Finite Group Databasebeta

← C2×#22626882 #34031967

Subgroups — C2×#22626882 #34031967

Order 18688.

1982 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626882C2116yesmaximal, direct product factor
2C2×#17989638C211yesmaximal, characteristic, complemented
3C2×Q8×#10596833C211yesmaximal, characteristic, complemented
4C2×#17989621C211yesmaximal, characteristic, complemented
5C2×#17989616C212yesmaximal, complemented
6C2×#17989615C211yesmaximal, characteristic, complemented
7C2×#17989614C212yesmaximal, complemented
8C2×#17989612C212yesmaximal, complemented
9C2×#17989604C211yesmaximal, characteristic, complemented
10C2×#17989594C212yesmaximal, complemented
11C22×C4⋊#10596833C211yesmaximal, characteristic, complemented
12C2×#17989123C211yesmaximal, characteristic
13C22×#11924038C2212yes—
14C22×#11924036C2212yes—
15C23×#10596833C2212yes—
16#17989638C2218yescomplemented
17Q8×#10596833C2218yescomplemented
18#17989621C2218yescomplemented
19#17989616C22116yescomplemented
20#17989615C2218yescomplemented
21#17989614C22116yescomplemented
22#17989612C22116yescomplemented
23#17989604C2218yescomplemented
24#17989594C22116yescomplemented
25C2×#14556658C2211yescharacteristic
26C2×#14556658C2212yes—
27C2×#14556658C2211yescharacteristic
28C2×C4⋊#10596833C2212yes—
29C2×C4⋊#10596833C2211yescharacteristic
30C2×C4⋊#10596833C2211yescharacteristic
31C2×C4⋊#10596833C2212yes—
32C2×C4⋊#10596833C2218yescomplemented
33C2×#14556648C2212yes—
34C2×#14556648C2212yes—
35C2×C22⋊#10596833C2211yescharacteristic
36C2×C22⋊#10596833C2211yescharacteristic
37C2×#14556646C2212yes—
38C2×C4×#10596833C2211yescharacteristic
39C2×C4×#10596833C2211yescharacteristic
40C2×C4×#10596833C2212yes—
41#17989123C418yes—
42#17989123C2218yes—
43C2×#14556586C412yes—
44C2×#14556582C411yescharacteristic
45C2×#14556580C411yescharacteristic
46C2×#14556575C412yes—
47C2×#14556565C411yescharacteristic
48C2×Q8×Dic73C2211yescharacteristic
49C2×#14556553C2211yescharacteristic
50C2×#14556551C2212yes—
51C2×#14556549C2212yes—
52C22×C4×Dic73C2211yescharacteristic
53C2×C22⋊Q8×C73C411yescharacteristic
54C2×#11924038—44no—
55C2×#11924038C2318yes—
56C2×#11924036—44no—
57C2×#11924036C2318yes—
58C22×#10596833—44no—
59C22×#10596833C2312yes—
60C22×#10596833C2312yes—
61C22×#10596833C2311yescharacteristic
62C22×#10596833C2311yescharacteristic
63C22×#10596833C2311yescharacteristic
64C22×#10596833C2311yescharacteristic
65C22×#10596833C2318yes—
66#14556658C2318yes—
67#14556658C2314yes—
68#14556658C2314yes—
69C4⋊#10596833C2314yes—
70C4⋊#10596833C2318yes—
71C4⋊#10596833C2314yes—
72C4⋊#10596833C2318yes—
73#14556648C2318yes—
