Finite Group Databasebeta

← C2×#22623554 #33993085

Subgroups — C2×#22623554 #33993085

Order 18688.

8418 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623554C2132yesmaximal, direct product factor
2C2×Q8×Dic146C211yesmaximal, characteristic, complemented
3C2×#17989302C213yesmaximal, complemented
4C2×#17989301C216yesmaximal, complemented
5C2×#17989257C213yesmaximal, complemented
6C2×#17989256C212yesmaximal, complemented
7C2×#17989135C213yesmaximal, complemented
8C2×#17989132C216yesmaximal, complemented
9C2×#41×Dic73C212yesmaximal, complemented
10C2×C4×#11924029C211yesmaximal, characteristic, complemented
11C2×C4×#11924027C213yesmaximal, complemented
12C2×#550×C73C211yesmaximal, characteristic, complemented
13C22×#11924029C2211yescharacteristic
14C22×#11924027C2213yes—
15C22×#11924027C2216yes—
16Q8×Dic146C22116yescomplemented
17#17989302C22148yescomplemented
18#17989301C22196yescomplemented
19#17989257C22148yescomplemented
20#17989256C22132yescomplemented
21#17989135C22148yescomplemented
22#17989132C22196yescomplemented
23C2×#14556591C2213yes—
24C2×#14556590C2216yes—
25C2×#14556587C2216yes—
26C2×#14556578C2216yes—
27C2×#14556577C22112yes—
28C2×#14556565C2216yes—
29C2×#14556563C22112yes—
30C2×#14556562C2216yes—
31C2×#14556562C2216yes—
32#41×Dic73C22132yescomplemented
33C4×#11924029C22116yescomplemented
34C4×#11924027C22148yescomplemented
35C2×Q8×Dic73C2212yes—
36C2×Q8×Dic73C2211yescharacteristic
37C2×C4×Dic146C2213yes—
38C2×#14556556C2211yescharacteristic
39C2×C4×D292C2213yes—
40C2×D4×Dic73C2213yes—
41C2×D4×Dic73C2216yes—
42C2×D4×Dic73C2216yes—
43C2×#14556553C2213yes—
44C2×C4×C22⋊D73C2216yes—
45C2×#14556551C2216yes—
46C2×#14556550C2216yes—
47C2×#14556549C2213yes—
48C2×#14556549C2213yes—
49C2×#14556547C2216yes—
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×#11924029—28no—
59C2×#11924029C2318yes—
60C2×#11924027—224no—
61C2×#11924027—424no—
62C2×#11924027C23124yes—
63C2×#11924027C23148yes—
64C22×Dic146C2316yes—
65C22×Dic146C2311yescharacteristic
66C22×D292C2313yes—
67C22×C22⋊D73C2316yes—
68C22×C22⋊D73C23112yes—
69C23×Dic73C2316yes—
70C23×Dic73C2316yes—
71C22×C4×D73C2313yes—
72C22×C4×D73C2313yes—
73C22×C4×D73C2313yes—
74#14556591C23124yes—
75#14556590C23148yes—
76#14556587C23148yes—
77#14556578C23148yes—
78#14556577C23196yes—
79#14556565C23148yes—
80#14556563C23196yes—
81#14556562C23148yes—
82#14556562C23148yes—
83Q8×Dic73C23116yes—
84Q8×Dic73C2318yes—
85C4×Dic146C23124yes—
86#14556556C2318yes—
87C4×D292C23124yes—
88D4×Dic73C23148yes—
89D4×Dic73C23124yes—
90D4×Dic73C23148yes—
91#14556553C23124yes—
92C4×C22⋊D73C23148yes—
