Finite Group Databasebeta

← C2×#22626764 #34031551

Subgroups — C2×#22626764 #34031551

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#22626764C2116yesmaximal, direct product factor
2C2×#17989638C211yesmaximal, characteristic, complemented
3C2×Q8×#10596833C211yesmaximal, characteristic, complemented
4C2×#17989621C211yesmaximal, characteristic, complemented
5C2×#17989620C212yesmaximal, complemented
6C2×#17989619C212yesmaximal, complemented
7C2×#17989615C211yesmaximal, characteristic, complemented
8C2×#17989612C212yesmaximal, complemented
9C2×#17989603C211yesmaximal, characteristic, complemented
10C2×#17989552C212yesmaximal, complemented
11C2×C4×#11924037C211yesmaximal, characteristic, complemented
12C2×#17989123C211yesmaximal, characteristic
13C22×#11924038C2212yes—
14C22×#11924037C2211yescharacteristic
15C22×#11924037C2211yescharacteristic
16C22×#11924036C2212yes—
17#17989638C2218yescomplemented
18Q8×#10596833C2218yescomplemented
19#17989621C2218yescomplemented
20#17989620C22116yescomplemented
21#17989619C22116yescomplemented
22#17989615C2218yescomplemented
23#17989612C22116yescomplemented
24#17989603C2218yescomplemented
25C2×#14556658C2212yes—
26C2×C4⋊#10596833C2211yescharacteristic
27C2×C4⋊#10596833C2211yescharacteristic
28C2×C4⋊#10596833C2212yes—
29C2×#14556648C2212yes—
30C2×#14556648C2212yes—
31C2×C22⋊#10596833C2212yes—
32#17989552C22116yescomplemented
33C4×#11924037C2218yescomplemented
34C2×#14556646C2211yescharacteristic
35C2×#14556646C2212yes—
36C2×#14556646C2211yescharacteristic
37C2×C4×#10596833C2212yes—
38C2×C4×#10596833C2211yescharacteristic
39C2×C4×#10596833C2212yes—
40C2×C4×#10596833C2211yescharacteristic
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—28no—
55C2×#11924038C2318yes—
56C2×#11924037—24no—
57C2×#11924037—24no—
58C2×#11924037C2314yes—
59C2×#11924037C2314yes—
60C2×#11924036—28no—
61C2×#11924036C2318yes—
62C22×#10596833C2312yes—
63C22×#10596833C2311yescharacteristic
64C22×#10596833C2311yescharacteristic
65C22×#10596833C2312yes—
66C22×#10596833C2311yescharacteristic
67C22×#10596833C2311yescharacteristic
68#14556658C2318yes—
69C4⋊#10596833C2314yes—
70C4⋊#10596833C2318yes—
71C4⋊#10596833C2314yes—
72#14556648C2318yes—
73#14556648C2318yes—
74C22⋊#10596833C2318yes—
75#14556646C2318yes—
76#14556646C2314yes—
77#14556646C2314yes—
78C4×#10596833C2314yes—
79C4×#10596833C2318yes—
80C4×#10596833C2314yes—
81C4×#10596833C2318yes—
82C22×D292C2×C411yescharacteristic
83C22×C22⋊D73C2×C412yes—
84C23×Dic73C2×C411yescharacteristic
85C23×Dic73C2311yescharacteristic
86C22×C4×D73C2312yes—
87#14556586C2×C4116yes—
88#14556582C2×C418yes—
89#14556580C2×C418yes—
90#14556575C2×C4116yes—
91#14556565C2×C418yes—
92Q8×Dic73C2×C414yes—
93Q8×Dic73C2314yes—