74#14556648C2318yes—
75C22⋊#10596833C2314yes—
76C22⋊#10596833C2314yes—
77#14556646C2318yes—
78C4×#10596833C2314yes—
79C4×#10596833C2318yes—
80C4×#10596833C2314yes—
81C22×D292C2×C411yescharacteristic
82C22×C22⋊D73C2×C412yes—
83C23×Dic73C2×C411yescharacteristic
84C23×Dic73C2311yescharacteristic
85C22×C4×D73C2312yes—
86#14556586C2×C4116yes—
87#14556582C2×C418yes—
88#14556580C2×C418yes—
89#14556575C2×C4116yes—
90#14556565C2×C418yes—
91Q8×Dic73C2×C414yes—
92Q8×Dic73C2314yes—
93#14556553C2×C414yes—
94#14556553C2314yes—
95#14556551C2×C418yes—
96#14556551C2318yes—
97#14556549C2×C418yes—
98#14556549C2318yes—
99C2×C4⋊Dic73C2×C412yes—
100C2×C4⋊Dic73C2×C411yescharacteristic
101C2×C4⋊Dic73C2×C412yes—
102C2×#11924007C2×C412yes—
103C2×#11924005C2×C412yes—
104C2×#11924005C2×C412yes—
105C2×#11924005C2×C412yes—
106C2×#11924005C2×C412yes—
107C2×C4×Dic73C2×C414yes—
108C2×C4×Dic73C2×C411yescharacteristic
109C2×C4×Dic73C2×C411yescharacteristic
110C2×C4×Dic73C2314yes—
111C2×C4×Dic73C2311yescharacteristic
112C2×C4×Dic73C2312yes—
113C2×C4×Dic73C2311yescharacteristic
114C22×Q8×C73C2×C411yescharacteristic
115C23×C4×C73C2×C411yescharacteristic
116C22⋊Q8×C73C2×C418yes—
117C2×C4⋊C4×C73C2×C412yes—
118C2×C4⋊C4×C73C2×C411yescharacteristic
119C2×C22⋊C4×C73C2×C412yes—
120#11924038—48no—
121#11924036—48no—
122C2×#10596833—24no—
123C2×#10596833—24no—
124C2×#10596833—22no—
125C2×#10596833—22no—
126C2×#10596833—22no—
127C2×#10596833—22no—
128C2×#10596833—48no—
129C2×#10596833C2414yes—
130C2×#10596833C2414yes—
131C2×#10596833C2412yes—
132C2×#10596833C2412yes—
133C2×#10596833C2412yes—
134C2×#10596833C2412yes—
135C23×D73C22×C412yes—
136C2×D292—24no—
137C2×D292C22×C414yes—
138C2×C22⋊D73—28no—
139C2×C22⋊D73C22×C418yes—
140C22×Dic73—24no—
141C22×Dic73—22no—
142C22×Dic73—22no—
143C22×Dic73C22×C411yescharacteristic
144C22×Dic73C22×C412yes—
145C22×Dic73C22×C411yescharacteristic
146C22×Dic73C22×C411yescharacteristic
147C22×Dic73C22×C414yes—
148C22×Dic73C22×C412yes—
149C22×Dic73C2412yes—
150C22×Dic73C2411yescharacteristic
151C2×C4×D73—24no—
152C2×C4×D73—24no—
153C2×C4×D73C22×C414yes—
154C2×C4×D73C2414yes—
155C4⋊Dic73C22×C418yes—
156C4⋊Dic73C22×C414yes—
157C4⋊Dic73C22×C418yes—
158#11924007C22×C418yes—
159#11924005C22×C418yes—
160#11924005C22×C418yes—
161#11924005C22×C418yes—
162#11924005C22×C418yes—
163C4×Dic73C22×C412yes—
164C4×Dic73C22×C412yes—
165C4×Dic73C22×C414yes—
166C4×Dic73C22×C414yes—
167C4×Dic73C22×C414yes—
168C4×Dic73C2412yes—
169C4×Dic73C2412yes—
170C4×Dic73C2414yes—
171C24×C73C22×C411yescharacteristic
172C2×Q8×C73—24no—
173C2×Q8×C73C22×C414yes—
174C22×C4×C73—24no—
175C22×C4×C73C22×C412yes—
