Finite Group Databasebeta

← #13370455

Subgroups — #13370455

Order 1792.

1022 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic14C211yesmaximal, characteristic
2C2×#89473C211yesmaximal, characteristic
3#11731087C211yesmaximal, characteristic
4C7×#1763C211yesmaximal, characteristic
5#11729381C211yesmaximal, characteristic
6#11727976C211yesmaximal, characteristic
7#11727975C211yesmaximal, characteristic
8#11727974C211yesmaximal, characteristic
9Dic7×#103C211yesmaximal, characteristic
10#11727582C211yesmaximal, characteristic
11#11727244C211yesmaximal, characteristic
12#11726560C211yesmaximal, characteristic
13#11726557C211yesmaximal, characteristic
14#11726554C211yesmaximal, characteristic
15#11726553C211yesmaximal, characteristic
16C2×C4×C7×Q8C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19Q8×Dic14—24no—
20Q8×Dic14C418yescomplemented
21C2×#6451—21no—
22C2×#6451—21no—
23C2×#6451C2211yescharacteristic
24C2×#6451C2211yescharacteristic
25C2×#6432—21no—
26C2×#6432C2211yescharacteristic
27C2×C4×Dic14—21no—
28C2×C4×Dic14C2211yescharacteristic
29C2×C7×#103C2211yescharacteristic
30C2×C7×#103C2211yescharacteristic
31C2×C7×#97C2211yescharacteristic
32C4×C7×C4⋊C4C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35C2×#6389C2211yescharacteristic
36C2×Dic7⋊C8C2211yescharacteristic
37C2×Dic7⋊C8C2211yescharacteristic
38C2×C8×Dic7C2211yescharacteristic
39#89749C2211yescharacteristic
40#89748C2211yescharacteristic
41C4⋊C4×Dic7C2211yescharacteristic
42C4⋊C4×Dic7C2211yescharacteristic
43#89708C2211yescharacteristic
44#89705—21no—
45C4×Dic7⋊C4—21no—
46C2×C4⋊(C7⋊C8)C2211yescharacteristic
47#89473—24no—
48#89426—21no—
49#89418—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#22525—71nomaximal, Sylow
63C22×C7×Q8C2311yescharacteristic
64C22×Dic14—21no—
65C22×Dic14—21no—
66C22×Dic14—21no—
67C22×Dic14D411yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C4×C7×Q8—22no—
71C4×C7×Q8C2×C414yes—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C42×C7—21no—
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79Q8×Dic7—22no—
80Q8×Dic7—22no—
81Q8×Dic7C2×C414yes—
82Q8×Dic7C2×C414yes—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7D411yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4D411yescharacteristic
96C2×Dic7⋊C4D411yescharacteristic
97C2×C4×Dic7—21no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7C2311yescharacteristic
102C2×C4×Dic7C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105#6451—28no—
106#6451—28no—
107#6451—22no—
108#6451—22no—
109#6451C2×C414yes—
110#6451C2×C414yes—
111#6432—22no—
112#6432—28no—
113#6432C2×C414yes—
114C4×Dic14—28no—
115C4×Dic14C2×C414yes—
116C4×Dic14D414yes—
117C7×#103—22no—
118C7×#103Q814yescomplemented
119C7×#97—22no—
120C7×#95—21no—
121#6409—22no—
122#6409—22no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95—21no—
128C7⋊#95—21no—
129#6389—22no—
130Dic7⋊C8—22no—
131Dic7⋊C8—22no—
132C8×Dic7—22no—
133C4⋊(C7⋊C8)—22no—
