Finite Group Databasebeta

← C2×#23773221 #35155807

Subgroups — C2×#23773221 #35155807

Order 20224.

2142 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579152C211yesmaximal, characteristic, complemented
2C2×#19099031C211yesmaximal, characteristic
3#23773221C2116yesmaximal, direct product factor
4C2×#19098873C212yesmaximal, complemented
5C2×#19098869C212yesmaximal, complemented
6C2×#19098866C212yesmaximal, complemented
7C2×#19098862C212yesmaximal, complemented
8C2×#19098606C211yesmaximal, characteristic, complemented
9C2×#19098586C211yesmaximal, characteristic, complemented
10C2×#19098565C211yesmaximal, characteristic
11C2×#19098562C211yesmaximal, characteristic
12C2×#428×C79C211yesmaximal, characteristic, complemented
13C22×#11929624C2211yescharacteristic
14C22×#11929608C2211yescharacteristic
15C2×#14579152—21no—
16C2×#14579152—21no—
17C2×#14579152—24no—
18C2×#14579152C2212yes—
19C2×#14579152C2212yes—
20C2×#14579152C2218yescomplemented
21#19099031—24no—
22#19099031C2218yes—
23C2×#14579126C2211yescharacteristic
24C2×#14579126C2211yescharacteristic
25C2×#14579123C2211yescharacteristic
26C2×#14579123C2211yescharacteristic
27C2×(C22⋊D79)⋊C4—21no—
28C2×(C22⋊D79)⋊C4—21no—
29C2×#14579118—21no—
30C22×C4⋊Dic79—21no—
31C22×C4⋊Dic79C2211yescharacteristic
32C22×Dic79⋊C4—21no—
33C22×C4×Dic79C2211yescharacteristic
34C2×C42×D79—21no—
35#19098873C22116yescomplemented
36#19098869C22116yescomplemented
37#19098866C22116yescomplemented
38#19098862C22116yescomplemented
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×#11929624Q818yes—
60C2×#11929624C2314yes—
61C22×Dic158C2311yescharacteristic
62C22×D316C2311yescharacteristic
63C22×C22⋊D79—21no—
64C23×Dic79—21no—
65C23×Dic79D411yescharacteristic
66C23×Dic79D411yescharacteristic
67C22×C4×D79—21no—
68C22×C4×D79—21no—
69C22×C4×D79—21no—
70C2×#11929608—24no—
71C2×#11929608C2314yes—
72C22×C79⋊C8C2312yes—
73#14579152—24no—
74#14579152—24no—
75#14579152—28no—
76#14579152—28no—
77#14579152C2318yes—
78#14579152C2318yes—
79#14579126—22no—
80#14579126—22no—
81#14579126C2314yes—
82#14579126C2314yes—
83#14579123—22no—
84#14579123—22no—
85#14579123C2314yes—
86#14579123C2314yes—
87(C22⋊D79)⋊C4—28no—
88(C22⋊D79)⋊C4—28no—
89#14579118—28no—
90C2×C4⋊Dic79—22no—
91C2×C4⋊Dic79—21no—
92C2×C4⋊Dic79—22no—
93C2×C4⋊Dic79—22no—
94C2×C4⋊Dic79—28no—
95C2×C4⋊Dic79C2312yes—
96C2×C4⋊Dic79C2314yes—
97C2×Dic79⋊C4—22no—
98C2×Dic79⋊C4—22no—
99C2×Dic79⋊C4—28no—
100C2×Dic79⋊C4—21no—
101C2×Dic79⋊C4—21no—
102C2×(C2×D79)⋊C4—21no—
103C2×(C2×D79)⋊C4—21no—
104C2×C4×Dic79—21no—
105C2×C4×Dic79—21no—
106C2×C4×Dic79D411yescharacteristic
107C2×C4×Dic79D411yescharacteristic
108C2×C4×Dic79D414yes—
109C2×C4×Dic79C2311yescharacteristic
