Finite Group Databasebeta

← C2×#22626762 #34031549

Subgroups — C2×#22626762 #34031549

Order 18688.

2014 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626762C2116yesmaximal, direct product factor
2C2×#17989637C211yesmaximal, characteristic, complemented
3C2×#17989636C211yesmaximal, characteristic, complemented
4C2×#17989622C212yesmaximal, complemented
5C2×#17989622C211yesmaximal, characteristic, complemented
6C2×#17989618C212yesmaximal, complemented
7C2×D4×#10596833C211yesmaximal, characteristic, complemented
8C2×#17989611C212yesmaximal, complemented
9C2×#17989603C211yesmaximal, characteristic, complemented
10C2×#17989551C212yesmaximal, complemented
11C2×C4×#11924037C211yesmaximal, characteristic, complemented
12C2×#17989118C211yesmaximal, characteristic
13C22×#11924037C2211yescharacteristic
14C22×#11924037C2211yescharacteristic
15C22×#11924037C2212yes—
16C23×#10596833C2212yes—
17#17989637C2218yescomplemented
18#17989636C2218yescomplemented
19#17989622C22116yescomplemented
20#17989622C2218yescomplemented
21#17989618C22116yescomplemented
22D4×#10596833C2218yescomplemented
23#17989611C22116yescomplemented
24#17989603C2218yescomplemented
25C2×#14556658C2212yes—
26C2×C4⋊#10596833C2211yescharacteristic
27C2×C4⋊#10596833C2211yescharacteristic
28C2×C4⋊#10596833C2212yes—
29C2×C22⋊#10596833C2212yes—
30C2×C22⋊#10596833C2212yes—
31C2×C22⋊#10596833C2212yes—
32#17989551C22116yescomplemented
33C4×#11924037C2218yescomplemented
34C2×#14556646C2211yescharacteristic
35C2×#14556646C2212yes—
36C2×#14556646C2211yescharacteristic
37C2×C4×#10596833C2212yes—
38C2×C4×#10596833C2211yescharacteristic
39C2×C4×#10596833C2211yescharacteristic
40C2×C4×#10596833C2212yes—
41#17989118C418yes—
42#17989118C2218yes—
43C2×#14556584C412yes—
44C2×#14556582C411yescharacteristic
45C2×#14556578C411yescharacteristic
46C2×#14556572C412yes—
47C2×#14556570C411yescharacteristic
48C2×#14556556C2211yescharacteristic
49C2×D4×Dic73C2211yescharacteristic
50C2×D4×Dic73C2212yes—
51C2×#14556547C2212yes—
52C22×C4×Dic73C2211yescharacteristic
53C2×#121×C73C411yescharacteristic
54C2×#11924037—24no—
55C2×#11924037—24no—
56C2×#11924037—28no—
57C2×#11924037C2314yes—
58C2×#11924037C2314yes—
59C2×#11924037C2318yes—
60C22×#10596833—28no—
61C22×#10596833C2312yes—
62C22×#10596833C2318yes—
63C22×#10596833C2311yescharacteristic
64C22×#10596833C2312yes—
65C22×#10596833C2311yescharacteristic
66C22×#10596833C2311yescharacteristic
67C22×#10596833C2311yescharacteristic
68#14556658C2318yes—
69C4⋊#10596833C2314yes—
70C4⋊#10596833C2314yes—
71C4⋊#10596833C2318yes—
72C22⋊#10596833C2318yes—
73C22⋊#10596833C2318yes—
