Finite Group Databasebeta

← C2×#22623530 #33993043

Subgroups — C2×#22623530 #33993043

Order 18688.

7802 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623530C2132yesmaximal, direct product factor
2C2×#17989323C213yesmaximal, complemented
3C2×#17989316C211yesmaximal, characteristic, complemented
4C2×#17989307C213yesmaximal, complemented
5C2×#17989284C216yesmaximal, complemented
6C2×#17989244C212yesmaximal, complemented
7C2×#17989135C213yesmaximal, complemented
8C2×#17989118C216yesmaximal, complemented
9C2×#41×Dic73C212yesmaximal, complemented
10C2×C4×#11924027C213yesmaximal, complemented
11C2×C4×Q8×D73C211yesmaximal, characteristic, complemented
12C2×#550×C73C211yesmaximal, characteristic, complemented
13C22×#11924027C2213yes—
14C22×Q8×D73C2211yescharacteristic
15#17989323C22148yescomplemented
16#17989316C22116yescomplemented
17#17989307C22148yescomplemented
18#17989284C22196yescomplemented
19#17989244C22132yescomplemented
20#17989135C22148yescomplemented
21C2×#14556591C2213yes—
22#17989118C22196yescomplemented
23C2×#14556584C22112yes—
24C2×#14556582C2216yes—
25C2×#14556578C2216yes—
26C2×#14556578C2216yes—
27C2×#14556572C22112yes—
28C2×#14556570C2216yes—
29C2×#14556562C2216yes—
30C2×#14556562C2216yes—
31#41×Dic73C22132yescomplemented
32C4×#11924027C22148yescomplemented
33C2×Q8×Dic73C2212yes—
34C2×Q8×Dic73C2211yescharacteristic
35C2×C4×Dic146C2213yes—
36C2×C4×Dic146C2213yes—
37C2×#14556556C2213yes—
38C2×#14556556C2213yes—
39C4×Q8×D73C22116yescomplemented
40C2×D4×Dic73C2213yes—
41C2×D4×Dic73C2216yes—
42C2×D4×Dic73C2216yes—
43C2×D4×Dic73C2216yes—
44C2×C4×C22⋊D73C2216yes—
45C2×#14556551C2216yes—
46C2×#14556550C2216yes—
47C2×#14556549C2211yescharacteristic
48C2×#14556547C2216yes—
49C2×C4⋊C4×D73C2213yes—
50C22×C4×Dic73C2216yes—
51C2×C42×D73C2213yes—
52C22×#41×C73C2212yes—
53#550×C73C22116yescomplemented
54C2×#127×C73C2213yes—
55C2×#121×C73C2216yes—
56C2×C4×Q8×C73C2211yescharacteristic
57C2×C4×D4×C73C2213yes—
58C2×#11924027—224no—
59C2×#11924027C23124yes—
60C22×Dic146C2313yes—
61C22×Dic146C2313yes—
62C2×Q8×D73—28no—
63C2×Q8×D73C2318yes—
64C22×C22⋊D73C2316yes—
65C23×Dic73C2316yes—
66C23×Dic73C2316yes—
67C22×C4×D73C2313yes—
68C22×C4×D73C2313yes—
69C22×C4×D73C2311yescharacteristic
70#14556591C23124yes—
71#14556584C23196yes—
72#14556582C23148yes—
73#14556578C23148yes—
74#14556578C23148yes—
75#14556572C23196yes—
76#14556570C23148yes—
77#14556562C23148yes—
78#14556562C23148yes—
79Q8×Dic73C23116yes—
80Q8×Dic73C2318yes—
81C4×Dic146C23124yes—
82C4×Dic146C23124yes—
83#14556556C23124yes—
84#14556556C23124yes—
