Finite Group Databasebeta

← C2×#22624327 #33993776

Subgroups — C2×#22624327 #33993776

Order 18688.

7810 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#22624327C2132yesmaximal, direct product factor
2C2×#17989335C211yesmaximal, characteristic, complemented
3C2×#17989331C212yesmaximal, complemented
4C2×#17989291C214yesmaximal, complemented
5C2×#17989255C214yesmaximal, complemented
6C2×#17989235C214yesmaximal, complemented
7C2×#17989189C214yesmaximal, complemented
8C2×#17989167C214yesmaximal, complemented
9C2×#17989164C214yesmaximal, complemented
10C2×#17989143C211yesmaximal, characteristic, complemented
11C2×#17989136C212yesmaximal, complemented
12C2×#562×C73C211yesmaximal, characteristic, complemented
13C22×#11924029C2212yes—
14C22×#11924028C2214yes—
15C22×#11924027C2212yes—
16C22×#11924027C2214yes—
17#17989335C22116yescomplemented
18#17989331C22132yescomplemented
19#17989291C22164yescomplemented
20#17989255C22164yescomplemented
21#17989235C22164yescomplemented
22#17989189C22164yescomplemented
23#17989167C22164yescomplemented
24#17989164C22164yescomplemented
25#17989143C22116yescomplemented
26#17989136C22132yescomplemented
27C2×#14556590C2212yes—
28C2×#14556589C2212yes—
29C2×#14556587C2214yes—
30C2×#14556587C2212yes—
31C2×#14556583C2211yescharacteristic
32C2×#14556581C2214yes—
33C2×#14556579C2212yes—
34C2×#14556576C2214yes—
35C2×#14556575C2214yes—
36C2×#14556573C2214yes—
37C2×#14556569C2214yes—
38C2×#14556569C2214yes—
39C2×#14556569C2214yes—
40C2×#14556568C2214yes—
41C2×#14556568C2214yes—
42C2×#14556567C2214yes—
43C2×#14556566C2214yes—
44C2×#14556566C2214yes—
45C2×#14556565C2214yes—
46C2×#14556564C2214yes—
47C2×#14556564C2214yes—
48C2×#14556564C2214yes—
49C2×#14556564C2214yes—
50C2×#14556564C2214yes—
51C2×#14556563C2214yes—
52C2×#14556563C2214yes—
53C2×#14556563C2214yes—
54C2×#14556556C2214yes—
55C2×C4×C22⋊D73C2214yes—
56C2×#14556551C2214yes—
57C2×#14556551C2214yes—
58C2×#14556551C2214yes—
59C2×#14556549C2212yes—
60C22×#11924007C2214yes—
61C2×C4⋊C4×D73C2214yes—
62C2×C42×D73C2211yescharacteristic
63#562×C73C22116yescomplemented
64C2×#125×C73C2211yescharacteristic
65C2×#124×C73C2212yes—
66C2×#123×C73C2214yes—
67C2×C22⋊Q8×C73C2214yes—
68C2×#121×C73C2214yes—
69C2×#11924029—48no—
70C2×#11924029C23116yes—
71C2×#11924028—416no—
72C2×#11924028C23132yes—
73C2×#11924027—48no—
74C2×#11924027—416no—
75C2×#11924027C23116yes—
76C2×#11924027C23132yes—
77C22×Dic146C2314yes—
78C22×Dic146C2312yes—
79C22×D292C2314yes—
80C22×D292C2312yes—
