Finite Group Databasebeta

← #13320679

Subgroups — #13320679

Order 1792.

970 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742893C211yesmaximal, characteristic
2C7×#2925C211yesmaximal, characteristic, complemented
3#41×C7⋊C8C211yesmaximal, characteristic, complemented
4#11728554C212yesmaximal, complemented
5#11728552C212yesmaximal, complemented
6#11728513C212yesmaximal, complemented
7#11728508C212yesmaximal, complemented
8#11728254C211yesmaximal, characteristic, complemented
9#11728252C211yesmaximal, characteristic, complemented
10C2×#89279C211yesmaximal, characteristic, complemented
11#11728180C211yesmaximal, characteristic, complemented
12C4×C7×#41C2211yescharacteristic
13#90523—21no—
14#90276C411yescharacteristic
15#90259—21no—
16#90258C411yescharacteristic
17#90256C412yes—
18#90250—21no—
19#90244C411yescharacteristic
20#90243C412yes—
21#90233C411yescharacteristic
22#90231—22no—
23#90219—22no—
24#90213—21no—
25#90170C2212yes—
26#90168C2211yescharacteristic
27C4×#1157C2212yes—
28C2×C4×Dic14C2211yescharacteristic
29C7×#433—22no—
30C7×#432C2212yes—
31C7×#431C2212yes—
32C7×#423C2211yescharacteristic
33C2×C7×C4⋊C8C2211yescharacteristic
34C7×#408—21no—
35C7×#405—21no—
36Q8×C7⋊C8—21no—
37D4×C7⋊C8—22no—
38D4×C7⋊C8—21no—
39#89697C2211yescharacteristic
40#89697C2212yes—
41C2×C4×C7⋊C8C2211yescharacteristic
42C2×C4×C7⋊C8C2212yes—
43#89279C2211yescharacteristic
44#89279C2211yescharacteristic
45#89279C2211yescharacteristic
46#89279C2211yescharacteristic
47#89279C2212yes—
48#89279C2212yes—
49#89278C2212yes—
50#89278C2212yes—
51#89278C2212yes—
52#89278C2212yes—
53#17510—71nomaximal, Sylow
54C2×C7×#41D411yescharacteristic
55C2×#1157—21no—
56C2×#1157C2×C412yes—
57C22×Dic14C2×C411yescharacteristic
58C2×C7×M16—21no—
59C22×C7×C8—21no—
60C4×C7×Q8C2×C411yescharacteristic
61C4×C7×Q8D411yescharacteristic
62C4×C7×D4C2×C411yescharacteristic
63C4×C7×D4C2×C412yes—
64C4×C7×D4D412yes—
65C4×C7×D4D411yescharacteristic
66C7×#117D411yescharacteristic
67C7×#117D412yes—
68C2×C42×C7C2311yescharacteristic
69C2×C42×C7C2312yes—
70#6510—21no—
71#6507—21no—
72#6503—21no—
73#6502—22no—
74#6502—22no—
75#6499—21no—
76#6493—21no—
77#6493C2×C412yes—
78#6493C2×C412yes—
79#6492—22no—
80C4×C22⋊D7—22no—
81C4×C22⋊D7—22no—
82C4×C22⋊D7—22no—
83#6489C2×C412yes—
84C2×C4⋊Dic7C2×C411yescharacteristic
85#6487C2×C412yes—
86C2×Dic7⋊C4—21no—
87C2×C4×Dic7—21no—
88C22×C7⋊C8—21no—
89C22×C7⋊C8—22no—
90#6461—22no—
91#6459—22no—
92#6457—22no—
93#6456—21no—
94#6456—21no—
95#6452—22no—
96#6450—21no—
97#6450—21no—
98#6448—22no—
99#6447—22no—
100#6446—22no—
101(C22⋊D7)⋊C4—22no—
102(C22⋊D7)⋊C4—22no—
103#6442—22no—
104#6439—22no—
105#6438—21no—
106#6438C2×C412yes—
107#6437C2×C411yescharacteristic