85D4×Dic73C23148yes—
86D4×Dic73C23148yes—
87D4×Dic73C23148yes—
88D4×Dic73C23124yes—
89C4×C22⋊D73C23148yes—
90#14556551C23148yes—
91#14556550C23148yes—
92#14556549C2318yes—
93#14556547C23148yes—
94C2×C4⋊Dic73C2316yes—
95C2×C4⋊Dic73C2316yes—
96C2×C4⋊Dic73C2313yes—
97C2×C4⋊Dic73C2311yescharacteristic
98C2×#11924007C2316yes—
99C2×#11924007C2316yes—
100C2×#11924007C2316yes—
101C4⋊C4×D73C23124yes—
102C2×C22⋊Dic73C2316yes—
103C2×C22⋊Dic73C23112yes—
104C2×C22⋊Dic73C23112yes—
105C2×#11924005C2312yes—
106C2×C4×Dic73C2313yes—
107C2×C4×Dic73C2313yes—
108C2×C4×Dic73C2313yes—
109C2×C4×Dic73C2312yes—
110C2×C4×Dic73C2316yes—
111C2×C4×Dic73C2316yes—
112C2×C4×Dic73C23148yes—
113C42×D73C23124yes—
114C2×#41×C73—216no—
115C2×#41×C73C23116yes—
116C22×Q8×C73C2311yescharacteristic
117C22×D4×C73C2313yes—
118C22×D4×C73C2316yes—
119C22×D4×C73C2316yes—
120C23×C4×C73C2316yes—
121#127×C73C23124yes—
122#121×C73C23148yes—
123C4×Q8×C73C2318yes—
124C4×D4×C73C23124yes—
125C2×C4⋊C4×C73C2311yescharacteristic
126C2×C4⋊C4×C73C2313yes—
127C2×C22⋊C4×C73C2316yes—
128C2×C42×C73C2313yes—
129C23×D73C2411yescharacteristic
130#11924027—296no—
131C2×Dic146—212no—
132C2×Dic146—212no—
133C2×Dic146C24112yes—
134C2×Dic146C24112yes—
135Q8×D73—232no—
136C2×C22⋊D73—224no—
137C2×C22⋊D73C24124yes—
138C22×Dic73—224no—
139C22×Dic73—412no—
140C22×Dic73C2412yes—
141C22×Dic73C2413yes—
142C22×Dic73C2413yes—
143C22×Dic73C2416yes—
144C22×Dic73C24124yes—
145C22×Dic73C24124yes—
146C22×Dic73C2411yescharacteristic
147C2×C4×D73—212no—
148C2×C4×D73—46no—
149C2×C4×D73#4118yescomplemented
150C2×C4×D73C24112yes—
151C2×C4×D73C24112yes—
152C2×C4×D73C2414yes—
153C4⋊Dic73C24112yes—
154C4⋊Dic73C2414yes—
155C4⋊Dic73C24124yes—
156C4⋊Dic73C24124yes—
157#11924007C24124yes—
158#11924007C24124yes—
159#11924007C24124yes—
160C22⋊Dic73C24124yes—
161C22⋊Dic73C24148yes—
162C22⋊Dic73C24148yes—
163#11924005C2418yes—
164C4×Dic73C24112yes—
165C4×Dic73C24112yes—
166C4×Dic73C24112yes—
167C4×Dic73C2418yes—
168C4×Dic73C24124yes—
169C4×Dic73C24124yes—
170C24×C73C2416yes—
171#41×C73—264no—
172C2×Q8×C73#4118yescomplemented
173C2×Q8×C73C2414yes—
174C2×D4×C73—224no—
175C2×D4×C73—212no—
176C2×D4×C73—412no—
177C2×D4×C73C24112yes—
178C2×D4×C73C24124yes—
179C2×D4×C73C24124yes—
180C22×C4×C73—224no—
181C22×C4×C73C2413yes—
182C22×C4×C73C2412yes—
183C22×C4×C73C24124yes—
184C22×C4×C73C2411yescharacteristic
185C22×C4×C73C2413yes—
186C4⋊C4×C73C2414yes—
187C4⋊C4×C73C24112yes—
188C22⋊C4×C73C24124yes—
189C42×C73C24112yes—
190C23×C73—212no—
191C23×C73—212no—
192C23×C73—212no—
193C23×C73C25112yes—
194C23×C73C2511yescharacteristic