110C2×C4×Dic79C2311yescharacteristic
111C2×C4×Dic79C2314yes—
112C42×D79—28no—
113#14579094C2318yes—
114#14579092C2318yes—
115Dic79⋊C8C2318yes—
116Dic79⋊C8C2318yes—
117#14579086C2318yes—
118#14579084C2318yes—
119#14579083C2318yes—
120#14579081C2318yes—
121C23×C4×C79—21no—
122C23×C4×C79C2311yescharacteristic
123C2×M16×C79—24no—
124C2×M16×C79C2314yes—
125C22×C8×C79C2312yes—
126#117×C79—22no—
127#117×C79C2314yes—
128C2×C4⋊C4×C79—22no—
129C2×C4⋊C4×C79—21no—
130C2×C4⋊C4×C79C2311yescharacteristic
131C2×C4⋊C4×C79C2312yes—
132C2×C4⋊C4×C79C2311yescharacteristic
133C2×C4⋊C4×C79C2314yes—
134C2×C22⋊C4×C79—21no—
135C2×C42×C79—21no—
136#107×C79C2318yes—
137#106×C79C2318yes—
138C23×D79—21no—
139#11929624—216no—
140#11929624—28no—
141#11929624C2×Q8116yes—
142C2×Dic158—21no—
143C2×Dic158—22no—
144C2×Dic158C2×Q814yes—
145C2×Dic158C2412yes—
146C2×D316—21no—
147C2×D316—22no—
148C2×D316C2×Q814yes—
149C2×D316C2412yes—
150C2×C22⋊D79—28no—
151C2×C22⋊D79—24no—
152C22×Dic79—21no—
153C22×Dic79—24no—
154C22×Dic79—22no—
155C22×Dic79—42no—
156C22×Dic79—42no—
157C22×Dic79—44no—
158C22×Dic79C2×D411yescharacteristic
159C22×Dic79C2×D411yescharacteristic
160C22×Dic79C2×D411yescharacteristic
161C22×Dic79C2×D411yescharacteristic
162C22×Dic79C2×D414yes—
163C22×Dic79C2×D414yes—
164C2×C4×D79—24no—
165C2×C4×D79—24no—
166C2×C4×D79—28no—
167C2×C4×D79—24no—
168C2×C4×D79—44no—
169C2×C4×D79—44no—
170#11929608—28no—
171C2×C79⋊C8—24no—
172C2×C79⋊C8C2414yes—
173C4⋊Dic79—28no—
174C4⋊Dic79—28no—
175C4⋊Dic79—24no—
176C4⋊Dic79—22no—
177C4⋊Dic79C2414yes—
178Dic79⋊C4—28no—
179Dic79⋊C4—28no—
180Dic79⋊C4—24no—
181Dic79⋊C4—24no—
182(C2×D79)⋊C4—24no—
183(C2×D79)⋊C4—24no—
184C4×Dic79—24no—
185C4×Dic79—24no—
186C4×Dic79C2×D414yes—
187C4×Dic79C2×D414yes—
188C4×Dic79C2×D412yes—
189C4×Dic79C2×D412yes—
190C4×Dic79C2412yes—
191C4×Dic79C2412yes—
192C24×C79C2×D411yescharacteristic
193C22×C4×C79—21no—
194C22×C4×C79—21no—
195C22×C4×C79—24no—
196C22×C4×C79—22no—
197C22×C4×C79—44no—
198C22×C4×C79C2×D412yes—
199C22×C4×C79C2×D411yescharacteristic
200C22×C4×C79C2×Q814yes—
201C22×C4×C79#4114yes—
202C22×C4×C79C2411yescharacteristic
203C22×C4×C79C2412yes—
204M16×C79—28no—
205C2×C8×C79—24no—
206C2×C8×C79C2414yes—
207C4⋊C4×C79—24no—
208C4⋊C4×C79—22no—
209C4⋊C4×C79—21no—
210C4⋊C4×C79—21no—
211C4⋊C4×C79C2412yes—
212C4⋊C4×C79C2414yes—
213C4⋊C4×C79C2412yes—
214C22⋊C4×C79—24no—
215C42×C79—24no—
216C23×C79—22no—
217C23×C79—22no—
218C23×C79C22⋊Q814yes—
219C23×C79C22×D412yes—
220C23×C79C22×D411yescharacteristic
221C22×D79—24no—
222C22×D79—22no—
223C22×D79—42no—
224C22×D79—42no—
225C2×C4×C79—22no—
226C2×C4×C79—22no—
227C2×C4×C79—24no—
228C2×C4×C79—48no—
229C2×C4×C79—42no—
230C2×C4×C79—42no—
231C2×C4×C79—44no—
232C2×C4×C79C22⋊Q814yes—
