Finite Group Databasebeta

← #34307743

Subgroups — #34307743

Order 20224.

740 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772198C211yesmaximal, characteristic
2#23771965C211yesmaximal, characteristic, complemented
3#23764114C211yesmaximal, characteristic, complemented
4#23764107C211yesmaximal, characteristic, complemented
5#23763999C211yesmaximal, characteristic, complemented
6#23763999C211yesmaximal, characteristic, complemented
7#23763730C211yesmaximal, characteristic
8#23763730C211yesmaximal, characteristic
9#23763540C211yesmaximal, characteristic, complemented
10#23763420C211yesmaximal, characteristic, complemented
11#23763417C211yesmaximal, characteristic, complemented
12#23763277C211yesmaximal, characteristic, complemented
13#23763246C211yesmaximal, characteristic, complemented
14#23763239C211yesmaximal, characteristic, complemented
15C79×#2925C211yesmaximal, characteristic, complemented
16C2×#14579181C2211yescharacteristic
17C2×#14579167C2211yescharacteristic
18C2×#14579167C2211yescharacteristic
19C4×#11929624C2211yescharacteristic
20#19098673C411yescharacteristic, complemented
21#19098673C411yescharacteristic, complemented
22#19098668C411yescharacteristic, complemented
23#19098668C411yescharacteristic, complemented
24#19098663C411yescharacteristic, complemented
25#19098663C411yescharacteristic, complemented
26#19098661C411yescharacteristic, complemented
27#19098661C411yescharacteristic, complemented
28#19098599C2211yescharacteristic
29#19098589C2211yescharacteristic
30#19098589C2211yescharacteristic
31#19098588—21no—
32#19098588—21no—
33#19098587C2211yescharacteristic
34#19098585—21no—
35C4⋊C8×D79—21no—
36C2×#14579088C2211yescharacteristic
37C2×#14579088C2211yescharacteristic
38C2×#14579087C2211yescharacteristic
39C2×#14579087C2211yescharacteristic
40#19098559C2211yescharacteristic
41#19098557C2211yescharacteristic
42C2×#14579072C2211yescharacteristic
43C2×#14579071C2211yescharacteristic
44C2×C8×Dic79C2211yescharacteristic
45C4×C4⋊Dic79C2211yescharacteristic
46#19098327C2211yescharacteristic
47#19098327C2211yescharacteristic
48#19098327C2211yescharacteristic
49#19098327C2211yescharacteristic
50#19098324C2211yescharacteristic
51#19098324C2211yescharacteristic
52#19098322C2211yescharacteristic
53#19098322C2211yescharacteristic
54#19098317C2211yescharacteristic
55#19098317C2211yescharacteristic
56C4×#41×C79C2211yescharacteristic
57#433×C79—21no—
58#433×C79—21no—
59#432×C79—21no—
60#431×C79—21no—
61#423×C79C2211yescharacteristic
62C2×C4⋊C8×C79C2211yescharacteristic
63#408×C79C2211yescharacteristic
64#408×C79C2211yescharacteristic
65#405×C79C2211yescharacteristic
66#405×C79C2211yescharacteristic
67C2×#11929624C2311yescharacteristic
68#14579181—22no—
69#14579181—22no—
70C2×#11929614C2×C411yescharacteristic
