Finite Group Databasebeta

← #13412654

Subgroups — #13412654

Order 1792.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740503C211yesmaximal, characteristic
2C2×#89837C211yesmaximal, characteristic, complemented
3C2×#89653C211yesmaximal, characteristic
4#11733443C211yesmaximal, characteristic
5C7×#2059C211yesmaximal, characteristic, complemented
6#11728373C211yesmaximal, characteristic, complemented
7#11728333C211yesmaximal, characteristic
8#11728308C211yesmaximal, characteristic, complemented
9#11727374C211yesmaximal, characteristic, complemented
10#11726630C211yesmaximal, characteristic
11#11726624C211yesmaximal, characteristic
12#11726613C211yesmaximal, characteristic, complemented
13#11726589C211yesmaximal, characteristic, complemented
14#11726404C211yesmaximal, characteristic
15#11726387C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17C22×#1131C2211yescharacteristic
18C2×#6459—21no—
19C2×#6455C2211yescharacteristic
20C2×D7×C4⋊C4C2211yescharacteristic
21C2×#6432C2211yescharacteristic
22C2×C7×#107C2211yescharacteristic
23C2×C7×#106C2211yescharacteristic
24C7×#389C2211yescharacteristic
25C2×(C2×D7)⋊C8C2211yescharacteristic
26C2×#6391C2211yescharacteristic
27C2×#6391C2211yescharacteristic
28C2×C8×Dic7C2211yescharacteristic
29#89837—24no—
30#89766—21no—
31#89764—21no—
32#89759C2211yescharacteristic
33#89759C2211yescharacteristic
34#89757C2211yescharacteristic
35#89751—21no—
36#89750—21no—
37#89748C2211yescharacteristic
38#89748C2211yescharacteristic
39#89745—21no—
40#89745C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46C2×#6381C2211yescharacteristic
47C4×(C2×D7)⋊C4C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×#6377C2211yescharacteristic
50#89653—24no—
51#89440—21no—
52#89429—21no—
53C7×#340C2211yescharacteristic
54C7×#340C2211yescharacteristic
55C7×#336C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89345C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#26078—71nomaximal, Sylow
62C22×C4×D7—21no—
63C22×C4×D7C2311yescharacteristic
64C22×Dic14—21no—
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C7×#125—22no—
68C7×#125—41no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C42×C7C2311yescharacteristic
77C2×(C2×D7)⋊C4—21no—
78C2×(C2×D7)⋊C4—21no—
79C2×(C2×D7)⋊C4—21no—
80C2×(C2×D7)⋊C4D411yescharacteristic
81C2×(C2×D7)⋊C4D411yescharacteristic
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C2×#1131—22no—
100C2×#1131—22no—
101C2×#1131Q814yescomplemented
102#6459—44no—
103#6455—22no—
104#6455—41no—
105D7×C4⋊C4—22no—
106D7×C4⋊C4—41no—
107#6432—22no—
108#6432—41no—
109C7×#107—22no—
110C7×#106—41no—
111C7×#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95D411yescharacteristic
118C7⋊#95D411yescharacteristic
119(C2×D7)⋊C8Q814yescomplemented
120#6391—22no—
121#6391—41no—
122C8×Dic7—22no—
123#6382—22no—
124#6382—22no—
125#6382—22no—
126#6381—22no—
127#6381—22no—
128#6381D414yes—
129#6377—22no—
130C2×#501—71no—
131C2×#428—71no—
132#2441—71no—
133#2440—71no—
134#2084—71no—
135#2083—71no—
136#2059D711yescharacteristic, complemented
137#2058—71no—
138#1943—71no—
139#1942—71no—
140#1854—71no—
141#1853—71no—
142#1844—71no—
143#1809—71no—
144#1753—71no—
145C23×D7C2×D411yescharacteristic
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2411yescharacteristic
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C2×C4×D7—21no—
161C2×C4×D7—21no—
162C2×C4×D7—21no—
163C2×C4×D7—21no—
164C2×C4×D7—42no—
165C2×C4×D7—42no—
166C2×C4×D7—81no—
167C2×C4×D7C2×Q812yes—
168C2×C4×D7C2×Q812yes—
169C2×Dic14—42no—
170C2×Dic14—42no—
171C2×Dic14—81no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8C2×Q812yes—
177C2×C7×C8C2×Q812yes—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—42no—
186C7×C4⋊C4—42no—
187C7×C4⋊C4—42no—
188C7×C4⋊C4C2×D412yes—
189C42×C7—21no—