108C4×D28—21no—
109C4×D28C2×C412yes—
110C4×D28C2312yes—
111#6435—22no—
112#6435—22no—
113C42×D7—22no—
114#6433C2×C412yes—
115#6432C2×C411yescharacteristic
116C4×Dic14—21no—
117C4×Dic14—21no—
118C4×Dic14C2×C411yescharacteristic
119C4×Dic14C2×C411yescharacteristic
120C4×Dic14C2×C412yes—
121C4×Dic14C2311yescharacteristic
122C4×Dic14C2311yescharacteristic
123C7×C4⋊C8—21no—
124C7×C4⋊C8—21no—
125C7×C4⋊C8C2311yescharacteristic
126C7×C4⋊C8C2311yescharacteristic
127C7×C4⋊C8C2312yes—
128C7×C22⋊C8—22no—
129C7×C22⋊C8—22no—
130C7×#97—22no—
131C4×C7×C8—22no—
132C22⋊(C7⋊C8)—21no—
133C22⋊(C7⋊C8)—22no—
134C4⋊(C7⋊C8)—21no—
135C4⋊(C7⋊C8)—22no—
136C4×C7⋊C8—21no—
137C4×C7⋊C8—22no—
138C4×C7⋊C8C2311yescharacteristic
139C4×C7⋊C8C2311yescharacteristic
140C4×C7⋊C8C2312yes—
141#3480—71no—
142C8×#41—71no—
143#2925D711yescharacteristic, complemented
144#1494—72no—
145#1492—71no—
146#1488—72no—
147#1487—71no—
148#1482—72no—
149#1480—71no—
150#1476—72no—
151C2×Q8⋊C8—71no—
152C7×#41—22no—
153C7×#41—22no—
154C2×C7×Q8C22⋊C411yescharacteristic
155C2×C7×D4C22⋊C411yescharacteristic
156C2×C7×D4C22⋊C412yes—
157C22×C4×C7C22×C412yes—
158C22×C4×C7C22×C411yescharacteristic
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D411yescharacteristic
163C2×C22⋊D7—22no—
164C2×C22⋊D7—22no—
165C2×C22⋊D7—22no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168#1157—24no—
169#1157—42no—
170#1157—42no—
171#1157—42no—
172C2×D28—21no—
173C2×D28—21no—
174C2×D28C22×C412yes—
175C2×C4×D7—22no—
176C2×C4×D7—22no—
177C2×C4×D7—22no—
178C2×Dic14—22no—
179C2×Dic14—21no—
180C2×Dic14—21no—
181C2×Dic14—41no—
182C2×Dic14C22×C411yescharacteristic
183C2×Dic14C22×C411yescharacteristic
184C7×M16—42no—
185C2×C7×C8—21no—
186C2×C7×C8—22no—
187C2×C7×C8—21no—
188C2×C7×C8—42no—
189C7×C4⋊C4C22⋊C411yescharacteristic
190C7×C4⋊C4C22⋊C411yescharacteristic
191C7×C4⋊C4C22⋊C412yes—
192C7×C4⋊C4C22⋊C412yes—
193C7×C22⋊C4C22⋊C412yes—
194C7×C22⋊C4C22⋊C412yes—
195C7×C22⋊C4C22⋊C412yes—
196C42×C7C22×C411yescharacteristic
197C42×C7C22×C412yes—
198C42×C7C2×D411yescharacteristic
199C42×C7C2×D412yes—
200C42×C7C2×D412yes—
201C42×C7C2×D411yescharacteristic
202C42×C7C2411yescharacteristic
203C22⋊Dic7—22no—
204C22⋊Dic7—22no—
205(C2×D7)⋊C4—22no—
206(C2×D7)⋊C4—22no—
207(C2×D7)⋊C4—22no—
208(C2×D7)⋊C4—22no—
209(C2×D7)⋊C4—22no—
210(C2×D7)⋊C4—22no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7C22×C411yescharacteristic
214C4⋊Dic7C22×C411yescharacteristic
215C4⋊Dic7C22×C412yes—
216Dic7⋊C4—21no—
217Dic7⋊C4—21no—
218Dic7⋊C4—22no—
219Dic7⋊C4—22no—
220Dic7⋊C4—22no—
221Dic7⋊C4—22no—
222Dic7⋊C4—21no—
223Dic7⋊C4—21no—
224C4×Dic7—21no—
225C4×Dic7—21no—
226C4×Dic7—22no—
227C4×Dic7—22no—
228C4×Dic7—21no—
