Finite Group Databasebeta

← #13372212

Subgroups — #13372212

Order 1792.

986 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×SD16C211yesmaximal, characteristic
2C2×SD16×Dic7C211yesmaximal, characteristic
3C2×SD16×Dic7C211yesmaximal, characteristic
4#11739906C211yesmaximal, characteristic, complemented
5#11739779C211yesmaximal, characteristic
6#11729382C211yesmaximal, characteristic, complemented
7#11729360C211yesmaximal, characteristic
8#11728313C211yesmaximal, characteristic, complemented
9#11728309C211yesmaximal, characteristic, complemented
10#11727989C211yesmaximal, characteristic, complemented
11#11727976C211yesmaximal, characteristic, complemented
12#11727921C211yesmaximal, characteristic
13#11727908C211yesmaximal, characteristic
14C8×Dic7⋊C4C211yesmaximal, characteristic, complemented
15#11727523C211yesmaximal, 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
23C4×C7×SD16—24no—
24C2×C7×#106C2211yescharacteristic
25C2×C7×#103C2211yescharacteristic
26C2×C7×#102C2211yescharacteristic
27C2×C4×C7×C8C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32#89982—21no—
33C2×C7⋊#95—21no—
34SD16×Dic7—24no—
35SD16×Dic7C418yescomplemented
36C2×#6391C2211yescharacteristic
37C2×#6391C2211yescharacteristic
38C2×Dic7⋊C8C2211yescharacteristic
39C2×C8×Dic7C2211yescharacteristic
40C2×C8×Dic7C2211yescharacteristic
41#89749—21no—
42#89749C2211yescharacteristic
43#89749C2211yescharacteristic
44#89745—21no—
45#89745C2211yescharacteristic
46#89745C2211yescharacteristic
47C2×#6381C2211yescharacteristic
48C2×#6381C2211yescharacteristic
49#89718—21no—
50#89714—21no—
51C4×Dic7⋊C4—21no—
52C4×Dic7⋊C4C2211yescharacteristic
53#89700—21no—
54#89700C2211yescharacteristic
55#89700C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89345C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89319C2211yescharacteristic
62#22683—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×Q8—41no—
75C4×C7×D4—22no—
76C4×C7×D4—41no—
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×Dic7C2×C414yescomplemented
89D4×Dic7—22no—
90D4×Dic7—22no—
91D4×Dic7—41no—
92D4×Dic7C2×C414yes—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4⋊Dic7C2311yescharacteristic
99C2×Dic7⋊C4—21no—
100C2×Dic7⋊C4—21no—
101C2×Dic7⋊C4—21no—
102C2×Dic7⋊C4—21no—
103C2×Dic7⋊C4D411yescharacteristic
104C2×Dic7⋊C4D411yescharacteristic
105C2×C4×Dic7—21no—
106C2×C4×Dic7—21no—
107C2×C4×Dic7C2311yescharacteristic
108C2×C4×Dic7C2311yescharacteristic
109C22×C7⋊C8C2311yescharacteristic
110C22×C7⋊C8C2311yescharacteristic
111C7×#106—22no—
112C7×#103—22no—
113C7×#102—22no—
114C4×C7×C8—22no—
115#6409—22no—
116#6409C2×C414yescomplemented
117#6406—22no—
118#6406C2×C414yes—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95D411yescharacteristic
128C7⋊#95D411yescharacteristic
129#6391—22no—
130#6391C2×C414yescomplemented
131Dic7⋊C8—22no—
132C8×Dic7—22no—
133C8×Dic7C2×C414yescomplemented
134#6381—41no—
135#6381—41no—
136C2×C4×SD16—71no—
137C2×C4×SD16—71no—
138C2×C4×SD16D711yescharacteristic
139#2351—71no—
140#2349—71no—
141#1941—71no—
142#1941—71no—
143#1863—71no—
144#1863—71no—
