Finite Group Databasebeta

← #13373159

Subgroups — #13373159

Order 1792.

866 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739779C211yesmaximal, characteristic
2C2×#89529C211yesmaximal, characteristic
3#11731086C211yesmaximal, characteristic, complemented
4C4×C7×#102C211yesmaximal, characteristic
5#11729366C211yesmaximal, characteristic
6#11727925C211yesmaximal, characteristic
7#11727907C211yesmaximal, characteristic
8#11727906C211yesmaximal, characteristic
9#11727537C211yesmaximal, characteristic, complemented
10#11727521C211yesmaximal, characteristic, complemented
11#11727215C211yesmaximal, characteristic, complemented
12#11726659C211yesmaximal, characteristic, complemented
13#11726608C211yesmaximal, characteristic, complemented
14#11726603C211yesmaximal, characteristic, complemented
15#11726541C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×D4×Dic7C2211yescharacteristic
18C2×D4×Dic7C2211yescharacteristic
19C22×#1142C2211yescharacteristic
20C2×#6450C2211yescharacteristic
21C2×C7×#102C2211yescharacteristic
22C2×C7×#102C2211yescharacteristic
23C2×C4×C7×C8C2211yescharacteristic
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×#6406C2211yescharacteristic
26#89982—21no—
27C2×C7⋊#95—21no—
28C2×#6390C2211yescharacteristic
29C2×#6390C2211yescharacteristic
30C2×#6389C2211yescharacteristic
31C2×#6389C2211yescharacteristic
32C2×Dic7⋊C8C2211yescharacteristic
33#89749C2211yescharacteristic
34#89748C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36#89746C2211yescharacteristic
37#89745C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40#89718—21no—
41#89714—21no—
42#89705—21no—
43C4×Dic7⋊C4—21no—
44C4×Dic7⋊C4C2211yescharacteristic
45#89701C2211yescharacteristic
46#89700C2211yescharacteristic
47#89529C418yescomplemented
48#89424—21no—
49#89422—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#22776—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C23×Dic7—21no—
64C23×Dic7—21no—
65C22×Dic14C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C4×C7×D4—22no—
69C4×C7×D4—41no—
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C22⋊C4—21no—
74C2×C42×C7—21no—
75C2×C42×C7C2311yescharacteristic
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—21no—
78D4×Dic7—22no—
79D4×Dic7—22no—
80D4×Dic7—41no—
81D4×Dic7C2×C414yes—
82C2×#1142—22no—
83C2×#1142—22no—
84C2×#1142C2×C414yes—
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⋊C4D411yescharacteristic
94C2×Dic7⋊C4D411yescharacteristic
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C2×C4×Dic7C2311yescharacteristic
98C2×C4×Dic7C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101#6450—22no—
102#6450C2×C414yescomplemented
103C7×#102—22no—
104C7×#102C2×C414yes—
105C4×C7×C8—41no—
106C7×#95—21no—
107#6406—22no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95D411yescharacteristic
118C7⋊#95D411yescharacteristic
119#6390—41no—
120#6390C2×C414yescomplemented
121#6389—41no—
122#6389C2×C414yescomplemented
123Dic7⋊C8—22no—
124#6384—41no—
125#6381C2×C414yescomplemented
126C2×#440—71no—
127#2355—71no—
128#2349—71no—
129#1935—71no—
130#1933—71no—
131#1921—71no—
132#1918—71no—
133#1877—71no—
134#1877—71no—
135#1836—71no—
136#1805—71no—
137#1802—71no—
138#1771—71no—
139#1762—71no—
140C4×#102D711yescharacteristic
141C24×C7—21no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4C22×C412yes—
146C2×C7×D4#4112yes—
147C2×C7×D4#4112yes—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—42no—
152C22×C4×C7—42no—
153C22×C4×C7—81no—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2411yescharacteristic
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—24no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—42no—
163C22×Dic7—42no—
164C22×Dic7—81no—
165C22×Dic7—81no—
166C22×Dic7—81no—
167C22×Dic7C2×D411yescharacteristic
168C22×Dic7C2×D411yescharacteristic
169C22×Dic7C2×D411yescharacteristic
170C22×Dic7C2×D411yescharacteristic