93#14556551C23148yes—
94#14556550C23148yes—
95#14556549C23124yes—
96#14556549C23124yes—
97#14556547C23148yes—
98C2×C4⋊Dic73C2316yes—
99C2×C4⋊Dic73C2313yes—
100C2×C4⋊Dic73C2313yes—
101C2×#11924007C2316yes—
102C2×#11924007C2312yes—
103C2×C22⋊Dic73C2316yes—
104C2×C22⋊Dic73C23112yes—
105C2×#11924005C2316yes—
106C2×#11924005C2316yes—
107C2×C4×Dic73C2316yes—
108C2×C4×Dic73C23148yes—
109C2×C4×Dic73C2311yescharacteristic
110C2×C4×Dic73C2313yes—
111C2×C4×Dic73C2313yes—
112C2×C4×Dic73C2312yes—
113C2×C4×Dic73C2316yes—
114C42×D73C23124yes—
115C2×#41×C73—216no—
116C2×#41×C73C23116yes—
117C22×Q8×C73C2311yescharacteristic
118C22×D4×C73C2316yes—
119C22×D4×C73C2313yes—
120C22×D4×C73C2316yes—
121C23×C4×C73C2316yes—
122#127×C73C23124yes—
123#121×C73C23148yes—
124C4×Q8×C73C2318yes—
125C4×D4×C73C23124yes—
126C2×C4⋊C4×C73C2313yes—
127C2×C4⋊C4×C73C2311yescharacteristic
128C2×C22⋊C4×C73C2316yes—
129C2×C42×C73C2313yes—
130C23×D73C2413yes—
131#11924029—232no—
132#11924027—296no—
133#11924027—496no—
134C2×Dic146—412no—
135C2×Dic146#4118yescomplemented
136C2×Dic146C24124yes—
137C2×Dic146C2414yes—
138C2×D292—212no—
139C2×D292C24112yes—
140C2×C22⋊D73—224no—
141C2×C22⋊D73—424no—
142C2×C22⋊D73C24124yes—
143C2×C22⋊D73C24148yes—
144C22×Dic73—224no—
145C22×Dic73—224no—
146C22×Dic73C2413yes—
147C22×Dic73C2411yescharacteristic
148C22×Dic73C2416yes—
149C22×Dic73C24124yes—
150C22×Dic73C24124yes—
151C22×Dic73C2412yes—
152C22×Dic73C2411yescharacteristic
153C2×C4×D73—212no—
154C2×C4×D73—212no—
155C2×C4×D73—212no—
156C2×C4×D73C24112yes—
157C2×C4×D73C24112yes—
158C2×C4×D73C24112yes—
159C4⋊Dic73C24112yes—
160C4⋊Dic73C24112yes—
161C4⋊Dic73C24124yes—
162#11924007C2418yes—
163#11924007C24124yes—
164C22⋊Dic73C24124yes—
165C22⋊Dic73C24148yes—
166#11924005C24124yes—
167#11924005C24124yes—
168C4×Dic73C2414yes—
169C4×Dic73C24112yes—
170C4×Dic73C24112yes—
171C4×Dic73C2418yes—
172C4×Dic73C24124yes—
173C4×Dic73C24124yes—
174C24×C73C2416yes—
175#41×C73—264no—
176C2×Q8×C73#4118yescomplemented
177C2×Q8×C73C2414yes—
178C2×D4×C73—224no—
179C2×D4×C73—212no—
180C2×D4×C73—412no—
181C2×D4×C73C24112yes—
182C2×D4×C73C24124yes—
183C2×D4×C73C24124yes—
184C22×C4×C73—224no—
185C22×C4×C73C2413yes—
186C22×C4×C73C2412yes—
187C22×C4×C73C24124yes—
188C22×C4×C73C2411yescharacteristic
189C22×C4×C73C2413yes—
190C4⋊C4×C73C2414yes—
191C4⋊C4×C73C24112yes—
192C22⋊C4×C73C24124yes—
193C42×C73C24112yes—
194C23×C73—212no—
195C23×C73—212no—
196C23×C73—212no—
197C23×C73C25112yes—
198C23×C73C2511yescharacteristic
199C22×D73—26no—
200C22×D73—26no—
201C22×D73—26no—
202C22×D73C2516yes—
203Q8×C73C2×#41116yescomplemented
