Finite Group Databasebeta

← #13355539

Subgroups — #13355539

Order 1792.

1818 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#131C211yesmaximal, characteristic, complemented
2C2×#90524C211yesmaximal, characteristic
3#11741532C211yesmaximal, characteristic, complemented
4C2×#89905C211yesmaximal, characteristic, complemented
5C2×#89885C211yesmaximal, characteristic, complemented
6#11729744C212yesmaximal, complemented
7#11729743C212yesmaximal, complemented
8#11729739C211yesmaximal, characteristic, complemented
9#11729727C212yesmaximal, complemented
10#11729726C212yesmaximal, complemented
11C2×#89346C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C22×#1157C2211yescharacteristic
14C2×C7×#131—24no—
15C2×C7×#131—24no—
16C2×C7×#131—24no—
17C22×C7×M16C2212yes—
18C22×C7×M16C2211yescharacteristic
19C23×C7×C8C2212yes—
20C23×C7×C8C2211yescharacteristic
21#90524—24no—
22#90524C418yescomplemented
23C2×#6509—21no—
24C2×#6502—22no—
25C2×#6501—21no—
26C2×#6493C2212yes—
27C2×#6489C2211yescharacteristic
28C22×C4⋊Dic7C2211yescharacteristic
29C22×C4⋊Dic7C2211yescharacteristic
30C2×#6487C2212yes—
31#90002—21no—
32#89993—21no—
33#89992—22no—
34#89984C2212yes—
35#89984C2212yes—
36#89905—24no—
37#89885—24no—
38C2×#6395C2211yescharacteristic
39C2×#6395C2211yescharacteristic
40C2×#6395C2212yes—
41C2×#6390C2211yescharacteristic
42C2×#6390C2211yescharacteristic
43C2×#6390C2212yes—
44#89346C2212yes—
45#89346C2212yes—
46#89346C2212yes—
47#89346C2212yes—
48#20976—71nomaximal, Sylow
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41—22no—
53C2×C7×#41C2×C414yes—
54C2×C7×#41D414yes—
55C22×C7×Q8D411yescharacteristic
56C22×C7×D4D411yescharacteristic
57C22×C7×D4D412yes—
58C23×C4×C7C2312yes—
59C23×C4×C7C2311yescharacteristic
60C22×C22⋊D7—21no—
61C22×C22⋊D7—22no—
62C23×Dic7—21no—
63C23×Dic7—21no—
64C2×#1157—22no—
65C2×#1157—22no—
66C2×#1157—22no—
67C2×#1157—22no—
68C2×#1157—22no—
69C2×#1157C2×C414yes—
70C22×D28C2311yescharacteristic
71C22×C4×D7—21no—
72C22×Dic14C2311yescharacteristic
73C7×#131—232no—
74C2×C7×M16—24no—
75C2×C7×M16—22no—
76C2×C7×M16—22no—
77C2×C7×M16—24no—
78C2×C7×M16—24no—
79C2×C7×M16D414yescomplemented
80C22×C7×C8—24no—
81C22×C7×C8—22no—
82C22×C7×C8—22no—
83C22×C7×C8—24no—
84C22×C7×C8—24no—
85C22×C7×C8D414yescomplemented
86C22×C7×C8C2311yescharacteristic
87C22×C7×C8C2312yes—
88C22×C7×C8C2311yescharacteristic
89C2×C22⋊Dic7—22no—
90C2×C22⋊Dic7—21no—
91C2×C22⋊Dic7—22no—
92C2×C22⋊Dic7—22no—
93#6509—28no—
94#6502—216no—
95#6501—28no—
96#6493—24no—
97#6493C2×C418yescomplemented
98C2×(C2×D7)⋊C4—22no—
99#6489—22no—
100#6489—22no—
101C2×C4⋊Dic7—21no—
102C2×C4⋊Dic7—22no—
103C2×C4⋊Dic7—22no—
104C2×C4⋊Dic7—22no—
105C2×C4⋊Dic7—21no—
106C2×C4⋊Dic7C2×C414yes—
107C2×C4⋊Dic7C2312yes—
108C2×C4⋊Dic7C2311yescharacteristic
109C2×C4⋊Dic7C2311yescharacteristic
110C2×C4⋊Dic7C2312yes—
111#6487—24no—
112#6487C2×C418yescomplemented
113C2×Dic7⋊C4—22no—
