Finite Group Databasebeta

← C2×#22622961 #33966951

Subgroups — C2×#22622961 #33966951

Order 18688.

1906 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989315C211yesmaximal, characteristic, complemented
2C2×#17989311C211yesmaximal, characteristic
3#22622961C2116yesmaximal, direct product factor
4C2×#17988980C212yesmaximal
5C2×#17988966C211yesmaximal, characteristic, complemented
6C2×#17988812C212yesmaximal
7C2×#17988795C212yesmaximal, complemented
8C2×#17988793C211yesmaximal, characteristic
9C2×#17988714C211yesmaximal, characteristic, complemented
10C2×#17988691C212yesmaximal, complemented
11C2×C4×#11924015C211yesmaximal, characteristic
12C2×#462×C73C211yesmaximal, characteristic, complemented
13C22×#11924018C2212yes—
14C22×#11924018C2212yes—
15C22×#11924017C2212yes—
16C22×#11924015C2211yescharacteristic
17#17989315—24no—
18#17989315C2218yescomplemented
19#17989311—24no—
20#17989311C2218yes—
21C2×#14556591C2211yescharacteristic
22C2×#14556589C2212yes—
23C2×#14556586—22no—
24C2×#14556583—21no—
25C2×#14556581—22no—
26C2×#14556580C2212yes—
27C2×#14556571—21no—
28C2×Q8×Dic73C2212yes—
29C2×C4×Dic146—21no—
30C2×C4×Dic146C2211yescharacteristic
31C2×#14556556C2212yes—
32C2×C4×D292C2211yescharacteristic
33C2×#14556549—21no—
34#17988980C22116yes—
35#17988966C2218yescomplemented
36#17988812C22116yes—
37#17988795C22116yescomplemented
38#17988793C2218yes—
39#17988714C2218yescomplemented
40#17988691C22116yescomplemented
41C4×#11924015C2218yes—
42C2×#14556522C2211yescharacteristic
43C2×#14556520C2212yes—
44C2×#14556516C2212yes—
45C2×#14556515C2211yescharacteristic
46C2×#14556511C2211yescharacteristic
47C2×#14556501C2212yes—
48C2×C4×#10596831C2211yescharacteristic
49C22×Q16×C73C2212yes—
50C2×C4⋊Q8×C73C2211yescharacteristic
51C2×C4×Q8×C73C2211yescharacteristic
52#462×C73C2218yescomplemented
53C2×C4⋊C8×C73C2211yescharacteristic
54C2×#103×C73C2212yes—
55C22×Dic146D411yescharacteristic
56C22×Dic146C2312yes—
57C22×D292C2311yescharacteristic
58C22×C4×D73—21no—
59C2×#11924018—28no—
60C2×#11924018—44no—
61C2×#11924018C2318yes—
62C2×#11924018C2318yes—
63C2×#11924017—44no—
64C2×#11924017C2318yes—
65C2×#11924015—24no—
66C2×#11924015C2314yes—
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69#14556591—22no—
70#14556591C2314yes—
71#14556589—24no—
72#14556589C2318yes—
73#14556586—216no—
74#14556583—28no—
75#14556581—216no—
76#14556580—24no—
77#14556580C2318yes—
78#14556571—28no—
79Q8×Dic73—24no—
80Q8×Dic73C2318yes—
81C4×Dic146—28no—
82C4×Dic146D414yes—
83C4×Dic146C2314yes—
84#14556556—24no—
85#14556556C2318yes—
86C4×D292—22no—
87C4×D292C2314yes—
88#14556549—28no—
89C2×C4⋊Dic73—22no—
90C2×C4⋊Dic73—21no—
91C2×C4⋊Dic73—21no—
92C2×C4⋊Dic73D411yescharacteristic
93C2×C4⋊Dic73C2311yescharacteristic
94C2×#11924007—22no—
95C2×#11924007—22no—
96C2×#11924007—22no—
97C2×#11924007D412yes—
98C2×#11924005—22no—
99C2×#11924005—22no—
100C2×#11924005—21no—
101C2×C4×Dic73—22no—
102C2×C4×Dic73—22no—
103C2×C4×Dic73C2312yes—
104#14556522C2314yes—
105#14556520C2318yes—
106#14556516C2318yes—
107#14556515C2314yes—
