Finite Group Databasebeta

← #2296×D79 #34446646

Subgroups — #2296×D79 #34446646

Order 20224.

3854 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D79C211yesmaximal, characteristic
2C2×#117×D79C211yesmaximal, characteristic
3C22×C42×D79C211yesmaximal, characteristic
4#23765884C211yesmaximal, characteristic, complemented
5#23765883C211yesmaximal, characteristic, complemented
6#23765882C211yesmaximal, characteristic, complemented
7#23765881C214yesmaximal, complemented
8#23765880C214yesmaximal, complemented
9#23765879C214yesmaximal, complemented
10#389×D79C214yesmaximal, complemented
11#385×D79C214yesmaximal
12C4×C4⋊C4×D79C214yesmaximal, complemented
13C79×#2296C211yesmaximal, characteristic, complemented
14C23×C4×D79C2212yes—
15C23×C4×D79C2211yescharacteristic
16C2×#14579120C2211yescharacteristic
17C2×#14579120C2211yescharacteristic
18C2×#14579119C2212yes—
19C2×#14579119C2212yes—
20C2×#14579118C2212yes—
21C2×#14579118C2212yes—
22C2×#14579117C2212yes—
23C2×#14579117C2212yes—
24#117×D79—28no—
25#117×D79C4116yescomplemented
26C2×C4⋊C4×D79C2218yes—
27C2×C4⋊C4×D79C2212yes—
28C2×C4⋊C4×D79C2212yes—
29C2×C22⋊C4×D79C2212yes—
30C2×C22⋊C4×D79C2212yes—
31C22×C4×Dic79C2212yes—
32C22×C4×Dic79C2211yescharacteristic
33C2×C42×D79—28no—
34C2×C42×D79C2212yes—
35C2×C42×D79C2212yes—
36C2×C42×D79C2212yes—
37C2×C42×D79C2212yes—
38C2×C42×D79C2212yes—
39C2×C42×D79C2212yes—
40#19098462C2214yescomplemented
41#19098461C2214yescomplemented
42#19098460C2218yescomplemented
43#19098459C2214yescomplemented
44#19098458C2218yescomplemented
45#19098438C2214yes—
46#19098437C2218yes—
47#19098436C2214yes—
48#19098435C2214yes—
49#19098434C2218yes—
50C4⋊C4×Dic79C2218yescomplemented
51C4×Dic79⋊C4C2214yescomplemented
52C4×Dic79⋊C4C2214yescomplemented
53C4×C4⋊Dic79C2214yescomplemented
54#19098414C2218yescomplemented
55#95×D79C2218yes—
56C2×#117×C79C2211yescharacteristic
57C2×#117×C79C2211yescharacteristic
58C22×C42×C79C2211yescharacteristic
59#389×C79C2214yescomplemented
60#385×C79C2214yes—
61C4×C4⋊C4×C79C2214yescomplemented
62C24×D79C2311yescharacteristic
63C23×Dic79C2312yes—
64C23×Dic79C2311yescharacteristic
65C22×C4×D79—28no—
66C22×C4×D79—216no—
67C22×C4×D79C2×C418yes—
68C22×C4×D79D418yes—
69C22×C4×D79Q818yes—
70C22×C4×D79C2314yes—
71C22×C4×D79C2314yes—
72C22×C4×D79C2314yes—
73C22×C4×D79C2314yes—
74C22×C4×D79C2311yescharacteristic
75C22×C4×D79C2311yescharacteristic
76C22×C4×D79C2311yescharacteristic
77C22×C4×D79C2311yescharacteristic
78#14579120—24no—
79#14579120C2×C418yescomplemented
80#14579119—28no—
81#14579119C2×C4116yescomplemented
