Finite Group Databasebeta

← #13405987

Subgroups — #13405987

Order 1792.

1474 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#532C211yesmaximal, characteristic, complemented
2C2×#89810C212yesmaximal, complemented
3#11739525C211yesmaximal, characteristic, complemented
4#11729546C211yesmaximal, characteristic, complemented
5#11729536C211yesmaximal, characteristic, complemented
6#11729521C211yesmaximal, characteristic, complemented
7#11729516C211yesmaximal, characteristic, complemented
8#11729435C212yesmaximal, complemented
9#11729367C212yesmaximal, complemented
10#11729360C212yesmaximal, complemented
11C2×#89319C211yesmaximal, characteristic, complemented
12C22×C7×#41—21no—
13C7×#532—28no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#127C2211yescharacteristic
16C2×C7×#124—21no—
17C2×C7×#121—21no—
18C2×C7×#121C2212yes—
19C2×C4×C7×D4—21no—
20C2×C4×C7×D4C2212yes—
21C22×C42×C7C2211yescharacteristic
22C22×#1141C2212yes—
23C2×#6493C2211yescharacteristic
24C2×#6492—21no—
25C23×C7⋊C8C2212yes—
26C2×#6406C2212yes—
27C2×#6406C2212yes—
28C2×#6406C2212yes—
29#89985—21no—
30#89984C2211yescharacteristic
31#89810—28no—
32C2×#6383C2212yes—
33C2×#6381C2212yes—
34#89712—21no—
35#89712—21no—
36#89711C2211yescharacteristic
37#89711C2211yescharacteristic
38#89700C2211yescharacteristic
39#89700C2211yescharacteristic
40#89324C2212yes—
41#89324C2212yes—
42#89319C2211yescharacteristic
43#89319C2211yescharacteristic
44#89319C2212yes—
45#25496—71nomaximal, Sylow
46C2×C7×#41—28no—
47C2×C7×#41—44no—
48C2×C7×#41—44no—
49C22×C7×Q8—21no—
50C22×C7×D4—22no—
51C22×C7×D4—21no—
52C22×C7×D4—21no—
53C22×C7×D4C2312yes—
54C23×C4×C7—22no—
55C23×C4×C7D411yescharacteristic
56C23×C4×C7D411yescharacteristic
57C23×C4×C7C2311yescharacteristic
58C22×C22⋊D7—21no—
59C22×D28C2311yescharacteristic
60C7×C4⋊Q8—24no—
61C7×#127—24no—
62C7×#124—28no—
63C7×#121—28no—
64C7×#121—24no—
65C7×#121D418yes—
66C4×C7×D4—28no—
67C4×C7×D4—24no—
68C4×C7×D4—24no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4C2312yes—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—22no—
73C2×C42×C7—24no—
74C2×C42×C7D411yescharacteristic
75C2×C42×C7D411yescharacteristic
76C2×C42×C7C2311yescharacteristic
77C2×C42×C7C2311yescharacteristic
78C2×C22⋊Dic7—21no—
79C2×#1141—24no—
80C2×#1141—24no—
81C2×#1141—42no—
82#6493—24no—
83#6492—44no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—21no—
86C2×(C2×D7)⋊C4—21no—
87C2×(C2×D7)⋊C4—21no—
88C2×(C2×D7)⋊C4—21no—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C22×C7⋊C8—24no—
95C22×C7⋊C8—24no—
96C22×C7⋊C8—42no—
97C22×C7⋊C8C2312yes—
98C22×C7⋊C8C2312yes—
99#6406—24no—
100#6406—24no—
101#6406—42no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105#6383—24no—
106#6381—24no—
107C2×#532D711yescharacteristic, complemented
108C2×#466—72no—
109#2854—71no—
110#2057—71no—
111#2041—71no—
112#2023—71no—
113#2011—71no—
114#1893—72no—
115#1879—72no—
116#1862—72no—
117C2×#336—71no—
118C24×C7—22no—
119C24×C7C2×D411yescharacteristic
120C23×D7—21no—
121C7×#41—432no—
122C2×C7×Q8—24no—
123C2×C7×Q8—42no—
124C2×C7×Q8—42no—
125C2×C7×D4—24no—
126C2×C7×D4—24no—
127C2×C7×D4—28no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—42no—
134C2×C7×D4—42no—
135C2×C7×D4—44no—
136C2×C7×D4—44no—
137C2×C7×D4—42no—
138C2×C7×D4—42no—
139C2×C7×D4C2×D414yes—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—28no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—41no—
150C22×C4×C7—44no—
151C22×C4×C7—44no—
152C22×C4×C7D814yescomplemented
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D414yes—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2411yescharacteristic
160C2×C22⋊D7—42no—
161C2×C22⋊D7—42no—
162C2×C22⋊D7—42no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C2×D28—22no—
167C2×D28—22no—
168C2×D28—21no—
169C2×D28—21no—
170C7×C4⋊C4—24no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4C2×D414yes—
173C7×C22⋊C4—28no—
174C7×C22⋊C4—28no—