171C2×Dic14—21no—
172C2×Dic14—21no—
173C2×Dic14—21no—
174C2×Dic14—21no—
175C2×Dic14—21no—
176C2×Dic14C22×C412yes—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8C22×C412yes—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4C22×C412yes—
189C7×C22⋊C4—42no—
190C42×C7—21no—
191C42×C7—21no—
192C42×C7—81no—
193C22⋊Dic7—24no—
194C22⋊Dic7—42no—
195#1142—28no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7C22×C412yes—
202Dic7⋊C4—22no—
203Dic7⋊C4—24no—
204Dic7⋊C4—41no—
205Dic7⋊C4—42no—
206Dic7⋊C4—81no—
207Dic7⋊C4—81no—
208Dic7⋊C4—81no—
209C4×Dic7—21no—
210C4×Dic7—21no—
211C4×Dic7—24no—
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×D4—71no—
224C2×C4×D4—71no—
225C2×C4×D4C2×D711yescharacteristic
226#440—78no—
227C2×#107—71no—
228C2×C4⋊C8—71no—
229C2×#103—71no—
230C2×#103—71no—
231C2×#103—71no—
232C2×#102—71no—
233C2×#102C2×D711yescharacteristic
234C2×#102C2×D711yescharacteristic
235C2×#97—71no—
236C2×#97—71no—
237C2×C4×C8C2×D711yescharacteristic
238#391—71no—
239#391—71no—
240#389—71no—
241#389—71no—
242#387—141no—
243#387—141no—
244#385—141no—
245#385—141no—
246#384—71no—
247#382—71no—
248#382—141no—
249#380—141no—
250C4×C4⋊C4—71no—
251C4×C4⋊C4—71no—
252C4×C4⋊C4—141no—
253C4×C4⋊C4C2×D711yescharacteristic
254C2×#95—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×C7—22no—
266C23×C7—22no—
267C23×C7—22no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7C22×D411yescharacteristic
273C7×D4—22no—
274C7×D4—22no—
275C7×D4—22no—
276C7×D4#11714yes—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—22no—
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—84no—
292C2×C4×C7#13712yes—
293C2×C4×C7C23×C411yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7H(5,𝔽2)11yescharacteristic
298C2×C4×C72−1+411yescharacteristic
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—22no—
304C2×Dic7—22no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
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×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—84no—
322C2×Dic7—84no—
323C2×Dic7C4×D412yes—
324C2×Dic7C4×D412yes—
325Dic14—22no—
326Dic14—22no—
327Dic14—41no—
328Dic14—41no—
329C7×C8—22no—
330C7×C8—41no—
331C7⋊C8—22no—
332C7⋊C8—41no—
333C22×Q8—71no—
334C22×D4C22×D711yescharacteristic
335C23×C4—21no—
336C23×C4—141no—
337C23×C4—141no—
338C2×SD16—74no—
339C2×SD16—142no—
340C2×SD16—142no—
341C22×C8—71no—
342C22×C8—71no—
343C22×C8C22×D711yescharacteristic
344C22×C8C22×D711yescharacteristic
345C4×Q8—74no—
346C4×Q8—142no—
347C4×D4—22no—
348C4×D4—41no—
349C4×D4—74no—
350C4×D4—142no—
351C4×D4—142no—
352C4×D4—281no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
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—21no—
367C2×C22⋊C4—141no—
368C2×C22⋊C4—141no—
369C2×C42—21no—
370C2×C42—71no—
371C2×C42—71no—
372C2×C42—141no—
373C2×C42—141no—
374C2×C42C22×D711yescharacteristic
375#107—74no—
376C4⋊C8—142no—
377#103—74no—
378#103—281no—
379#103—281no—
380#102—22no—
381#102—142no—
382#102C4×D714yes—
383#97—74no—
384#97—281no—
385C4×C8—41no—
386#95—21no—
387#95—71no—
388#95—71no—
389#95—141no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95—141no—
397#95—141no—
398C22×C7—24no—
399C22×C7—42no—
400C22×C7—42no—
401C22×C7—42no—
402C22×C7—42no—
403C22×C7—42no—
404C22×C7—42no—
405C22×C7C2×C4×D411yescharacteristic
406C22×C7#54611yescharacteristic
407C22×C7#54811yescharacteristic
408C22×C7C2×#13711yescharacteristic
409C22×C7#57511yescharacteristic
410C22×C7#57611yescharacteristic
411C22×C7#57711yescharacteristic
412C4×C7—22no—
413C4×C7—42no—
414C4×C7—41no—
415C4×C7—81no—
416C4×C7—81no—
417C4×C7C2×#11711yescharacteristic
418C4×C7#52011yescharacteristic
419C4×C7#55211yescharacteristic
420C4×C7#55311yescharacteristic
421Dic7—22no—
422Dic7—22no—
423Dic7—42no—
424Dic7—42no—
425Dic7—41no—
426Dic7—41no—
427Dic7—81no—
428Dic7—81no—
429C24—21no—
430C2×Q8—72no—
431C2×Q8—141no—