176C22×C4×C73C22×C412yes—
177C22×C4×C73C22×C411yescharacteristic
178C22×C4×C73C22×C414yes—
179C22×C4×C73C22×C411yescharacteristic
180C4⋊C4×C73C22×C418yes—
181C4⋊C4×C73C22×C414yes—
182C22⋊C4×C73C22×C418yes—
183C23×C73—22no—
184C23×C73—22no—
185C23×C73#13114yes—
186C23×C73C23×C412yes—
187C23×C73C23×C411yescharacteristic
188C22×D73—24no—
189C22×D73—24no—
190C22×D73—24no—
191C22×D73C23×C414yes—
192Q8×C73—28no—
193C2×C4×C73—24no—
194C2×C4×C73—22no—
195C2×C4×C73—24no—
196C2×C4×C73—28no—
197C2×C4×C73#13114yes—
198C2×C4×C73C23×C414yes—
199C2×C4×C73C23×C414yes—
200C2×C4×C73C23×C412yes—
201C2×C4×C73C23×C412yes—
202C22⋊D73—216no—
203C2×Dic73—22no—
204C2×Dic73—28no—
205C2×Dic73—22no—
206C2×Dic73—22no—
207C2×Dic73—24no—
208C2×Dic73—24no—
209C2×Dic73—22no—
210C2×Dic73—22no—
211C2×Dic73C23×C412yes—
212C2×Dic73C23×C412yes—
213C2×Dic73C23×C412yes—
214C2×Dic73C23×C414yes—
215C2×Dic73C23×C411yescharacteristic
216C2×Dic73H(5,𝔽2)12yes—
217C2×Dic732−1+412yes—
218C2×Dic73C2511yescharacteristic
219D292—28no—
220C4×D73—28no—
221C4×D73—24no—
222C4×D73—24no—
223#10596833—24no—
224#10596833—24no—
225#10596833—22no—
226#10596833—22no—
227#10596833—22no—
228#10596833—22no—
229C22×C73—22no—
230C22×C73—28no—
231C22×C73—22no—
232C22×C73#52012yes—
233C22×C73C2×#13112yes—
234C22×C73C2×#13114yes—
235C22×C73C2×#13112yes—
236C22×C73C24×C411yescharacteristic
237C2×D73—24no—
238C2×D73—24no—
239C2×D73—24no—
240C2×D73—48no—
241C4×C73—24no—
242C4×C73—22no—
243C4×C73—24no—
244C4×C73C2×#13114yes—
245Dic73—22no—
246Dic73—22no—
247Dic73—22no—
248Dic73—24no—
249Dic73#52012yes—
250Dic73C2×H(5,𝔽2)11yescharacteristic
251Dic73C2×2−1+411yescharacteristic
252C2×#2994—731nomaximal, Sylow
253C2×C73—24no—
254C2×C73#299414yescomplemented
255C2×C73C2×#52011yescharacteristic
256C2×C73C22×#13111yescharacteristic
257C2×C73C22×#13111yescharacteristic
258D73—44no—
259C2×#529—731no—
260#2994—7316no—
261C2×#446—731no—
262C2×C8×Q8—731no—
263C2×#436—731no—
264C2×#435—732no—
265C2×C8×D4—732no—
266C2×C8×D4—731no—
267C2×#433—732no—
268C2×#431—731no—
269C2×#424—732no—
270C22×C4⋊C8—731no—
271C73C2×#299411yescharacteristic, complemented, Sylow
272C22×M16—732no—
273C23×C8—732no—
274C23×C8—732no—
275#529—738no—
276#529—738no—
277C2×#126—732no—
278C2×#125—731no—
279C2×#124—731no—
280C2×#123—732no—
281C2×C22⋊Q8#6440811yescharacteristic
282C2×#121—731no—
283C2×C4×Q8—731no—
284C2×C4×D4—731no—
285C2×C4×D4—732no—
286C2×#117—732no—
287C22×C42—731no—
288#446—738no—
289C8×Q8—738no—
290#436—738no—
291#435—7316no—
292C8×D4—7316no—
293C8×D4—738no—
294#433—7316no—
295#431—738no—
296#424—7316no—
297C2×C4⋊C8—732no—
298C2×C4⋊C8—731no—
