Finite Group Databasebeta

← #13373226

Subgroups — #13373226

Order 1792.

866 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739906C211yesmaximal, characteristic, complemented
2C2×#89589C211yesmaximal, characteristic
3#11731084C211yesmaximal, characteristic
4C7×#1763C211yesmaximal, characteristic, complemented
5#11729387C211yesmaximal, characteristic
6#11727994C211yesmaximal, characteristic
7#11727975C211yesmaximal, characteristic, complemented
8Dic7×#103C211yesmaximal, characteristic, complemented
9#11727572C211yesmaximal, characteristic
10#11727565C211yesmaximal, characteristic, complemented
11#11727215C211yesmaximal, characteristic, complemented
12#11726647C211yesmaximal, characteristic
13#11726606C211yesmaximal, characteristic
14#11726603C211yesmaximal, characteristic, complemented
15#11726541C211yesmaximal, characteristic, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C22×#1143C2211yescharacteristic
20C2×#6456C2211yescharacteristic
21C2×#6435—21no—
22C2×C7×#103C2211yescharacteristic
23C2×C7×#103C2211yescharacteristic
24C2×C7×#97C2211yescharacteristic
25C4×C7×C4⋊C4C2211yescharacteristic
26C2×#6409C2211yescharacteristic
27C2×#6395C2211yescharacteristic
28C2×#6395C2211yescharacteristic
29C2×#6389C2211yescharacteristic
30C2×Dic7⋊C8C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32#89760C2211yescharacteristic
33#89749—21no—
34#89749C2211yescharacteristic
35#89748C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37#89745—21no—
38#89745C2211yescharacteristic
39C2×#6383C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41#89712—21no—
42#89711C2211yescharacteristic
43C4×Dic7⋊C4—21no—
44C4×Dic7⋊C4C2211yescharacteristic
45#89700—21no—
46#89700C2211yescharacteristic
47#89589C418yescomplemented
48#89441—21no—
49#89439—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#22781—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×D28C2311yescharacteristic
63C22×C4×D7—21no—
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C4×C7×Q8—22no—
67C4×C7×Q8—41no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C42×C7—21no—
73C2×C42×C7—21no—
74C2×C42×C7C2311yescharacteristic
75Q8×Dic7—22no—
76Q8×Dic7—22no—
77Q8×Dic7—41no—
78Q8×Dic7C2×C414yes—
79C2×#1143—22no—
80C2×#1143—22no—
81C2×#1143C2×C414yes—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—21no—
86C2×(C2×D7)⋊C4—21no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4D411yescharacteristic
97C2×Dic7⋊C4D411yescharacteristic
98C2×C4×Dic7—21no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7C2311yescharacteristic
102C2×C4×Dic7C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105#6456—22no—
106#6456C2×C414yes—
107#6435—44no—
108C7×#103—22no—
109C7×#103C2×C414yes—
110C7×#97—41no—
111C7×#95—21no—
112#6409—22no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95D411yescharacteristic
119C7⋊#95D411yescharacteristic
120#6395—41no—
121#6395C2×C414yes—
122#6389C2×C414yes—
123Dic7⋊C8—22no—
124C8×Dic7—41no—
125#6383—41no—
126#6381C2×C414yes—
127C2×#440—71no—
128#2354—71no—
129#2351—71no—
130#1936—71no—
131#1934—71no—
132#1921—71no—
133#1918—71no—
134#1880—71no—
135#1880—71no—
136#1836—71no—
137#1804—71no—
138#1803—71no—
139#1771—71no—
140#1763D711yescharacteristic, complemented
141C4×#103—71no—
142C23×D7—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8C22×C412yes—
147C2×C7×Q8#4112yes—
148C2×C7×Q8#4112yes—
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×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C22×Dic7C2×D411yescharacteristic
163C2×D28—21no—
164C2×D28—21no—
165C2×D28—21no—
166C2×D28—21no—
167C2×D28—21no—
168C2×D28C22×C412yes—
169C2×C4×D7—24no—
170C2×C4×D7—42no—
171C2×C4×D7—42no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8C22×C412yes—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4C22×C412yes—
185C42×C7—21no—
186C42×C7—21no—
187C42×C7—42no—
188C42×C7—42no—
189#1143—28no—
190(C2×D7)⋊C4—24no—
191(C2×D7)⋊C4—42no—
