Finite Group Databasebeta

← #13372079

Subgroups — #13372079

Order 1792.

1018 conjugacy classes of subgroups.

The subgroup lattice has 660 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×Dic7⋊C4C211yesmaximal, characteristic, complemented
5D4×Dic7⋊C4C211yesmaximal, characteristic
6#11729382C211yesmaximal, characteristic, complemented
7#11729360C211yesmaximal, characteristic
8#11728368C211yesmaximal, characteristic, complemented
9#11728360C211yesmaximal, characteristic, complemented
10#11727989C211yesmaximal, characteristic, complemented
11#11727975C211yesmaximal, characteristic, complemented
12#11727921C211yesmaximal, characteristic
13#11727907C211yesmaximal, characteristic
14#11727587C211yesmaximal, characteristic, complemented
15#11727574C211yesmaximal, 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
23C22×Dic7⋊C4—21no—
24C7×#440—24no—
25C2×C7×#107C2211yescharacteristic
26C2×C7×#103C2211yescharacteristic
27C2×C7×#102C2211yescharacteristic
28C2×C7×#97C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33#89982—21no—
34#89936—24no—
35SD16×Dic7C418yescomplemented
36C2×#6392C2211yescharacteristic
37C2×#6391C2211yescharacteristic
38C2×#6389C2211yescharacteristic
39C2×Dic7⋊C8C2211yescharacteristic
40C2×C8×Dic7C2211yescharacteristic
41#89748—21no—
42#89748C2211yescharacteristic
43#89748C2211yescharacteristic
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#22671—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⋊C4—21no—
104C2×Dic7⋊C4—21no—
105C2×Dic7⋊C4—21no—
106C2×Dic7⋊C4—21no—
107C2×Dic7⋊C4—44no—
108C2×Dic7⋊C4D411yescharacteristic
109C2×Dic7⋊C4D411yescharacteristic
110C2×C4×Dic7—21no—
111C2×C4×Dic7—21no—
112C2×C4×Dic7C2311yescharacteristic
113C2×C4×Dic7C2311yescharacteristic
114C22×C7⋊C8C2311yescharacteristic
115C22×C7⋊C8C2311yescharacteristic
116C7×#107—22no—
117C7×#103—22no—
118C7×#102—22no—
119C7×#97—22no—
120#6409—22no—
121#6409C2×C414yescomplemented
122#6406—22no—
123#6406C2×C414yes—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95—21no—
128C7⋊#95D411yescharacteristic
129C7⋊#95D411yescharacteristic
130#6392—22no—
131#6391C2×C414yescomplemented
132#6389—22no—
133Dic7⋊C8—22no—
134C8×Dic7C2×C414yescomplemented
135#6381—41no—
136#6381—41no—
137C2×#440—71no—
138C2×#440D711yescharacteristic
139C2×C4×SD16—71no—
140Q8×C4⋊C4—71no—
141D4×C4⋊C4—71no—
142#1944—71no—
143#1944—71no—
144#1863—71no—
145#1863—71no—
146#1862—71no—
147#1862—71no—
148#1845—71no—
149#1803—71no—
150#1802—71no—
151#1770—71no—
152C24×C7—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8C22×C412yes—
157C2×C7×Q8C2×D412yes—
158C2×C7×Q8C2×Q812yes—
159C2×C7×D4—21no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4C22×C412yes—
163C2×C7×D4C2×D412yes—
164C2×C7×D4C2×Q812yes—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—42no—
169C22×C4×C7—42no—
170C22×C4×C7—81no—
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2411yescharacteristic
174C22×Dic7—21no—
175C22×Dic7—21no—
176C22×Dic7—24no—
177C22×Dic7—21no—
178C22×Dic7—21no—
179C22×Dic7—42no—
180C22×Dic7—42no—
181C22×Dic7—42no—
182C22×Dic7—42no—
183C22×Dic7—81no—
184C22×Dic7C2×D411yescharacteristic
