Finite Group Databasebeta

← C2×#22619782 #33953565

Subgroups — C2×#22619782 #33953565

Order 18688.

2094 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989215C211yesmaximal, characteristic, complemented
2#22619782C2116yesmaximal, direct product factor
3C2×#17988894C212yesmaximal, complemented
4C2×#17988882C212yesmaximal, complemented
5C2×#17988887C212yesmaximal, complemented
6C2×#17988867C212yesmaximal, complemented
7C2×#17988627C211yesmaximal, characteristic, complemented
8C22×#14556524C211yesmaximal, characteristic, complemented
9C2×#17988614C211yesmaximal, characteristic, complemented
10C2×#17988588C211yesmaximal, characteristic
11C2×#17988583C211yesmaximal, characteristic
12C2×#520×C73C211yesmaximal, characteristic
13C22×#11924012C2211yescharacteristic
14C23×#10596831C2211yescharacteristic
15#17989215—24no—
16#17989215C2218yescomplemented
17C2×#14556591C2212yes—
18C2×#14556590—21no—
19C2×#14556583C2212yes—
20C2×#14556582—21no—
21C2×#14556568—22no—
22C2×#14556550C2211yescharacteristic
23C22×C4⋊Dic73C2211yescharacteristic
24#17988894C22116yescomplemented
25#17988887C22116yescomplemented
26#17988882C22116yescomplemented
27#17988867C22116yescomplemented
28#17988627C2218yescomplemented
29C2×#14556524C2212yes—
30C2×#14556524C2212yes—
31C2×#14556524C2212yes—
32C2×#14556524C2218yescomplemented
33C2×#14556522C2212yes—
34#17988614C2218yescomplemented
35C2×C4⋊#10596831C2212yes—
36C2×C4⋊#10596831C2212yes—
37#17988588C2218yes—
38#17988583C2218yes—
39C2×#14556515C2211yescharacteristic
40C2×#14556515C2212yes—
41C2×#14556515C2211yescharacteristic
42C2×#14556512C2211yescharacteristic
43C2×#14556512C2212yes—
44C2×#14556512C2211yescharacteristic
45C22×#41×C73C2211yescharacteristic
46#520×C73—24no—
47#520×C73C2218yes—
48C2×C4×Q8×C73C2212yes—
49C2×C4×D4×C73—22no—
50C2×C4×D4×C73C2212yes—
51C2×#117×C73—21no—
52C2×#117×C73C2211yescharacteristic
53C22×C4⋊C4×C73—21no—
54C22×C4⋊C4×C73C2211yescharacteristic
55C22×Dic146—22no—
56C23×Dic73—21no—
57C2×#11924012—24no—
58C2×#11924012C2314yes—
59C22×#10596831—24no—
60C22×#10596831C2312yes—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2314yes—
64#14556591—24no—
65#14556591C2318yes—
66#14556590—28no—
67#14556583—24no—
68#14556583C2318yes—
69#14556582—28no—
70#14556568—216no—
71#14556550—22no—
72#14556550C2314yes—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73—22no—
75C2×C4⋊Dic73—22no—
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2312yes—
79C2×C4⋊Dic73C2314yes—
80C2×#11924007—22no—
81C2×#11924007—22no—
82C2×C22⋊Dic73—21no—
83C2×C4×Dic73—21no—
84#14556524C2318yes—
85#14556524C2318yes—
86#14556524C2318yes—
