Finite Group Databasebeta

← #13353328

Subgroups — #13353328

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×#90195C211yesmaximal, characteristic
2C2×#89885C211yesmaximal, characteristic
3C2×#89874C211yesmaximal, characteristic, complemented
4#11739602C211yesmaximal, characteristic, complemented
5C2×#89738C211yesmaximal, characteristic
6C2×C7×#340C211yesmaximal, characteristic, complemented
7#11729312C211yesmaximal, characteristic, complemented
8#11726659C212yesmaximal, complemented
9#11726658C212yesmaximal, complemented
10#11726647C212yesmaximal, complemented
11#11726646C212yesmaximal, complemented
12C22×#1157C2211yescharacteristic
13C23×C7×C8C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C2×C4×C22⋊D7—22no—
17C2×#6489C2211yescharacteristic
18C2×#6489C2211yescharacteristic
19C22×#1136C2211yescharacteristic
20#90195—24no—
21#90195C418yescomplemented
22C2×#6456C2212yes—
23C2×#6455—21no—
24C2×D7×C4⋊C4—21no—
25C2×#6450C2212yes—
26#89982—22no—
27#89885—24no—
28C2×#6395C2211yescharacteristic
29C2×#6395C2211yescharacteristic
30C2×#6390C2211yescharacteristic
31C2×#6390C2211yescharacteristic
32#89874—24no—
33C2×Dic7⋊C8C2212yes—
34C2×Dic7⋊C8C2212yes—
35#89749C2212yes—
36#89748C2212yes—
37#89745—21no—
38#89745—21no—
39C2×#6384C2212yes—
40#89738—24no—
41C2×#6383C2212yes—
42C7×#340C2212yes—
43C7×#340C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#20652—71nomaximal, Sylow
47C23×C4×C7—21no—
48C23×C4×C7—21no—
49C23×C4×C7C2311yescharacteristic
50C22×C22⋊D7—21no—
51C2×#1157—22no—
52C2×#1157—22no—
53C2×#1157—22no—
54C2×#1157C2×C414yes—
55C2×#1157D414yes—
56C2×#1157Q814yes—
57C22×D28C2311yescharacteristic
58C22×C4×D7—21no—
59C22×C4×D7—22no—
60C22×Dic14C2311yescharacteristic
61C22×C7×C8—22no—
62C22×C7×C8—22no—
63C22×C7×C8—22no—
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—22no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4C2×C414yes—
72C2×C7×C4⋊C4C2312yes—
73C2×C7×C4⋊C4C2312yes—
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7D412yes—
76C4×C22⋊D7—216no—
77C2×(C2×D7)⋊C4—22no—
78#6489—22no—
79#6489—22no—
80#6489C2×C414yes—
81#6489D414yes—
82C2×C4⋊Dic7D411yescharacteristic
83C2×C4⋊Dic7D411yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×Dic7⋊C4—22no—
87C2×Dic7⋊C4—22no—
88C2×Dic7⋊C4—22no—
89C2×C4×Dic7—22no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7C2312yes—
92C2×#1136—22no—
93C2×#1136—22no—
94C2×#1136—22no—
95C22×C7⋊C8C2312yes—
96#6456—24no—
97#6456C2×C418yescomplemented
98#6455—28no—
99D7×C4⋊C4—28no—
100#6450—24no—
101#6450C2×C418yescomplemented
102C7⋊#95—22no—
103C7⋊#95—22no—
104#6395—22no—
105#6395—22no—
106#6390—22no—
107#6390—22no—
108Dic7⋊C8—24no—
109Dic7⋊C8—24no—
110#6384—24no—
111#6383—24no—
112C2×#520—71no—
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×#340D711yescharacteristic, complemented
123C24×C7C2×D411yescharacteristic
124C23×D7—21no—
125C22×C4×C7—22no—
126C22×C4×C7—24no—
127C22×C4×C7—22no—
128C22×C4×C7—42no—
129C22×C4×C7—81no—
130C22×C4×C7—81no—
131C22×C4×C7—81no—
132C22×C4×C7—81no—
133C22×C4×C7C22×C412yes—