195C22×D73—22no—
196C22×D73—22no—
197C22×D73C2×#4114yes—
198C22×D73C2512yes—
199Q8×C73C2×#41116yescomplemented
200D4×C73—248no—
201D4×C73—224no—
202D4×C73—424no—
203C2×C4×C73—24no—
204C2×C4×C73—26no—
205C2×C4×C73—248no—
206C2×C4×C73C2×#4114yes—
207C2×C4×C73C2×#41112yes—
208C2×C4×C73C2516yes—
209C2×C4×C73C2514yes—
210C2×C4×C73C2512yes—
211C2×C4×C73C2516yes—
212C22⋊D73—248no—
213C2×Dic73—26no—
214C2×Dic73—212no—
215C2×Dic73—26no—
216C2×Dic73—24no—
217C2×Dic73—248no—
218C2×Dic73—424no—
219C2×Dic73C2×#4114yes—
220C2×Dic73C2514yes—
221C2×Dic73C2516yes—
222C2×Dic73C2516yes—
223C2×Dic73C25112yes—
224C2×Dic73C2512yes—
225C4×D73—224no—
226C4×D73—412no—
227C4×D73C2×#41116yescomplemented
228Dic146—224no—
229Dic146—224no—
230C22×C73—212no—
231C22×C73—212no—
232C22×C73—212no—
233C22×C73—424no—
234C22×C73C22×#4112yes—
235C22×C73C22×#4112yes—
236C22×C73#58512yes—
237C22×C73C2611yescharacteristic
238C2×D73—22no—
239C2×D73—22no—
240C2×D73—28no—
241C2×D73C22×#4114yes—
242C4×C73—24no—
243C4×C73—26no—
244C4×C73C22×#4114yes—
245C4×C73C22×#41112yes—
246Dic73—26no—
247Dic73—212no—
248Dic73—26no—
249Dic73—24no—
250Dic73C22×#4114yes—
251C2×#3480—731nomaximal, Sylow
252C2×C73—412no—
253C2×C73#348014yescomplemented
254C2×C73C23×#4111yescharacteristic
255C2×C73C23×#4111yescharacteristic
256C2×C73C2×#58511yescharacteristic
257D73—24no—
258#3480—7332no—
259C2×#559—733no—
260C2×#557—731no—
261C2×#556—733no—
262C2×#553—736no—
263C2×#550D7311yescharacteristic, complemented
264C2×#548—732no—
265C2×#532—733no—
266C2×#529—736no—
267C2×C4×#41—732no—
268C2×C4×#41—733no—
269C22×C4×Q8—731no—
270C73C2×#348011yescharacteristic, complemented, Sylow
271C22×#41—733no—
272C22×#41C2×D7312yes—
273C23×Q8—731no—
274#559—7348no—
275#557—7316no—
276#556—7348no—
277#553—7396no—
278#550C2×D73116yescomplemented
279#548—7332no—
280#532—7348no—
281C2×C4⋊Q8—733no—
282C2×#127C2×D7313yes—
283#529—7396no—
284C2×#126—7312no—
285C2×#125—736no—
286C2×#124—736no—
287C2×#124—736no—
288C2×#123—7312no—
289C2×C22⋊Q8—736no—
290C2×#121—736no—
291C2×#121—736no—
292C2×#121C2×D7316yes—
293C4×#41—7332no—
294C4×#41—7348no—
295C2×C4×Q8—731no—
296C2×C4×Q8—732no—
297C2×C4×Q8—7316no—
298C2×C4×Q8—733no—
299C2×C4×Q8—733no—
300C2×C4×Q8—733no—
301C2×C4×Q8—733no—
302C2×C4×Q8C2×D7311yescharacteristic
303C2×C4×D4—736no—
304C2×C4×D4—736no—
305C2×C4×D4—733no—
306C2×C4×D4—736no—
307C2×C4×D4—736no—
308C2×C4×D4—736no—
309C2×C4×D4C2×D7313yes—
310C2×#117—736no—
311C2×#117—731no—
312C2×#117—736no—
313C22×C4⋊C4—733no—
314C22×C42—736no—
315C22×C42—733no—
316C2×#41—216no—
317C2×#41—7324no—
318C2×#41—14624no—
