Finite Group Databasebeta

← #13373136

Subgroups — #13373136

Order 1792.

838 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic7⋊C4C211yesmaximal, characteristic, complemented
2C2×#89589C211yesmaximal, characteristic
3#11731091C211yesmaximal, characteristic
4C7×#1763C211yesmaximal, characteristic, complemented
5#11729387C211yesmaximal, characteristic
6#11727994C211yesmaximal, characteristic
7#11727976C211yesmaximal, characteristic, complemented
8Dic7×#103C211yesmaximal, characteristic, complemented
9#11727572C211yesmaximal, characteristic
10#11727565C211yesmaximal, characteristic, complemented
11#11727212C211yesmaximal, characteristic, complemented
12#11726646C211yesmaximal, characteristic
13#11726606C211yesmaximal, characteristic
14#11726603C211yesmaximal, characteristic, complemented
15#11726544C211yesmaximal, characteristic, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C22×#1143C2211yescharacteristic
20C2×#6456C2211yescharacteristic
21C2×C7×#103C2211yescharacteristic
22C2×C7×#103C2211yescharacteristic
23C2×C7×#97C2211yescharacteristic
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×#6409C2211yescharacteristic
26C2×#6395C2211yescharacteristic
27C2×#6395C2211yescharacteristic
28C2×#6389C2211yescharacteristic
29C2×Dic7⋊C8C2211yescharacteristic
30C2×C8×Dic7C2211yescharacteristic
31#89760C2211yescharacteristic
32#89749C2211yescharacteristic
33#89748—21no—
34#89748C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36#89745—21no—
37#89745C2211yescharacteristic
38C2×#6383C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40#89712—21no—
41#89711C2211yescharacteristic
42C4×(C2×D7)⋊C4—21no—
43C4×Dic7⋊C4—21no—
44C4×Dic7⋊C4C2211yescharacteristic
45#89700—21no—
46#89700C2211yescharacteristic
47#89589C418yescomplemented
48#89441—21no—
49#89436—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#22775—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—
107C7×#103—22no—
108C7×#103C2×C414yes—
109C7×#97—41no—
110C7×#95—21no—
111#6409—22no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95D411yescharacteristic
118C7⋊#95D411yescharacteristic
119#6395—41no—
120#6395C2×C414yes—
121#6389C2×C414yes—
122Dic7⋊C8—22no—
123C8×Dic7—41no—
124#6383—41no—
125#6381C2×C414yes—
126C2×#440—71no—
127#2357—71no—
128Q8×C4⋊C4—71no—
129#1936—71no—
130#1934—71no—
131#1921—71no—
132#1918—71no—
133#1880—71no—
134#1880—71no—
135#1836—71no—
136#1805—71no—
137#1802—71no—
138#1771—71no—
139#1763D711yescharacteristic, complemented
140C4×#103—71no—
141C23×D7—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8C22×C412yes—
146C2×C7×Q8#4112yes—
147C2×C7×Q8#4112yes—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C2×D28—21no—
163C2×D28—21no—
164C2×D28—21no—
165C2×D28—21no—
166C2×D28—21no—
167C2×D28C22×C412yes—
168C2×C4×D7—24no—
169C2×C4×D7—81no—
170C2×C4×D7—81no—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8C22×C412yes—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—42no—
183C7×C4⋊C4C22×C412yes—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—42no—
187C42×C7—81no—
188#1143—28no—
189(C2×D7)⋊C4—24no—
190(C2×D7)⋊C4—42no—
191(C2×D7)⋊C4—42no—
192(C2×D7)⋊C4—81no—
193(C2×D7)⋊C4—81no—
194C4⋊Dic7—21no—
195C4⋊Dic7—24no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—42no—
201C4⋊Dic7C22×C412yes—
202Dic7⋊C4—22no—
203Dic7⋊C4—41no—
204Dic7⋊C4—42no—
205Dic7⋊C4—81no—
206Dic7⋊C4—81no—
207Dic7⋊C4—81no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7—24no—
211C4×Dic7—42no—
212C4×Dic7—81no—
213C4×Dic7C22×C412yes—
