Finite Group Databasebeta

← #13320530

Subgroups — #13320530

Order 1792.

958 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90269C211yesmaximal, characteristic
2#11736727C211yesmaximal, characteristic
3#11736483C211yesmaximal, characteristic
4C7×#1487C211yesmaximal, characteristic, complemented
5#11729376C211yesmaximal, characteristic, complemented
6#11728524C211yesmaximal, characteristic, complemented
7#11728254C211yesmaximal, characteristic, complemented
8#11725997C214yesmaximal, complemented
9#11725989C214yesmaximal, 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×#423C2211yescharacteristic
31#89898—21no—
32M16×Dic7—21no—
33#89697C2211yescharacteristic
34#89696C2211yescharacteristic
35#89609—21no—
36#89607—21no—
37#89607—21no—
38#89607C2212yes—
39#89605—21no—
40#89605C2212yes—
41#89498—21no—
42#89491—21no—
43C4×#1131C2212yes—
44#89457C2212yes—
45C7×Q8⋊C8C2214yescomplemented
46Q8⋊(C7⋊C8)C2214yescomplemented
47#89279C2214yescomplemented
48#89251C2214yescomplemented
49#17491—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×#1131—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⋊C8—21no—
125C7×C4⋊C8—21no—
126C7×C4⋊C8C2312yes—
127C7×C22⋊C8—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—
134#6389—22no—
135Dic7⋊C8—22no—
136C8×Dic7—22no—
137C4⋊(C7⋊C8)—21no—
138C4⋊(C7⋊C8)—21no—
139C4⋊(C7⋊C8)C2312yes—
140C4×C7⋊C8C2312yes—
141C2×#533—71no—
142#2920—71no—
143#2920—71no—
144#1497—74no—
145#1487—71no—
146#1487—71no—
147#1487—71no—
148#1487D711yescharacteristic, complemented
149#1485—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—22no—
165C2×C4×D7—22no—
166C2×C4×D7—42no—
167C2×C4×D7—42no—
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×D7C22×C411yescharacteristic
176C2×C4×D7C2×D411yescharacteristic
177C2×C4×D7C2×D411yescharacteristic
178C2×C4×D7C2×D411yescharacteristic
179C2×C4×D7C2×D411yescharacteristic
180C2×Dic14—21no—
181C2×Dic14—22no—
182C2×Dic14—22no—
183C2×Dic14—42no—
184C2×Dic14—44no—
185C2×Dic14C22×C412yes—
186C7×M16—42no—
187C2×C7×C8—22no—
188C2×C7×C8—22no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4C22×C412yes—
193C42×C7—22no—
194C42×C7—22no—
195C42×C7C22×C411yescharacteristic
196C42×C7C2×D411yescharacteristic
197C42×C7C2×D411yescharacteristic
198C42×C7C2411yescharacteristic
199(C2×D7)⋊C4—22no—
200(C2×D7)⋊C4—22no—
201(C2×D7)⋊C4—22no—
202(C2×D7)⋊C4—22no—
203C4⋊Dic7—21no—
204C4⋊Dic7—22no—
205C4⋊Dic7—22no—
206C4⋊Dic7C22×C412yes—
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—
216Dic7⋊C4—22no—
217C4×Dic7—22no—
218C4×Dic7—22no—
219C4×Dic7C22⋊C411yescharacteristic
220C4×Dic7C22⋊C411yescharacteristic
221C4×Dic7C22⋊C411yescharacteristic