229C4×Dic7—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—22no—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—44no—
234C2×C7⋊C8—42no—
235#559—71no—
236#559—141no—
237#557—71no—
238#556—71no—
239#556—141no—
240#553—72no—
241#553—142no—
242#550—71no—
243#548—72no—
244#532—71no—
245#532—141no—
246#529—72no—
247#529—142no—
248C4×#41—72no—
249C4×#41—141no—
250C4×#41C2×D711yescharacteristic
251C2×C4×Q8—71no—
252C8×Q8—141no—
253C8×D4—142no—
254C8×D4—141no—
255#433—22no—
256#432C2×D712yes—
257#431—71no—
258#431—72no—
259#431C2×D712yes—
260#423C2×D711yescharacteristic
261C2×C4⋊C8C2×D711yescharacteristic
262#408—21no—
263#405—21no—
264C2×C4×C8—71no—
265C2×C4×C8—72no—
266Q8⋊C8—71no—
267Q8⋊C8—71no—
268Q8⋊C8—71no—
269Q8⋊C8—71no—
270Q8⋊C8—72no—
271Q8⋊C8—72no—
272D4⋊C8—72no—
273D4⋊C8—72no—
274D4⋊C8—72no—
275D4⋊C8—72no—
276C23×C7C2×C22⋊C412yes—
277C23×C7C2×C22⋊C411yescharacteristic
278C22×D7—22no—
279C7×Q8—22no—
280C7×D4—24no—
281C7×D4—22no—
282C2×C4×C7—21no—
283C2×C4×C7—22no—
284C2×C4×C7—21no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—42no—
289C2×C4×C7—41no—
290C2×C4×C7C2×C22⋊C412yes—
291C2×C4×C7C2×C22⋊C412yes—
292C2×C4×C7C2×C22⋊C411yescharacteristic
293C2×C4×C7C2×C22⋊C412yes—
294C2×C4×C7C2×C22⋊C411yescharacteristic
295C2×C4×C7C2×C22⋊C411yescharacteristic
296C2×C4×C7#13512yes—
297C2×C4×C7#13512yes—
298C2×C4×C7C23×C411yescharacteristic
299C2×C4×C7C22×D411yescharacteristic
300C2×C4×C7C22×D411yescharacteristic
301C22⋊D7—44no—
302C22⋊D7—44no—
303C22⋊D7—82no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—22no—
308C2×Dic7—21no—
309C2×Dic7—42no—
310C2×Dic7—81no—
311D28—22no—
312D28—22no—
313D28—42no—
314D28—81no—
315C4×D7—44no—
316C4×D7—42no—
317C4×D7—42no—
318C4×D7—82no—
319Dic14—21no—
320Dic14—21no—
321Dic14—21no—
322Dic14—42no—
323Dic14—42no—
324Dic14#13112yes—
325C7×C8—41no—
326C7×C8—41no—
327C7×C8—42no—
328C7⋊C8—22no—
329C7⋊C8—42no—
330C7⋊C8—41no—
331C2×#41—72no—
332C2×#41—141no—
333C2×#41C22⋊D711yescharacteristic
334C22×Q8—71no—
335C2×M16—21no—
336C22×C8—21no—
337C22×C8—141no—
338C22×C8—142no—
339C4⋊Q8—71no—
340C4⋊Q8—141no—
341#127—71no—
342#127—141no—
343#126—142no—
344#126—142no—
345#126—142no—
346#125—72no—
347#125—142no—
348#124—72no—
349#124—141no—
350#124—141no—
351#124—142no—
352#124—141no—
353#123—142no—
354#123—142no—
355#123—142no—
356C22⋊Q8—72no—
357C22⋊Q8—142no—
358#121—72no—
359#121—72no—
360#121—142no—
361#121—142no—
362#121—141no—
363#121—142no—
364C4×Q8—71no—
365C4×Q8—71no—
366C4×Q8—72no—
367C4×Q8—71no—
368C4×Q8—71no—
369C4×Q8—141no—
370C4×Q8—141no—
371C4×Q8—141no—
372C4×Q8—141no—
373C4×Q8C4×D711yescharacteristic
374C4×Q8D2811yescharacteristic
375C4×D4—72no—
376C4×D4—72no—
377C4×D4—142no—
378C4×D4—142no—
379C4×D4—142no—
380C4×D4—141no—