108#14556511C2314yes—
109#14556501C2318yes—
110C4×#10596831C2314yes—
111C22×Q8×C73C2311yescharacteristic
112C22×Q8×C73C2312yes—
113C2×Q16×C73—44no—
114C2×Q16×C73C2318yes—
115C22×C8×C73C2312yes—
116C4⋊Q8×C73—22no—
117C4⋊Q8×C73C2314yes—
118C4×Q8×C73—22no—
119C4×Q8×C73C2314yes—
120C2×C4⋊C4×C73—21no—
121C2×C4⋊C4×C73—21no—
122C2×C4⋊C4×C73—21no—
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C42×C73—21no—
125C2×C42×C73C2311yescharacteristic
126C4⋊C8×C73C2314yes—
127#103×C73C2318yes—
128C23×D73—21no—
129C2×Dic146—24no—
130C2×Dic146—24no—
131C2×Dic146—24no—
132C2×Dic146—22no—
133C2×Dic146C2×D414yes—
134C2×Dic146C2414yes—
135C2×D292—22no—
136C2×D292—22no—
137C2×D292—21no—
138C2×D292C2412yes—
139C22×Dic73—21no—
140C22×Dic73—22no—
141C22×Dic73C2×D412yes—
142C22×Dic73C2×D412yes—
143C2×C4×D73—24no—
144C2×C4×D73—44no—
145#11924018—216no—
146#11924018—48no—
147#11924017—48no—
148#11924015—28no—
149C2×#10596831—22no—
150C2×#10596831#4114yes—
151C2×#10596831C2412yes—
152C2×#10596831C2412yes—
153C4⋊Dic73—24no—
154C4⋊Dic73—24no—
155C4⋊Dic73—21no—
156C4⋊Dic73—28no—
157C4⋊Dic73C2×D414yes—
158C4⋊Dic73C2412yes—
159#11924007—28no—
160#11924007—28no—
161#11924007—28no—
162#11924007C2×D418yes—
163#11924005—24no—
164#11924005—28no—
165#11924005—28no—
166C4×Dic73—28no—
167C4×Dic73—28no—
168C4×Dic73C2×D414yes—
169C4×Dic73C2414yes—
170C2×Q8×C73—21no—
171C2×Q8×C73—22no—
172C2×Q8×C73—22no—
173C2×Q8×C73—24no—
174C2×Q8×C73—22no—
175C2×Q8×C73—24no—
176C2×Q8×C73C2414yes—
177C2×Q8×C73C2412yes—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—22no—
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2411yescharacteristic
184Q16×C73—48no—
185C2×C8×C73—24no—
186C2×C8×C73C2414yes—
187C4⋊C4×C73—24no—
188C4⋊C4×C73—24no—
189C4⋊C4×C73—24no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73C2412yes—
192C42×C73—24no—
193C42×C73C2×D412yes—
194C42×C73C2412yes—
195C23×C73C22×D411yescharacteristic
196C22×D73—22no—
197C22×D73—42no—
198C22×D73—42no—
199C22×D73—42no—
200Q8×C73—22no—
201Q8×C73—24no—
202Q8×C73—22no—
203Q8×C73—24no—
204Q8×C73—42no—
205Q8×C73—44no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
208C2×C4×C73—22no—
209C2×C4×C73—24no—
210C2×C4×C73—42no—
211C2×C4×C73—44no—
212C2×C4×C73—42no—
213C2×C4×C73#13714yes—
214C2×C4×C73C22×D412yes—
215C2×C4×C73C22×D411yescharacteristic
216C2×C4×C73C22×D412yes—
217C2×C4×C73C2×#4112yes—
218C2×C4×C73H(5,𝔽2)12yes—
219C2×C4×C73C2511yescharacteristic
220C2×Dic73—24no—
221C2×Dic73—24no—
222C2×Dic73—22no—
223C2×Dic73—24no—
224C2×Dic73—44no—
225C2×Dic73—42no—
226C2×Dic73C22×D414yes—
227C2×Dic73C22×D414yes—
228D292—22no—
229D292—22no—
230D292—42no—
231C4×D73—48no—
232Dic146—28no—
233Dic146—24no—
234Dic146—24no—
235Dic146—44no—
236C8×C73—24no—
237#10596831—22no—
238#10596831C2×#4114yes—
239C22×C73#55012yes—
240C22×C73C2×#13712yes—
241C22×C73#57812yes—
242C22×C73C23×D411yescharacteristic
243C2×D73—42no—
244C2×D73—42no—
245C2×D73—42no—
246C2×D73—84no—
247C4×C73—22no—
248C4×C73—42no—
249C4×C73—44no—
250C4×C73—42no—
251C4×C73#55012yes—
252C4×C73C2×#13714yes—
