Finite Group Databasebeta

← #13365957

Subgroups — #13365957

Order 1792.

784 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#478C211yesmaximal, characteristic
2Dic7×#125C211yesmaximal, characteristic, complemented
3#11739764C211yesmaximal, characteristic
4#11729492C211yesmaximal, characteristic, complemented
5#11729359C211yesmaximal, characteristic
6#11728367C211yesmaximal, characteristic, complemented
7#11728363C211yesmaximal, characteristic, complemented
8#11728316C211yesmaximal, characteristic, complemented
9#11728311C211yesmaximal, characteristic, complemented
10#11728209C211yesmaximal, characteristic, complemented
11#11728195C211yesmaximal, characteristic, complemented
12#11727941C211yesmaximal, characteristic
13#11727941C211yesmaximal, characteristic
14#11727910C211yesmaximal, characteristic
15#11727909C211yesmaximal, characteristic
16C2×C7×#125C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×#6489—21no—
19C7×#478C418yescomplemented
20C2×C7×#107C2211yescharacteristic
21C2×C7×#106C2211yescharacteristic
22C2×C7×C4⋊C8C2211yescharacteristic
23C2×C7×#102C2211yescharacteristic
24C2×C7×#102C2211yescharacteristic
25#89993C2211yescharacteristic
26#89993C2211yescharacteristic
27C2×#6406C2211yescharacteristic
28C2×#6406C2211yescharacteristic
29C4×C22⋊Dic7—21no—
30C2×#6392C2211yescharacteristic
31C2×#6391C2211yescharacteristic
32C2×#6389C2211yescharacteristic
33C2×#6389C2211yescharacteristic
34#89757C2211yescharacteristic
35#89757C2211yescharacteristic
36#89757C2211yescharacteristic
37#89757C2211yescharacteristic
38#89753—21no—
39#89753—21no—
40C4⋊C4×Dic7—21no—
41C4⋊C4×Dic7—21no—
42C4⋊C4×Dic7C2211yescharacteristic
43C4⋊C4×Dic7C2211yescharacteristic
44#89723—21no—
45#89723—21no—
46#89708C2211yescharacteristic
47C4×C4⋊Dic7C2211yescharacteristic
48C42×Dic7C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89345C2211yescharacteristic
53#89345C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22120—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C7×#125—22no—
66C7×#125C2×C414yes—
67C4×C7×D4—22no—
68C4×C7×D4C2×C414yes—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C22⋊C4—21no—
75C2×C42×C7C2311yescharacteristic
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—21no—
78C2×C22⋊Dic7—21no—
79C2×C22⋊Dic7—21no—
80#6489—44no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7D411yescharacteristic
93C2×C4×Dic7D411yescharacteristic
94C2×C4×Dic7D411yescharacteristic
95C2×C4×Dic7D411yescharacteristic
96C2×C4×Dic7C2311yescharacteristic
97C2×C4×Dic7C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C7×#107C2×C414yes—
101C7×#106C2×C414yes—
102C7×C4⋊C8C2×C414yes—
103C7×#102C2×C414yes—
104C7×#102C2×C414yes—
105#6406—41no—
106#6406—41no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116#6392—41no—
117#6391—41no—
118#6389—41no—
119#6389—41no—
120C4⋊(C7⋊C8)—22no—
121C2×#478D711yescharacteristic
122C4×#125—71no—
123#2293—71no—
124#1989—71no—
125#1949—71no—
126#1944—71no—
127#1943—71no—
128#1936—71no—
129#1936—71no—
130#1926—71no—
131#1925—71no—
132#1919—71no—
133#1918—71no—
134#1804—71no—
135#1767—71no—
136C24×C7—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4C22×C412yes—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—24no—
147C22×C4×C7—42no—
148C22×C4×C7—42no—
149C22×C4×C7C2×D411yescharacteristic
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×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8C22×C412yes—
165C2×C7×C8C22×C412yes—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—24no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4C22×C412yes—
