Finite Group Databasebeta

← #13364350

Subgroups — #13364350

Order 1792.

858 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89607C211yesmaximal, characteristic
2D7×#382C211yesmaximal, characteristic
3#11731198C211yesmaximal, characteristic
4C7×#1949C211yesmaximal, characteristic, complemented
5#11728386C211yesmaximal, characteristic
6#11728362C211yesmaximal, characteristic, complemented
7#11728333C211yesmaximal, characteristic
8#11728310C211yesmaximal, characteristic, complemented
9#11728209C211yesmaximal, characteristic, complemented
10#11727294C211yesmaximal, characteristic, complemented
11#11727222C211yesmaximal, characteristic, complemented
12#11726629C211yesmaximal, characteristic
13#11726628C211yesmaximal, characteristic
14#11726603C211yesmaximal, characteristic, complemented
15#11726603C211yesmaximal, characteristic, complemented
16C2×D7×C4⋊C4C2211yescharacteristic
17C2×D7×C4⋊C4C2211yescharacteristic
18C2×#6435—21no—
19C2×C42×D7C2211yescharacteristic
20C2×C7×#107C2211yescharacteristic
21C2×C7×#106C2211yescharacteristic
22C2×C7×C4⋊C8C2211yescharacteristic
23C7×#382C2211yescharacteristic
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C8C2211yescharacteristic
26C2×(C2×D7)⋊C8C2211yescharacteristic
27C2×#6392C2211yescharacteristic
28C2×#6391C2211yescharacteristic
29C2×#6389C2211yescharacteristic
30C2×#6389C2211yescharacteristic
31#89759C2211yescharacteristic
32#89759C2211yescharacteristic
33#89749C2211yescharacteristic
34#89749C2211yescharacteristic
35#89748C2211yescharacteristic
36#89748C2211yescharacteristic
37C2×#6382C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41#89708C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43C4×Dic7⋊C4—21no—
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45#89607C418yescomplemented
46#89439—21no—
47D7×#95—21no—
48D7×#95—21no—
49#89422—21no—
50#89422—21no—
51#89422—21no—
52C7×#340C2211yescharacteristic
53C7×#340C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#21986—71nomaximal, Sylow
61C22×C4×D7—21no—
62C22×C4×D7—21no—
63C22×C4×D7D411yescharacteristic
64C22×C4×D7D411yescharacteristic
65C22×C4×D7C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C42×C7—21no—
73C2×C42×C7C2311yescharacteristic
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×C4⋊Dic7C2311yescharacteristic
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—21no—
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×C4×Dic7—21no—
87C2×C4×Dic7D411yescharacteristic
88C2×C4×Dic7D411yescharacteristic
89C2×C4×Dic7C2311yescharacteristic
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92D7×C4⋊C4—41no—
93D7×C4⋊C4—41no—
94D7×C4⋊C4—41no—
95D7×C4⋊C4—41no—
96#6435—44no—
97C42×D7C2×C414yes—
98C42×D7Q814yes—
99C7×#107—41no—
100C7×#106—41no—
101C7×C4⋊C8C2×C414yescomplemented
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114(C2×D7)⋊C8C2×C414yes—
115(C2×D7)⋊C8C2×C414yes—
116#6392—41no—
117#6391—41no—
118#6389C2×C414yescomplemented
119#6389C2×C414yescomplemented
120#6382—41no—
121#6382—41no—
122#6381—41no—
123#6381—41no—
124C4⋊(C7⋊C8)C2×C414yescomplemented
125C2×#432—71no—
126#2285—71no—
127C2×#382—71no—
128#1949—71no—
129#1949—71no—
130#1949—71no—
131#1949D711yescharacteristic, complemented
132#1853—71no—
133#1853—71no—
134#1852—71no—
135#1852—71no—
136#1847—71no—
137#1847—71no—
138#1836—71no—
139#1836—71no—
140C23×D7C2×D411yescharacteristic
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2411yescharacteristic
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7C2×D411yescharacteristic
153C22×Dic7C2×D411yescharacteristic
154C22×Dic7C2×D411yescharacteristic
155C2×C4×D7—22no—
156C2×C4×D7—22no—
157C2×C4×D7—21no—
158C2×C4×D7—21no—
159C2×C4×D7—21no—
160C2×C4×D7—21no—
161C2×C4×D7—41no—
162C2×C4×D7—41no—
163C2×C4×D7—42no—
164C2×C4×D7—42no—
165C2×C4×D7—42no—
166C2×C4×D7—42no—
167C2×C4×D7—42no—
168C2×C4×D7—42no—
169C2×C4×D7C4⋊C414yes—
170C2×C4×D7C4⋊C414yes—
