Finite Group Databasebeta

← #34183380

Subgroups — #34183380

Order 20224.

676 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#23772448C211yesmaximal, characteristic, complemented
2#23763344C211yesmaximal, characteristic, complemented
3#23763312C211yesmaximal, characteristic, complemented
4#23761891C211yesmaximal, characteristic, complemented
5#23761288C211yesmaximal, characteristic, complemented
6#23761246C211yesmaximal, characteristic, complemented
7C79×#3002C211yesmaximal, characteristic
8#19098641—21no—
9#19098635—21no—
10#19098635—21no—
11D4×C79⋊C8—21no—
12#19098624—21no—
13C2×#14579088—21no—
14C2×#14579088—21no—
15#19098530—21no—
16#19098527C2211yescharacteristic
17C2×C22⋊(C79⋊C8)—21no—
18C2×C22⋊(C79⋊C8)—21no—
19C2×C22⋊(C79⋊C8)C2211yescharacteristic
20C2×C22⋊(C79⋊C8)C2211yescharacteristic
21C4×C22⋊Dic79C2211yescharacteristic
22#19098300—21no—
23#19098297—21no—
24C22×#41×C79C2211yescharacteristic
25C2×#135×C79—21no—
26#525×C79C2211yescharacteristic
27#452×C79—21no—
28#451×C79C411yescharacteristic, complemented
29#450×C79C411yescharacteristic, complemented
30#449×C79—21no—
31#448×C79C411yescharacteristic, complemented
32#447×C79C411yescharacteristic, complemented
33#416×C79—21no—
34C2×C22⋊C8×C79C2211yescharacteristic
35C2×#11929608—21no—
36C2×#11929608D411yescharacteristic
37C2×#11929608D411yescharacteristic
38C22×C79⋊C8—21no—
39C22×C79⋊C8—21no—
40C22×C79⋊C8D411yescharacteristic
41C22×C79⋊C8D411yescharacteristic
42C2×C22⋊Dic79—21no—
43C2×C4×Dic79—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)—21no—
46#14579088—42no—
47#14579088—42no—
48C22⋊(C79⋊C8)—21no—
49C22⋊(C79⋊C8)—21no—
50C22⋊(C79⋊C8)—21no—
51C22⋊(C79⋊C8)—21no—
52C22⋊(C79⋊C8)—21no—
53C22⋊(C79⋊C8)—21no—
54C22⋊(C79⋊C8)—21no—
55C22⋊(C79⋊C8)—21no—
56C22⋊(C79⋊C8)—42no—
57C22⋊(C79⋊C8)—42no—
58#14579072—21no—
59C4×C79⋊C8—21no—
60C79⋊#95—21no—
61C2×#41×C79—21no—
62C2×#41×C79—21no—
63C2×#41×C79—21no—
64C2×#41×C79—21no—
65C2×#41×C79—42no—
66C2×#41×C79—42no—
67C2×#41×C79D411yescharacteristic
68C22×Q8×C79C2×C411yescharacteristic
69C22×D4×C79—21no—
70C22×D4×C79C2×C411yescharacteristic
71C23×C4×C79—21no—
72C23×C4×C79C2311yescharacteristic
73#135×C79—42no—
74#135×C79—42no—
75C2×Q16×C79—21no—
76C2×SD16×C79—21no—
77C2×SD16×C79—21no—
78C2×D8×C79—21no—
79C22×C8×C79—21no—
80#123×C79—21no—
81C22⋊Q8×C79C2×C411yescharacteristic
82C22⋊Q8×C79D411yescharacteristic
83#121×C79C2×C411yescharacteristic
84#121×C79D411yescharacteristic
85#120×C79—21no—
86C4×D4×C79—21no—
87C4×D4×C79—21no—
88#117×C79D411yescharacteristic
89#103×C79—21no—
90#103×C79—21no—
91#102×C79—21no—
92#102×C79—21no—
93C22⋊C8×C79—21no—
94C22⋊C8×C79C2×C411yescharacteristic
95C22⋊C8×C79C2×C411yescharacteristic
96C22×Dic79—41no—
97#11929608—41no—
98#11929608—41no—
99#11929608—81no—
100C2×C79⋊C8—21no—
101C2×C79⋊C8—21no—
102C2×C79⋊C8—21no—
103C2×C79⋊C8—21no—
104C2×C79⋊C8—41no—
105C2×C79⋊C8—41no—
106C2×C79⋊C8—41no—
107C2×C79⋊C8—41no—
108C2×C79⋊C8—41no—
109C2×C79⋊C8—81no—
110C22⋊Dic79—22no—
111C22⋊Dic79—41no—
112C4×Dic79—41no—