233C2×C4×C79C22⋊Q818yes—
234C2×C4×C79C22×D412yes—
235C2×C4×C79C22×D411yescharacteristic
236C2×C4×C79C22×Q812yes—
237C2×C4×C79C22×Q814yes—
238C2×C4×C79C2×#4112yes—
239C2×C4×C79C2×#4114yes—
240C2×C4×C79C2511yescharacteristic
241C22⋊D79—216no—
242C2×Dic79—24no—
243C2×Dic79—22no—
244C2×Dic79—22no—
245C2×Dic79—22no—
246C2×Dic79—22no—
247C2×Dic79—22no—
248C2×Dic79—24no—
249C2×Dic79—44no—
250C2×Dic79—44no—
251C2×Dic79—48no—
252C2×Dic79—44no—
253C2×Dic79C22×D412yes—
254C2×Dic79C22×D412yes—
255C2×Dic79C22×D412yes—
256C2×Dic79C22×D412yes—
257D316—24no—
258D316—22no—
259D316C22×Q814yes—
260C4×D79—216no—
261C4×D79—48no—
262C4×D79—48no—
263Dic158—24no—
264Dic158—22no—
265Dic158C22×Q814yes—
266C8×C79—24no—
267C79⋊C8—24no—
268C22×C79—28no—
269C22×C79—22no—
270C22×C79—22no—
271C22×C79C2×C22⋊Q814yes—
272C22×C79C2×C22⋊Q812yes—
273C22×C79#57512yes—
274C22×C79#57512yes—
275C22×C79C23×D411yescharacteristic
276C2×D79—24no—
277C2×D79—48no—
278C2×D79—42no—
279C2×D79—42no—
280C4×C79—44no—
281C4×C79—42no—
282C4×C79—42no—
283C4×C79C2×C22⋊Q812yes—
284C4×C79C2×C22⋊Q814yes—
285C4×C79C23×Q811yescharacteristic
286C4×C79C22×#4111yescharacteristic
287Dic79—24no—
288Dic79—22no—
289Dic79—22no—
290Dic79—22no—
291Dic79—22no—
292Dic79—44no—
293C2×#3080—791nomaximal, Sylow
294C2×C79—24no—
295C2×C79#308014yescomplemented
296C2×C79C22×C22⋊Q811yescharacteristic
297C2×C79C2×#57511yescharacteristic
298C2×C79C2×#57511yescharacteristic
299D79—44no—
300C22×#125—791no—
301C2×C4×#41—791no—
302#3080—7916no—
303C2×#479—792no—
304C2×#479—792no—
305C2×#478—792no—
306C2×#478—792no—
307C2×#428—791no—
308C2×#428D7911yescharacteristic, complemented
309C2×#423—791no—
310C2×#417—791no—
311C2×#417—791no—
312C79C2×#308011yescharacteristic, complemented, Sylow
313C22×#41—791no—
314C22×M16—791no—
315C22×M16C2×D7911yescharacteristic
316C2×#125—792no—
317C2×#125—792no—
318C2×#125—798no—
319C2×#125—1581no—
320C2×#125—1584no—
321C2×#125—1581no—
322C4×#41—798no—
323C4×#41—1584no—
324C2×C4×Q8—791no—
325C2×C4×Q8—791no—
326C2×C4×D4—791no—
327C2×C4×D4—791no—
328C2×C4×D4—1581no—
329C2×C4×D4—1581no—
330C2×#117—1581no—
331C2×#117C2×D7911yescharacteristic
332C22×C4⋊C4—791no—
333C22×C4⋊C4—1581no—
334C22×C4⋊C4—1581no—
335C22×C4⋊C4C2×D7911yescharacteristic
336C22×C42—791no—
337C22×C42—1581no—
338#479—7916no—
339#479—7916no—
340#478—7916no—
341#478—7916no—
342#428—798no—
343#428C2×D7918yescomplemented
344C2×#107—792no—
345C2×#107C2×D7912yes—
346C2×#106—792no—
347C2×#106C2×D7912yes—
348#423—798no—
349C2×C4⋊C8—792no—
350C2×C4⋊C8—792no—
351#417—798no—
352#417—798no—
353C2×#103—792no—
354C2×#103—792no—
355C2×#102—792no—
356C2×#102—792no—
357C2×#41—798no—
358C2×#41—794no—
359C2×#41—1582no—
360C2×#41—1584no—
361C22×Q8—791no—
