Finite Group Databasebeta

← #13370461

Subgroups — #13370461

Order 1792.

1066 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90270C211yesmaximal, characteristic
2C7×#2355C211yesmaximal, characteristic, complemented
3C2×#89796C211yesmaximal, characteristic, complemented
4#11727338C211yesmaximal, characteristic
5#11727337C211yesmaximal, characteristic, complemented
6#11727310C211yesmaximal, characteristic
7#11727310C211yesmaximal, characteristic
8#11727309C211yesmaximal, characteristic, complemented
9#11727308C211yesmaximal, characteristic, complemented
10#11727245C211yesmaximal, characteristic, complemented
11#11727245C211yesmaximal, characteristic, complemented
12#11727244C211yesmaximal, characteristic, complemented
13#11727228C211yesmaximal, characteristic
14#11727228C211yesmaximal, characteristic
15C4×#6383C211yesmaximal, characteristic
16C2×C4×C7×Q8C2211yescharacteristic
17C2×#6509—21no—
18#90270—24no—
19#90270C418yescomplemented
20C2×#6459—21no—
21C2×#6459—21no—
22C2×D7×C4⋊C4—21no—
23C2×C4×D28C2211yescharacteristic
24C2×C4×D28C2211yescharacteristic
25C2×C4×D28C2211yescharacteristic
26C2×#6433C2211yescharacteristic
27C2×#6433C2211yescharacteristic
28C2×#6432C2211yescharacteristic
29C7×#391—21no—
30C7×#387—21no—
31C7×#387—21no—
32C7×#384C2211yescharacteristic
33C7×#382C2211yescharacteristic
34C7×#382C2211yescharacteristic
35C4×C7×C4⋊C4—21no—
36C4×C7×C4⋊C4C2211yescharacteristic
37C4×C7×C4⋊C4C2211yescharacteristic
38C4×C7×C4⋊C4C2211yescharacteristic
39#89796—24no—
40C2×#6383C2211yescharacteristic
41C2×#6383C2211yescharacteristic
42C2×#6383C2211yescharacteristic
43C2×#6383C2211yescharacteristic
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4⋊(C7⋊C8)C2211yescharacteristic
47C2×#6377C2211yescharacteristic
48C2×#6377C2211yescharacteristic
49C2×C4×C7⋊C8C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#22526—71nomaximal, Sylow
63C22×C7×Q8D411yescharacteristic
64C22×D28C2311yescharacteristic
65C22×C4×D7—21no—
66C22×C4×D7—21no—
67C22×C4×D7—21no—
68C22×Dic14—21no—
69C4×C7×Q8C2×C414yes—
70C4×C7×Q8D414yes—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4D411yescharacteristic
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C4⋊C4D411yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C42×C7—21no—
81C2×C42×C7—21no—
82C2×C42×C7—21no—
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86#6509—28no—
87C2×(C2×D7)⋊C4—21no—
88C2×(C2×D7)⋊C4—21no—
89C2×(C2×D7)⋊C4—21no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102#6459—28no—
103#6459—28no—
104D7×C4⋊C4—28no—
105C4×D28—22no—
106C4×D28—22no—
107C4×D28—22no—
108C4×D28C2×C414yes—
109C4×D28C2×C414yes—
110C4×D28C2×C414yes—
111#6433—22no—
112#6433—22no—
113#6433C2×C414yes—
114#6433C2×C414yes—
115#6432—22no—
116#6432C2×C414yes—
117C7×#95—21no—
118C7×#95—21no—
119C7×#95—21no—
120C7×#95—21no—
121C7×#95—21no—
122C7×#95—21no—
123C7×#95—21no—
124#6383—22no—
125#6383—22no—
126#6383—22no—
127#6383Q814yescomplemented
128C4⋊(C7⋊C8)—22no—
129C4⋊(C7⋊C8)—22no—
130C4⋊(C7⋊C8)—22no—
131#6377—22no—
132#6377—22no—
133C4×C7⋊C8—22no—
134C2×#554—71no—
135C2×#446—71no—
136#2355D711yescharacteristic, complemented