114C2×C4×Dic7—21no—
115C7⋊#95—22no—
116C7⋊#95—22no—
117#6395—22no—
118#6395—22no—
119#6395—24no—
120#6390—22no—
121#6390—22no—
122#6390—24no—
123C22×#131D711yescharacteristic, complemented
124C2×#537—71no—
125C2×#417—71no—
126C2×#416—71no—
127#2311—71no—
128#1895—72no—
129#1894—72no—
130#1893—72no—
131#1892—72no—
132#1736—71no—
133C2×#340—71no—
134C24×C7C2×D412yes—
135C24×C7C2×D411yescharacteristic
136C23×D7—21no—
137C7×#41—28no—
138C7×#41—28no—
139C7×#41—28no—
140C7×#41—28no—
141C2×C7×Q8—22no—
142C2×C7×Q8—22no—
143C2×C7×Q8C22⋊C414yes—
144C2×C7×D4—24no—
145C2×C7×D4—24no—
146C2×C7×D4—22no—
147C2×C7×D4—22no—
148C2×C7×D4C22⋊C414yes—
149C2×C7×D4C22⋊C418yes—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7C22×C414yes—
155C22×C4×C7C22×C412yes—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D414yes—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D412yes—
162C22×C4×C7#4112yes—
163C22×C4×C7#4112yes—
164C22×C4×C7C2411yescharacteristic
165C2×C22⋊D7—24no—
166C2×C22⋊D7—28no—
167C2×C22⋊D7—42no—
168C2×C22⋊D7—42no—
169C2×C22⋊D7—82no—
170C2×C22⋊D7—82no—
171C22×Dic7—21no—
172C22×Dic7—22no—
173C22×Dic7—24no—
174C22×Dic7—21no—
175C22×Dic7—21no—
176C22×Dic7—22no—
177C22×Dic7—42no—
178C22×Dic7—81no—
179C22×Dic7—81no—
180C22×Dic7—81no—
181C22×Dic7—81no—
182#1157—28no—
183#1157—28no—
184#1157—44no—
185#1157—44no—
186C2×D28—21no—
187C2×D28—21no—
188C2×D28—21no—
189C2×D28—21no—
190C2×D28—21no—
191C2×D28C22×C412yes—
192C2×C4×D7—24no—
193C2×C4×D7—42no—
194C2×C4×D7—42no—
195C2×Dic14—21no—
196C2×Dic14—21no—
197C2×Dic14—21no—
198C2×Dic14—21no—
199C2×Dic14—21no—
200C2×Dic14C22×C412yes—
201C7×M16—216no—
202C7×M16—28no—
203C2×C7×C8—216no—
204C2×C7×C8—28no—
205C2×C7×C8—21no—
206C2×C7×C8—21no—
207C2×C7×C8—21no—
208C2×C7×C8—22no—
209C2×C7×C8—22no—
210C2×C7×C8—21no—
211C2×C7×C8C2×D412yes—
212C2×C7×C8C2×D414yes—
213C2×C7×C8C2×D412yes—
214C22⋊Dic7—28no—
215C22⋊Dic7—42no—
216C22⋊Dic7—82no—
217C22⋊Dic7—82no—
218(C2×D7)⋊C4—28no—
219C4⋊Dic7—24no—
220C4⋊Dic7—22no—
221C4⋊Dic7—22no—
222C4⋊Dic7—21no—
223C4⋊Dic7—21no—
224C4⋊Dic7—22no—
225C4⋊Dic7—21no—
226C4⋊Dic7—21no—
227C4⋊Dic7—42no—
228C4⋊Dic7C22×C414yes—
229Dic7⋊C4—28no—
230C4×Dic7—42no—
231C22×#41—71no—
232C22×#41C2×D711yescharacteristic
233C2×#131—24no—
234C2×#131—24no—
235C2×#131—24no—
236C22×M16C2×D712yes—
237C22×M16C2×D711yescharacteristic
238C23×C8C2×D712yes—
239C23×C8C2×D711yescharacteristic
240#537—78no—
241#537—144no—
242C2×C22⋊Q8—72no—
243C2×C22⋊Q8—141no—
244C2×#121—72no—
245C2×#121—141no—
246C2×#121—142no—
247C2×#117—71no—
248C22×C4⋊C4—71no—
249C22×C4⋊C4—71no—
250#417—144no—
251#416—144no—
252C2×#103—71no—
253C2×#103—71no—
254C2×#103—72no—
255C2×#102—71no—
256C2×#102—71no—
257C2×#102—72no—
258#391—141no—
259#390—141no—
260#386—142no—
261#380—72no—
262#380—72no—
263#340—72no—
264#340—72no—
265#340—72no—
266#340—72no—
267C23×C7—21no—
268C23×C7—21no—
269C23×C7—21no—
270C23×C7—21no—
271C23×C7—22no—
272C23×C7—22no—