82#14579118—28no—
83#14579118C2×C4116yescomplemented
84#14579117—28no—
85#14579117C2×C4116yescomplemented
86C2×C4⋊Dic79C2318yes—
87C2×C4⋊Dic79C2312yes—
88C2×C4⋊Dic79C2312yes—
89C2×Dic79⋊C4C2318yes—
90C2×Dic79⋊C4C2318yes—
91C2×Dic79⋊C4C2314yes—
92C2×Dic79⋊C4C2314yes—
93C4⋊C4×D79—28no—
94C4⋊C4×D79—232no—
95C4⋊C4×D79C2×C4116yescomplemented
96C2×C22⋊Dic79C2312yes—
97C2×C22⋊Dic79C2312yes—
98C2×(C2×D79)⋊C4C2314yes—
99C2×(C2×D79)⋊C4C2314yes—
100C22⋊C4×D79—28no—
101C22⋊C4×D79C2×C4116yes—
102C2×C4×Dic79—24no—
103C2×C4×Dic79—28no—
104C2×C4×Dic79C2314yes—
105C2×C4×Dic79C2314yes—
106C2×C4×Dic79C2312yes—
107C2×C4×Dic79C2312yes—
108C2×C4×Dic79C2314yes—
109C2×C4×Dic79C2312yes—
110C2×C4×Dic79C2314yes—
111C2×C4×Dic79C2312yes—
112C42×D79—28no—
113C42×D79—28no—
114C42×D79—28no—
115C42×D79—28no—
116C42×D79—28no—
117C42×D79C2×C4116yescomplemented
118C79⋊#95C2318yes—
119C79⋊#95C2318yes—
120C79⋊#95C2318yes—
121C23×C4×C79C2312yes—
122C23×C4×C79C2311yescharacteristic
123#117×C79—24no—
124#117×C79C2×C418yescomplemented
125C2×C4⋊C4×C79C2318yes—
126C2×C4⋊C4×C79C2312yes—
127C2×C4⋊C4×C79C2312yes—
128C2×C22⋊C4×C79C2312yes—
129C2×C22⋊C4×C79C2312yes—
130C2×C42×C79—24no—
131C2×C42×C79C2312yes—
132C2×C42×C79C2312yes—
133C2×C42×C79C2312yes—
134C2×C42×C79C2312yes—
135#95×C79C2318yes—
136C23×D79—28no—
137C23×D79C22×C414yes—
138C23×D79C2×D414yes—
139C23×D79C2×Q814yes—
140C23×D79C2411yescharacteristic
141C23×D79C2411yescharacteristic
142C22×Dic79—28no—
143C22×Dic79—24no—
144C22×Dic79C22×C414yes—
145C22×Dic79C2×D414yes—
146C22×Dic79C2×Q814yes—
147C22×Dic79C2414yes—
148C22×Dic79C2414yes—
149C22×Dic79C2414yes—
150C22×Dic79C2411yescharacteristic
151C22×Dic79C2411yescharacteristic
152C2×C4×D79—264no—
153C2×C4×D79—28no—
154C2×C4×D79—28no—
155C2×C4×D79—28no—
156C2×C4×D79—28no—
157C2×C4×D79—216no—
158C2×C4×D79—28no—
159C2×C4×D79—216no—
160C2×C4×D79—28no—
161C2×C4×D79C4⋊C4164yescomplemented
162C2×C4×D79C22×C4116yes—
163C2×C4×D79C22×C414yes—
164C2×C4×D79C22×C414yes—
165C2×C4×D79C22×C414yes—
166C2×C4×D79C22×C414yes—
167C2×C4×D79C2×D414yes—
168C2×C4×D79C2×D414yes—
169C2×C4×D79C2×D414yes—
170C2×C4×D79C2×D414yes—
171C2×C4×D79C2×Q814yes—
172C2×C4×D79C2×Q814yes—
173C2×C4×D79C2×Q814yes—
174C2×C4×D79C2×Q814yes—
175C2×C4×D79#41116yes—
176C4⋊Dic79—216no—
177C4⋊Dic79—24no—
178C4⋊Dic79C22×C418yescomplemented
179Dic79⋊C4—28no—
180Dic79⋊C4—216no—
181Dic79⋊C4—216no—
182Dic79⋊C4C22×C4116yescomplemented
183C22⋊Dic79—24no—
184C22⋊Dic79C22×C418yes—
185(C2×D79)⋊C4—28no—
186(C2×D79)⋊C4C22×C4116yes—
187C4×Dic79—28no—
188C4×Dic79—28no—
189C4×Dic79—28no—
190C4×Dic79—24no—
191C4×Dic79—24no—