137#1993—71no—
138#1992—71no—
139#1991—71no—
140#1989—71no—
141#1989—71no—
142#1988—71no—
143#1804—71no—
144#1804—71no—
145#1802—71no—
146#1762—71no—
147#1762—71no—
148C4×#102—71no—
149C23×D7—21no—
150C2×C7×Q8—41no—
151C2×C7×Q8—41no—
152C2×C7×Q8C22⋊C414yes—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2411yescharacteristic
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C2×D28—21no—
168C2×D28—21no—
169C2×D28—21no—
170C2×D28C22×C412yes—
171C2×D28C2×Q812yes—
172C2×D28#4112yes—
173C2×C4×D7—24no—
174C2×C4×D7—24no—
175C2×C4×D7—24no—
176C2×C4×D7—42no—
177C2×C4×D7—42no—
178C2×C4×D7—81no—
179C2×C4×D7—81no—
180C2×C4×D7—81no—
181C2×C4×D7—81no—
182C2×Dic14—24no—
183C2×Dic14—81no—
184C2×Dic14—81no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
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⋊C4—81no—
194C7×C4⋊C4C22⋊C414yes—
195C7×C4⋊C4C22⋊C414yes—
196C7×C4⋊C4C22⋊C414yes—
197C7×C4⋊C4C2×Q812yes—
198C42×C7—42no—
199C42×C7—81no—
200C42×C7—81no—
201C42×C7C22×C412yes—
202C42×C7C22×C412yes—
203C42×C7C22×C412yes—
204C42×C7C2×D412yes—
205C42×C7C2×D412yes—
206C42×C7C2×D412yes—
207(C2×D7)⋊C4—24no—
208(C2×D7)⋊C4—24no—
209(C2×D7)⋊C4—24no—
210C4⋊Dic7—24no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7C22×C412yes—
215C4⋊Dic7C22×C412yes—
216C4⋊Dic7C22×C412yes—
217Dic7⋊C4—24no—
218Dic7⋊C4—24no—
219Dic7⋊C4—24no—
220Dic7⋊C4—24no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8C2×Q812yes—
233#554—78no—
234#554—144no—
235C2×C4⋊Q8—71no—
236C2×#125—71no—
237C2×#125—71no—
238C2×C22⋊Q8—141no—
239C2×C22⋊Q8—141no—
240C2×C22⋊Q8—141no—
241C2×C4×Q8C2×D711yescharacteristic
242C2×C4×D4—71no—
243C2×C4×D4—71no—
244C2×C4×D4—71no—
245C22×C4⋊C4—141no—
246#446—144no—
247C2×C4⋊C8—71no—
248C2×C4⋊C8—71no—
249C2×C4⋊C8—71no—
250C2×#102—71no—
251C2×#102—71no—
252C2×#102—71no—
253C2×#102—71no—
254C2×#97—71no—
255C2×#97—71no—
256C2×C4×C8—71no—
257#391—21no—
258#387—21no—
259#387—21no—
260#384C2×D711yescharacteristic
261#382C2×D711yescharacteristic
262#382C2×D711yescharacteristic
263C4×C4⋊C4—21no—
264C4×C4⋊C4C2×D711yescharacteristic
265C4×C4⋊C4C2×D711yescharacteristic
266C4×C4⋊C4C2×D711yescharacteristic
267#340—71no—
268#340—71no—
269#340—71no—
270#340—71no—
271#340—71no—
272#340—71no—
273#340—71no—
274#340—71no—
275#340—71no—
276#340—71no—
277#340—71no—
278#340—71no—
279C23×C7C22×D411yescharacteristic
280C22×D7—22no—
281C22×D7—22no—
282C22×D7—22no—
283C22×D7—41no—
284C22×D7—41no—
285C22×D7—41no—
286C22×D7—41no—
287C7×Q8—44no—
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—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—22no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7C2×C22⋊C412yes—
313C2×C4×C7C2×C22⋊C412yes—
314C2×C4×C7C2×C22⋊C412yes—
315C2×C4×C7C2×C22⋊C412yes—
316C2×C4×C7C2×C22⋊C412yes—
317C2×C4×C7C2×C22⋊C412yes—
318C2×C4×C7C23×C411yescharacteristic
319C2×C4×C7C22×D411yescharacteristic
320C2×C4×C7C22×Q811yescharacteristic
321C2×C4×C7C2×#4111yescharacteristic
322C2×C4×C72−1+411yescharacteristic
323C2×C4×C72−1+411yescharacteristic
324C2×Dic7—22no—
325C2×Dic7—22no—
326C2×Dic7—22no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333D28—22no—
