Finite Group Databasebeta

← #13388527

Subgroups — #13388527

Order 1792.

1462 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#550C211yesmaximal, characteristic, complemented
2C2×#89844C211yesmaximal, characteristic, complemented
3#11739945C211yesmaximal, characteristic, complemented
4C2×C4×#1141C211yesmaximal, characteristic, complemented
5#11729521C212yesmaximal, complemented
6#11729518C211yesmaximal, characteristic, complemented
7#11729516C212yesmaximal, complemented
8#11729388C211yesmaximal, characteristic, complemented
9#11729367C212yesmaximal, complemented
10#11729360C212yesmaximal, complemented
11#11729358C211yesmaximal, characteristic, complemented
12C22×C7×#41—21no—
13C7×#550—24no—
14C7×#550—24no—
15C2×C7×#127C2212yes—
16C2×C7×#127C2211yescharacteristic
17C2×C7×#121—21no—
18C2×C7×#121—22no—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4C2212yes—
21C2×C4×C7×D4C2211yescharacteristic
22C2×#6509—21no—
23C22×#1141C2211yescharacteristic
24C22×#1141C2212yes—
25C2×C4×D28C2211yescharacteristic
26C2×#6406C2211yescharacteristic
27C2×#6406C2212yes—
28C2×#6406C2212yes—
29C2×#6406C2212yes—
30C2×#6406C2212yes—
31#89844—24no—
32C4×#1141—24no—
33#89766—22no—
34#89759—21no—
35C2×#6383C2211yescharacteristic
36C2×#6381C2211yescharacteristic
37#89711C2212yes—
38#89707C2211yescharacteristic
39#89700C2212yes—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2212yes—
43#89319C2212yes—
44#89319C2212yes—
45#24184—71nomaximal, Sylow
46C2×C7×#41—28no—
47C2×C7×#41—44no—
48C2×C7×#41—44no—
49C22×C7×Q8D411yescharacteristic
50C22×C7×D4—22no—
51C22×C7×D4—21no—
52C22×C7×D4—22no—
53C22×C7×D4—21no—
54C22×C7×D4C2312yes—
55C22×C7×D4C2311yescharacteristic
56C23×C4×C7—22no—
57C23×C4×C7—21no—
58C22×D28C2311yescharacteristic
59C22×C4×D7—21no—
60C7×#127—24no—
61C7×#127—24no—
62C7×#127—22no—
63C7×#127D414yes—
64C7×#121—28no—
65C7×#121—216no—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C4×C7×D4—24no—
69C4×C7×D4—24no—
70C4×C7×D4—22no—
71C4×C7×D4D414yes—
72C2×C7×C4⋊C4D411yescharacteristic
73C2×C7×C4⋊C4D412yes—
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4—21no—
77C2×C42×C7C2312yes—
78C2×C42×C7C2311yescharacteristic
79#6509—44no—
80C2×#1141—22no—
81C2×#1141—22no—
82C2×#1141—24no—
83C2×#1141—24no—
84C2×#1141—22no—
85C2×#1141—42no—
86C2×(C2×D7)⋊C4—21no—
87C2×(C2×D7)⋊C4—22no—
88C2×(C2×D7)⋊C4—22no—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2312yes—
93C2×Dic7⋊C4—22no—
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2312yes—
96C22×C7⋊C8C2311yescharacteristic
97C4×D28—22no—
98C4×D28—22no—
99#6406—22no—
100#6406—24no—
101#6406—24no—
102#6406—42no—
103#6406—42no—
104C7⋊#95—22no—
105#6383—22no—
106#6381—22no—
107C4⋊(C7⋊C8)—22no—
108C4×C7⋊C8—22no—
109C2×#550D711yescharacteristic, complemented
110C2×#459—71no—
111C2×C4×D8—71no—
112#2538—71no—
113#2041—72no—
114#2011—72no—
115#1984—71no—
116#1957—71no—
117#1879—72no—
118#1862—72no—
119#1802—71no—
120C24×C7—22no—
121C24×C7—21no—
122C23×D7—21no—
123C7×#41—432no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8D814yescomplemented
127C2×C7×D4—28no—
128C2×C7×D4—28no—
129C2×C7×D4—24no—
130C2×C7×D4—24no—
131C2×C7×D4—22no—
132C2×C7×D4—21no—
133C2×C7×D4—22no—
134C2×C7×D4—21no—
