Finite Group Databasebeta

← #13321515

Subgroups — #13321515

Order 1792.

1038 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742855C211yesmaximal, characteristic
2C7×#2353C211yesmaximal, characteristic, complemented
3#11739746C211yesmaximal, characteristic, complemented
4#11727319C212yesmaximal, complemented
5#11727286C212yesmaximal, complemented
6#11727264C211yesmaximal, characteristic, complemented
7#11727263C211yesmaximal, characteristic, complemented
8#11727197C212yesmaximal, complemented
9#11727185C212yesmaximal, complemented
10C2×#89247C211yesmaximal, characteristic, complemented
11C2×#89246C211yesmaximal, characteristic, complemented
12C2×C4×C7×D4C2211yescharacteristic
13#90495—21no—
14#90482C2212yes—
15C2×#6493C2211yescharacteristic
16C2×#6487C2211yescharacteristic
17#90256C412yescomplemented
18#90247—22no—
19#90243C412yescomplemented
20#90241C412yescomplemented
21#90232C412yescomplemented
22#90231—22no—
23#90206—21no—
24#90185—21no—
25#90177—21no—
26#90170C2212yes—
27C7×#388—21no—
28C7×#387—21no—
29C7×#386—21no—
30C7×#383C2211yescharacteristic, complemented
31C7×#381C2211yescharacteristic, complemented
32C7×#380C2211yescharacteristic
33C4×C7×C22⋊C4—21no—
34C4×C7×C22⋊C4C2211yescharacteristic, complemented
35C7×#376C2211yescharacteristic, complemented
36C2×C7×#95C2211yescharacteristic
37C2×C22⋊(C7⋊C8)C2211yescharacteristic
38C2×C22⋊(C7⋊C8)C2211yescharacteristic
39C2×C22⋊(C7⋊C8)C2212yescomplemented
40D4×C7⋊C8—22no—
41D4×C7⋊C8—22no—
42#89697C2212yescomplemented
43#89247C2212yescomplemented
44#89247C2212yescomplemented
45#89247C2212yescomplemented
46#89247C2212yescomplemented
47#89246C2212yes—
48#89246C2212yes—
49#89246C2212yes—
50#89246C2212yes—
51#17615—71nomaximal, Sylow
52C22×C7×D4D411yescharacteristic
53C23×C4×C7—21no—
54C23×C4×C7C2311yescharacteristic
55C23×C4×C7C2311yescharacteristic
56C22×C22⋊D7—21no—
57C2×#1157C2×C412yescomplemented
58C22×D28C2×C411yescharacteristic
59C22×Dic14C2×C411yescharacteristic
60C4×C7×D4C2×C412yes—
61C4×C7×D4C2×C412yes—
62C4×C7×D4D412yes—
63C4×C7×D4D412yes—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4D411yescharacteristic
66C2×C7×C22⋊C4—21no—
67C2×C7×C22⋊C4—21no—
68C2×C7×C22⋊C4—21no—
69C2×C7×C22⋊C4D411yescharacteristic
70C2×C7×C22⋊C4D411yescharacteristic
71C2×C42×C7—21no—
72C2×C42×C7—21no—
73C2×C42×C7C2311yescharacteristic
74#6516—22no—
75C2×C22⋊Dic7—21no—
76#6502—22no—
77#6502—22no—
78#6500—22no—
79#6498—22no—
80#6493—21no—
81#6493—21no—
82#6493C2×C412yes—
83#6493C2×C412yescomplemented
84#6493C2312yes—
85#6492—22no—
86#6492—22no—
87C4×C22⋊D7—22no—
88C4×C22⋊D7—22no—
89C4×C22⋊D7—22no—
90C2×(C2×D7)⋊C4—21no—
91#6489C2×C412yescomplemented
92C2×C4⋊Dic7C2×C411yescharacteristic
93C2×C4⋊Dic7C2×C411yescharacteristic
94#6487—21no—
95#6487—21no—
96#6487C2×C412yes—
97#6487C2×C412yescomplemented
98#6487C2312yes—
99C2×Dic7⋊C4—21no—
100C22×C7⋊C8—21no—
101C22×C7⋊C8—21no—
102C22×C7⋊C8—22no—
103#6461—22no—
104#6459—22no—
105#6457—22no—
106#6452—22no—
107#6448—22no—
108#6447—22no—
109#6446—22no—
110(C22⋊D7)⋊C4—22no—
111(C22⋊D7)⋊C4—22no—
112D7×C22⋊C4—22no—
113#6442—22no—
114#6440—22no—
115#6439—22no—
116#6438C2×C412yescomplemented
117C4×D28C2×C412yescomplemented
118#6435—22no—
119#6433C2×C412yescomplemented
120C4×Dic14C2×C412yescomplemented
121C7×#95—21no—
122C7×#95—21no—
123C7×#95—21no—
124C7×#95—21no—