87#14556522C2318yes—
88C4⋊#10596831C2318yes—
89C4⋊#10596831C2318yes—
90#14556515C2314yes—
91#14556515C2318yes—
92#14556515C2314yes—
93#14556512C2314yes—
94#14556512C2318yes—
95#14556512C2314yes—
96C2×#41×C73—22no—
97C2×#41×C73—24no—
98C2×#41×C73Q818yescomplemented
99C2×#41×C73C2314yes—
100C22×Q8×C73C2311yescharacteristic
101C22×D4×C73—21no—
102C22×D4×C73C2311yescharacteristic
103C23×C4×C73—21no—
104C23×C4×C73—21no—
105C23×C4×C73—22no—
106C23×C4×C73C2311yescharacteristic
107C4×Q8×C73—24no—
108C4×Q8×C73C2318yes—
109C4×D4×C73—216no—
110C4×D4×C73—24no—
111C4×D4×C73C2318yes—
112#117×C73—28no—
113#117×C73D414yes—
114#117×C73C2314yes—
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73—22no—
117C2×C4⋊C4×C73—22no—
118C2×C4⋊C4×C73—28no—
119C2×C4⋊C4×C73D411yescharacteristic
120C2×C4⋊C4×C73D411yescharacteristic
121C2×C4⋊C4×C73C2312yes—
122C2×C4⋊C4×C73C2314yes—
123C2×C22⋊C4×C73—22no—
124C2×C22⋊C4×C73D412yes—
125C2×C42×C73—22no—
126C2×C42×C73C2312yes—
127C2×Dic146—28no—
128C2×Dic146—84no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—22no—
132C22×Dic73—24no—
133C22×Dic73—82no—
134#11924012—28no—
135C2×#10596831—22no—
136C2×#10596831—22no—
137C2×#10596831—28no—
138C2×#10596831—24no—
139C2×#10596831C2414yes—
140C2×#10596831C2412yes—
141C2×#10596831C2412yes—
142C4⋊Dic73—24no—
143C4⋊Dic73—22no—
144C4⋊Dic73—28no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73C2412yes—
148C4⋊Dic73C2414yes—
149C4⋊Dic73C2412yes—
150#11924007—28no—
151#11924007—28no—
152C22⋊Dic73—24no—
153C4×Dic73—24no—
154C24×C73—21no—
155C24×C73C2×D411yescharacteristic
156#41×C73—216no—
157#41×C73—28no—
158#41×C73C2×Q8116yescomplemented
159C2×Q8×C73—21no—
160C2×Q8×C73—22no—
161C2×Q8×C73C2×Q814yes—
162C2×Q8×C73C2412yes—
163C2×D4×C73—28no—
164C2×D4×C73—24no—
165C2×D4×C73—21no—
166C2×D4×C73—22no—
167C2×D4×C73C2×Q814yes—
168C2×D4×C73C2412yes—
169C22×C4×C73—21no—
170C22×C4×C73—28no—
171C22×C4×C73—24no—
172C22×C4×C73—22no—
173C22×C4×C73—24no—
174C22×C4×C73—28no—
175C22×C4×C73—84no—
176C22×C4×C73—82no—
177C22×C4×C73C2×D412yes—
178C22×C4×C73C2×D412yes—
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D412yes—
181C22×C4×C73C2×Q814yes—
182C22×C4×C73#4114yes—
183C22×C4×C73C2411yescharacteristic
184C22×C4×C73C2412yes—
185C4⋊C4×C73—24no—
186C4⋊C4×C73—28no—
187C4⋊C4×C73—22no—
188C4⋊C4×C73C2×D414yes—
189C4⋊C4×C73C2×D414yes—
190C4⋊C4×C73C2414yes—
191C22⋊C4×C73—28no—
192C22⋊C4×C73C2×D418yes—
193C42×C73—28no—
194C42×C73C2×D414yes—
195C42×C73C2414yes—
196C23×C73—22no—
197C23×C73—22no—
198C23×C73—24no—
199C23×C73—22no—
200C23×C73—42no—
201C23×C73—42no—
202C23×C73C22⋊Q814yes—
203C23×C73C22×D412yes—