381C4×D4—141no—
382C4×D4—142no—
383C4×D4—142no—
384C4×D4—141no—
385C4×D4C4×D712yes—
386C4×D4C4×D711yescharacteristic
387C4×D4D2812yes—
388C4×D4D2811yescharacteristic
389#117—72no—
390#117—142no—
391#117—142no—
392#117C22⋊D711yescharacteristic
393#117C22⋊D712yes—
394C2×C4⋊C4—71no—
395C2×C4⋊C4—141no—
396C2×C42—141no—
397C2×C42—142no—
398C2×C42C22×D711yescharacteristic
399C2×C42C22×D712yes—
400C4⋊C8—21no—
401C4⋊C8—21no—
402C4⋊C8—142no—
403C4⋊C8—141no—
404C4⋊C8C22×D711yescharacteristic
405C4⋊C8C22×D711yescharacteristic
406C4⋊C8C22×D712yes—
407C22⋊C8—22no—
408C22⋊C8—22no—
409C22⋊C8—141no—
410C22⋊C8—142no—
411#97—22no—
412C4×C8—22no—
413C4×C8—71no—
414C4×C8—72no—
415C4×C8—71no—
416C4×C8—142no—
417C4×C8—141no—
418C22×C7—21no—
419C22×C7—21no—
420C22×C7—22no—
421C22×C7—22no—
422C22×C7—22no—
423C22×C7#41612yes—
424C22×C7C22×C22⋊C411yescharacteristic
425C2×D7—42no—
426C2×D7—42no—
427C2×D7—42no—
428C4×C7—21no—
429C4×C7—22no—
430C4×C7—22no—
431C4×C7—21no—
432C4×C7—22no—
433C4×C7—21no—
434C4×C7#41612yes—
435C4×C7C2×#13111yescharacteristic
436C4×C7#56811yescharacteristic
437C4×C7C2×#13511yescharacteristic
438C4×C7C2×#13511yescharacteristic
439Dic7—22no—
440Dic7—41no—
441Dic7—42no—
442Dic7—41no—
443Dic7—41no—
444#41—22no—
445#41—22no—
446#41—144no—
447#41—282no—
448#41—282no—
449#41—282no—
450C2×Q8—71no—
451C2×Q8—71no—
452C2×Q8—142no—
453C2×Q8—141no—
454C2×Q8—141no—
455C2×Q8—281no—
456C2×Q8(C2×D7)⋊C411yescharacteristic
457C2×D4—72no—
458C2×D4—142no—
459C2×D4—142no—
460C2×D4—142no—
461C2×D4—141no—
462C2×D4—141no—
463C2×D4(C2×D7)⋊C412yes—
464C2×D4(C2×D7)⋊C411yescharacteristic
465C22×C4—141no—
466C22×C4—141no—
467C22×C4—142no—
468C22×C4—142no—
469C22×C4—142no—
470C22×C4C2×C4×D711yescharacteristic
471C22×C4C2×C4×D712yes—
472C22×C4C2×D2811yescharacteristic
473C22×C4C2×D2812yes—
474C22×C4C2×C22⋊D712yes—
475C22×C4C2×C22⋊D711yescharacteristic
476M16—42no—
477C2×C8—21no—
478C2×C8—22no—
479C2×C8—21no—
480C2×C8—42no—
481C2×C8—141no—
482C2×C8—142no—
483C2×C8—141no—
484C2×C8—284no—
485C2×C8—282no—
486C4⋊C4—71no—
487C4⋊C4—71no—
488C4⋊C4—72no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—142no—
492C4⋊C4—142no—
493C4⋊C4—142no—
494C4⋊C4—142no—
495C4⋊C4—141no—
496C4⋊C4—141no—
497C4⋊C4—141no—
498C4⋊C4—141no—
499C4⋊C4(C2×D7)⋊C411yescharacteristic
500C4⋊C4(C2×D7)⋊C411yescharacteristic
501C4⋊C4(C2×D7)⋊C412yes—
502C4⋊C4(C2×D7)⋊C412yes—
503C22⋊C4—142no—
504C22⋊C4—142no—
505C22⋊C4—142no—
506C22⋊C4—142no—
507C22⋊C4—142no—
508C22⋊C4—142no—
509C22⋊C4—142no—
510C22⋊C4—142no—
511C22⋊C4(C2×D7)⋊C412yes—
512C22⋊C4(C2×D7)⋊C412yes—
513C22⋊C4(C2×D7)⋊C412yes—
514C42—142no—
515C42—142no—
516C42—141no—
517C42—141no—
518C42—141no—
519C42—141no—
520C42C2×C4×D712yes—
