Finite Group Databasebeta

← #13360948

Subgroups — #13360948

Order 1792.

972 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89657C211yesmaximal, characteristic
2#11733409C211yesmaximal, characteristic
3C4×#6455C211yesmaximal, characteristic, complemented
4C7×#1921C211yesmaximal, characteristic, complemented
5Dic7×#107C211yesmaximal, characteristic, complemented
6#11727356C211yesmaximal, characteristic, complemented
7#11727217C211yesmaximal, characteristic, complemented
8#11726635C211yesmaximal, characteristic
9#11726631C211yesmaximal, characteristic, complemented
10#11726567C211yesmaximal, characteristic, complemented
11#11726538C211yesmaximal, characteristic, complemented
12#11726534C211yesmaximal, characteristic, complemented
13#11726534C211yesmaximal, characteristic, complemented
14#11726379C211yesmaximal, characteristic, complemented
15#11726353C211yesmaximal, characteristic, complemented
16C2×#6458C2211yescharacteristic
17C2×#6456—21no—
18C2×#6456C2211yescharacteristic
19C2×#6455C2211yescharacteristic
20C2×#6455C2211yescharacteristic
21C2×C42×D7C2211yescharacteristic
22C2×C7×#107C2211yescharacteristic
23C7×#389C2211yescharacteristic
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×#6395C2211yescharacteristic
26C2×#6395C2211yescharacteristic
27C2×(C2×D7)⋊C8C2211yescharacteristic
28C2×(C2×D7)⋊C8C2211yescharacteristic
29C2×#6392C2211yescharacteristic
30C2×C8×Dic7C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32#89762—21no—
33#89759C2211yescharacteristic
34C4⋊C4×Dic7C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37C2×#6383C2211yescharacteristic
38C2×#6383C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41#89711C2211yescharacteristic
42C4×(C2×D7)⋊C4—21no—
43C4×C4⋊Dic7C2211yescharacteristic
44C42×Dic7—21no—
45#89657C418yescomplemented
46#89438—21no—
47#89438—21no—
48#89437—21no—
49#89420—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52C7×#340C2211yescharacteristic
53C7×#336C2211yescharacteristic
54#89346C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60C4×#482—71nomaximal, Sylow
61C22×D28—21no—
62C22×D28C2311yescharacteristic
63C22×C4×D7—21no—
64C22×C4×D7D411yescharacteristic
65C22×C4×D7D411yescharacteristic
66C22×C4×D7C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C42×C7—21no—
75C2×C42×C7C2311yescharacteristic
76C2×(C2×D7)⋊C4—21no—
77C2×(C2×D7)⋊C4—21no—
78C2×(C2×D7)⋊C4—21no—
79C2×(C2×D7)⋊C4—21no—
80C2×(C2×D7)⋊C4—21no—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4×Dic7—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7D411yescharacteristic
93C2×C4×Dic7D411yescharacteristic
94C2×C4×Dic7C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97#6458—22no—
98#6458C2×C414yes—
99#6456—22no—
100#6456—22no—
101#6456—44no—
102#6455—22no—
103#6455—22no—
104#6455—22no—
105#6455C2×C414yescomplemented
106C42×D7—22no—
107C42×D7—22no—
108C7×#107C2×C414yescomplemented
109C7×#95—21no—
110C7×#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116#6395—22no—
117#6395C2×C414yescomplemented
118(C2×D7)⋊C8—22no—
119(C2×D7)⋊C8C2×C414yescomplemented
120#6392—22no—
121C8×Dic7—22no—
122C8×Dic7—22no—
123#6383—22no—
124#6383C2×C414yescomplemented
125#6382—22no—
126#6382C2×C414yescomplemented
127C2×#482—71no—
128C4×#121—71no—
129C4×#117—71no—
130#1926—71no—
131#1925—71no—
132#1921D711yescharacteristic, complemented
133#1920—71no—
134#1875—71no—
135#1854—71no—
136#1825—71no—
137C4×#107—71no—
138C4×#106—71no—
