Finite Group Databasebeta

← #13387831

Subgroups — #13387831

Order 1792.

1400 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic, complemented
2C2×#89856C211yesmaximal, characteristic, complemented
3#11739860C211yesmaximal, characteristic, complemented
4C2×#89786C211yesmaximal, characteristic, complemented
5#11729542C211yesmaximal, characteristic, complemented
6#11729438C212yesmaximal, complemented
7#11729434C212yesmaximal, complemented
8#11729369C211yesmaximal, characteristic, complemented
9#11729359C211yesmaximal, characteristic, complemented
10#11729314C212yesmaximal, complemented
11#11729310C212yesmaximal, complemented
12C22×C7×#41—21no—
13C7×#547—24no—
14C7×#547—24no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#123—22no—
17C2×C7×C22⋊Q8—21no—
18C2×C7×#121C2212yes—
19C2×C4×C7×D4C2211yescharacteristic
20C2×C4×C7×D4C2211yescharacteristic
21C22×C7×C4⋊C4C2212yes—
22C22×#1141C2211yescharacteristic
23C2×#6493C2212yes—
24C2×C4×D28C2211yescharacteristic
25#89995—21no—
26C2×#6406C2211yescharacteristic
27C2×#6406C2212yes—
28#89984C2212yes—
29C2×C22⋊(C7⋊C8)C2212yes—
30C2×C22⋊(C7⋊C8)C2212yes—
31#89856—24no—
32#89786—24no—
33#89765—21no—
34C2×#6383C2212yes—
35C2×#6383C2211yescharacteristic
36C2×#6382C2212yes—
37C2×#6382C2211yescharacteristic
38#89728—22no—
39#89708C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×#6377C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24128—71nomaximal, Sylow
46C2×C7×#41—28no—
47C2×C7×#41—48no—
48C22×C7×Q8—21no—
49C22×C7×D4—22no—
50C22×C7×D4D411yescharacteristic
51C22×C7×D4C2311yescharacteristic
52C23×C4×C7—22no—
53C23×C4×C7—21no—
54C23×C4×C7C2312yes—
55C22×C22⋊D7—22no—
56C22×D28C2311yescharacteristic
57C22×C4×D7—21no—
58C7×C4⋊Q8—22no—
59C7×C4⋊Q8D414yescomplemented
60C7×#123—216no—
61C7×C22⋊Q8—28no—
62C7×#121—24no—
63C7×#121—24no—
64C4×C7×D4—22no—
65C4×C7×D4—22no—
66C4×C7×D4—22no—
67C4×C7×D4D414yescomplemented
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—24no—
71C2×C7×C4⋊C4—24no—
72C2×C7×C4⋊C4D411yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2312yes—
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4D412yes—
78C2×C42×C7C2311yescharacteristic
79C2×C22⋊Dic7—22no—
80C2×#1141—22no—
81C2×#1141—22no—
82C2×#1141—22no—
83#6493—24no—
84#6493—42no—
85C2×(C2×D7)⋊C4—21no—
86C2×(C2×D7)⋊C4—22no—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2312yes—
89C22×C7⋊C8C2311yescharacteristic
90C22×C7⋊C8C2312yes—
91C22×C7⋊C8C2311yescharacteristic
92C4×D28—22no—
93C4×D28—22no—
94#6406—22no—
95#6406—42no—
96C7⋊#95—21no—
97C7⋊#95—21no—
98C7⋊#95—22no—
99C22⋊(C7⋊C8)—24no—
100C22⋊(C7⋊C8)—42no—
101#6383—24no—
102#6383—22no—
103#6382—24no—
104#6382—22no—
105C4⋊(C7⋊C8)—22no—
106#6377—22no—
107C2×#547D711yescharacteristic, complemented
108C2×#476—71no—
109C2×#442—71no—
110#2535—71no—
111#2034—72no—
112#1999—72no—
113#1993—71no—
114#1955—71no—
115#1874—72no—
116#1853—72no—
117#1804—71no—
118C24×C7—21no—
119C24×C7C2×D412yes—
120C23×D7—21no—
121C7×#41—432no—
122C2×C7×Q8—24no—
123C2×C7×Q8—44no—
124C2×C7×D4—24no—
125C2×C7×D4—28no—
126C2×C7×D4—22no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—44no—
133C2×C7×D4—44no—
134C2×C7×D4C2×D412yes—
135C22×C4×C7—28no—
136C22×C4×C7—21no—
137C22×C4×C7—22no—
138C22×C4×C7—24no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—44no—
146C22×C4×C7—44no—
147C22×C4×C7—44no—
148C22×C4×C7C2×D412yes—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2411yescharacteristic
152C2×C22⋊D7—44no—
153C2×C22⋊D7—44no—
154C2×C22⋊D7—82no—
155C22×Dic7—22no—
156C22×Dic7—21no—
157C2×D28—21no—
158C2×D28—21no—
159C2×D28—21no—
160C2×D28—21no—
161C2×D28—21no—
162C2×D28#4112yescomplemented
163C2×C4×D7—42no—
164C2×C4×D7—82no—
165C7×C4⋊C4—28no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4C2×D412yes—
