Finite Group Databasebeta

← #13403278

Subgroups — #13403278

Order 1792.

828 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2548C211yesmaximal, characteristic
2C2×#89843C211yesmaximal, characteristic
3C2×#89794C211yesmaximal, characteristic, complemented
4#11731104C211yesmaximal, characteristic, complemented
5#11729515C211yesmaximal, characteristic
6#11729476C211yesmaximal, characteristic, complemented
7#11729476C211yesmaximal, characteristic, complemented
8#11729475C211yesmaximal, characteristic, complemented
9#11729475C211yesmaximal, characteristic, complemented
10#11729385C211yesmaximal, characteristic, complemented
11#11727407C211yesmaximal, characteristic, complemented
12#11727406C211yesmaximal, characteristic, complemented
13#11727391C211yesmaximal, characteristic, complemented
14#11727390C211yesmaximal, characteristic, complemented
15#11727239C211yesmaximal, characteristic, complemented
16C2×C7×#127C2211yescharacteristic
17C2×C7×#124C2211yescharacteristic
18C2×C7×#124C2211yescharacteristic
19C2×C4×C7×Q8C2211yescharacteristic
20C2×#6433C2211yescharacteristic
21C7×#394—21no—
22C7×#394—21no—
23C7×#390—21no—
24C7×#390—21no—
25C7×#390C2211yescharacteristic
26C7×#390C2211yescharacteristic
27C4×C7×C4⋊C4C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35C2×#6406C2211yescharacteristic
36C2×#6406C2211yescharacteristic
37C2×#6406C2211yescharacteristic
38C2×#6406C2211yescharacteristic
39C2×#6406C2211yescharacteristic
40#89843—24no—
41#89794—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#25291—71nomaximal, Sylow
62C22×C7×Q8C2311yescharacteristic
63C22×C7×D4—21no—
64C22×C7×D4—21no—
65C22×C7×D4C2311yescharacteristic
66C22×C7×D4C2311yescharacteristic
67C7×#127—22no—
68C7×#127—22no—
69C7×#124—22no—
70C7×#124—22no—
71C7×#124—41no—
72C7×#124—41no—
73C4×C7×Q8—22no—
74C4×C7×Q8—22no—
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×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—21no—
86C2×C7×C22⋊C4—21no—
87C2×C42×C7—21no—
88C2×C42×C7C2311yescharacteristic
89C2×C42×C7C2311yescharacteristic
90C2×C42×C7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106#6433—22no—
107#6433D414yes—
108C7×#95D411yescharacteristic
109C7×#95D411yescharacteristic
110#6409—22no—
111#6409—22no—
112#6409—22no—
113#6409—22no—
114#6406—22no—
115#6406—22no—
116#6406—22no—
117#6406—22no—
118#6406—22no—
119#6406—22no—
120#6406—41no—
121#6406—41no—
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×#490—71no—
134C2×#479—71no—
135#2548D711yescharacteristic
136#2370—71no—
137#2067—71no—
138#2065—71no—
139#2042—71no—
140#2042—71no—
141#1983—71no—
142#1983—71no—
143#1982—71no—
144#1940—71no—
145#1926—71no—
146#1925—71no—
147#1763—71no—
148C24×C7—21no—
149C24×C7—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8#4112yes—
156C2×C7×D4—21no—
157C2×C7×D4—21no—
158C2×C7×D4—21no—
159C2×C7×D4—21no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C2×C7×D4—21no—
165C2×C7×D4—21no—
166C2×C7×D4—21no—
167C2×C7×D4—21no—
168C2×C7×D4—42no—
169C2×C7×D4—42no—