74C22⋊#10596833C2318yes—
75#14556646C2318yes—
76#14556646C2314yes—
77#14556646C2314yes—
78C4×#10596833C2314yes—
79C4×#10596833C2318yes—
80C4×#10596833C2314yes—
81C4×#10596833C2318yes—
82C22×Dic146C2×C411yescharacteristic
83C23×Dic73C2×C411yescharacteristic
84C23×Dic73C2312yes—
85C23×Dic73C2311yescharacteristic
86#14556584C2×C4116yes—
87#14556582C2×C418yes—
88#14556578C2×C418yes—
89#14556572C2×C4116yes—
90#14556570C2×C418yes—
91#14556556C2×C414yes—
92#14556556C2314yes—
93D4×Dic73C2×C414yes—
94D4×Dic73C2×C418yes—
95D4×Dic73C2314yes—
96D4×Dic73C2318yes—
97#14556547C2×C418yes—
98#14556547C2318yes—
99C2×C4⋊Dic73C2×C411yescharacteristic
100C2×C4⋊Dic73C2×C412yes—
101C2×#11924007C2×C412yes—
102C2×#11924007C2×C412yes—
103C2×#11924007C2×C412yes—
104C2×C22⋊Dic73C2×C412yes—
105C2×C22⋊Dic73C2×C412yes—
106C2×C22⋊Dic73C2×C412yes—
107C2×C22⋊Dic73C2×C412yes—
108C2×C4×Dic73C2×C411yescharacteristic
109C2×C4×Dic73C2×C411yescharacteristic
110C2×C4×Dic73C2×C414yes—
111C2×C4×Dic73C2311yescharacteristic
112C2×C4×Dic73C2311yescharacteristic
113C2×C4×Dic73C2312yes—
114C2×C4×Dic73C2314yes—
115C22×D4×C73C2×C411yescharacteristic
116C22×D4×C73C2×C412yes—
117C23×C4×C73C2×C411yescharacteristic
118#121×C73C2×C418yes—
119C2×C4⋊C4×C73C2×C411yescharacteristic
120C2×C22⋊C4×C73C2×C412yes—
121#11924037—28no—
122#11924037—28no—
123#11924037—216no—
124C2×#10596833—24no—
125C2×#10596833—22no—
126C2×#10596833—24no—
127C2×#10596833—22no—
128C2×#10596833—216no—
129C2×#10596833#4114yes—
130C2×#10596833#4114yes—
131C2×#10596833C2414yes—
132C2×#10596833C2412yes—
133C2×#10596833C2414yes—
134C2×#10596833C2412yes—
135C2×#10596833C2412yes—
136C2×#10596833C2412yes—
137C2×Dic146—24no—
138C2×Dic146C22×C414yes—
139C22×Dic73—24no—
140C22×Dic73—24no—
141C22×Dic73—22no—
142C22×Dic73—22no—
143C22×Dic73—42no—
144C22×Dic73C22×C411yescharacteristic
145C22×Dic73C22×C412yes—
146C22×Dic73C22×C411yescharacteristic
147C22×Dic73C22×C411yescharacteristic
148C22×Dic73C22×C412yes—
149C22×Dic73C22×C412yes—
150C22×Dic73C22×C414yes—
151C22×Dic73C22×C414yes—
152C22×Dic73C2414yes—
153C22×Dic73C2412yes—
154C22×Dic73C2411yescharacteristic
155C4⋊Dic73C22×C414yes—
156C4⋊Dic73C22×C418yes—
157#11924007C22×C418yes—
158#11924007C22×C418yes—
159#11924007C22×C418yes—
160C22⋊Dic73C22×C418yes—
161C22⋊Dic73C22×C418yes—
162C22⋊Dic73C22×C418yes—
163C22⋊Dic73C22×C418yes—
164C4×Dic73C22×C412yes—
165C4×Dic73C22×C412yes—
166C4×Dic73C22×C414yes—
167C4×Dic73C22×C414yes—
168C4×Dic73C22×C414yes—
169C4×Dic73C2412yes—
170C4×Dic73C2412yes—
171C4×Dic73C2414yes—
172C24×C73C22×C412yes—
173C24×C73C22×C411yescharacteristic
174C2×D4×C73—24no—
175C2×D4×C73—44no—