125C7×#95—21no—
126C7×#95C2311yescharacteristic
127C7×#95C2311yescharacteristic
128C7×#95C2311yescharacteristic
129C7×#95C2311yescharacteristic
130C22⋊(C7⋊C8)—21no—
131C22⋊(C7⋊C8)—21no—
132C22⋊(C7⋊C8)—22no—
133C22⋊(C7⋊C8)C2312yes—
134C22⋊(C7⋊C8)C2312yes—
135C4⋊(C7⋊C8)—22no—
136C4×C7⋊C8—22no—
137#3505—71no—
138#2972—71no—
139#2353D711yescharacteristic, complemented
140#1521—72no—
141#1520—71no—
142#1519—71no—
143#1516—72no—
144#1501—72no—
145#1500—72no—
146C2×#328—71no—
147C2×#327—71no—
148C24×C7C2×D411yescharacteristic
149C24×C7C2×D411yescharacteristic
150C23×D7—21no—
151C2×C7×D4—22no—
152C2×C7×D4—22no—
153C2×C7×D4C22⋊C412yes—
154C2×C7×D4C22⋊C412yes—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—42no—
160C22×C4×C7—42no—
161C22×C4×C7—81no—
162C22×C4×C7—81no—
163C22×C4×C7—81no—
164C22×C4×C7C22×C411yescharacteristic
165C22×C4×C7C22×C412yes—
166C22×C4×C7C22×C411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D412yes—
174C22×C4×C7C2411yescharacteristic
175C2×C22⋊D7—22no—
176C2×C22⋊D7—22no—
177C2×C22⋊D7—22no—
178C2×C22⋊D7—22no—
179C2×C22⋊D7—42no—
180C2×C22⋊D7—42no—
181C22×Dic7—21no—
182C22×Dic7—21no—
183C22×Dic7—21no—
184#1157—82no—
185C2×D28—21no—
186C2×D28—81no—
187C2×D28C22×C412yes—
188C2×C4×D7—22no—
189C2×C4×D7—22no—
190C2×Dic14—21no—
191C2×Dic14—81no—
192C2×Dic14C22×C412yes—
193C7×C4⋊C4—81no—
194C7×C4⋊C4C22⋊C412yes—
195C7×C4⋊C4C22⋊C412yes—
196C7×C22⋊C4—42no—
197C7×C22⋊C4—42no—
198C7×C22⋊C4—81no—
199C7×C22⋊C4C22⋊C412yes—
200C7×C22⋊C4C22⋊C412yes—
201C7×C22⋊C4C22⋊C412yes—
202C7×C22⋊C4C22⋊C412yes—
203C42×C7—42no—
204C42×C7—81no—
205C42×C7C22×C412yes—
206C42×C7C2×D412yes—
207C22⋊Dic7—22no—
208C22⋊Dic7—22no—
209C22⋊Dic7—22no—
210C22⋊Dic7—22no—
211(C2×D7)⋊C4—22no—
212(C2×D7)⋊C4—22no—
213(C2×D7)⋊C4—22no—
214(C2×D7)⋊C4—22no—
215(C2×D7)⋊C4—22no—
216(C2×D7)⋊C4—22no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219C4⋊Dic7C22×C412yes—
220C4⋊Dic7C22×C412yes—
221Dic7⋊C4—22no—
222Dic7⋊C4—22no—
223Dic7⋊C4—22no—
224Dic7⋊C4—22no—
225Dic7⋊C4—22no—
226Dic7⋊C4—22no—
227C4×Dic7—22no—
228C4×Dic7—22no—
229C2×C7⋊C8—22no—
230C2×C7⋊C8—22no—
231C2×C7⋊C8—81no—
232C2×C7⋊C8—81no—
233C2×C7⋊C8—82no—
234#553—72no—
235#553—142no—
236#552—72no—
237#551—142no—
238#548—72no—
239#546—72no—
240#539—141no—
241#538—141no—
242#534—141no—
243#529—72no—
244#525—72no—
245C2×C22⋊Q8—71no—
246C2×#121—71no—
247#518—141no—
248C2×C4×D4C2×D711yescharacteristic
249C8×D4—142no—
250C8×D4—142no—
251#431—72no—
252C2×C22⋊C8—71no—
253C2×C22⋊C8—71no—
254C2×C22⋊C8—72no—
255#388—21no—
256#387—21no—
257#386—21no—
258#383C2×D711yescharacteristic, complemented
259#381C2×D711yescharacteristic, complemented
260#380C2×D711yescharacteristic
261C4×C22⋊C4—21no—
262C4×C22⋊C4C2×D711yescharacteristic, complemented
263#376C2×D711yescharacteristic, complemented
264C2×#95C2×D711yescharacteristic
265#328—72no—
266#328—72no—
267#328—72no—
268#328—72no—
269#327—72no—
270#327—72no—
271#327—72no—
272#327—72no—
273C23×C7—21no—
274C23×C7—21no—
275C23×C7—41no—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7—41no—
279C23×C7C2×C22⋊C412yes—
280C23×C7C2×C22⋊C411yescharacteristic
281C23×C7C2×C22⋊C411yescharacteristic
282C23×C7#13512yes—
283C23×C7#13512yes—
284C23×C7C22×D411yescharacteristic
285C22×D7—22no—
286C22×D7—42no—
287C22×D7—81no—
288C22×D7—81no—
289C7×D4—24no—
290C7×D4—24no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—22no—