94#14556553C2×C414yes—
95#14556553C2314yes—
96#14556551C2×C418yes—
97#14556551C2318yes—
98#14556549C2×C418yes—
99#14556549C2318yes—
100C2×C4⋊Dic73C2×C412yes—
101C2×C4⋊Dic73C2×C411yescharacteristic
102C2×C4⋊Dic73C2×C412yes—
103C2×#11924007C2×C412yes—
104C2×#11924005C2×C412yes—
105C2×#11924005C2×C412yes—
106C2×#11924005C2×C412yes—
107C2×#11924005C2×C412yes—
108C2×C4×Dic73C2×C414yes—
109C2×C4×Dic73C2×C411yescharacteristic
110C2×C4×Dic73C2×C411yescharacteristic
111C2×C4×Dic73C2314yes—
112C2×C4×Dic73C2311yescharacteristic
113C2×C4×Dic73C2311yescharacteristic
114C2×C4×Dic73C2312yes—
115C22×Q8×C73C2×C411yescharacteristic
116C23×C4×C73C2×C411yescharacteristic
117C22⋊Q8×C73C2×C418yes—
118C2×C4⋊C4×C73C2×C412yes—
119C2×C4⋊C4×C73C2×C411yescharacteristic
120C2×C22⋊C4×C73C2×C412yes—
121#11924038—216no—
122#11924037—28no—
123#11924037—28no—
124#11924036—216no—
125C2×#10596833—24no—
126C2×#10596833—22no—
127C2×#10596833—22no—
128C2×#10596833—24no—
129C2×#10596833#4114yes—
130C2×#10596833#4114yes—
131C2×#10596833C2414yes—
132C2×#10596833C2412yes—
133C2×#10596833C2412yes—
134C2×#10596833C2414yes—
135C2×#10596833C2412yes—
136C2×#10596833C2412yes—
137C23×D73C22×C412yes—
138C2×D292—24no—
139C2×D292C22×C414yes—
140C2×C22⋊D73—44no—
141C2×C22⋊D73C22×C418yes—
142C22×Dic73—22no—
143C22×Dic73—22no—
144C22×Dic73—42no—
145C22×Dic73C22×C411yescharacteristic
146C22×Dic73C22×C412yes—
147C22×Dic73C22×C411yescharacteristic
148C22×Dic73C22×C411yescharacteristic
149C22×Dic73C22×C412yes—
150C22×Dic73C22×C414yes—
151C22×Dic73C2412yes—
152C22×Dic73C2411yescharacteristic
153C2×C4×D73—24no—
154C2×C4×D73—24no—
155C2×C4×D73C22×C414yes—
156C2×C4×D73C2414yes—
157C4⋊Dic73C22×C418yes—
158C4⋊Dic73C22×C414yes—
159C4⋊Dic73C22×C418yes—
160#11924007C22×C418yes—
161#11924005C22×C418yes—
162#11924005C22×C418yes—
163#11924005C22×C418yes—
164#11924005C22×C418yes—
165C4×Dic73C22×C412yes—
166C4×Dic73C22×C412yes—
167C4×Dic73C22×C414yes—
168C4×Dic73C22×C414yes—
169C4×Dic73C22×C414yes—
170C4×Dic73C2412yes—
171C4×Dic73C2412yes—
172C4×Dic73C2414yes—
173C24×C73C22×C411yescharacteristic
174C2×Q8×C73—24no—
175C2×Q8×C73C22×C414yes—
176C22×C4×C73—24no—
177C22×C4×C73C22×C412yes—
178C22×C4×C73C22×C412yes—
179C22×C4×C73C22×C411yescharacteristic
180C22×C4×C73C22×C414yes—
181C22×C4×C73C22×C411yescharacteristic
182C4⋊C4×C73C22×C418yes—
183C4⋊C4×C73C22×C414yes—
184C22⋊C4×C73C22×C418yes—
185C23×C73—22no—
186C23×C73—22no—
187C23×C73—22no—
188C23×C73C23×C412yes—
189C23×C73C23×C411yescharacteristic
190C22×D73—24no—
191C22×D73—24no—
192C22×D73—24no—
193C22×D73C23×C414yes—
194Q8×C73—28no—
195C2×C4×C73—24no—
196C2×C4×C73—22no—
197C2×C4×C73—24no—
198C2×C4×C73—28no—
199C2×C4×C73#13114yes—
200C2×C4×C73C23×C414yes—
