Finite Group Databasebeta

← #13405852

Subgroups — #13405852

Order 1792.

790 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2580C211yesmaximal, characteristic
2#11739920C211yesmaximal, characteristic, complemented
3C2×#89796C211yesmaximal, characteristic, complemented
4C2×#89780C211yesmaximal, characteristic
5#11729363C211yesmaximal, characteristic
6#11727407C211yesmaximal, characteristic
7#11727406C211yesmaximal, characteristic
8#11727399C211yesmaximal, characteristic
9#11727398C211yesmaximal, characteristic
10#11727378C211yesmaximal, characteristic, complemented
11#11727374C211yesmaximal, characteristic, complemented
12#11727372C211yesmaximal, characteristic, complemented
13#11727364C211yesmaximal, characteristic, complemented
14#11727309C211yesmaximal, characteristic, complemented
15#11727294C211yesmaximal, characteristic, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×#6507—21no—
19C2×#6432C2211yescharacteristic
20C7×#399—21no—
21C7×#393—21no—
22C7×#390C2211yescharacteristic
23C7×#390C2211yescharacteristic
24C7×#389C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#385—21no—
27C7×#385—21no—
28C7×#382C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33#89796—24no—
34#89780—24no—
35#89755—21no—
36#89755—21no—
37#89746—21no—
38C2×#6382C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43#89700C2211yescharacteristic
44#89700C2211yescharacteristic
45#89700C2211yescharacteristic
46#89700C2211yescharacteristic
47C2×C4⋊(C7⋊C8)C2211yescharacteristic
48C2×C4⋊(C7⋊C8)C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×#6377C2211yescharacteristic
51C2×#6377C2211yescharacteristic
52C2×C4×C7⋊C8C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#25485—71nomaximal, Sylow
62C22×C7×Q8D411yescharacteristic
63C22×C7×D4C2311yescharacteristic
64C23×C4×C7—21no—
65C22×Dic14—21no—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C4×C7×D4—22no—
69C4×C7×D4—22no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4D411yescharacteristic
73C2×C7×C4⋊C4D411yescharacteristic
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86#6507—44no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×C4×Dic7—21no—
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101#6432—22no—
102#6432—22no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95—21no—
107#6406—22no—
108#6406—22no—
109#6406—22no—
110#6406Q814yescomplemented
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116#6382—22no—
117#6382—22no—
118#6382—22no—
119#6381—22no—
120C4⋊(C7⋊C8)—22no—
121C4⋊(C7⋊C8)—22no—
122C4⋊(C7⋊C8)D414yes—
123#6377—22no—
124#6377—22no—
125C4×C7⋊C8—22no—
126C2×#476—71no—
127C2×#446—71no—
128#2580D711yescharacteristic
129#2546—71no—
130#2088—71no—
131#2088—71no—
132#2067—71no—
133#2065—71no—
134#1993—71no—
135#1949—71no—
136#1943—71no—
137#1941—71no—
138#1936—71no—
139#1934—71no—
140#1762—71no—
141C24×C7—21no—
142C2×C7×Q8—22no—
143C2×C7×Q8—22no—
144C2×C7×Q8—41no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4C2×Q812yes—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—42no—
155C22×C4×C7—81no—
156C22×C4×C7—81no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2411yescharacteristic
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C2×Dic14—42no—
169C2×Dic14—42no—
170C2×Dic14—42no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—81no—
181C7×C4⋊C4—81no—
182C7×C22⋊C4—42no—
183C7×C22⋊C4—42no—
184C7×C22⋊C4—42no—
185C7×C22⋊C4—81no—
186C7×C22⋊C4—81no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7C2×D412yes—