253C4×C73C22×#4111yescharacteristic
254C4×C73C2×H(5,𝔽2)11yescharacteristic
255Dic73—24no—
256Dic73—24no—
257Dic73—44no—
258Dic73—42no—
259C2×#3398—731nomaximal, Sylow
260C2×C73#339814yescomplemented
261C2×C73C2×#55011yescharacteristic
262C2×C73C22×#13711yescharacteristic
263C2×C73C2×#57811yescharacteristic
264D73—82no—
265C2×#557—731no—
266C2×#556—731no—
267#3398—7316no—
268C2×#497—732no—
269C2×#494—731no—
270C2×#464—732no—
271C2×#462—732no—
272C2×#462D7311yescharacteristic, complemented
273C2×#461—731no—
274C2×#446—731no—
275C2×#441—732no—
276C2×C4×SD16—731no—
277C73C2×#339811yescharacteristic, complemented, Sylow
278C22×Q16—732no—
279C22×Q16—732no—
280C22×Q16C2×D7312yes—
281C22×SD16—731no—
282C22×SD16—732no—
283#557—738no—
284#557—1464no—
285#556—738no—
286#556—1464no—
287C2×C4⋊Q8—732no—
288C2×C4⋊Q8—731no—
289C2×C4⋊Q8C2×D7311yescharacteristic
290C2×#126—1462no—
291C2×#125—1461no—
292C2×#125—1462no—
293C2×#124—732no—
294C2×C22⋊Q8—1461no—
295C2×C4×Q8—731no—
296C2×C4×Q8—732no—
297C2×C4×Q8—732no—
298C2×C4×Q8—1461no—
299C2×C4×Q8C2×D7311yescharacteristic
300C2×C4×D4—731no—
301C2×#117—1461no—
302#497—7316no—
303#494—738no—
304#464—7316no—
305#462—7316no—
306#462C2×D7318yescomplemented
307#461—738no—
308#446—738no—
309#441—7316no—
310C4×SD16—738no—
311C2×#106—731no—
312C2×C4⋊C8—732no—
313C2×C4⋊C8C2×D7311yescharacteristic
314C2×#103—731no—
315C2×#103—732no—
316C2×#103C2×D7312yes—
317C2×#102—731no—
318C2×#97—732no—
319C2×C4×C8—731no—
320C22×Q8—731no—
321C22×Q8—732no—
322C22×Q8C22×D7312yes—
323C22×Q8C22×D7311yescharacteristic
324C22×D4—731no—
325C23×C4—1461no—
326C2×Q16—44no—
327C2×Q16—2924no—
328C2×Q16—738no—
329C2×Q16—738no—
330C2×Q16—1468no—
331C2×Q16C22×D7318yes—
332C2×SD16—2924no—
333C2×SD16—734no—
334C2×SD16—738no—
335C2×SD16—1464no—
336C22×C8—731no—
337C22×C8—731no—
338C22×C8C22×D7312yes—
339C4⋊Q8—22no—
340C4⋊Q8—738no—
341C4⋊Q8—734no—
342C4⋊Q8—1464no—
343C4⋊Q8—1462no—
344C4⋊Q8C22×D7314yes—
345#126—14616no—
346#125—1468no—
347#125—14616no—
348#124—738no—
349#124—1464no—
350C22⋊Q8—1468no—
351C4×Q8—22no—
352C4×Q8—734no—
353C4×Q8—734no—
354C4×Q8—738no—
355C4×Q8—738no—
356C4×Q8—1468no—
357C4×Q8—1464no—
358C4×Q8—1464no—
359C4×Q8C22×D7314yes—
360C4×D4—734no—
361C4×D4—1462no—
362#117—1468no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—732no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—1462no—
370C2×C4⋊C4—1462no—
371C2×C4⋊C4—1462no—
372C2×C4⋊C4—1462no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4C22×D7311yescharacteristic
376C2×C22⋊C4—1462no—
377C2×C22⋊C4—1462no—
378C2×C22⋊C4—1461no—
379C2×C42—21no—
380C2×C42—732no—
381C2×C42—1462no—
382C2×C42—1462no—
383C2×C42C22×D7311yescharacteristic
384#106—734no—
385C4⋊C8—738no—
386C4⋊C8C22×D7314yes—
387#103—734no—
388#103—738no—
389#103C22×D7318yes—
390#102—734no—
391#97—738no—
392C4×C8—734no—
393C24—1461no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—734no—
401C2×Q8—734no—
402C2×Q8—1464no—
403C2×Q8—1464no—
404C2×Q8—1464no—
405C2×Q8—1462no—
406C2×Q8C23×D7312yes—
407C2×Q8C23×D7314yes—
408C2×D4—732no—
409C2×D4—1461no—
