Finite Group Databasebeta

← C2×#23773213 #35155798

Subgroups — C2×#23773213 #35155798

Order 20224.

2358 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579160C211yesmaximal, characteristic, complemented
2C2×#19099031C211yesmaximal, characteristic
3#23773213C2116yesmaximal, direct product factor
4C2×#19098857C212yesmaximal, complemented
5C2×#19098847C212yesmaximal, complemented
6C2×#19098852C212yesmaximal, complemented
7C2×#19098842C212yesmaximal, complemented
8C2×#19098606C211yesmaximal, characteristic, complemented
9C2×#19098586C211yesmaximal, characteristic, complemented
10C2×#19098565C211yesmaximal, characteristic
11C2×#19098562C211yesmaximal, characteristic
12C2×#428×C79C211yesmaximal, characteristic, complemented
13C22×#11929624C2211yescharacteristic
14C23×Dic158—21no—
15C22×#11929608C2211yescharacteristic
16C2×#14579160—21no—
17C2×#14579160—21no—
18C2×#14579160—24no—
19C2×#14579160C2212yes—
20C2×#14579160C2212yes—
21C2×#14579160C2218yescomplemented
22#19099031—24no—
23#19099031C2218yes—
24C2×#14579126C2211yescharacteristic
25C2×#14579126C2211yescharacteristic
26C2×#14579123C2211yescharacteristic
27C2×#14579123C2211yescharacteristic
28C2×(C22⋊D79)⋊C4—21no—
29C2×(C22⋊D79)⋊C4—21no—
30C2×#14579118—21no—
31C22×C4⋊Dic79C2211yescharacteristic
32C22×Dic79⋊C4—21no—
33C22×C4×Dic79C2211yescharacteristic
34C2×C42×D79—21no—
35#19098857C22116yescomplemented
36#19098852C22116yescomplemented
37#19098847C22116yescomplemented
38#19098842C22116yescomplemented
39#19098606C2218yescomplemented
40C2×#14579094C2212yes—
41C2×#14579092C2212yes—
42#19098586C2218yescomplemented
43C2×Dic79⋊C8C2212yes—
44C2×Dic79⋊C8C2212yes—
45#19098565C2218yes—
46#19098562C2218yes—
47C2×#14579086C2212yes—
48C2×#14579084C2212yes—
49C2×#14579083C2212yes—
50C2×#14579081C2212yes—
51C22×M16×C79C2211yescharacteristic
52C2×#117×C79C2211yescharacteristic
53C22×C4⋊C4×C79C2211yescharacteristic
54#428×C79C2218yescomplemented
55C2×#107×C79C2212yes—
56C2×#106×C79C2212yes—
57C2×#11929624—22no—
58C2×#11929624—24no—
59C2×#11929624Q818yescomplemented
60C2×#11929624C2314yes—
61C22×Dic158—22no—
62C22×Dic158—22no—
63C22×Dic158—28no—
64C22×Dic158—48no—
65C22×Dic158C2311yescharacteristic
66C22×D316C2311yescharacteristic
67C22×C22⋊D79—21no—
68C23×Dic79—21no—
69C23×Dic79D411yescharacteristic
70C23×Dic79D411yescharacteristic
71C22×C4×D79—21no—
72C22×C4×D79—21no—
73C22×C4×D79—21no—
74C2×#11929608—24no—
75C2×#11929608C2314yes—
76C22×C79⋊C8C2312yes—
77#14579160—24no—
78#14579160—24no—
79#14579160—28no—
80#14579160—28no—
81#14579160C2318yes—
82#14579160C2318yes—
83#14579126—22no—
84#14579126—22no—
85#14579126C2314yes—
86#14579126C2314yes—
87#14579123—22no—
88#14579123—22no—
89#14579123C2314yes—
90#14579123C2314yes—
91(C22⋊D79)⋊C4—28no—
92(C22⋊D79)⋊C4—28no—
93#14579118—28no—
94C2×C4⋊Dic79—22no—
95C2×C4⋊Dic79—21no—
96C2×C4⋊Dic79C2312yes—
97C2×C4⋊Dic79C2314yes—
98C2×Dic79⋊C4—22no—