294C2×C4×C7—41no—
295C2×C4×C7—42no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—42no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—84no—
316C2×C4×C7C2×C22⋊C412yes—
317C2×C4×C7C2×C22⋊C412yes—
318C2×C4×C7C2×C22⋊C411yescharacteristic
319C2×C4×C7C2×C22⋊C412yes—
320C2×C4×C7C2×C22⋊C411yescharacteristic
321C2×C4×C7C23×C411yescharacteristic
322C2×C4×C7C22×D411yescharacteristic
323C22⋊D7—42no—
324C22⋊D7—42no—
325C22⋊D7—44no—
326C22⋊D7—82no—
327C22⋊D7—82no—
328C2×Dic7—22no—
329C2×Dic7—22no—
330C2×Dic7—22no—
331C2×Dic7—42no—
332C2×Dic7—81no—
333C2×Dic7—81no—
334D28—42no—
335D28—81no—
336C4×D7—82no—
337C4×D7—82no—
338Dic14—42no—
339Dic14—81no—
340C7⋊C8—42no—
341C7⋊C8—42no—
342C2×#41—72no—
343C22×Q8—71no—
344C22×D4—71no—
345C22×D4—141no—
346C22×D4C22⋊D711yescharacteristic
347C23×C4—21no—
348C23×C4C22×D711yescharacteristic
349C23×C4C22×D711yescharacteristic
350C22×C8—141no—
351C22×C8—141no—
352C22×C8—142no—
353#126—142no—
354#126—142no—
355#126—142no—
356#125—72no—
357#125—142no—
358#124—72no—
359#124—142no—
360#123—142no—
361#123—142no—
362#123—142no—
363#123—142no—
364#123—142no—
365C22⋊Q8—72no—
366C22⋊Q8—72no—
367C22⋊Q8—72no—
368C22⋊Q8—141no—
369C22⋊Q8—141no—
370C22⋊Q8—142no—
371#121—72no—
372#121—72no—
373#121—72no—
374#121—142no—
375#121—142no—
376#121—141no—
377#121—141no—
378#121—142no—
379#120—142no—
380#120—142no—
381C4×Q8—72no—
382C4×D4—72no—
383C4×D4—142no—
384C4×D4—142no—
385C4×D4—142no—
386C4×D4—142no—
387C4×D4—142no—
388C4×D4C4×D712yes—
389C4×D4C4×D712yes—
390C4×D4D2812yes—
391C4×D4D2812yes—
392#117—72no—
393#117—142no—
394#117—142no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—71no—
397C2×C4⋊C4—71no—
398C2×C4⋊C4—141no—
399C2×C4⋊C4C22⋊D711yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—141no—
404C2×C22⋊C4—141no—
405C2×C22⋊C4—142no—
406C2×C22⋊C4C22⋊D711yescharacteristic
407C2×C22⋊C4C22⋊D711yescharacteristic
408C2×C42—21no—
409C2×C42—21no—
410C2×C42C22×D711yescharacteristic
411C4⋊C8—142no—
412C22⋊C8—72no—
413C22⋊C8—72no—
414C22⋊C8—141no—
415C22⋊C8—141no—
416C22⋊C8—142no—
417C4×C8—142no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95C22×D711yescharacteristic
424#95C22×D711yescharacteristic
425#95C22×D711yescharacteristic
426#95C22×D711yescharacteristic
427C22×C7—21no—
428C22×C7—21no—
429C22×C7—21no—
430C22×C7—21no—
431C22×C7—21no—
432C22×C7—21no—
433C22×C7—22no—
434C22×C7—22no—
435C22×C7—22no—
436C22×C7—42no—
437C22×C7—42no—
438C22×C7—81no—
439C22×C7—81no—
440C22×C7#41612yes—
441C22×C7#41612yes—
442C22×C7C22×C22⋊C411yescharacteristic
443C22×C7C2×#13511yescharacteristic
444C22×C7C2×#13511yescharacteristic
445C2×D7—42no—
446C2×D7—42no—
447C2×D7—42no—
448C2×D7—162no—
449C4×C7—21no—
450C4×C7—21no—
451C4×C7—21no—
452C4×C7—22no—
453C4×C7—21no—
454C4×C7—22no—
455C4×C7—22no—
456C4×C7—42no—
457C4×C7—81no—
458C4×C7—81no—
459C4×C7—81no—
460Dic7—42no—
461Dic7—42no—
462Dic7—42no—
463C24—141no—
464C24C2×C22⋊D711yescharacteristic
465C24C2×C22⋊D711yescharacteristic
466#41—562no—
467C2×Q8—72no—
468C2×Q8—141no—
469C2×Q8—561no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—72no—
473C2×D4—142no—
474C2×D4—142no—
475C2×D4—142no—
476C2×D4—142no—
477C2×D4—141no—
478C2×D4—282no—
479C2×D4—282no—
480C2×D4—561no—
481C2×D4(C2×D7)⋊C412yes—