410C2×D4—1462no—
411C2×D4—1462no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—2924no—
416C22×C4—732no—
417C22×C4—732no—
418C22×C4—1461no—
419C22×C4—1462no—
420C22×C4—1464no—
421C22×C4D4×D7311yescharacteristic
422C22×C4D4×D7311yescharacteristic
423C22×C4C23×D7311yescharacteristic
424Q16—48no—
425Q16—2928no—
426Q16—14616no—
427SD16—2928no—
428SD16—1468no—
429C2×C8—24no—
430C2×C8—734no—
431C2×C8—732no—
432C2×C8—732no—
433C2×C8—1462no—
434C2×C8C23×D7314yes—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—21no—
439C4⋊C4—738no—
440C4⋊C4—734no—
441C4⋊C4—732no—
442C4⋊C4—1468no—
443C4⋊C4—1468no—
444C4⋊C4—1468no—
445C4⋊C4—1468no—
446C4⋊C4—1461no—
447C4⋊C4—1464no—
448C4⋊C4—1464no—
449C4⋊C4C23×D7312yes—
450C22⋊C4—1468no—
451C22⋊C4—1468no—
452C22⋊C4—1464no—
453C42—24no—
454C42—734no—
455C42—734no—
456C42—1468no—
457C42—1468no—
458C42D4×D7312yes—
459C42C23×D7312yes—
460C23—2922no—
461C23—2922no—
462C23—2922no—
463C23—1462no—
464C23C2×D4×D7311yescharacteristic
465Q8—22no—
466Q8—24no—
467Q8—22no—
468Q8—24no—
469Q8—42no—
470Q8—44no—
471Q8—2924no—
472Q8—1464no—
473Q8—1464no—
474Q8—1468no—
475D4—2922no—
476D4—1462no—
477D4—1462no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—2922no—
486C2×C4—2924no—
487C2×C4—2928no—
488C2×C4—734no—
489C2×C4—734no—
490C2×C4—1462no—
491C2×C4—1464no—
492C2×C4—1464no—
493C2×C4—1464no—
494C2×C4#1455662314yes—
495C2×C4C2×D4×D7312yes—
496C2×C4C2×D4×D7312yes—
497C2×C4C2×D4×D7311yescharacteristic
498C2×C4#41×D7312yes—
499C2×C4#1455663612yes—
500C2×C4C24×D7311yescharacteristic
501C8—24no—
502C8—734no—
503C8—1462no—
504C22—2922no—
505C22—2922no—
506C22—2922no—
507C22—5844no—
508C22#1798925512yes—
509C22C2×#1455662312yes—
510C22#1798940312yes—
511C22C22×D4×D7311yescharacteristic
512C4—22no—
513C4—42no—
514C4—44no—
515C4—42no—
516C4—2922no—
517C4—2924no—
518C4—1464no—
519C4—1464no—
520C4#1798925512yes—
521C4C2×#1455662314yes—
522C4C2×#41×D7311yescharacteristic
523C4C2×#1455663611yescharacteristic
524C2—5842no—
525C2#2262296114yesdirect product factor
526C2C2×#1798925511yescharacteristic
527C2C22×#1455662311yescharacteristic
528C2C2×#1798940311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC2×#462×C73C2
socleC23×C73C22×D4
prime radicalC22×D292C23
hypercenterC2×#462D73
hypocenterC73C2×#3398
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
1. C2×#17989315C2semidirect
3. #22622961C2direct (N × H)
5. C2×#17988966C2semidirect
7. C2×#17988795C2semidirect
9. C2×#17988714C2semidirect
10. C2×#17988691C2semidirect
12. C2×#462×C73C2semidirect
18. #17989315C22semidirect
35. #17988966C22semidirect
37. #17988795C22semidirect
39. #17988714C22semidirect
40. #17988691C22semidirect
52. #462×C73C22semidirect
260. C2×C73#3398semidirect
272. C2×#462D73semidirect
277. C73C2×#3398semidirect
306. #462C2×D73semidirect
525. C2#22622961direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179893152
C2×#179893112
#226229612
C2×#179889802
C2×#179889662
C2×#179888122
C2×#179887952
C2×#179887932
C2×#179887142
C2×#179886912
C2×C4×#119240152
C2×#462×C732
C2×#339873