Finite Group Databasebeta

← #13410913

Subgroups — #13410913

Order 1792.

994 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2339C211yesmaximal, characteristic
2C2×#90005C211yesmaximal, characteristic
3C2×#89779C211yesmaximal, characteristic, complemented
4#11730937C211yesmaximal, characteristic, complemented
5#11729889C211yesmaximal, characteristic
6#11729361C211yesmaximal, characteristic, complemented
7#11729360C211yesmaximal, characteristic, complemented
8#11729359C211yesmaximal, characteristic, complemented
9#11727440C211yesmaximal, characteristic, complemented
10#11727437C211yesmaximal, characteristic, complemented
11#11727437C211yesmaximal, characteristic, complemented
12#11727422C211yesmaximal, characteristic, complemented
13#11727266C211yesmaximal, characteristic, complemented
14#11727260C211yesmaximal, characteristic, complemented
15#11727203C211yesmaximal, characteristic, complemented
16C22×C7×#41C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C2×#6489C2211yescharacteristic
20C23×C7⋊C8C2211yescharacteristic
21C2×#6432C2211yescharacteristic
22C7×#391C2211yescharacteristic
23C7×#391C2211yescharacteristic
24C7×#386—21no—
25C7×#386—21no—
26C7×#381—21no—
27C7×#380C2211yescharacteristic
28C4×C7×C22⋊C4C2211yescharacteristic
29C2×C7×#95—21no—
30#90005—24no—
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35C2×#6406C2211yescharacteristic
36C2×#6406C2211yescharacteristic
37C2×#6406C2211yescharacteristic
38C2×#6406C2211yescharacteristic
39#89984C2211yescharacteristic
40C2×C22⋊(C7⋊C8)C2211yescharacteristic
41#89779—24no—
42C2×#6382C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46#89720—21no—
47#89719—21no—
48#89708C2211yescharacteristic
49#89700C2211yescharacteristic
50#89700C2211yescharacteristic
51C2×C4⋊(C7⋊C8)C2211yescharacteristic
52C2×C4⋊(C7⋊C8)C2211yescharacteristic
53#89427—21no—
54#89418—21no—
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#25954—71nomaximal, Sylow
62C2×C7×#41—22no—
63C2×C7×#41—22no—
64C2×C7×#41—22no—
65C2×C7×#41—41no—
66C2×C7×#41—41no—
67C2×C7×#41Q814yescomplemented
68C22×C7×Q8C2311yescharacteristic
69C22×C7×D4—21no—
70C22×C7×D4C2311yescharacteristic
71C23×C4×C7—21no—
72C23×C4×C7—21no—
73C23×C4×C7—21no—
74C23×C4×C7C2311yescharacteristic
75C22×Dic14—21no—
76C4×C7×D4—22no—
77C4×C7×D4—22no—
78C4×C7×D4—41no—
79C4×C7×D4—41no—
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4D411yescharacteristic
86C2×C7×C22⋊C4D411yescharacteristic
87C2×C42×C7C2311yescharacteristic
88C2×C42×C7C2311yescharacteristic
89C2×C22⋊Dic7—21no—
90C2×C22⋊Dic7—21no—
91#6489—22no—
92#6489—41no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×C4×Dic7—21no—
101C22×C7⋊C8—22no—
102C22×C7⋊C8—22no—
103C22×C7⋊C8—41no—
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108#6432—22no—
109#6432—41no—
110C7×#95—21no—
111C7×#95—21no—
112C7×#95—21no—
113C7×#95—21no—
