Finite Group Databasebeta

← #13403248

Subgroups — #13403248

Order 1792.

792 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2547C211yesmaximal, characteristic
2C2×#89854C211yesmaximal, characteristic, complemented
3C2×#89781C211yesmaximal, characteristic
4#11731104C211yesmaximal, characteristic, complemented
5#11729532C211yesmaximal, characteristic, complemented
6#11729476C211yesmaximal, characteristic
7#11729476C211yesmaximal, characteristic
8#11729475C211yesmaximal, characteristic
9#11729475C211yesmaximal, characteristic
10#11729363C211yesmaximal, characteristic
11#11727437C211yesmaximal, characteristic, complemented
12#11727436C211yesmaximal, characteristic, complemented
13#11727421C211yesmaximal, characteristic, complemented
14#11727420C211yesmaximal, characteristic, complemented
15#11727239C211yesmaximal, characteristic, complemented
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#124C2211yescharacteristic
18C2×C7×#124C2211yescharacteristic
19C2×C4×C7×D4C2211yescharacteristic
20C2×#6433C2211yescharacteristic
21C7×#397—21no—
22C7×#397—21no—
23C7×#391C2211yescharacteristic
24C7×#391C2211yescharacteristic
25C7×#380—21no—
26C7×#380—21no—
27C4×C7×C4⋊C4C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35C2×#6409C2211yescharacteristic
36C2×#6406C2211yescharacteristic
37C2×#6406C2211yescharacteristic
38C2×#6406C2211yescharacteristic
39C2×#6406C2211yescharacteristic
40#89854—24no—
41#89781—24no—
42#89750—21no—
43#89744—21no—
44C2×#6382C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46C4×C4⋊Dic7C2211yescharacteristic
47C4×C4⋊Dic7C2211yescharacteristic
48C4×C4⋊Dic7C2211yescharacteristic
49#89700C2211yescharacteristic
50#89700C2211yescharacteristic
51C2×C4⋊(C7⋊C8)C2211yescharacteristic
52C2×C4⋊(C7⋊C8)C2211yescharacteristic
53C2×C4⋊(C7⋊C8)C2211yescharacteristic
54C2×#6377C2211yescharacteristic
55#89424—21no—
56#89421—21no—
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#25289—71nomaximal, Sylow
62C22×C7×Q8C2311yescharacteristic
63C22×C7×Q8C2311yescharacteristic
64C22×C7×D4C2311yescharacteristic
65C23×C4×C7—21no—
66C7×C4⋊Q8—22no—
67C7×C4⋊Q8—22no—
68C7×#124—22no—
69C7×#124—22no—
70C7×#124—41no—
71C7×#124—41no—
72C4×C7×D4—22no—
73C4×C7×D4—22no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C22⋊C4—21no—
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C2×C4×Dic7—21no—
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102#6433—22no—
103#6433D414yes—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95—21no—
107C7×#95—21no—
108C7×#95D411yescharacteristic
109C7×#95D411yescharacteristic
110#6409—22no—
111#6409—22no—
112#6409—22no—
113#6409—22no—
114#6409—22no—
115#6409—22no—
116#6409—41no—
117#6409—41no—
118#6406—22no—
119#6406—22no—
120#6406—22no—
121#6406—22no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127#6382—22no—
128#6381—22no—
129C4⋊(C7⋊C8)—22no—
130C4⋊(C7⋊C8)—41no—
131C4⋊(C7⋊C8)—41no—
132#6377—22no—
133C2×#491—71no—
134C2×#478—71no—
135#2547D711yescharacteristic
136#2370—71no—
137#2068—71no—
138#2066—71no—
139#2042—71no—
140#2042—71no—
141#1983—71no—
142#1983—71no—
143#1981—71no—
144#1940—71no—
145#1928—71no—
146#1927—71no—
147#1762—71no—
148C24×C7—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—21no—
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C2×C7×D4—21no—
165C2×C7×D4—21no—
166C2×C7×D4#4112yes—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—42no—
171C22×C4×C7—42no—
172C22×C4×C7—42no—
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2411yescharacteristic
180C22×Dic7—21no—
181C22×Dic7—21no—
182C22×Dic7—21no—
183C22×Dic7—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—41no—
187C7×C4⋊C4—41no—
188C7×C4⋊C4—42no—
189C7×C4⋊C4—42no—
190C7×C22⋊C4—42no—
191C7×C22⋊C4—42no—
192C7×C22⋊C4—42no—
193C7×C22⋊C4—42no—
194C7×C22⋊C4—42no—