71C2×#11929612C2×C411yescharacteristic
72C2×#11929611C2×C411yescharacteristic
73C2×#11929609C2×C411yescharacteristic
74#14579167—22no—
75#14579167—22no—
76#14579167—41no—
77#14579167—41no—
78C2×#11929608C2311yescharacteristic
79C2×#11929608C2311yescharacteristic
80C2×#11929607—21no—
81C2×#11929607—21no—
82C22×C79⋊C8C2311yescharacteristic
83C22×C79⋊C8C2311yescharacteristic
84C2×C8×D79—21no—
85C2×C8×D79—21no—
86C4×Dic158C2×C411yescharacteristic
87C4×Dic158C2×C411yescharacteristic
88C4×D316C2×C411yescharacteristic
89C4×D316C2×C411yescharacteristic
90C4×C22⋊D79—21no—
91C4×C22⋊D79—21no—
92#14579120C2311yescharacteristic
93#14579118—21no—
94C2×C4⋊Dic79—21no—
95C2×C4×Dic79C2311yescharacteristic
96C2×C4×Dic79C2311yescharacteristic
97C42×D79—21no—
98#14579094C2×C411yescharacteristic
99#14579094C2×C411yescharacteristic
100#14579092C2×C411yescharacteristic
101#14579092C2×C411yescharacteristic
102C4⋊(C79⋊C8)—21no—
103C4⋊(C79⋊C8)—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106Dic79⋊C8—21no—
107Dic79⋊C8—21no—
108#14579088—41no—
109#14579088—41no—
110#14579087—22no—
111#14579087—22no—
112#14579085C2×C411yescharacteristic
113#14579085C2×C411yescharacteristic
114#14579084C2×C411yescharacteristic
115#14579084C2×C411yescharacteristic
116#14579082C2×C411yescharacteristic
117#14579082C2×C411yescharacteristic
118#14579081C2×C411yescharacteristic
119#14579081C2×C411yescharacteristic
120#14579072C2×C411yescharacteristic
121#14579072C2×C411yescharacteristic
122#14579072C2×C411yescharacteristic
123#14579072C2×C411yescharacteristic
124#14579071—41no—
125C8×Dic79—41no—
126C79⋊#95—21no—
127C2×#41×C79C2311yescharacteristic
128C2×M16×C79C2311yescharacteristic
129C2×M16×C79C2311yescharacteristic
130C22×C8×C79C2311yescharacteristic
131C22×C8×C79C2311yescharacteristic
132C4×Q8×C79C2×C411yescharacteristic
133C4×Q8×C79C2×C411yescharacteristic
134C4×D4×C79—21no—
135C4×D4×C79—21no—
136C4×D4×C79C2×C411yescharacteristic
137C4×D4×C79C2×C411yescharacteristic
138#117×C79—21no—
139#117×C79C2311yescharacteristic
140C2×C42×C79—21no—
141C2×C42×C79C2311yescharacteristic
142C4⋊C8×C79—21no—
143C4⋊C8×C79—21no—
144C4⋊C8×C79D411yescharacteristic
145C4⋊C8×C79D411yescharacteristic
146C4⋊C8×C79D411yescharacteristic
147C4⋊C8×C79D411yescharacteristic
148C22⋊C8×C79—21no—
149C22⋊C8×C79—21no—
150C22⋊C8×C79—21no—
151C22⋊C8×C79—21no—
152#97×C79—21no—
153#97×C79—21no—
154C4×C8×C79—21no—
155C4×C8×C79—21no—
156#11929624—21no—
157#11929624—21no—
158#11929624#4112yes—
159#11929624#4112yes—
160C2×Dic158C22×C411yescharacteristic
161C2×D316C22×C411yescharacteristic
162C2×C22⋊D79—21no—
163C22×Dic79—21no—
164C22×Dic79—21no—
165C2×C4×D79—21no—
166C2×C4×D79—21no—
167C2×C4×D79—21no—
168#11929614—22no—
169#11929612—22no—
170#11929611—22no—
171#11929609—22no—
172#11929608—21no—
173#11929608—21no—
174#11929608—21no—