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×D412yes—
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×Q812yes—
138C22×C4×C7#4112yes—
139C22×C4×C7#4112yes—
140C22×C4×C7C2411yescharacteristic
141C2×C22⋊D7—24no—
142C2×C22⋊D7—24no—
143C2×C22⋊D7—24no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7C2×D412yes—
148C22×Dic7C2×D412yes—
149#1157—28no—
150#1157—28no—
151#1157—28no—
152#1157C4⋊C4116yescomplemented
153C2×D28—21no—
154C2×D28—21no—
155C2×D28—21no—
156C2×D28C22×C412yes—
157C2×D28C2×D412yes—
158C2×D28C2×Q812yes—
159C2×C4×D7—24no—
160C2×C4×D7—24no—
161C2×C4×D7—24no—
162C2×C4×D7—28no—
163C2×C4×D7—82no—
164C2×C4×D7—82no—
165C2×Dic14—21no—
166C2×Dic14—21no—
167C2×Dic14—21no—
168C2×Dic14C22×C412yes—
169C2×Dic14C2×D412yes—
170C2×Dic14C2×Q812yes—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—44no—
178C7×C4⋊C4—24no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—42no—
183C7×C4⋊C4C22×C414yes—
184C22⋊Dic7—42no—
185C22⋊Dic7C22⋊C418yes—
186(C2×D7)⋊C4—28no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7C22⋊C414yes—
192C4⋊Dic7C22⋊C414yes—
193Dic7⋊C4—28no—
194Dic7⋊C4—82no—
195Dic7⋊C4—82no—
196C4×Dic7—28no—
197C4×Dic7—42no—
198C4×Dic7C22×C414yes—
199C4×Dic7C2×D414yes—
200#1136—44no—
201C2×C7⋊C8—22no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C22×#41—71no—
205C22×M16—71no—
206C23×C8C2×D711yescharacteristic
207#520—78no—
208#520—144no—
209C2×C4×Q8—72no—
210C2×C4×D4—72no—
211C2×C4×D4—142no—
212C2×#117—71no—
213C2×#117—71no—
214C2×#117—141no—
215C22×C4⋊C4—141no—
216C22×C4⋊C4C2×D711yescharacteristic
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—71no—
225C2×#103—72no—
226C2×#102—71no—
227C2×#102—71no—
228C2×#102—72no—
229#389—141no—
230#389—141no—
231#385—142no—
232#384—72no—
233#382—72no—
234#340—72no—
235#340—72no—
236#340C2×D712yes—
237#340C2×D712yes—
238C23×C7—21no—
239C23×C7—21no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7C2×C22⋊C412yes—
243C23×C7#12112yes—
244C23×C7C22⋊Q812yes—
245C23×C7C22×D411yescharacteristic
246C22×D7—22no—
247C22×D7—22no—
248C22×D7—22no—
249C22×D7—41no—
250C22×D7—41no—
251C22×D7—41no—
252C22×D7—41no—
253C2×C4×C7—24no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—84no—
260C2×C4×C7—84no—
261C2×C4×C7C2×C22⋊C412yes—
262C2×C4×C7C2×C4⋊C414yes—
263C2×C4×C7#11714yes—
264C2×C4×C7#12114yes—
265C2×C4×C7#12112yes—
266C2×C4×C7C22⋊Q814yes—
267C2×C4×C7C22⋊Q812yes—
268C2×C4×C7C23×C411yescharacteristic
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×Q811yescharacteristic
272C2×C4×C7C2×#4111yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C22⋊D7—216no—
275C2×Dic7—22no—
276C2×Dic7—22no—
277C2×Dic7—22no—
278C2×Dic7—22no—
279C2×Dic7—22no—
280C2×Dic7—22no—
281C2×Dic7—22no—
282C2×Dic7—41no—
283C2×Dic7—41no—
284C2×Dic7—41no—
285C2×Dic7—41no—
286C2×Dic7—41no—
287C2×Dic7—41no—
288C2×Dic7C2×C22⋊C414yes—
289C2×Dic7C2×C22⋊C414yes—
290D28—22no—
291D28—22no—
292D28—22no—
293D28C2×C4⋊C414yes—
294C4×D7—216no—
295C4×D7—88no—
296Dic14—22no—
297Dic14—22no—
298Dic14—22no—
299Dic14C2×C4⋊C414yes—
