Finite Group Databasebeta

← C2×#23773350 #35156077

Subgroups — C2×#23773350 #35156077

Order 20224.

2326 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579141C211yesmaximal, characteristic
2C2×#117×D79C211yesmaximal, characteristic
3#23773350C2116yesmaximal, direct product factor
4C2×#19098900C212yesmaximal, complemented
5C2×#19098907C212yesmaximal, complemented
6C2×#19098906C212yesmaximal, complemented
7C2×#19098895C212yesmaximal, complemented
8C2×#19098603C211yesmaximal, characteristic, complemented
9C2×#19098573C211yesmaximal, characteristic, complemented
10C2×#19098571C211yesmaximal, characteristic, complemented
11C2×#19098526C211yesmaximal, characteristic
12C2×#428×C79C211yesmaximal, characteristic, complemented
13C23×Dic158C2211yescharacteristic
14C23×C4×D79C2211yescharacteristic
15C22×#11929608C2211yescharacteristic
16C2×#14579141—21no—
17C2×#14579141—24no—
18C2×#14579141—21no—
19C2×#14579141C2212yes—
20C2×#14579141C2212yes—
21C2×#14579141C2218yes—
22C2×#14579120C2211yescharacteristic
23C2×#14579119C2211yescharacteristic
24C2×#14579119C2211yescharacteristic
25C2×#14579118—21no—
26C2×#14579117—21no—
27#117×D79—24no—
28#117×D79C2218yes—
29C22×C4⋊Dic79—21no—
30C2×C4⋊C4×D79C2211yescharacteristic
31C2×C4⋊C4×D79C2211yescharacteristic
32C22×(C2×D79)⋊C4—21no—
33C2×C22⋊C4×D79—21no—
34C2×C42×D79—21no—
35#19098907C22116yescomplemented
36#19098906C22116yescomplemented
37#19098900C22116yescomplemented
38#19098895C22116yescomplemented
39#19098603C2218yescomplemented
40C2×#14579094C2212yescomplemented
41C2×#14579092C2212yescomplemented
42#19098573C2218yescomplemented
43#19098571C2218yescomplemented
44C2×#14579086C2212yescomplemented
45C2×#14579086C2212yescomplemented
46C2×#14579084C2212yescomplemented
47C2×#14579084C2212yescomplemented
48#19098526C2218yes—
49C2×(C2×D79)⋊C8C2212yes—
50C2×(C2×D79)⋊C8C2212yes—
51C22×M16×C79C2211yescharacteristic
52C2×#117×C79C2211yescharacteristic
53C22×C4⋊C4×C79C2211yescharacteristic
54#428×C79C2218yescomplemented
55C2×#107×C79C2212yescomplemented
56C2×#106×C79C2212yescomplemented
57C24×D79D411yescharacteristic
58C22×Dic158—21no—
59C22×Dic158—22no—
60C22×Dic158—24no—
61C22×Dic158—24no—
62C22×Dic158C2312yes—
63C22×Dic158C2314yes—
64C23×Dic79—21no—
65C23×Dic79D411yescharacteristic
66C22×C4×D79—24no—
67C22×C4×D79—24no—
68C22×C4×D79—21no—
69C22×C4×D79—21no—
70C22×C4×D79D411yescharacteristic
71C22×C4×D79D414yes—
72C22×C4×D79D411yescharacteristic
73C22×C4×D79C2311yescharacteristic
74C22×C4×D79C2314yes—
75C22×C4×D79C2311yescharacteristic
76C2×#11929608—24no—
77C2×#11929608C2314yes—
78C22×C79⋊C8C2312yes—