171C2×C4×D7C22×C412yes—
172C2×C4×D7C2×Q812yes—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8C22×C412yes—
178C2×C7×C8C22×C412yes—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C42×C7—42no—
188C42×C7C22×C412yes—
189C42×C7C2×Q812yes—
190(C2×D7)⋊C4—42no—
191(C2×D7)⋊C4—42no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200Dic7⋊C4—42no—
201Dic7⋊C4—42no—
202Dic7⋊C4—42no—
203Dic7⋊C4—42no—
204Dic7⋊C4—42no—
205Dic7⋊C4—42no—
206C4×Dic7—42no—
207C4×Dic7C4⋊C414yes—
208C4×Dic7C4⋊C414yes—
209C4×Dic7C22×C412yes—
210C4×Dic7C2×Q812yes—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8C22×C412yes—
216C2×C7⋊C8C22×C412yes—
217C2×#117—141no—
218C22×C4⋊C4—71no—
219C22×C4⋊C4—71no—
220C22×C42—71no—
221#432—78no—
222C2×#107—71no—
223C2×#107—71no—
224C2×#107—71no—
225C2×#107C2×D711yescharacteristic
226C2×#106—71no—
227C2×#106—71no—
228C2×#106—71no—
229C2×#106C2×D711yescharacteristic
230C2×C4⋊C8—71no—
231C2×C4⋊C8C2×D711yescharacteristic
232C2×C22⋊C8—71no—
233C2×C22⋊C8—71no—
234C2×#97—71no—
235C2×#97—71no—
236#384—71no—
237#384—71no—
238#382—71no—
239#382—71no—
240#382—71no—
241#382—141no—
242#382—141no—
243#382—141no—
244#382C2×D711yescharacteristic
245#380—71no—
246#380—71no—
247#380—141no—
248C4×C4⋊C4—71no—
249C4×C4⋊C4—141no—
250C4×C4⋊C4C2×D711yescharacteristic
251C2×#95—141no—
252C2×#95—141no—
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340C2×D711yescharacteristic
260#340C2×D711yescharacteristic
261C23×C7C22×D411yescharacteristic
262C22×D7—21no—
263C22×D7—21no—
264C22×D7—21no—
265C22×D7—21no—
266C22×D7C2×C4⋊C412yes—
267C22×D7AGL(1,ℤ/8)12yes—
268C22×D7#13712yes—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7C2×C4⋊C412yes—
286C2×C4×C7C2×C4⋊C412yes—
287C2×C4×C7C23×C411yescharacteristic
288C2×C4×C7C22×Q811yescharacteristic
289C2×C4×C7H(5,𝔽2)11yescharacteristic
290C2×C4×C7H(5,𝔽2)11yescharacteristic
291C2×C4×C72−1+411yescharacteristic
292C2×C4×C72−1+411yescharacteristic
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7C2×C4⋊C412yes—
312C2×Dic7C2×C4⋊C412yes—
313C2×Dic7C2×C4⋊C412yes—
314C4×D7—28no—
315C4×D7—22no—
316C4×D7—22no—
317C4×D7—22no—
318C4×D7—22no—
319C4×D7—44no—
320C4×D7—84no—
321C4×D7—84no—
322C7×C8—22no—
323C7×C8—22no—
324C7⋊C8—22no—
325C7⋊C8—22no—
326C23×C4—71no—
327C23×C4—71no—
328C23×C4—71no—
329C23×C4—141no—
330C23×C4—141no—
331C22×C8—71no—
332C22×C8—71no—
333C22×C8C22×D711yescharacteristic
334C22×C8C22×D711yescharacteristic
335#117—284no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—281no—
347C2×C4⋊C4—281no—
348C2×C4⋊C4—281no—
349C2×C4⋊C4—281no—
350C2×C4⋊C4C22×D711yescharacteristic
351C2×C4⋊C4C22×D711yescharacteristic
352C2×C4⋊C4C22×D711yescharacteristic
353C2×C4⋊C4C22×D711yescharacteristic
354C2×C22⋊C4—141no—
355C2×C22⋊C4—141no—
356C2×C42—21no—
357C2×C42—71no—
358C2×C42—71no—
359C2×C42—71no—
360C2×C42—74no—
361C2×C42—74no—
362C2×C42—141no—
363C2×C42C22×D711yescharacteristic
364#107—41no—
365#107—281no—
366#107—281no—
367#107—281no—
368#106—41no—
369#106—281no—
370#106—281no—
371#106—281no—
372C4⋊C8—74no—
373C4⋊C8C4×D714yescomplemented
374C22⋊C8—74no—
375C22⋊C8—74no—
376#97—74no—
377#97—74no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—141no—
382#95—141no—
383#95—141no—
384#95—141no—
385#95—141no—
386#95—141no—
387#95—141no—
388#95—141no—
389#95—141no—
390C22×C7C22×C4⋊C411yescharacteristic
391C22×C7#53511yescharacteristic
392C22×C7#53611yescharacteristic
393C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
394C22×C7C2×#13711yescharacteristic
395C22×C7#57511yescharacteristic
396C22×C7#57511yescharacteristic
397C2×D7—22no—
398C2×D7—22no—
399C2×D7—22no—
400C2×D7—22no—
401C2×D7—22no—
402C2×D7—22no—
403C2×D7#42814yes—
404C4×C7—22no—
405C4×C7—22no—
406C4×C7—81no—
407C4×C7—81no—
408C4×C7—81no—
409C4×C7—81no—
410C4×C7#52011yescharacteristic
411C4×C7#52011yescharacteristic
412C4×C7#54411yescharacteristic
413C4×C7#54411yescharacteristic
414Dic7—22no—
415Dic7—22no—
