Finite Group Databasebeta

← C2×#22622322 #33964444

Subgroups — C2×#22622322 #33964444

Order 18688.

1942 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989298C211yesmaximal, characteristic
2C2×#17989297C211yesmaximal, characteristic
3#22622322C2116yesmaximal, direct product factor
4C2×#17988990C211yesmaximal, characteristic, complemented
5C2×#17988924C212yesmaximal, complemented
6C2×#17988921C212yesmaximal, complemented
7C2×#17988883C211yesmaximal, characteristic, complemented
8C2×#17988827C212yesmaximal, complemented
9C2×C4×#11924018C211yesmaximal, characteristic, complemented
10C2×#17988652C211yesmaximal, characteristic
11C2×#107×D73C212yesmaximal, complemented
12C2×#477×C73C211yesmaximal, characteristic, complemented
13C22×#11924018C2211yescharacteristic
14C22×C8×D73C2212yes—
15#17989298—24no—
16#17989298C2218yes—
17#17989297—24no—
18#17989297C2218yes—
19C2×#14556591C2211yescharacteristic
20C2×#14556583—21no—
21C2×#14556582—21no—
22C2×#14556571C2212yes—
23C2×#14556569—22no—
24C2×#14556567—22no—
25C2×#14556567C2212yes—
26C2×C4×Dic146C2211yescharacteristic
27C2×C4×D292—21no—
28C2×C4×D292C2211yescharacteristic
29C2×#14556553—21no—
30C2×C4⋊C4×D73C2212yes—
31C2×C4⋊C4×D73C2212yes—
32#17988990C2218yescomplemented
33#17988924C22116yescomplemented
34#17988921C22116yescomplemented
35#17988883C2218yescomplemented
36#17988827C22116yescomplemented
37C4×#11924018C2218yescomplemented
38#17988652C2218yes—
39C2×#14556525C2212yes—
40C2×#14556524C2212yes—
41C2×#14556524C2212yes—
42C2×#14556524C2211yescharacteristic
43#107×D73C22116yescomplemented
44C2×#14556516C2212yes—
45C2×#14556515C2211yescharacteristic
46C2×#14556514C2211yescharacteristic
47C2×#14556503C2212yes—
48C2×C4×#10596831C2211yescharacteristic
49C2×C4⋊Q8×C73C2211yescharacteristic
50C2×C4×Q8×C73C2211yescharacteristic
51#477×C73C2218yescomplemented
52C2×#107×C73C2212yes—
53C2×C4⋊C8×C73C2211yescharacteristic
54C2×#103×C73C2212yes—
55C22×Dic146—21no—
56C22×Dic146C2311yescharacteristic
57C22×D292D411yescharacteristic
58C22×C4×D73—22no—
59C22×C4×D73—22no—
60C22×C4×D73C2312yes—
61C2×#11924018—24no—
62C2×#11924018C2314yes—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C2×C8×D73—28no—
66C2×C8×D73C2318yes—
67#14556591—22no—
68#14556591C2314yes—
69#14556583—28no—
70#14556582—28no—
71#14556571—24no—
72#14556571C2318yes—
73#14556569—216no—
74#14556567—24no—
75#14556567—216no—
76#14556567C2318yes—
77C4×Dic146—22no—
78C4×Dic146C2314yes—
79C4×D292—28no—
80C4×D292D414yes—
81C4×D292C2314yes—
82#14556553—28no—
83C2×C4⋊Dic73—22no—
84C2×C4⋊Dic73—21no—
85C2×C4⋊Dic73D411yescharacteristic
86C2×C4⋊Dic73C2312yes—
87C2×C4⋊Dic73C2311yescharacteristic
88C2×#11924007—22no—
89C2×#11924007—22no—
90C2×#11924007—22no—
91C2×#11924007—21no—
92C4⋊C4×D73—24no—
93C4⋊C4×D73—24no—
94C4⋊C4×D73C2318yes—
95C4⋊C4×D73C2318yes—
96C2×#11924005—22no—
97C2×#11924005—22no—
98C2×#11924005D412yes—
99C2×C4×Dic73—21no—
100#14556525C2318yes—
101#14556524C2314yes—
102#14556524C2318yes—
103#14556524C2318yes—
104#14556516C2318yes—
105#14556515C2314yes—
106#14556514C2314yes—
107#14556503C2318yes—