175#11929608—21no—
176#11929607—42no—
177#11929607—42no—
178C2×C79⋊C8—21no—
179C2×C79⋊C8—21no—
180C2×C79⋊C8—21no—
181C2×C79⋊C8—21no—
182C2×C79⋊C8—21no—
183C2×C79⋊C8—21no—
184C2×C79⋊C8C22×C411yescharacteristic
185C2×C79⋊C8C22×C411yescharacteristic
186C2×C79⋊C8C22×C411yescharacteristic
187C2×C79⋊C8C22×C411yescharacteristic
188C8×D79—42no—
189C8×D79—42no—
190C4⋊Dic79—42no—
191C4⋊Dic79C22×C411yescharacteristic
192C4⋊Dic79C22×C411yescharacteristic
193Dic79⋊C4—21no—
194Dic79⋊C4—21no—
195C22⋊Dic79—21no—
196(C2×D79)⋊C4—21no—
197(C2×D79)⋊C4—21no—
198C4×Dic79—21no—
199C4×Dic79—21no—
200C4×Dic79—21no—
201C4×Dic79—21no—
202C4×Dic79—21no—
203C4×Dic79—21no—
204C4×Dic79—21no—
205#41×C79—21no—
206#41×C79—21no—
207#41×C79—21no—
208#41×C79—21no—
209C2×Q8×C79C22×C411yescharacteristic
210C2×D4×C79—21no—
211C2×D4×C79C22×C411yescharacteristic
212C22×C4×C79—21no—
213C22×C4×C79—21no—
214C22×C4×C79—21no—
215C22×C4×C79#4111yescharacteristic
216C22×C4×C79#4111yescharacteristic
217C22×C4×C79C2411yescharacteristic
218M16×C79—21no—
219M16×C79—21no—
220M16×C79—21no—
221M16×C79—21no—
222C2×C8×C79—21no—
223C2×C8×C79—21no—
224C2×C8×C79—21no—
225C2×C8×C79—21no—
226C2×C8×C79—21no—
227C2×C8×C79—21no—
228C2×C8×C79C2×D411yescharacteristic
229C2×C8×C79C2×D411yescharacteristic
230C2×C8×C79C2×D411yescharacteristic
231C2×C8×C79C2×D411yescharacteristic
232C4⋊C4×C79—21no—
233C4⋊C4×C79—21no—
234C4⋊C4×C79C22×C411yescharacteristic
235C4⋊C4×C79C22×C411yescharacteristic
236C22⋊C4×C79—21no—
237C22⋊C4×C79—21no—
238C22⋊C4×C79—21no—
239C42×C79—21no—
240C42×C79—21no—
241C42×C79—21no—
242C42×C79C22×C411yescharacteristic
243C42×C79C22×C411yescharacteristic
244C42×C79C2×D411yescharacteristic
245C42×C79C2×D411yescharacteristic
246C23×C79—21no—
247C23×C79C2×#4111yescharacteristic
248C22×D79—21no—
249Q8×C79—21no—
250Q8×C79—21no—
251D4×C79—21no—
252D4×C79—21no—
253D4×C79—41no—
254D4×C79—41no—
255C2×C4×C79—21no—
256C2×C4×C79—21no—
257C2×C4×C79—21no—
258C2×C4×C79—41no—
259C2×C4×C79—41no—
260C2×C4×C79—42no—
261C2×C4×C79—41no—
262C2×C4×C79—41no—
263C2×C4×C79—81no—
264C2×C4×C79C4×D411yescharacteristic
265C2×C4×C79C4×D411yescharacteristic
266C2×C4×C79C4×D411yescharacteristic
267C2×C4×C79C4×D411yescharacteristic
268C2×C4×C79C23×C411yescharacteristic
269C2×C4×C79C22×D411yescharacteristic
270C2×C4×C79C2×#4111yescharacteristic
271C2×C4×C79C2×#4111yescharacteristic
272C2×C4×C79H(5,𝔽2)11yescharacteristic
273C2×C4×C792−1+411yescharacteristic
274C22⋊D79—22no—
275C22⋊D79—22no—
276C2×Dic79—21no—
277C2×Dic79—22no—
278C2×Dic79—21no—
279C2×Dic79—21no—
280C2×Dic79—41no—
281C2×Dic79—41no—
282C2×Dic79—41no—
283C2×Dic79—41no—
284C2×Dic79—41no—
285D316—21no—
286D316#11712yes—
287C4×D79—22no—
288C4×D79—22no—
289C4×D79—42no—
290C4×D79—81no—
291Dic158—21no—
292Dic158#11712yes—
293C8×C79—21no—
294C8×C79—21no—
295C8×C79—21no—
