Finite Group Databasebeta

← #13426511

Subgroups — #13426511

Order 1792.

728 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11730957C211yesmaximal, characteristic
2#11730910C211yesmaximal, characteristic
3C7×#1923C211yesmaximal, characteristic
4#11728242C211yesmaximal, characteristic, complemented
5#11728208C211yesmaximal, characteristic, complemented
6#11727690C211yesmaximal, characteristic
7#11727690C211yesmaximal, characteristic
8#11727682C211yesmaximal, characteristic
9#11727679C211yesmaximal, characteristic
10#11727523C211yesmaximal, characteristic, complemented
11C4×#6390C211yesmaximal, characteristic, complemented
12#11726438C211yesmaximal, characteristic, complemented
13#11726390C211yesmaximal, characteristic, complemented
14#11726377C211yesmaximal, characteristic, complemented
15#11726376C211yesmaximal, characteristic, complemented
16C22×C4⋊Dic7C2211yescharacteristic
17C2×#6487C2211yescharacteristic
18C2×#6442—21no—
19C2×C4×Dic14C2211yescharacteristic
20C2×C7×C4⋊C8C2211yescharacteristic
21C2×C7×C22⋊C8C2211yescharacteristic
22C2×C7×C22⋊C8C2211yescharacteristic
23C2×C4×C7×C8C2211yescharacteristic
24C4×C7×C22⋊C4C2211yescharacteristic
25#89984C2211yescharacteristic
26#89984C2211yescharacteristic
27C2×#6391C2211yescharacteristic
28C2×#6391C2211yescharacteristic
29C2×#6391C2211yescharacteristic
30C2×#6391C2211yescharacteristic
31C2×#6390C2211yescharacteristic
32C2×#6390C2211yescharacteristic
33C2×#6390C2211yescharacteristic
34C2×#6390C2211yescharacteristic
35#89725—21no—
36#89718—21no—
37C22⋊C4×Dic7—21no—
38#89707C2211yescharacteristic
39C4×C4⋊Dic7C2211yescharacteristic
40C4×C4⋊Dic7C2211yescharacteristic
41#89701C2211yescharacteristic
42#89700C2211yescharacteristic
43#89700C2211yescharacteristic
44#89700C2211yescharacteristic
45#89431—21no—
46#89427—21no—
47#89425—21no—
48#89419—21no—
49C7×#336C2211yescharacteristic
50C7×#336C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#27034—71nomaximal, Sylow
60C23×C4×C7C2311yescharacteristic
61C23×Dic7—21no—
62C22×Dic14C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C2×C7×C22⋊C4D411yescharacteristic
68C2×C7×C22⋊C4D411yescharacteristic
69C2×C42×C7C2311yescharacteristic
70C2×C42×C7C2311yescharacteristic
71C2×C22⋊Dic7—21no—
72C2×C22⋊Dic7—21no—
73C2×C22⋊Dic7—21no—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—41no—
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86#6487—22no—
87#6487—22no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7—21no—
96#6442—44no—
97C4×Dic14—22no—
98C4×Dic14—41no—
99C7×C4⋊C8—22no—
100C7×C22⋊C8—22no—
101C7×C22⋊C8—41no—
102C4×C7×C8—41no—
103C7×#95D411yescharacteristic
104C7×#95D411yescharacteristic
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114#6391—22no—
115#6391—22no—
116#6391—22no—
117#6391—41no—
118#6390—22no—
119#6390—22no—
120#6390—22no—
121#6390—41no—
122#2496—71no—
123#2360—71no—
124#2072—71no—
125#2066—71no—
126#2035—71no—
127#2035—71no—
128#1948—71no—
129#1941—71no—
130#1923D711yescharacteristic
131#1920—71no—
132#1919—71no—
133#1865—71no—
134#1853—71no—
135#1824—71no—
136C4×#103—71no—
137C24×C7C2×D411yescharacteristic
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2411yescharacteristic
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—42no—
159C22×Dic7—42no—
160C22×Dic7—81no—
161C2×Dic14—21no—
162C2×Dic14—21no—
163C2×Dic14—21no—
164C2×Dic14—21no—
165C2×Dic14—21no—
166C2×Dic14#4112yes—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
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—21no—
178C2×C7×C8#4112yes—
179C7×C22⋊C4—41no—
180C7×C22⋊C4SD1614yes—
