Finite Group Databasebeta

← C2×#22622737 #33965698

Subgroups — C2×#22622737 #33965698

Order 18688.

1778 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989315C211yesmaximal, characteristic
2C2×#17989314C211yesmaximal, characteristic
3#22622737C2116yesmaximal, direct product factor
4C2×#17988957C211yesmaximal, characteristic
5C2×#17988954C211yesmaximal, characteristic
6C2×#17988936C211yesmaximal, characteristic
7C2×#17988893C211yesmaximal, characteristic
8C2×#17988871C211yesmaximal, characteristic
9C2×#17988798C211yesmaximal, characteristic
10C2×#17988796C211yesmaximal, characteristic
11C2×#17988796C211yesmaximal, characteristic
12C2×#17988716C211yesmaximal, characteristic
13C2×#17988693C211yesmaximal, characteristic
14C2×#17988692C211yesmaximal, characteristic
15C2×#17988681C211yesmaximal, characteristic
16C2×#479×C73C211yesmaximal, characteristic
17C22×Dic292C2211yescharacteristic
18C22×Dic292C2211yescharacteristic
19C22×#11924018C2211yescharacteristic
20#17989315—24no—
21#17989315C2218yes—
22#17989314—24no—
23#17989314C2218yes—
24C2×#14556591C2211yescharacteristic
25C2×#14556590—21no—
26C2×#14556590C2211yescharacteristic
27C2×#14556589C2211yescharacteristic
28C2×#14556589C2211yescharacteristic
29C2×#14556583—21no—
30C2×#14556582C2211yescharacteristic
31C2×#14556581—21no—
32C2×#14556581—21no—
33C2×#14556581—21no—
34C2×#14556581—21no—
35C2×C4×Dic146—21no—
36C2×C4×Dic146C2211yescharacteristic
37C2×C4×Dic146C2211yescharacteristic
38C2×#14556556—21no—
39C2×#14556556C2211yescharacteristic
40C2×#14556556C2211yescharacteristic
41C2×#14556556C2211yescharacteristic
42C2×#14556556C2211yescharacteristic
43#17988957C2218yes—
44#17988954C2218yes—
45#17988936C2218yes—
46#17988893C2218yes—
47#17988871C2218yes—
48#17988798C2218yes—
49#17988796C2218yes—
50#17988796C2218yes—
51#17988716C2218yes—
52#17988693C2218yes—
53#17988692C2218yes—
54#17988681C2218yes—
55C2×#14556522C2211yescharacteristic
56C2×#14556520C2211yescharacteristic
57C2×#14556520C2211yescharacteristic
58C2×#14556516C2211yescharacteristic
59C2×#14556516C2211yescharacteristic
60C2×#14556515C2211yescharacteristic
61C2×#14556514C2211yescharacteristic
62C2×#14556514C2211yescharacteristic
63C2×#14556514C2211yescharacteristic
64C2×#14556514C2211yescharacteristic
65C2×#14556514C2211yescharacteristic
66C2×#14556502C2211yescharacteristic
67C2×#14556501C2211yescharacteristic
68C2×C8×Dic73C2211yescharacteristic
69C2×#125×C73C2211yescharacteristic
70C2×C4×Q8×C73C2211yescharacteristic
71#479×C73C2218yes—
72C2×#107×C73C2211yescharacteristic
73C2×#106×C73C2211yescharacteristic
74C2×C4⋊C8×C73C2211yescharacteristic
75C2×#103×C73C2211yescharacteristic
76C2×#103×C73C2211yescharacteristic
77C22×Dic146—21no—
