Finite Group Databasebeta

← #13361368

Subgroups — #13361368

Order 1792.

1000 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89642C211yesmaximal, characteristic
2#11733412C211yesmaximal, characteristic, complemented
3C4×#6456C211yesmaximal, characteristic
4C7×#1920C211yesmaximal, characteristic, complemented
5Dic7×#106C211yesmaximal, characteristic, complemented
6#11727380C211yesmaximal, characteristic
7#11727214C211yesmaximal, characteristic, complemented
8#11726646C211yesmaximal, characteristic
9#11726553C211yesmaximal, characteristic, complemented
10#11726550C211yesmaximal, characteristic, complemented
11#11726538C211yesmaximal, characteristic, complemented
12#11726534C211yesmaximal, characteristic
13#11726534C211yesmaximal, characteristic
14#11726425C211yesmaximal, characteristic
15#11726356C211yesmaximal, characteristic, complemented
16C2×#6456C2211yescharacteristic
17C2×#6456C2211yescharacteristic
18C2×#6451C2211yescharacteristic
19C2×#6450—21no—
20C2×C4×D28C2211yescharacteristic
21C2×C42×D7—21no—
22C2×C7×#106C2211yescharacteristic
23C7×#389C2211yescharacteristic
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×#6395C2211yescharacteristic
26C2×#6395C2211yescharacteristic
27C2×#6391C2211yescharacteristic
28C2×Dic7⋊C8C2211yescharacteristic
29C2×Dic7⋊C8C2211yescharacteristic
30C2×C8×Dic7C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32#89748C2211yescharacteristic
33C4⋊C4×Dic7C2211yescharacteristic
34C4⋊C4×Dic7C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37#89744—21no—
38C2×#6383C2211yescharacteristic
39C2×#6383C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C4×(C2×D7)⋊C4—21no—
43C42×Dic7C2211yescharacteristic
44#89700C2211yescharacteristic
45#89642C418yescomplemented
46#89438—21no—
47#89438—21no—
48#89421—21no—
49#89418—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52C7×#340C2211yescharacteristic
53C7×#336C2211yescharacteristic
54#89346C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60C4×#476—71nomaximal, Sylow
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C22×C4×D7—21no—
64C22×C4×D7—21no—
65C22×Dic14—21no—
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C42×C7—21no—
74C2×C42×C7C2311yescharacteristic
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×(C2×D7)⋊C4—21no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×Dic7⋊C4—21no—
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×C4×Dic7—21no—
86C2×C4×Dic7—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7D411yescharacteristic
90C2×C4×Dic7D411yescharacteristic
91C2×C4×Dic7D411yescharacteristic
92C2×C4×Dic7D411yescharacteristic
93C2×C4×Dic7C2311yescharacteristic
94C2×C4×Dic7C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97#6456—22no—
98#6456—22no—
99#6456C2×C414yes—
100#6456Q814yescomplemented
101#6451—22no—
102#6451C2×C414yescomplemented
103#6450—44no—
104C4×D28—22no—
105C4×D28—22no—
106C42×D7—44no—
107C7×#106C2×C414yescomplemented
108C7×#95—21no—
109C7×#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115#6395—22no—
116#6395C2×C414yes—
117#6391—22no—
118Dic7⋊C8—22no—
119Dic7⋊C8C2×C414yescomplemented
120C8×Dic7—22no—
121C8×Dic7—22no—
122#6383—22no—
