×#41D414yes—
56C22×C7×Q8D411yescharacteristic
57C22×C7×Q8C2311yescharacteristic
58C22×C7×D4—21no—
59C22×C7×D4—22no—
60C22×C7×D4D412yes—
61C22×C7×D4D411yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C23×C4×C7—21no—
65C23×C4×C7C2312yes—
66C23×C4×C7C2311yescharacteristic
67C23×Dic7—21no—
68C22×Dic14—22no—
69C22×Dic14—22no—
70C22×Dic14—22no—
71C22×Dic14—22no—
72C22×Dic14—21no—
73C22×Dic14—22no—
74C22×Dic14D414yescomplemented
75C22×Dic14C2312yes—
76C7×C22⋊Q8—28no—
77C7×C22⋊Q8—24no—
78C7×C22⋊Q8—24no—
79C7×#121—216no—
80C7×#121—28no—
81C7×#121—24no—
82C7×#121—24no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4—22no—
85C2×C7×C4⋊C4—22no—
86C2×C7×C4⋊C4D414yes—
87C2×C7×C4⋊C4C2312yes—
88C2×C7×C22⋊C4—22no—
89C2×C7×C22⋊C4—22no—
90C2×C22⋊Dic7—24no—
91C2×C22⋊Dic7—21no—
92#6514—232no—
93#6507—28no—
94C2×#1144—24no—
95C2×#1144—24no—
96C2×#1144—24no—
97#6499—28no—
98#6498—216no—
99C2×#1142—24no—
100C2×#1142—24no—
101C2×#1142—24no—
102#6489—22no—
103#6489D414yescomplemented
104C2×C4⋊Dic7C2312yes—
105#6487—28no—
106#6487—28no—
107C2×Dic7⋊C4—24no—
108C2×C4×Dic7—21no—
109C2×#1136—22no—
110C2×#1136—22no—
111C2×#1136D414yescomplemented
112C22×C7⋊C8—22no—
113C22×C7⋊C8—22no—
114C22×C7⋊C8—22no—
115C22×C7⋊C8C2312yes—
116C22×C7⋊C8C2312yes—
117#6409—24no—
118#6406—24no—
119C22⋊(C7⋊C8)—28no—
120#6384—28no—
121C22×#137—71no—
122C2×#537D711yescharacteristic
123C2×#536—71no—
124C2×#416—71no—
125C22×#103—71no—
126C2×#408—71no—
127#2001—74no—
128#2000—74no—
129#1736—71no—
130C24×C7—21no—
131C24×C7C2×D412yes—
132C24×C7C2×D411yescharacteristic
133C7×#41—28no—
134C7×#41—28no—
135C7×#41—44no—
136C7×#41—44no—
137C7×#41—44no—
138C7×#41—44no—
139C7×#41—44no—
140C7×#41—44no—
141C2×C7×Q8—22no—
142C2×C7×Q8—22no—
143C2×C7×Q8—22no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8—21no—
148C2×C7×Q8—21no—
149C2×C7×Q8C2×D412yes—
150C2×C7×D4—24no—
151C2×C7×D4—24no—
152C2×C7×D4—24no—
153C2×C7×D4—24no—
154C2×C7×D4—28no—
155C2×C7×D4—22no—
156C2×C7×D4—22no—
157C2×C7×D4—22no—
158C2×C7×D4—21no—
159C2×C7×D4—21no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—42no—
164C2×C7×D4—42no—
165C2×C7×D4—84no—
166C2×C7×D4C2×D412yes—
167C22×C4×C7—22no—
168C22×C4×C7—21no—
169C22×C4×C7—24no—
170C22×C4×C7—24no—
171C22×C4×C7—22no—
172C22×C4×C7—22no—
173C22×C4×C7—22no—
174C22×C4×C7—22no—
175C22×C4×C7—22no—
176C22×C4×C7—22no—
177C22×C4×C7—42no—
178C22×C4×C7—42no—
179C22×C4×C7—82no—
180C22×C4×C7C2×D412yes—
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2×D412yes—
183C22×C4×C7C2×D412yes—
184C22×C4×C7C2×D412yes—
185C22×C4×C7C2×D412yes—
186C22×C4×C7C2×D412yes—
187C22×C4×C7#4112yes—
188C22×C4×C7C2411yescharacteristic
189C22×Dic7—24no—
190C22×Dic7—22no—
191C22×Dic7—22no—
192C22×Dic7—42no—
193C22×Dic7—42no—
194C2×Dic14—28no—
195C2×Dic14—28no—
196C2×Dic14—22no—
197C2×Dic14—22no—
