Finite Group Databasebeta

← #13323849

Subgroups — #13323849

Order 1792.

648 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11737037C211yesmaximal, characteristic
2#11736871C211yesmaximal, characteristic, complemented
3#11736853C211yesmaximal, characteristic, complemented
4C7×C8×D8C211yesmaximal, characteristic, complemented
5#11728634C211yesmaximal, characteristic, complemented
6#11728594C211yesmaximal, characteristic, complemented
7#11728593C211yesmaximal, characteristic, complemented
8#11726329C211yesmaximal, characteristic, complemented
9#11726315C211yesmaximal, characteristic, complemented
10#11725926C211yesmaximal, characteristic, complemented
11#11725923C211yesmaximal, characteristic, complemented
12#11725908C211yesmaximal, characteristic, complemented
13#11725907C211yesmaximal, characteristic, complemented
14C8×#1132C211yesmaximal, characteristic, complemented
15#11725803C211yesmaximal, characteristic, complemented
16#90248C2211yescharacteristic
17C4×D4×D7C2211yescharacteristic
18C4×C7×D8C2211yescharacteristic
19C7×C8×D4C2211yescharacteristic
20C7×C8×D4C2211yescharacteristic, complemented
21#89928C411yescharacteristic
22#89925C411yescharacteristic
23#89883—21no—
24C8×C22⋊D7—21no—
25C2×(C2×D7)⋊C8—21no—
26#89874—21no—
27C4×#1142C2211yescharacteristic
28C4×#1141C2211yescharacteristic
29#89784C2211yescharacteristic, complemented
30D4×C7⋊C8C2211yescharacteristic
31#89663C411yescharacteristic
32#89656C411yescharacteristic
33#89609C2211yescharacteristic
34#89606C2211yescharacteristic
35#89605C2211yescharacteristic
36#89603C2211yescharacteristic, complemented
37#89559C411yescharacteristic
38#89549C411yescharacteristic
39#89548C411yescharacteristic
40#89535C411yescharacteristic
41#89501—21no—
42(C22⋊D7)⋊C8—21no—
43D7×C22⋊C8—21no—
44#89492—21no—
45C4×#1132C2211yescharacteristic
46C8×D28C2211yescharacteristic
47#89457C2211yescharacteristic
48#89457C2211yescharacteristic
49#89457C2211yescharacteristic
50C8×Dic14C2211yescharacteristic, complemented
51C7×#335C2211yescharacteristic, complemented
52C7×D4⋊C8C2211yescharacteristic
53C7×D4⋊C8C2211yescharacteristic, complemented
54C7×C82C2211yescharacteristic, complemented
55D4⋊(C7⋊C8)C2211yescharacteristic
56D4⋊(C7⋊C8)C2211yescharacteristic, complemented
57#89279C2211yescharacteristic, complemented
58#89278C2211yescharacteristic
59#89275C2211yescharacteristic, complemented
60#89274C2211yescharacteristic, complemented
61#89254C2211yescharacteristic
62#89252C2211yescharacteristic, complemented
63#89251C2211yescharacteristic, complemented
64#89249C2211yescharacteristic, complemented
65#89249C2211yescharacteristic, complemented
66#89249C2211yescharacteristic, complemented
67#17918—71nomaximal, Sylow
68C2×D4×D7C2×C411yescharacteristic
69C22×C4×D7—21no—
70C2×C7×D8C2×C411yescharacteristic
71C22×C7×C8—21no—
72C22×C7×C8—21no—
73C4×C7×D4C2311yescharacteristic
74C4×C7×D4C2311yescharacteristic
75#6501C2×C411yescharacteristic
76#6499C2×C411yescharacteristic
77D4×Dic7C2×C411yescharacteristic
78C2×#1142C2×C411yescharacteristic
79C2×#1141C2×C411yescharacteristic
80C4×C22⋊D7—21no—
81C4×C22⋊D7—21no—
82#6489—21no—
83C2×#1136—21no—
84C22×C7⋊C8—21no—
85C2×#1132C2×C411yescharacteristic
86C2×#1131—21no—
87C2×C8×D7—21no—
88C2×C8×D7—21no—
89#6460C2×C411yescharacteristic
90#6456C2×C411yescharacteristic
91D7×C4⋊C4C2×C411yescharacteristic
92#6453C2×C411yescharacteristic
93#6446—21no—
