Finite Group Databasebeta

← #13360196

Subgroups — #13360196

Order 1792.

1008 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#461C211yesmaximal, characteristic
2C2×#89936C211yesmaximal, characteristic
3C2×SD16×Dic7C211yesmaximal, characteristic
4#11740672C211yesmaximal, characteristic, complemented
5#11739762C211yesmaximal, characteristic
6#11729801C211yesmaximal, characteristic
7#11729801C211yesmaximal, characteristic
8#11729532C211yesmaximal, characteristic, complemented
9#11729359C211yesmaximal, characteristic
10#11728209C211yesmaximal, characteristic, complemented
11#11728196C211yesmaximal, characteristic, complemented
12#11727978C211yesmaximal, characteristic, complemented
13#11727977C211yesmaximal, characteristic, complemented
14#11727974C211yesmaximal, characteristic, complemented
15Dic7×#103C211yesmaximal, characteristic, complemented
16C22×C7×SD16C2211yescharacteristic
17C22×C7×SD16C2211yescharacteristic
18C2×C7×C4⋊Q8C2211yescharacteristic
19C2×C4×C7×D4C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C2×Q8×Dic7C2211yescharacteristic
22C2×D4×Dic7C2211yescharacteristic
23C2×D4×Dic7C2211yescharacteristic
24C7×#461C418yescomplemented
25C2×C7×C4⋊C8C2211yescharacteristic
26C2×C7×#103C2211yescharacteristic
27C2×C7×#103C2211yescharacteristic
28#90002C2211yescharacteristic
29#90002C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35C2×#6406C2211yescharacteristic
36C4×C22⋊Dic7—21no—
37C2×C7⋊#95—21no—
38#89936—24no—
39SD16×Dic7—24no—
40C2×#6392C2211yescharacteristic
41C2×#6391C2211yescharacteristic
42C2×#6389C2211yescharacteristic
43C2×C8×Dic7C2211yescharacteristic
44#89753—21no—
45#89753—21no—
46#89753—21no—
47#89753—21no—
48C4⋊C4×Dic7C2211yescharacteristic
49C4⋊C4×Dic7C2211yescharacteristic
50C22⋊C4×Dic7—21no—
51C22⋊C4×Dic7—21no—
52#89708C2211yescharacteristic
53C4×C4⋊Dic7C2211yescharacteristic
54#89704C2211yescharacteristic
55C2×C4⋊(C7⋊C8)C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89345C2211yescharacteristic
59#89345C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#21514—71nomaximal, Sylow
63C22×C7×Q8C2311yescharacteristic
64C22×C7×Q8C2311yescharacteristic
65C22×C7×D4C2311yescharacteristic
66C23×C4×C7—21no—
67C23×Dic7—21no—
68C23×Dic7—21no—
69C2×C7×SD16—22no—
70C2×C7×SD16—22no—
71C2×C7×SD16—22no—
72C2×C7×SD16—22no—
73C2×C7×SD16C2×C414yes—
74C2×C7×SD16C2×C414yes—
75C22×C7×C8C2311yescharacteristic
76C22×C7×C8C2311yescharacteristic
77C7×C4⋊Q8—22no—
78C7×C4⋊Q8C2×C414yescomplemented
79C4×C7×D4—22no—
80C4×C7×D4C2×C414yes—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C7×C22⋊C4—21no—
85C2×C42×C7C2311yescharacteristic
86C2×C22⋊Dic7—21no—
87C2×C22⋊Dic7—21no—
88Q8×Dic7—22no—
89Q8×Dic7—22no—
90Q8×Dic7—22no—
91Q8×Dic7—22no—
92D4×Dic7—22no—
93D4×Dic7—22no—
94D4×Dic7—22no—
95D4×Dic7D414yes—
96C2×C4⋊Dic7—21no—
97C2×C4⋊Dic7—21no—
