Finite Group Databasebeta

← C2×#22622627 #33965608

Subgroups — C2×#22622627 #33965608

Order 18688.

1894 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989322C211yesmaximal, characteristic
2#22622627C2116yesmaximal, direct product factor
3C2×#17988944C212yesmaximal, complemented
4C2×#17988935C211yesmaximal, characteristic
5C2×#17988869C211yesmaximal, characteristic, complemented
6C2×#17988884C212yesmaximal, complemented
7C2×#17988809C212yesmaximal, complemented
8C2×#17988787C211yesmaximal, characteristic
9C2×Q8×#10596831C211yesmaximal, characteristic, complemented
10C2×C4×#11924018C211yesmaximal, characteristic, complemented
11C2×C4×#11924015C212yesmaximal, complemented
12C2×#557×C73C211yesmaximal, characteristic, complemented
13C22×#11924018C2211yescharacteristic
14C22×#11924015C2212yes—
15#17989322—24no—
16#17989322C2218yes—
17C2×#14556588C2211yescharacteristic
18C2×#14556580—21no—
19C2×#14556579C2212yes—
20C2×#14556576—22no—
21C2×C4×Dic146C2211yescharacteristic
22C2×C4×D292C2212yes—
23C2×#14556553—21no—
24#17988944C22116yescomplemented
25#17988935C2218yes—
26#17988884C22116yescomplemented
27#17988869C2218yescomplemented
28#17988809C22116yescomplemented
29#17988787C2218yes—
30Q8×#10596831C2218yescomplemented
31C4×#11924018C2218yescomplemented
32C4×#11924015C22116yescomplemented
33C2×#14556524C2211yescharacteristic
34C2×#14556522C2212yes—
35C2×C4⋊#10596831C2212yes—
36C2×C4⋊#10596831C2211yescharacteristic
37C2×#14556515C2211yescharacteristic
38C2×#14556514C2212yes—
39C2×#14556514C2211yescharacteristic
40C2×#14556511C2212yes—
41C2×#14556511C2212yes—
42C2×#14556511C2212yes—
43C2×C4×#10596831C2212yes—
44C2×C4×#10596831C2211yescharacteristic
45#557×C73—24no—
46#557×C73C2218yescomplemented
47C2×C4⋊Q8×C73—21no—
48C2×C4⋊Q8×C73C2211yescharacteristic
49C2×#125×C73—22no—
50C2×#125×C73C2212yes—
51C2×C4×Q8×C73—21no—
52C2×C4×Q8×C73C2212yes—
53C2×C4×Q8×C73C2211yescharacteristic
54C2×C4×Q8×C73C2211yescharacteristic
55C22×Dic146C2311yescharacteristic
56C22×D292—21no—
57C22×D292—21no—
58C22×D292C2312yes—
59C22×C4×D73—22no—
60C2×#11924018—24no—
61C2×#11924018C2314yes—
62C2×#11924015—28no—
63C2×#11924015C2318yes—
64C22×#10596831C2312yes—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556588—22no—
68#14556588C2314yes—
69#14556580—28no—
70#14556579—24no—
71#14556579C2318yes—
72#14556576—216no—
73C4×Dic146—22no—
74C4×Dic146C2314yes—
75C4×D292—24no—
76C4×D292C2318yes—
77#14556553—28no—
78C2×C4⋊Dic73C2311yescharacteristic
79C2×#11924007—21no—
80C2×#11924005—22no—
81C2×#11924005—22no—
82C2×#11924005—22no—
83C2×C4×Dic73—21no—
84#14556524C2314yes—
85#14556522C2318yes—
86C4⋊#10596831C2314yes—
87C4⋊#10596831C2318yes—
88#14556515C2314yes—
89#14556514C2318yes—
90#14556514C2314yes—
91#14556511C2318yes—
92#14556511C2318yes—
93#14556511C2318yes—
94C4×#10596831C2318yes—
95C4×#10596831C2314yes—
96C22×Q8×C73—21no—
97C22×Q8×C73D411yescharacteristic
98C22×Q8×C73C2311yescharacteristic
99C4⋊Q8×C73—28no—
100C4⋊Q8×C73—22no—
101C4⋊Q8×C73C2314yes—
102#125×C73—216no—
103#125×C73—24no—
104#125×C73C2318yes—
105C4×Q8×C73—28no—
106C4×Q8×C73—24no—
107C4×Q8×C73—22no—