170C2×C7×D4—42no—
171C2×C7×D4—42no—
172C2×C7×D4—81no—
173C2×C7×D4—81no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2411yescharacteristic
183C22×Dic7—21no—
184C22×Dic7—21no—
185C22×Dic7—21no—
186C22×Dic7—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—41no—
190C7×C4⋊C4—41no—
191C7×C4⋊C4—42no—
192C7×C22⋊C4—42no—
193C7×C22⋊C4—42no—
194C7×C22⋊C4—42no—
195C7×C22⋊C4—42no—
196C7×C22⋊C4—81no—
197C7×C22⋊C4—81no—
198C7×C22⋊C4—81no—
199C7×C22⋊C4—81no—
200C42×C7—21no—
201C42×C7—21no—
202C42×C7—21no—
203C42×C7—21no—
204C42×C7—42no—
205C42×C7C2×D412yes—
206C42×C7#4112yes—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7C2×D412yes—
213C4⋊Dic7C2×D412yes—
214C4⋊Dic7#4112yes—
215Dic7⋊C4—24no—
216Dic7⋊C4—24no—
217Dic7⋊C4—42no—
218Dic7⋊C4—81no—
219C4×Dic7—42no—
220C4×Dic7—81no—
221C4×Dic7—81no—
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×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×#127C2×D711yescharacteristic
235C2×#125—71no—
236C2×#124C2×D711yescharacteristic
237C2×#124C2×D711yescharacteristic
238C2×C4×Q8C2×D711yescharacteristic
239#490—144no—
240#479—144no—
241C2×#107—71no—
242C2×#106—71no—
243C2×C4⋊C8—71no—
244C2×C4⋊C8—71no—
245C2×C4⋊C8—71no—
246C2×#103—71no—
247C2×#103—71no—
248C2×#103—71no—
249C2×#103—71no—
250C2×#102—71no—
251C2×#102—71no—
252C2×#102—71no—
253C2×#102—71no—
254C2×#102—71no—
255C2×#102—71no—
256C2×#102—71no—
257C2×#102—71no—
258C2×#97—71no—
259#394—21no—
260#394—21no—
261#390—21no—
262#390—21no—
263#390C2×D711yescharacteristic
264#390C2×D711yescharacteristic
265#389—71no—
266#389—71no—
267#389—141no—
268#387—141no—
269#387—141no—
270C4×C4⋊C4—71no—
271C4×C4⋊C4—71no—
272C4×C4⋊C4—71no—
273C4×C4⋊C4—141no—
274C4×C4⋊C4C2×D711yescharacteristic
275#340—71no—
276#340—71no—
277#336—71no—
278#336—71no—
279C23×C7—41no—
280C23×C7—41no—
281C23×C7—41no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7—41no—
285C23×C7—41no—
286C23×C7—41no—
287C23×C7—41no—
288C23×C7—41no—
289C23×C7—41no—
290C23×C7—41no—
291C23×C7—41no—
292C23×C7—41no—
293C23×C7C22×D411yescharacteristic
294C7×Q8—22no—
295C7×Q8—22no—
296C7×Q8—41no—
297C7×Q8—41no—
298C7×D4—41no—
299C7×D4—41no—
300C7×D4—41no—
301C7×D4—41no—
302C7×D4—41no—
303C7×D4—41no—
304C7×D4—41no—
305C7×D4—41no—
306C7×D4—84no—
307C7×D4—84no—
308C2×C4×C7—21no—
309C2×C4×C7—21no—
310C2×C4×C7—21no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—21no—
314C2×C4×C7—21no—
315C2×C4×C7—21no—
316C2×C4×C7—22no—
317C2×C4×C7—21no—
318C2×C4×C7—21no—
319C2×C4×C7—21no—
320C2×C4×C7—21no—
321C2×C4×C7—21no—
322C2×C4×C7—21no—
323C2×C4×C7—21no—
324C2×C4×C7—21no—
325C2×C4×C7—41no—
326C2×C4×C7—41no—
327C2×C4×C7—41no—
328C2×C4×C7—41no—
329C2×C4×C7—41no—
330C2×C4×C7#12112yes—
331C2×C4×C7#12112yes—
332C2×C4×C7C22×D411yescharacteristic
333C2×C4×C7C2×#4111yescharacteristic
334C2×C4×C7C2×#4111yescharacteristic
335C2×C4×C7C2×#4111yescharacteristic
336C2×C4×C7H(5,𝔽2)11yescharacteristic
337C2×C4×C72−1+411yescharacteristic
338C2×Dic7—22no—
339C2×Dic7—22no—
340C2×Dic7—22no—
341C2×Dic7—41no—
342C2×Dic7—41no—