175C42×C7—22no—
176C42×C7—22no—
177C42×C7—22no—
178C42×C7—22no—
179C22⋊Dic7—42no—
180#1141—48no—
181(C2×D7)⋊C4—42no—
182(C2×D7)⋊C4—42no—
183(C2×D7)⋊C4—42no—
184(C2×D7)⋊C4—42no—
185(C2×D7)⋊C4—42no—
186C4⋊Dic7—22no—
187C4⋊Dic7—22no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190Dic7⋊C4—42no—
191Dic7⋊C4—42no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—48no—
198C2×C7⋊C8#4114yescomplemented
199C22×#41—21no—
200C22×D8—72no—
201C23×C8—72no—
202#532—28no—
203C2×C4⋊Q8C2×D711yescharacteristic
204C2×#127C2×D711yescharacteristic
205C2×#124—21no—
206C2×#123—141no—
207C2×#121—21no—
208C2×#121—71no—
209C2×#121C2×D712yes—
210C2×C4×D4—21no—
211C2×C4×D4C2×D712yes—
212C22×C42C2×D711yescharacteristic
213#466—148no—
214C2×#107—72no—
215C2×#102—72no—
216C2×#102—72no—
217C2×#102—72no—
218C2×#102—72no—
219#390—71no—
220#390—71no—
221#389—71no—
222#389—71no—
223#388—141no—
224#388—141no—
225#386—141no—
226#380—71no—
227#340—72no—
228#340—72no—
229#336—71no—
230#336—72no—
231#336—71no—
232C23×C7—21no—
233C23×C7—21no—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7—24no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7#12012yes—
244C23×C7C2×D812yes—
245C23×C7AGL(1,ℤ/8)12yes—
246C23×C7C22×D411yescharacteristic
247C22×D7—41no—
248C22×D7—41no—
249C22×D7—41no—
250C22×D7—41no—
251C22×D7—41no—
252C22×D7—41no—
253C22×D7—41no—
254C7×Q8—48no—
255C7×D4—24no—
256C7×D4—24no—
257C7×D4—48no—
258C7×D4—48no—
259C7×D4—416no—
260C7×D4—42no—
261C7×D4—42no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—28no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—21no—
274C2×C4×C7—24no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—44no—
278C2×C4×C7—416no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7#12012yes—
282C2×C4×C7C2×D812yes—
283C2×C4×C7C2×D812yes—
284C2×C4×C7#13512yes—
285C2×C4×C7C22×D412yes—
286C2×C4×C7C2×#4112yes—
287C2×C4×C7H(5,𝔽2)11yescharacteristic
288C2×C4×C72−1+411yescharacteristic
289C22⋊D7—84no—
290C2×Dic7—41no—
291C2×Dic7—41no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299D28—42no—
300D28—42no—
301C7⋊C8—24no—
302C7⋊C8—42no—
303C2×#41—28no—
304C2×#41—44no—
305C2×#41—44no—
306C22×Q8—21no—
307C22×D4—22no—
308C22×D4—21no—
309C22×D4—21no—
310C22×D4—71no—
311C22×D4—141no—
312C22×D4C22×D712yes—
313C23×C4—22no—
314C23×C4C22⋊D711yescharacteristic
315C23×C4C22⋊D711yescharacteristic
316C23×C4C22×D711yescharacteristic
317C2×D8—144no—
318C2×D8—144no—
319C2×D8—282no—
320C22×C8—72no—
321C22×C8—72no—
322C22×C8—144no—
323C22×C8—144no—
324C22×C8—282no—
325C4⋊Q8—24no—
326#127—24no—
327#124—28no—
328#123—284no—
329#121—28no—
330#121—24no—
331#121—144no—
332#121C22⋊D718yes—
333C4×D4—28no—
334C4×D4—24no—
335C4×D4—24no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4C22×D712yes—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—141no—
346C2×C22⋊C4—141no—
347C2×C22⋊C4—141no—
348C2×C22⋊C4—141no—
349C2×C22⋊C4—141no—
350C2×C22⋊C4—141no—
351C2×C42—24no—
352C2×C42C22⋊D711yescharacteristic
353C2×C42C22⋊D711yescharacteristic
354C2×C42C22×D711yescharacteristic
355C2×C42C22×D711yescharacteristic
356#107—144no—
357#102—144no—
358#102—144no—
359#102—144no—
360#102—282no—
361#95—141no—
362#95—141no—
363#95—141no—
364C22×C7—22no—
365C22×C7—22no—
366C22×C7—22no—
367C22×C7—22no—
368C22×C7—22no—
369C22×C7—22no—
370C22×C7—44no—
371C22×C7—44no—
372C22×C7—44no—
373C22×C7—82no—
374C22×C7—82no—
375C22×C7—82no—
376C22×C7—82no—
377C22×C7#44714yes—
378C22×C7C2×#12011yescharacteristic
379C22×C7#53211yescharacteristic
380C22×C7#53711yescharacteristic
381C22×C7C22×D811yescharacteristic
382C22×C7C2×#13511yescharacteristic
383C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
384C22×C7#57511yescharacteristic
385C2×D7—81no—
386C2×D7—81no—
387C2×D7—81no—
388C2×D7—81no—
389C2×D7—81no—
390C2×D7—81no—
391C2×D7—81no—
392C4×C7—22no—
393C4×C7—22no—
394C4×C7—22no—
395C4×C7—22no—
396C4×C7—44no—
397C4×C7—41no—
398C4×C7#54612yes—