201C2×C4×C73C23×C414yes—
202C2×C4×C73C23×C412yes—
203C2×C4×C73C23×C412yes—
204C22⋊D73—48no—
205C2×Dic73—22no—
206C2×Dic73—22no—
207C2×Dic73—22no—
208C2×Dic73—24no—
209C2×Dic73—24no—
210C2×Dic73—22no—
211C2×Dic73—22no—
212C2×Dic73—44no—
213C2×Dic73C23×C412yes—
214C2×Dic73C23×C412yes—
215C2×Dic73C23×C412yes—
216C2×Dic73C23×C414yes—
217C2×Dic73C23×C411yescharacteristic
218C2×Dic73C2×#4112yes—
219C2×Dic73C2×#4112yes—
220C2×Dic73C2511yescharacteristic
221D292—28no—
222C4×D73—28no—
223C4×D73—24no—
224C4×D73—24no—
225#10596833—24no—
226#10596833—22no—
227#10596833—22no—
228#10596833—24no—
229#10596833C2×#4114yes—
230#10596833C2×#4114yes—
231C22×C73—22no—
232C22×C73—22no—
233C22×C73—22no—
234C22×C73—44no—
235C22×C73C4×#4112yes—
236C22×C73C2×#13112yes—
237C22×C73#56812yes—
238C22×C73C24×C411yescharacteristic
239C2×D73—24no—
240C2×D73—24no—
241C2×D73—24no—
242C2×D73—48no—
243C4×C73—24no—
244C4×C73—22no—
245C4×C73—24no—
246C4×C73C2×#13114yes—
247Dic73—22no—
248Dic73—22no—
249Dic73—22no—
250Dic73—24no—
251Dic73C4×#4112yes—
252Dic73C22×#4111yescharacteristic
253Dic73C22×#4111yescharacteristic
254C2×#2969—731nomaximal, Sylow
255C2×C73—42no—
256C2×C73#296914yescomplemented
257C2×C73C2×C4×#4111yescharacteristic
258C2×C73C22×#13111yescharacteristic
259C2×C73C2×#56811yescharacteristic
260D73—44no—
261C2×#529—731no—
262#2969—7316no—
263C2×#446—731no—
264C2×C8×Q8—731no—
265C2×#436—732no—
266C2×#436—731no—
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#6440811yescharacteristic
285C2×#121—731no—
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—7316no—
294#436—738no—
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×Q8C2×#6440811yescharacteristic
316C22×D4—732no—
317C22×D4—731no—
318C23×C4—731no—
319C23×C4—731no—
320C23×C4—732no—
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—731no—
332C22×C8—731no—
333C22×C8—732no—
334C22×C8—738no—
335C22×C8—1468no—
336#126—7316no—
337#125—738no—
338#124—738no—
339#123—7316no—
340C22⋊Q8C2×#6440818yes—
341#121—738no—
342C4×Q8—734no—
343C4×Q8—734no—
344C4×D4—734no—
345C4×D4—734no—
346C4×D4—738no—
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⋊C4C2×#6440812yes—
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—
382C24—732no—
383C24C22×#6440811yescharacteristic
384C2×Q8—24no—
385C2×Q8C22×#6440814yes—
386C2×D4—2924no—
387C2×D4—738no—
388C2×D4—734no—
389C2×D4—1464no—
390C22×C4—24no—
391C22×C4—2922no—
392C22×C4—734no—
393C22×C4—731no—
394C22×C4—732no—
395C22×C4—732no—
396C22×C4—731no—
397C22×C4—731no—
398C22×C4—731no—
399C22×C4—732no—
400C22×C4—734no—
401C22×C4—734no—
402C22×C4—1464no—
403C22×C4—1464no—
404C22×C4—1462no—
405C22×C4—1462no—
406C22×C4C22×#6440814yes—
407C22×C4C22×#6440811yescharacteristic
408C22×C4C22×#6440812yes—