114C7×#95—21no—
115C7×#95D411yescharacteristic
116C7×#95D411yescharacteristic
117#6409—22no—
118#6409—22no—
119#6409—41no—
120#6409—41no—
121#6406—22no—
122#6406—22no—
123#6406—22no—
124#6406D414yes—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95—21no—
128C7⋊#95—21no—
129C7⋊#95—21no—
130C22⋊(C7⋊C8)—22no—
131#6382—41no—
132#6381—22no—
133#6381—22no—
134#6381—41no—
135C4⋊(C7⋊C8)—22no—
136C4⋊(C7⋊C8)—41no—
137C2×#474—71no—
138C2×#416—71no—
139#2439—71no—
140#2339D711yescharacteristic
141#2097—71no—
142#2068—71no—
143#2068—71no—
144#2024—71no—
145#1864—71no—
146#1862—71no—
147#1854—71no—
148#1850—71no—
149#1808—71no—
150#1804—71no—
151#1788—71no—
152C24×C7—21no—
153C24×C7C2×D411yescharacteristic
154C7×#41—28no—
155C7×#41—44no—
156C7×#41—44no—
157C7×#41—44no—
158C2×C7×Q8—21no—
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
161C2×C7×Q8—21no—
162C2×C7×Q8—21no—
163C2×C7×Q8C2×Q812yes—
164C2×C7×D4—24no—
165C2×C7×D4—21no—
166C2×C7×D4—21no—
167C2×C7×D4—21no—
168C2×C7×D4—42no—
169C2×C7×D4—42no—
170C2×C7×D4C2×D412yes—
171C2×C7×D4C2×Q812yes—
172C2×C7×D4#4112yes—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—21no—
177C22×C4×C7—21no—
178C22×C4×C7—24no—
179C22×C4×C7—21no—
180C22×C4×C7—21no—
181C22×C4×C7—42no—
182C22×C4×C7—42no—
183C22×C4×C7—42no—
184C22×C4×C7—42no—
185C22×C4×C7—42no—
186C22×C4×C7—42no—
187C22×C4×C7—81no—
188C22×C4×C7—81no—
189C22×C4×C7C2×D411yescharacteristic
190C22×C4×C7C2×D411yescharacteristic
191C22×C4×C7C2×D411yescharacteristic
192C22×C4×C7C2×D411yescharacteristic
193C22×C4×C7C2×D411yescharacteristic
194C22×C4×C7C2×Q812yes—
195C22×C4×C7#4112yes—
196C22×C4×C7C2411yescharacteristic
197C22×Dic7—21no—
198C22×Dic7—21no—
199C22×Dic7—21no—
200C22×Dic7—21no—
201C2×Dic14—42no—
202C2×Dic14—42no—
203C2×Dic14—81no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—41no—
210C7×C22⋊C4—42no—
211C7×C22⋊C4—42no—
212C7×C22⋊C4—81no—
213C42×C7—21no—
214C42×C7—21no—
215C42×C7—21no—
216C42×C7—21no—
217C22⋊Dic7—42no—
218C22⋊Dic7—81no—
219C4⋊Dic7—21no—
220C4⋊Dic7—21no—
221C4⋊Dic7—21no—
222C4⋊Dic7—21no—
223C4⋊Dic7—21no—
224C4⋊Dic7—21no—
225C4⋊Dic7—21no—
226C4⋊Dic7—42no—
227C4⋊Dic7C2×D412yes—
228Dic7⋊C4—42no—
229Dic7⋊C4—81no—
230C4×Dic7—42no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—21no—
236C2×C7⋊C8—21no—
237C2×C7⋊C8—21no—
238C2×C7⋊C8—21no—
239C2×C7⋊C8—21no—
240C2×C7⋊C8—21no—
241C2×C7⋊C8—21no—
242C2×C7⋊C8—44no—
243C2×C7⋊C8C2×D412yes—
244C22×#41C2×D711yescharacteristic
245C23×C8—71no—
246C2×C4⋊Q8—71no—
247C2×C4×D4C2×D711yescharacteristic
248C2×C4×D4C2×D711yescharacteristic
249C2×#117—71no—
250#474—144no—
251C2×#107—71no—
252C2×#107—71no—
253C2×#107—71no—
254C2×#106—71no—
255C2×C4⋊C8—71no—
256C2×C4⋊C8—71no—
257#416—144no—
258C2×#103—71no—
259C2×#103—71no—
260C2×#103—71no—
261C2×#103—71no—
262C2×#102—71no—
263C2×#102—71no—
264C2×#102—71no—
265C2×#102—71no—
266C2×C22⋊C8—71no—
267#396—141no—
