Finite Group Databasebeta

← C2×#22619781 #33953564

Subgroups — C2×#22619781 #33953564

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
2C2×#17989068C211yesmaximal, characteristic
3#22619781C2116yesmaximal, direct product factor
4C2×#17988897C212yesmaximal, complemented
5C2×#17988881C212yesmaximal, complemented
6C2×#17988886C212yesmaximal, complemented
7C2×#17988866C212yesmaximal, complemented
8C22×#14556524C211yesmaximal, characteristic, complemented
9C2×#17988615C211yesmaximal, characteristic, complemented
10C2×#17988589C211yesmaximal, characteristic
11C2×#17988581C211yesmaximal, characteristic
12C2×#428×C73C211yesmaximal, characteristic, complemented
13C22×#11924028C2211yescharacteristic
14C23×#10596831C2211yescharacteristic
15#17989215—24no—
16#17989215C2218yescomplemented
17C2×#14556591—21no—
18C2×#14556590C2212yes—
19C2×#14556583—21no—
20C2×#14556582C2212yes—
21C2×#14556568—22no—
22#17989068—24no—
23#17989068C2218yes—
24C2×#14556556C2212yes—
25C2×#14556553C2212yes—
26C2×C4×C22⋊D73—22no—
27C2×#14556550C2211yescharacteristic
28C2×#14556549—21no—
29C22×C4⋊Dic73C2211yescharacteristic
30C2×C4⋊C4×D73—21no—
31#17988897C22116yescomplemented
32#17988886C22116yescomplemented
33#17988881C22116yescomplemented
34#17988866C22116yescomplemented
35C2×#14556524C2212yes—
36C2×#14556524C2212yes—
37C2×#14556524C2218yescomplemented
38#17988615C2218yescomplemented
39C2×#14556520C2212yes—
40C2×#14556520C2212yes—
41#17988589C2218yes—
42#17988581C2218yes—
43C2×#14556516C2212yes—
44C2×#14556514C2211yescharacteristic
45C2×#14556514C2211yescharacteristic
46C2×#14556513C2212yes—
47C2×#14556511C2211yescharacteristic
48C2×#14556511C2211yescharacteristic
49C22×M16×C73C2211yescharacteristic
50C2×#117×C73C2211yescharacteristic
51C22×C4⋊C4×C73C2211yescharacteristic
52#428×C73C2218yescomplemented
53C2×#107×C73C2212yes—
54C2×#106×C73C2212yes—
55C2×#11924028—22no—
56C2×#11924028—24no—
57C2×#11924028Q818yescomplemented
58C2×#11924028C2314yes—
59C22×Dic146—22no—
60C22×Dic146C2311yescharacteristic
61C22×D292C2311yescharacteristic
62C22×C22⋊D73—21no—
63C23×Dic73—21no—
64C22×C4×D73—22no—
65C22×C4×D73—21no—
66C22×#10596831—24no—
67C22×#10596831C2314yes—
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70#14556591—28no—
71#14556590—24no—
72#14556590C2318yes—
73#14556583—28no—
74#14556582—24no—
75#14556582C2318yes—
76#14556568—216no—
77#14556556—24no—
78#14556556C2318yes—
79#14556553—24no—
80#14556553C2318yes—
81C4×C22⋊D73—216no—
82#14556550D414yes—
83#14556550C2314yes—
84#14556549—28no—
85C2×C4⋊Dic73—22no—
86C2×C4⋊Dic73—21no—
87C2×C4⋊Dic73—22no—
88C2×C4⋊Dic73D411yescharacteristic
89C2×C4⋊Dic73D411yescharacteristic
90C2×C4⋊Dic73C2312yes—
91C2×C4⋊Dic73C2314yes—
92C2×#11924007—22no—
93C2×#11924007—22no—
94C2×#11924007—22no—
95C4⋊C4×D73—28no—
96C2×C22⋊Dic73D412yes—
97C2×#11924005—22no—
98C2×C4×Dic73—22no—
99C2×C4×Dic73C2312yes—
100#14556524C2318yes—
101#14556524C2318yes—
102#14556520C2318yes—
103#14556520C2318yes—
104#14556516C2318yes—
105#14556514C2314yes—
106#14556514C2314yes—
107#14556513C2318yes—
108#14556511C2314yes—
109#14556511C2314yes—
110C23×C4×C73—21no—
111C23×C4×C73C2311yescharacteristic
112C2×M16×C73—24no—
113C2×M16×C73C2314yes—