99C2×Dic79⋊C4—22no—
100C2×Dic79⋊C4—28no—
101C2×Dic79⋊C4—21no—
102C2×Dic79⋊C4—21no—
103C2×(C2×D79)⋊C4—21no—
104C2×(C2×D79)⋊C4—21no—
105C2×C4×Dic79—21no—
106C2×C4×Dic79—21no—
107C2×C4×Dic79D411yescharacteristic
108C2×C4×Dic79D411yescharacteristic
109C2×C4×Dic79D414yes—
110C2×C4×Dic79C2311yescharacteristic
111C2×C4×Dic79C2311yescharacteristic
112C2×C4×Dic79C2314yes—
113C42×D79—28no—
114#14579094C2318yes—
115#14579092C2318yes—
116Dic79⋊C8C2318yes—
117Dic79⋊C8C2318yes—
118#14579086C2318yes—
119#14579084C2318yes—
120#14579083C2318yes—
121#14579081C2318yes—
122C23×C4×C79—21no—
123C23×C4×C79C2311yescharacteristic
124C2×M16×C79—24no—
125C2×M16×C79C2314yes—
126C22×C8×C79C2312yes—
127#117×C79—22no—
128#117×C79C2314yes—
129C2×C4⋊C4×C79—22no—
130C2×C4⋊C4×C79—21no—
131C2×C4⋊C4×C79C2311yescharacteristic
132C2×C4⋊C4×C79C2312yes—
133C2×C4⋊C4×C79C2311yescharacteristic
134C2×C4⋊C4×C79C2314yes—
135C2×C22⋊C4×C79—21no—
136C2×C42×C79—21no—
137#107×C79C2318yes—
138#106×C79C2318yes—
139C23×D79—21no—
140#11929624—216no—
141#11929624—28no—
142#11929624C2×Q8116yescomplemented
143C2×Dic158—28no—
144C2×Dic158—28no—
145C2×Dic158—21no—
146C2×Dic158—22no—
147C2×Dic158—432no—
148C2×Dic158—48no—
149C2×Dic158—48no—
150C2×Dic158C2×Q814yescomplemented
151C2×Dic158C2412yes—
152C2×D316—21no—
153C2×D316—22no—
154C2×D316C2×Q814yescomplemented
155C2×D316C2412yes—
156C2×C22⋊D79—28no—
157C2×C22⋊D79—24no—
158C22×Dic79—24no—
159C22×Dic79—24no—
160C22×Dic79—21no—
161C22×Dic79—24no—
162C22×Dic79—22no—
163C22×Dic79—44no—
164C22×Dic79C2×D411yescharacteristic
165C22×Dic79C2×D414yes—
166C22×Dic79C2×D411yescharacteristic
167C22×Dic79C2×D414yes—
168C22×Dic79C2×D411yescharacteristic
169C22×Dic79C2×D411yescharacteristic
170C2×C4×D79—24no—
171C2×C4×D79—24no—
172C2×C4×D79—28no—
173C2×C4×D79—24no—
174C2×C4×D79—44no—
175C2×C4×D79—44no—
176#11929608—28no—
177C2×C79⋊C8—24no—
178C2×C79⋊C8C2414yes—
179C4⋊Dic79—24no—
180C4⋊Dic79—22no—
181C4⋊Dic79C2414yes—
182Dic79⋊C4—28no—
183Dic79⋊C4—28no—
184Dic79⋊C4—24no—
185Dic79⋊C4—24no—
186(C2×D79)⋊C4—24no—
187(C2×D79)⋊C4—24no—
188C4×Dic79—24no—
189C4×Dic79—24no—
190C4×Dic79C2×D414yes—
191C4×Dic79C2×D414yes—
192C4×Dic79C2×D412yes—
193C4×Dic79C2×D412yes—
194C4×Dic79C2412yes—
195C4×Dic79C2412yes—
196C24×C79C2×D411yescharacteristic
197C22×C4×C79—24no—
198C22×C4×C79—22no—
199C22×C4×C79—21no—
200C22×C4×C79—21no—
201C22×C4×C79—44no—
202C22×C4×C79C2×D412yes—
203C22×C4×C79C2×D411yescharacteristic
204C22×C4×C79C2×Q814yes—
205C22×C4×C79#4114yes—
206C22×C4×C79C2411yescharacteristic
207C22×C4×C79C2412yes—
208M16×C79—28no—
209C2×C8×C79—24no—
210C2×C8×C79C2414yes—
211C4⋊C4×C79—24no—
212C4⋊C4×C79—22no—
213C4⋊C4×C79—21no—
214C4⋊C4×C79—21no—
215C4⋊C4×C79C2412yes—
216C4⋊C4×C79C2414yes—
217C4⋊C4×C79C2412yes—
218C22⋊C4×C79—24no—
219C42×C79—24no—
220C23×C79—22no—
221C23×C79—22no—
222C23×C79C22⋊Q814yes—
223C23×C79C22×D412yes—
224C23×C79C22×D411yescharacteristic
225C22×D79—24no—