399C4×C7#54712yes—
400Dic7—81no—
401Dic7—81no—
402Dic7—81no—
403C24—22no—
404C24—141no—
405C24C2×C22⋊D711yescharacteristic
406#41—432no—
407C2×Q8—24no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—28no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—142no—
425C2×D4—141no—
426C2×D4—141no—
427C2×D4—142no—
428C2×D4—282no—
429C2×D4—282no—
430C2×D4—282no—
431C2×D4C2×C22⋊D714yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—28no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4#114114yescomplemented
448C22×C4C2×C22⋊D711yescharacteristic
449C22×C4C2×C22⋊D711yescharacteristic
450C22×C4C2×C22⋊D711yescharacteristic
451C22×C4C2×C22⋊D711yescharacteristic
452C22×C4C2×C22⋊D711yescharacteristic
453C22×C4C2×C22⋊D714yes—
454C22×C4C23×D711yescharacteristic
455D8—288no—
456C2×C8—74no—
457C2×C8—142no—
458C2×C8—142no—
459C2×C8—142no—
460C2×C8—142no—
461C2×C8—142no—
462C2×C8—288no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—142no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—142no—
469C4⋊C4—282no—
470C4⋊C4—282no—
471C4⋊C4C2×C22⋊D714yes—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—282no—
475C22⋊C4—282no—
476C22⋊C4—282no—
477C22⋊C4—282no—
478C22⋊C4—282no—
479C22⋊C4—282no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C2×C7—24no—
485C2×C7—84no—
486C2×C7#243011yescharacteristic
487C2×C7C2×#44711yescharacteristic
488C2×C7#300211yescharacteristic
489C2×C7#316911yescharacteristic
490C2×C7#318011yescharacteristic
491C2×C7#324511yescharacteristic
492C2×C7#324711yescharacteristic
493D7—161no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—24no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—281no—
506C23—281no—
507C23—281no—
508C23—281no—
509C23—281no—
510C23—281no—
511C23—281no—
512C23C2×#114112yes—
513C23#649512yes—
514C23#651612yes—
515C23C22×C22⋊D711yescharacteristic
516Q8—48no—
517D4—24no—
518D4—24no—
519D4—48no—
520D4—48no—
521D4—416no—
522D4—42no—
523D4—42no—
524D4—282no—
525D4—282no—
526D4—564no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—44no—
543C2×C4—416no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4C2×#114112yes—
556C2×C4C2×#114112yes—
557C2×C4#651312yes—
558C2×C4#651612yes—
559C2×C4D7×#4112yes—
560C2×C4#655311yescharacteristic
561C2×C4#655411yescharacteristic
562C2×C4C22×C22⋊D712yes—
563C8—144no—
564C8—282no—
565C7#2549611yescharacteristic, complemented, Sylow
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—561no—
580C22—561no—
581C22—561no—
582C22—561no—
583C22—561no—
584C22—561no—
585C22—561no—
586C22#8998914yes—
587C22C22×#114111yescharacteristic
588C22C2×#649511yescharacteristic
589C22C2×#651311yescharacteristic
590C22#9051311yescharacteristic
591C22#9052311yescharacteristic
592C22#9052411yescharacteristic
593C22C2×#651611yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—44no—
599C4—41no—
600C4—561no—
601C4—561no—
602C4—561no—
603C4#9051512yes—
604C4#9052012yes—
605C2—24no—
606C2—84no—
607C2—1121no—
608C2#1174073611yescharacteristic
609C2#1174073811yescharacteristic
610C2#1174075511yescharacteristic
611C2#1174075611yescharacteristic
612C2C2×#8998911yescharacteristic
613C2#1174152511yescharacteristic
614C2#1174153611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#532C2
socleC23×C7C22×D4
hypercenterC2×#532D7
hypocenterC7#25496
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×C7×#532C2semidirect
2. C2×#89810C2semidirect
3. #11739525C2semidirect
4. #11729546C2semidirect
5. #11729536C2semidirect
6. #11729521C2semidirect
7. #11729516C2semidirect
8. #11729435C2semidirect
9. #11729367C2semidirect
10. #11729360C2semidirect
11. C2×#89319C2semidirect
107. C2×#532D7semidirect
152. C22×C4×C7D8semidirect
198. C2×C7⋊C8#41semidirect
447. C22×C4#1141semidirect
565. C7#25496semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5322
C2×#898102
#117395252
#117295462
#117295362
#117295212
#117295162
#117294352
#117293672
#117293602
C2×#893192
#254967