Finite Group Databasebeta

← #13371029

Subgroups — #13371029

Order 1792.

1010 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11741533C211yesmaximal, characteristic
2C2×#90004C211yesmaximal, characteristic
3#11731324C211yesmaximal, characteristic, complemented
4C7×#1788C211yesmaximal, characteristic
5#11728010C211yesmaximal, characteristic, complemented
6#11728008C211yesmaximal, characteristic, complemented
7#11727996C211yesmaximal, characteristic, complemented
8#11727995C211yesmaximal, characteristic, complemented
9#11727923C211yesmaximal, characteristic, complemented
10#11727921C211yesmaximal, characteristic, complemented
11#11727906C211yesmaximal, characteristic, complemented
12Dic7×#102C211yesmaximal, characteristic, complemented
13#11727703C211yesmaximal, characteristic, complemented
14#11727678C211yesmaximal, characteristic, complemented
15#11727267C211yesmaximal, characteristic, complemented
16C22×C7×#41C2211yescharacteristic
17C2×D4×Dic7C2211yescharacteristic
18C2×D4×Dic7C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
20C22×#1136C2211yescharacteristic
21C2×C7×#103C2211yescharacteristic
22C2×C7×#103C2211yescharacteristic
23C2×C7×#102C2211yescharacteristic
24C2×C7×#102C2211yescharacteristic
25C2×C7×C22⋊C8C2211yescharacteristic
26C7×#380C2211yescharacteristic
27#90004C418yescomplemented
28#90002C2211yescharacteristic
29#90002C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32#89992—21no—
33#89992—21no—
34C2×#6406C2211yescharacteristic
35C2×#6406C2211yescharacteristic
36#89984C2211yescharacteristic
37#89983—21no—
38C4×C22⋊Dic7C2211yescharacteristic
39C2×C7⋊#95—21no—
40C2×#6389C2211yescharacteristic
41C2×C8×Dic7C2211yescharacteristic
42C4⋊C4×Dic7C2211yescharacteristic
43C4⋊C4×Dic7C2211yescharacteristic
44#89745C2211yescharacteristic
45#89745C2211yescharacteristic
46C2×#6382C2211yescharacteristic
47C2×#6381C2211yescharacteristic
48#89714—21no—
49C22⋊C4×Dic7—21no—
50#89425—21no—
51#89425—21no—
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89345C2211yescharacteristic
57#89345C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22578—71nomaximal, Sylow
61C2×C7×#41—22no—
62C2×C7×#41—22no—
63C2×C7×#41—22no—
64C2×C7×#41—41no—
65C2×C7×#41—41no—
66C2×C7×#41C2×C414yes—
67C22×C7×Q8C2311yescharacteristic
68C22×C7×D4—21no—
69C22×C7×D4C2311yescharacteristic
70C23×C4×C7—21no—
71C23×C4×C7C2311yescharacteristic
72C23×Dic7—21no—
73C23×Dic7—21no—
74C23×Dic7—21no—
75C22×C7×C8C2311yescharacteristic
76C22×C7×C8C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C22⋊C4—21no—
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7—21no—
82C2×C22⋊Dic7—21no—
83C2×C22⋊Dic7D411yescharacteristic
84C2×C22⋊Dic7D411yescharacteristic
85D4×Dic7—22no—
86D4×Dic7—22no—
87D4×Dic7—41no—
88D4×Dic7—41no—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7—22no—
91C2×C4⋊Dic7C2×C414yescomplemented
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7C2311yescharacteristic
