Finite Group Databasebeta

← #13360993

Subgroups — #13360993

Order 1792.

964 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89577C211yesmaximal, characteristic
2#11733924C211yesmaximal, characteristic
3C4×D7×C4⋊C4C211yesmaximal, characteristic
4C7×#1928C211yesmaximal, characteristic, complemented
5#11728386C211yesmaximal, characteristic
6#11728359C211yesmaximal, characteristic, complemented
7#11728022C211yesmaximal, characteristic
8#11727974C211yesmaximal, characteristic, complemented
9#11727974C211yesmaximal, characteristic, complemented
10#11727421C211yesmaximal, characteristic, complemented
11#11727213C211yesmaximal, characteristic, complemented
12#11726625C211yesmaximal, characteristic
13#11726540C211yesmaximal, characteristic, complemented
14#11726376C211yesmaximal, characteristic, complemented
15#11726354C211yesmaximal, characteristic
16C2×#6509C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C2×D7×C4⋊C4C2211yescharacteristic
19C2×D7×C4⋊C4C2211yescharacteristic
20C2×C42×D7—21no—
21C2×C42×D7C2211yescharacteristic
22C2×C7×#107C2211yescharacteristic
23C2×C7×#103C2211yescharacteristic
24C2×C7×#103C2211yescharacteristic
25C7×#391C2211yescharacteristic
26C4×C7×C4⋊C4C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×(C2×D7)⋊C8C2211yescharacteristic
30C2×(C2×D7)⋊C8C2211yescharacteristic
31C2×#6391C2211yescharacteristic
32C2×#6389C2211yescharacteristic
33C2×#6389C2211yescharacteristic
34#89759C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37C4⋊C4×Dic7C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40C4×C4⋊Dic7C2211yescharacteristic
41C4×Dic7⋊C4—21no—
42#89700C2211yescharacteristic
43#89577C418yescomplemented
44#89440—21no—
45#89438—21no—
46D7×#95—21no—
47#89427—21no—
48#89421—21no—
49#89421—21no—
50C7×#340C2211yescharacteristic
51C7×#336C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#21579—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×C4×D7—21no—
63C22×C4×D7—21no—
64C22×C4×D7D411yescharacteristic
65C22×C4×D7D411yescharacteristic
66C22×C4×D7C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C42×C7—21no—
73C2×C42×C7C2311yescharacteristic
74#6509—22no—
75#6509C2×C414yes—
76Q8×Dic7—22no—
77Q8×Dic7—22no—
78C2×(C2×D7)⋊C4—21no—
79C2×(C2×D7)⋊C4—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7D411yescharacteristic
95C2×C4×Dic7D411yescharacteristic
96C2×C4×Dic7C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99D7×C4⋊C4—22no—
100D7×C4⋊C4—22no—
101D7×C4⋊C4—22no—
102D7×C4⋊C4C2×C414yes—
103C42×D7—22no—
104C42×D7—22no—
105C42×D7—44no—
106C7×#107—22no—
107C7×#103—22no—
108C7×#103C2×C414yescomplemented
109C7×#95—21no—
110C7×#95—21no—
111C7×#95—21no—
112#6409—22no—
113#6409C2×C414yescomplemented
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120(C2×D7)⋊C8—22no—
121(C2×D7)⋊C8C2×C414yes—
122#6391C2×C414yescomplemented
123#6389—22no—
124#6389—22no—
125#6382C2×C414yescomplemented
126#6381—22no—
127C2×#483—71no—
128C4×C22⋊Q8—71no—
129C2×C4×C4⋊C4—71no—
130#1928—71no—
131#1928D711yescharacteristic, complemented
132#1919—71no—
133#1919—71no—
134#1872—71no—
135#1852—71no—