139C4×#102—71no—
140C4×#102—71no—
141C4×C22⋊C8—71no—
142C23×D7—21no—
143C23×D7C2×D411yescharacteristic
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2411yescharacteristic
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7C2×D411yescharacteristic
155C22×Dic7C2×D411yescharacteristic
156C22×Dic7C2×D411yescharacteristic
157C2×D28—21no—
158C2×D28—21no—
159C2×D28—21no—
160C2×D28—21no—
161C2×D28—24no—
162C2×D28—21no—
163C2×D28—42no—
164C2×D28—42no—
165C2×D28C22×C412yes—
166C2×C4×D7—21no—
167C2×C4×D7—21no—
168C2×C4×D7—21no—
169C2×C4×D7—21no—
170C2×C4×D7—22no—
171C2×C4×D7—22no—
172C2×C4×D7—22no—
173C2×C4×D7—22no—
174C2×C4×D7—22no—
175C2×C4×D7—41no—
176C2×C4×D7—42no—
177C2×C4×D7—42no—
178C2×C4×D7—42no—
179C2×C4×D7C22×C412yes—
180C2×C4×D7#4112yes—
181C2×C7×C8—21no—
182C2×C7×C8—21no—
183C2×C7×C8—21no—
184C2×C7×C8—21no—
185C2×C7×C8—21no—
186C2×C7×C8C22×C412yes—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—42no—
193C7×C4⋊C4C22×C412yes—
194C7×C4⋊C4C22×C412yes—
195C7×C4⋊C4#4112yes—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7—42no—
199(C2×D7)⋊C4—24no—
200(C2×D7)⋊C4—24no—
201(C2×D7)⋊C4—24no—
202(C2×D7)⋊C4—42no—
203(C2×D7)⋊C4—42no—
204(C2×D7)⋊C4—42no—
205(C2×D7)⋊C4—42no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7C22×C412yes—
211C4⋊Dic7#4112yes—
212C4×Dic7—24no—
213C4×Dic7—24no—
214C4×Dic7—22no—
215C4×Dic7—22no—
216C4×Dic7—42no—
217C4×Dic7—42no—
218C4×Dic7—42no—
219C4×Dic7—42no—
220C4×Dic7#4112yes—
221C4×Dic7#4112yes—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8C22×C412yes—
228C2×#121—71no—
229C2×C4×D4—71no—
230C2×C4×D4—141no—
231C2×#117—71no—
232C2×#117—71no—
233C22×C42—71no—
234#482—78no—
235C2×#107—71no—
236C2×#107C2×D711yescharacteristic
237C2×#106—71no—
238C2×#106—71no—
239C2×#102—71no—
240C2×#102—71no—
241C2×#102—71no—
242C2×#102—71no—
243C2×C22⋊C8—71no—
244C2×C22⋊C8—71no—
245C2×C4×C8—71no—
246C2×C4×C8—71no—
247#390—71no—
248#389C2×D711yescharacteristic
249#388—141no—
250#386—141no—
251#380—71no—
252C4×C4⋊C4—71no—
253C4×C4⋊C4—71no—
254C4×C4⋊C4—71no—
255C4×C4⋊C4—71no—
256C4×C4⋊C4C2×D711yescharacteristic
257C4×C22⋊C4—141no—
258C4×C22⋊C4—141no—
259C4×C22⋊C4—141no—
260#376—141no—
261C43—141no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#340—71no—
267#340C2×D711yescharacteristic
268#340C2×D711yescharacteristic
269#340C2×D711yescharacteristic
270#336—71no—
271#336C2×D711yescharacteristic
272C23×C7C22×D411yescharacteristic
273C22×D7—21no—
274C22×D7—21no—
275C22×D7—21no—
276C22×D7—21no—
277C22×D7—21no—
278C22×D7—21no—
279C22×D7—22no—
280C22×D7—41no—
281C22×D7—41no—
282C22×D7—41no—
283C22×D7—41no—
284C22×D7—41no—
285C22×D7—41no—
286C22×D7C4×D412yes—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—21no—
292C2×C4×C7—22no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
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×C7#13512yes—
305C2×C4×C7C23×C411yescharacteristic
306C2×C4×C7C2×#4111yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×C4×C7C2×#4111yescharacteristic
310C2×C4×C7C2×#4111yescharacteristic
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—21no—
314C2×Dic7—21no—
315C2×Dic7—22no—
316C2×Dic7—22no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7C4×D412yes—
325C2×Dic7#12312yes—
326C2×Dic7#12312yes—
327C2×Dic7#13512yes—
328C2×Dic7AGL(1,ℤ/8)12yes—
329D28—22no—
330D28—22no—
331D28—48no—
332D28—41no—
333D28—41no—
334C4×D7—22no—
335C4×D7—22no—
336C4×D7—22no—
337C4×D7—22no—