81C22×D292C2314yes—
82C22×C22⋊D73C2314yes—
83C22×C22⋊D73C2314yes—
84C22×C22⋊D73C2314yes—
85C22×C22⋊D73C2314yes—
86C22×C22⋊D73C2314yes—
87C23×Dic73C2314yes—
88C23×Dic73C2314yes—
89C22×C4×D73C2312yes—
90C22×C4×D73C2314yes—
91C22×C4×D73C2311yescharacteristic
92C22×C4×D73C2314yes—
93C22×C4×D73C2314yes—
94#14556590C23116yes—
95#14556589C23116yes—
96#14556587C23116yes—
97#14556587C23132yes—
98#14556583C2318yes—
99#14556581C23132yes—
100#14556579C23116yes—
101#14556576C23132yes—
102#14556575C23132yes—
103#14556573C23132yes—
104#14556569C23132yes—
105#14556569C23132yes—
106#14556569C23132yes—
107#14556568C23132yes—
108#14556568C23132yes—
109#14556567C23132yes—
110#14556566C23132yes—
111#14556566C23132yes—
112#14556565C23132yes—
113#14556564C23132yes—
114#14556564C23132yes—
115#14556564C23132yes—
116#14556564C23132yes—
117#14556564C23132yes—
118#14556563C23132yes—
119#14556563C23132yes—
120#14556563C23132yes—
121#14556556C23132yes—
122C4×C22⋊D73C23132yes—
123#14556551C23132yes—
124#14556551C23132yes—
125#14556551C23132yes—
126#14556549C23116yes—
127C2×C4⋊Dic73C2312yes—
128C2×C4⋊Dic73C2314yes—
129C2×#11924007C2314yes—
130C2×#11924007C2314yes—
131C2×#11924007C2314yes—
132C2×#11924007C23132yes—
133C2×#11924007C2314yes—
134C2×#11924007C2314yes—
135C2×#11924007C2314yes—
136C4⋊C4×D73C23132yes—
137C2×C22⋊Dic73C2314yes—
138C2×#11924005C2314yes—
139C2×#11924005C2314yes—
140C2×#11924005C2314yes—
141C2×#11924005C2314yes—
142C2×#11924005C2314yes—
143C2×C4×Dic73C2312yes—
144C2×C4×Dic73C2314yes—
145C2×C4×Dic73C2311yescharacteristic
146C42×D73C2318yes—
147C22×Q8×C73C2312yes—
148C22×D4×C73C2314yes—
149C22×D4×C73C2312yes—
150C23×C4×C73C2314yes—
151#125×C73C2318yes—
152#124×C73C23116yes—
153#123×C73C23132yes—
154C22⋊Q8×C73C23132yes—
155#121×C73C23132yes—
156C2×C4⋊C4×C73C2312yes—
157C2×C4⋊C4×C73C2314yes—
158C2×C4⋊C4×C73C2314yes—
159C2×C22⋊C4×C73C2314yes—
160C2×C22⋊C4×C73C2314yes—
161C2×C22⋊C4×C73C2314yes—
162C2×C42×C73C2311yescharacteristic
163C23×D73C2411yescharacteristic
164C23×D73C2414yes—
165#11924029—432no—
166#11924028—464no—
167#11924027—464no—
168#11924027—432no—
169C2×Dic146—216no—
170C2×Dic146—44no—
171C2×Dic146C24116yes—
172C2×Dic146C2418yes—
173C2×D292—48no—
174C2×D292—48no—
175C2×D292—44no—
176C2×D292C24116yes—
177C2×D292C2418yes—
178C2×D292C24116yes—
179C2×C22⋊D73—216no—
180C2×C22⋊D73—216no—
181C2×C22⋊D73—48no—
182C2×C22⋊D73—48no—
183C2×C22⋊D73—48no—
184C2×C22⋊D73C24116yes—
185C2×C22⋊D73C24116yes—
186C2×C22⋊D73C24116yes—