136#1823—71no—
137#1776—71no—
138#1776—71no—
139#1763—71no—
140#1763—71no—
141#1749—71no—
142C23×D7C2×D411yescharacteristic
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8—21no—
148C2×C7×Q8C22×C412yes—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2411yescharacteristic
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C22×Dic7C2×D411yescharacteristic
163C2×C4×D7—22no—
164C2×C4×D7—22no—
165C2×C4×D7—21no—
166C2×C4×D7—21no—
167C2×C4×D7—21no—
168C2×C4×D7—21no—
169C2×C4×D7—22no—
170C2×C4×D7—24no—
171C2×C4×D7—24no—
172C2×C4×D7—22no—
173C2×C4×D7—22no—
174C2×C4×D7—41no—
175C2×C4×D7—42no—
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⋊C4C22×C412yes—
192C7×C4⋊C4#4112yes—
193C42×C7—21no—
194C42×C7—21no—
195C42×C7—42no—
196(C2×D7)⋊C4—24no—
197(C2×D7)⋊C4—42no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—42no—
204C4⋊Dic7C22×C412yes—
205C4⋊Dic7C22×C412yes—
206C4⋊Dic7#4112yes—
207Dic7⋊C4—24no—
208Dic7⋊C4—24no—
209Dic7⋊C4—24no—
210Dic7⋊C4—42no—
211C4×Dic7—22no—
212C4×Dic7—22no—
213C4×Dic7—42no—
214C4×Dic7—42no—
215C4×Dic7—42no—
216C4×Dic7—42no—
217C4×Dic7—42no—
218C4×Dic7#4112yes—
219C4×Dic7#4112yes—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8C22×C412yes—
226C2×C22⋊Q8—71no—
227C2×C4×Q8—71no—
228C22×C4⋊C4—71no—
229C22×C4⋊C4—71no—
230C22×C42—71no—
231C22×C42—141no—
232#483—78no—
233C2×#107—71no—
234C2×#107C2×D711yescharacteristic
235C2×#106—71no—
236C2×#106—71no—
237C2×#103—71no—
238C2×#103—71no—
239C2×#103C2×D711yescharacteristic
240C2×#103C2×D711yescharacteristic
241C2×C22⋊C8—71no—
242C2×C22⋊C8—71no—
243C2×#97—71no—
244C2×#97—71no—
245#391C2×D711yescharacteristic
246#389—71no—
247#387—141no—
248#385—141no—
249#380—71no—
250C4×C4⋊C4—71no—
251C4×C4⋊C4—71no—
252C4×C4⋊C4—71no—
253C4×C4⋊C4—71no—
254C4×C4⋊C4—141no—
255C4×C4⋊C4—141no—
256C4×C4⋊C4—141no—
257C4×C4⋊C4C2×D711yescharacteristic
258C4×C22⋊C4—141no—
259C2×#95—141no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#340—71no—
267#340C2×D711yescharacteristic
268#336—71no—
269#336C2×D711yescharacteristic
270C23×C7C22×D411yescharacteristic
271C22×D7—21no—
272C22×D7—21no—
273C22×D7—21no—
274C22×D7—21no—
275C22×D7C4×D412yes—
276C22×D7AGL(1,ℤ/8)12yes—
277C22×D7#13712yes—
278C7×Q8—22no—
279C7×Q8—22no—
280C7×Q8—41no—
281C7×Q8—41no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—21no—
287C2×C4×C7—22no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7#13712yes—
300C2×C4×C7C23×C411yescharacteristic
301C2×C4×C7C2×#4111yescharacteristic
302C2×C4×C7C2×#4111yescharacteristic
303C2×C4×C7C2×#4111yescharacteristic
304C2×C4×C7C2×#4111yescharacteristic
305C2×C4×C7C2×#4111yescharacteristic
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—22no—
313C2×Dic7—22no—
314C2×Dic7—22no—
315C2×Dic7—41no—
316C2×Dic7—41no—
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—
327C4×D7—28no—
328C4×D7—22no—
329C4×D7—22no—
330C4×D7—22no—
331C4×D7—22no—
332C4×D7—44no—
333C4×D7—48no—
334C4×D7—48no—
335C7×C8—22no—
336C7×C8—41no—
337C7⋊C8—22no—
338C7⋊C8—41no—
339C22×Q8C22×D711yescharacteristic
340C23×C4—71no—
341C23×C4—71no—
342C23×C4—71no—