204C23×C73C22×D411yescharacteristic
205Q8×C73—24no—
206Q8×C73—22no—
207Q8×C73C22×Q814yes—
208D4×C73—216no—
209D4×C73—24no—
210D4×C73—22no—
211D4×C73C22×Q814yes—
212C2×C4×C73—24no—
213C2×C4×C73—24no—
214C2×C4×C73—24no—
215C2×C4×C73—22no—
216C2×C4×C73—216no—
217C2×C4×C73—24no—
218C2×C4×C73—44no—
219C2×C4×C73—88no—
220C2×C4×C73—84no—
221C2×C4×C73C22⋊Q814yes—
222C2×C4×C73C22⋊Q818yes—
223C2×C4×C73C22×D414yes—
224C2×C4×C73C22×D414yes—
225C2×C4×C73C22×D411yescharacteristic
226C2×C4×C73C22×D412yes—
227C2×C4×C73C22×Q812yes—
228C2×C4×C73C22×Q814yes—
229C2×C4×C73C2×#4112yes—
230C2×C4×C73C2×#4114yes—
231C2×C4×C73C2511yescharacteristic
232C2×Dic73—22no—
233C2×Dic73—22no—
234C2×Dic73—24no—
235C2×Dic73—44no—
236C2×Dic73—42no—
237C2×Dic73—42no—
238C2×Dic73—84no—
239Dic146—88no—
240#10596831—22no—
241#10596831—22no—
242#10596831—24no—
243C22×C73—28no—
244C22×C73—22no—
245C22×C73—22no—
246C22×C73—24no—
247C22×C73—48no—
248C22×C73—42no—
249C22×C73—42no—
250C22×C73C2×C22⋊Q814yes—
251C22×C73C2×C22⋊Q812yes—
252C22×C73#57612yes—
253C22×C73#57812yes—
254C22×C73C23×D411yescharacteristic
255C4×C73—24no—
256C4×C73—24no—
257C4×C73—24no—
258C4×C73—44no—
259C4×C73C2×C22⋊Q814yes—
260C4×C73C2×C22⋊Q812yes—
261C4×C73C23×Q811yescharacteristic
262C4×C73C22×#4111yescharacteristic
263Dic73—42no—
264Dic73—44no—
265Dic73—42no—
266C2×#3083—731nomaximal, Sylow
267C2×C73—24no—
268C2×C73—44no—
269C2×C73#308314yescomplemented
270C2×C73C22×C22⋊Q811yescharacteristic
271C2×C73C2×#57611yescharacteristic
272C2×C73C2×#57811yescharacteristic
273C2×#544—731no—
274C2×#520D7311yescharacteristic
275#3083—7316no—
276C2×#479—732no—
277C2×#477—732no—
278C2×#478—732no—
279C2×#474—732no—
280C2×#428—731no—
281C22×#107—731no—
282C2×#424—731no—
283C2×#417—731no—
284C2×#416—731no—
285C73C2×#308311yescharacteristic, complemented, Sylow
286C22×#41C2×D7311yescharacteristic
287C22×M16—731no—
288C23×C8—731no—
289#544—738no—
290#544—1464no—
291C2×C4⋊Q8—732no—
292C2×C4⋊Q8—1461no—
293C2×#125—732no—
294C2×#125—1461no—
295C2×C22⋊Q8—1462no—
296#520—24no—
297#520C2×D7318yes—
298C2×C4×Q8C2×D7312yes—
299C2×C4×D4—22no—
300C2×C4×D4C2×D7312yes—
301C2×#117—21no—
302C2×#117—731no—
303C2×#117C2×D7311yescharacteristic
304C22×C4⋊C4—21no—
305C22×C4⋊C4—731no—
306C22×C4⋊C4C2×D7311yescharacteristic
307#479—7316no—
308#478—7316no—
309#477—7316no—
310#474—7316no—
311#428—738no—
312C2×#107—732no—
313C2×#107—732no—
314C2×#107—732no—
315C2×#107—738no—
316C2×#106—732no—
317#424—738no—
318C2×C4⋊C8—732no—
319C2×C4⋊C8—732no—
320#417—738no—
321#416—738no—
322C2×#103—731no—
323C2×#103—732no—
324C2×#103—731no—
325C2×#102—731no—
326C2×#102—732no—
327C2×#102—731no—
