Finite Group Databasebeta

← #34300475

Subgroups — #34300475

Order 20224.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766149C211yesmaximal, characteristic, complemented
2#23766043C211yesmaximal, characteristic
3#23764353C211yesmaximal, characteristic, complemented
4#23764170C211yesmaximal, characteristic, complemented
5#23764144C211yesmaximal, characteristic, complemented
6#23764129C211yesmaximal, characteristic, complemented
7#23764099C211yesmaximal, characteristic
8#23764085C211yesmaximal, characteristic
9#23764029C211yesmaximal, characteristic
10#23764022C211yesmaximal, characteristic
11#23763987C211yesmaximal, characteristic, complemented
12#23763980C211yesmaximal, characteristic, complemented
13#23763410C211yesmaximal, characteristic
14#23763259C211yesmaximal, characteristic, complemented
15C2×#476×C79C211yesmaximal, characteristic
16C2×#14579095C2211yescharacteristic
17C2×#14579093C2211yescharacteristic
18C2×C4⋊(C79⋊C8)C2211yescharacteristic
19C2×#14579082C2211yescharacteristic
20C2×#14579082C2211yescharacteristic
21C2×#14579071C2211yescharacteristic
22C2×C8×Dic79C2211yescharacteristic
23#19098470—21no—
24#19098470—21no—
25#19098466C2211yescharacteristic
26#19098466C2211yescharacteristic
27#19098462C2211yescharacteristic
28#19098462C2211yescharacteristic
29#19098462C2211yescharacteristic
30#19098462C2211yescharacteristic
31#19098456—21no—
32#19098456—21no—
33#19098449—21no—
34#19098449—21no—
35#19098429C2211yescharacteristic
36#19098426—21no—
37#19098423—21no—
38C4⋊C4×Dic79C2211yescharacteristic
39C4⋊C4×Dic79C2211yescharacteristic
40C4×C4⋊Dic79C2211yescharacteristic
41#19098409C2211yescharacteristic
42#19098326C2211yescharacteristic
43#19098326C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098317C2211yescharacteristic
51#19098317C2211yescharacteristic
52C2×C4⋊Q8×C79C2211yescharacteristic
53C2×C4×D4×C79C2211yescharacteristic
54#476×C79C418yescomplemented
55C2×#106×C79C2211yescharacteristic
56C2×#106×C79C2211yescharacteristic
57C2×C4⋊C8×C79C2211yescharacteristic
58C2×#102×C79C2211yescharacteristic
59C2×#102×C79C2211yescharacteristic
60C22×C79⋊C8C2311yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62C2×C4⋊Dic79—21no—
63C2×C4⋊Dic79—21no—
64C2×C4⋊Dic79—21no—
65C2×C4⋊Dic79—21no—
66C2×C4⋊Dic79C2311yescharacteristic
67C2×C4⋊Dic79C2311yescharacteristic
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C22⋊Dic79—21no—
71C2×C22⋊Dic79—21no—
72C2×C22⋊Dic79—21no—
73C2×C22⋊Dic79—21no—
74C2×C4×Dic79—21no—
75C2×C4×Dic79—21no—
76C2×C4×Dic79—21no—
77C2×C4×Dic79C2311yescharacteristic
78C2×C4×Dic79C2311yescharacteristic
79#14579095—41no—
80#14579093—41no—
81C4⋊(C79⋊C8)—22no—
82#14579082—41no—
83#14579082—41no—
84#14579071—41no—
85C8×Dic79—41no—
86C79⋊#95—21no—
87C79⋊#95—21no—
88C79⋊#95—21no—
89C79⋊#95—21no—
90C79⋊#95—21no—
91C79⋊#95—21no—
92C79⋊#95—21no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C79⋊#95D411yescharacteristic
96C79⋊#95D411yescharacteristic
97C79⋊#95D411yescharacteristic
98C79⋊#95D411yescharacteristic
99C22×Q8×C79—21no—
100C22×D4×C79C2311yescharacteristic
