Finite Group Databasebeta

← #34210541

Subgroups — #34210541

Order 20224.

1774 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
1C2×#19098460C212yesmaximal
2C2×#19098458C212yesmaximal
3C2×#19098432C212yesmaximal
4C2×#19098432C211yesmaximal, characteristic
5C2×#19098414C211yesmaximal, characteristic
6C2×#19098414C211yesmaximal, characteristic
7#23761665C211yesmaximal, characteristic
8#23761664C211yesmaximal, characteristic
9#23761663C211yesmaximal, characteristic
10Dic79⋊#95C212yesmaximal
11C79×#1444C211yesmaximal, characteristic
12C22×C4⋊Dic79C2212yes—
13C22×C4⋊Dic79C2211yescharacteristic
14C22×Dic79⋊C4C2212yes—
15C22×Dic79⋊C4C2212yes—
16C22×Dic79⋊C4C2212yes—
17C22×Dic79⋊C4C2212yes—
18C22×Dic79⋊C4C2211yescharacteristic
19C22×Dic79⋊C4C2211yescharacteristic
20C22×Dic79⋊C4C2212yes—
21C22×C4×Dic79C2211yescharacteristic
22C22×C4×Dic79C2211yescharacteristic
23C22×C4×Dic79C2212yes—
24C22×C4×Dic79C2211yescharacteristic
25#19098460—28no—
26#19098458—28no—
27#19098432—28no—
28#19098432—24no—
29#19098414—24no—
30#19098414C418yescomplemented
31C2×C79⋊#95C2212yes—
32C2×C79⋊#95C2212yes—
33C2×C79⋊#95C2211yescharacteristic
34C2×C79⋊#95C2212yes—
35C2×C79⋊#95C2211yescharacteristic
36C22×C4⋊C4×C79C2212yes—
37C22×C4⋊C4×C79C2211yescharacteristic
38C2×#95×C79C2211yescharacteristic
39C2×#95×C79C2212yes—
40C2×#95×C79C2211yescharacteristic
41C23×Dic79C2312yes—
42C23×Dic79C2311yescharacteristic
43C23×Dic79C2311yescharacteristic
44C23×Dic79C2312yes—
45C23×Dic79C2311yescharacteristic
46C23×Dic79C2311yescharacteristic
47C2×C4⋊Dic79—22no—
48C2×C4⋊Dic79—24no—
49C2×C4⋊Dic79—24no—
50C2×C4⋊Dic79—42no—
51C2×C4⋊Dic79—42no—
52C2×Dic79⋊C4—22no—
53C2×Dic79⋊C4—22no—
54C2×Dic79⋊C4—22no—
55C2×Dic79⋊C4—22no—
56C2×Dic79⋊C4—24no—
57C2×Dic79⋊C4—24no—
58C2×Dic79⋊C4—24no—
59C2×Dic79⋊C4—24no—
60C2×Dic79⋊C4—24no—
61C2×Dic79⋊C4—24no—
62C2×Dic79⋊C4—24no—
63C2×Dic79⋊C4—24no—
64C2×Dic79⋊C4—24no—
65C2×Dic79⋊C4—24no—
66C2×Dic79⋊C4—42no—
67C2×Dic79⋊C4—42no—
68C2×Dic79⋊C4C2×C418yes—
69C2×Dic79⋊C4D414yes—
70C2×Dic79⋊C4D418yes—
71C2×Dic79⋊C4Q814yes—
72C2×Dic79⋊C4Q818yes—
73C2×C4×Dic79—24no—
74C2×C4×Dic79—24no—
75C2×C4×Dic79—22no—
76C2×C4×Dic79—22no—
77C2×C4×Dic79—24no—
78C2×C4×Dic79—24no—
79C2×C4×Dic79C2×C418yes—
80C2×C4×Dic79C2×C414yes—
81C2×C4×Dic79D414yes—
82C2×C4×Dic79Q814yes—
83C79⋊#95—22no—
84C79⋊#95—24no—
85C79⋊#95—24no—
86C79⋊#95—24no—
87C79⋊#95C2×C414yes—
88C23×C4×C79C2312yes—
89C23×C4×C79C2311yescharacteristic
90C23×C4×C79C2311yescharacteristic
91C23×C4×C79C2311yescharacteristic
92C23×C4×C79C2312yes—
93C2×C4⋊C4×C79—24no—
94C2×C4⋊C4×C79—22no—
95C2×C4⋊C4×C79—24no—
96C2×C4⋊C4×C79—42no—
97C2×C4⋊C4×C79—42no—