192C42×C7#4112yes—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—42no—
200C4⋊Dic7C2×Q812yes—
201C4⋊Dic7#4112yes—
202Dic7⋊C4—42no—
203Dic7⋊C4—42no—
204Dic7⋊C4—42no—
205Dic7⋊C4—42no—
206C4×Dic7—42no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8C2×D412yes—
217C2×C7⋊C8C2×D412yes—
218C2×C7⋊C8C2×Q812yes—
219C2×C4⋊Q8—71no—
220C2×C4⋊Q8—141no—
221C2×C4×Q8C2×D711yescharacteristic
222C2×C4×D4C2×D711yescharacteristic
223#476—144no—
224#446—144no—
225C2×#107—71no—
226C2×#106—71no—
227C2×#106—71no—
228C2×#106—71no—
229C2×C4⋊C8—71no—
230C2×C4⋊C8—71no—
231C2×C4⋊C8—71no—
232C2×#102—71no—
233C2×#102—71no—
234C2×#102—71no—
235C2×#102—71no—
236C2×#97—71no—
237C2×#97—71no—
238C2×C4×C8—71no—
239#399—21no—
240#398—141no—
241#398—141no—
242#393—21no—
243#391—141no—
244#390C2×D711yescharacteristic
245#390C2×D711yescharacteristic
246#389—71no—
247#389—71no—
248#389—71no—
249#389—71no—
250#389C2×D711yescharacteristic
251#389C2×D711yescharacteristic
252#385—21no—
253#385—21no—
254#382C2×D711yescharacteristic
255C4×C4⋊C4—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264C23×C7—22no—
265C23×C7—41no—
266C23×C7—41no—
267C23×C7—41no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7C22×D411yescharacteristic
272C7×Q8—44no—
273C7×D4—22no—
274C7×D4—22no—
275C7×D4—41no—
276C7×D4—41no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—84no—
302C2×C4×C7#12112yes—
303C2×C4×C7#12112yes—
304C2×C4×C7AGL(1,ℤ/8)12yes—
305C2×C4×C7C22×D411yescharacteristic
306C2×C4×C7C22×Q811yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×C4×C72−1+411yescharacteristic
310C2×C4×C72−1+411yescharacteristic
311C2×Dic7—22no—
312C2×Dic7—22no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323Dic14—84no—
324C7⋊C8—22no—
325C7⋊C8—22no—
326C7⋊C8—22no—
327C7⋊C8—41no—
328C22×Q8—141no—
329C22×Q8C22⋊D711yescharacteristic
330C22×D4C22×D711yescharacteristic
331C23×C4—21no—
332C22×C8—71no—
333C22×C8—71no—
334C22×C8—71no—
335C22×C8—71no—
336C4⋊Q8—142no—
337C4⋊Q8—142no—
338C4⋊Q8—284no—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×D4—22no—
342C4×D4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—71no—
347C2×C4⋊C4—71no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—141no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4C22⋊D711yescharacteristic
355C2×C4⋊C4C22⋊D711yescharacteristic
356C2×C4⋊C4C22⋊D711yescharacteristic
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C4⋊C4C22×D711yescharacteristic
359C2×C4⋊C4C22×D711yescharacteristic
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C42—141no—
366C2×C42C22×D711yescharacteristic
367C2×C42C22×D711yescharacteristic
368C2×C42C22×D711yescharacteristic
369#107—142no—
370#106—142no—
371#106—142no—
372#106—142no—
373C4⋊C8—74no—
374C4⋊C8—142no—
375C4⋊C8—142no—
376#102—74no—
377#102—142no—
378#102—142no—
379#102—142no—
380#97—142no—
381#97—142no—
382C4×C8—142no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—141no—
388#95—141no—
389#95—141no—
390#95—141no—
391#95—141no—
392C22×C7—42no—
393C22×C7—42no—
394C22×C7—42no—
395C22×C7—81no—
396C22×C7—81no—
397C22×C7—81no—
398C22×C7—81no—
399C22×C7C2×#12111yescharacteristic
400C22×C7#54811yescharacteristic
401C22×C7D4×Q811yescharacteristic
402C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
403C22×C7#57511yescharacteristic
404C22×C7#57711yescharacteristic
405C22×C7#57711yescharacteristic
406C4×C7—22no—
407C4×C7—22no—
408C4×C7—22no—
409C4×C7—41no—
410C4×C7—41no—
411C4×C7—41no—
412C4×C7—81no—
413C4×C7—81no—
414C4×C7—81no—
