Finite Group Databasebeta

← #34210941

Subgroups — #34210941

Order 20224.

1630 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098460C212yesmaximal
2C2×#19098458C212yesmaximal
3C2×#19098427C211yesmaximal, characteristic
4C2×C4⋊C4×Dic79C211yesmaximal, characteristic
5C2×#19098414C211yesmaximal, characteristic
6#23761665C211yesmaximal, characteristic
7#23761664C211yesmaximal, characteristic
8#23761663C211yesmaximal, characteristic
9#23761629C212yesmaximal
10Dic79⋊#95C212yesmaximal
11C79×#1444C211yesmaximal, characteristic
12C22×C4⋊Dic79C2211yescharacteristic
13C22×C4⋊Dic79C2212yes—
14C22×Dic79⋊C4C2211yescharacteristic
15C22×Dic79⋊C4C2211yescharacteristic
16C22×Dic79⋊C4C2212yes—
17C22×Dic79⋊C4C2212yes—
18C22×Dic79⋊C4C2212yes—
19C22×C4×Dic79C2212yes—
20C22×C4×Dic79C2212yes—
21C22×C4×Dic79C2211yescharacteristic
22#19098460—28no—
23#19098458—28no—
24#19098427—24no—
25C4⋊C4×Dic79—24no—
26#19098414C418yescomplemented
27C2×C79⋊#95C2212yes—
28C2×C79⋊#95C2211yescharacteristic
29C2×C79⋊#95C2212yes—
30C2×C79⋊#95C2211yescharacteristic
31C2×C79⋊#95C2211yescharacteristic
32C2×C79⋊#95C2212yes—
33C2×C79⋊#95C2212yes—
34C2×C79⋊#95C2211yescharacteristic
35C22×C4⋊C4×C79C2211yescharacteristic
36C22×C4⋊C4×C79C2212yes—
37C2×#95×C79C2211yescharacteristic
38C2×#95×C79C2212yes—
39C2×#95×C79C2211yescharacteristic
40C23×Dic79C2311yescharacteristic
41C23×Dic79C2311yescharacteristic
42C23×Dic79C2312yes—
43C23×Dic79C2311yescharacteristic
44C23×Dic79C2312yes—
45C23×Dic79C2311yescharacteristic
46C2×C4⋊Dic79—22no—
47C2×C4⋊Dic79—24no—
48C2×C4⋊Dic79—24no—
49C2×C4⋊Dic79—24no—
50C2×C4⋊Dic79—42no—
51C2×Dic79⋊C4—24no—
52C2×Dic79⋊C4—22no—
53C2×Dic79⋊C4—22no—
54C2×Dic79⋊C4—24no—
55C2×Dic79⋊C4—24no—
56C2×Dic79⋊C4—24no—
57C2×Dic79⋊C4—24no—
58C2×Dic79⋊C4—24no—
59C2×Dic79⋊C4—42no—
60C2×Dic79⋊C4—42no—
61C2×Dic79⋊C4C2×C418yes—
62C2×Dic79⋊C4D418yes—
63C2×Dic79⋊C4Q818yes—
64C2×C4×Dic79—24no—
65C2×C4×Dic79—24no—
66C2×C4×Dic79—24no—
67C2×C4×Dic79—42no—
68C2×C4×Dic79—42no—
69C2×C4×Dic79C2×C418yes—
70C2×C4×Dic79C2×C414yes—
71C2×C4×Dic79D414yes—
72C2×C4×Dic79Q814yes—
73C79⋊#95—22no—
74C79⋊#95—24no—
75C79⋊#95—24no—
76C79⋊#95—22no—
77C79⋊#95—22no—
78C79⋊#95—24no—
79C79⋊#95—24no—
80C79⋊#95C2×C414yes—
81C23×C4×C79C2311yescharacteristic
82C23×C4×C79C2311yescharacteristic
83C23×C4×C79C2312yes—
84C23×C4×C79C2311yescharacteristic
85C23×C4×C79C2312yes—
86C2×C4⋊C4×C79—22no—
87C2×C4⋊C4×C79—24no—
88C2×C4⋊C4×C79—24no—
89C2×C4⋊C4×C79—24no—
90C2×C4⋊C4×C79—42no—
91#95×C79—22no—
92#95×C79—24no—
93#95×C79C2×C414yes—