416Dic7—22no—
417Dic7—81no—
418Dic7—81no—
419Dic7—81no—
420Dic7—81no—
421C24—71no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—71no—
427C22×C4—71no—
428C22×C4—71no—
429C22×C4—74no—
430C22×C4—72no—
431C22×C4—72no—
432C22×C4—74no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—141no—
436C22×C4—141no—
437C22×C4—141no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—141no—
441C22×C4—141no—
442C22×C4—141no—
443C22×C4—281no—
444C22×C4—281no—
445C22×C4—282no—
446C22×C4—282no—
447C22×C4—282no—
448C22×C4—282no—
449C22×C4—282no—
450C22×C4—282no—
451C22×C4D4×D711yescharacteristic
452C22×C4D4×D711yescharacteristic
453C22×C4C23×D711yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—72no—
459C2×C8—72no—
460C2×C8—141no—
461C2×C8—141no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8C2×C4×D712yes—
465C2×C8C2×C4×D712yes—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—282no—
483C4⋊C4—282no—
484C4⋊C4—282no—
485C4⋊C4—282no—
486C4⋊C4—282no—
487C4⋊C4—282no—
488C22⋊C4—282no—
489C22⋊C4—282no—
490C42—42no—
491C42—74no—
492C42—74no—
493C42—72no—
494C42—72no—
495C42—282no—
496C42C2×C4×D712yes—
497C42Q8×D712yes—
498C2×C7#231711yescharacteristic
499C2×C7C2×#42811yescharacteristic
500C2×C7#291111yescharacteristic
501C2×C7#321311yescharacteristic
502C2×C7#321511yescharacteristic
503C2×C7#323411yescharacteristic
504C2×C7#323611yescharacteristic
505D7—24no—
506C23—72no—
507C23—72no—
508C23—72no—
509C23—141no—
510C23—141no—
511C23—141no—
512C23—141no—
513C23C2×D4×D711yescharacteristic
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—72no—
531C2×C4—72no—
532C2×C4—72no—
533C2×C4—148no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—142no—
541C2×C4—142no—
542C2×C4—142no—
543C2×C4—142no—
544C2×C4—284no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—564no—
558C2×C4—564no—
559C2×C4D7×C4⋊C412yes—
560C2×C4D7×C4⋊C412yes—
561C2×C4C22×C4×D711yescharacteristic
562C2×C4#654811yescharacteristic
563C2×C4C2×Q8×D711yescharacteristic
564C2×C4#655111yescharacteristic
565C2×C4#655311yescharacteristic
566C2×C4#655411yescharacteristic
567C8—22no—
568C8—22no—
569C8—142no—
570C8—142no—
571C7#2198611yescharacteristic, complemented, Sylow
572C22—74no—
573C22—142no—
574C22—142no—
575C22—142no—
576C22—142no—
577C22—142no—
578C22—142no—
579C22C2×D7×C4⋊C411yescharacteristic
580C22#9029811yescharacteristic
581C22#9031911yescharacteristic
582C22#9045911yescharacteristic
583C22#9045911yescharacteristic
584C22D7×AGL(1,ℤ/8)11yescharacteristic
585C22D7×#13711yescharacteristic
586C4—22no—
587C4—22no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—142no—
593C4—142no—
594C4—142no—
595C4—561no—
596C4—561no—
597C4—561no—
598C4—561no—
599C4#9023711yescharacteristic
600C4#9026511yescharacteristic
601C4#9038011yescharacteristic
602C4#9042011yescharacteristic
603C2—144no—
604C2#1173341311yescharacteristic
605C2#1173662211yescharacteristic
606C2D7×#42811yescharacteristic
607C2#1173798711yescharacteristic
608C2#1173805111yescharacteristic
609C2#1173861911yescharacteristic
610C2#1173868311yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1949C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1949D7
hypocenterC7#21986
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. C7×#1949C2semidirect
6. #11728362C2semidirect
8. #11728310C2semidirect
9. #11728209C2semidirect
10. #11727294C2semidirect
11. #11727222C2semidirect
14. #11726603C2semidirect
15. #11726603C2semidirect
45. #89607C4semidirect
101. C7×C4⋊C8C2×C4semidirect
118. #6389C2×C4semidirect
119. #6389C2×C4semidirect
124. C4⋊(C7⋊C8)C2×C4semidirect
131. #1949D7semidirect
373. C4⋊C8C4×D7semidirect
571. C7#21986semidirect

Maximal subgroups

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

subgroupindex
C2×#896072
D7×#3822
#117311982
C7×#19492
#117283862
#117283622
#117283332
#117283102
#117282092
#117272942
#117272222
#117266292
#117266282
#117266032
#219867