296C8×C79—21no—
297C8×C79—41no—
298C8×C79—41no—
299C79⋊C8—21no—
300C79⋊C8—21no—
301C79⋊C8—21no—
302C79⋊C8—21no—
303C79⋊C8—41no—
304C79⋊C8—41no—
305C22×C79—21no—
306C22×C79—21no—
307C22×C79—41no—
308C22×C79—41no—
309C22×C79—41no—
310C22×C79C2×C4×D411yescharacteristic
311C22×C79#55211yescharacteristic
312C22×C79#55311yescharacteristic
313C2×D79—22no—
314C2×D79—41no—
315C2×D79—41no—
316C4×C79—21no—
317C4×C79—21no—
318C4×C79—21no—
319C4×C79—21no—
320C4×C79—41no—
321C4×C79—41no—
322C4×C79—41no—
323C4×C79C2×#11711yescharacteristic
324C4×C79#52011yescharacteristic
325C4×C79#54611yescharacteristic
326C4×C79#54811yescharacteristic
327Dic79—22no—
328Dic79—42no—
329Dic79—41no—
330Dic79—41no—
331Dic79—81no—
332#22784—791nomaximal, Sylow
333C2×C79—21no—
334C2×C79—21no—
335C2×C79—81no—
336C2×C79#235211yescharacteristic
337C2×C79#295511yescharacteristic
338C2×C79#295611yescharacteristic
339D79—42no—
340#2943—791no—
341#2925D7911yescharacteristic, complemented
342#2906—791no—
343#1938—791no—
344#1937—791no—
345#1922—791no—
346#1922—791no—
347#1876—791no—
348#1876—791no—
349#1838—791no—
350#1806—791no—
351#1806—791no—
352#1771—791no—
353#1765—791no—
354#1764—791no—
355C79#2278411yescharacteristic, complemented, Sylow
356C2×#135—791no—
357C2×#131—791no—
358C2×#131—791no—
359C4×#41—791no—
360C4×#41C2×D7911yescharacteristic
361#442—791no—
362#442—791no—
363#441—791no—
364#441—791no—
365#440—791no—
366#440—791no—
367#440—791no—
368#440—791no—
369#433—21no—
370#433—21no—
371#432—21no—
372#431—21no—
373#427—791no—
374#424—791no—
375#424—791no—
376#424—1581no—
377#424—1581no—
378#423—791no—
379#423—1581no—
380#423C2×D7911yescharacteristic
381C2×C4⋊C8—1581no—
382C2×C4⋊C8C2×D7911yescharacteristic
383C2×#104—791no—
384C2×#104—791no—
385C2×#104—791no—
386C2×#104—791no—
387#416—791no—
388#416—791no—
389#408C2×D7911yescharacteristic
390#408C2×D7911yescharacteristic
391#405C2×D7911yescharacteristic
392#405C2×D7911yescharacteristic
393C2×#97—791no—
394C2×#97—791no—
395C2×C4×C8—791no—
396C4×C4⋊C4—791no—
397#341—791no—
398#341—791no—
399#341—791no—
400#341—791no—
401#339—791no—
402#339—791no—
403#339—791no—
404#339—791no—
405#336—791no—
406#336—791no—
407C2×#41—791no—
408C2×#41C22×D7911yescharacteristic
409#135—1582no—
410#135—1582no—
411C2×Q16—791no—
412C2×SD16—791no—
413C2×SD16—791no—
414C2×D8—791no—
415#131—3161no—
416#131—3161no—
417#131—1582no—
418#131—1582no—
419C2×M16—791no—
420C2×M16—791no—
421C2×M16—1581no—
422C2×M16—1581no—
423C2×M16C22×D7911yescharacteristic
424C2×M16C22×D7911yescharacteristic
425C22×C8—791no—
426C22×C8—791no—
427C22×C8—1581no—
428C22×C8—1581no—
429C22×C8C22×D7911yescharacteristic
430C22×C8C22×D7911yescharacteristic
431C4×Q8—791no—
432C4×Q8—791no—
433C4×Q8C4×D7911yescharacteristic
434C4×Q8C4×D7911yescharacteristic
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—791no—
