Finite Group Databasebeta

← C2×#22610389 #33087771

Subgroups — C2×#22610389 #33087771

Order 18688.

2066 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556574C211yesmaximal, characteristic
2C22×#14556505C212yesmaximal
3C2×#17988450C211yesmaximal, characteristic, complemented
4#22610389C218yesmaximal, direct product factor
5C2×#17988353C212yesmaximal, complemented
6C22×#117×C73C211yesmaximal, characteristic, complemented
7C23×C22⋊D73C2211yescharacteristic
8C2×#14556574—22no—
9C2×#14556574—22no—
10C2×#14556574—21no—
11C2×#14556574—21no—
12C2×#14556574—22no—
13C2×#14556574C414yescomplemented
14C2×#14556574C2214yes—
15C22×#11924007C412yescomplemented
16C22×C22⋊Dic73C2212yes—
17C22×C22⋊Dic73C2211yescharacteristic
18C22×#11924005C412yescomplemented
19C2×#14556505—28no—
20C2×#14556505C2218yes—
21#17988450—22no—
22#17988450C2214yescomplemented
23C2×#14556498—22no—
24#17988353C2218yescomplemented
25C24×C4×C73C2211yescharacteristic
26C2×#117×C73—22no—
27C2×#117×C73—28no—
28C2×#117×C73C2214yescomplemented
29C22×C4⋊C4×C73—21no—
30C22×C22⋊C4×C73C2212yes—
31C22×C42×C73—21no—
32C24×D73C2×C411yescharacteristic
33C22×C22⋊D73—24no—
34C22×C22⋊D73—21no—
35C22×C22⋊D73—21no—
36C22×C22⋊D73—21no—
37C22×C22⋊D73—21no—
38C22×C22⋊D73—24no—
39C22×C22⋊D73C2×C412yes—
40C22×C22⋊D73C2312yes—
41C23×Dic73—22no—
42C23×Dic73C2×C411yescharacteristic
43C23×Dic73C2×C412yes—
44#14556574—28no—
45#14556574—24no—
46#14556574—24no—
47#14556574—24no—
48#14556574—24no—
49#14556574—28no—
50C2×#11924007—24no—
51C2×#11924007—22no—
52C2×#11924007—22no—
53C2×#11924007C2×C418yescomplemented
54C2×C22⋊Dic73—21no—
55C2×C22⋊Dic73—21no—
56C2×C22⋊Dic73—21no—
57C2×C22⋊Dic73—21no—
58C2×C22⋊Dic73—24no—
59C2×C22⋊Dic73—22no—
60C2×C22⋊Dic73—24no—
61C2×C22⋊Dic73C2×C412yes—
62C2×C22⋊Dic73C2314yes—
63C2×C22⋊Dic73C2312yes—
64C2×#11924005—24no—
65C2×#11924005—22no—
66C2×#11924005—22no—
67C2×#11924005C2×C418yescomplemented
68#14556505—216no—
69#14556498—28no—
70C25×C73C2311yescharacteristic
71C23×C4×C73—24no—
72C23×C4×C73—24no—
73C23×C4×C73—22no—
74C23×C4×C73C2×C412yes—
75C23×C4×C73C2×C412yes—
76C23×C4×C73D412yes—
77C23×C4×C73D412yes—
78C23×C4×C73D411yescharacteristic
79C23×C4×C73D411yescharacteristic
80C23×C4×C73D412yes—
81C23×C4×C73C2312yes—
82#117×C73—216no—
83#117×C73—216no—
84C2×C4⋊C4×C73—28no—
85C2×C4⋊C4×C73—24no—
86C2×C22⋊C4×C73—22no—
87C2×C22⋊C4×C73—24no—
88C2×C22⋊C4×C73D418yes—
89C2×C22⋊C4×C73C2314yes—
90C2×C42×C73—28no—
91C2×C42×C73—24no—
92C23×D73—21no—
93C23×D73—21no—
94C23×D73—24no—
95C23×D73—44no—
96C23×D73C22×C412yes—