300C7×C8—41no—
301C7×C8—41no—
302C7⋊C8—42no—
303C2×#41—74no—
304C2×#41—74no—
305C2×#41—74no—
306C2×#41—142no—
307C2×#41—142no—
308C2×#41—142no—
309C22×Q8—71no—
310C22×D4—71no—
311C22×D4—141no—
312C23×C4—21no—
313C23×C4—21no—
314C23×C4—142no—
315C23×C4—141no—
316C23×C4C22×D711yescharacteristic
317C2×M16—142no—
318C2×M16—142no—
319C2×M16—142no—
320C22×C8—22no—
321C22×C8—22no—
322C22×C8—22no—
323C22×C8—72no—
324C22×C8C22×D711yescharacteristic
325C22×C8C22×D711yescharacteristic
326C4×Q8—78no—
327C4×Q8—144no—
328C4×D4—78no—
329C4×D4—144no—
330C4×D4—1416no—
331#117—74no—
332#117—74no—
333#117—148no—
334#117—142no—
335#117—142no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—148no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—142no—
348C2×C4⋊C4—142no—
349C2×C4⋊C4C4×D714yes—
350C2×C4⋊C4C22×D712yes—
351C2×C4⋊C4C22×D712yes—
352C2×C22⋊C4—72no—
353C2×C22⋊C4—142no—
354C2×C22⋊C4—141no—
355C2×C42—72no—
356C2×C42—142no—
357C2×C42—141no—
358C4⋊C8—144no—
359C4⋊C8—144no—
360#103—144no—
361#103—142no—
362#103—142no—
363#102—144no—
364#102—142no—
365#102—142no—
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#38014yes—
375C22×C7C22×C22⋊C411yescharacteristic
376C22×C7C2×#12111yescharacteristic
377C22×C7C2×C22⋊Q811yescharacteristic
378C22×C7#57611yescharacteristic
379C22×C7#57711yescharacteristic
380C22×C7#57711yescharacteristic
381C22×C7#57811yescharacteristic
382C2×D7—24no—
383C2×D7—42no—
384C2×D7—42no—
385C2×D7—42no—
386C2×D7—42no—
387C2×D7—42no—
388C2×D7—42no—
389C4×C7—44no—
390C4×C7—82no—
391C4×C7#38014yes—
392C4×C7C22×C4⋊C411yescharacteristic
393C4×C7C2×#11711yescharacteristic
394C4×C7C2×#12111yescharacteristic
395C4×C7C2×C22⋊Q811yescharacteristic
396Dic7—24no—
397Dic7—24no—
398Dic7—24no—
399Dic7—81no—
400Dic7—81no—
401C24—141no—
402C24D4×D711yescharacteristic
403#41—716no—
404#41—148no—
405#41—148no—
406#41—148no—
407C2×Q8—72no—
408C2×Q8—72no—
409C2×Q8—72no—
410C2×Q8—141no—
411C2×Q8—141no—
412C2×Q8—141no—
413C2×D4—72no—
414C2×D4—72no—
415C2×D4—72no—
416C2×D4—144no—
417C2×D4—144no—
418C2×D4—144no—
419C2×D4—141no—
420C2×D4—141no—
421C2×D4—141no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—42no—
426C22×C4—72no—
427C22×C4—72no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—148no—
433C22×C4—141no—
434C22×C4—144no—
435C22×C4—144no—
436C22×C4—144no—
437C22×C4—141no—
438C22×C4—141no—
439C22×C4—562no—
440C22×C4—562no—
441C22×C4C2×C4×D712yes—
442C22×C4#115712yes—
443C22×C4#115712yes—
444C22×C4D4×D711yescharacteristic
445C22×C4D4×D712yes—
446C22×C4D4×D712yes—
447C22×C4Q8×D712yes—
448C22×C4C23×D711yescharacteristic
449M16—284no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—44no—
457C2×C8—142no—
458C2×C8—142no—
459C2×C8—142no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—42no—
465C4⋊C4—74no—
466C4⋊C4—74no—
467C4⋊C4—148no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—562no—
473C4⋊C4—562no—
474C4⋊C4C2×C4×D714yes—
475C22⋊C4—78no—
476C22⋊C4—148no—
477C22⋊C4—282no—
478C42—74no—
479C42—74no—
480C42—148no—
481C42—282no—
482C2×C7—24no—