338C4×D7—44no—
339C4×D7—44no—
340C4×D7—84no—
341C7×C8—22no—
342C7×C8—41no—
343C7⋊C8—22no—
344C7⋊C8—41no—
345C22×D4—71no—
346C22×D4—141no—
347C23×C4—71no—
348C23×C4—71no—
349C23×C4—71no—
350C23×C4—141no—
351C22×C8—71no—
352C22×C8—71no—
353C22×C8C22×D711yescharacteristic
354C22×C8C22×D711yescharacteristic
355#121—74no—
356#121—142no—
357C4×D4—142no—
358C4×D4—142no—
359C4×D4—284no—
360#117—74no—
361#117—142no—
362#117—142no—
363#117—142no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—71no—
366C2×C4⋊C4—71no—
367C2×C4⋊C4—71no—
368C2×C4⋊C4C22×D711yescharacteristic
369C2×C4⋊C4C22×D711yescharacteristic
370C2×C4⋊C4C22×D711yescharacteristic
371C2×C4⋊C4C22×D711yescharacteristic
372C2×C22⋊C4—141no—
373C2×C22⋊C4—141no—
374C2×C22⋊C4—141no—
375C2×C22⋊C4—141no—
376C2×C22⋊C4—141no—
377C2×C22⋊C4—141no—
378C2×C22⋊C4—141no—
379C2×C42—21no—
380C2×C42—71no—
381C2×C42—71no—
382C2×C42—71no—
383C2×C42—142no—
384C2×C42—142no—
385C2×C42—141no—
386C2×C42—141no—
387C2×C42—141no—
388C2×C42—141no—
389C2×C42—141no—
390C2×C42—141no—
391C2×C42C22×D711yescharacteristic
392#107—142no—
393#107C4×D714yescomplemented
394#106—74no—
395#106—142no—
396#102—74no—
397#102—74no—
398#102—142no—
399#102—142no—
400C22⋊C8—74no—
401C22⋊C8—142no—
402C4×C8—142no—
403C4×C8—142no—
404#95—21no—
405#95—21no—
406#95—141no—
407#95—141no—
408#95—141no—
409#95—141no—
410#95—141no—
411C22×C7C2×C4×D411yescharacteristic
412C22×C7C2×#12311yescharacteristic
413C22×C7#52511yescharacteristic
414C22×C7C2×#13511yescharacteristic
415C22×C7C2×#13511yescharacteristic
416C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
417C22×C7#57511yescharacteristic
418C2×D7—22no—
419C2×D7—22no—
420C2×D7—22no—
421C2×D7—41no—
422C2×D7—41no—
423C2×D7—41no—
424C2×D7—41no—
425C2×D7—42no—
426C2×D7—42no—
427C2×D7—42no—
428C2×D7—81no—
429C2×D7—81no—
430C2×D7—81no—
431C2×D7—81no—
432C4×C7—22no—
433C4×C7—41no—
434C4×C7—42no—
435C4×C7—42no—
436C4×C7—42no—
437C4×C7—81no—
438C4×C7C4×#4111yescharacteristic
439C4×C7C4×#4111yescharacteristic
440C4×C7#52911yescharacteristic
441C4×C7#52911yescharacteristic
442Dic7—22no—
443Dic7—22no—
444Dic7—42no—
445Dic7—42no—
446Dic7—81no—
447Dic7#48214yescomplemented
448C24—71no—
449C24—141no—
450C2×D4—72no—
451C2×D4—144no—
452C2×D4—141no—
453C2×D4—141no—
454C2×D4—141no—
455C2×D4—141no—
456C2×D4—141no—
457C2×D4—282no—
458C2×D4—282no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—71no—
464C22×C4—71no—
465C22×C4—71no—
466C22×C4—72no—
467C22×C4—72no—
468C22×C4—142no—
469C22×C4—142no—
470C22×C4—142no—
471C22×C4—142no—
472C22×C4—141no—
473C22×C4—141no—
474C22×C4—141no—
475C22×C4—141no—
476C22×C4—141no—
477C22×C4—141no—
478C22×C4—142no—
479C22×C4—141no—
480C22×C4—281no—
481C22×C4—282no—
482C22×C4—282no—
483C22×C4—282no—
484C22×C4D4×D711yescharacteristic
485C22×C4D4×D711yescharacteristic
486C22×C4C23×D711yescharacteristic
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—72no—
493C2×C8—141no—
494C2×C8—141no—
495C2×C8—141no—
496C2×C8—141no—
497C2×C8—141no—
498C2×C8C2×C4×D712yes—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—72no—
506C4⋊C4—72no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—141no—
510C4⋊C4—141no—
511C4⋊C4C2×C4×D712yes—
512C4⋊C4C2×C4×D712yes—
513C4⋊C4#116112yes—
514C22⋊C4—144no—
515C22⋊C4—144no—
516C22⋊C4—144no—
517C22⋊C4—282no—
518C22⋊C4—282no—
519C22⋊C4—282no—
520C22⋊C4—282no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—72no—
525C42—72no—
526C42—142no—
