Finite Group Databasebeta

← #13320341

Subgroups — #13320341

Order 1792.

988 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90269C211yesmaximal, characteristic
2#11736727C211yesmaximal, characteristic
3D7×#431C211yesmaximal, characteristic
4C7×#1479C211yesmaximal, characteristic, complemented
5#11729375C211yesmaximal, characteristic, complemented
6#11728521C211yesmaximal, characteristic, complemented
7#11728520C211yesmaximal, characteristic, complemented
8#11725989C214yesmaximal, 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×#431C2211yescharacteristic
30C7×#431C2211yescharacteristic
31#89697C2211yescharacteristic
32#89697C2211yescharacteristic
33#89609—21no—
34#89607—21no—
35#89605C2212yes—
36D7×C4⋊C8C2212yes—
37#89498—21no—
38D7×C22⋊C8—21no—
39#89491—21no—
40#89490—21no—
41C4×#1131C2212yes—
42#89457—21no—
43#89457C2212yes—
44C4×C8×D7—21no—
45C7×Q8⋊C8C2214yescomplemented
46Q8⋊(C7⋊C8)C2214yescomplemented
47#89279C2214yescomplemented
48#89251C2214yescomplemented
49#17464—71nomaximal, Sylow
50C22×C7×Q8C2×C411yescharacteristic
51C22×C4×D7C2×C411yescharacteristic
52C22×C4×D7D411yescharacteristic
53C22×C4×D7D411yescharacteristic
54C22×Dic14—21no—
55C22×Dic14C2×C411yescharacteristic
56C4×C7×Q8—21no—
57C4×C7×Q8—21no—
58C4×C7×Q8C2×C412yes—
59C4×C7×Q8C2312yes—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4C2×C411yescharacteristic
62C2×C42×C7—21no—
63C2×C42×C7C2311yescharacteristic
64#6509—21no—
65#6509—21no—
66#6509C2×C412yes—
67#6509C2×C412yes—
68#6507—21no—
69#6507—21no—
70#6507C2×C412yes—
71#6507C2×C412yes—
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—21no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7C2×C411yescharacteristic
76C2×Dic7⋊C4—21no—
77C2×Dic7⋊C4—21no—
78C2×Dic7⋊C4—21no—
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—21no—
81C2×C4×Dic7—21no—
82C2×C4×Dic7C2×C411yescharacteristic
83C2×C4×Dic7D411yescharacteristic
84C2×C4×Dic7D411yescharacteristic
85C2×#1131—22no—
86C2×C8×D7—22no—
87#6460—21no—
88#6460—21no—
89#6460C2×C412yes—
90#6460C2×C412yes—
91#6459—22no—
92#6459—22no—
93#6459—22no—
94#6459—22no—
95#6452—22no—
96#6452—22no—
97#6452—22no—
98#6452—22no—
99#6451—21no—
100#6451—21no—
101#6451C2×C412yes—
102#6451C2×C412yes—
103#6435—22no—
104#6435—22no—
105#6435—22no—
106#6435—22no—
107C42×D7C2×C411yescharacteristic
108C42×D7C2×C411yescharacteristic
109C42×D7D411yescharacteristic
110C42×D7D411yescharacteristic
111C42×D7D411yescharacteristic
112C42×D7D411yescharacteristic
113C42×D7C2311yescharacteristic
114C42×D7C2311yescharacteristic
115C4×Dic14—21no—
116C4×Dic14—21no—
117C4×Dic14—22no—
118C4×Dic14—22no—
119C4×Dic14—22no—
120C4×Dic14—22no—
