Finite Group Databasebeta

← #13364927

Subgroups — #13364927

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×#390C211yesmaximal, characteristic, complemented
2C2×#89989C211yesmaximal, characteristic
3C4×#6493C211yesmaximal, characteristic, complemented
4#11729882C211yesmaximal, characteristic, complemented
5#11729314C211yesmaximal, characteristic, complemented
6#11729309C211yesmaximal, characteristic, complemented
7C4×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
8#11727411C212yesmaximal, complemented
9#11727401C212yesmaximal, complemented
10#11727398C212yesmaximal, complemented
11#11727390C212yesmaximal, complemented
12C23×C7×D4C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4—21no—
15C22×C7×C22⋊C4—21no—
16C22×C42×C7C2211yescharacteristic
17C22×#1141C2212yes—
18C2×#6493C2211yescharacteristic
19C2×C4×C22⋊D7—21no—
20C2×C4×D28C2211yescharacteristic, complemented
21C7×#390—21no—
22C7×#390—21no—
23C7×#390C2212yes—
24C7×#390C2212yes—
25#89989C418yescomplemented
26C2×#6406C2212yes—
27C2×#6406C2212yes—
28C2×#6406C2212yes—
29#89985—21no—
30#89984C2211yescharacteristic
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×#6383C2212yescomplemented
34C2×#6381C2212yescomplemented
35#89712—21no—
36#89711C2211yescharacteristic, complemented
37C4×(C2×D7)⋊C4—21no—
38C4×C4⋊Dic7C2211yescharacteristic, complemented
39#89700C2211yescharacteristic, complemented
40C2×C4×C7⋊C8C2212yescomplemented
41#89324C2212yescomplemented
42#89324C2212yescomplemented
43#89319C2212yescomplemented
44#22040—71nomaximal, Sylow
45C25×C7—21no—
46C22×C7×D4—21no—
47C22×C7×D4—22no—
48C22×C7×D4—42no—
49C22×C7×D4—42no—
50C22×C7×D4C2×C414yescomplemented
51C22×C7×D4C2312yes—
52C23×C4×C7—21no—
53C23×C4×C7D411yescharacteristic
54C23×C4×C7D411yescharacteristic
55C23×C4×C7C2311yescharacteristic
56C22×C22⋊D7—21no—
57C22×D28C2311yescharacteristic
58C22×C4×D7—21no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4—42no—
61C2×C7×C4⋊C4C2312yes—
62C2×C7×C22⋊C4—22no—
63C2×C7×C22⋊C4—22no—
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4—44no—
67C2×C7×C22⋊C4—44no—
68C2×C42×C7—42no—
69C2×C42×C7D411yescharacteristic
70C2×C42×C7D411yescharacteristic
71C2×C42×C7C2311yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×C22⋊Dic7—21no—
74C2×#1141—42no—
75C2×#1141—42no—
76C2×#1141C2×C418yescomplemented
77#6493—22no—
78#6493C2×C414yes—
79C4×C22⋊D7—44no—
80C2×(C2×D7)⋊C4—21no—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—21no—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×Dic7⋊C4—21no—
88C2×C4×Dic7—21no—
89C22×C7⋊C8C2312yes—
90C22×C7⋊C8C2312yes—
91C4×D28—42no—
92#6406—42no—
93#6406—42no—
94#6406C2×C418yescomplemented
95C7⋊#95—21no—
96C7⋊#95—21no—
97C7⋊#95—21no—
98C22⋊(C7⋊C8)—22no—
99C22⋊(C7⋊C8)C2×C414yescomplemented
100#6383—42no—
101#6381—42no—
102C4×C7⋊C8—42no—
103C2×#447—71no—
104C4×#121—71no—
105C2×#390D711yescharacteristic, complemented
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—42no—
117C24×C7—42no—
118C24×C7—42no—
119C24×C7—42no—
120C24×C7—42no—
121C24×C7—42no—
122C24×C7C2×D411yescharacteristic
123C23×D7—21no—
124C2×C7×D4—24no—
125C2×C7×D4—22no—
126C2×C7×D4—22no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—48no—
131C2×C7×D4—42no—
132C2×C7×D4—42no—
133C2×C7×D4—48no—
134C2×C7×D4C22×C414yes—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—41no—
141C22×C4×C7—41no—
142C22×C4×C7—41no—
143C22×C4×C7—42no—
144C22×C4×C7—42no—
145C22×C4×C7—42no—
146C22×C4×C7D814yescomplemented
147C22×C4×C7C22×C412yes—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7#4112yes—
154C22×C4×C7C2411yescharacteristic
155C2×C22⋊D7—24no—
156C2×C22⋊D7—42no—
157C2×C22⋊D7—42no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C2×D28—22no—
162C2×D28—22no—
163C2×D28—21no—
164C2×D28C22×C412yes—
165C2×C4×D7—42no—
166C2×C4×D7—42no—
167C2×C4×D7—42no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—42no—
171C7×C22⋊C4—44no—
172C7×C22⋊C4—44no—
173C7×C22⋊C4—44no—
174C7×C22⋊C4—44no—
175C42×C7—22no—