108C4×#10596831C2314yes—
109C22×Q8×C73—21no—
110C22×Q8×C73C2311yescharacteristic
111C22×C8×C73C2312yes—
112C4⋊Q8×C73—22no—
113C4⋊Q8×C73C2314yes—
114C4×Q8×C73—22no—
115C4×Q8×C73C2314yes—
116C2×C4⋊C4×C73—21no—
117C2×C4⋊C4×C73—21no—
118C2×C4⋊C4×C73C2312yes—
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C42×C73—21no—
121C2×C42×C73C2311yescharacteristic
122#107×C73C2318yes—
123C4⋊C8×C73C2314yes—
124#103×C73C2318yes—
125C23×D73C2×D412yes—
126C2×Dic146—22no—
127C2×Dic146—22no—
128C2×Dic146—21no—
129C2×Dic146—24no—
130C2×Dic146—82no—
131C2×Dic146C2412yes—
132C2×D292Q1618yescomplemented
133C2×D292C2×D414yes—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—22no—
137C22×Dic73C2×D412yes—
138C2×C4×D73—24no—
139C2×C4×D73—24no—
140C2×C4×D73—28no—
141C2×C4×D73—28no—
142C2×C4×D73—84no—
143C2×C4×D73—84no—
144C2×C4×D73C2×D414yes—
145C2×C4×D73C2414yes—
146#11924018—28no—
147C2×#10596831—22no—
148C2×#10596831#4114yes—
149C2×#10596831C2412yes—
150C2×#10596831C2412yes—
151C8×D73—216no—
152C4⋊Dic73—21no—
153C4⋊Dic73—22no—
154C4⋊Dic73—24no—
155C4⋊Dic73—28no—
156C4⋊Dic73C2×D414yes—
157C4⋊Dic73C2412yes—
158C4⋊Dic73C2414yes—
159#11924007—24no—
160#11924007—28no—
161#11924007—28no—
162#11924007—28no—
163#11924005—28no—
164#11924005—28no—
165#11924005C2×D418yes—
166C4×Dic73—24no—
167C2×Q8×C73—21no—
168C2×Q8×C73—24no—
169C2×Q8×C73—22no—
170C2×Q8×C73—22no—
171C2×Q8×C73—82no—
172C2×Q8×C73C2412yes—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—22no—
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2411yescharacteristic
179C2×C8×C73—24no—
180C2×C8×C73C2414yes—
181C4⋊C4×C73—24no—
182C4⋊C4×C73—24no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—22no—
185C4⋊C4×C73C2414yes—
186C4⋊C4×C73C2412yes—
187C42×C73—24no—
188C42×C73C2×D412yes—
189C42×C73C2412yes—
190C23×C73C22×D411yescharacteristic
191C22×D73—24no—
192C22×D73—24no—
193C22×D73C2×Q1618yes—
194C22×D73C22×D414yes—
195Q8×C73—22no—
196Q8×C73—22no—
197Q8×C73—42no—
198Q8×C73—84no—
199C2×C4×C73—22no—
200C2×C4×C73—22no—
201C2×C4×C73—22no—
202C2×C4×C73—24no—
203C2×C4×C73—42no—
204C2×C4×C73—44no—
205C2×C4×C73—42no—
206C2×C4×C73C2×Q1614yes—
207C2×C4×C73C22×D412yes—
208C2×C4×C73C22×D411yescharacteristic
209C2×C4×C73C22×D412yes—
210C2×C4×C73C2×#4112yes—
211C2×C4×C732−1+412yes—
212C2×C4×C73C2511yescharacteristic
213C2×Dic73—24no—
214C2×Dic73—22no—
215C2×Dic73—22no—
216C2×Dic73—24no—
217C2×Dic73—42no—
218C2×Dic73—42no—
219C2×Dic73—44no—
220C2×Dic73C22×D414yes—
221D292C2×Q16116yescomplemented
222C4×D73—28no—
223C4×D73—24no—
224C4×D73—24no—
225C4×D73—88no—
226C4×D73—88no—
227Dic146—22no—
228Dic146—22no—
229Dic146—42no—
230Dic146—84no—
231C8×C73—24no—
232#10596831—22no—
233#10596831C2×#4114yes—
234C22×C73#54712yes—
235C22×C73C22×Q1612yes—
236C22×C73#57612yes—
237C22×C73C23×D411yescharacteristic
238C2×D73—216no—
239C2×D73—24no—
240C2×D73—24no—
241C2×D73C22×Q1618yes—
242C4×C73—22no—
243C4×C73—42no—
244C4×C73—42no—
245C4×C73—44no—
246C4×C73#54712yes—
247C4×C73C22×Q1614yes—
248C4×C73C22×#4111yescharacteristic
249C4×C73C2×2−1+411yescharacteristic
250Dic73—24no—
251Dic73—42no—
252Dic73—44no—
