Finite Group Databasebeta

← C2×#22626856 #34031936

Subgroups — C2×#22626856 #34031936

Order 18688.

2210 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626856C2116yesmaximal, direct product factor
2C2×#17989639C212yesmaximal, complemented
3C2×#17989619C212yesmaximal, complemented
4C2×#17989618C212yesmaximal, complemented
5C2×#17989614C212yesmaximal, complemented
6C2×#17989609C211yesmaximal, characteristic, complemented
7C2×#17989608C211yesmaximal, characteristic, complemented
8C2×#17989603C211yesmaximal, characteristic, complemented
9C2×#17989590C211yesmaximal, characteristic, complemented
10C2×#17989588C211yesmaximal, characteristic, complemented
11C22×C4⋊#10596833C211yesmaximal, characteristic, complemented
12C2×C4×#11924027C211yesmaximal, characteristic
13C22×#11924037C2212yes—
14C22×#11924036C2212yes—
15C23×#10596833C2212yes—
16#17989639C22116yescomplemented
17#17989619C22116yescomplemented
18#17989618C22116yescomplemented
19#17989614C22116yescomplemented
20#17989609C2218yescomplemented
21#17989608C2218yescomplemented
22#17989603C2218yescomplemented
23#17989590C2218yescomplemented
24C2×#14556658C2212yes—
25C2×#14556658C2212yes—
26#17989588C2218yescomplemented
27C2×C4⋊#10596833C2211yescharacteristic
28C2×C4⋊#10596833C2211yescharacteristic
29C2×C4⋊#10596833C2212yes—
30C2×C4⋊#10596833C2211yescharacteristic
31C2×C4⋊#10596833C2211yescharacteristic
32C2×C4⋊#10596833C2218yescomplemented
33C2×#14556648C2212yes—
34C2×C22⋊#10596833C2212yes—
35C2×C22⋊#10596833C2212yes—
36C2×#14556646C2212yes—
37C2×#14556646C2212yes—
38C2×C4×#10596833C2212yes—
39C22×#11924027C411yescharacteristic
40C4×#11924027C418yes—
41C4×#11924027C2218yes—
42C2×C4×Dic146C411yescharacteristic
43C2×#14556556C2211yescharacteristic
44C2×D4×Dic73C2211yescharacteristic
45C2×C4×C22⋊D73C411yescharacteristic
46C2×C4×C22⋊D73C411yescharacteristic
47C2×#14556551C2211yescharacteristic
48C2×#14556551C2211yescharacteristic
49C2×#14556549C411yescharacteristic
50C2×#14556547C411yescharacteristic
51C2×#14556547C411yescharacteristic
52C22×C4×Dic73C2211yescharacteristic
53C22×C4×Dic73C2211yescharacteristic
54C2×C42×D73C2211yescharacteristic
55C2×C4×D4×C73C411yescharacteristic
56C2×#11924037—44no—
57C2×#11924037C2318yes—
58C2×#11924036—44no—
59C2×#11924036C2318yes—
60C22×#10596833—44no—
61C22×#10596833C2312yes—
62C22×#10596833C2312yes—
63C22×#10596833C2312yes—
64C22×#10596833C2312yes—
65C22×#10596833C2318yes—
66#14556658C2318yes—
67#14556658C2318yes—
68C4⋊#10596833C2318yes—
69C4⋊#10596833C2314yes—
70C4⋊#10596833C2314yes—
71C4⋊#10596833C2314yes—
72C4⋊#10596833C2314yes—
73#14556648C2318yes—
74C22⋊#10596833C2318yes—
75C22⋊#10596833C2318yes—
76#14556646C2318yes—