79#14579141—28no—
80#14579141—24no—
81#14579141—28no—
82#14579141—24no—
83#14579141C2318yes—
84#14579141C2318yes—
85#14579120—22no—
86#14579120C2314yes—
87#14579119—22no—
88#14579119—22no—
89#14579119C2314yes—
90#14579119C2314yes—
91#14579118—28no—
92#14579117—28no—
93C2×C4⋊Dic79—22no—
94C2×C4⋊Dic79—22no—
95C2×C4⋊Dic79—28no—
96C2×C4⋊Dic79C2311yescharacteristic
97C2×C4⋊Dic79C2311yescharacteristic
98C2×Dic79⋊C4—21no—
99C2×Dic79⋊C4—21no—
100C4⋊C4×D79—22no—
101C4⋊C4×D79—22no—
102C4⋊C4×D79C2314yes—
103C4⋊C4×D79C2314yes—
104C2×C22⋊Dic79—21no—
105C2×(C2×D79)⋊C4—21no—
106C2×(C2×D79)⋊C4—21no—
107C2×(C2×D79)⋊C4—22no—
108C2×(C2×D79)⋊C4—28no—
109C2×(C2×D79)⋊C4—22no—
110C22⋊C4×D79—28no—
111C2×C4×Dic79—21no—
112C2×C4×Dic79—21no—
113C2×C4×Dic79—21no—
114C42×D79—28no—
115#14579094C2318yescomplemented
116#14579092C2318yescomplemented
117#14579086C2318yescomplemented
118#14579086C2318yescomplemented
119#14579084C2318yescomplemented
120#14579084C2318yescomplemented
121(C2×D79)⋊C8C2318yes—
122(C2×D79)⋊C8C2318yes—
123C23×C4×C79—21no—
124C23×C4×C79C2311yescharacteristic
125C2×M16×C79—24no—
126C2×M16×C79C2314yes—
127C22×C8×C79C2312yes—
128#117×C79—22no—
129#117×C79C2314yes—
130C2×C4⋊C4×C79—21no—
131C2×C4⋊C4×C79—22no—
132C2×C4⋊C4×C79C2311yescharacteristic
133C2×C4⋊C4×C79C2312yes—
134C2×C4⋊C4×C79C2311yescharacteristic
135C2×C4⋊C4×C79C2314yes—
136C2×C22⋊C4×C79—21no—
137C2×C42×C79—21no—
138#107×C79C2318yescomplemented
139#106×C79C2318yescomplemented
140C23×D79—24no—
141C23×D79—24no—
142C23×D79—42no—
143C23×D79C2×D411yescharacteristic
144C23×D79C2×D414yes—
145C23×D79C2×D411yescharacteristic
146C2×Dic158—24no—
147C2×Dic158—22no—
148C2×Dic158—24no—
149C2×Dic158—28no—
150C2×Dic158—24no—
151C2×Dic158—28no—
152C2×Dic158—48no—
153C2×Dic158—44no—
154C2×Dic158C2414yes—
155C22×Dic79—24no—
156C22×Dic79—24no—
157C22×Dic79—22no—
158C22×Dic79—21no—
159C22×Dic79—21no—
160C22×Dic79—44no—
161C22×Dic79C2×D411yescharacteristic
162C22×Dic79C2×D411yescharacteristic
163C22×Dic79C2×D414yes—
164C2×C4×D79—24no—
165C2×C4×D79—24no—
166C2×C4×D79—24no—
167C2×C4×D79—216no—
168C2×C4×D79—24no—
169C2×C4×D79—22no—
170C2×C4×D79—22no—
171C2×C4×D79—28no—
172C2×C4×D79—22no—
173C2×C4×D79—22no—
174C2×C4×D79—28no—
175C2×C4×D79—44no—
176C2×C4×D79—44no—
177C2×C4×D79C2×D414yes—
178C2×C4×D79C2×D412yes—
179C2×C4×D79C2×D414yes—
180C2×C4×D79C2×D412yes—
181C2×C4×D79C2412yes—
182C2×C4×D79C2412yes—
183#11929608—28no—
184C2×C79⋊C8—24no—
185C2×C79⋊C8C2414yes—
186C4⋊Dic79—28no—
187C4⋊Dic79—28no—
188C4⋊Dic79—21no—
189C4⋊Dic79—21no—
190C4⋊Dic79C2412yes—
191C4⋊Dic79C2412yes—
192Dic79⋊C4—24no—
193Dic79⋊C4—24no—
194C22⋊Dic79—24no—