94(C22⋊D7)⋊C4—21no—
95D7×C22⋊C4—21no—
96#6442—21no—
97C4×D28C2311yescharacteristic
98C42×D7C2311yescharacteristic
99C4×Dic14C2311yescharacteristic
100C7×#107C2×C411yescharacteristic
101C7×C4⋊C8C2311yescharacteristic
102C7×C4⋊C8C2311yescharacteristic
103C7×#102C2×C411yescharacteristic
104C7×#102C2×C411yescharacteristic
105C7×C22⋊C8—21no—
106C7×C22⋊C8—21no—
107C4×C7×C8C2311yescharacteristic
108C4×C7×C8C2311yescharacteristic
109C4×C7×C8C2311yescharacteristic
110#6406C2×C411yescharacteristic
111#6406C2×C411yescharacteristic
112C22⋊(C7⋊C8)—21no—
113C22⋊(C7⋊C8)—21no—
114#6395C2×C411yescharacteristic
115(C2×D7)⋊C8—21no—
116(C2×D7)⋊C8—21no—
117(C2×D7)⋊C8—21no—
118(C2×D7)⋊C8—21no—
119(C2×D7)⋊C8C2×C411yescharacteristic
120(C2×D7)⋊C8C2×C411yescharacteristic
121(C2×D7)⋊C8D411yescharacteristic
122(C2×D7)⋊C8D411yescharacteristic
123#6391C2×C411yescharacteristic
124#6390C2×C411yescharacteristic
125#6389—21no—
126Dic7⋊C8—21no—
127Dic7⋊C8—21no—
128Dic7⋊C8—21no—
129Dic7⋊C8—21no—
130Dic7⋊C8C2×C411yescharacteristic
131Dic7⋊C8C2×C411yescharacteristic
132Dic7⋊C8D411yescharacteristic
133Dic7⋊C8D411yescharacteristic
134C8×Dic7—21no—
135C8×Dic7—21no—
136#6384C2×C411yescharacteristic
137#6383C2×C411yescharacteristic
138#6382C2×C411yescharacteristic
139#6381C2×C411yescharacteristic
140C4⋊(C7⋊C8)C2311yescharacteristic
141C4⋊(C7⋊C8)C2311yescharacteristic
142C4×C7⋊C8C2311yescharacteristic
143C4×C7⋊C8C2311yescharacteristic
144C4×C7⋊C8C2311yescharacteristic
145#3105—71no—
146#2993—71no—
147#2972—71no—
148#1586—71no—
149#1585—71no—
150#1582—71no—
151#1581—71no—
152#1580—71no—
153SD16⋊C8—71no—
154C8×SD16—71no—
155C8×D8D711yescharacteristic, complemented
156#1574—71no—
157#1486—71no—
158#1482—71no—
159#1454—71no—
160C23×D7—21no—
161C2×C7×D4C22×C411yescharacteristic
162C2×C7×D4C22×C411yescharacteristic
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C2×C22⋊D7—21no—
166C2×C22⋊D7—21no—
167C22×Dic7—21no—
168D4×D7—22no—
169D4×D7—22no—
170C2×D28C22×C411yescharacteristic
171C2×C4×D7—21no—
172C2×C4×D7—21no—
173C2×C4×D7—21no—
174C2×C4×D7—81no—
175C2×C4×D7—81no—
176C2×C4×D7C22×C411yescharacteristic
177C2×C4×D7C2×D411yescharacteristic
178C2×C4×D7#4111yescharacteristic
179C2×Dic14C22×C411yescharacteristic
180C7×D8—22no—
181C2×C7×C8—21no—
182C2×C7×C8—21no—
183C2×C7×C8—21no—
184C2×C7×C8—81no—
185C2×C7×C8—81no—
186C2×C7×C8C22×C411yescharacteristic
187C2×C7×C8C22×C411yescharacteristic
188C2×C7×C8C2×D411yescharacteristic
189C2×C7×C8C2×D411yescharacteristic
190C7×C4⋊C4C22×C411yescharacteristic
191C7×C4⋊C4C22×C411yescharacteristic
192C7×C22⋊C4—21no—
193C7×C22⋊C4—21no—
194C42×C7C2411yescharacteristic
195C22⋊Dic7—21no—
196C22⋊Dic7—21no—
197C22⋊Dic7—21no—
198C22⋊Dic7—21no—
199#1142—22no—
200#1141—22no—
201(C2×D7)⋊C4—21no—
202(C2×D7)⋊C4—21no—
203(C2×D7)⋊C4—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7C22×C411yescharacteristic
207C4⋊Dic7C22×C411yescharacteristic
208Dic7⋊C4—21no—
209Dic7⋊C4—21no—
210Dic7⋊C4—21no—
211C4×Dic7—21no—
212C4×Dic7C22×C411yescharacteristic
213C4×Dic7C2×D411yescharacteristic
214C4×Dic7#4111yescharacteristic
215#1136—81no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—81no—
220C2×C7⋊C8C22×C411yescharacteristic
221C2×C7⋊C8C22×C411yescharacteristic
222C2×C7⋊C8C2×D411yescharacteristic
223C2×C7⋊C8C2×D411yescharacteristic
224#1132—22no—
225#1131—81no—
226C8×D7—81no—
227C8×D7—81no—
228#529—71no—
229C2×C4×D4—71no—
230#484—71no—