145#1862—71no—
146#1862—71no—
147#1846—71no—
148#1805—71no—
149#1804—71no—
150C8×C4⋊C4—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—42no—
168C22×C4×C7—42no—
169C22×C4×C7—81no—
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2411yescharacteristic
173C22×Dic7—21no—
174C22×Dic7—21no—
175C22×Dic7—24no—
176C22×Dic7—21no—
177C22×Dic7—21no—
178C22×Dic7—42no—
179C22×Dic7—42no—
180C22×Dic7—81no—
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×C8—21no—
191C2×C7×C8C22×C412yes—
192C2×C7×C8C2×D412yes—
193C2×C7×C8C2×Q812yes—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—42no—
199C7×C22⋊C4—42no—
200C42×C7—21no—
201C42×C7—21no—
202C42×C7—42no—
203C22⋊Dic7—24no—
204C22⋊Dic7—42no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—24no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—42no—
213C4⋊Dic7C22×C412yes—
214C4⋊Dic7C22×C412yes—
215Dic7⋊C4—22no—
216Dic7⋊C4—22no—
217Dic7⋊C4—42no—
218Dic7⋊C4—42no—
219Dic7⋊C4—81no—
220Dic7⋊C4—81no—
221C4×Dic7—21no—
222C4×Dic7—21no—
223C4×Dic7—24no—
224C4×Dic7—42no—
225C4×Dic7C22×C412yes—
226C4×Dic7#4112yes—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8C22×C412yes—
233C22×SD16C2×D711yescharacteristic
234C2×C4×Q8—71no—
235C2×C4×Q8—71no—
236C2×C4×Q8C2×D711yescharacteristic
237C2×C4×D4—71no—
238C2×C4×D4—71no—
239C2×C4×D4C2×D711yescharacteristic
240C4×SD16—24no—
241C4×SD16—78no—
242C4×SD16—144no—
243C2×#107—71no—
244C2×#107—71no—
245C2×#106—71no—
246C2×#106—71no—
247C2×#106C2×D711yescharacteristic
248C2×C4⋊C8—71no—
249C2×#103—71no—
250C2×#103—71no—
251C2×#103C2×D711yescharacteristic
252C2×#102—71no—
253C2×#102—71no—
254C2×#102C2×D711yescharacteristic
255C2×C4×C8—71no—
256C2×C4×C8—71no—
257C2×C4×C8C2×D711yescharacteristic
258#389—71no—
259#389—71no—
260#389—71no—
261#389—71no—
262#389—141no—
263#389—141no—
264#385—141no—
265#385—141no—
266#382—71no—
267#382—71no—
268#382—141no—
269#380—141no—
270C4×C4⋊C4—71no—
271C4×C4⋊C4—141no—
272C2×#95—141no—
273#340—71no—
274#340—71no—
275#340—71no—
276#340—71no—
277#336—71no—
278#336—71no—
279C23×C7—22no—
280C23×C7—22no—
281C23×C7—22no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7—41no—
285C23×C7—41no—
286C23×C7C22×D411yescharacteristic
287C7×Q8—22no—
288C7×Q8—22no—
289C7×Q8—22no—
290C7×Q8C2×C4⋊C414yes—
291C7×D4—22no—
292C7×D4—22no—
293C7×D4—22no—
294C7×D4C2×C4⋊C414yes—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—21no—
298C2×C4×C7—22no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—84no—
310C2×C4×C7#13512yes—
311C2×C4×C7C23×C411yescharacteristic
312C2×C4×C7C22×D411yescharacteristic
313C2×C4×C7C22×Q811yescharacteristic
314C2×C4×C7C2×#4111yescharacteristic
315C2×C4×C7H(5,𝔽2)11yescharacteristic
316C2×C4×C72−1+411yescharacteristic
317C2×Dic7—21no—
318C2×Dic7—21no—
319C2×Dic7—21no—
320C2×Dic7—21no—
321C2×Dic7—21no—
322C2×Dic7—21no—
323C2×Dic7—22no—
324C2×Dic7—22no—
325C2×Dic7—21no—
326C2×Dic7—21no—
327C2×Dic7—21no—
328C2×Dic7—41no—
329C2×Dic7—41no—
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—84no—
339C2×Dic7—84no—
340C2×Dic7C4×D412yes—
341C2×Dic7C4×D412yes—
342C2×Dic7#13512yes—
343C7×C8—41no—