101C2×C4×Dic7C2311yescharacteristic
102C2×#1136—41no—
103C2×#1136—41no—
104C2×#1136C2×C414yescomplemented
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C7×#103—41no—
108C7×#103—41no—
109C7×#102—22no—
110C7×#102—22no—
111C7×C22⋊C8—22no—
112C7×#95—21no—
113#6409C2×C414yescomplemented
114#6409C2×C414yescomplemented
115#6406C2×C414yes—
116#6406C2×C414yes—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95D411yescharacteristic
127C7⋊#95D411yescharacteristic
128#6389—41no—
129C8×Dic7—41no—
130#6382—41no—
131#6381—41no—
132C2×#417—71no—
133#2345—71no—
134#2339—71no—
135#1935—71no—
136#1933—71no—
137#1928—71no—
138#1927—71no—
139#1864—71no—
140#1862—71no—
141#1823—71no—
142#1809—71no—
143#1788D711yescharacteristic
144#1762—71no—
145C4×#102—71no—
146#1753—71no—
147C24×C7—21no—
148C24×C7C2×D411yescharacteristic
149C7×#41—28no—
150C7×#41—44no—
151C7×#41—44no—
152C7×#41—44no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8C22×C412yes—
159C2×C7×D4—24no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—42no—
164C2×C7×D4—42no—
165C2×C7×D4C22×C412yes—
166C2×C7×D4#4112yes—
167C2×C7×D4#4112yes—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—21no—
171C22×C4×C7—21no—
172C22×C4×C7—21no—
173C22×C4×C7—24no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—42no—
177C22×C4×C7—42no—
178C22×C4×C7C22×C412yes—
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7#4112yes—
181C22×C4×C7C2411yescharacteristic
182C22×Dic7—21no—
183C22×Dic7—21no—
184C22×Dic7—24no—
185C22×Dic7—21no—
186C22×Dic7—21no—
187C22×Dic7—42no—
188C22×Dic7—42no—
189C22×Dic7—42no—
190C22×Dic7—42no—
191C22×Dic7—81no—
192C22×Dic7—81no—
193C22×Dic7—81no—
194C22×Dic7—81no—
195C22×Dic7C2×D411yescharacteristic
196C22×Dic7C2×D411yescharacteristic
197C22×Dic7C2×D411yescharacteristic
198C22×Dic7C2×D411yescharacteristic
199C2×C7×C8—21no—
200C2×C7×C8—21no—
201C2×C7×C8—21no—
202C2×C7×C8—21no—
203C2×C7×C8—21no—
204C2×C7×C8—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C22⋊C4—42no—
210C22⋊Dic7—22no—
211C22⋊Dic7—41no—
212C22⋊Dic7—42no—
213C22⋊Dic7—42no—
214C22⋊Dic7—81no—
215C4⋊Dic7—24no—
216C4⋊Dic7—21no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219C4⋊Dic7—21no—
220C4⋊Dic7—21no—
221C4⋊Dic7—21no—
222C4⋊Dic7C22×C412yes—
223C4⋊Dic7C22×C412yes—
224Dic7⋊C4—81no—
225Dic7⋊C4—81no—
226C4×Dic7—21no—
227C4×Dic7—21no—
228C4×Dic7—21no—
229C4×Dic7—21no—
230C4×Dic7—81no—
231C4×Dic7—81no—
232#1136—44no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—21no—
236C2×C7⋊C8—21no—
237C2×C7⋊C8C22×C412yes—
238C2×C7⋊C8C22×C412yes—
239C22×#41C2×D711yescharacteristic
240C22×M16—71no—
241C2×C4×D4—71no—
242C2×C4×D4—71no—
243C22×C4⋊C4—71no—
244C2×#107—71no—
245C2×#106—71no—
246#417—78no—
247C2×#103—71no—
248C2×#103—71no—
249C2×#103C2×D711yescharacteristic
250C2×#103C2×D711yescharacteristic
251C2×#102—71no—
252C2×#102—71no—
253C2×#102C2×D711yescharacteristic