273C23×C7—22no—
274C23×C7—22no—
275C23×C7—22no—
276C23×C7—22no—
277C23×C7C2×C22⋊C414yes—
278C23×C7C2×C22⋊C412yes—
279C23×C7#12112yes—
280C23×C7#12112yes—
281C23×C7C22×D411yescharacteristic
282C22×D7—22no—
283C22×D7—41no—
284C22×D7—41no—
285C22×D7—41no—
286C22×D7—41no—
287C22×D7—41no—
288C22×D7—41no—
289C7×Q8—28no—
290C7×D4—216no—
291C7×D4—28no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—24no—
295C2×C4×C7—24no—
296C2×C4×C7—24no—
297C2×C4×C7—24no—
298C2×C4×C7C2×C22⋊C414yes—
299C2×C4×C7C2×C22⋊C412yes—
300C2×C4×C7C4×D414yes—
301C2×C4×C7C4×D414yes—
302C2×C4×C7#12014yes—
303C2×C4×C7#12112yes—
304C2×C4×C7#12112yes—
305C2×C4×C7#12314yes—
306C2×C4×C7C23×C411yescharacteristic
307C2×C4×C7C22×D411yescharacteristic
308C2×C4×C7C22×D411yescharacteristic
309C2×C4×C7C22×D411yescharacteristic
310C2×C4×C7C2×#4111yescharacteristic
311C2×C4×C7C2×#4111yescharacteristic
312C22⋊D7—48no—
313C22⋊D7—88no—
314C2×Dic7—22no—
315C2×Dic7—24no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—42no—
319C2×Dic7—42no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—42no—
323C2×Dic7—41no—
324C2×Dic7—42no—
325C2×Dic7—41no—
326C2×Dic7—42no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—84no—
330C2×Dic7—84no—
331D28—22no—
332D28—22no—
333D28—41no—
334D28—41no—
335C4×D7—48no—
336Dic14—22no—
337Dic14—22no—
338Dic14—41no—
339Dic14—41no—
340C7×C8—22no—
341C7×C8—24no—
342C7×C8—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—74no—
348C2×#41—142no—
349C2×#41—142no—
350C2×#41—142no—
351C2×#41—142no—
352C2×#41—142no—
353C2×#41C4×D714yes—
354C2×#41C22⋊D714yes—
355C22×Q8—71no—
356C22×Q8D2811yescharacteristic
357C22×D4—71no—
358C22×D4—141no—
359C22×D4—142no—
360C22×D4D2812yes—
361C22×D4D2811yescharacteristic
362C23×C4—141no—
363C23×C4—141no—
364C23×C4—141no—
365C23×C4C22×D712yes—
366C23×C4C22×D711yescharacteristic
367#131—232no—
368C2×M16—24no—
369C2×M16—22no—
370C2×M16—22no—
371C2×M16—24no—
372C2×M16—24no—
373C2×M16C22⋊D714yescomplemented
374C22×C8—24no—
375C22×C8—22no—
376C22×C8—22no—
377C22×C8—24no—
378C22×C8—24no—
379C22×C8C22⋊D714yescomplemented
380C22×C8C22×D711yescharacteristic
381C22×C8C22×D712yes—
382C22×C8C22×D711yescharacteristic
383C22⋊Q8—78no—
384C22⋊Q8—148no—
385C22⋊Q8—144no—
386#121—78no—
387#121—148no—
388#121—1416no—
389#121—144no—
390#117—142no—
391#117—142no—
392C2×C4⋊C4—74no—
393C2×C4⋊C4—72no—
394C2×C4⋊C4—72no—
395C2×C4⋊C4—71no—
396C2×C4⋊C4—71no—
397C2×C4⋊C4—141no—
398C2×C4⋊C4—142no—
399C2×C4⋊C4—142no—
400C2×C4⋊C4—142no—
401C2×C4⋊C4—142no—
402C2×C4⋊C4—141no—
403C2×C22⋊C4—142no—
404C2×C22⋊C4—142no—
405C2×C22⋊C4—141no—
406C2×C22⋊C4—142no—
407C2×C22⋊C4—142no—
408C2×C42—141no—
409#103—142no—
410#103—142no—
411#103—144no—
412#102—142no—
413#102—142no—
414#102—144no—
415#95—142no—
416#95—142no—
417C22×C7—22no—
418C22×C7—22no—
419C22×C7—22no—
420C22×C7—22no—
421C22×C7—22no—
422C22×C7—22no—
423C22×C7—24no—
424C22×C7—24no—
425C22×C7—24no—