77#14556646C2318yes—
78C4×#10596833C2318yes—
79C2×#11924027—28no—
80C2×#11924027C2×C418yes—
81C22×Dic146C2×C411yescharacteristic
82C22×C22⋊D73C2×C411yescharacteristic
83C22×C22⋊D73C2×C411yescharacteristic
84C23×Dic73C2×C411yescharacteristic
85C23×Dic73C2×C411yescharacteristic
86C23×Dic73C2311yescharacteristic
87C23×Dic73C2311yescharacteristic
88C22×C4×D73C2×C411yescharacteristic
89C22×C4×D73C2×C411yescharacteristic
90C22×C4×D73C2311yescharacteristic
91C4×Dic146C2×C418yes—
92#14556556C2×C414yes—
93#14556556C2314yes—
94D4×Dic73C2×C414yes—
95D4×Dic73C2314yes—
96C4×C22⋊D73C2×C418yes—
97C4×C22⋊D73C2×C418yes—
98#14556551C2×C414yes—
99#14556551C2×C414yes—
100#14556551C2314yes—
101#14556551C2314yes—
102#14556549C2×C418yes—
103#14556547C2×C418yes—
104#14556547C2×C418yes—
105C2×C4⋊Dic73C2×C411yescharacteristic
106C2×#11924007C2×C411yescharacteristic
107C2×#11924007C2×C411yescharacteristic
108C2×#11924007C2×C411yescharacteristic
109C2×#11924007C2×C411yescharacteristic
110C2×C22⋊Dic73C2×C411yescharacteristic
111C2×C22⋊Dic73C2×C411yescharacteristic
112C2×#11924005C2×C411yescharacteristic
113C2×#11924005C2×C411yescharacteristic
114C2×C4×Dic73C2×C411yescharacteristic
115C2×C4×Dic73C2×C411yescharacteristic
116C2×C4×Dic73C2×C411yescharacteristic
117C2×C4×Dic73C2×C411yescharacteristic
118C2×C4×Dic73C2×C411yescharacteristic
119C2×C4×Dic73C2×C414yes—
120C2×C4×Dic73C2×C414yes—
121C2×C4×Dic73C2311yescharacteristic
122C2×C4×Dic73C2311yescharacteristic
123C2×C4×Dic73C2311yescharacteristic
124C2×C4×Dic73C2311yescharacteristic
125C2×C4×Dic73C2314yes—
126C2×C4×Dic73C2314yes—
127C42×D73C2×C414yes—
128C42×D73C2314yes—
129C22×D4×C73C2×C411yescharacteristic
130C23×C4×C73C2×C411yescharacteristic
131C23×C4×C73C2×C411yescharacteristic
132C4×D4×C73C2×C418yes—
133C2×C4⋊C4×C73C2×C411yescharacteristic
134C2×C22⋊C4×C73C2×C411yescharacteristic
135C2×C22⋊C4×C73C2×C411yescharacteristic
136C2×C42×C73C2×C411yescharacteristic
137#11924037—48no—
138#11924036—48no—
139C2×#10596833—24no—
140C2×#10596833—24no—
141C2×#10596833—24no—
142C2×#10596833—24no—
143C2×#10596833—48no—
144C2×#10596833C2414yes—
145C2×#10596833C2414yes—
146C2×#10596833C2414yes—
147C2×#10596833C2414yes—
148C23×D73C22×C411yescharacteristic
149#11924027—232no—
150C2×Dic146—24no—
151C2×Dic146C22×C414yes—
152C2×C22⋊D73—24no—
153C2×C22⋊D73M1618yescomplemented
154C2×C22⋊D73C22×C414yes—
155C2×C22⋊D73C22×C414yes—
156C22×Dic73—24no—
157C22×Dic73—24no—
158C22×Dic73—22no—
159C22×Dic73—22no—
160C22×Dic73—22no—
161C22×Dic73—22no—
162C22×Dic73C22×C411yescharacteristic
163C22×Dic73C22×C411yescharacteristic
164C22×Dic73C22×C411yescharacteristic
165C22×Dic73C22×C412yes—
166C22×Dic73C22×C412yes—
167C22×Dic73C22×C411yescharacteristic