171C7×C4⋊C4C2×D414yes—
172C7×C22⋊C4—28no—
173C7×C22⋊C4—24no—
174C7×C22⋊C4—24no—
175C42×C7C2×D412yes—
176C42×C7C2×D412yes—
177C22⋊Dic7—82no—
178#1141—28no—
179(C2×D7)⋊C4—44no—
180(C2×D7)⋊C4—42no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—22no—
184C4⋊Dic7—22no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8—21no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—21no—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—21no—
193C2×C7⋊C8#4112yescomplemented
194C22×#41—21no—
195C22×D8—71no—
196#547—24no—
197#547—24no—
198C2×C4⋊Q8C2×D711yescharacteristic
199C2×#123—22no—
200C2×C22⋊Q8—21no—
201C2×#121—72no—
202C2×#121C2×D712yes—
203C2×C4×D4—71no—
204C2×C4×D4C2×D711yescharacteristic
205C2×C4×D4C2×D711yescharacteristic
206C22×C4⋊C4C2×D712yes—
207#476—144no—
208#442—144no—
209C2×#106—71no—
210C2×#106—72no—
211C2×C4⋊C8—71no—
212C2×#102—71no—
213C2×#102—71no—
214C2×#102—72no—
215C2×#102—72no—
216C2×C22⋊C8—72no—
217C2×C22⋊C8—72no—
218C2×#97—71no—
219#397—141no—
220#392—141no—
221#386—142no—
222#382—71no—
223#380—72no—
224#340—72no—
225#340—72no—
226#340—72no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—41no—
235C23×C7—41no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7AGL(1,ℤ/8)14yes—
239C23×C7C22×D411yescharacteristic
240C22×D7—22no—
241C22×D7—41no—
242C22×D7—41no—
243C22×D7—42no—
244C22×D7—42no—
245C7×Q8—48no—
246C7×D4—28no—
247C7×D4—22no—
248C7×D4—22no—
249C7×D4—416no—
250C7×D4—41no—
251C7×D4—41no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—24no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—48no—
260C2×C4×C7—416no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7#12012yes—
269C2×C4×C7#12012yes—
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C2×#4111yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C2×C4×C7H(5,𝔽2)11yescharacteristic
275C2×C4×C72−1+411yescharacteristic
276C22⋊D7—88no—
277C2×Dic7—41no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7—42no—
282D28—22no—
283D28—22no—
284D28—41no—
285D28—41no—
286C4×D7—84no—
287C7⋊C8—22no—
288C7⋊C8—41no—
289C7⋊C8—42no—
290C2×#41—28no—
291C2×#41—48no—
292C22×Q8—21no—
293C22×D4—22no—
294C22×D4—71no—
295C22×D4—142no—
296C22×D4C22⋊D711yescharacteristic
297C22×D4C22×D711yescharacteristic
298C23×C4—22no—
299C23×C4—21no—
300C23×C4—141no—
301C23×C4C22×D712yes—
302C2×D8—142no—
303C2×D8—142no—
304C2×D8—142no—
305C22×C8—71no—
306C22×C8—71no—
307C22×C8—72no—
308C4⋊Q8—22no—
309C4⋊Q8C22⋊D714yescomplemented
310#123—216no—
311C22⋊Q8—28no—
312#121—24no—
313#121—24no—
314#121—144no—
315#121—282no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—142no—
320C4×D4—142no—
321C4×D4C22⋊D714yescomplemented
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—72no—
328C2×C4⋊C4C22⋊D711yescharacteristic
329C2×C4⋊C4C22×D711yescharacteristic
330C2×C4⋊C4C22×D712yes—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—141no—
334C2×C22⋊C4—142no—
335C2×C22⋊C4—142no—
336C2×C22⋊C4C22⋊D712yes—
337C2×C42C22×D711yescharacteristic
338#106—142no—
339#106—144no—
340C4⋊C8—142no—
341#102—142no—
342#102—142no—
343#102—144no—
344#102—282no—
345C22⋊C8—144no—
346C22⋊C8—282no—
347#97—142no—
348#95—142no—
349#95—141no—
350#95—141no—
351C22×C7—24no—
352C22×C7—24no—
353C22×C7—24no—
354C22×C7—42no—
355C22×C7—42no—
356C22×C7—42no—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7—42no—
361C22×C7—82no—
362C22×C7—82no—
363C22×C7C2×#12011yescharacteristic
364C22×C7#54611yescharacteristic
365C22×C7#54711yescharacteristic
366C22×C7C2×AGL(1,ℤ/8)12yes—
367C22×C7#57712yes—
368C2×D7—42no—
369C2×D7—44no—
370C2×D7—82no—
371C2×D7—81no—
372C2×D7—81no—
373C4×C7—22no—
374C4×C7—22no—
375C4×C7—42no—
376C4×C7—42no—
377C4×C7—44no—
378C4×C7#52511yescharacteristic
379C4×C7#53711yescharacteristic
380C4×C7#54811yescharacteristic