344C7×C8C2×C4⋊C414yes—
345C7⋊C8—22no—
346C7⋊C8—41no—
347C22×Q8C22×D711yescharacteristic
348C22×D4C22×D711yescharacteristic
349C23×C4—21no—
350C23×C4—141no—
351C23×C4—141no—
352C2×SD16Dic1414yes—
353C2×SD16C4×D714yes—
354C2×SD16C22⋊D714yes—
355C22×C8—71no—
356C22×C8—71no—
357C22×C8C22×D711yescharacteristic
358C22×C8C22×D711yescharacteristic
359C4×Q8—22no—
360C4×Q8—41no—
361C4×Q8—74no—
362C4×Q8—142no—
363C4×Q8—142no—
364C4×Q8—281no—
365C4×D4—22no—
366C4×D4—41no—
367C4×D4—74no—
368C4×D4—142no—
369C4×D4—142no—
370C4×D4—281no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—71no—
373C2×C4⋊C4—71no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—141no—
383C2×C4⋊C4—141no—
384C2×C4⋊C4C22×D711yescharacteristic
385C2×C4⋊C4C22×D711yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—141no—
388C2×C22⋊C4—141no—
389C2×C42—21no—
390C2×C42—71no—
391C2×C42—71no—
392C2×C42—141no—
393C2×C42—141no—
394C2×C42C22×D711yescharacteristic
395#107—281no—
396#107—281no—
397#106—22no—
398#106—74no—
399#106—142no—
400C4⋊C8—142no—
401#103—22no—
402#103—74no—
403#103—142no—
404#102—22no—
405#102—74no—
406#102—142no—
407C4×C8—22no—
408C4×C8—74no—
409C4×C8—142no—
410#95—71no—
411#95—71no—
412#95—141no—
413#95—141no—
414#95—141no—
415#95—141no—
416#95—141no—
417#95—141no—
418#95—141no—
419#95—141no—
420C22×C7—24no—
421C22×C7—42no—
422C22×C7—42no—
423C22×C7—42no—
424C22×C7—42no—
425C22×C7—42no—
426C22×C7—42no—
427C22×C7C2×C4×D411yescharacteristic
428C22×C7D4211yescharacteristic
429C22×C7D4×Q811yescharacteristic
430C22×C7C2×#13511yescharacteristic
431C22×C7C2×#13511yescharacteristic
432C22×C7#57711yescharacteristic
433C22×C7#57711yescharacteristic
434C4×C7—24no—
435C4×C7—22no—
436C4×C7—41no—
437C4×C7—81no—
438C4×C7—81no—
439C4×C7C22×C4⋊C411yescharacteristic
440C4×C7#52011yescharacteristic
441C4×C7#54711yescharacteristic
442C4×C7#55411yescharacteristic
443Dic7—22no—
444Dic7—22no—
445Dic7—22no—
446Dic7—42no—
447Dic7—42no—
448Dic7—41no—
449Dic7—81no—
450Dic7—81no—
451C24—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8C2×Dic1412yes—
456C2×Q8C2×C4×D712yes—
457C2×Q8C2×C22⋊D712yes—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4C2×Dic1412yes—
462C2×D4C2×C4×D712yes—
463C2×D4C2×C22⋊D712yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—71no—
470C22×C4—71no—
471C22×C4—71no—
472C22×C4—71no—
473C22×C4—81no—
474C22×C4—144no—
475C22×C4—141no—
476C22×C4—141no—
477C22×C4—141no—
478C22×C4—141no—
479C22×C4—282no—
480C22×C4—282no—
481C22×C4—561no—
482C22×C4—561no—
483C22×C4—561no—
484C22×C4D4×D711yescharacteristic
485C22×C4D4×D711yescharacteristic
486C22×C4C23×D711yescharacteristic
487SD16Dic7⋊C4116yescomplemented
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—72no—
492C2×C8—141no—
493C2×C8—141no—
494C2×C8—141no—
495C2×C8—141no—
496C2×C8—141no—
497C2×C8C2×Dic1412yes—
498C2×C8C2×C4×D712yes—
499C2×C8C2×C22⋊D712yes—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—72no—
506C4⋊C4—72no—
507C4⋊C4—142no—
508C4⋊C4—144no—
509C4⋊C4—142no—
510C4⋊C4—141no—
511C4⋊C4—141no—
512C4⋊C4—141no—
513C4⋊C4—141no—
514C4⋊C4—141no—
515C4⋊C4—141no—