192(C2×D7)⋊C4—42no—
193(C2×D7)⋊C4—81no—
194(C2×D7)⋊C4—81no—
195C4⋊Dic7—21no—
196C4⋊Dic7—24no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—42no—
202C4⋊Dic7C22×C412yes—
203Dic7⋊C4—22no—
204Dic7⋊C4—41no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207Dic7⋊C4—42no—
208Dic7⋊C4—81no—
209C4×Dic7—21no—
210C4×Dic7—21no—
211C4×Dic7—24no—
212C4×Dic7—42no—
213C4×Dic7—42no—
214C4×Dic7C22×C412yes—
215C4×Dic7#4112yes—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8C22×C412yes—
222C22×SD16—71no—
223C2×C4×Q8—71no—
224C2×C4×Q8—71no—
225C2×C4×Q8C2×D711yescharacteristic
226C2×C4×D4—71no—
227C2×#117—141no—
228#440—78no—
229C2×#107—71no—
230C2×C4⋊C8—71no—
231C2×#103—71no—
232C2×#103C2×D711yescharacteristic
233C2×#103C2×D711yescharacteristic
234C2×#102—71no—
235C2×#102—71no—
236C2×#102—71no—
237C2×#97—71no—
238C2×#97C2×D711yescharacteristic
239C2×C4×C8—71no—
240#390—71no—
241#390—71no—
242#389—71no—
243#389—71no—
244#389—141no—
245#389—141no—
246#388—141no—
247#388—141no—
248#384—71no—
249#382—71no—
250#382—141no—
251#380—141no—
252C4×C4⋊C4—71no—
253C4×C4⋊C4—71no—
254C4×C4⋊C4—141no—
255C4×C4⋊C4C2×D711yescharacteristic
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340C2×D711yescharacteristic
263#340C2×D711yescharacteristic
264#336—71no—
265#336—71no—
266C23×C7C22×D411yescharacteristic
267C22×D7—22no—
268C22×D7—41no—
269C22×D7—41no—
270C22×D7—41no—
271C22×D7—41no—
272C22×D7—41no—
273C22×D7—41no—
274C7×Q8—22no—
275C7×Q8—22no—
276C7×Q8—22no—
277C7×Q8#11714yescomplemented
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
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×C7—41no—
296C2×C4×C7C23×C411yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7H(5,𝔽2)11yescharacteristic
301C2×C4×C72−1+411yescharacteristic
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—22no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7C4×D412yes—
321C2×Dic7C4×D412yes—
322C2×Dic7AGL(1,ℤ/8)12yes—
323D28—22no—
324D28—22no—
325D28—41no—
326D28—41no—
327C4×D7—48no—
328C7×C8—22no—
329C7×C8—41no—
330C7⋊C8—22no—
331C7⋊C8—41no—
332C22×Q8C22×D711yescharacteristic
333C22×D4—71no—
334C23×C4—141no—
335C2×SD16—74no—
336C2×SD16—142no—
337C2×SD16—142no—
338C22×C8—71no—
339C22×C8—71no—
340C22×C8C22×D711yescharacteristic
341C22×C8C22×D711yescharacteristic
342C4×Q8—22no—
343C4×Q8—41no—
344C4×Q8—74no—
345C4×Q8—142no—
346C4×Q8—142no—
347C4×Q8—281no—
348C4×D4—74no—
349C4×D4—142no—
350#117—284no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
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⋊C4C22×D711yescharacteristic
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C4⋊C4C22×D711yescharacteristic
366C2×C22⋊C4—141no—
367C2×C22⋊C4—141no—
368C2×C22⋊C4—141no—
369C2×C22⋊C4—141no—
370C2×C22⋊C4—141no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—71no—
374C2×C42—71no—
375C2×C42—141no—
376C2×C42—141no—
377C2×C42—141no—
378C2×C42C22×D711yescharacteristic
379#107—74no—
380C4⋊C8—142no—
381#103—22no—
382#103—142no—
383#103C4×D714yes—
384#102—74no—
385#102—281no—
386#102—281no—
387#97—41no—
388#97—74no—
389C4×C8—281no—
390#95—21no—
391#95—71no—
392#95—71no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95—141no—
397#95—141no—
398C22×C7C2×C4×D411yescharacteristic
399C22×C7#54611yescharacteristic
400C22×C7#54811yescharacteristic
401C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
402C22×C7#57511yescharacteristic
403C22×C7#57711yescharacteristic
404C22×C7#57811yescharacteristic
405C2×D7—42no—
406C2×D7—42no—
407C2×D7—42no—
408C2×D7—81no—
409C2×D7—81no—
410C2×D7—81no—
411C2×D7—81no—
412C4×C7—24no—
413C4×C7—42no—
414C4×C7—41no—
415C4×C7—41no—
416C4×C7—81no—
417C4×C7—81no—
418C4×C7C2×#11711yescharacteristic
419C4×C7#52011yescharacteristic
420C4×C7#55211yescharacteristic
421C4×C7#55311yescharacteristic
422Dic7—22no—
423Dic7—22no—
424Dic7—22no—
425Dic7—42no—
426Dic7—41no—
427Dic7—81no—
428Dic7—81no—
