Finite Group Databasebeta

← #34210248

Subgroups — #34210248

Order 20224.

2006 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579119C211yesmaximal, characteristic
2C2×#19098432C211yesmaximal, characteristic, complemented
3C2×#19098420C211yesmaximal, characteristic
4C2×C4×(C2×D79)⋊C4C211yesmaximal, characteristic
5#23761644C214yesmaximal
6#23761631C214yesmaximal, complemented
7#23761630C211yesmaximal, characteristic, complemented
8#23761629C211yesmaximal, characteristic, complemented
9C2×#382×C79C211yesmaximal, characteristic, complemented
10C23×C4×D79C2211yescharacteristic
11C2×#14579119—24no—
12C2×#14579119—24no—
13C2×#14579119—24no—
14C22×C4⋊Dic79C2211yescharacteristic
15C22×C4⋊Dic79C2211yescharacteristic
16C22×Dic79⋊C4C2214yes—
17C22×(C2×D79)⋊C4C2212yes—
18C22×(C2×D79)⋊C4C2212yes—
19C22×(C2×D79)⋊C4C2212yes—
20C22×C4×Dic79C2212yes—
21C22×C4×Dic79C2211yescharacteristic
22#19098432—24no—
23#19098420—24no—
24C4×(C2×D79)⋊C4C418yescomplemented
25C2×C79⋊#95C2212yes—
26C2×C79⋊#95C2214yes—
27C2×C79⋊#95C2214yes—
28C2×C79⋊#95C2212yes—
29C22×C4⋊C4×C79C2211yescharacteristic
30C22×C4⋊C4×C79C2211yescharacteristic
31C22×C42×C79C2211yescharacteristic
32#382×C79—24no—
33C2×#95×C79C2214yes—
34C24×D79C2311yescharacteristic
35C23×Dic79C2312yes—
36C23×Dic79C2312yes—
37C23×Dic79C2311yescharacteristic
38C23×Dic79C2312yes—
39C22×C4×D79—22no—
40C22×C4×D79—22no—
41C22×C4×D79—22no—
42C22×C4×D79—22no—
43C22×C4×D79C2×C414yes—
44C22×C4×D79D414yes—
45C22×C4×D79Q814yes—
46#14579119—232no—
47C2×C4⋊Dic79—22no—
48C2×C4⋊Dic79—22no—
49C2×C4⋊Dic79—22no—
50C2×C4⋊Dic79—22no—
51C2×C4⋊Dic79—22no—
52C2×C4⋊Dic79D414yes—
53C2×Dic79⋊C4—28no—
54C2×Dic79⋊C4—28no—
55C2×Dic79⋊C4—44no—
56C2×(C2×D79)⋊C4—24no—
57C2×(C2×D79)⋊C4—24no—
58C2×(C2×D79)⋊C4—24no—
59C2×(C2×D79)⋊C4—24no—
60C2×(C2×D79)⋊C4—24no—
61C2×(C2×D79)⋊C4—24no—
62C2×(C2×D79)⋊C4—44no—
63C2×(C2×D79)⋊C4C2×C418yes—
64C2×C4×Dic79—22no—
65C2×C4×Dic79—24no—
66C2×C4×Dic79—22no—
67C2×C4×Dic79—24no—
68C2×C4×Dic79—24no—
69C2×C4×Dic79C2×C414yes—
70C79⋊#95—24no—
71C79⋊#95—28no—
72C79⋊#95—44no—
73C79⋊#95C2×C418yes—
74C23×C4×C79C2312yes—
75C23×C4×C79C2312yes—
76C23×C4×C79C2312yes—
77C23×C4×C79C2311yescharacteristic
78C2×C4⋊C4×C79—22no—
79C2×C4⋊C4×C79—22no—
80C2×C4⋊C4×C79—22no—
81C2×C4⋊C4×C79—22no—
82C2×C4⋊C4×C79—22no—
83C2×C4⋊C4×C79D414yes—
84C2×C42×C79—22no—
85C2×C42×C79—22no—
86C2×C42×C79C2×C414yes—
87#95×C79—28no—
88C23×D79—21no—
89C23×D79—21no—
90C23×D79—22no—
91C23×D79—22no—
92C23×D79—21no—
93C23×D79—21no—
94C23×D79—22no—
95C23×D79—22no—
96C23×D79C22×C412yes—
97C23×D79C2×D412yes—
98C23×D79C2×Q812yes—
99C22×Dic79—21no—
100C22×Dic79—21no—
101C22×Dic79—22no—
102C22×Dic79—22no—
103C22×Dic79—22no—