343C23×C4—141no—
344C23×C4—141no—
345C22×C8—71no—
346C22×C8—71no—
347C22×C8C22×D711yescharacteristic
348C22×C8C22×D711yescharacteristic
349C22⋊Q8—74no—
350C22⋊Q8—142no—
351C4×Q8—142no—
352C4×Q8—142no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—74no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—142no—
359C2×C4⋊C4—142no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—142no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—141no—
366C2×C4⋊C4C22×D711yescharacteristic
367C2×C4⋊C4C22×D711yescharacteristic
368C2×C4⋊C4C22×D711yescharacteristic
369C2×C22⋊C4—141no—
370C2×C22⋊C4—141no—
371C2×C42—21no—
372C2×C42—71no—
373C2×C42—71no—
374C2×C42—71no—
375C2×C42—142no—
376C2×C42—142no—
377C2×C42—141no—
378C2×C42—141no—
379C2×C42—141no—
380C2×C42—141no—
381C2×C42—141no—
382C2×C42—284no—
383C2×C42C22×D711yescharacteristic
384#107—22no—
385#107—142no—
386#106—74no—
387#106—74no—
388#103—22no—
389#103—74no—
390#103—142no—
391#103C4×D714yescomplemented
392C22⋊C8—74no—
393C22⋊C8—142no—
394#97—142no—
395#97—142no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—141no—
400#95—141no—
401#95—141no—
402#95—141no—
403#95—141no—
404#95—141no—
405C22×C7C2×C4×D411yescharacteristic
406C22×C7C2×#12311yescharacteristic
407C22×C7#52511yescharacteristic
408C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
409C22×C7C2×#13711yescharacteristic
410C22×C7C2×#13711yescharacteristic
411C22×C7#57511yescharacteristic
412C2×D7—22no—
413C2×D7—22no—
414C2×D7—22no—
415C2×D7—22no—
416C2×D7—22no—
417C2×D7—22no—
418C2×D7#44014yes—
419C4×C7—22no—
420C4×C7—41no—
421C4×C7—42no—
422C4×C7—42no—
423C4×C7—42no—
424C4×C7—81no—
425C4×C7C4×#4111yescharacteristic
426C4×C7C4×#4111yescharacteristic
427C4×C7#52911yescharacteristic
428C4×C7#52911yescharacteristic
429Dic7—22no—
430Dic7—22no—
431Dic7—22no—
432Dic7—42no—
433Dic7—42no—
434Dic7—42no—
435Dic7—81no—
436C24—71no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D712yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—71no—
448C22×C4—71no—
449C22×C4—71no—
450C22×C4—72no—
451C22×C4—72no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—144no—
456C22×C4—142no—
457C22×C4—144no—
458C22×C4—142no—
459C22×C4—141no—
460C22×C4—141no—
461C22×C4—141no—
462C22×C4—141no—
463C22×C4—141no—
464C22×C4—141no—
465C22×C4—141no—
466C22×C4—141no—
467C22×C4—281no—
468C22×C4—282no—
469C22×C4—282no—
470C22×C4—282no—
471C22×C4—282no—
472C22×C4D4×D711yescharacteristic
473C22×C4D4×D711yescharacteristic
474C22×C4C23×D711yescharacteristic
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—72no—
481C2×C8—141no—
482C2×C8—141no—
483C2×C8—141no—
484C2×C8—141no—
485C2×C8—141no—
486C2×C8C2×C4×D712yes—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—72no—
492C4⋊C4—72no—
493C4⋊C4—72no—
494C4⋊C4—144no—
495C4⋊C4—144no—
496C4⋊C4—144no—
497C4⋊C4—141no—
498C4⋊C4—141no—
499C4⋊C4—141no—
500C4⋊C4—141no—
501C4⋊C4—141no—
502C4⋊C4—282no—
503C4⋊C4—282no—
504C4⋊C4C2×C4×D712yes—
505C4⋊C4#115912yes—
506C22⋊C4—144no—
507C22⋊C4—282no—
508C42—21no—
509C42—21no—
510C42—42no—
511C42—72no—
512C42—72no—
513C42—142no—
