Finite Group Databasebeta

← #13416668

Subgroups — #13416668

Order 1792.

860 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2442C211yesmaximal, characteristic, complemented
2C2×#89838C211yesmaximal, characteristic, complemented
3C2×#89736C211yesmaximal, characteristic, complemented
4#11730914C211yesmaximal, characteristic
5#11729493C211yesmaximal, characteristic, complemented
6#11729314C211yesmaximal, characteristic
7#11727383C211yesmaximal, characteristic, complemented
8#11727379C211yesmaximal, characteristic, complemented
9#11727364C211yesmaximal, characteristic, complemented
10#11727354C211yesmaximal, characteristic, complemented
11#11727312C211yesmaximal, characteristic, complemented
12#11727294C211yesmaximal, characteristic, complemented
13#11727271C211yesmaximal, characteristic
14#11727260C211yesmaximal, characteristic, complemented
15#11727204C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C2×#6493C2211yescharacteristic
19C2×#6489C2211yescharacteristic
20C22×#1136C2211yescharacteristic
21C2×C4×D28C2211yescharacteristic
22C7×#400—21no—
23C7×#398—21no—
24C7×#389C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#385—21no—
27C7×#382C2211yescharacteristic
28C7×#380—21no—
29C7×#380C2211yescharacteristic
30C4×C7×C22⋊C4C2211yescharacteristic
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32#89838—24no—
33C2×#6383C2211yescharacteristic
34C2×#6383C2211yescharacteristic
35C2×#6383C2211yescharacteristic
36C2×#6383C2211yescharacteristic
37#89736—24no—
38C2×#6382C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89729—21no—
45#89728—21no—
46#89711C2211yescharacteristic
47C4×C4⋊Dic7C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4⋊(C7⋊C8)C2211yescharacteristic
51#89441—21no—
52#89438—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26363—71nomaximal, Sylow
62C23×C4×C7—21no—
63C23×C4×C7C2311yescharacteristic
64C22×C22⋊D7—21no—
65C22×D28C2311yescharacteristic
66C22×C4×D7—21no—
67C7×#125—22no—
68C7×#125—41no—
69C2×C7×C4⋊C4—22no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—41no—
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4D411yescharacteristic
82C2×C7×C22⋊C4D411yescharacteristic
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C22⋊Dic7—21no—
86#6493—22no—
87#6493Q814yes—
88C2×(C2×D7)⋊C4—21no—
89C2×(C2×D7)⋊C4—21no—
90C2×(C2×D7)⋊C4—21no—
91C2×(C2×D7)⋊C4—21no—
92#6489—22no—
93#6489—41no—
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×Dic7⋊C4—21no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7—21no—
100C2×#1136—22no—
101C2×#1136—22no—
102C2×#1136—22no—
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C4×D28—22no—
108C4×D28—41no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95—21no—
112C7×#95—21no—
113C7×#95—21no—
114C7×#95D411yescharacteristic
115C7×#95D411yescharacteristic
116C7⋊#95—21no—
117C7⋊#95—21no—
118C22⋊(C7⋊C8)—22no—
119#6383—22no—
120#6383—22no—
121#6383—22no—
122#6383—41no—
123#6382—22no—
124#6382—22no—
125#6382—22no—
126#6381—22no—
127#6381—41no—
128#6381D414yescomplemented
129C4⋊(C7⋊C8)—22no—