299C2×C4⋊C8—731no—
300C2×C4⋊C8—738no—
301C2×C4⋊C8—731no—
302C2×C4⋊C8—732no—
303C2×C4⋊C8—732no—
304C2×C4⋊C8—731no—
305C2×C22⋊C8—731no—
306C2×C22⋊C8—731no—
307C2×C22⋊C8—732no—
308C2×C22⋊C8—732no—
309C2×#97—732no—
310C2×C4×C8—731no—
311C2×C4×C8—731no—
312C2×C4×C8—732no—
313C22×Q8C2×#6440811yescharacteristic
314C22×D4—732no—
315C22×D4—731no—
316C23×C4—731no—
317C23×C4—731no—
318C23×C4—732no—
319C23×C4C2×#6440811yescharacteristic
320C2×M16—2924no—
321C2×M16—738no—
322C22×C8—2924no—
323C22×C8—2924no—
324C22×C8—738no—
325C22×C8—738no—
326C22×C8—732no—
327C22×C8—731no—
328C22×C8—732no—
329C22×C8—731no—
330C22×C8—731no—
331C22×C8—731no—
332#126—7316no—
333#125—738no—
334#124—738no—
335#123—7316no—
336C22⋊Q8C2×#6440818yes—
337#121—738no—
338C4×Q8—734no—
339C4×Q8—734no—
340C4×D4—734no—
341C4×D4—734no—
342C4×D4—738no—
343C4×D4—738no—
344#117—738no—
345#117—738no—
346C2×C4⋊C4—732no—
347C2×C4⋊C4—732no—
348C2×C4⋊C4—731no—
349C2×C4⋊C4—732no—
350C2×C4⋊C4C2×#6440812yes—
351C2×C4⋊C4C2×#6440811yescharacteristic
352C2×C22⋊C4—732no—
353C2×C22⋊C4—732no—
354C2×C22⋊C4—732no—
355C2×C22⋊C4—732no—
356C2×C22⋊C4C2×#6440812yes—
357C2×C42—731no—
358C2×C42—732no—
359C2×C42—731no—
360C2×C42—731no—
361C2×C42—731no—
362C2×C42—734no—
363C2×C42—734no—
364C4⋊C8—738no—
365C4⋊C8—734no—
366C4⋊C8—734no—
367C4⋊C8—734no—
368C4⋊C8—738no—
369C4⋊C8—738no—
370C4⋊C8—734no—
371C22⋊C8—734no—
372C22⋊C8—734no—
373C22⋊C8—738no—
374C22⋊C8—738no—
375#97—738no—
376C4×C8—734no—
377C4×C8—734no—
378C4×C8—738no—
379C24—732no—
380C24C22×#6440811yescharacteristic
381C2×Q8—24no—
382C2×Q8C22×#6440814yes—
383C2×D4—738no—
384C2×D4—734no—
385C2×D4—1468no—
386C2×D4—1464no—
387C22×C4—24no—
388C22×C4—731no—
389C22×C4—732no—
390C22×C4—734no—
391C22×C4—732no—
392C22×C4—731no—
393C22×C4—731no—
394C22×C4—731no—
395C22×C4—732no—
396C22×C4—734no—
397C22×C4—734no—
398C22×C4—1464no—
399C22×C4—1464no—
400C22×C4—1462no—
401C22×C4—1462no—
402C22×C4—1464no—
403C22×C4C22×#6440814yes—
404C22×C4C22×#6440811yescharacteristic
405C22×C4C22×#6440812yes—
406C22×C4C22×#6440812yes—
407C22×C4C22×#6440811yescharacteristic
408M16—2928no—
409C2×C8—2928no—
410C2×C8—2928no—
411C2×C8—732no—
412C2×C8—732no—
413C2×C8—734no—
414C2×C8—732no—
415C2×C8—732no—
416C2×C8—734no—
417C2×C8—1464no—
418C2×C8—1462no—
419C2×C8—1464no—
420C2×C8—1462no—
421C2×C8—1462no—
422C2×C8—1462no—
423C4⋊C4—738no—
424C4⋊C4—738no—
425C4⋊C4—734no—
426C4⋊C4—738no—
427C4⋊C4C22×#6440818yes—
428C4⋊C4C22×#6440814yes—
429C22⋊C4—738no—
430C22⋊C4—738no—
431C22⋊C4—738no—
432C22⋊C4—738no—
433C22⋊C4C22×#6440818yes—
434C42—732no—
435C42—732no—
436C42—734no—
437C42—734no—
438C42—734no—