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—732no—
421C2×C8—734no—
422C2×C8—14616no—
423C2×C8—1464no—
424C2×C8—1462no—
425C2×C8—1462no—
426C2×C8—1464no—
427C4⋊C4—738no—
428C4⋊C4—738no—
429C4⋊C4—734no—
430C4⋊C4—738no—
431C4⋊C4C22×#6440818yes—
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—22no—
447C23—22no—
448C23—22no—
449C23—734no—
450C23—1464no—
451C23—1464no—
452C23—1464no—
453C23C23×#6440812yes—
454C23C23×#6440811yescharacteristic
455Q8—28no—
456D4—2928no—
457D4—1468no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—28no—
462C2×C4—2924no—
463C2×C4—732no—
464C2×C4—732no—
465C2×C4—731no—
466C2×C4—731no—
467C2×C4—732no—
468C2×C4—732no—
469C2×C4—732no—
470C2×C4—734no—
471C2×C4—1464no—
472C2×C4—1464no—
473C2×C4—1468no—
474C2×C4—1462no—
475C2×C4—1462no—
476C2×C4—1464no—
477C2×C4—1462no—
478C2×C4—1462no—
479C2×C4—1462no—
480C2×C4—1464no—
481C2×C4#1455667814yes—
482C2×C4C23×#6440812yes—
483C2×C4C23×#6440814yes—
484C2×C4C23×#6440814yes—
485C2×C4C23×#6440812yes—
486C8—734no—
487C8—734no—
488C8—1464no—
489C8—1462no—
490C8—1462no—
491C8—1464no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—44no—
496C22—2928no—
497C22—1464no—
498C22—1464no—
499C22—1464no—
500C22#41×#6440812yes—
501C22C2×#1455667812yes—
502C22#1798966712yes—
503C22C24×#6440811yescharacteristic
504C4—24no—
505C4—22no—
506C4—24no—
507C4—731no—
508C4—731no—
509C4—732no—
510C4—1462no—
511C4—1462no—
512C4—1462no—
513C4—1464no—
514C4C2×#1455667814yes—
515C2—42no—
516C2—2924no—
517C2#2262676414yesdirect product factor
518C2C2×#41×#6440811yescharacteristic
519C2C22×#1455667811yescharacteristic
520C2C2×#1798966711yescharacteristic
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×#2969
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22626764C2direct (N × H)
2. C2×#17989638C2semidirect
3. C2×Q8×#10596833C2semidirect
4. C2×#17989621C2semidirect
5. C2×#17989620C2semidirect
6. C2×#17989619C2semidirect
7. C2×#17989615C2semidirect
8. C2×#17989612C2semidirect
9. C2×#17989603C2semidirect
10. C2×#17989552C2semidirect
11. C2×C4×#11924037C2semidirect
17. #17989638C22semidirect
18. Q8×#10596833C22semidirect
19. #17989621C22semidirect
20. #17989620C22semidirect
21. #17989619C22semidirect
22. #17989615C22semidirect
23. #17989612C22semidirect
24. #17989603C22semidirect
32. #17989552C22semidirect
33. C4×#11924037C22semidirect
256. C2×C73#2969semidirect
273. C73C2×#2969semidirect
517. C2#22626764direct (N × H)

Maximal subgroups

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

subgroupindex
#226267642
C2×#179896382
C2×Q8×#105968332
C2×#179896212
C2×#179896202
C2×#179896192
C2×#179896152
C2×#179896122
C2×#179896032
C2×#179895522
C2×C4×#119240372
C2×#179891232
C2×#296973