Finite Group Databasebeta

← #34347374

Subgroups — #34347374

Order 20224.

708 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766347C211yesmaximal, characteristic
2#23764929C211yesmaximal, characteristic
3#23764924C211yesmaximal, characteristic
4#23764785C211yesmaximal, characteristic
5#23764771C211yesmaximal, characteristic
6#23764166C211yesmaximal, characteristic, complemented
7#23764158C211yesmaximal, characteristic, complemented
8#23764156C211yesmaximal, characteristic, complemented
9#23764124C211yesmaximal, characteristic, complemented
10#23763989C211yesmaximal, characteristic, complemented
11#23763982C211yesmaximal, characteristic, complemented
12#23763569C211yesmaximal, characteristic, complemented
13#23763407C211yesmaximal, characteristic
14#23763228C211yesmaximal, characteristic
15C79×#2570C211yesmaximal, characteristic, complemented
16C2×C4×D316C2211yescharacteristic
17C2×#14579119—21no—
18C2×#14579094C2211yescharacteristic
19C2×#14579094C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×#14579094C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×C4⋊(C79⋊C8)C2211yescharacteristic
25C2×C4⋊(C79⋊C8)C2211yescharacteristic
26C2×C4⋊(C79⋊C8)C2211yescharacteristic
27C2×#14579081C2211yescharacteristic
28C2×#14579081C2211yescharacteristic
29C2×#14579081C2211yescharacteristic
30C2×#14579081C2211yescharacteristic
31C2×#14579072C2211yescharacteristic
32#19098465C2211yescharacteristic
33#19098465C2211yescharacteristic
34#19098455—21no—
35#19098455—21no—
36#19098433C2211yescharacteristic
37#19098420—21no—
38C4×C4⋊Dic79C2211yescharacteristic
39C4×C4⋊Dic79C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48#400×C79—21no—
49#398×C79—21no—
50#389×C79C2211yescharacteristic
51#389×C79C2211yescharacteristic
52#387×C79—21no—
53#387×C79—21no—
54#384×C79C2211yescharacteristic
55#382×C79C2211yescharacteristic
56#382×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58C4×C4⋊C4×C79C2211yescharacteristic
59C22×D316C2311yescharacteristic
60C22×C4×D79—21no—
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C4×D316—22no—
66C4×D316Q814yes—
67#14579119—44no—
68C2×C4⋊Dic79—21no—
69C2×C4⋊Dic79—21no—
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×(C2×D79)⋊C4—21no—
74C2×(C2×D79)⋊C4—21no—
75C2×(C2×D79)⋊C4—21no—
76C2×(C2×D79)⋊C4—21no—
77C2×(C2×D79)⋊C4—21no—
78C2×C4×Dic79—21no—
79C2×C4×Dic79—21no—
80#14579094—22no—
81#14579094—41no—
82#14579094—41no—
83#14579094—41no—
84#14579092—22no—
85#14579092—41no—
86C4⋊(C79⋊C8)—22no—
87C4⋊(C79⋊C8)—22no—
88C4⋊(C79⋊C8)—22no—
89#14579081—22no—
90#14579081—22no—
91#14579081—41no—
92#14579081—41no—
93#14579072—22no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C2×C4⋊C4×C79—21no—
97C2×C4⋊C4×C79—21no—
98C2×C4⋊C4×C79—21no—
99C2×C4⋊C4×C79—21no—
100C2×C4⋊C4×C79D411yescharacteristic
101C2×C4⋊C4×C79D411yescharacteristic
102C2×C4⋊C4×C79C2311yescharacteristic
103C2×C4⋊C4×C79C2311yescharacteristic
104C2×C4⋊C4×C79C2311yescharacteristic
105C2×C4⋊C4×C79C2311yescharacteristic
106C2×C42×C79—21no—
107C2×C42×C79C2311yescharacteristic
108C2×C42×C79C2311yescharacteristic