222C4×Dic7C22⋊C411yescharacteristic
223C4×Dic7C22⋊C411yescharacteristic
224C4×Dic7C22⋊C411yescharacteristic
225C4×Dic7C22×C411yescharacteristic
226C4×Dic7C22×C411yescharacteristic
227C4×Dic7C2×D411yescharacteristic
228C4×Dic7C2×D411yescharacteristic
229C4×Dic7C2×D411yescharacteristic
230C4×Dic7C2×D411yescharacteristic
231#1136—42no—
232C2×C7⋊C8—22no—
233C2×C7⋊C8—22no—
234#1131—44no—
235#533—72no—
236#533—72no—
237#533—72no—
238#533—72no—
239#533—141no—
240#533—141no—
241#533—141no—
242#533—141no—
243C2×C4⋊Q8—71no—
244C2×C4⋊Q8—71no—
245C2×#125—141no—
246C2×C22⋊Q8—71no—
247C2×C22⋊Q8—71no—
248C2×C22⋊Q8—141no—
249C2×C4×Q8—71no—
250C2×C4×Q8—141no—
251C2×C4×Q8C2×D711yescharacteristic
252C2×#117—141no—
253C22×C42—71no—
254#432—72no—
255#432—141no—
256#432—141no—
257#432—141no—
258#431—71no—
259#431—72no—
260#431—72no—
261#431—141no—
262#431C2×D711yescharacteristic
263#423—71no—
264#423—141no—
265#423C2×D711yescharacteristic
266#407—141no—
267#407—141no—
268C4×M16—72no—
269C4×M16—141no—
270Q8⋊C8—74no—
271Q8⋊C8—74no—
272Q8⋊C8—74no—
273Q8⋊C8C2×D714yescomplemented
274C23×C7C2×C22⋊C411yescharacteristic
275C22×D7—22no—
276C22×D7—41no—
277C22×D7—41no—
278C22×D7C2×C22⋊C411yescharacteristic
279C22×D7C2×C22⋊C411yescharacteristic
280C7×Q8—24no—
281C7×Q8—42no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—41no—
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×C7#13712yes—
292C2×C4×C7#13712yes—
293C2×C4×C7C23×C411yescharacteristic
294C2×C4×C7C22×D411yescharacteristic
295C2×C4×C7C22×D411yescharacteristic
296C2×Dic7—22no—
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—22no—
300C2×Dic7—41no—
301C2×Dic7—42no—
302C2×Dic7—42no—
303C2×Dic7C2×C22⋊C411yescharacteristic
304C2×Dic7C2×C22⋊C411yescharacteristic
305C2×Dic7C2×C22⋊C411yescharacteristic
306C2×Dic7C2×C22⋊C411yescharacteristic
307C2×Dic7C2×C22⋊C411yescharacteristic
308C2×Dic7C2×C22⋊C411yescharacteristic
309C4×D7—22no—
310C4×D7—22no—
311C4×D7—21no—
312C4×D7—21no—
313C4×D7—21no—
314C4×D7—21no—
315C4×D7—22no—
316C4×D7—22no—
317C4×D7—22no—
318C4×D7—22no—
319C4×D7—41no—
320C4×D7—41no—
321C4×D7—41no—
322C4×D7—41no—
323Dic14—24no—
324Dic14—44no—
325Dic14—42no—
326Dic14—44no—
327C7×C8—42no—
328C7×C8—42no—
329C7⋊C8—42no—
330C7⋊C8—42no—
331C22×Q8—71no—
332C22×Q8—141no—
333C22×Q8C4×D711yescharacteristic
334C23×C4—71no—
335C23×C4—71no—
336C23×C4—71no—
337C2×M16—21no—
338C2×M16—142no—
339C2×M16—141no—
340C4⋊Q8—72no—
341C4⋊Q8—72no—
342C4⋊Q8—72no—
343C4⋊Q8—72no—
344C4⋊Q8—141no—
345C4⋊Q8—141no—
346C4⋊Q8—141no—
347C4⋊Q8—141no—
348#125—142no—
349#125—142no—
350#125—142no—
351#125—142no—
352C22⋊Q8—72no—
353C22⋊Q8—72no—
354C22⋊Q8—72no—
355C22⋊Q8—72no—
356C22⋊Q8—141no—
357C22⋊Q8—141no—