113C4×Dic79—41no—
114C24×C79—21no—
115C24×C79#4111yescharacteristic
116#41×C79—22no—
117#41×C79—22no—
118#41×C79—22no—
119#41×C79—22no—
120#41×C79—41no—
121#41×C79—41no—
122#41×C79—41no—
123#41×C79—41no—
124#41×C79—41no—
125#41×C79—41no—
126#41×C79—42no—
127#41×C79—42no—
128#41×C79—42no—
129#41×C79—42no—
130C2×Q8×C79—21no—
131C2×Q8×C79—21no—
132C2×Q8×C79—42no—
133C2×Q8×C79C22⋊C411yescharacteristic
134C2×D4×C79—21no—
135C2×D4×C79—21no—
136C2×D4×C79—21no—
137C2×D4×C79—21no—
138C2×D4×C79—21no—
139C2×D4×C79—42no—
140C2×D4×C79—42no—
141C2×D4×C79—42no—
142C2×D4×C79—41no—
143C2×D4×C79—41no—
144C2×D4×C79C22⋊C411yescharacteristic
145C22×C4×C79—22no—
146C22×C4×C79—22no—
147C22×C4×C79—21no—
148C22×C4×C79—21no—
149C22×C4×C79—41no—
150C22×C4×C79—41no—
151C22×C4×C79—41no—
152C22×C4×C79—41no—
153C22×C4×C79—41no—
154C22×C4×C79—41no—
155C22×C4×C79C22×C411yescharacteristic
156C22×C4×C79C2×D411yescharacteristic
157C22×C4×C79C2×D411yescharacteristic
158C22×C4×C79C2×D411yescharacteristic
159C22×C4×C79C2×D411yescharacteristic
160C22×C4×C79#4111yescharacteristic
161Q16×C79—42no—
162SD16×C79—42no—
163SD16×C79—42no—
164D8×C79—42no—
165C2×C8×C79—21no—
166C2×C8×C79—21no—
167C2×C8×C79—41no—
168C2×C8×C79—41no—
169C4⋊C4×C79—21no—
170C4⋊C4×C79—21no—
171C4⋊C4×C79C22⋊C411yescharacteristic
172C4⋊C4×C79C22⋊C411yescharacteristic
173C22⋊C4×C79—21no—
174C22⋊C4×C79—21no—
175C22⋊C4×C79—21no—
176C22⋊C4×C79—21no—
177C22⋊C4×C79—21no—
178C42×C79—21no—
179C23×C79—21no—
180C23×C79—21no—
181C23×C79—22no—
182C23×C79—21no—
183C23×C79—41no—
184C23×C79—41no—
185C23×C79—41no—
186C23×C79—41no—
187C23×C79—41no—
188C23×C79—41no—
189C23×C79—41no—
190C23×C79—41no—
191C23×C79C4×D411yescharacteristic
192C23×C79#12111yescharacteristic
193C23×C79#12311yescharacteristic
194Q8×C79—22no—
195Q8×C79—22no—
196Q8×C79—41no—
197Q8×C79—41no—
198Q8×C79—41no—
199D4×C79—22no—
200D4×C79—22no—
201D4×C79—41no—
202D4×C79—41no—
203D4×C79—41no—
204D4×C79—42no—
205D4×C79—42no—
206D4×C79—42no—
207D4×C79—42no—
208D4×C79—41no—
209D4×C79—41no—
210D4×C79—81no—
211D4×C79—81no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—22no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79—42no—
226C2×C4×C79—42no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—42no—
232C2×C4×C79—42no—
233C2×C4×C79C2×C22⋊C411yescharacteristic
234C2×C4×C79C4×D411yescharacteristic
235C2×C4×C79#12011yescharacteristic
236C2×C4×C79#12111yescharacteristic
237C2×Dic79—42no—
238C2×Dic79—81no—
239C2×Dic79—81no—
240C8×C79—41no—
241C8×C79—41no—
242C79⋊C8—41no—
243C79⋊C8—41no—
244C79⋊C8—41no—
245C79⋊C8—41no—
246C79⋊C8—81no—
247C22×C79—21no—
248C22×C79—21no—
249C22×C79—21no—
250C22×C79—21no—
251C22×C79—21no—
252C22×C79—21no—
253C22×C79—21no—
254C22×C79—21no—
255C22×C79—42no—
256C22×C79—42no—
257C22×C79—41no—
258C22×C79—41no—
259C22×C79—42no—
260C22×C79—41no—
261C22×C79—81no—
262C22×C79—81no—
263C22×C79—81no—
264C22×C79—81no—
265C22×C79—81no—
266C22×C79—81no—
267C22×C79#38611yescharacteristic
268C22×C79#44311yescharacteristic
269C22×C79#44411yescharacteristic
