Finite Group Databasebeta

← #34356759

Subgroups — #34356759

Order 20224.

762 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23767752C211yesmaximal, characteristic
2#23764715C211yesmaximal, characteristic, complemented
3#23764593C211yesmaximal, characteristic, complemented
4#23764551C211yesmaximal, characteristic, complemented
5#23764401C211yesmaximal, characteristic
6#23764357C211yesmaximal, characteristic, complemented
7#23764096C211yesmaximal, characteristic, complemented
8#23764027C211yesmaximal, characteristic, complemented
9#23764015C211yesmaximal, characteristic, complemented
10#23763982C211yesmaximal, characteristic, complemented
11#23763755C211yesmaximal, characteristic, complemented
12#23763722C211yesmaximal, characteristic
13#23763495C211yesmaximal, characteristic, complemented
14#23763230C211yesmaximal, characteristic, complemented
15C79×#2362C211yesmaximal, characteristic, complemented
16C22×#11929611C2211yescharacteristic
17C2×#14579153C2211yescharacteristic
18C2×#14579135C2211yescharacteristic
19C2×#14579092C2211yescharacteristic
20C2×C4⋊(C79⋊C8)C2211yescharacteristic
21C2×#14579085C2211yescharacteristic
22C2×#14579085C2211yescharacteristic
23C2×#14579085C2211yescharacteristic
24C2×#14579081C2211yescharacteristic
25C2×#14579081C2211yescharacteristic
26C2×#14579081C2211yescharacteristic
27C2×C22⋊(C79⋊C8)C2211yescharacteristic
28C2×C22⋊(C79⋊C8)C2211yescharacteristic
29C2×#14579072C2211yescharacteristic
30#19098492—21no—
31#19098478—21no—
32#19098465C2211yescharacteristic
33#19098465C2211yescharacteristic
34#19098463—21no—
35#19098452—21no—
36#19098423C2211yescharacteristic
37C4×C4⋊Dic79C2211yescharacteristic
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C2×C22⋊Q8×C79C2211yescharacteristic
49C2×C4×Q8×C79C2211yescharacteristic
50C2×#117×C79C2211yescharacteristic
51#391×C79C2211yescharacteristic
52#389×C79C2211yescharacteristic
53#388×C79—21no—
54#388×C79—21no—
55#387×C79—21no—
56C4×C4⋊C4×C79—21no—
57C4×C4⋊C4×C79C2211yescharacteristic
58C4×C22⋊C4×C79C2211yescharacteristic
59C22×D316C2311yescharacteristic
60C22×C22⋊D79—21no—
61C2×#11929611—22no—
62C2×#11929611—41no—
63C2×#11929611—41no—
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68#14579153—22no—
69#14579153—41no—
70#14579135—22no—
71#14579135—41no—
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—21no—
77C2×C22⋊Dic79—21no—
78C2×(C2×D79)⋊C4—21no—
79C2×(C2×D79)⋊C4—21no—
80C2×(C2×D79)⋊C4—21no—
81C2×(C2×D79)⋊C4—21no—
82C2×(C2×D79)⋊C4—21no—
83C2×C4×Dic79—21no—
84#14579092—22no—
85C4⋊(C79⋊C8)—41no—
86#14579085—22no—
87#14579085—22no—
88#14579085—41no—
89#14579081—22no—
90#14579081—22no—
91#14579081—41no—
92C22⋊(C79⋊C8)—41no—
93C22⋊(C79⋊C8)—41no—
94#14579072—41no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C22×Q8×C79C2311yescharacteristic
