Finite Group Databasebeta

← #34258327

Subgroups — #34258327

Order 20224.

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#23773449C211yesmaximal, characteristic
2#23772545C211yesmaximal, characteristic, complemented
3#23772423C211yesmaximal, characteristic, complemented
4#23762263C211yesmaximal, characteristic, complemented
5#23762256C211yesmaximal, characteristic, complemented
6#23762235C211yesmaximal, characteristic, complemented
7#23762228C211yesmaximal, characteristic, complemented
8#23762223C211yesmaximal, characteristic, complemented
9#23762211C211yesmaximal, characteristic, complemented
10#23762199C211yesmaximal, characteristic, complemented
11#23762194C211yesmaximal, characteristic, complemented
12#23761824C211yesmaximal, characteristic, complemented
13#23761806C211yesmaximal, characteristic, complemented
14#23761700C211yesmaximal, characteristic, complemented
15C79×#1574C211yesmaximal, characteristic, complemented
16#19099097C2211yescharacteristic
17C2×C4×D316C2211yescharacteristic
18#19098899C411yescharacteristic
19#19098882C411yescharacteristic
20#19098864C411yescharacteristic
21#19098864C411yescharacteristic
22#19098779C411yescharacteristic
23#19098779C411yescharacteristic
24#19098716C411yescharacteristic
25#19098699C411yescharacteristic
26#19098690C2211yescharacteristic, complemented
27#19098687C2211yescharacteristic, complemented
28C4×#11929613C2211yescharacteristic
29C4×#11929613C2211yescharacteristic
30C4×D632C2211yescharacteristic
31C4×D632C2211yescharacteristic
32#19098640C2211yescharacteristic, complemented
33#19098639—21no—
34#19098632—21no—
35C8×D316—21no—
36C8×D316—21no—
37#19098628C2211yescharacteristic
38#19098628C2211yescharacteristic
39#19098628C2211yescharacteristic
40#19098628C2211yescharacteristic
41#19098628C2211yescharacteristic, complemented
42#19098623—21no—
43#19098616C2211yescharacteristic
44#19098616C2211yescharacteristic
45#19098588—21no—
46C22⋊C8×D79—21no—
47C22⋊C8×D79—21no—
48#19098293C2211yescharacteristic, complemented
49#19098293C2211yescharacteristic, complemented
50#19098290C2211yescharacteristic
51#19098290C2211yescharacteristic
52#19098290C2211yescharacteristic
53#19098290C2211yescharacteristic
54#19098290C2211yescharacteristic, complemented
55#19098290C2211yescharacteristic, complemented
56#19098285C2211yescharacteristic, complemented
57#19098285C2211yescharacteristic, complemented
58#19098284C2211yescharacteristic, complemented
59C8×C79⋊C8C2211yescharacteristic, complemented
60#431×C79C2211yescharacteristic
61#431×C79C2211yescharacteristic
62#423×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic, complemented
64#335×C79C2211yescharacteristic, complemented
65#334×C79C2211yescharacteristic, complemented
66#334×C79C2211yescharacteristic, complemented
67C22×D316C2×C411yescharacteristic
68C22×C4×D79—21no—
69C2×#11929613C2×C411yescharacteristic
70C2×#11929613C2×C411yescharacteristic
71C2×D632C2×C411yescharacteristic
72C2×D632C2×C411yescharacteristic
73C2×#11929607—21no—
74C2×C8×D79—21no—
75C2×C8×D79—21no—
76C2×C8×D79—21no—
77C2×C8×D79—21no—