270C4×C79—21no—
271C4×C79—21no—
272C4×C79—21no—
273C4×C79—41no—
274C4×C79—41no—
275C4×C79—42no—
276C4×C79—41no—
277C4×C79B(4,𝔽2)11yescharacteristic
278C4×C79B(4,𝔽2)11yescharacteristic
279Dic79—82no—
280#9037—791nomaximal, Sylow
281C2×C79—21no—
282C2×C79—21no—
283C2×C79—41no—
284C2×C79—41no—
285C2×C79—81no—
286C2×C79—82no—
287C2×C79#187611yescharacteristic
288C2×C79#188111yescharacteristic
289C2×C79#188911yescharacteristic
290#3002D7911yescharacteristic
291#2976—791no—
292#1789—791no—
293#1780—791no—
294#1501—791no—
295#1298—791no—
296#1282—791no—
297C79#903711yescharacteristic, complemented, Sylow
298C22×#41C2×D7911yescharacteristic
299C2×#135—21no—
300#525C2×D7911yescharacteristic
301#452—21no—
302#451Dic7911yescharacteristic, complemented
303#450Dic7911yescharacteristic, complemented
304#449—21no—
305#448Dic7911yescharacteristic, complemented
306#447Dic7911yescharacteristic, complemented
307#436—1581no—
308#435—1581no—
309#435—1581no—
310C8×D4—1581no—
311#433—1581no—
312C2×#104—1581no—
313C2×#104—1581no—
314#416—21no—
315#408—1581no—
316#407—791no—
317C2×C22⋊C8—791no—
318C2×C22⋊C8—791no—
319C2×C22⋊C8—1581no—
320C2×C22⋊C8—1581no—
321C2×C22⋊C8C2×D7911yescharacteristic
322C4×C22⋊C4—791no—
323#328—1581no—
324#327—1581no—
325C2×#41—21no—
326C2×#41—21no—
327C2×#41—21no—
328C2×#41—21no—
329C2×#41—42no—
330C2×#41—42no—
331C2×#41C22⋊D7911yescharacteristic
332C22×Q8C2×Dic7911yescharacteristic
333C22×D4—21no—
334C22×D4C2×Dic7911yescharacteristic
335C23×C4—21no—
336C23×C4C22×D7911yescharacteristic
337#135—42no—
338#135—42no—
339C2×Q16—21no—
340C2×SD16—21no—
341C2×SD16—21no—
342C2×D8—21no—
343C2×M16—791no—
344C2×M16—791no—
345C2×M16—1581no—
346C22×C8—21no—
347C22×C8—791no—
348C22×C8—791no—
349C22×C8—1581no—
350C22×C8—1581no—
351#123—21no—
352C22⋊Q8C2×Dic7911yescharacteristic
353C22⋊Q8C22⋊D7911yescharacteristic
354#121C2×Dic7911yescharacteristic
355#121C22⋊D7911yescharacteristic
356#120—21no—
357C4×D4—21no—
358C4×D4—21no—
359#117C22⋊D7911yescharacteristic
360C2×C22⋊C4—1581no—
361C2×C42—1581no—
362C4⋊C8—1581no—
363C4⋊C8—1581no—
364#104—3162no—
365#104—3162no—
366#103—21no—
367#103—21no—
368#102—21no—
369#102—21no—
370C22⋊C8—21no—
371C22⋊C8—3162no—
372C22⋊C8—3162no—
373C22⋊C8—1581no—
374C22⋊C8—1581no—
375C22⋊C8—1581no—
376C22⋊C8—1581no—
377C22⋊C8—1581no—
378C22⋊C8—1581no—
379C22⋊C8—1581no—
380C22⋊C8—1581no—
381C22⋊C8C2×Dic7911yescharacteristic
382C22⋊C8C2×Dic7911yescharacteristic
383#97—1581no—
384C4×C8—1581no—
385#95—1581no—
386C24—21no—
387C24#1192962311yescharacteristic
388#41—22no—
389#41—22no—
390#41—22no—
391#41—22no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398#41—42no—
399#41—42no—
400#41—42no—
401#41—42no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—42no—
405C2×Q8C22⋊Dic7911yescharacteristic
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4C22⋊Dic7911yescharacteristic
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—3161no—
428C22×C4C22×Dic7911yescharacteristic
429C22×C4D4×D7911yescharacteristic
