Finite Group Databasebeta

← #13353323

Subgroups — #13353323

Order 1792.

1462 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#520C211yesmaximal, characteristic
2C2×#90004C211yesmaximal, characteristic
3C2×#90005C211yesmaximal, characteristic
4#11739678C211yesmaximal, characteristic, complemented
5C2×#89771C211yesmaximal, characteristic, complemented
6#11729381C212yesmaximal, complemented
7#11729380C212yesmaximal, complemented
8#11729359C212yesmaximal, complemented
9#11729358C212yesmaximal, complemented
10#11729312C211yesmaximal, characteristic, complemented
11C2×#89324C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C7×#520—24no—
14C7×#520C418yescomplemented
15C2×C4×C7×Q8C2212yes—
16C2×C4×C7×D4—22no—
17C2×C4×C7×D4C2212yes—
18C2×C7×#117—21no—
19C2×C7×#117C2211yescharacteristic
20C22×C7×C4⋊C4—21no—
21C22×C7×C4⋊C4C2211yescharacteristic
22C2×#6489C2211yescharacteristic
23C22×C4⋊Dic7C2211yescharacteristic
24C22×#1136C2211yescharacteristic
25C23×C7⋊C8C2211yescharacteristic
26#90005—24no—
27#90004—24no—
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2212yes—
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2212yes—
34#89771—24no—
35#89757—21no—
36#89757—21no—
37#89722—22no—
38#89708C2212yes—
39#89707C2212yes—
40C2×C4⋊(C7⋊C8)C2212yes—
41C2×C4⋊(C7⋊C8)C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#20652—71nomaximal, Sylow
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41C2×C414yes—
51C2×C7×#41D414yes—
52C2×C7×#41Q814yes—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4C2311yescharacteristic
56C23×C4×C7—21no—
57C23×C4×C7—21no—
58C23×C4×C7—22no—
59C23×C4×C7C2311yescharacteristic
60C23×Dic7—21no—
61C4×C7×Q8—24no—
62C4×C7×Q8C2×C418yescomplemented
63C4×C7×D4—216no—
64C4×C7×D4—24no—
65C4×C7×D4C2×C418yescomplemented
66C7×#117—28no—
67C7×#117C2×C414yes—
68C7×#117D414yes—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—22no—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4—28no—
73C2×C7×C4⋊C4C2×C414yes—
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C4⋊C4D411yescharacteristic
76C2×C7×C4⋊C4C2312yes—
77C2×C7×C22⋊C4—22no—
78C2×C7×C22⋊C4D412yes—
79C2×C42×C7—22no—
80C2×C42×C7C2312yes—
81C2×C22⋊Dic7—21no—
82#6489—22no—
83#6489—22no—
84C2×C4⋊Dic7—22no—
85C2×C4⋊Dic7—22no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7—22no—
88C2×C4⋊Dic7—22no—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2312yes—
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4×Dic7—21no—
93C2×#1136—22no—
94C2×#1136—22no—
95C2×#1136—22no—
96C22×C7⋊C8—22no—
97C22×C7⋊C8—22no—
98C22×C7⋊C8—22no—
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2312yes—
102#6409—24no—
103#6409—22no—
104#6409—22no—
105#6406—24no—
106#6406—22no—
107#6406—22no—
108C7⋊#95—22no—
109C7⋊#95—22no—
110C4⋊(C7⋊C8)—24no—
111C4⋊(C7⋊C8)—24no—
112C2×#520D711yescharacteristic
113C2×#424—71no—
114C2×#417—71no—
115C2×#416—71no—
116#2317—71no—
117#1805—72no—
118#1804—72no—
119#1803—72no—
120#1802—72no—
121#1736—71no—
122C2×#340—71no—
123C24×C7—21no—
124C24×C7C2×D411yescharacteristic
125C7×#41—28no—
126C7×#41—28no—
127C7×#41—28no—
128C7×#41C4⋊C4116yescomplemented
129C2×C7×Q8—21no—
130C2×C7×Q8—21no—
131C2×C7×Q8—21no—
132C2×C7×Q8C22×C412yes—
133C2×C7×Q8C2×D412yes—
