Finite Group Databasebeta

← #13355544

Subgroups — #13355544

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
1C2×#90524C211yesmaximal, characteristic
2C22×#6470C211yesmaximal, characteristic, complemented
3C2×C7×#417C211yesmaximal, characteristic, complemented
4C2×#90005C211yesmaximal, characteristic, complemented
5#11739631C211yesmaximal, characteristic, complemented
6#11729739C211yesmaximal, characteristic, complemented
7C2×#89324C211yesmaximal, characteristic, complemented
8#11728025C212yesmaximal, complemented
9#11728024C212yesmaximal, complemented
10#11727957C212yesmaximal, complemented
11#11727956C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C22×#1157C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16#90524—24no—
17#90524C418yescomplemented
18C2×#6509C2212yes—
19C2×#6502—22no—
20C2×#6501C2212yes—
21C2×#6493—21no—
22C2×#6489C2211yescharacteristic
23C22×C4⋊Dic7C2211yescharacteristic
24C2×#6487—21no—
25C23×C7⋊C8C2211yescharacteristic
26C2×#6470—24no—
27C2×#6470—24no—
28C2×#6470—24no—
29C22×#1131C2212yes—
30C22×C8×D7C2212yes—
31C7×#417—24no—
32C2×C7×#103C2212yes—
33C2×C7×#102C2212yes—
34#90005—24no—
35C2×#6409C2211yescharacteristic
36C2×#6409C2211yescharacteristic
37C2×#6406C2211yescharacteristic
38C2×#6406C2211yescharacteristic
39#89985—22no—
40#89760—21no—
41#89759C2212yes—
42#89759C2212yes—
43#89746—21no—
44#89346C2212yes—
45#89346C2212yes—
46#89324C2212yes—
47#89324C2212yes—
48#20976—71nomaximal, Sylow
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41—22no—
53C2×C7×#41—22no—
54C2×C7×#41C2×C414yes—
55C22×C7×Q8C2311yescharacteristic
56C22×C7×D4—21no—
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C23×C4×C7—21no—
60C23×C4×C7C2311yescharacteristic
61C22×C22⋊D7—22no—
62C22×C22⋊D7D412yes—
63C23×Dic7—21no—
64C2×#1157—22no—
65C2×#1157—22no—
66C2×#1157—22no—
67C2×#1157—22no—
68C2×#1157C2×C414yes—
69C2×#1157D414yes—
70C22×D28D411yescharacteristic
71C22×C4×D7C2312yes—
72C22×Dic14D411yescharacteristic
73C2×C7×M16—22no—
74C2×C7×M16—22no—
75C2×C7×M16D414yescomplemented
76C22×C7×C8C2312yes—
77C2×C7×C4⋊C4—22no—
78C2×C7×C4⋊C4—22no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4C2312yes—
81C2×C22⋊Dic7—22no—
82C2×C22⋊Dic7—21no—
83#6509—24no—
84#6509C2×C418yescomplemented
85#6502—216no—
86#6501—24no—
87#6501C2×C418yescomplemented
88#6493—28no—
89C2×(C2×D7)⋊C4—22no—
90C2×(C2×D7)⋊C4—22no—
91C2×(C2×D7)⋊C4—22no—
92#6489—22no—
93#6489—22no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7—22no—
96C2×C4⋊Dic7C2×C414yes—
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4⋊Dic7C2312yes—
100#6487—28no—
101C2×Dic7⋊C4—22no—
102C2×C4×Dic7—21no—
103C22×C7⋊C8—22no—
104C22×C7⋊C8—22no—
105C22×C7⋊C8D414yescomplemented
106C22×C7⋊C8C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108#6470—232no—
109C2×#1131—24no—
110C2×#1131—24no—
111C2×#1131—24no—
112C2×C8×D7—24no—
113C2×C8×D7—24no—
114C2×C8×D7—24no—
115C7×#103—24no—
116C7×#102—24no—
117#6409—22no—
118#6409—22no—
119#6406—22no—
120#6406—22no—
121C7⋊#95—22no—
122C7⋊#95—22no—
123C22×#131—71no—
124C2×#537—71no—
125C2×#417D711yescharacteristic, complemented
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×C7—21no—
135C24×C7C2×D411yescharacteristic
136C23×D7C2×D412yes—
137C7×#41—28no—
138C7×#41—28no—
139C7×#41—44no—
140C7×#41—44no—
141C2×C7×Q8—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8C22×C412yes—