362C22×D4—791no—
363C22×D4—1581no—
364C23×C4—21no—
365C23×C4—791no—
366C23×C4—791no—
367C23×C4—1581no—
368C23×C4—1581no—
369C23×C4—1581no—
370C23×C4—1581no—
371C23×C4C22×D7911yescharacteristic
372C2×M16—24no—
373C2×M16—794no—
374C2×M16—1584no—
375C2×M16C22×D7914yes—
376C22×C8—792no—
377C22×C8C22×D7912yes—
378#125—798no—
379#125—798no—
380#125—1588no—
381#125—1588no—
382#125—1584no—
383#125—1584no—
384C4×Q8—794no—
385C4×Q8—794no—
386C4×Q8—1582no—
387C4×Q8—1582no—
388C4×D4—794no—
389C4×D4—794no—
390C4×D4—1582no—
391C4×D4—1582no—
392C4×D4—1588no—
393C4×D4—1588no—
394#117—22no—
395#117—1588no—
396#117C22×D7914yes—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—792no—
400C2×C4⋊C4—794no—
401C2×C4⋊C4—1588no—
402C2×C4⋊C4—1582no—
403C2×C4⋊C4—1582no—
404C2×C4⋊C4—1588no—
405C2×C4⋊C4—1582no—
406C2×C4⋊C4—1582no—
407C2×C4⋊C4—1581no—
408C2×C4⋊C4—1581no—
409C2×C4⋊C4—1582no—
410C2×C4⋊C4—1581no—
411C2×C4⋊C4C22×D7912yes—
412C2×C4⋊C4C22×D7914yes—
413C2×C4⋊C4C22×D7911yescharacteristic
414C2×C4⋊C4C22×D7911yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—1581no—
417C2×C22⋊C4—1581no—
418C2×C42—21no—
419C2×C42—791no—
420C2×C42—794no—
421C2×C42—791no—
422C2×C42—791no—
423C2×C42—791no—
424C2×C42—794no—
425C2×C42—1588no—
426C2×C42—1581no—
427C2×C42—1581no—
428#107—798no—
429#107C22×D7918yes—
430#106—798no—
431#106C22×D7918yes—
432C4⋊C8—798no—
433C4⋊C8—798no—
434#103—798no—
435#103—798no—
436#102—798no—
437#102—798no—
438C24—1581no—
439C24D4×D7911yescharacteristic
440#41—7916no—
441#41—1588no—
442#41—15816no—
443C2×Q8—794no—
444C2×Q8—792no—
445C2×Q8—1582no—
446C2×Q8—1581no—
447C2×D4—794no—
448C2×D4—792no—
449C2×D4—1588no—
450C2×D4—1584no—
451C2×D4—1582no—
452C2×D4—1581no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—44no—
458C22×C4—3164no—
459C22×C4—3164no—
460C22×C4—3164no—
461C22×C4—3162no—
462C22×C4—3162no—
463C22×C4—794no—
464C22×C4—794no—
465C22×C4—791no—
466C22×C4—791no—
467C22×C4—791no—
468C22×C4—791no—
469C22×C4—1584no—
470C22×C4—1584no—
471C22×C4—1584no—
472C22×C4—1582no—
473C22×C4—1581no—
474C22×C4—1588no—
475C22×C4—1584no—
476C22×C4D4×D7912yes—
477C22×C4D4×D7911yescharacteristic
478C22×C4Q8×D7914yes—
479C22×C4#1192962414yes—
480C22×C4C23×D7912yes—
481C22×C4C23×D7911yescharacteristic
482M16—28no—
483M16—1588no—
484C2×C8—24no—
485C2×C8—794no—
486C2×C8—1584no—
487C2×C8C23×D7914yes—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—794no—
493C4⋊C4—1588no—
494C4⋊C4—1588no—
495C4⋊C4—1584no—
496C4⋊C4—1584no—
497C4⋊C4—1588no—
498C4⋊C4—1588no—
499C4⋊C4—1584no—
500C4⋊C4—1582no—
501C4⋊C4C23×D7914yes—
502C4⋊C4C23×D7912yes—
503C4⋊C4C23×D7912yes—
504C22⋊C4—24no—
505C22⋊C4—1584no—
506C22⋊C4—1584no—
507C42—24no—
508C42—792no—
509C42—792no—