94C22×Dic79—22no—
95C22×Dic79—22no—
96C22×Dic79—22no—
97C22×Dic79—21no—
98C22×Dic79—21no—
99C22×Dic79—22no—
100C22×Dic79—22no—
101C22×Dic79—22no—
102C22×Dic79—22no—
103C22×Dic79—22no—
104C22×Dic79—22no—
105C22×Dic79—22no—
106C22×Dic79—21no—
107C22×Dic79—21no—
108C22×Dic79—22no—
109C22×Dic79—22no—
110C22×Dic79—22no—
111C22×Dic79—21no—
112C22×Dic79—21no—
113C22×Dic79—21no—
114C22×Dic79—21no—
115C22×Dic79—22no—
116C22×Dic79—21no—
117C22×Dic79—21no—
118C22×Dic79C22×C414yes—
119C22×Dic79C22×C412yes—
120C22×Dic79C22×C412yes—
121C22×Dic79C2×D412yes—
122C22×Dic79C2×D414yes—
123C22×Dic79C2×D412yes—
124C22×Dic79C2×D414yes—
125C22×Dic79C2×Q812yes—
126C22×Dic79C2×Q814yes—
127C22×Dic79C2×Q812yes—
128C22×Dic79C2×Q814yes—
129C22×Dic79#4114yes—
130C4⋊Dic79—28no—
131C4⋊Dic79—48no—
132Dic79⋊C4—48no—
133Dic79⋊C4—48no—
134Dic79⋊C4—44no—
135Dic79⋊C4—44no—
136Dic79⋊C4C4⋊C4132yescomplemented
137C4×Dic79—48no—
138C4×Dic79—48no—
139C4×Dic79C4⋊C4116yescomplemented
140C24×C79C2411yescharacteristic
141C22×C4×C79—22no—
142C22×C4×C79—22no—
143C22×C4×C79—22no—
144C22×C4×C79—22no—
145C22×C4×C79—22no—
146C22×C4×C79—22no—
147C22×C4×C79—22no—
148C22×C4×C79—22no—
149C22×C4×C79—22no—
150C22×C4×C79—22no—
151C22×C4×C79—22no—
152C22×C4×C79—22no—
153C22×C4×C79—21no—
154C22×C4×C79—21no—
155C22×C4×C79—22no—
156C22×C4×C79—22no—
157C22×C4×C79—22no—
158C22×C4×C79—22no—
159C22×C4×C79—21no—
160C22×C4×C79—21no—
161C22×C4×C79—21no—
162C22×C4×C79—21no—
163C22×C4×C79C22×C412yes—
164C22×C4×C79C22×C414yes—
165C22×C4×C79C2×D412yes—
166C22×C4×C79C2×D414yes—
167C22×C4×C79C2×Q812yes—
168C22×C4×C79C2×Q814yes—
169C22×C4×C79#4114yes—
170C4⋊C4×C79—28no—
171C4⋊C4×C79—48no—
172C23×C79C23×C411yescharacteristic
173C23×C79C22×D411yescharacteristic
174C23×C79C22×D412yes—
175C23×C79C22×Q811yescharacteristic
176C23×C79C22×Q812yes—
177C23×C79C2×#4112yes—
178C23×C79C2×#4112yes—
179C23×C79H(5,𝔽2)11yescharacteristic
180C23×C79H(5,𝔽2)11yescharacteristic
181C23×C792−1+411yescharacteristic
182C23×C792−1+411yescharacteristic
183C2×C4×C79—24no—
184C2×C4×C79—24no—
185C2×C4×C79—24no—
186C2×C4×C79—22no—
187C2×C4×C79—22no—
188C2×C4×C79—24no—
189C2×C4×C79—24no—
190C2×C4×C79—24no—
191C2×C4×C79—24no—
192C2×C4×C79—24no—
193C2×C4×C79—24no—
194C2×C4×C79—24no—
195C2×C4×C79—24no—
196C2×C4×C79—42no—
197C2×C4×C79—42no—
198C2×C4×C79—42no—
199C2×C4×C79—42no—
200C2×C4×C79—41no—
201C2×C4×C79—41no—
202C2×C4×C79—41no—
203C2×C4×C79—41no—
204C2×C4×C79—41no—
205C2×C4×C79—41no—
206C2×C4×C79—42no—
207C2×C4×C79—42no—
208C2×C4×C79—42no—
