Finite Group Databasebeta

← #13322432

Subgroups — #13322432

Order 1792.

996 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
1D7×#533C211yesmaximal, characteristic
2#11736483C211yesmaximal, characteristic
3C2×#89609C211yesmaximal, characteristic
4C7×#1550C211yesmaximal, characteristic, complemented
5#11729584C211yesmaximal, characteristic, complemented
6#11728263C211yesmaximal, characteristic, complemented
7#11728192C211yesmaximal, characteristic, complemented
8#11726242C212yesmaximal, complemented
9#11726237C212yesmaximal, complemented
10#11726184C212yesmaximal, complemented
11#11726174C212yesmaximal, complemented
12C7×#533C2211yescharacteristic
13C22×#1131—21no—
14#90419C2211yescharacteristic
15#90419C2211yescharacteristic
16D7×C4⋊Q8C2211yescharacteristic
17D7×C4⋊Q8C2211yescharacteristic
18#90379—21no—
19D7×#125—21no—
20D7×C22⋊Q8—21no—
21D7×C22⋊Q8C412yes—
22#90313—21no—
23#90313C412yes—
24#90269—21no—
25#90269C412yes—
26C4×Q8×D7—21no—
27C4×Q8×D7C412yes—
28D7×#117—21no—
29#90208—21no—
30C2×C42×D7C2211yescharacteristic
31#90161C2211yescharacteristic
32C7×#423C2211yescharacteristic
33C2×C7×C4⋊C8C2211yescharacteristic
34#89898—21no—
35M16×Dic7—21no—
36C2×Dic7⋊C8—21no—
37#89696C2211yescharacteristic
38C2×C4⋊(C7⋊C8)C2211yescharacteristic
39#89609—21no—
40#89609—21no—
41#89609C2212yes—
42#89609C2212yes—
43#89607—21no—
44#89607C2212yes—
45#89605—21no—
46#89605C2212yes—
47C7×#333C2212yescomplemented
48C7×#332C2212yescomplemented
49#89342C2212yescomplemented
50#89341C2212yescomplemented
51#89281C2212yescomplemented
52#89277C2212yescomplemented
53#89276C2212yescomplemented
54#89276C2212yescomplemented
55#17749—71nomaximal, Sylow
56C2×Q8×D7—21no—
57C2×Q8×D7C2×C412yes—
58C22×C4×D7C2×C411yescharacteristic
59C22×C4×D7D411yescharacteristic
60C22×C4×D7D411yescharacteristic
61C2×C7×M16—21no—
62C22×C7×C8—21no—
63C7×C4⋊Q8C2311yescharacteristic
64C7×C4⋊Q8C2311yescharacteristic
65C7×#125—21no—
66C7×C22⋊Q8—21no—
67C7×C22⋊Q8C2×C412yes—
68C4×C7×Q8—21no—
69C4×C7×Q8C2×C412yes—
70C7×#117—21no—
71C2×C42×C7C2311yescharacteristic
72#6509—21no—
73#6509C2×C412yes—
74Q8×Dic7—21no—
75Q8×Dic7C2×C412yes—
76#6507—21no—
77#6507C2×C412yes—
78#6489—21no—
79#6487—21no—
80#6487C2×C412yes—
81C2×C4×Dic7C2×C411yescharacteristic
82C2×C4×Dic7D411yescharacteristic
83C2×C4×Dic7D411yescharacteristic
84C2×#1136—21no—
85C22×C7⋊C8—21no—
86C2×#1131—22no—
87C2×#1131—22no—
88C2×#1131—44no—
89#6460—21no—
90#6460C2×C412yes—
91#6459—22no—
92#6459—22no—
93#6455—21no—
94#6455—21no—
95D7×C4⋊C4—22no—
96D7×C4⋊C4—22no—
97D7×C4⋊C4—21no—
98D7×C4⋊C4—21no—
99D7×C4⋊C4—21no—
100D7×C4⋊C4C2×C412yes—
101#6453—21no—
102#6453—21no—
103#6452—22no—
104#6452—22no—
105#6451—21no—
106#6451—21no—
107#6451—21no—
108#6451C2×C412yes—
109#6450—22no—
110#6450—22no—
111#6450—21no—
112#6450C2×C412yes—
113D7×C22⋊C4—22no—
114#6441—22no—
115#6441—22no—
116#6440—22no—
117#6435—22no—
