Finite Group Databasebeta

← #13364930

Subgroups — #13364930

Order 1792.

1308 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#447C211yesmaximal, characteristic
2C2×#89993C211yesmaximal, characteristic, complemented
3Dic7×#121C211yesmaximal, characteristic, complemented
4#11729882C211yesmaximal, characteristic, complemented
5#11729792C212yesmaximal, complemented
6#11729791C212yesmaximal, complemented
7#11729431C211yesmaximal, characteristic, complemented
8#11727939C212yesmaximal, complemented
9#11727909C212yesmaximal, complemented
10#11727681C211yesmaximal, characteristic, complemented
11Dic7×C22⋊C8C211yesmaximal, characteristic, complemented
12C23×C7×D4C2211yescharacteristic
13C22×C7×D8C2212yes—
14C2×C7×#121C2211yescharacteristic
15C22×C22⋊Dic7—21no—
16C22×C22⋊Dic7—21no—
17C2×D4×Dic7—21no—
18C2×D4×Dic7C2211yescharacteristic, complemented
19C22×C4⋊Dic7C2211yescharacteristic
20C22×C4×Dic7C2211yescharacteristic
21C7×#447C418yescomplemented
22C2×C7×#102C2212yes—
23C2×C7×C22⋊C8C2211yescharacteristic
24#89993—21no—
25#89993—21no—
26#89993C2212yes—
27#89993C2212yes—
28#89993C2211yescharacteristic, complemented
29#89992—21no—
30C2×#6406C2212yes—
31C2×#6406C2212yes—
32C2×#6406C2212yescomplemented
33#89984C2211yescharacteristic
34C2×C22⋊(C7⋊C8)C2211yescharacteristic
35C2×#6392C2212yescomplemented
36C2×C8×Dic7C2212yescomplemented
37#89757C2211yescharacteristic, complemented
38C4⋊C4×Dic7C2211yescharacteristic, complemented
39#89723—21no—
40C22⋊C4×Dic7—21no—
41#89346C2212yescomplemented
42#89345C2212yescomplemented
43#89324C2212yescomplemented
44#22040—71nomaximal, Sylow
45C25×C7—21no—
46C22×C7×D4—21no—
47C22×C7×D4—21no—
48C22×C7×D4—22no—
49C22×C7×D4—42no—
50C22×C7×D4—42no—
51C22×C7×D4C2×C414yescomplemented
52C22×C7×D4C2312yes—
53C22×C7×D4C2311yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C23×Dic7—21no—
56C23×Dic7—21no—
57C23×Dic7D411yescharacteristic
58C23×Dic7D411yescharacteristic
59C2×C7×D8—42no—
60C2×C7×D8—42no—
61C2×C7×D8C2×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—
71C2×C22⋊Dic7—22no—
72C2×C22⋊Dic7—22no—
73C2×C22⋊Dic7—22no—
74C2×C22⋊Dic7—22no—
75C2×C22⋊Dic7—44no—
76C2×C22⋊Dic7—44no—
77D4×Dic7—42no—
78D4×Dic7—44no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—42no—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2312yes—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4×Dic7—21no—
86C2×C4×Dic7—42no—
87C2×C4×Dic7D411yescharacteristic
88C2×C4×Dic7D411yescharacteristic
89C2×C4×Dic7C2311yescharacteristic
90C2×C4×Dic7C2311yescharacteristic
91C22×C7⋊C8C2312yes—
92C7×#102C2×C418yescomplemented
93C7×C22⋊C8C2×C414yescomplemented
94#6406—42no—
95#6406—42no—
96#6406—42no—
97C7⋊#95—21no—
98C7⋊#95—21no—
99C7⋊#95—21no—
100C22⋊(C7⋊C8)—22no—
101#6392—42no—
102C8×Dic7—42no—
103C2×#447D711yescharacteristic
104C4×#121—71no—
105C2×#390—71no—
106#1970—72no—
107#1934—72no—
108#1925—72no—
109#1879—72no—
110#1874—71no—
111#1852—71no—
112#1786—71no—
113C4×C22⋊C8—71no—
114C24×C7—24no—
115C24×C7—22no—
116C24×C7—21no—
117C24×C7—42no—
118C24×C7—42no—
119C24×C7—42no—
120C24×C7—42no—
121C24×C7—42no—
122C24×C7—42no—
123C24×C7C2×D411yescharacteristic
124C2×C7×D4—24no—
125C2×C7×D4—24no—
126C2×C7×D4—21no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—48no—
135C2×C7×D4—42no—
136C2×C7×D4—42no—