78C2×C8×D79—21no—
79#14579155—21no—
80#14579153C2×C411yescharacteristic
81#14579149C2×C411yescharacteristic
82#14579144—21no—
83#14579140C2×C411yescharacteristic
84#14579135C2×C411yescharacteristic
85C4×Dic158C2311yescharacteristic
86C4×D316—21no—
87C4×D316—21no—
88C4×D316C2×C411yescharacteristic
89C4×D316C2×C411yescharacteristic
90C4×D316C2311yescharacteristic
91C4×D316C2311yescharacteristic
92C4×D316C2311yescharacteristic
93C4×C22⋊D79—21no—
94#14579118—21no—
95C2×C4⋊Dic79C2×C411yescharacteristic
96C2×(C2×D79)⋊C4—21no—
97#14579095C2×C411yescharacteristic
98#14579095C2×C411yescharacteristic
99#14579093C2×C411yescharacteristic
100#14579093C2×C411yescharacteristic
101C4⋊(C79⋊C8)—21no—
102C4⋊(C79⋊C8)D411yescharacteristic
103C4⋊(C79⋊C8)D411yescharacteristic
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106#14579086C2×C411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579083C2×C411yescharacteristic
109#14579083C2×C411yescharacteristic
110#14579083C2×C411yescharacteristic
111#14579083C2×C411yescharacteristic
112#14579083C2×C411yescharacteristic
113#14579083C2×C411yescharacteristic
114C22⋊(C79⋊C8)—21no—
115C22⋊(C79⋊C8)D411yescharacteristic
116C22⋊(C79⋊C8)D411yescharacteristic
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8—21no—
120(C2×D79)⋊C8—21no—
121(C2×D79)⋊C8—21no—
122(C2×D79)⋊C8—21no—
123#14579071—21no—
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C4×C79⋊C8C2311yescharacteristic
128C8×Dic79—21no—
129C2×M16×C79—21no—
130C2×C42×C79C2311yescharacteristic
131C4⋊C8×C79—21no—
132C4⋊C8×C79C2×C411yescharacteristic
133C4⋊C8×C79C2×C411yescharacteristic
134C4⋊C8×C79C2311yescharacteristic
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138C22⋊C8×C79—21no—
139C22⋊C8×C79C2×C411yescharacteristic
140C22⋊C8×C79C2×C411yescharacteristic
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C23×D79—21no—
145C2×Dic158C22×C411yescharacteristic
146C2×D316—22no—
147C2×D316—22no—
148C2×D316—21no—
149C2×D316C22×C411yescharacteristic
150C2×D316C22×C411yescharacteristic
151C2×D316C22×C411yescharacteristic
152C2×C22⋊D79—21no—
153C22×Dic79—21no—
154C2×C4×D79—21no—
155C2×C4×D79—21no—
156C2×C4×D79—21no—
157C2×C4×D79—21no—
158C2×C4×D79—21no—
159C2×C4×D79—81no—
160C2×C4×D79—81no—
161#11929613—22no—
162#11929613—22no—
163D632—22no—
164D632—22no—
165#11929607—81no—
166C2×C79⋊C8—21no—
167C2×C79⋊C8—21no—
168C2×C79⋊C8C2×D411yescharacteristic
169C2×C79⋊C8C2×D411yescharacteristic
170C2×C79⋊C8C2×D411yescharacteristic
171C2×C79⋊C8C2×D411yescharacteristic
172C8×D79—81no—
173C8×D79—81no—
174C8×D79—81no—
175C8×D79—81no—
176C8×D79—81no—
177C4⋊Dic79—21no—
178C4⋊Dic79C22×C411yescharacteristic
179C4⋊Dic79C22×C411yescharacteristic
180C4⋊Dic79C22×C411yescharacteristic
181C4⋊Dic79C22×C411yescharacteristic
182Dic79⋊C4—21no—
183Dic79⋊C4—21no—
184Dic79⋊C4—21no—
185Dic79⋊C4—21no—
186C22⋊Dic79—21no—
187(C2×D79)⋊C4—21no—
188(C2×D79)⋊C4—21no—
189(C2×D79)⋊C4—21no—
190(C2×D79)⋊C4—21no—
191(C2×D79)⋊C4—21no—
192(C2×D79)⋊C4—21no—
193(C2×D79)⋊C4—21no—
194(C2×D79)⋊C4—21no—
195C4×Dic79—21no—
196C22×C4×C79C22×C411yescharacteristic