78C22×Dic146D411yescharacteristic
79C22×Dic146C2311yescharacteristic
80C22×Dic146C2311yescharacteristic
81C22×Dic146C2311yescharacteristic
82C2×Dic292—24no—
83C2×Dic292—24no—
84C2×Dic292C2314yes—
85C2×Dic292C2314yes—
86C2×#11924018—24no—
87C2×#11924018C2314yes—
88C22×#10596831C2311yescharacteristic
89C22×#10596831C2311yescharacteristic
90#14556591—22no—
91#14556591C2314yes—
92#14556590—28no—
93#14556590—22no—
94#14556590C2314yes—
95#14556589—22no—
96#14556589—22no—
97#14556589C2314yes—
98#14556589C2314yes—
99#14556583—28no—
100#14556582—22no—
101#14556582C2314yes—
102#14556581—28no—
103#14556581—28no—
104#14556581—28no—
105#14556581—28no—
106C4×Dic146—28no—
107C4×Dic146—22no—
108C4×Dic146D414yes—
109C4×Dic146C2314yes—
110C4×Dic146C2314yes—
111#14556556—28no—
112#14556556—22no—
113#14556556—22no—
114#14556556—22no—
115#14556556—22no—
116#14556556C2314yes—
117#14556556C2314yes—
118#14556556C2314yes—
119#14556556C2314yes—
120C2×C4⋊Dic73—21no—
121C2×C4⋊Dic73—21no—
122C2×C4⋊Dic73—21no—
123C2×C4⋊Dic73D411yescharacteristic
124C2×C4⋊Dic73C2311yescharacteristic
125C2×#11924007—21no—
126C2×#11924007—21no—
127C2×#11924007—21no—
128C2×#11924007—21no—
129C2×#11924007—21no—
130C2×#11924007—21no—
131C2×#11924007—21no—
132C2×#11924007—21no—
133C2×#11924007—21no—
134C2×#11924007—21no—
135C2×#11924007—21no—
136C2×#11924007D411yescharacteristic
137C2×#11924007D411yescharacteristic
138C2×C4×Dic73—21no—
139C2×C4×Dic73—21no—
140C2×C4×Dic73—21no—
141C2×C4×Dic73—21no—
142C2×C4×Dic73—21no—
143C2×C4×Dic73C2311yescharacteristic
144C2×C4×Dic73C2311yescharacteristic
145#14556522C2314yes—
146#14556520C2314yes—
147#14556520C2314yes—
148#14556516C2314yes—
149#14556516C2314yes—
150#14556515C2314yes—
151#14556514C2314yes—
152#14556514C2314yes—
153#14556514C2314yes—
154#14556514C2314yes—
155#14556514C2314yes—
156#14556502C2314yes—
157#14556501C2314yes—
158C8×Dic73C2314yes—
159C22×Q8×C73C2311yescharacteristic
160C22×C8×C73C2311yescharacteristic
161C22×C8×C73C2311yescharacteristic
162#125×C73—22no—
163#125×C73C2314yes—
164C4×Q8×C73—22no—
165C4×Q8×C73C2314yes—
166C2×C4⋊C4×C73—21no—
167C2×C4⋊C4×C73—21no—
168C2×C4⋊C4×C73—21no—
169C2×C4⋊C4×C73C2311yescharacteristic
170C2×C4⋊C4×C73C2311yescharacteristic
171C2×C4⋊C4×C73C2311yescharacteristic
172C2×C42×C73—21no—
173C2×C42×C73C2311yescharacteristic
174#107×C73C2314yes—
175#106×C73C2314yes—
176C4⋊C8×C73C2314yes—
177#103×C73C2314yes—
178#103×C73C2314yes—
179C2×Dic146—21no—
180C2×Dic146—22no—
181C2×Dic146—24no—
182C2×Dic146—22no—
183C2×Dic146—22no—
184C2×Dic146—22no—
185C2×Dic146—22no—
186C2×Dic146—21no—
187C2×Dic146—21no—
188C2×Dic146—24no—
189C2×Dic146—44no—
190C2×Dic146C2×D414yes—
191C2×Dic146#4114yes—
192C2×Dic146C2412yes—
193C2×Dic146C2412yes—