181C42×C7—21no—
182C42×C7—21no—
183C42×C7—21no—
184C42×C7—21no—
185C22⋊Dic7—24no—
186C22⋊Dic7—42no—
187C22⋊Dic7—42no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—42no—
201C4⋊Dic7—42no—
202C4⋊Dic7—42no—
203C4⋊Dic7#4112yes—
204C4⋊Dic7#4112yes—
205Dic7⋊C4—24no—
206Dic7⋊C4—42no—
207Dic7⋊C4—42no—
208Dic7⋊C4—42no—
209Dic7⋊C4—81no—
210C4×Dic7—42no—
211C4×Dic7—42no—
212C4×Dic7—81no—
213C2×#126—141no—
214C2×C22⋊Q8—71no—
215C2×C4×Q8—71no—
216C22×C4⋊C4—71no—
217C2×#106—71no—
218C2×#106—71no—
219C2×#106—71no—
220C2×#106—71no—
221C2×C4⋊C8C2×D711yescharacteristic
222C2×#103—71no—
223C2×#103—71no—
224C2×#103—71no—
225C2×#103—71no—
226C2×C22⋊C8C2×D711yescharacteristic
227C2×C22⋊C8C2×D711yescharacteristic
228C2×C4×C8C2×D711yescharacteristic
229#400—141no—
230#397—141no—
231#391—71no—
232#389—71no—
233#389—71no—
234#389—71no—
235#389—141no—
236#387—141no—
237#387—141no—
238#385—141no—
239#384—71no—
240#380—71no—
241#380—71no—
242C4×C4⋊C4—71no—
243C4×C4⋊C4—71no—
244C4×C22⋊C4—141no—
245C4×C22⋊C4C2×D711yescharacteristic
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#340—71no—
254#336C2×D711yescharacteristic
255#336C2×D711yescharacteristic
256C23×C7—21no—
257C23×C7—21no—
258C23×C7—21no—
259C23×C7—21no—
260C23×C7—21no—
261C23×C7—21no—
262C23×C7C2×SD1612yes—
263C23×C7C22×D411yescharacteristic
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7C2×SD1612yes—
282C2×C4×C7C2×SD1612yes—
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7C2×#4111yescharacteristic
285C2×C4×C7C2×#4111yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C2×C4×C7H(5,𝔽2)11yescharacteristic
288C2×C4×C72−1+411yescharacteristic
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—22no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—84no—
314Dic14—22no—
315Dic14—22no—
316Dic14—41no—
317Dic14—41no—
318C7×C8—22no—
319C7×C8—41no—
320C7×C8—41no—
321C7×C8—41no—
322C22×Q8—71no—
323C23×C4—141no—
324C23×C4C22×D711yescharacteristic
325C22×C8C22×D711yescharacteristic
326C22×C8C22×D711yescharacteristic
327C22×C8C22×D711yescharacteristic
328C22×C8C22×D711yescharacteristic
329#126—284no—
330C22⋊Q8—142no—
331C22⋊Q8—142no—
332C4×Q8—142no—
333C4×Q8—281no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—71no—
341C2×C4⋊C4—142no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—141no—
350C2×C4⋊C4—281no—
351C2×C22⋊C4—141no—
352C2×C22⋊C4—141no—
353C2×C22⋊C4—141no—
354C2×C22⋊C4D2811yescharacteristic
355C2×C22⋊C4D2811yescharacteristic
356C2×C42—141no—
357C2×C42—141no—
358C2×C42—141no—
359C2×C42C22×D711yescharacteristic
360C2×C42C22×D711yescharacteristic
361#106—142no—
362#106—142no—
363#106—142no—
364#106—281no—
365C4⋊C8—22no—
366#103—142no—
367#103—142no—
368#103—142no—
369#103—281no—
370C22⋊C8—22no—
371C22⋊C8—41no—
372C4×C8—41no—
373#95—141no—
374#95—141no—
375#95—141no—
376#95—141no—
377#95—141no—
378#95—141no—
379#95—141no—
380#95—141no—
381#95—141no—
382#95D2811yescharacteristic
383#95D2811yescharacteristic
384C22×C7—22no—
385C22×C7—22no—
386C22×C7—22no—
387C22×C7—41no—
388C22×C7—41no—
389C22×C7—41no—
390C22×C7—41no—
391C22×C7#52511yescharacteristic
392C22×C7#54611yescharacteristic
393C22×C7#54711yescharacteristic
394C22×C7C22×SD1611yescharacteristic
395C22×C7#57511yescharacteristic
396C22×C7#57711yescharacteristic
397C22×C7#57811yescharacteristic
398C4×C7—22no—
399C4×C7—22no—
400C4×C7—41no—
401C4×C7—41no—
402C4×C7—41no—
403C4×C7#52911yescharacteristic
404C4×C7#53911yescharacteristic
405C4×C7#55311yescharacteristic
406C4×C7#55611yescharacteristic
407Dic7—42no—
408Dic7—42no—
409Dic7—42no—
410Dic7—81no—