254C2×#102C2×D711yescharacteristic
255C2×C22⋊C8C2×D711yescharacteristic
256C2×#97—71no—
257C2×C4×C8—71no—
258#391—71no—
259#391—71no—
260#389—71no—
261#389—71no—
262#387—141no—
263#387—141no—
264#386—141no—
265#386—141no—
266#381—141no—
267#380—71no—
268#380—141no—
269#380C2×D711yescharacteristic
270C4×C4⋊C4—71no—
271C4×C4⋊C4—71no—
272C4×C22⋊C4—71no—
273C4×C22⋊C4—141no—
274C2×#95—141no—
275#340—71no—
276#340—71no—
277#340—71no—
278#340—71no—
279#340—71no—
280#340—71no—
281#336—71no—
282#336—71no—
283C23×C7—22no—
284C23×C7—22no—
285C23×C7—22no—
286C23×C7—21no—
287C23×C7—21no—
288C23×C7—21no—
289C23×C7—21no—
290C23×C7—21no—
291C23×C7—21no—
292C23×C7—41no—
293C23×C7—41no—
294C23×C7—41no—
295C23×C7—41no—
296C23×C7C4×D412yes—
297C23×C7C22×D411yescharacteristic
298C7×Q8—22no—
299C7×Q8—22no—
300C7×Q8—41no—
301C7×Q8—41no—
302C7×D4—22no—
303C7×D4—22no—
304C7×D4—22no—
305C7×D4—48no—
306C7×D4#11714yes—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—22no—
310C2×C4×C7—22no—
311C2×C4×C7—22no—
312C2×C4×C7—22no—
313C2×C4×C7—22no—
314C2×C4×C7—48no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
317C2×C4×C7—41no—
318C2×C4×C7—41no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7—41no—
322C2×C4×C7—41no—
323C2×C4×C7C4×D412yes—
324C2×C4×C7C23×C411yescharacteristic
325C2×C4×C7C2×#4111yescharacteristic
326C2×C4×C7C2×#4111yescharacteristic
327C2×C4×C7C2×#4111yescharacteristic
328C2×C4×C7H(5,𝔽2)11yescharacteristic
329C2×C4×C7H(5,𝔽2)11yescharacteristic
330C2×Dic7—21no—
331C2×Dic7—21no—
332C2×Dic7—22no—
333C2×Dic7—22no—
334C2×Dic7—21no—
335C2×Dic7—21no—
336C2×Dic7—21no—
337C2×Dic7—21no—
338C2×Dic7—21no—
339C2×Dic7—21no—
340C2×Dic7—21no—
341C2×Dic7—21no—
342C2×Dic7—21no—
343C2×Dic7—41no—
344C2×Dic7—41no—
345C2×Dic7—41no—
346C2×Dic7—41no—
347C2×Dic7—41no—
348C2×Dic7—41no—
349C2×Dic7—41no—
350C2×Dic7—41no—
351C2×Dic7—41no—
352C2×Dic7—41no—
353C2×Dic7—84no—
354C2×Dic7—84no—
355C2×Dic7—84no—
356C2×Dic7#13712yes—
357C7×C8—41no—
358C7×C8—41no—
359C7⋊C8—22no—
360C7⋊C8—22no—
361C2×#41—22no—
362C2×#41—22no—
363C2×#41—22no—
364C2×#41—41no—
365C2×#41—41no—
366C2×#41C4×D714yes—
367C22×Q8C22×D711yescharacteristic
368C22×D4—21no—
369C22×D4C22×D711yescharacteristic
370C23×C4—21no—
371C23×C4—141no—
372C23×C4—141no—
373C23×C4—141no—
374C23×C4C22×D711yescharacteristic
375C2×M16—74no—
376C2×M16—281no—
377C2×M16—281no—
378C22×C8—71no—
379C22×C8—71no—
380C22×C8C22×D711yescharacteristic
381C22×C8C22×D711yescharacteristic
382C4×D4—142no—
383C4×D4—142no—
384C4×D4—281no—
385C4×D4—281no—
386C2×C4⋊C4—74no—
387C2×C4⋊C4—71no—
388C2×C4⋊C4—71no—
389C2×C4⋊C4—71no—
390C2×C4⋊C4—71no—
391C2×C4⋊C4—141no—
392C2×C4⋊C4—142no—
393C2×C4⋊C4—141no—
394C2×C4⋊C4—141no—
395C2×C4⋊C4C22×D711yescharacteristic
396C2×C4⋊C4C22×D711yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—71no—
399C2×C22⋊C4—71no—
400C2×C22⋊C4—141no—
401C2×C22⋊C4—141no—
402C2×C22⋊C4—141no—
403C2×C42—71no—
404C2×C42—71no—
405C2×C42—141no—