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4⋊Dic7C2311yescharacteristic
100C2×C4⋊Dic7C2311yescharacteristic
101C2×C4⋊Dic7C2311yescharacteristic
102C2×C4×Dic7—21no—
103C2×C4×Dic7—21no—
104C2×C4×Dic7—21no—
105C2×C4×Dic7—21no—
106C2×C4×Dic7—21no—
107C2×C4×Dic7C2311yescharacteristic
108C2×C4×Dic7C2311yescharacteristic
109C22×C7⋊C8C2311yescharacteristic
110C22×C7⋊C8C2311yescharacteristic
111C7×C4⋊C8C2×C414yescomplemented
112C7×#103C2×C414yescomplemented
113C7×#103C2×C414yescomplemented
114#6409—22no—
115#6409—22no—
116#6409—22no—
117#6409—22no—
118#6406—22no—
119#6406—22no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95—21no—
128C7⋊#95—21no—
129C7⋊#95D411yescharacteristic
130C7⋊#95D411yescharacteristic
131C7⋊#95D411yescharacteristic
132C7⋊#95D411yescharacteristic
133#6392—22no—
134#6391—22no—
135#6389—22no—
136C8×Dic7—22no—
137C4⋊(C7⋊C8)—22no—
138C2×#461D711yescharacteristic
139C2×#440—71no—
140C2×C4×SD16—71no—
141#2327—71no—
142#2275—71no—
143#1981—71no—
144#1949—71no—
145#1928—71no—
146#1927—71no—
147#1877—71no—
148#1877—71no—
149#1804—71no—
150#1768—71no—
151#1763—71no—
152C4×#103—71no—
153C24×C7—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—21no—
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
161C2×C7×Q8—21no—
162C2×C7×Q8—21no—
163C2×C7×Q8—21no—
164C2×C7×Q8C22×C412yes—
165C2×C7×Q8C22×C412yes—
166C2×C7×D4—21no—
167C2×C7×D4—21no—
168C2×C7×D4—21no—
169C2×C7×D4C22×C412yes—
170C2×C7×D4C2×D412yes—
171C2×C7×D4#4112yes—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—24no—
176C22×C4×C7—81no—
177C22×C4×C7—81no—
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2411yescharacteristic
181C22×Dic7—21no—
182C22×Dic7—21no—
183C22×Dic7—24no—
184C22×Dic7—24no—
185C22×Dic7—21no—
186C22×Dic7—21no—
187C22×Dic7—81no—
188C22×Dic7—81no—
189C22×Dic7—81no—
190C22×Dic7—81no—
191C22×Dic7C2×D411yescharacteristic
192C22×Dic7C2×D411yescharacteristic
193C22×Dic7C2×D411yescharacteristic
194C22×Dic7C2×D411yescharacteristic
195C7×SD16—28no—
196C7×SD16—28no—
197C2×C7×C8—21no—
198C2×C7×C8—21no—
199C2×C7×C8—21no—
200C2×C7×C8—21no—
201C2×C7×C8C22×C412yes—
202C2×C7×C8C22×C412yes—
203C7×C4⋊C4—24no—
204C7×C4⋊C4—24no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4C22×C412yes—
207C7×C22⋊C4—24no—
208C42×C7C22×C412yes—
209C42×C7#4112yes—
210C22⋊Dic7—24no—
211C22⋊Dic7—24no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7—21no—
216C4⋊Dic7—21no—
217C4⋊Dic7—21no—
218C4⋊Dic7—42no—
219C4⋊Dic7—42no—
220C4⋊Dic7C2×D412yes—
221C4⋊Dic7#4112yes—
222C4×Dic7—21no—
223C4×Dic7—21no—
224C4×Dic7—42no—
225C4×Dic7—42no—
226C4×Dic7—81no—
227C4×Dic7—81no—
228C4×Dic7—81no—
229C4×Dic7C2×D412yes—
230C4×Dic7#4112yes—
231C2×C7⋊C8—21no—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—21no—
236C2×C7⋊C8—21no—
237C22×SD16C2×D711yescharacteristic
238C22×SD16C2×D711yescharacteristic
239C2×C4⋊Q8C2×D711yescharacteristic
