Finite Group Databasebeta

← #13358176

Subgroups — #13358176

Order 1792.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#440C211yesmaximal, characteristic
2#11739899C211yesmaximal, characteristic, complemented
3#11739770C211yesmaximal, characteristic
4#11729804C211yesmaximal, characteristic
5#11729803C211yesmaximal, characteristic
6C4×#6409C211yesmaximal, characteristic, complemented
7C4×#6406C211yesmaximal, characteristic
8#11728367C211yesmaximal, characteristic, complemented
9#11728366C211yesmaximal, characteristic, complemented
10#11728009C211yesmaximal, characteristic, complemented
11#11728007C211yesmaximal, characteristic, complemented
12#11727941C211yesmaximal, characteristic
13#11727939C211yesmaximal, characteristic
14#11727568C211yesmaximal, characteristic, complemented
15#11727558C211yesmaximal, characteristic, complemented
16C22×C7×SD16C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C7×#440C418yescomplemented
20C2×C7×#107C2211yescharacteristic
21C2×C7×#103C2211yescharacteristic
22C2×C7×#102C2211yescharacteristic
23C2×C7×#97C2211yescharacteristic
24#90002C2211yescharacteristic
25#90002C2211yescharacteristic
26C2×#6409C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28#89993C2211yescharacteristic
29#89993C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32#89984—21no—
33#89983—21no—
34C2×#6392C2211yescharacteristic
35C2×#6391C2211yescharacteristic
36C2×#6389C2211yescharacteristic
37C2×#6389C2211yescharacteristic
38C2×C8×Dic7C2211yescharacteristic
39C2×C8×Dic7C2211yescharacteristic
40#89757C2211yescharacteristic
41#89757C2211yescharacteristic
42#89757C2211yescharacteristic
43#89757C2211yescharacteristic
44#89753—21no—
45#89753—21no—
46#89723—21no—
47#89723—21no—
48#89709—21no—
49#89708—21no—
50C4×C4⋊Dic7C2211yescharacteristic
51C4×C4⋊Dic7C2211yescharacteristic
52#89704C2211yescharacteristic
53C2×C4×C7⋊C8C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#21291—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C2×C7×SD16—22no—
65C2×C7×SD16—22no—
66C2×C7×SD16C2×C414yes—
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C4×C7×Q8—22no—
70C4×C7×Q8C2×C414yes—
71C4×C7×D4—22no—
72C4×C7×D4C2×C414yes—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C22⋊C4—21no—
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79C2×C22⋊Dic7—21no—
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7—21no—
82C2×C22⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7C2311yescharacteristic
94C2×C4×Dic7C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C7×#107C2×C414yes—
98C7×#103C2×C414yes—
99C7×#102C2×C414yes—
100C7×#97C2×C414yes—
101#6409—22no—
102#6409—22no—
103#6406—22no—
104#6406—22no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95D411yescharacteristic
116C7⋊#95D411yescharacteristic
117C7⋊#95D411yescharacteristic
118C7⋊#95D411yescharacteristic
119#6392—22no—
120#6391—22no—
121#6389—22no—
122#6389—22no—
123C8×Dic7—22no—
124C8×Dic7—22no—
125C4×C7⋊C8—22no—
126C2×#440D711yescharacteristic
127#2320—71no—
128#2319—71no—
129#1973—71no—
130#1968—71no—
131#1943—71no—
132#1942—71no—
133#1936—71no—
134#1935—71no—
135#1934—71no—
136#1933—71no—
137#1776—71no—
138C4×#103—71no—
139C4×#102—71no—
140#1726—71no—
141C24×C7—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8C22×C412yes—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4—21no—
153C2×C7×D4C22×C412yes—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—24no—
158C22×C4×C7—81no—
159C22×C4×C7—81no—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2411yescharacteristic
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7C2×D411yescharacteristic
168C22×Dic7C2×D411yescharacteristic
169C22×Dic7C2×D411yescharacteristic