176C2×D4×C73C22×C414yes—
177C2×D4×C73C22×C418yes—
178C22×C4×C73—24no—
179C22×C4×C73C22×C411yescharacteristic
180C22×C4×C73C22×C412yes—
181C22×C4×C73C22×C414yes—
182C22×C4×C73C22×C411yescharacteristic
183C4⋊C4×C73C22×C414yes—
184C22⋊C4×C73C22×C418yes—
185C23×C73—24no—
186C23×C73—22no—
187C23×C73—22no—
188C23×C73—24no—
189C23×C73—22no—
190C23×C73—24no—
191C23×C73C23×C414yes—
192C23×C73C23×C412yes—
193C23×C73C23×C411yescharacteristic
194D4×C73—28no—
195D4×C73—48no—
196C2×C4×C73—22no—
197C2×C4×C73—24no—
198C2×C4×C73—28no—
199C2×C4×C73#13114yes—
200C2×C4×C73C23×C412yes—
201C2×C4×C73C23×C414yes—
202C2×C4×C73C23×C412yes—
203C2×Dic73—24no—
204C2×Dic73—22no—
205C2×Dic73—24no—
206C2×Dic73—22no—
207C2×Dic73—22no—
208C2×Dic73—28no—
209C2×Dic73—24no—
210C2×Dic73—24no—
211C2×Dic73—24no—
212C2×Dic73—22no—
213C2×Dic73—22no—
214C2×Dic73—44no—
215C2×Dic73C23×C412yes—
216C2×Dic73C23×C412yes—
217C2×Dic73C23×C412yes—
218C2×Dic73C23×C414yes—
219C2×Dic73C23×C414yes—
220C2×Dic73C23×C411yescharacteristic
221C2×Dic73C2×#4112yes—
222C2×Dic73C2×#4112yes—
223C2×Dic73C2511yescharacteristic
224Dic146—28no—
225#10596833—24no—
226#10596833—22no—
227#10596833—24no—
228#10596833—22no—
229#10596833C2×#4114yes—
230#10596833C2×#4114yes—
231C22×C73—24no—
232C22×C73—22no—
233C22×C73—22no—
234C22×C73—24no—
235C22×C73—22no—
236C22×C73—24no—
237C22×C73—44no—
238C22×C73—48no—
239C22×C73C4×#4112yes—
240C22×C73C2×#13112yes—
241C22×C73#56812yes—
242C22×C73C24×C411yescharacteristic
243C4×C73—22no—
244C4×C73—24no—
245C4×C73C2×#13114yes—
246Dic73—24no—
247Dic73—22no—
248Dic73—24no—
249Dic73—22no—
250Dic73—22no—
251Dic73C4×#4112yes—
252Dic73C22×#4111yescharacteristic
253Dic73C22×#4111yescharacteristic
254C2×#2969—731nomaximal, Sylow
255C2×C73—42no—
256C2×C73—44no—
257C2×C73#296914yescomplemented
258C2×C73C2×C4×#4111yescharacteristic
259C2×C73C22×#13111yescharacteristic
260C2×C73C2×#56811yescharacteristic
261C2×#529—731no—
262#2969—7316no—
263C2×#446—731no—
264C2×C8×Q8—731no—
265C2×#436—731no—
266C2×#436—732no—
267C2×#435—732no—
268C2×C8×D4—731no—
269C2×#433—732no—
270C2×#431—731no—
271C2×#405—732no—
272C2×C4×M16—731no—
273C73C2×#296911yescharacteristic, complemented, Sylow
274C22×M16—731no—
275C22×M16—731no—
276C22×M16—732no—
277C23×C8—732no—
278#529—738no—
279#529—738no—
280C2×#126—732no—
281C2×#125—731no—
282C2×#124—731no—
283C2×#123—732no—
284C2×C22⋊Q8—731no—
285C2×#121#6440811yescharacteristic
286C2×C4×Q8—731no—
287C2×C4×D4—731no—
288C2×C4×D4—732no—
289C2×#117—732no—
290C22×C42—731no—
291#446—738no—
292C8×Q8—738no—
293#436—738no—
294#436—7316no—
295#435—7316no—
296C8×D4—738no—
297#433—7316no—