121C4×Dic14C2×C412yes—
122C4×Dic14C2312yes—
123C7×C4⋊C8—21no—
124C7×C4⋊C8C2312yes—
125C7×C22⋊C8—21no—
126C7×C22⋊C8—21no—
127C4×C7×C8—21no—
128C4×C7×C8C2312yes—
129C22⋊(C7⋊C8)—21no—
130C22⋊(C7⋊C8)—21no—
131(C2×D7)⋊C8—22no—
132(C2×D7)⋊C8—22no—
133#6389—22no—
134Dic7⋊C8—22no—
135Dic7⋊C8—22no—
136C8×Dic7—22no—
137C4⋊(C7⋊C8)—21no—
138C4⋊(C7⋊C8)C2312yes—
139C4×C7⋊C8—21no—
140C4×C7⋊C8C2312yes—
141C2×#533—71no—
142#2920—71no—
143C2×#431—71no—
144#1485—74no—
145#1481—74no—
146#1479—71no—
147#1479—71no—
148#1479—71no—
149#1479D711yescharacteristic, complemented
150C23×D7C22⋊C411yescharacteristic
151C2×C7×Q8—21no—
152C2×C7×Q8—22no—
153C2×C7×Q8—22no—
154C2×C7×Q8C22×C412yes—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7C22×C411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7C22⋊C411yescharacteristic
163C22×Dic7C22⋊C411yescharacteristic
164C22×Dic7C22⋊C411yescharacteristic
165C2×C4×D7—22no—
166C2×C4×D7—22no—
167C2×C4×D7—24no—
168C2×C4×D7—42no—
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×D7C22×C411yescharacteristic
176C2×C4×D7C22×C411yescharacteristic
177C2×C4×D7C2×D411yescharacteristic
178C2×C4×D7C2×D411yescharacteristic
179C2×C4×D7C2×D411yescharacteristic
180C2×C4×D7C2×D411yescharacteristic
181C2×Dic14—22no—
182C2×Dic14—22no—
183C2×Dic14—21no—
184C2×Dic14—22no—
185C2×Dic14—22no—
186C2×Dic14—44no—
187C2×Dic14C22×C412yes—
188C2×C7×C8—22no—
189C2×C7×C8—22no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4C22×C412yes—
194C42×C7—22no—
195C42×C7—22no—
196C42×C7C22×C411yescharacteristic
197C42×C7C2×D411yescharacteristic
198C42×C7C2×D411yescharacteristic
199C42×C7C2411yescharacteristic
200(C2×D7)⋊C4—22no—
201(C2×D7)⋊C4—22no—
202(C2×D7)⋊C4—22no—
203(C2×D7)⋊C4—22no—
204C4⋊Dic7—21no—
205C4⋊Dic7—22no—
206C4⋊Dic7—22no—
207C4⋊Dic7C22×C412yes—
208Dic7⋊C4—22no—
209Dic7⋊C4—22no—
210Dic7⋊C4—22no—
211Dic7⋊C4—22no—
212Dic7⋊C4—22no—
213Dic7⋊C4—22no—
214Dic7⋊C4—22no—
215Dic7⋊C4—22no—
216Dic7⋊C4—22no—
217Dic7⋊C4—22no—
218C4×Dic7—22no—
219C4×Dic7—22no—
220C4×Dic7C22⋊C411yescharacteristic
221C4×Dic7C22⋊C411yescharacteristic
222C4×Dic7C22⋊C411yescharacteristic
223C4×Dic7C22⋊C411yescharacteristic
224C4×Dic7C22⋊C411yescharacteristic
225C4×Dic7C22⋊C411yescharacteristic
226C4×Dic7C22×C411yescharacteristic
227C4×Dic7C22×C411yescharacteristic
228C4×Dic7C2×D411yescharacteristic
229C4×Dic7C2×D411yescharacteristic
230C4×Dic7C2×D411yescharacteristic
231C4×Dic7C2×D411yescharacteristic
232C2×C7⋊C8—22no—
233C2×C7⋊C8—22no—
234#1131—44no—
235C8×D7—44no—
236#533—72no—
237#533—72no—
238#533—72no—
239#533—72no—
240#533—141no—
241#533—141no—
242#533—141no—
243#533—141no—
244C2×C4⋊Q8—71no—
245C2×C4⋊Q8—71no—