426C22×C7—42no—
427C22×C7—42no—
428C22×C7—42no—
429C22×C7—42no—
430C22×C7#38614yes—
431C22×C7C22×C22⋊C411yescharacteristic
432C22×C7C2×#12111yescharacteristic
433C22×C7C2×#12111yescharacteristic
434C22×C7#57611yescharacteristic
435C22×C7#57711yescharacteristic
436C22×C7#57711yescharacteristic
437C22×C7#57811yescharacteristic
438C2×D7—42no—
439C2×D7—42no—
440C2×D7—42no—
441C2×D7—81no—
442C2×D7—81no—
443C2×D7—81no—
444C2×D7—81no—
445C4×C7—24no—
446C4×C7#38614yes—
447C4×C7C2×C4×D411yescharacteristic
448C4×C7C2×C4×D411yescharacteristic
449C4×C7C2×#12011yescharacteristic
450C4×C7C2×#12311yescharacteristic
451Dic7—44no—
452Dic7—42no—
453Dic7—81no—
454Dic7—81no—
455Dic7—82no—
456C24—141no—
457C24C2×D2812yes—
458C24C2×D2811yescharacteristic
459#41—28no—
460#41—28no—
461#41—28no—
462#41—28no—
463#41—148no—
464#41—148no—
465#41—284no—
466#41—284no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—72no—
470C2×Q8—141no—
471C2×Q8—141no—
472C2×Q8—141no—
473C2×Q8—141no—
474C2×Q8—141no—
475C2×Q8(C2×D7)⋊C414yes—
476C2×D4—24no—
477C2×D4—24no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—72no—
481C2×D4—144no—
482C2×D4—148no—
483C2×D4—141no—
484C2×D4—141no—
485C2×D4—141no—
486C2×D4—141no—
487C2×D4—141no—
488C2×D4—282no—
489C2×D4—282no—
490C2×D4—562no—
491C2×D4—562no—
492C2×D4(C2×D7)⋊C418yes—
493C2×D4(C2×D7)⋊C414yes—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—142no—
499C22×C4—141no—
500C22×C4—144no—
501C22×C4—144no—
502C22×C4—141no—
503C22×C4—141no—
504C22×C4—142no—
505C22×C4—282no—
506C22×C4—282no—
507C22×C4—282no—
508C22×C4—561no—
509C22×C4—561no—
510C22×C4—561no—
511C22×C4—561no—
512C22×C4C2×C4×D714yes—
513C22×C4C2×C4×D712yes—
514C22×C4C2×D2812yes—
515C22×C4C2×D2811yescharacteristic
516C22×C4#115712yes—
517C22×C4#115712yes—
518C22×C4C2×C22⋊D714yes—
519C22×C4C2×C22⋊D712yes—
520C22×C4C2×C22⋊D712yes—
521C22×C4C2×C22⋊D712yes—
522C22×C4C23×D711yescharacteristic
523M16—216no—
524M16—28no—
525C2×C8—216no—
526C2×C8—28no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—21no—
533C2×C8C2×C22⋊D712yes—
534C2×C8C2×C22⋊D714yes—
535C2×C8C2×C22⋊D712yes—
536C4⋊C4—74no—
537C4⋊C4—148no—
538C4⋊C4—144no—
539C4⋊C4—142no—
540C4⋊C4—142no—
541C4⋊C4—141no—
542C4⋊C4—141no—
543C4⋊C4—142no—
544C4⋊C4—141no—
545C4⋊C4—141no—
546C4⋊C4—282no—
547C22⋊C4—148no—
548C22⋊C4—148no—
549C22⋊C4—282no—
550C22⋊C4—562no—
551C22⋊C4—562no—
552C42—282no—
553C2×C7—24no—
554C2×C7—44no—
555C2×C7C2×#38611yescharacteristic
556C2×C7#289711yescharacteristic
557C2×C7#289811yescharacteristic
558C2×C7#306011yescharacteristic
559C2×C7#306111yescharacteristic
560C2×C7#306211yescharacteristic
561C2×C7#306311yescharacteristic
562D7—82no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—142no—
574C23—281no—
575C23—281no—
576C23—281no—
577C23—281no—
578C23—281no—
579C23—281no—
580C23C2×(C2×D7)⋊C414yes—
581C23C2×(C2×D7)⋊C412yes—
582C23#649312yes—
583C23#649312yes—