190C42×C7—21no—
191(C2×D7)⋊C4—22no—
192(C2×D7)⋊C4—22no—
193(C2×D7)⋊C4—42no—
194(C2×D7)⋊C4—42no—
195(C2×D7)⋊C4—81no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—42no—
204C4⋊Dic7—42no—
205C4⋊Dic7C2×D412yes—
206Dic7⋊C4—42no—
207Dic7⋊C4—42no—
208Dic7⋊C4—42no—
209Dic7⋊C4—42no—
210Dic7⋊C4—81no—
211Dic7⋊C4—81no—
212Dic7⋊C4—81no—
213C4×Dic7—21no—
214C4×Dic7—21no—
215C4×Dic7—42no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8C2×D412yes—
219C2×C7⋊C8C2×Q812yes—
220C2×C7⋊C8C2×Q812yes—
221C2×C7⋊C8#4112yes—
222#1131—28no—
223C22×M16—71no—
224C2×C4⋊Q8—71no—
225C2×#125C2×D711yescharacteristic
226C2×C22⋊Q8—141no—
227C2×#117—71no—
228C22×C4⋊C4—71no—
229#501—144no—
230#428—144no—
231C2×#107—71no—
232C2×#107—71no—
233C2×#107—71no—
234C2×#107C2×D711yescharacteristic
235C2×#106—71no—
236C2×#106—71no—
237C2×#106—71no—
238C2×#106—71no—
239C2×#106—71no—
240C2×#106C2×D711yescharacteristic
241C2×C22⋊C8—71no—
242C2×#97—71no—
243C2×C4×C8—71no—
244#399—141no—
245#398—141no—
246#396—141no—
247#395—141no—
248#389—71no—
249#389—71no—
250#389—71no—
251#389—141no—
252#389C2×D711yescharacteristic
253#387—141no—
254#385—141no—
255#384—71no—
256#384—71no—
257#380—71no—
258#380—71no—
259C4×C22⋊C4—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#340C2×D711yescharacteristic
265#340C2×D711yescharacteristic
266#336—71no—
267#336C2×D711yescharacteristic
268C23×C7C22×D411yescharacteristic
269C22×D7—21no—
270C22×D7—21no—
271C22×D7—21no—
272C22×D7—21no—
273C22×D7—21no—
274C22×D7—21no—
275C22×D7C4⋊Q812yes—
276C2×C4×C7—22no—
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—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7C22×D411yescharacteristic
295C2×C4×C7C22×Q811yescharacteristic
296C2×C4×C7C22×Q811yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7H(5,𝔽2)11yescharacteristic
299C2×C4×C72−1+411yescharacteristic
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—22no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7C4⋊Q812yes—
319C2×Dic7AGL(1,ℤ/8)12yes—
320C2×Dic7#13712yes—
321C4×D7—22no—
322C4×D7—22no—
323C4×D7—22no—
324C4×D7—22no—
325C4×D7—84no—
326Dic14—84no—
327C7×C8—22no—
328C7×C8—22no—
329C7⋊C8—22no—
330C7⋊C8C22⋊Q814yescomplemented
331C22×Q8—141no—
332C23×C4—71no—
333C23×C4—141no—
334C2×M16—74no—
335C2×M16—142no—
336C2×M16—142no—
337C22×C8—71no—
338C22×C8—71no—
339C22×C8C22×D711yescharacteristic
340C22×C8C22×D711yescharacteristic
341C4⋊Q8—142no—
342C4⋊Q8—281no—
343#125—22no—
344#125—41no—
345C22⋊Q8—284no—
346#117—142no—
347#117—281no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—142no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—281no—
366C2×C4⋊C4C22×D711yescharacteristic
367C2×C4⋊C4C22×D711yescharacteristic
368C2×C4⋊C4C22×D711yescharacteristic
369C2×C4⋊C4C22×D711yescharacteristic
370C2×C22⋊C4—71no—
371C2×C22⋊C4—71no—
372C2×C22⋊C4—141no—
373C2×C22⋊C4—141no—
374C2×C22⋊C4—141no—
375C2×C42—71no—
376C2×C42—141no—
377C2×C42C22×D711yescharacteristic
378#107—22no—
379#107—74no—
380#107—142no—
381#107—142no—
382#106—41no—
383#106—142no—
384#106—142no—
385#106—142no—
386#106—142no—
387#106—281no—
388C22⋊C8—74no—
389#97—142no—
390C4×C8—142no—
391#95—21no—
392#95—71no—
393#95—71no—
394#95—141no—
395#95—141no—
396#95—141no—
397#95—141no—
398#95—141no—
399C22×C7C2×C4⋊Q811yescharacteristic
400C22×C7D4211yescharacteristic
401C22×C7#54711yescharacteristic
402C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
403C22×C7C2×#13711yescharacteristic
404C22×C7#57711yescharacteristic
405C22×C7#57711yescharacteristic
406C2×D7—22no—
407C2×D7—22no—
408C2×D7—22no—
409C2×D7—41no—
410C2×D7—41no—
411C2×D7—41no—
412C2×D7—41no—
413C4×C7—41no—
414C4×C7—42no—
415C4×C7—41no—
416C4×C7—81no—
417C4×C7—81no—
418C4×C7—81no—
419C4×C7C2×C22⋊Q811yescharacteristic
420C4×C7#54411yescharacteristic
