Finite Group Databasebeta

← #13320563

Subgroups — #13320563

Order 1792.

960 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90269C211yesmaximal, characteristic
2D7×#432C211yesmaximal, characteristic
3#11735962C211yesmaximal, characteristic
4C7×#1491C211yesmaximal, characteristic, complemented
5#11729377C211yesmaximal, characteristic, complemented
6#11728556C211yesmaximal, characteristic, complemented
7#11727611C211yesmaximal, characteristic, complemented
8#11725997C214yesmaximal, complemented
9#11725986C214yesmaximal, complemented
10C2×C4×C7×Q8C2211yescharacteristic
11C2×#6509C411yescharacteristic
12C2×#6507C411yescharacteristic
13#90269—21no—
14#90269—21no—
15#90269—21no—
16#90269—21no—
17#90269C412yes—
18#90269C412yes—
19#90269C2212yes—
20#90269C2212yes—
21C2×#6460C411yescharacteristic
22C2×#6459—21no—
23C2×#6452—21no—
24C2×#6451C411yescharacteristic
25C2×#6435—21no—
26C2×C42×D7C2211yescharacteristic
27C2×C4×Dic14—21no—
28C2×C4×Dic14C2211yescharacteristic
29C7×#432C2211yescharacteristic
30C4×C7×M16C2211yescharacteristic
31#89898—21no—
32#89890—21no—
33#89698C2211yescharacteristic
34C4×#1136C2211yescharacteristic
35#89607C2212yes—
36D7×C4⋊C8C2212yes—
37D7×C22⋊C8—21no—
38#89491—21no—
39#89477—21no—
40#89477—21no—
41#89477—21no—
42D7×#97—21no—
43#89457C2212yes—
44#89457C2212yes—
45C7×Q8⋊C8C2214yescomplemented
46Q8⋊(C7⋊C8)C2214yescomplemented
47#89279C2214yescomplemented
48#89251C2214yescomplemented
49#17496—71nomaximal, Sylow
50C22×C7×Q8C2×C411yescharacteristic
51C22×C4×D7C2×C411yescharacteristic
52C22×C4×D7D411yescharacteristic
53C22×C4×D7D411yescharacteristic
54C22×Dic14—21no—
55C22×Dic14C2×C411yescharacteristic
56C2×C7×M16—21no—
57C4×C7×Q8—21no—
58C4×C7×Q8—21no—
59C4×C7×Q8C2×C412yes—
60C4×C7×Q8C2312yes—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4C2×C411yescharacteristic
63C2×C42×C7—21no—
64C2×C42×C7C2311yescharacteristic
65#6509—21no—
66#6509—21no—
67#6509C2×C412yes—
68#6509C2×C412yes—
69#6507—21no—
70#6507—21no—
71#6507C2×C412yes—
72#6507C2×C412yes—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7C2×C411yescharacteristic
77C2×Dic7⋊C4—21no—
78C2×Dic7⋊C4—21no—
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—21no—
82C2×C4×Dic7—21no—
83C2×C4×Dic7C2×C411yescharacteristic
84C2×C4×Dic7D411yescharacteristic
85C2×C4×Dic7D411yescharacteristic
86C2×#1136—21no—
87C2×C8×D7—22no—
88#6460—21no—
89#6460—21no—
90#6460C2×C412yes—
91#6460C2×C412yes—
92#6459—22no—
93#6459—22no—
94#6459—22no—
95#6459—22no—
96#6452—22no—
97#6452—22no—
98#6452—22no—
99#6452—22no—
100#6451—21no—
101#6451—21no—
102#6451C2×C412yes—
103#6451C2×C412yes—
104#6435—22no—
105#6435—22no—
106#6435—22no—