209C2×C4×C79—42no—
210C2×C4×C79C2×C4⋊C414yes—
211C2×C4×C79C2×C4⋊C418yes—
212C2×Dic79—24no—
213C2×Dic79—22no—
214C2×Dic79—24no—
215C2×Dic79—22no—
216C2×Dic79—22no—
217C2×Dic79—22no—
218C2×Dic79—22no—
219C2×Dic79—22no—
220C2×Dic79—24no—
221C2×Dic79—24no—
222C2×Dic79—24no—
223C2×Dic79—24no—
224C2×Dic79—24no—
225C2×Dic79—24no—
226C2×Dic79—24no—
227C2×Dic79—42no—
228C2×Dic79—42no—
229C2×Dic79—42no—
230C2×Dic79—42no—
231C2×Dic79—41no—
232C2×Dic79—41no—
233C2×Dic79—41no—
234C2×Dic79—41no—
235C2×Dic79—41no—
236C2×Dic79—41no—
237C2×Dic79—42no—
238C2×Dic79—42no—
239C2×Dic79—42no—
240C2×Dic79—42no—
241C2×Dic79C2×C4⋊C414yes—
242C2×Dic79C2×C4⋊C418yes—
243C2×Dic79C2×C4⋊C414yes—
244C2×Dic79C2×C4⋊C418yes—
245C2×Dic79#12714yes—
246C2×Dic79C4⋊Q814yes—
247C2×Dic79C4⋊Q818yes—
248C22×C79C22×C4⋊C411yescharacteristic
249C22×C79C22×C4⋊C412yes—
250C22×C79C4×#4112yes—
251C22×C79#51811yescharacteristic
252C22×C79#51911yescharacteristic
253C22×C79#52511yescharacteristic
254C22×C79#52511yescharacteristic
255C22×C79C2×#12711yescharacteristic
256C22×C79C2×C4⋊Q811yescharacteristic
257C22×C79C2×C4⋊Q812yes—
258C22×C79#53312yes—
259C22×C79#53711yescharacteristic
260C22×C79#53711yescharacteristic
261C22×C79#54411yescharacteristic
262C22×C79#54411yescharacteristic
263C22×C79#54612yes—
264C22×C79#54612yes—
265C22×C79#54812yes—
266C22×C79#54812yes—
267C22×C79#55412yes—
268C22×C79#55412yes—
269C22×C79#55712yes—
270C22×C79#55712yes—
271C4×C79—28no—
272C4×C79—24no—
273C4×C79—44no—
274C4×C79—81no—
275C4×C79—81no—
276Dic79—24no—
277Dic79—28no—
278Dic79—44no—
279Dic79—81no—
280Dic79—81no—
281Dic79#38418yes—
282#13954—791nomaximal, Sylow
283C2×C79C2×#38411yescharacteristic
284C2×C79#229611yescharacteristic
285C2×C79#231611yescharacteristic
286C2×C79#234912yes—
287C2×C79#235112yes—
288C2×C79#254011yescharacteristic
289C2×C79#254011yescharacteristic
290C2×C79#255012yes—
291C2×C79#255012yes—
292C2×C79#258411yescharacteristic
293C2×C79#258411yescharacteristic
294C2×#389—792no—
295C2×#389—792no—
296C2×#382—791no—
297C2×C4×C4⋊C4—791no—
298C2×C4×C4⋊C4—791no—
299#1444—791no—
300#1444—791no—
301#1444—791no—
302#1444D7911yescharacteristic
303C4⋊#95—792no—
304C4⋊#95—792no—
305C79#1395411yescharacteristic, complemented, Sylow
306C22×C4⋊C4—792no—
307C22×C4⋊C4—792no—
308C22×C4⋊C4—792no—
309C22×C4⋊C4—792no—
310C22×C4⋊C4—791no—
311C22×C4⋊C4—791no—
312C22×C4⋊C4—791no—
313C22×C4⋊C4C2×D7911yescharacteristic
314C22×C4⋊C4C2×D7912yes—
315C22×C42—792no—
316C22×C42—791no—
317C22×C42—792no—
318#389—1588no—