214C4×Dic7#4112yes—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8C22×C412yes—
221C22×SD16—71no—
222C2×C4×Q8—71no—
223C2×C4×Q8—71no—
224C2×C4×Q8C2×D711yescharacteristic
225C2×C4×D4—71no—
226#440—78no—
227C2×#107—71no—
228C2×C4⋊C8—71no—
229C2×#103—71no—
230C2×#103C2×D711yescharacteristic
231C2×#103C2×D711yescharacteristic
232C2×#102—71no—
233C2×#102—71no—
234C2×#102—71no—
235C2×#97—71no—
236C2×#97C2×D711yescharacteristic
237C2×C4×C8—71no—
238#390—71no—
239#390—71no—
240#389—71no—
241#389—71no—
242#389—141no—
243#389—141no—
244#388—141no—
245#388—141no—
246#384—71no—
247#384—141no—
248#382—71no—
249#381—141no—
250C4×C4⋊C4—71no—
251C4×C4⋊C4—71no—
252C4×C4⋊C4—141no—
253C4×C4⋊C4C2×D711yescharacteristic
254C4×C22⋊C4—141no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340C2×D711yescharacteristic
262#340C2×D711yescharacteristic
263#336—71no—
264#336—71no—
265C23×C7C22×D411yescharacteristic
266C22×D7—22no—
267C22×D7—41no—
268C22×D7—41no—
269C22×D7—41no—
270C22×D7—41no—
271C22×D7—41no—
272C22×D7—41no—
273C7×Q8—22no—
274C7×Q8—22no—
275C7×Q8—22no—
276C7×Q8#11714yes—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
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×C7C23×C411yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
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—22no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
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×D412yes—
320C2×Dic7C4×D412yes—
321C2×Dic7#13712yes—
322D28—22no—
323D28—22no—
324D28—41no—
325D28—41no—
326C4×D7—84no—
327C7×C8—22no—
328C7×C8—41no—
329C7⋊C8—22no—
330C7⋊C8—41no—
331C22×Q8C22×D711yescharacteristic
332C22×D4—71no—
333C23×C4—141no—
334C2×SD16—74no—
335C2×SD16—142no—
336C2×SD16—142no—
337C22×C8—71no—
338C22×C8—71no—
339C22×C8C22×D711yescharacteristic
340C22×C8C22×D711yescharacteristic
341C4×Q8—22no—
342C4×Q8—41no—
343C4×Q8—74no—
344C4×Q8—142no—
345C4×Q8—142no—
346C4×Q8—281no—
347C4×D4—74no—
348C4×D4—142no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—141no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4C22×D711yescharacteristic
362C2×C4⋊C4C22×D711yescharacteristic
363C2×C4⋊C4C22×D711yescharacteristic
364C2×C22⋊C4—141no—
365C2×C22⋊C4—141no—
366C2×C22⋊C4—141no—
367C2×C22⋊C4—141no—
368C2×C22⋊C4—141no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—71no—
372C2×C42—71no—
373C2×C42—141no—
374C2×C42—141no—
375C2×C42—141no—
376C2×C42C22×D711yescharacteristic
377#107—74no—
378C4⋊C8—142no—
379#103—22no—
380#103—142no—
381#103C4×D714yes—
382#102—74no—
383#102—281no—
384#102—281no—
385#97—41no—
386#97—74no—
387C4×C8—281no—
388#95—21no—
389#95—71no—
390#95—71no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396C22×C7C2×C4×D411yescharacteristic
397C22×C7#54611yescharacteristic
398C22×C7#54811yescharacteristic
399C22×C7C2×#13711yescharacteristic
400C22×C7#57511yescharacteristic
401C22×C7#57611yescharacteristic
402C22×C7#57711yescharacteristic
403C2×D7—42no—
404C2×D7—42no—
405C2×D7—42no—
406C2×D7—81no—
407C2×D7—81no—
408C2×D7—81no—
409C2×D7—81no—
410C4×C7—24no—
411C4×C7—42no—
412C4×C7—41no—
413C4×C7—41no—
414C4×C7—81no—
415C4×C7—81no—
416C4×C7C2×#11711yescharacteristic
417C4×C7#52011yescharacteristic
418C4×C7#55211yescharacteristic
419C4×C7#55311yescharacteristic
420Dic7—22no—
421Dic7—22no—
422Dic7—22no—
423Dic7—42no—
424Dic7—41no—
425Dic7—81no—
426Dic7—81no—
427C24—141no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8C2×C4×D712yes—
432C2×Q8#115712yes—
433C2×Q8#115712yes—
434C2×D4—72no—
435C2×D4—141no—
436C2×D4—141no—
437C2×D4—141no—