104C22×Dic79—22no—
105C22×Dic79—22no—
106C22×Dic79—21no—
107C22×Dic79—21no—
108C22×Dic79—24no—
109C22×Dic79—24no—
110C22×Dic79—22no—
111C22×Dic79—22no—
112C22×Dic79—22no—
113C22×Dic79—24no—
114C22×Dic79—24no—
115C22×Dic79—22no—
116C22×Dic79C22×C412yes—
117C22×Dic79C22×C414yes—
118C22×Dic79C2×D414yes—
119C22×Dic79C2×D412yes—
120C22×Dic79C2×Q812yes—
121C22×Dic79#4114yes—
122C22×Dic79#4114yes—
123C2×C4×D79—28no—
124C2×C4×D79—28no—
125C2×C4×D79—28no—
126C2×C4×D79—28no—
127C2×C4×D79—28no—
128C2×C4×D79—28no—
129C2×C4×D79C4⋊C4116yescomplemented
130C4⋊Dic79—28no—
131C4⋊Dic79—44no—
132Dic79⋊C4—416no—
133(C2×D79)⋊C4—216no—
134(C2×D79)⋊C4—216no—
135(C2×D79)⋊C4—48no—
136C4×Dic79—216no—
137C4×Dic79—28no—
138C24×C79C2411yescharacteristic
139C22×C4×C79—22no—
140C22×C4×C79—22no—
141C22×C4×C79—22no—
142C22×C4×C79—22no—
143C22×C4×C79—24no—
144C22×C4×C79—22no—
145C22×C4×C79—22no—
146C22×C4×C79—22no—
147C22×C4×C79—22no—
148C22×C4×C79—24no—
149C22×C4×C79—22no—
150C22×C4×C79—22no—
151C22×C4×C79—22no—
152C22×C4×C79—24no—
153C22×C4×C79—24no—
154C22×C4×C79C22×C414yes—
155C22×C4×C79C22×C412yes—
156C22×C4×C79C2×D414yes—
157C22×C4×C79C2×D412yes—
158C22×C4×C79C2×D412yes—
159C22×C4×C79C2×D412yes—
160C22×C4×C79C2×Q812yes—
161C22×C4×C79#4112yes—
162C22×C4×C79#4112yes—
163C4⋊C4×C79—28no—
164C4⋊C4×C79—44no—
165C42×C79—28no—
166C23×C79C23×C411yescharacteristic
167C23×C79C22×D411yescharacteristic
168C23×C79C22×D411yescharacteristic
169C23×C79C22×D411yescharacteristic
170C23×C79C22×Q811yescharacteristic
171C23×C79C2×#4112yes—
172C23×C79C2×#4112yes—
173C23×C79C2×#4111yescharacteristic
174C23×C79C2×#4111yescharacteristic
175C23×C79H(5,𝔽2)12yes—
176C23×C792−1+412yes—
177C22×D79—24no—
178C22×D79—22no—
179C22×D79—22no—
180C22×D79—24no—
181C22×D79—22no—
182C22×D79—22no—
183C22×D79—22no—
184C22×D79—22no—
185C22×D79—22no—
186C22×D79—22no—
187C22×D79—24no—
188C22×D79—22no—
189C22×D79—22no—
190C22×D79—24no—
191C22×D79—42no—
192C22×D79—42no—
193C22×D79—42no—
194C22×D79—42no—
195C22×D79—42no—
196C22×D79—42no—
197C22×D79—42no—
198C22×D79—42no—
199C22×D79C2×C4⋊C414yes—
200C2×C4×C79—24no—
201C2×C4×C79—24no—
202C2×C4×C79—24no—
203C2×C4×C79—24no—
204C2×C4×C79—28no—
205C2×C4×C79—42no—
206C2×C4×C79—42no—
207C2×C4×C79—42no—
208C2×C4×C79—44no—
209C2×C4×C79—42no—
210C2×C4×C79—42no—
211C2×C4×C79—42no—
212C2×C4×C79—42no—
213C2×C4×C79—42no—
214C2×C4×C79—42no—
215C2×C4×C79—44no—
216C2×C4×C79—44no—
217C2×C4×C79C2×C4⋊C414yes—
218C2×C4×C79C4×D414yes—
219C2×C4×C79C4×D414yes—
220C2×C4×C79#12014yes—
221C2×C4×C79C22⋊Q814yes—
222C2×C4×C79C22⋊Q814yes—
223C2×C4×C79#12314yes—
224C2×Dic79—24no—
225C2×Dic79—22no—
226C2×Dic79—22no—
227C2×Dic79—24no—
228C2×Dic79—22no—