147C2×C7×D4—24no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4—21no—
153C2×C7×D4—42no—
154C2×C7×D4—42no—
155C2×C7×D4C22×C412yes—
156C22×C4×C7—21no—
157C22×C4×C7—24no—
158C22×C4×C7—22no—
159C22×C4×C7—42no—
160C22×C4×C7—42no—
161C22×C4×C7—42no—
162C22×C4×C7—81no—
163C22×C4×C7—81no—
164C22×C4×C7C22×C412yes—
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D412yes—
167C22×C4×C7C2×D412yes—
168C22×C4×C7C2×D412yes—
169C22×C4×C7#4112yes—
170C22×C4×C7#4112yes—
171C22×C4×C7C2411yescharacteristic
172C2×C22⋊D7—24no—
173C2×C22⋊D7—24no—
174C2×C22⋊D7—28no—
175C2×C22⋊D7—82no—
176C2×C22⋊D7—82no—
177C2×C22⋊D7C22⋊C418yes—
178C22×Dic7—22no—
179C22×Dic7—24no—
180C22×Dic7—21no—
181C22×Dic7—21no—
182C22×Dic7—81no—
183C22×Dic7—81no—
184C22×Dic7C2×D412yes—
185#1157—28no—
186#1157—28no—
187#1157—28no—
188#1157—28no—
189C2×D28—22no—
190C2×D28—22no—
191C2×D28C22⋊C414yes—
192C2×C4×D7—22no—
193C2×C4×D7—22no—
194C2×C4×D7—22no—
195C2×C4×D7—22no—
196C2×C4×D7C22×C414yes—
197C2×C4×D7C2×D414yes—
198C2×Dic14—22no—
199C2×Dic14—22no—
200C2×Dic14C22⋊C414yes—
201C7×M16—28no—
202C2×C7×C8—22no—
203C2×C7×C8—22no—
204C2×C7×C8C2×D414yes—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—42no—
208C22⋊Dic7—28no—
209C22⋊Dic7—42no—
210(C2×D7)⋊C4—28no—
211(C2×D7)⋊C4—82no—
212(C2×D7)⋊C4—82no—
213C4⋊Dic7—24no—
214C4⋊Dic7—22no—
215C4⋊Dic7—21no—
216C4⋊Dic7—21no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219C4⋊Dic7C22×C414yes—
220Dic7⋊C4—28no—
221C4×Dic7—42no—
222C2×C7⋊C8—28no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8C2×D412yes—
228C2×C7⋊C8C2×D412yes—
229#1131—216no—
230C8×D7—216no—
231C22×#41—71no—
232C22×#41C2×D711yescharacteristic
233C2×#131—144no—
234C2×#131—144no—
235C2×#131—144no—
236C22×M16—72no—
237C22×M16C2×D711yescharacteristic
238C23×C8—71no—
239C23×C8—72no—
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⋊C4C2×D711yescharacteristic
250#417—24no—
251#416—144no—
252C2×#103—71no—
253C2×#103—71no—
254C2×#103C2×D712yes—
255C2×#102—71no—
256C2×#102—71no—
257C2×#102C2×D712yes—
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—41no—
273C23×C7—41no—
274C23×C7—41no—
275C23×C7—41no—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7C2×C22⋊C412yes—
279C23×C7#12112yes—
280C23×C7#12112yes—
281C23×C7C22×D411yescharacteristic
282C22×D7—22no—
283C22×D7—22no—
284C22×D7—22no—
285C22×D7—22no—
286C22×D7—22no—
287C22×D7—22no—
288C22×D7C2×C22⋊C414yes—
289C7×Q8—22no—
290C7×Q8—22no—
291C7×Q8—41no—
292C7×Q8—41no—
293C7×D4—22no—
294C7×D4—22no—
295C7×D4—48no—
296C7×D4—41no—
297C7×D4—41no—
298C2×C4×C7—22no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—48no—
302C2×C4×C7—42no—
303C2×C4×C7—42no—
304C2×C4×C7—42no—
305C2×C4×C7—84no—
306C2×C4×C7C2×C22⋊C412yes—
307C2×C4×C7C4×D414yes—
308C2×C4×C7C4×D414yes—
309C2×C4×C7#12014yes—
310C2×C4×C7#12112yes—
311C2×C4×C7#12112yes—
312C2×C4×C7#12314yes—
313C2×C4×C7C23×C411yescharacteristic
314C2×C4×C7C22×D411yescharacteristic
315C2×C4×C7C22×D411yescharacteristic
316C2×C4×C7C22×D411yescharacteristic
317C2×C4×C7C2×#4111yescharacteristic
318C2×C4×C7C2×#4111yescharacteristic
319C22⋊D7—216no—
320C22⋊D7—88no—
321C2×Dic7—22no—
322C2×Dic7—22no—
323C2×Dic7—24no—
324C2×Dic7—42no—
325C2×Dic7—42no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—84no—
333C2×Dic7C2×C22⋊C414yes—
334D28—28no—
335C4×D7—24no—
336C4×D7—24no—
337C4×D7—24no—