114C22×C8×C73C2312yes—
115#117×C73—22no—
116#117×C73C2314yes—
117C2×C4⋊C4×C73—21no—
118C2×C4⋊C4×C73—22no—
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C4⋊C4×C73C2312yes—
121C2×C4⋊C4×C73C2314yes—
122C2×C4⋊C4×C73C2311yescharacteristic
123C2×C22⋊C4×C73—21no—
124C2×C42×C73—21no—
125#107×C73C2318yes—
126#106×C73C2318yes—
127C23×D73—21no—
128#11924028—216no—
129#11924028—28no—
130#11924028C2×Q8116yescomplemented
131C2×Dic146—21no—
132C2×Dic146—22no—
133C2×Dic146—28no—
134C2×Dic146—84no—
135C2×Dic146C2×Q814yes—
136C2×Dic146C2412yes—
137C2×D292—21no—
138C2×D292—22no—
139C2×D292C2×Q814yes—
140C2×D292C2412yes—
141C2×C22⋊D73—28no—
142C2×C22⋊D73—24no—
143C22×Dic73—21no—
144C22×Dic73—22no—
145C22×Dic73—24no—
146C22×Dic73—82no—
147C22×Dic73C2×D412yes—
148C22×Dic73C2×D412yes—
149C2×C4×D73—28no—
150C2×C4×D73—24no—
151C2×C4×D73—28no—
152C2×C4×D73—84no—
153C2×#10596831—22no—
154C2×#10596831—22no—
155C2×#10596831—28no—
156C2×#10596831C2412yes—
157C2×#10596831C2412yes—
158C4⋊Dic73—22no—
159C4⋊Dic73—24no—
160C4⋊Dic73—28no—
161C4⋊Dic73C2×D414yes—
162C4⋊Dic73C2×D414yes—
163C4⋊Dic73C2414yes—
164#11924007—28no—
165#11924007—28no—
166#11924007—28no—
167C22⋊Dic73C2×D418yes—
168#11924005—28no—
169C4×Dic73—28no—
170C4×Dic73C2×D414yes—
171C4×Dic73C2414yes—
172C24×C73C2×D411yescharacteristic
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—22no—
176C22×C4×C73—24no—
177C22×C4×C73—82no—
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D412yes—
180C22×C4×C73C2×Q814yes—
181C22×C4×C73#4114yes—
182C22×C4×C73C2411yescharacteristic
183C22×C4×C73C2412yes—
184M16×C73—28no—
185C2×C8×C73—24no—
186C2×C8×C73C2414yes—
187C4⋊C4×C73—24no—
188C4⋊C4×C73—22no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73C2412yes—
192C4⋊C4×C73C2414yes—
193C4⋊C4×C73C2412yes—
194C22⋊C4×C73—24no—
195C42×C73—24no—
196C23×C73—22no—
197C23×C73—22no—
198C23×C73C22⋊Q814yes—
199C23×C73C22×D412yes—
200C23×C73C22×D411yescharacteristic
201C22×D73—24no—
202C22×D73—22no—
203C22×D73—42no—
204C22×D73—42no—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
207C2×C4×C73—24no—
208C2×C4×C73—44no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—84no—
212C2×C4×C73C22⋊Q814yes—
213C2×C4×C73C22⋊Q818yes—
214C2×C4×C73C22×D412yes—
215C2×C4×C73C22×D411yescharacteristic
216C2×C4×C73C22×Q812yes—
217C2×C4×C73C22×Q814yes—
218C2×C4×C73C2×#4112yes—
219C2×C4×C73C2×#4114yes—
220C2×C4×C73C2511yescharacteristic
221C22⋊D73—216no—
222C2×Dic73—24no—
223C2×Dic73—24no—
224C2×Dic73—24no—
225C2×Dic73—22no—
226C2×Dic73—24no—
227C2×Dic73—44no—
228C2×Dic73—84no—
229C2×Dic73C22×D414yes—
230C2×Dic73C22×D414yes—
231D292—24no—
232D292—22no—
233D292C22×Q814yes—
234C4×D73—216no—
235C4×D73—88no—
236Dic146—24no—
237Dic146—22no—
238Dic146—88no—
239Dic146C22×Q814yes—
240C8×C73—24no—
241#10596831—22no—
242#10596831—22no—
243C22×C73—28no—
244C22×C73—22no—
245C22×C73—22no—
246C22×C73C2×C22⋊Q814yes—
247C22×C73C2×C22⋊Q812yes—
248C22×C73#57612yes—
249C22×C73#57812yes—
250C22×C73C23×D411yescharacteristic
251C2×D73—24no—
252C2×D73—48no—
253C2×D73—42no—
254C2×D73—42no—
255C4×C73—42no—
256C4×C73—44no—
257C4×C73—42no—
258C4×C73C2×C22⋊Q812yes—
259C4×C73C2×C22⋊Q814yes—
260C4×C73C23×Q811yescharacteristic