246C2×#125—141no—
247C2×C22⋊Q8—71no—
248C2×C22⋊Q8—71no—
249C2×C22⋊Q8—141no—
250C2×C4×Q8—71no—
251C2×C4×Q8—141no—
252C2×C4×Q8C2×D711yescharacteristic
253C2×#117—141no—
254C22×C42—71no—
255#432—141no—
256#432—141no—
257#431—71no—
258#431—72no—
259#431—71no—
260#431—72no—
261#431—141no—
262#431—141no—
263#431C2×D711yescharacteristic
264#431C2×D711yescharacteristic
265#423—141no—
266C2×C4⋊C8—72no—
267#407—141no—
268C2×C22⋊C8—141no—
269C4×M16—72no—
270C2×C4×C8—141no—
271Q8⋊C8—74no—
272Q8⋊C8—74no—
273Q8⋊C8—74no—
274Q8⋊C8C2×D714yescomplemented
275C23×C7C2×C22⋊C411yescharacteristic
276C22×D7—22no—
277C22×D7—22no—
278C22×D7—41no—
279C22×D7C2×C22⋊C411yescharacteristic
280C22×D7C2×C22⋊C411yescharacteristic
281C7×Q8—22no—
282C7×Q8—22no—
283C7×Q8—42no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—41no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7C2×C22⋊C411yescharacteristic
291C2×C4×C7C2×C22⋊C411yescharacteristic
292C2×C4×C7C2×C22⋊C411yescharacteristic
293C2×C4×C7#13112yes—
294C2×C4×C7#13112yes—
295C2×C4×C7C23×C411yescharacteristic
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C22×D411yescharacteristic
298C2×Dic7—22no—
299C2×Dic7—22no—
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—22no—
303C2×Dic7—42no—
304C2×Dic7—42no—
305C2×Dic7C2×C22⋊C411yescharacteristic
306C2×Dic7C2×C22⋊C411yescharacteristic
307C2×Dic7C2×C22⋊C411yescharacteristic
308C2×Dic7C2×C22⋊C411yescharacteristic
309C2×Dic7C2×C22⋊C411yescharacteristic
310C2×Dic7C2×C22⋊C411yescharacteristic
311C4×D7—22no—
312C4×D7—21no—
313C4×D7—21no—
314C4×D7—21no—
315C4×D7—21no—
316C4×D7—22no—
317C4×D7—22no—
318C4×D7—22no—
319C4×D7—22no—
320C4×D7—22no—
321C4×D7—22no—
322C4×D7—41no—
323C4×D7—41no—
324C4×D7#13714yes—
325Dic14—22no—
326Dic14—22no—
327Dic14—44no—
328Dic14—42no—
329Dic14—44no—
330C7×C8—42no—
331C7×C8—42no—
332C7⋊C8—42no—
333C7⋊C8—42no—
334C22×Q8—71no—
335C22×Q8—141no—
336C22×Q8C4×D711yescharacteristic
337C23×C4—71no—
338C23×C4—71no—
339C23×C4—71no—
340C2×M16—142no—
341C22×C8—142no—
342C4⋊Q8—72no—
343C4⋊Q8—72no—
344C4⋊Q8—72no—
345C4⋊Q8—72no—
346C4⋊Q8—141no—
347C4⋊Q8—141no—
348C4⋊Q8—141no—
349C4⋊Q8—141no—
350#125—142no—
351#125—142no—
352#125—142no—
353#125—142no—
354C22⋊Q8—72no—
355C22⋊Q8—72no—
356C22⋊Q8—72no—
357C22⋊Q8—72no—
358C22⋊Q8—141no—
359C22⋊Q8—141no—
360C22⋊Q8—142no—
361C22⋊Q8—142no—
362C22⋊Q8—142no—
363C22⋊Q8—142no—
364C22⋊Q8—141no—
365C22⋊Q8—141no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—72no—
369C4×Q8—72no—
370C4×Q8—142no—
371C4×Q8—142no—
372C4×Q8—142no—
373C4×Q8—142no—
374C4×Q8—141no—
375C4×Q8—141no—
376C4×Q8C4×D712yes—
377C4×Q8C22×D712yes—
378#117—142no—
379#117—142no—
380#117—142no—