134C2×Q82—71no—
135C2×#446—71no—
136#2355—71no—
137#1993—71no—
138#1993—71no—
139#1990—71no—
140#1988—71no—
141#1988—71no—
142#1987—71no—
143#1805—71no—
144#1805—71no—
145#1803—71no—
146#1763—71no—
147#1763D711yescharacteristic
148C4×#103—71no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8C22×C412yes—
153C2×C7×Q8C2×Q812yes—
154C2×C7×Q8#4112yes—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2411yescharacteristic
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7C2×D411yescharacteristic
167C22×Dic7C2×D411yescharacteristic
168C22×Dic7C2×D411yescharacteristic
169C22×Dic7C2×D411yescharacteristic
170C2×Dic14—22no—
171C2×Dic14—22no—
172C2×Dic14—24no—
173C2×Dic14—24no—
174C2×Dic14—24no—
175C2×Dic14—81no—
176C2×Dic14—81no—
177C2×Dic14—81no—
178C2×Dic14—81no—
179C2×Dic14—81no—
180C2×Dic14—81no—
181C2×Dic14C22⋊C414yes—
182C2×C7×C8—21no—
183C2×C7×C8—21no—
184C2×C7×C8—21no—
185C2×C7×C8—21no—
186C2×C7×C8—21no—
187C2×C7×C8C2×Q812yes—
188C7×C4⋊C4—24no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4C22×C412yes—
194C7×C4⋊C4C2×Q812yes—
195C42×C7—24no—
196C42×C7—81no—
197C42×C7C22×C412yes—
198C42×C7C2×D412yes—
199C4⋊Dic7—24no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—24no—
203C4⋊Dic7—24no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7C22⋊C414yes—
209C4⋊Dic7C22×C412yes—
210C4⋊Dic7C22×C412yes—
211Dic7⋊C4—24no—
212Dic7⋊C4—24no—
213Dic7⋊C4—42no—
214Dic7⋊C4—81no—
215Dic7⋊C4C22⋊C414yes—
216Dic7⋊C4C22⋊C414yes—
217C4×Dic7—24no—
218C4×Dic7—24no—
219C4×Dic7—42no—
220C4×Dic7—81no—
221C4×Dic7C22×C412yes—
222C4×Dic7C22×C412yes—
223C4×Dic7C2×D412yes—
224C4×Dic7C2×D412yes—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231Q82—78no—
232Q82—144no—
233C2×C4⋊Q8—71no—
234C2×C4⋊Q8—71no—
235C2×C4⋊Q8—71no—
236C2×C4⋊Q8—141no—
237C2×C4⋊Q8—141no—
238C2×C4⋊Q8—141no—
239C2×C4×Q8—71no—
240C2×C4×Q8—71no—
241C2×C4×Q8—71no—
242C2×C4×Q8—141no—
243C2×C4×Q8C2×D711yescharacteristic
244#446—144no—
245C2×C4⋊C8—71no—
246C2×C4⋊C8—71no—
247C2×C4⋊C8—71no—
248C2×#103—71no—
249C2×#103—71no—
250C2×#103C2×D711yescharacteristic
251C2×#103C2×D711yescharacteristic
252C2×#97—71no—
253C2×#97C2×D711yescharacteristic
254C2×C4×C8—71no—
255#391—141no—
256#387—141no—
257#387—141no—
258#384—71no—
259#382—71no—
260#382—71no—
261C4×C4⋊C4—71no—
262C4×C4⋊C4—71no—
263C4×C4⋊C4—141no—
264C4×C4⋊C4C2×D711yescharacteristic
265#340—71no—
266#340—71no—
267#340—71no—
268#340—71no—
269#340—71no—
270#340—71no—
271#340—71no—
272#340—71no—
273#340—71no—
274#340—71no—
275#340C2×D711yescharacteristic
276#340C2×D711yescharacteristic
277C23×C7C22×D411yescharacteristic
278C7×Q8—22no—
279C7×Q8—22no—
280C7×Q8—22no—
281C7×Q8C4×Q814yes—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7C2×C22⋊C412yes—
299C2×C4×C7C2×C22⋊C412yes—
300C2×C4×C7C23×C411yescharacteristic
301C2×C4×C7C22×D411yescharacteristic
302C2×C4×C7C22×Q811yescharacteristic
303C2×C4×C7C2×#4111yescharacteristic
304C2×C4×C72−1+411yescharacteristic
305C2×C4×C72−1+411yescharacteristic
