Finite Group Databasebeta

← #34300430

Subgroups — #34300430

Order 20224.

784 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#125×Dic79C211yesmaximal, characteristic, complemented
2#23765865C211yesmaximal, characteristic
3#23764358C211yesmaximal, characteristic, complemented
4#23764170C211yesmaximal, characteristic, complemented
5#23764144C211yesmaximal, characteristic, complemented
6#23764139C211yesmaximal, characteristic, complemented
7#23764099C211yesmaximal, characteristic
8#23764099C211yesmaximal, characteristic
9#23764029C211yesmaximal, characteristic
10#23764022C211yesmaximal, characteristic
11#23763980C211yesmaximal, characteristic, complemented
12#23763973C211yesmaximal, characteristic, complemented
13#23763410C211yesmaximal, characteristic
14#23763255C211yesmaximal, characteristic, complemented
15C2×#478×C79C211yesmaximal, characteristic
16C2×#14579120—21no—
17C2×#14579095C2211yescharacteristic
18C2×#14579093C2211yescharacteristic
19C2×C4⋊(C79⋊C8)C2211yescharacteristic
20C2×#14579082C2211yescharacteristic
21C2×#14579082C2211yescharacteristic
22C2×#14579071C2211yescharacteristic
23C2×#14579071C2211yescharacteristic
24#19098466C2211yescharacteristic
25#19098466C2211yescharacteristic
26#19098462C2211yescharacteristic
27#19098462C2211yescharacteristic
28#19098462C2211yescharacteristic
29#19098462C2211yescharacteristic
30#19098456—21no—
31#19098456—21no—
32#19098449—21no—
33#19098449—21no—
34#19098429C2211yescharacteristic
35C4⋊C4×Dic79—21no—
36C4⋊C4×Dic79—21no—
37C4⋊C4×Dic79C2211yescharacteristic
38C4⋊C4×Dic79C2211yescharacteristic
39C4×C4⋊Dic79C2211yescharacteristic
40C4×C22⋊Dic79—21no—
41C42×Dic79C2211yescharacteristic
42#19098326C2211yescharacteristic
43#19098326C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098317C2211yescharacteristic
51#19098317C2211yescharacteristic
52C2×#125×C79C2211yescharacteristic
53C2×C4×D4×C79C2211yescharacteristic
54#478×C79C418yescomplemented
55C2×#107×C79C2211yescharacteristic
56C2×#106×C79C2211yescharacteristic
57C2×C4⋊C8×C79C2211yescharacteristic
58C2×#102×C79C2211yescharacteristic
59C2×#102×C79C2211yescharacteristic
60C22×C79⋊C8C2311yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62#14579120—44no—
63C2×C4⋊Dic79—21no—
64C2×C4⋊Dic79—21no—
65C2×C4⋊Dic79—21no—
66C2×C4⋊Dic79—21no—
67C2×C4⋊Dic79C2311yescharacteristic
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
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×Dic79D411yescharacteristic
79C2×C4×Dic79D411yescharacteristic
80C2×C4×Dic79D411yescharacteristic
81C2×C4×Dic79D411yescharacteristic
82C2×C4×Dic79C2311yescharacteristic
83C2×C4×Dic79C2311yescharacteristic
84#14579095—41no—
85#14579093—41no—
86C4⋊(C79⋊C8)—22no—
87#14579082—41no—
88#14579082—41no—
89#14579071—41no—
90#14579071—41no—
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—
100C22×D4×C79C2311yescharacteristic
101C23×C4×C79—21no—
102C22×C8×C79C2311yescharacteristic
103C22×C8×C79C2311yescharacteristic
104#125×C79—22no—
105#125×C79C2×C414yes—
106C4×D4×C79—22no—
107C4×D4×C79C2×C414yes—
108C2×C4⋊C4×C79—21no—
109C2×C4⋊C4×C79—21no—
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C22⋊C4×C79—21no—