204D4×C73—248no—
205D4×C73—224no—
206D4×C73—424no—
207C2×C4×C73—24no—
208C2×C4×C73—26no—
209C2×C4×C73—248no—
210C2×C4×C73C2×#4114yes—
211C2×C4×C73C2×#41112yes—
212C2×C4×C73C2516yes—
213C2×C4×C73C2514yes—
214C2×C4×C73C2512yes—
215C2×C4×C73C2516yes—
216C22⋊D73—248no—
217C22⋊D73—448no—
218C2×Dic73—22no—
219C2×Dic73—26no—
220C2×Dic73—212no—
221C2×Dic73—248no—
222C2×Dic73—248no—
223C2×Dic73C2×#4118yes—
224C2×Dic73H(5,𝔽2)14yes—
225C2×Dic73C2516yes—
226C2×Dic73C2512yes—
227C2×Dic73C25112yes—
228C2×Dic73C2514yes—
229C2×Dic73C2512yes—
230D292—224no—
231C4×D73—224no—
232C4×D73—224no—
233C4×D73—224no—
234Dic146—424no—
235Dic146C2×#41116yescomplemented
236C22×C73—212no—
237C22×C73—212no—
238C22×C73—212no—
239C22×C73—424no—
240C22×C73C22×#4112yes—
241C22×C73C22×#4112yes—
242C22×C73C2×H(5,𝔽2)12yes—
243C22×C73C2611yescharacteristic
244C2×D73—26no—
245C2×D73—26no—
246C2×D73—26no—
247C2×D73—412no—
248C4×C73—24no—
249C4×C73—26no—
250C4×C73C22×#4114yes—
251C4×C73C22×#41112yes—
252Dic73—22no—
253Dic73—26no—
254Dic73—212no—
255Dic73C22×#4118yes—
256Dic73C2×H(5,𝔽2)14yes—
257C2×#3483—731nomaximal, Sylow
258C2×C73—412no—
259C2×C73#348314yescomplemented
260C2×C73C23×#4111yescharacteristic
261C2×C73C23×#4111yescharacteristic
262C2×C73C22×H(5,𝔽2)11yescharacteristic
263D73—46no—
264#3483—7332no—
265C2×Q82—731no—
266C2×#555—736no—
267C2×#555—733no—
268C2×#550—732no—
269C2×#550—733no—
270C2×#550D7311yescharacteristic, complemented
271C2×#532—733no—
272C2×#532—736no—
273C2×C4×#41—732no—
274C2×C4×#41—731no—
275C2×C4×#41—733no—
276C73C2×#348311yescharacteristic, complemented, Sylow
277C22×#41—731no—
278C22×#41—736no—
279C22×#41—733no—
280C22×#41C2×D7312yes—
281Q82—7316no—
282#555—7396no—
283#555—7348no—
284#550—7332no—
285#550—7348no—
286#550C2×D73116yescomplemented
287#532—7348no—
288#532—7396no—
289C2×C4⋊Q8—736no—
290C2×C4⋊Q8—733no—
291C2×#127—736no—
292C2×#127C2×D7313yes—
293C2×#124—736no—
294C2×#124—7312no—
295C2×#121—736no—
296C2×#121—736no—
297C2×#121—7312no—
298C2×#121—736no—
299C2×#121C2×D7316yes—
300C4×#41—7332no—
301C4×#41—7316no—
302C4×#41—7348no—
303C2×C4×Q8—731no—
304C2×C4×Q8—732no—
305C2×C4×Q8—731no—
306C2×C4×Q8—733no—
307C2×C4×Q8C2×D7311yescharacteristic
308C2×C4×D4—736no—
309C2×C4×D4—733no—
310C2×C4×D4—736no—
311C2×C4×D4—736no—
312C2×C4×D4—733no—
313C2×C4×D4—736no—
314C2×C4×D4—733no—
315C2×C4×D4C2×D7313yes—
316C2×#117—736no—
317C2×#117—733no—
318C2×#117—733no—
319C2×#117—736no—
320C22×C42—736no—
321C22×C42—733no—
322C2×#41—216no—
323C2×#41—29224no—
324C2×#41—738no—
325C2×#41—7348no—
326C2×#41—7324no—
327C2×#41—14624no—
328C2×#41—1468no—