98#95×C79—22no—
99#95×C79—24no—
100#95×C79C2×C414yes—
101C22×Dic79—22no—
102C22×Dic79—22no—
103C22×Dic79—21no—
104C22×Dic79—21no—
105C22×Dic79—22no—
106C22×Dic79—22no—
107C22×Dic79—22no—
108C22×Dic79—22no—
109C22×Dic79—22no—
110C22×Dic79—22no—
111C22×Dic79—21no—
112C22×Dic79—21no—
113C22×Dic79—22no—
114C22×Dic79—22no—
115C22×Dic79—22no—
116C22×Dic79—22no—
117C22×Dic79—22no—
118C22×Dic79—22no—
119C22×Dic79—22no—
120C22×Dic79C22×C414yes—
121C22×Dic79C22×C412yes—
122C22×Dic79C22×C412yes—
123C22×Dic79C2×D412yes—
124C22×Dic79C2×D414yes—
125C22×Dic79C2×D412yes—
126C22×Dic79C2×D412yes—
127C22×Dic79C2×D414yes—
128C22×Dic79C2×D412yes—
129C22×Dic79C2×Q812yes—
130C22×Dic79C2×Q814yes—
131C22×Dic79C2×Q812yes—
132C22×Dic79C2×Q812yes—
133C22×Dic79C2×Q814yes—
134C22×Dic79C2×Q812yes—
135C22×Dic79#4112yes—
136C22×Dic79#4112yes—
137C4⋊Dic79—48no—
138C4⋊Dic79—44no—
139Dic79⋊C4—28no—
140Dic79⋊C4—28no—
141Dic79⋊C4—48no—
142Dic79⋊C4—48no—
143Dic79⋊C4—48no—
144Dic79⋊C4—48no—
145Dic79⋊C4C4⋊C4132yescomplemented
146C4×Dic79—216no—
147C4×Dic79—44no—
148C4×Dic79—44no—
149C4×Dic79C4⋊C4116yescomplemented
150C24×C79C2411yescharacteristic
151C22×C4×C79—22no—
152C22×C4×C79—22no—
153C22×C4×C79—22no—
154C22×C4×C79—22no—
155C22×C4×C79—22no—
156C22×C4×C79—22no—
157C22×C4×C79—22no—
158C22×C4×C79—21no—
159C22×C4×C79—21no—
160C22×C4×C79—22no—
161C22×C4×C79—21no—
162C22×C4×C79—21no—
163C22×C4×C79—22no—
164C22×C4×C79—22no—
165C22×C4×C79—22no—
166C22×C4×C79—22no—
167C22×C4×C79—22no—
168C22×C4×C79—22no—
169C22×C4×C79—22no—
170C22×C4×C79—22no—
171C22×C4×C79—22no—
172C22×C4×C79C22×C412yes—
173C22×C4×C79C22×C414yes—
174C22×C4×C79C2×D412yes—
175C22×C4×C79C2×D412yes—
176C22×C4×C79C2×D414yes—
177C22×C4×C79C2×Q812yes—
178C22×C4×C79C2×Q812yes—
179C22×C4×C79C2×Q814yes—
180C4⋊C4×C79—48no—
181C4⋊C4×C79—44no—
182C23×C79C23×C411yescharacteristic
183C23×C79C22×D411yescharacteristic
184C23×C79C22×D412yes—
185C23×C79C22×D411yescharacteristic
186C23×C79C22×Q811yescharacteristic
187C23×C79C22×Q812yes—
188C23×C79C22×Q811yescharacteristic
189C23×C79C2×#4111yescharacteristic
190C23×C79C2×#4111yescharacteristic
191C23×C79H(5,𝔽2)12yes—
192C23×C792−1+412yes—
193C2×C4×C79—24no—
194C2×C4×C79—24no—
195C2×C4×C79—24no—
196C2×C4×C79—22no—
197C2×C4×C79—22no—
198C2×C4×C79—24no—
199C2×C4×C79—24no—
200C2×C4×C79—24no—
201C2×C4×C79—24no—
202C2×C4×C79—24no—
203C2×C4×C79—22no—
204C2×C4×C79—22no—
205C2×C4×C79—24no—
206C2×C4×C79—24no—
207C2×C4×C79—41no—
208C2×C4×C79—41no—
209C2×C4×C79—41no—
210C2×C4×C79—41no—
211C2×C4×C79—42no—
212C2×C4×C79—42no—
213C2×C4×C79—42no—
214C2×C4×C79—42no—
215C2×C4×C79—42no—
216C2×C4×C79—42no—
217C2×C4×C79—42no—
218C2×C4×C79—42no—
