Finite Group Databasebeta

← #34347350

Subgroups — #34347350

Order 20224.

678 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23768635C211yesmaximal, characteristic, complemented
2#23766373C211yesmaximal, characteristic
3#23764931C211yesmaximal, characteristic
4#23764919C211yesmaximal, characteristic
5#23764784C211yesmaximal, characteristic
6#23764784C211yesmaximal, characteristic
7#23764170C211yesmaximal, characteristic, complemented
8#23764167C211yesmaximal, characteristic, complemented
9#23764157C211yesmaximal, characteristic, complemented
10#23764116C211yesmaximal, characteristic, complemented
11#23763988C211yesmaximal, characteristic, complemented
12#23763981C211yesmaximal, characteristic, complemented
13#23763562C211yesmaximal, characteristic, complemented
14C4×#14579082C211yesmaximal, characteristic
15C79×#1804C211yesmaximal, characteristic
16C2×#14579095C2211yescharacteristic
17C2×#14579094C2211yescharacteristic
18C2×#14579094C2211yescharacteristic
19C2×#14579093C2211yescharacteristic
20C2×#14579092C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×Dic79⋊C8C2211yescharacteristic
23C2×Dic79⋊C8C2211yescharacteristic
24C2×#14579082C2211yescharacteristic
25C2×#14579082C2211yescharacteristic
26C2×C4×C79⋊C8C2211yescharacteristic
27#19098505—21no—
28#19098495—21no—
29#19098466C2211yescharacteristic
30#19098466C2211yescharacteristic
31#19098462C2211yescharacteristic
32#19098462C2211yescharacteristic
33#19098456—21no—
34#19098456—21no—
35#19098450—21no—
36#19098446—21no—
37#19098432C2211yescharacteristic
38#19098429C2211yescharacteristic
39#19098428C2211yescharacteristic
40#19098426—21no—
41C4⋊C4×Dic79C2211yescharacteristic
42C4⋊C4×Dic79C2211yescharacteristic
43C4×C4⋊Dic79C2211yescharacteristic
44C4×C4⋊Dic79C2211yescharacteristic
45C4×C22⋊Dic79—21no—
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098317C2211yescharacteristic
51#19098317C2211yescharacteristic
52C2×C4×D4×C79C2211yescharacteristic
53C2×C4⋊C8×C79C2211yescharacteristic
54C2×#102×C79C2211yescharacteristic
55C2×#102×C79C2211yescharacteristic
56#382×C79C2211yescharacteristic
57#340×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59C22×C79⋊C8C2311yescharacteristic
60C22×C79⋊C8C2311yescharacteristic
61C2×C4⋊Dic79—21no—
62C2×C4⋊Dic79—21no—
63C2×C4⋊Dic79D411yescharacteristic
64C2×C4⋊Dic79D411yescharacteristic
65C2×C4⋊Dic79C2311yescharacteristic
66C2×C4⋊Dic79C2311yescharacteristic
67C2×C4⋊Dic79C2311yescharacteristic
68C2×C4⋊Dic79C2311yescharacteristic
69C2×Dic79⋊C4—21no—
70C2×Dic79⋊C4—21no—
71C2×C22⋊Dic79—21no—
72C2×C22⋊Dic79—21no—
73C2×C22⋊Dic79—21no—
74C2×C22⋊Dic79—21no—
75C2×C4×Dic79—21no—
76C2×C4×Dic79—21no—
77C2×C4×Dic79—21no—
78C2×C4×Dic79C2311yescharacteristic
79C2×C4×Dic79C2311yescharacteristic
80#14579095—41no—
81#14579094—41no—
82#14579094—41no—
83#14579093—41no—
84#14579092—22no—
85#14579092—22no—
86Dic79⋊C8—22no—
87Dic79⋊C8—22no—
88#14579082—22no—
89#14579082—22no—
90C4×C79⋊C8—22no—
91C79⋊#95—21no—
92C79⋊#95—21no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95D411yescharacteristic
101C79⋊#95D411yescharacteristic
102C22×D4×C79C2311yescharacteristic
103C23×C4×C79—21no—
104C22×C8×C79C2311yescharacteristic
105C22×C8×C79C2311yescharacteristic
