Finite Group Databasebeta

← #34347342

Subgroups — #34347342

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#23768634C211yesmaximal, characteristic, complemented
2#23764933C211yesmaximal, characteristic
3#23764921C211yesmaximal, characteristic
4#23764788C211yesmaximal, characteristic
5#23764788C211yesmaximal, characteristic
6#23764168C211yesmaximal, characteristic, complemented
7#23764166C211yesmaximal, characteristic, complemented
8#23764158C211yesmaximal, characteristic, complemented
9#23764119C211yesmaximal, characteristic, complemented
10#23763985C211yesmaximal, characteristic, complemented
11#23763978C211yesmaximal, characteristic, complemented
12#23763562C211yesmaximal, characteristic, complemented
13#23763410C211yesmaximal, characteristic
14C4×#14579082C211yesmaximal, characteristic
15C79×#2357C211yesmaximal, characteristic
16C2×#14579094C2211yescharacteristic
17C2×#14579094C2211yescharacteristic
18C2×#14579094C2211yescharacteristic
19C2×#14579092C2211yescharacteristic
20C2×#14579092C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×C4⋊(C79⋊C8)C2211yescharacteristic
23C2×C4⋊(C79⋊C8)C2211yescharacteristic
24C2×C4⋊(C79⋊C8)C2211yescharacteristic
25C2×#14579082C2211yescharacteristic
26C2×#14579082C2211yescharacteristic
27C2×#14579082C2211yescharacteristic
28C2×#14579082C2211yescharacteristic
29C2×C4×C79⋊C8C2211yescharacteristic
30#19098502—21no—
31#19098496—21no—
32#19098461C2211yescharacteristic
33#19098461C2211yescharacteristic
34#19098450—21no—
35#19098446—21no—
36#19098433C2211yescharacteristic
37#19098429C2211yescharacteristic
38#19098429C2211yescharacteristic
39C4×C4⋊Dic79C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C2×C4×D4×C79C2211yescharacteristic
49#390×C79C2211yescharacteristic
50#390×C79C2211yescharacteristic
51#388×C79—21no—
52#388×C79—21no—
53#382×C79C2211yescharacteristic
54#381×C79—21no—
55C4×C4⋊C4×C79C2211yescharacteristic
56C4×C4⋊C4×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58C4×C22⋊C4×C79—21no—
59C22×C79⋊C8C2311yescharacteristic
60C22×C79⋊C8C2311yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63C2×C4⋊Dic79—21no—
64C2×C4⋊Dic79—21no—
65C2×C4⋊Dic79C2311yescharacteristic
66C2×C4⋊Dic79C2311yescharacteristic
67C2×C4⋊Dic79C2311yescharacteristic
68C2×C4⋊Dic79C2311yescharacteristic
69C2×Dic79⋊C4—21no—
70C2×Dic79⋊C4—21no—
71C2×C4×Dic79—21no—
72#14579094—41no—
73#14579094—41no—
74#14579094—41no—
75#14579092—22no—
76#14579092—22no—
77#14579092—41no—
78C4⋊(C79⋊C8)—22no—
79C4⋊(C79⋊C8)—22no—
80C4⋊(C79⋊C8)—22no—
81#14579082—22no—
82#14579082—22no—
83#14579082—41no—
84#14579082—41no—
85C4×C79⋊C8—22no—
86C79⋊#95—21no—
87C79⋊#95—21no—
88C79⋊#95—21no—
89C79⋊#95—21no—
90C79⋊#95—21no—
91C79⋊#95—21no—
92C79⋊#95—21no—
93C22×D4×C79C2311yescharacteristic
94C23×C4×C79—21no—
95C4×D4×C79—22no—
96C4×D4×C79Q814yes—
97C2×C4⋊C4×C79D411yescharacteristic
98C2×C4⋊C4×C79D411yescharacteristic
99C2×C4⋊C4×C79C2311yescharacteristic
100C2×C4⋊C4×C79C2311yescharacteristic
101C2×C4⋊C4×C79C2311yescharacteristic
102C2×C22⋊C4×C79—21no—
103C2×C22⋊C4×C79—21no—
104C2×C22⋊C4×C79—21no—
105C2×C22⋊C4×C79—21no—
106C2×C22⋊C4×C79—21no—
107C2×C42×C79—21no—
108C2×C42×C79—21no—
109C2×C42×C79C2311yescharacteristic
110C2×C42×C79C2311yescharacteristic
111C2×C42×C79C2311yescharacteristic
112#95×C79—21no—
113#95×C79—21no—
114#95×C79—21no—
115#95×C79—21no—
116#95×C79D411yescharacteristic
117#95×C79D411yescharacteristic
118C22×Dic79—21no—
119C22×Dic79—21no—
120C22×Dic79—21no—
121C22×Dic79—21no—
122C2×C79⋊C8—21no—
123C2×C79⋊C8—21no—
124C2×C79⋊C8—21no—
125C2×C79⋊C8—21no—
126C2×C79⋊C8—21no—
127C2×C79⋊C8—21no—
128C2×C79⋊C8—21no—