219C2×C4×C79—42no—
220C2×C4×C79C2×C4⋊C414yes—
221C2×C4×C79C2×C4⋊C418yes—
222C2×Dic79—24no—
223C2×Dic79—24no—
224C2×Dic79—24no—
225C2×Dic79—24no—
226C2×Dic79—24no—
227C2×Dic79—24no—
228C2×Dic79—22no—
229C2×Dic79—22no—
230C2×Dic79—24no—
231C2×Dic79—24no—
232C2×Dic79—41no—
233C2×Dic79—41no—
234C2×Dic79—41no—
235C2×Dic79—41no—
236C2×Dic79—42no—
237C2×Dic79—42no—
238C2×Dic79—42no—
239C2×Dic79—42no—
240C2×Dic79—42no—
241C2×Dic79—42no—
242C2×Dic79—42no—
243C2×Dic79—42no—
244C2×Dic79—42no—
245C2×Dic79C2×C4⋊C418yes—
246C2×Dic79C2×C4⋊C414yes—
247C2×Dic79C2×C4⋊C418yes—
248C2×Dic79C2×C4⋊C414yes—
249C2×Dic79C4×D414yes—
250C2×Dic79C4×Q814yes—
251C2×Dic79#12114yes—
252C2×Dic79C22⋊Q814yes—
253C2×Dic79#12514yes—
254C2×Dic79#12714yes—
255C2×Dic79C4⋊Q814yes—
256C2×Dic79C4⋊Q818yes—
257C2×Dic79C4⋊Q814yes—
258C22×C79C22×C4⋊C411yescharacteristic
259C22×C79C22×C4⋊C412yes—
260C22×C79C2×C4×D411yescharacteristic
261C22×C79C2×C4×Q811yescharacteristic
262C22×C79#52012yes—
263C22×C79C2×#12111yescharacteristic
264C22×C79C2×C22⋊Q811yescharacteristic
265C22×C79C2×#12511yescharacteristic
266C22×C79C2×#12711yescharacteristic
267C22×C79C2×C4⋊Q811yescharacteristic
268C22×C79C2×C4⋊Q812yes—
269C22×C79C2×C4⋊Q811yescharacteristic
270C22×C79#53511yescharacteristic
271C22×C79#53611yescharacteristic
272C22×C79#54412yes—
273C22×C79D4212yes—
274C22×C79#54712yes—
275C22×C79D4×Q812yes—
276C22×C79D4×Q812yes—
277C22×C79#55012yes—
278C22×C79#55412yes—
279C22×C79#55712yes—
280C22×C79Q8212yes—
281C4×C79—24no—
282C4×C79—28no—
283C4×C79—42no—
284C4×C79—42no—
285C4×C79—82no—
286Dic79—28no—
287Dic79—42no—
288Dic79—42no—
289Dic79—82no—
290Dic79#38418yescomplemented
291Dic79#38918yescomplemented
292#13910—791nomaximal, Sylow
293C2×C79C2×#38411yescharacteristic
294C2×C79C2×#38911yescharacteristic
295C2×C79#231711yescharacteristic
296C2×C79D4×C4⋊C412yes—
297C2×C79Q8×C4⋊C412yes—
298C2×C79#254111yescharacteristic
299C2×C79#254211yescharacteristic
300C2×C79#254611yescharacteristic
301C2×C79#255112yes—
302C2×C79#255212yes—
303C2×C79#256711yescharacteristic
304C2×#389—792no—
305C2×#389—792no—
306C2×#384—791no—
307C2×#384—792no—
308C2×C4×C4⋊C4—791no—
309C2×C4×C4⋊C4—791no—
310#1444—791no—
311#1444—791no—
312#1444—791no—
313#1444D7911yescharacteristic
314C4⋊#95—792no—
315C79#1391011yescharacteristic, complemented, Sylow
316C22×C4⋊C4—791no—
317C22×C4⋊C4—792no—
318C22×C4⋊C4—792no—
319C22×C4⋊C4—792no—
320C22×C4⋊C4—792no—
321C22×C4⋊C4—791no—
322C22×C4⋊C4—791no—
323C22×C4⋊C4—792no—
324C22×C4⋊C4—792no—
325C22×C4⋊C4C2×D7912yes—
326C22×C4⋊C4C2×D7911yescharacteristic
327C22×C42—791no—
328C22×C42—792no—
329C22×C42—791no—
330C22×C42—791no—