358C22⋊Q8—142no—
359C22⋊Q8—142no—
360C22⋊Q8—142no—
361C22⋊Q8—142no—
362C22⋊Q8—141no—
363C22⋊Q8—141no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—72no—
367C4×Q8—72no—
368C4×Q8—142no—
369C4×Q8—142no—
370C4×Q8—142no—
371C4×Q8—142no—
372C4×Q8—141no—
373C4×Q8—141no—
374C4×Q8C4×D712yes—
375C4×Q8C22×D712yes—
376#117—142no—
377#117—142no—
378#117—142no—
379#117—142no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—71no—
382C2×C4⋊C4—141no—
383C2×C4⋊C4—141no—
384C2×C4⋊C4—141no—
385C2×C4⋊C4—141no—
386C2×C4⋊C4—141no—
387C2×C4⋊C4—141no—
388C2×C4⋊C4C4×D711yescharacteristic
389C2×C22⋊C4—141no—
390C2×C22⋊C4—141no—
391C2×C42—21no—
392C2×C42—71no—
393C2×C42—71no—
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—141no—
404C2×C42C22×D711yescharacteristic
405C4⋊C8—21no—
406C4⋊C8—21no—
407C4⋊C8—72no—
408C4⋊C8—142no—
409C4⋊C8—141no—
410C4⋊C8—141no—
411C4⋊C8C22×D712yes—
412C22⋊C8—21no—
413C22⋊C8—142no—
414C22⋊C8—142no—
415C22⋊C8—142no—
416C22⋊C8—141no—
417#97—142no—
418#97—142no—
419C4×C8—72no—
420C4×C8—142no—
421C4×C8C22×D712yes—
422C22×C7—21no—
423C22×C7—21no—
424C22×C7#41912yes—
425C22×C7C22×C22⋊C411yescharacteristic
426C2×D7—21no—
427C2×D7—21no—
428C2×D7—21no—
429C2×D7—21no—
430C2×D7—21no—
431C2×D7—21no—
432C2×D7—44no—
433C4×C7—21no—
434C4×C7—21no—
435C4×C7—21no—
436C4×C7—42no—
437C4×C7—42no—
438C4×C7#41912yes—
439C4×C7#56812yes—
440C4×C7C2×#13711yescharacteristic
441C4×C7C2×#13711yescharacteristic
442Dic7—21no—
443Dic7—21no—
444Dic7—21no—
445Dic7—21no—
446Dic7—21no—
447Dic7—42no—
448Dic7—42no—
449Dic7#42112yes—
450C24—71no—
451C2×Q8—21no—
452C2×Q8—42no—
453C2×Q8—72no—
454C2×Q8—142no—
455C2×Q8—142no—
456C2×Q8—141no—
457C2×Q8—282no—
458C2×Q8—284no—
459C2×Q8C2×C4×D712yes—
460C22×C4—21no—
461C22×C4—21no—
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—71no—
473C22×C4—71no—
474C22×C4—71no—
475C22×C4—71no—
476C22×C4—71no—
477C22×C4—142no—
478C22×C4—142no—
479C22×C4—141no—
480C22×C4—141no—
481C22×C4—282no—
482C22×C4—282no—
483C22×C4C2×C4×D711yescharacteristic
484C22×C4D4×D711yescharacteristic
485C22×C4D4×D711yescharacteristic
486M16—42no—
487M16—284no—
488M16—282no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—142no—
492C2×C8—142no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—72no—
497C4⋊C4—142no—
498C4⋊C4—142no—
499C4⋊C4—142no—
500C4⋊C4—142no—
501C4⋊C4—142no—
502C4⋊C4—142no—
503C4⋊C4—142no—
504C4⋊C4—142no—
505C4⋊C4—142no—
506C4⋊C4—142no—
507C4⋊C4—142no—
508C4⋊C4—141no—
509C4⋊C4—142no—
510C4⋊C4C2×C4×D712yes—
511C22⋊C4—142no—
512C22⋊C4—142no—
513C22⋊C4—142no—
514C22⋊C4—142no—
515C42—22no—
516C42—22no—
517C42—71no—
518C42—71no—
519C42—71no—
520C42—71no—
521C42—71no—
522C42—71no—
523C42—71no—
