Finite Group Databasebeta

← #13364941

Subgroups — #13364941

Order 1792.

1092 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#448C211yesmaximal, characteristic
2C2×#90002C211yesmaximal, characteristic, complemented
3Dic7×#121C211yesmaximal, characteristic
4#11729886C211yesmaximal, characteristic, complemented
5#11729803C212yesmaximal, complemented
6#11729801C212yesmaximal, complemented
7#11729431C211yesmaximal, characteristic
8#11728007C212yesmaximal, complemented
9#11727977C212yesmaximal, complemented
10#11727682C211yesmaximal, characteristic, complemented
11Dic7×C22⋊C8C211yesmaximal, characteristic, complemented
12C23×C7×Q8C2211yescharacteristic
13C22×C7×SD16C2212yes—
14C2×C7×#121C2211yescharacteristic
15C2×D4×Dic7—21no—
16C2×D4×Dic7C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C22×C4×Dic7C2211yescharacteristic
19C7×#448C418yescomplemented
20C2×C7×#103C2212yes—
21C2×C7×C22⋊C8C2211yescharacteristic
22#90002—21no—
23#90002—21no—
24#90002C2212yes—
25#90002C2212yes—
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28#89993C2211yescharacteristic
29#89992—21no—
30C2×#6406C2212yes—
31#89984C2211yescharacteristic
32C2×C7⋊#95—21no—
33C2×C7⋊#95—21no—
34C2×C22⋊(C7⋊C8)C2211yescharacteristic
35C2×#6391C2212yes—
36C2×C8×Dic7C2212yes—
37#89757C2211yescharacteristic
38C4⋊C4×Dic7C2211yescharacteristic
39#89723—21no—
40C22⋊C4×Dic7—21no—
41#89346C2212yes—
42#89345C2212yes—
43#89324C2212yes—
44#22041—71nomaximal, Sylow
45C22×C7×Q8—21no—
46C22×C7×Q8—22no—
47C22×C7×Q8—42no—
48C22×C7×Q8—42no—
49C22×C7×Q8C2×C414yescomplemented
50C22×C7×Q8C2312yes—
51C22×C7×D4—21no—
52C22×C7×D4C2311yescharacteristic
53C23×C4×C7—21no—
54C23×C4×C7C2311yescharacteristic
55C23×Dic7—21no—
56C23×Dic7—21no—
57C23×Dic7D411yescharacteristic
58C23×Dic7D411yescharacteristic
59C2×C7×SD16—42no—
60C2×C7×SD16—42no—
61C2×C7×SD16C2×C418yescomplemented
62C22×C7×C8C2312yes—
63C7×#121—22no—
64C7×#121C2×C414yes—
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C22⋊C4—21no—
67C2×C22⋊Dic7—21no—
68C2×C22⋊Dic7—21no—
69C2×C22⋊Dic7—21no—
70C2×C22⋊Dic7—21no—
71D4×Dic7—42no—
72D4×Dic7—44no—
73C2×C4⋊Dic7—21no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7—42no—
76C2×C4⋊Dic7C2311yescharacteristic
77C2×C4⋊Dic7C2312yes—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4×Dic7—21no—
80C2×C4×Dic7—42no—
81C2×C4×Dic7D411yescharacteristic
82C2×C4×Dic7D411yescharacteristic
83C2×C4×Dic7C2311yescharacteristic
84C2×C4×Dic7C2311yescharacteristic
85C22×C7⋊C8C2312yes—
86C7×#103C2×C418yescomplemented
87C7×C22⋊C8C2×C414yescomplemented
88#6409—42no—
89#6409—42no—
90#6406—42no—
91C7⋊#95—21no—
92C7⋊#95—21no—
93C7⋊#95—21no—
94C7⋊#95—22no—
95C7⋊#95—22no—
96C7⋊#95—22no—
97C7⋊#95—22no—
98C22⋊(C7⋊C8)—22no—
99#6391—42no—
100C8×Dic7—42no—
101C2×#448D711yescharacteristic
102C4×#121—71no—
103C2×#391—71no—
104#1968—72no—
105#1933—72no—
106#1927—72no—
107#1877—72no—
108#1874—71no—
109#1853—71no—
110#1787—71no—
111C4×C22⋊C8—71no—
112C24×C7—21no—
113C24×C7C2×D411yescharacteristic