98C23×C4×C79C2311yescharacteristic
99C22⋊Q8×C79—22no—
100C22⋊Q8×C79—22no—
101C4×Q8×C79—22no—
102C4×Q8×C79—41no—
103#117×C79—22no—
104#117×C79—41no—
105C2×C4⋊C4×C79—21no—
106C2×C4⋊C4×C79—21no—
107C2×C4⋊C4×C79—21no—
108C2×C4⋊C4×C79C2311yescharacteristic
109C2×C4⋊C4×C79C2311yescharacteristic
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C22⋊C4×C79—21no—
112C2×C22⋊C4×C79—21no—
113C2×C22⋊C4×C79D411yescharacteristic
114C2×C22⋊C4×C79D411yescharacteristic
115C2×C42×C79—21no—
116C2×C42×C79—21no—
117C2×C42×C79—21no—
118C2×C42×C79C2311yescharacteristic
119C2×C42×C79C2311yescharacteristic
120#95×C79—21no—
121#95×C79—21no—
122#95×C79—21no—
123#95×C79—21no—
124#95×C79D411yescharacteristic
125#95×C79D411yescharacteristic
126C23×D79—21no—
127C2×D316—21no—
128C2×D316—21no—
129C2×D316—21no—
130C2×D316—21no—
131C2×D316—21no—
132C2×D316—21no—
133C2×C22⋊D79—42no—
134C2×C22⋊D79—42no—
135C2×C22⋊D79—81no—
136C22×Dic79—21no—
137C22×Dic79—21no—
138C22×Dic79—21no—
139#11929611—44no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8—21no—
149C2×C79⋊C8—21no—
150C2×C79⋊C8—21no—
151C2×C79⋊C8—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—21no—
157C4⋊Dic79#4112yes—
158Dic79⋊C4—42no—
159Dic79⋊C4—42no—
160C22⋊Dic79—81no—
161(C2×D79)⋊C4—42no—
162(C2×D79)⋊C4—42no—
163(C2×D79)⋊C4—81no—
164(C2×D79)⋊C4—81no—
165(C2×D79)⋊C4—81no—
166C4×Dic79—81no—
167C24×C79C2×D411yescharacteristic
168C2×Q8×C79—21no—
169C2×Q8×C79—21no—
170C2×Q8×C79—21no—
171C2×Q8×C79—21no—
172C2×Q8×C79—21no—
173C2×Q8×C79#4112yes—
174C22×C4×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79—21no—
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79—21no—
181C22×C4×C79—21no—
182C22×C4×C79—21no—
183C22×C4×C79—21no—
184C22×C4×C79C2×D411yescharacteristic
185C22×C4×C79C2×D411yescharacteristic
186C22×C4×C79C2×D411yescharacteristic
187C22×C4×C79C2×D411yescharacteristic
188C22×C4×C79C2×D411yescharacteristic
189C22×C4×C79C2411yescharacteristic
190C4⋊C4×C79—24no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—42no—
196C4⋊C4×C79—42no—
197C4⋊C4×C79#4112yes—
198C4⋊C4×C79#4112yes—
199C22⋊C4×C79—24no—
200C22⋊C4×C79—41no—
201C22⋊C4×C79—41no—
202C22⋊C4×C79—42no—
203C42×C79—21no—
204C42×C79—21no—
205C42×C79—21no—
206C42×C79—21no—
207C42×C79—42no—
208C42×C79—42no—
209C42×C79—42no—
210C23×C79—21no—
211C23×C79—21no—
212C23×C79—21no—
213C23×C79—21no—
214C23×C79—21no—
215C23×C79—21no—
216C23×C79—21no—
217C23×C79C22×D411yescharacteristic
218C22×D79—41no—
219C22×D79—41no—
220C22×D79—41no—
221C22×D79—41no—
222C22×D79—41no—
223C22×D79—41no—
224C22×D79—41no—
225Q8×C79—22no—
226Q8×C79—22no—
227Q8×C79—41no—
228Q8×C79—41no—
229C2×C4×C79—22no—
230C2×C4×C79—22no—
231C2×C4×C79—21no—
232C2×C4×C79—21no—
233C2×C4×C79—21no—
234C2×C4×C79—21no—
235C2×C4×C79—21no—
236C2×C4×C79—21no—
237C2×C4×C79—22no—
238C2×C4×C79—21no—
239C2×C4×C79—21no—