524C42—71no—
525C42—71no—
526C42—71no—
527C42—71no—
528C42—71no—
529C42—142no—
530C42—142no—
531C42C2×C4×D711yescharacteristic
532C42D4×D711yescharacteristic
533C42D4×D711yescharacteristic
534C42C23×D711yescharacteristic
535C2×C7—22no—
536C2×C7#287911yescharacteristic
537C2×C7C2×#41911yescharacteristic
538C2×C7C2×#42111yescharacteristic
539D7—41no—
540D7—41no—
541C23—71no—
542C23—71no—
543C23—142no—
544C23—281no—
545C23—281no—
546C23D7×C22⋊C411yescharacteristic
547Q8—24no—
548Q8—42no—
549Q8—144no—
550Q8—282no—
551Q8—284no—
552Q8—284no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—71no—
560C2×C4—71no—
561C2×C4—71no—
562C2×C4—71no—
563C2×C4—71no—
564C2×C4—71no—
565C2×C4—142no—
566C2×C4—142no—
567C2×C4—142no—
568C2×C4—142no—
569C2×C4—142no—
570C2×C4—142no—
571C2×C4—141no—
572C2×C4—141no—
573C2×C4—141no—
574C2×C4—141no—
575C2×C4—142no—
576C2×C4—142no—
577C2×C4—142no—
578C2×C4—142no—
579C2×C4—281no—
580C2×C4—281no—
581C2×C4—281no—
582C2×C4—281no—
583C2×C4—281no—
584C2×C4—282no—
585C2×C4—282no—
586C2×C4D7×C22⋊C411yescharacteristic
587C2×C4D7×C22⋊C411yescharacteristic
588C2×C4D7×C22⋊C411yescharacteristic
589C2×C4#647812yes—
590C2×C4#648112yes—
591C2×C4C22×C4×D711yescharacteristic
592C2×C4C2×D4×D711yescharacteristic
593C2×C4C2×D4×D711yescharacteristic
594C8—42no—
595C8—42no—
596C8—282no—
597C8—282no—
598C7#1749111yescharacteristic, complemented, Sylow
599C22—21no—
600C22—21no—
601C22—141no—
602C22—141no—
603C22—141no—
604C22—141no—
605C22—141no—
606C22—141no—
607C22—284no—
608C22#8957412yes—
609C22C2×D7×C22⋊C411yescharacteristic
610C4—21no—
611C4—21no—
612C4—21no—
613C4—42no—
614C4—42no—
615C4—72no—
616C4—141no—
617C4—141no—
618C4—141no—
619C4—141no—
620C4—141no—
621C4—282no—
622C4—282no—
623C4#8957412yes—
624C4#9043612yes—
625C4C2×#647811yescharacteristic
626C4C2×#648111yescharacteristic
627C2—22no—
628C2—281no—
629C2—281no—
630C2#1173607511yescharacteristic
631C2C2×#8957411yescharacteristic
632C2D7×#42111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1487C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1487D7
hypocenterC7#17491
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
4. C7×#1487C2semidirect
5. #11729376C2semidirect
6. #11728524C2semidirect
7. #11728254C2semidirect
8. #11725997C2semidirect
9. #11725989C2semidirect
45. C7×Q8⋊C8C22semidirect
46. Q8⋊(C7⋊C8)C22semidirect
47. #89279C22semidirect
48. #89251C22semidirect
148. #1487D7semidirect
273. Q8⋊C8C2×D7semidirect
598. C7#17491semidirect

Maximal subgroups

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

subgroupindex
C2×#902692
#117367272
#117364832
C7×#14872
#117293762
#117285242
#117282542
#117259972
#117259892
#174917