334D28—22no—
335D28—22no—
336D28C4×Q814yes—
337C4×D7—48no—
338C4×D7—84no—
339C4×D7—84no—
340Dic14—84no—
341C7⋊C8—22no—
342C7⋊C8—41no—
343C7⋊C8—41no—
344C7⋊C8—41no—
345C22×Q8—141no—
346C22×Q8C22⋊D711yescharacteristic
347C22×D4—71no—
348C23×C4—141no—
349C23×C4—141no—
350C23×C4—141no—
351C22×C8—71no—
352C22×C8—71no—
353C22×C8—71no—
354C22×C8—71no—
355C4⋊Q8—74no—
356C4⋊Q8—142no—
357#125—74no—
358#125—74no—
359#125—142no—
360#125—142no—
361C22⋊Q8—148no—
362C22⋊Q8—148no—
363C22⋊Q8—148no—
364C4×Q8C4×D714yes—
365C4×Q8D2814yes—
366C4×D4—74no—
367C4×D4—74no—
368C4×D4—74no—
369C4×D4—142no—
370C4×D4—142no—
371C4×D4—142no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—141no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—148no—
383C2×C4⋊C4C22⋊D711yescharacteristic
384C2×C4⋊C4C22⋊D711yescharacteristic
385C2×C4⋊C4C22⋊D711yescharacteristic
386C2×C4⋊C4C22×D711yescharacteristic
387C2×C4⋊C4C22×D711yescharacteristic
388C2×C4⋊C4C22×D711yescharacteristic
389C2×C4⋊C4C22×D711yescharacteristic
390C2×C22⋊C4—141no—
391C2×C22⋊C4—141no—
392C2×C22⋊C4—141no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42C22×D711yescharacteristic
397C2×C42C22×D711yescharacteristic
398C2×C42C22×D711yescharacteristic
399C4⋊C8—142no—
400C4⋊C8—142no—
401C4⋊C8—142no—
402#102—74no—
403#102—142no—
404#102—142no—
405#102—142no—
406#97—142no—
407#97—142no—
408C4×C8—142no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416C22×C7C22×C22⋊C411yescharacteristic
417C22×C7#54811yescharacteristic
418C22×C7D4×Q811yescharacteristic
419C22×C7#57511yescharacteristic
420C22×C7#57511yescharacteristic
421C22×C7#57611yescharacteristic
422C22×C7#57711yescharacteristic
423C2×D7—24no—
424C2×D7—42no—
425C2×D7—42no—
426C2×D7—42no—
427C2×D7—42no—
428C2×D7—42no—
429C2×D7—42no—
430C4×C7—22no—
431C4×C7—22no—
432C4×C7—22no—
433C4×C7—22no—
434C4×C7—22no—
435C4×C7—22no—
436C4×C7—42no—
437C4×C7—81no—
438C4×C7—81no—
439C4×C7—81no—
440C4×C7C2×C4×Q811yescharacteristic
441C4×C7#51911yescharacteristic
442C4×C7#54811yescharacteristic
443C4×C7D4×Q811yescharacteristic
444Dic7—42no—
445Dic7—42no—
446Dic7—42no—
447C24—141no—
448C2×Q8—41no—
449C2×Q8—41no—
450C2×Q8—144no—
451C2×Q8—561no—
452C2×Q8—561no—
453C2×Q8(C2×D7)⋊C414yes—
454C2×D4—72no—
455C2×D4—72no—
456C2×D4—72no—
457C2×D4—141no—
458C2×D4—141no—
459C2×D4—141no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—141no—
465C22×C4—141no—
466C22×C4—141no—
467C22×C4—144no—
468C22×C4—144no—
469C22×C4—144no—
470C22×C4—282no—
471C22×C4—282no—
472C22×C4—561no—
473C22×C4—561no—
474C22×C4—561no—
475C22×C4—561no—
476C22×C4C2×C22⋊D711yescharacteristic
477C22×C4C2×C22⋊D711yescharacteristic
478C22×C4C2×C22⋊D711yescharacteristic
479C22×C4C2×C22⋊D711yescharacteristic
480C22×C4C2×C22⋊D711yescharacteristic
481C22×C4C2×C22⋊D711yescharacteristic
482C22×C4C23×D711yescharacteristic
483C2×C8—72no—
484C2×C8—141no—
485C2×C8—141no—
486C2×C8—141no—
487C2×C8—141no—
488C2×C8—141no—
489C2×C8—141no—
490C2×C8—141no—
491C2×C8—141no—
492C2×C8—141no—
493C2×C8—141no—
494C2×C8—141no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—72no—
504C4⋊C4—72no—
505C4⋊C4—72no—
506C4⋊C4—81no—