135C2×C7×D4—22no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—22no—
139C2×C7×D4—21no—
140C2×C7×D4—22no—
141C2×C7×D4—44no—
142C2×C7×D4—44no—
143C2×C7×D4—42no—
144C2×C7×D4—42no—
145C2×C7×D4—81no—
146C2×C7×D4—81no—
147C2×C7×D4—82no—
148C2×C7×D4—82no—
149C2×C7×D4C2×D414yes—
150C2×C7×D4C2×D412yes—
151C22×C4×C7—28no—
152C22×C4×C7—21no—
153C22×C4×C7—24no—
154C22×C4×C7—44no—
155C22×C4×C7—44no—
156C22×C4×C7—42no—
157C22×C4×C7—42no—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2411yescharacteristic
163C22×Dic7—21no—
164C22×Dic7—22no—
165C2×D28—21no—
166C2×D28—21no—
167C2×D28—21no—
168C2×D28—21no—
169C2×D28—21no—
170C2×D28#4112yescomplemented
171C2×C4×D7—42no—
172C2×C4×D7—42no—
173C2×C4×D7—42no—
174C7×C4⋊C4—24no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4C2×D412yes—
178C7×C22⋊C4—28no—
179C7×C22⋊C4—24no—
180C42×C7—22no—
181C42×C7—22no—
182C42×C7C2×D412yes—
183C42×C7C2×D412yes—
184#1141—28no—
185#1141—48no—
186(C2×D7)⋊C4—42no—
187(C2×D7)⋊C4—44no—
188(C2×D7)⋊C4—82no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—22no—
192C4⋊Dic7—22no—
193C4⋊Dic7—42no—
194C4⋊Dic7—81no—
195Dic7⋊C4—44no—
196C2×C7⋊C8—21no—
197C2×C7⋊C8—21no—
198C2×C7⋊C8—22no—
199C2×C7⋊C8—21no—
200C2×C7⋊C8—22no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8#4112yescomplemented
205C22×#41—21no—
206C22×D8—71no—
207C22×D8—72no—
208#550—24no—
209#550—24no—
210C2×#127C2×D712yes—
211C2×#127C2×D711yescharacteristic
212C2×C22⋊Q8—141no—
213C2×#121—21no—
214C2×#121—22no—
215C2×C4×Q8C2×D711yescharacteristic
216C2×C4×D4—71no—
217C2×C4×D4C2×D712yes—
218C2×C4×D4C2×D711yescharacteristic
219#459—144no—
220C4×D8—144no—
221C2×#107—71no—
222C2×C4⋊C8—71no—
223C2×#102—72no—
224C2×#102—72no—
225C2×#102—71no—
226C2×#102—71no—
227C2×#102—72no—
228C2×#102—72no—
229C2×C4×C8—71no—
230#396—142no—
231#390—72no—
232#389—72no—
233#384—71no—
234#380—141no—
235#340—72no—
236#336—72no—
237#336—72no—
238C23×C7—24no—
239C23×C7—22no—
240C23×C7—41no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—41no—
244C23×C7—42no—
245C23×C7—41no—
246C23×C7—42no—
247C23×C7—41no—
248C23×C7—42no—
249C23×C7—41no—
250C23×C7—42no—
251C23×C7—41no—
252C23×C7C22×D411yescharacteristic
253C22×D7—22no—
254C22×D7—41no—
255C22×D7—41no—
256C22×D7—41no—
257C22×D7—41no—
258C22×D7—41no—
259C22×D7—41no—
260C7×Q8—28no—
261C7×D4—22no—
262C7×D4—22no—
263C7×D4—24no—
264C7×D4—24no—
265C7×D4—416no—
266C7×D4—42no—
267C7×D4—42no—
268C7×D4—41no—
269C7×D4—41no—
270C7×D4—48no—
271C7×D4—84no—
272C7×D4—88no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—416no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—48no—
285C2×C4×C7#12012yes—
286C2×C4×C7#12012yes—
287C2×C4×C7C2×D814yes—
288C2×C4×C7C2×D812yes—
289C2×C4×C7AGL(1,ℤ/8)12yes—
290C2×C4×C7C22×D411yescharacteristic
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7H(5,𝔽2)11yescharacteristic
295C2×C4×C72−1+411yescharacteristic
296C2×Dic7—41no—
297C2×Dic7—42no—
298C2×Dic7—41no—
299C2×Dic7—42no—
300C2×Dic7—41no—
301C2×Dic7—42no—
302D28—22no—
303D28—22no—
304D28—41no—
305D28—41no—
306C4×D7—84no—
307C7⋊C8—22no—
308C7⋊C8—41no—
309C7⋊C8—42no—
310C2×#41—28no—
311C2×#41—44no—
312C2×#41—44no—
313C22×Q8C22⋊D711yescharacteristic
314C22×D4—22no—
315C22×D4—21no—