319C2×#41C22×D73116yes—
320C22×Q8—738no—
321C22×Q8—733no—
322C22×Q8—733no—
323C22×Q8—1468no—
324C22×Q8C22×D7311yescharacteristic
325C22×D4—736no—
326C22×D4C22×D7316yes—
327C22×D4C22×D7313yes—
328C22×D4C22×D7316yes—
329C23×C4—736no—
330C23×C4—733no—
331C23×C4—731no—
332C23×C4—736no—
333C23×C4—733no—
334C23×C4C22×D7316yes—
335C4⋊Q8—7324no—
336#127C22×D73124yes—
337#126—7396no—
338#125—7348no—
339#124—7348no—
340#124—7348no—
341#123—7396no—
342C22⋊Q8—7348no—
343#121—7348no—
344#121—7348no—
345#121C22×D73148yes—
346C4×Q8—738no—
347C4×Q8—7316no—
348C4×Q8—7324no—
349C4×Q8—7324no—
350C4×Q8—7324no—
351C4×Q8—7324no—
352C4×Q8C22×D7318yes—
353C4×D4—7348no—
354C4×D4—7348no—
355C4×D4—7324no—
356C4×D4—7348no—
357C4×D4—7348no—
358C4×D4—7348no—
359C4×D4C22×D73124yes—
360#117—7348no—
361#117—738no—
362#117—7348no—
363C2×C4⋊C4—733no—
364C2×C4⋊C4—736no—
365C2×C4⋊C4—731no—
366C2×C4⋊C4—736no—
367C2×C4⋊C4—736no—
368C2×C4⋊C4—736no—
369C2×C4⋊C4—736no—
370C2×C4⋊C4—7324no—
371C2×C4⋊C4C22×D7313yes—
372C2×C4⋊C4C22×D7311yescharacteristic
373C2×C22⋊C4—7312no—
374C2×C22⋊C4—736no—
375C2×C22⋊C4—7312no—
376C2×C22⋊C4—732no—
377C2×C22⋊C4C22×D7316yes—
378C2×C42—732no—
379C2×C42—733no—
380C2×C42—736no—
381C2×C42—736no—
382C2×C42—733no—
383C2×C42—733no—
384C2×C42—7348no—
385C2×C42—7324no—
386C2×C42C22×D7313yes—
387C24—731no—
388C24C23×D7316yes—
389#41—264no—
390#41—14696no—
391C2×Q8—7312no—
392C2×Q8—7312no—
393C2×Q8—14632no—
394C2×Q8—14612no—
395C2×Q8—14612no—
396C2×Q8#1192402718yescomplemented
397C2×Q8C23×D7314yes—
398C2×D4—212no—
399C2×D4—224no—
400C2×D4—412no—
401C2×D4—7324no—
402C2×D4—14624no—
403C2×D4C23×D73124yes—
404C2×D4C23×D73112yes—
405C2×D4C23×D73124yes—
406C22×C4—224no—
407C22×C4—2926no—
408C22×C4—29212no—
409C22×C4—7324no—
410C22×C4—7312no—
411C22×C4—731no—
412C22×C4—738no—
413C22×C4—734no—
414C22×C4—7324no—
415C22×C4—732no—
416C22×C4—733no—
417C22×C4—733no—
418C22×C4—736no—
419C22×C4—7312no—
420C22×C4—14624no—
421C22×C4—14612no—
422C22×C4C23×D73124yes—
423C22×C4C23×D7312yes—
424C22×C4C23×D7313yes—
425C22×C4C23×D7311yescharacteristic
426C22×C4C23×D7313yes—
427C4⋊C4—7324no—
428C4⋊C4—7324no—
429C4⋊C4—7324no—
430C4⋊C4—7312no—
431C4⋊C4—734no—
432C4⋊C4—7324no—
433C4⋊C4—7324no—
434C4⋊C4C23×D73112yes—
435C4⋊C4C23×D7314yes—
436C22⋊C4—7348no—
437C22⋊C4—7324no—
438C22⋊C4—7348no—
439C22⋊C4—738no—
440C22⋊C4C23×D73124yes—
441C42—738no—
442C42—7312no—
443C42—7312no—
444C42—7324no—
445C42—7324no—
446C42—7312no—
447C42C23×D73112yes—
448C23—212no—
449C23—212no—
450C23—212no—
451C23—734no—
452C23—732no—
453C23—1462no—
454C23—1462no—
455C23C24×D73112yes—