298#431—738no—
299C2×C4⋊C8—731no—
300C2×C4⋊C8—731no—
301C2×C4⋊C8—732no—
302C2×C4⋊C8—732no—
303C2×C22⋊C8—732no—
304C2×C22⋊C8—732no—
305C2×C22⋊C8—732no—
306#405—7316no—
307C4×M16—738no—
308C2×#97—731no—
309C2×#97—731no—
310C2×#97—732no—
311C2×C4×C8—731no—
312C2×C4×C8—731no—
313C2×C4×C8—732no—
314C2×C4×C8—732no—
315C22×Q8—731no—
316C22×D4C2×#6440811yescharacteristic
317C22×D4C2×#6440812yes—
318C23×C4—731no—
319C23×C4—732no—
320C23×C4—731no—
321C23×C4C2×#6440811yescharacteristic
322C2×M16—734no—
323C2×M16—734no—
324C2×M16—738no—
325C2×M16—1464no—
326C2×M16—1464no—
327C2×M16—1468no—
328C22×C8—731no—
329C22×C8—731no—
330C22×C8—732no—
331C22×C8—738no—
332C22×C8—731no—
333C22×C8—732no—
334C22×C8—731no—
335C22×C8—1468no—
336#126—7316no—
337#125—738no—
338#124—738no—
339#123—7316no—
340C22⋊Q8—738no—
341#121C2×#6440818yes—
342C4×Q8—734no—
343C4×Q8—734no—
344C4×D4—734no—
345C4×D4—738no—
346C4×D4—734no—
347C4×D4—738no—
348#117—738no—
349#117—738no—
350C2×C4⋊C4—732no—
351C2×C4⋊C4—732no—
352C2×C4⋊C4—731no—
353C2×C4⋊C4—732no—
354C2×C4⋊C4—732no—
355C2×C4⋊C4C2×#6440811yescharacteristic
356C2×C22⋊C4—732no—
357C2×C22⋊C4—732no—
358C2×C22⋊C4—732no—
359C2×C22⋊C4—732no—
360C2×C22⋊C4C2×#6440812yes—
361C2×C42—731no—
362C2×C42—732no—
363C2×C42—731no—
364C2×C42—731no—
365C2×C42—731no—
366C2×C42—734no—
367C2×C42—734no—
368C4⋊C8—734no—
369C4⋊C8—734no—
370C4⋊C8—738no—
371C4⋊C8—738no—
372C22⋊C8—738no—
373C22⋊C8—738no—
374C22⋊C8—738no—
375#97—734no—
376#97—734no—
377#97—738no—
378C4×C8—734no—
379C4×C8—734no—
380C4×C8—738no—
381C4×C8—738no—
382C24C22×#6440812yes—
383C24C22×#6440811yescharacteristic
384C2×Q8—734no—
385C2×Q8—1464no—
386C2×D4—24no—
387C2×D4—44no—
388C2×D4C22×#6440814yes—
389C2×D4C22×#6440818yes—
390C22×C4—24no—
391C22×C4—2922no—
392C22×C4—734no—
393C22×C4—731no—
394C22×C4—732no—
395C22×C4—734no—
396C22×C4—732no—
397C22×C4—734no—
398C22×C4—731no—
399C22×C4—731no—
400C22×C4—731no—
401C22×C4—732no—
402C22×C4—732no—
403C22×C4—1464no—
404C22×C4—1464no—
405C22×C4—1462no—
406C22×C4—1462no—
407C22×C4C22×#6440814yes—
408C22×C4C22×#6440811yescharacteristic
409C22×C4C22×#6440812yes—
410C22×C4C22×#6440811yescharacteristic
411M16—1468no—
412M16—1468no—
413M16—14616no—
414C2×C8—734no—
415C2×C8—734no—
416C2×C8—732no—
417C2×C8—732no—
418C2×C8—734no—
419C2×C8—732no—
420C2×C8—734no—
421C2×C8—732no—
422C2×C8—14616no—
423C2×C8—1464no—
424C2×C8—1462no—
425C2×C8—1462no—
426C2×C8—1464no—
427C4⋊C4—738no—
428C4⋊C4—738no—
429C4⋊C4—738no—
430C4⋊C4—734no—
431C4⋊C4—738no—
432C4⋊C4C22×#6440814yes—
433C22⋊C4—738no—
434C22⋊C4—738no—
435C22⋊C4—738no—
436C22⋊C4—738no—
437C22⋊C4C22×#6440818yes—
438C42—732no—