185C22×Dic7C2×D411yescharacteristic
186C22×Dic7C2×D411yescharacteristic
187C22×Dic7C2×D411yescharacteristic
188C7×SD16C4⋊C4116yescomplemented
189C2×C7×C8—21no—
190C2×C7×C8—21no—
191C2×C7×C8—21no—
192C2×C7×C8C22×C412yes—
193C2×C7×C8C2×D412yes—
194C2×C7×C8C2×Q812yes—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—42no—
200C7×C22⋊C4—42no—
201C42×C7—21no—
202C42×C7—21no—
203C42×C7—42no—
204C22⋊Dic7—24no—
205C22⋊Dic7—42no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—24no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—42no—
214C4⋊Dic7C22×C412yes—
215C4⋊Dic7C22×C412yes—
216Dic7⋊C4—22no—
217Dic7⋊C4—22no—
218Dic7⋊C4—42no—
219Dic7⋊C4—42no—
220Dic7⋊C4—42no—
221Dic7⋊C4—42no—
222Dic7⋊C4—81no—
223Dic7⋊C4—81no—
224Dic7⋊C4—81no—
225Dic7⋊C4—81no—
226C4×Dic7—21no—
227C4×Dic7—21no—
228C4×Dic7—24no—
229C4×Dic7—42no—
230C4×Dic7C22×C412yes—
231C4×Dic7#4112yes—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—21no—
236C2×C7⋊C8—21no—
237C2×C7⋊C8C22×C412yes—
238C22×SD16C2×D711yescharacteristic
239C2×C4×Q8—71no—
240C2×C4×Q8—71no—
241C2×C4×Q8C2×D711yescharacteristic
242C2×C4×D4—71no—
243C2×C4×D4—71no—
244C2×C4×D4C2×D711yescharacteristic
245C22×C4⋊C4—141no—
246#440—24no—
247#440—144no—
248C4×SD16—78no—
249C2×#107—71no—
250C2×#107—71no—
251C2×#107—71no—
252C2×#107C2×D711yescharacteristic
253C2×#106—71no—
254C2×C4⋊C8—71no—
255C2×#103—71no—
256C2×#103—71no—
257C2×#103C2×D711yescharacteristic
258C2×#102—71no—
259C2×#102—71no—
260C2×#102C2×D711yescharacteristic
261C2×#97—71no—
262C2×#97C2×D711yescharacteristic
263C2×C4×C8—71no—
264#389—71no—
265#389—71no—
266#389—71no—
267#389—71no—
268#389—141no—
269#389—141no—
270#385—141no—
271#385—141no—
272#384—71no—
273#384—71no—
274#384—141no—
275#380—141no—
276C4×C4⋊C4—71no—
277C4×C4⋊C4—141no—
278#340—71no—
279#340—71no—
280#340—71no—
281#340—71no—
282#336—71no—
283#336—71no—
284C23×C7—22no—
285C23×C7—22no—
286C23×C7—22no—
287C23×C7—41no—
288C23×C7—41no—
289C23×C7—41no—
290C23×C7—41no—
291C23×C7C22×D411yescharacteristic
292C7×Q8—22no—
293C7×Q8—22no—
294C7×Q8—22no—
295C7×Q8C2×C4⋊C414yescomplemented
296C7×D4—22no—
297C7×D4—22no—
298C7×D4—22no—
299C7×D4C2×C4⋊C414yes—
300C2×C4×C7—22no—
301C2×C4×C7—22no—
302C2×C4×C7—22no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—84no—
316C2×C4×C7C23×C411yescharacteristic
317C2×C4×C7C22×D411yescharacteristic
318C2×C4×C7C22×Q811yescharacteristic
319C2×C4×C7C2×#4111yescharacteristic
320C2×C4×C7H(5,𝔽2)11yescharacteristic
321C2×C4×C72−1+411yescharacteristic
322C2×Dic7—21no—
323C2×Dic7—21no—
324C2×Dic7—22no—
325C2×Dic7—22no—
326C2×Dic7—21no—
327C2×Dic7—21no—
328C2×Dic7—21no—
329C2×Dic7—21no—
330C2×Dic7—21no—
331C2×Dic7—21no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—48no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C2×Dic7—41no—
342C2×Dic7—41no—
343C2×Dic7—84no—
344C2×Dic7C4×D412yes—
345C2×Dic7C4×D412yes—
346C2×Dic7AGL(1,ℤ/8)12yes—
347C2×Dic7#13712yes—
348C7×C8—41no—
349C7×C8C2×C4⋊C414yescomplemented
350C7⋊C8—22no—
351C7⋊C8—41no—
352C22×Q8C22×D711yescharacteristic
353C22×D4C22×D711yescharacteristic
354C23×C4—21no—
355C23×C4—141no—
356C23×C4—141no—
357C2×SD16Dic1414yes—
358C2×SD16C4×D714yes—