253Dic73—42no—
254C2×#3338—731nomaximal, Sylow
255C2×C73#333814yescomplemented
256C2×C73C2×#54711yescharacteristic
257C2×C73C23×Q1611yescharacteristic
258C2×C73C2×#57611yescharacteristic
259D73—28no—
260C2×#554—731no—
261C2×#554—731no—
262#3338—7316no—
263C2×#500—731no—
264C2×#484—732no—
265C2×#484—732no—
266C2×#477—731no—
267C2×#477D7311yescharacteristic, complemented
268C2×#467—732no—
269C2×C4×Q16—731no—
270C2×C8×D4—731no—
271C22×#107—732no—
272C73C2×#333811yescharacteristic, complemented, Sylow
273C22×Q16—731no—
274C23×C8—732no—
275#554—738no—
276#554—738no—
277#554—1464no—
278#554—1464no—
279C2×C4⋊Q8—731no—
280C2×C4⋊Q8C2×D7311yescharacteristic
281C2×#125—1461no—
282C2×#125—1461no—
283C2×C22⋊Q8—732no—
284C2×C22⋊Q8—732no—
285C2×C22⋊Q8—1462no—
286C2×C22⋊Q8—1462no—
287C2×C4×Q8—731no—
288C2×C4×Q8C2×D7311yescharacteristic
289C2×C4×D4—731no—
290C2×C4×D4—1461no—
291C2×C4×D4—1461no—
292C22×C4⋊C4—732no—
293C22×C4⋊C4—732no—
294#500—738no—
295#484—7316no—
296#484—7316no—
297#477—738no—
298#477C2×D7318yescomplemented
299#467—7316no—
300C4×Q16—738no—
301C8×D4—738no—
302C2×#107—732no—
303C2×#107—732no—
304C2×#107—732no—
305C2×#107—731no—
306C2×#107—7316no—
307C2×#107C2×D7312yes—
308C2×C4⋊C8C2×D7311yescharacteristic
309C2×#103—731no—
310C2×#103—732no—
311C2×#103—731no—
312C2×#103C2×D7312yes—
313C2×C22⋊C8—732no—
314C2×C4×C8—731no—
315C22×Q8—21no—
316C22×Q8—731no—
317C22×Q8—1461no—
318C22×Q8C22×D7311yescharacteristic
319C22×D4—731no—
320C23×C4—732no—
321C23×C4—1462no—
322C23×C4—1462no—
323C2×Q16—734no—
324C2×Q16—1464no—
325C22×C8—731no—
326C22×C8—731no—
327C22×C8—738no—
328C22×C8—1468no—
329C22×C8C22×D7312yes—
330C4⋊Q8—22no—
331C4⋊Q8—734no—
332C4⋊Q8—1462no—
333C4⋊Q8C22×D7314yes—
334#125—1468no—
335#125—1468no—
336C22⋊Q8—738no—
337C22⋊Q8—738no—
338C22⋊Q8—14616no—
339C22⋊Q8—14616no—
340C22⋊Q8—1464no—
341C22⋊Q8—1464no—
342C4×Q8—22no—
343C4×Q8—734no—
344C4×Q8—1462no—
345C4×Q8C22×D7314yes—
346C4×D4—734no—
347C4×D4—734no—
348C4×D4—1468no—
349C4×D4—1468no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—731no—
353C2×C4⋊C4—731no—
354C2×C4⋊C4—732no—
355C2×C4⋊C4—738no—
356C2×C4⋊C4—738no—
357C2×C4⋊C4—1464no—
358C2×C4⋊C4—1464no—
359C2×C4⋊C4—1462no—
360C2×C4⋊C4—1462no—
361C2×C4⋊C4—1462no—
362C2×C4⋊C4—1462no—
363C2×C4⋊C4—1461no—
364C2×C4⋊C4—1461no—
365C2×C4⋊C4C22×D7311yescharacteristic
366C2×C4⋊C4C22×D7312yes—
367C2×C22⋊C4—732no—
368C2×C22⋊C4—1462no—
369C2×C22⋊C4—1462no—
370C2×C42—21no—
371C2×C42—1461no—
372C2×C42C22×D7311yescharacteristic
373#107—738no—
374#107—734no—
375#107—738no—
376#107—738no—
377#107C22×D7318yes—
378C4⋊C8C22×D7314yes—
379#103—734no—
380#103—738no—
381#103—734no—
382#103C22×D7318yes—
383C22⋊C8—738no—
384C4×C8—734no—
385C24—732no—
386C2×Q8—21no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—82no—
391C2×Q8—5842no—
392C2×Q8—732no—
393C2×Q8—1462no—
394C2×Q8—1462no—
395C2×Q8—1461no—
396C2×Q8—1464no—
397C2×Q8C23×D7312yes—
398C2×D4—738no—
399C2×D4—734no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—5844no—
404C22×C4—5844no—
405C22×C4—732no—
406C22×C4—734no—
407C22×C4—734no—
408C22×C4—1468no—
