Finite Group Databasebeta

← #13373158

Subgroups — #13373158

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
1C7×#2355C211yesmaximal, characteristic, complemented
2C2×#89798C211yesmaximal, characteristic
3#11731215C211yesmaximal, characteristic
4#11729381C211yesmaximal, characteristic, complemented
5#11727441C211yesmaximal, characteristic
6#11727441C211yesmaximal, characteristic
7#11727429C211yesmaximal, characteristic, complemented
8#11727428C211yesmaximal, characteristic, complemented
9#11727338C211yesmaximal, characteristic
10#11727329C211yesmaximal, characteristic, complemented
11#11727232C211yesmaximal, characteristic, complemented
12#11727228C211yesmaximal, characteristic
13C4×#6383C211yesmaximal, characteristic
14#11727215C211yesmaximal, characteristic, complemented
15#11727212C211yesmaximal, characteristic, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17C22×#1143C2211yescharacteristic
18C2×C4×D28C2211yescharacteristic
19C2×C4×D28C2211yescharacteristic
20C2×C4×D28C2211yescharacteristic
21C7×#391C2211yescharacteristic
22C7×#391C2211yescharacteristic
23C7×#387—21no—
24C7×#387—21no—
25C7×#384C2211yescharacteristic
26C7×#382—21no—
27C4×C7×C4⋊C4—21no—
28C4×C7×C4⋊C4C2211yescharacteristic
29C4×C7×C4⋊C4C2211yescharacteristic
30C4×C7×C4⋊C4C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34#89798C418yescomplemented
35#89761—21no—
36#89761—21no—
37C2×#6383C2211yescharacteristic
38C2×#6383C2211yescharacteristic
39C2×#6383C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41#89708C2211yescharacteristic
42#89700C2211yescharacteristic
43#89700C2211yescharacteristic
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45C2×#6377C2211yescharacteristic
46C2×#6377C2211yescharacteristic
47C2×C4×C7⋊C8C2211yescharacteristic
48#89439—21no—
49D7×#95—21no—
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#22776—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×D28C2311yescharacteristic
63C22×C4×D7—21no—
64C22×C4×D7—21no—
65C22×C4×D7—21no—
66C4×C7×Q8—22no—
67C4×C7×Q8C2×C414yescomplemented
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4D411yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C42×C7—21no—
77C2×C42×C7—21no—
78C2×C42×C7—21no—
79C2×C42×C7C2311yescharacteristic
80C2×C42×C7C2311yescharacteristic
81C2×C42×C7C2311yescharacteristic
82C2×#1143—22no—
83C2×#1143—22no—
84C2×#1143C2×C414yes—
85C2×(C2×D7)⋊C4—21no—
86C2×(C2×D7)⋊C4—21no—
87C2×(C2×D7)⋊C4—21no—
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C4×D28—22no—
98C4×D28—22no—
99C4×D28—22no—
100C4×D28—41no—
101C4×D28—41no—
102C4×D28C2×C414yes—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95—21no—
107C7×#95—21no—
108C7×#95—21no—
109C7×#95D411yescharacteristic
110C7×#95D411yescharacteristic
111#6409—41no—
112#6409—41no—
113#6409C2×C414yescomplemented
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118#6383—22no—
119#6383—22no—
120#6383C2×C414yes—
121#6381C2×C414yescomplemented
122C4⋊(C7⋊C8)—22no—
123#6377—41no—
124#6377C2×C414yescomplemented
125C4×C7⋊C8—41no—
126C2×#440—71no—
127#2355D711yescharacteristic, complemented
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×#102—71no—
141C23×D7—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8C22×C412yes—
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×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2411yescharacteristic
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C2×D28—21no—
163C2×D28—21no—
164C2×D28—21no—
165C2×D28C22×C412yes—
166C2×D28#4112yes—
167C2×D28#4112yes—
168C2×C4×D7—24no—
169C2×C4×D7—42no—
170C2×C4×D7—42no—
171C2×C4×D7—42no—
172C2×C4×D7—42no—
173C2×C4×D7—81no—
174C2×C4×D7—81no—
175C2×C4×D7—81no—
176C2×C4×D7—81no—
177C7×C4⋊C4—24no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—41no—
185C7×C4⋊C4—42no—
186C7×C4⋊C4—81no—
187C7×C4⋊C4C22×C412yes—