343C2×Dic7—41no—
344C2×Dic7—41no—
345C2×Dic7—41no—
346C2×Dic7—41no—
347C2×Dic7—41no—
348C2×Dic7—41no—
349C2×Dic7—41no—
350C7⋊C8—41no—
351C7⋊C8—41no—
352C7⋊C8—41no—
353C7⋊C8—41no—
354C22×Q8C22×D711yescharacteristic
355C22×D4—21no—
356C22×D4—21no—
357C22×D4C22×D711yescharacteristic
358C22×D4C22×D711yescharacteristic
359C22×C8—71no—
360C22×C8—71no—
361C22×C8—71no—
362C22×C8—71no—
363#127—22no—
364#127—22no—
365#125—74no—
366#125—142no—
367#124—22no—
368#124—22no—
369#124—41no—
370#124—41no—
371C4×Q8—22no—
372C4×Q8—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4C22⋊D711yescharacteristic
383C2×C4⋊C4C22⋊D711yescharacteristic
384C2×C4⋊C4C22×D711yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C42—21no—
394C2×C42—141no—
395C2×C42—141no—
396C2×C42—141no—
397C2×C42C22×D711yescharacteristic
398C2×C42C22×D711yescharacteristic
399C2×C42C22×D711yescharacteristic
400#107—142no—
401#106—142no—
402C4⋊C8—142no—
403C4⋊C8—281no—
404C4⋊C8—281no—
405#103—142no—
406#103—142no—
407#103—142no—
408#103—142no—
409#102—142no—
410#102—142no—
411#102—142no—
412#102—142no—
413#102—142no—
414#102—142no—
415#102—281no—
416#102—281no—
417#97—142no—
418#95—141no—
419#95—141no—
420#95—141no—
421#95—141no—
422#95—141no—
423#95C22⋊D711yescharacteristic
424#95C22⋊D711yescharacteristic
425C22×C7—81no—
426C22×C7—81no—
427C22×C7—81no—
428C22×C7—81no—
429C22×C7—81no—
430C22×C7—81no—
431C22×C7—81no—
432C22×C7—81no—
433C22×C7—81no—
434C22×C7—81no—
435C22×C7—81no—
436C22×C7—81no—
437C22×C7—81no—
438C22×C7—81no—
439C22×C7C2×#12111yescharacteristic
440C22×C7#54611yescharacteristic
441C22×C7#54811yescharacteristic
442C22×C7#57511yescharacteristic
443C22×C7#57511yescharacteristic
444C22×C7#57611yescharacteristic
445C22×C7#57711yescharacteristic
446C4×C7—22no—
447C4×C7—22no—
448C4×C7—42no—
449C4×C7—41no—
450C4×C7—41no—
451C4×C7—41no—
452C4×C7—41no—
453C4×C7—81no—
454C4×C7#52911yescharacteristic
455C4×C7#54211yescharacteristic
456C4×C7#54811yescharacteristic
457C4×C7#55011yescharacteristic
458Dic7—42no—
459Dic7—42no—
460Dic7—42no—
461Dic7—81no—
462C24—21no—
463C24—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8#115912yes—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—42no—
486C2×D4—81no—
487C2×D4—81no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—141no—
491C22×C4—141no—
492C22×C4—141no—
493C22×C4—141no—
494C22×C4C2×C22⋊D711yescharacteristic
495C22×C4C2×C22⋊D711yescharacteristic
496C22×C4C2×C22⋊D711yescharacteristic
497C22×C4C2×C22⋊D711yescharacteristic
498C22×C4C2×C22⋊D711yescharacteristic
499C22×C4C2×C22⋊D711yescharacteristic
500C22×C4C23×D711yescharacteristic
501C2×C8—141no—
502C2×C8—141no—
503C2×C8—141no—
504C2×C8—141no—
505C2×C8—141no—
506C2×C8—141no—
507C2×C8—141no—
508C2×C8—141no—
509C2×C8—141no—
510C2×C8—141no—
511C2×C8—141no—
512C2×C8—141no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—42no—
518C4⋊C4—72no—
519C4⋊C4—72no—
520C4⋊C4—72no—
521C4⋊C4—144no—
522C4⋊C4—144no—
523C4⋊C4—141no—
524C4⋊C4—141no—
525C4⋊C4—141no—
526C4⋊C4—141no—