421C4×C7D4×Q811yescharacteristic
422C4×C7#55411yescharacteristic
423Dic7—22no—
424Dic7—22no—
425Dic7—22no—
426Dic7—42no—
427Dic7—81no—
428Dic7—81no—
429Dic7—81no—
430C24—71no—
431C2×Q8—282no—
432C2×Q8—282no—
433C2×Q8—561no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—71no—
439C22×C4—71no—
440C22×C4—71no—
441C22×C4—72no—
442C22×C4—72no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—282no—
452C22×C4—282no—
453C22×C4—561no—
454C22×C4D4×D711yescharacteristic
455C22×C4D4×D711yescharacteristic
456C22×C4C23×D711yescharacteristic
457M16—148no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—72no—
463C2×C8—72no—
464C2×C8—72no—
465C2×C8—72no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8Q8×D712yes—
469C2×C8Q8×D712yes—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—72no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—282no—
489C4⋊C4—282no—
490C4⋊C4—282no—
491C4⋊C4—282no—
492C4⋊C4—282no—
493C4⋊C4—282no—
494C4⋊C4—561no—
495C4⋊C4—561no—
496C4⋊C4—561no—
497C4⋊C4D4×D712yes—
498C22⋊C4—142no—
499C22⋊C4—142no—
500C22⋊C4—282no—
501C22⋊C4—282no—
502C22⋊C4—561no—
503C42—21no—
504C42—21no—
505C42—141no—
506C42—141no—
507C42—282no—
508C2×C7#244011yescharacteristic
509C2×C7#316311yescharacteristic
510C2×C7#316511yescharacteristic
511C2×C7#327211yescharacteristic
512C2×C7#327811yescharacteristic
513C2×C7#334411yescharacteristic
514C2×C7#334511yescharacteristic
515D7—42no—
516C23—72no—
517C23—141no—
518C23—141no—
519C23—141no—
520C23—141no—
521C23—141no—
522C23—141no—
523C23C2×D4×D711yescharacteristic
524Q8—564no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—72no—
544C2×C4—72no—
545C2×C4—72no—
546C2×C4—141no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—142no—
553C2×C4—142no—
554C2×C4—142no—
555C2×C4—142no—
556C2×C4—142no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—564no—
569C2×C4C2×D4×D711yescharacteristic
570C2×C4#654811yescharacteristic
571C2×C4C2×Q8×D711yescharacteristic
572C2×C4C2×Q8×D711yescharacteristic
573C2×C4#655111yescharacteristic
574C2×C4D7×#4111yescharacteristic
575C8—22no—
576C8—22no—
577C8—74no—
578C8—142no—
579C7#2607811yescharacteristic, complemented, Sylow
580C22—142no—
581C22—142no—
582C22—142no—
583C22—281no—
584C22—281no—
585C22—281no—
586C22—281no—
587C22#9028111yescharacteristic
588C22#9032211yescharacteristic
589C22D7×C4⋊Q811yescharacteristic
590C22#9046111yescharacteristic
591C22#9046111yescharacteristic
592C22D7×AGL(1,ℤ/8)11yescharacteristic
593C22D7×#13711yescharacteristic
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—142no—
601C4—142no—
602C4—142no—
603C4—282no—
604C4—561no—
605C4—561no—
606C4—561no—
607C4D7×C22⋊Q811yescharacteristic
608C4#9032711yescharacteristic
609C4#9041811yescharacteristic
610C4#9042011yescharacteristic
611C2—282no—
612C2#1173461911yescharacteristic
613C2#1173756911yescharacteristic
614C2#1173772011yescharacteristic
615C2#1173864211yescharacteristic
616C2#1173870011yescharacteristic
617C2#1173936211yescharacteristic
618C2#1173942111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2059C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenter#2059D7
hypocenterC7#26078
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
2. C2×#89837C2semidirect
5. C7×#2059C2semidirect
6. #11728373C2semidirect
8. #11728308C2semidirect
9. #11727374C2semidirect
12. #11726613C2semidirect
13. #11726589C2semidirect
15. #11726387C2semidirect
101. C2×#1131Q8semidirect
119. (C2×D7)⋊C8Q8semidirect
136. #2059D7semidirect
330. C7⋊C8C22⋊Q8semidirect
579. C7#26078semidirect

Maximal subgroups

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

subgroupindex
#117405032
C2×#898372
C2×#896532
#117334432
C7×#20592
#117283732
#117283332
#117283082
#117273742
#117266302
#117266242
#117266132
#117265892
#117264042
#117263872
#260787