134C2×C7×Q8C2×Q812yes—
135C2×C7×D4—24no—
136C2×C7×D4—24no—
137C2×C7×D4—24no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4C22×C412yes—
142C2×C7×D4C2×D412yes—
143C2×C7×D4C2×Q812yes—
144C22×C4×C7—21no—
145C22×C4×C7—24no—
146C22×C4×C7—24no—
147C22×C4×C7—24no—
148C22×C4×C7—22no—
149C22×C4×C7—24no—
150C22×C4×C7—28no—
151C22×C4×C7—82no—
152C22×C4×C7—82no—
153C22×C4×C7—81no—
154C22×C4×C7—81no—
155C22×C4×C7C22×C412yes—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×Q812yes—
162C22×C4×C7#4112yes—
163C22×C4×C7#4112yes—
164C22×C4×C7C2411yescharacteristic
165C22×Dic7—21no—
166C22×Dic7—22no—
167C22×Dic7—21no—
168C22×Dic7—42no—
169C22×Dic7—81no—
170C22×Dic7—81no—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—28no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4C22⋊C414yes—
175C7×C4⋊C4C22⋊C414yes—
176C7×C4⋊C4C22×C414yes—
177C7×C22⋊C4—28no—
178C7×C22⋊C4C22⋊C418yes—
179C42×C7—28no—
180C42×C7C22×C414yes—
181C42×C7C2×D414yes—
182C22⋊Dic7—42no—
183C4⋊Dic7—22no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—22no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—42no—
190C4⋊Dic7—82no—
191C4⋊Dic7—82no—
192C4×Dic7—42no—
193#1136—44no—
194C2×C7⋊C8—21no—
195C2×C7⋊C8—21no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—21no—
198C2×C7⋊C8—21no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—44no—
204C22×#41C2×D711yescharacteristic
205C22×M16—71no—
206C23×C8—71no—
207#520—24no—
208#520Dic718yescomplemented
209C2×C4×Q8C2×D712yes—
210C2×C4×D4—22no—
211C2×C4×D4C2×D712yes—
212C2×#117—21no—
213C2×#117—71no—
214C2×#117C2×D711yescharacteristic
215C22×C4⋊C4—21no—
216C22×C4⋊C4—71no—
217C22×C4⋊C4C2×D711yescharacteristic
218#424—144no—
219C2×C4⋊C8—72no—
220C2×C4⋊C8—72no—
221#417—144no—
222#416—144no—
223C2×#103—71no—
224C2×#103—72no—
225C2×#103—71no—
226C2×#102—71no—
227C2×#102—72no—
228C2×#102—71no—
229#389—141no—
230#389—141no—
231#385—142no—
232#384—72no—
233#382—72no—
234#340—72no—
235#340—72no—
236#340—72no—
237#340—72no—
238C23×C7—21no—
239C23×C7—21no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7—22no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—41no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—41no—
249C23×C7C2×C22⋊C412yes—
250C23×C7#12112yes—
251C23×C7C22⋊Q812yes—
252C23×C7C22×D411yescharacteristic
253C7×Q8—22no—
254C7×Q8—22no—
255C7×Q8—22no—
256C7×Q8C2×C4⋊C414yes—
257C7×D4—216no—
258C7×D4—22no—
259C7×D4—22no—
260C7×D4—22no—
261C7×D4C2×C4⋊C414yes—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—216no—
270C2×C4×C7—24no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—88no—
274C2×C4×C7—84no—
275C2×C4×C7C2×C22⋊C414yes—
276C2×C4×C7C2×C22⋊C414yes—
277C2×C4×C7C2×C22⋊C412yes—
278C2×C4×C7C2×C4⋊C414yes—
279C2×C4×C7#11714yes—
280C2×C4×C7#12114yes—
281C2×C4×C7#12112yes—
282C2×C4×C7C22⋊Q814yes—
283C2×C4×C7C22⋊Q812yes—
284C2×C4×C7C23×C411yescharacteristic
285C2×C4×C7C22×D411yescharacteristic
286C2×C4×C7C22×D411yescharacteristic
287C2×C4×C7C22×Q811yescharacteristic
288C2×C4×C7C2×#4111yescharacteristic
289C2×C4×C7C2×#4111yescharacteristic
290C2×Dic7—41no—
291C2×Dic7—41no—
292C2×Dic7—42no—
293C2×Dic7—41no—
294C2×Dic7—42no—
295C2×Dic7—41no—
296C2×Dic7—42no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—84no—