114C2×C42×C79C2311yescharacteristic
115#107×C79C2×C414yes—
116#106×C79C2×C414yes—
117C4⋊C8×C79C2×C414yes—
118#102×C79C2×C414yes—
119#102×C79C2×C414yes—
120C22×Dic79—21no—
121C22×Dic79—21no—
122C22×Dic79—21no—
123C22×Dic79—21no—
124C22×Dic79C2×D411yescharacteristic
125C22×Dic79C2×D411yescharacteristic
126C22×Dic79C2×D411yescharacteristic
127C22×Dic79C2×D411yescharacteristic
128C2×C79⋊C8—21no—
129C2×C79⋊C8—21no—
130C2×C79⋊C8—21no—
131C2×C79⋊C8—21no—
132C2×C79⋊C8—21no—
133C2×C79⋊C8—21no—
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—42no—
144C4⋊Dic79—42no—
145C4⋊Dic79—42no—
146C22⋊Dic79—42no—
147C22⋊Dic79—42no—
148C22⋊Dic79—42no—
149C22⋊Dic79—42no—
150C4×Dic79—22no—
151C4×Dic79—21no—
152C4×Dic79—21no—
153C4×Dic79—21no—
154C4×Dic79—21no—
155C4×Dic79—41no—
156C4×Dic79—41no—
157C4×Dic79—41no—
158C4×Dic79—42no—
159C4×Dic79—42no—
160C4×Dic79—42no—
161C24×C79—21no—
162C2×D4×C79—21no—
163C2×D4×C79—21no—
164C2×D4×C79—21no—
165C2×D4×C79—21no—
166C2×D4×C79—21no—
167C2×D4×C79C22×C412yes—
168C22×C4×C79—21no—
169C22×C4×C79—21no—
170C22×C4×C79—21no—
171C22×C4×C79—24no—
172C22×C4×C79—42no—
173C22×C4×C79—42no—
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2411yescharacteristic
177C2×C8×C79—21no—
178C2×C8×C79—21no—
179C2×C8×C79—21no—
180C2×C8×C79—21no—
181C2×C8×C79C22×C412yes—
182C2×C8×C79C22×C412yes—
183C4⋊C4×C79—24no—
184C4⋊C4×C79—24no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79C22×C412yes—
189C4⋊C4×C79C22×C412yes—
190C4⋊C4×C79C22×C412yes—
191C22⋊C4×C79—24no—
192C42×C79C22×C412yes—
193C42×C79#4112yes—
194C23×C79—22no—
195C23×C79—41no—
196C23×C79—41no—
197C23×C79—41no—
198C23×C79—41no—
199C23×C79—41no—
200C23×C79—41no—
201C23×C79C22×D411yescharacteristic
202D4×C79—22no—
203D4×C79—22no—
204D4×C79—41no—
205D4×C79—41no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—22no—
211C2×C4×C79—22no—
212C2×C4×C79—22no—
213C2×C4×C79—48no—
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×C79C4×D412yes—
221C2×C4×C79C4×D412yes—
222C2×C4×C79C23×C411yescharacteristic
223C2×C4×C79C2×#4111yescharacteristic
224C2×C4×C79H(5,𝔽2)11yescharacteristic
225C2×C4×C792−1+411yescharacteristic
226C2×C4×C792−1+411yescharacteristic
227C2×C4×C792−1+411yescharacteristic
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—21no—
237C2×Dic79—21no—
238C2×Dic79—21no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79AGL(1,ℤ/8)12yes—
252C8×C79—22no—
253C8×C79—22no—
254C79⋊C8—41no—
255C79⋊C8—41no—
256C22×C79—42no—
257C22×C79—42no—
258C22×C79—42no—
259C22×C79—81no—
260C22×C79—81no—
261C22×C79—81no—
262C22×C79—81no—
263C22×C79C2×C4×D411yescharacteristic
264C22×C79#53611yescharacteristic
265C22×C79#53711yescharacteristic
266C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C79#57511yescharacteristic
268C22×C79#57511yescharacteristic
269C22×C79#57511yescharacteristic
270C4×C79—22no—
271C4×C79—22no—
272C4×C79—42no—
273C4×C79—42no—
274C4×C79—42no—
275C4×C79#51911yescharacteristic
276C4×C79#52011yescharacteristic
277C4×C79#54111yescharacteristic
278C4×C79#54211yescharacteristic
279Dic79—22no—
280Dic79—41no—
281Dic79—41no—
282Dic79—41no—
283Dic79—81no—
