Finite Group Databasebeta

← C2×#23774347 #35159459

Subgroups — C2×#23774347 #35159459

Order 20224.

2134 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099117C211yesmaximal, characteristic
2C2×C4⋊Q8×D79C211yesmaximal, characteristic
3#23774347C2116yesmaximal, direct product factor
4C2×#19098941C212yesmaximal, complemented
5C2×#19098912C212yesmaximal, complemented
6C2×#19098774C211yesmaximal, characteristic, complemented
7C2×#19098773C211yesmaximal, characteristic, complemented
8C2×#19098770C211yesmaximal, characteristic, complemented
9C2×#19098720C212yesmaximal
10C2×#19098717C212yesmaximal
11C2×#19098615C211yesmaximal, characteristic
12C2×#462×C79C211yesmaximal, characteristic, complemented
13C22×Q8×D79C2212yes—
14C22×Dic316C2212yes—
15C22×#11929614C2212yes—
16C22×#11929614C2212yes—
17#19099117—24no—
18#19099117C2218yes—
19C2×#14579161C2211yescharacteristic
20C2×#14579160—21no—
21C2×#14579160—21no—
22C2×#14579160C2211yescharacteristic
23C2×#14579159C2212yes—
24C2×#14579159C2211yescharacteristic
25C4⋊Q8×D79—24no—
26C4⋊Q8×D79C2218yes—
27C2×#14579151—21no—
28C2×#14579141C2211yescharacteristic
29C2×#14579139—21no—
30C2×#14579137C2211yescharacteristic
31C2×C4×Dic158—21no—
32C2×C4×Dic158C2211yescharacteristic
33C2×#14579118—21no—
34C2×C4⋊C4×D79—21no—
35C2×C4⋊C4×D79—21no—
36C2×C42×D79C2211yescharacteristic
37#19098941C22116yescomplemented
38#19098912C22116yescomplemented
39#19098774C2218yescomplemented
40#19098773C2218yescomplemented
41#19098770C2218yescomplemented
42#19098720C22116yes—
43#19098717C22116yes—
44#19098615C2218yes—
45C2×C4⋊(C79⋊C8)C2211yescharacteristic
46C2×#14579086C2212yes—
47C2×#14579085C2212yes—
48C2×#14579084C2212yes—
49C2×(C2×D79)⋊C8C2212yes—
50C2×C8×Dic79C2212yes—
51C22×Q16×C79C2212yes—
52C2×C4⋊Q8×C79C2211yescharacteristic
53C2×C4×Q8×C79C2211yescharacteristic
54#462×C79C2218yescomplemented
55C2×C4⋊C8×C79C2211yescharacteristic
56C2×#103×C79C2212yes—
57C22×Dic158—21no—
58C22×Dic158—22no—
59C22×Dic158C2311yescharacteristic
60C22×Dic158C2312yes—
61C2×Q8×D79—24no—
62C2×Q8×D79—44no—
63C2×Q8×D79—44no—
64C2×Q8×D79C2318yes—
65C22×C4×D79—22no—
66C22×C4×D79D411yescharacteristic
67C22×C4×D79D411yescharacteristic
68C22×C4×D79C2311yescharacteristic
69C2×Dic316—44no—
70C2×Dic316C2318yes—
71C2×#11929614—44no—
72C2×#11929614—44no—
73C2×#11929614C2318yes—
74C2×#11929614C2318yes—
75C22×C79⋊C8C2312yes—
76#14579161—22no—
77#14579161C2314yes—
78#14579160—28no—
79#14579160—28no—
80#14579160—22no—
81#14579160C2314yes—
82#14579159—24no—
83#14579159—22no—
84#14579159C2318yes—
85#14579159C2314yes—
86#14579151—28no—
87#14579141—22no—
88#14579141C2314yes—
89#14579139—28no—
90#14579137—22no—
91#14579137C2314yes—
92C4×Dic158—22no—
93C4×Dic158—28no—
94C4×Dic158C2314yes—
95#14579118—28no—