409C22×C4—1468no—
410C22×C4—1461no—
411C22×C4—1461no—
412C22×C4—1462no—
413C22×C4—1464no—
414C22×C4—1464no—
415C22×C4D4×D7311yescharacteristic
416C22×C4D4×D7311yescharacteristic
417C22×C4C23×D7311yescharacteristic
418Q16—1468no—
419C2×C8—24no—
420C2×C8—734no—
421C2×C8—732no—
422C2×C8—732no—
423C2×C8—14616no—
424C2×C8—1462no—
425C2×C8C23×D7314yes—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—734no—
431C4⋊C4—732no—
432C4⋊C4—734no—
433C4⋊C4—1468no—
434C4⋊C4—1468no—
435C4⋊C4—1468no—
436C4⋊C4—1468no—
437C4⋊C4—1464no—
438C4⋊C4—1461no—
439C4⋊C4—1462no—
440C4⋊C4—1464no—
441C4⋊C4C23×D7312yes—
442C4⋊C4C23×D7314yes—
443C22⋊C4—738no—
444C22⋊C4—1468no—
445C22⋊C4—1468no—
446C42—24no—
447C42—1464no—
448C42D4×D7312yes—
449C42C23×D7312yes—
450C23—734no—
451C23—738no—
452C23—1464no—
453C23—1464no—
454C23C2×D4×D7311yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—42no—
458Q8—84no—
459Q8—2922no—
460Q8—5844no—
461Q8—1462no—
462Q8—1462no—
463D4—7316no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—2922no—
472C2×C4—2924no—
473C2×C4—2922no—
474C2×C4—5848no—
475C2×C4—5848no—
476C2×C4—734no—
477C2×C4—1462no—
478C2×C4—1462no—
479C2×C4—1464no—
480C2×C4—1464no—
481C2×C4—1464no—
482C2×C4—1464no—
483C2×C4—1468no—
484C2×C4Q16×D7314yes—
485C2×C4C2×D4×D7312yes—
486C2×C4C2×D4×D7311yescharacteristic
487C2×C4C2×D4×D7312yes—
488C2×C4#41×D7312yes—
489C2×C4#1455663912yes—
490C2×C4C24×D7311yescharacteristic
491C8—24no—
492C8—734no—
493C8—1462no—
494C22—738no—
495C22—14616no—
496C22—1464no—
497C22—1464no—
498C22#1798923912yes—
499C22C2×Q16×D7312yes—
500C22#1798939112yes—
501C22C22×D4×D7311yescharacteristic
502C4—22no—
503C4—42no—
504C4—42no—
505C4—44no—
506C4—2922no—
507C4—2922no—
508C4—2924no—
509C4—1464no—
510C4#1798923912yes—
511C4C2×Q16×D7314yes—
512C4C2×#41×D7311yescharacteristic
513C4C2×#1455663911yescharacteristic
514C2—1468no—
515C2#2262232214yesdirect product factor
516C2C2×#1798923911yescharacteristic
517C2C22×Q16×D7311yescharacteristic
518C2C2×#1798939111yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC2×#477×C73C2
socleC23×C73C22×D4
prime radicalC22×D292D4
hypercenterC2×#477D73
hypocenterC73C2×#3338
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #22622322C2direct (N × H)
4. C2×#17988990C2semidirect
5. C2×#17988924C2semidirect
6. C2×#17988921C2semidirect
7. C2×#17988883C2semidirect
8. C2×#17988827C2semidirect
9. C2×C4×#11924018C2semidirect
11. C2×#107×D73C2semidirect
12. C2×#477×C73C2semidirect
32. #17988990C22semidirect
33. #17988924C22semidirect
34. #17988921C22semidirect
35. #17988883C22semidirect
36. #17988827C22semidirect
37. C4×#11924018C22semidirect
43. #107×D73C22semidirect
51. #477×C73C22semidirect
132. C2×D292Q16semidirect
221. D292C2×Q16semidirect
255. C2×C73#3338semidirect
267. C2×#477D73semidirect
272. C73C2×#3338semidirect
298. #477C2×D73semidirect
515. C2#22622322direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179892982
C2×#179892972
#226223222
C2×#179889902
C2×#179889242
C2×#179889212
C2×#179888832
C2×#179888272
C2×C4×#119240182
C2×#179886522
C2×#107×D732
C2×#477×C732
C2×#333873