97C2×C22⋊D73—22no—
98C2×C22⋊D73—22no—
99C2×C22⋊D73—28no—
100C2×C22⋊D73—24no—
101C2×C22⋊D73—22no—
102C2×C22⋊D73—22no—
103C2×C22⋊D73—24no—
104C2×C22⋊D73—42no—
105C2×C22⋊D73—42no—
106C2×C22⋊D73—44no—
107C2×C22⋊D73—44no—
108C2×C22⋊D73—42no—
109C2×C22⋊D73—42no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—21no—
113C22×Dic73—21no—
114C22×Dic73—24no—
115C22×Dic73—24no—
116C22×Dic73—44no—
117C22×Dic73—48no—
118C22×Dic73—44no—
119C22×Dic73C22×C412yes—
120C22×Dic73C22×C414yes—
121#11924007—28no—
122#11924007—28no—
123C22⋊Dic73—22no—
124C22⋊Dic73—22no—
125C22⋊Dic73—22no—
126C22⋊Dic73—22no—
127C22⋊Dic73—24no—
128C22⋊Dic73—24no—
129C22⋊Dic73—44no—
130#11924005—28no—
131#11924005—28no—
132C24×C73—44no—
133C24×C73C22×C411yescharacteristic
134C24×C73C22×C411yescharacteristic
135C24×C73C2×D411yescharacteristic
136C24×C73C2×D411yescharacteristic
137C24×C73C2×D414yes—
138C24×C73C2×D411yescharacteristic
139C24×C73C2×D411yescharacteristic
140C24×C73#4114yes—
141C24×C73C2411yescharacteristic
142C22×C4×C73—28no—
143C22×C4×C73—28no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—24no—
147C22×C4×C73—28no—
148C22×C4×C73—28no—
149C22×C4×C73—22no—
150C22×C4×C73—22no—
151C22×C4×C73—24no—
152C22×C4×C73—24no—
153C22×C4×C73—24no—
154C22×C4×C73—28no—
155C22×C4×C73—24no—
156C22×C4×C73—48no—
157C22×C4×C73C22⋊C414yes—
158C22×C4×C73C22⋊C412yes—
159C22×C4×C73C22⋊C412yes—
160C22×C4×C73C22⋊C414yes—
161C22×C4×C73C22⋊C412yes—
162C22×C4×C73C22⋊C412yes—
163C22×C4×C73C22×C414yes—
164C22×C4×C73C2×D412yes—
165C22×C4×C73C2×D412yes—
166C22×C4×C73C2×D414yes—
167C4⋊C4×C73—216no—
168C22⋊C4×C73—28no—
169C22⋊C4×C73—24no—
170C22⋊C4×C73C2×D418yes—
171C42×C73—216no—
172C23×C73—24no—
173C23×C73—24no—
174C23×C73—24no—
175C23×C73—42no—
176C23×C73—42no—
177C23×C73—48no—
178C23×C73—42no—
179C23×C73—48no—
180C23×C73—42no—
181C23×C73C2×C22⋊C411yescharacteristic
182C23×C73C2×C22⋊C411yescharacteristic
183C23×C73C2×C22⋊C411yescharacteristic
184C23×C73C2×C22⋊C411yescharacteristic
185C23×C73C4×D414yes—
186C23×C73C4×D414yes—
187C23×C73#12012yes—
188C23×C73#12012yes—
189C23×C73#12114yes—
190C23×C73#12114yes—
191C23×C73#12312yes—
192C23×C73#12312yes—
193C23×C73C23×C411yescharacteristic
194C23×C73C22×D411yescharacteristic
195C23×C73C22×D412yes—
196C23×C73C22×D411yescharacteristic
197C23×C73C2×#4112yes—
198C22×D73—22no—
199C22×D73—24no—
200C22×D73—44no—
201C22×D73—42no—
202C22×D73—42no—
203C22×D73—44no—
204C22×D73—44no—
205C22×D73—88no—
206C2×C4×C73—28no—
207C2×C4×C73—22no—
208C2×C4×C73—22no—
209C2×C4×C73—28no—
210C2×C4×C73—22no—
211C2×C4×C73—22no—