192C4×Dic79—24no—
193C4×Dic79C22×C4116yescomplemented
194C4×Dic79C22×C418yescomplemented
195C24×C79C2411yescharacteristic
196C22×C4×C79—24no—
197C22×C4×C79—28no—
198C22×C4×C79C22×C414yes—
199C22×C4×C79C2×D414yes—
200C22×C4×C79C2×Q814yes—
201C22×C4×C79C2414yes—
202C22×C4×C79C2414yes—
203C22×C4×C79C2414yes—
204C22×C4×C79C2411yescharacteristic
205C22×C4×C79C2411yescharacteristic
206C4⋊C4×C79—24no—
207C4⋊C4×C79—216no—
208C4⋊C4×C79C22×C418yescomplemented
209C22⋊C4×C79—24no—
210C22⋊C4×C79C22×C418yes—
211C42×C79—24no—
212C42×C79—24no—
213C42×C79—24no—
214C42×C79C22×C418yescomplemented
215C23×C79—24no—
216C23×C79C23×C412yes—
217C23×C79C22×D412yes—
218C23×C79C22×Q812yes—
219C23×C79C2511yescharacteristic
220C22×D79—24no—
221C22×D79—216no—
222C22×D79—216no—
223C22×D79—216no—
224C22×D79C2×C4⋊C4116yes—
225C22×D79C23×C412yes—
226C22×D79C23×C412yes—
227C22×D79C22×D412yes—
228C22×D79C22×D412yes—
229C22×D79C22×Q812yes—
230C22×D79C22×Q812yes—
231C22×D79C2×#4118yes—
232C2×C4×C79—216no—
233C2×C4×C79—24no—
234C2×C4×C79—28no—
235C2×C4×C79—24no—
236C2×C4×C79—24no—
237C2×C4×C79—28no—
238C2×C4×C79C2×C4⋊C4116yescomplemented
239C2×C4×C79C23×C418yes—
240C2×C4×C79C23×C412yes—
241C2×C4×C79C23×C412yes—
242C2×C4×C79C22×D412yes—
243C2×C4×C79C22×D412yes—
244C2×C4×C79C22×Q812yes—
245C2×C4×C79C22×Q812yes—
246C2×C4×C79C2×#4118yes—
247C2×Dic79—216no—
248C2×Dic79—24no—
249C2×Dic79—28no—
250C2×Dic79—24no—
251C2×Dic79—24no—
252C2×Dic79—28no—
253C2×Dic79C2×C4⋊C4116yescomplemented
254C2×Dic79C23×C418yes—
255C2×Dic79C23×C412yes—
256C2×Dic79C23×C412yes—
257C2×Dic79C22×D412yes—
258C2×Dic79C22×D412yes—
259C2×Dic79C22×Q812yes—
260C2×Dic79C22×Q812yes—
261C2×Dic79C2×#4118yes—
262C4×D79—232no—
263C4×D79—232no—
264C4×D79—232no—
265C4×D79—416no—
266C4×D79C2×C4⋊C4116yes—
267C4×D79C2×C4⋊C4116yes—
268C4×D79C2×C4⋊C4116yescomplemented
269C4×D79C2×C4⋊C4116yescomplemented
270C22×C79—24no—
271C22×C79—24no—
272C22×C79—24no—
273C22×C79C22×C4⋊C414yes—
274C22×C79C24×C411yescharacteristic
275C22×C79C23×D411yescharacteristic
276C22×C79C23×Q811yescharacteristic
277C22×C79C22×#4114yes—
278C2×D79—24no—
279C2×D79—24no—
280C2×D79—24no—
281C2×D79—232no—
282C2×D79C22×C4⋊C414yes—
283C2×D79C22×C4⋊C414yes—
284C2×D79C4×#4118yes—
285C2×D79#52518yes—
286C2×D79#53318yes—
287C4×C79—28no—
288C4×C79—28no—
289C4×C79—44no—
290C4×C79C22×C4⋊C414yes—
291C4×C79C22×C4⋊C414yes—
292Dic79—28no—
293Dic79—28no—
294Dic79—44no—
295Dic79C22×C4⋊C414yes—
296Dic79C22×C4⋊C414yes—
297C2×#2296—791nomaximal, Sylow
298C2×C79—24no—
299C2×C79C23×C4⋊C411yescharacteristic
300C2×C79C2×C4×#4112yes—
301C2×C79C2×#52512yes—
