Finite Group Databasebeta

← C2×#23773951 #35157924

Subgroups — C2×#23773951 #35157924

Order 20224.

1902 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099118C211yesmaximal, characteristic
2C2×#125×D79C211yesmaximal, characteristic
3#23773951C2116yesmaximal, direct product factor
4C2×#19098973C211yesmaximal, characteristic, complemented
5C2×#19098972C211yesmaximal, characteristic, complemented
6C2×#19098969C211yesmaximal, characteristic, complemented
7C2×#19098964C212yesmaximal, complemented
8C2×#19098963C212yesmaximal, complemented
9C2×#19098604C212yesmaximal
10C2×#19098602C212yesmaximal
11C2×#19098515C211yesmaximal, characteristic
12C2×#501×C79C211yesmaximal, characteristic, complemented
13C22×#11929607C2212yes—
14#19099118—24no—
15#19099118C2218yes—
16C2×#14579161C2211yescharacteristic
17C2×#14579153C2211yescharacteristic
18C2×#14579152C2211yescharacteristic
19C2×#14579152C2211yescharacteristic
20C2×#14579152C2212yes—
21C2×#14579151—21no—
22C2×#14579151—21no—
23#125×D79—24no—
24#125×D79C2218yes—
25C2×#14579141—21no—
26C2×#14579137—21no—
27C2×Q8×Dic79—21no—
28C2×#14579126—21no—
29C2×#14579119C2212yes—
30C2×C4⋊C4×D79—21no—
31C2×C4⋊C4×D79—21no—
32C2×C4⋊C4×D79C2211yescharacteristic
33C2×C4⋊C4×D79C2211yescharacteristic
34C2×C42×D79C2211yescharacteristic
35#19098973C2218yescomplemented
36#19098972C2218yescomplemented
37#19098969C2218yescomplemented
38#19098964C22116yescomplemented
39#19098963C22116yescomplemented
40#19098604C22116yes—
41#19098602C22116yes—
42C2×#14579095C2212yes—
43C2×#14579094C2212yes—
44C2×#14579094C2212yes—
45C2×#14579093C2212yes—
46C2×#14579092C2212yes—
47C2×#14579092C2212yes—
48#19098515C2218yes—
49C2×#14579072C2211yescharacteristic
50C2×C8×Dic79C2212yes—
51C2×C4⋊Q8×C79C2211yescharacteristic
52C2×#125×C79C2211yescharacteristic
53#501×C79C2218yescomplemented
54C2×#107×C79C2212yes—
55C2×#106×C79C2212yes—
56C2×#97×C79C2211yescharacteristic
57C22×Dic158—21no—
58C22×C4×D79—21no—
59C22×C4×D79—21no—
60C22×C4×D79D411yescharacteristic
61C22×C4×D79D411yescharacteristic
62C22×C4×D79C2311yescharacteristic
63C2×#11929607Q8116yescomplemented
64C2×#11929607C2318yes—
65C22×C79⋊C8C2312yes—
66#14579161—22no—
67#14579161C2314yes—
68#14579153—22no—
69#14579153C2314yes—
70#14579152—22no—
71#14579152—22no—
72#14579152—24no—
73#14579152C2314yes—
74#14579152C2318yes—
75#14579152C2314yes—
76#14579151—28no—
77#14579151—28no—
78#14579141—28no—
79#14579137—28no—
80Q8×Dic79—28no—
81#14579126—28no—
82#14579119—24no—
83#14579119C2318yes—