129C2×C79⋊C8—21no—
130C2×C79⋊C8—21no—
131C2×C79⋊C8—21no—
132C2×C79⋊C8#4112yes—
133C2×C79⋊C8#4112yes—
134C4⋊Dic79—21no—
135C4⋊Dic79—21no—
136C4⋊Dic79—21no—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—42no—
143C4⋊Dic79—81no—
144Dic79⋊C4—42no—
145Dic79⋊C4—81no—
146C4×Dic79—81no—
147C24×C79—21no—
148C2×D4×C79—21no—
149C2×D4×C79—21no—
150C2×D4×C79—21no—
151C2×D4×C79—21no—
152C2×D4×C79—21no—
153C2×D4×C79C2×Q812yes—
154C22×C4×C79—21no—
155C22×C4×C79—24no—
156C22×C4×C79—21no—
157C22×C4×C79—21no—
158C22×C4×C79—81no—
159C22×C4×C79—81no—
160C22×C4×C79C2×D411yescharacteristic
161C22×C4×C79C2×D411yescharacteristic
162C22×C4×C79C2×D411yescharacteristic
163C22×C4×C79C2×D411yescharacteristic
164C22×C4×C79C2×D411yescharacteristic
165C22×C4×C79C2×D411yescharacteristic
166C22×C4×C79C2411yescharacteristic
167C4⋊C4×C79—22no—
168C4⋊C4×C79—21no—
169C4⋊C4×C79—21no—
170C4⋊C4×C79—21no—
171C4⋊C4×C79—21no—
172C4⋊C4×C79—21no—
173C4⋊C4×C79—41no—
174C4⋊C4×C79C2×Q812yes—
175C22⋊C4×C79—24no—
176C22⋊C4×C79—42no—
177C22⋊C4×C79—42no—
178C22⋊C4×C79—81no—
179C22⋊C4×C79—81no—
180C42×C79—21no—
181C42×C79—21no—
182C42×C79—21no—
183C42×C79—21no—
184C42×C79—81no—
185C42×C79—81no—
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—21no—
205C2×C4×C79—21no—
206C2×C4×C79—22no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—41no—
217C2×C4×C79—41no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—84no—
225C2×C4×C79C22⋊Q812yes—
226C2×C4×C79C22⋊Q812yes—
227C2×C4×C79#13512yes—
228C2×C4×C79C22×Q811yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C2×C4×C79H(5,𝔽2)11yescharacteristic
233C2×C4×C792−1+411yescharacteristic
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—41no—
246C79⋊C8—22no—
247C79⋊C8—22no—
248C79⋊C8—41no—
249C79⋊C8—41no—
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—22no—
267C4×C79—42no—
268C4×C79—41no—
269C4×C79—41no—
270C4×C79—41no—
271C4×C79—81no—
272C4×C79—81no—
273C4×C79#53311yescharacteristic
274C4×C79#54411yescharacteristic
275C4×C79#55211yescharacteristic
276C4×C79#55311yescharacteristic
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#2357D7911yescharacteristic
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#1804—791no—
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⋊C8—791no—
316C2×#102—791no—
317C2×#102—791no—
318C2×#102—791no—
319C2×#102—791no—
320C2×C4×C8—791no—
321#400—1581no—
322#398—1581no—
323#390C2×D7911yescharacteristic
324#390C2×D7911yescharacteristic
325#389—791no—
326#389—791no—
327#388—21no—
328#388—21no—
329#387—1581no—
330#387—1581no—
331#384—791no—
332#382—791no—
333#382—791no—
334#382C2×D7911yescharacteristic
335#381—21no—
336C4×C4⋊C4—791no—
337C4×C4⋊C4C2×D7911yescharacteristic
338C4×C4⋊C4C2×D7911yescharacteristic
339C4×C4⋊C4C2×D7911yescharacteristic
340C4×C22⋊C4—21no—
341#340—791no—
342#340—791no—
343#340—791no—
344#340—791no—
345#340—791no—
346#340—791no—
347#336—791no—
348#336—791no—
349C22×D4C22×D7911yescharacteristic
350C23×C4—21no—
351C22×C8—791no—
352C22×C8—791no—
353C22×C8—791no—
354C22×C8—791no—
355C4×D4—22no—
356C4×D4Dic15814yes—
357C2×C4⋊C4—791no—
358C2×C4⋊C4—791no—
359C2×C4⋊C4—791no—
360C2×C4⋊C4—791no—
361C2×C4⋊C4—1581no—
362C2×C4⋊C4—1581no—
363C2×C4⋊C4—1581no—
364C2×C4⋊C4—1581no—
365C2×C4⋊C4C22⋊D7911yescharacteristic
366C2×C4⋊C4C22⋊D7911yescharacteristic
367C2×C4⋊C4C22×D7911yescharacteristic
368C2×C4⋊C4C22×D7911yescharacteristic
369C2×C4⋊C4C22×D7911yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—1581no—
378C2×C42C22×D7911yescharacteristic
379C2×C42C22×D7911yescharacteristic
380C2×C42C22×D7911yescharacteristic
381#107—3161no—
382#107—3161no—