109C2×C42×C79C2311yescharacteristic
110#95×C79—21no—
111#95×C79—21no—
112#95×C79—21no—
113#95×C79—21no—
114#95×C79—21no—
115#95×C79—21no—
116#95×C79—21no—
117#95×C79D411yescharacteristic
118#95×C79D411yescharacteristic
119C23×D79—21no—
120C2×D316—21no—
121C2×D316—21no—
122C2×D316—21no—
123C2×D316—21no—
124C2×D316—21no—
125C2×D316C2×Q812yes—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C2×C4×D79—24no—
130C2×C4×D79—42no—
131C2×C4×D79—42no—
132C2×C79⋊C8—21no—
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C2×C79⋊C8—21no—
136C2×C79⋊C8—21no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8#4112yes—
143C2×C79⋊C8#4112yes—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—42no—
150C4⋊Dic79—81no—
151C4⋊Dic79C2×Q812yes—
152(C2×D79)⋊C4—24no—
153(C2×D79)⋊C4—42no—
154(C2×D79)⋊C4—42no—
155(C2×D79)⋊C4—81no—
156(C2×D79)⋊C4—81no—
157C4×Dic79—42no—
158C4×Dic79—42no—
159C22×C4×C79—21no—
160C22×C4×C79—21no—
161C22×C4×C79—21no—
162C22×C4×C79—21no—
163C22×C4×C79C2×D411yescharacteristic
164C22×C4×C79C2×D411yescharacteristic
165C22×C4×C79C2×D411yescharacteristic
166C22×C4×C79C2×D411yescharacteristic
167C22×C4×C79C2×D411yescharacteristic
168C22×C4×C79C2×D411yescharacteristic
169C22×C4×C79C2411yescharacteristic
170C4⋊C4×C79—22no—
171C4⋊C4×C79—21no—
172C4⋊C4×C79—21no—
173C4⋊C4×C79—21no—
174C4⋊C4×C79—21no—
175C4⋊C4×C79—21no—
176C4⋊C4×C79—21no—
177C4⋊C4×C79—21no—
178C4⋊C4×C79—21no—
179C4⋊C4×C79—41no—
180C4⋊C4×C79—42no—
181C4⋊C4×C79—42no—
182C4⋊C4×C79—81no—
183C4⋊C4×C79—81no—
184C42×C79—21no—
185C42×C79—21no—
186C42×C79—21no—
187C42×C79—21no—
188C42×C79—81no—
189C42×C79C2×Q812yes—
190C42×C79#4112yes—
191C23×C79C22×D411yescharacteristic
192C22×D79—22no—
193C22×D79—41no—
194C22×D79—41no—
195C22×D79—41no—
196C22×D79—41no—
197C22×D79—41no—
198C22×D79—41no—
199C2×C4×C79—21no—
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—21no—
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—41no—
215C2×C4×C79—41no—
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—41no—
225C2×C4×C79—41no—
226C2×C4×C79C22⋊Q812yes—
227C2×C4×C79C22⋊Q812yes—
228C2×C4×C79AGL(1,ℤ/8)12yes—
229C2×C4×C79C22×Q811yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C2×C4×C79H(5,𝔽2)11yescharacteristic
234C2×C4×C792−1+411yescharacteristic
235C2×Dic79—22no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244D316—22no—
245D316—22no—
246D316—41no—
247D316—41no—
248C4×D79—48no—
249C79⋊C8—22no—
250C79⋊C8—22no—
251C79⋊C8—41no—
252C79⋊C8—41no—
253C22×C79C2×C22⋊Q811yescharacteristic
254C22×C79#54711yescharacteristic
255C22×C79#55011yescharacteristic
256C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C79#57511yescharacteristic
258C22×C79#57611yescharacteristic
259C22×C79#57811yescharacteristic
260C2×D79—42no—
261C2×D79—42no—
262C2×D79—42no—
263C2×D79—81no—
264C2×D79—81no—
265C2×D79—81no—
266C2×D79—81no—
267C4×C79—22no—
268C4×C79—22no—
269C4×C79—22no—
270C4×C79—41no—
271C4×C79—41no—
272C4×C79—41no—
273C4×C79—81no—
274C4×C79—81no—
275C4×C79—81no—
276C4×C79—81no—
277C4×C79#53311yescharacteristic
278C4×C79#54411yescharacteristic
279C4×C79#55211yescharacteristic