284Dic79—81no—
285Dic79—81no—
286Dic79—81no—
287#22120—791nomaximal, Sylow
288C2×C79—82no—
289C2×C79#232311yescharacteristic
290C2×C79#294311yescharacteristic
291C2×C79C4×AGL(1,ℤ/8)11yescharacteristic
292C2×C79#323411yescharacteristic
293C2×C79#324011yescharacteristic
294C2×C79#325111yescharacteristic
295C2×C79#325611yescharacteristic
296C2×#478D7911yescharacteristic
297C4×#125—791no—
298#2293—791no—
299#1989—791no—
300#1949—791no—
301#1944—791no—
302#1943—791no—
303#1936—791no—
304#1936—791no—
305#1926—791no—
306#1925—791no—
307#1919—791no—
308#1918—791no—
309#1804—791no—
310#1767—791no—
311C79#2212011yescharacteristic, complemented, Sylow
312C2×#125C2×D7911yescharacteristic
313C2×C4×D4C2×D7911yescharacteristic
314C2×#117—1581no—
315#478Dic7918yescomplemented
316C2×#107—791no—
317C2×#107C2×D7911yescharacteristic
318C2×#106—791no—
319C2×#106C2×D7911yescharacteristic
320C2×C4⋊C8—791no—
321C2×C4⋊C8C2×D7911yescharacteristic
322C2×#102—791no—
323C2×#102—791no—
324C2×#102C2×D7911yescharacteristic
325C2×#102C2×D7911yescharacteristic
326C2×#97—791no—
327C2×#97—791no—
328#390—791no—
329#390—791no—
330#389—791no—
331#389—791no—
332#389—791no—
333#389—791no—
334#388—1581no—
335#388—1581no—
336#387—1581no—
337#387—1581no—
338#382—791no—
339C4×C4⋊C4—791no—
340C4×C4⋊C4—791no—
341C4×C4⋊C4—791no—
342C4×C4⋊C4—1581no—
343C4×C4⋊C4—1581no—
344C4×C22⋊C4—1581no—
345C43—791no—
346#340—791no—
347#340—791no—
348#340—791no—
349#340—791no—
350#340—791no—
351#340—791no—
352#340—791no—
353#340—791no—
354#336—791no—
355#336—791no—
356C22×D4C22×D7911yescharacteristic
357C23×C4—21no—
358C22×C8—791no—
359C22×C8—791no—
360C22×C8C22×D7911yescharacteristic
361C22×C8C22×D7911yescharacteristic
362#125—22no—
363#125C2×Dic7914yes—
364C4×D4—22no—
365C4×D4C2×Dic7914yes—
366#117—3164no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—791no—
370C2×C4⋊C4—791no—
371C2×C4⋊C4—791no—
372C2×C4⋊C4—791no—
373C2×C4⋊C4—1581no—
374C2×C4⋊C4—1581no—
375C2×C4⋊C4—1581no—
376C2×C4⋊C4—1581no—
377C2×C4⋊C4C22×D7911yescharacteristic
378C2×C4⋊C4C22×D7911yescharacteristic
379C2×C4⋊C4C22×D7911yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—1581no—
382C2×C22⋊C4—1581no—
383C2×C22⋊C4—1581no—
384C2×C22⋊C4—1581no—
385C2×C42—791no—
386C2×C42—791no—
387C2×C42—791no—
388C2×C42—791no—
389C2×C42—791no—
390C2×C42—791no—
391C2×C42—1581no—
392C2×C42—1581no—
393C2×C42—1581no—
394C2×C42C22×D7911yescharacteristic
395#107—3161no—
396#107C2×Dic7914yes—
397#106—3161no—
398#106C2×Dic7914yes—
399C4⋊C8—1582no—
400C4⋊C8C2×Dic7914yes—
401#102—3161no—
402#102—3161no—
403#102C2×Dic7914yes—
404#102C2×Dic7914yes—
405#97—3161no—
406#97—3161no—
407#95—1581no—
408#95—1581no—
409#95—1581no—
410#95—1581no—
411#95—1581no—
412#95—1581no—
413#95—1581no—
414#95—1581no—
415#95—1581no—
416C24—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C22×Dic7912yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—791no—
430C22×C4—791no—
431C22×C4—791no—
432C22×C4—791no—
433C22×C4—1581no—
434C22×C4—1581no—
435C22×C4—1581no—
436C22×C4—1581no—
437C22×C4D4×D7911yescharacteristic
438C22×C4D4×D7911yescharacteristic
439C22×C4C23×D7911yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—1581no—