510C42—794no—
511C42—794no—
512C42—792no—
513C42—792no—
514C42—1584no—
515C42—1584no—
516C23—22no—
517C23—22no—
518C23—3162no—
519C23—3162no—
520C23—1584no—
521C23—1582no—
522C23#1457913914yes—
523C23C2×D4×D7912yes—
524C23C2×D4×D7911yescharacteristic
525Q8—794no—
526Q8—1584no—
527Q8—1582no—
528D4—794no—
529D4—15816no—
530D4—1584no—
531D4—1582no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—48no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—3168no—
540C2×C4—3168no—
541C2×C4—3168no—
542C2×C4—3164no—
543C2×C4—3164no—
544C2×C4—3164no—
545C2×C4—792no—
546C2×C4—792no—
547C2×C4—792no—
548C2×C4—792no—
549C2×C4—1584no—
550C2×C4—1584no—
551C2×C4—1582no—
552C2×C4—1582no—
553C2×C4—1582no—
554C2×C4—1582no—
555C2×C4—1582no—
556C2×C4—15816no—
557C2×C4#1457913914yes—
558C2×C4#1457913918yes—
559C2×C4C2×D4×D7912yes—
560C2×C4C2×D4×D7911yescharacteristic
561C2×C4C2×Q8×D7912yes—
562C2×C4C2×Q8×D7914yes—
563C2×C4C2×#1192962412yes—
564C2×C4C2×#1192962414yes—
565C2×C4C24×D7911yescharacteristic
566C8—24no—
567C8—1584no—
568C22—28no—
569C22—22no—
570C22—22no—
571C22—3168no—
572C22—3162no—
573C22—3162no—
574C22—1584no—
575C22C2×#1457913914yes—
576C22C2×#1457913912yes—
577C22#1909935912yes—
578C22#1909936212yes—
579C22C22×D4×D7911yescharacteristic
580C4—44no—
581C4—42no—
582C4—42no—
583C4—3164no—
584C4—1582no—
585C4—1582no—
586C4—1582no—
587C4—1582no—
588C4—1584no—
589C4C2×#1457913914yes—
590C4C2×#1457913912yes—
591C4C22×Q8×D7911yescharacteristic
592C4C22×#1192962411yescharacteristic
593C2—24no—
594C2—3164no—
595C2#2377322114yesdirect product factor
596C2C22×#1457913911yescharacteristic
597C2C2×#1909935911yescharacteristic
598C2C2×#1909936211yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×#428×C79C2
socleC23×C79C22×D4
prime radicalC22×#11929624C22
hypercenterC2×#428D79
hypocenterC79C2×#3080
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×#14579152C2semidirect
3. #23773221C2direct (N × H)
4. C2×#19098873C2semidirect
5. C2×#19098869C2semidirect
6. C2×#19098866C2semidirect
7. C2×#19098862C2semidirect
8. C2×#19098606C2semidirect
9. C2×#19098586C2semidirect
12. C2×#428×C79C2semidirect
20. C2×#14579152C22semidirect
35. #19098873C22semidirect
36. #19098869C22semidirect
37. #19098866C22semidirect
38. #19098862C22semidirect
39. #19098606C22semidirect
42. #19098586C22semidirect
54. #428×C79C22semidirect
295. C2×C79#3080semidirect
308. C2×#428D79semidirect
312. C79C2×#3080semidirect
343. #428C2×D79semidirect
595. C2#23773221direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145791522
C2×#190990312
#237732212
C2×#190988732
C2×#190988692
C2×#190988662
C2×#190988622
C2×#190986062
C2×#190985862
C2×#190985652
C2×#190985622
C2×#428×C792
C2×#308079