280C4×C79#55311yescharacteristic
281Dic79—42no—
282Dic79—81no—
283Dic79—81no—
284#26098—791nomaximal, Sylow
285C2×C79#244111yescharacteristic
286C2×C79#307711yescharacteristic
287C2×C79#308311yescharacteristic
288C2×C79#334311yescharacteristic
289C2×C79#334711yescharacteristic
290C2×C79#339911yescharacteristic
291C2×C79#340211yescharacteristic
292D79—82no—
293#2570D7911yescharacteristic, complemented
294#2354—791no—
295#2093—791no—
296#2092—791no—
297#2067—791no—
298#2065—791no—
299#1949—791no—
300#1947—791no—
301#1947—791no—
302#1940—791no—
303#1921—791no—
304#1920—791no—
305#1847—791no—
306#1804—791no—
307#1762—791no—
308C79#2609811yescharacteristic, complemented, Sylow
309C2×C4×D4—791no—
310C2×#117—1581no—
311C2×#107—791no—
312C2×#107—791no—
313C2×#107—791no—
314C2×#107—791no—
315C2×#106—791no—
316C2×#106—791no—
317C2×C4⋊C8—791no—
318C2×C4⋊C8—791no—
319C2×C4⋊C8—791no—
320C2×#102—791no—
321C2×#102—791no—
322C2×#102—791no—
323C2×#102—791no—
324C2×#97—791no—
325#400—21no—
326#398—21no—
327#390—791no—
328#390—791no—
329#389C2×D7911yescharacteristic
330#389C2×D7911yescharacteristic
331#388—1581no—
332#388—1581no—
333#387—21no—
334#387—21no—
335#384—791no—
336#384C2×D7911yescharacteristic
337#382C2×D7911yescharacteristic
338#382C2×D7911yescharacteristic
339#380—1581no—
340C4×C4⋊C4—791no—
341C4×C4⋊C4—791no—
342C4×C4⋊C4C2×D7911yescharacteristic
343C4×C4⋊C4C2×D7911yescharacteristic
344#340—791no—
345#340—791no—
346#340—791no—
347#340—791no—
348#340—791no—
349#340—791no—
350#336—791no—
351#336—791no—
352C22×D4—791no—
353C23×C4—1581no—
354C22×C8—791no—
355C22×C8—791no—
356C22×C8—791no—
357C22×C8—791no—
358C4×D4—794no—
359C4×D4—1582no—
360#117—3164no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—791no—
366C2×C4⋊C4—791no—
367C2×C4⋊C4—791no—
368C2×C4⋊C4—1581no—
369C2×C4⋊C4—1581no—
370C2×C4⋊C4C22⋊D7911yescharacteristic
371C2×C4⋊C4C22⋊D7911yescharacteristic
372C2×C4⋊C4C22×D7911yescharacteristic
373C2×C4⋊C4C22×D7911yescharacteristic
374C2×C4⋊C4C22×D7911yescharacteristic
375C2×C4⋊C4C22×D7911yescharacteristic
376C2×C22⋊C4—1581no—
377C2×C22⋊C4—1581no—
378C2×C22⋊C4—1581no—
379C2×C22⋊C4—1581no—
380C2×C22⋊C4—1581no—
381C2×C42—21no—
382C2×C42—1581no—
383C2×C42—1581no—
384C2×C42C22×D7911yescharacteristic
385C2×C42C22×D7911yescharacteristic
386C2×C42C22×D7911yescharacteristic
387#107—3161no—
388#107—3161no—
389#107—3161no—
390#107—1582no—
391#106—3161no—
392#106—1582no—
393C4⋊C8—1582no—
394C4⋊C8—1582no—
395C4⋊C8—1582no—
396#102—3161no—
397#102—3161no—
398#102—1582no—
399#102—1582no—
400#97—1582no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—1581no—
409#95—1581no—
410#95C22⋊D7911yescharacteristic
411#95C22⋊D7911yescharacteristic
412C24—1581no—
413C2×D4—792no—
414C2×D4—1581no—
415C2×D4—1581no—
416C2×D4—1581no—
417C2×D4—1581no—
418C2×D4—1581no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—3162no—
424C22×C4—3162no—
425C22×C4—1581no—
426C22×C4—1581no—
427C22×C4—1581no—
428C22×C4—1584no—
429C22×C4C2×C22⋊D7911yescharacteristic
430C22×C4C2×C22⋊D7911yescharacteristic
431C22×C4C2×C22⋊D7911yescharacteristic