168C22×Dic73C22×C411yescharacteristic
169C22×Dic73C22×C414yes—
170C22×Dic73C22×C414yes—
171C22×Dic73C22×C411yescharacteristic
172C22×Dic73C2412yes—
173C22×Dic73C2412yes—
174C22×Dic73C2411yescharacteristic
175C2×C4×D73—24no—
176C2×C4×D73—24no—
177C2×C4×D73—22no—
178C2×C4×D73—22no—
179C2×C4×D73C22×C412yes—
180C2×C4×D73C22×C414yes—
181C2×C4×D73C22×C414yes—
182C2×C4×D73C2412yes—
183C4⋊Dic73C22×C414yes—
184#11924007C22×C414yes—
185#11924007C22×C414yes—
186#11924007C22×C414yes—
187#11924007C22×C414yes—
188C22⋊Dic73C22×C414yes—
189C22⋊Dic73C22×C414yes—
190#11924005C22×C414yes—
191#11924005C22×C414yes—
192C4×Dic73C22×C412yes—
193C4×Dic73C22×C412yes—
194C4×Dic73C22×C414yes—
195C4×Dic73C22×C414yes—
196C4×Dic73C22×C414yes—
197C4×Dic73C22×C412yes—
198C4×Dic73C22×C412yes—
199C4×Dic73C22×C414yes—
200C4×Dic73C22×C414yes—
201C4×Dic73C2412yes—
202C4×Dic73C2412yes—
203C4×Dic73C2412yes—
204C4×Dic73C2412yes—
205C24×C73C22×C411yescharacteristic
206C24×C73C22×C411yescharacteristic
207C2×D4×C73—24no—
208C2×D4×C73C22×C414yes—
209C22×C4×C73—24no—
210C22×C4×C73—24no—
211C22×C4×C73C22×C411yescharacteristic
212C22×C4×C73C22×C411yescharacteristic
213C22×C4×C73C22×C411yescharacteristic
214C22×C4×C73C22×C411yescharacteristic
215C22×C4×C73C22×C414yes—
216C22×C4×C73C22×C414yes—
217C22×C4×C73C22×C411yescharacteristic
218C4⋊C4×C73C22×C414yes—
219C22⋊C4×C73C22×C414yes—
220C22⋊C4×C73C22×C414yes—
221C42×C73C22×C414yes—
222C23×C73—22no—
223C23×C73—22no—
224C23×C73—22no—
225C23×C73—22no—
226C23×C73—22no—
227C23×C73C2×M1614yes—
228C23×C73C23×C412yes—
229C23×C73C23×C412yes—
230C23×C73C23×C411yescharacteristic
231C22×D73—22no—
232C22×D73—22no—
233C22×D73C2×M1614yes—
234C22×D73C23×C412yes—
235D4×C73—28no—
236C2×C4×C73—22no—
237C2×C4×C73—22no—
238C2×C4×C73—22no—
239C2×C4×C73—22no—
240C2×C4×C73—28no—
241C2×C4×C73—28no—
242C2×C4×C73#13114yes—
243C2×C4×C73C23×C412yes—
244C2×C4×C73C23×C412yes—
245C2×C4×C73C23×C412yes—
246C2×C4×C73C23×C412yes—
247C2×C4×C73C23×C412yes—
248C22⋊D73—28no—
249C22⋊D73C2×M16116yescomplemented
250C2×Dic73—22no—
251C2×Dic73—22no—
252C2×Dic73—22no—
253C2×Dic73—22no—
254C2×Dic73—28no—
255C2×Dic73—22no—
256C2×Dic73—24no—
257C2×Dic73—24no—
258C2×Dic73—28no—
259C2×Dic73—22no—
260C2×Dic73—22no—
261C2×Dic73—22no—
262C2×Dic73—22no—
263C2×Dic73C2×M1614yes—
264C2×Dic73C23×C412yes—
265C2×Dic73C23×C412yes—
266C2×Dic73C23×C412yes—
267C2×Dic73C23×C412yes—
268C2×Dic73C23×C412yes—
269C2×Dic73C23×C411yescharacteristic
270C2×Dic73C23×C412yes—
271C2×Dic73H(5,𝔽2)12yes—
272C2×Dic732−1+412yes—
273C2×Dic73C2511yescharacteristic
274C4×D73—28no—
275C4×D73—28no—
276C4×D73—24no—
277C4×D73—22no—
278C4×D73—22no—
279Dic146—28no—
280#10596833—24no—