101C23×C4×C79—21no—
102C22×C8×C79C2311yescharacteristic
103C22×C8×C79C2311yescharacteristic
104C4⋊Q8×C79—22no—
105C4⋊Q8×C79C2×C414yes—
106C4×D4×C79—22no—
107C4×D4×C79C2×C414yes—
108C2×C4⋊C4×C79—21no—
109C2×C4⋊C4×C79C2311yescharacteristic
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C22⋊C4×C79—21no—
113C2×C42×C79C2311yescharacteristic
114#106×C79C2×C414yes—
115#106×C79C2×C414yes—
116C4⋊C8×C79C2×C414yes—
117#102×C79C2×C414yes—
118#102×C79C2×C414yes—
119C22×Dic79—21no—
120C22×Dic79—21no—
121C22×Dic79—21no—
122C22×Dic79—21no—
123C22×Dic79C2×D411yescharacteristic
124C22×Dic79C2×D411yescharacteristic
125C22×Dic79C2×D411yescharacteristic
126C22×Dic79C2×D411yescharacteristic
127C2×C79⋊C8—21no—
128C2×C79⋊C8—21no—
129C2×C79⋊C8—21no—
130C2×C79⋊C8—21no—
131C2×C79⋊C8—21no—
132C2×C79⋊C8—21no—
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—81no—
142C4⋊Dic79—81no—
143C4⋊Dic79—81no—
144C4⋊Dic79—81no—
145C22⋊Dic79—81no—
146C22⋊Dic79—81no—
147C22⋊Dic79—81no—
148C22⋊Dic79—81no—
149C4×Dic79—21no—
150C4×Dic79—21no—
151C4×Dic79—21no—
152C4×Dic79—21no—
153C4×Dic79—81no—
154C4×Dic79—81no—
155C4×Dic79—81no—
156C24×C79—21no—
157C2×Q8×C79—24no—
158C2×Q8×C79—81no—
159C2×Q8×C79—81no—
160C2×D4×C79—21no—
161C2×D4×C79—21no—
162C2×D4×C79—21no—
163C2×D4×C79—21no—
164C2×D4×C79—21no—
165C2×D4×C79C22×C412yes—
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79—21no—
169C22×C4×C79—24no—
170C22×C4×C79—81no—
171C22×C4×C79—81no—
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2411yescharacteristic
175C2×C8×C79—21no—
176C2×C8×C79—21no—
177C2×C8×C79—21no—
178C2×C8×C79—21no—
179C2×C8×C79C22×C412yes—
180C2×C8×C79C22×C412yes—
181C4⋊C4×C79—24no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79C22×C412yes—
186C4⋊C4×C79C22×C412yes—
187C4⋊C4×C79C22×C412yes—
188C22⋊C4×C79—24no—
189C42×C79C22×C412yes—
190C42×C79#4112yes—
191C23×C79—22no—
192C23×C79—41no—
193C23×C79—41no—
194C23×C79—41no—
195C23×C79—41no—
196C23×C79—41no—
197C23×C79—41no—
198C23×C79C22×D411yescharacteristic
199Q8×C79—84no—
200D4×C79—22no—
201D4×C79—22no—
202D4×C79—41no—
203D4×C79—41no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—22no—
209C2×C4×C79—22no—
210C2×C4×C79—22no—
211C2×C4×C79—41no—
212C2×C4×C79—41no—
213C2×C4×C79—41no—
214C2×C4×C79—41no—
215C2×C4×C79—41no—
216C2×C4×C79—41no—
217C2×C4×C79—84no—
218C2×C4×C79C4×D412yes—
219C2×C4×C79C4×D412yes—
220C2×C4×C79C23×C411yescharacteristic
221C2×C4×C79C2×#4111yescharacteristic
222C2×C4×C79H(5,𝔽2)11yescharacteristic
223C2×C4×C792−1+411yescharacteristic
224C2×C4×C792−1+411yescharacteristic
225C2×C4×C792−1+411yescharacteristic
226C2×Dic79—21no—
227C2×Dic79—21no—
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—41no—
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—
250C8×C79—22no—
251C8×C79—22no—
252C79⋊C8—41no—
253C79⋊C8—41no—
254C22×C79—42no—
255C22×C79—42no—
256C22×C79—42no—
257C22×C79—81no—
258C22×C79—81no—
259C22×C79—81no—
260C22×C79—81no—
261C22×C79C2×C4×D411yescharacteristic
262C22×C79#53611yescharacteristic
263C22×C79#53711yescharacteristic
264C22×C79#57611yescharacteristic