521C42C2×C4×D711yescharacteristic
522C42C2×D2812yes—
523C42C2×D2811yescharacteristic
524C42C2×C22⋊D712yes—
525C42C2×C22⋊D711yescharacteristic
526C42C23×D711yescharacteristic
527C2×C7—22no—
528C2×C7—42no—
529C2×C7#288011yescharacteristic
530C2×C7C2×#41611yescharacteristic
531C2×C7#290111yescharacteristic
532D7—82no—
533C23—142no—
534C23C2×(C2×D7)⋊C412yes—
535C23C2×(C2×D7)⋊C411yescharacteristic
536Q8—22no—
537Q8—72no—
538Q8—141no—
539Q8—141no—
540Q8—141no—
541Q8—282no—
542Q8—282no—
543D4—24no—
544D4—22no—
545D4—142no—
546D4—142no—
547D4—284no—
548D4—284no—
549D4—282no—
550D4—561no—
551D4—562no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—141no—
561C2×C4—141no—
562C2×C4—141no—
563C2×C4—142no—
564C2×C4—141no—
565C2×C4—284no—
566C2×C4—282no—
567C2×C4—282no—
568C2×C4—282no—
569C2×C4—562no—
570C2×C4—561no—
571C2×C4C2×(C2×D7)⋊C412yes—
572C2×C4C2×(C2×D7)⋊C412yes—
573C2×C4C2×(C2×D7)⋊C412yes—
574C2×C4C2×(C2×D7)⋊C411yescharacteristic
575C2×C4C2×(C2×D7)⋊C411yescharacteristic
576C2×C4C2×(C2×D7)⋊C411yescharacteristic
577C2×C4#651312yes—
578C2×C4#651312yes—
579C2×C4C22×C4×D711yescharacteristic
580C2×C4C22×D2811yescharacteristic
581C2×C4C22×C22⋊D711yescharacteristic
582C8—41no—
583C8—41no—
584C8—42no—
585C8—142no—
586C8—282no—
587C8—281no—
588C7#1751011yescharacteristic, complemented, Sylow
589C22—21no—
590C22—21no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—282no—
595C22—282no—
596C22—282no—
597C22#8977412yes—
598C22C22×(C2×D7)⋊C411yescharacteristic
599C4—22no—
600C4—22no—
601C4—21no—
602C4—21no—
603C4—22no—
604C4—21no—
605C4—142no—
606C4—282no—
607C4—281no—
608C4—281no—
609C4—281no—
610C4#8977412yes—
611C4D7×#13111yescharacteristic
612C4#9044111yescharacteristic
613C4C2×#651311yescharacteristic
614C4C2×#651311yescharacteristic
615C2—22no—
616C2—42no—
617C2—562no—
618C2C2×#8977411yescharacteristic
619C2#1174123811yescharacteristic
620C2#1174126211yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#2925C2
socleC22×C7C22×C22⋊C4
prime radical#11742893C2
cosocleC42×C7C24
hypercenter#2925D7
hypocenterC7#17510
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2925

Complements

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

NH ≅ G/Ntype
2. C7×#2925C2semidirect
3. #41×C7⋊C8C2semidirect
4. #11728554C2semidirect
5. #11728552C2semidirect
6. #11728513C2semidirect
7. #11728508C2semidirect
8. #11728254C2semidirect
9. #11728252C2semidirect
10. C2×#89279C2semidirect
11. #11728180C2semidirect
143. #2925D7semidirect
588. C7#17510semidirect

Maximal subgroups

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

subgroupindex
#117428932
C7×#29252
#41×C7⋊C82
#117285542
#117285522
#117285132
#117285082
#117282542
#117282522
C2×#892792
#117281802
#175107