338C4×D7—24no—
339Dic14—28no—
340C7×C8—24no—
341C7⋊C8—22no—
342C7⋊C8—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—74no—
349C2×#41—74no—
350C2×#41—142no—
351C2×#41—142no—
352C2×#41—142no—
353C2×#41—142no—
354C2×#41C4×D714yes—
355C22×Q8—71no—
356C22×Q8C22×D711yescharacteristic
357C22×D4—21no—
358C22×D4—72no—
359C22×D4—71no—
360C22×D4—142no—
361C22×D4C22×D711yescharacteristic
362C23×C4—21no—
363C23×C4—21no—
364C23×C4—72no—
365C23×C4—141no—
366C23×C4C22×D711yescharacteristic
367#131—1432no—
368C2×M16—22no—
369C2×M16—22no—
370C2×M16—144no—
371C2×M16—144no—
372C2×M16—144no—
373C2×M16C22⋊D714yescomplemented
374C22×C8—74no—
375C22×C8—71no—
376C22×C8—71no—
377C22×C8—144no—
378C22×C8—144no—
379C22×C8—142no—
380C22×C8—142no—
381C22×C8—144no—
382C22×C8C22×D712yes—
383C22⋊Q8—78no—
384C22⋊Q8—148no—
385C22⋊Q8—144no—
386#121—78no—
387#121—148no—
388#121—144no—
389#121—1416no—
390#117—142no—
391#117—142no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—74no—
396C2×C4⋊C4—72no—
397C2×C4⋊C4—71no—
398C2×C4⋊C4—71no—
399C2×C4⋊C4—142no—
400C2×C4⋊C4—141no—
401C2×C4⋊C4—142no—
402C2×C4⋊C4C22×D712yes—
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—24no—
410#103—142no—
411#103—142no—
412#102—24no—
413#102—142no—
414#102—142no—
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—42no—
424C22×C7—42no—
425C22×C7—42no—
426C22×C7—81no—
427C22×C7—81no—
428C22×C7—81no—
429C22×C7—81no—
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—24no—
439C2×D7—24no—
440C2×D7—24no—
441C2×D7—42no—
442C2×D7—42no—
443C2×D7—42no—
444C2×D7—42no—
445C4×C7—42no—
446C4×C7—82no—
447C4×C7#38614yes—
448C4×C7C2×C4×D411yescharacteristic
449C4×C7C2×C4×D411yescharacteristic
450C4×C7C2×#12011yescharacteristic
451C4×C7C2×#12311yescharacteristic
452Dic7—24no—
453Dic7—44no—
454Dic7—81no—
455Dic7—81no—
456C24—21no—
457C24—72no—
458C24D4×D711yescharacteristic
459#41—28no—
460#41—28no—
461#41—44no—
462#41—44no—
463#41—148no—
464#41—148no—
465#41—148no—
466#41—148no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—74no—
473C2×Q8—142no—
474C2×Q8—142no—
475C2×Q8C2×C4×D712yes—
476C2×D4—24no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—78no—
485C2×D4—74no—
486C2×D4—144no—
487C2×D4—144no—
488C2×D4—148no—
489C2×D4—142no—
490C2×D4—142no—
491C2×D4—562no—
492C2×D4—562no—
493C2×D4C2×C4×D712yes—
494C22×C4—21no—
495C22×C4—24no—
496C22×C4—22no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—72no—
501C22×C4—74no—
502C22×C4—74no—
503C22×C4—81no—
504C22×C4—81no—
505C22×C4—142no—
506C22×C4—142no—
507C22×C4—142no—
508C22×C4—142no—
509C22×C4—142no—
510C22×C4—144no—
511C22×C4—141no—
512C22×C4—141no—
513C22×C4—561no—
514C22×C4—561no—
515C22×C4C2×C4×D712yes—
516C22×C4#115712yes—
517C22×C4D4×D711yescharacteristic
518C22×C4D4×D712yes—
519C22×C4D4×D712yes—
520C22×C4#115912yes—
521C22×C4C2×C22⋊D712yes—
522C22×C4C23×D711yescharacteristic
523M16—28no—
524M16—1416no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—72no—
528C2×C8—72no—
529C2×C8—1416no—
530C2×C8—148no—
531C2×C8—141no—
532C2×C8—141no—
533C2×C8—141no—
534C2×C8—141no—
535C2×C8C2×C22⋊D714yes—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—42no—
539C4⋊C4—74no—
540C4⋊C4—148no—
541C4⋊C4—144no—
542C4⋊C4—142no—
543C4⋊C4—141no—
544C4⋊C4—141no—
545C4⋊C4—141no—
546C4⋊C4—141no—
547C22⋊C4—148no—
548C22⋊C4—148no—
549C22⋊C4—282no—
550C22⋊C4—562no—
551C22⋊C4—562no—