197C22×C4×C79C2×D411yescharacteristic
198C22×C4×C79#4111yescharacteristic
199M16×C79—81no—
200C2×C8×C79—21no—
201C2×C8×C79—21no—
202C2×C8×C79—21no—
203C2×C8×C79—21no—
204C2×C8×C79C22×C411yescharacteristic
205C2×C8×C79C22×C411yescharacteristic
206C2×C8×C79C22×C411yescharacteristic
207C2×C8×C79C22×C411yescharacteristic
208C42×C79C22×C411yescharacteristic
209C42×C79C2×D411yescharacteristic
210C42×C79#4111yescharacteristic
211C42×C79C2411yescharacteristic
212C23×C79C4×D411yescharacteristic
213C22×D79—21no—
214C22×D79—21no—
215C22×D79—24no—
216C22×D79—21no—
217C22×D79—81no—
218C22×D79—81no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—41no—
224C2×C4×C79C4×D411yescharacteristic
225C2×C4×C79C4×D411yescharacteristic
226C2×C4×C79C4×D411yescharacteristic
227C2×C4×C79C23×C411yescharacteristic
228C2×C4×C79C22×D411yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C22⋊D79—81no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—81no—
237D316—24no—
238D316—21no—
239D316—21no—
240D316—21no—
241D316—21no—
242D316#13112yes—
243D316#13112yes—
244C4×D79—41no—
245C4×D79—41no—
246C4×D79—41no—
247C4×D79—41no—
248C4×D79—41no—
249C4×D79—41no—
250C4×D79—81no—
251C4×D79—81no—
252Dic158—21no—
253Dic158—21no—
254C8×C79—21no—
255C8×C79—21no—
256C8×C79—21no—
257C8×C79—21no—
258C8×C79—41no—
259C8×C79—41no—
260C8×C79—41no—
261C8×C79—41no—
262C79⋊C8—21no—
263C79⋊C8—21no—
264C79⋊C8—21no—
265C79⋊C8—21no—
266C79⋊C8—41no—
267C79⋊C8—41no—
268C22×C79—21no—
269C22×C79—21no—
270C22×C79—21no—
271C22×C79C2×C4×D411yescharacteristic
272C2×D79—22no—
273C2×D79—22no—
274C2×D79—41no—
275C2×D79—41no—
276C2×D79—41no—
277C2×D79—41no—
278C2×D79—41no—
279C2×D79—41no—
280C2×D79—41no—
281C2×D79—84no—
282C4×C79—21no—
283C4×C79—21no—
284C4×C79—21no—
285C4×C79—21no—
286C4×C79C2×#13111yescharacteristic
287C4×C79C2×#13111yescharacteristic
288C4×C79#57611yescharacteristic
289C4×C79#57711yescharacteristic
290Dic79—41no—
291Dic79—41no—
292Dic79—41no—
293Dic79—41no—
294Dic79—41no—
295#17918—791nomaximal, Sylow
296C2×C79—41no—
297C2×C79#292911yescharacteristic
298C2×C79#294611yescharacteristic
299C2×C79#295711yescharacteristic
300D79—42no—
301D79—42no—
302D79—81no—
303#3105—791no—
304#2993—791no—
305#2972—791no—
306#1586—791no—
307#1585—791no—
308#1582—791no—
309#1581—791no—
310#1580—791no—
311SD16⋊C8—791no—
312C8×SD16—791no—
313C8×D8—791no—
314#1574D7911yescharacteristic, complemented
315#1486—791no—
316#1482—791no—
317#1454—791no—
318C79#1791811yescharacteristic, complemented, Sylow
319#529—791no—
320C2×C4×D4—791no—
321#484—791no—
322#481—791no—
323#478—791no—
324#478—791no—
325#463—791no—
326#463—791no—
327#450—791no—
328#447—791no—
329#446—791no—
330C8×Q8—791no—
331C4×SD16—791no—
332C4×SD16—791no—
333C4×D8—791no—
334C4×D8—791no—
335#436—791no—
336#436—1581no—
337#435—1581no—
338C8×D4—791no—
339C8×D4—791no—
340C8×D4—791no—
341C8×D4—791no—
342C8×D4—791no—
343C8×D4—1581no—
344C8×D4—1581no—
345#433—1581no—
346#431—791no—
347#431—791no—
348#431C2×D7911yescharacteristic
349#431C2×D7911yescharacteristic
350#424—1581no—
351#423C2×D7911yescharacteristic