137C2×C7×D4—48no—
138C2×C7×D4—42no—
139C2×C7×D4—42no—
140C2×C7×D4C22×C414yes—
141C2×C7×D4C22×C412yes—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—21no—
145C22×C4×C7C22×C412yes—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7#4112yes—
148C22×C4×C7C2411yescharacteristic
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—22no—
154C22×Dic7—41no—
155C22×Dic7—41no—
156C22×Dic7—41no—
157C22×Dic7—42no—
158C22×Dic7—42no—
159C22×Dic7—42no—
160C22×Dic7—42no—
161C22×Dic7—42no—
162C22×Dic7—42no—
163C22×Dic7D814yescomplemented
164C22×Dic7C2×D411yescharacteristic
165C22×Dic7C2×D411yescharacteristic
166C22×Dic7C2×D411yescharacteristic
167C22×Dic7C2×D411yescharacteristic
168C7×D8—48no—
169C2×C7×C8—22no—
170C2×C7×C8—22no—
171C2×C7×C8C22×C414yes—
172C7×C4⋊C4—21no—
173C7×C4⋊C4C22×C412yes—
174C7×C22⋊C4—24no—
175C22⋊Dic7—42no—
176C22⋊Dic7—44no—
177C22⋊Dic7—42no—
178C22⋊Dic7—42no—
179C22⋊Dic7—44no—
180C22⋊Dic7—44no—
181C22⋊Dic7—42no—
182C22⋊Dic7—44no—
183C4⋊Dic7—22no—
184C4⋊Dic7—22no—
185C4⋊Dic7—22no—
186C4⋊Dic7—22no—
187C4⋊Dic7—42no—
188C4⋊Dic7—42no—
189C4×Dic7—22no—
190C4×Dic7—22no—
191C4×Dic7—41no—
192C4×Dic7—41no—
193C4×Dic7—42no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C23×D4C2×D711yescharacteristic
198C22×D8C2×D712yes—
199C2×#121C2×D711yescharacteristic
200C2×C4×D4—71no—
201C2×C4×D4—141no—
202C22×C4⋊C4—71no—
203C22×C22⋊C4—141no—
204C22×C22⋊C4—141no—
205C22×C42—71no—
206#447Dic718yescomplemented
207C2×#107—72no—
208C2×#102—72no—
209C2×#102—72no—
210C2×#102—72no—
211C2×#102C2×D712yes—
212C2×C22⋊C8—71no—
213C2×C22⋊C8C2×D711yescharacteristic
214C2×C4×C8—72no—
215#390—72no—
216#390—72no—
217#390—71no—
218#390—141no—
219#390—141no—
220#389—71no—
221#388—141no—
222#386—141no—
223#380—71no—
224C4×C4⋊C4—71no—
225C4×C22⋊C4—141no—
226#340—72no—
227#340—72no—
228#336—72no—
229C23×C7—21no—
230C23×C7—21no—
231C23×C7—21no—
232C23×C7—21no—
233C23×C7—22no—
234C23×C7—24no—
235C23×C7—48no—
236C23×C7—48no—
237C23×C7—41no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—41no—
241C23×C7—42no—
242C23×C7—41no—
243C23×C7—41no—
244C23×C7—42no—
245C23×C7—42no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—42no—
249C23×C7—48no—
250C23×C7—84no—
251C23×C7—84no—
252C23×C7—84no—
253C23×C7—84no—
254C23×C7C4×D412yes—
255C23×C7C2×D812yes—
256C23×C7#13512yes—
257C23×C7C22×D411yescharacteristic
258C7×D4—24no—
259C7×D4—24no—
260C7×D4—24no—
261C7×D4—48no—
262C7×D4—48no—
263C7×D4—42no—
264C7×D4—42no—
265C7×D4—42no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—24no—
269C2×C4×C7—24no—
270C2×C4×C7—22no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7C4×D412yes—
274C2×C4×C7C23×C411yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C7H(5,𝔽2)12yes—
277C2×C4×C7H(5,𝔽2)12yes—
278C2×Dic7—21no—
279C2×Dic7—21no—
280C2×Dic7—21no—
281C2×Dic7—21no—
282C2×Dic7—21no—
283C2×Dic7—21no—
284C2×Dic7—21no—
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—44no—
288C2×Dic7—44no—
289C2×Dic7—41no—
290C2×Dic7—42no—
291C2×Dic7—41no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—42no—
295C2×Dic7—42no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—84no—
299C2×Dic7—84no—
300C2×Dic7C2×D812yes—
301C2×Dic7C2×D812yes—
302C2×Dic7AGL(1,ℤ/8)12yes—
303C7×C8—24no—
304C7⋊C8—42no—
305C25—21no—
306C22×D4—21no—
