Finite Group Databasebeta

← #13360150

Subgroups — #13360150

Order 1792.

980 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2276C211yesmaximal, characteristic, complemented
2C2×#89800C211yesmaximal, characteristic
3C2×#89802C211yesmaximal, characteristic, complemented
4C2×C4×#1143C211yesmaximal, characteristic
5#11730832C211yesmaximal, characteristic
6#11729387C211yesmaximal, characteristic
7#11729386C211yesmaximal, characteristic, complemented
8#11729380C211yesmaximal, characteristic, complemented
9#11727242C211yesmaximal, characteristic
10#11727228C211yesmaximal, characteristic
11C4×#6384C211yesmaximal, characteristic, complemented
12#11727221C211yesmaximal, characteristic, complemented
13#11727218C211yesmaximal, characteristic, complemented
14#11727217C211yesmaximal, characteristic
15#11727190C211yesmaximal, characteristic, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C22×#1144C2211yescharacteristic
20C22×#1143C2211yescharacteristic
21C2×#6438C2211yescharacteristic
22C2×C4×D28C2211yescharacteristic
23C2×C4×Dic14C2211yescharacteristic
24C4×C7×C4⋊C4—21no—
25C4×C7×C4⋊C4—21no—
26C4×C7×C4⋊C4—21no—
27C4×C7×C4⋊C4C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29C4×C7×C4⋊C4C2211yescharacteristic
30C7×#376—21no—
31C7×#376C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34#89802—24no—
35#89800C418yescomplemented
36C4×#1143—24no—
37C2×#6384C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6384C2211yescharacteristic
40C2×#6383C2211yescharacteristic
41C2×#6383C2211yescharacteristic
42C2×#6383C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6382C2211yescharacteristic
45#89711C2211yescharacteristic
46#89707C2211yescharacteristic
47#89701C2211yescharacteristic
48C2×C4⋊(C7⋊C8)C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×#6377C2211yescharacteristic
51C2×C4×C7⋊C8C2211yescharacteristic
52#89442—21no—
53#89442—21no—
54#89440—21no—
55#89440—21no—
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89319C2211yescharacteristic
61#89319C2211yescharacteristic
62#21511—71nomaximal, Sylow
63C22×C7×Q8C2311yescharacteristic
64C22×D28C2311yescharacteristic
65C22×C4×D7—21no—
66C22×Dic14C2311yescharacteristic
67C4×C7×Q8—22no—
68C4×C7×Q8—22no—
69C4×C7×Q8—22no—
70C4×C7×Q8—22no—
71C4×C7×Q8C2×C414yescomplemented
72C4×C7×Q8D414yes—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C42×C7—21no—
80C2×C42×C7—21no—
81C2×C42×C7—21no—
82C2×C42×C7—21no—
83C2×C42×C7—21no—
84C2×C42×C7—21no—
85C2×C42×C7C2311yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C2×C42×C7C2311yescharacteristic
88C2×#1144—22no—
89C2×#1144—22no—
90C2×#1144C2×C414yescomplemented
91C2×#1143—22no—
92C2×#1143—22no—
93C2×#1143C2×C414yes—
94C2×(C2×D7)⋊C4—21no—
95C2×(C2×D7)⋊C4—21no—
96C2×(C2×D7)⋊C4—21no—
97C2×(C2×D7)⋊C4—21no—
98C2×(C2×D7)⋊C4—21no—
99C2×C4⋊Dic7—21no—
100C2×C4⋊Dic7—21no—
101C2×C4⋊Dic7C2311yescharacteristic
102C2×C4⋊Dic7C2311yescharacteristic
103C2×Dic7⋊C4—21no—
104C2×C4×Dic7—21no—
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108C22×C7⋊C8C2311yescharacteristic
109#6438—22no—
110#6438C2×C414yes—
111C4×D28—22no—
112C4×D28—22no—
113C4×Dic14—22no—
114C4×Dic14—22no—
115C7×#95—21no—
116C7×#95—21no—
117C7×#95—21no—
118C7×#95D411yescharacteristic
119C7×#95D411yescharacteristic
120C7×#95D411yescharacteristic
121C7×#95D411yescharacteristic
122#6409—22no—
123#6409—22no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126#6384—22no—