130C4⋊(C7⋊C8)—22no—
131C2×#478—71no—
132C2×#428—71no—
133#2442D711yescharacteristic, complemented
134#2363—71no—
135#2092—71no—
136#2088—71no—
137#2065—71no—
138#2059—71no—
139#1949—71no—
140#1918—71no—
141#1895—71no—
142#1874—71no—
143#1862—71no—
144#1854—71no—
145#1851—71no—
146C24×C7C2×D411yescharacteristic
147C23×D7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—42no—
157C22×C4×C7—42no—
158C22×C4×C7—81no—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×Q812yes—
165C22×C4×C7#4112yes—
166C22×C4×C7C2411yescharacteristic
167C2×C22⋊D7—24no—
168C2×C22⋊D7—81no—
169C2×C22⋊D7—81no—
170C22×Dic7—21no—
171C22×Dic7—21no—
172C22×Dic7—21no—
173C2×D28—21no—
174C2×D28—21no—
175C2×D28—21no—
176C2×D28—21no—
177C2×D28—21no—
178C2×D28C2×Q812yes—
179C2×C4×D7—42no—
180C2×C4×D7—42no—
181C2×C4×D7—81no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—42no—
190C7×C4⋊C4—42no—
191C7×C4⋊C4—42no—
192C7×C4⋊C4—81no—
193C7×C4⋊C4—81no—
194C7×C4⋊C4C2×D412yes—
195C7×C22⋊C4—41no—
196C7×C22⋊C4—41no—
197C7×C22⋊C4—42no—
198C42×C7—21no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—21no—
202C22⋊Dic7—42no—
203(C2×D7)⋊C4—24no—
204(C2×D7)⋊C4—42no—
205(C2×D7)⋊C4—42no—
206(C2×D7)⋊C4—81no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7C2×D412yes—
212C4⋊Dic7C2×Q812yes—
213Dic7⋊C4—81no—
214C4×Dic7—42no—
215C4×Dic7—81no—
216#1136—28no—
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⋊C8C2×D412yes—
228C2×C7⋊C8#4112yes—
229C22×M16—71no—
230C2×#125C2×D711yescharacteristic
231C2×#121—71no—
232C2×C4×D4—71no—
233C2×#117—71no—
234C22×C4⋊C4C2×D711yescharacteristic
235#478—144no—
236#428—144no—
237C2×#107—71no—
238C2×#107—71no—
239C2×#107—71no—
240C2×#106—71no—
241C2×#106—71no—
242C2×#106—71no—
243C2×C4⋊C8—71no—
244C2×C4⋊C8—71no—
245C2×#102—71no—
246C2×#102—71no—
247C2×#102—71no—
248C2×#102—71no—
249C2×C22⋊C8—71no—
250#400—21no—
251#398—21no—
252#390—71no—
253#390—141no—
254#389—71no—
255#389C2×D711yescharacteristic
256#389C2×D711yescharacteristic
257#388—141no—
258#386—141no—
259#385—21no—
260#382C2×D711yescharacteristic
261#380—21no—
262#380C2×D711yescharacteristic
263C4×C4⋊C4—71no—
264C4×C22⋊C4—141no—
265C4×C22⋊C4C2×D711yescharacteristic
266#340—71no—
267#340—71no—
268#340—71no—
269#340—71no—
270#340—71no—
271#340—71no—
272#336—71no—
273#336—71no—
274C23×C7—21no—
275C23×C7—21no—
276C23×C7—21no—
277C23×C7—21no—
278C23×C7—21no—
279C23×C7—21no—
280C23×C7C22⋊Q812yes—
281C23×C7C22×D411yescharacteristic
282C22×D7—22no—
283C22×D7—41no—
284C22×D7—41no—
285C22×D7—41no—
286C22×D7—41no—
287C22×D7—41no—
288C22×D7—41no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—22no—
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—41no—
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—41no—
319C2×C4×C7—84no—
320C2×C4×C7C22⋊Q812yes—
321C2×C4×C7C22×D411yescharacteristic
322C2×C4×C7C22×Q811yescharacteristic
323C2×C4×C7C2×#4111yescharacteristic
324C2×C4×C7C2×#4111yescharacteristic
325C2×C4×C7H(5,𝔽2)11yescharacteristic
326C2×C4×C72−1+411yescharacteristic
327C22⋊D7—84no—
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—