281#10596833—24no—
282#10596833—24no—
283#10596833—24no—
284C22×C73—22no—
285C22×C73—28no—
286C22×C73—22no—
287C22×C73—22no—
288C22×C73—22no—
289C22×C73—22no—
290C22×C73—44no—
291C22×C73#52012yes—
292C22×C73C22×M1614yes—
293C22×C73C22×M1612yes—
294C22×C73C2×#13112yes—
295C22×C73C24×C411yescharacteristic
296C2×D73—22no—
297C2×D73—28no—
298C2×D73—22no—
299C2×D73C22×M1614yes—
300C4×C73—22no—
301C4×C73—22no—
302C4×C73—22no—
303C4×C73—22no—
304C4×C73C2×#13114yes—
305Dic73—22no—
306Dic73—22no—
307Dic73—22no—
308Dic73—22no—
309Dic73—22no—
310Dic73#52012yes—
311Dic73C22×M1614yes—
312Dic73C2×H(5,𝔽2)11yescharacteristic
313Dic73C2×2−1+411yescharacteristic
314C2×#2990—731nomaximal, Sylow
315C2×C73—24no—
316C2×C73—42no—
317C2×C73#299014yescomplemented
318C2×C73C2×#52011yescharacteristic
319C2×C73C23×M1611yescharacteristic
320C2×C73C22×#13111yescharacteristic
321D73—24no—
322C2×C4×#41—731no—
323#2990—7316no—
324C2×#446—732no—
325C2×#435—732no—
326C2×#435—732no—
327C2×C8×D4—732no—
328C2×#432—731no—
329C2×#432—731no—
330C2×#431—731no—
331C2×#423—731no—
332C22×C4⋊C8—731no—
333C22×C4⋊C8—731no—
334C73C2×#299011yescharacteristic, complemented, Sylow
335C22×#41—731no—
336C22×M16—732no—
337C23×C8—732no—
338C23×C8—732no—
339C4×#41—738no—
340C4×#41—738no—
341C2×C4×Q8—731no—
342C2×C4×Q8—731no—
343C2×C4×D4—731no—
344C2×C4×D4—731no—
345C2×C4×D4—731no—
346C2×C4×D4—731no—
347C2×C4×D4—731no—
348C2×C4×D4#6440811yescharacteristic
349C2×#117—731no—
350C2×#117—731no—
351C2×#117—731no—
352C22×C42—731no—
353C22×C42—731no—
354C22×C42—731no—
355#446—7316no—
356#435—7316no—
357#435—7316no—
358C8×D4—7316no—
359#432—738no—
360#432—738no—
361#431—738no—
362#423—738no—
363C2×C4⋊C8—731no—
364C2×C4⋊C8—731no—
365C2×C4⋊C8—731no—
366C2×C4⋊C8—732no—
367C2×C4⋊C8—732no—
368C2×C4⋊C8—738no—
369C2×C4⋊C8—738no—
370C2×C4⋊C8—732no—
371C2×C4⋊C8—731no—
372C2×C22⋊C8—732no—
373C2×C22⋊C8—732no—
374C2×C22⋊C8—732no—
375C2×#97—732no—
376C2×#97—732no—
377C2×C4×C8—732no—
378C2×#41—738no—
379C2×#41—1468no—
380C22×Q8—731no—
381C22×D4—731no—
382C22×D4—731no—
383C22×D4C2×#6440811yescharacteristic
384C23×C4—731no—
385C23×C4—731no—
386C23×C4—731no—
387C23×C4—731no—
388C23×C4—731no—
389C23×C4—731no—
390C23×C4—731no—
391C23×C4C2×#6440811yescharacteristic
392C23×C4C2×#6440811yescharacteristic
393C2×M16—2924no—
394C2×M16—738no—
395C22×C8—2924no—
396C22×C8—2924no—
397C22×C8—738no—
398C22×C8—738no—
399C22×C8—732no—
400C22×C8—732no—
401C22×C8—732no—
402C22×C8—732no—
403C4×Q8—734no—
404C4×Q8—734no—
405C4×Q8—738no—
406C4×D4—734no—
407C4×D4—734no—
408C4×D4—738no—
409C4×D4—738no—
410C4×D4—734no—
411C4×D4—734no—
412C4×D4—734no—
413C4×D4—734no—
414C4×D4C2×#6440818yes—
415#117—738no—
416#117—738no—