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—22no—
313C2×Dic7—22no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7C2×C22⋊C412yes—
325C2×Dic7C2×C22⋊C412yes—
326C2×Dic7C2×C22⋊C412yes—
327C2×Dic7C2×C22⋊C412yes—
328C2×Dic7#13712yes—
329C2×Dic7#13712yes—
330Dic14—28no—
331Dic14—84no—
332Dic14—84no—
333Dic14—84no—
334C7×C8—22no—
335C7×C8—41no—
336C7⋊C8—41no—
337C7⋊C8—41no—
338C22×Q8—71no—
339C22×Q8—141no—
340C22×Q8—141no—
341C22×Q8—141no—
342C22×Q8C22×D711yescharacteristic
343C22×C8—71no—
344C22×C8—71no—
345C22×C8C22×D711yescharacteristic
346C22×C8C22×D711yescharacteristic
347C4⋊Q8—74no—
348C4⋊Q8—74no—
349C4⋊Q8—74no—
350C4⋊Q8—148no—
351C4⋊Q8—142no—
352C4⋊Q8—142no—
353C4⋊Q8—148no—
354C4⋊Q8—148no—
355C4⋊Q8—142no—
356C4×Q8—22no—
357C4×Q8—74no—
358C4×Q8—74no—
359C4×Q8—74no—
360C4×Q8—74no—
361C4×Q8—148no—
362C4×Q8—142no—
363C4×Q8—142no—
364C4×Q8C4×D714yes—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—71no—
367C2×C4⋊C4—71no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—71no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—71no—
372C2×C4⋊C4—71no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4—141no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4C22×D711yescharacteristic
381C2×C4⋊C4C22×D711yescharacteristic
382C2×C4⋊C4C22×D711yescharacteristic
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—71no—
386C2×C42—71no—
387C2×C42—141no—
388C2×C42—141no—
389C2×C42—141no—
390C2×C42—141no—
391C2×C42C22×D711yescharacteristic
392C4⋊C8—142no—
393C4⋊C8—142no—
394C4⋊C8—142no—
395#103—22no—
396#103—142no—
397#103—142no—
398#103Dic1414yescomplemented
399#97—22no—
400#97—142no—
401C4×C8—142no—
402#95—21no—
403#95—141no—
404#95—141no—
405#95—141no—
406#95—141no—
407#95—141no—
408#95—141no—
409C22×C7C22×C22⋊C411yescharacteristic
410C22×C7#54811yescharacteristic
411C22×C7D4×Q811yescharacteristic
412C22×C7C2×#13711yescharacteristic
413C22×C7C2×#13711yescharacteristic
414C22×C7#57611yescharacteristic
415C22×C7#57711yescharacteristic
416C4×C7—24no—
417C4×C7—22no—
418C4×C7—22no—
419C4×C7—42no—
420C4×C7—42no—
421C4×C7—81no—
422C4×C7C2×C4×Q811yescharacteristic
423C4×C7#51911yescharacteristic
424C4×C7#54811yescharacteristic
425C4×C7D4×Q811yescharacteristic
426Dic7—22no—
427Dic7—22no—
428Dic7—22no—
429Dic7—42no—
430Dic7—42no—
431Dic7—81no—
432Dic7—81no—
433Dic7#41914yes—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—74no—
438C2×Q8—142no—
439C2×Q8—142no—
440C2×Q8—144no—
441C2×Q8—144no—
442C2×Q8—144no—
443C2×Q8—561no—
444C2×Q8—561no—
445C2×Q8—561no—
446C2×Q8—561no—
447C2×Q8—561no—
448C2×Q8—561no—
449C2×Q8C2×Dic1412yes—
450C2×Q8C2×C4×D712yes—
451C2×Q8#115712yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—71no—
457C22×C4—71no—
458C22×C4—71no—
459C22×C4—71no—
460C22×C4—141no—
461C22×C4—141no—
462C22×C4—141no—
463C22×C4—141no—
464C22×C4D4×D711yescharacteristic
465C22×C4D4×D711yescharacteristic
466C22×C4C23×D711yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8—141no—