261C4×C73C22×#4111yescharacteristic
262Dic73—24no—
263Dic73—24no—
264Dic73—24no—
265Dic73—44no—
266C2×#3083—731nomaximal, Sylow
267C2×C73—24no—
268C2×C73#308314yescomplemented
269C2×C73C22×C22⋊Q811yescharacteristic
270C2×C73C2×#57611yescharacteristic
271C2×C73C2×#57811yescharacteristic
272D73—44no—
273C2×#544—731no—
274C2×#520—731no—
275#3083—7316no—
276C2×#479—732no—
277C2×#477—732no—
278C2×#478—732no—
279C2×#474—732no—
280C2×#428D7311yescharacteristic, complemented
281C22×#107—731no—
282C2×#424—731no—
283C2×#417—731no—
284C2×#416—731no—
285C73C2×#308311yescharacteristic, complemented, Sylow
286C22×#41—731no—
287C22×M16C2×D7311yescharacteristic
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—738no—
297#520—1464no—
298C2×C4×Q8—732no—
299C2×C4×D4—732no—
300C2×C4×D4—1462no—
301C2×#117—731no—
302C2×#117—1461no—
303C2×#117C2×D7311yescharacteristic
304C22×C4⋊C4—731no—
305C22×C4⋊C4—1461no—
306C22×C4⋊C4C2×D7311yescharacteristic
307#479—7316no—
308#478—7316no—
309#477—7316no—
310#474—7316no—
311#428C2×D7318yescomplemented
312C2×#107—732no—
313C2×#107—732no—
314C2×#107—738no—
315C2×#107C2×D7312yes—
316C2×#106C2×D7312yes—
317#424—738no—
318C2×C4⋊C8—732no—
319C2×C4⋊C8—732no—
320#417—738no—
321#416—738no—
322C2×#103—732no—
323C2×#103—731no—
324C2×#103—731no—
325C2×#102—732no—
326C2×#102—731no—
327C2×#102—731no—
328C2×#41—738no—
329C2×#41—734no—
330C2×#41—1462no—
331C2×#41—1464no—
332C22×Q8—731no—
333C22×Q8—1462no—
334C22×D4—731no—
335C22×D4—1461no—
336C23×C4—21no—
337C23×C4—1462no—
338C23×C4—1461no—
339C23×C4—1461no—
340C23×C4C22×D7311yescharacteristic
341C2×M16—24no—
342C2×M16C22×D7314yes—
343C22×C8—731no—
344C22×C8—731no—
345C22×C8—734no—
346C22×C8—1464no—
347C22×C8C22×D7312yes—
348C4⋊Q8—738no—
349C4⋊Q8—1468no—
350C4⋊Q8—1464no—
351#125—738no—
352#125—1468no—
353#125—1464no—
354C22⋊Q8—14616no—
355C4×Q8—738no—
356C4×Q8—1464no—
357C4×D4—738no—
358C4×D4—1464no—
359C4×D4—14616no—
360#117—22no—
361#117—734no—
362#117—734no—
363#117—1468no—
364#117C22×D7314yes—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—732no—
370C2×C4⋊C4—734no—
371C2×C4⋊C4—1468no—
372C2×C4⋊C4—1462no—
373C2×C4⋊C4—1462no—
374C2×C4⋊C4—1462no—
375C2×C4⋊C4—1462no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4—1462no—
378C2×C4⋊C4C22×D7312yes—
379C2×C4⋊C4C22×D7314yes—
380C2×C4⋊C4C22×D7311yescharacteristic
381C2×C4⋊C4C22×D7311yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—732no—
384C2×C22⋊C4—1462no—
385C2×C42—21no—
386C2×C42—732no—
387C2×C42—1462no—
388#107—738no—
389#107—738no—
390#107C22×D7318yes—
391#106C22×D7318yes—
392C4⋊C8—738no—
393C4⋊C8—738no—
394#103—738no—
395#103—734no—
396#103—734no—
397#102—738no—
398#102—734no—
399#102—734no—
400C24—1461no—
401C24D4×D7311yescharacteristic
402#41—7316no—
403#41—14616no—
404#41—1468no—
405C2×Q8—5844no—
406C2×Q8—734no—
407C2×Q8—732no—
408C2×Q8—1468no—
409C2×Q8—1461no—
410C2×Q8—1462no—
411C2×D4—734no—
412C2×D4—732no—
413C2×D4—1468no—
414C2×D4—1464no—
415C2×D4—1461no—
416C2×D4—1462no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—82no—
422C22×C4—5844no—
423C22×C4—5842no—
424C22×C4—732no—
425C22×C4—732no—
426C22×C4—1468no—
427C22×C4—1462no—
428C22×C4—1464no—
429C22×C4—1461no—
430C22×C4—1468no—
431C22×C4—1464no—