84C2×C4⋊Dic79—21no—
85C2×C4⋊Dic79—21no—
86C2×C4⋊Dic79—21no—
87C2×C4⋊Dic79—22no—
88C2×C4⋊Dic79C2311yescharacteristic
89C2×C4⋊Dic79C2311yescharacteristic
90C2×C4⋊Dic79C2312yes—
91C2×Dic79⋊C4—21no—
92C2×Dic79⋊C4—21no—
93C2×Dic79⋊C4—21no—
94C2×Dic79⋊C4—21no—
95C2×Dic79⋊C4—21no—
96C2×Dic79⋊C4—21no—
97C2×Dic79⋊C4—22no—
98C4⋊C4×D79—28no—
99C4⋊C4×D79—28no—
100C4⋊C4×D79—22no—
101C4⋊C4×D79—22no—
102C4⋊C4×D79C2314yes—
103C4⋊C4×D79C2314yes—
104C2×(C2×D79)⋊C4—22no—
105C2×C4×Dic79—22no—
106C2×C4×Dic79D411yescharacteristic
107C2×C4×Dic79D411yescharacteristic
108C2×C4×Dic79C2311yescharacteristic
109C42×D79D414yes—
110C42×D79C2314yes—
111#14579095C2318yes—
112#14579094C2318yes—
113#14579094C2318yes—
114#14579093C2318yes—
115#14579092C2318yes—
116#14579092C2318yes—
117#14579072C2314yes—
118C8×Dic79C2318yes—
119C22×Q8×C79—21no—
120C22×C8×C79C2312yes—
121C4⋊Q8×C79—22no—
122C4⋊Q8×C79C2314yes—
123#125×C79—22no—
124#125×C79C2314yes—
125C2×C4⋊C4×C79—21no—
126C2×C4⋊C4×C79—21no—
127C2×C4⋊C4×C79—21no—
128C2×C4⋊C4×C79C2311yescharacteristic
129C2×C4⋊C4×C79C2312yes—
130C2×C4⋊C4×C79C2311yescharacteristic
131C2×C42×C79C2311yescharacteristic
132#107×C79C2318yes—
133#106×C79C2318yes—
134#97×C79C2314yes—
135C23×D79C2×D411yescharacteristic
136C2×Dic158—24no—
137C2×Dic158—44no—
138C22×Dic79—22no—
139C22×Dic79—21no—
140C22×Dic79—21no—
141C22×Dic79C2×D411yescharacteristic
142C22×Dic79C2×D411yescharacteristic
143C22×Dic79C2×D411yescharacteristic
144C2×C4×D79—24no—
145C2×C4×D79—24no—
146C2×C4×D79—42no—
147C2×C4×D79—42no—
148C2×C4×D79—44no—
149C2×C4×D79—44no—
150C2×C4×D79C2×D414yes—
151C2×C4×D79C2×D414yes—
152C2×C4×D79C2×D412yes—
153C2×C4×D79C2×Q818yes—
154C2×C4×D79C2412yes—
155#11929607C2×Q8132yescomplemented
156C2×C79⋊C8C2×Q818yes—
157C2×C79⋊C8C2414yes—
158C4⋊Dic79—24no—
159C4⋊Dic79—24no—
160C4⋊Dic79—28no—
161C4⋊Dic79—21no—
162C4⋊Dic79—21no—
163C4⋊Dic79—24no—
164C4⋊Dic79—22no—
165C4⋊Dic79C2412yes—
166C4⋊Dic79C2414yes—
167C4⋊Dic79C2412yes—
168Dic79⋊C4—24no—
169Dic79⋊C4—24no—
170Dic79⋊C4—24no—
171Dic79⋊C4—24no—
172Dic79⋊C4—28no—
173Dic79⋊C4—24no—
174Dic79⋊C4—24no—
175(C2×D79)⋊C4—28no—
176C4×Dic79—28no—
177C4×Dic79C2×D412yes—
178C4×Dic79C2×D414yes—
179C4×Dic79C2×D414yes—
180C4×Dic79C2412yes—
181C2×Q8×C79—24no—
182C2×Q8×C79—44no—
183C22×C4×C79—21no—
184C22×C4×C79—21no—
185C22×C4×C79—22no—
186C22×C4×C79C2×D411yescharacteristic
187C22×C4×C79C2×D411yescharacteristic
188C22×C4×C79C2411yescharacteristic
189C2×C8×C79C2×Q818yes—
190C2×C8×C79C2414yes—
191C4⋊C4×C79—24no—
192C4⋊C4×C79—24no—