482C2×D4(C2×D7)⋊C412yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—81no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—141no—
493C22×C4—141no—
494C22×C4—141no—
495C22×C4—142no—
496C22×C4—142no—
497C22×C4C2×C4×D711yescharacteristic
498C22×C4C2×C4×D711yescharacteristic
499C22×C4C2×C4×D712yes—
500C22×C4C2×D2811yescharacteristic
501C22×C4C2×D2811yescharacteristic
502C22×C4C2×D2812yes—
503C22×C4C2×C22⋊D711yescharacteristic
504C22×C4C2×C22⋊D711yescharacteristic
505C22×C4C2×C22⋊D711yescharacteristic
506C22×C4C2×C22⋊D711yescharacteristic
507C22×C4C23×D711yescharacteristic
508C2×C8—142no—
509C2×C8—142no—
510C2×C8—562no—
511C2×C8—561no—
512C2×C8—561no—
513C4⋊C4—72no—
514C4⋊C4—72no—
515C4⋊C4—81no—
516C4⋊C4—142no—
517C4⋊C4—142no—
518C4⋊C4—142no—
519C4⋊C4—142no—
520C4⋊C4—142no—
521C4⋊C4—141no—
522C4⋊C4—141no—
523C4⋊C4—142no—
524C4⋊C4(C2×D7)⋊C412yes—
525C4⋊C4(C2×D7)⋊C412yes—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—81no—
529C22⋊C4—142no—
530C22⋊C4—142no—
531C22⋊C4—142no—
532C22⋊C4—142no—
533C22⋊C4—142no—
534C22⋊C4—142no—
535C22⋊C4—142no—
536C22⋊C4—142no—
537C22⋊C4—142no—
538C22⋊C4—142no—
539C22⋊C4(C2×D7)⋊C412yes—
540C22⋊C4(C2×D7)⋊C412yes—
541C22⋊C4(C2×D7)⋊C412yes—
542C22⋊C4(C2×D7)⋊C412yes—
543C42—42no—
544C42—81no—
545C42—142no—
546C42—142no—
547C42C2×C4×D712yes—
548C42C2×D2812yes—
549C2×C7—21no—
550C2×C7—21no—
551C2×C7—22no—
552C2×C7—22no—
553C2×C7—42no—
554C2×C7#288311yescharacteristic
555C2×C7C2×#41611yescharacteristic
556C2×C7#290111yescharacteristic
557D7—82no—
558C23—21no—
559C23—21no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—142no—
565C23—282no—
566C23—561no—
567C23—561no—
568C23C2×(C2×D7)⋊C412yes—
569C23C2×(C2×D7)⋊C411yescharacteristic
570C23C2×(C2×D7)⋊C411yescharacteristic
571C23#651312yes—
572C23#651312yes—
573C23C22×C22⋊D711yescharacteristic
574Q8—282no—
575Q8—561no—
576D4—24no—
577D4—24no—
578D4—282no—
579D4—282no—
580D4—282no—
581D4—284no—
582D4—561no—
583D4—562no—
584D4—562no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—84no—
610C2×C4—142no—
611C2×C4—142no—
612C2×C4—142no—
613C2×C4—282no—
614C2×C4—562no—
615C2×C4—562no—
616C2×C4—561no—
617C2×C4—561no—
618C2×C4C2×(C2×D7)⋊C412yes—
619C2×C4C2×(C2×D7)⋊C412yes—
620C2×C4C2×(C2×D7)⋊C412yes—
621C2×C4C2×(C2×D7)⋊C411yescharacteristic
622C2×C4C2×(C2×D7)⋊C411yescharacteristic
623C2×C4C22×C4×D711yescharacteristic
624C2×C4C22×D2811yescharacteristic
625C8—282no—
626C8—282no—
627C7#1761511yescharacteristic, complemented, Sylow
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—42no—
638C22—42no—
639C22—81no—
640C22—81no—
641C22—282no—
642C22—282no—
643C22—282no—
644C22—1122no—
645C22#8977412yes—
646C22#8977412yes—
647C22C22×(C2×D7)⋊C411yescharacteristic
648C22C2×#651311yescharacteristic
649C22C2×#651311yescharacteristic
650C4—22no—
651C4—22no—
652C4—21no—
653C4—21no—
654C4—22no—
655C4—21no—
656C4—21no—
657C4—42no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—282no—
662C4—282no—
663C4—282no—
664C2—21no—
665C2—21no—
666C2—22no—
667C2—22no—
668C2—42no—
669C2—562no—
670C2C2×#8977411yescharacteristic
671C2#1173998111yescharacteristic
672C2#1173999111yescharacteristic
subgroupG/H
centerC22C22×(C2×D7)⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2353C2
socleC22×C7C22×C22⋊C4
prime radical#11742855C2
cosocleC22×C4×C7C24
hypercenter#2353D7
hypocenterC7#17615
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2353

