Finite Group Databasebeta

← #13372161

Subgroups — #13372161

Order 1792.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#440C211yesmaximal, characteristic
2C2×#89936C211yesmaximal, characteristic
3C2×SD16×Dic7C211yesmaximal, characteristic
4Q8×C4⋊Dic7C211yesmaximal, characteristic, complemented
5#11739778C211yesmaximal, characteristic
6#11729381C211yesmaximal, characteristic, complemented
7#11729358C211yesmaximal, characteristic
8#11728367C211yesmaximal, characteristic, complemented
9#11728366C211yesmaximal, characteristic, complemented
10#11727992C211yesmaximal, characteristic, complemented
11#11727990C211yesmaximal, characteristic, complemented
12#11727924C211yesmaximal, characteristic
13#11727922C211yesmaximal, characteristic
14#11727570C211yesmaximal, characteristic, complemented
15#11727564C211yesmaximal, characteristic, complemented
16C22×C7×SD16C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C2×Q8×Dic7C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C2×D4×Dic7C2211yescharacteristic
22C2×D4×Dic7C2211yescharacteristic
23C7×#440C418yescomplemented
24C2×C7×#107C2211yescharacteristic
25C2×C7×#103C2211yescharacteristic
26C2×C7×#102C2211yescharacteristic
27C2×C7×#97C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32#89984—21no—
33C2×C7⋊#95—21no—
34#89936—24no—
35SD16×Dic7—24no—
36C2×#6392C2211yescharacteristic
37C2×#6392C2211yescharacteristic
38C2×#6391C2211yescharacteristic
39C2×#6391C2211yescharacteristic
40C2×#6389C2211yescharacteristic
41C2×C8×Dic7C2211yescharacteristic
42#89757—21no—
43#89757—21no—
44#89757C2211yescharacteristic
45#89757C2211yescharacteristic
46#89757C2211yescharacteristic
47#89757C2211yescharacteristic
48#89722—21no—
49#89722—21no—
50#89708C2211yescharacteristic
51#89707—21no—
52#89707C2211yescharacteristic
53C4×C4⋊Dic7—21no—
54C4×C4⋊Dic7C2211yescharacteristic
55C2×C4⋊(C7⋊C8)C2211yescharacteristic
56#89345C2211yescharacteristic
57#89345C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#22678—71nomaximal, Sylow
63C22×C7×Q8C2311yescharacteristic
64C22×C7×D4C2311yescharacteristic
65C23×C4×C7—21no—
66C23×Dic7—21no—
67C23×Dic7—21no—
68C2×C7×SD16C2×C414yes—
69C2×C7×SD16D414yes—
70C2×C7×SD16Q814yes—
71C22×C7×C8C2311yescharacteristic
72C22×C7×C8C2311yescharacteristic
73C4×C7×Q8—22no—
74C4×C7×Q8C2×C414yescomplemented
75C4×C7×D4—22no—
76C4×C7×D4C2×C414yes—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4—21no—
81C2×C42×C7—21no—
82C2×C42×C7C2311yescharacteristic
83C2×C22⋊Dic7—21no—
84C2×C22⋊Dic7—21no—
85Q8×Dic7—22no—
86Q8×Dic7—22no—
87Q8×Dic7—41no—
88Q8×Dic7—41no—
89D4×Dic7—22no—
90D4×Dic7—22no—
91D4×Dic7—41no—
92D4×Dic7—41no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7—21no—
96C2×C4⋊Dic7—21no—
97C2×C4⋊Dic7—21no—
98C2×C4⋊Dic7—21no—
99C2×C4⋊Dic7—21no—
100C2×C4⋊Dic7—21no—
101C2×C4⋊Dic7D411yescharacteristic
102C2×C4⋊Dic7D411yescharacteristic
103C2×C4⋊Dic7C2311yescharacteristic
104C2×C4⋊Dic7C2311yescharacteristic
105C2×C4⋊Dic7C2311yescharacteristic
106C2×C4⋊Dic7C2311yescharacteristic
107C2×C4×Dic7—21no—
108C2×C4×Dic7—21no—
109C2×C4×Dic7C2311yescharacteristic
110C2×C4×Dic7C2311yescharacteristic
111C22×C7⋊C8C2311yescharacteristic
112C22×C7⋊C8C2311yescharacteristic
113C7×#107C2×C414yescomplemented
114C7×#103C2×C414yescomplemented
115C7×#102C2×C414yes—
116C7×#97C2×C414yescomplemented
117#6409—22no—
118#6409—22no—
119#6406—22no—
120#6406—22no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95D411yescharacteristic