432C22×C4C2×C22⋊D7911yescharacteristic
433C22×C4C2×C22⋊D7911yescharacteristic
434C22×C4C2×C22⋊D7911yescharacteristic
435C22×C4C23×D7911yescharacteristic
436C2×C8—792no—
437C2×C8—792no—
438C2×C8—1581no—
439C2×C8—1581no—
440C2×C8—1581no—
441C2×C8—1581no—
442C2×C8—1581no—
443C2×C8—1581no—
444C2×C8—1581no—
445C2×C8—1581no—
446C2×C8—1581no—
447C2×C8—1581no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—3162no—
463C4⋊C4—792no—
464C4⋊C4—6321no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C22⋊C4—3162no—
471C22⋊C4—3162no—
472C22⋊C4—6321no—
473C22⋊C4—6321no—
474C22⋊C4—1584no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—81no—
480C42—3162no—
481C42—3162no—
482C42Q8×D7912yes—
483C42#1192962512yes—
484C23—3161no—
485C23—3161no—
486C23—3161no—
487C23—3161no—
488C23—3161no—
489C23—3161no—
490C23—1582no—
491C23C22×C22⋊D7911yescharacteristic
492D4—3161no—
493D4—3161no—
494D4—1582no—
495D4—1582no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—3168no—
524C2×C4—3161no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—3161no—
531C2×C4—3161no—
532C2×C4—1582no—
533C2×C4#1457913712yes—
534C2×C4#1457913712yes—
535C2×C4#1457918612yes—
536C2×C4C2×Q8×D7911yescharacteristic
537C2×C4#41×D7911yescharacteristic
538C2×C4#41×D7911yescharacteristic
539C2×C4C2×#1192962511yescharacteristic
540C2×C4#1457920711yescharacteristic
541C2×C4#1457920811yescharacteristic
542C8—3161no—
543C8—3161no—
544C8—1582no—
545C8—1582no—
546C22—3162no—
547C22—3162no—
548C22—3162no—
549C22—6321no—
550C22—6321no—
551C22—6321no—
552C22—6321no—
553C22C2×#1457913711yescharacteristic
554C22#1909921211yescharacteristic
555C22#1909923211yescharacteristic
556C22C2×#1457918611yescharacteristic
557C22#1909936111yescharacteristic
558C22#1909936611yescharacteristic
559C22#1909937811yescharacteristic
560C4—22no—
561C4—22no—
562C4—22no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—3162no—
571C4—6321no—
572C4—6321no—
573C4#1909911911yescharacteristic
574C4#1909919311yescharacteristic
575C4#1909925111yescharacteristic
576C4#1909926611yescharacteristic
577C2—6322no—
578C2#2376710911yescharacteristic
579C2#2377319311yescharacteristic
580C2#2377324711yescharacteristic
581C2#2377584011yescharacteristic
582C2#2377589411yescharacteristic
583C2#2377643811yescharacteristic
584C2#2377647511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2570C2
socleC23×C79C22×D4
prime radicalC22×D316C23
hypercenter#2570D79
hypocenterC79#26098
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2570

Complements

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

NH ≅ G/Ntype
6. #23764166C2semidirect
7. #23764158C2semidirect
8. #23764156C2semidirect
9. #23764124C2semidirect
10. #23763989C2semidirect
11. #23763982C2semidirect
12. #23763569C2semidirect
15. C79×#2570C2semidirect
293. #2570D79semidirect
308. C79#26098semidirect

Maximal subgroups

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

subgroupindex
#237663472
#237649292
#237649242
#237647852
#237647712
#237641662
#237641582
#237641562
#237641242
#237639892
#237639822
#237635692
#237634072
#237632282
C79×#25702
#2609879