Finite Group Databasebeta

← #13426557

Subgroups — #13426557

Order 1792.

700 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
1#11739607C211yesmaximal, characteristic
2#11731363C211yesmaximal, characteristic
3C7×#1823C211yesmaximal, characteristic
4#11729310C211yesmaximal, characteristic
5#11727691C211yesmaximal, characteristic
6#11727666C211yesmaximal, characteristic
7#11727280C211yesmaximal, characteristic
8#11726669C211yesmaximal, characteristic, complemented
9#11726666C211yesmaximal, characteristic, complemented
10#11726661C211yesmaximal, characteristic, complemented
11#11726612C211yesmaximal, characteristic, complemented
12#11726574C211yesmaximal, characteristic, complemented
13#11726552C211yesmaximal, characteristic, complemented
14#11726543C211yesmaximal, characteristic, complemented
15#11726535C211yesmaximal, characteristic, complemented
16C22×C7×C4⋊C4C2211yescharacteristic
17C2×#6487C2211yescharacteristic
18C2×#6450C2211yescharacteristic
19C2×C7×C22⋊C8C2211yescharacteristic
20C7×#380C2211yescharacteristic
21#89984C2211yescharacteristic
22#89982—21no—
23C4×C22⋊Dic7—21no—
24C4×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊(C7⋊C8)C2211yescharacteristic
26C2×#6392C2211yescharacteristic
27C2×#6390C2211yescharacteristic
28C2×#6390C2211yescharacteristic
29C2×Dic7⋊C8C2211yescharacteristic
30C2×C8×Dic7C2211yescharacteristic
31#89757C2211yescharacteristic
32#89750—21no—
33#89749C2211yescharacteristic
34C4⋊C4×Dic7C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36#89746C2211yescharacteristic
37#89745C2211yescharacteristic
38#89745C2211yescharacteristic
39#89744—21no—
40C2×#6384C2211yescharacteristic
41C2×#6384C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6382C2211yescharacteristic
45#89724—21no—
46#89723—21no—
47C7⋊#401—21no—
48#89425—21no—
49C7×#340C2211yescharacteristic
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52C7×#340C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#27037—71nomaximal, Sylow
60C23×C4×C7—21no—
61C23×C4×C7C2311yescharacteristic
62C22×Dic14C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—41no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C22⋊Dic7—21no—
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7D411yescharacteristic
76C2×C22⋊Dic7D411yescharacteristic
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81#6487—22no—
82#6487—22no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7C2311yescharacteristic
91C2×C4×Dic7C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94#6450—22no—
95#6450—41no—
96C7×C22⋊C8—41no—
97C7×#95—21no—
98C7⋊#95—21no—
99C7⋊#95—21no—
100C7⋊#95—21no—
101C7⋊#95—21no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95D411yescharacteristic
109C7⋊#95D411yescharacteristic
110C22⋊(C7⋊C8)—22no—
111#6392—22no—
112#6390—22no—
113#6390—41no—
114Dic7⋊C8—22no—
115C8×Dic7—41no—
116#6384—22no—
117#6384—22no—
118#6382—22no—
119#6382—22no—
120#6382—41no—
121#2494—71no—
122#2360—71no—
123#2074—71no—
124#2066—71no—
125#2036—71no—
126#2035—71no—
127#1949—71no—
128#1941—71no—
129#1923—71no—
130#1921—71no—
131#1920—71no—
132#1865—71no—
133#1853—71no—
134#1823D711yescharacteristic
135C4×#103—71no—
136C24×C7C2×D411yescharacteristic
137C22×C4×C7—21no—
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×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—42no—
148C22×C4×C7—42no—
149C22×C4×C7—81no—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2411yescharacteristic
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7C2×D411yescharacteristic