328C2×#41—22no—
329C2×#41—24no—
330C2×#41Dic14618yescomplemented
331C2×#41C22×D7314yes—
332C22×Q8—1462no—
333C22×Q8C22×D7311yescharacteristic
334C22×D4—21no—
335C22×D4C22×D7311yescharacteristic
336C23×C4—21no—
337C23×C4—21no—
338C23×C4—22no—
339C23×C4—1461no—
340C23×C4C22×D7311yescharacteristic
341C2×M16—734no—
342C2×M16—1464no—
343C22×C8—731no—
344C22×C8—731no—
345C22×C8—734no—
346C22×C8—732no—
347C22×C8—1464no—
348C4⋊Q8—738no—
349C4⋊Q8—1468no—
350C4⋊Q8—1464no—
351#125—738no—
352#125—1468no—
353#125—1464no—
354C22⋊Q8—14616no—
355C4×Q8—24no—
356C4×Q8C22×D7318yes—
357C4×D4—216no—
358C4×D4—24no—
359C4×D4C22×D7318yes—
360#117—28no—
361#117—734no—
362#117—1462no—
363#117C22⋊D7314yes—
364#117C22×D7314yes—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—28no—
369C2×C4⋊C4—732no—
370C2×C4⋊C4—734no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1462no—
375C2×C4⋊C4—1462no—
376C2×C4⋊C4—1462no—
377C2×C4⋊C4—1462no—
378C2×C4⋊C4C22⋊D7311yescharacteristic
379C2×C4⋊C4C22⋊D7311yescharacteristic
380C2×C4⋊C4C22×D7312yes—
381C2×C4⋊C4C22×D7314yes—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—1461no—
384C2×C22⋊C4C22⋊D7312yes—
385C2×C42—22no—
386C2×C42—1461no—
387C2×C42C22×D7312yes—
388#107—738no—
389#107—738no—
390#107—738no—
391#106—738no—
392C4⋊C8—738no—
393C4⋊C8—738no—
394#103—734no—
395#103—734no—
396#103—738no—
397#102—734no—
398#102—734no—
399#102—738no—
400C24—21no—
401C24C2×C22⋊D7311yescharacteristic
402#41—216no—
403#41—28no—
404#41C2×Dic146116yescomplemented
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—5844no—
408C2×Q8—1468no—
409C2×Q8C2×Dic14614yes—
410C2×Q8C23×D7312yes—
411C2×D4—28no—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4C2×Dic14614yes—
416C2×D4C23×D7312yes—
417C22×C4—21no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—84no—
424C22×C4—82no—
425C22×C4—5842no—
426C22×C4—1461no—
427C22×C4—1461no—
428C22×C4—1462no—
429C22×C4—1464no—
430C22×C4C2×C22⋊D7312yes—
431C22×C4C2×C22⋊D7312yes—
432C22×C4C2×C22⋊D7312yes—
433C22×C4C2×C22⋊D7311yescharacteristic
434C22×C4C2×Dic14614yes—
435C22×C4#1192402814yes—
436C22×C4C23×D7312yes—
437C22×C4C23×D7311yescharacteristic
438M16—1468no—
439C2×C8—734no—
440C2×C8—732no—
441C2×C8—732no—
442C2×C8—1462no—
443C2×C8—1462no—
444C2×C8—1468no—
445C2×C8—1464no—
446C4⋊C4—24no—
447C4⋊C4—28no—
448C4⋊C4—22no—
449C4⋊C4—732no—
450C4⋊C4—734no—
451C4⋊C4—732no—
452C4⋊C4—1468no—
453C4⋊C4—1468no—
454C4⋊C4—1464no—
455C4⋊C4—1462no—
456C4⋊C4—1461no—
457C4⋊C4—1461no—
458C4⋊C4—1468no—
459C4⋊C4C2×C22⋊D7314yes—
460C4⋊C4C2×C22⋊D7314yes—
461C4⋊C4C23×D7314yes—
462C22⋊C4—28no—
463C22⋊C4—1464no—
464C22⋊C4C2×C22⋊D7318yes—
465C42—28no—