172C7×C4⋊C4C22×C412yes—
173C7×C4⋊C4C22×C412yes—
174C7×C22⋊C4—24no—
175C42×C7C22×C412yes—
176C42×C7#4112yes—
177C22⋊Dic7—42no—
178C22⋊Dic7—42no—
179C22⋊Dic7—42no—
180C22⋊Dic7—42no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—42no—
190C4⋊Dic7—42no—
191C4⋊Dic7—42no—
192C4⋊Dic7—42no—
193C4×Dic7—21no—
194C4×Dic7—21no—
195C4×Dic7—21no—
196C4×Dic7—21no—
197C4×Dic7—22no—
198C4×Dic7—41no—
199C4×Dic7—41no—
200C4×Dic7—41no—
201C4×Dic7—42no—
202C4×Dic7—42no—
203C4×Dic7—42no—
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×#125C2×D711yescharacteristic
211C2×C4×D4C2×D711yescharacteristic
212C2×#117—141no—
213#478Dic718yescomplemented
214C2×#107—71no—
215C2×#107C2×D711yescharacteristic
216C2×#106—71no—
217C2×#106C2×D711yescharacteristic
218C2×C4⋊C8—71no—
219C2×C4⋊C8C2×D711yescharacteristic
220C2×#102—71no—
221C2×#102—71no—
222C2×#102C2×D711yescharacteristic
223C2×#102C2×D711yescharacteristic
224C2×#97—71no—
225C2×#97—71no—
226#390—71no—
227#390—71no—
228#389—71no—
229#389—71no—
230#389—71no—
231#389—71no—
232#388—141no—
233#388—141no—
234#387—141no—
235#387—141no—
236#382—71no—
237C4×C4⋊C4—71no—
238C4×C4⋊C4—71no—
239C4×C4⋊C4—71no—
240C4×C4⋊C4—141no—
241C4×C4⋊C4—141no—
242C4×C22⋊C4—141no—
243C43—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340—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×D4—22no—
263C7×D4—22no—
264C7×D4—41no—
265C7×D4—41no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—48no—
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×C7C4×D412yes—
281C2×C4×C7C4×D412yes—
282C2×C4×C7C23×C411yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7H(5,𝔽2)11yescharacteristic
285C2×C4×C72−1+411yescharacteristic
286C2×C4×C72−1+411yescharacteristic
287C2×C4×C72−1+411yescharacteristic
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
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—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×Dic7AGL(1,ℤ/8)12yes—
312C7×C8—22no—
313C7×C8—22no—
314C7⋊C8—41no—
315C7⋊C8—41no—
316C22×D4C22×D711yescharacteristic
317C23×C4—21no—
318C22×C8—71no—
319C22×C8—71no—
320C22×C8C22×D711yescharacteristic
321C22×C8C22×D711yescharacteristic
322#125—22no—
323#125C2×Dic714yes—
324C4×D4—22no—
325C4×D4C2×Dic714yes—
326#117—284no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C4⋊C4C22×D711yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—141no—
342C2×C22⋊C4—141no—
343C2×C22⋊C4—141no—
344C2×C22⋊C4—141no—
345C2×C42—71no—
346C2×C42—71no—
347C2×C42—71no—
348C2×C42—71no—
349C2×C42—71no—
350C2×C42—71no—
351C2×C42—141no—
352C2×C42—141no—
353C2×C42—141no—
354C2×C42C22×D711yescharacteristic
355#107—281no—
356#107C2×Dic714yes—
357#106—281no—
358#106C2×Dic714yes—
359C4⋊C8—142no—
360C4⋊C8C2×Dic714yes—
361#102—281no—
362#102—281no—
363#102C2×Dic714yes—
364#102C2×Dic714yes—
365#97—281no—
366#97—281no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95—141no—
376C22×C7—42no—
377C22×C7—42no—
378C22×C7—42no—
379C22×C7—81no—
380C22×C7—81no—
381C22×C7—81no—
382C22×C7—81no—
383C22×C7C2×C4×D411yescharacteristic
384C22×C7#53611yescharacteristic
385C22×C7#53711yescharacteristic
386C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
387C22×C7#57511yescharacteristic
388C22×C7#57511yescharacteristic
389C22×C7#57511yescharacteristic
390C4×C7—22no—
391C4×C7—22no—
392C4×C7—42no—
393C4×C7—42no—
394C4×C7—42no—
395C4×C7#51911yescharacteristic
396C4×C7#52011yescharacteristic
397C4×C7#54111yescharacteristic
398C4×C7#54211yescharacteristic
399Dic7—22no—
400Dic7—41no—
401Dic7—41no—
402Dic7—41no—
403Dic7—81no—