584C23C22×D2811yescharacteristic
585Q8—28no—
586Q8—142no—
587Q8—142no—
588Q8—281no—
589Q8—281no—
590D4—216no—
591D4—28no—
592D4—142no—
593D4—142no—
594D4—288no—
595D4—281no—
596D4—281no—
597D4—568no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—144no—
605C2×C4—142no—
606C2×C4—282no—
607C2×C4—282no—
608C2×C4—281no—
609C2×C4—281no—
610C2×C4—288no—
611C2×C4—281no—
612C2×C4—282no—
613C2×C4—281no—
614C2×C4—281no—
615C2×C4—282no—
616C2×C4—281no—
617C2×C4—281no—
618C2×C4—281no—
619C2×C4—282no—
620C2×C4—564no—
621C2×C4—564no—
622C2×C4C2×(C2×D7)⋊C414yes—
623C2×C4C2×(C2×D7)⋊C412yes—
624C2×C4C4×C22⋊D714yes—
625C2×C4C4×C22⋊D714yes—
626C2×C4#649214yes—
627C2×C4#649312yes—
628C2×C4#649312yes—
629C2×C4#651614yes—
630C2×C4C22×C4×D711yescharacteristic
631C2×C4C2×#115711yescharacteristic
632C2×C4C2×#115711yescharacteristic
633C2×C4C22×C22⋊D711yescharacteristic
634C2×C4C22×C22⋊D711yescharacteristic
635C2×C4C22×C22⋊D711yescharacteristic
636C8—22no—
637C8—24no—
638C8—22no—
639C7#2097611yescharacteristic, complemented, Sylow
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—282no—
654C22—282no—
655C22—282no—
656C22—561no—
657C22—561no—
658C22—561no—
659C22—561no—
660C22#8998514yes—
661C22#9044211yescharacteristic
662C22#9044211yescharacteristic
663C22#9044311yescharacteristic
664C22#9044411yescharacteristic
665C22C22×(C2×D7)⋊C411yescharacteristic
666C22C2×#649311yescharacteristic
667C22C2×#649311yescharacteristic
668C4—24no—
669C4—282no—
670C4—284no—
671C4—561no—
672C4—561no—
673C4—562no—
674C4#8998514yes—
675C4C2×C4×C22⋊D711yescharacteristic
676C4C2×C4×C22⋊D711yescharacteristic
677C4C2×#649211yescharacteristic
678C4C2×#651611yescharacteristic
679C2—24no—
680C2—44no—
681C2—562no—
682C2#1174124211yescharacteristic
683C2#1174124511yescharacteristic
684C2#1174124611yescharacteristic
685C2#1174124711yescharacteristic
686C2#1174125111yescharacteristic
687C2#1174125311yescharacteristic
688C2C2×#8998511yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC22×C7×#131C2
socleC23×C7C22×D4
prime radicalC2×#90524C2
cosocleC22×C4×C7C24
hypercenterC22×#131D7
hypocenterC7#20976
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×#41 ◁ C22×#131

Complements

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

NH ≅ G/Ntype
1. C22×C7×#131C2semidirect
3. #11741532C2semidirect
4. C2×#89905C2semidirect
5. C2×#89885C2semidirect
6. #11729744C2semidirect
7. #11729743C2semidirect
8. #11729739C2semidirect
9. #11729727C2semidirect
10. #11729726C2semidirect
11. C2×#89346C2semidirect
22. #90524C4semidirect
79. C2×C7×M16D4semidirect
85. C22×C7×C8D4semidirect
97. #6493C2×C4semidirect
112. #6487C2×C4semidirect
123. C22×#131D7semidirect
373. C2×M16C22⋊D7semidirect
379. C22×C8C22⋊D7semidirect
639. C7#20976semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1312
C2×#905242
#117415322
C2×#899052
C2×#898852
#117297442
#117297432
#117297392
#117297272
#117297262
C2×#893462
#209767