108C4×Q8×C73D414yes—
109C4×Q8×C73C2318yes—
110C4×Q8×C73C2314yes—
111C4×Q8×C73C2314yes—
112C2×C4⋊C4×C73—22no—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73—22no—
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73—22no—
117C2×C4⋊C4×C73D411yescharacteristic
118C2×C4⋊C4×C73D412yes—
119C2×C4⋊C4×C73C2312yes—
120C2×C4⋊C4×C73C2311yescharacteristic
121C2×C42×C73—22no—
122C2×C42×C73—21no—
123C2×C42×C73C2311yescharacteristic
124C2×C42×C73C2312yes—
125C23×D73—22no—
126C2×Dic146—22no—
127C2×Dic146—22no—
128C2×Dic146—21no—
129C2×Dic146C2412yes—
130C2×D292—24no—
131C2×D292—24no—
132C2×D292—24no—
133C2×D292—22no—
134C2×D292—24no—
135C2×D292—44no—
136C2×D292—82no—
137C2×D292C2414yes—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C2×C4×D73—28no—
141C2×C4×D73—84no—
142#11924018—28no—
143#11924015—216no—
144C2×#10596831—24no—
145C2×#10596831—22no—
146C2×#10596831—22no—
147C2×#10596831C2412yes—
148C2×#10596831C2412yes—
149C2×#10596831C2414yes—
150C4⋊Dic73—21no—
151C4⋊Dic73C2412yes—
152#11924007—24no—
153#11924005—28no—
154#11924005—28no—
155#11924005—28no—
156C4×Dic73—24no—
157C2×Q8×C73—24no—
158C2×Q8×C73—21no—
159C2×Q8×C73—24no—
160C2×Q8×C73—22no—
161C2×Q8×C73—44no—
162C2×Q8×C73C2×D414yes—
163C2×Q8×C73#4114yes—
164C2×Q8×C73C2412yes—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—22no—
168C22×C4×C73C2×D412yes—
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D412yes—
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2411yescharacteristic
173C4⋊C4×C73—28no—
174C4⋊C4×C73—24no—
175C4⋊C4×C73—28no—
176C4⋊C4×C73—24no—
177C4⋊C4×C73—28no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—22no—
180C4⋊C4×C73C2×D414yes—
181C4⋊C4×C73C2×D418yes—
182C4⋊C4×C73C2412yes—
183C4⋊C4×C73C2414yes—
184C42×C73—28no—
185C42×C73—24no—
186C42×C73C2×D414yes—
187C42×C73C2×D412yes—
188C42×C73C2414yes—
189C42×C73C2412yes—
190C23×C73C22×D411yescharacteristic
191C22×D73—24no—
192C22×D73—44no—
193C22×D73—44no—
194C22×D73—44no—
195Q8×C73—24no—
196Q8×C73—28no—
197Q8×C73—22no—
198Q8×C73—48no—
199Q8×C73C2×#4114yes—
200C2×C4×C73—24no—
201C2×C4×C73—24no—
202C2×C4×C73—24no—
203C2×C4×C73—22no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—24no—
207C2×C4×C73—44no—
208C2×C4×C73—42no—
209C2×C4×C73#13514yes—
210C2×C4×C73C22×D414yes—
211C2×C4×C73C22×D412yes—
212C2×C4×C73C22×D414yes—
213C2×C4×C73C22×D411yescharacteristic
214C2×C4×C73C22×D412yes—
215C2×C4×C73C2×#4112yes—
216C2×C4×C732−1+412yes—
217C2×C4×C73C2511yescharacteristic
218C2×Dic73—22no—
219C2×Dic73—22no—
220C2×Dic73—42no—
221C2×Dic73—42no—
222D292—24no—
223D292—24no—
224D292—44no—
225D292—48no—
226D292—84no—
227C4×D73—88no—
228Dic146—22no—
229Dic146—22no—
230Dic146—42no—
231#10596831—24no—
232#10596831—22no—
233#10596831—22no—
234C22×C73#54812yes—
235C22×C73C2×#13512yes—
236C22×C73#57712yes—
237C22×C73C23×D411yescharacteristic
238C2×D73—44no—
239C2×D73—44no—
240C2×D73—44no—
241C2×D73—88no—
242C4×C73—24no—
243C4×C73—24no—
244C4×C73—24no—
245C4×C73—22no—
246C4×C73—44no—
247C4×C73—42no—
248C4×C73#54812yes—
249C4×C73C2×#13514yes—
250C4×C73C22×#4111yescharacteristic
251C4×C73C2×2−1+411yescharacteristic