438C4×D4—791no—
439C4×D4—1581no—
440C4×D4—1581no—
441C4×D4C4×D7911yescharacteristic
442C4×D4C4×D7911yescharacteristic
443#117—21no—
444#117—791no—
445#117—1581no—
446#117C22×D7911yescharacteristic
447C2×C4⋊C4—1581no—
448C2×C42—21no—
449C2×C42—791no—
450C2×C42—791no—
451C2×C42—1581no—
452C2×C42C22×D7911yescharacteristic
453#107—791no—
454#107—791no—
455#106—791no—
456#106—791no—
457C4⋊C8—21no—
458C4⋊C8—21no—
459C4⋊C8—1581no—
460C4⋊C8—1581no—
461C4⋊C8—1581no—
462C4⋊C8—1581no—
463C4⋊C8—1581no—
464C4⋊C8—1581no—
465C4⋊C8C22⋊D7911yescharacteristic
466C4⋊C8C22⋊D7911yescharacteristic
467C4⋊C8C22⋊D7911yescharacteristic
468C4⋊C8C22⋊D7911yescharacteristic
469#104—3161no—
470#104—3161no—
471#104—1582no—
472#104—1582no—
473#103—791no—
474#103—791no—
475#103—791no—
476#103—791no—
477#102—791no—
478#102—791no—
479#102—791no—
480#102—791no—
481C22⋊C8—21no—
482C22⋊C8—21no—
483C22⋊C8—21no—
484C22⋊C8—21no—
485#97—21no—
486#97—21no—
487#97—3161no—
488#97—791no—
489#97—791no—
490#97—791no—
491#97—791no—
492C4×C8—21no—
493C4×C8—21no—
494C4×C8—3161no—
495#95—1581no—
496#41—21no—
497#41—21no—
498#41—21no—
499#41—21no—
500#41—792no—
501#41—792no—
502#41—1581no—
503#41—1581no—
504C2×Q8—791no—
505C2×Q8C2×C4×D7911yescharacteristic
506C2×D4—21no—
507C2×D4—791no—
508C2×D4—1581no—
509C2×D4C2×C4×D7911yescharacteristic
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—1581no—
514C22×C4—1581no—
515C22×C4—1581no—
516C22×C4—1581no—
517C22×C4—1581no—
518C22×C4#1192962411yescharacteristic
519C22×C4#1192962411yescharacteristic
520C22×C4C23×D7911yescharacteristic
521Q16—1582no—
522SD16—1582no—
523SD16—1582no—
524D8—1582no—
525M16—21no—
526M16—21no—
527M16—21no—
528M16—21no—
529M16—3162no—
530M16—3162no—
531M16—1581no—
532M16—1581no—
533M16—1581no—
534M16—1581no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—3162no—
542C2×C8—3162no—
543C2×C8—791no—
544C2×C8—791no—
545C2×C8—791no—
546C2×C8—791no—
547C2×C8—1581no—
548C2×C8—1581no—
549C2×C8—1581no—
550C2×C8—1581no—
551C2×C8—1581no—
552C2×C8—1581no—
553C2×C8C2×C22⋊D7911yescharacteristic
554C2×C8C2×C22⋊D7911yescharacteristic
555C2×C8C2×C22⋊D7911yescharacteristic
556C2×C8C2×C22⋊D7911yescharacteristic
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—3162no—
560C4⋊C4—791no—
561C4⋊C4—791no—
562C4⋊C4—1581no—
563C4⋊C4—1581no—
564C4⋊C4C2×C4×D7911yescharacteristic
565C4⋊C4C2×C4×D7911yescharacteristic
566C22⋊C4—21no—
567C22⋊C4—21no—
568C22⋊C4—21no—
569C22⋊C4—1581no—
570C22⋊C4—1581no—
571C22⋊C4—1581no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42—1581no—
576C42—1581no—
577C42—1581no—
578C42—1581no—
579C42—1581no—
580C42—1581no—
581C42—1581no—
582C42C2×C4×D7911yescharacteristic
583C42C2×C4×D7911yescharacteristic
584C42C2×C22⋊D7911yescharacteristic
585C42C2×C22⋊D7911yescharacteristic
586C23—21no—