439C42—732no—
440C42—734no—
441C42—732no—
442C23—22no—
443C23—22no—
444C23—734no—
445C23—1464no—
446C23—1464no—
447C23—1464no—
448C23#1455667914yes—
449C23C23×#6440812yes—
450C23C23×#6440811yescharacteristic
451Q8—28no—
452D4—14616no—
453D4—1468no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—732no—
459C2×C4—732no—
460C2×C4—731no—
461C2×C4—731no—
462C2×C4—732no—
463C2×C4—732no—
464C2×C4—732no—
465C2×C4—734no—
466C2×C4—1464no—
467C2×C4—1464no—
468C2×C4—1468no—
469C2×C4—1462no—
470C2×C4—1462no—
471C2×C4—1464no—
472C2×C4—1462no—
473C2×C4—1468no—
474C2×C4—1462no—
475C2×C4—1462no—
476C2×C4—1464no—
477C2×C4#1455667814yes—
478C2×C4C23×#6440812yes—
479C2×C4C23×#6440814yes—
480C2×C4C23×#6440814yes—
481C2×C4C23×#6440812yes—
482C8—1464no—
483C8—1462no—
484C8—1464no—
485C8—1462no—
486C8—1462no—
487C8—1462no—
488C22—28no—
489C22—22no—
490C22—22no—
491C22—2928no—
492C22—1464no—
493C22—1464no—
494C22—1464no—
495C22#1798965612yes—
496C22C2×#1455667812yes—
497C22C2×#1455667914yes—
498C22C2×#1455667912yes—
499C22C24×#6440811yescharacteristic
500C4—24no—
501C4—22no—
502C4—24no—
503C4—731no—
504C4—731no—
505C4—732no—
506C4—1462no—
507C4—1462no—
508C4—1462no—
509C4—1464no—
510C4C2×#1455667814yes—
511C2—24no—
512C2—2924no—
513C2#2262688214yesdirect product factor
514C2C2×#1798965611yescharacteristic
515C2C22×#1455667811yescharacteristic
516C2C22×#1455667911yescharacteristic
subgroupG/H
centerC23C23×#64408
commutatorC22×C73C24×C4
FrattiniC22C24×#64408
FittingC2×C22⋊Q8×C73C4
socleC23×C73C23×C4
prime radicalC2×#14556565C4
cosocleC2×Dic73C25
hypercenterC2×C22⋊Q8#64408
hypocenterC73C2×#2994
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22626882C2direct (N × H)
2. C2×#17989638C2semidirect
3. C2×Q8×#10596833C2semidirect
4. C2×#17989621C2semidirect
5. C2×#17989616C2semidirect
6. C2×#17989615C2semidirect
7. C2×#17989614C2semidirect
8. C2×#17989612C2semidirect
9. C2×#17989604C2semidirect
10. C2×#17989594C2semidirect
11. C22×C4⋊#10596833C2semidirect
16. #17989638C22semidirect
17. Q8×#10596833C22semidirect
18. #17989621C22semidirect
19. #17989616C22semidirect
20. #17989615C22semidirect
21. #17989614C22semidirect
22. #17989612C22semidirect
23. #17989604C22semidirect
24. #17989594C22semidirect
32. C2×C4⋊#10596833C22semidirect
254. C2×C73#2994semidirect
271. C73C2×#2994semidirect
513. C2#22626882direct (N × H)

Maximal subgroups

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

subgroupindex
#226268822
C2×#179896382
C2×Q8×#105968332
C2×#179896212
C2×#179896162
C2×#179896152
C2×#179896142
C2×#179896122
C2×#179896042
C2×#179895942
C22×C4⋊#105968332
C2×#179891232
C2×#299473