300C7⋊C8—41no—
301C7⋊C8—41no—
302C7⋊C8—42no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41—22no—
306C2×#41Dic1414yes—
307C2×#41D2814yes—
308C2×#41C2×Dic714yes—
309C22×Q8C22×D711yescharacteristic
310C22×D4—21no—
311C22×D4C22×D711yescharacteristic
312C23×C4—21no—
313C23×C4—21no—
314C23×C4—22no—
315C23×C4—141no—
316C23×C4C22×D711yescharacteristic
317C2×M16—142no—
318C2×M16—142no—
319C2×M16—142no—
320C22×C8—71no—
321C22×C8—71no—
322C22×C8—72no—
323C22×C8—142no—
324C22×C8—142no—
325C22×C8—142no—
326C4×Q8—24no—
327C4×Q8C2×Dic718yescomplemented
328C4×D4—216no—
329C4×D4—24no—
330C4×D4C2×Dic718yescomplemented
331#117—28no—
332#117—142no—
333#117—142no—
334#117C2×Dic714yes—
335#117C22⋊D714yes—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—28no—
340C2×C4⋊C4—72no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—142no—
344C2×C4⋊C4—142no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—142no—
348C2×C4⋊C4C2×Dic714yes—
349C2×C4⋊C4C22⋊D711yescharacteristic
350C2×C4⋊C4C22⋊D711yescharacteristic
351C2×C4⋊C4C22×D712yes—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—141no—
354C2×C22⋊C4C22⋊D712yes—
355C2×C42—22no—
356C2×C42—141no—
357C2×C42C22×D712yes—
358C4⋊C8—144no—
359C4⋊C8—144no—
360#103—142no—
361#103—142no—
362#103—144no—
363#102—142no—
364#102—142no—
365#102—144no—
366#95—142no—
367#95—142no—
368C22×C7—22no—
369C22×C7—22no—
370C22×C7—22no—
371C22×C7—22no—
372C22×C7—22no—
373C22×C7—22no—
374C22×C7—24no—
375C22×C7—42no—
376C22×C7—42no—
377C22×C7—42no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7—42no—
381C22×C7#38014yes—
382C22×C7C22×C22⋊C411yescharacteristic
383C22×C7C2×#12111yescharacteristic
384C22×C7C2×C22⋊Q811yescharacteristic
385C22×C7#57611yescharacteristic
386C22×C7#57711yescharacteristic
387C22×C7#57711yescharacteristic
388C22×C7#57811yescharacteristic
389C4×C7—24no—
390C4×C7—24no—
391C4×C7—24no—
392C4×C7—44no—
393C4×C7#38014yes—
394C4×C7C22×C4⋊C411yescharacteristic
395C4×C7C2×#11711yescharacteristic
396C4×C7C2×#12111yescharacteristic
397C4×C7C2×C22⋊Q811yescharacteristic
398Dic7—81no—
399Dic7—82no—
400Dic7—81no—
401C24—21no—
402C24C2×C22⋊D711yescharacteristic
403#41—28no—
404#41—28no—
405#41—28no—
406#41C4⋊Dic7116yescomplemented
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8C2×Dic1412yes—
411C2×Q8C2×D2812yes—
412C2×Q8C22×Dic712yes—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4C2×Dic1412yes—
420C2×D4C2×D2812yes—
421C2×D4C22×Dic712yes—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—141no—
434C22×C4—142no—
435C22×C4—141no—
436C22×C4—282no—
437C22×C4—561no—
438C22×C4—561no—
439C22×C4C2×Dic1412yes—
440C22×C4C2×D2812yes—
441C22×C4#115712yes—
442C22×C4#115712yes—
443C22×C4C22×Dic712yes—
444C22×C4C2×C22⋊D712yes—
445C22×C4C2×C22⋊D712yes—
446C22×C4C2×C22⋊D711yescharacteristic
447C22×C4C2×C22⋊D712yes—
448C22×C4C23×D711yescharacteristic
449M16—284no—
450C2×C8—141no—
451C2×C8—141no—
452C2×C8—142no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—142no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8—142no—
459C2×C8—284no—
460C4⋊C4—24no—
461C4⋊C4—28no—
462C4⋊C4—22no—
463C4⋊C4—142no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—142no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—282no—