411Dic7—81no—
412Dic7—81no—
413Dic7—81no—
414Dic7—81no—
415C24C2×D2811yescharacteristic
416C2×Q8—72no—
417C2×Q8—141no—
418C2×Q8—141no—
419C2×Q8—141no—
420C2×Q8—141no—
421C2×Q8—141no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—141no—
429C22×C4—141no—
430C22×C4—141no—
431C22×C4—141no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4—141no—
436C22×C4—282no—
437C22×C4—282no—
438C22×C4—561no—
439C22×C4C2×D2811yescharacteristic
440C22×C4C2×D2811yescharacteristic
441C22×C4C2×D2811yescharacteristic
442C22×C4C2×D2811yescharacteristic
443C22×C4C2×D2811yescharacteristic
444C22×C4C23×D711yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8#115912yes—
457C4⋊C4—72no—
458C4⋊C4—72no—
459C4⋊C4—144no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—282no—
473C4⋊C4—282no—
474C4⋊C4—282no—
475C4⋊C4—282no—
476C4⋊C4—282no—
477C4⋊C4—282no—
478C4⋊C4—561no—
479C22⋊C4—41no—
480C22⋊C4—144no—
481C22⋊C4—282no—
482C22⋊C4—282no—
483C22⋊C4#113214yes—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—282no—
489C42—282no—
490C42—561no—
491C2×C7—42no—
492C2×C7#246611yescharacteristic
493C2×C7#310311yescharacteristic
494C2×C7#312011yescharacteristic
495C2×C7#329811yescharacteristic
496C2×C7#331111yescharacteristic
497C2×C7#333511yescharacteristic
498C2×C7#335211yescharacteristic
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23C2×#113212yes—
506C23C22×D2811yescharacteristic
507Q8—142no—
508Q8—142no—
509Q8—281no—
510Q8—281no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—142no—
529C2×C4—142no—
530C2×C4—142no—
531C2×C4—281no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—281no—
540C2×C4—281no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—281no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—564no—
553C2×C4C2×#113212yes—
554C2×C4C2×#113212yes—
555C2×C4C2×#115911yescharacteristic
556C2×C4#654811yescharacteristic
557C2×C4D7×#4111yescharacteristic
558C2×C4D7×#4111yescharacteristic
559C2×C4D7×#4111yescharacteristic
560C2×C4#655411yescharacteristic
561C8—22no—
562C8—41no—
563C8—41no—
564C8—41no—
565C7#2703411yescharacteristic, complemented, Sylow
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#9021511yescharacteristic
574C22#9024511yescharacteristic
575C22#9024611yescharacteristic
576C22C22×#113211yescharacteristic
577C22#9043911yescharacteristic
578C22#9044211yescharacteristic
579C22#9044411yescharacteristic
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—282no—
586C4—282no—
587C4—282no—
588C4—561no—
589C4—561no—
590C4—561no—
591C4—561no—
592C4—561no—
593C4#9029211yescharacteristic
594C4#9035111yescharacteristic
595C4#9037111yescharacteristic
596C4#9039311yescharacteristic
597C2—42no—
598C2#1173296111yescharacteristic
599C2#1173675411yescharacteristic
600C2#1173683011yescharacteristic
601C2#1173685711yescharacteristic
602C2#1173689211yescharacteristic
603C2#1173694711yescharacteristic
604C2#1173697511yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1923C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#1923D7
hypocenterC7#27034
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
4. #11728242C2semidirect
5. #11728208C2semidirect
10. #11727523C2semidirect
11. C4×#6390C2semidirect
12. #11726438C2semidirect
13. #11726390C2semidirect
14. #11726377C2semidirect
15. #11726376C2semidirect
565. C7#27034semidirect

Maximal subgroups

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

subgroupindex
#117309572
#117309102
C7×#19232
#117282422
#117282082
#117276902
#117276822
#117276792
#117275232
C4×#63902
#117264382
#117263902
#117263772
#117263762
#270347