229C2×Dic79—22no—
230C2×Dic79—22no—
231C2×Dic79—22no—
232C2×Dic79—24no—
233C2×Dic79—24no—
234C2×Dic79—24no—
235C2×Dic79—24no—
236C2×Dic79—24no—
237C2×Dic79—28no—
238C2×Dic79—42no—
239C2×Dic79—42no—
240C2×Dic79—44no—
241C2×Dic79—42no—
242C2×Dic79—42no—
243C2×Dic79—42no—
244C2×Dic79—44no—
245C2×Dic79—44no—
246C2×Dic79C2×C4⋊C414yes—
247C2×Dic79#11718yescomplemented
248C4×D79—232no—
249C22×C79C22×C4⋊C411yescharacteristic
250C22×C79C2×#11712yes—
251C22×C79C2×C4×D411yescharacteristic
252C22×C79C2×C4×D411yescharacteristic
253C22×C79#52012yes—
254C22×C79C2×#12011yescharacteristic
255C22×C79C2×C22⋊Q811yescharacteristic
256C22×C79C2×C22⋊Q811yescharacteristic
257C22×C79C2×#12311yescharacteristic
258C22×C79#53211yescharacteristic
259C22×C79#53211yescharacteristic
260C22×C79#53312yes—
261C22×C79#53511yescharacteristic
262C22×C79#53611yescharacteristic
263C22×C79#54412yes—
264C22×C79#54612yes—
265C22×C79#54712yes—
266C22×C79#54812yes—
267C22×C79#55012yes—
268C22×C79#55212yes—
269C22×C79#55312yes—
270C22×C79#55512yes—
271C22×C79#55612yes—
272C2×D79—24no—
273C2×D79—24no—
274C2×D79—24no—
275C2×D79—44no—
276C2×D79—44no—
277C2×D79—42no—
278C2×D79—42no—
279C2×D79—44no—
280C2×D79—44no—
281C2×D79—42no—
282C2×D79—42no—
283C4×C79—44no—
284C4×C79—44no—
285C4×C79—82no—
286C4×C79#38518yes—
287Dic79—28no—
288Dic79—24no—
289Dic79—44no—
290Dic79—82no—
291#13879—791nomaximal, Sylow
292C2×C79#228511yescharacteristic
293C2×C79C2×#38511yescharacteristic
294C2×C79#231711yescharacteristic
295C2×C79#235212yes—
296C2×C79#235412yes—
297C2×C79#243111yescharacteristic
298C2×C79#243211yescharacteristic
299C2×C79#243912yes—
300C2×C79#244112yes—
301C2×C79#249911yescharacteristic
302C2×C79#250011yescharacteristic
303D79—44no—
304C22×#117—791no—
305C2×#384—791no—
306C2×#382D7911yescharacteristic, complemented
307C2×#380—791no—
308C2×C4×C22⋊C4—791no—
309#1439—794no—
310#1436—794no—
311C4⋊#95—791no—
312C4⋊#95—791no—
313C79#1387911yescharacteristic, complemented, Sylow
314C24×C4—791no—
315C2×#117—1584no—
316C2×#117—1584no—
317C2×#117—1584no—
318C22×C4⋊C4—794no—
319C22×C4⋊C4—791no—
320C22×C4⋊C4—791no—
321C22×C4⋊C4C2×D7911yescharacteristic
322C22×C4⋊C4C2×D7911yescharacteristic
323C22×C22⋊C4—792no—
324C22×C22⋊C4—792no—
325C22×C22⋊C4—792no—
326C22×C42—791no—
327C22×C42—792no—
328C22×C42C2×D7911yescharacteristic
329#384—1584no—
330#382—24no—
331#380—1584no—
332C4×C22⋊C4—798no—
333C2×#95—794no—
334C2×#95—792no—
335C2×#95—794no—
336C2×#95—792no—
337C2×#95C2×D7914yes—
338C25—791no—
339C23×C4—792no—
340C23×C4—791no—
341C23×C4—794no—
342C23×C4—794no—
343C23×C4—794no—
344C23×C4—792no—
345C23×C4—792no—
346C23×C4—1582no—
347C23×C4—1582no—
348C23×C4—1582no—