128C7⋊#95D411yescharacteristic
129#6392—22no—
130#6392—41no—
131#6391—22no—
132#6391—41no—
133#6389—22no—
134C8×Dic7—22no—
135C4⋊(C7⋊C8)—22no—
136C2×#440—71no—
137C2×#440D711yescharacteristic
138C2×C4×SD16—71no—
139Q8×C4⋊C4—71no—
140#2349—71no—
141#1943—71no—
142#1942—71no—
143#1865—71no—
144#1864—71no—
145#1863—71no—
146#1862—71no—
147#1847—71no—
148#1805—71no—
149#1802—71no—
150#1771—71no—
151C24×C7—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8C22×C412yes—
156C2×C7×Q8C2×D412yes—
157C2×C7×Q8C2×Q812yes—
158C2×C7×D4—21no—
159C2×C7×D4—21no—
160C2×C7×D4—21no—
161C2×C7×D4C22×C412yes—
162C2×C7×D4C2×D412yes—
163C2×C7×D4C2×Q812yes—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—24no—
168C22×C4×C7—81no—
169C22×C4×C7—81no—
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2411yescharacteristic
173C22×Dic7—21no—
174C22×Dic7—21no—
175C22×Dic7—21no—
176C22×Dic7—21no—
177C22×Dic7—42no—
178C22×Dic7—42no—
179C22×Dic7—42no—
180C22×Dic7—42no—
181C22×Dic7—81no—
182C22×Dic7—81no—
183C22×Dic7C2×D411yescharacteristic
184C22×Dic7C2×D411yescharacteristic
185C22×Dic7C2×D411yescharacteristic
186C22×Dic7C2×D411yescharacteristic
187C7×SD16C4⋊C4116yescomplemented
188C2×C7×C8—21no—
189C2×C7×C8—21no—
190C2×C7×C8C22×C412yes—
191C2×C7×C8C22×C412yes—
192C2×C7×C8C2×D412yes—
193C2×C7×C8C2×Q812yes—
194C7×C4⋊C4—24no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4C22×C412yes—
198C7×C4⋊C4C22×C412yes—
199C7×C22⋊C4—24no—
200C42×C7—24no—
201C42×C7C22×C412yes—
202C42×C7#4112yes—
203C22⋊Dic7—42no—
204C22⋊Dic7—42no—
205C4⋊Dic7—22no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—41no—
215C4⋊Dic7—42no—
216C4⋊Dic7—42no—
217C4⋊Dic7—42no—
218C4⋊Dic7—81no—
219C4⋊Dic7—81no—
220C4⋊Dic7—81no—
221C4⋊Dic7—81no—
222C4⋊Dic7—81no—
223C4×Dic7—21no—
224C4×Dic7—21no—
225C4×Dic7—21no—
226C4×Dic7—21no—
227C4×Dic7—42no—
228C4×Dic7—42no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C22×SD16C2×D711yescharacteristic
236C2×C4×Q8—71no—
237C2×C4×Q8—71no—
238C2×C4×Q8C2×D711yescharacteristic
239C2×C4×D4—71no—
240C2×C4×D4—71no—
241C2×C4×D4C2×D711yescharacteristic
242#440—144no—
243#440Dic718yescomplemented
244C4×SD16—144no—
245C2×#107—71no—
246C2×#107—71no—
247C2×#107C2×D711yescharacteristic
248C2×#106—71no—
249C2×#106—71no—
250C2×C4⋊C8—71no—
251C2×#103—71no—
252C2×#103—71no—
253C2×#103C2×D711yescharacteristic
254C2×#102—71no—
255C2×#102—71no—
256C2×#102C2×D711yescharacteristic
257C2×#97—71no—
258C2×#97C2×D711yescharacteristic
259C2×C4×C8—71no—
260#389—71no—
261#389—71no—
262#389—71no—
263#389—71no—
264#389—141no—
265#389—141no—
266#385—141no—
267#385—141no—
268#384—71no—
269#384—141no—
270#382—71no—
271#380—141no—
272C4×C4⋊C4—71no—
273C4×C4⋊C4—141no—
274C2×#95—141no—
275#340—71no—
276#340—71no—
277#340—71no—
278#340—71no—
279#336—71no—
280#336—71no—
281C23×C7—22no—
282C23×C7—22no—
283C23×C7—22no—
284C23×C7—41no—
285C23×C7—41no—
286C23×C7—41no—
287C23×C7—41no—
288C23×C7C22×D411yescharacteristic
289C7×Q8—22no—
290C7×Q8—22no—
291C7×Q8—22no—
292C7×Q8C2×C4⋊C414yes—
293C7×D4—22no—
294C7×D4—22no—
295C7×D4—22no—
296C7×D4C2×C4⋊C414yes—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—22no—
305C2×C4×C7—22no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—84no—
311C2×C4×C7C4×D412yes—
312C2×C4×C7C4×D412yes—
313C2×C4×C7C23×C411yescharacteristic
314C2×C4×C7C22×D411yescharacteristic
315C2×C4×C7C22×Q811yescharacteristic
316C2×C4×C7C2×#4111yescharacteristic