240C2×C4×Q8—71no—
241C2×C4×Q8—71no—
242C2×C4×D4—71no—
243C2×C4×D4—71no—
244C2×C4×D4C2×D711yescharacteristic
245#461Dic718yescomplemented
246#440—144no—
247C4×SD16—144no—
248C2×#107—71no—
249C2×#106—71no—
250C2×C4⋊C8—71no—
251C2×C4⋊C8C2×D711yescharacteristic
252C2×#103—71no—
253C2×#103—71no—
254C2×#103—71no—
255C2×#103—71no—
256C2×#103C2×D711yescharacteristic
257C2×#103C2×D711yescharacteristic
258C2×#102—71no—
259C2×#102—71no—
260C2×#97—71no—
261C2×C4×C8—71no—
262#391—71no—
263#391—71no—
264#387—141no—
265#387—141no—
266#387—141no—
267#387—141no—
268#382—71no—
269C4×C4⋊C4—71no—
270C4×C4⋊C4—71no—
271C4×C4⋊C4—71no—
272C4×C22⋊C4—141no—
273C4×C22⋊C4—141no—
274C4×C22⋊C4—141no—
275#376—71no—
276C2×#95—141no—
277#340—71no—
278#340—71no—
279#340—71no—
280#340—71no—
281#336—71no—
282#336—71no—
283C23×C7—22no—
284C23×C7—22no—
285C23×C7—22no—
286C23×C7—41no—
287C23×C7—41no—
288C23×C7—41no—
289C23×C7—41no—
290C23×C7C22×D411yescharacteristic
291C7×Q8—22no—
292C7×Q8—22no—
293C7×Q8—22no—
294C7×Q8—22no—
295C7×Q8—41no—
296C7×Q8—41no—
297C7×Q8—41no—
298C7×Q8—41no—
299C7×D4—22no—
300C7×D4—22no—
301C7×D4—22no—
302C7×D4C4×D414yes—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—22no—
308C2×C4×C7—22no—
309C2×C4×C7—22no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—84no—
317C2×C4×C7C4×D412yes—
318C2×C4×C7C4×D412yes—
319C2×C4×C7C23×C411yescharacteristic
320C2×C4×C7C22×D411yescharacteristic
321C2×C4×C7C2×#4111yescharacteristic
322C2×C4×C7C2×#4111yescharacteristic
323C2×C4×C7C2×#4111yescharacteristic
324C2×C4×C7C2×#4111yescharacteristic
325C2×Dic7—21no—
326C2×Dic7—21no—
327C2×Dic7—21no—
328C2×Dic7—21no—
329C2×Dic7—22no—
330C2×Dic7—21no—
331C2×Dic7—21no—
332C2×Dic7—22no—
333C2×Dic7—21no—
334C2×Dic7—21no—
335C2×Dic7—21no—
336C2×Dic7—21no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C2×Dic7—41no—
342C2×Dic7—41no—
343C2×Dic7—41no—
344C2×Dic7—41no—
345C2×Dic7—41no—
346C2×Dic7—41no—
347C2×Dic7—84no—
348C2×Dic7—84no—
349C2×Dic7#12112yes—
350C2×Dic7#12112yes—
351C7×C8—22no—
352C7×C8—22no—
353C7⋊C8—41no—
354C7⋊C8—41no—
355C22×Q8C22×D711yescharacteristic
356C22×Q8C22×D711yescharacteristic
357C22×D4C22×D711yescharacteristic
358C23×C4—21no—
359C23×C4—141no—
360C23×C4—141no—
361C2×SD16—22no—
362C2×SD16—22no—
363C2×SD16—22no—
364C2×SD16—22no—
365C2×SD16C2×Dic714yes—
366C2×SD16C2×Dic714yes—
367C22×C8—71no—
368C22×C8—71no—
369C22×C8C22×D711yescharacteristic
370C22×C8C22×D711yescharacteristic
371C4⋊Q8—22no—
372C4⋊Q8C2×Dic714yescomplemented
373C4×Q8—142no—
374C4×Q8—142no—
375C4×Q8—142no—
376C4×Q8—142no—
377C4×D4—22no—
378C4×D4—74no—
379C4×D4—142no—
380C4×D4—142no—
381C4×D4—142no—
382C4×D4C2×Dic714yes—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—71no—
386C2×C4⋊C4—71no—
387C2×C4⋊C4—71no—
388C2×C4⋊C4—71no—
389C2×C4⋊C4—141no—
390C2×C4⋊C4—141no—
391C2×C4⋊C4C22×D711yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—141no—