514C42—142no—
515C42—282no—
516C42—282no—
517C42—282no—
518C42—282no—
519C42—282no—
520C2×C7C4×#12311yescharacteristic
521C2×C7C2×#44011yescharacteristic
522C2×C7C4×#13711yescharacteristic
523C2×C7#309211yescharacteristic
524C2×C7#309511yescharacteristic
525C2×C7#311311yescharacteristic
526C2×C7#311411yescharacteristic
527D7—24no—
528C23—72no—
529C23—72no—
530C23—72no—
531C23—141no—
532C23—141no—
533C23—141no—
534C23—141no—
535C23C2×D4×D711yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—72no—
558C2×C4—72no—
559C2×C4—72no—
560C2×C4—148no—
561C2×C4—141no—
562C2×C4—141no—
563C2×C4—141no—
564C2×C4—141no—
565C2×C4—141no—
566C2×C4—141no—
567C2×C4—142no—
568C2×C4—142no—
569C2×C4—142no—
570C2×C4—142no—
571C2×C4—142no—
572C2×C4—142no—
573C2×C4—142no—
574C2×C4—284no—
575C2×C4—288no—
576C2×C4—288no—
577C2×C4—281no—
578C2×C4—281no—
579C2×C4—281no—
580C2×C4—281no—
581C2×C4—281no—
582C2×C4—281no—
583C2×C4—281no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4#648112yes—
587C2×C4C22×C4×D711yescharacteristic
588C2×C4C2×#115911yescharacteristic
589C2×C4D7×#4111yescharacteristic
590C2×C4D7×#4111yescharacteristic
591C2×C4D7×#4111yescharacteristic
592C2×C4D7×#4111yescharacteristic
593C8—22no—
594C8—41no—
595C8—142no—
596C8—281no—
597C7#2157911yescharacteristic, complemented, Sylow
598C22—74no—
599C22—142no—
600C22—142no—
601C22—142no—
602C22—142no—
603C22—142no—
604C22—142no—
605C22C4×D4×D711yescharacteristic
606C22#9029711yescharacteristic
607C22D7×#12311yescharacteristic
608C22C2×#648111yescharacteristic
609C22#9045911yescharacteristic
610C22D7×AGL(1,ℤ/8)11yescharacteristic
611C22D7×#13711yescharacteristic
612C4—22no—
613C4—41no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—142no—
619C4—142no—
620C4—142no—
621C4—282no—
622C4—282no—
623C4—282no—
624C4—561no—
625C4#9021311yescharacteristic
626C4C4×#115911yescharacteristic
627C4#9037111yescharacteristic
628C4#9039711yescharacteristic
629C2—144no—
630C2#1173289811yescharacteristic
631C2C4×#648111yescharacteristic
632C2D7×#44011yescharacteristic
633C2#1173790311yescharacteristic
634C2#1173795211yescharacteristic
635C2#1173891311yescharacteristic
636C2#1173896211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1928C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1928D7
hypocenterC7#21579
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

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

NH ≅ G/Ntype
4. C7×#1928C2semidirect
6. #11728359C2semidirect
8. #11727974C2semidirect
9. #11727974C2semidirect
10. #11727421C2semidirect
11. #11727213C2semidirect
13. #11726540C2semidirect
14. #11726376C2semidirect
43. #89577C4semidirect
108. C7×#103C2×C4semidirect
113. #6409C2×C4semidirect
122. #6391C2×C4semidirect
125. #6382C2×C4semidirect
131. #1928D7semidirect
391. #103C4×D7semidirect
597. C7#21579semidirect

Maximal subgroups

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

subgroupindex
C2×#895772
#117339242
C4×D7×C4⋊C42
C7×#19282
#117283862
#117283592
#117280222
#117279742
#117274212
#117272132
#117266252
#117265402
#117263762
#117263542
#215797