195(C2×D79)⋊C4—24no—
196(C2×D79)⋊C4—24no—
197(C2×D79)⋊C4—28no—
198(C2×D79)⋊C4—28no—
199C4×Dic79—24no—
200C4×Dic79—24no—
201C4×Dic79—24no—
202C24×C79C2×D411yescharacteristic
203C22×C4×C79—21no—
204C22×C4×C79—21no—
205C22×C4×C79—24no—
206C22×C4×C79—22no—
207C22×C4×C79—44no—
208C22×C4×C79C2×D411yescharacteristic
209C22×C4×C79C2×D412yes—
210C22×C4×C79#4114yes—
211C22×C4×C79#4114yes—
212C22×C4×C79C2411yescharacteristic
213C22×C4×C79C2412yes—
214M16×C79—28no—
215C2×C8×C79—24no—
216C2×C8×C79C2414yes—
217C4⋊C4×C79—24no—
218C4⋊C4×C79—22no—
219C4⋊C4×C79—21no—
220C4⋊C4×C79—21no—
221C4⋊C4×C79C2412yes—
222C4⋊C4×C79C2414yes—
223C4⋊C4×C79C2412yes—
224C22⋊C4×C79—24no—
225C42×C79—24no—
226C23×C79—22no—
227C23×C79—22no—
228C23×C79#12314yes—
229C23×C79C22×D412yes—
230C23×C79C22×D411yescharacteristic
231C22×D79—22no—
232C22×D79—22no—
233C22×D79—28no—
234C22×D79—22no—
235C22×D79—22no—
236C22×D79—22no—
237C22×D79—28no—
238C22×D79—44no—
239C22×D79—416no—
240C22×D79#13714yes—
241C22×D79C22×D412yes—
242C22×D79C22×D412yes—
243C2×C4×C79—22no—
244C2×C4×C79—22no—
245C2×C4×C79—24no—
246C2×C4×C79—48no—
247C2×C4×C79—42no—
248C2×C4×C79—42no—
249C2×C4×C79—44no—
250C2×C4×C79#12318yes—
251C2×C4×C79#12314yes—
252C2×C4×C79C22×D412yes—
253C2×C4×C79C22×D411yescharacteristic
254C2×C4×C79C2×#4112yes—
255C2×C4×C79C2×#4114yes—
256C2×C4×C79C2×#4112yes—
257C2×C4×C79C2×#4114yes—
258C2×C4×C79C2511yescharacteristic
259C2×Dic79—28no—
260C2×Dic79—22no—
261C2×Dic79—22no—
262C2×Dic79—22no—
263C2×Dic79—22no—
264C2×Dic79—24no—
265C2×Dic79—48no—
266C2×Dic79—44no—
267C2×Dic79—42no—
268C2×Dic79—42no—
269C2×Dic79C22×D412yes—
270C2×Dic79C22×D412yes—
271C4×D79—28no—
272C4×D79—24no—
273C4×D79—28no—
274C4×D79—24no—
275C4×D79—22no—
276C4×D79—22no—
277C4×D79—22no—
278C4×D79—22no—
279C4×D79—48no—
280C4×D79—48no—
281Dic158—24no—
282Dic158—24no—
283Dic158—48no—
284Dic158—44no—
285C8×C79—24no—
286C79⋊C8—24no—
287C22×C79—28no—
288C22×C79—22no—
289C22×C79—22no—
290C22×C79C2×#12314yes—
291C22×C79C2×#12312yes—
292C22×C79C2×#13712yes—
293C22×C79#57512yes—
294C22×C79C23×D411yescharacteristic
295C2×D79—28no—
296C2×D79—22no—
297C2×D79—22no—
298C2×D79—22no—
299C2×D79—22no—
300C2×D79—22no—
301C2×D79—416no—
302C2×D79—44no—
303C2×D79C2×#13714yes—
304C4×C79—42no—
305C4×C79—44no—
306C4×C79—42no—
307C4×C79C2×#12314yes—
308C4×C79C2×#12312yes—
309C4×C79C22×#4111yescharacteristic
310C4×C79C22×#4111yescharacteristic
311Dic79—22no—
312Dic79—22no—
313Dic79—44no—
314Dic79—42no—
315Dic79—42no—
316C2×#3095—791nomaximal, Sylow
317C2×C79—24no—
318C2×C79#309514yescomplemented
319C2×C79C22×#12311yescharacteristic
320C2×C79C22×#13711yescharacteristic