302C2×C79C2×#53312yes—
303D79—24no—
304D79#229618yesdirect product factor
305C22×#117—791no—
306C22×#117—791no—
307C23×C42—791no—
308#2296—791no—
309#2296—794no—
310#2296—794no—
311#2296—794no—
312#2296—791no—
313#2296—791no—
314#2296D7911yescharacteristic, direct product factor
315C2×#389—794no—
316C2×#385—794no—
317C2×C4×C4⋊C4—794no—
318C79C2×#229611yescharacteristic, complemented, Sylow
319C24×C4—791no—
320C24×C4—792no—
321C2×#117—791no—
322C2×#117—7916no—
323C2×#117—791no—
324C2×#117—792no—
325C2×#117—792no—
326C2×#117—792no—
327C2×#117—792no—
328C2×#117—792no—
329C2×#117—792no—
330C2×#117—1588no—
331C2×#117C2×D7911yescharacteristic
332C2×#117C2×D7911yescharacteristic
333C22×C4⋊C4—792no—
334C22×C4⋊C4—792no—
335C22×C4⋊C4—798no—
336C22×C22⋊C4—792no—
337C22×C22⋊C4—792no—
338C22×C42—792no—
339C22×C42—792no—
340C22×C42—791no—
341C22×C42—792no—
342C22×C42—792no—
343C22×C42—792no—
344C22×C42—792no—
345C22×C42—792no—
346C22×C42—1588no—
347C22×C42C2×D7911yescharacteristic
348#389—798no—
349#389—794no—
350#389—794no—
351#389—794no—
352#389—798no—
353#389C2×D7914yescomplemented
354#385—794no—
355#385—798no—
356#385—794no—
357#385—794no—
358#385—798no—
359#385C2×D7914yes—
360C4×C4⋊C4—798no—
361C4×C4⋊C4—794no—
362C4×C4⋊C4—794no—
363C4×C4⋊C4—794no—
364C4×C4⋊C4—798no—
365C4×C4⋊C4C2×D7914yescomplemented
366C2×#95—798no—
367C25—791no—
368C23×C4—791no—
369C23×C4—798no—
370C23×C4—798no—
371C23×C4—798no—
372C23×C4—791no—
373C23×C4—791no—
374C23×C4—791no—
375C23×C4—791no—
376C23×C4—794no—
377C23×C4—794no—
378C23×C4—792no—
379C23×C4—794no—
380C23×C4—794no—
381C23×C4—15816no—
382C23×C4—1588no—
383C23×C4C22×D7912yes—
384C23×C4C22×D7911yescharacteristic
385#117—24no—
386#117—798no—
387#117—7916no—
388#117—7916no—
389#117—7916no—
390#117—1588no—
391#117—1588no—
392#117—1588no—
393#117—1584no—
394#117C4×D7918yescomplemented
395C2×C4⋊C4—794no—
396C2×C4⋊C4—792no—
397C2×C4⋊C4—7916no—
398C2×C4⋊C4—794no—
399C2×C4⋊C4—792no—
400C2×C4⋊C4—798no—
401C2×C4⋊C4—798no—
402C2×C4⋊C4—798no—
403C2×C4⋊C4—1588no—
404C2×C4⋊C4—15832no—
405C2×C4⋊C4C22×D7912yes—
406C2×C4⋊C4C22×D7918yes—
407C2×C4⋊C4C22×D7912yes—
408C2×C22⋊C4—792no—
409C2×C22⋊C4—794no—
410C2×C22⋊C4—7916no—
411C2×C22⋊C4—792no—
412C2×C22⋊C4—794no—
413C2×C22⋊C4—1588no—
414C2×C22⋊C4C22×D7912yes—
415C2×C22⋊C4C22×D7912yes—
416C2×C42—24no—
417C2×C42—794no—
418C2×C42—792no—
419C2×C42—7916no—
420C2×C42—794no—
421C2×C42—794no—
422C2×C42—794no—
423C2×C42—792no—
424C2×C42—792no—
425C2×C42—792no—
426C2×C42—1588no—
427C2×C42—1588no—
428C2×C42—1588no—
429C2×C42—1588no—
430C2×C42—1588no—