359C2×SD16C22⋊D714yes—
360C22×C8—71no—
361C22×C8—71no—
362C22×C8C22×D711yescharacteristic
363C22×C8C22×D711yescharacteristic
364C4×Q8—22no—
365C4×Q8—41no—
366C4×Q8—74no—
367C4×Q8—142no—
368C4×Q8—142no—
369C4×Q8—281no—
370C4×D4—22no—
371C4×D4—41no—
372C4×D4—74no—
373C4×D4—142no—
374C4×D4—142no—
375C4×D4—281no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—71no—
379C2×C4⋊C4—71no—
380C2×C4⋊C4—71no—
381C2×C4⋊C4—71no—
382C2×C4⋊C4—71no—
383C2×C4⋊C4—141no—
384C2×C4⋊C4—141no—
385C2×C4⋊C4—141no—
386C2×C4⋊C4—141no—
387C2×C4⋊C4—141no—
388C2×C4⋊C4—141no—
389C2×C4⋊C4—141no—
390C2×C4⋊C4—141no—
391C2×C4⋊C4—141no—
392C2×C4⋊C4—141no—
393C2×C4⋊C4—284no—
394C2×C4⋊C4C22×D711yescharacteristic
395C2×C4⋊C4C22×D711yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—141no—
398C2×C22⋊C4—141no—
399C2×C42—21no—
400C2×C42—71no—
401C2×C42—71no—
402C2×C42—141no—
403C2×C42—141no—
404C2×C42C22×D711yescharacteristic
405#107—22no—
406#107—142no—
407#107—281no—
408#107—281no—
409#106—74no—
410C4⋊C8—142no—
411#103—22no—
412#103—74no—
413#103—142no—
414#102—22no—
415#102—74no—
416#102—142no—
417#97—22no—
418#97—142no—
419C4×C8—74no—
420#95—71no—
421#95—71no—
422#95—141no—
423#95—141no—
424#95—141no—
425#95—141no—
426C22×C7—24no—
427C22×C7—42no—
428C22×C7—42no—
429C22×C7—42no—
430C22×C7—42no—
431C22×C7—42no—
432C22×C7—42no—
433C22×C7C2×C4×D411yescharacteristic
434C22×C7D4211yescharacteristic
435C22×C7D4×Q811yescharacteristic
436C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
437C22×C7C2×#13711yescharacteristic
438C22×C7#57611yescharacteristic
439C22×C7#57811yescharacteristic
440C4×C7—24no—
441C4×C7—41no—
442C4×C7—41no—
443C4×C7—81no—
444C4×C7—81no—
445C4×C7C22×C4⋊C411yescharacteristic
446C4×C7#52011yescharacteristic
447C4×C7#54711yescharacteristic
448C4×C7#55411yescharacteristic
449Dic7—22no—
450Dic7—42no—
451Dic7—42no—
452Dic7—41no—
453Dic7—41no—
454Dic7—81no—
455Dic7—81no—
456Dic7#44014yescomplemented
457C24—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8C2×Dic1412yes—
462C2×Q8C2×C4×D712yes—
463C2×Q8C2×C22⋊D712yes—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4C2×Dic1412yes—
468C2×D4C2×C4×D712yes—
469C2×D4C2×C22⋊D712yes—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—71no—
476C22×C4—71no—
477C22×C4—71no—
478C22×C4—71no—
479C22×C4—81no—
480C22×C4—144no—
481C22×C4—141no—
482C22×C4—141no—
483C22×C4—141no—
484C22×C4—141no—
485C22×C4—282no—
486C22×C4—282no—
487C22×C4—282no—
488C22×C4—282no—
489C22×C4—561no—
490C22×C4D4×D711yescharacteristic
491C22×C4D4×D711yescharacteristic
492C22×C4C23×D711yescharacteristic
493SD16Dic7⋊C4116yescomplemented
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—72no—
498C2×C8—141no—
499C2×C8—141no—
500C2×C8—141no—
501C2×C8—141no—
502C2×C8—141no—
503C2×C8C2×Dic1412yes—
504C2×C8C2×C4×D712yes—
505C2×C8C2×C22⋊D712yes—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—42no—
511C4⋊C4—72no—
512C4⋊C4—72no—
513C4⋊C4—142no—
514C4⋊C4—142no—
515C4⋊C4—144no—
516C4⋊C4—141no—
517C4⋊C4—141no—
518C4⋊C4—141no—
519C4⋊C4—141no—
520C4⋊C4—141no—
521C4⋊C4—141no—
522C4⋊C4—282no—
523C4⋊C4—282no—
524C4⋊C4—282no—
525C4⋊C4—282no—
526C4⋊C4—282no—
527C4⋊C4—561no—