114C2×C7×Q8—24no—
115C2×C7×Q8—22no—
116C2×C7×Q8—22no—
117C2×C7×Q8—22no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—48no—
121C2×C7×Q8—42no—
122C2×C7×Q8—42no—
123C2×C7×Q8—48no—
124C2×C7×Q8C22×C414yes—
125C2×C7×D4—24no—
126C2×C7×D4—21no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—42no—
130C2×C7×D4—42no—
131C2×C7×D4C22×C412yes—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—24no—
135C22×C4×C7—22no—
136C22×C4×C7—21no—
137C22×C4×C7—42no—
138C22×C4×C7—42no—
139C22×C4×C7C22×C412yes—
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7#4112yes—
142C22×C4×C7C2411yescharacteristic
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—41no—
149C22×Dic7—41no—
150C22×Dic7—41no—
151C22×Dic7—42no—
152C22×Dic7—42no—
153C22×Dic7—42no—
154C22×Dic7—42no—
155C22×Dic7—42no—
156C22×Dic7—42no—
157C22×Dic7SD1614yescomplemented
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C7×SD16—48no—
163C2×C7×C8—22no—
164C2×C7×C8—22no—
165C2×C7×C8C22×C414yes—
166C7×C4⋊C4—21no—
167C7×C4⋊C4C22×C412yes—
168C7×C22⋊C4—24no—
169C22⋊Dic7—42no—
170C22⋊Dic7—42no—
171C22⋊Dic7—42no—
172C22⋊Dic7—42no—
173C4⋊Dic7—22no—
174C4⋊Dic7—22no—
175C4⋊Dic7—22no—
176C4⋊Dic7—22no—
177C4⋊Dic7—42no—
178C4⋊Dic7—42no—
179C4×Dic7—22no—
180C4×Dic7—22no—
181C4×Dic7—41no—
182C4×Dic7—41no—
183C4×Dic7—42no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C23×Q8C2×D711yescharacteristic
188C22×SD16C2×D712yes—
189C2×#121C2×D711yescharacteristic
190C2×C4×D4—71no—
191C2×C4×D4—141no—
192C22×C4⋊C4—71no—
193C22×C42—71no—
194#448Dic718yescomplemented
195C2×#106—72no—
196C2×#103—72no—
197C2×#103—72no—
198C2×#103C2×D712yes—
199C2×#102—72no—
200C2×C22⋊C8—71no—
201C2×C22⋊C8C2×D711yescharacteristic
202C2×C4×C8—72no—
203#391—72no—
204#391—72no—
205#391—141no—
206#391—141no—
207#390—71no—
208#389—71no—
209#388—141no—
210#386—141no—
211#380—71no—
212C4×C4⋊C4—71no—
213C4×C22⋊C4—141no—
214C2×#95—141no—
215C2×#95—141no—
216#340—72no—
217#340—72no—
218#336—72no—
219C23×C7—21no—
220C23×C7—21no—
221C23×C7—21no—
222C23×C7—21no—
223C23×C7—22no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7C4×D412yes—
231C23×C7C2×SD1612yes—
232C23×C7#13512yes—
233C23×C7C22×D411yescharacteristic
234C7×Q8—24no—
235C7×Q8—24no—
236C7×Q8—48no—
237C7×Q8—42no—
238C7×Q8—42no—
239C7×D4—24no—
240C7×D4—48no—
241C7×D4—42no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—24no—
245C2×C4×C7—24no—
246C2×C4×C7—22no—
247C2×C4×C7—24no—
248C2×C4×C7—48no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7C4×D412yes—
254C2×C4×C7C23×C411yescharacteristic
255C2×C4×C7C2×#4111yescharacteristic
256C2×C4×C7H(5,𝔽2)12yes—
257C2×C4×C7H(5,𝔽2)12yes—
258C2×Dic7—21no—
259C2×Dic7—21no—
260C2×Dic7—21no—
261C2×Dic7—21no—
262C2×Dic7—21no—
263C2×Dic7—21no—
264C2×Dic7—21no—
265C2×Dic7—21no—
266C2×Dic7—21no—
267C2×Dic7—44no—
268C2×Dic7—44no—
269C2×Dic7—41no—
270C2×Dic7—42no—
271C2×Dic7—41no—
272C2×Dic7—41no—
273C2×Dic7—41no—
274C2×Dic7—42no—
275C2×Dic7—41no—
276C2×Dic7—41no—
277C2×Dic7—42no—
278C2×Dic7—84no—