316C22×D4—22no—
317C22×D4—21no—
318C22×D4—71no—
319C22×D4C22×D712yes—
320C22×D4C22×D711yescharacteristic
321C23×C4—22no—
322C23×C4—21no—
323C23×C4—141no—
324C2×D8—142no—
325C2×D8—142no—
326C2×D8—142no—
327C2×D8—144no—
328C2×D8—144no—
329C2×D8—282no—
330C22×C8—71no—
331C22×C8—72no—
332C22×C8—71no—
333#127—24no—
334#127—24no—
335#127—22no—
336#127C22⋊D714yes—
337C22⋊Q8—284no—
338#121—28no—
339#121—216no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×D4—24no—
343C4×D4—24no—
344C4×D4—22no—
345C4×D4—142no—
346C4×D4—142no—
347C4×D4C22⋊D714yes—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—72no—
350C2×C4⋊C4—142no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4C22⋊D711yescharacteristic
354C2×C4⋊C4C22⋊D712yes—
355C2×C4⋊C4C22×D711yescharacteristic
356C2×C22⋊C4—22no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—141no—
359C2×C22⋊C4—142no—
360C2×C22⋊C4—142no—
361C2×C42C22×D712yes—
362C2×C42C22×D711yescharacteristic
363#107—142no—
364C4⋊C8—142no—
365#102—144no—
366#102—142no—
367#102—144no—
368#102—142no—
369#102—282no—
370#102—282no—
371C4×C8—142no—
372#95—142no—
373C22×C7—42no—
374C22×C7—44no—
375C22×C7—44no—
376C22×C7—42no—
377C22×C7—44no—
378C22×C7—42no—
379C22×C7—81no—
380C22×C7—82no—
381C22×C7—82no—
382C22×C7—81no—
383C22×C7—82no—
384C22×C7—81no—
385C22×C7—82no—
386C22×C7—81no—
387C22×C7C2×#12011yescharacteristic
388C22×C7#54811yescharacteristic
389C22×C7#55011yescharacteristic
390C22×C7C22×D811yescharacteristic
391C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
392C22×C7#57611yescharacteristic
393C22×C7#57711yescharacteristic
394C2×D7—42no—
395C2×D7—42no—
396C2×D7—42no—
397C2×D7—81no—
398C2×D7—81no—
399C2×D7—81no—
400C2×D7—81no—
401C4×C7—24no—
402C4×C7—22no—
403C4×C7—42no—
404C4×C7—42no—
405C4×C7#44714yes—
406C4×C7#52511yescharacteristic
407C4×C7#53711yescharacteristic
408C4×C7#54611yescharacteristic
409C4×C7#54711yescharacteristic
410Dic7—82no—
411Dic7—81no—
412C24—22no—
413C24—21no—
414C24—141no—
415#41—432no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8#114114yescomplemented
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—72no—
438C2×D4—81no—
439C2×D4—81no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—141no—
443C2×D4—141no—
444C2×D4—141no—
445C2×D4—141no—
446C2×D4—141no—
447C2×D4C2×C22⋊D714yes—
448C2×D4C2×C22⋊D712yes—
449C22×C4—28no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—141no—
457C22×C4—142no—
458C22×C4—282no—
459C22×C4—282no—
460C22×C4—282no—
461C22×C4C2×C22⋊D712yes—
462C22×C4C2×C22⋊D712yes—
463C22×C4C2×C22⋊D711yescharacteristic
464C22×C4C2×C22⋊D711yescharacteristic
465C22×C4C23×D711yescharacteristic
466D8—148no—
467D8—288no—
468C2×C8—72no—
469C2×C8—141no—
470C2×C8—141no—
471C2×C8—142no—
472C2×C8—141no—
473C2×C8—142no—
474C2×C8—141no—
475C2×C8—141no—
476C2×C8—142no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—142no—
481C4⋊C4—142no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—284no—
485C4⋊C4—282no—
486C4⋊C4—561no—
487C4⋊C4C2×C22⋊D712yes—
488C22⋊C4—28no—
489C22⋊C4—24no—
490C22⋊C4—284no—
491C22⋊C4—282no—
492C22⋊C4—562no—
493C42—22no—
494C42—22no—
495C42C2×C22⋊D712yes—
496C42C2×C22⋊D712yes—
497C2×C7—82no—
498C2×C7—84no—