527C4⋊C4—141no—
528C4⋊C4—282no—
529C4⋊C4—561no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—81no—
535C22⋊C4—81no—
536C22⋊C4—81no—
537C22⋊C4—81no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—42no—
543C42—282no—
544C42—561no—
545C42—561no—
546C42D4×D712yes—
547C42#115912yes—
548C2×C7—161no—
549C2×C7—161no—
550C2×C7#242611yescharacteristic
551C2×C7#304011yescharacteristic
552C2×C7#304511yescharacteristic
553C2×C7#330911yescharacteristic
554C2×C7#331711yescharacteristic
555C2×C7#337011yescharacteristic
556C2×C7#337111yescharacteristic
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23C22×C22⋊D711yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—41no—
575Q8—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—84no—
585D4—84no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—142no—
609C2×C4—142no—
610C2×C4—142no—
611C2×C4—281no—
612C2×C4—281no—
613C2×C4—281no—
614C2×C4—281no—
615C2×C4—281no—
616C2×C4—281no—
617C2×C4—281no—
618C2×C4—281no—
619C2×C4—281no—
620C2×C4#650112yes—
621C2×C4#650112yes—
622C2×C4C2×D4×D711yescharacteristic
623C2×C4C2×#115911yescharacteristic
624C2×C4C2×#115911yescharacteristic
625C2×C4#654811yescharacteristic
626C2×C4D7×#4111yescharacteristic
627C2×C4#655411yescharacteristic
628C8—281no—
629C8—281no—
630C8—281no—
631C8—281no—
632C7#2529111yescharacteristic, complemented, Sylow
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22C2×#650111yescharacteristic
648C22#9049711yescharacteristic
649C22#9051311yescharacteristic
650C22#9051311yescharacteristic
651C22#9052211yescharacteristic
652C22#9052611yescharacteristic
653C22#9052711yescharacteristic
654C4—22no—
655C4—22no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—42no—
661C4—81no—
662C4—282no—
663C4—282no—
664C4—282no—
665C4—561no—
666C4#9028911yescharacteristic
667C4#9029011yescharacteristic
668C4#9029111yescharacteristic
669C4#9029411yescharacteristic
670C2—161no—
671C2—161no—
672C2#1174009111yescharacteristic
673C2#1174039011yescharacteristic
674C2#1174083011yescharacteristic
675C2#1174101711yescharacteristic
676C2#1174101811yescharacteristic
677C2#1174115911yescharacteristic
678C2#1174116311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2548C2
socleC23×C7C22×D4
prime radicalC2×C7×#127C22
hypercenter#2548D7
hypocenterC7#25291
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2548

Complements

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

NH ≅ G/Ntype
3. C2×#89794C2semidirect
4. #11731104C2semidirect
6. #11729476C2semidirect
7. #11729476C2semidirect
8. #11729475C2semidirect
9. #11729475C2semidirect
10. #11729385C2semidirect
11. #11727407C2semidirect
12. #11727406C2semidirect
13. #11727391C2semidirect
14. #11727390C2semidirect
15. #11727239C2semidirect
632. C7#25291semidirect

Maximal subgroups

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

subgroupindex
C7×#25482
C2×#898432
C2×#897942
#117311042
#117295152
#117294762
#117294752
#117293852
#117274072
#117274062
#117273912
#117273902
#117272392
#252917