279C2×Dic7—84no—
280C2×Dic7C2×SD1612yes—
281C2×Dic7C2×SD1612yes—
282C2×Dic7#13712yes—
283C7×C8—24no—
284C7⋊C8—42no—
285C22×Q8—21no—
286C22×Q8—22no—
287C22×Q8—42no—
288C22×Q8—42no—
289C22×Q8C2×Dic714yescomplemented
290C22×Q8C22×D712yes—
291C22×D4—21no—
292C22×D4C22×D711yescharacteristic
293C23×C4—21no—
294C23×C4—71no—
295C23×C4—71no—
296C23×C4—141no—
297C23×C4—141no—
298C23×C4C22×D711yescharacteristic
299C2×SD16—42no—
300C2×SD16—42no—
301C2×SD16C2×Dic718yescomplemented
302C22×C8—72no—
303C22×C8C22×D712yes—
304#121—22no—
305#121C2×Dic714yes—
306C4×D4—282no—
307C4×D4—284no—
308C2×C4⋊C4—72no—
309C2×C4⋊C4—71no—
310C2×C4⋊C4—71no—
311C2×C4⋊C4—141no—
312C2×C4⋊C4—141no—
313C2×C4⋊C4—282no—
314C2×C4⋊C4C22×D711yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—141no—
317C2×C22⋊C4—141no—
318C2×C22⋊C4—141no—
319C2×C22⋊C4—141no—
320C2×C42—71no—
321C2×C42—71no—
322C2×C42—71no—
323C2×C42—71no—
324C2×C42—141no—
325C2×C42—282no—
326#106—282no—
327#103—282no—
328#103—282no—
329#103C2×Dic718yescomplemented
330#102—282no—
331C22⋊C8—142no—
332C22⋊C8C2×Dic714yescomplemented
333C4×C8—282no—
334#95—142no—
335#95—142no—
336#95—142no—
337#95—142no—
338#95—141no—
339#95—141no—
340#95—141no—
341C22×C7—22no—
342C22×C7—22no—
343C22×C7—22no—
344C22×C7—22no—
345C22×C7—22no—
346C22×C7—22no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—42no—
350C22×C7—81no—
351C22×C7—81no—
352C22×C7—81no—
353C22×C7—81no—
354C22×C7C4×SD1614yes—
355C22×C7C2×C4×D411yescharacteristic
356C22×C7#53411yescharacteristic
357C22×C7#53511yescharacteristic
358C22×C7C22×SD1611yescharacteristic
359C22×C7C2×#13511yescharacteristic
360C22×C7C2×#13711yescharacteristic
361C22×C7#57511yescharacteristic
362C4×C7—22no—
363C4×C7—44no—
364C4×C7—42no—
365C4×C7#51812yes—
366C4×C7#53812yes—
367Dic7—22no—
368Dic7—22no—
369Dic7—22no—
370Dic7—41no—
371Dic7—81no—
372Dic7—81no—
373Dic7—82no—
374C24—21no—
375C24D4×D711yescharacteristic
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—48no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—48no—
386C2×Q8C22×Dic714yes—
387C2×D4—24no—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4C22×Dic712yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—71no—
402C22×C4—71no—
403C22×C4—71no—
404C22×C4—74no—
405C22×C4—71no—
406C22×C4—142no—
407C22×C4—142no—
408C22×C4—142no—
409C22×C4—141no—
410C22×C4—141no—
411C22×C4—282no—
412C22×C4—281no—
413C22×C4—281no—
414C22×C4—281no—
415C22×C4—282no—
416C22×C4—282no—
417C22×C4—282no—
418C22×C4—282no—
419C22×C4—282no—
420C22×C4D4×D711yescharacteristic
421C22×C4#115912yes—
422C22×C4C22×Dic712yes—
423C22×C4C23×D711yescharacteristic
424SD16—48no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—142no—
428C2×C8—142no—
429C2×C8—142no—
430C2×C8C22×Dic714yes—
431C4⋊C4—21no—
432C4⋊C4—142no—
433C4⋊C4—142no—
434C4⋊C4—142no—
435C4⋊C4—142no—
436C4⋊C4—282no—
437C4⋊C4—282no—
438C4⋊C4C22×Dic712yes—
439C22⋊C4—24no—
440C22⋊C4—282no—
441C22⋊C4—282no—
442C22⋊C4—282no—
443C22⋊C4—282no—
444C42—142no—