381C4×C7#55011yescharacteristic
382Dic7—81no—
383Dic7—82no—
384C24—21no—
385C24—141no—
386C24C2×C22⋊D712yes—
387#41—432no—
388C2×Q8—24no—
389C2×Q8—44no—
390C2×D4—24no—
391C2×D4—28no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—72no—
401C2×D4—141no—
402C2×D4—141no—
403C2×D4—141no—
404C2×D4—141no—
405C2×D4—141no—
406C2×D4—284no—
407C2×D4—284no—
408C2×D4—562no—
409C2×D4C2×C22⋊D712yes—
410C22×C4—28no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—142no—
424C22×C4—141no—
425C22×C4—282no—
426C22×C4—562no—
427C22×C4C2×C22⋊D712yes—
428C22×C4C2×C22⋊D711yescharacteristic
429C22×C4C2×C22⋊D711yescharacteristic
430C22×C4C23×D711yescharacteristic
431D8—148no—
432C2×C8—72no—
433C2×C8—141no—
434C2×C8—141no—
435C2×C8—142no—
436C2×C8—141no—
437C2×C8—142no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8—142no—
441C4⋊C4—28no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—142no—
447C4⋊C4—142no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4C2×C22⋊D712yes—
451C4⋊C4C2×C22⋊D714yes—
452C22⋊C4—28no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—284no—
456C22⋊C4—282no—
457C22⋊C4—562no—
458C42C2×C22⋊D712yes—
459C42C2×C22⋊D712yes—
460C2×C7—44no—
461C2×C7—82no—
462C2×C7#240711yescharacteristic
463C2×C7#300311yescharacteristic
464C2×C7#301011yescharacteristic
465C2×C7#330612yes—
466C2×C7#334512yes—
467D7—82no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—142no—
480C23—282no—
481C23—281no—
482C23—282no—
483C23—281no—
484C23#649514yes—
485C23C22×C22⋊D711yescharacteristic
486Q8—48no—
487D4—28no—
488D4—22no—
489D4—22no—
490D4—416no—
491D4—41no—
492D4—41no—
493D4—142no—
494D4—142no—
495D4—281no—
496D4—281no—
497D4—568no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—48no—
506C2×C4—416no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—282no—
515C2×C4—282no—
516C2×C4—282no—
517C2×C4—282no—
518C2×C4—281no—
519C2×C4—564no—
520C2×C4#651612yes—
521C2×C4#651612yes—
522C2×C4D7×#4111yescharacteristic
523C2×C4D7×#4111yescharacteristic
524C2×C4#655311yescharacteristic
525C2×C4#655411yescharacteristic
526C2×C4C22×C22⋊D711yescharacteristic
527C2×C4C22×C22⋊D711yescharacteristic
528C8—142no—
529C8—281no—
530C8—282no—
531C7#2412811yescharacteristic, complemented, Sylow
532C22—24no—
533C22—24no—
534C22—24no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—82no—
543C22—82no—
544C22—284no—
545C22—282no—
546C22—562no—
547C22—561no—
548C22—561no—
549C22C2×#649512yes—
550C22#9051511yescharacteristic
551C22#9052011yescharacteristic
552C22#9052812yes—
553C22C2×#651611yescharacteristic
554C4—22no—
555C4—22no—
556C4—42no—
557C4—44no—
558C4—42no—
559C4—562no—
560C4—561no—
561C4#9051911yescharacteristic
562C4#9052211yescharacteristic
563C4#9052411yescharacteristic
564C4#9052511yescharacteristic
565C2—44no—
566C2—82no—
567C2—562no—
568C2#1174096412yes—
569C2#1174097112yes—
570C2#1174148311yescharacteristic
571C2#1174154211yescharacteristic
572C2#1174155811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#547C2
socleC23×C7C22×D4
hypercenterC2×#547D7
hypocenterC7#24128
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×C7×#547C2semidirect
2. C2×#89856C2semidirect
3. #11739860C2semidirect
4. C2×#89786C2semidirect
5. #11729542C2semidirect
6. #11729438C2semidirect
7. #11729434C2semidirect
8. #11729369C2semidirect
9. #11729359C2semidirect
10. #11729314C2semidirect
11. #11729310C2semidirect
59. C7×C4⋊Q8D4semidirect
67. C4×C7×D4D4semidirect
107. C2×#547D7semidirect
162. C2×D28#41semidirect
193. C2×C7⋊C8#41semidirect
309. C4⋊Q8C22⋊D7semidirect
321. C4×D4C22⋊D7semidirect
531. C7#24128semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5472
C2×#898562
#117398602
C2×#897862
#117295422
#117294382
#117294342
#117293692
#117293592
#117293142
#117293102
#241287