240C2×C4×C79—21no—
241C2×C4×C79—21no—
242C2×C4×C79—21no—
243C2×C4×C79—21no—
244C2×C4×C79—21no—
245C2×C4×C79—21no—
246C2×C4×C79—21no—
247C2×C4×C79—41no—
248C2×C4×C79—41no—
249C2×C4×C79—41no—
250C2×C4×C79—41no—
251C2×C4×C79—41no—
252C2×C4×C79—41no—
253C2×C4×C79—41no—
254C2×C4×C79—41no—
255C2×C4×C79—41no—
256C2×C4×C79—41no—
257C2×C4×C79—41no—
258C2×C4×C79—41no—
259C2×C4×C79—41no—
260C2×C4×C79C2×#4111yescharacteristic
261C2×C4×C79C2×#4111yescharacteristic
262C2×C4×C79C2×#4111yescharacteristic
263C2×C4×C79C2×#4111yescharacteristic
264C2×C4×C79H(5,𝔽2)11yescharacteristic
265C2×C4×C79H(5,𝔽2)11yescharacteristic
266C22⋊D79—84no—
267C2×Dic79—22no—
268C2×Dic79—41no—
269C2×Dic79—41no—
270C2×Dic79—41no—
271C2×Dic79—41no—
272C2×Dic79—41no—
273C2×Dic79—41no—
274C2×Dic79—41no—
275C2×Dic79—41no—
276D316—41no—
277D316—41no—
278D316—41no—
279D316—41no—
280C79⋊C8—41no—
281C79⋊C8—41no—
282C79⋊C8—41no—
283C79⋊C8—41no—
284C22×C79—41no—
285C22×C79—41no—
286C22×C79—41no—
287C22×C79—41no—
288C22×C79—41no—
289C22×C79—41no—
290C22×C79—41no—
291C22×C79#52511yescharacteristic
292C22×C79#54611yescharacteristic
293C22×C79#54611yescharacteristic
294C22×C79#57511yescharacteristic
295C22×C79#57511yescharacteristic
296C22×C79#57711yescharacteristic
297C22×C79#57711yescharacteristic
298C2×D79—81no—
299C2×D79—81no—
300C2×D79—81no—
301C2×D79—81no—
302C2×D79—81no—
303C2×D79—81no—
304C2×D79—81no—
305C4×C79—42no—
306C4×C79—41no—
307C4×C79—42no—
308C4×C79—41no—
309C4×C79—42no—
310C4×C79—41no—
311C4×C79—41no—
312C4×C79—41no—
313C4×C79—81no—
314C4×C79#52911yescharacteristic
315C4×C79#53811yescharacteristic
316C4×C79#55511yescharacteristic
317C4×C79#55911yescharacteristic
318Dic79—42no—
319Dic79—81no—
320Dic79—81no—
321#26751—791nomaximal, Sylow
322C2×C79—81no—
323C2×C79#245911yescharacteristic
324C2×C79#311111yescharacteristic
325C2×C79#312611yescharacteristic
326C2×C79#330711yescharacteristic
327C2×C79#331111yescharacteristic
328C2×C79#331711yescharacteristic
329C2×C79#332611yescharacteristic
330D79—161no—
331#2490—791no—
332#2362D7911yescharacteristic, complemented
333#2055—791no—
334#2035—791no—
335#2026—791no—
336#2000—791no—
337#1989—791no—
338#1936—791no—
339#1926—791no—
340#1924—791no—
341#1920—791no—
342#1880—791no—
343#1875—791no—
344#1825—791no—
345#1763—791no—
346C79#2675111yescharacteristic, complemented, Sylow
347C22×SD16—791no—
348C2×#125—791no—
349C2×C22⋊Q8C2×D7911yescharacteristic
350C2×#121—791no—
351C2×C4×Q8C2×D7911yescharacteristic
352C2×#117C2×D7911yescharacteristic
353C2×#106—791no—
354C2×C4⋊C8—791no—
355C2×#103—791no—
356C2×#103—791no—
357C2×#103—791no—
358C2×#102—791no—
359C2×#102—791no—
360C2×#102—791no—
361C2×C22⋊C8—791no—
362C2×C22⋊C8—791no—
363C2×#97—791no—
364#397—1581no—
365#394—1581no—
366#391C2×D7911yescharacteristic
367#390—791no—
368#390—791no—