349C23×C4—1582no—
350C23×C4C22×D7912yes—
351C23×C4C22×D7912yes—
352C23×C4C22×D7912yes—
353C23×C4C22×D7911yescharacteristic
354#117—15832no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—3164no—
361C2×C4⋊C4—794no—
362C2×C4⋊C4—1588no—
363C2×C4⋊C4—1588no—
364C2×C4⋊C4—1582no—
365C2×C4⋊C4—1582no—
366C2×C4⋊C4—1582no—
367C2×C4⋊C4—1582no—
368C2×C4⋊C4—1582no—
369C2×C4⋊C4C22⋊D7914yes—
370C2×C22⋊C4—3164no—
371C2×C22⋊C4—798no—
372C2×C22⋊C4—1584no—
373C2×C22⋊C4—1584no—
374C2×C22⋊C4—1584no—
375C2×C22⋊C4—1584no—
376C2×C22⋊C4—1584no—
377C2×C22⋊C4—1584no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—794no—
381C2×C42—1584no—
382C2×C42—1584no—
383C2×C42—1582no—
384C2×C42—1584no—
385C2×C42—1582no—
386C2×C42C4×D7914yes—
387#95—28no—
388#95—3164no—
389#95—798no—
390#95—1584no—
391#95—1588no—
392C24—792no—
393C24—792no—
394C24—792no—
395C24—1582no—
396C24—1582no—
397C24—1581no—
398C24—1581no—
399C24—1582no—
400C24—1582no—
401C24—1581no—
402C24—1581no—
403C24C23×D7911yescharacteristic
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—792no—
420C22×C4—794no—
421C22×C4—794no—
422C22×C4—792no—
423C22×C4—794no—
424C22×C4—792no—
425C22×C4—7916no—
426C22×C4—794no—
427C22×C4—1584no—
428C22×C4—1582no—
429C22×C4—1582no—
430C22×C4—1584no—
431C22×C4—1582no—
432C22×C4—1584no—
433C22×C4—1584no—
434C22×C4—1582no—
435C22×C4—1582no—
436C22×C4—1581no—
437C22×C4—1582no—
438C22×C4—1581no—
439C22×C4—1582no—
440C22×C4—1582no—
441C22×C4—1581no—
442C22×C4—1582no—
443C22×C4—1581no—
444C22×C4—1588no—
445C22×C4—1588no—
446C22×C4—1588no—
447C22×C4—1588no—
448C22×C4—1588no—
449C22×C4—1588no—
450C22×C4C2×C4×D7914yes—
451C22×C4C2×C4×D7912yes—
452C22×C4D4×D7912yes—
453C22×C4D4×D7912yes—
454C22×C4C2×C22⋊D7914yes—
455C22×C4C2×C22⋊D7912yes—
456C22×C4Q8×D7912yes—
457C22×C4#1192962412yes—
458C22×C4#1192962512yes—
459C4⋊C4—28no—
460C4⋊C4—44no—
461C4⋊C4—31616no—
462C4⋊C4—3164no—
463C4⋊C4—1588no—
464C22⋊C4—3168no—
465C22⋊C4—15816no—
466C22⋊C4—15816no—
467C42—28no—
468C42—15816no—
469C42—1588no—
470C23—3162no—
471C23—3162no—
472C23—3162no—
473C23—3162no—
474C23—3162no—
475C23—3162no—
476C23—3162no—
477C23—3162no—
478C23—794no—
479C23—1584no—
480C23—1584no—
481C23—1582no—
482C23—1582no—
483C23—1584no—
484C23—1582no—
485C23—1582no—
486C23—1582no—
487C23—1582no—
488C23—1582no—
489C23—1582no—
490C23—1584no—
491C23—1582no—
492C23—1582no—
493C23C22×C4×D7911yescharacteristic
494C23C2×D4×D7911yescharacteristic
495C23C2×D4×D7911yescharacteristic
496C23C22×C22⋊D7911yescharacteristic
497C23C2×Q8×D7911yescharacteristic
498C23#41×D7912yes—
499C23#41×D7912yes—
500C23C2×#1192962411yescharacteristic
501C23C2×#1192962511yescharacteristic
502C23#1457920712yes—