195C42×C7—21no—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7—21no—
199C42×C7C2×D412yes—
200C42×C7#4112yes—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7C2×D412yes—
207C4⋊Dic7C2×D412yes—
208C4⋊Dic7#4112yes—
209Dic7⋊C4—24no—
210Dic7⋊C4—24no—
211Dic7⋊C4—42no—
212Dic7⋊C4—81no—
213C4×Dic7—42no—
214C4×Dic7—81no—
215C4×Dic7—81no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C4⋊Q8C2×D711yescharacteristic
229C2×#125—71no—
230C2×#124C2×D711yescharacteristic
231C2×#124C2×D711yescharacteristic
232C2×C4×D4C2×D711yescharacteristic
233#491—144no—
234#478—144no—
235C2×#107—71no—
236C2×#106—71no—
237C2×C4⋊C8—71no—
238C2×C4⋊C8—71no—
239C2×C4⋊C8—71no—
240C2×#103—71no—
241C2×#103—71no—
242C2×#103—71no—
243C2×#103—71no—
244C2×#103—71no—
245C2×#103—71no—
246C2×#103—71no—
247C2×#103—71no—
248C2×#102—71no—
249C2×#102—71no—
250C2×#102—71no—
251C2×#102—71no—
252C2×#97—71no—
253#397—21no—
254#397—21no—
255#391C2×D711yescharacteristic
256#391C2×D711yescharacteristic
257#389—71no—
258#389—71no—
259#389—141no—
260#387—141no—
261#387—141no—
262#380—21no—
263#380—21no—
264C4×C4⋊C4—71no—
265C4×C4⋊C4—71no—
266C4×C4⋊C4—71no—
267C4×C4⋊C4—141no—
268C4×C4⋊C4C2×D711yescharacteristic
269#340—71no—
270#340—71no—
271#336—71no—
272#336—71no—
273C23×C7—22no—
274C23×C7—41no—
275C23×C7—41no—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7—41no—
279C23×C7—41no—
280C23×C7C22×D411yescharacteristic
281C7×Q8—41no—
282C7×Q8—41no—
283C7×Q8—41no—
284C7×Q8—41no—
285C7×Q8—41no—
286C7×Q8—41no—
287C7×Q8—41no—
288C7×Q8—41no—
289C7×D4—22no—
290C7×D4—22no—
291C7×D4—41no—
292C7×D4—41no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
317C2×C4×C7—41no—
318C2×C4×C7—84no—
319C2×C4×C7#12112yes—
320C2×C4×C7#12112yes—
321C2×C4×C7C22×D411yescharacteristic
322C2×C4×C7C2×#4111yescharacteristic
323C2×C4×C7C2×#4111yescharacteristic
324C2×C4×C7C2×#4111yescharacteristic
325C2×C4×C7H(5,𝔽2)11yescharacteristic
326C2×C4×C72−1+411yescharacteristic
327C2×Dic7—22no—
328C2×Dic7—22no—
329C2×Dic7—22no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C7⋊C8—41no—
340C7⋊C8—41no—
341C7⋊C8—41no—
342C7⋊C8—41no—
343C22×Q8C22×D711yescharacteristic
344C22×Q8C22×D711yescharacteristic
345C22×D4C22×D711yescharacteristic
346C23×C4—21no—
347C22×C8—71no—
348C22×C8—71no—
349C22×C8—71no—
350C22×C8—71no—
351C4⋊Q8—22no—
352C4⋊Q8—22no—
353#125—74no—
354#125—142no—
355#124—22no—
356#124—22no—
357#124—41no—
358#124—41no—
359C4×D4—22no—
360C4×D4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—71no—
364C2×C4⋊C4—71no—
365C2×C4⋊C4—71no—
366C2×C4⋊C4—71no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4—141no—
369C2×C4⋊C4—141no—
370C2×C4⋊C4—141no—
371C2×C4⋊C4C22⋊D711yescharacteristic
372C2×C4⋊C4C22⋊D711yescharacteristic
373C2×C4⋊C4C22×D711yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C42—141no—
380C2×C42—141no—
381C2×C42—141no—
382C2×C42C22×D711yescharacteristic
383C2×C42C22×D711yescharacteristic
384C2×C42C22×D711yescharacteristic
385#107—142no—
386#106—142no—
387C4⋊C8—142no—
388C4⋊C8—281no—
389C4⋊C8—281no—
390#103—142no—
391#103—142no—
392#103—142no—
393#103—142no—
394#103—142no—
395#103—142no—
396#103—281no—
397#103—281no—
398#102—142no—
399#102—142no—
400#102—142no—
401#102—142no—
402#97—142no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—141no—
408#95—141no—
409#95—141no—
410#95—141no—
411#95—141no—
412#95C22⋊D711yescharacteristic
413#95C22⋊D711yescharacteristic
414C22×C7—42no—
415C22×C7—42no—
416C22×C7—42no—
417C22×C7—81no—
418C22×C7—81no—
419C22×C7—81no—
420C22×C7—81no—
421C22×C7C2×#12111yescharacteristic
422C22×C7#54611yescharacteristic
423C22×C7#54811yescharacteristic
424C22×C7#57511yescharacteristic
425C22×C7#57511yescharacteristic
426C22×C7#57711yescharacteristic
427C22×C7#57811yescharacteristic
428C4×C7—22no—
429C4×C7—22no—
430C4×C7—41no—
431C4×C7—41no—
432C4×C7—41no—