381#117—142no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—71no—
384C2×C4⋊C4—141no—
385C2×C4⋊C4—141no—
386C2×C4⋊C4—141no—
387C2×C4⋊C4—141no—
388C2×C4⋊C4—141no—
389C2×C4⋊C4—141no—
390C2×C4⋊C4C4×D711yescharacteristic
391C2×C22⋊C4—141no—
392C2×C22⋊C4—141no—
393C2×C42—21no—
394C2×C42—71no—
395C2×C42—71no—
396C2×C42—71no—
397C2×C42—71no—
398C2×C42—71no—
399C2×C42—71no—
400C2×C42—71no—
401C2×C42—71no—
402C2×C42—71no—
403C2×C42—71no—
404C2×C42—71no—
405C2×C42—141no—
406C2×C42C22×D711yescharacteristic
407C4⋊C8—21no—
408C4⋊C8—72no—
409C4⋊C8—142no—
410C4⋊C8—142no—
411C4⋊C8—141no—
412C4⋊C8C22×D712yes—
413C22⋊C8—21no—
414C22⋊C8—21no—
415C22⋊C8—142no—
416C22⋊C8—142no—
417C22⋊C8—141no—
418C22⋊C8—141no—
419#97—142no—
420C4×C8—21no—
421C4×C8—72no—
422C4×C8—142no—
423C4×C8—141no—
424C4×C8C22×D712yes—
425C22×C7—21no—
426C22×C7—21no—
427C22×C7#40812yes—
428C22×C7C22×C22⋊C411yescharacteristic
429C2×D7—21no—
430C2×D7—21no—
431C2×D7—21no—
432C2×D7—21no—
433C2×D7—21no—
434C2×D7—44no—
435C2×D7#41912yes—
436C4×C7—21no—
437C4×C7—21no—
438C4×C7—21no—
439C4×C7—42no—
440C4×C7—42no—
441C4×C7#40812yes—
442C4×C7C2×#13111yescharacteristic
443C4×C7C2×#13111yescharacteristic
444C4×C7C2×#13712yes—
445Dic7—21no—
446Dic7—21no—
447Dic7—21no—
448Dic7—21no—
449Dic7—42no—
450Dic7—42no—
451Dic7#41912yes—
452Dic7#42112yes—
453C24—71no—
454C2×Q8—21no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—72no—
458C2×Q8—142no—
459C2×Q8—142no—
460C2×Q8—141no—
461C2×Q8—142no—
462C2×Q8—142no—
463C2×Q8—284no—
464C2×Q8C2×C4×D712yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—71no—
468C22×C4—71no—
469C22×C4—71no—
470C22×C4—71no—
471C22×C4—71no—
472C22×C4—71no—
473C22×C4—71no—
474C22×C4—71no—
475C22×C4—71no—
476C22×C4—71no—
477C22×C4—71no—
478C22×C4—71no—
479C22×C4—71no—
480C22×C4—71no—
481C22×C4—71no—
482C22×C4—144no—
483C22×C4—142no—
484C22×C4—142no—
485C22×C4—141no—
486C22×C4—141no—
487C22×C4—282no—
488C22×C4C2×C4×D711yescharacteristic
489C22×C4D4×D711yescharacteristic
490C22×C4D4×D711yescharacteristic
491M16—284no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—142no—
495C2×C8—142no—
496C2×C8—284no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—72no—
501C4⋊C4—142no—
502C4⋊C4—142no—
503C4⋊C4—142no—
504C4⋊C4—142no—
505C4⋊C4—142no—
506C4⋊C4—142no—
507C4⋊C4—142no—
508C4⋊C4—142no—
509C4⋊C4—142no—
510C4⋊C4—142no—
511C4⋊C4—142no—
512C4⋊C4—141no—
513C4⋊C4—142no—
514C4⋊C4C2×C4×D712yes—
515C22⋊C4—142no—
516C22⋊C4—142no—
517C22⋊C4—142no—
518C22⋊C4—142no—
519C42—22no—
520C42—22no—
521C42—71no—
522C42—71no—
523C42—71no—
524C42—71no—
525C42—71no—
526C42—71no—
527C42—71no—
528C42—71no—
529C42—71no—
530C42—71no—
