Finite Group Databasebeta

← #13360395

Subgroups — #13360395

Order 1792.

954 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2323C211yesmaximal, characteristic
2C2×#89823C211yesmaximal, characteristic
3C2×#89802C211yesmaximal, characteristic, complemented
4#11739680C211yesmaximal, characteristic
5C2×#89769C211yesmaximal, characteristic
6#11729451C211yesmaximal, characteristic
7C4×#6409C211yesmaximal, characteristic, complemented
8#11727432C211yesmaximal, characteristic, complemented
9#11727428C211yesmaximal, characteristic, complemented
10#11727374C211yesmaximal, characteristic, complemented
11#11727368C211yesmaximal, characteristic, complemented
12#11727270C211yesmaximal, characteristic
13#11727265C211yesmaximal, characteristic
14#11727235C211yesmaximal, characteristic, complemented
15C4×#6384C211yesmaximal, characteristic, complemented
16C2×C7×C22⋊Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C7×#117C2211yescharacteristic
19C22×#1144C2211yescharacteristic
20C2×#6487C2211yescharacteristic
21C22×#1136C2211yescharacteristic
22C23×C7⋊C8C2211yescharacteristic
23C2×C4×Dic14C2211yescharacteristic
24C7×#391C2211yescharacteristic
25C7×#389—21no—
26C7×#389C2211yescharacteristic
27C7×#388—21no—
28C7×#388—21no—
29C7×#387—21no—
30C7×#380C2211yescharacteristic
31C4×C7×C4⋊C4C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35#89984C2211yescharacteristic
36#89823C418yescomplemented
37#89802—24no—
38#89769—24no—
39#89750—21no—
40#89744—21no—
41C2×#6384C2211yescharacteristic
42C2×#6384C2211yescharacteristic
43C2×#6384C2211yescharacteristic
44C2×#6382C2211yescharacteristic
45C2×#6382C2211yescharacteristic
46C2×#6381C2211yescharacteristic
47#89717—21no—
48#89716—21no—
49C4×C4⋊Dic7C2211yescharacteristic
50#89701C2211yescharacteristic
51#89700C2211yescharacteristic
52C2×#6377C2211yescharacteristic
53C2×#6377C2211yescharacteristic
54C2×C4×C7⋊C8C2211yescharacteristic
55C2×C4×C7⋊C8C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#21530—71nomaximal, Sylow
63C22×C7×Q8C2311yescharacteristic
64C23×C4×C7C2311yescharacteristic
65C22×Dic14C2311yescharacteristic
66C7×C22⋊Q8—22no—
67C7×C22⋊Q8C2×C414yes—
68C4×C7×Q8—22no—
69C4×C7×Q8—22no—
70C7×#117—22no—
71C7×#117—22no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4D411yescharacteristic
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4D411yescharacteristic
83C2×C7×C22⋊C4D411yescharacteristic
84C2×C42×C7—21no—
85C2×C42×C7—21no—
86C2×C42×C7C2311yescharacteristic
87C2×C42×C7C2311yescharacteristic
88C2×C22⋊Dic7—21no—
89C2×C22⋊Dic7—21no—
90C2×#1144—22no—
91C2×#1144—22no—
92C2×#1144C2×C414yes—
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96#6487—22no—
97#6487C2×C414yes—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×Dic7⋊C4—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—21no—
103C2×#1136—22no—
104C2×#1136—22no—
105C2×#1136—22no—
106C22×C7⋊C8—22no—
107C22×C7⋊C8—22no—
108C22×C7⋊C8C2×C414yes—
109C22×C7⋊C8C2311yescharacteristic
110C22×C7⋊C8C2311yescharacteristic
111C22×C7⋊C8C2311yescharacteristic
112C22×C7⋊C8C2311yescharacteristic
113C4×Dic14—22no—
114C4×Dic14—22no—
115C7×#95—21no—
116C7×#95—21no—
117C7×#95—21no—
118C7×#95—21no—