417#117—738no—
418C2×C4⋊C4—731no—
419C2×C4⋊C4—731no—
420C2×C4⋊C4—731no—
421C2×C4⋊C4—731no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4C2×#6440811yescharacteristic
424C2×C22⋊C4—731no—
425C2×C22⋊C4—731no—
426C2×C22⋊C4—731no—
427C2×C22⋊C4—731no—
428C2×C22⋊C4C2×#6440811yescharacteristic
429C2×C22⋊C4C2×#6440811yescharacteristic
430C2×C42—731no—
431C2×C42—731no—
432C2×C42—731no—
433C2×C42—731no—
434C2×C42—731no—
435C2×C42—731no—
436C2×C42—731no—
437C2×C42—731no—
438C2×C42—734no—
439C2×C42—734no—
440C2×C42—734no—
441C2×C42—734no—
442C2×C42—731no—
443C2×C42—734no—
444C2×C42—734no—
445C2×C42C2×#6440811yescharacteristic
446C4⋊C8—734no—
447C4⋊C8—734no—
448C4⋊C8—734no—
449C4⋊C8—738no—
450C4⋊C8—738no—
451C4⋊C8—738no—
452C4⋊C8—734no—
453C22⋊C8—738no—
454C22⋊C8—738no—
455C22⋊C8—738no—
456#97—738no—
457#97—738no—
458C4×C8—738no—
459C24—731no—
460C24C22×#6440811yescharacteristic
461C24C22×#6440811yescharacteristic
462#41—14632no—
463C2×Q8—734no—
464C2×Q8—1464no—
465C2×D4—24no—
466C2×D4—738no—
467C2×D4—734no—
468C2×D4—734no—
469C2×D4—1464no—
470C2×D4C22×#6440814yes—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—731no—
474C22×C4—731no—
475C22×C4—734no—
476C22×C4—734no—
477C22×C4—732no—
478C22×C4—732no—
479C22×C4—732no—
480C22×C4—732no—
481C22×C4—731no—
482C22×C4—731no—
483C22×C4—731no—
484C22×C4—731no—
485C22×C4—731no—
486C22×C4—734no—
487C22×C4—732no—
488C22×C4—734no—
489C22×C4—732no—
490C22×C4—1462no—
491C22×C4—1462no—
492C22×C4—1464no—
493C22×C4—1462no—
494C22×C4—1462no—
495C22×C4—1462no—
496C22×C4—1462no—
497C22×C4—1464no—
498C22×C4—1464no—
499C22×C4—1464no—
500C22×C4C22×#6440814yes—
501C22×C4C22×#6440814yes—
502C22×C4C22×#6440811yescharacteristic
503C22×C4C22×#6440811yescharacteristic
504C22×C4C22×#6440811yescharacteristic
505C22×C4C22×#6440811yescharacteristic
506C22×C4C22×#6440811yescharacteristic
507M16—2928no—
508C2×C8—2928no—
509C2×C8—2928no—
510C2×C8—734no—
511C2×C8—734no—
512C2×C8—734no—
513C2×C8—734no—
514C2×C8—1464no—
515C2×C8—1464no—
516C2×C8—1464no—
517C2×C8—1464no—
518C4⋊C4—734no—
519C4⋊C4—734no—
520C4⋊C4—734no—
521C4⋊C4—734no—
522C4⋊C4—734no—
523C4⋊C4C22×#6440814yes—
524C22⋊C4—734no—
525C22⋊C4—734no—
526C22⋊C4—734no—
527C22⋊C4—734no—
528C22⋊C4C22×#6440814yes—
529C22⋊C4C22×#6440814yes—
530C42—732no—
531C42—734no—
532C42—734no—
533C42—734no—
534C42—732no—
535C42—732no—
536C42—732no—
537C42—734no—
538C42—734no—
539C42—732no—
540C42—732no—
541C42—732no—
542C42—732no—
543C42C22×#6440814yes—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—734no—
550C23—732no—
551C23—1462no—
552C23—1462no—
553C23D73⋊M1614yes—
554C23C23×#6440812yes—
555C23C23×#6440812yes—
556C23C23×#6440811yescharacteristic
557Q8—1468no—