212C2×C4×C73—28no—
213C2×C4×C73—24no—
214C2×C4×C73—24no—
215C2×C4×C73—28no—
216C2×C4×C73—24no—
217C2×C4×C73—24no—
218C2×C4×C73—28no—
219C2×C4×C73—48no—
220C2×C4×C73—416no—
221C2×C4×C73C2×C22⋊C414yes—
222C2×C4×C73C2×C22⋊C414yes—
223C22⋊D73—42no—
224C22⋊D73—44no—
225C22⋊D73—42no—
226C22⋊D73—44no—
227C22⋊D73—44no—
228C2×Dic73—24no—
229C2×Dic73—22no—
230C2×Dic73—24no—
231C2×Dic73—44no—
232C2×Dic73—42no—
233C2×Dic73—42no—
234C2×Dic73—44no—
235C2×Dic73—44no—
236C2×Dic73—48no—
237C2×Dic73—44no—
238C2×Dic73—88no—
239C22×C73—22no—
240C22×C73—22no—
241C22×C73—22no—
242C22×C73—22no—
243C22×C73—22no—
244C22×C73—22no—
245C22×C73—28no—
246C22×C73—24no—
247C22×C73—28no—
248C22×C73—24no—
249C22×C73—44no—
250C22×C73—42no—
251C22×C73—44no—
252C22×C73—42no—
253C22×C73—42no—
254C22×C73—42no—
255C22×C73#38612yes—
256C22×C73#38612yes—
257C22×C73#38612yes—
258C22×C73#38612yes—
259C22×C73#41014yes—
260C22×C73C22×C22⋊C411yescharacteristic
261C22×C73C2×C4×D412yes—
262C22×C73C2×#12011yescharacteristic
263C22×C73C2×#12112yes—
264C22×C73C2×#12311yescharacteristic
265C2×D73—44no—
266C2×D73—42no—
267C2×D73—84no—
268C2×D73—84no—
269C2×D73—84no—
270C4×C73—24no—
271C4×C73—24no—
272C4×C73—48no—
273Dic73—44no—
274Dic73—42no—
275Dic73—84no—
276C2×#1899—731nomaximal, Sylow
277C2×C73—22no—
278C2×C73—21no—
279C2×C73—22no—
280C2×C73—22no—
281C2×C73—21no—
282C2×C73—22no—
283C2×C73—22no—
284C2×C73#189914yescomplemented
285C2×C73C2×#38611yescharacteristic
286C2×C73C2×#41012yes—
287D73—82no—
288C22×#123—731no—
289C22×#117D7311yescharacteristic, complemented
290C22×#99—732no—
291C2×#381—731no—
292#1899—738no—
293C2×#342—732no—
294C73C2×#189911yescharacteristic, complemented, Sylow
295C23×D4—731no—
296C24×C4C2×D7311yescharacteristic
297C2×#123—734no—
298C2×#123—734no—
299C2×#123—1462no—
300C2×#123—1462no—
301C2×#123—1461no—
302C2×#123—1462no—
303C2×#123—1461no—
304C2×#117—22no—
305C2×#117—28no—
306C2×#117C2×D7314yescomplemented
307C22×C4⋊C4—21no—
308C22×C4⋊C4—732no—
309C22×C22⋊C4—731no—
310C22×C22⋊C4—732no—
311C22×C22⋊C4—732no—
312C22×C22⋊C4C2×D7312yes—
313C22×C42—21no—
314C2×#99—738no—
315C2×#99—1468no—
316#381—734no—
317#381—1462no—
318C2×#95—1462no—
319#342—738no—
320C25—731no—
321C25C22×D7311yescharacteristic
322C22×D4—732no—
323C22×D4—732no—
324C22×D4—1461no—
325C22×D4—1461no—
326C22×D4—1464no—
327C22×D4—1461no—
328C22×D4—1461no—
329C22×D4—1464no—
330C23×C4—24no—
331C23×C4—24no—
332C23×C4—22no—
333C23×C4—732no—
334C23×C4—731no—
335C23×C4—1462no—
336C23×C4C4×D7312yes—
337C23×C4C4×D7312yes—
338C23×C4D29212yes—