176C42×C7—22no—
177C42×C7—41no—
178C42×C7—41no—
179C22⋊Dic7—42no—
180#1141—48no—
181(C2×D7)⋊C4—24no—
182(C2×D7)⋊C4—42no—
183(C2×D7)⋊C4—42no—
184(C2×D7)⋊C4—42no—
185C4⋊Dic7—21no—
186C4⋊Dic7—22no—
187C4⋊Dic7—22no—
188C4⋊Dic7C22×C412yes—
189Dic7⋊C4—42no—
190C4×Dic7—42no—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8C22×C414yes—
197C23×D4C2×D711yescharacteristic
198C22×D8—72no—
199C2×#121—71no—
200C2×C4×D4—71no—
201C2×C4×D4—141no—
202C22×C4⋊C4C2×D711yescharacteristic
203C22×C22⋊C4—21no—
204C22×C22⋊C4—21no—
205C22×C42C2×D711yescharacteristic
206#447—78no—
207C2×#107—72no—
208C2×#102—72no—
209C2×#102—72no—
210C2×#102—72no—
211C2×#102—72no—
212C2×C22⋊C8—71no—
213C2×C22⋊C8—71no—
214C2×C4×C8—72no—
215#390—21no—
216#390—21no—
217#390—71no—
218#390C2×D712yes—
219#390C2×D712yes—
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—24no—
234C23×C7—48no—
235C23×C7—48no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—48no—
243C23×C7—84no—
244C23×C7—84no—
245C23×C7—84no—
246C23×C7—84no—
247C23×C7C4×D412yes—
248C23×C7C2×D812yes—
249C23×C7#13512yes—
250C23×C7C22×D411yescharacteristic
251C22×D7—22no—
252C22×D7—41no—
253C22×D7—41no—
254C22×D7—41no—
255C22×D7—41no—
256C22×D7—41no—
257C22×D7—41no—
258C7×D4—24no—
259C7×D4—24no—
260C7×D4—48no—
261C7×D4—42no—
262C7×D4—42no—
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—21no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—44no—
277C2×C4×C7—44no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—84no—
282C2×C4×C7C4×D412yes—
283C2×C4×C7C2×D812yes—
284C2×C4×C7C2×D812yes—
285C2×C4×C7AGL(1,ℤ/8)12yes—
286C2×C4×C7C23×C411yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C2×C4×C7H(5,𝔽2)12yes—
289C2×C4×C7H(5,𝔽2)12yes—
290C22⋊D7—48no—
291C2×Dic7—22no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300D28—24no—
301D28—42no—
302C4×D7—84no—
303C7⋊C8—24no—
304C7⋊C8—42no—
305C25—21no—
306C22×D4—22no—
307C22×D4—21no—
308C22×D4—42no—
309C22×D4—42no—
310C22×D4—71no—
311C22×D4—141no—
312C22×D4C4×D714yescomplemented
313C22×D4C22×D712yes—
314C23×C4—21no—
315C23×C4—141no—
316C23×C4C22⋊D711yescharacteristic
317C23×C4C22⋊D711yescharacteristic
318C23×C4C22×D711yescharacteristic
319C2×D8—78no—
320C2×D8—282no—
321C2×D8—282no—
322C22×C8—72no—
323C22×C8—72no—
324#121—74no—
325#121—142no—
326C4×D4—282no—
327C4×D4—284no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4C22×D712yes—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—141no—
342C2×C22⋊C4—141no—
343C2×C22⋊C4—141no—
344C2×C22⋊C4—141no—
345C2×C22⋊C4—141no—
346C2×C42—42no—
347C2×C42—141no—
348C2×C42C22⋊D711yescharacteristic
349C2×C42C22⋊D711yescharacteristic
350C2×C42C22×D711yescharacteristic
351C2×C42C22×D711yescharacteristic
352#107—282no—
353#102—78no—
354#102—282no—
355#102—282no—
356#102—282no—
357C22⋊C8—74no—
358C22⋊C8—142no—
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—44no—
371C22×C7—44no—
372C22×C7—816no—
373C22×C7—82no—
374C22×C7—82no—
375C22×C7—82no—
376C22×C7—82no—
377C22×C7C4×D814yes—
378C22×C7C2×C4×D411yescharacteristic
379C22×C7#53411yescharacteristic
380C22×C7#53511yescharacteristic
381C22×C7C22×D811yescharacteristic
382C22×C7C2×#13511yescharacteristic
383C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
384C22×C7#57511yescharacteristic
385C2×D7—42no—
386C2×D7—42no—
387C2×D7—42no—
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—41no—
397C4×C7—82no—
398C4×C7#51812yes—
399C4×C7#53812yes—
400Dic7—42no—
401Dic7—81no—
402Dic7—81no—
403C24—24no—
404C24—22no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24—141no—
412C24C2×C22⋊D711yescharacteristic
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—48no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—48no—
423C2×D4—72no—
424C2×D4—144no—
425C2×D4—141no—
426C2×D4—142no—
427C2×D4—142no—
428C2×D4—282no—
429C2×D4—282no—
430C2×D4C2×C4×D714yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—141no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—282no—
446C22×C4—282no—