226C22×D79—22no—
227C22×D79—42no—
228C22×D79—42no—
229C2×C4×C79—22no—
230C2×C4×C79—22no—
231C2×C4×C79—24no—
232C2×C4×C79—48no—
233C2×C4×C79—42no—
234C2×C4×C79—44no—
235C2×C4×C79—42no—
236C2×C4×C79C22⋊Q814yes—
237C2×C4×C79C22⋊Q818yes—
238C2×C4×C79C22×D412yes—
239C2×C4×C79C22×D411yescharacteristic
240C2×C4×C79C22×Q812yes—
241C2×C4×C79C22×Q814yes—
242C2×C4×C79C2×#4112yes—
243C2×C4×C79C2×#4114yes—
244C2×C4×C79C2511yescharacteristic
245C22⋊D79—216no—
246C2×Dic79—24no—
247C2×Dic79—22no—
248C2×Dic79—22no—
249C2×Dic79—22no—
250C2×Dic79—28no—
251C2×Dic79—28no—
252C2×Dic79—24no—
253C2×Dic79—48no—
254C2×Dic79—44no—
255C2×Dic79AGL(1,ℤ/8)14yes—
256C2×Dic79#13714yes—
257C2×Dic79C22×D412yes—
258C2×Dic79C22×D412yes—
259C2×Dic79C22×D412yes—
260C2×Dic79C22×D412yes—
261D316—24no—
262D316—22no—
263D316C22×Q814yescomplemented
264C4×D79—216no—
265C4×D79—48no—
266C4×D79—48no—
267Dic158—24no—
268Dic158—22no—
269Dic158—416no—
270Dic158—416no—
271Dic158C22×Q814yescomplemented
272C8×C79—24no—
273C79⋊C8—24no—
274C22×C79—28no—
275C22×C79—22no—
276C22×C79—22no—
277C22×C79C2×C22⋊Q814yes—
278C22×C79C2×C22⋊Q812yes—
279C22×C79C2×AGL(1,ℤ/8)12yes—
280C22×C79C2×#13712yes—
281C22×C79C23×D411yescharacteristic
282C2×D79—24no—
283C2×D79—48no—
284C2×D79—42no—
285C2×D79—42no—
286C4×C79—44no—
287C4×C79—42no—
288C4×C79—42no—
289C4×C79C2×C22⋊Q812yes—
290C4×C79C2×C22⋊Q814yes—
291C4×C79C23×Q811yescharacteristic
292C4×C79C22×#4111yescharacteristic
293Dic79—24no—
294Dic79—22no—
295Dic79—22no—
296Dic79—44no—
297Dic79C2×AGL(1,ℤ/8)14yes—
298Dic79C2×#13714yes—
299C2×#3079—791nomaximal, Sylow
300C2×C79—24no—
301C2×C79#307914yescomplemented
302C2×C79C22×C22⋊Q811yescharacteristic
303C2×C79C22×AGL(1,ℤ/8)11yescharacteristic
304C2×C79C22×#13711yescharacteristic
305D79—44no—
306C22×C4⋊Q8—791no—
307C2×C4×#41—791no—
308#3079—7916no—
309C2×#477—792no—
310C2×#475—792no—
311C2×#476—792no—
312C2×#474—792no—
313C2×#428—791no—
314C2×#428D7911yescharacteristic, complemented
315C2×#423—791no—
316C2×#417—791no—
317C2×#417—791no—
318C79C2×#307911yescharacteristic, complemented, Sylow
319C22×#41—791no—
320C23×Q8—1581no—
321C22×M16—791no—
322C22×M16C2×D7911yescharacteristic
323C2×C4⋊Q8—792no—
324C2×C4⋊Q8—792no—
325C2×C4⋊Q8—798no—
326C2×C4⋊Q8—1581no—
327C2×C4⋊Q8—1584no—
328C2×C4⋊Q8—1581no—
329C4×#41—798no—
330C4×#41—1584no—
331C2×C4×Q8—791no—
332C2×C4×Q8—791no—
333C2×C4×D4—791no—
334C2×C4×D4—791no—
335C2×C4×D4—1581no—
336C2×C4×D4—1581no—
337C2×#117—1581no—
338C2×#117C2×D7911yescharacteristic
339C22×C4⋊C4—791no—
340C22×C4⋊C4—1581no—
341C22×C4⋊C4C2×D7911yescharacteristic
342C22×C42—791no—
343C22×C42—1581no—
344#477—7916no—
345#476—7916no—
346#475—7916no—
347#474—7916no—
348#428—798no—
349#428C2×D7918yescomplemented
350C2×#107—792no—
351C2×#107C2×D7912yes—
352C2×#106—792no—
353C2×#106C2×D7912yes—
354#423—798no—
355C2×C4⋊C8—792no—
356C2×C4⋊C8—792no—
357#417—798no—
358#417—798no—
359C2×#103—792no—
360C2×#103—792no—