317C2×C4×C7H(5,𝔽2)11yescharacteristic
318C2×C4×C72−1+411yescharacteristic
319C2×Dic7—21no—
320C2×Dic7—21no—
321C2×Dic7—21no—
322C2×Dic7—21no—
323C2×Dic7—21no—
324C2×Dic7—21no—
325C2×Dic7—21no—
326C2×Dic7—21no—
327C2×Dic7—21no—
328C2×Dic7—21no—
329C2×Dic7—21no—
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—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C2×Dic7—41no—
342C2×Dic7—84no—
343C2×Dic7—84no—
344C2×Dic7#13712yes—
345C7×C8—22no—
346C7×C8C2×C4⋊C414yes—
347C7⋊C8—41no—
348C7⋊C8—41no—
349C22×Q8C22×D711yescharacteristic
350C22×D4C22×D711yescharacteristic
351C23×C4—21no—
352C23×C4—141no—
353C23×C4—141no—
354C2×SD16Dic1414yes—
355C2×SD16D2814yes—
356C2×SD16C2×Dic714yes—
357C22×C8—71no—
358C22×C8—71no—
359C22×C8C22×D711yescharacteristic
360C22×C8C22×D711yescharacteristic
361C4×Q8—22no—
362C4×Q8—142no—
363C4×Q8—142no—
364C4×Q8—281no—
365C4×Q8—281no—
366C4×Q8C2×Dic714yescomplemented
367C4×D4—22no—
368C4×D4—142no—
369C4×D4—142no—
370C4×D4—281no—
371C4×D4—281no—
372C4×D4C2×Dic714yes—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—71no—
379C2×C4⋊C4—71no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—141no—
383C2×C4⋊C4—141no—
384C2×C4⋊C4—141no—
385C2×C4⋊C4—141no—
386C2×C4⋊C4—141no—
387C2×C4⋊C4—141no—
388C2×C4⋊C4C22×D711yescharacteristic
389C2×C4⋊C4C22×D711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—141no—
392C2×C22⋊C4—141no—
393C2×C42—21no—
394C2×C42—71no—
395C2×C42—71no—
396C2×C42—141no—
397C2×C42—141no—
398C2×C42C22×D711yescharacteristic
399#107—142no—
400#107—281no—
401#107C2×Dic714yescomplemented
402#106—142no—
403#106—281no—
404C4⋊C8—142no—
405#103—142no—
406#103—142no—
407#103C2×Dic714yescomplemented
408#102—142no—
409#102—142no—
410#102C2×Dic714yes—
411#97—142no—
412#97C2×Dic714yescomplemented
413C4×C8—142no—
414#95—71no—
415#95—71no—
416#95—141no—
417#95—141no—
418#95—141no—
419#95—141no—
420#95—141no—
421#95—141no—
422C22×C7—24no—
423C22×C7—42no—
424C22×C7—42no—
425C22×C7—42no—
426C22×C7—42no—
427C22×C7—42no—
428C22×C7—42no—
429C22×C7C2×C4×D411yescharacteristic
430C22×C7D4211yescharacteristic
431C22×C7D4×Q811yescharacteristic
432C22×C7C2×#13711yescharacteristic
433C22×C7#57511yescharacteristic
434C22×C7#57611yescharacteristic
435C22×C7#57711yescharacteristic
436C4×C7—24no—
437C4×C7—22no—
438C4×C7—22no—
439C4×C7—42no—
440C4×C7—42no—
441C4×C7C22×C4⋊C411yescharacteristic
442C4×C7#52011yescharacteristic
443C4×C7#54711yescharacteristic
444C4×C7#55411yescharacteristic
445Dic7—22no—
446Dic7—41no—
447Dic7—41no—
448Dic7—41no—
449Dic7—81no—
450Dic7—81no—
451Dic7—81no—
452Dic7—81no—
453C24—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8C2×Dic1412yes—
458C2×Q8C2×D2812yes—
459C2×Q8C22×Dic712yes—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4C2×Dic1412yes—
464C2×D4C2×D2812yes—
465C2×D4C22×Dic712yes—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—24no—
470C22×C4—71no—
471C22×C4—71no—
472C22×C4—71no—
473C22×C4—71no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4—141no—
477C22×C4—141no—
478C22×C4—141no—
479C22×C4—141no—
480C22×C4—282no—
481C22×C4—282no—
482C22×C4—282no—
483C22×C4—282no—
484C22×C4—561no—
485C22×C4—561no—
486C22×C4D4×D711yescharacteristic
487C22×C4D4×D711yescharacteristic
488C22×C4C23×D711yescharacteristic
489SD16C4⋊Dic7116yescomplemented
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×C8C2×Dic1412yes—
499C2×C8C2×D2812yes—
500C2×C8C22×Dic712yes—
501C2×C8C22×Dic712yes—