527C42—142no—
528C42—144no—
529C42—144no—
530C42—282no—
531C42—282no—
532C42—282no—
533C42—282no—
534C2×C7C4×#12311yescharacteristic
535C2×C7C4×#13511yescharacteristic
536C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
537C2×C7C2×#48211yescharacteristic
538C2×C7#309111yescharacteristic
539C2×C7#310211yescharacteristic
540C2×C7#310611yescharacteristic
541D7—42no—
542D7—82no—
543C23—72no—
544C23—141no—
545C23—141no—
546C23—141no—
547C23—141no—
548C23—141no—
549C23—141no—
550C23—142no—
551C23—281no—
552C23—281no—
553C23—281no—
554C23—281no—
555C23—281no—
556C23—281no—
557C23C2×D4×D711yescharacteristic
558D4—142no—
559D4—142no—
560D4—288no—
561D4—281no—
562D4—281no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—72no—
581C2×C4—72no—
582C2×C4—72no—
583C2×C4—72no—
584C2×C4—72no—
585C2×C4—141no—
586C2×C4—141no—
587C2×C4—141no—
588C2×C4—141no—
589C2×C4—142no—
590C2×C4—142no—
591C2×C4—142no—
592C2×C4—142no—
593C2×C4—142no—
594C2×C4—142no—
595C2×C4—284no—
596C2×C4—284no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4—564no—
605C2×C4#648212yes—
606C2×C4C22×C4×D711yescharacteristic
607C2×C4C2×#116111yescharacteristic
608C2×C4D7×#4111yescharacteristic
609C2×C4D7×#4111yescharacteristic
610C2×C4D7×#4111yescharacteristic
611C2×C4D7×#4111yescharacteristic
612C8—22no—
613C8—41no—
614C8—142no—
615C8—281no—
616C7C4×#48211yescharacteristic, complemented, Sylow
617C22—142no—
618C22—142no—
619C22—142no—
620C22—281no—
621C22—281no—
622C22—281no—
623C22—281no—
624C22—282no—
625C22—282no—
626C22—282no—
627C22—561no—
628C22—561no—
629C22—561no—
630C22—561no—
631C22C4×D4×D711yescharacteristic
632C22#9031811yescharacteristic
633C22D7×#12311yescharacteristic
634C22C2×#648211yescharacteristic
635C22D7×#13511yescharacteristic
636C22D7×AGL(1,ℤ/8)11yescharacteristic
637C22#9046411yescharacteristic
638C4—22no—
639C4—41no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—74no—
644C4—81no—
645C4—142no—
646C4—142no—
647C4—282no—
648C4—282no—
649C4—561no—
650C4#9021311yescharacteristic
651C4C4×#116111yescharacteristic
652C4#9038211yescharacteristic
653C4#9039711yescharacteristic
654C2—282no—
655C2—562no—
656C2#1173238511yescharacteristic
657C2C4×#648211yescharacteristic
658C2#1173731011yescharacteristic
659C2#1173853811yescharacteristic
660C2#1173858511yescharacteristic
661C2D7×#48211yescharacteristic
662C2#1173891211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1921C2
socleC23×C7C22×D4
prime radicalC2×#6458C22
cosocleC22×C4×C7C24
hypercenter#1921D7
hypocenterC7C4×#482
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
3. C4×#6455C2semidirect
4. C7×#1921C2semidirect
5. Dic7×#107C2semidirect
6. #11727356C2semidirect
7. #11727217C2semidirect
9. #11726631C2semidirect
10. #11726567C2semidirect
11. #11726538C2semidirect
12. #11726534C2semidirect
13. #11726534C2semidirect
14. #11726379C2semidirect
15. #11726353C2semidirect
45. #89657C4semidirect
105. #6455C2×C4semidirect
108. C7×#107C2×C4semidirect
117. #6395C2×C4semidirect
119. (C2×D7)⋊C8C2×C4semidirect
124. #6383C2×C4semidirect
126. #6382C2×C4semidirect
132. #1921D7semidirect
393. #107C4×D7semidirect
447. Dic7#482semidirect
616. C7C4×#482semidirect

Maximal subgroups

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

subgroupindex
C2×#896572
#117334092
C4×#64552
C7×#19212
Dic7×#1072
#117273562
#117272172
#117266352
#117266312
#117265672
#117265382
#117265342
#117263792
#117263532
C4×#4827