268#393—141no—
269#391—141no—
270#391C2×D711yescharacteristic
271#391C2×D711yescharacteristic
272#389—71no—
273#389—71no—
274#387—141no—
275#386—21no—
276#386—21no—
277#382—71no—
278#381—21no—
279#380—71no—
280#380C2×D711yescharacteristic
281C4×C22⋊C4C2×D711yescharacteristic
282C2×#95—21no—
283#340—71no—
284#340—71no—
285#340—71no—
286#340—71no—
287#336—71no—
288#336—71no—
289C23×C7—22no—
290C23×C7—21no—
291C23×C7—22no—
292C23×C7—22no—
293C23×C7—21no—
294C23×C7—21no—
295C23×C7—21no—
296C23×C7—21no—
297C23×C7—21no—
298C23×C7—41no—
299C23×C7—41no—
300C23×C7—41no—
301C23×C7—41no—
302C23×C7C22⋊Q812yes—
303C23×C7C22×D411yescharacteristic
304C7×Q8—22no—
305C7×Q8—22no—
306C7×Q8—41no—
307C7×Q8—41no—
308C7×D4—22no—
309C7×D4—22no—
310C7×D4—22no—
311C7×D4—48no—
312C7×D4C22⋊Q814yes—
313C2×C4×C7—21no—
314C2×C4×C7—21no—
315C2×C4×C7—21no—
316C2×C4×C7—21no—
317C2×C4×C7—22no—
318C2×C4×C7—22no—
319C2×C4×C7—22no—
320C2×C4×C7—22no—
321C2×C4×C7—22no—
322C2×C4×C7—21no—
323C2×C4×C7—21no—
324C2×C4×C7—21no—
325C2×C4×C7—21no—
326C2×C4×C7—21no—
327C2×C4×C7—21no—
328C2×C4×C7—21no—
329C2×C4×C7—21no—
330C2×C4×C7—21no—
331C2×C4×C7—48no—
332C2×C4×C7—41no—
333C2×C4×C7—41no—
334C2×C4×C7—41no—
335C2×C4×C7—41no—
336C2×C4×C7—41no—
337C2×C4×C7—41no—
338C2×C4×C7—41no—
339C2×C4×C7—41no—
340C2×C4×C7—84no—
341C2×C4×C7—84no—
342C2×C4×C7C22⋊Q812yes—
343C2×C4×C7#13512yes—
344C2×C4×C7C22×D411yescharacteristic
345C2×C4×C7C22×Q811yescharacteristic
346C2×C4×C7C2×#4111yescharacteristic
347C2×C4×C7C2×#4111yescharacteristic
348C2×C4×C7H(5,𝔽2)11yescharacteristic
349C2×C4×C7H(5,𝔽2)11yescharacteristic
350C2×Dic7—22no—
351C2×Dic7—41no—
352C2×Dic7—41no—
353C2×Dic7—41no—
354C2×Dic7—41no—
355C2×Dic7—41no—
356C2×Dic7—41no—
357C2×Dic7—41no—
358C2×Dic7—41no—
359C2×Dic7—41no—
360C2×Dic7—41no—
361C2×Dic7—41no—
362Dic14—84no—
363C7⋊C8—22no—
364C7⋊C8—41no—
365C7⋊C8—41no—
366C7⋊C8—41no—
367C2×#41—22no—
368C2×#41—22no—
369C2×#41—22no—
370C2×#41—41no—
371C2×#41—41no—
372C2×#41Dic1414yescomplemented
373C22×Q8—141no—
374C22×Q8C22×D711yescharacteristic
375C22×D4—21no—
376C22×D4C22×D711yescharacteristic
377C23×C4—21no—
378C23×C4—21no—
379C23×C4—21no—
380C23×C4C22×D711yescharacteristic
381C22×C8—71no—
382C22×C8—71no—
383C22×C8—71no—
384C22×C8—71no—
385C22×C8—142no—
386C22×C8—142no—
387C22×C8—281no—
388C4⋊Q8—142no—
389C4⋊Q8—281no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—41no—
393C4×D4—41no—
394#117—142no—
395#117—281no—
396C2×C4⋊C4—71no—
397C2×C4⋊C4—71no—
398C2×C4⋊C4—71no—
399C2×C4⋊C4—71no—
400C2×C4⋊C4—141no—
401C2×C4⋊C4—141no—
402C2×C4⋊C4—141no—
403C2×C4⋊C4C22×D711yescharacteristic
404C2×C4⋊C4C22×D711yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—141no—
409C2×C22⋊C4—141no—
410C2×C22⋊C4C22⋊D711yescharacteristic
411C2×C22⋊C4C22⋊D711yescharacteristic
412C2×C42—141no—
413C2×C42C22×D711yescharacteristic
414C2×C42C22×D711yescharacteristic
415#107—142no—