198C2×Dic14—22no—
199C2×Dic14—22no—
200C2×Dic14—24no—
201C2×Dic14—22no—
202C2×Dic14—42no—
203C2×Dic14—44no—
204C2×Dic14—42no—
205C2×Dic14—44no—
206C2×Dic14C2×D414yes—
207C7×C4⋊C4—28no—
208C7×C4⋊C4—24no—
209C7×C4⋊C4—22no—
210C7×C4⋊C4C2×D414yes—
211C7×C22⋊C4—28no—
212C7×C22⋊C4—28no—
213C22⋊Dic7—216no—
214C22⋊Dic7—24no—
215#1144—216no—
216#1142—216no—
217C4⋊Dic7—22no—
218C4⋊Dic7C2×D414yes—
219Dic7⋊C4—216no—
220C4×Dic7—24no—
221#1136—28no—
222C2×C7⋊C8—22no—
223C2×C7⋊C8—22no—
224C2×C7⋊C8—22no—
225C2×C7⋊C8—22no—
226C2×C7⋊C8—22no—
227C2×C7⋊C8—44no—
228C2×C7⋊C8C2×D414yes—
229C22×#41C2×D711yescharacteristic
230C22×#41C2×D711yescharacteristic
231C23×Q8—71no—
232C2×#137—144no—
233C2×#137—144no—
234C2×#137—144no—
235C22×Q16—72no—
236C22×SD16—72no—
237C22×M16—71no—
238C23×C8—71no—
239#537—24no—
240#537—24no—
241#536—144no—
242#536—144no—
243C2×C4⋊Q8—141no—
244C2×#124—141no—
245C2×#123—142no—
246C2×C22⋊Q8—21no—
247C2×C22⋊Q8—74no—
248C2×C22⋊Q8C2×D712yes—
249C2×#121—21no—
250C2×#121—22no—
251C2×#121C2×D712yes—
252C2×#117—71no—
253C22×C4⋊C4C2×D711yescharacteristic
254#416—144no—
255C2×#103—72no—
256C2×#103—74no—
257C2×#103—144no—
258C2×#102—72no—
259#408—144no—
260C2×C22⋊C8—74no—
261#340—74no—
262C23×C7—22no—
263C23×C7—22no—
264C23×C7—22no—
265C23×C7—22no—
266C23×C7—22no—
267C23×C7—22no—
268C23×C7—24no—
269C23×C7—24no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—42no—
273C23×C7—42no—
274C23×C7#12012yes—
275C23×C7#12012yes—
276C23×C7#12112yes—
277C23×C7C22×D411yescharacteristic
278C7×Q8—22no—
279C7×Q8—22no—
280C7×Q8—41no—
281C7×Q8—41no—
282C7×Q8—44no—
283C7×D4—22no—
284C7×D4—22no—
285C7×D4—48no—
286C7×D4—48no—
287C7×D4—41no—
288C7×D4—41no—
289C7×D4—44no—
290C7×D4—88no—
291C2×C4×C7—22no—
292C2×C4×C7—24no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—22no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—44no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—48no—
304C2×C4×C7—84no—
305C2×C4×C7#12014yes—
306C2×C4×C7#12012yes—
307C2×C4×C7#12012yes—
308C2×C4×C7#12114yes—
309C2×C4×C7#12114yes—
310C2×C4×C7#12112yes—
311C2×C4×C7#12714yes—
312C2×C4×C7C22×D411yescharacteristic
313C2×C4×C7C22×D411yescharacteristic
314C2×C4×C7C22×D411yescharacteristic
315C2×C4×C7C22×D411yescharacteristic
316C2×C4×C7C22×D411yescharacteristic
317C2×C4×C7C2×#4111yescharacteristic
318C2×Dic7—24no—
319C2×Dic7—24no—
320C2×Dic7—42no—
321C2×Dic7—42no—
322C2×Dic7—42no—
323C2×Dic7—42no—
324C2×Dic7—48no—
325Dic14—24no—
326Dic14—24no—
327Dic14—42no—
328Dic14—42no—
329Dic14—48no—
330C7⋊C8—24no—
331C7⋊C8—42no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41C22⋊D714yes—
344C22×Q8—74no—
345C22×Q8—72no—
346C22×Q8—142no—
347C22×Q8—141no—
348C22×Q8—142no—
349C22×Q8—142no—
350C22×Q8—142no—
351C22×Q8—142no—
352C22×Q8C22⋊D711yescharacteristic