456C23C24×D7311yescharacteristic
457Q8—14624no—
458Q8—14624no—
459Q8C2×#11924027116yescomplemented
460D4—224no—
461D4—248no—
462D4—424no—
463D4—14648no—
464C2×C4—24no—
465C2×C4—26no—
466C2×C4—248no—
467C2×C4—29212no—
468C2×C4—29224no—
469C2×C4—734no—
470C2×C4—732no—
471C2×C4—7316no—
472C2×C4—734no—
473C2×C4—736no—
474C2×C4—736no—
475C2×C4—7312no—
476C2×C4—14648no—
477C2×C4—14612no—
478C2×C4—1466no—
479C2×C4—1466no—
480C2×C4—1464no—
481C2×C4—14624no—
482C2×C4#41×D7314yes—
483C2×C4C2×#11924027112yes—
484C2×C4C24×D7314yes—
485C2×C4C24×D7316yes—
486C2×C4C24×D7312yes—
487C2×C4C24×D7316yes—
488C22—212no—
489C22—212no—
490C22—212no—
491C22—424no—
492C22—734no—
493C22—1468no—
494C22—1462no—
495C22—1462no—
496C22C2×#41×D7312yes—
497C22C22×#1192402712yes—
498C22#1798942212yes—
499C22C25×D7311yescharacteristic
500C4—24no—
501C4—26no—
502C4—734no—
503C4—14612no—
504C4—1466no—
505C4—1466no—
506C4—1464no—
507C4C2×#41×D7314yes—
508C4C22×#11924027112yes—
509C2—412no—
510C2—1464no—
511C2#2262353014yesdirect product factor
512C2C22×#41×D7311yescharacteristic
513C2C23×#1192402711yescharacteristic
514C2C2×#1798942211yescharacteristic
subgroupG/H
centerC23C24×D73
commutator, cosocleC22×C73C26
FrattiniC22C25×D73
FittingC2×#550×C73C2
socleC23×C73C25
hypercenterC2×#550D73
hypocenterC73C2×#3480
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #22623530C2direct (N × H)
2. C2×#17989323C2semidirect
3. C2×#17989316C2semidirect
4. C2×#17989307C2semidirect
5. C2×#17989284C2semidirect
6. C2×#17989244C2semidirect
7. C2×#17989135C2semidirect
8. C2×#17989118C2semidirect
9. C2×#41×Dic73C2semidirect
10. C2×C4×#11924027C2semidirect
11. C2×C4×Q8×D73C2semidirect
12. C2×#550×C73C2semidirect
15. #17989323C22semidirect
16. #17989316C22semidirect
17. #17989307C22semidirect
18. #17989284C22semidirect
19. #17989244C22semidirect
20. #17989135C22semidirect
22. #17989118C22semidirect
31. #41×Dic73C22semidirect
32. C4×#11924027C22semidirect
39. C4×Q8×D73C22semidirect
53. #550×C73C22semidirect
149. C2×C4×D73#41semidirect
172. C2×Q8×C73#41semidirect
199. Q8×C73C2×#41semidirect
227. C4×D73C2×#41semidirect
253. C2×C73#3480semidirect
263. C2×#550D73semidirect
270. C73C2×#3480semidirect
278. #550C2×D73semidirect
396. C2×Q8#11924027semidirect
459. Q8C2×#11924027semidirect
511. C2#22623530direct (N × H)

Maximal subgroups

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

subgroupindex
#226235302
C2×#179893232
C2×#179893162
C2×#179893072
C2×#179892842
C2×#179892442
C2×#179891352
C2×#179891182
C2×#41×Dic732
C2×C4×#119240272
C2×C4×Q8×D732
C2×#550×C732
C2×#348073