96C2×C4⋊Dic79—21no—
97C2×C4⋊Dic79—21no—
98C2×C4⋊Dic79—21no—
99C2×C4⋊Dic79C2311yescharacteristic
100C2×Dic79⋊C4—21no—
101C2×Dic79⋊C4—22no—
102C2×Dic79⋊C4—22no—
103C2×Dic79⋊C4—21no—
104C2×Dic79⋊C4—21no—
105C2×Dic79⋊C4—21no—
106C2×Dic79⋊C4—21no—
107C4⋊C4×D79—28no—
108C4⋊C4×D79—28no—
109C2×(C2×D79)⋊C4—21no—
110C2×(C2×D79)⋊C4—21no—
111C2×C4×Dic79—21no—
112C2×C4×Dic79D411yescharacteristic
113C2×C4×Dic79D411yescharacteristic
114C2×C4×Dic79C2311yescharacteristic
115C42×D79D414yes—
116C42×D79C2314yes—
117C4⋊(C79⋊C8)C2314yes—
118#14579086C2318yes—
119#14579085C2318yes—
120#14579084C2318yes—
121(C2×D79)⋊C8C2318yes—
122C8×Dic79C2318yes—
123C22×Q8×C79C2311yescharacteristic
124C22×Q8×C79C2312yes—
125C2×Q16×C79—44no—
126C2×Q16×C79C2318yes—
127C22×C8×C79C2312yes—
128C4⋊Q8×C79—22no—
129C4⋊Q8×C79C2314yes—
130C4×Q8×C79—22no—
131C4×Q8×C79C2314yes—
132C2×C4⋊C4×C79—21no—
133C2×C4⋊C4×C79—21no—
134C2×C4⋊C4×C79—21no—
135C2×C4⋊C4×C79C2311yescharacteristic
136C2×C42×C79—21no—
137C2×C42×C79C2311yescharacteristic
138C4⋊C8×C79C2314yes—
139#103×C79C2318yes—
140C23×D79C2×D411yescharacteristic
141C2×Dic158—28no—
142C2×Dic158—24no—
143C2×Dic158—22no—
144C2×Dic158—21no—
145C2×Dic158—24no—
146C2×Dic158—24no—
147C2×Dic158—22no—
148C2×Dic158—22no—
149C2×Dic158—48no—
150C2×Dic158—44no—
151C2×Dic158C2414yes—
152C2×Dic158C2412yes—
153Q8×D79—48no—
154Q8×D79—48no—
155Q8×D79—416no—
156C22×Dic79—22no—
157C22×Dic79—21no—
158C22×Dic79—21no—
159C22×Dic79C2×D411yescharacteristic
160C22×Dic79C2×D411yescharacteristic
161C22×Dic79C2×D411yescharacteristic
162C2×C4×D79—24no—
163C2×C4×D79—28no—
164C2×C4×D79—22no—
165C2×C4×D79—22no—
166C2×C4×D79—48no—
167C2×C4×D79Q1618yescomplemented
168C2×C4×D79C2×D412yes—
169C2×C4×D79C2×D414yes—
170C2×C4×D79C2×D414yes—
171C2×C4×D79C2412yes—
172Dic316—48no—
173#11929614—48no—
174#11929614—48no—
175C2×C79⋊C8—24no—
176C2×C79⋊C8C2414yes—
177C4⋊Dic79—24no—
178C4⋊Dic79—24no—
179C4⋊Dic79—24no—
180C4⋊Dic79—21no—
181C4⋊Dic79C2412yes—
182Dic79⋊C4—28no—
183Dic79⋊C4—28no—
184Dic79⋊C4—24no—
185Dic79⋊C4—24no—
186Dic79⋊C4—24no—
187Dic79⋊C4—24no—
188Dic79⋊C4—24no—
189(C2×D79)⋊C4—24no—
190(C2×D79)⋊C4—24no—
191C4×Dic79—24no—
192C4×Dic79C2×D412yes—
193C4×Dic79C2×D414yes—
194C4×Dic79C2×D414yes—
195C4×Dic79C2412yes—
196C2×Q8×C79—22no—
197C2×Q8×C79—21no—
198C2×Q8×C79—24no—
199C2×Q8×C79—24no—
200C2×Q8×C79—22no—
201C2×Q8×C79—22no—
202C2×Q8×C79C2414yes—
203C2×Q8×C79C2412yes—
204C22×C4×C79—21no—
205C22×C4×C79—22no—
206C22×C4×C79—21no—
207C22×C4×C79C2×D411yescharacteristic
208C22×C4×C79C2×D411yescharacteristic
209C22×C4×C79C2411yescharacteristic
210Q16×C79—48no—
211C2×C8×C79—24no—
212C2×C8×C79C2414yes—
213C4⋊C4×C79—24no—
214C4⋊C4×C79—24no—
215C4⋊C4×C79—24no—
216C4⋊C4×C79—21no—
217C4⋊C4×C79C2412yes—
218C42×C79—24no—
219C42×C79C2×D412yes—