118C42×D7—22no—
119C42×D7C2×C411yescharacteristic
120C42×D7C2×C411yescharacteristic
121C42×D7D411yescharacteristic
122C42×D7D411yescharacteristic
123C42×D7D411yescharacteristic
124C42×D7D411yescharacteristic
125C42×D7C2311yescharacteristic
126C42×D7C2311yescharacteristic
127#6433—21no—
128#6432C2311yescharacteristic
129#6432C2311yescharacteristic
130C4×Dic14—21no—
131C4×Dic14—22no—
132C4×Dic14—22no—
133C4×Dic14C2×C412yes—
134C7×C4⋊C8—21no—
135C7×C4⋊C8—21no—
136C7×C4⋊C8C2312yes—
137C7×C4⋊C8C2312yes—
138(C2×D7)⋊C8—22no—
139(C2×D7)⋊C8—22no—
140#6389—22no—
141Dic7⋊C8—22no—
142Dic7⋊C8—22no—
143C8×Dic7—22no—
144C4⋊(C7⋊C8)—21no—
145C4⋊(C7⋊C8)—21no—
146C4⋊(C7⋊C8)C2312yes—
147C4⋊(C7⋊C8)C2312yes—
148C2×#533—71no—
149#2920—71no—
150C2×#423—71no—
151#1559—72no—
152#1559—72no—
153#1550—71no—
154#1550—71no—
155#1550—71no—
156#1550D711yescharacteristic, complemented
157#1548—72no—
158#1546—72no—
159C23×D7C22⋊C411yescharacteristic
160C2×C7×Q8—21no—
161C2×C7×Q8C22×C412yes—
162C22×C4×C7C22×C411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×Dic7C22⋊C411yescharacteristic
166C22×Dic7C22⋊C411yescharacteristic
167C22×Dic7C22⋊C411yescharacteristic
168Q8×D7—42no—
169Q8×D7—42no—
170Q8×D7—44no—
171C2×C4×D7—22no—
172C2×C4×D7—22no—
173C2×C4×D7—22no—
174C2×C4×D7—22no—
175C2×C4×D7—22no—
176C2×C4×D7—22no—
177C2×C4×D7—42no—
178C2×C4×D7C22⋊C411yescharacteristic
179C2×C4×D7C22⋊C411yescharacteristic
180C2×C4×D7C22⋊C411yescharacteristic
181C2×C4×D7C22⋊C411yescharacteristic
182C2×C4×D7C22⋊C411yescharacteristic
183C2×C4×D7C22⋊C411yescharacteristic
184C2×C4×D7C22×C411yescharacteristic
185C2×C4×D7C22×C411yescharacteristic
186C2×C4×D7C2×D411yescharacteristic
187C2×C4×D7C2×D411yescharacteristic
188C2×C4×D7C2×D411yescharacteristic
189C2×C4×D7C2×D411yescharacteristic
190C2×Dic14—21no—
191C2×Dic14—22no—
192C2×Dic14—22no—
193C2×Dic14C22×C412yes—
194C7×M16—42no—
195C2×C7×C8—22no—
196C2×C7×C8—22no—
197C2×C7×C8—42no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4C22×C412yes—
204C7×C22⋊C4—22no—
205C42×C7—22no—
206C42×C7C22×C411yescharacteristic
207C42×C7C2×D411yescharacteristic
208C42×C7C2×D411yescharacteristic
209C42×C7C2411yescharacteristic
210C22⋊Dic7—22no—
211(C2×D7)⋊C4—22no—
212(C2×D7)⋊C4—22no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7—21no—
216C4⋊Dic7—22no—
217C4⋊Dic7—22no—
218C4⋊Dic7C22×C412yes—
219Dic7⋊C4—22no—
220Dic7⋊C4—22no—
221Dic7⋊C4—22no—
222Dic7⋊C4—22no—
223Dic7⋊C4—22no—
224Dic7⋊C4—22no—
225Dic7⋊C4—22no—
226Dic7⋊C4—22no—
227C4×Dic7—22no—
228C4×Dic7—22no—
229C4×Dic7—22no—
230C4×Dic7C22⋊C411yescharacteristic
231C4×Dic7C22⋊C411yescharacteristic
232C4×Dic7C22⋊C411yescharacteristic
233C4×Dic7C22⋊C411yescharacteristic
234C4×Dic7C22⋊C411yescharacteristic
235C4×Dic7C22⋊C411yescharacteristic
236C4×Dic7C22×C411yescharacteristic
237C4×Dic7C22×C411yescharacteristic
238C4×Dic7C2×D411yescharacteristic
239C4×Dic7C2×D411yescharacteristic