445C2×C8—1581no—
446C2×C8—1581no—
447C2×C8—1581no—
448C2×C8—1581no—
449C2×C8—1581no—
450C2×C8C22×Dic7912yes—
451C2×C8C22×Dic7912yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—3162no—
458C4⋊C4—3162no—
459C4⋊C4—3162no—
460C4⋊C4—3162no—
461C4⋊C4—1581no—
462C4⋊C4—1581no—
463C4⋊C4—1581no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4C22×Dic7912yes—
470C4⋊C4C22×Dic7912yes—
471C4⋊C4C22×Dic7912yes—
472C22⋊C4—24no—
473C22⋊C4—3162no—
474C22⋊C4—3162no—
475C22⋊C4—3162no—
476C22⋊C4—3162no—
477C42—3162no—
478C42—3161no—
479C42—3161no—
480C42—3161no—
481C42—3162no—
482C42—3162no—
483C42—1582no—
484C42—1581no—
485C42—1581no—
486C42—1581no—
487C42—1581no—
488C42C22×Dic7912yes—
489C42#1192962312yes—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×D4×D7911yescharacteristic
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—48no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
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—3161no—
528C2×C4—792no—
529C2×C4—1581no—
530C2×C4—1581no—
531C2×C4—1581no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4—1581no—
537C2×C4—1581no—
538C2×C4—1581no—
539C2×C4—1581no—
540C2×C4D4×Dic7912yes—
541C2×C4D4×Dic7912yes—
542C2×C4C23×Dic7911yescharacteristic
543C2×C4C2×#1192962311yescharacteristic
544C2×C4#1457920711yescharacteristic
545C2×C4#1457920811yescharacteristic
546C2×C4#1457920911yescharacteristic
547C2×C4#1457920911yescharacteristic
548C8—22no—
549C8—22no—
550C8—3161no—
551C8—3161no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×D4×Dic7911yescharacteristic
560C22#1909913611yescharacteristic
561C22#1909914311yescharacteristic
562C22AGL(1,ℤ/8)×D7911yescharacteristic
563C22#1909935911yescharacteristic
564C22#1909936011yescharacteristic
565C22#1909936011yescharacteristic
566C4—22no—
567C4—22no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—3161no—
572C4—3161no—
573C4—3161no—
574C4—6321no—
575C4—6321no—
576C4—6321no—
577C4—6321no—
578C4—1582no—
579C4#1909904111yescharacteristic
580C4#1909904611yescharacteristic
581C4#1909917311yescharacteristic
582C4#1909918611yescharacteristic
583C2—82no—
584C2#2376608411yescharacteristic
585C2#2377220111yescharacteristic
586C2AGL(1,ℤ/8)×Dic7911yescharacteristic
587C2#2377458811yescharacteristic
588C2#2377463911yescharacteristic
589C2#2377476111yescharacteristic
590C2#2377480011yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC2×#478×C79C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
cosocleC22×C4×C79C24
hypercenterC2×#478D79
hypocenterC79#22120
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. #125×Dic79C2semidirect
3. #23764358C2semidirect
4. #23764170C2semidirect
5. #23764144C2semidirect
6. #23764139C2semidirect
11. #23763980C2semidirect
12. #23763973C2semidirect
14. #23763255C2semidirect
54. #478×C79C4semidirect
311. C79#22120semidirect
315. #478Dic79semidirect

Maximal subgroups

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

subgroupindex
#125×Dic792
#237658652
#237643582
#237641702
#237641442
#237641392
#237640992
#237640292
#237640222
#237639802
#237639732
#237634102
#237632552
C2×#478×C792
#2212079