265C22×C79#57711yescharacteristic
266C22×C79#57711yescharacteristic
267C22×C79#57711yescharacteristic
268C4×C79—22no—
269C4×C79—22no—
270C4×C79—42no—
271C4×C79—42no—
272C4×C79—42no—
273C4×C79#51911yescharacteristic
274C4×C79#52011yescharacteristic
275C4×C79#54111yescharacteristic
276C4×C79#54211yescharacteristic
277Dic79—41no—
278Dic79—41no—
279Dic79—41no—
280Dic79—41no—
281Dic79—81no—
282Dic79—81no—
283Dic79—81no—
284Dic79—81no—
285#22123—791nomaximal, Sylow
286C2×C79—82no—
287C2×C79#232311yescharacteristic
288C2×C79#294611yescharacteristic
289C2×C79#294911yescharacteristic
290C2×C79#324111yescharacteristic
291C2×C79#324411yescharacteristic
292C2×C79#325711yescharacteristic
293C2×C79#326311yescharacteristic
294C2×#476D7911yescharacteristic
295#2331—791no—
296#2319—791no—
297#1988—791no—
298#1949—791no—
299#1944—791no—
300#1941—791no—
301#1936—791no—
302#1934—791no—
303#1926—791no—
304#1925—791no—
305#1920—791no—
306#1919—791no—
307#1804—791no—
308#1768—791no—
309C79#2212311yescharacteristic, complemented, Sylow
310C2×C4⋊Q8C2×D7911yescharacteristic
311C2×C4×D4C2×D7911yescharacteristic
312#476Dic7918yescomplemented
313C2×#107—791no—
314C2×#106—791no—
315C2×#106C2×D7911yescharacteristic
316C2×#106C2×D7911yescharacteristic
317C2×C4⋊C8—791no—
318C2×C4⋊C8C2×D7911yescharacteristic
319C2×#102—791no—
320C2×#102—791no—
321C2×#102C2×D7911yescharacteristic
322C2×#102C2×D7911yescharacteristic
323C2×#97—791no—
324C2×C4×C8—791no—
325#391—1581no—
326#391—1581no—
327#390—791no—
328#390—791no—
329#389—791no—
330#389—791no—
331#389—791no—
332#389—791no—
333#388—1581no—
334#388—1581no—
335#387—1581no—
336#387—1581no—
337#382—791no—
338#381—1581no—
339#380—1581no—
340C4×C4⋊C4—791no—
341C4×C4⋊C4—791no—
342C4×C4⋊C4—791no—
343#376—791no—
344#340—791no—
345#340—791no—
346#340—791no—
347#340—791no—
348#340—791no—
349#340—791no—
350#340—791no—
351#340—791no—
352#336—791no—
353#336—791no—
354C22×Q8—21no—
355C22×D4C22×D7911yescharacteristic
356C23×C4—21no—
357C22×C8—791no—
358C22×C8—791no—
359C22×C8C22×D7911yescharacteristic
360C22×C8C22×D7911yescharacteristic
361C4⋊Q8—22no—
362C4⋊Q8C2×Dic7914yes—
363C4×D4—22no—
364C4×D4C2×Dic7914yes—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—791no—
367C2×C4⋊C4—791no—
368C2×C4⋊C4—791no—
369C2×C4⋊C4—791no—
370C2×C4⋊C4—1581no—
371C2×C4⋊C4—1581no—
372C2×C4⋊C4—1581no—
373C2×C4⋊C4—1581no—
374C2×C4⋊C4C22×D7911yescharacteristic
375C2×C4⋊C4C22×D7911yescharacteristic
376C2×C4⋊C4C22×D7911yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—1581no—
379C2×C22⋊C4—1581no—
380C2×C22⋊C4—1581no—
381C2×C22⋊C4—1581no—
382C2×C42—791no—
383C2×C42—791no—
384C2×C42—1581no—
385C2×C42—1581no—
386C2×C42—1581no—
387C2×C42C22×D7911yescharacteristic
388#107—3161no—
389#106—3161no—
390#106C2×Dic7914yes—
391#106C2×Dic7914yes—
392C4⋊C8—1582no—
393C4⋊C8C2×Dic7914yes—
394#102—3161no—
395#102—3161no—
396#102C2×Dic7914yes—
397#102C2×Dic7914yes—
398#97—3161no—
399C4×C8—3161no—
400#95—791no—
401#95—791no—
402#95—791no—
403#95—791no—
404#95—1581no—
405#95—1581no—
406#95—1581no—
407#95—1581no—
408#95—1581no—
409#95—1581no—