528C4⋊C4—561no—
529C4⋊C4—561no—
530C4⋊C4—561no—
531C22⋊C4—42no—
532C22⋊C4—144no—
533C22⋊C4—282no—
534C42—21no—
535C42—21no—
536C42—42no—
537C42—72no—
538C42—72no—
539C42—144no—
540C42—141no—
541C42—141no—
542C42—282no—
543C2×C7—44no—
544C2×C7D4×C4⋊C411yescharacteristic
545C2×C7C2×#44011yescharacteristic
546C2×C7#295111yescharacteristic
547C2×C7#327411yescharacteristic
548C2×C7#327611yescharacteristic
549C2×C7#338511yescharacteristic
550C2×C7#338611yescharacteristic
551C23—22no—
552C23—22no—
553C23—22no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23C2×D4×D711yescharacteristic
559Q8—22no—
560Q8—22no—
561Q8—22no—
562Q8C2×Dic7⋊C414yescomplemented
563D4—22no—
564D4—22no—
565D4—22no—
566D4C2×Dic7⋊C414yes—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—72no—
583C2×C4—72no—
584C2×C4—72no—
585C2×C4—72no—
586C2×C4—84no—
587C2×C4—141no—
588C2×C4—141no—
589C2×C4—141no—
590C2×C4—141no—
591C2×C4—141no—
592C2×C4—141no—
593C2×C4—142no—
594C2×C4—142no—
595C2×C4—141no—
596C2×C4—141no—
597C2×C4—288no—
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—281no—
607C2×C4—281no—
608C2×C4—564no—
609C2×C4C22×Dic1411yescharacteristic
610C2×C4C22×C4×D711yescharacteristic
611C2×C4#654811yescharacteristic
612C2×C4D7×#4111yescharacteristic
613C2×C4#655411yescharacteristic
614C2×C4C22×C22⋊D711yescharacteristic
615C8—41no—
616C8—142no—
617C8—281no—
618C8C2×Dic7⋊C414yescomplemented
619C7#2267111yescharacteristic, complemented, Sylow
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22D4×Dic1411yescharacteristic
628C22C4×D4×D711yescharacteristic
629C22D7×AGL(1,ℤ/8)11yescharacteristic
630C22#9046511yescharacteristic
631C22D7×#13711yescharacteristic
632C22#9047011yescharacteristic
633C22D4×C22⋊D711yescharacteristic
634C4—24no—
635C4—41no—
636C4—41no—
637C4—74no—
638C4—81no—
639C4—81no—
640C4—142no—
641C4—281no—
642C4—282no—
643C4—282no—
644C4—281no—
645C4—561no—
646C4—561no—
647C4#9023011yescharacteristic
648C4#9023711yescharacteristic
649C4C22×Dic7⋊C411yescharacteristic
650C4#9051511yescharacteristic
651C2—44no—
652C2#1173720411yescharacteristic
653C2#1173720511yescharacteristic
654C2D7×#44011yescharacteristic
655C2#1173722311yescharacteristic
656C2D4×Dic7⋊C411yescharacteristic
657C2#1174140411yescharacteristic
658C2#1174144111yescharacteristic
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#22671
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×Dic7⋊C4C2semidirect
6. #11729382C2semidirect
8. #11728368C2semidirect
9. #11728360C2semidirect
10. #11727989C2semidirect
11. #11727975C2semidirect
14. #11727587C2semidirect
15. #11727574C2semidirect
35. SD16×Dic7C4semidirect
88. Q8×Dic7C2×C4semidirect
121. #6409C2×C4semidirect
131. #6391C2×C4semidirect
134. C8×Dic7C2×C4semidirect
188. C7×SD16C4⋊C4semidirect
295. C7×Q8C2×C4⋊C4semidirect
349. C7×C8C2×C4⋊C4semidirect
456. Dic7#440semidirect
493. SD16Dic7⋊C4semidirect
562. Q8C2×Dic7⋊C4semidirect
618. C8C2×Dic7⋊C4semidirect
619. C7#22671semidirect

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×Dic7⋊C42
D4×Dic7⋊C42
#117293822
#117293602
#117283682
#117283602
#117279892
#117279752
#117279212
#117279072
#117275872
#117275742
#226717