331#389—1588no—
332#389—1588no—
333#384—1584no—
334#384—1588no—
335C4×C4⋊C4—798no—
336C4×C4⋊C4—1584no—
337C2×#95—791no—
338C2×#95—792no—
339C2×#95—792no—
340C2×#95—792no—
341C2×#95—791no—
342C2×#95C2×D7912yes—
343C2×#95C2×D7911yescharacteristic
344C2×#95C2×D7911yescharacteristic
345C23×C4—791no—
346C23×C4—791no—
347C23×C4—792no—
348C23×C4—791no—
349C23×C4—791no—
350C23×C4—792no—
351C23×C4C22×D7912yes—
352C23×C4C22×D7911yescharacteristic
353C23×C4C22×D7911yescharacteristic
354C23×C4C22×D7912yes—
355C23×C4C22×D7911yescharacteristic
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—42no—
360C2×C4⋊C4—42no—
361C2×C4⋊C4—3162no—
362C2×C4⋊C4—3162no—
363C2×C4⋊C4—3162no—
364C2×C4⋊C4—3162no—
365C2×C4⋊C4—798no—
366C2×C4⋊C4—798no—
367C2×C4⋊C4—798no—
368C2×C4⋊C4—794no—
369C2×C4⋊C4—794no—
370C2×C4⋊C4—1584no—
371C2×C4⋊C4—1584no—
372C2×C4⋊C4—1584no—
373C2×C4⋊C4—1584no—
374C2×C4⋊C4—1584no—
375C2×C4⋊C4—1584no—
376C2×C4⋊C4—1584no—
377C2×C4⋊C4—1584no—
378C2×C4⋊C4—1584no—
379C2×C4⋊C4—1584no—
380C2×C4⋊C4—1584no—
381C2×C4⋊C4—1582no—
382C2×C4⋊C4—1584no—
383C2×C4⋊C4—1582no—
384C2×C4⋊C4—1582no—
385C2×C4⋊C4—1582no—
386C2×C4⋊C4—1582no—
387C2×C42—794no—
388C2×C42—794no—
389C2×C42—794no—
390C2×C42—798no—
391C2×C42—1582no—
392C2×C42—1582no—
393C2×C42—1584no—
394C2×C42—1584no—
395C2×C42—1584no—
396C2×C42—1584no—
397#95—22no—
398#95—24no—
399#95—794no—
400#95—1584no—
401#95—1584no—
402#95—1582no—
403#95—1584no—
404#95C4×D7914yes—
405C24C23×D7911yescharacteristic
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—792no—
428C22×C4—792no—
429C22×C4—794no—
430C22×C4—792no—
431C22×C4—792no—
432C22×C4—792no—
433C22×C4—792no—
434C22×C4—792no—
435C22×C4—792no—
436C22×C4—792no—
437C22×C4—794no—
438C22×C4—792no—
439C22×C4—794no—
440C22×C4—794no—
441C22×C4—794no—
442C22×C4—792no—
443C22×C4—792no—
444C22×C4—1582no—
445C22×C4—1582no—
446C22×C4—1582no—
447C22×C4—1582no—
448C22×C4—1582no—
449C22×C4—1582no—
450C22×C4—1582no—
451C22×C4—1581no—
452C22×C4—1581no—
453C22×C4—1582no—
454C22×C4—1582no—
455C22×C4—1582no—
456C22×C4—1581no—
457C22×C4—1581no—
458C22×C4—1582no—
459C22×C4—1582no—
460C22×C4—1582no—
461C22×C4—1582no—
462C22×C4—1582no—
463C22×C4C2×C4×D7912yes—
464C22×C4C2×C4×D7914yes—
465C22×C4D4×D7912yes—
466C22×C4D4×D7912yes—
467C22×C4D4×D7914yes—
468C22×C4Q8×D7912yes—
469C22×C4Q8×D7912yes—
470C22×C4Q8×D7914yes—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—3164no—
474C4⋊C4—3168no—
475C4⋊C4—3168no—
476C4⋊C4—3168no—
477C4⋊C4—3168no—
478C4⋊C4—3168no—
479C4⋊C4—7932no—
480C4⋊C4—1588no—
481C4⋊C4—1588no—
482C42—3164no—
483C42—3164no—
484C42—7916no—
485C42—15816no—
486C23C22×C4×D7911yescharacteristic