353C22×Q8C22×D711yescharacteristic
354C22×D4—21no—
355C22×D4—22no—
356C22×D4C22⋊D712yes—
357C22×D4C22⋊D711yescharacteristic
358C22×D4C22×D711yescharacteristic
359C23×C4—21no—
360C23×C4—21no—
361C23×C4—141no—
362C23×C4C22×D712yes—
363C23×C4C22×D711yescharacteristic
364#137—1432no—
365C2×Q16—144no—
366C2×Q16—144no—
367C2×Q16—144no—
368C2×SD16—144no—
369C2×SD16—144no—
370C2×SD16—144no—
371C2×M16—74no—
372C2×M16—142no—
373C2×M16—142no—
374C22×C8—72no—
375C22×C8—72no—
376C22×C8—142no—
377C22×C8—142no—
378C22×C8—142no—
379C4⋊Q8—148no—
380#124—148no—
381#123—1416no—
382C22⋊Q8—28no—
383C22⋊Q8—24no—
384C22⋊Q8—24no—
385C22⋊Q8—148no—
386C22⋊Q8—148no—
387#121—216no—
388#121—28no—
389#121—24no—
390#121—24no—
391#117—74no—
392#117—142no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—72no—
397C2×C4⋊C4—144no—
398C2×C4⋊C4C22⋊D714yes—
399C2×C4⋊C4C22×D712yes—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—141no—
403C2×C22⋊C4—144no—
404C2×C42—141no—
405#103—144no—
406#103—148no—
407#102—144no—
408C22⋊C8—148no—
409C22×C7—22no—
410C22×C7—22no—
411C22×C7—24no—
412C22×C7—24no—
413C22×C7—42no—
414C22×C7—42no—
415C22×C7—42no—
416C22×C7—42no—
417C22×C7—42no—
418C22×C7—42no—
419C22×C7—44no—
420C22×C7—44no—
421C22×C7—82no—
422C22×C7—82no—
423C22×C7#39214yes—
424C22×C7C2×#12011yescharacteristic
425C22×C7C2×#12011yescharacteristic
426C22×C7C2×#12111yescharacteristic
427C22×C7#57712yes—
428C22×C7#57812yes—
429C4×C7—42no—
430C4×C7—42no—
431C4×C7—44no—
432C4×C7#39214yes—
433C4×C7C2×#12011yescharacteristic
434C4×C7C2×#12111yescharacteristic
435C4×C7C2×#12111yescharacteristic
436C4×C7C2×#12711yescharacteristic
437Dic7—44no—
438Dic7—44no—
439C24—21no—
440C24C2×C22⋊D712yes—
441C24C2×C22⋊D711yescharacteristic
442#41—28no—
443#41—28no—
444#41—44no—
445#41—44no—
446#41—44no—
447#41—44no—
448#41—44no—
449#41—44no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—74no—
459C2×Q8—144no—
460C2×Q8—142no—
461C2×Q8—148no—
462C2×Q8—148no—
463C2×Q8—142no—
464C2×Q8—142no—
465C2×Q8—142no—
466C2×Q8—142no—
467C2×Q8—282no—
468C2×Q8—284no—
469C2×Q8—284no—
470C2×Q8—282no—
471C2×Q8C2×C22⋊D712yes—
472C2×D4—24no—
473C2×D4—24no—
474C2×D4—24no—
475C2×D4—24no—
476C2×D4—28no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—84no—
488C2×D4C2×C22⋊D712yes—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—82no—
502C22×C4—144no—
503C22×C4—142no—
504C22×C4—142no—
505C22×C4—282no—
506C22×C4—282no—
507C22×C4D4×D712yes—
508C22×C4D4×D712yes—
509C22×C4D4×D712yes—
510C22×C4#115912yes—
511C22×C4C2×C22⋊D712yes—
512C22×C4C2×C22⋊D711yescharacteristic
513C22×C4C2×C22⋊D712yes—
514C22×C4C2×C22⋊D712yes—
515C22×C4C23×D711yescharacteristic
516Q16—1416no—
517SD16—1416no—
518M16—148no—
519C2×C8—74no—
520C2×C8—142no—
521C2×C8—142no—
522C2×C8—142no—
523C2×C8—142no—
524C2×C8—142no—
525C2×C8—284no—