406C2×C42—141no—
407#107—281no—
408#106—281no—
409#103—41no—
410#103—41no—
411#103—74no—
412#103—74no—
413#102—22no—
414#102—22no—
415#102—74no—
416#102—74no—
417C22⋊C8—22no—
418#97—281no—
419C4×C8—281no—
420#95—21no—
421#95—71no—
422#95—71no—
423#95—141no—
424#95—141no—
425#95—141no—
426#95—141no—
427#95—141no—
428#95—141no—
429#95—141no—
430#95—141no—
431#95—141no—
432C22×C7—22no—
433C22×C7—24no—
434C22×C7—22no—
435C22×C7—22no—
436C22×C7—41no—
437C22×C7—42no—
438C22×C7—42no—
439C22×C7—42no—
440C22×C7—41no—
441C22×C7—42no—
442C22×C7—42no—
443C22×C7—41no—
444C22×C7—42no—
445C22×C7—41no—
446C22×C7C2×C4×D411yescharacteristic
447C22×C7#54611yescharacteristic
448C22×C7#54611yescharacteristic
449C22×C7C2×#13711yescharacteristic
450C22×C7#57511yescharacteristic
451C22×C7#57611yescharacteristic
452C22×C7#57711yescharacteristic
453C4×C7—22no—
454C4×C7—42no—
455C4×C7—81no—
456C4×C7—81no—
457C4×C7C2×#11711yescharacteristic
458C4×C7#51811yescharacteristic
459C4×C7#55111yescharacteristic
460C4×C7#55111yescharacteristic
461Dic7—22no—
462Dic7—41no—
463Dic7—42no—
464Dic7—42no—
465Dic7—41no—
466Dic7—41no—
467Dic7—81no—
468Dic7—81no—
469C24—21no—
470C24D4×D711yescharacteristic
471#41—28no—
472#41—44no—
473#41—44no—
474#41—44no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8C2×C4×D712yes—
481C2×D4—24no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4C2×C4×D712yes—
488C2×D4#115912yes—
489C2×D4#115912yes—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—24no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—71no—
501C22×C4—71no—
502C22×C4—71no—
503C22×C4—71no—
504C22×C4—141no—
505C22×C4—141no—
506C22×C4—144no—
507C22×C4—141no—
508C22×C4—141no—
509C22×C4—282no—
510C22×C4—282no—
511C22×C4—282no—
512C22×C4—282no—
513C22×C4—561no—
514C22×C4—561no—
515C22×C4—561no—
516C22×C4—561no—
517C22×C4C2×C4×D712yes—
518C22×C4D4×D711yescharacteristic
519C22×C4#115912yes—
520C22×C4C23×D711yescharacteristic
521M16—284no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—72no—
529C2×C8—72no—
530C2×C8—141no—
531C2×C8—141no—
532C2×C8—141no—
533C2×C8—141no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—72no—
539C4⋊C4—72no—
540C4⋊C4—144no—
541C4⋊C4—141no—
542C4⋊C4—141no—
543C4⋊C4—141no—
544C4⋊C4—141no—
545C4⋊C4—141no—
546C4⋊C4—141no—
547C4⋊C4—561no—
548C4⋊C4—561no—
549C22⋊C4—42no—
550C22⋊C4—142no—
551C22⋊C4—281no—
552C22⋊C4—282no—
553C22⋊C4—282no—
554C22⋊C4—561no—
555C42—141no—
556C42—141no—
557C42—141no—
558C42—141no—
559C42—561no—
560C42—561no—
561C2×C7—42no—
562C2×C7—44no—
563C2×C7#234211yescharacteristic
564C2×C7C4×#13711yescharacteristic
565C2×C7#294611yescharacteristic
566C2×C7#330711yescharacteristic
567C2×C7#331411yescharacteristic
568C2×C7#332211yescharacteristic
569C2×C7#333111yescharacteristic
570C23—22no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—22no—
577C23—21no—
578C23—21no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23(C22⋊D7)⋊C412yes—