415C4×C7#53311yescharacteristic
416C4×C7#54111yescharacteristic
417C4×C7#54711yescharacteristic
418C4×C7D4×Q811yescharacteristic
419Dic7—42no—
420Dic7—42no—
421Dic7—81no—
422Dic7—81no—
423C24—21no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—41no—
427C2×Q8—282no—
428C2×Q8—282no—
429C2×Q8—282no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4Q8×D712yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—141no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4C2×C22⋊D711yescharacteristic
447C22×C4C2×C22⋊D711yescharacteristic
448C22×C4C2×C22⋊D711yescharacteristic
449C22×C4C2×C22⋊D711yescharacteristic
450C22×C4C2×C22⋊D711yescharacteristic
451C22×C4C2×C22⋊D711yescharacteristic
452C22×C4C23×D711yescharacteristic
453C2×C8—72no—
454C2×C8—72no—
455C2×C8—72no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8—141no—
459C2×C8—141no—
460C2×C8—141no—
461C2×C8—141no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—72no—
475C4⋊C4—72no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—282no—
485C4⋊C4—282no—
486C4⋊C4—282no—
487C4⋊C4—282no—
488C4⋊C4—282no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—282no—
499C42D4×D712yes—
500C42#115912yes—
501C2×C7—82no—
502C2×C7#242911yescharacteristic
503C2×C7#305511yescharacteristic
504C2×C7#305711yescharacteristic
505C2×C7#337111yescharacteristic
506C2×C7#337311yescharacteristic
507C2×C7#338411yescharacteristic
508C2×C7#338811yescharacteristic
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C22×C22⋊D711yescharacteristic
517Q8—44no—
518Q8—564no—
519D4—22no—
520D4—22no—
521D4—41no—
522D4—41no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—84no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4#650212yes—
561C2×C4#650212yes—
562C2×C4#651212yes—
563C2×C4C2×D4×D711yescharacteristic
564C2×C4C2×#115911yescharacteristic
565C2×C4C2×Q8×D711yescharacteristic
566C2×C4#655111yescharacteristic
567C2×C4D7×#4111yescharacteristic
568C2×C4#655411yescharacteristic
569C8—142no—
570C8—142no—
571C8—142no—
572C8—281no—
573C7#2548511yescharacteristic, complemented, Sylow
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×#650211yescharacteristic
582C22Q8×C22⋊D711yescharacteristic
583C22C2×#651211yescharacteristic
584C22#9051311yescharacteristic
585C22#9052211yescharacteristic
586C22#9052711yescharacteristic
587C22#9052711yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—282no—
598C4—282no—
599C4—561no—
600C4—561no—
601C4#9031311yescharacteristic
602C4#9031611yescharacteristic
603C4#9032211yescharacteristic
604C4#9032311yescharacteristic
605C2—82no—
606C2#1174020311yescharacteristic
607C2#1174036911yescharacteristic
608C2#1174082211yescharacteristic
609C2#1174107511yescharacteristic
610C2#1174107711yescharacteristic
611C2#1174116011yescharacteristic
612C2#1174116311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2580C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2580D7
hypocenterC7#25485
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
2. #11739920C2semidirect
3. C2×#89796C2semidirect
10. #11727378C2semidirect
11. #11727374C2semidirect
12. #11727372C2semidirect
13. #11727364C2semidirect
14. #11727309C2semidirect
15. #11727294C2semidirect
110. #6406Q8semidirect
573. C7#25485semidirect

Maximal subgroups

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

subgroupindex
C7×#25802
#117399202
C2×#897962
C2×#897802
#117293632
#117274072
#117274062
#117273992
#117273982
#117273782
#117273742
#117273722
#117273642
#117273092
#117272942
#254857