107#6435—22no—
108C42×D7C2×C411yescharacteristic
109C42×D7C2×C411yescharacteristic
110C42×D7D411yescharacteristic
111C42×D7D411yescharacteristic
112C42×D7D411yescharacteristic
113C42×D7D411yescharacteristic
114C42×D7C2311yescharacteristic
115C42×D7C2311yescharacteristic
116C4×Dic14—21no—
117C4×Dic14—21no—
118C4×Dic14—22no—
119C4×Dic14—22no—
120C4×Dic14—22no—
121C4×Dic14—22no—
122C4×Dic14C2×C412yes—
123C4×Dic14C2312yes—
124C7×C4⋊C8C2312yes—
125C7×C22⋊C8—21no—
126C7×#97—21no—
127C7×#97—21no—
128C4×C7×C8C2312yes—
129C22⋊(C7⋊C8)—21no—
130(C2×D7)⋊C8—22no—
131(C2×D7)⋊C8—22no—
132(C2×D7)⋊C8—22no—
133#6389—22no—
134Dic7⋊C8—22no—
135Dic7⋊C8—22no—
136Dic7⋊C8—22no—
137C4⋊(C7⋊C8)C2312yes—
138#6377—21no—
139#6377—21no—
140C4×C7⋊C8C2312yes—
141C2×#533—71no—
142#2920—71no—
143C2×#432—71no—
144#1497—74no—
145#1491—71no—
146#1491—71no—
147#1491—71no—
148#1491D711yescharacteristic, complemented
149#1481—74no—
150C23×D7C22⋊C411yescharacteristic
151C2×C7×Q8—21no—
152C2×C7×Q8—42no—
153C2×C7×Q8C22×C412yes—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7C22×C411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7C22⋊C411yescharacteristic
162C22×Dic7C22⋊C411yescharacteristic
163C22×Dic7C22⋊C411yescharacteristic
164C2×C4×D7—24no—
165C2×C4×D7—42no—
166C2×C4×D7—42no—
167C2×C4×D7C22⋊C411yescharacteristic
168C2×C4×D7C22⋊C411yescharacteristic
169C2×C4×D7C22⋊C411yescharacteristic
170C2×C4×D7C22⋊C411yescharacteristic
171C2×C4×D7C22⋊C411yescharacteristic
172C2×C4×D7C22⋊C411yescharacteristic
173C2×C4×D7C22×C411yescharacteristic
174C2×C4×D7C22×C411yescharacteristic
175C2×C4×D7C2×D411yescharacteristic
176C2×C4×D7C2×D411yescharacteristic
177C2×C4×D7C2×D411yescharacteristic
178C2×C4×D7C2×D411yescharacteristic
179C2×Dic14—21no—
180C2×Dic14—22no—
181C2×Dic14—22no—
182C2×Dic14—42no—
183C2×Dic14—44no—
184C2×Dic14C22×C412yes—
185C7×M16—42no—
186C2×C7×C8—22no—
187C2×C7×C8—22no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4C22×C412yes—
192C42×C7—22no—
193C42×C7—22no—
194C42×C7C22×C411yescharacteristic
195C42×C7C2×D411yescharacteristic
196C42×C7C2×D411yescharacteristic
197C42×C7C2411yescharacteristic
198(C2×D7)⋊C4—22no—
199(C2×D7)⋊C4—22no—
200(C2×D7)⋊C4—22no—
201(C2×D7)⋊C4—22no—
202C4⋊Dic7—21no—
203C4⋊Dic7—22no—
204C4⋊Dic7—22no—
205C4⋊Dic7C22×C412yes—
206Dic7⋊C4—22no—
207Dic7⋊C4—22no—
208Dic7⋊C4—22no—
209Dic7⋊C4—22no—
210Dic7⋊C4—22no—
211Dic7⋊C4—22no—
212Dic7⋊C4—22no—
213Dic7⋊C4—22no—
214Dic7⋊C4—22no—
215Dic7⋊C4—22no—
216C4×Dic7—22no—
217C4×Dic7—22no—
218C4×Dic7C22⋊C411yescharacteristic
219C4×Dic7C22⋊C411yescharacteristic
220C4×Dic7C22⋊C411yescharacteristic
221C4×Dic7C22⋊C411yescharacteristic