193C4⋊C4×C79—24no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—22no—
197C4⋊C4×C79C2412yes—
198C4⋊C4×C79C2414yes—
199C4⋊C4×C79C2412yes—
200C42×C79C2×D412yes—
201C42×C79C2412yes—
202C23×C79C22×D411yescharacteristic
203C22×D79—22no—
204C22×D79—22no—
205C22×D79C4⋊Q814yes—
206C22×D79C22×D412yes—
207Q8×C79—48no—
208C2×C4×C79—22no—
209C2×C4×C79—22no—
210C2×C4×C79—24no—
211C2×C4×C79—22no—
212C2×C4×C79—22no—
213C2×C4×C79—42no—
214C2×C4×C79—42no—
215C2×C4×C79—44no—
216C2×C4×C79C22×D412yes—
217C2×C4×C79C22×D412yes—
218C2×C4×C79C22×D411yescharacteristic
219C2×C4×C79C22×Q814yes—
220C2×C4×C79C2511yescharacteristic
221C2×Dic79—22no—
222C2×Dic79—22no—
223C2×Dic79—24no—
224C2×Dic79—22no—
225C2×Dic79—22no—
226C2×Dic79—42no—
227C2×Dic79—42no—
228C2×Dic79—44no—
229C2×Dic79C4⋊Q814yes—
230C2×Dic79C22×D412yes—
231C2×Dic79C22×D412yes—
232C2×Dic79C22×D412yes—
233C4×D79—44no—
234C4×D79—44no—
235C4×D79—48no—
236C4×D79—48no—
237C4×D79C4⋊Q818yes—
238C4×D79C22×Q818yes—
239Dic158—48no—
240C8×C79C22×Q818yes—
241C79⋊C8C22×Q818yes—
242C22×C79C2×C4⋊Q812yes—
243C22×C79#57512yes—
244C22×C79#57512yes—
245C22×C79C23×D411yescharacteristic
246C2×D79—28no—
247C2×D79—22no—
248C2×D79—22no—
249C2×D79C2×C4⋊Q814yes—
250C4×C79—22no—
251C4×C79—22no—
252C4×C79—44no—
253C4×C79—42no—
254C4×C79—42no—
255C4×C79C2×C4⋊Q812yes—
256C4×C79C23×Q812yes—
257Dic79—22no—
258Dic79—22no—
259Dic79—44no—
260Dic79—42no—
261Dic79—42no—
262Dic79C2×C4⋊Q814yes—
263C2×#3166—791nomaximal, Sylow
264C2×C79#316614yescomplemented
265C2×C79C22×C4⋊Q811yescharacteristic
266C2×C79C2×#57511yescharacteristic
267C2×C79C2×#57511yescharacteristic
268D79—24no—
269C2×#533—791no—
270C22×#125—791no—
271#3166—7916no—
272C2×#501—791no—
273C2×#501—791no—
274C2×#501—791no—
275C2×#501D7911yescharacteristic, complemented
276C2×#499—792no—
277C2×#499—792no—
278C2×#428—792no—
279C2×#428—792no—
280C2×C4×M16—791no—
281C79C2×#316611yescharacteristic, complemented, Sylow
282C22×M16—792no—
283#533—798no—
284#533—1584no—
285C2×C4⋊Q8—791no—
286C2×C4⋊Q8C2×D7911yescharacteristic
287C2×#125—798no—
288C2×#125—791no—
289C2×#125—791no—
290C2×#125—792no—
291C2×#125—791no—
292C2×#125—1581no—
293C2×#125—1581no—
294C2×#125—1584no—
295C2×#125C2×D7911yescharacteristic
296C2×C22⋊Q8—1581no—
297C2×C22⋊Q8—1581no—
298C2×C4×Q8—1581no—
299C2×C4×Q8—1581no—
300C2×#117—792no—
301C22×C4⋊C4—791no—
302C22×C4⋊C4—791no—
303C22×C4⋊C4—1581no—
304C22×C4⋊C4—1581no—
305C22×C42—791no—
306#501—798no—
307#501—798no—
308#501—798no—
309#501C2×D7918yescomplemented
310#499—7916no—