470C4⋊C4—562no—
471C4⋊C4—562no—
472C4⋊C4C22⋊Dic714yes—
473C4⋊C4C22⋊Dic714yes—
474C4⋊C4C22×Dic714yes—
475C22⋊C4—28no—
476C22⋊C4—282no—
477C22⋊C4C22⋊Dic718yes—
478C42—28no—
479C42—282no—
480C42C22×Dic714yes—
481C42C2×C22⋊D714yes—
482C2×C7—24no—
483C2×C7—44no—
484C2×C7C2×#38011yescharacteristic
485C2×C7#289711yescharacteristic
486C2×C7#289811yescharacteristic
487C2×C7#304511yescharacteristic
488C2×C7#304611yescharacteristic
489C2×C7#308311yescharacteristic
490C2×C7#308411yescharacteristic
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23#648712yes—
503C23#649312yes—
504C23C2×C22⋊Dic712yes—
505C23C22×C22⋊D711yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8C2×C4⋊Dic714yes—
510D4—216no—
511D4—22no—
512D4—22no—
513D4—22no—
514D4C2×C4⋊Dic714yes—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—216no—
523C2×C4—24no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—88no—
527C2×C4—84no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—282no—
531C2×C4—281no—
532C2×C4—282no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—282no—
536C2×C4—281no—
537C2×C4—564no—
538C2×C4#648714yes—
539C2×C4#648712yes—
540C2×C4C2×C4⋊Dic714yes—
541C2×C4#648914yes—
542C2×C4#649314yes—
543C2×C4#649312yes—
544C2×C4C2×C22⋊Dic714yes—
545C2×C4C2×C22⋊Dic714yes—
546C2×C4C2×C22⋊Dic712yes—
547C2×C4C22×Dic1411yescharacteristic
548C2×C4C22×D2811yescharacteristic
549C2×C4C2×#115711yescharacteristic
550C2×C4C2×#115711yescharacteristic
551C2×C4C23×Dic711yescharacteristic
552C2×C4C22×C22⋊D711yescharacteristic
553C8—281no—
554C8—281no—
555C8—282no—
556C7#2065211yescharacteristic, complemented, Sylow
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22#8998414yes—
571C22C2×#648711yescharacteristic
572C22C2×#649311yescharacteristic
573C22#9052611yescharacteristic
574C22#9052711yescharacteristic
575C22#9052811yescharacteristic
576C22#9052911yescharacteristic
577C22C22×C22⋊Dic711yescharacteristic
578C4—24no—
579C4—24no—
580C4—24no—
581C4—44no—
582C4—562no—
583C4—561no—
584C4—561no—
585C4#8998414yes—
586C4C2×#648711yescharacteristic
587C4C22×C4⋊Dic711yescharacteristic
588C4C2×#648911yescharacteristic
589C4C2×#649311yescharacteristic
590C2—24no—
591C2—44no—
592C2#1174000411yescharacteristic
593C2#1174000511yescharacteristic
594C2#1174001011yescharacteristic
595C2#1174001711yescharacteristic
596C2C2×#8998411yescharacteristic
597C2#1174154411yescharacteristic
598C2#1174155611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#520C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenterC2×#520D7
hypocenterC7#20652
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
4. #11739678C2semidirect
5. C2×#89771C2semidirect
6. #11729381C2semidirect
7. #11729380C2semidirect
8. #11729359C2semidirect
9. #11729358C2semidirect
10. #11729312C2semidirect
11. C2×#89324C2semidirect
14. C7×#520C4semidirect
62. C4×C7×Q8C2×C4semidirect
65. C4×C7×D4C2×C4semidirect
128. C7×#41C4⋊C4semidirect
208. #520Dic7semidirect
327. C4×Q8C2×Dic7semidirect
330. C4×D4C2×Dic7semidirect
406. #41C4⋊Dic7semidirect
556. C7#20652semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5202
C2×#900042
C2×#900052
#117396782
C2×#897712
#117293812
#117293802
#117293592
#117293582
#117293122
C2×#893242
#206527