222C4×Dic7C22⋊C411yescharacteristic
223C4×Dic7C22⋊C411yescharacteristic
224C4×Dic7C22×C411yescharacteristic
225C4×Dic7C22×C411yescharacteristic
226C4×Dic7C2×D411yescharacteristic
227C4×Dic7C2×D411yescharacteristic
228C4×Dic7C2×D411yescharacteristic
229C4×Dic7C2×D411yescharacteristic
230#1136—42no—
231C2×C7⋊C8—22no—
232C2×C7⋊C8—22no—
233C8×D7—44no—
234#533—72no—
235#533—72no—
236#533—72no—
237#533—72no—
238#533—141no—
239#533—141no—
240#533—141no—
241#533—141no—
242C2×C4⋊Q8—71no—
243C2×C4⋊Q8—71no—
244C2×#125—141no—
245C2×C22⋊Q8—71no—
246C2×C22⋊Q8—71no—
247C2×C22⋊Q8—141no—
248C2×C4×Q8—71no—
249C2×C4×Q8—141no—
250C2×C4×Q8C2×D711yescharacteristic
251C2×#117—141no—
252C22×C42—71no—
253#432—71no—
254#432—72no—
255#432—141no—
256#432—141no—
257#432—141no—
258#432—141no—
259#432C2×D711yescharacteristic
260#431—72no—
261#431—72no—
262#423—141no—
263C2×C4⋊C8—72no—
264#407—141no—
265C2×C22⋊C8—141no—
266C4×M16—71no—
267C4×M16C2×D711yescharacteristic
268C2×#97—141no—
269Q8⋊C8—74no—
270Q8⋊C8—74no—
271Q8⋊C8—74no—
272Q8⋊C8C2×D714yescomplemented
273C23×C7C2×C22⋊C411yescharacteristic
274C22×D7—22no—
275C22×D7—22no—
276C22×D7—41no—
277C22×D7C2×C22⋊C411yescharacteristic
278C22×D7C2×C22⋊C411yescharacteristic
279C7×Q8—24no—
280C7×Q8—42no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7C2×C22⋊C411yescharacteristic
289C2×C4×C7C2×C22⋊C411yescharacteristic
290C2×C4×C7C2×C22⋊C411yescharacteristic
291C2×C4×C7C23×C411yescharacteristic
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C2×Dic7—22no—
295C2×Dic7—22no—
296C2×Dic7—22no—
297C2×Dic7—22no—
298C2×Dic7—41no—
299C2×Dic7—42no—
300C2×Dic7—42no—
301C2×Dic7C2×C22⋊C411yescharacteristic
302C2×Dic7C2×C22⋊C411yescharacteristic
303C2×Dic7C2×C22⋊C411yescharacteristic
304C2×Dic7C2×C22⋊C411yescharacteristic
305C2×Dic7C2×C22⋊C411yescharacteristic
306C2×Dic7C2×C22⋊C411yescharacteristic
307C4×D7—22no—
308C4×D7—22no—
309C4×D7—22no—
310C4×D7—22no—
311C4×D7—22no—
312C4×D7—41no—
313C4×D7—41no—
314C4×D7—41no—
315C4×D7—41no—
316C4×D7—41no—
317C4×D7—41no—
318C4×D7#13714yes—
319Dic14—24no—
320Dic14—44no—
321Dic14—44no—
322Dic14—42no—
323C7×C8—42no—
324C7×C8—42no—
325C7⋊C8—42no—
326C7⋊C8—42no—
327C22×Q8—71no—
328C22×Q8—141no—
329C22×Q8C4×D711yescharacteristic
330C23×C4—71no—
331C23×C4—71no—
332C23×C4—71no—
333C2×M16—21no—
334C2×M16—141no—
335C22×C8—142no—
336C4⋊Q8—72no—
337C4⋊Q8—72no—
338C4⋊Q8—72no—
339C4⋊Q8—72no—
340C4⋊Q8—141no—
341C4⋊Q8—141no—
342C4⋊Q8—141no—
343C4⋊Q8—141no—
344#125—142no—
345#125—142no—
346#125—142no—
347#125—142no—
348C22⋊Q8—72no—
349C22⋊Q8—72no—
350C22⋊Q8—72no—
351C22⋊Q8—72no—
352C22⋊Q8—141no—
353C22⋊Q8—141no—
354C22⋊Q8—142no—
355C22⋊Q8—142no—
356C22⋊Q8—142no—