240C4×Dic7C2×D411yescharacteristic
241C4×Dic7C2×D411yescharacteristic
242#1136—42no—
243C2×C7⋊C8—22no—
244C2×C7⋊C8—22no—
245C2×C7⋊C8—42no—
246#1131—44no—
247#1131—44no—
248C22×M16—141no—
249#533—71no—
250#533—71no—
251#533—72no—
252#533—72no—
253#533—71no—
254#533—141no—
255#533—141no—
256#533—141no—
257#533—141no—
258#533C2×D711yescharacteristic
259C2×C4⋊Q8—71no—
260C2×C4⋊Q8—71no—
261C2×#125—141no—
262C2×C22⋊Q8—72no—
263C2×C22⋊Q8—141no—
264C2×C4×Q8—72no—
265C2×C4×Q8—141no—
266C2×#117—141no—
267C22×C42—71no—
268#432—72no—
269#432—141no—
270#431—72no—
271#431—141no—
272#423—71no—
273#423—72no—
274#423—72no—
275#423—141no—
276#423—141no—
277#423C2×D711yescharacteristic
278C2×C4⋊C8—71no—
279C2×C4⋊C8—141no—
280C2×C4⋊C8C2×D711yescharacteristic
281#407—141no—
282C4×M16—141no—
283#333—72no—
284#333—72no—
285#333—72no—
286#333C2×D712yescomplemented
287#332—72no—
288#332—72no—
289#332—72no—
290#332C2×D712yescomplemented
291C23×C7C2×C22⋊C411yescharacteristic
292C22×D7—22no—
293C22×D7—22no—
294C22×D7—41no—
295C22×D7C2×C22⋊C411yescharacteristic
296C22×D7C2×C22⋊C411yescharacteristic
297C7×Q8—42no—
298C7×Q8—41no—
299C7×Q8—41no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—41no—
305C2×C4×C7C2×C22⋊C411yescharacteristic
306C2×C4×C7C2×C22⋊C411yescharacteristic
307C2×C4×C7C2×C22⋊C411yescharacteristic
308C2×C4×C7#13712yes—
309C2×C4×C7#13712yes—
310C2×C4×C7C23×C411yescharacteristic
311C2×C4×C7C22×D411yescharacteristic
312C2×C4×C7C22×D411yescharacteristic
313C2×Dic7—22no—
314C2×Dic7—22no—
315C2×Dic7—22no—
316C2×Dic7—22no—
317C2×Dic7—22no—
318C2×Dic7C2×C22⋊C411yescharacteristic
319C2×Dic7C2×C22⋊C411yescharacteristic
320C2×Dic7C2×C22⋊C411yescharacteristic
321C2×Dic7C2×C22⋊C411yescharacteristic
322C2×Dic7C2×C22⋊C411yescharacteristic
323C2×Dic7C2×C22⋊C411yescharacteristic
324C4×D7—22no—
325C4×D7—21no—
326C4×D7—21no—
327C4×D7—21no—
328C4×D7—21no—
329C4×D7—21no—
330C4×D7—21no—
331C4×D7—22no—
332C4×D7—22no—
333C4×D7—22no—
334C4×D7—22no—
335C4×D7—22no—
336C4×D7—41no—
337C4×D7—41no—
338C4×D7—44no—
339C4×D7—44no—
340C4×D7AGL(1,ℤ/8)12yes—
341C4×D7#13712yes—
342Dic14—41no—
343Dic14—41no—
344Dic14—44no—
345Dic14—44no—
346Dic14—42no—
347C7×C8—42no—
348C7×C8—42no—
349C7⋊C8—42no—
350C7⋊C8—42no—
351C22×Q8—72no—
352C22×Q8—141no—
353C23×C4—71no—
354C23×C4—71no—
355C23×C4—71no—
356C2×M16—21no—
357C2×M16—142no—
358C2×M16—142no—
359C2×M16—141no—
360C2×M16—284no—
361C22×C8—21no—
362C22×C8—141no—
363C4⋊Q8—72no—
364C4⋊Q8—72no—
365C4⋊Q8—71no—
366C4⋊Q8—71no—
367C4⋊Q8—141no—
368C4⋊Q8—141no—
369C4⋊Q8—141no—
370C4⋊Q8—141no—
371C4⋊Q8C22×D711yescharacteristic
372C4⋊Q8C22×D711yescharacteristic
373#125—21no—
374#125—142no—
375#125—142no—
376#125—141no—
377#125—141no—
378#125—141no—
379C22⋊Q8—21no—
380C22⋊Q8—72no—
381C22⋊Q8—72no—
382C22⋊Q8—72no—
383C22⋊Q8—142no—
384C22⋊Q8—142no—
385C22⋊Q8—142no—
386C22⋊Q8—142no—
387C22⋊Q8—141no—