531C42—71no—
532C42—71no—
533C42—142no—
534C42—142no—
535C42C2×C4×D711yescharacteristic
536C42D4×D711yescharacteristic
537C42D4×D711yescharacteristic
538C42C23×D711yescharacteristic
539C2×C7—22no—
540C2×C7C2×#40811yescharacteristic
541C2×C7C2×#41911yescharacteristic
542C2×C7C2×#42111yescharacteristic
543D7—22no—
544D7—41no—
545C23—71no—
546C23—71no—
547C23—142no—
548C23—142no—
549C23—281no—
550C23D7×C22⋊C411yescharacteristic
551Q8—22no—
552Q8—22no—
553Q8—42no—
554Q8—142no—
555Q8—142no—
556Q8—284no—
557Q8—284no—
558Q8—282no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—71no—
566C2×C4—71no—
567C2×C4—74no—
568C2×C4—71no—
569C2×C4—71no—
570C2×C4—71no—
571C2×C4—71no—
572C2×C4—142no—
573C2×C4—142no—
574C2×C4—142no—
575C2×C4—142no—
576C2×C4—142no—
577C2×C4—142no—
578C2×C4—142no—
579C2×C4—141no—
580C2×C4—141no—
581C2×C4—141no—
582C2×C4—141no—
583C2×C4—142no—
584C2×C4—142no—
585C2×C4—142no—
586C2×C4—142no—
587C2×C4—142no—
588C2×C4—281no—
589C2×C4—281no—
590C2×C4—282no—
591C2×C4—282no—
592C2×C4D7×C22⋊C411yescharacteristic
593C2×C4D7×C22⋊C411yescharacteristic
594C2×C4D7×C22⋊C411yescharacteristic
595C2×C4#646412yes—
596C2×C4#647012yes—
597C2×C4C22×C4×D711yescharacteristic
598C2×C4C2×D4×D711yescharacteristic
599C2×C4C2×D4×D711yescharacteristic
600C8—42no—
601C8—42no—
602C8—282no—
603C8—282no—
604C7#1746411yescharacteristic, complemented, Sylow
605C22—21no—
606C22—21no—
607C22—72no—
608C22—141no—
609C22—141no—
610C22—141no—
611C22—141no—
612C22—141no—
613C22—284no—
614C22#8949912yes—
615C22C2×D7×C22⋊C411yescharacteristic
616C4—21no—
617C4—21no—
618C4—21no—
619C4—42no—
620C4—42no—
621C4—72no—
622C4—72no—
623C4—141no—
624C4—141no—
625C4—141no—
626C4—141no—
627C4—282no—
628C4—282no—
629C4#8949912yes—
630C4C2×#646411yescharacteristic
631C4C2×#647011yescharacteristic
632C4D7×#13712yes—
633C2—22no—
634C2—142no—
635C2—281no—
636C2C2×#8949911yescharacteristic
637C2D7×#41911yescharacteristic
638C2D7×#42111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1479C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1479D7
hypocenterC7#17464
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
4. C7×#1479C2semidirect
5. #11729375C2semidirect
6. #11728521C2semidirect
7. #11728520C2semidirect
8. #11725989C2semidirect
9. #11725986C2semidirect
45. C7×Q8⋊C8C22semidirect
46. Q8⋊(C7⋊C8)C22semidirect
47. #89279C22semidirect
48. #89251C22semidirect
149. #1479D7semidirect
274. Q8⋊C8C2×D7semidirect
604. C7#17464semidirect

Maximal subgroups

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

subgroupindex
C2×#902692
#117367272
D7×#4312
C7×#14792
#117293752
#117285212
#117285202
#117259892
#117259862
#174647