337D28—22no—
338D28—22no—
339D28—41no—
340D28—41no—
341C4×D7—84no—
342C7⋊C8—22no—
343C7⋊C8—22no—
344C7⋊C8—41no—
345C7⋊C8—41no—
346C22×D4—71no—
347C22×D4—141no—
348C23×C4—21no—
349C23×C4—141no—
350C23×C4C22×D711yescharacteristic
351C2×M16—142no—
352C2×M16—142no—
353C2×M16—142no—
354C22×C8—71no—
355C22×C8—71no—
356C22×C8—71no—
357C22×C8—71no—
358#125—22no—
359#125—41no—
360#121—74no—
361#121—142no—
362C4×D4—142no—
363C4×D4—281no—
364#117—142no—
365#117—281no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—41no—
373C2×C4⋊C4—71no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C4⋊C4C22×D711yescharacteristic
379C2×C4⋊C4C22×D711yescharacteristic
380C2×C4⋊C4C22×D711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—141no—
383C2×C22⋊C4—141no—
384C2×C22⋊C4—141no—
385C2×C22⋊C4—141no—
386C2×C22⋊C4—141no—
387C2×C22⋊C4C22⋊D711yescharacteristic
388C2×C22⋊C4C22⋊D711yescharacteristic
389C2×C42—141no—
390C2×C42—141no—
391C2×C42C22×D711yescharacteristic
392C2×C42C22×D711yescharacteristic
393#107—74no—
394#107—142no—
395#107—281no—
396#106—142no—
397#106—142no—
398#106—142no—
399C4⋊C8—142no—
400C4⋊C8—142no—
401#102—142no—
402#102—142no—
403#102—142no—
404#102—281no—
405C22⋊C8—142no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—141no—
412#95—141no—
413#95C22⋊D711yescharacteristic
414#95C22⋊D711yescharacteristic
415C22×C7—22no—
416C22×C7—22no—
417C22×C7—22no—
418C22×C7—41no—
419C22×C7—41no—
420C22×C7—41no—
421C22×C7—41no—
422C22×C7C2×C22⋊Q811yescharacteristic
423C22×C7D4211yescharacteristic
424C22×C7#54711yescharacteristic
425C22×C7#57511yescharacteristic
426C22×C7#57511yescharacteristic
427C22×C7#57611yescharacteristic
428C22×C7#57711yescharacteristic
429C2×D7—42no—
430C2×D7—42no—
431C2×D7—42no—
432C2×D7—81no—
433C2×D7—81no—
434C2×D7—81no—
435C2×D7—81no—
436C4×C7—22no—
437C4×C7—41no—
438C4×C7—41no—
439C4×C7—41no—
440C4×C7—41no—
441C4×C7—42no—
442C4×C7—81no—
443C4×C7—81no—
444C4×C7—81no—
445C4×C7#52511yescharacteristic
446C4×C7#53911yescharacteristic
447C4×C7D4×Q811yescharacteristic
448C4×C7#55411yescharacteristic
449Dic7—42no—
450Dic7—42no—
451Dic7—81no—
452C24—141no—
453C24C2×C22⋊D711yescharacteristic
454C2×D4—72no—
455C2×D4—144no—
456C2×D4—141no—
457C2×D4—141no—
458C2×D4—141no—
459C2×D4—141no—
460C2×D4—141no—
461C2×D4—561no—
462C2×D4—561no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—141no—
475C22×C4—141no—
476C22×C4—141no—
477C22×C4—282no—
478C22×C4—282no—
479C22×C4—561no—
480C22×C4Q8×D712yes—
481C22×C4#116112yes—
482C22×C4C2×C22⋊D711yescharacteristic
483C22×C4C2×C22⋊D711yescharacteristic
484C22×C4C2×C22⋊D711yescharacteristic
485C22×C4C2×C22⋊D711yescharacteristic
486C22×C4C2×C22⋊D711yescharacteristic
487C22×C4C23×D711yescharacteristic
488M16—148no—
489C2×C8—72no—
490C2×C8—72no—
491C2×C8—141no—
492C2×C8—141no—
493C2×C8—141no—
494C2×C8—141no—
495C2×C8—141no—
496C2×C8—141no—
497C2×C8—141no—
498C2×C8—141no—
499C2×C8—141no—
500C2×C8—141no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—72no—
512C4⋊C4—72no—
513C4⋊C4—81no—
514C4⋊C4—81no—