339C23×C4D29212yes—
340C23×C4C22⋊D7311yescharacteristic
341C23×C4C22⋊D7312yes—
342C23×C4C22⋊D7311yescharacteristic
343C23×C4C22×D7312yes—
344#123—1468no—
345#123—1464no—
346#123—1464no—
347#123—1464no—
348#123—1464no—
349#123—1468no—
350#117—216no—
351#117—216no—
352C2×C4⋊C4—28no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—738no—
355C2×C4⋊C4—1464no—
356C2×C4⋊C4—1462no—
357C2×C4⋊C4—1462no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—732no—
361C2×C22⋊C4—732no—
362C2×C22⋊C4—738no—
363C2×C22⋊C4—734no—
364C2×C22⋊C4—1462no—
365C2×C22⋊C4—1464no—
366C2×C22⋊C4—1464no—
367C2×C22⋊C4—1461no—
368C2×C22⋊C4—1461no—
369C2×C22⋊C4—1461no—
370C2×C22⋊C4—1461no—
371C2×C22⋊C4—1464no—
372C2×C22⋊C4—1462no—
373C2×C22⋊C4—1462no—
374C2×C22⋊C4C22⋊D7318yes—
375C2×C22⋊C4C22×D7314yes—
376C2×C42—28no—
377C2×C42—24no—
378#99—14616no—
379#95—1468no—
380C24—44no—
381C24—2924no—
382C24—732no—
383C24—1461no—
384C24—1461no—
385C24—1464no—
386C24C2×C4×D7311yescharacteristic
387C24C2×C4×D7311yescharacteristic
388C24C2×C22⋊D7311yescharacteristic
389C24C2×C22⋊D7311yescharacteristic
390C24C2×C22⋊D7314yes—
391C24C2×D29211yescharacteristic
392C24C2×D29211yescharacteristic
393C24#1192402814yes—
394C24C23×D7311yescharacteristic
395C2×D4—2922no—
396C2×D4—2922no—
397C2×D4—2924no—
398C2×D4—2924no—
399C2×D4—2922no—
400C2×D4—2922no—
401C2×D4—1462no—
402C2×D4—1462no—
403C2×D4—1468no—
404C2×D4—1464no—
405C2×D4—1462no—
406C2×D4—1462no—
407C2×D4—1464no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—48no—
423C22×C4—2924no—
424C22×C4—2928no—
425C22×C4—2924no—
426C22×C4—734no—
427C22×C4—732no—
428C22×C4—1464no—
429C22×C4—1462no—
430C22×C4—1461no—
431C22×C4—1462no—
432C22×C4—1461no—
433C22×C4—1464no—
434C22×C4#1192400514yes—
435C22×C4#1192400514yes—
436C22×C4#1192400512yes—
437C22×C4#1192400512yes—
438C22×C4#1192400512yes—
439C22×C4#1192400512yes—
440C22×C4C2×C4×D7314yes—
441C22×C4C2×C22⋊D7312yes—
442C22×C4C2×C22⋊D7312yes—
443C22×C4C2×D29214yes—
444C4⋊C4—216no—
445C4⋊C4—1468no—
446C4⋊C4—1468no—
447C22⋊C4—28no—
448C22⋊C4—24no—
449C22⋊C4—2924no—
450C22⋊C4—1464no—
451C22⋊C4—1464no—
452C22⋊C4—1462no—
453C22⋊C4—1462no—
454C22⋊C4—1462no—
455C22⋊C4—1462no—
456C22⋊C4—1468no—
457C22⋊C4—1468no—
458C22⋊C4C2×C22⋊D7318yes—
459C42—216no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—42no—
464C23—42no—
465C23—48no—
466C23—42no—
467C23—48no—
468C23—42no—
469C23—2924no—
470C23—2924no—
471C23—2922no—
472C23—2922no—
473C23—2924no—
474C23—5848no—
475C23—1462no—
476C23—1464no—
477C23C2×#1192400511yescharacteristic