429C24—141no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8C2×C4×D712yes—
434C2×Q8#115712yes—
435C2×Q8#115712yes—
436C2×D4—72no—
437C2×D4—141no—
438C2×D4—141no—
439C2×D4—141no—
440C2×D4—141no—
441C2×D4—141no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—71no—
447C22×C4—71no—
448C22×C4—71no—
449C22×C4—71no—
450C22×C4—144no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—282no—
455C22×C4—282no—
456C22×C4D4×D711yescharacteristic
457C22×C4D4×D711yescharacteristic
458C22×C4C23×D711yescharacteristic
459SD16—148no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—72no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8—141no—
471C2×C8C2×C4×D712yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—72no—
479C4⋊C4—142no—
480C4⋊C4—144no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—281no—
487C4⋊C4—282no—
488C4⋊C4—282no—
489C4⋊C4—282no—
490C4⋊C4—282no—
491C4⋊C4—561no—
492C4⋊C4C2×C4×D712yes—
493C22⋊C4—144no—
494C22⋊C4—282no—
495C22⋊C4—282no—
496C22⋊C4—561no—
497C22⋊C4—561no—
498C42—21no—
499C42—21no—
500C42—42no—
501C42—42no—
502C42—72no—
503C42—72no—
504C42—144no—
505C42—141no—
506C42—141no—
507C42—282no—
508C42—282no—
509C2×C7#235211yescharacteristic
510C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
511C2×C7#294711yescharacteristic
512C2×C7#330811yescharacteristic
513C2×C7#332111yescharacteristic
514C2×C7#336811yescharacteristic
515C2×C7#337711yescharacteristic
516D7—82no—
517C23—142no—
518C23—281no—
519C23—281no—
520C23—281no—
521C23—281no—
522C23—281no—
523C23—281no—
524C23C2×D4×D711yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—22no—
528Q8#643514yescomplemented
529D4—142no—
530D4—142no—
531D4—281no—
532D4—281no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—72no—
552C2×C4—72no—
553C2×C4—72no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—141no—
558C2×C4—141no—
559C2×C4—141no—
560C2×C4—141no—
561C2×C4—141no—
562C2×C4—141no—
563C2×C4—142no—
564C2×C4—288no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4C22×C4×D711yescharacteristic
574C2×C4C2×#115711yescharacteristic
575C2×C4C2×#115711yescharacteristic
576C2×C4#654811yescharacteristic
577C2×C4D7×#4111yescharacteristic
578C2×C4#655411yescharacteristic
579C8—22no—
580C8—41no—
581C8—142no—
582C8—281no—
583C7#2278111yescharacteristic, complemented, Sylow
584C22—282no—
585C22—282no—
586C22—282no—
587C22—561no—
588C22—561no—
589C22—561no—
590C22—561no—
591C22#9018111yescharacteristic
592C22C4×D4×D711yescharacteristic
593C22#9024311yescharacteristic
594C22D7×AGL(1,ℤ/8)11yescharacteristic
595C22#9046811yescharacteristic
596C22#9046911yescharacteristic
597C22#9047011yescharacteristic
598C4—24no—
599C4—41no—
600C4—41no—
601C4—42no—
602C4—81no—
603C4—81no—
604C4—142no—
605C4—142no—
606C4—142no—
607C4—281no—
608C4—282no—
609C4—561no—
610C4—561no—
611C4C2×#643511yescharacteristic
612C4#9022211yescharacteristic
613C4#9023111yescharacteristic
614C4#9023711yescharacteristic
615C2—562no—
616C2#1173090611yescharacteristic
617C2#1173634111yescharacteristic
618C2#1173641411yescharacteristic
619C2#1173719711yescharacteristic
620C2#1173721811yescharacteristic
621C2#1173726311yescharacteristic
622C2#1173727511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1763C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1763D7
hypocenterC7#22781
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
1. #11739906C2semidirect
4. C7×#1763C2semidirect
7. #11727975C2semidirect
8. Dic7×#103C2semidirect
10. #11727565C2semidirect
11. #11727215C2semidirect
14. #11726603C2semidirect
15. #11726541C2semidirect
47. #89589C4semidirect
140. #1763D7semidirect
277. C7×Q8#117semidirect
528. Q8#6435semidirect
583. C7#22781semidirect

Maximal subgroups

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

subgroupindex
#117399062
C2×#895892
#117310842
C7×#17632
#117293872
#117279942
#117279752
Dic7×#1032
#117275722
#117275652
#117272152
#117266472
#117266062
#117266032
#117265412
#227817