187C2×C22⋊D73C24116yes—
188C2×C22⋊D73C24116yes—
189C22×Dic73—216no—
190C22×Dic73—48no—
191C22×Dic73C2414yes—
192C22×Dic73C2414yes—
193C22×Dic73C24116yes—
194C22×Dic73C24116yes—
195C22×Dic73C2412yes—
196C22×Dic73C2411yescharacteristic
197C2×C4×D73—28no—
198C2×C4×D73—216no—
199C2×C4×D73—24no—
200C2×C4×D73—48no—
201C2×C4×D73—48no—
202C2×C4×D73C2414yes—
203C2×C4×D73C2418yes—
204C2×C4×D73C24116yes—
205C2×C4×D73C24116yes—
206C2×C4×D73C24116yes—
207C4⋊Dic73C24116yes—
208C4⋊Dic73C2418yes—
209#11924007C24116yes—
210#11924007C24116yes—
211#11924007C24116yes—
212#11924007C24116yes—
213#11924007C24116yes—
214#11924007C24116yes—
215C22⋊Dic73C24116yes—
216#11924005C24116yes—
217#11924005C24116yes—
218#11924005C24116yes—
219#11924005C24116yes—
220#11924005C24116yes—
221C4×Dic73C2414yes—
222C4×Dic73C2418yes—
223C4×Dic73C24116yes—
224C24×C73C2414yes—
225C2×Q8×C73—44no—
226C2×Q8×C73C2418yes—
227C2×D4×C73—48no—
228C2×D4×C73—44no—
229C2×D4×C73C24116yes—
230C2×D4×C73C2418yes—
231C22×C4×C73—48no—
232C22×C4×C73C2411yescharacteristic
233C22×C4×C73C2412yes—
234C22×C4×C73C2414yes—
235C22×C4×C73C2414yes—
236C22×C4×C73C24116yes—
237C4⋊C4×C73C24116yes—
238C4⋊C4×C73C24116yes—
239C4⋊C4×C73C2418yes—
240C22⋊C4×C73C24116yes—
241C22⋊C4×C73C24116yes—
242C22⋊C4×C73C24116yes—
243C42×C73C2414yes—
244C23×C73—28no—
245C23×C73—28no—
246C23×C73—28no—
247C23×C73C2518yes—
248C23×C73C2511yescharacteristic
249C22×D73—24no—
250C22×D73—28no—
251C22×D73—28no—
252C22×D73—22no—
253C22×D73—28no—
254C22×D73C2518yes—
255C22×D73C2512yes—
256Q8×C73—48no—
257D4×C73—416no—
258D4×C73—48no—
259C2×C4×C73—28no—
260C2×C4×C73—28no—
261C2×C4×C73—24no—
262C2×C4×C73—22no—
263C2×C4×C73—416no—
264C2×C4×C73C2512yes—
265C2×C4×C73C2514yes—
266C2×C4×C73C2518yes—
267C2×C4×C73C2518yes—
268C22⋊D73—232no—
269C22⋊D73—232no—
270C22⋊D73—416no—
271C22⋊D73—416no—
272C22⋊D73—416no—
273C2×Dic73—28no—
274C2×Dic73—28no—
275C2×Dic73—232no—
276C2×Dic73—416no—
277C2×Dic73H(5,𝔽2)18yes—
278C2×Dic732−1+414yes—
279C2×Dic73C2518yes—
280C2×Dic73C2518yes—
281C2×Dic73C2514yes—
282C2×Dic73C2512yes—
283D292—416no—
284D292—48no—
285D292—416no—
286C4×D73—216no—
287C4×D73—232no—
288C4×D73—28no—
289C4×D73—416no—
290C4×D73—416no—
291Dic146—232no—
292Dic146—48no—
293C22×C73—28no—
294C22×C73—28no—
295C22×C73—28no—
296C22×C73—416no—
297C22×C73C2×H(5,𝔽2)14yes—
298C22×C73C2×2−1+412yes—
299C22×C73C2611yescharacteristic
300C2×D73—28no—
301C2×D73—24no—
302C2×D73—28no—
303C2×D73—22no—
304C2×D73—28no—
305C2×D73—44no—
306C2×D73—416no—
307C4×C73—28no—