170C22×Dic7C2×D411yescharacteristic
171C7×SD16—28no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8C22×C412yes—
175C2×C7×C8C22×C412yes—
176C2×C7×C8#4112yes—
177C2×C7×C8#4112yes—
178C7×C4⋊C4—24no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4C22×C412yes—
182C7×C4⋊C4C22×C412yes—
183C7×C22⋊C4—24no—
184C42×C7—24no—
185C42×C7C22×C412yes—
186C42×C7#4112yes—
187C22⋊Dic7—81no—
188C22⋊Dic7—81no—
189C22⋊Dic7—81no—
190C22⋊Dic7—81no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—81no—
200C4⋊Dic7—81no—
201C4⋊Dic7—81no—
202C4⋊Dic7—81no—
203C4×Dic7—21no—
204C4×Dic7—21no—
205C4×Dic7—21no—
206C4×Dic7—21no—
207C4×Dic7—81no—
208C4×Dic7—81no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8#4112yes—
214C2×C7⋊C8#4112yes—
215C22×SD16C2×D711yescharacteristic
216C2×C4×Q8C2×D711yescharacteristic
217C2×C4×D4C2×D711yescharacteristic
218#440Dic718yescomplemented
219C2×#107—71no—
220C2×#107C2×D711yescharacteristic
221C2×#106—71no—
222C2×#103—71no—
223C2×#103—71no—
224C2×#103C2×D711yescharacteristic
225C2×#102—71no—
226C2×#102—71no—
227C2×#102C2×D711yescharacteristic
228C2×#97—71no—
229C2×#97—71no—
230C2×#97C2×D711yescharacteristic
231C2×C4×C8—71no—
232C2×C4×C8—71no—
233C2×C4×C8—71no—
234#391—71no—
235#391—71no—
236#390—71no—
237#390—71no—
238#389—71no—
239#389—71no—
240#389—71no—
241#389—71no—
242#388—141no—
243#388—141no—
244#387—141no—
245#387—141no—
246#383—141no—
247#382—141no—
248#381—141no—
249#380—141no—
250C4×C4⋊C4—71no—
251C4×C4⋊C4—71no—
252#376—71no—
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259C23×C7—22no—
260C23×C7—41no—
261C23×C7—41no—
262C23×C7—41no—
263C23×C7—41no—
264C23×C7—41no—
265C23×C7—41no—
266C23×C7C22×D411yescharacteristic
267C7×Q8—22no—
268C7×Q8—22no—
269C7×Q8—41no—
270C7×Q8—41no—
271C7×D4—22no—
272C7×D4—22no—
273C7×D4—41no—
274C7×D4—41no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—84no—
289C2×C4×C7C4×D412yes—
290C2×C4×C7C4×D412yes—
291C2×C4×C7C23×C411yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C7×C8—22no—
322C7×C8#11714yes—
323C7⋊C8—22no—
324C7⋊C8—22no—
325C22×Q8C22×D711yescharacteristic
326C22×D4C22×D711yescharacteristic
327C23×C4—21no—
328C2×SD16—22no—
329C2×SD16—22no—
330C2×SD16C2×Dic714yes—
331C22×C8—71no—
332C22×C8—71no—
333C22×C8C22×D711yescharacteristic
334C22×C8C22×D711yescharacteristic
335C4×Q8—22no—
336C4×Q8C2×Dic714yes—
337C4×D4—22no—
338C4×D4C2×Dic714yes—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—71no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C4⋊C4C22×D711yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—141no—
352C2×C22⋊C4—141no—
353C2×C22⋊C4—141no—
354C2×C22⋊C4—141no—
355C2×C42—21no—
356C2×C42—71no—
357C2×C42—71no—
358C2×C42—141no—
359C2×C42—141no—
360C2×C42C22×D711yescharacteristic
361#107—142no—
362#107C2×Dic714yes—
363#106—142no—
364#103—142no—
365#103—142no—
366#103C2×Dic714yes—
367#102—142no—
368#102—142no—
369#102C2×Dic714yes—
370#97—142no—
371#97—142no—
372#97C2×Dic714yes—
373C4×C8—142no—
374C4×C8—142no—
375C4×C8—142no—
376#95—71no—
377#95—71no—
378#95—71no—
379#95—71no—
380#95—141no—
381#95—141no—
382#95—141no—
383#95—141no—
384#95—141no—
385#95—141no—
386#95—141no—
387#95—141no—
388#95—141no—
389#95—141no—
390C22×C7—42no—
391C22×C7—42no—
392C22×C7—42no—
393C22×C7—81no—
394C22×C7—81no—
395C22×C7—81no—
396C22×C7—81no—
397C22×C7C2×C4×D411yescharacteristic
398C22×C7#53211yescharacteristic
399C22×C7#53211yescharacteristic
400C22×C7#57611yescharacteristic
401C22×C7#57711yescharacteristic
402C22×C7#57711yescharacteristic
403C22×C7#57811yescharacteristic
404C4×C7—22no—
405C4×C7—22no—
406C4×C7—42no—
407C4×C7—42no—
408C4×C7—42no—
409C4×C7C2×#11711yescharacteristic
410C4×C7C4×#4111yescharacteristic