231#481—71no—
232#478—71no—
233#478—71no—
234#463—71no—
235#463—71no—
236#450—71no—
237#447—71no—
238#446—71no—
239C8×Q8—71no—
240C4×SD16—71no—
241C4×SD16—71no—
242C4×D8—71no—
243C4×D8C2×D711yescharacteristic
244#436—71no—
245#436—141no—
246#435—141no—
247C8×D4—71no—
248C8×D4—71no—
249C8×D4—71no—
250C8×D4—141no—
251C8×D4—141no—
252C8×D4C2×D711yescharacteristic
253C8×D4C2×D711yescharacteristic, complemented
254#433—141no—
255#431—71no—
256#431—71no—
257#431—71no—
258#431—71no—
259#424—141no—
260#423—71no—
261C2×C22⋊C8—141no—
262C2×C22⋊C8—141no—
263#335—71no—
264#335C2×D711yescharacteristic, complemented
265#334—71no—
266#334—71no—
267Q8⋊C8—71no—
268Q8⋊C8—71no—
269D4⋊C8—71no—
270D4⋊C8—71no—
271D4⋊C8—71no—
272D4⋊C8—71no—
273D4⋊C8C2×D711yescharacteristic
274D4⋊C8C2×D711yescharacteristic, complemented
275#322—71no—
276#322—71no—
277#322—71no—
278C82C2×D711yescharacteristic, complemented
279C23×C7—21no—
280C23×C7—21no—
281C22×D7—21no—
282C22×D7—24no—
283C22×D7—81no—
284C22×D7—81no—
285C22×D7C4×D411yescharacteristic
286C7×D4—21no—
287C7×D4—21no—
288C7×D4—21no—
289C7×D4#13112yes—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7C23×C411yescharacteristic
297C2×C4×C7C22×D411yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C22⋊D7—24no—
300C22⋊D7—81no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—81no—
305C2×Dic7C4×D411yescharacteristic
306C2×Dic7C4×D411yescharacteristic
307C2×Dic7C4×D411yescharacteristic
308D28—21no—
309D28#13112yes—
310C4×D7—21no—
311C4×D7—21no—
312C4×D7—21no—
313C4×D7—21no—
314C4×D7—41no—
315C4×D7—41no—
316C4×D7—41no—
317C4×D7—81no—
318C4×D7—81no—
319Dic14—21no—
320Dic14—21no—
321C7×C8—21no—
322C7×C8—21no—
323C7×C8—21no—
324C7×C8—21no—
325C7×C8—41no—
326C7×C8—41no—
327C7×C8—41no—
328C7⋊C8—21no—
329C7⋊C8—21no—
330C7⋊C8—21no—
331C7⋊C8—21no—
332C7⋊C8—41no—
333C7⋊C8—41no—
334C7⋊C8—41no—
335C22×D4—71no—
336C23×C4—141no—
337C2×SD16—71no—
338C2×SD16—71no—
339C2×D8—71no—
340C2×D8C4×D711yescharacteristic
341C2×M16—141no—
342C2×M16—141no—
343C22×C8—21no—
344C22×C8—21no—
345C22×C8—141no—
346C22×C8—141no—
347C22×C8—141no—
348#126—141no—
349#125—71no—
350#124—71no—
351#123—141no—
352C22⋊Q8—71no—
353#121—71no—
354C4×Q8—71no—
355C4×D4—71no—
356C4×D4—71no—
357C4×D4—71no—
358C4×D4—141no—
359C4×D4—141no—
360C4×D4—141no—
361C4×D4C22×D711yescharacteristic
362C4×D4C22×D711yescharacteristic
363#117—141no—
364C2×C4⋊C4—71no—
365C2×C22⋊C4—141no—
366C2×C42—71no—
367#107—71no—
368#107C4×D711yescharacteristic
369#106—71no—
370#106—71no—
371C4⋊C8—71no—
372C4⋊C8—71no—
373C4⋊C8—71no—
374C4⋊C8—71no—
375C4⋊C8—71no—
376C4⋊C8—71no—
377C4⋊C8—141no—
378C4⋊C8—141no—
379C4⋊C8—141no—
380C4⋊C8—141no—
381C4⋊C8C22×D711yescharacteristic
382C4⋊C8C22×D711yescharacteristic
383#103—71no—
384#103—71no—
385#102—71no—
386#102—71no—
387#102—71no—
388#102—71no—
389#102C4×D711yescharacteristic
390#102C4×D711yescharacteristic
391C22⋊C8—21no—
392C22⋊C8—21no—
393C22⋊C8—71no—
394C22⋊C8—71no—
395C22⋊C8—71no—
396C22⋊C8—71no—
397C22⋊C8—141no—
398C22⋊C8—141no—
399C22⋊C8—141no—
400C22⋊C8—141no—
401C22⋊C8—141no—
402C22⋊C8—141no—
403#97—141no—
404C4×C8—71no—
405C4×C8—71no—
406C4×C8—71no—
407C4×C8—141no—
408C4×C8—141no—
409C4×C8C22×D711yescharacteristic
410C4×C8C22×D711yescharacteristic
411C4×C8C22×D711yescharacteristic
412C22×C7—22no—
413C22×C7—41no—
414C22×C7—41no—
415C22×C7—41no—
416C22×C7—41no—
417C22×C7—41no—
418C22×C7C2×C4×D411yescharacteristic