503C23#1457920812yes—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—3164no—
522C2×C4—3162no—
523C2×C4—3162no—
524C2×C4—3164no—
525C2×C4—3164no—
526C2×C4—3162no—
527C2×C4—3162no—
528C2×C4—3162no—
529C2×C4—794no—
530C2×C4—798no—
531C2×C4—1588no—
532C2×C4—1584no—
533C2×C4—1584no—
534C2×C4—1584no—
535C2×C4—1584no—
536C2×C4—1584no—
537C2×C4—1582no—
538C2×C4—1584no—
539C2×C4—1582no—
540C2×C4—1584no—
541C2×C4—1582no—
542C2×C4—1582no—
543C2×C4—1582no—
544C2×C4—1582no—
545C2×C4—15832no—
546C2×C4C4⋊C4×D7914yes—
547C2×C4C4×C22⋊D7914yes—
548C2×C4#1457912314yes—
549C2×C4#1457912914yes—
550C2×C4#1457913714yes—
551C2×C4#1457913914yes—
552C2×C4#1457914614yes—
553C22—3162no—
554C22—3164no—
555C22—3164no—
556C22—3162no—
557C22—3164no—
558C22—3164no—
559C22—3162no—
560C22—3162no—
561C22—1584no—
562C22—1584no—
563C22—1584no—
564C22C2×C4⋊C4×D7911yescharacteristic
565C22#117×D7912yes—
566C22C2×C4×C22⋊D7911yescharacteristic
567C22C2×#1457912311yescharacteristic
568C22#1909904312yes—
569C22C2×#1457912911yescharacteristic
570C22C2×#1457913711yescharacteristic
571C22C2×#1457913911yescharacteristic
572C22C2×#1457914611yescharacteristic
573C22#1909911211yescharacteristic
574C22#1909911211yescharacteristic
575C22#1909911912yes—
576C22#1909913111yescharacteristic
577C22#1909913311yescharacteristic
578C22#1909919312yes—
579C22#1909920612yes—
580C22#1909921212yes—
581C22#1909922112yes—
582C22#1909923212yes—
583C22#1909925112yes—
584C22#1909926612yes—
585C22#1909927612yes—
586C22#1909928112yes—
587C4—44no—
588C4—44no—
589C4—82no—
590C4—3164no—
591C4—6322no—
592C4—1584no—
593C4—1588no—
594C4#1909843418yes—
595C2—3164no—
596C2#2376580611yescharacteristic
597C2C2×#1909843411yescharacteristic
598C2#2376602511yescharacteristic
599C2#2376632612yes—
600C2#2376635012yes—
601C2#2376702211yescharacteristic
602C2#2376703111yescharacteristic
603C2#2376709412yes—
604C2#2376710912yes—
605C2#2376785611yescharacteristic
606C2#2376786511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D79
commutatorC23×C79C23×C4
FittingC2×#382×C79C2
socle, cosocleC24×C79C24
prime radicalC24×D79C23
hypercenterC2×#382D79
hypocenterC79#13879
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. C2×#19098432C2semidirect
6. #23761631C2semidirect
7. #23761630C2semidirect
8. #23761629C2semidirect
9. C2×#382×C79C2semidirect
24. C4×(C2×D79)⋊C4C4semidirect
129. C2×C4×D79C4⋊C4semidirect
247. C2×Dic79#117semidirect
306. C2×#382D79semidirect
313. C79#13879semidirect

Maximal subgroups

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

subgroupindex
C22×#145791192
C2×#190984322
C2×#190984202
C2×C4×(C2×D79)⋊C42
#237616442
#237616312
#237616302
#237616292
C2×#382×C792
#1387979