188C42×C7—24no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—21no—
192C42×C7—21no—
193C42×C7—81no—
194C42×C7—81no—
195C42×C7C22×C412yes—
196C42×C7#4112yes—
197#1143—28no—
198(C2×D7)⋊C4—24no—
199(C2×D7)⋊C4—42no—
200(C2×D7)⋊C4—42no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7C22×C412yes—
207Dic7⋊C4—81no—
208Dic7⋊C4—81no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8C22×C412yes—
220C2×C7⋊C8C22×C412yes—
221C22×SD16—71no—
222C2×C4×Q8C2×D711yescharacteristic
223C2×C4×D4—71no—
224C2×C4×D4—71no—
225C2×C4×D4—71no—
226#440—78no—
227C2×#107—71no—
228C2×C4⋊C8—71no—
229C2×#103—71no—
230C2×#103—71no—
231C2×#103—71no—
232C2×#102—71no—
233C2×#102—71no—
234C2×#102—71no—
235C2×#97—71no—
236C2×#97—71no—
237C2×C4×C8—71no—
238#391C2×D711yescharacteristic
239#391C2×D711yescharacteristic
240#389—71no—
241#389—71no—
242#387—21no—
243#387—21no—
244#385—141no—
245#385—141no—
246#384C2×D711yescharacteristic
247#382—21no—
248#382—71no—
249#380—141no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4C2×D711yescharacteristic
252C4×C4⋊C4C2×D711yescharacteristic
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#340—71no—
262#340—71no—
263#336—71no—
264#336—71no—
265C23×C7C22×D411yescharacteristic
266C22×D7—22no—
267C22×D7—22no—
268C22×D7—22no—
269C22×D7—41no—
270C22×D7—41no—
271C22×D7—41no—
272C22×D7—41no—
273C7×Q8—22no—
274C7×Q8—22no—
275C7×Q8—41no—
276C7×Q8—41no—
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—22no—
284C2×C4×C7—22no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7C4×D412yes—
305C2×C4×C7C4×D412yes—
306C2×C4×C7#13712yes—
307C2×C4×C7C23×C411yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×C4×C7C2×#4111yescharacteristic
310C2×C4×C7C2×#4111yescharacteristic
311C2×C4×C7H(5,𝔽2)11yescharacteristic
312C2×C4×C72−1+411yescharacteristic
313C2×Dic7—22no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322D28—22no—
323D28—22no—
324D28—22no—
325D28#11714yes—
326C4×D7—84no—
327C4×D7—84no—
328C4×D7—84no—
329C7⋊C8—22no—
330C7⋊C8—22no—
331C7⋊C8—41no—
332C7⋊C8—41no—
333C22×Q8C22×D711yescharacteristic
334C22×D4—71no—
335C23×C4—141no—
336C23×C4—141no—
337C23×C4—141no—
338C2×SD16—74no—
339C2×SD16—142no—
340C2×SD16—142no—
341C22×C8—71no—
342C22×C8—71no—
343C22×C8—71no—
344C22×C8—71no—
345C4×Q8—22no—
346C4×Q8C4×D714yescomplemented
347C4×D4—74no—
348C4×D4—142no—
349C4×D4—142no—
350C4×D4—142no—
351C4×D4—281no—
352C4×D4—281no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—71no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4C22⋊D711yescharacteristic
362C2×C4⋊C4C22⋊D711yescharacteristic
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×C42—21no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42C22×D711yescharacteristic
373C2×C42C22×D711yescharacteristic
374C2×C42C22×D711yescharacteristic
375#107—74no—
376C4⋊C8—142no—
377#103—74no—
378#103—281no—
379#103—281no—
380#102—74no—
381#102—142no—
382#102—142no—
383#97—74no—
384#97—281no—
385C4×C8—281no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95C22⋊D711yescharacteristic
397#95C22⋊D711yescharacteristic
398C22×C7C2×C4×D411yescharacteristic
399C22×C7#54611yescharacteristic
400C22×C7#54811yescharacteristic
401C22×C7C2×#13711yescharacteristic
402C22×C7#57511yescharacteristic
403C22×C7#57611yescharacteristic
404C22×C7#57711yescharacteristic
405C2×D7—24no—
406C2×D7—42no—
407C2×D7—42no—
408C2×D7—42no—
409C2×D7—42no—
410C2×D7—42no—
411C2×D7—42no—
412C4×C7—22no—
413C4×C7—22no—
414C4×C7—22no—
415C4×C7—42no—
416C4×C7—42no—
417C4×C7—41no—
418C4×C7—41no—
419C4×C7—41no—
420C4×C7—81no—
421C4×C7—81no—
422C4×C7C2×#11711yescharacteristic
423C4×C7#52011yescharacteristic
424C4×C7#55211yescharacteristic
425C4×C7#55311yescharacteristic
426Dic7—42no—
427Dic7—81no—
428Dic7—81no—
429C24—141no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8C2×C4×D712yes—
436C2×D4—72no—
437C2×D4—72no—
438C2×D4—72no—
439C2×D4—141no—