388C22⋊Q8—141no—
389C22⋊Q8—141no—
390C22⋊Q8C4×D712yes—
391C4×Q8—21no—
392C4×Q8—72no—
393C4×Q8—72no—
394C4×Q8—72no—
395C4×Q8—142no—
396C4×Q8—142no—
397C4×Q8—142no—
398C4×Q8—142no—
399C4×Q8—141no—
400C4×Q8—141no—
401C4×Q8—141no—
402C4×Q8C4×D712yes—
403#117—21no—
404#117—142no—
405#117—142no—
406#117—141no—
407#117—141no—
408#117—141no—
409C2×C4⋊C4—72no—
410C2×C4⋊C4—141no—
411C2×C4⋊C4—141no—
412C2×C4⋊C4—142no—
413C2×C4⋊C4—142no—
414C2×C4⋊C4—141no—
415C2×C22⋊C4—142no—
416C2×C42—71no—
417C2×C42—71no—
418C2×C42—71no—
419C2×C42—71no—
420C2×C42—71no—
421C2×C42—71no—
422C2×C42—71no—
423C2×C42—71no—
424C2×C42—71no—
425C2×C42—71no—
426C2×C42—71no—
427C2×C42—142no—
428C2×C42C22×D711yescharacteristic
429C4⋊C8—21no—
430C4⋊C8—21no—
431C4⋊C8—72no—
432C4⋊C8—72no—
433C4⋊C8—142no—
434C4⋊C8—142no—
435C4⋊C8—141no—
436C4⋊C8—141no—
437C4⋊C8C22×D712yes—
438C4⋊C8C22×D712yes—
439C22⋊C8—142no—
440C22⋊C8—142no—
441#97—142no—
442C4×C8—142no—
443C22×C7—21no—
444C22×C7—21no—
445C22×C7#41912yes—
446C22×C7C22×C22⋊C411yescharacteristic
447C2×D7—21no—
448C2×D7—21no—
449C2×D7—21no—
450C2×D7—21no—
451C2×D7—21no—
452C2×D7—44no—
453C2×D7#41712yes—
454C4×C7—21no—
455C4×C7—21no—
456C4×C7—21no—
457C4×C7—42no—
458C4×C7—42no—
459C4×C7#41912yes—
460C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
461C4×C7C2×#13711yescharacteristic
462C4×C7C2×#13711yescharacteristic
463C4×C7C2×#13711yescharacteristic
464Dic7—21no—
465Dic7—21no—
466Dic7—21no—
467Dic7—21no—
468Dic7—42no—
469Dic7—42no—
470Dic7#41312yes—
471Dic7#41712yes—
472C24—71no—
473C2×Q8—21no—
474C2×Q8—72no—
475C2×Q8—142no—
476C2×Q8—142no—
477C2×Q8—141no—
478C2×Q8—282no—
479C2×Q8—282no—
480C2×Q8—284no—
481C2×Q8C2×C4×D712yes—
482C22×C4—71no—
483C22×C4—71no—
484C22×C4—71no—
485C22×C4—71no—
486C22×C4—71no—
487C22×C4—71no—
488C22×C4—71no—
489C22×C4—71no—
490C22×C4—71no—
491C22×C4—71no—
492C22×C4—71no—
493C22×C4—71no—
494C22×C4—71no—
495C22×C4—71no—
496C22×C4—71no—
497C22×C4—142no—
498C22×C4—142no—
499C22×C4—142no—
500C22×C4—142no—
501C22×C4—142no—
502C22×C4—142no—
503C22×C4—282no—
504C22×C4C2×C4×D711yescharacteristic
505C22×C4D4×D711yescharacteristic
506C22×C4D4×D711yescharacteristic
507M16—42no—
508M16—284no—
509M16—284no—
510M16—282no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—42no—
514C2×C8—142no—
515C2×C8—142no—
516C2×C8—282no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—72no—
523C4⋊C4—142no—
524C4⋊C4—142no—
525C4⋊C4—142no—
526C4⋊C4—142no—
527C4⋊C4—142no—
528C4⋊C4—142no—
529C4⋊C4—142no—
530C4⋊C4—142no—
531C4⋊C4—142no—
532C4⋊C4—142no—
533C4⋊C4—141no—
534C4⋊C4—141no—
535C4⋊C4—141no—
536C4⋊C4C2×C4×D712yes—
537C22⋊C4—22no—
538C22⋊C4—142no—
539C22⋊C4—142no—
540C22⋊C4—142no—
541C42—22no—
542C42—71no—
543C42—71no—
544C42—71no—
545C42—71no—
546C42—71no—
547C42—71no—
548C42—71no—
549C42—71no—
550C42—71no—
551C42—71no—
552C42—71no—
553C42—71no—
554C42—142no—
555C42—142no—
556C42—142no—