445C42—142no—
446C42—281no—
447C42—281no—
448C42—282no—
449C2×C7—24no—
450C2×C7—82no—
451C2×C7#232111yescharacteristic
452C2×C7C2×C4×SD1611yescharacteristic
453C2×C7C4×#13711yescharacteristic
454C2×C7#318711yescharacteristic
455C2×C7#319011yescharacteristic
456C2×C7#320611yescharacteristic
457C2×C7#321111yescharacteristic
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—22no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23D7×SD1612yes—
470C23#647912yes—
471C23D4×Dic712yes—
472C23C2×D4×D711yescharacteristic
473Q8—24no—
474Q8—24no—
475Q8—48no—
476Q8—42no—
477Q8—42no—
478D4—24no—
479D4—48no—
480D4—42no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—48no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—72no—
493C2×C4—72no—
494C2×C4—72no—
495C2×C4—141no—
496C2×C4—141no—
497C2×C4—141no—
498C2×C4—141no—
499C2×C4—141no—
500C2×C4—141no—
501C2×C4—141no—
502C2×C4—141no—
503C2×C4—141no—
504C2×C4—284no—
505C2×C4—284no—
506C2×C4—281no—
507C2×C4—281no—
508C2×C4—282no—
509C2×C4—281no—
510C2×C4—281no—
511C2×C4—282no—
512C2×C4—281no—
513C2×C4—282no—
514C2×C4—281no—
515C2×C4—564no—
516C2×C4—564no—
517C2×C4D4×Dic712yes—
518C2×C4C2×#115911yescharacteristic
519C2×C4#654812yes—
520C2×C4#654812yes—
521C2×C4C23×Dic711yescharacteristic
522C8—24no—
523C8—282no—
524C7#2204111yescharacteristic, complemented, Sylow
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22SD16×Dic714yes—
539C22#9028011yescharacteristic
540C22#9028311yescharacteristic
541C22C2×D7×SD1611yescharacteristic
542C22C2×#647911yescharacteristic
543C22D7×#13711yescharacteristic
544C22#9046811yescharacteristic
545C22C2×D4×Dic711yescharacteristic
546C4—22no—
547C4—42no—
548C4—44no—
549C4—142no—
550C4—142no—
551C4—142no—
552C4—281no—
553C4—561no—
554C4—561no—
555C4—562no—
556C4#9028412yes—
557C4#9048912yes—
558C2—24no—
559C2—82no—
560C2#1173747911yescharacteristic
561C2#1173748811yescharacteristic
562C2#1173748911yescharacteristic
563C2#1173750211yescharacteristic
564C2#1174002311yescharacteristic
565C2C2×SD16×Dic711yescharacteristic
566C2Dic7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#448C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
cosocleC22×C4×C7C24
hypercenterC2×#448D7
hypocenterC7#22041
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
2. C2×#90002C2semidirect
4. #11729886C2semidirect
5. #11729803C2semidirect
6. #11729801C2semidirect
8. #11728007C2semidirect
9. #11727977C2semidirect
10. #11727682C2semidirect
11. Dic7×C22⋊C8C2semidirect
19. C7×#448C4semidirect
49. C22×C7×Q8C2×C4semidirect
61. C2×C7×SD16C2×C4semidirect
86. C7×#103C2×C4semidirect
87. C7×C22⋊C8C2×C4semidirect
157. C22×Dic7SD16semidirect
194. #448Dic7semidirect
289. C22×Q8C2×Dic7semidirect
301. C2×SD16C2×Dic7semidirect
329. #103C2×Dic7semidirect
332. C22⋊C8C2×Dic7semidirect
524. C7#22041semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4482
C2×#900022
Dic7×#1212
#117298862
#117298032
#117298012
#117294312
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