352C2×C22⋊C8—1581no—
353C2×C22⋊C8—1581no—
354#335C2×D7911yescharacteristic, complemented
355#335C2×D7911yescharacteristic, complemented
356#334C2×D7911yescharacteristic, complemented
357#334C2×D7911yescharacteristic, complemented
358Q8⋊C8—791no—
359Q8⋊C8—791no—
360D4⋊C8—791no—
361D4⋊C8—791no—
362D4⋊C8—791no—
363D4⋊C8—791no—
364D4⋊C8—791no—
365D4⋊C8—791no—
366#322—791no—
367#322—791no—
368#322—791no—
369C82—791no—
370C22×D4—791no—
371C23×C4—1581no—
372C2×SD16—791no—
373C2×SD16—791no—
374C2×D8—791no—
375C2×D8—791no—
376C2×M16—21no—
377C2×M16—1581no—
378C22×C8—1581no—
379C22×C8—1581no—
380C22×C8—1581no—
381C22×C8—1581no—
382C22×C8—1581no—
383#126—1581no—
384#125—791no—
385#124—791no—
386#123—1581no—
387C22⋊Q8—791no—
388#121—791no—
389C4×Q8—791no—
390C4×D4—791no—
391C4×D4—791no—
392C4×D4—791no—
393C4×D4—791no—
394C4×D4—791no—
395C4×D4—1581no—
396C4×D4—1581no—
397C4×D4—1581no—
398#117—1581no—
399C2×C4⋊C4—791no—
400C2×C22⋊C4—1581no—
401C2×C42C22×D7911yescharacteristic
402#107—791no—
403#107—791no—
404#106—791no—
405#106—791no—
406C4⋊C8—21no—
407C4⋊C8—791no—
408C4⋊C8—791no—
409C4⋊C8—1581no—
410C4⋊C8—1581no—
411C4⋊C8—1581no—
412C4⋊C8C4×D7911yescharacteristic
413C4⋊C8C4×D7911yescharacteristic
414C4⋊C8C22×D7911yescharacteristic
415C4⋊C8C22×D7911yescharacteristic
416C4⋊C8C22×D7911yescharacteristic
417C4⋊C8C22×D7911yescharacteristic
418#103—791no—
419#103—791no—
420#102—791no—
421#102—791no—
422#102—791no—
423#102—791no—
424#102—791no—
425#102—791no—
426C22⋊C8—21no—
427C22⋊C8—791no—
428C22⋊C8—791no—
429C22⋊C8—1581no—
430C22⋊C8—1581no—
431C22⋊C8—1581no—
432C22⋊C8—1581no—
433C22⋊C8—1581no—
434C22⋊C8—1581no—
435C22⋊C8—1581no—
436C22⋊C8C4×D7911yescharacteristic
437C22⋊C8C4×D7911yescharacteristic
438#97—1581no—
439C4×C8—21no—
440C4×C8—791no—
441C4×C8—791no—
442C4×C8—791no—
443C4×C8—791no—
444C4×C8—1581no—
445C4×C8C22×D7911yescharacteristic
446C4×C8C22×D7911yescharacteristic
447C24—1581no—
448C2×Q8—791no—
449C2×D4—791no—
450C2×D4—791no—
451C2×D4—791no—
452C2×D4—1582no—
453C2×D4—1582no—
454C2×D4—1581no—
455C2×D4—1581no—
456C22×C4—6321no—
457C22×C4—6321no—
458C22×C4—1581no—
459C22×C4—1581no—
460C22×C4—1581no—
461C22×C4—1581no—
462C22×C4—1581no—
463C22×C4—1581no—
464C22×C4C2×C4×D7911yescharacteristic
465C22×C4D4×D7911yescharacteristic
466C22×C4#1192962511yescharacteristic
467SD16—1582no—
468SD16—1582no—
469D8—1582no—
470D8—1582no—
471M16—81no—
472M16—6321no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—791no—
478C2×C8—791no—
479C2×C8—791no—
480C2×C8—791no—
481C2×C8—6321no—
482C2×C8—6321no—
483C2×C8—6321no—
484C2×C8—6321no—
485C2×C8—6321no—
486C2×C8—1581no—
487C2×C8—1581no—
488C2×C8C2×C4×D7911yescharacteristic
489C2×C8C2×C4×D7911yescharacteristic
490C2×C8C2×C4×D7911yescharacteristic
491C2×C8C2×C4×D7911yescharacteristic
492C4⋊C4—791no—
493C4⋊C4—791no—
494C4⋊C4—791no—
495C4⋊C4—791no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C4⋊C4—1581no—
501C22⋊C4—1581no—
502C22⋊C4—1581no—
503C22⋊C4—1581no—
504C22⋊C4—1581no—
505C22⋊C4—1581no—
506C22⋊C4—1581no—
507C22⋊C4—1581no—
508C22⋊C4—1581no—
509C22⋊C4—1581no—
510C42—1581no—