416#107—142no—
417#107—281no—
418#106—281no—
419C4⋊C8—142no—
420C4⋊C8—281no—
421#103—142no—
422#103—142no—
423#103—281no—
424#103—281no—
425#102—74no—
426#102—142no—
427#102—142no—
428#102—142no—
429C22⋊C8—142no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—141no—
436#95—141no—
437#95—141no—
438#95—141no—
439#95—141no—
440#95C22⋊D711yescharacteristic
441#95C22⋊D711yescharacteristic
442C22×C7—22no—
443C22×C7—22no—
444C22×C7—22no—
445C22×C7—24no—
446C22×C7—42no—
447C22×C7—42no—
448C22×C7—41no—
449C22×C7—42no—
450C22×C7—42no—
451C22×C7—41no—
452C22×C7—41no—
453C22×C7—42no—
454C22×C7—42no—
455C22×C7—41no—
456C22×C7C2×C22⋊Q811yescharacteristic
457C22×C7D4211yescharacteristic
458C22×C7#54611yescharacteristic
459C22×C7C2×#13511yescharacteristic
460C22×C7#57511yescharacteristic
461C22×C7#57611yescharacteristic
462C22×C7#57811yescharacteristic
463C4×C7—22no—
464C4×C7—22no—
465C4×C7—42no—
466C4×C7—41no—
467C4×C7—41no—
468C4×C7—41no—
469C4×C7—81no—
470C4×C7—81no—
471C4×C7C2×C22⋊Q811yescharacteristic
472C4×C7#54311yescharacteristic
473C4×C7#54611yescharacteristic
474C4×C7#55111yescharacteristic
475Dic7—42no—
476Dic7—81no—
477Dic7—81no—
478Dic7—81no—
479C24—21no—
480C24C2×C22⋊D711yescharacteristic
481#41—28no—
482#41—44no—
483#41—44no—
484#41—44no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—282no—
491C2×Q8—282no—
492C2×Q8—561no—
493C2×Q8C2×Dic1412yes—
494C2×D4—24no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—42no—
499C2×D4—42no—
500C2×D4C2×Dic1412yes—
501C2×D4D4×D712yes—
502C2×D4#115912yes—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—24no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—42no—
517C22×C4—81no—
518C22×C4—81no—
519C22×C4—141no—
520C22×C4—141no—
521C22×C4—141no—
522C22×C4—141no—
523C22×C4C2×Dic1412yes—
524C22×C4#115712yes—
525C22×C4C2×C22⋊D711yescharacteristic
526C22×C4C2×C22⋊D711yescharacteristic
527C22×C4C2×C22⋊D711yescharacteristic
528C22×C4C2×C22⋊D711yescharacteristic
529C22×C4C2×C22⋊D711yescharacteristic
530C22×C4C23×D711yescharacteristic
531C2×C8—72no—
532C2×C8—141no—
533C2×C8—141no—
534C2×C8—141no—
535C2×C8—141no—
536C2×C8—141no—
537C2×C8—141no—
538C2×C8—141no—
539C2×C8—141no—
540C2×C8—141no—
541C2×C8—141no—
542C2×C8—141no—
543C2×C8—284no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—72no—
549C4⋊C4—141no—
550C4⋊C4—141no—
551C4⋊C4—141no—
552C4⋊C4—141no—
553C4⋊C4—141no—
554C4⋊C4—141no—
555C4⋊C4—141no—
556C4⋊C4—282no—
557C4⋊C4—282no—
558C4⋊C4—561no—
559C22⋊C4—22no—
560C22⋊C4—41no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—81no—
564C22⋊C4—282no—
565C22⋊C4—561no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—282no—
571C2×C7—42no—
572C2×C7—44no—
573C2×C7#243711yescharacteristic
574C2×C7#307611yescharacteristic
575C2×C7#308311yescharacteristic
576C2×C7#326911yescharacteristic
577C2×C7#328011yescharacteristic
578C2×C7#330211yescharacteristic
