Finite Group Databasebeta

← #13373217

Subgroups — #13373217

Order 1792.

942 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×C7×#442C211yesmaximal, characteristic
2#11739782C211yesmaximal, characteristic, complemented
3#11739778C211yesmaximal, characteristic, complemented
4#11729791C211yesmaximal, characteristic
5#11729791C211yesmaximal, characteristic
6#11729359C211yesmaximal, characteristic, complemented
7#11729358C211yesmaximal, characteristic, complemented
8#11728312C211yesmaximal, characteristic, complemented
9#11728311C211yesmaximal, characteristic, complemented
10#11727941C211yesmaximal, characteristic, complemented
11#11727939C211yesmaximal, characteristic, complemented
12#11727906C211yesmaximal, characteristic, complemented
13Dic7×#102C211yesmaximal, characteristic, complemented
14#11727569C211yesmaximal, characteristic, complemented
15#11727564C211yesmaximal, characteristic, complemented
16C22×C7×D8C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C2×D4×Dic7C2211yescharacteristic
20C2×D4×Dic7C2211yescharacteristic
21C2×#6489—21no—
22C7×#442C418yescomplemented
23C2×C7×#106C2211yescharacteristic
24C2×C7×#102C2211yescharacteristic
25C2×C7×#102C2211yescharacteristic
26C2×C7×#97C2211yescharacteristic
27#89993C2211yescharacteristic
28#89993C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33#89984—21no—
34#89984—21no—
35C2×C7⋊#95—21no—
36C2×#6389C2211yescharacteristic
37C2×C8×Dic7C2211yescharacteristic
38#89757C2211yescharacteristic
39#89757C2211yescharacteristic
40C4⋊C4×Dic7C2211yescharacteristic
41C4⋊C4×Dic7C2211yescharacteristic
42#89723—21no—
43#89723—21no—
44#89722—21no—
45#89722—21no—
46#89708C2211yescharacteristic
47#89707C2211yescharacteristic
48C4×C4⋊Dic7C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22780—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C23×C4×C7—21no—
65C23×Dic7—21no—
66C23×Dic7—21no—
67C2×C7×D8—22no—
68C2×C7×D8—22no—
69C2×C7×D8C2×C414yes—
70C22×C7×C8C2311yescharacteristic
71C22×C7×C8C2311yescharacteristic
72C4×C7×D4—22no—
73C4×C7×D4—22no—
74C4×C7×D4C2×C414yes—
75C4×C7×D4C2×C414yescomplemented
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C42×C7C2311yescharacteristic
81C2×C22⋊Dic7—21no—
82C2×C22⋊Dic7—21no—
83C2×C22⋊Dic7—21no—
84C2×C22⋊Dic7—21no—
85C2×C22⋊Dic7—21no—
86C2×C22⋊Dic7—21no—
87D4×Dic7—22no—
88D4×Dic7—22no—
89D4×Dic7—41no—
90D4×Dic7—41no—
91#6489—44no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7—21no—
96C2×C4⋊Dic7D411yescharacteristic
97C2×C4⋊Dic7D411yescharacteristic
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4⋊Dic7C2311yescharacteristic
100C2×C4⋊Dic7C2311yescharacteristic
101C2×C4⋊Dic7C2311yescharacteristic
102C2×C4×Dic7—21no—
103C2×C4×Dic7—21no—
104C2×C4×Dic7C2311yescharacteristic
105C2×C4×Dic7C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108C7×#106C2×C414yescomplemented
109C7×#102C2×C414yes—
110C7×#102C2×C414yescomplemented
111C7×#97C2×C414yescomplemented
112#6406—22no—
113#6406—22no—
114#6406—41no—
115#6406—41no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95D411yescharacteristic
123C7⋊#95D411yescharacteristic
124#6389—41no—
125C8×Dic7—41no—
126C4⋊(C7⋊C8)—22no—
127C2×#442D711yescharacteristic
128#2354—71no—
129#2349—71no—
130#1936—71no—
131#1934—71no—
132#1920—71no—
133#1919—71no—
134#1879—71no—
135#1879—71no—
136#1836—71no—
137#1804—71no—
138#1802—71no—