516C4⋊C4—282no—
517C4⋊C4—282no—
518C4⋊C4—282no—
519C4⋊C4—561no—
520C4⋊C4—561no—
521C22⋊C4—42no—
522C22⋊C4—144no—
523C22⋊C4—282no—
524C42—21no—
525C42—21no—
526C42—42no—
527C42—72no—
528C42—72no—
529C42—144no—
530C42—141no—
531C42—141no—
532C42—282no—
533C2×C7—44no—
534C2×C7D4×C4⋊C411yescharacteristic
535C2×C7C4×#13511yescharacteristic
536C2×C7#294911yescharacteristic
537C2×C7#328211yescharacteristic
538C2×C7#328211yescharacteristic
539C2×C7#338111yescharacteristic
540C2×C7#338111yescharacteristic
541C23—22no—
542C23—22no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C2×D4×D711yescharacteristic
549Q8—22no—
550Q8—22no—
551Q8—22no—
552Q8C2×Dic7⋊C414yes—
553D4—22no—
554D4—22no—
555D4—22no—
556D4C2×Dic7⋊C414yes—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—72no—
572C2×C4—72no—
573C2×C4—72no—
574C2×C4—84no—
575C2×C4—141no—
576C2×C4—141no—
577C2×C4—141no—
578C2×C4—141no—
579C2×C4—141no—
580C2×C4—141no—
581C2×C4—141no—
582C2×C4—141no—
583C2×C4—142no—
584C2×C4—142no—
585C2×C4—141no—
586C2×C4—281no—
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—564no—
597C2×C4—564no—
598C2×C4#647912yes—
599C2×C4C22×Dic1411yescharacteristic
600C2×C4C22×C4×D711yescharacteristic
601C2×C4#654811yescharacteristic
602C2×C4D7×#4111yescharacteristic
603C2×C4#655411yescharacteristic
604C2×C4C22×C22⋊D711yescharacteristic
605C8—41no—
606C8—142no—
607C8—281no—
608C8C2×Dic7⋊C414yes—
609C7#2268311yescharacteristic, complemented, Sylow
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22D4×Dic1411yescharacteristic
618C22C4×D4×D711yescharacteristic
619C22C2×#647911yescharacteristic
620C22#9045311yescharacteristic
621C22D7×#13511yescharacteristic
622C22#9046111yescharacteristic
623C22D4×C22⋊D711yescharacteristic
624C4—22no—
625C4—24no—
626C4—41no—
627C4—81no—
628C4—81no—
629C4—142no—
630C4—142no—
631C4—142no—
632C4—282no—
633C4—282no—
634C4—281no—
635C4—561no—
636C4—561no—
637C4#9023011yescharacteristic
638C4#9023711yescharacteristic
639C4C22×Dic7⋊C411yescharacteristic
640C4#9051511yescharacteristic
641C2—44no—
642C2#1173707711yescharacteristic
643C2#1173707811yescharacteristic
644C2C4×#647911yescharacteristic
645C2#1173709411yescharacteristic
646C2D4×Dic7⋊C411yescharacteristic
647C2#1174131211yescharacteristic
648C2#1174136411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C4×C7×SD16C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC2×C4×SD16D7
hypocenterC7#22683
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
4. #11739906C2semidirect
6. #11729382C2semidirect
8. #11728313C2semidirect
9. #11728309C2semidirect
10. #11727989C2semidirect
11. #11727976C2semidirect
14. C8×Dic7⋊C4C2semidirect
15. #11727523C2semidirect
35. SD16×Dic7C4semidirect
88. Q8×Dic7C2×C4semidirect
116. #6409C2×C4semidirect
130. #6391C2×C4semidirect
133. C8×Dic7C2×C4semidirect
187. C7×SD16C4⋊C4semidirect
487. SD16Dic7⋊C4semidirect
609. C7#22683semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×SD162
C2×SD16×Dic72
#117399062
#117397792
#117293822
#117293602
#117283132
#117283092
#117279892
#117279762
#117279212
#117279082
C8×Dic7⋊C42
#117275232
#226837