558D4—28no—
559D4—7316no—
560D4—1468no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—732no—
568C2×C4—732no—
569C2×C4—734no—
570C2×C4—731no—
571C2×C4—731no—
572C2×C4—732no—
573C2×C4—732no—
574C2×C4—732no—
575C2×C4—732no—
576C2×C4—732no—
577C2×C4—732no—
578C2×C4—1462no—
579C2×C4—1462no—
580C2×C4—1464no—
581C2×C4—1468no—
582C2×C4—1462no—
583C2×C4—1462no—
584C2×C4—1468no—
585C2×C4—1462no—
586C2×C4—1462no—
587C2×C4—1464no—
588C2×C4—1464no—
589C2×C4—1462no—
590C2×C4—1462no—
591C2×C4—1462no—
592C2×C4—1462no—
593C2×C4—1468no—
594C2×C4—1462no—
595C2×C4—1468no—
596C2×C4#1455667814yes—
597C2×C4C23×#6440812yes—
598C2×C4C23×#6440812yes—
599C2×C4C23×#6440812yes—
600C2×C4C23×#6440812yes—
601C2×C4C23×#6440812yes—
602C8—1464no—
603C8—1464no—
604C8—1464no—
605C8—1464no—
606C22—28no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—44no—
613C22—734no—
614C22—1468no—
615C22—1462no—
616C22—1462no—
617C22#1798965612yes—
618C22C2×D73⋊M1614yes—
619C22C2×D73⋊M1612yes—
620C22C2×#1455667812yes—
621C22C24×#6440811yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—731no—
627C4—731no—
628C4—732no—
629C4—734no—
630C4—1462no—
631C4—1462no—
632C4—1462no—
633C4—1462no—
634C4—1462no—
635C4C2×#1455667814yes—
636C2—24no—
637C2—42no—
638C2—1464no—
639C2#2262685614yesdirect product factor
640C2C2×#1798965611yescharacteristic
641C2C22×D73⋊M1611yescharacteristic
642C2C22×#1455667811yescharacteristic
subgroupG/H
centerC23C23×#64408
commutatorC22×C73C24×C4
FrattiniC22C24×#64408
FittingC2×C4×D4×C73C4
socleC23×C73C23×C4
prime radicalC22×#11924027C4
cosocleC2×Dic73C25
hypercenterC2×C4×D4#64408
hypocenterC73C2×#2990
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #22626856C2direct (N × H)
2. C2×#17989639C2semidirect
3. C2×#17989619C2semidirect
4. C2×#17989618C2semidirect
5. C2×#17989614C2semidirect
6. C2×#17989609C2semidirect
7. C2×#17989608C2semidirect
8. C2×#17989603C2semidirect
9. C2×#17989590C2semidirect
10. C2×#17989588C2semidirect
11. C22×C4⋊#10596833C2semidirect
16. #17989639C22semidirect
17. #17989619C22semidirect
18. #17989618C22semidirect
19. #17989614C22semidirect
20. #17989609C22semidirect
21. #17989608C22semidirect
22. #17989603C22semidirect
23. #17989590C22semidirect
26. #17989588C22semidirect
32. C2×C4⋊#10596833C22semidirect
153. C2×C22⋊D73M16semidirect
249. C22⋊D73C2×M16semidirect
317. C2×C73#2990semidirect
334. C73C2×#2990semidirect
639. C2#22626856direct (N × H)

Maximal subgroups

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

subgroupindex
#226268562
C2×#179896392
C2×#179896192
C2×#179896182
C2×#179896142
C2×#179896092
C2×#179896082
C2×#179896032
C2×#179895902
C2×#179895882
C22×C4⋊#105968332
C2×C4×#119240272
C2×#299073