430C22×C4D4×D7911yescharacteristic
431C22×C4C2×C22⋊D7911yescharacteristic
432C22×C4C2×C22⋊D7911yescharacteristic
433C22×C4#1192962311yescharacteristic
434Q16—42no—
435SD16—42no—
436SD16—42no—
437D8—42no—
438M16—3161no—
439M16—3161no—
440M16—6321no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—3161no—
446C2×C8—3161no—
447C2×C8—3161no—
448C2×C8—3161no—
449C2×C8—3161no—
450C2×C8—6321no—
451C2×C8—1581no—
452C2×C8—1581no—
453C2×C8—1581no—
454C2×C8—1581no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4C22⋊Dic7911yescharacteristic
458C4⋊C4C22⋊Dic7911yescharacteristic
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—21no—
462C22⋊C4—21no—
463C22⋊C4—21no—
464C22⋊C4—3161no—
465C22⋊C4—1582no—
466C42—21no—
467C42—3161no—
468C42—3161no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—21no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23D4×Dic7911yescharacteristic
482C23#1457913311yescharacteristic
483C23#1457914211yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—41no—
487Q8—41no—
488Q8—41no—
489D4—22no—
490D4—22no—
491D4—41no—
492D4—41no—
493D4—41no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—42no—
498D4—41no—
499D4—41no—
500D4—81no—
501D4—81no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—3162no—
524C2×C4—6321no—
525C2×C4—6321no—
526C2×C4C2×C22⋊Dic7911yescharacteristic
527C2×C4D4×Dic7911yescharacteristic
528C2×C4#1457912911yescharacteristic
529C2×C4#1457913311yescharacteristic
530C8—41no—
531C8—41no—
532C8—3161no—
533C8—3161no—
534C8—3161no—
535C8—3161no—
536C8—6321no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—42no—
546C22—42no—
547C22—41no—
548C22—41no—
549C22—42no—
550C22—41no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22#1909844311yescharacteristic
558C22#1909868011yescharacteristic
559C22#1909868511yescharacteristic
560C4—21no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—42no—
566C4—41no—
567C4—6322no—
568C4#1909874911yescharacteristic
569C4#1909874911yescharacteristic
570C2—21no—
571C2—21no—
572C2—41no—
573C2—41no—
574C2—82no—
575C2—81no—
576C2#2376372911yescharacteristic
577C2#2376376411yescharacteristic
578C2#2376380911yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
Fitting, prime radicalC79×#3002C2
socleC22×C79#386
cosocleC23×C4×C79C23
hypercenter#3002D79
hypocenterC79#9037
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #3002

Complements

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

NH ≅ G/Ntype
1. #23772448C2semidirect
2. #23763344C2semidirect
3. #23763312C2semidirect
4. #23761891C2semidirect
5. #23761288C2semidirect
6. #23761246C2semidirect
28. #451×C79C4semidirect
29. #450×C79C4semidirect
31. #448×C79C4semidirect
32. #447×C79C4semidirect
297. C79#9037semidirect
302. #451Dic79semidirect
303. #450Dic79semidirect
305. #448Dic79semidirect
306. #447Dic79semidirect

Maximal subgroups

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

subgroupindex
#237724482
#237633442
#237633122
#237618912
#237612882
#237612462
C79×#30022
#903779