394C2×C22⋊C4—141no—
395C2×C42—71no—
396C2×C42—71no—
397C2×C42—141no—
398C2×C42—141no—
399C2×C42—141no—
400C2×C42—141no—
401C2×C42—141no—
402C2×C42C22×D711yescharacteristic
403#107—142no—
404#106—142no—
405C4⋊C8—142no—
406C4⋊C8C2×Dic714yescomplemented
407#103—142no—
408#103—142no—
409#103—142no—
410#103—142no—
411#103C2×Dic714yescomplemented
412#103C2×Dic714yescomplemented
413#102—142no—
414#102—142no—
415#97—142no—
416C4×C8—142no—
417#95—71no—
418#95—71no—
419#95—71no—
420#95—71no—
421#95—141no—
422#95—141no—
423#95—141no—
424#95—141no—
425#95—141no—
426#95—141no—
427#95—141no—
428#95—141no—
429#95—141no—
430C22×C7—24no—
431C22×C7—42no—
432C22×C7—42no—
433C22×C7—42no—
434C22×C7—42no—
435C22×C7—42no—
436C22×C7—42no—
437C22×C7C2×C4×D411yescharacteristic
438C22×C7C2×#12111yescharacteristic
439C22×C7#52511yescharacteristic
440C22×C7#57711yescharacteristic
441C22×C7#57711yescharacteristic
442C22×C7#57711yescharacteristic
443C22×C7#57811yescharacteristic
444C4×C7—22no—
445C4×C7—22no—
446C4×C7—42no—
447C4×C7—42no—
448C4×C7—42no—
449C4×C7C2×C4×D411yescharacteristic
450C4×C7C4×#4111yescharacteristic
451C4×C7#52911yescharacteristic
452C4×C7#53211yescharacteristic
453Dic7—22no—
454Dic7—22no—
455Dic7—41no—
456Dic7—41no—
457Dic7—42no—
458Dic7—42no—
459Dic7—81no—
460Dic7—81no—
461C24—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8C22×Dic712yes—
473C2×Q8C22×Dic712yes—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4D4×D712yes—
478C2×D4#115912yes—
479C2×D4C22×Dic712yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—24no—
484C22×C4—71no—
485C22×C4—71no—
486C22×C4—71no—
487C22×C4—71no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—144no—
491C22×C4—141no—
492C22×C4—141no—
493C22×C4—144no—
494C22×C4—141no—
495C22×C4—141no—
496C22×C4—561no—
497C22×C4—561no—
498C22×C4—561no—
499C22×C4—561no—
500C22×C4D4×D711yescharacteristic
501C22×C4D4×D711yescharacteristic
502C22×C4C23×D711yescharacteristic
503SD16—28no—
504SD16—28no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—141no—
510C2×C8—141no—
511C2×C8—141no—
512C2×C8—141no—
513C2×C8—141no—
514C2×C8—141no—
515C2×C8C22×Dic712yes—
516C2×C8C22×Dic712yes—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—21no—
520C4⋊C4—72no—
521C4⋊C4—72no—
522C4⋊C4—141no—
523C4⋊C4—141no—
524C4⋊C4—141no—
525C4⋊C4—141no—
526C4⋊C4—141no—
527C4⋊C4—141no—
528C4⋊C4—282no—
529C4⋊C4—282no—
530C4⋊C4C22×Dic712yes—
531C22⋊C4—24no—
532C22⋊C4—144no—
533C22⋊C4—144no—
534C42—72no—
535C42—72no—
536C42—141no—
537C42—141no—
538C42—282no—
539C42—282no—
540C42—561no—
541C42—561no—
542C42—561no—
543C42#115912yes—
544C42C22×Dic712yes—
545C2×C7—44no—
546C2×C7C4×#12111yescharacteristic
547C2×C7#294711yescharacteristic
548C2×C7#294911yescharacteristic
549C2×C7#304211yescharacteristic
550C2×C7#304611yescharacteristic