487C23C2×D4×D7911yescharacteristic
488C23C2×D4×D7912yes—
489C23C2×D4×D7911yescharacteristic
490C23C2×Q8×D7911yescharacteristic
491C23C2×Q8×D7911yescharacteristic
492C23C2×Q8×D7912yes—
493C23#41×D7911yescharacteristic
494C23#41×D7911yescharacteristic
495C23#1457920612yes—
496C23#1457920812yes—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—3162no—
525C2×C4—3162no—
526C2×C4—3161no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—3162no—
531C2×C4—3162no—
532C2×C4—3162no—
533C2×C4—3162no—
534C2×C4—3162no—
535C2×C4—3162no—
536C2×C4—3162no—
537C2×C4—794no—
538C2×C4—794no—
539C2×C4—798no—
540C2×C4—794no—
541C2×C4—794no—
542C2×C4—794no—
543C2×C4—794no—
544C2×C4—794no—
545C2×C4—794no—
546C2×C4—794no—
547C2×C4—798no—
548C2×C4—794no—
549C2×C4—798no—
550C2×C4—1584no—
551C2×C4—1584no—
552C2×C4—1582no—
553C2×C4—1582no—
554C2×C4—1584no—
555C2×C4—1584no—
556C2×C4—1584no—
557C2×C4—1584no—
558C2×C4—1584no—
559C2×C4—1584no—
560C2×C4C4⋊C4×D7914yes—
561C2×C4C4⋊C4×D7918yes—
562C22C2×C4⋊C4×D7911yescharacteristic
563C22C2×C4⋊C4×D7912yes—
564C22C4×D4×D7911yescharacteristic
565C22C4×Q8×D7911yescharacteristic
566C22#1909904212yes—
567C22#121×D7911yescharacteristic
568C22C22⋊Q8×D7911yescharacteristic
569C22#125×D7911yescharacteristic
570C22#127×D7911yescharacteristic
571C22C4⋊Q8×D7911yescharacteristic
572C22C4⋊Q8×D7912yes—
573C22C4⋊Q8×D7911yescharacteristic
574C22#1909912811yescharacteristic
575C22#1909913511yescharacteristic
576C22#1909919212yes—
577C22#1909919512yes—
578C22#1909921112yes—
579C22#1909922612yes—
580C22#1909922612yes—
581C22#1909923112yes—
582C22#1909926712yes—
583C22#1909929012yes—
584C22#1909929312yes—
585C4—24no—
586C4—28no—
587C4—42no—
588C4—42no—
589C4—82no—
590C4—3162no—
591C4—3162no—
592C4—798no—
593C4—798no—
594C4—6322no—
595C4—1588no—
596C2#384×D7911yescharacteristic
597C2#389×D7911yescharacteristic
598C2#2376601911yescharacteristic
599C2#2376629812yes—
600C2#2376631412yes—
601C2#2376834111yescharacteristic
602C2#2376834811yescharacteristic
603C2#2376838411yescharacteristic
604C2#2376842512yes—
605C2#2376843012yes—
606C2#2376860411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D79
commutatorC23×C79C23×C4
FittingC79×#1444C2
socle, prime radical, cosocleC24×C79C24
hypercenter#1444D79
hypocenterC79#13910
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
30. #19098414C4semidirect
145. Dic79⋊C4C4⋊C4semidirect
149. C4×Dic79C4⋊C4semidirect
290. Dic79#384semidirect
291. Dic79#389semidirect
315. C79#13910semidirect

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×#190984322
C2×#190984142
#237616652
#237616642
#237616632
Dic79⋊#952
C79×#14442
#1391079