123#6383C2×C414yes—
124#6382—22no—
125#6382C2×C414yescomplemented
126C2×#476—71no—
127C4×C4⋊Q8—71no—
128C42×D4—71no—
129#1987—71no—
130#1948—71no—
131#1920—71no—
132#1920D711yescharacteristic, complemented
133#1864—71no—
134#1864—71no—
135#1802—71no—
136C4×#106—71no—
137C4×#106—71no—
138C4×C4⋊C8—71no—
139C4×#102—71no—
140C4×#102—71no—
141C23×D7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2411yescharacteristic
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7C2×D411yescharacteristic
153C22×Dic7C2×D411yescharacteristic
154C22×Dic7C2×D411yescharacteristic
155C22×Dic7C2×D411yescharacteristic
156C2×D28—21no—
157C2×D28—21no—
158C2×D28—21no—
159C2×D28C22×C412yes—
160C2×D28C2×Q812yes—
161C2×D28#4112yes—
162C2×C4×D7—24no—
163C2×C4×D7—24no—
164C2×C4×D7—24no—
165C2×C4×D7—42no—
166C2×C4×D7—42no—
167C2×C4×D7—42no—
168C2×C4×D7—42no—
169C2×C4×D7—42no—
170C2×C4×D7—42no—
171C2×Dic14—24no—
172C2×Dic14—42no—
173C2×Dic14—42no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8C22×C412yes—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—42no—
186C7×C4⋊C4C22×C412yes—
187C7×C4⋊C4C22×C412yes—
188C7×C4⋊C4C2×Q812yes—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—42no—
192(C2×D7)⋊C4—24no—
193(C2×D7)⋊C4—24no—
194(C2×D7)⋊C4—24no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7C22×C412yes—
200C4⋊Dic7#4112yes—
201Dic7⋊C4—24no—
202Dic7⋊C4—42no—
203Dic7⋊C4—42no—
204Dic7⋊C4—42no—
205C4×Dic7—22no—
206C4×Dic7—22no—
207C4×Dic7—41no—
208C4×Dic7—42no—
209C4×Dic7—42no—
210C4×Dic7—42no—
211C4×Dic7—42no—
212C4×Dic7SD1614yes—
213C4×Dic7C22×C412yes—
214C4×Dic7C2×Q812yes—
215C4×Dic7#4112yes—
216C4×Dic7#4112yes—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8C22×C412yes—
223C2×C4⋊Q8—71no—
224C2×C4×Q8—141no—
225C2×C4×D4—71no—
226C2×C4×D4—71no—
227C2×C4×D4—71no—
228C22×C42—141no—
229#476—78no—
230C2×#106—71no—
231C2×#106—71no—
232C2×#106—71no—
233C2×#106C2×D711yescharacteristic
234C2×C4⋊C8—71no—
235C2×C4⋊C8—71no—
236C2×#102—71no—
237C2×#102—71no—
238C2×#102—71no—
239C2×#102—71no—
240C2×C4×C8—71no—
241C2×C4×C8—71no—
242#391—141no—
243#389—71no—
244#389—141no—
245#389C2×D711yescharacteristic
246#384—71no—
247C4×C4⋊C4—71no—
248C4×C4⋊C4—71no—
249C4×C4⋊C4—71no—
250C4×C4⋊C4—71no—
251C4×C4⋊C4—141no—
252C4×C4⋊C4C2×D711yescharacteristic
253C4×C22⋊C4—141no—
254C4×C22⋊C4—141no—
255C4×C22⋊C4—141no—
256C43—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340C2×D711yescharacteristic
263#340C2×D711yescharacteristic
264#340C2×D711yescharacteristic
265#336—71no—
266#336C2×D711yescharacteristic
267C23×C7C22×D411yescharacteristic
268C22×D7—22no—
269C22×D7—22no—
270C22×D7—22no—
271C22×D7—41no—
272C22×D7—41no—
273C22×D7—41no—
274C22×D7—41no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
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—41no—
292C2×C4×C7C2×SD1612yes—
293C2×C4×C7C23×C411yescharacteristic
294C2×C4×C7C22×Q811yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—22no—
304C2×Dic7—22no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7C4×D412yes—
313C2×Dic7C4×D412yes—
314C2×Dic7C22⋊Q812yes—
315C2×Dic7C22⋊Q812yes—
316C2×Dic7C2×SD1612yes—
317C2×Dic7C2×SD1612yes—
318C2×Dic7#13512yes—