307C22×D4—21no—
308C22×D4—22no—
309C22×D4—42no—
310C22×D4—42no—
311C22×D4C2×Dic714yescomplemented
312C22×D4C22×D711yescharacteristic
313C22×D4C22×D712yes—
314C23×C4—71no—
315C23×C4—71no—
316C23×C4—141no—
317C23×C4—141no—
318C23×C4C22×D711yescharacteristic
319C2×D8—42no—
320C2×D8—42no—
321C2×D8C2×Dic718yescomplemented
322C22×C8—72no—
323C22×C8C22×D712yes—
324#121—22no—
325#121C2×Dic714yes—
326C4×D4—282no—
327C4×D4—284no—
328C2×C4⋊C4—72no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—282no—
334C2×C4⋊C4C22×D711yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—142no—
337C2×C22⋊C4—142no—
338C2×C22⋊C4—142no—
339C2×C22⋊C4—142no—
340C2×C22⋊C4—141no—
341C2×C22⋊C4—141no—
342C2×C22⋊C4—141no—
343C2×C22⋊C4—141no—
344C2×C22⋊C4—284no—
345C2×C22⋊C4—284no—
346C2×C42—71no—
347C2×C42—71no—
348C2×C42—71no—
349C2×C42—71no—
350C2×C42—141no—
351C2×C42—282no—
352#107—282no—
353#102—282no—
354#102—282no—
355#102—282no—
356#102C2×Dic718yescomplemented
357C22⋊C8—142no—
358C22⋊C8C2×Dic714yescomplemented
359C4×C8—282no—
360#95—141no—
361#95—141no—
362#95—141no—
363C22×C7—22no—
364C22×C7—22no—
365C22×C7—22no—
366C22×C7—22no—
367C22×C7—22no—
368C22×C7—22no—
369C22×C7—44no—
370C22×C7—42no—
371C22×C7—42no—
372C22×C7—44no—
373C22×C7—42no—
374C22×C7—44no—
375C22×C7—816no—
376C22×C7—81no—
377C22×C7—82no—
378C22×C7—81no—
379C22×C7—82no—
380C22×C7—81no—
381C22×C7—82no—
382C22×C7—81no—
383C22×C7—82no—
384C22×C7C4×D814yes—
385C22×C7C2×C4×D411yescharacteristic
386C22×C7#53411yescharacteristic
387C22×C7#53511yescharacteristic
388C22×C7C22×D811yescharacteristic
389C22×C7C2×#13511yescharacteristic
390C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
391C22×C7#57511yescharacteristic
392C4×C7—22no—
393C4×C7—42no—
394C4×C7#51812yes—
395C4×C7#53812yes—
396Dic7—22no—
397Dic7—22no—
398Dic7—22no—
399Dic7—41no—
400Dic7—81no—
401Dic7—81no—
402Dic7—82no—
403C24—24no—
404C24—21no—
405C24—22no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24D4×D711yescharacteristic
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—48no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—48no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4C22×Dic714yes—
430C2×D4C22×Dic712yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—71no—
435C22×C4—71no—
436C22×C4—71no—
437C22×C4—74no—
438C22×C4—71no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—141no—
442C22×C4—141no—
443C22×C4—142no—
444C22×C4—282no—
445C22×C4—281no—
446C22×C4—281no—
447C22×C4—281no—
448C22×C4—282no—
449C22×C4—282no—
450C22×C4—282no—
451C22×C4—282no—
452C22×C4—282no—
453C22×C4D4×D711yescharacteristic
454C22×C4#115912yes—
455C22×C4C22×Dic712yes—
456C22×C4C23×D711yescharacteristic
457D8—48no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—142no—
461C2×C8—142no—
462C2×C8—142no—
463C2×C8C22×Dic714yes—
464C4⋊C4—21no—
465C4⋊C4—142no—
466C4⋊C4—142no—
467C4⋊C4—142no—
468C4⋊C4—142no—
469C4⋊C4—282no—
470C4⋊C4—282no—
471C4⋊C4C22×Dic712yes—
472C22⋊C4—24no—
473C22⋊C4—284no—
474C22⋊C4—284no—
475C22⋊C4—284no—
476C22⋊C4—282no—
477C22⋊C4—282no—
478C22⋊C4—284no—
479C22⋊C4—282no—
480C22⋊C4—282no—
481C42—142no—
482C42—142no—
483C42—281no—
484C42—281no—
485C42—282no—
486C2×C7—24no—
487C2×C7—82no—
488C2×C7—84no—
489C2×C7#232111yescharacteristic
490C2×C7C2×C4×D811yescharacteristic