502C4⋊C4—24no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—142no—
506C4⋊C4—141no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—141no—
510C4⋊C4—141no—
511C4⋊C4—141no—
512C4⋊C4—141no—
513C4⋊C4—141no—
514C4⋊C4—281no—
515C4⋊C4—282no—
516C4⋊C4—282no—
517C4⋊C4—282no—
518C4⋊C4—561no—
519C4⋊C4—561no—
520C4⋊C4—561no—
521C4⋊C4—561no—
522C4⋊C4—561no—
523C4⋊C4C22×Dic712yes—
524C4⋊C4C22×Dic712yes—
525C22⋊C4—24no—
526C22⋊C4—282no—
527C22⋊C4—282no—
528C42—24no—
529C42—141no—
530C42—141no—
531C42—141no—
532C42—141no—
533C42—282no—
534C42—282no—
535C42#115912yes—
536C42C22×Dic712yes—
537C2×C7—44no—
538C2×C7D4×C4⋊C411yescharacteristic
539C2×C7C4×#13711yescharacteristic
540C2×C7#294611yescharacteristic
541C2×C7#326611yescharacteristic
542C2×C7#326811yescharacteristic
543C2×C7#338511yescharacteristic
544C2×C7#338811yescharacteristic
545C23—22no—
546C23—22no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23C2×D4×D711yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8C2×C4⋊Dic714yes—
557D4—22no—
558D4—22no—
559D4—22no—
560D4C2×C4⋊Dic714yes—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—72no—
575C2×C4—84no—
576C2×C4—141no—
577C2×C4—141no—
578C2×C4—141no—
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—281no—
588C2×C4—281no—
589C2×C4—281no—
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—564no—
600C2×C4—564no—
601C2×C4D4×Dic712yes—
602C2×C4D4×Dic712yes—
603C2×C4C22×Dic1411yescharacteristic
604C2×C4C22×D2811yescharacteristic
605C2×C4C2×#115911yescharacteristic
606C2×C4#655311yescharacteristic
607C2×C4#655411yescharacteristic
608C2×C4C23×Dic711yescharacteristic
609C8—22no—
610C8—281no—
611C8—281no—
612C8C2×C4⋊Dic714yes—
613C7#2267811yescharacteristic, complemented, Sylow
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22D4×Dic1411yescharacteristic
622C22D4×D2811yescharacteristic
623C22#9046511yescharacteristic
624C22#9046611yescharacteristic
625C22D7×#13711yescharacteristic
626C22#9046811yescharacteristic
627C22C2×D4×Dic711yescharacteristic
628C4—24no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—42no—
633C4—142no—
634C4—281no—
635C4—281no—
636C4—281no—
637C4—561no—
638C4—561no—
639C4—561no—
640C4—561no—
641C4#9023411yescharacteristic
642C4#9024611yescharacteristic
643C4C22×C4⋊Dic711yescharacteristic
644C4#9051811yescharacteristic
645C2—44no—
646C2#1173720411yescharacteristic
647C2#1173720611yescharacteristic
648C2#1173724011yescharacteristic
649C2#1173724111yescharacteristic
650C2D4×C4⋊Dic711yescharacteristic
651C2#1174137511yescharacteristic
652C2Dic7×#13711yescharacteristic
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#22678
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
4. Q8×C4⋊Dic7C2semidirect
6. #11729381C2semidirect
8. #11728367C2semidirect
9. #11728366C2semidirect
10. #11727992C2semidirect
11. #11727990C2semidirect
14. #11727570C2semidirect
15. #11727564C2semidirect
23. C7×#440C4semidirect
74. C4×C7×Q8C2×C4semidirect
113. C7×#107C2×C4semidirect
114. C7×#103C2×C4semidirect
116. C7×#97C2×C4semidirect
187. C7×SD16C4⋊C4semidirect
243. #440Dic7semidirect
366. C4×Q8C2×Dic7semidirect
401. #107C2×Dic7semidirect
407. #103C2×Dic7semidirect
412. #97C2×Dic7semidirect
489. SD16C4⋊Dic7semidirect
613. C7#22678semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4402
C2×#899362
C2×SD16×Dic72
Q8×C4⋊Dic72
#117397782
#117293812
#117293582
#117283672
#117283662
#117279922
#117279902
#117279242
#117279222
#117275702
#117275642
#226787