308C4×C73—24no—
309C4×C73—28no—
310C4×C73—22no—
311Dic73—28no—
312Dic73—28no—
313Dic73C2×H(5,𝔽2)18yes—
314Dic73C2×2−1+414yes—
315C2×#3548—731nomaximal, Sylow
316C2×C73—48no—
317C2×C73#354814yescomplemented
318C2×C73C22×H(5,𝔽2)12yes—
319C2×C73C22×2−1+411yescharacteristic
320D73—48no—
321D73—42no—
322#3548—7332no—
323C2×#562—731no—
324C2×#562—732no—
325C2×#562D7311yescharacteristic, complemented
326C2×#554—734no—
327C2×#550—734no—
328C2×#547—734no—
329C2×#540—734no—
330C2×#537—734no—
331C2×#537—734no—
332C2×#533—731no—
333C2×#532—732no—
334C73C2×#354811yescharacteristic, complemented, Sylow
335C22×#41—732no—
336C22×#41—732no—
337C22×#41—734no—
338C22×#41—734no—
339#562—7316no—
340#562—7332no—
341#562C2×D73116yescomplemented
342#554—7364no—
343#550—7364no—
344#547—7364no—
345#540—7364no—
346#537—7364no—
347#537—7364no—
348#533—7316no—
349#532—7332no—
350C2×C4⋊Q8—732no—
351C2×C4⋊Q8—732no—
352C2×#127—732no—
353C2×#127—734no—
354C2×#125—734no—
355C2×#125—731no—
356C2×#125C2×D7311yescharacteristic
357C2×#124—732no—
358C2×#124C2×D7312yes—
359C2×#123—734no—
360C2×#123—734no—
361C2×#123—734no—
362C2×#123C2×D7314yes—
363C2×C22⋊Q8—734no—
364C2×C22⋊Q8—734no—
365C2×C22⋊Q8—734no—
366C2×C22⋊Q8—734no—
367C2×C22⋊Q8—734no—
368C2×C22⋊Q8—734no—
369C2×C22⋊Q8C2×D7314yes—
370C2×#121—734no—
371C2×#121—734no—
372C2×#121—734no—
373C2×#121—734no—
374C2×#121—734no—
375C2×#121—734no—
376C2×#121—734no—
377C2×#121—734no—
378C2×#121—734no—
379C2×#121—734no—
380C2×#121—734no—
381C2×#121C2×D7314yes—
382C2×C4×Q8—734no—
383C2×C4×D4—734no—
384C2×C4×D4—734no—
385C2×C4×D4—734no—
386C2×C4×D4—734no—
387C2×#117—732no—
388C22×C4⋊C4—734no—
389C22×C4⋊C4—734no—
390C22×C42—731no—
391C2×#41—29216no—
392C2×#41—29216no—
393C2×#41—2928no—
394C2×#41—2928no—
395C2×#41—7316no—
396C2×#41—7316no—
397C2×#41—7332no—
398C2×#41—7332no—
399C22×Q8—732no—
400C22×Q8—734no—
401C22×Q8C22×D7312yes—
402C22×D4—734no—
403C22×D4—734no—
404C22×D4—734no—
405C22×D4—734no—
406C22×D4—734no—
407C22×D4—734no—
408C22×D4—734no—
409C22×D4—732no—
410C22×D4C22×D7314yes—
411C22×D4C22×D7312yes—
412C23×C4—734no—
413C23×C4—734no—
414C23×C4—734no—
415C23×C4—734no—
416C23×C4—734no—
417C23×C4—731no—
418C23×C4—732no—
419C23×C4C22×D7314yes—
420C4⋊Q8—7316no—
421C4⋊Q8—7316no—
422#127—7316no—
423#127—7332no—
424#125—7332no—
425#125—738no—
426#125C22×D7318yes—
427#124—7316no—
428#124C22×D73116yes—
429#123—7332no—
430#123—7332no—
431#123—7332no—
432#123C22×D73132yes—
433C22⋊Q8—7332no—
434C22⋊Q8—7332no—
435C22⋊Q8—7332no—