252Dic73—42no—
253Dic73—42no—
254C2×#3365—731nomaximal, Sylow
255C2×C73#336514yescomplemented
256C2×C73C2×#54811yescharacteristic
257C2×C73C22×#13511yescharacteristic
258C2×C73C2×#57711yescharacteristic
259D73—84no—
260C2×#559—731no—
261C2×#557D7311yescharacteristic, complemented
262#3365—7316no—
263C2×#488—732no—
264C2×#486—731no—
265C2×#475—731no—
266C2×#478—732no—
267C2×#464—732no—
268C2×#460—731no—
269C2×C8×Q8—731no—
270C2×C4×Q16—731no—
271C2×C4×SD16—732no—
272C73C2×#336511yescharacteristic, complemented, Sylow
273C22×Q16—731no—
274C22×SD16—732no—
275#559—738no—
276#559—1464no—
277#557—24no—
278#557C2×D7318yescomplemented
279C2×C4⋊Q8—21no—
280C2×C4⋊Q8C2×D7311yescharacteristic
281C2×#127—731no—
282C2×#125—22no—
283C2×#125C2×D7312yes—
284C2×#124—732no—
285C2×#124—1461no—
286C2×#123—1462no—
287C2×C4×Q8—21no—
288C2×C4×Q8—731no—
289C2×C4×Q8C2×D7312yes—
290C2×C4×Q8C2×D7311yescharacteristic
291C2×C4×Q8C2×D7311yescharacteristic
292C2×C4×D4—732no—
293C2×C4×D4—1461no—
294#488—7316no—
295#486—738no—
296#478—7316no—
297#475—738no—
298#464—7316no—
299#460—738no—
300C8×Q8—738no—
301C4×Q16—738no—
302C4×SD16—7316no—
303C2×#107—731no—
304C2×#106—732no—
305C2×C4⋊C8—732no—
306C2×C4⋊C8—731no—
307C2×#103—731no—
308C2×#103—732no—
309C2×#103—731no—
310C2×#102—732no—
311C2×#102—732no—
312C2×#102—732no—
313C2×C4×C8—731no—
314C2×C4×C8—732no—
315C22×Q8—21no—
316C22×Q8—731no—
317C22×Q8C22⋊D7311yescharacteristic
318C22×Q8C22×D7311yescharacteristic
319C22×D4—732no—
320C22×D4—1461no—
321C22×D4—1461no—
322C23×C4—1462no—
323C2×Q16—734no—
324C2×Q16—1464no—
325C2×SD16—738no—
326C2×SD16—1468no—
327C22×C8—732no—
328C22×C8—731no—
329C22×C8—731no—
330C4⋊Q8—28no—
331C4⋊Q8—22no—
332C4⋊Q8C22×D7314yes—
333#127—734no—
334#127—1462no—
335#125—216no—
336#125—24no—
337#125C22×D7318yes—
338#124—738no—
339#124—1468no—
340#124—1464no—
341#123—14616no—
342C4×Q8—28no—
343C4×Q8—24no—
344C4×Q8—22no—
345C4×Q8—734no—
346C4×Q8—1462no—
347C4×Q8C22⋊D7314yes—
348C4×Q8C22×D7318yes—
349C4×Q8C22×D7314yes—
350C4×Q8C22×D7314yes—
351C4×D4—738no—
352C4×D4—1468no—
353C4×D4—1464no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—731no—
360C2×C4⋊C4—1461no—
361C2×C4⋊C4C22⋊D7311yescharacteristic
362C2×C4⋊C4C22⋊D7312yes—
363C2×C4⋊C4C22×D7311yescharacteristic
364C2×C4⋊C4C22×D7312yes—
365C2×C22⋊C4—1462no—
366C2×C22⋊C4—1462no—
367C2×C22⋊C4—1462no—
368C2×C42—22no—
369C2×C42—21no—
370C2×C42—1461no—
371C2×C42C22×D7312yes—
372C2×C42C22×D7311yescharacteristic
373#107—734no—
374#106—738no—
375C4⋊C8—734no—
376C4⋊C8—738no—
377#103—734no—
378#103—734no—
379#103—738no—
380#102—738no—
381#102—738no—
382#102—738no—
383C4×C8—738no—
384C4×C8—734no—
385C24—1462no—
386C2×Q8—24no—
387C2×Q8—21no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—44no—
391C2×Q8—732no—
392C2×Q8—1462no—
393C2×Q8—1462no—
394C2×Q8—1461no—
395C2×Q8C2×C22⋊D7314yes—
396C2×Q8#1192402914yes—
397C2×Q8C23×D7312yes—
398C2×D4—2924no—
399C2×D4—5842no—
400C2×D4—734no—
401C2×D4—1464no—
402C2×D4—1462no—
403C2×D4—1464no—
404C2×D4—1464no—
405C2×D4—1464no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—5844no—
410C22×C4—1461no—