119C7×#95—21no—
120#6409—22no—
121#6409—22no—
122#6409C2×C414yes—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95—21no—
128#6384—22no—
129#6384—22no—
130#6384C2×C414yes—
131#6382—22no—
132#6382—22no—
133#6381C2×C414yes—
134#6377—22no—
135#6377D414yes—
136C4×C7⋊C8—22no—
137C4×C7⋊C8—22no—
138C2×#467—71no—
139C2×#441—71no—
140C2×#405—71no—
141#2323—71no—
142#2323D711yescharacteristic
143#1971—71no—
144#1943—71no—
145#1933—71no—
146#1933—71no—
147#1894—71no—
148#1894—71no—
149#1807—71no—
150#1776—71no—
151C4×#103—71no—
152C4×#103—71no—
153C24×C7C2×D411yescharacteristic
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—21no—
159C2×C7×Q8C22×C412yes—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7C22×C412yes—
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7#4112yes—
175C22×C4×C7C2411yescharacteristic
176C22×Dic7—21no—
177C22×Dic7—21no—
178C22×Dic7—21no—
179C22×Dic7—21no—
180C2×Dic14—21no—
181C2×Dic14—21no—
182C2×Dic14—21no—
183C2×Dic14—21no—
184C2×Dic14—21no—
185C2×Dic14C22×C412yes—
186C7×C4⋊C4—24no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—42no—
195C7×C4⋊C4—81no—
196C7×C4⋊C4C22×C412yes—
197C7×C22⋊C4—24no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—81no—
201C42×C7—21no—
202C42×C7—21no—
203C42×C7—42no—
204C42×C7—81no—
205C42×C7C2×D412yes—
206C42×C7#4112yes—
207C22⋊Dic7—24no—
208C22⋊Dic7—81no—
209#1144—28no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7C22×C412yes—
216Dic7⋊C4—24no—
217Dic7⋊C4—42no—
218Dic7⋊C4—81no—
219C4×Dic7—42no—
220C4×Dic7—81no—
221#1136—28no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—28no—
229C2×C7⋊C8C22×C412yes—
230C2×C7⋊C8C22×C412yes—
231C2×C7⋊C8C2×D412yes—
232C2×C7⋊C8C2×D412yes—
233C2×C7⋊C8#4112yes—
234C2×C7⋊C8#4112yes—
235C22×Q16—71no—
236C22×M16—71no—
237C23×C8—71no—
238C2×C22⋊Q8—71no—
239C2×C22⋊Q8C2×D711yescharacteristic
240C2×C4×Q8—71no—
241C2×C4×Q8C2×D711yescharacteristic
242C2×#117C2×D711yescharacteristic
243#467—78no—
244#441—144no—
245C2×#107—71no—
246C2×#106—71no—
247C2×#106—71no—
248C2×#103—71no—
249C2×#103—71no—
250C2×#103—71no—
251C2×#103—71no—
252C2×#103—71no—
253C2×#103—71no—
254#405—144no—
255C2×#97—71no—
256C2×#97—71no—
257C2×C4×C8—71no—
258C2×C4×C8—71no—
259#391—71no—
260#391C2×D711yescharacteristic
261#389—21no—
262#389—71no—
263#389—141no—
264#389C2×D711yescharacteristic
265#388—21no—
266#388—21no—
267#388—141no—
268#388—141no—
269#387—21no—
270#387—141no—
271#380—71no—
272#380C2×D711yescharacteristic
273C4×C4⋊C4—71no—
274C4×C4⋊C4C2×D711yescharacteristic
275#340—71no—
276#340—71no—
277#340—71no—
278#340—71no—
279#340—71no—
280#340—71no—
281C23×C7—21no—
282C23×C7—21no—
283C23×C7—21no—
284C23×C7—21no—
285C23×C7—21no—
286C23×C7—21no—
287C23×C7C4×D412yes—
288C23×C7C22×D411yescharacteristic
289C7×Q8—22no—
290C7×Q8—22no—
291C7×Q8—41no—
292C7×Q8—41no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
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—22no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—22no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—21no—
310C2×C4×C7—21no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
317C2×C4×C7—41no—