319C2×Dic7AGL(1,ℤ/8)12yes—
320D28—22no—
321D28—22no—
322D28—22no—
323D28C4×Q814yescomplemented
324C4×D7—48no—
325C4×D7—48no—
326C4×D7—48no—
327Dic14—48no—
328C7×C8—22no—
329C7×C8—41no—
330C7⋊C8—22no—
331C7⋊C8—41no—
332C22×Q8—141no—
333C22×D4—71no—
334C23×C4—141no—
335C23×C4—141no—
336C23×C4—141no—
337C22×C8—71no—
338C22×C8—71no—
339C22×C8C22×D711yescharacteristic
340C22×C8C22×D711yescharacteristic
341C4⋊Q8—74no—
342C4⋊Q8—142no—
343C4×Q8—284no—
344C4×D4—74no—
345C4×D4—74no—
346C4×D4—142no—
347C4×D4—142no—
348C4×D4—142no—
349C4×D4—142no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—141no—
355C2×C4⋊C4—141no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4C22×D711yescharacteristic
359C2×C4⋊C4C22×D711yescharacteristic
360C2×C4⋊C4C22×D711yescharacteristic
361C2×C4⋊C4C22×D711yescharacteristic
362C2×C22⋊C4—141no—
363C2×C22⋊C4—141no—
364C2×C22⋊C4—141no—
365C2×C42—21no—
366C2×C42—71no—
367C2×C42—71no—
368C2×C42—71no—
369C2×C42—71no—
370C2×C42—71no—
371C2×C42—71no—
372C2×C42—141no—
373C2×C42—141no—
374C2×C42—141no—
375C2×C42—141no—
376C2×C42—284no—
377C2×C42C22×D711yescharacteristic
378#106—74no—
379#106—142no—
380#106—142no—
381#106C4×D714yescomplemented
382C4⋊C8—74no—
383C4⋊C8—142no—
384#102—74no—
385#102—74no—
386#102—142no—
387#102—142no—
388C4×C8—142no—
389C4×C8—142no—
390#95—21no—
391#95—21no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95—141no—
397C22×C7C2×C4×D411yescharacteristic
398C22×C7C2×C22⋊Q811yescharacteristic
399C22×C7#52511yescharacteristic
400C22×C7C22×SD1611yescharacteristic
401C22×C7C2×#13511yescharacteristic
402C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
403C22×C7#57511yescharacteristic
404C2×D7—24no—
405C2×D7—42no—
406C2×D7—42no—
407C2×D7—42no—
408C2×D7—42no—
409C2×D7—42no—
410C2×D7—42no—
411C4×C7—22no—
412C4×C7—41no—
413C4×C7—42no—
414C4×C7—42no—
415C4×C7—42no—
416C4×C7—81no—
417C4×C7C2×C4×Q811yescharacteristic
418C4×C7C4×#4111yescharacteristic
419C4×C7#52911yescharacteristic
420C4×C7#53311yescharacteristic
421Dic7—22no—
422Dic7—22no—
423Dic7—42no—
424Dic7—42no—
425Dic7—81no—
426Dic7C4×SD1614yes—
427Dic7#47614yescomplemented
428C24—141no—
429C2×Q8—144no—
430C2×Q8—282no—
431C2×Q8—282no—
432C2×D4—72no—
433C2×D4—72no—
434C2×D4—72no—
435C2×D4—141no—
436C2×D4—141no—
437C2×D4—141no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—71no—
443C22×C4—71no—
444C22×C4—71no—
445C22×C4—71no—
446C22×C4—144no—
447C22×C4—144no—
448C22×C4—144no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—282no—
453C22×C4—282no—
454C22×C4—282no—
455C22×C4—282no—
456C22×C4—282no—
457C22×C4—282no—
458C22×C4D4×D711yescharacteristic
459C22×C4D4×D711yescharacteristic
460C22×C4C23×D711yescharacteristic
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—72no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8C2×C4×D712yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—72no—
480C4⋊C4—72no—
481C4⋊C4—144no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—282no—
487C4⋊C4—282no—
488C4⋊C4—282no—
489C4⋊C4C2×C4×D712yes—
490C4⋊C4C2×C4×D712yes—
491C4⋊C4Q8×D712yes—
492C22⋊C4—144no—
493C22⋊C4—144no—
494C22⋊C4—144no—
495C42—21no—
496C42—21no—
497C42—42no—
498C42—74no—