431C2×C42—1588no—
432C2×C42—1584no—
433C2×C42C22×D7912yes—
434C2×C42C22×D7912yes—
435C2×C42C22×D7912yes—
436C2×C42C22×D7912yes—
437#95—798no—
438#95—798no—
439#95—798no—
440#95C22×D7918yes—
441C24—791no—
442C24—794no—
443C24—791no—
444C24—794no—
445C24—794no—
446C24—1588no—
447C24C23×D7911yescharacteristic
448C22×C4—24no—
449C22×C4—28no—
450C22×C4—794no—
451C22×C4—791no—
452C22×C4—791no—
453C22×C4—794no—
454C22×C4—794no—
455C22×C4—794no—
456C22×C4—794no—
457C22×C4—794no—
458C22×C4—794no—
459C22×C4—7964no—
460C22×C4—794no—
461C22×C4—794no—
462C22×C4—794no—
463C22×C4—794no—
464C22×C4—794no—
465C22×C4—794no—
466C22×C4—794no—
467C22×C4—794no—
468C22×C4—794no—
469C22×C4—794no—
470C22×C4—7916no—
471C22×C4—7916no—
472C22×C4—794no—
473C22×C4—1588no—
474C22×C4—15816no—
475C22×C4—1588no—
476C22×C4—1588no—
477C22×C4—1588no—
478C22×C4—15816no—
479C22×C4—1588no—
480C22×C4—15864no—
481C22×C4—1588no—
482C22×C4—1584no—
483C22×C4—1588no—
484C22×C4C2×C4×D7914yes—
485C22×C4D4×D7914yes—
486C22×C4Q8×D7914yes—
487C22×C4C23×D7914yes—
488C22×C4C23×D7914yes—
489C22×C4C23×D7914yes—
490C22×C4C23×D7911yescharacteristic
491C22×C4C23×D7911yescharacteristic
492C4⋊C4—216no—
493C4⋊C4—24no—
494C4⋊C4—7916no—
495C4⋊C4—798no—
496C4⋊C4—15816no—
497C4⋊C4—1588no—
498C4⋊C4—15816no—
499C4⋊C4—15816no—
500C4⋊C4—1584no—
501C4⋊C4C2×C4×D7918yescomplemented
502C22⋊C4—24no—
503C22⋊C4—798no—
504C22⋊C4—7916no—
505C22⋊C4—1588no—
506C22⋊C4—1584no—
507C22⋊C4C2×C4×D7918yes—
508C42—24no—
509C42—24no—
510C42—24no—
511C42—7916no—
512C42—798no—
513C42—1588no—
514C42—1588no—
515C42—1588no—
516C42—1584no—
517C42—1584no—
518C42—1584no—
519C42C2×C4×D7918yescomplemented
520C23—24no—
521C23—792no—
522C23—792no—
523C23—792no—
524C23—798no—
525C23—7916no—
526C23—792no—
527C23—792no—
528C23—792no—
529C23—15816no—
530C23—15816no—
531C23—15816no—
532C23—1584no—
533C23C22×C4×D7912yes—
534C23C2×D4×D7912yes—
535C23C2×Q8×D7912yes—
536C23C24×D7911yescharacteristic
537C2×C4—216no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—31616no—
544C2×C4—7916no—
545C2×C4—792no—
546C2×C4—792no—
547C2×C4—792no—
548C2×C4—792no—
549C2×C4—792no—
550C2×C4—792no—
551C2×C4—7916no—
552C2×C4—7916no—
553C2×C4—7916no—
554C2×C4—7916no—
555C2×C4—798no—
556C2×C4—798no—
557C2×C4—15832no—
558C2×C4—15832no—
559C2×C4—15832no—
560C2×C4—1584no—
561C2×C4—1588no—
562C2×C4—15816no—
563C2×C4—1584no—
564C2×C4—1588no—
565C2×C4—1584no—
566C2×C4C4⋊C4×D79116yescomplemented
567C2×C4C22×C4×D7918yes—
568C2×C4C22×C4×D7912yes—
569C2×C4C22×C4×D7912yes—
570C2×C4C2×D4×D7912yes—
571C2×C4C2×D4×D7912yes—
572C2×C4C2×Q8×D7912yes—