551C2×C7#312011yescharacteristic
552C2×C7#312311yescharacteristic
553C23—22no—
554C23—22no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23C2×D4×D711yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—41no—
568Q8—41no—
569D4—22no—
570D4—22no—
571D4—22no—
572D4D4×Dic714yes—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—72no—
587C2×C4—72no—
588C2×C4—84no—
589C2×C4—141no—
590C2×C4—141no—
591C2×C4—141no—
592C2×C4—141no—
593C2×C4—141no—
594C2×C4—141no—
595C2×C4—141no—
596C2×C4—141no—
597C2×C4—142no—
598C2×C4—141no—
599C2×C4—141no—
600C2×C4—142no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4—281no—
605C2×C4—281no—
606C2×C4—281no—
607C2×C4—281no—
608C2×C4—281no—
609C2×C4—281no—
610C2×C4—281no—
611C2×C4—564no—
612C2×C4—564no—
613C2×C4D4×Dic712yes—
614C2×C4D4×Dic712yes—
615C2×C4C2×D4×D711yescharacteristic
616C2×C4C2×#115911yescharacteristic
617C2×C4C2×#115911yescharacteristic
618C2×C4D7×#4111yescharacteristic
619C2×C4D7×#4111yescharacteristic
620C2×C4C23×Dic711yescharacteristic
621C8—22no—
622C8—22no—
623C8—281no—
624C8—281no—
625C7#2151411yescharacteristic, complemented, Sylow
626C22—24no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22D7×#12111yescharacteristic
634C22#9029711yescharacteristic
635C22#9045311yescharacteristic
636C22#9045311yescharacteristic
637C22#9046911yescharacteristic
638C22#9047011yescharacteristic
639C22C2×D4×Dic711yescharacteristic
640C4—22no—
641C4—22no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—142no—
646C4—142no—
647C4—281no—
648C4—281no—
649C4—282no—
650C4—282no—
651C4—561no—
652C4—561no—
653C4#9028711yescharacteristic
654C4#9029111yescharacteristic
655C4C2×D4×Dic711yescharacteristic
656C4#41×Dic711yescharacteristic
657C2—44no—
658C2#1173785711yescharacteristic
659C2#1173785811yescharacteristic
660C2#1173787511yescharacteristic
661C2#1173787611yescharacteristic
662C2Dic7×#12111yescharacteristic
663C2#1174128811yescharacteristic
664C2#1174141011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#461C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC2×#461D7
hypocenterC7#21514
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
4. #11740672C2semidirect
8. #11729532C2semidirect
10. #11728209C2semidirect
11. #11728196C2semidirect
12. #11727978C2semidirect
13. #11727977C2semidirect
14. #11727974C2semidirect
15. Dic7×#103C2semidirect
24. C7×#461C4semidirect
78. C7×C4⋊Q8C2×C4semidirect
111. C7×C4⋊C8C2×C4semidirect
112. C7×#103C2×C4semidirect
113. C7×#103C2×C4semidirect
245. #461Dic7semidirect
372. C4⋊Q8C2×Dic7semidirect
406. C4⋊C8C2×Dic7semidirect
411. #103C2×Dic7semidirect
412. #103C2×Dic7semidirect
625. C7#21514semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4612
C2×#899362
C2×SD16×Dic72
#117406722
#117397622
#117298012
#117295322
#117293592
#117282092
#117281962
#117279782
#117279772
#117279742
Dic7×#1032
#215147