106C4×D4×C79—22no—
107C4×D4×C79Q814yes—
108C2×C4⋊C4×C79C2311yescharacteristic
109C2×C4⋊C4×C79C2311yescharacteristic
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C22⋊C4×C79—21no—
112C2×C42×C79C2311yescharacteristic
113C4⋊C8×C79—22no—
114#102×C79—41no—
115#102×C79—41no—
116#95×C79—21no—
117#95×C79—21no—
118C22×Dic79—21no—
119C22×Dic79—21no—
120C22×Dic79—21no—
121C22×Dic79—21no—
122C22×Dic79C2×D411yescharacteristic
123C22×Dic79C2×D411yescharacteristic
124C22×Dic79C2×D411yescharacteristic
125C22×Dic79C2×D411yescharacteristic
126C2×C79⋊C8—21no—
127C2×C79⋊C8—21no—
128C2×C79⋊C8—21no—
129C2×C79⋊C8—21no—
130C2×C79⋊C8#4112yes—
131C2×C79⋊C8#4112yes—
132C4⋊Dic79—22no—
133C4⋊Dic79—21no—
134C4⋊Dic79—21no—
135C4⋊Dic79—21no—
136C4⋊Dic79—21no—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—41no—
142C4⋊Dic79—42no—
143C4⋊Dic79—81no—
144Dic79⋊C4—42no—
145Dic79⋊C4—81no—
146C22⋊Dic79—42no—
147C22⋊Dic79—42no—
148C22⋊Dic79—81no—
149C22⋊Dic79—81no—
150C4×Dic79—21no—
151C4×Dic79—21no—
152C4×Dic79—21no—
153C4×Dic79—21no—
154C4×Dic79—81no—
155C4×Dic79—81no—
156C4×Dic79—81no—
157C24×C79—21no—
158C2×D4×C79—21no—
159C2×D4×C79—21no—
160C2×D4×C79—21no—
161C2×D4×C79—21no—
162C2×D4×C79—21no—
163C2×D4×C79C2×Q812yes—
164C22×C4×C79—21no—
165C22×C4×C79—24no—
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79—81no—
169C22×C4×C79—81no—
170C22×C4×C79C2×D411yescharacteristic
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2411yescharacteristic
173C2×C8×C79—21no—
174C2×C8×C79—21no—
175C2×C8×C79—21no—
176C2×C8×C79—21no—
177C2×C8×C79—21no—
178C2×C8×C79—21no—
179C4⋊C4×C79—21no—
180C4⋊C4×C79—21no—
181C4⋊C4×C79—21no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79C2×Q812yes—
185C22⋊C4×C79—24no—
186C42×C79C2×Q812yes—
187C42×C79#4112yes—
188C23×C79—22no—
189C23×C79—41no—
190C23×C79—41no—
191C23×C79—41no—
192C23×C79—41no—
193C23×C79—41no—
194C23×C79—41no—
195C23×C79C22×D411yescharacteristic
196D4×C79—22no—
197D4×C79—22no—
198D4×C79—41no—
199D4×C79—41no—
200C2×C4×C79—21no—
201C2×C4×C79—21no—
202C2×C4×C79—21no—
203C2×C4×C79—21no—
204C2×C4×C79—22no—
205C2×C4×C79—41no—
206C2×C4×C79—41no—
207C2×C4×C79—41no—
208C2×C4×C79—41no—
209C2×C4×C79—41no—
210C2×C4×C79—41no—
211C2×C4×C79—41no—
212C2×C4×C79—41no—
213C2×C4×C79—84no—
214C2×C4×C79C22⋊Q812yes—
215C2×C4×C79C22⋊Q812yes—
216C2×C4×C79C22×Q811yescharacteristic
217C2×C4×C79C2×#4111yescharacteristic
218C2×C4×C79C2×#4111yescharacteristic
219C2×C4×C79C2×#4111yescharacteristic
220C2×C4×C79H(5,𝔽2)11yescharacteristic
221C2×C4×C792−1+411yescharacteristic
222C2×Dic79—21no—
223C2×Dic79—21no—
224C2×Dic79—21no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—41no—
234C2×Dic79—41no—
235C2×Dic79—41no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79#13512yes—
246C8×C79—41no—
247C8×C79—41no—
248C79⋊C8—22no—
249C79⋊C8—22no—
250C22×C79—42no—
251C22×C79—42no—
252C22×C79—42no—
253C22×C79—81no—
254C22×C79—81no—
255C22×C79—81no—
256C22×C79—81no—
257C22×C79C2×C22⋊Q811yescharacteristic
258C22×C79#54711yescharacteristic
259C22×C79#55011yescharacteristic
260C22×C79C2×#13511yescharacteristic
261C22×C79#57511yescharacteristic
262C22×C79#57611yescharacteristic
263C22×C79#57711yescharacteristic
264C4×C79—22no—
265C4×C79—22no—