311#499—7916no—
312#428—7916no—
313#428—7916no—
314C2×#107—792no—
315C2×#107—792no—
316C2×#107—792no—
317C2×#107C2×D7912yes—
318C2×#106—792no—
319C2×#106—792no—
320C2×#106—792no—
321C2×#106C2×D7912yes—
322C4×M16—798no—
323C2×#97—791no—
324C2×#97C2×D7911yescharacteristic
325C2×C4×C8—792no—
326C22×Q8—21no—
327C22×Q8—1581no—
328C23×C4—791no—
329C23×C4—791no—
330C23×C4—791no—
331C23×C4—1581no—
332C23×C4—1581no—
333C2×M16—7916no—
334C2×M16—798no—
335C22×C8—792no—
336C22×C8C22×D7912yes—
337C4⋊Q8—22no—
338C4⋊Q8—794no—
339C4⋊Q8—1582no—
340C4⋊Q8C22×D7914yes—
341#125—22no—
342#125—794no—
343#125—794no—
344#125—798no—
345#125—794no—
346#125—1588no—
347#125—1588no—
348#125—1582no—
349#125—1582no—
350#125—1584no—
351#125—1582no—
352#125C22×D7914yes—
353C22⋊Q8—1588no—
354C22⋊Q8—1588no—
355C4×Q8—1588no—
356C4×Q8—1588no—
357#117—798no—
358#117—1584no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—791no—
363C2×C4⋊C4—791no—
364C2×C4⋊C4—792no—
365C2×C4⋊C4—794no—
366C2×C4⋊C4—794no—
367C2×C4⋊C4—1582no—
368C2×C4⋊C4—1582no—
369C2×C4⋊C4—1588no—
370C2×C4⋊C4—1588no—
371C2×C4⋊C4—1581no—
372C2×C4⋊C4—1581no—
373C2×C4⋊C4—1581no—
374C2×C4⋊C4—1581no—
375C2×C4⋊C4—1581no—
376C2×C4⋊C4—1581no—
377C2×C4⋊C4—1582no—
378C2×C4⋊C4—1582no—
379C2×C4⋊C4—1581no—
380C2×C4⋊C4—1581no—
381C2×C4⋊C4—1581no—
382C2×C4⋊C4C22×D7911yescharacteristic
383C2×C4⋊C4C22×D7911yescharacteristic
384C2×C4⋊C4C22×D7912yes—
385C2×C22⋊C4—1582no—
386C2×C42—791no—
387C2×C42—791no—
388C2×C42—791no—
389C2×C42—794no—
390C2×C42—794no—
391C2×C42—1582no—
392C2×C42C22×D7911yescharacteristic
393#107—798no—
394#107—798no—
395#107—798no—
396#107C22×D7918yes—
397#106—798no—
398#106—798no—
399#106—798no—
400#106C22×D7918yes—
401#97—794no—
402#97C22×D7914yes—
403C4×C8—798no—
404C24—791no—
405C2×Q8—24no—
406C2×Q8—44no—
407C2×Q8—3164no—
408C2×Q8—1584no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—3162no—
413C22×C4—3162no—
414C22×C4—3164no—
415C22×C4—3164no—
416C22×C4—794no—
417C22×C4—794no—
418C22×C4—791no—
419C22×C4—798no—
420C22×C4—792no—
421C22×C4—792no—
422C22×C4—791no—
423C22×C4—791no—
424C22×C4—1584no—
425C22×C4—1584no—
426C22×C4—1582no—
427C22×C4—1581no—
428C22×C4—1581no—
429C22×C4D4×D7911yescharacteristic
430C22×C4D4×D7911yescharacteristic
431C22×C4C23×D7911yescharacteristic
432M16—7932no—
433C2×C8—798no—
434C2×C8—794no—
435C2×C8Q8×D7918yes—
436C2×C8C23×D7914yes—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—792no—
444C4⋊C4—792no—
445C4⋊C4—794no—
446C4⋊C4—1584no—
447C4⋊C4—1584no—
448C4⋊C4—1584no—