436C22⋊Q8—7332no—
437C22⋊Q8—7332no—
438C22⋊Q8—7332no—
439C22⋊Q8C22×D73132yes—
440#121—7332no—
441#121—7332no—
442#121—7332no—
443#121—7332no—
444#121—7332no—
445#121—7332no—
446#121—7332no—
447#121—7332no—
448#121—7332no—
449#121—7332no—
450#121—7332no—
451#121C22×D73132yes—
452C4×Q8—7332no—
453C4×D4—7332no—
454C4×D4—7332no—
455C4×D4—7332no—
456C4×D4—7332no—
457#117—7316no—
458C2×C4⋊C4—734no—
459C2×C4⋊C4—734no—
460C2×C4⋊C4—734no—
461C2×C4⋊C4—734no—
462C2×C4⋊C4—734no—
463C2×C4⋊C4—7332no—
464C2×C4⋊C4—732no—
465C2×C4⋊C4—734no—
466C2×C4⋊C4—734no—
467C2×C4⋊C4—7332no—
468C2×C4⋊C4C22×D7314yes—
469C2×C4⋊C4C22×D7312yes—
470C2×C4⋊C4C22×D7314yes—
471C2×C22⋊C4—734no—
472C2×C22⋊C4—734no—
473C2×C22⋊C4—734no—
474C2×C22⋊C4—734no—
475C2×C22⋊C4—734no—
476C2×C22⋊C4—734no—
477C2×C22⋊C4C22×D7314yes—
478C2×C22⋊C4C22×D7314yes—
479C2×C22⋊C4C22×D7314yes—
480C2×C42—734no—
481C2×C42—731no—
482C2×C42—732no—
483C2×C42—738no—
484C2×C42C22×D7311yescharacteristic
485C24—734no—
486C24—731no—
487C24C23×D7314yes—
488#41—29264no—
489#41—29264no—
490#41—29232no—
491#41—29232no—
492C2×Q8—44no—
493C2×Q8—2924no—
494C2×Q8—738no—
495C2×Q8—7316no—
496C2×Q8—14616no—
497C2×Q8C23×D7318yes—
498C2×D4—48no—
499C2×D4—44no—
500C2×D4—2928no—
501C2×D4—2928no—
502C2×D4—2924no—
503C2×D4—2928no—
504C2×D4—2928no—
505C2×D4—2928no—
506C2×D4—7316no—
507C2×D4—7316no—
508C2×D4—7316no—
509C2×D4—7316no—
510C2×D4—7316no—
511C2×D4—7316no—
512C2×D4—7316no—
513C2×D4—738no—
514C2×D4—14616no—
515C2×D4—14616no—
516C2×D4C23×D73116yes—
517C2×D4C23×D7318yes—
518C22×C4—48no—
519C22×C4—2928no—
520C22×C4—2928no—
521C22×C4—2928no—
522C22×C4—7316no—
523C22×C4—7316no—
524C22×C4—7316no—
525C22×C4—731no—
526C22×C4—732no—
527C22×C4—7316no—
528C22×C4—734no—
529C22×C4—734no—
530C22×C4—734no—
531C22×C4—738no—
532C22×C4—7316no—
533C22×C4—1464no—
534C22×C4—14616no—
535C22×C4—1468no—
536C22×C4—14616no—
537C22×C4C23×D7311yescharacteristic
538C22×C4C23×D7312yes—
539C22×C4C23×D7314yes—
540C22×C4C23×D7314yes—
541C22×C4C23×D73116yes—
542C4⋊C4—7316no—
543C4⋊C4—7316no—
544C4⋊C4—7316no—
545C4⋊C4—7316no—
546C4⋊C4—7316no—
547C4⋊C4—7316no—
548C4⋊C4—7316no—
549C4⋊C4—738no—
550C4⋊C4C23×D73116yes—
551C4⋊C4C23×D7318yes—
552C4⋊C4C23×D73116yes—
553C22⋊C4—7316no—
554C22⋊C4—7316no—
555C22⋊C4—7316no—
556C22⋊C4—7316no—
557C22⋊C4—7316no—
558C22⋊C4—7316no—
559C22⋊C4C23×D73116yes—
560C22⋊C4C23×D73116yes—
561C22⋊C4C23×D73116yes—
562C42—734no—
563C42—738no—
564C42—7316no—
565C42C23×D7314yes—
566C23—28no—
567C23—28no—
568C23—28no—