266C4×C79—42no—
267C4×C79—81no—
268C4×C79—81no—
269C4×C79#53311yescharacteristic
270C4×C79#54411yescharacteristic
271C4×C79#55211yescharacteristic
272C4×C79#55311yescharacteristic
273Dic79—22no—
274Dic79—41no—
275Dic79—41no—
276Dic79—41no—
277Dic79—81no—
278Dic79—81no—
279Dic79—81no—
280Dic79—81no—
281#26096—791nomaximal, Sylow
282C2×C79—82no—
283C2×C79#244111yescharacteristic
284C2×C79#307611yescharacteristic
285C2×C79#308111yescharacteristic
286C2×C79#333911yescharacteristic
287C2×C79#335111yescharacteristic
288C2×C79#339411yescharacteristic
289C2×C79#340311yescharacteristic
290#2570—791no—
291#2357—791no—
292#2094—791no—
293#2092—791no—
294#2067—791no—
295#2067—791no—
296#1949—791no—
297#1949—791no—
298#1947—791no—
299#1939—791no—
300#1920—791no—
301#1919—791no—
302#1846—791no—
303#1804D7911yescharacteristic
304C4×#102—791no—
305C79#2609611yescharacteristic, complemented, Sylow
306C2×C4×D4C2×D7911yescharacteristic
307C2×#107—791no—
308C2×#107—791no—
309C2×#107—791no—
310C2×#106—791no—
311C2×#106—791no—
312C2×#106—791no—
313C2×C4⋊C8—791no—
314C2×C4⋊C8—791no—
315C2×C4⋊C8C2×D7911yescharacteristic
316C2×#102—791no—
317C2×#102—791no—
318C2×#102C2×D7911yescharacteristic
319C2×#102C2×D7911yescharacteristic
320C2×C4×C8—791no—
321#400—1581no—
322#398—1581no—
323#390—791no—
324#390—791no—
325#389—791no—
326#389—791no—
327#388—1581no—
328#388—1581no—
329#387—1581no—
330#387—1581no—
331#384—791no—
332#382—791no—
333#382—791no—
334#382C2×D7911yescharacteristic
335#381—1581no—
336C4×C4⋊C4—791no—
337C4×C4⋊C4—791no—
338C4×C4⋊C4—791no—
339C4×C4⋊C4—791no—
340C4×C22⋊C4—1581no—
341#340—791no—
342#340—791no—
343#340—791no—
344#340—791no—
345#340C2×D7911yescharacteristic
346#340C2×D7911yescharacteristic
347#336—791no—
348#336—791no—
349C22×D4C22×D7911yescharacteristic
350C23×C4—21no—
351C22×C8—791no—
352C22×C8—791no—
353C22×C8C22×D7911yescharacteristic
354C22×C8C22×D7911yescharacteristic
355C4×D4—22no—
356C4×D4Dic15814yes—
357C2×C4⋊C4—791no—
358C2×C4⋊C4—791no—
359C2×C4⋊C4—791no—
360C2×C4⋊C4—791no—
361C2×C4⋊C4—791no—
362C2×C4⋊C4—791no—
363C2×C4⋊C4—1581no—
364C2×C4⋊C4—1581no—
365C2×C4⋊C4—1581no—
366C2×C4⋊C4—1581no—
367C2×C4⋊C4C22×D7911yescharacteristic
368C2×C4⋊C4C22×D7911yescharacteristic
369C2×C4⋊C4C22×D7911yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—1581no—
372C2×C22⋊C4—1581no—
373C2×C22⋊C4—1581no—
374C2×C22⋊C4—1581no—
375C2×C42—791no—
376C2×C42—791no—
377C2×C42—1581no—
378C2×C42—1581no—
379C2×C42—1581no—
380C2×C42C22×D7911yescharacteristic
381#107—3161no—
382#107—3161no—
383#107—3161no—
384#106—3161no—
385#106—1582no—
386#106—1582no—
387C4⋊C8—22no—
388C4⋊C8—1582no—
389C4⋊C8—1582no—
390#102—41no—
391#102—41no—
392#102—1582no—
393#102—1582no—
394C4×C8—1582no—
395#95—21no—
396#95—21no—
397#95—791no—
398#95—791no—
399#95—1581no—
400#95—1581no—
401#95—1581no—
402#95—1581no—
403#95—1581no—
404#95—1581no—
405#95—1581no—
406#95—1581no—
407#95—1581no—
408C24—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4C2×Dic15812yes—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—791no—
422C22×C4—791no—
423C22×C4—791no—
424C22×C4—791no—
425C22×C4—1581no—
426C22×C4—1581no—
427C22×C4—1581no—
428C22×C4—1581no—
429C22×C4D4×D7911yescharacteristic