511C42C2×C4×D7911yescharacteristic
512C42D4×D7911yescharacteristic
513C42#1192962511yescharacteristic
514C42C23×D7911yescharacteristic
515C23—6321no—
516C23—6321no—
517C23—1584no—
518C23—1581no—
519C23—1581no—
520C23—1581no—
521C23#1457912311yescharacteristic
522Q8—1581no—
523Q8—1581no—
524D4—792no—
525D4—792no—
526D4—6321no—
527D4—1584no—
528D4—1581no—
529D4—1581no—
530D4—1581no—
531D4—1581no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—6321no—
544C2×C4—6321no—
545C2×C4—6321no—
546C2×C4—1581no—
547C2×C4—1581no—
548C2×C4—1581no—
549C2×C4—1581no—
550C2×C4—1581no—
551C2×C4#1457912311yescharacteristic
552C2×C4#1457912311yescharacteristic
553C2×C4#1457912311yescharacteristic
554C2×C4C22×C4×D7911yescharacteristic
555C2×C4C2×D4×D7911yescharacteristic
556C2×C4C2×#1192962511yescharacteristic
557C8—21no—
558C8—21no—
559C8—21no—
560C8—21no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—41no—
565C8—3161no—
566C8—3161no—
567C8—1581no—
568C8—1581no—
569C8—1581no—
570C8—1581no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—3161no—
575C22—3161no—
576C22—3161no—
577C22—3161no—
578C22—3161no—
579C22—3161no—
580C22—3161no—
581C22—6324no—
582C22—1582no—
583C22—1582no—
584C22C2×#1457912311yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—3161no—
590C4—3161no—
591C4—3161no—
592C4—3161no—
593C4—3161no—
594C4#131×D7911yescharacteristic
595C4#131×D7911yescharacteristic
596C4#1909936711yescharacteristic
597C4#1909937211yescharacteristic
598C2—41no—
599C2—3162no—
600C2—3162no—
601C2—6321no—
602C2#2377209311yescharacteristic
603C2#2377223011yescharacteristic
604C2#2377231611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1574C2
socleC22×C79C2×C4×D4
prime radical#19098699C4
cosocleC42×C79C24
hypercenter#1574D79
hypocenterC79#17918
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1574

Complements

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

NH ≅ G/Ntype
2. #23772545C2semidirect
3. #23772423C2semidirect
4. #23762263C2semidirect
5. #23762256C2semidirect
6. #23762235C2semidirect
7. #23762228C2semidirect
8. #23762223C2semidirect
9. #23762211C2semidirect
10. #23762199C2semidirect
11. #23762194C2semidirect
12. #23761824C2semidirect
13. #23761806C2semidirect
14. #23761700C2semidirect
15. C79×#1574C2semidirect
26. #19098690C22semidirect
27. #19098687C22semidirect
32. #19098640C22semidirect
41. #19098628C22semidirect
48. #19098293C22semidirect
49. #19098293C22semidirect
54. #19098290C22semidirect
55. #19098290C22semidirect
56. #19098285C22semidirect
57. #19098285C22semidirect
58. #19098284C22semidirect
59. C8×C79⋊C8C22semidirect
63. #335×C79C22semidirect
64. #335×C79C22semidirect
65. #334×C79C22semidirect
66. #334×C79C22semidirect
314. #1574D79semidirect
318. C79#17918semidirect
354. #335C2×D79semidirect
355. #335C2×D79semidirect
356. #334C2×D79semidirect
357. #334C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237734492
#237725452
#237724232
#237622632
#237622562
#237622352
#237622282
#237622232
#237622112
#237621992
#237621942
#237618242
#237618062
#237617002
C79×#15742
#1791879