Finite Group Databasebeta

← #13412649

Subgroups — #13412649

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
1C2×C7×#501C211yesmaximal, characteristic, complemented
2#11740678C211yesmaximal, characteristic
3#11740503C211yesmaximal, characteristic
4C2×#89653C211yesmaximal, characteristic
5#11729537C211yesmaximal, characteristic, complemented
6#11729493C211yesmaximal, characteristic, complemented
7#11728387C211yesmaximal, characteristic
8#11728373C211yesmaximal, characteristic, complemented
9#11728366C211yesmaximal, characteristic, complemented
10#11728333C211yesmaximal, characteristic
11#11728332C211yesmaximal, characteristic
12#11728321C211yesmaximal, characteristic, complemented
13#11728308C211yesmaximal, characteristic, complemented
14#11727573C211yesmaximal, characteristic, complemented
15#11727562C211yesmaximal, characteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#125C2211yescharacteristic
18C2×#6509—21no—
19C22×#1131C2211yescharacteristic
20C2×#6455C2211yescharacteristic
21C2×D7×C4⋊C4C2211yescharacteristic
22C7×#501—24no—
23C2×C7×#107C2211yescharacteristic
24C2×C7×#107C2211yescharacteristic
25C2×C7×#106C2211yescharacteristic
26C2×C7×#106C2211yescharacteristic
27C2×C7×#97C2211yescharacteristic
28#90001—21no—
29C2×(C2×D7)⋊C8C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6391C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33#89766—21no—
34#89764—21no—
35#89761—21no—
36#89759C2211yescharacteristic
37#89759C2211yescharacteristic
38#89757C2211yescharacteristic
39#89757C2211yescharacteristic
40#89756—21no—
41#89750—21no—
42#89748C2211yescharacteristic
43#89748C2211yescharacteristic
44#89745—21no—
45C2×#6382C2211yescharacteristic
46C2×#6382C2211yescharacteristic
47C2×#6382C2211yescharacteristic
48C2×#6381C2211yescharacteristic
49C4×(C2×D7)⋊C4C2211yescharacteristic
50#89700C2211yescharacteristic
51#89700C2211yescharacteristic
52#89653—24no—
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89319C2211yescharacteristic
61#26078—71nomaximal, Sylow
62C22×C7×Q8—21no—
63C22×C4×D7—21no—
64C22×C4×D7C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C7×C4⋊Q8—22no—
68C7×C4⋊Q8—41no—
69C7×#125—22no—
70C7×#125—41no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C42×C7C2311yescharacteristic
79#6509—44no—
80C2×(C2×D7)⋊C4—21no—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4D411yescharacteristic
84C2×(C2×D7)⋊C4D411yescharacteristic
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—21no—
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×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C2×#1131—22no—
103C2×#1131—22no—
104C2×#1131Q814yescomplemented
105#6455—22no—
106#6455—41no—
107D7×C4⋊C4—22no—
108D7×C4⋊C4—41no—
109C7×#107—22no—
110C7×#107D414yes—
111C7×#106—22no—
112C7×#106—22no—
113C7×#97—22no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95D411yescharacteristic
121C7⋊#95D411yescharacteristic
122(C2×D7)⋊C8Q814yescomplemented
123#6392—22no—
124#6391—22no—
125C8×Dic7—22no—
126#6382—22no—
127#6382—41no—
128#6382—41no—
129#6381—22no—
130C2×#501D711yescharacteristic, complemented
131C2×#428—71no—
132#2441—71no—
133#2440—71no—
134#2084—71no—
135#2083—71no—
136#2059—71no—
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
146C2×C7×Q8—42no—
147C2×C7×Q8—42no—
148C2×C7×Q8—81no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2411yescharacteristic
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C22×Dic7C2×D411yescharacteristic
163C2×C4×D7—21no—
164C2×C4×D7—21no—
165C2×C4×D7—21no—
166C2×C4×D7—21no—
167C2×C4×D7—42no—
168C2×C4×D7—42no—
169C2×C4×D7—81no—
170C2×C4×D7C2×Q812yes—
171C2×C4×D7C2×Q812yes—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8C2×D412yes—
175C2×C7×C8C2×Q812yes—
176C2×C7×C8C2×Q812yes—
177C2×C7×C8#4112yes—
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—42no—
185C7×C4⋊C4—42no—
186C7×C4⋊C4—42no—
187C7×C4⋊C4C2×D412yes—
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—21no—
204C4⋊Dic7—42no—
205C4⋊Dic7—42no—
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⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8C2×Q812yes—
221C2×C7⋊C8C2×Q812yes—
222#1131—28no—
223C22×M16—71no—
224C2×C4⋊Q8C2×D711yescharacteristic
225C2×#125C2×D711yescharacteristic
226C2×C22⋊Q8—141no—
227C2×#117—71no—
228C22×C4⋊C4—71no—
229#501—24no—
230#428—144no—
231C2×#107—71no—
232C2×#107—71no—
233C2×#107C2×D711yescharacteristic
234C2×#107C2×D711yescharacteristic
235C2×#106—71no—
236C2×#106—71no—
237C2×#106—71no—
238C2×#106—71no—