439C42—732no—
440C42—734no—
441C42—734no—
442C42—734no—
443C42—732no—
444C42—732no—
445C42—734no—
446C23—24no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—22no—
451C23—24no—
452C23C23×#6440814yes—
453C23C23×#6440812yes—
454C23C23×#6440811yescharacteristic
455Q8—1468no—
456D4—28no—
457D4—48no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—2924no—
462C2×C4—732no—
463C2×C4—732no—
464C2×C4—731no—
465C2×C4—731no—
466C2×C4—732no—
467C2×C4—732no—
468C2×C4—732no—
469C2×C4—734no—
470C2×C4—734no—
471C2×C4—1464no—
472C2×C4—1462no—
473C2×C4—1462no—
474C2×C4—1464no—
475C2×C4—1464no—
476C2×C4—1468no—
477C2×C4—1462no—
478C2×C4—1464no—
479C2×C4—1464no—
480C2×C4—1462no—
481C2×C4—1462no—
482C2×C4#1455667814yes—
483C2×C4C23×#6440812yes—
484C2×C4C23×#6440814yes—
485C2×C4C23×#6440812yes—
486C8—734no—
487C8—734no—
488C8—1464no—
489C8—1462no—
490C8—1462no—
491C8—1464no—
492C22—24no—
493C22—22no—
494C22—22no—
495C22—24no—
496C22—22no—
497C22—24no—
498C22—44no—
499C22—48no—
500C22#41×#6440812yes—
501C22C2×#1455667812yes—
502C22#1798966512yes—
503C22C24×#6440811yescharacteristic
504C4—22no—
505C4—24no—
506C4—731no—
507C4—731no—
508C4—732no—
509C4—1462no—
510C4—1464no—
511C4—1464no—
512C4—1462no—
513C4—1462no—
514C4C2×#1455667814yes—
515C2—42no—
516C2—44no—
517C2#2262676214yesdirect product factor
518C2C2×#41×#6440811yescharacteristic
519C2C22×#1455667811yescharacteristic
520C2C2×#1798966511yescharacteristic
subgroupG/H
centerC23C23×#64408
commutatorC22×C73C24×C4
FrattiniC22C24×#64408
Fitting, prime radicalC2×#121×C73C4
socleC23×C73C23×C4
cosocleC2×Dic73C25
hypercenterC2×#121#64408
hypocenterC73C2×#2969
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #22626762C2direct (N × H)
2. C2×#17989637C2semidirect
3. C2×#17989636C2semidirect
4. C2×#17989622C2semidirect
5. C2×#17989622C2semidirect
6. C2×#17989618C2semidirect
7. C2×D4×#10596833C2semidirect
8. C2×#17989611C2semidirect
9. C2×#17989603C2semidirect
10. C2×#17989551C2semidirect
11. C2×C4×#11924037C2semidirect
17. #17989637C22semidirect
18. #17989636C22semidirect
19. #17989622C22semidirect
20. #17989622C22semidirect
21. #17989618C22semidirect
22. D4×#10596833C22semidirect
23. #17989611C22semidirect
24. #17989603C22semidirect
32. #17989551C22semidirect
33. C4×#11924037C22semidirect
257. C2×C73#2969semidirect
273. C73C2×#2969semidirect
517. C2#22626762direct (N × H)

Maximal subgroups

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

subgroupindex
#226267622
C2×#179896372
C2×#179896362
C2×#179896222
C2×#179896182
C2×D4×#105968332
C2×#179896112
C2×#179896032
C2×#179895512
C2×C4×#119240372
C2×#179891182
C2×#296973