357C22⋊Q8—142no—
358C22⋊Q8—141no—
359C22⋊Q8—141no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8—72no—
363C4×Q8—72no—
364C4×Q8—142no—
365C4×Q8—142no—
366C4×Q8—142no—
367C4×Q8—142no—
368C4×Q8—141no—
369C4×Q8—141no—
370C4×Q8C4×D712yes—
371C4×Q8C22×D712yes—
372#117—142no—
373#117—142no—
374#117—142no—
375#117—142no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—141no—
383C2×C4⋊C4—141no—
384C2×C4⋊C4C4×D711yescharacteristic
385C2×C22⋊C4—141no—
386C2×C22⋊C4—141no—
387C2×C42—21no—
388C2×C42—71no—
389C2×C42—71no—
390C2×C42—71no—
391C2×C42—71no—
392C2×C42—71no—
393C2×C42—71no—
394C2×C42—71no—
395C2×C42—71no—
396C2×C42—71no—
397C2×C42—71no—
398C2×C42—71no—
399C2×C42—141no—
400C2×C42C22×D711yescharacteristic
401C4⋊C8—72no—
402C4⋊C8—142no—
403C4⋊C8—142no—
404C4⋊C8—142no—
405C4⋊C8C22×D712yes—
406C22⋊C8—21no—
407C22⋊C8—142no—
408C22⋊C8—142no—
409C22⋊C8—142no—
410C22⋊C8—141no—
411#97—21no—
412#97—21no—
413#97—142no—
414#97—141no—
415#97—141no—
416C4×C8—72no—
417C4×C8C22×D712yes—
418C22×C7—21no—
419C22×C7—21no—
420C22×C7#42112yes—
421C22×C7C22×C22⋊C411yescharacteristic
422C2×D7—21no—
423C2×D7—21no—
424C2×D7—21no—
425C2×D7—21no—
426C2×D7—21no—
427C2×D7—44no—
428C2×D7#41912yes—
429C4×C7—21no—
430C4×C7—21no—
431C4×C7—21no—
432C4×C7—21no—
433C4×C7—42no—
434C4×C7—42no—
435C4×C7#56812yes—
436C4×C7C2×#13712yes—
437Dic7—21no—
438Dic7—21no—
439Dic7—21no—
440Dic7—21no—
441Dic7—21no—
442Dic7—42no—
443Dic7—42no—
444Dic7#41912yes—
445C24—71no—
446C2×Q8—21no—
447C2×Q8—42no—
448C2×Q8—72no—
449C2×Q8—142no—
450C2×Q8—142no—
451C2×Q8—141no—
452C2×Q8—282no—
453C2×Q8—284no—
454C2×Q8C2×C4×D712yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—71no—
458C22×C4—71no—
459C22×C4—71no—
460C22×C4—71no—
461C22×C4—71no—
462C22×C4—71no—
463C22×C4—71no—
464C22×C4—71no—
465C22×C4—71no—
466C22×C4—71no—
467C22×C4—71no—
468C22×C4—71no—
469C22×C4—71no—
470C22×C4—71no—
471C22×C4—71no—
472C22×C4—144no—
473C22×C4—141no—
474C22×C4—141no—
475C22×C4—282no—
476C22×C4—282no—
477C22×C4C2×C4×D711yescharacteristic
478C22×C4D4×D711yescharacteristic
479C22×C4D4×D711yescharacteristic
480M16—42no—
481M16—282no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—142no—
485C2×C8—142no—
486C2×C8—284no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—72no—
491C4⋊C4—142no—
492C4⋊C4—142no—
493C4⋊C4—142no—
494C4⋊C4—142no—
495C4⋊C4—142no—
496C4⋊C4—142no—
497C4⋊C4—142no—
498C4⋊C4—142no—
499C4⋊C4—142no—
500C4⋊C4—142no—
501C4⋊C4—142no—
502C4⋊C4—141no—
503C4⋊C4—142no—
504C4⋊C4C2×C4×D712yes—
505C22⋊C4—142no—
506C22⋊C4—142no—
507C22⋊C4—142no—
508C22⋊C4—142no—
509C42—22no—
510C42—22no—
511C42—71no—
512C42—71no—
513C42—71no—
514C42—71no—
515C42—71no—