515C4⋊C4—141no—
516C4⋊C4—141no—
517C4⋊C4—141no—
518C4⋊C4—141no—
519C4⋊C4—561no—
520C4⋊C4D4×D712yes—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—42no—
524C22⋊C4—144no—
525C22⋊C4—282no—
526C22⋊C4—282no—
527C22⋊C4—282no—
528C22⋊C4—561no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—282no—
534C42—561no—
535C2×C7—42no—
536C2×C7#244911yescharacteristic
537C2×C7#308011yescharacteristic
538C2×C7#308111yescharacteristic
539C2×C7#327111yescharacteristic
540C2×C7#327311yescharacteristic
541C2×C7#334811yescharacteristic
542C2×C7#335511yescharacteristic
543D7—82no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—142no—
551C23—281no—
552C23—281no—
553C23—281no—
554C23—281no—
555C23—281no—
556C23—281no—
557C23#650912yes—
558C23C22×C22⋊D711yescharacteristic
559D4—142no—
560D4—142no—
561D4—281no—
562D4—281no—
563D4—564no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—84no—
595C2×C4—142no—
596C2×C4—142no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4—564no—
605C2×C4#650912yes—
606C2×C4C2×D4×D711yescharacteristic
607C2×C4#654811yescharacteristic
608C2×C4C2×Q8×D711yescharacteristic
609C2×C4C2×#116111yescharacteristic
610C2×C4D7×#4111yescharacteristic
611C2×C4#655411yescharacteristic
612C8—142no—
613C8—142no—
614C8—281no—
615C8—281no—
616C7#2636311yescharacteristic, complemented, Sylow
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—282no—
625C22—282no—
626C22—282no—
627C22—561no—
628C22—561no—
629C22—561no—
630C22—561no—
631C22D4×C22⋊D711yescharacteristic
632C22C2×#650911yescharacteristic
633C22#9051311yescharacteristic
634C22#9051311yescharacteristic
635C22#9051511yescharacteristic
636C22#9052611yescharacteristic
637C22#9052811yescharacteristic
638C4—22no—
639C4—41no—
640C4—42no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—282no—
648C4—282no—
649C4—561no—
650C4#9031811yescharacteristic
651C4#9032311yescharacteristic
652C4#9032711yescharacteristic
653C4#9033611yescharacteristic
654C2—42no—
655C2—562no—
656C2#1174020911yescharacteristic
657C2#1174046211yescharacteristic
658C2#1174091911yescharacteristic
659C2#1174094911yescharacteristic
660C2#1174099211yescharacteristic
661C2#1174103911yescharacteristic
662C2#1174105811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2442C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
hypercenter#2442D7
hypocenterC7#26363
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
1. C7×#2442C2semidirect
2. C2×#89838C2semidirect
3. C2×#89736C2semidirect
5. #11729493C2semidirect
7. #11727383C2semidirect
8. #11727379C2semidirect
9. #11727364C2semidirect
10. #11727354C2semidirect
11. #11727312C2semidirect
12. #11727294C2semidirect
14. #11727260C2semidirect
15. #11727204C2semidirect
128. #6381D4semidirect
133. #2442D7semidirect
616. C7#26363semidirect

Maximal subgroups

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

subgroupindex
C7×#24422
C2×#898382
C2×#897362
#117309142
#117294932
#117293142
#117273832
#117273792
#117273642
#117273542
#117273122
#117272942
#117272712
#117272602
#117272042
#263637