319#389—1588no—
320#382—1584no—
321C4×C4⋊C4—798no—
322C4×C4⋊C4—1584no—
323C2×#95—791no—
324C2×#95—792no—
325C2×#95—791no—
326C2×#95—791no—
327C2×#95—792no—
328C2×#95—792no—
329C2×#95—792no—
330C2×#95—791no—
331C2×#95C2×D7912yes—
332C2×#95C2×D7911yescharacteristic
333C2×#95C2×D7911yescharacteristic
334C23×C4—791no—
335C23×C4—792no—
336C23×C4—791no—
337C23×C4—792no—
338C23×C4—791no—
339C23×C4—791no—
340C23×C4C22×D7911yescharacteristic
341C23×C4C22×D7911yescharacteristic
342C23×C4C22×D7912yes—
343C23×C4C22×D7912yes—
344C23×C4C22×D7911yescharacteristic
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—3162no—
351C2×C4⋊C4—3162no—
352C2×C4⋊C4—3162no—
353C2×C4⋊C4—798no—
354C2×C4⋊C4—798no—
355C2×C4⋊C4—798no—
356C2×C4⋊C4—1582no—
357C2×C4⋊C4—1582no—
358C2×C4⋊C4—1584no—
359C2×C4⋊C4—1584no—
360C2×C4⋊C4—1584no—
361C2×C4⋊C4—1584no—
362C2×C4⋊C4—1584no—
363C2×C4⋊C4—1584no—
364C2×C4⋊C4—1584no—
365C2×C4⋊C4—1584no—
366C2×C4⋊C4—1584no—
367C2×C4⋊C4—1582no—
368C2×C42—3162no—
369C2×C42—3162no—
370C2×C42—798no—
371C2×C42—794no—
372C2×C42—794no—
373C2×C42—794no—
374C2×C42—1584no—
375C2×C42—1584no—
376C2×C42—1584no—
377#95—22no—
378#95—24no—
379#95—794no—
380#95—1584no—
381#95—1582no—
382#95—1582no—
383#95—1584no—
384#95—1584no—
385#95—1582no—
386#95—1584no—
387#95C4×D7914yes—
388C24C23×D7911yescharacteristic
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—792no—
412C22×C4—794no—
413C22×C4—792no—
414C22×C4—792no—
415C22×C4—794no—
416C22×C4—792no—
417C22×C4—794no—
418C22×C4—794no—
419C22×C4—792no—
420C22×C4—792no—
421C22×C4—794no—
422C22×C4—794no—
423C22×C4—1582no—
424C22×C4—1582no—
425C22×C4—1581no—
426C22×C4—1581no—
427C22×C4—1581no—
428C22×C4—1581no—
429C22×C4—1581no—
430C22×C4—1581no—
431C22×C4—1582no—
432C22×C4—1582no—
433C22×C4—1582no—
434C22×C4—1582no—
435C22×C4—1582no—
436C22×C4—1582no—
437C22×C4—1582no—
438C22×C4—1582no—
439C22×C4—1582no—
440C22×C4—1582no—
441C22×C4—1581no—
442C22×C4—1582no—
443C22×C4—1581no—
444C22×C4—1582no—
445C22×C4—1581no—
446C22×C4—1581no—
447C22×C4C2×C4×D7912yes—
448C22×C4C2×C4×D7914yes—
449C22×C4D4×D7912yes—
450C22×C4D4×D7914yes—
451C22×C4Q8×D7912yes—
452C22×C4Q8×D7914yes—
453C22×C4#1192962314yes—
454C4⋊C4—28no—
455C4⋊C4—48no—
456C4⋊C4—3168no—
457C4⋊C4—3168no—
458C4⋊C4—3164no—
459C4⋊C4—3164no—
460C4⋊C4—3168no—
461C4⋊C4—7932no—
462C4⋊C4—1588no—
463C42—3168no—
464C42—3168no—
465C42—7916no—
466C23C22×C4×D7911yescharacteristic
467C23C2×D4×D7911yescharacteristic
468C23C2×D4×D7912yes—
469C23C2×Q8×D7911yescharacteristic
470C23C2×Q8×D7912yes—