499C42—72no—
500C42—72no—
501C42—72no—
502C42—72no—
503C42—142no—
504C42—142no—
505C42—281no—
506C42—282no—
507C42—282no—
508C42—282no—
509C42—282no—
510C2×C7C4×C22⋊Q811yescharacteristic
511C2×C7C2×C4×SD1611yescharacteristic
512C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
513C2×C7C2×#47611yescharacteristic
514C2×C7#307611yescharacteristic
515C2×C7#310111yescharacteristic
516C2×C7#310611yescharacteristic
517D7—44no—
518C23—142no—
519C23—142no—
520C23—142no—
521C23—281no—
522C23—281no—
523C23—281no—
524C23—281no—
525C23C2×D4×D711yescharacteristic
526Q8—288no—
527D4—74no—
528D4—142no—
529D4—142no—
530D4—142no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
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—41no—
548C2×C4—72no—
549C2×C4—72no—
550C2×C4—72no—
551C2×C4—72no—
552C2×C4—72no—
553C2×C4—72no—
554C2×C4—72no—
555C2×C4—72no—
556C2×C4—141no—
557C2×C4—141no—
558C2×C4—141no—
559C2×C4—141no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—288no—
563C2×C4—288no—
564C2×C4—288no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4D7×SD1612yes—
573C2×C4C22×C4×D711yescharacteristic
574C2×C4C2×Q8×D711yescharacteristic
575C2×C4D7×#4111yescharacteristic
576C2×C4D7×#4111yescharacteristic
577C2×C4D7×#4111yescharacteristic
578C2×C4D7×#4111yescharacteristic
579C8—22no—
580C8—41no—
581C8—142no—
582C8—281no—
583C7C4×#47611yescharacteristic, complemented, Sylow
584C22—144no—
585C22—282no—
586C22—282no—
587C22—282no—
588C22—282no—
589C22—282no—
590C22—282no—
591C22C4×D4×D711yescharacteristic
592C22D7×C22⋊Q811yescharacteristic
593C22#9034711yescharacteristic
594C22C2×D7×SD1611yescharacteristic
595C22D7×#13511yescharacteristic
596C22D7×AGL(1,ℤ/8)11yescharacteristic
597C22#9046811yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—74no—
604C4—74no—
605C4—81no—
606C4—142no—
607C4—142no—
608C4—282no—
609C4—282no—
610C4—561no—
611C4#9021311yescharacteristic
612C4C4×Q8×D711yescharacteristic
613C4#9037911yescharacteristic
614C4#9039711yescharacteristic
615C2—284no—
616C2#1173238611yescharacteristic
617C2C4×D7×SD1611yescharacteristic
618C2#1173721811yescharacteristic
619C2D7×#47611yescharacteristic
620C2#1173867411yescharacteristic
621C2#1173879411yescharacteristic
622C2#1173885711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1920C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1920D7
hypocenterC7C4×#476
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
2. #11733412C2semidirect
4. C7×#1920C2semidirect
5. Dic7×#106C2semidirect
7. #11727214C2semidirect
9. #11726553C2semidirect
10. #11726550C2semidirect
11. #11726538C2semidirect
15. #11726356C2semidirect
45. #89642C4semidirect
100. #6456Q8semidirect
102. #6451C2×C4semidirect
107. C7×#106C2×C4semidirect
119. Dic7⋊C8C2×C4semidirect
125. #6382C2×C4semidirect
132. #1920D7semidirect
323. D28C4×Q8semidirect
381. #106C4×D7semidirect
427. Dic7#476semidirect
583. C7C4×#476semidirect

Maximal subgroups

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

subgroupindex
C2×#896422
#117334122
C4×#64562
C7×#19202
Dic7×#1062
#117273802
#117272142
#117266462
#117265532
#117265502
#117265382
#117265342
#117264252
#117263562
C4×#4767