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C2×Dic14—21no—
161C2×Dic14—21no—
162C2×Dic14—21no—
163C2×Dic14—21no—
164C2×Dic14—21no—
165C2×Dic14#4112yes—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—42no—
180C7×C4⋊C4#4112yes—
181C7×C22⋊C4—42no—
182C22⋊Dic7—24no—
183C22⋊Dic7—22no—
184C22⋊Dic7—22no—
185C22⋊Dic7—81no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—42no—
192C4⋊Dic7#4112yes—
193Dic7⋊C4—24no—
194Dic7⋊C4—42no—
195Dic7⋊C4—81no—
196Dic7⋊C4—81no—
197C4×Dic7—21no—
198C4×Dic7—21no—
199C4×Dic7—21no—
200C4×Dic7—21no—
201C4×Dic7—42no—
202C4×Dic7—81no—
203C4×Dic7—81no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8#4112yes—
210C2×C22⋊Q8—71no—
211C2×C4×Q8—71no—
212C22×C4⋊C4C2×D711yescharacteristic
213C2×#107—71no—
214C2×#106—71no—
215C2×#106—71no—
216C2×#106—71no—
217C2×C4⋊C8—71no—
218C2×#103—71no—
219C2×#103—71no—
220C2×#103—71no—
221C2×#103—71no—
222C2×C22⋊C8—71no—
223C2×C22⋊C8C2×D711yescharacteristic
224C2×C4×C8—71no—
225#401—141no—
226#397—141no—
227#391—71no—
228#389—71no—
229#389—71no—
230#389—71no—
231#389—141no—
232#388—141no—
233#387—141no—
234#387—141no—
235#385—141no—
236#382—71no—
237#380—71no—
238#380C2×D711yescharacteristic
239C4×C4⋊C4—71no—
240C4×C4⋊C4—71no—
241C4×C22⋊C4—71no—
242C4×C22⋊C4—141no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340C2×D711yescharacteristic
248#340C2×D711yescharacteristic
249#340C2×D711yescharacteristic
250#340C2×D711yescharacteristic
251#336—71no—
252#336—71no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7—21no—
259C23×C7#13512yes—
260C23×C7C22×D411yescharacteristic
261C2×C4×C7—21no—
262C2×C4×C7—22no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—84no—
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7C2×#4111yescharacteristic
285C2×C4×C7H(5,𝔽2)11yescharacteristic
286C2×C4×C72−1+411yescharacteristic
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—21no—
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#13712yes—
311Dic14—22no—
312Dic14—22no—
313Dic14—41no—
314Dic14—41no—
315C7×C8—41no—
316C7×C8—41no—
317C7⋊C8—22no—
318C7⋊C8—41no—
319C22×Q8—71no—
320C23×C4—21no—
321C23×C4C22×D711yescharacteristic
322C22×C8—71no—
323C22×C8—71no—
324C22×C8C22×D711yescharacteristic
325C22×C8C22×D711yescharacteristic
326C22⋊Q8—142no—
327C22⋊Q8—142no—
328C4×Q8—142no—
329C4×Q8—281no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—41no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C4⋊C4C22×D711yescharacteristic
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—71no—
347C2×C22⋊C4—71no—
348C2×C22⋊C4—141no—
349C2×C22⋊C4—141no—
350C2×C42—71no—
351C2×C42—71no—
352C2×C42—141no—
353C2×C42—141no—
354C2×C42—141no—
355#107—142no—
356#106—142no—
357#106—142no—
358#106—281no—
359C4⋊C8—142no—
360#103—142no—
361#103—142no—
362#103—142no—
363#103—281no—
364C22⋊C8—41no—
365C22⋊C8—142no—
366C4×C8—281no—
367#95—21no—
368#95—71no—
369#95—71no—
370#95—141no—
371#95—141no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95—141no—
376#95—141no—
377#95—141no—
378#95—141no—
379#95—141no—
380C22×C7—22no—
381C22×C7—22no—
382C22×C7—22no—
383C22×C7—41no—
384C22×C7—41no—
385C22×C7—41no—
386C22×C7—41no—
387C22×C7#52511yescharacteristic
388C22×C7#54611yescharacteristic
389C22×C7#54711yescharacteristic
390C22×C7C2×#13511yescharacteristic
391C22×C7C2×#13711yescharacteristic
392C22×C7#57711yescharacteristic
393C22×C7#57711yescharacteristic
394C4×C7—41no—
395C4×C7—42no—
396C4×C7—81no—
397C4×C7—81no—
398C4×C7—81no—
399C4×C7#52911yescharacteristic
400C4×C7#53911yescharacteristic
401C4×C7#55311yescharacteristic
402C4×C7#55611yescharacteristic