432C2×Q8—141no—
433C2×Q8—141no—
434C2×Q8—141no—
435C2×Q8—141no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4C2×C4×D712yes—
440C2×D4#115712yes—
441C2×D4#115712yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—71no—
448C22×C4—71no—
449C22×C4—71no—
450C22×C4—71no—
451C22×C4—81no—
452C22×C4—144no—
453C22×C4—141no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—141no—
457C22×C4—282no—
458C22×C4—282no—
459C22×C4—561no—
460C22×C4—561no—
461C22×C4—561no—
462C22×C4D4×D711yescharacteristic
463C22×C4D4×D711yescharacteristic
464C22×C4C23×D711yescharacteristic
465SD16—148no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—72no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8—141no—
476C2×C8—141no—
477C2×C8C2×C4×D712yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—72no—
484C4⋊C4—144no—
485C4⋊C4—142no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—281no—
492C4⋊C4—282no—
493C4⋊C4—561no—
494C4⋊C4—561no—
495C4⋊C4—561no—
496C4⋊C4C2×C4×D712yes—
497C22⋊C4—42no—
498C22⋊C4—144no—
499C22⋊C4—282no—
500C42—21no—
501C42—21no—
502C42—72no—
503C42—72no—
504C42—81no—
505C42—144no—
506C42—141no—
507C42—141no—
508C42—561no—
509C2×C7—44no—
510C2×C7#235211yescharacteristic
511C2×C7C4×#13711yescharacteristic
512C2×C7#294611yescharacteristic
513C2×C7#331111yescharacteristic
514C2×C7#331411yescharacteristic
515C2×C7#337011yescharacteristic
516C2×C7#337211yescharacteristic
517C23—22no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23C2×D4×D711yescharacteristic
525Q8—142no—
526Q8—142no—
527Q8—281no—
528Q8—281no—
529D4—22no—
530D4—22no—
531D4—22no—
532D4#643514yes—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—72no—
548C2×C4—72no—
549C2×C4—84no—
550C2×C4—141no—
551C2×C4—141no—
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—142no—
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—281no—
571C2×C4—281no—
572C2×C4—564no—
573C2×C4—564no—
574C2×C4#647812yes—
575C2×C4C22×C4×D711yescharacteristic
576C2×C4C2×#115711yescharacteristic
577C2×C4C2×#115711yescharacteristic
578C2×C4#654811yescharacteristic
579C2×C4D7×#4111yescharacteristic
580C2×C4#655411yescharacteristic
581C8—22no—
582C8—41no—
583C8—142no—
584C8—281no—
585C7#2277611yescharacteristic, complemented, Sylow
586C22—24no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22#9018111yescharacteristic
594C22C4×D4×D711yescharacteristic
595C22#9024311yescharacteristic
596C22#9044811yescharacteristic
597C22C2×#647811yescharacteristic
598C22#9045911yescharacteristic
599C22#9046111yescharacteristic
600C4—22no—
601C4—42no—
602C4—41no—
603C4—81no—
604C4—81no—
605C4—142no—
606C4—142no—
607C4—281no—
608C4—282no—
609C4—282no—
610C4—281no—
611C4—561no—
612C4—561no—
613C4C2×#643511yescharacteristic
614C4#9022211yescharacteristic
615C4#9023111yescharacteristic
616C4#9023711yescharacteristic
617C2—44no—
618C2#1173090611yescharacteristic
619C2#1173620811yescharacteristic
620C2#1173630011yescharacteristic
621C2#1173701711yescharacteristic
622C2#1173703211yescharacteristic
623C2#1173707311yescharacteristic
624C2C4×#647811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC4×C7×#102C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC4×#102D7
hypocenterC7#22776
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
3. #11731086C2semidirect
9. #11727537C2semidirect
10. #11727521C2semidirect
11. #11727215C2semidirect
12. #11726659C2semidirect
13. #11726608C2semidirect
14. #11726603C2semidirect
15. #11726541C2semidirect
47. #89529C4semidirect
102. #6450C2×C4semidirect
120. #6390C2×C4semidirect
122. #6389C2×C4semidirect
125. #6381C2×C4semidirect
585. C7#22776semidirect

Maximal subgroups

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

subgroupindex
#117397792
C2×#895292
#117310862
C4×C7×#1022
#117293662
#117279252
#117279072
#117279062
#117275372
#117275212
#117272152
#117266592
#117266082
#117266032
#117265412
#227767