410#95—1581no—
411#95—1581no—
412#95—1581no—
413C24—21no—
414C2×Q8—24no—
415C2×Q8—81no—
416C2×Q8—81no—
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—81no—
428C22×C4—81no—
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—6321no—
457C4⋊C4—6321no—
458C4⋊C4—6321no—
459C4⋊C4—6321no—
460C4⋊C4—1581no—
461C4⋊C4—1581no—
462C4⋊C4—1581no—
463C4⋊C4—1581no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4C22×Dic7912yes—
469C4⋊C4C22×Dic7912yes—
470C4⋊C4C22×Dic7912yes—
471C22⋊C4—24no—
472C22⋊C4—6321no—
473C22⋊C4—6321no—
474C22⋊C4—6321no—
475C22⋊C4—6321no—
476C42—6321no—
477C42—6321no—
478C42—6321no—
479C42—1581no—
480C42—1581no—
481C42—1581no—
482C42—1581no—
483C42C22×Dic7912yes—
484C42#1192962312yes—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C2×D4×D7911yescharacteristic
493Q8—84no—
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—84no—
512C2×C4—3161no—
513C2×C4—3161no—
514C2×C4—3161no—
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—1581no—
525C2×C4—1581no—
526C2×C4—1581no—
527C2×C4—1581no—
528C2×C4—1581no—
529C2×C4—1581no—
530C2×C4—1581no—
531C2×C4—1581no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4D4×Dic7912yes—
537C2×C4D4×Dic7912yes—
538C2×C4C23×Dic7911yescharacteristic
539C2×C4C2×#1192962311yescharacteristic
540C2×C4#1457920711yescharacteristic
541C2×C4#1457920811yescharacteristic
542C2×C4#1457920911yescharacteristic
543C2×C4#1457920911yescharacteristic
544C8—22no—
545C8—22no—
546C8—3161no—
547C8—3161no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22C2×D4×Dic7911yescharacteristic
556C22#1909913611yescharacteristic
557C22#1909914311yescharacteristic
558C22#1909936711yescharacteristic
559C22#1909936911yescharacteristic
560C22#1909936911yescharacteristic
561C22#1909937311yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—3161no—
568C4—3161no—
569C4—3161no—
570C4—3161no—
571C4—6321no—
572C4—6321no—
573C4—6321no—
574C4—6321no—
575C4#1909904111yescharacteristic
576C4#1909904611yescharacteristic
577C4#1909917311yescharacteristic
578C4#1909918611yescharacteristic
579C2—82no—
580C2#2376608411yescharacteristic
581C2#2377223311yescharacteristic
582C2#2377226011yescharacteristic
583C2#2377465211yescharacteristic
584C2#2377469511yescharacteristic
585C2#2377481011yescharacteristic
586C2#2377487211yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC2×#476×C79C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
cosocleC22×C4×C79C24
hypercenterC2×#476D79
hypocenterC79#22123
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. #23766149C2semidirect
3. #23764353C2semidirect
4. #23764170C2semidirect
5. #23764144C2semidirect
6. #23764129C2semidirect
11. #23763987C2semidirect
12. #23763980C2semidirect
14. #23763259C2semidirect
54. #476×C79C4semidirect
309. C79#22123semidirect
312. #476Dic79semidirect

Maximal subgroups

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

subgroupindex
#237661492
#237660432
#237643532
#237641702
#237641442
#237641292
#237640992
#237640852
#237640292
#237640222
#237639872
#237639802
#237634102
#237632592
C2×#476×C792
#2212379