569C23—738no—
570C23—732no—
571C23—1468no—
572C23—1468no—
573C23—1464no—
574C23—1462no—
575C23—1468no—
576C23C24×D7318yes—
577C23C24×D7311yescharacteristic
578Q8—48no—
579Q8—2928no—
580Q8—14632no—
581D4—416no—
582D4—48no—
583D4—29216no—
584D4—29216no—
585D4—2928no—
586D4—29216no—
587D4—29216no—
588D4—29216no—
589D4—14632no—
590D4—14632no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—416no—
596C2×C4—29216no—
597C2×C4—29216no—
598C2×C4—29216no—
599C2×C4—738no—
600C2×C4—734no—
601C2×C4—732no—
602C2×C4—734no—
603C2×C4—738no—
604C2×C4—738no—
605C2×C4—14632no—
606C2×C4—14616no—
607C2×C4—1468no—
608C2×C4—14632no—
609C2×C4—1468no—
610C2×C4—1468no—
611C2×C4C24×D7312yes—
612C2×C4C24×D7314yes—
613C2×C4C24×D7318yes—
614C2×C4C24×D7318yes—
615C22—28no—
616C22—28no—
617C22—28no—
618C22—416no—
619C22—29216no—
620C22—2924no—
621C22—1468no—
622C22—1468no—
623C22—1464no—
624C22—1462no—
625C22—1468no—
626C22H(5,𝔽2)×D7314yes—
627C222−1+4×D7312yes—
628C22C25×D7311yescharacteristic
629C4—28no—
630C4—28no—
631C4—24no—
632C4—22no—
633C4—738no—
634C4—734no—
635C4—1468no—
636C4—1468no—
637C2—48no—
638C2—2928no—
639C2—2922no—
640C2#2262432714yesdirect product factor
641C2C2×H(5,𝔽2)×D7312yes—
642C2C2×2−1+4×D7311yescharacteristic
subgroupG/H
centerC23C24×D73
commutator, cosocleC22×C73C26
FrattiniC22C25×D73
FittingC2×#562×C73C2
socleC23×C73C25
hypercenterC2×#562D73
hypocenterC73C2×#3548
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #22624327C2direct (N × H)
2. C2×#17989335C2semidirect
3. C2×#17989331C2semidirect
4. C2×#17989291C2semidirect
5. C2×#17989255C2semidirect
6. C2×#17989235C2semidirect
7. C2×#17989189C2semidirect
8. C2×#17989167C2semidirect
9. C2×#17989164C2semidirect
10. C2×#17989143C2semidirect
11. C2×#17989136C2semidirect
12. C2×#562×C73C2semidirect
17. #17989335C22semidirect
18. #17989331C22semidirect
19. #17989291C22semidirect
20. #17989255C22semidirect
21. #17989235C22semidirect
22. #17989189C22semidirect
23. #17989167C22semidirect
24. #17989164C22semidirect
25. #17989143C22semidirect
26. #17989136C22semidirect
63. #562×C73C22semidirect
317. C2×C73#3548semidirect
325. C2×#562D73semidirect
334. C73C2×#3548semidirect
341. #562C2×D73semidirect
640. C2#22624327direct (N × H)

Maximal subgroups

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

subgroupindex
#226243272
C2×#179893352
C2×#179893312
C2×#179892912
C2×#179892552
C2×#179892352
C2×#179891892
C2×#179891672
C2×#179891642
C2×#179891432
C2×#179891362
C2×#562×C732
C2×#354873