404Dic7—81no—
405Dic7—81no—
406Dic7—81no—
407C24—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4C22×Dic712yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—71no—
421C22×C4—71no—
422C22×C4—71no—
423C22×C4—71no—
424C22×C4—141no—
425C22×C4—141no—
426C22×C4—141no—
427C22×C4—141no—
428C22×C4D4×D711yescharacteristic
429C22×C4D4×D711yescharacteristic
430C22×C4C23×D711yescharacteristic
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8—141no—
441C2×C8C22×Dic712yes—
442C2×C8C22×Dic712yes—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—282no—
457C4⋊C4—282no—
458C4⋊C4—282no—
459C4⋊C4—282no—
460C4⋊C4C22×Dic712yes—
461C4⋊C4C22×Dic712yes—
462C4⋊C4C22×Dic712yes—
463C22⋊C4—24no—
464C22⋊C4—282no—
465C22⋊C4—282no—
466C22⋊C4—282no—
467C22⋊C4—282no—
468C42—142no—
469C42—141no—
470C42—141no—
471C42—141no—
472C42—141no—
473C42—281no—
474C42—281no—
475C42—281no—
476C42—282no—
477C42—282no—
478C42—282no—
479C42#115912yes—
480C42C22×Dic712yes—
481C2×C7—82no—
482C2×C7#232311yescharacteristic
483C2×C7#294311yescharacteristic
484C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
485C2×C7#323411yescharacteristic
486C2×C7#324011yescharacteristic
487C2×C7#325111yescharacteristic
488C2×C7#325611yescharacteristic
489C23—22no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23C2×D4×D711yescharacteristic
497D4—22no—
498D4—22no—
499D4—41no—
500D4—41no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—48no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—72no—
516C2×C4—141no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—141no—
524C2×C4—141no—
525C2×C4—141no—
526C2×C4—141no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
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×C4D4×Dic712yes—
540C2×C4D4×Dic712yes—
541C2×C4C2×#115911yescharacteristic
542C2×C4#655111yescharacteristic
543C2×C4#655111yescharacteristic
544C2×C4#655311yescharacteristic
545C2×C4#655411yescharacteristic
546C2×C4C23×Dic711yescharacteristic
547C8—22no—
548C8—22no—
549C8—281no—
550C8—281no—
551C7#2212011yescharacteristic, complemented, Sylow
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22#9031911yescharacteristic
560C22#9032011yescharacteristic
561C22#9045911yescharacteristic
562C22#9045911yescharacteristic
563C22D7×AGL(1,ℤ/8)11yescharacteristic
564C22#9046411yescharacteristic
565C22C2×D4×Dic711yescharacteristic
566C4—22no—
567C4—22no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—142no—
572C4—281no—
573C4—281no—
574C4—281no—
575C4—561no—
576C4—561no—
577C4—561no—
578C4—561no—
579C4#9031611yescharacteristic
580C4#9033411yescharacteristic
581C4#9050311yescharacteristic
582C4#9051811yescharacteristic
583C2—82no—
584C2#1173859411yescharacteristic
585C2#1173861911yescharacteristic
586C2#1173862011yescharacteristic
587C2#1173862611yescharacteristic
588C2#1174018911yescharacteristic
589C2#1174134211yescharacteristic
590C2Dic7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#478C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC2×#478D7
hypocenterC7#22120
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
2. Dic7×#125C2semidirect
4. #11729492C2semidirect
6. #11728367C2semidirect
7. #11728363C2semidirect
8. #11728316C2semidirect
9. #11728311C2semidirect
10. #11728209C2semidirect
11. #11728195C2semidirect
19. C7×#478C4semidirect
213. #478Dic7semidirect
551. C7#22120semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4782
Dic7×#1252
#117397642
#117294922
#117293592
#117283672
#117283632
#117283162
#117283112
#117282092
#117281952
#117279412
#117279102
#117279092
#221207