432C22×C4D4×D7312yes—
433C22×C4D4×D7311yescharacteristic
434C22×C4Q8×D7314yes—
435C22×C4#1192402814yes—
436C22×C4C23×D7312yes—
437C22×C4C23×D7311yescharacteristic
438M16—28no—
439C2×C8—24no—
440C2×C8—732no—
441C2×C8—732no—
442C2×C8—1462no—
443C2×C8—1462no—
444C2×C8—1468no—
445C2×C8C23×D7314yes—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—734no—
451C4⋊C4—734no—
452C4⋊C4—734no—
453C4⋊C4—1468no—
454C4⋊C4—1468no—
455C4⋊C4—1468no—
456C4⋊C4—1468no—
457C4⋊C4—1464no—
458C4⋊C4—1462no—
459C4⋊C4C23×D7312yes—
460C4⋊C4C23×D7312yes—
461C4⋊C4C23×D7314yes—
462C22⋊C4—24no—
463C22⋊C4—738no—
464C22⋊C4—1468no—
465C42—24no—
466C42—734no—
467C42—734no—
468C42—1468no—
469C23—22no—
470C23—22no—
471C23—2922no—
472C23—2922no—
473C23—1462no—
474C23—1464no—
475C23#1455656914yes—
476C23C2×D4×D7312yes—
477C23C2×D4×D7311yescharacteristic
478Q8—5848no—
479Q8—734no—
480Q8—1462no—
481Q8—1464no—
482D4—734no—
483D4—14616no—
484D4—1462no—
485D4—1464no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—84no—
493C2×C4—2924no—
494C2×C4—5848no—
495C2×C4—5844no—
496C2×C4—734no—
497C2×C4—734no—
498C2×C4—1464no—
499C2×C4—1464no—
500C2×C4—1464no—
501C2×C4—1462no—
502C2×C4—1464no—
503C2×C4—14616no—
504C2×C4#1455656914yes—
505C2×C4#1455656918yes—
506C2×C4C2×D4×D7312yes—
507C2×C4C2×D4×D7311yescharacteristic
508C2×C4C2×Q8×D7312yes—
509C2×C4C2×Q8×D7314yes—
510C2×C4C2×#1192402812yes—
511C2×C4C2×#1192402814yes—
512C2×C4C24×D7311yescharacteristic
513C8—24no—
514C8—1462no—
515C8—1462no—
516C22—28no—
517C22—22no—
518C22—22no—
519C22—2928no—
520C22—2922no—
521C22—2922no—
522C22—1464no—
523C22C2×#1455656914yes—
524C22C2×#1455656912yes—
525C22#1798939112yes—
526C22#1798940312yes—
527C22C22×D4×D7311yescharacteristic
528C4—44no—
529C4—42no—
530C4—42no—
531C4—2924no—
532C4—1464no—
533C4—1464no—
534C4—1464no—
535C4C2×#1455656914yes—
536C4C2×#1455656912yes—
537C4C22×Q8×D7311yescharacteristic
538C4C22×#1192402811yescharacteristic
539C2—24no—
540C2—2924no—
541C2#2261978114yesdirect product factor
542C2C22×#1455656911yescharacteristic
543C2C2×#1798939111yescharacteristic
544C2C2×#1798940311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC2×#428×C73C2
socleC23×C73C22×D4
prime radicalC22×#11924028C22
hypercenterC2×#428D73
hypocenterC73C2×#3083
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#17989215C2semidirect
3. #22619781C2direct (N × H)
4. C2×#17988897C2semidirect
5. C2×#17988881C2semidirect
6. C2×#17988886C2semidirect
7. C2×#17988866C2semidirect
8. C22×#14556524C2semidirect
9. C2×#17988615C2semidirect
12. C2×#428×C73C2semidirect
16. #17989215C22semidirect
31. #17988897C22semidirect
32. #17988886C22semidirect
33. #17988881C22semidirect
34. #17988866C22semidirect
37. C2×#14556524C22semidirect
38. #17988615C22semidirect
52. #428×C73C22semidirect
57. C2×#11924028Q8semidirect
130. #11924028C2×Q8semidirect
268. C2×C73#3083semidirect
280. C2×#428D73semidirect
285. C73C2×#3083semidirect
311. #428C2×D73semidirect
541. C2#22619781direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179892152
C2×#179890682
#226197812
C2×#179888972
C2×#179888812
C2×#179888862
C2×#179888662
C22×#145565242
C2×#179886152
C2×#179885892
C2×#179885812
C2×#428×C732
C2×#308373