321C2×C79C2×#57511yescharacteristic
322D79—24no—
323D79—42no—
324C22×C22⋊Q8—791no—
325C22×#117—791no—
326#3095—7916no—
327C2×#484—792no—
328C2×#485—792no—
329C2×#485—792no—
330C2×#483—792no—
331C2×#428—791no—
332C2×#428D7911yescharacteristic, complemented
333C2×#419—791no—
334C2×#419—791no—
335C2×#407—791no—
336C79C2×#309511yescharacteristic, complemented, Sylow
337C23×Q8—791no—
338C24×C4—791no—
339C22×M16—791no—
340C22×M16C2×D7911yescharacteristic
341C2×C22⋊Q8—792no—
342C2×C22⋊Q8—792no—
343C2×C22⋊Q8—798no—
344C2×C22⋊Q8—1584no—
345C2×C22⋊Q8—1581no—
346C2×C22⋊Q8—1581no—
347C2×#117—791no—
348C2×#117—798no—
349C2×#117—791no—
350C2×#117—791no—
351C2×#117—1581no—
352C2×#117—1581no—
353C2×#117—1584no—
354C2×#117C2×D7911yescharacteristic
355C22×C4⋊C4—791no—
356C22×C4⋊C4—791no—
357C22×C4⋊C4—1581no—
358C22×C4⋊C4C2×D7911yescharacteristic
359C22×C22⋊C4—1581no—
360C22×C22⋊C4—1581no—
361C22×C42—1581no—
362#485—7916no—
363#485—7916no—
364#484—7916no—
365#483—7916no—
366#428—798no—
367#428C2×D7918yescomplemented
368C2×#107—792no—
369C2×#107C2×D7912yescomplemented
370C2×#106—792no—
371C2×#106C2×D7912yescomplemented
372#419—798no—
373#419—798no—
374C2×#103—792no—
375C2×#103—792no—
376C2×#103—792no—
377C2×#103—792no—
378#407—798no—
379C2×C22⋊C8—792no—
380C2×C22⋊C8—792no—
381C25—791no—
382C22×Q8—792no—
383C22×Q8—794no—
384C22×Q8—1584no—
385C22×Q8—1584no—
386C22×Q8—1582no—
387C22×Q8—1581no—
388C23×C4—21no—
389C23×C4—791no—
390C23×C4—794no—
391C23×C4—791no—
392C23×C4—791no—
393C23×C4—791no—
394C23×C4—791no—
395C23×C4—794no—
396C23×C4—1581no—
397C23×C4—1581no—
398C23×C4—1581no—
399C23×C4—1584no—
400C23×C4—1584no—
401C23×C4C22×D7911yescharacteristic
402C2×M16—24no—
403C2×M16—794no—
404C2×M16—1584no—
405C2×M16C22×D7914yes—
406C22×C8—792no—
407C22×C8C22×D7912yes—
408C22⋊Q8—798no—
409C22⋊Q8—798no—
410C22⋊Q8—1588no—
411C22⋊Q8—1584no—
412C22⋊Q8—1584no—
413C22⋊Q8—1588no—
414#117—22no—
415#117—794no—
416#117—794no—
417#117—794no—
418#117—1588no—
419#117—1588no—
420#117—1582no—
421#117—1582no—
422#117—1582no—
423#117C22×D7914yes—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—791no—
427C2×C4⋊C4—791no—
428C2×C4⋊C4—794no—
429C2×C4⋊C4—794no—
430C2×C4⋊C4—1582no—
431C2×C4⋊C4—1582no—
432C2×C4⋊C4—1581no—
433C2×C4⋊C4—1581no—
434C2×C4⋊C4—1588no—
435C2×C4⋊C4—1582no—
436C2×C4⋊C4—1582no—
437C2×C4⋊C4C22×D7911yescharacteristic
438C2×C4⋊C4C22×D7911yescharacteristic
439C2×C4⋊C4C22×D7912yes—
440C2×C4⋊C4C22×D7914yes—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1588no—
443C2×C22⋊C4—1581no—
444C2×C22⋊C4—1581no—
445C2×C22⋊C4—1588no—
446C2×C22⋊C4—1582no—