361C2×#102—792no—
362C2×#102—792no—
363C2×#41—798no—
364C2×#41—794no—
365C2×#41—1582no—
366C2×#41—1584no—
367C22×Q8—3168no—
368C22×Q8—791no—
369C22×Q8—1588no—
370C22×Q8—1582no—
371C22×Q8—1582no—
372C22×D4—791no—
373C22×D4—1581no—
374C23×C4—21no—
375C23×C4—791no—
376C23×C4—791no—
377C23×C4—1581no—
378C23×C4—1581no—
379C23×C4—1581no—
380C23×C4—1581no—
381C23×C4C22×D7911yescharacteristic
382C2×M16—24no—
383C2×M16—794no—
384C2×M16—1584no—
385C2×M16C22×D7914yes—
386C22×C8—792no—
387C22×C8C22×D7912yes—
388C4⋊Q8—798no—
389C4⋊Q8—798no—
390C4⋊Q8—1588no—
391C4⋊Q8—1588no—
392C4⋊Q8—1584no—
393C4⋊Q8—1584no—
394C4×Q8—794no—
395C4×Q8—794no—
396C4×Q8—1582no—
397C4×Q8—1582no—
398C4×D4—794no—
399C4×D4—794no—
400C4×D4—1582no—
401C4×D4—1582no—
402C4×D4—1588no—
403C4×D4—1588no—
404#117—22no—
405#117—1588no—
406#117C22×D7914yes—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—792no—
410C2×C4⋊C4—794no—
411C2×C4⋊C4—1588no—
412C2×C4⋊C4—1582no—
413C2×C4⋊C4—1582no—
414C2×C4⋊C4—1581no—
415C2×C4⋊C4—1581no—
416C2×C4⋊C4—1582no—
417C2×C4⋊C4—1581no—
418C2×C4⋊C4C22×D7912yes—
419C2×C4⋊C4C22×D7914yes—
420C2×C4⋊C4C22×D7911yescharacteristic
421C2×C4⋊C4C22×D7911yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1581no—
424C2×C22⋊C4—1581no—
425C2×C42—21no—
426C2×C42—791no—
427C2×C42—794no—
428C2×C42—791no—
429C2×C42—791no—
430C2×C42—791no—
431C2×C42—794no—
432C2×C42—1588no—
433C2×C42—1581no—
434C2×C42—1581no—
435#107—798no—
436#107C22×D7918yes—
437#106—798no—
438#106C22×D7918yes—
439C4⋊C8—798no—
440C4⋊C8—798no—
441#103—798no—
442#103—798no—
443#102—798no—
444#102—798no—
445C24—1581no—
446C24D4×D7911yescharacteristic
447#41—7916no—
448#41—1588no—
449#41—15816no—
450C2×Q8—3168no—
451C2×Q8—31632no—
452C2×Q8—3168no—
453C2×Q8—794no—
454C2×Q8—792no—
455C2×Q8—1588no—
456C2×Q8—1588no—
457C2×Q8—1582no—
458C2×Q8—1581no—
459C2×D4—794no—
460C2×D4—792no—
461C2×D4—1588no—
462C2×D4—1584no—
463C2×D4—1582no—
464C2×D4—1581no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—44no—
470C22×C4—3164no—
471C22×C4—3164no—
472C22×C4—3164no—
473C22×C4—791no—
474C22×C4—791no—
475C22×C4—794no—
476C22×C4—794no—
477C22×C4—791no—
478C22×C4—791no—
479C22×C4—1584no—
480C22×C4—1584no—
481C22×C4—1584no—
482C22×C4—1582no—
483C22×C4—1584no—
484C22×C4—1584no—
485C22×C4—1581no—
486C22×C4—1588no—
487C22×C4—1584no—
488C22×C4D4×D7912yes—
489C22×C4D4×D7911yescharacteristic
490C22×C4Q8×D7914yes—
491C22×C4#1192962414yes—
492C22×C4C23×D7912yes—
493C22×C4C23×D7911yescharacteristic
494M16—28no—
495M16—1588no—
496C2×C8—24no—
497C2×C8—794no—
498C2×C8—1584no—
499C2×C8C23×D7914yes—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—794no—
505C4⋊C4—1588no—
506C4⋊C4—1588no—
507C4⋊C4—1584no—
508C4⋊C4—1584no—
509C4⋊C4—1584no—
510C4⋊C4—1582no—
511C4⋊C4C23×D7914yes—
512C4⋊C4C23×D7912yes—
513C4⋊C4C23×D7912yes—
514C22⋊C4—24no—
515C22⋊C4—1584no—
516C22⋊C4—1584no—
517C42—24no—
518C42—792no—
519C42—792no—
520C42—794no—