573C2×C4C2×Q8×D7912yes—
574C2×C4#41×D7918yes—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—794no—
579C22—798no—
580C22—798no—
581C22—798no—
582C22—794no—
583C22—15832no—
584C22—1584no—
585C22—1584no—
586C22—1584no—
587C22C2×C4⋊C4×D7914yes—
588C22C23×C4×D7911yescharacteristic
589C22C22×D4×D7911yescharacteristic
590C22C22×Q8×D7911yescharacteristic
591C22C2×#41×D7914yes—
592C4—28no—
593C4—28no—
594C4—44no—
595C4—3164no—
596C4—794no—
597C4—794no—
598C4—1588no—
599C4—1588no—
600C4C2×C4⋊C4×D7914yes—
601C4C2×C4⋊C4×D7914yes—
602C2—24no—
603C2—798no—
604C2—1584no—
605C2C22×C4⋊C4×D7911yescharacteristic
606C2C4×#41×D7912yes—
607C2#525×D7912yes—
608C2#533×D7912yes—
subgroupG/H
centerC22×C4C23×D79
commutatorC22×C79C24×C4
FrattiniC23C24×D79
FittingC79×#2296C2
socle, cosocleC23×C79C25
prime radicalC24×D79C23
hypercenter#2296D79
hypocenterC79C2×#2296
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C22×C4 ◁ #2296

Complements

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

NH ≅ G/Ntype
4. #23765884C2semidirect
5. #23765883C2semidirect
6. #23765882C2semidirect
7. #23765881C2semidirect
8. #23765880C2semidirect
9. #23765879C2semidirect
10. #389×D79C2semidirect
12. C4×C4⋊C4×D79C2semidirect
13. C79×#2296C2semidirect
25. #117×D79C4semidirect
40. #19098462C22semidirect
41. #19098461C22semidirect
42. #19098460C22semidirect
43. #19098459C22semidirect
44. #19098458C22semidirect
50. C4⋊C4×Dic79C22semidirect
51. C4×Dic79⋊C4C22semidirect
52. C4×Dic79⋊C4C22semidirect
53. C4×C4⋊Dic79C22semidirect
54. #19098414C22semidirect
59. #389×C79C22semidirect
61. C4×C4⋊C4×C79C22semidirect
79. #14579120C2×C4semidirect
81. #14579119C2×C4semidirect
83. #14579118C2×C4semidirect
85. #14579117C2×C4semidirect
95. C4⋊C4×D79C2×C4semidirect
117. C42×D79C2×C4semidirect
124. #117×C79C2×C4semidirect
161. C2×C4×D79C4⋊C4semidirect
178. C4⋊Dic79C22×C4semidirect
182. Dic79⋊C4C22×C4semidirect
193. C4×Dic79C22×C4semidirect
194. C4×Dic79C22×C4semidirect
208. C4⋊C4×C79C22×C4semidirect
214. C42×C79C22×C4semidirect
238. C2×C4×C79C2×C4⋊C4semidirect
253. C2×Dic79C2×C4⋊C4semidirect
268. C4×D79C2×C4⋊C4semidirect
269. C4×D79C2×C4⋊C4semidirect
304. D79#2296direct (N × H)
314. #2296D79direct (N × H)
318. C79C2×#2296semidirect
353. #389C2×D79semidirect
365. C4×C4⋊C4C2×D79semidirect
394. #117C4×D79semidirect
501. C4⋊C4C2×C4×D79semidirect
519. C42C2×C4×D79semidirect
566. C2×C4C4⋊C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D792
C22×C42×D792
#237658842
#237658832
#237658822
#237658812
#237658802
#237658792
#389×D792
#385×D792
C4×C4⋊C4×D792
C79×#22962
C2×#229679