127#6384—22no—
128#6384C2×C414yescomplemented
129#6383—22no—
130#6383—22no—
131#6383C2×C414yes—
132#6382—22no—
133#6382—22no—
134C4⋊(C7⋊C8)—22no—
135C4⋊(C7⋊C8)C2×C414yescomplemented
136#6377—22no—
137C4×C7⋊C8—22no—
138C2×#464—71no—
139C2×#441—71no—
140C2×C4×SD16—71no—
141#2326—71no—
142#2276D711yescharacteristic, complemented
143#1983—71no—
144#1948—71no—
145#1927—71no—
146#1926—71no—
147#1880—71no—
148#1878—71no—
149#1803—71no—
150#1768—71no—
151#1762—71no—
152C4×#103—71no—
153C23×D7—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8C22×C412yes—
158C2×C7×Q8C2×D412yes—
159C2×C7×Q8#4112yes—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2411yescharacteristic
171C22×Dic7—21no—
172C22×Dic7—21no—
173C22×Dic7—21no—
174C2×D28—21no—
175C2×D28—21no—
176C2×D28—21no—
177C2×D28—21no—
178C2×D28—21no—
179C2×D28C22×C412yes—
180C2×C4×D7—42no—
181C2×C4×D7—81no—
182C2×C4×D7—81no—
183C2×Dic14—21no—
184C2×Dic14—21no—
185C2×Dic14—21no—
186C2×Dic14—21no—
187C2×Dic14—21no—
188C2×Dic14C22×C412yes—
189C7×C4⋊C4—24no—
190C7×C4⋊C4—24no—
191C7×C4⋊C4—24no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4C22×C412yes—
196C7×C4⋊C4C2×D412yes—
197C7×C4⋊C4#4112yes—
198C42×C7—24no—
199C42×C7—24no—
200C42×C7—24no—
201C42×C7—21no—
202C42×C7—21no—
203C42×C7—81no—
204C42×C7—81no—
205C42×C7—81no—
206C42×C7C22×C412yes—
207C42×C7C2×D412yes—
208C42×C7#4112yes—
209C42×C7#4112yes—
210#1144—28no—
211#1143—28no—
212(C2×D7)⋊C4—24no—
213(C2×D7)⋊C4—24no—
214(C2×D7)⋊C4—42no—
215(C2×D7)⋊C4—81no—
216(C2×D7)⋊C4—81no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219C4⋊Dic7—21no—
220C4⋊Dic7—21no—
221C4⋊Dic7—81no—
222C4⋊Dic7—81no—
223Dic7⋊C4—42no—
224C4×Dic7—42no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8C22×C412yes—
236C2×C7⋊C8C22×C412yes—
237C22×Q16—71no—
238C22×SD16—71no—
239C2×#124—71no—
240C2×C4×Q8—71no—
241C2×C4×Q8C2×D711yescharacteristic
242C2×C4×Q8C2×D711yescharacteristic
243C2×C4×Q8C2×D711yescharacteristic
244C2×C4×D4—71no—
245#464—78no—
246#441—144no—
247C4×SD16—144no—
248C2×#106—71no—
249C2×#106—71no—
250C2×C4⋊C8—71no—
251C2×C4⋊C8—71no—
252C2×#103—71no—
253C2×#103—71no—
254C2×#103—71no—
255C2×#103—71no—
256C2×#103—71no—
257C2×#102—71no—
258C2×#102—71no—
259C2×#102—71no—
260C2×#97—71no—
261C2×C4×C8—71no—
262#391—71no—
263#390—71no—
264#388—141no—
265#388—141no—
266#385—141no—
267#385—141no—
268#384—71no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4C2×D711yescharacteristic
273C4×C4⋊C4C2×D711yescharacteristic
274C4×C4⋊C4C2×D711yescharacteristic
275#376—21no—
276#376C2×D711yescharacteristic
277#340—71no—
278#340—71no—
279#340—71no—
280#340—71no—
281#336—71no—
282#336—71no—
283C23×C7C22×D411yescharacteristic
284C22×D7—22no—
285C22×D7—41no—
286C22×D7—41no—
287C22×D7—41no—
288C22×D7—41no—
289C22×D7—41no—
290C22×D7—41no—
291C7×Q8—22no—
292C7×Q8—22no—
293C7×Q8—22no—
294C7×Q8C4×D414yes—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—22no—
307C2×C4×C7—22no—
308C2×C4×C7—21no—
309C2×C4×C7—21no—
310C2×C4×C7—22no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—21no—
314C2×C4×C7—21no—
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×C7C4×D412yes—
323C2×C4×C7#12112yes—
324C2×C4×C7#12112yes—
325C2×C4×C7C23×C411yescharacteristic
326C2×C4×C7C22×D411yescharacteristic
327C2×C4×C7C2×#4111yescharacteristic
328C2×C4×C7C2×#4111yescharacteristic
329C2×C4×C7C2×#4111yescharacteristic