433C4×C7—41no—
434C4×C7—81no—
435C4×C7—81no—
436C4×C7—81no—
437C4×C7#52911yescharacteristic
438C4×C7#54211yescharacteristic
439C4×C7#54811yescharacteristic
440C4×C7#55011yescharacteristic
441Dic7—42no—
442Dic7—42no—
443Dic7—42no—
444Dic7—81no—
445C24—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4#115912yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—141no—
471C22×C4—141no—
472C22×C4—141no—
473C22×C4—141no—
474C22×C4C2×C22⋊D711yescharacteristic
475C22×C4C2×C22⋊D711yescharacteristic
476C22×C4C2×C22⋊D711yescharacteristic
477C22×C4C2×C22⋊D711yescharacteristic
478C22×C4C2×C22⋊D711yescharacteristic
479C22×C4C2×C22⋊D711yescharacteristic
480C22×C4C23×D711yescharacteristic
481C2×C8—141no—
482C2×C8—141no—
483C2×C8—141no—
484C2×C8—141no—
485C2×C8—141no—
486C2×C8—141no—
487C2×C8—141no—
488C2×C8—141no—
489C2×C8—141no—
490C2×C8—141no—
491C2×C8—141no—
492C2×C8—141no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—72no—
500C4⋊C4—72no—
501C4⋊C4—72no—
502C4⋊C4—144no—
503C4⋊C4—144no—
504C4⋊C4—141no—
505C4⋊C4—141no—
506C4⋊C4—141no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—282no—
510C4⋊C4—561no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—282no—
521C42—561no—
522C42—561no—
523C42D4×D712yes—
524C42#115912yes—
525C2×C7—82no—
526C2×C7#242611yescharacteristic
527C2×C7#304011yescharacteristic
528C2×C7#304611yescharacteristic
529C2×C7#330711yescharacteristic
530C2×C7#331511yescharacteristic
531C2×C7#337611yescharacteristic
532C2×C7#337711yescharacteristic
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C22×C22⋊D711yescharacteristic
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8—41no—
548Q8—41no—
549D4—22no—
550D4—22no—
551D4—41no—
552D4—41no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—84no—
579C2×C4—142no—
580C2×C4—142no—
581C2×C4—142no—
582C2×C4—281no—
583C2×C4—281no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4—281no—
587C2×C4—281no—
588C2×C4—281no—
589C2×C4—281no—
590C2×C4—281no—
591C2×C4#650112yes—
592C2×C4#650112yes—
593C2×C4C2×D4×D711yescharacteristic
594C2×C4C2×#115911yescharacteristic
595C2×C4C2×#115911yescharacteristic
596C2×C4#654811yescharacteristic
597C2×C4D7×#4111yescharacteristic
598C2×C4#655411yescharacteristic
599C8—281no—
600C8—281no—
601C8—281no—
602C8—281no—
603C7#2528911yescharacteristic, complemented, Sylow
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22C2×#650111yescharacteristic
612C22#9049711yescharacteristic
613C22#9051311yescharacteristic
614C22#9051311yescharacteristic
615C22#9052211yescharacteristic
616C22#9052811yescharacteristic
617C22#9052911yescharacteristic
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—282no—
628C4—282no—
629C4—282no—
630C4—561no—
631C4#9028911yescharacteristic
632C4#9029011yescharacteristic
633C4#9029111yescharacteristic
634C4#9029411yescharacteristic
635C2—82no—
636C2#1174009111yescharacteristic
637C2#1174039011yescharacteristic
638C2#1174085111yescharacteristic
639C2#1174095711yescharacteristic
640C2#1174095811yescharacteristic
641C2#1174111411yescharacteristic
642C2#1174112311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2547C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2547D7
hypocenterC7#25289
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2547

Complements

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

NH ≅ G/Ntype
2. C2×#89854C2semidirect
4. #11731104C2semidirect
5. #11729532C2semidirect
11. #11727437C2semidirect
12. #11727436C2semidirect
13. #11727421C2semidirect
14. #11727420C2semidirect
15. #11727239C2semidirect
603. C7#25289semidirect

Maximal subgroups

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

subgroupindex
C7×#25472
C2×#898542
C2×#897812
#117311042
#117295322
#117294762
#117294752
#117293632
#117274372
#117274362
#117274212
#117274202
#117272392
#252897