552C42—282no—
553C2×C7—24no—
554C2×C7—82no—
555C2×C7C2×#38611yescharacteristic
556C2×C7#289711yescharacteristic
557C2×C7#289811yescharacteristic
558C2×C7#306011yescharacteristic
559C2×C7#306111yescharacteristic
560C2×C7#306211yescharacteristic
561C2×C7#306311yescharacteristic
562D7—44no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—74no—
575C23—142no—
576C23—142no—
577C23—142no—
578C23—142no—
579C23—142no—
580C23—142no—
581C23D7×C22⋊C412yes—
582C23#644712yes—
583C23#650112yes—
584C23C2×D4×D711yescharacteristic
585Q8—22no—
586Q8—22no—
587Q8—41no—
588Q8—41no—
589Q8—148no—
590D4—22no—
591D4—22no—
592D4—48no—
593D4—41no—
594D4—41no—
595D4—1416no—
596D4—148no—
597D4—568no—
598C2×C4—22no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—48no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—74no—
606C2×C4—84no—
607C2×C4—142no—
608C2×C4—142no—
609C2×C4—144no—
610C2×C4—144no—
611C2×C4—144no—
612C2×C4—144no—
613C2×C4—144no—
614C2×C4—282no—
615C2×C4—282no—
616C2×C4—281no—
617C2×C4—281no—
618C2×C4—281no—
619C2×C4—281no—
620C2×C4—281no—
621C2×C4—281no—
622C2×C4—564no—
623C2×C4D7×C22⋊C412yes—
624C2×C4(C22⋊D7)⋊C414yes—
625C2×C4#644614yes—
626C2×C4#644712yes—
627C2×C4C4×C22⋊D714yes—
628C2×C4#650014yes—
629C2×C4#650112yes—
630C2×C4C22×C4×D711yescharacteristic
631C2×C4C2×#115711yescharacteristic
632C2×C4C2×D4×D711yescharacteristic
633C2×C4C2×D4×D711yescharacteristic
634C2×C4C2×#115911yescharacteristic
635C2×C4C22×C22⋊D711yescharacteristic
636C8—24no—
637C8—142no—
638C8—142no—
639C7#2097611yescharacteristic, complemented, Sylow
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—144no—
654C22—144no—
655C22—144no—
656C22—282no—
657C22—282no—
658C22—282no—
659C22—282no—
660C22#8972814yes—
661C22C2×D7×C22⋊C411yescharacteristic
662C22C2×#644711yescharacteristic
663C22#9046511yescharacteristic
664C22#9046611yescharacteristic
665C22#9046911yescharacteristic
666C22#9047011yescharacteristic
667C22C2×#650111yescharacteristic
668C4—42no—
669C4—82no—
670C4—144no—
671C4—284no—
672C4—561no—
673C4—561no—
674C4#8972814yes—
675C4C2×(C22⋊D7)⋊C411yescharacteristic
676C4C2×#644611yescharacteristic
677C4C2×C4×C22⋊D711yescharacteristic
678C4C2×#650011yescharacteristic
679C2—24no—
680C2—82no—
681C2—284no—
682C2#1173632711yescharacteristic
683C2#1173632811yescharacteristic
684C2#1173633511yescharacteristic
685C2#1173633611yescharacteristic
686C2C2×#8972811yescharacteristic
687C2#1174139611yescharacteristic
688C2#1174143111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#417C2
socleC23×C7C22×D4
prime radicalC2×#90524C2
cosocleC22×C4×C7C24
hypercenterC2×#417D7
hypocenterC7#20976
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#6470C2semidirect
3. C2×C7×#417C2semidirect
4. C2×#90005C2semidirect
5. #11739631C2semidirect
6. #11729739C2semidirect
7. C2×#89324C2semidirect
8. #11728025C2semidirect
9. #11728024C2semidirect
10. #11727957C2semidirect
11. #11727956C2semidirect
17. #90524C4semidirect
75. C2×C7×M16D4semidirect
84. #6509C2×C4semidirect
87. #6501C2×C4semidirect
105. C22×C7⋊C8D4semidirect
125. C2×#417D7semidirect
373. C2×M16C22⋊D7semidirect
639. C7#20976semidirect

Maximal subgroups

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

subgroupindex
C2×#905242
C22×#64702
C2×C7×#4172
C2×#900052
#117396312
#117297392
C2×#893242
#117280252
#117280242
#117279572
#117279562
#209767