516C42—71no—
517C42—71no—
518C42—71no—
519C42—71no—
520C42—71no—
521C42—71no—
522C42—71no—
523C42—142no—
524C42—142no—
525C42C2×C4×D711yescharacteristic
526C42D4×D711yescharacteristic
527C42D4×D711yescharacteristic
528C42C23×D711yescharacteristic
529C2×C7—22no—
530C2×C7#287911yescharacteristic
531C2×C7C2×#41911yescharacteristic
532C2×C7C2×#42111yescharacteristic
533D7—22no—
534D7—41no—
535C23—71no—
536C23—71no—
537C23—142no—
538C23—142no—
539C23—281no—
540C23D7×C22⋊C411yescharacteristic
541Q8—24no—
542Q8—42no—
543Q8—144no—
544Q8—282no—
545Q8—284no—
546Q8—284no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—71no—
555C2×C4—74no—
556C2×C4—71no—
557C2×C4—71no—
558C2×C4—71no—
559C2×C4—71no—
560C2×C4—71no—
561C2×C4—142no—
562C2×C4—142no—
563C2×C4—142no—
564C2×C4—142no—
565C2×C4—142no—
566C2×C4—142no—
567C2×C4—142no—
568C2×C4—142no—
569C2×C4—142no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4—281no—
574C2×C4—281no—
575C2×C4—281no—
576C2×C4—281no—
577C2×C4—282no—
578C2×C4—282no—
579C2×C4D7×C22⋊C411yescharacteristic
580C2×C4D7×C22⋊C411yescharacteristic
581C2×C4D7×C22⋊C411yescharacteristic
582C2×C4C22×C4×D711yescharacteristic
583C2×C4C2×D4×D711yescharacteristic
584C2×C4C2×D4×D711yescharacteristic
585C8—42no—
586C8—42no—
587C8—282no—
588C8—282no—
589C7#1749611yescharacteristic, complemented, Sylow
590C22—21no—
591C22—21no—
592C22—72no—
593C22—141no—
594C22—141no—
595C22—141no—
596C22—141no—
597C22—141no—
598C22—284no—
599C22#8959312yes—
600C22C2×D7×C22⋊C411yescharacteristic
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—42no—
606C4—42no—
607C4—72no—
608C4—141no—
609C4—141no—
610C4—141no—
611C4—141no—
612C4—141no—
613C4—282no—
614C4—282no—
615C4#9043612yes—
616C4D7×#13712yes—
617C2—22no—
618C2—142no—
619C2—281no—
620C2#1173607511yescharacteristic
621C2D7×#41911yescharacteristic
622C2C2×#8959311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1491C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1491D7
hypocenterC7#17496
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1491

Complements

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

NH ≅ G/Ntype
4. C7×#1491C2semidirect
5. #11729377C2semidirect
6. #11728556C2semidirect
7. #11727611C2semidirect
8. #11725997C2semidirect
9. #11725986C2semidirect
45. C7×Q8⋊C8C22semidirect
46. Q8⋊(C7⋊C8)C22semidirect
47. #89279C22semidirect
48. #89251C22semidirect
148. #1491D7semidirect
272. Q8⋊C8C2×D7semidirect
589. C7#17496semidirect

Maximal subgroups

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

subgroupindex
C2×#902692
D7×#4322
#117359622
C7×#14912
#117293772
#117285562
#117276112
#117259972
#117259862
#174967