587C23—1581no—
588C23C2×#1192962411yescharacteristic
589Q8—21no—
590Q8—21no—
591Q8—792no—
592Q8—1581no—
593D4—21no—
594D4—21no—
595D4—41no—
596D4—41no—
597D4—792no—
598D4—1582no—
599D4—1582no—
600D4—1581no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—81no—
610C2×C4—3162no—
611C2×C4—3161no—
612C2×C4—3161no—
613C2×C4—3161no—
614C2×C4—3161no—
615C2×C4—3161no—
616C2×C4—6321no—
617C2×C4—1582no—
618C2×C4—1581no—
619C2×C4—1581no—
620C2×C4—1581no—
621C2×C4—1582no—
622C2×C4—1582no—
623C2×C4C4×C22⋊D7911yescharacteristic
624C2×C4C4×C22⋊D7911yescharacteristic
625C2×C4C4×C22⋊D7911yescharacteristic
626C2×C4C4×C22⋊D7911yescharacteristic
627C2×C4C22×C4×D7911yescharacteristic
628C2×C4C22×C22⋊D7911yescharacteristic
629C2×C4#41×D7911yescharacteristic
630C2×C4#41×D7911yescharacteristic
631C2×C4#1457920711yescharacteristic
632C2×C4#1457920811yescharacteristic
633C8—21no—
634C8—21no—
635C8—21no—
636C8—21no—
637C8—41no—
638C8—41no—
639C8—3161no—
640C8—3161no—
641C8—1581no—
642C8—1581no—
643C8—1581no—
644C8—1581no—
645C22—21no—
646C22—21no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—3161no—
651C22—3161no—
652C22—1582no—
653C22C2×C4×C22⋊D7911yescharacteristic
654C22#1909925211yescharacteristic
655C22#1909925711yescharacteristic
656C4—21no—
657C4—21no—
658C4—21no—
659C4—21no—
660C4—41no—
661C4—41no—
662C4—41no—
663C4—3162no—
664C4—3161no—
665C4—3161no—
666C4—6321no—
667C4—1582no—
668C4#117×D7911yescharacteristic
669C4#1909904311yescharacteristic
670C4#1909920111yescharacteristic
671C4#1909922011yescharacteristic
672C2—21no—
673C2—21no—
674C2—81no—
675C2—3162no—
676C2#2376633011yescharacteristic
677C2#2377230111yescharacteristic
678C2#2377230711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#2925C2
socleC22×C79C2×C4×D4
prime radicalC2×#14579181C22
cosocleC22×C4×C79C24
hypercenter#2925D79
hypocenterC79#22784
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ #2925

Complements

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

NH ≅ G/Ntype
2. #23771965C2semidirect
3. #23764114C2semidirect
4. #23764107C2semidirect
5. #23763999C2semidirect
6. #23763999C2semidirect
9. #23763540C2semidirect
10. #23763420C2semidirect
11. #23763417C2semidirect
12. #23763277C2semidirect
13. #23763246C2semidirect
14. #23763239C2semidirect
15. C79×#2925C2semidirect
20. #19098673C4semidirect
21. #19098673C4semidirect
22. #19098668C4semidirect
23. #19098668C4semidirect
24. #19098663C4semidirect
25. #19098663C4semidirect
26. #19098661C4semidirect
27. #19098661C4semidirect
341. #2925D79semidirect
355. C79#22784semidirect

Maximal subgroups

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

subgroupindex
#237721982
#237719652
#237641142
#237641072
#237639992
#237637302
#237635402
#237634202
#237634172
#237632772
#237632462
#237632392
C79×#29252
#2278479