447C2×C22⋊C4—1582no—
448C2×C22⋊C4—1581no—
449C2×C42—21no—
450C2×C42—1588no—
451C2×C42—1581no—
452C2×C42—1581no—
453C2×C42—1581no—
454#107—798no—
455#107C22×D7918yescomplemented
456#106—798no—
457#106C22×D7918yescomplemented
458#103—798no—
459#103—798no—
460#103—798no—
461#103—798no—
462C22⋊C8—798no—
463C22⋊C8—798no—
464C24—3162no—
465C24—794no—
466C24—791no—
467C24—791no—
468C24—1584no—
469C24—1584no—
470C24D4×D7911yescharacteristic
471C2×Q8—3168no—
472C2×Q8—3164no—
473C2×Q8—794no—
474C2×Q8—1584no—
475C2×Q8—1588no—
476C2×Q8—1584no—
477C2×Q8—1588no—
478C2×Q8—1584no—
479C2×Q8—1582no—
480C22×C4—24no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—44no—
485C22×C4—3164no—
486C22×C4—3164no—
487C22×C4—3164no—
488C22×C4—794no—
489C22×C4—791no—
490C22×C4—791no—
491C22×C4—792no—
492C22×C4—792no—
493C22×C4—794no—
494C22×C4—792no—
495C22×C4—792no—
496C22×C4—794no—
497C22×C4—1584no—
498C22×C4—1584no—
499C22×C4—1584no—
500C22×C4—1581no—
501C22×C4—1581no—
502C22×C4—1582no—
503C22×C4—1584no—
504C22×C4—1582no—
505C22×C4—1582no—
506C22×C4—1588no—
507C22×C4—1582no—
508C22×C4—1582no—
509C22×C4—1588no—
510C22×C4—15816no—
511C22×C4—1584no—
512C22×C4—1584no—
513C22×C4D4×D7912yes—
514C22×C4D4×D7911yescharacteristic
515C22×C4#1192962414yes—
516C22×C4#1192962514yes—
517C22×C4C23×D7912yes—
518C22×C4C23×D7911yescharacteristic
519M16—28no—
520M16—1588no—
521C2×C8—24no—
522C2×C8—794no—
523C2×C8—1584no—
524C2×C8C23×D7914yes—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—792no—
530C4⋊C4—792no—
531C4⋊C4—1584no—
532C4⋊C4—1584no—
533C4⋊C4—1588no—
534C4⋊C4—1588no—
535C4⋊C4—1581no—
536C4⋊C4—1581no—
537C4⋊C4C23×D7912yes—
538C4⋊C4C23×D7912yes—
539C4⋊C4C23×D7914yes—
540C22⋊C4—24no—
541C22⋊C4—1584no—
542C22⋊C4—1584no—
543C22⋊C4—1588no—
544C22⋊C4—1588no—
545C22⋊C4—1584no—
546C42—24no—
547C42—1584no—
548C42—1584no—
549C42—1584no—
550C23—22no—
551C23—22no—
552C23—31616no—
553C23—3164no—
554C23—794no—
555C23—792no—
556C23—792no—
557C23—1588no—
558C23—1588no—
559C23—1582no—
560C23—1582no—
561C23—1582no—
562C23—1582no—
563C23—1582no—
564C23#1457914614yes—
565C23C2×D4×D7912yes—
566C23C2×D4×D7911yescharacteristic
567Q8—3164no—
568Q8—3168no—
569Q8—1584no—
570Q8—1584no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—3168no—
579C2×C4—3168no—
580C2×C4—3168no—
581C2×C4—3164no—
582C2×C4—3162no—
583C2×C4—3162no—
584C2×C4—792no—
585C2×C4—792no—
586C2×C4—1582no—
587C2×C4—1582no—
588C2×C4—1584no—
589C2×C4—1588no—
590C2×C4—1582no—
591C2×C4—1582no—
592C2×C4—1584no—
593C2×C4—1588no—
594C2×C4—1588no—
595C2×C4—1584no—
596C2×C4—1582no—
597C2×C4—1582no—
598C2×C4—1582no—
599C2×C4—1582no—