139#1771—71no—
140#1762—71no—
141C4×#102—71no—
142C24×C7—21no—
143C24×C7—21no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4C22×C412yes—
153C2×C7×D4C22×C412yes—
154C2×C7×D4#4112yes—
155C2×C7×D4#4112yes—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—24no—
159C22×C4×C7—24no—
160C22×C4×C7—42no—
161C22×C4×C7—42no—
162C22×C4×C7—81no—
163C22×C4×C7—81no—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2411yescharacteristic
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—21no—
171C22×Dic7—42no—
172C22×Dic7—42no—
173C22×Dic7—42no—
174C22×Dic7—42no—
175C22×Dic7—81no—
176C22×Dic7—81no—
177C22×Dic7C2×D411yescharacteristic
178C22×Dic7C2×D411yescharacteristic
179C22×Dic7C2×D411yescharacteristic
180C22×Dic7C2×D411yescharacteristic
181C7×D8—28no—
182C2×C7×C8—21no—
183C2×C7×C8—21no—
184C2×C7×C8—21no—
185C2×C7×C8—21no—
186C2×C7×C8C22×C412yes—
187C2×C7×C8C22×C412yes—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4C22×C412yes—
191C7×C4⋊C4C22×C412yes—
192C7×C22⋊C4—24no—
193C7×C22⋊C4—24no—
194C42×C7C22×C412yes—
195C42×C7#4112yes—
196C22⋊Dic7—42no—
197C22⋊Dic7—42no—
198C22⋊Dic7—42no—
199C22⋊Dic7—42no—
200C22⋊Dic7—81no—
201C22⋊Dic7—81no—
202C4⋊Dic7—22no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—41no—
212C4⋊Dic7—42no—
213C4⋊Dic7—81no—
214C4⋊Dic7—81no—
215C4⋊Dic7—81no—
216C4×Dic7—21no—
217C4×Dic7—21no—
218C4×Dic7—21no—
219C4×Dic7—21no—
220C4×Dic7—42no—
221C4×Dic7—42no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C22×D8C2×D711yescharacteristic
229C2×C4×D4—71no—
230C2×C4×D4—71no—
231C2×C4×D4C2×D711yescharacteristic
232C2×C4×D4C2×D711yescharacteristic
233C2×#117—141no—
234#442Dic718yescomplemented
235C2×#106C2×D711yescharacteristic
236C2×C4⋊C8—71no—
237C2×#102—71no—
238C2×#102—71no—
239C2×#102—71no—
240C2×#102—71no—
241C2×#102C2×D711yescharacteristic
242C2×#102C2×D711yescharacteristic
243C2×#97—71no—
244C2×#97C2×D711yescharacteristic
245C2×C4×C8—71no—
246#390—71no—
247#390—71no—
248#389—71no—
249#389—71no—
250#388—141no—
251#388—141no—
252#385—141no—
253#385—141no—
254#384—71no—
255#382—71no—
256#380—141no—
257#380—141no—
258C4×C4⋊C4—71no—
259C4×C4⋊C4—71no—
260C4×C4⋊C4—71no—
261C2×#95—141no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#340—71no—
267#340—71no—
268#340—71no—
269#340—71no—
270#336—71no—
271#336—71no—
272C23×C7—22no—
273C23×C7—22no—
274C23×C7—22no—
275C23×C7—22no—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7—41no—
279C23×C7—41no—
280C23×C7—41no—
281C23×C7—41no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7—41no—
285C23×C7—41no—
286C23×C7C22×D411yescharacteristic
287C7×D4—22no—
288C7×D4—22no—
289C7×D4—22no—
290C7×D4—22no—
291C7×D4—22no—
292C7×D4—41no—
293C7×D4—41no—
294C7×D4#11714yescomplemented
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—48no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—84no—
307C2×C4×C7C4×D412yes—
308C2×C4×C7C4×D412yes—
309C2×C4×C7C23×C411yescharacteristic
310C2×C4×C7C2×#4111yescharacteristic
311C2×C4×C7C2×#4111yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×C4×C7H(5,𝔽2)11yescharacteristic
314C2×C4×C72−1+411yescharacteristic
315C2×Dic7—21no—
316C2×Dic7—21no—
317C2×Dic7—21no—
318C2×Dic7—21no—
319C2×Dic7—21no—
320C2×Dic7—21no—
321C2×Dic7—21no—
322C2×Dic7—21no—
323C2×Dic7—21no—
324C2×Dic7—21no—
325C2×Dic7—21no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—84no—