220C42×C79C2412yes—
221C23×C79C22×D411yescharacteristic
222C22×D79—22no—
223C22×D79—22no—
224C22×D79C2×Q1614yes—
225C22×D79C22×D412yes—
226Q8×C79—22no—
227Q8×C79—24no—
228Q8×C79—24no—
229Q8×C79—22no—
230Q8×C79—44no—
231Q8×C79—42no—
232C2×C4×C79—22no—
233C2×C4×C79—22no—
234C2×C4×C79—22no—
235C2×C4×C79—24no—
236C2×C4×C79—44no—
237C2×C4×C79—42no—
238C2×C4×C79—42no—
239C2×C4×C79C2×Q1614yes—
240C2×C4×C79C22×D412yes—
241C2×C4×C79C22×D411yescharacteristic
242C2×C4×C79C22×D412yes—
243C2×C4×C79H(5,𝔽2)12yes—
244C2×C4×C79H(5,𝔽2)12yes—
245C2×C4×C79C2511yescharacteristic
246C2×Dic79—22no—
247C2×Dic79—22no—
248C2×Dic79—22no—
249C2×Dic79—24no—
250C2×Dic79—44no—
251C2×Dic79—42no—
252C2×Dic79—42no—
253C2×Dic79C2×Q1614yes—
254C2×Dic79#13714yes—
255C2×Dic79C22×D412yes—
256C2×Dic79C22×D412yes—
257C2×Dic79C22×D412yes—
258C4×D79—24no—
259C4×D79—22no—
260C4×D79—22no—
261C4×D79—28no—
262C4×D79—416no—
263C4×D79C2×Q16116yescomplemented
264Dic158—22no—
265Dic158—24no—
266Dic158—22no—
267Dic158—24no—
268Dic158—416no—
269Dic158—44no—
270Dic158—42no—
271Dic158—48no—
272C8×C79—24no—
273C79⋊C8—24no—
274C22×C79#53512yes—
275C22×C79C22×Q1612yes—
276C22×C79C2×#13712yes—
277C22×C79C23×D411yescharacteristic
278C2×D79—22no—
279C2×D79—28no—
280C2×D79—22no—
281C2×D79C22×Q1614yes—
282C4×C79—22no—
283C4×C79—42no—
284C4×C79—44no—
285C4×C79—42no—
286C4×C79#53512yes—
287C4×C79C22×Q1614yes—
288C4×C79C2×H(5,𝔽2)11yescharacteristic
289C4×C79C2×H(5,𝔽2)11yescharacteristic
290Dic79—22no—
291Dic79—44no—
292Dic79—42no—
293Dic79—42no—
294Dic79C22×Q1614yes—
295Dic79C2×#13714yes—
296C2×#3209—791nomaximal, Sylow
297C2×C79#320914yescomplemented
298C2×C79C2×#53511yescharacteristic
299C2×C79C23×Q1611yescharacteristic
300C2×C79C22×#13711yescharacteristic
301D79—24no—
302C2×#533—791no—
303C22×C4⋊Q8—791no—
304#3209—7916no—
305C2×#494—792no—
306C2×#487—792no—
307C2×#462—791no—
308C2×#462—791no—
309C2×#462—791no—
310C2×#462D7911yescharacteristic, complemented
311C2×#451—792no—
312C2×#451—792no—
313C2×#431—791no—
314C79C2×#320911yescharacteristic, complemented, Sylow
315C23×Q8—792no—
316C22×Q16—792no—
317C22×Q16—792no—
318C22×Q16—792no—
319C22×Q16C2×D7912yes—
320#533—798no—
321#533—1584no—
322C2×C4⋊Q8—791no—
323C2×C4⋊Q8—792no—
324C2×C4⋊Q8—791no—
325C2×C4⋊Q8—798no—
326C2×C4⋊Q8—791no—
327C2×C4⋊Q8—1581no—
328C2×C4⋊Q8—1581no—
329C2×C4⋊Q8—1584no—
330C2×C4⋊Q8C2×D7911yescharacteristic
331C2×#125—1581no—
332C2×C22⋊Q8—791no—
333C2×C22⋊Q8—791no—
334C2×C22⋊Q8—1581no—
335C2×C4×Q8—791no—
336C2×C4×Q8—1581no—
337C2×C4×Q8C2×D7911yescharacteristic
338C2×#117—1581no—
339C22×C4⋊C4—1581no—
340C22×C4⋊C4—1581no—
341C22×C42—791no—
342#494—7916no—
343#487—7916no—
344#462—798no—
345#462—798no—
346#462—798no—
347#462C2×D7918yescomplemented
348#451—7916no—
349#451—7916no—