478C23C2×#1192400511yescharacteristic
479C23C2×#1192400511yescharacteristic
480C23C2×#1192400511yescharacteristic
481C23C4×C22⋊D7314yes—
482C23C4×C22⋊D7314yes—
483C23#1455656012yes—
484C23#1455656012yes—
485C23#1455656514yes—
486C23#1455656514yes—
487C23#1455657412yes—
488C23#1455657412yes—
489C23C22×C4×D7311yescharacteristic
490C23C22×C22⋊D7311yescharacteristic
491C23C22×C22⋊D7312yes—
492C23C22×D29211yescharacteristic
493C23C2×#1192402812yes—
494D4—2924no—
495D4—2924no—
496D4—2922no—
497D4—2924no—
498D4—2922no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—48no—
513C2×C4—416no—
514C2×C4—2924no—
515C2×C4—2922no—
516C2×C4—2922no—
517C2×C4—2924no—
518C2×C4—2924no—
519C2×C4—2928no—
520C2×C4—2924no—
521C2×C4—5848no—
522C2×C4—1464no—
523C2×C4—1462no—
524C2×C4—1464no—
525C2×C4C2×#1192400514yes—
526C2×C4C2×#1192400514yes—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—24no—
534C22—28no—
535C22—28no—
536C22—24no—
537C22—44no—
538C22—42no—
539C22—44no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—2924no—
544C22—2922no—
545C22—5844no—
546C22—5844no—
547C22—5844no—
548C22#1798846612yes—
549C22#1798846612yes—
550C22#1798846612yes—
551C22#1798846612yes—
552C22#1798856014yes—
553C22C22×#1192400511yescharacteristic
554C22C2×C4×C22⋊D7312yes—
555C22C2×#1455656011yescharacteristic
556C22C2×#1455656512yes—
557C22C2×#1455657411yescharacteristic
558C4—24no—
559C4—24no—
560C4—48no—
561C4—2924no—
562C4—2922no—
563C4—5844no—
564C2—21no—
565C2—22no—
566C2—22no—
567C2—22no—
568C2—22no—
569C2—22no—
570C2—21no—
571C2—5842no—
572C2#2261038914yesdirect product factor
573C2C2×#1798846611yescharacteristic
574C2C2×#1798856012yes—
subgroupG/H
centerC23C2×#11924005
commutatorC23×C73C23×C4
FrattiniC24C23×D73
FittingC22×#117×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC23×C22⋊D73C22
cosocleC24×C73C24
hypercenterC22×#117D73
hypocenterC73C2×#1899
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. C2×#17988450C2semidirect
4. #22610389C2direct (N × H)
5. C2×#17988353C2semidirect
6. C22×#117×C73C2semidirect
13. C2×#14556574C4semidirect
15. C22×#11924007C4semidirect
18. C22×#11924005C4semidirect
22. #17988450C22semidirect
24. #17988353C22semidirect
28. C2×#117×C73C22semidirect
53. C2×#11924007C2×C4semidirect
67. C2×#11924005C2×C4semidirect
284. C2×C73#1899semidirect
289. C22×#117D73semidirect
294. C73C2×#1899semidirect
306. C2×#117C2×D73semidirect
572. C2#22610389direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145565742
C22×#145565052
C2×#179884502
#226103892
C2×#179883532
C22×#117×C732
C2×#189973