Finite Group Databasebeta

← #34347371

Subgroups — #34347371

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#23768630C211yesmaximal, characteristic, complemented
2#23766348C211yesmaximal, characteristic
3#23764923C211yesmaximal, characteristic
4#23764784C211yesmaximal, characteristic
5#23764774C211yesmaximal, characteristic
6#23764165C211yesmaximal, characteristic, complemented
7#23764156C211yesmaximal, characteristic, complemented
8#23764155C211yesmaximal, characteristic, complemented
9#23764122C211yesmaximal, characteristic, complemented
10#23763994C211yesmaximal, characteristic, complemented
11#23763986C211yesmaximal, characteristic, complemented
12#23763567C211yesmaximal, characteristic, complemented
13#23763408C211yesmaximal, characteristic
14#23763227C211yesmaximal, characteristic
15C79×#2093C211yesmaximal, characteristic
16C2×D4×Dic79C2211yescharacteristic
17C2×#14579117—21no—
18C2×#14579094C2211yescharacteristic
19C2×#14579094C2211yescharacteristic
20C2×#14579093C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×C4⋊(C79⋊C8)C2211yescharacteristic
23C2×Dic79⋊C8C2211yescharacteristic
24C2×Dic79⋊C8C2211yescharacteristic
25C2×#14579082C2211yescharacteristic
26C2×#14579082C2211yescharacteristic
27C2×#14579071C2211yescharacteristic
28#19098501—21no—
29#19098493—21no—
30#19098466C2211yescharacteristic
31#19098462C2211yescharacteristic
32#19098461C2211yescharacteristic
33#19098457—21no—
34#19098451—21no—
35#19098449—21no—
36#19098444—21no—
37#19098432C2211yescharacteristic
38#19098428C2211yescharacteristic
39#19098428C2211yescharacteristic
40#19098421—21no—
41C4⋊C4×Dic79C2211yescharacteristic
42C4⋊C4×Dic79C2211yescharacteristic
43C4×Dic79⋊C4C2211yescharacteristic
44C4×C4⋊Dic79C2211yescharacteristic
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098317C2211yescharacteristic
52C2×#107×C79C2211yescharacteristic
53C2×#107×C79C2211yescharacteristic
54C2×#102×C79C2211yescharacteristic
55C2×#102×C79C2211yescharacteristic
56#390×C79C2211yescharacteristic
57#384×C79C2211yescharacteristic
58#336×C79C2211yescharacteristic
59C23×Dic79—21no—
60C22×C79⋊C8C2311yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62D4×Dic79—22no—
63D4×Dic79Q814yes—
64#14579117—44no—
65C2×C4⋊Dic79—21no—
66C2×C4⋊Dic79C2311yescharacteristic
67C2×C4⋊Dic79C2311yescharacteristic
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×Dic79⋊C4—21no—
71C2×Dic79⋊C4—21no—
72C2×Dic79⋊C4—21no—
73C2×Dic79⋊C4D411yescharacteristic
74C2×Dic79⋊C4D411yescharacteristic
75C2×C22⋊Dic79—21no—
76C2×C22⋊Dic79—21no—
77C2×C22⋊Dic79—21no—
78C2×C4×Dic79—21no—
79C2×C4×Dic79—21no—
80C2×C4×Dic79—21no—
81C2×C4×Dic79C2311yescharacteristic
82C2×C4×Dic79C2311yescharacteristic
83#14579094—41no—
84#14579094—41no—
85#14579093—22no—
86#14579092—41no—
87C4⋊(C79⋊C8)—22no—
88Dic79⋊C8—22no—
89Dic79⋊C8—22no—
90#14579082—22no—
91#14579082—41no—
92#14579071—22no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95D411yescharacteristic
103C79⋊#95D411yescharacteristic
104C22×D4×C79C2311yescharacteristic
105C22×C8×C79C2311yescharacteristic
106C22×C8×C79C2311yescharacteristic
107C2×C4⋊C4×C79—21no—
108C2×C4⋊C4×C79—21no—
109C2×C4⋊C4×C79C2311yescharacteristic
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C22⋊C4×C79—21no—
113C2×C22⋊C4×C79—21no—
114C2×C42×C79C2311yescharacteristic
115#107×C79—22no—
116#107×C79—41no—
117#102×C79—22no—
118#102×C79—41no—
119C22×Dic79—24no—
120C22×Dic79—21no—
121C22×Dic79—21no—
122C22×Dic79—21no—
123C22×Dic79—21no—
124C22×Dic79—42no—
125C22×Dic79—42no—
126C22×Dic79C2×D411yescharacteristic
127C22×Dic79C2×D411yescharacteristic
128C22×Dic79C2×D411yescharacteristic
129C22×Dic79C2×D411yescharacteristic
130C2×C79⋊C8—21no—
131C2×C79⋊C8—21no—
132C2×C79⋊C8—21no—
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C2×C79⋊C8#4112yes—
136C4⋊Dic79—21no—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—81no—
144C4⋊Dic79C2×Q812yes—
145Dic79⋊C4—22no—
146Dic79⋊C4—41no—
147Dic79⋊C4—42no—
148Dic79⋊C4—42no—
149Dic79⋊C4—81no—
150C22⋊Dic79—24no—