350#431—798no—
351C2×C4⋊C8—791no—
352C2×C4⋊C8C2×D7911yescharacteristic
353C2×#103—792no—
354C2×#103—792no—
355C2×#103—792no—
356C2×#103C2×D7912yes—
357C2×C22⋊C8—792no—
358C2×C4×C8—792no—
359C22×Q8—3164no—
360C22×Q8—3164no—
361C22×Q8—798no—
362C22×Q8—791no—
363C22×Q8—792no—
364C22×Q8—1581no—
365C22×Q8—1582no—
366C22×Q8—1584no—
367C22×Q8C22×D7911yescharacteristic
368C22×Q8C22×D7912yes—
369C23×C4—791no—
370C23×C4—791no—
371C23×C4—791no—
372C23×C4—1582no—
373C2×Q16—44no—
374C2×Q16—3164no—
375C2×Q16—3164no—
376C2×Q16—3164no—
377C2×Q16—798no—
378C2×Q16—798no—
379C2×Q16—798no—
380C2×Q16C22×D7918yes—
381C22×C8—792no—
382C22×C8C22×D7912yes—
383C4⋊Q8—22no—
384C4⋊Q8—794no—
385C4⋊Q8—798no—
386C4⋊Q8—794no—
387C4⋊Q8—794no—
388C4⋊Q8—1588no—
389C4⋊Q8—1588no—
390C4⋊Q8—1582no—
391C4⋊Q8—1584no—
392C4⋊Q8—1582no—
393C4⋊Q8—1582no—
394C4⋊Q8C22×D7914yes—
395#125—1588no—
396C22⋊Q8—794no—
397C22⋊Q8—794no—
398C22⋊Q8—1582no—
399C22⋊Q8—1588no—
400C22⋊Q8—1582no—
401C4×Q8—22no—
402C4×Q8—794no—
403C4×Q8—1588no—
404C4×Q8—1582no—
405C4×Q8C22×D7914yes—
406#117—1588no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—791no—
411C2×C4⋊C4—1588no—
412C2×C4⋊C4—1588no—
413C2×C4⋊C4—1582no—
414C2×C4⋊C4—1582no—
415C2×C4⋊C4—1581no—
416C2×C4⋊C4—1581no—
417C2×C4⋊C4—1581no—
418C2×C4⋊C4—1581no—
419C2×C4⋊C4—1581no—
420C2×C4⋊C4—1581no—
421C2×C4⋊C4—1581no—
422C2×C4⋊C4—1581no—
423C2×C4⋊C4C22×D7911yescharacteristic
424C2×C22⋊C4—1581no—
425C2×C22⋊C4—1581no—
426C2×C42—21no—
427C2×C42—791no—
428C2×C42—791no—
429C2×C42—791no—
430C2×C42—794no—
431C2×C42—794no—
432C2×C42—1581no—
433C2×C42C22×D7911yescharacteristic
434C4⋊C8—794no—
435C4⋊C8C22×D7914yes—
436#103—798no—
437#103—798no—
438#103—798no—
439#103C22×D7918yes—
440C22⋊C8—798no—
441C4×C8—798no—
442C24—791no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—22no—
448C2×Q8—24no—
449C2×Q8—3164no—
450C2×Q8—3168no—
451C2×Q8—3168no—
452C2×Q8—3168no—
453C2×Q8—31616no—
454C2×Q8—794no—
455C2×Q8—792no—
456C2×Q8—1584no—
457C2×Q8—1588no—
458C2×Q8—1582no—
459C2×Q8—1584no—
460C2×Q8—1582no—
461C2×Q8—1584no—
462C2×Q8—1581no—
463C2×Q8—1582no—
464C2×Q8C23×D7914yes—
465C2×Q8C23×D7912yes—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—3168no—
470C22×C4—791no—
471C22×C4—791no—
472C22×C4—798no—
473C22×C4—794no—
474C22×C4—791no—
475C22×C4—792no—
476C22×C4—794no—
477C22×C4—792no—
478C22×C4—1588no—
479C22×C4—1584no—
480C22×C4—1581no—
481C22×C4—1581no—
482C22×C4—1582no—
483C22×C4—1582no—
484C22×C4—1582no—
485C22×C4D4×D7911yescharacteristic
486C22×C4D4×D7911yescharacteristic
487C22×C4C23×D7911yescharacteristic
488Q16—48no—
489Q16—3168no—
490Q16—3168no—
491Q16—3168no—
492C2×C8—24no—
493C2×C8—794no—
494C2×C8—1584no—
495C2×C8C23×D7914yes—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—21no—
500C4⋊C4—792no—