239C2×#106C2×D711yescharacteristic
240C2×#106C2×D711yescharacteristic
241C2×C22⋊C8—71no—
242C2×#97C2×D711yescharacteristic
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—71no—
252#389—141no—
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#340—71no—
265#340—71no—
266#336—71no—
267#336—71no—
268C23×C7C22×D411yescharacteristic
269C22×D7—21no—
270C22×D7—21no—
271C22×D7—21no—
272C22×D7—21no—
273C22×D7—21no—
274C22×D7—21no—
275C22×D7C4⋊Q812yes—
276C7×Q8—84no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
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—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×C7—41no—
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C22×Q811yescharacteristic
297C2×C4×C7C22×Q811yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7H(5,𝔽2)11yescharacteristic
300C2×C4×C72−1+411yescharacteristic
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
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×Dic7—41no—
319C2×Dic7C4⋊Q812yes—
320C2×Dic7AGL(1,ℤ/8)12yes—
321C2×Dic7#13712yes—
322C4×D7—22no—
323C4×D7—22no—
324C4×D7—22no—
325C4×D7—22no—
326C4×D7—84no—
327C7×C8—22no—
328C7×C8C22⋊Q814yescomplemented
329C7⋊C8—22no—
330C7⋊C8—22no—
331C22×Q8—21no—
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—22no—
342C4⋊Q8—41no—
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—142no—
380#107—142no—
381#107C22⋊D714yes—
382#106—22no—
383#106—22no—
384#106—142no—
385#106—142no—
386#106—281no—
387#106—281no—
388C22⋊C8—74no—
389#97—22no—
390C4×C8—142no—
391#95—71no—
392#95—71no—
393#95—141no—
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—42no—
414C4×C7—41no—
415C4×C7—42no—
416C4×C7—41no—
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—81no—
427Dic7—81no—
428Dic7—81no—
429Dic7—81no—
430C24—71no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—81no—
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—72no—
461C2×C8—72no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8Q8×D712yes—
467C2×C8Q8×D712yes—
468C2×C8#116112yes—
469C2×C8C2×C22⋊D712yes—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—282no—
488C4⋊C4—282no—
489C4⋊C4—282no—
490C4⋊C4—282no—
491C4⋊C4—282no—
492C4⋊C4—282no—
493C4⋊C4—561no—
494C4⋊C4—561no—
495C4⋊C4—561no—
496C4⋊C4C2×C22⋊D712yes—
497C4⋊C4C2×C22⋊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—84no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
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—281no—
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×C4#654811yescharacteristic
570C2×C4C2×Q8×D711yescharacteristic
571C2×C4C2×Q8×D711yescharacteristic
572C2×C4C2×#116111yescharacteristic
573C2×C4#655111yescharacteristic
574C2×C4C22×C22⋊D711yescharacteristic
575C8—22no—
576C8—142no—
577C8—142no—
578C8#650914yescomplemented
579C7#2607811yescharacteristic, complemented, Sylow
580C22—142no—
581C22—142no—
582C22—142no—
583C22—281no—
584C22—281no—
585C22—281no—
586C22—281no—
587C22D7×C4⋊Q811yescharacteristic
588C22#9041711yescharacteristic
589C22D7×AGL(1,ℤ/8)11yescharacteristic
590C22#9046611yescharacteristic
591C22D7×#13711yescharacteristic
592C22#9046911yescharacteristic
593C22D4×C22⋊D711yescharacteristic
594C4—41no—
595C4—41no—
596C4—42no—
597C4—42no—
598C4—81no—
599C4—81no—
600C4—142no—
601C4—142no—
602C4—142no—
603C4—561no—
604C4—561no—
605C4—561no—
606C4—561no—
607C4#9042011yescharacteristic
608C4#9042111yescharacteristic
609C4C2×#650911yescharacteristic
610C4Q8×C22⋊D711yescharacteristic
611C2—282no—
612C2#1173941911yescharacteristic
613C2#1173942111yescharacteristic
614C2#1173943611yescharacteristic
615C2#1173943711yescharacteristic
616C2#1174067811yescharacteristic
617C2#1174140311yescharacteristic
618C2#1174144211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#501C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenterC2×#501D7
hypocenterC7#26078
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
1. C2×C7×#501C2semidirect
5. #11729537C2semidirect
6. #11729493C2semidirect
8. #11728373C2semidirect
9. #11728366C2semidirect
12. #11728321C2semidirect
13. #11728308C2semidirect
14. #11727573C2semidirect
104. C2×#1131Q8semidirect
122. (C2×D7)⋊C8Q8semidirect
130. C2×#501D7semidirect
328. C7×C8C22⋊Q8semidirect
578. C8#6509semidirect
579. C7#26078semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5012
#117406782
#117405032
C2×#896532
#117295372
#117294932
#117283872
#117283732
#117283662
#117283332
#117283322
#117283212
#117283082
#117275732
#117275622
#260787