449C4⋊C4—1584no—
450C4⋊C4—1588no—
451C4⋊C4—1584no—
452C4⋊C4—1584no—
453C4⋊C4—1588no—
454C4⋊C4—1584no—
455C4⋊C4—1584no—
456C4⋊C4—1581no—
457C4⋊C4—1581no—
458C4⋊C4—1582no—
459C4⋊C4—1584no—
460C4⋊C4C23×D7912yes—
461C4⋊C4C23×D7914yes—
462C4⋊C4C23×D7912yes—
463C22⋊C4—1588no—
464C42—792no—
465C42—794no—
466C42—794no—
467C42—792no—
468C42—1588no—
469C42D4×D7912yes—
470C42C23×D7912yes—
471C23—794no—
472C23—792no—
473C23—1582no—
474C23—1582no—
475C23C2×D4×D7911yescharacteristic
476Q8—48no—
477Q8—3168no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—3164no—
487C2×C4—3164no—
488C2×C4—3168no—
489C2×C4—3168no—
490C2×C4—3162no—
491C2×C4—3162no—
492C2×C4—3164no—
493C2×C4—794no—
494C2×C4—792no—
495C2×C4—798no—
496C2×C4—798no—
497C2×C4—792no—
498C2×C4—792no—
499C2×C4—1584no—
500C2×C4—1582no—
501C2×C4—1582no—
502C2×C4—1582no—
503C2×C4—1582no—
504C2×C4C2×D4×D7912yes—
505C2×C4C2×D4×D7912yes—
506C2×C4C2×D4×D7911yescharacteristic
507C2×C4C2×Q8×D7914yes—
508C2×C4C24×D7911yescharacteristic
509C8—798no—
510C8C2×Q8×D7918yes—
511C22—794no—
512C22—1582no—
513C22—1582no—
514C22—1588no—
515C22C4⋊Q8×D7912yes—
516C22#1909935912yes—
517C22#1909936212yes—
518C22C22×D4×D7911yescharacteristic
519C4—22no—
520C4—22no—
521C4—42no—
522C4—44no—
523C4—42no—
524C4—3162no—
525C4—3162no—
526C4—3164no—
527C4—794no—
528C4—1582no—
529C4—1582no—
530C4C4⋊Q8×D7912yes—
531C4C22×Q8×D7912yes—
532C2—1584no—
533C2#2377395114yesdirect product factor
534C2C2×C4⋊Q8×D7911yescharacteristic
535C2C2×#1909935911yescharacteristic
536C2C2×#1909936211yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×#501×C79C2
socleC23×C79C22×D4
prime radicalC23×D79C2×D4
hypercenterC2×#501D79
hypocenterC79C2×#3166
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. #23773951C2direct (N × H)
4. C2×#19098973C2semidirect
5. C2×#19098972C2semidirect
6. C2×#19098969C2semidirect
7. C2×#19098964C2semidirect
8. C2×#19098963C2semidirect
12. C2×#501×C79C2semidirect
35. #19098973C22semidirect
36. #19098972C22semidirect
37. #19098969C22semidirect
38. #19098964C22semidirect
39. #19098963C22semidirect
53. #501×C79C22semidirect
63. C2×#11929607Q8semidirect
155. #11929607C2×Q8semidirect
264. C2×C79#3166semidirect
275. C2×#501D79semidirect
281. C79C2×#3166semidirect
309. #501C2×D79semidirect
533. C2#23773951direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190991182
C2×#125×D792
#237739512
C2×#190989732
C2×#190989722
C2×#190989692
C2×#190989642
C2×#190989632
C2×#190986042
C2×#190986022
C2×#190985152
C2×#501×C792
C2×#316679