419C2×D7—21no—
420C2×D7—22no—
421C2×D7—21no—
422C2×D7—21no—
423C2×D7—41no—
424C2×D7—41no—
425C2×D7—84no—
426C4×C7—21no—
427C4×C7—41no—
428C4×C7—41no—
429C4×C7C2×#13111yescharacteristic
430C4×C7C2×#13111yescharacteristic
431C4×C7#57611yescharacteristic
432C4×C7#57711yescharacteristic
433Dic7—21no—
434Dic7—21no—
435Dic7—21no—
436Dic7—41no—
437Dic7—41no—
438Dic7—41no—
439C24—141no—
440C2×Q8—71no—
441C2×D4—71no—
442C2×D4—141no—
443C2×D4—142no—
444C2×D4—142no—
445C2×D4—141no—
446C2×D4C2×C4×D711yescharacteristic
447C2×D4C2×C4×D711yescharacteristic
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—71no—
451C22×C4—71no—
452C22×C4—71no—
453C22×C4—141no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—141no—
457C22×C4—561no—
458C22×C4—561no—
459SD16—142no—
460SD16—142no—
461D8—22no—
462D8—142no—
463M16—561no—
464M16—561no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—71no—
469C2×C8—71no—
470C2×C8—71no—
471C2×C8—71no—
472C2×C8—81no—
473C2×C8—81no—
474C2×C8—141no—
475C2×C8—141no—
476C2×C8—141no—
477C2×C8—561no—
478C2×C8—561no—
479C2×C8—561no—
480C2×C8C2×C4×D711yescharacteristic
481C2×C8C2×C4×D711yescharacteristic
482C2×C8D4×D711yescharacteristic
483C2×C8D4×D711yescharacteristic
484C4⋊C4—71no—
485C4⋊C4—71no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4C2×C4×D711yescharacteristic
492C4⋊C4C2×C4×D711yescharacteristic
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—141no—
496C22⋊C4—141no—
497C22⋊C4—141no—
498C22⋊C4—141no—
499C22⋊C4—141no—
500C22⋊C4—141no—
501C22⋊C4—141no—
502C42—71no—
503C42—71no—
504C42—71no—
505C42—141no—
506C42C23×D711yescharacteristic
507C2×C7—42no—
508C2×C7—81no—
509C2×C7#292911yescharacteristic
510C2×C7#294611yescharacteristic
511C2×C7#295711yescharacteristic
512D7—41no—
513D7—42no—
514C23—21no—
515C23—21no—
516C23—71no—
517C23—144no—
518C23—141no—
519C23—561no—
520C23—561no—
521Q8—141no—
522Q8—141no—
523D4—21no—
524D4—21no—
525D4—21no—
526D4—72no—
527D4—144no—
528D4—141no—
529D4—561no—
530D4#646412yes—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—71no—
538C2×C4—71no—
539C2×C4—71no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—141no—
544C2×C4—141no—
545C2×C4—141no—
546C2×C4—141no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—561no—
551C2×C4—561no—
552C2×C4—561no—
553C2×C4C22×C4×D711yescharacteristic
554C2×C4C2×D4×D711yescharacteristic
555C2×C4D7×#4111yescharacteristic
556C8—21no—
557C8—21no—
558C8—21no—
559C8—21no—
560C8—41no—
561C8—41no—
562C8—41no—
563C8—141no—
564C8—141no—
565C8—141no—
566C8—141no—
567C8—281no—
568C8—281no—
569C8—281no—
570C7#1791811yescharacteristic, complemented, Sylow
571C22—22no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—141no—
578C22—142no—
579C22—141no—
580C22—141no—
581C22—281no—
582C22—281no—
583C22—564no—
584C22C4×D4×D711yescharacteristic
585C4—21no—
586C4—41no—
587C4—41no—
588C4—141no—
589C4—141no—
590C4—141no—
591C4—281no—
592C4—281no—
593C4—281no—
594C4C2×#646411yescharacteristic
595C4D7×#13111yescharacteristic
596C4#9044811yescharacteristic
597C4#9046111yescharacteristic
598C2—42no—
599C2—81no—
600C2—281no—
601C2—282no—
602C2#1173686611yescharacteristic
603C2#1173703211yescharacteristic
604C2#1173733511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×C8×D8C2
socleC22×C7C2×C4×D4
prime radical#89928C4
cosocleC42×C7C24
hypercenterC8×D8D7
hypocenterC7#17918
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×D8