491C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
492C2×C7#318611yescharacteristic
493C2×C7#318811yescharacteristic
494C2×C7#320511yescharacteristic
495C2×C7#321111yescharacteristic
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—24no—
502C23—48no—
503C23—48no—
504C23—48no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—84no—
518C23—84no—
519C23—84no—
520C23—84no—
521C23D7×D812yes—
522C23#647512yes—
523C23D4×Dic712yes—
524C23C2×D4×D711yescharacteristic
525D4—24no—
526D4—24no—
527D4—24no—
528D4—48no—
529D4—48no—
530D4—42no—
531D4—42no—
532D4—42no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—72no—
541C2×C4—72no—
542C2×C4—72no—
543C2×C4—141no—
544C2×C4—141no—
545C2×C4—141no—
546C2×C4—141no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—284no—
553C2×C4—284no—
554C2×C4—281no—
555C2×C4—282no—
556C2×C4—281no—
557C2×C4—282no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—282no—
562C2×C4—281no—
563C2×C4—564no—
564C2×C4—564no—
565C2×C4D4×Dic712yes—
566C2×C4C2×#115911yescharacteristic
567C2×C4#654812yes—
568C2×C4#654812yes—
569C2×C4C23×Dic711yescharacteristic
570C8—24no—
571C8—282no—
572C7#2204011yescharacteristic, complemented, Sylow
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—42no—
583C22—44no—
584C22—42no—
585C22—816no—
586C22—82no—
587C22—81no—
588C22—82no—
589C22—81no—
590C22—81no—
591C22—82no—
592C22—82no—
593C22—81no—
594C22D8×Dic714yes—
595C22#9028011yescharacteristic
596C22#9028311yescharacteristic
597C22C2×D7×D811yescharacteristic
598C22C2×#647511yescharacteristic
599C22D7×AGL(1,ℤ/8)11yescharacteristic
600C22#9046411yescharacteristic
601C22C2×D4×Dic711yescharacteristic
602C4—22no—
603C4—42no—
604C4—142no—
605C4—142no—
606C4—142no—
607C4—281no—
608C4—562no—
609C4—561no—
610C4—561no—
611C4#9028412yes—
612C4#9048912yes—
613C2—24no—
614C2—84no—
615C2—82no—
616C2#1173743911yescharacteristic
617C2#1173744011yescharacteristic
618C2#1173745111yescharacteristic
619C2#1173745211yescharacteristic
620C2#1174002311yescharacteristic
621C2C2×D8×Dic711yescharacteristic
622C2Dic7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#447C2
socleC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenterC2×#447D7
hypocenterC7#22040
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
2. C2×#89993C2semidirect
3. Dic7×#121C2semidirect
4. #11729882C2semidirect
5. #11729792C2semidirect
6. #11729791C2semidirect
7. #11729431C2semidirect
8. #11727939C2semidirect
9. #11727909C2semidirect
10. #11727681C2semidirect
11. Dic7×C22⋊C8C2semidirect
18. C2×D4×Dic7C22semidirect
21. C7×#447C4semidirect
28. #89993C22semidirect
32. C2×#6406C22semidirect
35. C2×#6392C22semidirect
36. C2×C8×Dic7C22semidirect
37. #89757C22semidirect
38. C4⋊C4×Dic7C22semidirect
41. #89346C22semidirect
42. #89345C22semidirect
43. #89324C22semidirect
51. C22×C7×D4C2×C4semidirect
61. C2×C7×D8C2×C4semidirect
92. C7×#102C2×C4semidirect
93. C7×C22⋊C8C2×C4semidirect
163. C22×Dic7D8semidirect
206. #447Dic7semidirect
311. C22×D4C2×Dic7semidirect
321. C2×D8C2×Dic7semidirect
356. #102C2×Dic7semidirect
358. C22⋊C8C2×Dic7semidirect
572. C7#22040semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4472
C2×#899932
Dic7×#1212
#117298822
#117297922
#117297912
#117294312
#117279392
#117279092
#117276812
Dic7×C22⋊C82
#220407