466C42—1464no—
467C42C2×C22⋊D7314yes—
468C42C23×D7314yes—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—42no—
474C23—42no—
475C23#1455657014yes—
476C23C22×C22⋊D7312yes—
477C23C22×C22⋊D7311yescharacteristic
478Q8—22no—
479Q8—24no—
480Q8—5848no—
481Q8C22×Dic14614yes—
482D4—216no—
483D4—22no—
484D4—24no—
485D4C22×Dic14614yes—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—216no—
491C2×C4—24no—
492C2×C4—44no—
493C2×C4—88no—
494C2×C4—84no—
495C2×C4—2924no—
496C2×C4—2922no—
497C2×C4—2922no—
498C2×C4—5844no—
499C2×C4—1462no—
500C2×C4—1462no—
501C2×C4—1464no—
502C2×C4#1455657014yes—
503C2×C4#1455657018yes—
504C2×C4C22×C22⋊D7314yes—
505C2×C4C22×C22⋊D7314yes—
506C2×C4C22×C22⋊D7312yes—
507C2×C4C22×C22⋊D7311yescharacteristic
508C2×C4C22×Dic14612yes—
509C2×C4C22×Dic14614yes—
510C2×C4C2×#1192402812yes—
511C2×C4C2×#1192402814yes—
512C2×C4C24×D7311yescharacteristic
513C8—1462no—
514C8—1462no—
515C8—1464no—
516C22—28no—
517C22—22no—
518C22—22no—
519C22—24no—
520C22—48no—
521C22—42no—
522C22—42no—
523C22C2×#1455657014yes—
524C22C2×#1455657012yes—
525C22#1798939012yes—
526C22#1798940212yes—
527C22C23×C22⋊D7311yescharacteristic
528C4—24no—
529C4—24no—
530C4—24no—
531C4—44no—
532C4—2924no—
533C4—2922no—
534C4—2922no—
535C4C2×#1455657014yes—
536C4C2×#1455657012yes—
537C4C23×Dic14611yescharacteristic
538C4C22×#1192402811yescharacteristic
539C2—24no—
540C2—44no—
541C2#2261978214yesdirect product factor
542C2C22×#1455657011yescharacteristic
543C2C2×#1798939011yescharacteristic
544C2C2×#1798940211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC2×#520×C73C2
socleC23×C73C22×D4
prime radicalC22×#41×C73C22
hypercenterC2×#520D73
hypocenterC73C2×#3083
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#17989215C2semidirect
2. #22619782C2direct (N × H)
3. C2×#17988894C2semidirect
4. C2×#17988882C2semidirect
5. C2×#17988887C2semidirect
6. C2×#17988867C2semidirect
7. C2×#17988627C2semidirect
8. C22×#14556524C2semidirect
9. C2×#17988614C2semidirect
16. #17989215C22semidirect
24. #17988894C22semidirect
25. #17988887C22semidirect
26. #17988882C22semidirect
27. #17988867C22semidirect
28. #17988627C22semidirect
32. C2×#14556524C22semidirect
34. #17988614C22semidirect
98. C2×#41×C73Q8semidirect
158. #41×C73C2×Q8semidirect
269. C2×C73#3083semidirect
285. C73C2×#3083semidirect
330. C2×#41Dic146semidirect
404. #41C2×Dic146semidirect
541. C2#22619782direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179892152
#226197822
C2×#179888942
C2×#179888822
C2×#179888872
C2×#179888672
C2×#179886272
C22×#145565242
C2×#179886142
C2×#179885882
C2×#179885832
C2×#520×C732
C2×#308373