499C2×C7#240711yescharacteristic
500C2×C7C2×#44711yescharacteristic
501C2×C7#301211yescharacteristic
502C2×C7#335811yescharacteristic
503C2×C7#336611yescharacteristic
504C2×C7#339111yescharacteristic
505C2×C7#339711yescharacteristic
506D7—82no—
507C23—24no—
508C23—22no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—41no—
513C23—42no—
514C23—41no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23—41no—
519C23—42no—
520C23—41no—
521C23—142no—
522C23—281no—
523C23—281no—
524C23—281no—
525C23—281no—
526C23—281no—
527C23—281no—
528C23C22×C22⋊D711yescharacteristic
529Q8—28no—
530D4—24no—
531D4—22no—
532D4—24no—
533D4—22no—
534D4—416no—
535D4—48no—
536D4—41no—
537D4—42no—
538D4—41no—
539D4—42no—
540D4—84no—
541D4—88no—
542D4—142no—
543D4—142no—
544D4—281no—
545D4—281no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—416no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—48no—
558C2×C4—282no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—282no—
562C2×C4—282no—
563C2×C4—281no—
564C2×C4—564no—
565C2×C4C2×#114114yes—
566C2×C4C2×#114112yes—
567C2×C4#649512yes—
568C2×C4#651612yes—
569C2×C4#651612yes—
570C2×C4D7×#4111yescharacteristic
571C2×C4D7×#4111yescharacteristic
572C2×C4#655311yescharacteristic
573C2×C4#655411yescharacteristic
574C2×C4C22×C22⋊D711yescharacteristic
575C2×C4C22×C22⋊D711yescharacteristic
576C8—142no—
577C8—282no—
578C8—281no—
579C7#2418411yescharacteristic, complemented, Sylow
580C22—44no—
581C22—42no—
582C22—44no—
583C22—42no—
584C22—44no—
585C22—42no—
586C22—82no—
587C22—81no—
588C22—82no—
589C22—81no—
590C22—82no—
591C22—81no—
592C22—82no—
593C22—81no—
594C22—282no—
595C22—282no—
596C22—282no—
597C22—561no—
598C22—561no—
599C22—561no—
600C22—561no—
601C22C22×#114111yescharacteristic
602C22C2×#649511yescharacteristic
603C22#9052211yescharacteristic
604C22#9052511yescharacteristic
605C22#9052611yescharacteristic
606C22#9052711yescharacteristic
607C22C2×#651611yescharacteristic
608C4—24no—
609C4—22no—
610C4—42no—
611C4—42no—
612C4—561no—
613C4—562no—
614C4#8998914yes—
615C4#9051511yescharacteristic
616C4#9051911yescharacteristic
617C4#9052011yescharacteristic
618C4#9052411yescharacteristic
619C2—84no—
620C2—82no—
621C2—562no—
622C2#1174102111yescharacteristic
623C2#1174102511yescharacteristic
624C2#1174102611yescharacteristic
625C2#1174102711yescharacteristic
626C2C2×#8998911yescharacteristic
627C2#1174154211yescharacteristic
628C2#1174154711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#550C2
socleC23×C7C22×D4
hypercenterC2×#550D7
hypocenterC7#24184
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. C2×C7×#550C2semidirect
2. C2×#89844C2semidirect
3. #11739945C2semidirect
4. C2×C4×#1141C2semidirect
5. #11729521C2semidirect
6. #11729518C2semidirect
7. #11729516C2semidirect
8. #11729388C2semidirect
9. #11729367C2semidirect
10. #11729360C2semidirect
11. #11729358C2semidirect
109. C2×#550D7semidirect
126. C2×C7×Q8D8semidirect
170. C2×D28#41semidirect
204. C2×C7⋊C8#41semidirect
418. C2×Q8#1141semidirect
579. C7#24184semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5502
C2×#898442
#117399452
C2×C4×#11412
#117295212
#117295182
#117295162
#117293882
#117293672
#117293602
#117293582
#241847