471C23#41×D7912yes—
472C23C2×#1192962312yes—
473C23#1457920611yescharacteristic
474C23#1457920711yescharacteristic
475C23#1457920811yescharacteristic
476C23#1457920911yescharacteristic
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—3162no—
505C2×C4—3162no—
506C2×C4—3162no—
507C2×C4—3162no—
508C2×C4—3161no—
509C2×C4—3161no—
510C2×C4—3162no—
511C2×C4—3162no—
512C2×C4—3162no—
513C2×C4—3162no—
514C2×C4—3161no—
515C2×C4—3161no—
516C2×C4—3161no—
517C2×C4—3161no—
518C2×C4—794no—
519C2×C4—798no—
520C2×C4—794no—
521C2×C4—798no—
522C2×C4—794no—
523C2×C4—794no—
524C2×C4—798no—
525C2×C4—1584no—
526C2×C4—1584no—
527C2×C4—1584no—
528C2×C4—1582no—
529C2×C4—1582no—
530C2×C4—1582no—
531C2×C4—1582no—
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×C4C4⋊C4×D7914yes—
541C2×C4C4⋊C4×D7918yes—
542C22C2×C4⋊C4×D7912yes—
543C22C2×C4⋊C4×D7911yescharacteristic
544C22C4×#1192962312yes—
545C22#1909903511yescharacteristic
546C22#1909903911yescharacteristic
547C22#1909906911yescharacteristic
548C22#1909907111yescharacteristic
549C22#127×D7911yescharacteristic
550C22C4⋊Q8×D7912yes—
551C22C4⋊Q8×D7911yescharacteristic
552C22#1909911512yes—
553C22#1909914211yescharacteristic
554C22#1909914311yescharacteristic
555C22#1909919011yescharacteristic
556C22#1909919311yescharacteristic
557C22#1909920012yes—
558C22#1909920512yes—
559C22#1909921912yes—
560C22#1909922312yes—
561C22#1909926712yes—
562C22#1909927112yes—
563C22#1909928812yes—
564C22#1909929012yes—
565C4—28no—
566C4—24no—
567C4—44no—
568C4—81no—
569C4—81no—
570C4—3164no—
571C4—798no—
572C4—6321no—
573C4—6321no—
574C4—1588no—
575C4—1584no—
576C2#384×D7911yescharacteristic
577C2#2376587911yescharacteristic
578C2#2376601511yescharacteristic
579C2#2376630612yes—
580C2#2376631912yes—
581C2#2376833311yescharacteristic
582C2#2376833511yescharacteristic
583C2#2376841412yes—
584C2#2376841612yes—
585C2#2376878211yescharacteristic
586C2#2376878411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D79
commutatorC23×C79C23×C4
FittingC79×#1444C2
socle, prime radical, cosocleC24×C79C24
hypercenter#1444D79
hypocenterC79#13954
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
26. #19098414C4semidirect
136. Dic79⋊C4C4⋊C4semidirect
139. C4×Dic79C4⋊C4semidirect
305. C79#13954semidirect

Maximal subgroups

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

subgroupindex
C2×#190984602
C2×#190984582
C2×#190984272
C2×C4⋊C4×Dic792
C2×#190984142
#237616652
#237616642
#237616632
#237616292
Dic79⋊#952
C79×#14442
#1395479