600C2×C4#1457914618yes—
601C2×C4#1457914614yes—
602C2×C4C2×D4×D7912yes—
603C2×C4C2×D4×D7911yescharacteristic
604C2×C4C2×#1192962412yes—
605C2×C4C2×#1192962414yes—
606C2×C4C2×#1192962512yes—
607C2×C4C2×#1192962514yes—
608C2×C4C24×D7911yescharacteristic
609C8—24no—
610C8—1584no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—31616no—
615C22—3164no—
616C22—794no—
617C22—1582no—
618C22—1582no—
619C22—1582no—
620C22—1582no—
621C22—1582no—
622C22—1588no—
623C22C2×#1457914614yes—
624C22C2×#1457914612yes—
625C22#137×D7912yes—
626C22#1909935912yes—
627C22C22×D4×D7911yescharacteristic
628C4—42no—
629C4—44no—
630C4—42no—
631C4—3162no—
632C4—3162no—
633C4—3164no—
634C4—1582no—
635C4—1582no—
636C4C2×#1457914612yes—
637C4C2×#1457914614yes—
638C4C22×#1192962411yescharacteristic
639C4C22×#1192962511yescharacteristic
640C2—24no—
641C2—3162no—
642C2—1584no—
643C2#2377335014yesdirect product factor
644C2C22×#1457914611yescharacteristic
645C2C2×#137×D7911yescharacteristic
646C2C2×#1909935911yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×#428×C79C2
socleC23×C79C22×D4
prime radicalC24×D79D4
hypercenterC2×#428D79
hypocenterC79C2×#3095
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
3. #23773350C2direct (N × H)
4. C2×#19098900C2semidirect
5. C2×#19098907C2semidirect
6. C2×#19098906C2semidirect
7. C2×#19098895C2semidirect
8. C2×#19098603C2semidirect
9. C2×#19098573C2semidirect
10. C2×#19098571C2semidirect
12. C2×#428×C79C2semidirect
35. #19098907C22semidirect
36. #19098906C22semidirect
37. #19098900C22semidirect
38. #19098895C22semidirect
39. #19098603C22semidirect
40. C2×#14579094C22semidirect
41. C2×#14579092C22semidirect
42. #19098573C22semidirect
43. #19098571C22semidirect
44. C2×#14579086C22semidirect
45. C2×#14579086C22semidirect
46. C2×#14579084C22semidirect
47. C2×#14579084C22semidirect
54. #428×C79C22semidirect
55. C2×#107×C79C22semidirect
56. C2×#106×C79C22semidirect
115. #14579094C23semidirect
116. #14579092C23semidirect
117. #14579086C23semidirect
118. #14579086C23semidirect
119. #14579084C23semidirect
120. #14579084C23semidirect
138. #107×C79C23semidirect
139. #106×C79C23semidirect
318. C2×C79#3095semidirect
332. C2×#428D79semidirect
336. C79C2×#3095semidirect
367. #428C2×D79semidirect
369. C2×#107C2×D79semidirect
371. C2×#106C2×D79semidirect
455. #107C22×D79semidirect
457. #106C22×D79semidirect
643. C2#23773350direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145791412
C2×#117×D792
#237733502
C2×#190989002
C2×#190989072
C2×#190989062
C2×#190988952
C2×#190986032
C2×#190985732
C2×#190985712
C2×#190985262
C2×#428×C792
C2×#309579