584C23C2×D4×D711yescharacteristic
585Q8—22no—
586Q8—22no—
587Q8—41no—
588Q8—41no—
589D4—22no—
590D4—22no—
591D4—22no—
592D4—48no—
593D4#644014yes—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—48no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—72no—
611C2×C4—141no—
612C2×C4—141no—
613C2×C4—142no—
614C2×C4—142no—
615C2×C4—141no—
616C2×C4—141no—
617C2×C4—141no—
618C2×C4—141no—
619C2×C4—141no—
620C2×C4—141no—
621C2×C4—141no—
622C2×C4—141no—
623C2×C4—141no—
624C2×C4—281no—
625C2×C4—281no—
626C2×C4—281no—
627C2×C4—281no—
628C2×C4—281no—
629C2×C4—281no—
630C2×C4—281no—
631C2×C4—281no—
632C2×C4—281no—
633C2×C4—281no—
634C2×C4—564no—
635C2×C4—564no—
636C2×C4—564no—
637C2×C4(C22⋊D7)⋊C412yes—
638C2×C4C22×C4×D711yescharacteristic
639C2×C4C2×#115911yescharacteristic
640C2×C4C2×#115911yescharacteristic
641C2×C4C2×#115911yescharacteristic
642C2×C4#654811yescharacteristic
643C2×C4#654811yescharacteristic
644C8—41no—
645C8—41no—
646C8—142no—
647C8—142no—
648C7#2257811yescharacteristic, complemented, Sylow
649C22—22no—
650C22—24no—
651C22—22no—
652C22—22no—
653C22—42no—
654C22—41no—
655C22—41no—
656C22—42no—
657C22—42no—
658C22—41no—
659C22—42no—
660C22—42no—
661C22—41no—
662C22—42no—
663C22C2×(C22⋊D7)⋊C411yescharacteristic
664C22#9028211yescharacteristic
665C22#9028211yescharacteristic
666C22#9046011yescharacteristic
667C22#9046111yescharacteristic
668C22#9046411yescharacteristic
669C22D7×#13711yescharacteristic
670C4—22no—
671C4—42no—
672C4—81no—
673C4—81no—
674C4—142no—
675C4—281no—
676C4—282no—
677C4—282no—
678C4—281no—
679C4—281no—
680C4—561no—
681C4—561no—
682C4C2×#644011yescharacteristic
683C4#9017711yescharacteristic
684C4#9027811yescharacteristic
685C4#9027811yescharacteristic
686C2—42no—
687C2—44no—
688C2#1173126711yescharacteristic
689C2#1173626511yescharacteristic
690C2#1173631211yescharacteristic
691C2#1173755311yescharacteristic
692C2#1173755911yescharacteristic
693C2#1173769711yescharacteristic
694C2#1173770511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1788C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenter#1788D7
hypocenterC7#22578
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
3. #11731324C2semidirect
5. #11728010C2semidirect
6. #11728008C2semidirect
7. #11727996C2semidirect
8. #11727995C2semidirect
9. #11727923C2semidirect
10. #11727921C2semidirect
11. #11727906C2semidirect
12. Dic7×#102C2semidirect
13. #11727703C2semidirect
14. #11727678C2semidirect
15. #11727267C2semidirect
27. #90004C4semidirect
91. C2×C4⋊Dic7C2×C4semidirect
104. C2×#1136C2×C4semidirect
113. #6409C2×C4semidirect
114. #6409C2×C4semidirect
648. C7#22578semidirect

Maximal subgroups

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

subgroupindex
#117415332
C2×#900042
#117313242
C7×#17882
#117280102
#117280082
#117279962
#117279952
#117279232
#117279212
#117279062
Dic7×#1022
#117277032
#117276782
#117272672
#225787