483C2×C7C2×#38011yescharacteristic
484C2×C7#289711yescharacteristic
485C2×C7#289811yescharacteristic
486C2×C7#304511yescharacteristic
487C2×C7#304611yescharacteristic
488C2×C7#308311yescharacteristic
489C2×C7#308411yescharacteristic
490D7—44no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—142no—
496C23—142no—
497C23—142no—
498C23—281no—
499C23—281no—
500C23—281no—
501C23—281no—
502C23D7×C22⋊C412yes—
503C23#644712yes—
504C23#645912yes—
505C23C2×D4×D711yescharacteristic
506Q8—74no—
507Q8—142no—
508Q8—142no—
509Q8—142no—
510D4—74no—
511D4—1416no—
512D4—142no—
513D4—142no—
514D4—142no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—74no—
522C2×C4—74no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4—142no—
526C2×C4—142no—
527C2×C4—142no—
528C2×C4—142no—
529C2×C4—142no—
530C2×C4—142no—
531C2×C4—142no—
532C2×C4—1416no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—568no—
540C2×C4#643514yes—
541C2×C4D7×C22⋊C412yes—
542C2×C4#644714yes—
543C2×C4#644712yes—
544C2×C4D7×C4⋊C414yes—
545C2×C4#645914yes—
546C2×C4#645912yes—
547C2×C4C22×C4×D711yescharacteristic
548C2×C4C2×#115711yescharacteristic
549C2×C4C2×#115711yescharacteristic
550C2×C4C2×D4×D711yescharacteristic
551C2×C4C2×D4×D711yescharacteristic
552C2×C4C2×Q8×D711yescharacteristic
553C8—41no—
554C8—41no—
555C8—282no—
556C7#2065211yescharacteristic, complemented, Sylow
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—144no—
564C22—282no—
565C22—282no—
566C22—282no—
567C22—282no—
568C22—282no—
569C22—282no—
570C22#8943914yes—
571C22C2×D7×C22⋊C411yescharacteristic
572C22C2×#644711yescharacteristic
573C22C2×#645911yescharacteristic
574C22#9044811yescharacteristic
575C22#9045311yescharacteristic
576C22#9045311yescharacteristic
577C22#9045711yescharacteristic
578C4—44no—
579C4—82no—
580C4—144no—
581C4—144no—
582C4—144no—
583C4—561no—
584C4—561no—
585C4#8943914yes—
586C4C2×#643511yescharacteristic
587C4C2×#644711yescharacteristic
588C4C2×D7×C4⋊C411yescharacteristic
589C4C2×#645911yescharacteristic
590C2—24no—
591C2—284no—
592C2C2×#8943911yescharacteristic
593C2#1173618711yescharacteristic
594C2#1173621211yescharacteristic
595C2#1173623511yescharacteristic
596C2#1173626011yescharacteristic
597C2#1173657511yescharacteristic
598C2#1173660711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#340C2
socleC23×C7C22×D4
prime radicalC22×#1157C22
cosocleC22×C4×C7C24
hypercenterC2×#340D7
hypocenterC7#20652
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. C2×#89874C2semidirect
4. #11739602C2semidirect
6. C2×C7×#340C2semidirect
7. #11729312C2semidirect
8. #11726659C2semidirect
9. #11726658C2semidirect
10. #11726647C2semidirect
11. #11726646C2semidirect
21. #90195C4semidirect
97. #6456C2×C4semidirect
101. #6450C2×C4semidirect
122. C2×#340D7semidirect
152. #1157C4⋊C4semidirect
556. C7#20652semidirect

Maximal subgroups

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

subgroupindex
C2×#901952
C2×#898852
C2×#898742
#117396022
C2×#897382
C2×C7×#3402
#117293122
#117266592
#117266582
#117266472
#117266462
#206527