Finite Group Databasebeta

← #13422421

Subgroups — #13422421

Order 1792.

704 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2367C211yesmaximal, characteristic
2#11731131C211yesmaximal, characteristic, complemented
3#11729476C211yesmaximal, characteristic
4#11729476C211yesmaximal, characteristic
5#11729383C211yesmaximal, characteristic, complemented
6#11729381C211yesmaximal, characteristic, complemented
7C4×#6406C211yesmaximal, characteristic
8#11729361C211yesmaximal, characteristic
9#11727428C211yesmaximal, characteristic, complemented
10#11727420C211yesmaximal, characteristic, complemented
11#11727406C211yesmaximal, characteristic
12#11727395C211yesmaximal, characteristic
13#11727236C211yesmaximal, characteristic, complemented
14#11727214C211yesmaximal, characteristic, complemented
15#11727213C211yesmaximal, characteristic, complemented
16C2×C7×#124C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C7×#391C2211yescharacteristic
20C7×#390C2211yescharacteristic
21C7×#388—21no—
22C7×#388—21no—
23C7×#385—21no—
24C4×C7×C4⋊C4C2211yescharacteristic
25C4×C7×C4⋊C4C2211yescharacteristic
26C4×C7×C22⋊C4—21no—
27C2×#6409C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35#89752—21no—
36#89750—21no—
37C2×#6382C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39#89708C2211yescharacteristic
40C4×C4⋊Dic7C2211yescharacteristic
41#89700C2211yescharacteristic
42#89700C2211yescharacteristic
43#89700C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89430—21no—
48#89422—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89319C2211yescharacteristic
56#89319C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26760—71nomaximal, Sylow
60C22×C7×Q8C2311yescharacteristic
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C7×#124—22no—
64C7×#124—22no—
65C4×C7×Q8—22no—
66C4×C7×Q8—41no—
67C4×C7×D4—22no—
68C4×C7×D4—41no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C42×C7—21no—
80C2×C42×C7—21no—
81C2×C42×C7C2311yescharacteristic
82C2×C42×C7C2311yescharacteristic
83C2×C42×C7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×C4×Dic7—21no—
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C7×#95—21no—
98C7×#95—21no—
99C7×#95—21no—
100C7×#95D411yescharacteristic
101C7×#95D411yescharacteristic
102#6409—22no—
103#6409—22no—
104#6409—22no—
105#6409—41no—
106#6406—22no—
107#6406—22no—
108#6406—22no—
109#6406—41no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117#6382—22no—
118#6382—22no—
119C4⋊(C7⋊C8)—41no—
120C4×C7⋊C8—41no—
121#2508—71no—
122#2367D711yescharacteristic
123#2073—71no—
124#2065—71no—
125#2042—71no—
126#2042—71no—
127#1933—71no—
128#1927—71no—
129#1920—71no—
130#1919—71no—
131#1916—71no—
132#1865—71no—
133#1864—71no—
134#1805—71no—
135C4×#102—71no—
136C24×C7—21no—
137C2×C7×Q8—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—21no—
141C2×C7×Q8—21no—
142C2×C7×Q8#4112yes—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4#4112yes—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—42no—
153C22×C4×C7—42no—
154C22×C4×C7—81no—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2411yescharacteristic
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—41no—
171C7×C4⋊C4—42no—
172C7×C4⋊C4#4112yes—
173C7×C22⋊C4—24no—
174C7×C22⋊C4—24no—
175C7×C22⋊C4—42no—
176C7×C22⋊C4—42no—
177C7×C22⋊C4—81no—
178C42×C7—21no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—21no—
182C42×C7—21no—
183C42×C7—21no—
184C42×C7—42no—
185C42×C7—81no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7#4112yes—
194Dic7⋊C4—42no—
195Dic7⋊C4—81no—
196Dic7⋊C4—81no—
197Dic7⋊C4—81no—
198C4×Dic7—42no—
199C2×C7⋊C8—21no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×#124C2×D711yescharacteristic
212C2×C4×Q8C2×D711yescharacteristic
213C2×C4×D4C2×D711yescharacteristic
214C2×#106—71no—
215C2×#106—71no—
216C2×C4⋊C8—71no—
217C2×#103—71no—
218C2×#103—71no—
219C2×#103—71no—
220C2×#103—71no—
221C2×#102—71no—
222C2×#102—71no—
223C2×#102—71no—
224C2×#102—71no—
225C2×C4×C8—71no—
226#400—141no—
227#398—141no—
228#391C2×D711yescharacteristic
229#390C2×D711yescharacteristic
230#389—71no—
231#389—71no—
232#389—71no—
233#389—71no—
234#388—21no—
235#388—21no—
236#387—141no—
237#385—21no—
238#382—71no—
239#382—141no—
240C4×C4⋊C4—71no—
241C4×C4⋊C4C2×D711yescharacteristic
242C4×C4⋊C4C2×D711yescharacteristic
243C4×C22⋊C4—21no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#336—71no—
251#336—71no—
252#336—71no—
253#336—71no—
254C23×C7—22no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—41no—
258C23×C7—41no—
259C23×C7—41no—
260C23×C7—41no—
261C23×C7C22×D411yescharacteristic
262C7×Q8—22no—
263C7×Q8—22no—
264C7×Q8—41no—
265C7×Q8—41no—
266C7×D4—22no—
267C7×D4—22no—
268C7×D4—41no—
269C7×D4—41no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—22no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—22no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—84no—
297C2×C4×C7#13512yes—
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C2×C4×C7C2×#4111yescharacteristic
302C2×C4×C7H(5,𝔽2)11yescharacteristic
303C2×C4×C72−1+411yescharacteristic
304C2×Dic7—22no—