369#390—1581no—
370#389C2×D7911yescharacteristic
371#388—21no—
372#388—21no—
373#388—1581no—
374#387—21no—
375#380—791no—
376C4×C4⋊C4—21no—
377C4×C4⋊C4—791no—
378C4×C4⋊C4C2×D7911yescharacteristic
379C4×C22⋊C4C2×D7911yescharacteristic
380#340—791no—
381#340—791no—
382#340—791no—
383#340—791no—
384#340—791no—
385#340—791no—
386#340—791no—
387#340—791no—
388#336—791no—
389#336—791no—
390C22×Q8C22×D7911yescharacteristic
391C22×D4—791no—
392C22×D4—1581no—
393C23×C4C22×D7911yescharacteristic
394C2×SD16—3161no—
395C2×SD16—3161no—
396C2×SD16—1582no—
397C22×C8—791no—
398C22×C8—791no—
399C22×C8—791no—
400C22×C8—791no—
401#125—3161no—
402#125—1582no—
403C22⋊Q8—22no—
404C22⋊Q8—22no—
405#121—3161no—
406#121—1582no—
407C4×Q8—22no—
408C4×Q8—41no—
409#117—22no—
410#117—41no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—791no—
415C2×C4⋊C4—791no—
416C2×C4⋊C4—791no—
417C2×C4⋊C4—1581no—
418C2×C4⋊C4—1581no—
419C2×C4⋊C4C22×D7911yescharacteristic
420C2×C4⋊C4C22×D7911yescharacteristic
421C2×C4⋊C4C22×D7911yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1581no—
425C2×C22⋊C4—1581no—
426C2×C22⋊C4—1581no—
427C2×C22⋊C4—1581no—
428C2×C22⋊C4—1581no—
429C2×C22⋊C4—1581no—
430C2×C22⋊C4C22⋊D7911yescharacteristic
431C2×C22⋊C4C22⋊D7911yescharacteristic
432C2×C42—21no—
433C2×C42—21no—
434C2×C42—21no—
435C2×C42—1581no—
436C2×C42C22×D7911yescharacteristic
437C2×C42C22×D7911yescharacteristic
438#106—1582no—
439C4⋊C8—3161no—
440#103—3161no—
441#103—1582no—
442#103—1582no—
443#102—3161no—
444#102—1582no—
445#102—1582no—
446C22⋊C8—3161no—
447C22⋊C8—3161no—
448#97—3161no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—1581no—
454#95—1581no—
455#95C22⋊D7911yescharacteristic
456#95C22⋊D7911yescharacteristic
457C24—1581no—
458C24C2×C22⋊D7911yescharacteristic
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8#1192962412yes—
465C2×D4—3162no—
466C2×D4—3162no—
467C2×D4—6321no—
468C2×D4—1581no—
469C2×D4—1581no—
470C2×D4—1581no—
471C2×D4—1581no—
472C2×D4—1581no—
473C2×D4—1581no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—1581no—
485C22×C4—1581no—
486C22×C4—1581no—
487C22×C4C2×C22⋊D7911yescharacteristic
488C22×C4C2×C22⋊D7911yescharacteristic
489C22×C4C2×C22⋊D7911yescharacteristic
490C22×C4C2×C22⋊D7911yescharacteristic
491C22×C4C2×C22⋊D7911yescharacteristic
492C22×C4C23×D7911yescharacteristic
493SD16—3164no—
494C2×C8—1581no—
495C2×C8—1581no—
496C2×C8—1581no—
497C2×C8—1581no—
498C2×C8—1581no—
499C2×C8—1581no—
500C2×C8—1581no—
501C2×C8—1581no—
502C2×C8—1581no—
503C2×C8—1581no—
504C2×C8—1581no—
505C2×C8—1581no—
506C4⋊C4—24no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—3162no—
514C4⋊C4—3162no—
515C4⋊C4—792no—
516C4⋊C4—1581no—
517C4⋊C4—1581no—
518C4⋊C4—1581no—
519C4⋊C4—1581no—
520C4⋊C4—1581no—
521C4⋊C4#1192962312yes—
522C4⋊C4#1192962412yes—