440C2×D4—141no—
441C2×D4—141no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—141no—
447C22×C4—144no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—282no—
451C22×C4—282no—
452C22×C4—282no—
453C22×C4—282no—
454C22×C4—561no—
455C22×C4—561no—
456C22×C4—561no—
457C22×C4—561no—
458C22×C4C2×C22⋊D711yescharacteristic
459C22×C4C2×C22⋊D711yescharacteristic
460C22×C4C2×C22⋊D711yescharacteristic
461C22×C4C2×C22⋊D711yescharacteristic
462C22×C4C2×C22⋊D711yescharacteristic
463C22×C4C2×C22⋊D711yescharacteristic
464C22×C4C23×D711yescharacteristic
465SD16—148no—
466C2×C8—72no—
467C2×C8—72no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8—141no—
476C2×C8—141no—
477C2×C8—141no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—72no—
488C4⋊C4—81no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—141no—
492C4⋊C4—141no—
493C4⋊C4—141no—
494C4⋊C4—561no—
495C4⋊C4—561no—
496C4⋊C4C2×C4×D712yes—
497C22⋊C4—144no—
498C22⋊C4—282no—
499C22⋊C4—282no—
500C42—24no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—81no—
506C42—81no—
507C42C2×C4×D712yes—
508C42#115712yes—
509C2×C7#235211yescharacteristic
510C2×C7C4×#13711yescharacteristic
511C2×C7#294611yescharacteristic
512C2×C7#331111yescharacteristic
513C2×C7#331411yescharacteristic
514C2×C7#337011yescharacteristic
515C2×C7#337211yescharacteristic
516D7—44no—
517C23—142no—
518C23—142no—
519C23—142no—
520C23—281no—
521C23—281no—
522C23—281no—
523C23—281no—
524C23C22×C22⋊D711yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529D4—74no—
530D4—142no—
531D4—142no—
532D4—142no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—142no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—564no—
570C2×C4—564no—
571C2×C4—564no—
572C2×C4C4×C22⋊D712yes—
573C2×C4C4×C22⋊D712yes—
574C2×C4#650512yes—
575C2×C4C22×C4×D711yescharacteristic
576C2×C4C2×#115711yescharacteristic
577C2×C4D7×#4111yescharacteristic
578C2×C4D7×#4111yescharacteristic
579C2×C4#655311yescharacteristic
580C2×C4#655411yescharacteristic
581C8—142no—
582C8—142no—
583C8—281no—
584C8—281no—
585C7#2277611yescharacteristic, complemented, Sylow
586C22—144no—
587C22—282no—
588C22—282no—
589C22—282no—
590C22—282no—
591C22—282no—
592C22—282no—
593C22C2×C4×C22⋊D711yescharacteristic
594C22C2×#650511yescharacteristic
595C22#9051311yescharacteristic
596C22#9052011yescharacteristic
597C22#9052211yescharacteristic
598C22#9052611yescharacteristic
599C22#9052811yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—42no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—282no—
611C4—561no—
612C4—561no—
613C4D7×#11711yescharacteristic
614C4#9021411yescharacteristic
615C4#9022011yescharacteristic
616C4#9022111yescharacteristic
617C2—284no—
618C2#1173971711yescharacteristic
619C2C4×#650511yescharacteristic
620C2#1174000911yescharacteristic
621C2#1174108011yescharacteristic
622C2#1174111011yescharacteristic
623C2#1174113711yescharacteristic
624C2#1174115911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2355C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#2355D7
hypocenterC7#22776
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2355

Complements

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

NH ≅ G/Ntype
1. C7×#2355C2semidirect
4. #11729381C2semidirect
7. #11727429C2semidirect
8. #11727428C2semidirect
10. #11727329C2semidirect
11. #11727232C2semidirect
14. #11727215C2semidirect
15. #11727212C2semidirect
34. #89798C4semidirect
67. C4×C7×Q8C2×C4semidirect
113. #6409C2×C4semidirect
121. #6381C2×C4semidirect
124. #6377C2×C4semidirect
127. #2355D7semidirect
346. C4×Q8C4×D7semidirect
585. C7#22776semidirect

Maximal subgroups

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

subgroupindex
C7×#23552
C2×#897982
#117312152
#117293812
#117274412
#117274292
#117274282
#117273382
#117273292
#117272322
#117272282
C4×#63832
#117272152
#117272122
#227767