411C4×C7#52911yescharacteristic
412C4×C7#52911yescharacteristic
413Dic7—41no—
414Dic7—41no—
415Dic7—41no—
416Dic7—41no—
417Dic7—81no—
418Dic7—81no—
419Dic7—81no—
420Dic7—81no—
421C24—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8C22×Dic712yes—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4C22×Dic712yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—71no—
439C22×C4—71no—
440C22×C4—71no—
441C22×C4—71no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4D4×D711yescharacteristic
449C22×C4D4×D711yescharacteristic
450C22×C4C23×D711yescharacteristic
451SD16—28no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—72no—
455C2×C8—72no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8—141no—
459C2×C8—141no—
460C2×C8#115712yes—
461C2×C8#115712yes—
462C2×C8C22×Dic712yes—
463C2×C8C22×Dic712yes—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—561no—
476C4⋊C4—561no—
477C4⋊C4—561no—
478C4⋊C4—561no—
479C4⋊C4C22×Dic712yes—
480C4⋊C4C22×Dic712yes—
481C22⋊C4—24no—
482C22⋊C4—561no—
483C22⋊C4—561no—
484C22⋊C4—561no—
485C22⋊C4—561no—
486C42—24no—
487C42—141no—
488C42—141no—
489C42—141no—
490C42—141no—
491C42—561no—
492C42—561no—
493C42#115912yes—
494C42C22×Dic712yes—
495C2×C7—82no—
496C2×C7#229311yescharacteristic
497C2×C7#294611yescharacteristic
498C2×C7#294711yescharacteristic
499C2×C7#317411yescharacteristic
500C2×C7#317511yescharacteristic
501C2×C7#317811yescharacteristic
502C2×C7#317911yescharacteristic
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C2×D4×D711yescharacteristic
511Q8—22no—
512Q8—22no—
513Q8—41no—
514Q8—41no—
515D4—22no—
516D4—22no—
517D4—41no—
518D4—41no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—84no—
533C2×C4—141no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—141no—
544C2×C4—141no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4D4×Dic712yes—
558C2×C4D4×Dic712yes—
559C2×C4C2×#115711yescharacteristic
560C2×C4C2×#115711yescharacteristic
561C2×C4C2×#115911yescharacteristic
562C2×C4D7×#4111yescharacteristic
563C2×C4D7×#4111yescharacteristic
564C2×C4C23×Dic711yescharacteristic
565C8—22no—
566C8—142no—
567C8—142no—
568C8#648914yes—
569C7#2129111yescharacteristic, complemented, Sylow
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22#9023911yescharacteristic
578C22#9023911yescharacteristic
579C22#9046511yescharacteristic
580C22#9046611yescharacteristic
581C22#9046911yescharacteristic
582C22#9047011yescharacteristic
583C22C2×D4×Dic711yescharacteristic
584C4—22no—
585C4—22no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—281no—
590C4—281no—
591C4—281no—
592C4—281no—
593C4—561no—
594C4—561no—
595C4—561no—
596C4—561no—
597C4#9024811yescharacteristic
598C4#9024811yescharacteristic
599C4C2×#648911yescharacteristic
600C4#41×Dic711yescharacteristic
601C2—82no—
602C2#1173721211yescharacteristic
603C2#1173721311yescharacteristic
604C2#1173721411yescharacteristic
605C2#1173721511yescharacteristic
606C2#1173976411yescharacteristic
607C2#1174137511yescharacteristic
608C2#1174141011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#440C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC2×#440D7
hypocenterC7#21291
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
2. #11739899C2semidirect
6. C4×#6409C2semidirect
8. #11728367C2semidirect
9. #11728366C2semidirect
10. #11728009C2semidirect
11. #11728007C2semidirect
14. #11727568C2semidirect
15. #11727558C2semidirect
19. C7×#440C4semidirect
218. #440Dic7semidirect
569. C7#21291semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4402
#117398992
#117397702
#117298042
#117298032
C4×#64092
C4×#64062
#117283672
#117283662
#117280092
#117280072
#117279412
#117279392
#117275682
#117275582
#212917