330C2×C4×C7C2×#4111yescharacteristic
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—
340D28—22no—
341D28—22no—
342D28—41no—
343D28—41no—
344C4×D7—84no—
345Dic14—22no—
346Dic14—22no—
347Dic14—41no—
348Dic14—41no—
349C7⋊C8—22no—
350C7⋊C8—22no—
351C7⋊C8—41no—
352C7⋊C8—41no—
353C22×Q8—71no—
354C22×Q8C22×D711yescharacteristic
355C22×D4—71no—
356C23×C4—141no—
357C2×Q16—74no—
358C2×Q16—142no—
359C2×Q16—142no—
360C2×SD16—74no—
361C2×SD16—142no—
362C2×SD16—142no—
363C22×C8—71no—
364C22×C8—71no—
365C22×C8—71no—
366C22×C8—71no—
367#124—74no—
368#124—142no—
369C4×Q8—22no—
370C4×Q8—22no—
371C4×Q8—22no—
372C4×Q8—22no—
373C4×Q8—142no—
374C4×Q8—142no—
375C4×Q8C4×D714yescomplemented
376C4×Q8D2814yes—
377C4×D4—142no—
378C4×D4—142no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—71no—
383C2×C4⋊C4—71no—
384C2×C4⋊C4—141no—
385C2×C4⋊C4—141no—
386C2×C4⋊C4—141no—
387C2×C4⋊C4C22×D711yescharacteristic
388C2×C4⋊C4C22×D711yescharacteristic
389C2×C4⋊C4C22×D711yescharacteristic
390C2×C22⋊C4—141no—
391C2×C22⋊C4—141no—
392C2×C22⋊C4—141no—
393C2×C22⋊C4—141no—
394C2×C22⋊C4—141no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42—141no—
402C2×C42C22×D711yescharacteristic
403C2×C42C22×D711yescharacteristic
404C2×C42C22×D711yescharacteristic
405#106—142no—
406#106—142no—
407C4⋊C8—74no—
408C4⋊C8—142no—
409#103—74no—
410#103—142no—
411#103—142no—
412#103—142no—
413#103—142no—
414#102—74no—
415#102—142no—
416#102—142no—
417#97—142no—
418C4×C8—142no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—141no—
423#95—141no—
424#95C22⋊D711yescharacteristic
425#95C22⋊D711yescharacteristic
426#95C22⋊D711yescharacteristic
427#95C22⋊D711yescharacteristic
428C22×C7C2×C4×D411yescharacteristic
429C22×C7C2×#12111yescharacteristic
430C22×C7#52511yescharacteristic
431C22×C7#57711yescharacteristic
432C22×C7#57711yescharacteristic
433C22×C7#57711yescharacteristic
434C22×C7#57811yescharacteristic
435C2×D7—42no—
436C2×D7—42no—
437C2×D7—42no—
438C2×D7—81no—
439C2×D7—81no—
440C2×D7—81no—
441C2×D7—81no—
442C4×C7—22no—
443C4×C7—22no—
444C4×C7—22no—
445C4×C7—22no—
446C4×C7—24no—
447C4×C7—41no—
448C4×C7—41no—
449C4×C7—42no—
450C4×C7—42no—
451C4×C7—42no—
452C4×C7C2×C4×D411yescharacteristic
453C4×C7C4×#4111yescharacteristic
454C4×C7#52911yescharacteristic
455C4×C7#53211yescharacteristic
456Dic7—42no—
457Dic7—81no—
458Dic7—81no—
459C24—141no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—72no—
464C2×Q8—141no—
465C2×Q8—141no—
466C2×Q8—141no—
467C2×Q8—141no—
468C2×Q8—141no—
469C2×Q8C2×C4×D712yes—
470C2×Q8C2×D2812yes—
471C2×Q8#115712yes—
472C2×D4—72no—
473C2×D4—141no—
474C2×D4—141no—
475C2×D4—141no—
476C2×D4—141no—
477C2×D4—141no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—141no—
483C22×C4—141no—
484C22×C4—141no—
485C22×C4—282no—
486C22×C4—561no—
487C22×C4—561no—
488C22×C4C2×C22⋊D711yescharacteristic
489C22×C4C2×C22⋊D711yescharacteristic
490C22×C4C2×C22⋊D711yescharacteristic
491C22×C4C2×C22⋊D711yescharacteristic
492C22×C4C2×C22⋊D711yescharacteristic
493C22×C4C2×C22⋊D711yescharacteristic
494C22×C4C23×D711yescharacteristic
495Q16—148no—
496SD16—148no—
497C2×C8—72no—
498C2×C8—72no—
499C2×C8—141no—
500C2×C8—141no—
501C2×C8—141no—
502C2×C8—141no—
503C2×C8—141no—
504C2×C8—141no—
505C2×C8—141no—
506C2×C8—141no—
507C2×C8—141no—
508C2×C8—141no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—141no—
516C4⋊C4—141no—
517C4⋊C4—141no—