521C42—794no—
522C42—792no—
523C42—792no—
524C42—1584no—
525C42—1584no—
526C23—22no—
527C23—22no—
528C23—3162no—
529C23—3162no—
530C23—1584no—
531C23—1582no—
532C23#1457913914yes—
533C23C2×D4×D7912yes—
534C23C2×D4×D7911yescharacteristic
535Q8—31616no—
536Q8—31616no—
537Q8—794no—
538Q8—1584no—
539Q8—1582no—
540D4—794no—
541D4—15816no—
542D4—1584no—
543D4—1582no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—48no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—3168no—
552C2×C4—3168no—
553C2×C4—3168no—
554C2×C4—3164no—
555C2×C4—794no—
556C2×C4—794no—
557C2×C4—792no—
558C2×C4—792no—
559C2×C4—792no—
560C2×C4—792no—
561C2×C4—1584no—
562C2×C4—1588no—
563C2×C4—1588no—
564C2×C4—1584no—
565C2×C4—1582no—
566C2×C4—1582no—
567C2×C4—1582no—
568C2×C4—15816no—
569C2×C4#1457913914yes—
570C2×C4#1457913918yes—
571C2×C4C2×D4×D7912yes—
572C2×C4C2×D4×D7911yescharacteristic
573C2×C4C2×Q8×D7912yes—
574C2×C4C2×Q8×D7914yes—
575C2×C4C2×#1192962412yes—
576C2×C4C2×#1192962414yes—
577C2×C4C24×D7911yescharacteristic
578C8—24no—
579C8—1584no—
580C22—28no—
581C22—22no—
582C22—22no—
583C22—3168no—
584C22—3162no—
585C22—3162no—
586C22—1584no—
587C22C2×#1457913914yes—
588C22C2×#1457913912yes—
589C22AGL(1,ℤ/8)×D7912yes—
590C22#137×D7912yes—
591C22C22×D4×D7911yescharacteristic
592C4—44no—
593C4—42no—
594C4—42no—
595C4—3164no—
596C4—794no—
597C4—794no—
598C4—1582no—
599C4—1582no—
600C4—1584no—
601C4C2×#1457913914yes—
602C4C2×#1457913912yes—
603C4C22×Q8×D7911yescharacteristic
604C4C22×#1192962411yescharacteristic
605C2—24no—
606C2—3164no—
607C2#2377321314yesdirect product factor
608C2C22×#1457913911yescharacteristic
609C2C2×AGL(1,ℤ/8)×D7911yescharacteristic
610C2C2×#137×D7911yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×#428×C79C2
socleC23×C79C22×D4
prime radicalC22×#11929624C22
hypercenterC2×#428D79
hypocenterC79C2×#3079
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
1. C22×#14579160C2semidirect
3. #23773213C2direct (N × H)
4. C2×#19098857C2semidirect
5. C2×#19098847C2semidirect
6. C2×#19098852C2semidirect
7. C2×#19098842C2semidirect
8. C2×#19098606C2semidirect
9. C2×#19098586C2semidirect
12. C2×#428×C79C2semidirect
21. C2×#14579160C22semidirect
35. #19098857C22semidirect
36. #19098852C22semidirect
37. #19098847C22semidirect
38. #19098842C22semidirect
39. #19098606C22semidirect
42. #19098586C22semidirect
54. #428×C79C22semidirect
59. C2×#11929624Q8semidirect
142. #11929624C2×Q8semidirect
150. C2×Dic158C2×Q8semidirect
154. C2×D316C2×Q8semidirect
263. D316C22×Q8semidirect
271. Dic158C22×Q8semidirect
301. C2×C79#3079semidirect
314. C2×#428D79semidirect
318. C79C2×#3079semidirect
349. #428C2×D79semidirect
607. C2#23773213direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145791602
C2×#190990312
#237732132
C2×#190988572
C2×#190988472
C2×#190988522
C2×#190988422
C2×#190986062
C2×#190985862
C2×#190985652
C2×#190985622
C2×#428×C792
C2×#307979