447C22×C4—282no—
448C22×C4#114114yescomplemented
449C22×C4C2×C4×D712yes—
450C22×C4#115712yes—
451C22×C4C2×C22⋊D711yescharacteristic
452C22×C4C2×C22⋊D711yescharacteristic
453C22×C4C2×C22⋊D711yescharacteristic
454C22×C4C2×C22⋊D711yescharacteristic
455C22×C4C2×C22⋊D711yescharacteristic
456C22×C4C23×D711yescharacteristic
457D8—288no—
458C2×C8—74no—
459C2×C8—142no—
460C2×C8—142no—
461C2×C8—142no—
462C2×C8—142no—
463C2×C8—142no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—72no—
468C4⋊C4—141no—
469C4⋊C4—142no—
470C4⋊C4—142no—
471C4⋊C4—282no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—144no—
477C22⋊C4—282no—
478C22⋊C4—282no—
479C22⋊C4—282no—
480C22⋊C4—282no—
481C42—22no—
482C42—22no—
483C42—41no—
484C42—41no—
485C42—282no—
486C2×C7—24no—
487C2×C7—84no—
488C2×C7#232111yescharacteristic
489C2×C7C2×C4×D811yescharacteristic
490C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
491C2×C7#318611yescharacteristic
492C2×C7#318811yescharacteristic
493C2×C7#320511yescharacteristic
494C2×C7#321111yescharacteristic
495D7—82no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—24no—
501C23—48no—
502C23—48no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—48no—
510C23—84no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23—142no—
515C23—281no—
516C23—281no—
517C23—281no—
518C23—281no—
519C23—281no—
520C23—281no—
521C23C4×C22⋊D712yes—
522C23C2×#114112yes—
523C23#651312yes—
524C23C22×C22⋊D711yescharacteristic
525D4—24no—
526D4—24no—
527D4—48no—
528D4—42no—
529D4—42no—
530D4—144no—
531D4—288no—
532D4—282no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—142no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—564no—
562C2×C4C4×C22⋊D712yes—
563C2×C4C2×#114112yes—
564C2×C4C2×#114112yes—
565C2×C4#649512yes—
566C2×C4C22×C4×D711yescharacteristic
567C2×C4C2×#115711yescharacteristic
568C2×C4#654812yes—
569C2×C4#654812yes—
570C8—144no—
571C8—282no—
572C7#2204011yescharacteristic, complemented, Sylow
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—816no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—282no—
588C22—282no—
589C22—282no—
590C22—561no—
591C22—561no—
592C22—561no—
593C22—561no—
594C22C4×#114114yes—
595C22C2×C4×C22⋊D711yescharacteristic
596C22C22×#114111yescharacteristic
597C22C2×#649511yescharacteristic
598C22#9049511yescharacteristic
599C22#9049611yescharacteristic
600C22C2×#651311yescharacteristic
601C22#9051311yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—41no—
607C4—82no—
608C4—282no—
609C4—561no—
610C4—561no—
611C4#9017712yes—
612C4#9018512yes—
613C2—24no—
614C2—84no—
615C2—562no—
616C2#1173954111yescharacteristic
617C2C2×C4×#114111yescharacteristic
618C2C4×#649511yescharacteristic
619C2#1174007011yescharacteristic
620C2#1174007111yescharacteristic
621C2#1174039111yescharacteristic
622C2#1174061111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#390C2
socleC23×C7C22×D4
prime radicalC2×#89989C2
cosocleC22×C4×C7C24
hypercenterC2×#390D7
hypocenterC7#22040
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C2×C7×#390C2semidirect
3. C4×#6493C2semidirect
4. #11729882C2semidirect
5. #11729314C2semidirect
6. #11729309C2semidirect
7. C4×C22⋊(C7⋊C8)C2semidirect
8. #11727411C2semidirect
9. #11727401C2semidirect
10. #11727398C2semidirect
11. #11727390C2semidirect
20. C2×C4×D28C22semidirect
25. #89989C4semidirect
33. C2×#6383C22semidirect
34. C2×#6381C22semidirect
36. #89711C22semidirect
38. C4×C4⋊Dic7C22semidirect
39. #89700C22semidirect
40. C2×C4×C7⋊C8C22semidirect
41. #89324C22semidirect
42. #89324C22semidirect
43. #89319C22semidirect
50. C22×C7×D4C2×C4semidirect
76. C2×#1141C2×C4semidirect
94. #6406C2×C4semidirect
99. C22⋊(C7⋊C8)C2×C4semidirect
105. C2×#390D7semidirect
146. C22×C4×C7D8semidirect
312. C22×D4C4×D7semidirect
448. C22×C4#1141semidirect
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×#3902
C2×#899892
C4×#64932
#117298822
#117293142
#117293092
C4×C22⋊(C7⋊C8)2
#117274112
#117274012
#117273982
#117273902
#220407