476C2×C8—141no—
477C2×C8—141no—
478C2×C8C2×Dic1412yes—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—74no—
485C4⋊C4—74no—
486C4⋊C4—74no—
487C4⋊C4—72no—
488C4⋊C4—72no—
489C4⋊C4—144no—
490C4⋊C4—144no—
491C4⋊C4—144no—
492C4⋊C4—144no—
493C4⋊C4—144no—
494C4⋊C4—141no—
495C4⋊C4—141no—
496C4⋊C4—141no—
497C4⋊C4—141no—
498C4⋊C4—141no—
499C4⋊C4—141no—
500C4⋊C4—282no—
501C4⋊C4—561no—
502C4⋊C4C2×Dic1412yes—
503C4⋊C4C2×C4×D712yes—
504C42—24no—
505C42—72no—
506C42—72no—
507C42—72no—
508C42—72no—
509C42—81no—
510C42—144no—
511C42—144no—
512C42—282no—
513C42—561no—
514C42C2×C4×D712yes—
515C42D4×D712yes—
516C2×C7Q8×C22⋊C411yescharacteristic
517C2×C7C2×#41911yescharacteristic
518C2×C7#289711yescharacteristic
519C2×C7#336911yescharacteristic
520C2×C7#337511yescharacteristic
521C2×C7#338511yescharacteristic
522C2×C7#338711yescharacteristic
523C23C2×D4×D711yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—22no—
527Q8—148no—
528Q8—564no—
529Q8—564no—
530Q8—564no—
531Q8C4×Dic1414yes—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—72no—
549C2×C4—72no—
550C2×C4—72no—
551C2×C4—72no—
552C2×C4—72no—
553C2×C4—72no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—141no—
558C2×C4—141no—
559C2×C4—141no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4D7×C22⋊C412yes—
573C2×C4D7×C22⋊C412yes—
574C2×C4C22×Dic1411yescharacteristic
575C2×C4C22×C4×D711yescharacteristic
576C2×C4C2×#115711yescharacteristic
577C2×C4C2×D4×D711yescharacteristic
578C2×C4#655411yescharacteristic
579C2×C4#655411yescharacteristic
580C8—22no—
581C8—41no—
582C8—281no—
583C8—281no—
584C7#2252511yescharacteristic, complemented, Sylow
585C22C2×D7×C22⋊C411yescharacteristic
586C22D4×Dic1411yescharacteristic
587C22#9024311yescharacteristic
588C22#9046511yescharacteristic
589C22D7×#13711yescharacteristic
590C22D7×#13711yescharacteristic
591C22#9046911yescharacteristic
592C4—22no—
593C4—22no—
594C4—24no—
595C4—42no—
596C4—42no—
597C4—74no—
598C4—81no—
599C4—142no—
600C4—142no—
601C4—142no—
602C4—282no—
603C4—282no—
604C4—561no—
605C4—561no—
606C4C2×C4×Dic1411yescharacteristic
607C4#9020711yescharacteristic
608C4D4×Dic1411yescharacteristic
609C4#9024311yescharacteristic
610C2C22⋊C4×Dic1411yescharacteristic
611C2#1173632711yescharacteristic
612C2D7×#41911yescharacteristic
613C2#1173720111yescharacteristic
614C2#1173720411yescharacteristic
615C2#1173726011yescharacteristic
616C2#1173726411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1763C2
socle, prime radicalC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenter#1763D7
hypocenterC7#22525
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
20. Q8×Dic14C4semidirect
118. C7×#103Q8semidirect
398. #103Dic14semidirect
584. C7#22525semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic142
C2×#894732
#117310872
C7×#17632
#117293812
#117279762
#117279752
#117279742
Dic7×#1032
#117275822
#117272442
#117265602
#117265572
#117265542
#117265532
#225257