Complements

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

NH ≅ G/Ntype
2. #11736871C2semidirect
3. #11736853C2semidirect
4. C7×C8×D8C2semidirect
5. #11728634C2semidirect
6. #11728594C2semidirect
7. #11728593C2semidirect
8. #11726329C2semidirect
9. #11726315C2semidirect
10. #11725926C2semidirect
11. #11725923C2semidirect
12. #11725908C2semidirect
13. #11725907C2semidirect
14. C8×#1132C2semidirect
15. #11725803C2semidirect
20. C7×C8×D4C22semidirect
29. #89784C22semidirect
36. #89603C22semidirect
50. C8×Dic14C22semidirect
51. C7×#335C22semidirect
53. C7×D4⋊C8C22semidirect
54. C7×C82C22semidirect
56. D4⋊(C7⋊C8)C22semidirect
57. #89279C22semidirect
59. #89275C22semidirect
60. #89274C22semidirect
62. #89252C22semidirect
63. #89251C22semidirect
64. #89249C22semidirect
65. #89249C22semidirect
66. #89249C22semidirect
155. C8×D8D7semidirect
253. C8×D4C2×D7semidirect
264. #335C2×D7semidirect
274. D4⋊C8C2×D7semidirect
278. C82C2×D7semidirect
570. C7#17918semidirect

Maximal subgroups

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

subgroupindex
#117370372
#117368712
#117368532
C7×C8×D82
#117286342
#117285942
#117285932
#117263292
#117263152
#117259262
#117259232
#117259082
#117259072
C8×#11322
#117258032
#179187