518C4⋊C4—141no—
519C4⋊C4—282no—
520C4⋊C4—561no—
521C4⋊C4—561no—
522C4⋊C4C2×C4×D712yes—
523C4⋊C4C2×D2812yes—
524C4⋊C4#115712yes—
525C22⋊C4—144no—
526C22⋊C4—144no—
527C22⋊C4—282no—
528C22⋊C4—561no—
529C22⋊C4—561no—
530C42—24no—
531C42—24no—
532C42—24no—
533C42—21no—
534C42—21no—
535C42—81no—
536C42—81no—
537C42—81no—
538C42—282no—
539C42C2×C4×D712yes—
540C42C2×D2812yes—
541C42#115712yes—
542C42#115712yes—
543C2×C7C4×#12111yescharacteristic
544C2×C7#294711yescharacteristic
545C2×C7#295211yescharacteristic
546C2×C7#304311yescharacteristic
547C2×C7#304611yescharacteristic
548C2×C7#312111yescharacteristic
549C2×C7#312511yescharacteristic
550D7—82no—
551C23—142no—
552C23—281no—
553C23—281no—
554C23—281no—
555C23—281no—
556C23—281no—
557C23—281no—
558C23C22×C22⋊D711yescharacteristic
559Q8—22no—
560Q8—22no—
561Q8—22no—
562Q8—142no—
563Q8—142no—
564Q8—281no—
565Q8—281no—
566Q8C4×D2814yes—
567D4—142no—
568D4—142no—
569D4—281no—
570D4—281no—
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—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—142no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4—281no—
605C2×C4—281no—
606C2×C4—564no—
607C2×C4C4×C22⋊D712yes—
608C2×C4C4×C22⋊D712yes—
609C2×C4#649312yes—
610C2×C4#649312yes—
611C2×C4C22×C4×D711yescharacteristic
612C2×C4C22×D2811yescharacteristic
613C2×C4C2×#115711yescharacteristic
614C2×C4C2×#115711yescharacteristic
615C2×C4C2×#115711yescharacteristic
616C2×C4C2×#115711yescharacteristic
617C8—142no—
618C8—142no—
619C8—281no—
620C8—281no—
621C7#2151111yescharacteristic, complemented, Sylow
622C22—282no—
623C22—282no—
624C22—282no—
625C22—561no—
626C22—561no—
627C22—561no—
628C22—561no—
629C22C2×C4×C22⋊D711yescharacteristic
630C22C2×#649311yescharacteristic
631C22#9048211yescharacteristic
632C22#9052711yescharacteristic
633C22#9052811yescharacteristic
634C22#9052811yescharacteristic
635C22#9052911yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—24no—
641C4—41no—
642C4—42no—
643C4—42no—
644C4—41no—
645C4—42no—
646C4—282no—
647C4—561no—
648C4—561no—
649C4C2×C4×D2811yescharacteristic
650C4C4×#115711yescharacteristic
651C4#9016811yescharacteristic
652C4#9017011yescharacteristic
653C2—562no—
654C2C4×#649311yescharacteristic
655C2#1173999211yescharacteristic
656C2#1173999611yescharacteristic
657C2#1173999711yescharacteristic
658C2#1174000311yescharacteristic
659C2#1174001711yescharacteristic
660C2#1174002111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2276C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#2276D7
hypocenterC7#21511
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ #376 ◁ #2276

Complements

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

NH ≅ G/Ntype
1. C7×#2276C2semidirect
3. C2×#89802C2semidirect
7. #11729386C2semidirect
8. #11729380C2semidirect
11. C4×#6384C2semidirect
12. #11727221C2semidirect
13. #11727218C2semidirect
15. #11727190C2semidirect
35. #89800C4semidirect
71. C4×C7×Q8C2×C4semidirect
90. C2×#1144C2×C4semidirect
128. #6384C2×C4semidirect
135. C4⋊(C7⋊C8)C2×C4semidirect
142. #2276D7semidirect
375. C4×Q8C4×D7semidirect
621. C7#21511semidirect

Maximal subgroups

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

subgroupindex
C7×#22762
C2×#898002
C2×#898022
C2×C4×#11432
#117308322
#117293872
#117293862
#117293802
#117272422
#117272282
C4×#63842
#117272212
#117272182
#117272172
#117271902
#215117