411C22×C4—1461no—
412C22×C4—1468no—
413C22×C4C2×C22⋊D7312yes—
414C22×C4C2×C22⋊D7312yes—
415C22×C4C2×C22⋊D7311yescharacteristic
416C22×C4C2×C22⋊D7311yescharacteristic
417C22×C4C23×D7311yescharacteristic
418Q16—1468no—
419SD16—14616no—
420C2×C8—732no—
421C2×C8—732no—
422C2×C8—734no—
423C2×C8—1464no—
424C2×C8—1462no—
425C2×C8—1462no—
426C4⋊C4—28no—
427C4⋊C4—24no—
428C4⋊C4—28no—
429C4⋊C4—24no—
430C4⋊C4—28no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—732no—
434C4⋊C4—1464no—
435C4⋊C4—1461no—
436C4⋊C4C2×C22⋊D7314yes—
437C4⋊C4C2×C22⋊D7318yes—
438C4⋊C4C23×D7314yes—
439C4⋊C4C23×D7312yes—
440C22⋊C4—1468no—
441C22⋊C4—1468no—
442C22⋊C4—1468no—
443C42—28no—
444C42—24no—
445C42—1464no—
446C42C2×C22⋊D7314yes—
447C42C2×C22⋊D7312yes—
448C42C23×D7314yes—
449C42C23×D7312yes—
450C23—2924no—
451C23—2924no—
452C23—2924no—
453C23—1464no—
454C23C22×C22⋊D7311yescharacteristic
455Q8—22no—
456Q8—24no—
457Q8—28no—
458Q8—48no—
459Q8—2922no—
460Q8—1462no—
461Q8—1462no—
462Q8C2×#1192402914yes—
463D4—2924no—
464D4—2928no—
465D4—5844no—
466D4—1464no—
467D4—1464no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—2922no—
478C2×C4—2922no—
479C2×C4—5848no—
480C2×C4—1462no—
481C2×C4—1462no—
482C2×C4#1455661114yes—
483C2×C4C22×C22⋊D7314yes—
484C2×C4C22×C22⋊D7314yes—
485C2×C4C22×C22⋊D7312yes—
486C2×C4C22×C22⋊D7311yescharacteristic
487C2×C4C22×C22⋊D7312yes—
488C2×C4C2×#1192402912yes—
489C2×C4#1455663912yes—
490C2×C4C24×D7311yescharacteristic
491C8—1464no—
492C8—1462no—
493C8—1462no—
494C22—2924no—
495C22—2924no—
496C22—2924no—
497C22—5848no—
498C22#1798924812yes—
499C22C2×#1455661112yes—
500C22#1798939512yes—
501C22C23×C22⋊D7311yescharacteristic
502C4—24no—
503C4—24no—
504C4—24no—
505C4—22no—
506C4—44no—
507C4—42no—
508C4—2922no—
509C4—2922no—
510C4#1798924812yes—
511C4C2×#1455661114yes—
512C4C22×#1192402911yescharacteristic
513C4C2×#1455663911yescharacteristic
514C2—5844no—
515C2#2262262714yesdirect product factor
516C2C2×#1798924811yescharacteristic
517C2C22×#1455661111yescharacteristic
518C2C2×#1798939511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC2×#557×C73C2
socleC23×C73C22×D4
prime radicalC2×#14556588C22
hypercenterC2×#557D73
hypocenterC73C2×#3365
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
2. #22622627C2direct (N × H)
3. C2×#17988944C2semidirect
5. C2×#17988869C2semidirect
6. C2×#17988884C2semidirect
7. C2×#17988809C2semidirect
9. C2×Q8×#10596831C2semidirect
10. C2×C4×#11924018C2semidirect
11. C2×C4×#11924015C2semidirect
12. C2×#557×C73C2semidirect
24. #17988944C22semidirect
26. #17988884C22semidirect
27. #17988869C22semidirect
28. #17988809C22semidirect
30. Q8×#10596831C22semidirect
31. C4×#11924018C22semidirect
32. C4×#11924015C22semidirect
46. #557×C73C22semidirect
255. C2×C73#3365semidirect
261. C2×#557D73semidirect
272. C73C2×#3365semidirect
278. #557C2×D73semidirect
515. C2#22622627direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179893222
#226226272
C2×#179889442
C2×#179889352
C2×#179888692
C2×#179888842
C2×#179888092
C2×#179887872
C2×Q8×#105968312
C2×C4×#119240182
C2×C4×#119240152
C2×#557×C732
C2×#336573