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—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C7⋊C8—41no—
317C7⋊C8—41no—
318C7⋊C8—41no—
319C7⋊C8—41no—
320C22×Q8C22×D711yescharacteristic
321C22×D4C22×D711yescharacteristic
322C23×C4—21no—
323C22×C8—71no—
324C22×C8—71no—
325C22×C8—71no—
326C22×C8—71no—
327#124—22no—
328#124—22no—
329C4×Q8—22no—
330C4×Q8—41no—
331C4×D4—22no—
332C4×D4—41no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4C22⋊D711yescharacteristic
343C2×C4⋊C4C22⋊D711yescharacteristic
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C4⋊C4C22×D711yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—141no—
354C2×C42C22×D711yescharacteristic
355C2×C42C22×D711yescharacteristic
356C2×C42C22×D711yescharacteristic
357#106—142no—
358#106—142no—
359C4⋊C8—281no—
360#103—142no—
361#103—142no—
362#103—142no—
363#103—281no—
364#102—142no—
365#102—142no—
366#102—142no—
367#102—281no—
368C4×C8—281no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95—141no—
376#95—141no—
377#95—141no—
378#95—141no—
379#95C22⋊D711yescharacteristic
380#95C22⋊D711yescharacteristic
381C22×C7—42no—
382C22×C7—42no—
383C22×C7—42no—
384C22×C7—81no—
385C22×C7—81no—
386C22×C7—81no—
387C22×C7—81no—
388C22×C7#52511yescharacteristic
389C22×C7#54611yescharacteristic
390C22×C7#54811yescharacteristic
391C22×C7C2×#13511yescharacteristic
392C22×C7#57511yescharacteristic
393C22×C7#57711yescharacteristic
394C22×C7#57711yescharacteristic
395C4×C7—22no—
396C4×C7—41no—
397C4×C7—41no—
398C4×C7—42no—
399C4×C7—41no—
400C4×C7—42no—
401C4×C7—41no—
402C4×C7—41no—
403C4×C7—81no—
404C4×C7#52911yescharacteristic
405C4×C7#53911yescharacteristic
406C4×C7#55211yescharacteristic
407C4×C7#55311yescharacteristic
408Dic7—42no—
409Dic7—81no—
410Dic7—81no—
411Dic7—81no—
412C24—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8#115712yes—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4#115712yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—81no—
431C22×C4—141no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4C2×C22⋊D711yescharacteristic
436C22×C4C2×C22⋊D711yescharacteristic
437C22×C4C2×C22⋊D711yescharacteristic
438C22×C4C2×C22⋊D711yescharacteristic
439C22×C4C2×C22⋊D711yescharacteristic
440C22×C4C2×C22⋊D711yescharacteristic
441C22×C4C23×D711yescharacteristic
442C2×C8—141no—
443C2×C8—141no—
444C2×C8—141no—
445C2×C8—141no—
446C2×C8—141no—
447C2×C8—141no—
448C2×C8—141no—
449C2×C8—141no—
450C2×C8—141no—
451C2×C8—141no—
452C2×C8—141no—
453C2×C8—141no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—42no—
460C4⋊C4—72no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—282no—
469C4⋊C4—561no—
470C4⋊C4—561no—
471C4⋊C4—561no—
472C4⋊C4#115912yes—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—81no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—42no—
485C42—81no—
486C42—282no—
487C2×C7—82no—
488C2×C7#246011yescharacteristic
489C2×C7#310211yescharacteristic
490C2×C7#312511yescharacteristic
491C2×C7#329611yescharacteristic
492C2×C7#332111yescharacteristic
493C2×C7#336011yescharacteristic
494C2×C7#337111yescharacteristic
495C23—22no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23C22×C22⋊D711yescharacteristic
503Q8—22no—
504Q8—22no—
505Q8—41no—
506Q8—41no—
507D4—22no—
508D4—22no—
509D4—41no—
510D4—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—142no—
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#651312yes—
551C2×C4C2×#115711yescharacteristic
552C2×C4C2×#115711yescharacteristic
553C2×C4C2×#115911yescharacteristic
554C2×C4#654811yescharacteristic
555C2×C4D7×#4111yescharacteristic
556C2×C4#655411yescharacteristic
557C8—281no—
558C8—281no—
559C8—281no—
560C8—281no—
561C7#2676011yescharacteristic, complemented, Sylow
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22#9048211yescharacteristic
570C22#9049711yescharacteristic
571C22C2×#651311yescharacteristic
572C22#9051311yescharacteristic
573C22#9052211yescharacteristic
574C22#9052711yescharacteristic
575C22#9052811yescharacteristic
576C4—22no—
577C4—41no—
578C4—42no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—42no—
584C4—81no—
585C4—282no—
586C4—561no—
587C4—561no—
588C4—561no—
589C4#9022911yescharacteristic
590C4#9023111yescharacteristic
591C4#9023211yescharacteristic
592C4#9025511yescharacteristic
593C2—82no—
594C2#1173981111yescharacteristic
595C2#1173997311yescharacteristic
596C2#1174002111yescharacteristic
597C2#1174093911yescharacteristic
598C2#1174098411yescharacteristic
599C2#1174111211yescharacteristic
600C2#1174116311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2367C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2367D7
hypocenterC7#26760
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
2. #11731131C2semidirect
5. #11729383C2semidirect
6. #11729381C2semidirect
9. #11727428C2semidirect
10. #11727420C2semidirect
13. #11727236C2semidirect
14. #11727214C2semidirect
15. #11727213C2semidirect
561. C7#26760semidirect

Maximal subgroups

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

subgroupindex
C7×#23672
#117311312
#117294762
#117293832
#117293812
C4×#64062
#117293612
#117274282
#117274202
#117274062
#117273952
#117272362
#117272142
#117272132
#267607