329C2×#41C22×D73116yes—
330C22×Q8—736no—
331C22×Q8—731no—
332C22×Q8C22×D7311yescharacteristic
333C22×D4—7312no—
334C22×D4—736no—
335C22×D4—733no—
336C22×D4C22×D7316yes—
337C22×D4C22×D7313yes—
338C22×D4C22×D7316yes—
339C23×C4—736no—
340C23×C4—736no—
341C23×C4—733no—
342C23×C4—733no—
343C23×C4—733no—
344C23×C4C22×D7316yes—
345C4⋊Q8—7348no—
346C4⋊Q8—7324no—
347#127—7348no—
348#127C22×D73124yes—
349#124—7348no—
350#124—7396no—
351#121—7348no—
352#121—7348no—
353#121—7396no—
354#121—7348no—
355#121C22×D73148yes—
356C4×Q8—738no—
357C4×Q8—7316no—
358C4×Q8—738no—
359C4×Q8—7324no—
360C4×Q8C22×D7318yes—
361C4×D4—7348no—
362C4×D4—7324no—
363C4×D4—7348no—
364C4×D4—7348no—
365C4×D4—7324no—
366C4×D4—7348no—
367C4×D4—7324no—
368C4×D4C22×D73124yes—
369#117—7348no—
370#117—7324no—
371#117—7324no—
372#117—7348no—
373C2×C4⋊C4—732no—
374C2×C4⋊C4—733no—
375C2×C4⋊C4—736no—
376C2×C4⋊C4—733no—
377C2×C4⋊C4—736no—
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C4⋊C4C22×D7313yes—
380C2×C22⋊C4—7312no—
381C2×C22⋊C4—736no—
382C2×C22⋊C4—736no—
383C2×C22⋊C4—736no—
384C2×C22⋊C4C22×D7316yes—
385C2×C42—733no—
386C2×C42—736no—
387C2×C42—732no—
388C2×C42—731no—
389C2×C42—733no—
390C2×C42—736no—
391C2×C42—7348no—
392C2×C42—7324no—
393C2×C42C22×D7313yes—
394C24—733no—
395C24C23×D7316yes—
396#41—264no—
397#41—29296no—
398#41—14696no—
399#41—14632no—
400C2×Q8—29212no—
401C2×Q8—738no—
402C2×Q8—7324no—
403C2×Q8—734no—
404C2×Q8#1192402718yescomplemented
405C2×Q8C23×D7314yes—
406C2×D4—212no—
407C2×D4—224no—
408C2×D4—412no—
409C2×D4—29224no—
410C2×D4—7348no—
411C2×D4—7324no—
412C2×D4—7312no—
413C2×D4—14624no—
414C2×D4—14612no—
415C2×D4C23×D73124yes—
416C2×D4C23×D73112yes—
417C2×D4C23×D73124yes—
418C22×C4—224no—
419C22×C4—731no—
420C22×C4—732no—
421C22×C4—7324no—
422C22×C4—7324no—
423C22×C4—733no—
424C22×C4—731no—
425C22×C4—736no—
426C22×C4—7312no—
427C22×C4—7312no—
428C22×C4—7312no—
429C22×C4—14612no—
430C22×C4—14624no—
431C22×C4—14624no—
432C22×C4—14612no—
433C22×C4—14612no—
434C22×C4C23×D73124yes—
435C22×C4C23×D7312yes—
436C22×C4C23×D7313yes—
437C22×C4C23×D7311yescharacteristic
438C22×C4C23×D7313yes—
439C4⋊C4—738no—
440C4⋊C4—7324no—
441C4⋊C4—7312no—
442C4⋊C4—7312no—
443C4⋊C4—7324no—
444C4⋊C4C23×D7314yes—
445C4⋊C4C23×D73112yes—
446C22⋊C4—7324no—
447C22⋊C4—7348no—
448C22⋊C4—7324no—
449C22⋊C4—7324no—
450C22⋊C4C23×D73124yes—
451C42—738no—
452C42—734no—
453C42—7312no—
454C42—7312no—
455C42—7324no—
456C42—7324no—
457C42C23×D73112yes—
458C23—212no—
459C23—212no—
460C23—212no—
461C23—736no—
462C23—1466no—
463C23—1466no—
464C23—1466no—
465C23C24×D73112yes—
466C23C24×D7311yescharacteristic