438C2×D4—141no—
439C2×D4—141no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—71no—
445C22×C4—71no—
446C22×C4—71no—
447C22×C4—71no—
448C22×C4—144no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—561no—
453C22×C4—561no—
454C22×C4D4×D711yescharacteristic
455C22×C4D4×D711yescharacteristic
456C22×C4C23×D711yescharacteristic
457SD16—148no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—72no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8C2×C4×D712yes—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—42no—
476C4⋊C4—72no—
477C4⋊C4—142no—
478C4⋊C4—144no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—281no—
485C4⋊C4—282no—
486C4⋊C4—282no—
487C4⋊C4—561no—
488C4⋊C4—561no—
489C4⋊C4—561no—
490C4⋊C4C2×C4×D712yes—
491C22⋊C4—144no—
492C22⋊C4—282no—
493C22⋊C4—282no—
494C22⋊C4—561no—
495C22⋊C4—561no—
496C42—21no—
497C42—21no—
498C42—42no—
499C42—72no—
500C42—72no—
501C42—81no—
502C42—144no—
503C42—141no—
504C42—141no—
505C42—282no—
506C42—561no—
507C2×C7#235211yescharacteristic
508C2×C7C4×#13711yescharacteristic
509C2×C7#294611yescharacteristic
510C2×C7#330911yescharacteristic
511C2×C7#332011yescharacteristic
512C2×C7#337111yescharacteristic
513C2×C7#337511yescharacteristic
514D7—82no—
515C23—142no—
516C23—281no—
517C23—281no—
518C23—281no—
519C23—281no—
520C23—281no—
521C23—281no—
522C23C2×D4×D711yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—22no—
526Q8#643514yes—
527D4—142no—
528D4—142no—
529D4—281no—
530D4—281no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—72no—
550C2×C4—72no—
551C2×C4—72no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—141no—
558C2×C4—141no—
559C2×C4—141no—
560C2×C4—141no—
561C2×C4—142no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—564no—
571C2×C4C22×C4×D711yescharacteristic
572C2×C4C2×#115711yescharacteristic
573C2×C4C2×#115711yescharacteristic
574C2×C4#654811yescharacteristic
575C2×C4D7×#4111yescharacteristic
576C2×C4#655411yescharacteristic
577C8—22no—
578C8—41no—
579C8—142no—
580C8—281no—
581C7#2277511yescharacteristic, complemented, Sylow
582C22—282no—
583C22—282no—
584C22—282no—
585C22—561no—
586C22—561no—
587C22—561no—
588C22—561no—
589C22#9018111yescharacteristic
590C22C4×D4×D711yescharacteristic
591C22#9024311yescharacteristic
592C22#9046511yescharacteristic
593C22D7×#13711yescharacteristic
594C22#9046811yescharacteristic
595C22#9046911yescharacteristic
596C4—24no—
597C4—41no—
598C4—41no—
599C4—42no—
600C4—81no—
601C4—81no—
602C4—142no—
603C4—142no—
604C4—142no—
605C4—281no—
606C4—282no—
607C4—561no—
608C4—561no—
609C4C2×#643511yescharacteristic
610C4#9022211yescharacteristic
611C4#9023111yescharacteristic
612C4#9023711yescharacteristic
613C2—562no—
614C2#1173090611yescharacteristic
615C2#1173634211yescharacteristic
616C2#1173641311yescharacteristic
617C2#1173720111yescharacteristic
618C2#1173722011yescharacteristic
619C2#1173726111yescharacteristic
620C2#1173727411yescharacteristic
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#22775
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. Q8×Dic7⋊C4C2semidirect
4. C7×#1763C2semidirect
7. #11727976C2semidirect
8. Dic7×#103C2semidirect
10. #11727565C2semidirect
11. #11727212C2semidirect
14. #11726603C2semidirect
15. #11726544C2semidirect
47. #89589C4semidirect
139. #1763D7semidirect
581. C7#22775semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic7⋊C42
C2×#895892
#117310912
C7×#17632
#117293872
#117279942
#117279762
Dic7×#1032
#117275722
#117275652
#117272122
#117266462
#117266062
#117266032
#117265442
#227757