318C2×C4×C7—41no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7C4×D412yes—
322C2×C4×C7#12112yes—
323C2×C4×C7#12112yes—
324C2×C4×C7C23×C411yescharacteristic
325C2×C4×C7C22×D411yescharacteristic
326C2×C4×C7C2×#4111yescharacteristic
327C2×C4×C7C2×#4111yescharacteristic
328C2×C4×C7C2×#4111yescharacteristic
329C2×C4×C7C2×#4111yescharacteristic
330C2×Dic7—22no—
331C2×Dic7—22no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C2×Dic7—41no—
342Dic14—22no—
343Dic14—22no—
344Dic14—41no—
345Dic14—41no—
346C7⋊C8—22no—
347C7⋊C8—22no—
348C7⋊C8—22no—
349C7⋊C8C4×D414yes—
350C22×Q8—71no—
351C22×Q8C22×D711yescharacteristic
352C23×C4C22×D711yescharacteristic
353C2×Q16—74no—
354C2×Q16—142no—
355C2×Q16—142no—
356C2×M16—142no—
357C2×M16—142no—
358C2×M16—142no—
359C22×C8—71no—
360C22×C8—74no—
361C22×C8—71no—
362C22×C8—71no—
363C22×C8—71no—
364C22×C8—142no—
365C22×C8—142no—
366C22⋊Q8—22no—
367C22⋊Q8—74no—
368C22⋊Q8—142no—
369C22⋊Q8C4×D714yes—
370C4×Q8—22no—
371C4×Q8—22no—
372C4×Q8—142no—
373C4×Q8—142no—
374#117—22no—
375#117—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—71no—
380C2×C4⋊C4—71no—
381C2×C4⋊C4—71no—
382C2×C4⋊C4—141no—
383C2×C4⋊C4—141no—
384C2×C4⋊C4—141no—
385C2×C4⋊C4C22⋊D711yescharacteristic
386C2×C4⋊C4C22⋊D711yescharacteristic
387C2×C4⋊C4C22×D711yescharacteristic
388C2×C4⋊C4C22×D711yescharacteristic
389C2×C4⋊C4C22×D711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—141no—
393C2×C22⋊C4—141no—
394C2×C22⋊C4C22⋊D711yescharacteristic
395C2×C22⋊C4C22⋊D711yescharacteristic
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—141no—
399C2×C42—141no—
400C2×C42C22×D711yescharacteristic
401C2×C42C22×D711yescharacteristic
402#107—74no—
403#106—142no—
404#106—142no—
405#103—74no—
406#103—74no—
407#103—142no—
408#103—142no—
409#103—142no—
410#103—142no—
411#97—74no—
412#97—142no—
413C4×C8—142no—
414C4×C8—142no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—141no—
421#95—141no—
422#95—141no—
423#95—141no—
424#95—141no—
425C22×C7—22no—
426C22×C7—22no—
427C22×C7—22no—
428C22×C7—41no—
429C22×C7—41no—
430C22×C7—41no—
431C22×C7—41no—
432C22×C7C2×C4×D411yescharacteristic
433C22×C7C2×#12111yescharacteristic
434C22×C7#53211yescharacteristic
435C22×C7#57711yescharacteristic
436C22×C7#57711yescharacteristic
437C22×C7#57811yescharacteristic
438C22×C7#57811yescharacteristic
439C4×C7—22no—
440C4×C7—22no—
441C4×C7—22no—
442C4×C7—42no—
443C4×C7—41no—
444C4×C7—41no—
445C4×C7—42no—
446C4×C7—81no—
447C4×C7—81no—
448C4×C7C2×C4×D411yescharacteristic
449C4×C7C4×#4111yescharacteristic
450C4×C7#52511yescharacteristic
451C4×C7#52911yescharacteristic
452Dic7—42no—
453Dic7—42no—
454Dic7—81no—
455Dic7—81no—
456C24C2×C22⋊D711yescharacteristic
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—72no—
463C2×Q8—141no—
464C2×Q8—141no—
465C2×Q8—141no—
466C2×Q8—141no—
467C2×Q8—141no—
468C2×Q8C2×C4×D712yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—141no—
478C22×C4—141no—
479C22×C4—141no—
480C22×C4—141no—
481C22×C4C2×C4×D712yes—
482C22×C4#115712yes—
483C22×C4C2×C22⋊D711yescharacteristic
484C22×C4C2×C22⋊D711yescharacteristic
485C22×C4C2×C22⋊D711yescharacteristic
486C22×C4C2×C22⋊D711yescharacteristic
487C22×C4C2×C22⋊D711yescharacteristic