403Dic7—22no—
404Dic7—41no—
405Dic7—41no—
406Dic7—42no—
407Dic7—42no—
408Dic7—41no—
409Dic7—81no—
410Dic7—81no—
411C24D4×D711yescharacteristic
412C2×Q8—72no—
413C2×Q8—141no—
414C2×Q8—141no—
415C2×Q8—141no—
416C2×Q8—141no—
417C2×Q8—141no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—71no—
431C22×C4—71no—
432C22×C4—71no—
433C22×C4—71no—
434C22×C4—81no—
435C22×C4—141no—
436C22×C4—141no—
437C22×C4—141no—
438C22×C4—141no—
439C22×C4D4×D711yescharacteristic
440C22×C4C23×D711yescharacteristic
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—72no—
448C2×C8—141no—
449C2×C8—141no—
450C2×C8—141no—
451C2×C8—141no—
452C2×C8—141no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—72no—
462C4⋊C4—144no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—282no—
469C4⋊C4—282no—
470C4⋊C4—561no—
471C4⋊C4—561no—
472C4⋊C4#115712yes—
473C22⋊C4—42no—
474C22⋊C4—142no—
475C22⋊C4—144no—
476C22⋊C4—142no—
477C22⋊C4—561no—
478C42—141no—
479C42—141no—
480C42—141no—
481C42—141no—
482C42—282no—
483C42—561no—
484C42—561no—
485C2×C7—42no—
486C2×C7#246611yescharacteristic
487C2×C7#310811yescharacteristic
488C2×C7#312511yescharacteristic
489C2×C7#329611yescharacteristic
490C2×C7#332011yescharacteristic
491C2×C7#333711yescharacteristic
492C2×C7#335611yescharacteristic
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23#647512yes—
500C23C2×D4×D711yescharacteristic
501Q8—142no—
502Q8—142no—
503Q8—281no—
504Q8—281no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—72no—
525C2×C4—84no—
526C2×C4—141no—
527C2×C4—141no—
528C2×C4—141no—
529C2×C4—141no—
530C2×C4—141no—
531C2×C4—141no—
532C2×C4—141no—
533C2×C4—141no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—142no—
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×C4C2×#115711yescharacteristic
550C2×C4#654811yescharacteristic
551C2×C4D7×#4111yescharacteristic
552C2×C4D7×#4111yescharacteristic
553C2×C4D7×#4111yescharacteristic
554C2×C4#655411yescharacteristic
555C8—41no—
556C8—41no—
557C8—142no—
558C8—281no—
559C7#2703711yescharacteristic, complemented, Sylow
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#9018111yescharacteristic
568C22#9034611yescharacteristic
569C22#9034711yescharacteristic
570C22C2×#647511yescharacteristic
571C22#9045311yescharacteristic
572C22D7×#13711yescharacteristic
573C22#9046911yescharacteristic
574C4—41no—
575C4—42no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—142no—
580C4—281no—
581C4—282no—
582C4—281no—
583C4—281no—
584C4—282no—
585C4—561no—
586C4—561no—
587C4#9021911yescharacteristic
588C4#9022411yescharacteristic
589C4#9035111yescharacteristic
590C4#9035211yescharacteristic
591C2—42no—
592C2#1173151611yescharacteristic
593C2#1173620411yescharacteristic
594C2#1173641311yescharacteristic
595C2#1173871411yescharacteristic
596C2#1173873711yescharacteristic
597C2#1173886211yescharacteristic
598C2#1173888611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1823C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#1823D7
hypocenterC7#27037
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
8. #11726669C2semidirect
9. #11726666C2semidirect
10. #11726661C2semidirect
11. #11726612C2semidirect
12. #11726574C2semidirect
13. #11726552C2semidirect
14. #11726543C2semidirect
15. #11726535C2semidirect
559. C7#27037semidirect

Maximal subgroups

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

subgroupindex
#117396072
#117313632
C7×#18232
#117293102
#117276912
#117276662
#117272802
#117266692
#117266662
#117266612
#117266122
#117265742
#117265522
#117265432
#117265352
#270377