579C2×C7#331911yescharacteristic
580C23—22no—
581C23—22no—
582C23—21no—
583C23—22no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23#648712yes—
594C23C22×C22⋊D711yescharacteristic
595Q8—22no—
596Q8—22no—
597Q8—41no—
598Q8—41no—
599Q8—564no—
600D4—22no—
601D4—22no—
602D4—22no—
603D4—48no—
604D4#644114yes—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—48no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—142no—
635C2×C4—281no—
636C2×C4—281no—
637C2×C4—281no—
638C2×C4—281no—
639C2×C4—281no—
640C2×C4—281no—
641C2×C4—281no—
642C2×C4—281no—
643C2×C4—281no—
644C2×C4—281no—
645C2×C4—281no—
646C2×C4#648712yes—
647C2×C4#651312yes—
648C2×C4C22×Dic1411yescharacteristic
649C2×C4C2×#115711yescharacteristic
650C2×C4C2×D4×D711yescharacteristic
651C2×C4C2×#115911yescharacteristic
652C2×C4#654811yescharacteristic
653C2×C4#654811yescharacteristic
654C8—142no—
655C8—281no—
656C8—281no—
657C8—281no—
658C7#2595411yescharacteristic, complemented, Sylow
659C22—22no—
660C22—22no—
661C22—22no—
662C22—24no—
663C22—42no—
664C22—41no—
665C22—42no—
666C22—41no—
667C22—41no—
668C22—42no—
669C22—41no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22C2×#648711yescharacteristic
674C22D4×C22⋊D711yescharacteristic
675C22#9049711yescharacteristic
676C22C2×#651311yescharacteristic
677C22#9051311yescharacteristic
678C22#9052611yescharacteristic
679C22#9052911yescharacteristic
680C4—22no—
681C4—22no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—42no—
686C4—81no—
687C4—81no—
688C4—282no—
689C4—561no—
690C4—561no—
691C4—561no—
692C4C2×#644111yescharacteristic
693C4#9017411yescharacteristic
694C4#9018111yescharacteristic
695C4#9018611yescharacteristic
696C2—42no—
697C2—44no—
698C2#1173955511yescharacteristic
699C2#1173995211yescharacteristic
700C2#1174000411yescharacteristic
701C2#1174094411yescharacteristic
702C2#1174095011yescharacteristic
703C2#1174095511yescharacteristic
704C2#1174097311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2339C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
hypercenter#2339D7
hypocenterC7#25954
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2339

Complements

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

NH ≅ G/Ntype
3. C2×#89779C2semidirect
4. #11730937C2semidirect
6. #11729361C2semidirect
7. #11729360C2semidirect
8. #11729359C2semidirect
9. #11727440C2semidirect
10. #11727437C2semidirect
11. #11727437C2semidirect
12. #11727422C2semidirect
13. #11727266C2semidirect
14. #11727260C2semidirect
15. #11727203C2semidirect
67. C2×C7×#41Q8semidirect
372. C2×#41Dic14semidirect
658. C7#25954semidirect

Maximal subgroups

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

subgroupindex
C7×#23392
C2×#900052
C2×#897792
#117309372
#117298892
#117293612
#117293602
#117293592
#117274402
#117274372
#117274222
#117272662
#117272602
#117272032
#259547