557C42C2×C4×D711yescharacteristic
558C42D4×D711yescharacteristic
559C42D4×D711yescharacteristic
560C42C23×D711yescharacteristic
561C2×C7—22no—
562C2×C7C2×#41311yescharacteristic
563C2×C7C2×#41911yescharacteristic
564C2×C7C2×#41711yescharacteristic
565D7—22no—
566D7—41no—
567C23—71no—
568C23—71no—
569C23—142no—
570C23—142no—
571C23—281no—
572C23D7×C22⋊C411yescharacteristic
573Q8—42no—
574Q8—41no—
575Q8—41no—
576Q8—282no—
577Q8—284no—
578Q8—284no—
579Q8—281no—
580Q8—281no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—71no—
587C2×C4—71no—
588C2×C4—72no—
589C2×C4—72no—
590C2×C4—71no—
591C2×C4—71no—
592C2×C4—71no—
593C2×C4—71no—
594C2×C4—142no—
595C2×C4—142no—
596C2×C4—142no—
597C2×C4—142no—
598C2×C4—142no—
599C2×C4—142no—
600C2×C4—142no—
601C2×C4—141no—
602C2×C4—141no—
603C2×C4—141no—
604C2×C4—141no—
605C2×C4—141no—
606C2×C4—141no—
607C2×C4—142no—
608C2×C4—142no—
609C2×C4—142no—
610C2×C4—142no—
611C2×C4—281no—
612C2×C4—281no—
613C2×C4—284no—
614C2×C4—284no—
615C2×C4D7×C22⋊C411yescharacteristic
616C2×C4D7×C22⋊C411yescharacteristic
617C2×C4D7×C22⋊C411yescharacteristic
618C2×C4#647812yes—
619C2×C4#648112yes—
620C2×C4C22×C4×D711yescharacteristic
621C2×C4C2×D4×D711yescharacteristic
622C2×C4C2×D4×D711yescharacteristic
623C8—42no—
624C8—42no—
625C8—282no—
626C8—282no—
627C7#1774911yescharacteristic, complemented, Sylow
628C22—21no—
629C22—21no—
630C22—72no—
631C22—141no—
632C22—141no—
633C22—141no—
634C22—141no—
635C22—141no—
636C22—284no—
637C22#8957412yes—
638C22C2×D7×C22⋊C411yescharacteristic
639C4—21no—
640C4—21no—
641C4—21no—
642C4—42no—
643C4—42no—
644C4—72no—
645C4—72no—
646C4—141no—
647C4—141no—
648C4—141no—
649C4—141no—
650C4—282no—
651C4—282no—
652C4#8957412yes—
653C4C2×#647811yescharacteristic
654C4C2×#648111yescharacteristic
655C4D7×AGL(1,ℤ/8)11yescharacteristic
656C4D7×#13711yescharacteristic
657C2—22no—
658C2—142no—
659C2—281no—
660C2D7×#41311yescharacteristic
661C2C2×#8957411yescharacteristic
662C2D7×#41711yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1550C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1550D7
hypocenterC7#17749
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
4. C7×#1550C2semidirect
5. #11729584C2semidirect
6. #11728263C2semidirect
7. #11728192C2semidirect
8. #11726242C2semidirect
9. #11726237C2semidirect
10. #11726184C2semidirect
11. #11726174C2semidirect
47. C7×#333C22semidirect
48. C7×#332C22semidirect
49. #89342C22semidirect
50. #89341C22semidirect
51. #89281C22semidirect
52. #89277C22semidirect
53. #89276C22semidirect
54. #89276C22semidirect
156. #1550D7semidirect
286. #333C2×D7semidirect
290. #332C2×D7semidirect
627. C7#17749semidirect

Maximal subgroups

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

subgroupindex
D7×#5332
#117364832
C2×#896092
C7×#15502
#117295842
#117282632
#117281922
#117262422
#117262372
#117261842
#117261742
#177497