526C4⋊C4—28no—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—74no—
530C4⋊C4—1416no—
531C4⋊C4—142no—
532C4⋊C4C2×C22⋊D714yes—
533C22⋊C4—28no—
534C22⋊C4—28no—
535C22⋊C4—1416no—
536C22⋊C4—144no—
537C42—144no—
538C2×C7—24no—
539C2×C7—82no—
540C2×C7—82no—
541C2×C7C2×#39211yescharacteristic
542C2×C7#300911yescharacteristic
543C2×C7#301111yescharacteristic
544C2×C7#301312yes—
545C2×C7#306312yes—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—24no—
552C23—22no—
553C23—22no—
554C23—41no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23#650012yes—
559C23#650212yes—
560C23#651612yes—
561C23C22×C22⋊D711yescharacteristic
562Q8—22no—
563Q8—22no—
564Q8—41no—
565Q8—41no—
566Q8—44no—
567Q8—144no—
568Q8—144no—
569Q8—282no—
570Q8—282no—
571Q8—288no—
572D4—22no—
573D4—22no—
574D4—48no—
575D4—48no—
576D4—41no—
577D4—41no—
578D4—44no—
579D4—88no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—48no—
593C2×C4—84no—
594C2×C4—144no—
595C2×C4—144no—
596C2×C4—282no—
597C2×C4—288no—
598C2×C4—282no—
599C2×C4—282no—
600C2×C4—282no—
601C2×C4#650014yes—
602C2×C4#650012yes—
603C2×C4#650114yes—
604C2×C4#650214yes—
605C2×C4#650212yes—
606C2×C4#650314yes—
607C2×C4#651612yes—
608C2×C4C2×D4×D711yescharacteristic
609C2×C4C2×D4×D711yescharacteristic
610C2×C4C2×D4×D711yescharacteristic
611C2×C4C2×#115911yescharacteristic
612C2×C4C22×C22⋊D711yescharacteristic
613C2×C4C22×C22⋊D711yescharacteristic
614C8—144no—
615C8—282no—
616C7#2339411yescharacteristic, complemented, Sylow
617C22—22no—
618C22—22no—
619C22—24no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—44no—
628C22—44no—
629C22—82no—
630C22—82no—
631C22#8999514yes—
632C22C2×#650011yescharacteristic
633C22C2×#650211yescharacteristic
634C22#9052712yes—
635C22#9052912yes—
636C22C2×#651611yescharacteristic
637C4—42no—
638C4—42no—
639C4—44no—
640C4—284no—
641C4—284no—
642C4#8999514yes—
643C4C2×#650011yescharacteristic
644C4C2×#650111yescharacteristic
645C4C2×#650211yescharacteristic
646C4C2×#650311yescharacteristic
647C2—24no—
648C2—82no—
649C2—82no—
650C2#1174086812yes—
651C2#1174087212yes—
652C2C2×#8999511yescharacteristic
653C2#1174155011yescharacteristic
654C2#1174156211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#537C2
socleC23×C7C22×D4
hypercenterC2×#537D7
hypocenterC7#23394
LCSG ▷ C22×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. C2×#90516C2semidirect
3. C22×#6514C2semidirect
4. C2×#90006C2semidirect
5. C2×#90005C2semidirect
6. C22×#6384C2semidirect
7. #11729454C2semidirect
8. #11729439C2semidirect
9. #11729312C2semidirect
74. C22×Dic14D4semidirect
103. #6489D4semidirect
111. C2×#1136D4semidirect
616. C7#23394semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5372
C2×#905162
C22×#65142
C2×#900062
C2×#900052
C22×#63842
#117294542
#117294392
#117293122
#233947