151C22⋊Dic79—42no—
152C22⋊Dic79—81no—
153C4×Dic79—21no—
154C4×Dic79—21no—
155C4×Dic79—42no—
156C4×Dic79—42no—
157C4×Dic79—81no—
158C4×Dic79C2×Q812yes—
159C4×Dic79#4112yes—
160C24×C79—21no—
161C2×D4×C79—21no—
162C2×D4×C79—21no—
163C2×D4×C79—21no—
164C2×D4×C79—21no—
165C2×D4×C79—21no—
166C2×D4×C79C2×Q812yes—
167C22×C4×C79—21no—
168C22×C4×C79—21no—
169C22×C4×C79—21no—
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#4112yes—
179C4⋊C4×C79—21no—
180C4⋊C4×C79—21no—
181C4⋊C4×C79—21no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—42no—
186C4⋊C4×C79—81no—
187C22⋊C4×C79—42no—
188C22⋊C4×C79—81no—
189C42×C79—21no—
190C42×C79—21no—
191C23×C79—22no—
192C23×C79—41no—
193C23×C79—41no—
194C23×C79—41no—
195C23×C79—41no—
196C23×C79—41no—
197C23×C79—41no—
198C23×C79C22×D411yescharacteristic
199D4×C79—22no—
200D4×C79—22no—
201D4×C79—41no—
202D4×C79—41no—
203C2×C4×C79—21no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—41no—
209C2×C4×C79—41no—
210C2×C4×C79—41no—
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×C79AGL(1,ℤ/8)12yes—
218C2×C4×C79C22×Q811yescharacteristic
219C2×C4×C79C2×#4111yescharacteristic
220C2×C4×C79C2×#4111yescharacteristic
221C2×C4×C79C2×#4111yescharacteristic
222C2×C4×C79H(5,𝔽2)11yescharacteristic
223C2×C4×C792−1+411yescharacteristic
224C2×Dic79—21no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—22no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—48no—
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—
246C2×Dic79—41no—
247C2×Dic79C22⋊Q812yes—
248C2×Dic79C22⋊Q812yes—
249C8×C79—22no—
250C8×C79—41no—
251C79⋊C8—22no—
252C79⋊C8—41no—
253C22×C79—42no—
254C22×C79—42no—
255C22×C79—42no—
256C22×C79—81no—
257C22×C79—81no—
258C22×C79—81no—
259C22×C79—81no—
260C22×C79C2×C22⋊Q811yescharacteristic
261C22×C79#54711yescharacteristic
262C22×C79#55011yescharacteristic
263C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C79#57511yescharacteristic
265C22×C79#57611yescharacteristic
266C22×C79#57811yescharacteristic
267C4×C79—22no—
268C4×C79—41no—
269C4×C79—81no—
270C4×C79—81no—
271C4×C79—81no—
272C4×C79#53311yescharacteristic
273C4×C79#54411yescharacteristic
274C4×C79#55211yescharacteristic
275C4×C79#55311yescharacteristic
276Dic79—22no—
277Dic79—22no—
278Dic79—42no—
279Dic79—41no—
280Dic79—41no—
281Dic79—81no—
282Dic79—81no—
283Dic79—81no—
284#26098—791nomaximal, Sylow
285C2×C79—82no—
286C2×C79#244111yescharacteristic
287C2×C79#307711yescharacteristic
288C2×C79#308311yescharacteristic
289C2×C79#334311yescharacteristic
290C2×C79#334711yescharacteristic
291C2×C79#339911yescharacteristic
292C2×C79#340211yescharacteristic
293#2570—791no—
294#2354—791no—
295#2093D7911yescharacteristic
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×#107C2×D7911yescharacteristic
314C2×#107C2×D7911yescharacteristic
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×#102C2×D7911yescharacteristic
323C2×#102C2×D7911yescharacteristic
324C2×#97—791no—
325#400—1581no—
326#398—1581no—
327#390—791no—
328#390C2×D7911yescharacteristic
329#389—791no—
330#389—791no—
331#388—1581no—
332#388—1581no—
333#387—1581no—
334#387—1581no—
335#384—791no—
336#384C2×D7911yescharacteristic
337#382—791no—
338#382—791no—
339#380—1581no—
340C4×C4⋊C4—791no—
341C4×C4⋊C4—791no—
342C4×C4⋊C4—791no—
343C4×C4⋊C4—791no—
344#340—791no—
345#340—791no—
346#340—791no—
347#340—791no—
348#340—791no—
349#340—791no—
350#336—791no—
351#336C2×D7911yescharacteristic
352C22×D4C22×D7911yescharacteristic
353C23×C4—1581no—
354C22×C8—791no—
355C22×C8—791no—
356C22×C8C22×D7911yescharacteristic
357C22×C8C22×D7911yescharacteristic
358C4×D4—794no—
359C4×D4—1582no—
360#117—3164no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—791no—
364C2×C4⋊C4—791no—
365C2×C4⋊C4—791no—
366C2×C4⋊C4—791no—
367C2×C4⋊C4—791no—
368C2×C4⋊C4—791no—
369C2×C4⋊C4—1581no—
370C2×C4⋊C4—1581no—
371C2×C4⋊C4—1581no—
372C2×C4⋊C4—1581no—
373C2×C4⋊C4C22×D7911yescharacteristic
374C2×C4⋊C4C22×D7911yescharacteristic
375C2×C4⋊C4C22×D7911yescharacteristic