339C2×Dic7—84no—
340C2×Dic7AGL(1,ℤ/8)12yes—
341C7×C8—22no—
342C7×C8—22no—
343C7⋊C8—41no—
344C7⋊C8—41no—
345C22×D4C22×D711yescharacteristic
346C22×D4C22×D711yescharacteristic
347C23×C4—21no—
348C23×C4—21no—
349C23×C4—141no—
350C23×C4—141no—
351C2×D8—22no—
352C2×D8—22no—
353C2×D8C2×Dic714yes—
354C22×C8—71no—
355C22×C8—71no—
356C22×C8C22×D711yescharacteristic
357C22×C8C22×D711yescharacteristic
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—142no—
361C4×D4—142no—
362C4×D4—281no—
363C4×D4—281no—
364C4×D4C2×Dic714yes—
365C4×D4C2×Dic714yescomplemented
366#117—284no—
367C2×C4⋊C4—71no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—71no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—71no—
372C2×C4⋊C4—71no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C4⋊C4C22×D711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—141no—
382C2×C22⋊C4—141no—
383C2×C22⋊C4—141no—
384C2×C22⋊C4—141no—
385C2×C22⋊C4—141no—
386C2×C22⋊C4—141no—
387C2×C42—71no—
388C2×C42—71no—
389C2×C42—141no—
390C2×C42—141no—
391C2×C42C22×D711yescharacteristic
392#106C2×Dic714yescomplemented
393C4⋊C8—142no—
394#102—142no—
395#102—142no—
396#102—281no—
397#102—281no—
398#102C2×Dic714yes—
399#102C2×Dic714yescomplemented
400#97—281no—
401#97C2×Dic714yescomplemented
402C4×C8—281no—
403#95—71no—
404#95—71no—
405#95—141no—
406#95—141no—
407#95—141no—
408#95—141no—
409#95—141no—
410#95—141no—
411C22×C7—24no—
412C22×C7—42no—
413C22×C7—42no—
414C22×C7—42no—
415C22×C7—42no—
416C22×C7—42no—
417C22×C7—42no—
418C22×C7—42no—
419C22×C7—42no—
420C22×C7—42no—
421C22×C7—81no—
422C22×C7—81no—
423C22×C7—81no—
424C22×C7—81no—
425C22×C7C2×C4×D411yescharacteristic
426C22×C7#54611yescharacteristic
427C22×C7#54811yescharacteristic
428C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
429C22×C7#57511yescharacteristic
430C22×C7#57611yescharacteristic
431C22×C7#57711yescharacteristic
432C4×C7—22no—
433C4×C7—22no—
434C4×C7—42no—
435C4×C7—42no—
436C4×C7C2×#11711yescharacteristic
437C4×C7#52011yescharacteristic
438C4×C7#55211yescharacteristic
439C4×C7#55311yescharacteristic
440Dic7—22no—
441Dic7—41no—
442Dic7—41no—
443Dic7—41no—
444Dic7—81no—
445Dic7—81no—
446Dic7—81no—
447Dic7—81no—
448C24—21no—
449C24—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4#115712yes—
459C2×D4#115712yes—
460C2×D4C22×Dic712yes—
461C2×D4C22×Dic712yes—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—71no—
469C22×C4—71no—
470C22×C4—71no—
471C22×C4—71no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—141no—
475C22×C4—141no—
476C22×C4—141no—
477C22×C4—141no—
478C22×C4—282no—
479C22×C4—282no—
480C22×C4—282no—
481C22×C4—282no—
482C22×C4—561no—
483C22×C4—561no—
484C22×C4D4×D711yescharacteristic
485C22×C4D4×D711yescharacteristic
486C22×C4C23×D711yescharacteristic
487D8—28no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—141no—
493C2×C8—141no—
494C2×C8—141no—
495C2×C8—141no—
496C2×C8—141no—
497C2×C8—141no—
498C2×C8C22×Dic712yes—
499C2×C8C22×Dic712yes—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—142no—
503C4⋊C4—141no—
504C4⋊C4—141no—
505C4⋊C4—141no—
506C4⋊C4—141no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—141no—
510C4⋊C4—141no—
511C4⋊C4—281no—
512C4⋊C4—282no—
513C4⋊C4—561no—
514C4⋊C4—561no—
515C4⋊C4—561no—
516C4⋊C4C22×Dic712yes—
517C4⋊C4C22×Dic712yes—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—282no—