488C22×C4C23×D711yescharacteristic
489Q16—148no—
490M16—148no—
491C2×C8—72no—
492C2×C8—72no—
493C2×C8—72no—
494C2×C8—72no—
495C2×C8—72no—
496C2×C8—72no—
497C2×C8—148no—
498C2×C8—141no—
499C2×C8—141no—
500C2×C8—141no—
501C2×C8—141no—
502C2×C8—141no—
503C2×C8—141no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—42no—
513C4⋊C4—72no—
514C4⋊C4—81no—
515C4⋊C4—144no—
516C4⋊C4—141no—
517C4⋊C4—141no—
518C4⋊C4—141no—
519C4⋊C4—141no—
520C4⋊C4—141no—
521C4⋊C4—282no—
522C4⋊C4—561no—
523C4⋊C4C2×C4×D712yes—
524C22⋊C4—24no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—81no—
528C22⋊C4—144no—
529C22⋊C4—561no—
530C42—21no—
531C42—21no—
532C42—42no—
533C42—81no—
534C42—282no—
535C42—561no—
536C42D4×D712yes—
537C42#115912yes—
538C2×C7—42no—
539C2×C7C4×#12111yescharacteristic
540C2×C7#295011yescharacteristic
541C2×C7#295211yescharacteristic
542C2×C7#306311yescharacteristic
543C2×C7#306311yescharacteristic
544C2×C7#317811yescharacteristic
545C2×C7#317811yescharacteristic
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23C4×C22⋊D712yes—
553C23C22×C22⋊D711yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—41no—
557Q8—41no—
558Q8—142no—
559Q8—142no—
560Q8—281no—
561Q8—281no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—142no—
591C2×C4—142no—
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×C4C4×C22⋊D712yes—
603C2×C4#650212yes—
604C2×C4#650212yes—
605C2×C4C22×C4×D711yescharacteristic
606C2×C4C2×#115711yescharacteristic
607C2×C4C2×D4×D711yescharacteristic
608C2×C4C2×#115911yescharacteristic
609C2×C4D7×#4111yescharacteristic
610C2×C4D7×#4111yescharacteristic
611C8—74no—
612C8—142no—
613C8—142no—
614C8—142no—
615C7#2153011yescharacteristic, complemented, Sylow
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×C22⋊D711yescharacteristic
624C22C2×#650211yescharacteristic
625C22#9052311yescharacteristic
626C22#9052711yescharacteristic
627C22#9052811yescharacteristic
628C22#9052911yescharacteristic
629C22#9052911yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—41no—
636C4—41no—
637C4—81no—
638C4—81no—
639C4—282no—
640C4—282no—
641C4—561no—
642C4—561no—
643C4C4×#115911yescharacteristic
644C4#9022911yescharacteristic
645C4C4×D4×D711yescharacteristic
646C4#9023811yescharacteristic
647C2—42no—
648C2C4×#650211yescharacteristic
649C2#1173999411yescharacteristic
650C2#1174000311yescharacteristic
651C2#1174084511yescharacteristic
652C2#1174086811yescharacteristic
653C2#1174089011yescharacteristic
654C2#1174090311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2323C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#2323D7
hypocenterC7#21530
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
3. C2×#89802C2semidirect
7. C4×#6409C2semidirect
8. #11727432C2semidirect
9. #11727428C2semidirect
10. #11727374C2semidirect
11. #11727368C2semidirect
14. #11727235C2semidirect
15. C4×#6384C2semidirect
36. #89823C4semidirect
615. C7#21530semidirect

Maximal subgroups

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

subgroupindex
C7×#23232
C2×#898232
C2×#898022
#117396802
C2×#897692
#117294512
C4×#64092
#117274322
#117274282
#117273742
#117273682
#117272702
#117272652
#117272352
C4×#63842
#215307