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—1581no—
379C2×C22⋊C4—1581no—
380C2×C22⋊C4—1581no—
381C2×C42—791no—
382C2×C42—791no—
383C2×C42—1581no—
384C2×C42—1581no—
385C2×C42—1581no—
386C2×C42C22×D7911yescharacteristic
387#107—22no—
388#107—41no—
389#107—3161no—
390#107—3161no—
391#106—3161no—
392#106—1582no—
393C4⋊C8—1582no—
394C4⋊C8—1582no—
395C4⋊C8—1582no—
396#102—22no—
397#102—41no—
398#102—3161no—
399#102—1582no—
400#97—1582no—
401#95—791no—
402#95—791no—
403#95—1581no—
404#95—1581no—
405#95—1581no—
406#95—1581no—
407#95—1581no—
408#95—1581no—
409#95—1581no—
410#95—1581no—
411#95—1581no—
412C24—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4Q8×D7912yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—3162no—
423C22×C4—3162no—
424C22×C4—791no—
425C22×C4—791no—
426C22×C4—791no—
427C22×C4—791no—
428C22×C4—1581no—
429C22×C4—1581no—
430C22×C4—1581no—
431C22×C4—1584no—
432C22×C4—1581no—
433C22×C4D4×D7911yescharacteristic
434C22×C4D4×D7911yescharacteristic
435C22×C4C23×D7911yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—792no—
442C2×C8—1581no—
443C2×C8—1581no—
444C2×C8—1581no—
445C2×C8—1581no—
446C2×C8—1581no—
447C2×C8#1192962312yes—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—42no—
455C4⋊C4—81no—
456C4⋊C4—3162no—
457C4⋊C4—3161no—
458C4⋊C4—3162no—
459C4⋊C4—792no—
460C4⋊C4—6321no—
461C4⋊C4—6321no—
462C4⋊C4—1581no—
463C4⋊C4—1581no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1582no—
470C22⋊C4—42no—
471C22⋊C4—81no—
472C22⋊C4—3162no—
473C22⋊C4—6321no—
474C22⋊C4—1584no—
475C42—21no—
476C42—21no—
477C42—3162no—
478C42—3162no—
479C42—792no—
480C42—792no—
481C42—6321no—
482C42—1581no—
483C42—1581no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×D4×D7911yescharacteristic
492D4—22no—
493D4—22no—
494D4—41no—
495D4—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—3161no—
511C2×C4—3161no—
512C2×C4—3168no—
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—792no—
523C2×C4—792no—
524C2×C4—1581no—
525C2×C4—1581no—
526C2×C4—1582no—
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#1457918512yes—
536C2×C4C2×Q8×D7911yescharacteristic
537C2×C4#41×D7911yescharacteristic
538C2×C4#41×D7911yescharacteristic
539C2×C4C2×#1192962311yescharacteristic
540C2×C4#1457920611yescharacteristic
541C2×C4#1457920811yescharacteristic
542C8—22no—
543C8—41no—
544C8—3161no—
545C8—1582no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C22⋊Q8×D7911yescharacteristic
554C22#1909921111yescharacteristic
555C22#1909923311yescharacteristic
556C22C2×#1457918511yescharacteristic
557C22#1909936211yescharacteristic
558C22#1909936411yescharacteristic
559C22#1909937911yescharacteristic
560C4—22no—
561C4—41no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—3162no—
566C4—3161no—
567C4—3161no—
568C4—6321no—
569C4—6321no—
570C4—6321no—
571C4—1582no—
572C4—1582no—
573C4#1909911511yescharacteristic
574C4#1909919211yescharacteristic
575C4#1909925011yescharacteristic
576C4#1909925811yescharacteristic
577C2—82no—
578C2#2376710611yescharacteristic
579C2#2377319211yescharacteristic
580C2#2377324611yescharacteristic
581C2#2377584411yescharacteristic
582C2#2377590311yescharacteristic
583C2#2377644711yescharacteristic
584C2#2377647611yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2093C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
hypercenter#2093D79
hypocenterC79#26098
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2093

Complements

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

NH ≅ G/Ntype
1. #23768630C2semidirect
6. #23764165C2semidirect
7. #23764156C2semidirect
8. #23764155C2semidirect
9. #23764122C2semidirect
10. #23763994C2semidirect
11. #23763986C2semidirect
12. #23763567C2semidirect
308. C79#26098semidirect

Maximal subgroups

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

subgroupindex
#237686302
#237663482
#237649232
#237647842
#237647742
#237641652
#237641562
#237641552
#237641222
#237639942
#237639862
#237635672
#237634082
#237632272
C79×#20932
#2609879