501C4⋊C4—1588no—
502C4⋊C4—1588no—
503C4⋊C4—1584no—
504C4⋊C4—1584no—
505C4⋊C4—1584no—
506C4⋊C4—1584no—
507C4⋊C4—1584no—
508C4⋊C4—1584no—
509C4⋊C4—1581no—
510C4⋊C4—1584no—
511C4⋊C4—1584no—
512C4⋊C4C23×D7912yes—
513C22⋊C4—1584no—
514C22⋊C4—1584no—
515C42—24no—
516C42—792no—
517C42—794no—
518C42—794no—
519C42—792no—
520C42—1584no—
521C42D4×D7912yes—
522C42C23×D7912yes—
523C23—794no—
524C23—792no—
525C23—1582no—
526C23—1582no—
527C23C2×D4×D7911yescharacteristic
528Q8—22no—
529Q8—24no—
530Q8—22no—
531Q8—24no—
532Q8—42no—
533Q8—44no—
534Q8—3168no—
535Q8—31616no—
536Q8—3164no—
537Q8—3162no—
538Q8—1582no—
539Q8—1584no—
540Q8—1582no—
541Q8—1584no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—31616no—
550C2×C4—3162no—
551C2×C4—3164no—
552C2×C4—3162no—
553C2×C4—794no—
554C2×C4—794no—
555C2×C4—792no—
556C2×C4—792no—
557C2×C4—7916no—
558C2×C4—792no—
559C2×C4—1588no—
560C2×C4—1582no—
561C2×C4—1582no—
562C2×C4—1584no—
563C2×C4—1582no—
564C2×C4—1582no—
565C2×C4—1582no—
566C2×C4—1584no—
567C2×C4Q16×D7914yes—
568C2×C4C2×D4×D7912yes—
569C2×C4C2×D4×D7912yes—
570C2×C4C2×D4×D7911yescharacteristic
571C2×C4#1457920612yes—
572C2×C4#1457920712yes—
573C2×C4C24×D7911yescharacteristic
574C8—24no—
575C8—1584no—
576C22—794no—
577C22—1582no—
578C22—1588no—
579C22—1582no—
580C22#1909912912yes—
581C22C2×Q16×D7912yes—
582C22#137×D7912yes—
583C22C22×D4×D7911yescharacteristic
584C4—22no—
585C4—42no—
586C4—44no—
587C4—42no—
588C4—3162no—
589C4—3164no—
590C4—3162no—
591C4—794no—
592C4—794no—
593C4—1582no—
594C4#1909912912yes—
595C4C2×Q16×D7914yes—
596C4C2×#1457920611yescharacteristic
597C4C2×#1457920711yescharacteristic
598C2—1584no—
599C2#2377434714yesdirect product factor
600C2C2×#1909912911yescharacteristic
601C2C22×Q16×D7911yescharacteristic
602C2C2×#137×D7911yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×#462×C79C2
socleC23×C79C22×D4
prime radicalC23×D79C2×D4
hypercenterC2×#462D79
hypocenterC79C2×#3209
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
3. #23774347C2direct (N × H)
4. C2×#19098941C2semidirect
5. C2×#19098912C2semidirect
6. C2×#19098774C2semidirect
7. C2×#19098773C2semidirect
8. C2×#19098770C2semidirect
12. C2×#462×C79C2semidirect
37. #19098941C22semidirect
38. #19098912C22semidirect
39. #19098774C22semidirect
40. #19098773C22semidirect
41. #19098770C22semidirect
54. #462×C79C22semidirect
167. C2×C4×D79Q16semidirect
263. C4×D79C2×Q16semidirect
297. C2×C79#3209semidirect
310. C2×#462D79semidirect
314. C79C2×#3209semidirect
347. #462C2×D79semidirect
599. C2#23774347direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190991172
C2×C4⋊Q8×D792
#237743472
C2×#190989412
C2×#190989122
C2×#190987742
C2×#190987732
C2×#190987702
C2×#190987202
C2×#190987172
C2×#190986152
C2×#462×C792
C2×#320979