383#107—3161no—
384#106—3161no—
385#106—1582no—
386#106—1582no—
387C4⋊C8—1582no—
388C4⋊C8—1582no—
389C4⋊C8—1582no—
390#102—3161no—
391#102—3161no—
392#102—1582no—
393#102—1582no—
394C4×C8—1582no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—1581no—
400#95—1581no—
401#95—1581no—
402#95—1581no—
403#95—1581no—
404#95—1581no—
405#95—1581no—
406#95C22⋊D7911yescharacteristic
407#95C22⋊D7911yescharacteristic
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—1581no—
422C22×C4—1581no—
423C22×C4—1581no—
424C22×C4—1581no—
425C22×C4C2×C22⋊D7911yescharacteristic
426C22×C4C2×C22⋊D7911yescharacteristic
427C22×C4C2×C22⋊D7911yescharacteristic
428C22×C4C2×C22⋊D7911yescharacteristic
429C22×C4C2×C22⋊D7911yescharacteristic
430C22×C4C2×C22⋊D7911yescharacteristic
431C22×C4C23×D7911yescharacteristic
432C2×C8—792no—
433C2×C8—792no—
434C2×C8—1581no—
435C2×C8—1581no—
436C2×C8—1581no—
437C2×C8—1581no—
438C2×C8—1581no—
439C2×C8—1581no—
440C2×C8—1581no—
441C2×C8—1581no—
442C2×C8—1581no—
443C2×C8—1581no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—41no—
451C4⋊C4—3162no—
452C4⋊C4—3162no—
453C4⋊C4—6321no—
454C4⋊C4—6321no—
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—42no—
466C22⋊C4—42no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—81no—
474C42—81no—
475C42—6321no—
476C42C2×Dic15812yes—
477C42#1192962412yes—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23C22×C22⋊D7911yescharacteristic
486D4—22no—
487D4—22no—
488D4—41no—
489D4—41no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—84no—
515C2×C4—3161no—
516C2×C4—3161no—
517C2×C4—3161no—
518C2×C4—3161no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—3161no—
523C2×C4—3161no—
524C2×C4—3161no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4#1457914012yes—
528C2×C4#1457914012yes—
529C2×C4#1457918112yes—
530C2×C4C22×Dic15811yescharacteristic
531C2×C4#41×D7911yescharacteristic
532C2×C4#41×D7911yescharacteristic
533C2×C4C2×#1192962411yescharacteristic
534C2×C4#1457920711yescharacteristic
535C2×C4#1457920811yescharacteristic
536C8—3161no—
537C8—3161no—
538C8—1582no—
539C8—1582no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22C2×#1457914011yescharacteristic
548C22#1909921211yescharacteristic
549C22#1909923211yescharacteristic
550C22C2×#1457918111yescharacteristic
551C22#1909936111yescharacteristic
552C22#1909936611yescharacteristic
553C22#1909937411yescharacteristic
554C4—22no—
555C4—22no—
556C4—22no—
557C4—42no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4—6321no—
564C4—6321no—
565C4—6321no—
566C4—6321no—
567C4#1909911611yescharacteristic
568C4#1909919111yescharacteristic
569C4#1909925211yescharacteristic
570C4#1909925711yescharacteristic
571C2—82no—
572C2#2376710711yescharacteristic
573C2#2377318811yescharacteristic
574C2#2377323111yescharacteristic
575C2#2377579611yescharacteristic
576C2#2377595411yescharacteristic
577C2#2377639211yescharacteristic
578C2#2377648411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2357C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
hypercenter#2357D79
hypocenterC79#26096
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. #23768634C2semidirect
6. #23764168C2semidirect
7. #23764166C2semidirect
8. #23764158C2semidirect
9. #23764119C2semidirect
10. #23763985C2semidirect
11. #23763978C2semidirect
12. #23763562C2semidirect
305. C79#26096semidirect

Maximal subgroups

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

subgroupindex
#237686342
#237649332
#237649212
#237647882
#237641682
#237641662
#237641582
#237641192
#237639852
#237639782
#237635622
#237634102
C4×#145790822
C79×#23572
#2609679