521C22⋊C4—282no—
522C22⋊C4—282no—
523C22⋊C4—282no—
524C22⋊C4—561no—
525C22⋊C4—561no—
526C42—141no—
527C42—141no—
528C42—141no—
529C42—141no—
530C42—282no—
531C42—282no—
532C42#115912yes—
533C42C22×Dic712yes—
534C2×C7—44no—
535C2×C7—82no—
536C2×C7#235211yescharacteristic
537C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
538C2×C7#294611yescharacteristic
539C2×C7#330611yescharacteristic
540C2×C7#331911yescharacteristic
541C2×C7#336611yescharacteristic
542C2×C7#337611yescharacteristic
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C2×D4×D711yescharacteristic
558D4—22no—
559D4—22no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4—41no—
564D4—41no—
565D4#648914yescomplemented
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—72no—
578C2×C4—84no—
579C2×C4—141no—
580C2×C4—141no—
581C2×C4—141no—
582C2×C4—141no—
583C2×C4—141no—
584C2×C4—141no—
585C2×C4—141no—
586C2×C4—141no—
587C2×C4—141no—
588C2×C4—141no—
589C2×C4—141no—
590C2×C4—281no—
591C2×C4—281no—
592C2×C4—281no—
593C2×C4—281no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—564no—
603C2×C4—564no—
604C2×C4D4×Dic712yes—
605C2×C4D4×Dic712yes—
606C2×C4C2×#115711yescharacteristic
607C2×C4C2×#115711yescharacteristic
608C2×C4C2×#115911yescharacteristic
609C2×C4#655311yescharacteristic
610C2×C4#655411yescharacteristic
611C2×C4C23×Dic711yescharacteristic
612C8—22no—
613C8—22no—
614C8—281no—
615C8—281no—
616C7#2278011yescharacteristic, complemented, Sylow
617C22—24no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22#9024111yescharacteristic
632C22#9024311yescharacteristic
633C22D7×AGL(1,ℤ/8)11yescharacteristic
634C22#9046411yescharacteristic
635C22#9046511yescharacteristic
636C22#9046611yescharacteristic
637C22C2×D4×Dic711yescharacteristic
638C4—22no—
639C4—22no—
640C4—42no—
641C4—42no—
642C4—142no—
643C4—281no—
644C4—281no—
645C4—281no—
646C4—561no—
647C4—561no—
648C4—561no—
649C4—561no—
650C4#9023211yescharacteristic
651C4#9025611yescharacteristic
652C4C2×#648911yescharacteristic
653C4#9051811yescharacteristic
654C2—44no—
655C2—82no—
656C2#1173729611yescharacteristic
657C2#1173729911yescharacteristic
658C2#1173732511yescharacteristic
659C2#1173732811yescharacteristic
660C2#1173978011yescharacteristic
661C2Dic7×AGL(1,ℤ/8)11yescharacteristic
662C2#1174137511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#442C2
socleC23×C7C22×D4
prime radicalC22×C7×D8C22
cosocleC22×C4×C7C24
hypercenterC2×#442D7
hypocenterC7#22780
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
2. #11739782C2semidirect
3. #11739778C2semidirect
6. #11729359C2semidirect
7. #11729358C2semidirect
8. #11728312C2semidirect
9. #11728311C2semidirect
10. #11727941C2semidirect
11. #11727939C2semidirect
12. #11727906C2semidirect
13. Dic7×#102C2semidirect
14. #11727569C2semidirect
15. #11727564C2semidirect
22. C7×#442C4semidirect
75. C4×C7×D4C2×C4semidirect
108. C7×#106C2×C4semidirect
110. C7×#102C2×C4semidirect
111. C7×#97C2×C4semidirect
234. #442Dic7semidirect
294. C7×D4#117semidirect
365. C4×D4C2×Dic7semidirect
392. #106C2×Dic7semidirect
399. #102C2×Dic7semidirect
401. #97C2×Dic7semidirect
565. D4#6489semidirect
616. C7#22780semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4422
#117397822
#117397782
#117297912
#117293592
#117293582
#117283122
#117283112
#117279412
#117279392
#117279062
Dic7×#1022
#117275692
#117275642
#227807