430C22×C4D4×D7911yescharacteristic
431C22×C4C23×D7911yescharacteristic
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—792no—
439C2×C8—792no—
440C2×C8—1581no—
441C2×C8—1581no—
442C2×C8—1581no—
443C2×C8—1581no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—3161no—
450C4⋊C4—3162no—
451C4⋊C4—3162no—
452C4⋊C4—6321no—
453C4⋊C4—6321no—
454C4⋊C4—1582no—
455C4⋊C4—1581no—
456C4⋊C4—1581no—
457C4⋊C4—1581no—
458C4⋊C4—1581no—
459C4⋊C4—1581no—
460C4⋊C4—1581no—
461C4⋊C4—1581no—
462C4⋊C4—1581no—
463C4⋊C4C2×Dic15812yes—
464C22⋊C4—24no—
465C22⋊C4—3162no—
466C22⋊C4—3162no—
467C22⋊C4—6321no—
468C22⋊C4—6321no—
469C42—6321no—
470C42—6321no—
471C42—6321no—
472C42—1581no—
473C42—1581no—
474C42—1581no—
475C42—1581no—
476C42C2×Dic15812yes—
477C42#1192962312yes—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23C2×D4×D7911yescharacteristic
486D4—22no—
487D4—22no—
488D4—41no—
489D4—41no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—84no—
504C2×C4—3161no—
505C2×C4—3161no—
506C2×C4—3161no—
507C2×C4—3161no—
508C2×C4—3161no—
509C2×C4—3161no—
510C2×C4—3161no—
511C2×C4—3161no—
512C2×C4—3161no—
513C2×C4—3161no—
514C2×C4—3161no—
515C2×C4—3161no—
516C2×C4—792no—
517C2×C4—1581no—
518C2×C4—1581no—
519C2×C4—1581no—
520C2×C4—1581no—
521C2×C4—1581no—
522C2×C4—1581no—
523C2×C4—1581no—
524C2×C4—1581no—
525C2×C4—1581no—
526C2×C4—1581no—
527C2×C4—1581no—
528C2×C4#1457913812yes—
529C2×C4#1457913812yes—
530C2×C4C22×Dic15811yescharacteristic
531C2×C4#41×D7911yescharacteristic
532C2×C4#41×D7911yescharacteristic
533C2×C4C2×#1192962311yescharacteristic
534C2×C4#1457920611yescharacteristic
535C2×C4#1457920911yescharacteristic
536C8—41no—
537C8—41no—
538C8—1582no—
539C8—1582no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22C2×#1457913811yescharacteristic
548C22#1909921511yescharacteristic
549C22#1909923111yescharacteristic
550C22#135×D7911yescharacteristic
551C22#1909936011yescharacteristic
552C22#1909936711yescharacteristic
553C22#1909937311yescharacteristic
554C4—22no—
555C4—22no—
556C4—42no—
557C4—81no—
558C4—81no—
559C4—3161no—
560C4—3161no—
561C4—3161no—
562C4—6321no—
563C4—6321no—
564C4—6321no—
565C4—6321no—
566C4—1582no—
567C4#1909911611yescharacteristic
568C4#1909919411yescharacteristic
569C4#1909924711yescharacteristic
570C4#1909926011yescharacteristic
571C2—82no—
572C2#2376711411yescharacteristic
573C2#2377318411yescharacteristic
574C2#2377323311yescharacteristic
575C2#2377579911yescharacteristic
576C2#2377596711yescharacteristic
577C2#2377639911yescharacteristic
578C2#2377648811yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#1804C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
hypercenter#1804D79
hypocenterC79#26096
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
1. #23768635C2semidirect
7. #23764170C2semidirect
8. #23764167C2semidirect
9. #23764157C2semidirect
10. #23764116C2semidirect
11. #23763988C2semidirect
12. #23763981C2semidirect
13. #23763562C2semidirect
305. C79#26096semidirect

Maximal subgroups

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

subgroupindex
#237686352
#237663732
#237649312
#237649192
#237647842
#237641702
#237641672
#237641572
#237641162
#237639882
#237639812
#237635622
C4×#145790822
C79×#18042
#2609679