Complements

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

NH ≅ G/Ntype
2. C7×#2353C2semidirect
3. #11739746C2semidirect
4. #11727319C2semidirect
5. #11727286C2semidirect
6. #11727264C2semidirect
7. #11727263C2semidirect
8. #11727197C2semidirect
9. #11727185C2semidirect
10. C2×#89247C2semidirect
11. C2×#89246C2semidirect
17. #90256C4semidirect
19. #90243C4semidirect
20. #90241C4semidirect
21. #90232C4semidirect
30. C7×#383C22semidirect
31. C7×#381C22semidirect
34. C4×C7×C22⋊C4C22semidirect
35. C7×#376C22semidirect
39. C2×C22⋊(C7⋊C8)C22semidirect
42. #89697C22semidirect
43. #89247C22semidirect
44. #89247C22semidirect
45. #89247C22semidirect
46. #89247C22semidirect
57. C2×#1157C2×C4semidirect
83. #6493C2×C4semidirect
91. #6489C2×C4semidirect
97. #6487C2×C4semidirect
116. #6438C2×C4semidirect
117. C4×D28C2×C4semidirect
119. #6433C2×C4semidirect
120. C4×Dic14C2×C4semidirect
139. #2353D7semidirect
258. #383C2×D7semidirect
259. #381C2×D7semidirect
262. C4×C22⋊C4C2×D7semidirect
263. #376C2×D7semidirect
627. C7#17615semidirect

Maximal subgroups

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

subgroupindex
#117428552
C7×#23532
#117397462
#117273192
#117272862
#117272642
#117272632
#117271972
#117271852
C2×#892472
C2×#892462
#176157