523C22⋊C4—24no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—42no—
527C22⋊C4—3162no—
528C22⋊C4—3162no—
529C22⋊C4—6321no—
530C22⋊C4—6321no—
531C22⋊C4—6321no—
532C22⋊C4—6321no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—42no—
538C42—42no—
539C42—42no—
540C42—6321no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—3161no—
549C23—3161no—
550C23—3161no—
551C23—3161no—
552C23—3161no—
553C23—3161no—
554C23—3161no—
555C23C22×C22⋊D7911yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—41no—
560D4—3161no—
561D4—3161no—
562D4—3161no—
563D4—3161no—
564D4—6324no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—3161no—
597C2×C4—3161no—
598C2×C4—3161no—
599C2×C4—3161no—
600C2×C4—3161no—
601C2×C4—3161no—
602C2×C4—3161no—
603C2×C4—3161no—
604C2×C4—1582no—
605C2×C4#41×D7911yescharacteristic
606C2×C4C2×#1192962311yescharacteristic
607C2×C4C2×#1192962411yescharacteristic
608C2×C4C2×#1192962411yescharacteristic
609C2×C4#1457920611yescharacteristic
610C2×C4#1457920711yescharacteristic
611C8—3161no—
612C8—3161no—
613C8—3161no—
614C8—3161no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—6321no—
623C22—6321no—
624C22—6321no—
625C22—6321no—
626C22—6321no—
627C22—6321no—
628C22—6321no—
629C22#1909907411yescharacteristic
630C22#1909920111yescharacteristic
631C22#1909920711yescharacteristic
632C22#1909936111yescharacteristic
633C22#1909936111yescharacteristic
634C22#1909937111yescharacteristic
635C22#1909937111yescharacteristic
636C4—41no—
637C4—41no—
638C4—42no—
639C4—42no—
640C4—41no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—81no—
645C4—3162no—
646C4—6321no—
647C4—6321no—
648C4#1909909311yescharacteristic
649C4#1909914611yescharacteristic
650C4#1909927811yescharacteristic
651C4#1909929611yescharacteristic
652C2—81no—
653C2—12641no—
654C2#2376733111yescharacteristic
655C2#2377350911yescharacteristic
656C2#2377369111yescharacteristic
657C2#2377537611yescharacteristic
658C2#2377543011yescharacteristic
659C2#2377552011yescharacteristic
660C2#2377565311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2362C2
socleC23×C79C22×D4
prime radicalC2×#14579135C22
hypercenter#2362D79
hypocenterC79#26751
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
2. #23764715C2semidirect
3. #23764593C2semidirect
4. #23764551C2semidirect
6. #23764357C2semidirect
7. #23764096C2semidirect
8. #23764027C2semidirect
9. #23764015C2semidirect
10. #23763982C2semidirect
11. #23763755C2semidirect
13. #23763495C2semidirect
14. #23763230C2semidirect
15. C79×#2362C2semidirect
332. #2362D79semidirect
346. C79#26751semidirect

Maximal subgroups

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

subgroupindex
#237677522
#237647152
#237645932
#237645512
#237644012
#237643572
#237640962
#237640272
#237640152
#237639822
#237637552
#237637222
#237634952
#237632302
C79×#23622
#2675179