507C4⋊C4—144no—
508C4⋊C4—144no—
509C4⋊C4—144no—
510C4⋊C4—144no—
511C4⋊C4—144no—
512C4⋊C4—141no—
513C4⋊C4—141no—
514C4⋊C4—141no—
515C4⋊C4(C2×D7)⋊C414yes—
516C4⋊C4(C2×D7)⋊C414yes—
517C4⋊C4(C2×D7)⋊C414yes—
518C4⋊C4Q8×D712yes—
519C22⋊C4—144no—
520C22⋊C4—144no—
521C22⋊C4—144no—
522C42—42no—
523C42—81no—
524C42—81no—
525C42C2×C4×D712yes—
526C42C2×C4×D712yes—
527C42C2×C4×D712yes—
528C42C2×D2812yes—
529C42C2×D2812yes—
530C42C2×D2812yes—
531C2×C7Q8×C22⋊C411yescharacteristic
532C2×C7#289611yescharacteristic
533C2×C7#289711yescharacteristic
534C2×C7#337011yescharacteristic
535C2×C7#337611yescharacteristic
536C2×C7#338811yescharacteristic
537C2×C7#338911yescharacteristic
538D7—44no—
539C23—142no—
540C23—142no—
541C23—142no—
542C23—281no—
543C23—281no—
544C23—281no—
545C23—281no—
546C23C22×C22⋊D711yescharacteristic
547Q8—44no—
548Q8—564no—
549D4—74no—
550D4—142no—
551D4—142no—
552D4—142no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—142no—
578C2×C4—142no—
579C2×C4—142no—
580C2×C4—288no—
581C2×C4—281no—
582C2×C4—281no—
583C2×C4—281no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4—281no—
587C2×C4—564no—
588C2×C4—564no—
589C2×C4C2×(C2×D7)⋊C412yes—
590C2×C4C2×(C2×D7)⋊C412yes—
591C2×C4C2×(C2×D7)⋊C412yes—
592C2×C4C2×(C2×D7)⋊C412yes—
593C2×C4C2×(C2×D7)⋊C412yes—
594C2×C4C2×(C2×D7)⋊C412yes—
595C2×C4C22×C4×D711yescharacteristic
596C2×C4C22×D2811yescharacteristic
597C2×C4C2×Q8×D711yescharacteristic
598C2×C4#655111yescharacteristic
599C2×C4D7×#4111yescharacteristic
600C2×C4#655411yescharacteristic
601C8—142no—
602C8—281no—
603C8—281no—
604C8—281no—
605C7#2252611yescharacteristic, complemented, Sylow
606C22—144no—
607C22—282no—
608C22—282no—
609C22—282no—
610C22—282no—
611C22—282no—
612C22—282no—
613C22C22×(C2×D7)⋊C411yescharacteristic
614C22Q8×C22⋊D711yescharacteristic
615C22#9051311yescharacteristic
616C22#9051311yescharacteristic
617C22#9052211yescharacteristic
618C22#9052611yescharacteristic
619C22#9052811yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—282no—
631C4—282no—
632C4—282no—
633C4C4×Q8×D711yescharacteristic
634C4#9026311yescharacteristic
635C4Q8×D2811yescharacteristic
636C4#9026711yescharacteristic
637C2—284no—
638C2#1173965811yescharacteristic
639C2Q8×(C2×D7)⋊C411yescharacteristic
640C2#1174000811yescharacteristic
641C2#1174107811yescharacteristic
642C2#1174111411yescharacteristic
643C2#1174114011yescharacteristic
644C2#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#22526
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2355

Complements

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

NH ≅ G/Ntype
2. C7×#2355C2semidirect
3. C2×#89796C2semidirect
5. #11727337C2semidirect
8. #11727309C2semidirect
9. #11727308C2semidirect
10. #11727245C2semidirect
11. #11727245C2semidirect
12. #11727244C2semidirect
19. #90270C4semidirect
127. #6383Q8semidirect
136. #2355D7semidirect
605. C7#22526semidirect

Maximal subgroups

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

subgroupindex
C2×#902702
C7×#23552
C2×#897962
#117273382
#117273372
#117273102
#117273092
#117273082
#117272452
#117272442
#117272282
C4×#63832
#225267