467Q8—29224no—
468Q8—7316no—
469Q8C2×#11924027116yescomplemented
470D4—224no—
471D4—248no—
472D4—424no—
473D4—29248no—
474D4—14648no—
475D4—14624no—
476C2×C4—24no—
477C2×C4—26no—
478C2×C4—248no—
479C2×C4—738no—
480C2×C4—734no—
481C2×C4—732no—
482C2×C4—734no—
483C2×C4—736no—
484C2×C4—732no—
485C2×C4—7312no—
486C2×C4—14624no—
487C2×C4—14648no—
488C2×C4—14648no—
489C2×C4—1466no—
490C2×C4—1462no—
491C2×C4—14612no—
492C2×C4—14624no—
493C2×C4—14624no—
494C2×C4#41×D7314yes—
495C2×C4C2×#11924027112yes—
496C2×C4C24×D7314yes—
497C2×C4C24×D7316yes—
498C2×C4C24×D7312yes—
499C2×C4C24×D7316yes—
500C22—212no—
501C22—212no—
502C22—212no—
503C22—424no—
504C22—29212no—
505C22—1466no—
506C22—1466no—
507C22—1466no—
508C22C2×#41×D7312yes—
509C22C22×#1192402712yes—
510C22H(5,𝔽2)×D7312yes—
511C22C25×D7311yescharacteristic
512C4—24no—
513C4—26no—
514C4—738no—
515C4—734no—
516C4—1466no—
517C4—1462no—
518C4—14612no—
519C4C2×#41×D7314yes—
520C4C22×#11924027112yes—
521C2—412no—
522C2—2926no—
523C2#2262355414yesdirect product factor
524C2C22×#41×D7311yescharacteristic
525C2C23×#1192402711yescharacteristic
526C2C2×H(5,𝔽2)×D7311yescharacteristic
subgroupG/H
centerC23C24×D73
commutator, cosocleC22×C73C26
FrattiniC22C25×D73
FittingC2×#550×C73C2
socleC23×C73C25
hypercenterC2×#550D73
hypocenterC73C2×#3483
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #22623554C2direct (N × H)
2. C2×Q8×Dic146C2semidirect
3. C2×#17989302C2semidirect
4. C2×#17989301C2semidirect
5. C2×#17989257C2semidirect
6. C2×#17989256C2semidirect
7. C2×#17989135C2semidirect
8. C2×#17989132C2semidirect
9. C2×#41×Dic73C2semidirect
10. C2×C4×#11924029C2semidirect
11. C2×C4×#11924027C2semidirect
12. C2×#550×C73C2semidirect
16. Q8×Dic146C22semidirect
17. #17989302C22semidirect
18. #17989301C22semidirect
19. #17989257C22semidirect
20. #17989256C22semidirect
21. #17989135C22semidirect
22. #17989132C22semidirect
32. #41×Dic73C22semidirect
33. C4×#11924029C22semidirect
34. C4×#11924027C22semidirect
53. #550×C73C22semidirect
135. C2×Dic146#41semidirect
176. C2×Q8×C73#41semidirect
203. Q8×C73C2×#41semidirect
235. Dic146C2×#41semidirect
259. C2×C73#3483semidirect
270. C2×#550D73semidirect
276. C73C2×#3483semidirect
286. #550C2×D73semidirect
404. C2×Q8#11924027semidirect
469. Q8C2×#11924027semidirect
523. C2#22623554direct (N × H)

Maximal subgroups

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

subgroupindex
#226235542
C2×Q8×Dic1462
C2×#179893022
C2×#179893012
C2×#179892572
C2×#179892562
C2×#179891352
C2×#179891322
C2×#41×Dic732
C2×C4×#119240292
C2×C4×#119240272
C2×#550×C732
C2×#348373