194C2×Dic146C2412yes—
195C22×Dic73—21no—
196C22×Dic73—21no—
197C22×Dic73—21no—
198C22×Dic73—21no—
199C22×Dic73C2×D411yescharacteristic
200C22×Dic73C2×D411yescharacteristic
201C22×Dic73C2×D411yescharacteristic
202C22×Dic73C2×D411yescharacteristic
203Dic292—28no—
204Dic292—28no—
205#11924018—28no—
206C2×#10596831—22no—
207C2×#10596831—22no—
208C2×#10596831C2412yes—
209C2×#10596831C2412yes—
210C4⋊Dic73—24no—
211C4⋊Dic73—24no—
212C4⋊Dic73—21no—
213C4⋊Dic73—24no—
214C4⋊Dic73C2×D414yes—
215C4⋊Dic73C2412yes—
216#11924007—24no—
217#11924007—24no—
218#11924007—24no—
219#11924007—24no—
220#11924007—24no—
221#11924007—24no—
222#11924007—24no—
223#11924007—24no—
224#11924007—24no—
225#11924007—24no—
226#11924007—24no—
227#11924007C2×D414yes—
228#11924007C2×D414yes—
229C4×Dic73—24no—
230C4×Dic73—24no—
231C4×Dic73—24no—
232C4×Dic73—24no—
233C4×Dic73—24no—
234C4×Dic73C2×D412yes—
235C4×Dic73C2×D412yes—
236C4×Dic73C2412yes—
237C4×Dic73C2412yes—
238C2×Q8×C73—21no—
239C2×Q8×C73—22no—
240C2×Q8×C73—22no—
241C2×Q8×C73C2412yes—
242C22×C4×C73—21no—
243C22×C4×C73—21no—
244C22×C4×C73—21no—
245C22×C4×C73—21no—
246C22×C4×C73C2×D411yescharacteristic
247C22×C4×C73C2×D411yescharacteristic
248C22×C4×C73C2411yescharacteristic
249C2×C8×C73—22no—
250C2×C8×C73—22no—
251C2×C8×C73C2412yes—
252C2×C8×C73C2412yes—
253C4⋊C4×C73—24no—
254C4⋊C4×C73—24no—
255C4⋊C4×C73—24no—
256C4⋊C4×C73—21no—
257C4⋊C4×C73—21no—
258C4⋊C4×C73—21no—
259C4⋊C4×C73C2412yes—
260C4⋊C4×C73C2412yes—
261C4⋊C4×C73C2412yes—
262C42×C73—24no—
263C42×C73C2×D412yes—
264C42×C73C2412yes—
265C23×C73C22×D411yescharacteristic
266Q8×C73—22no—
267Q8×C73—22no—
268Q8×C73—42no—
269C2×C4×C73—22no—
270C2×C4×C73—22no—
271C2×C4×C73—22no—
272C2×C4×C73—22no—
273C2×C4×C73—22no—
274C2×C4×C73—22no—
275C2×C4×C73—42no—
276C2×C4×C73—42no—
277C2×C4×C73—42no—
278C2×C4×C73—42no—
279C2×C4×C73C22×D412yes—
280C2×C4×C73C22×D412yes—
281C2×C4×C73C22×D411yescharacteristic
282C2×C4×C73C2×#4112yes—
283C2×C4×C732−1+412yes—
284C2×C4×C73C2511yescharacteristic
285C2×Dic73—24no—
286C2×Dic73—22no—
287C2×Dic73—22no—
288C2×Dic73—22no—
289C2×Dic73—22no—
290C2×Dic73—22no—
291C2×Dic73—22no—
292C2×Dic73—22no—
293C2×Dic73—42no—
294C2×Dic73—42no—
295C2×Dic73—42no—
296C2×Dic73#13714yes—
297C2×Dic73C22×D412yes—
298C2×Dic73C22×D412yes—
299C2×Dic73C22×D412yes—
300C2×Dic73C22×D412yes—
301Dic146—24no—
302Dic146—22no—
303Dic146—28no—
304Dic146—22no—
305Dic146—22no—
306Dic146—22no—
307Dic146—22no—
308Dic146—42no—
309Dic146—42no—
310Dic146—48no—
311Dic146C2×#4114yes—
312C8×C73—22no—
313C8×C73—22no—
314#10596831—22no—
315#10596831—22no—
316C22×C73#54812yes—
317C22×C73C2×#13712yes—
318C22×C73#57812yes—
319C22×C73C23×D411yescharacteristic
320C4×C73—22no—
321C4×C73—22no—
322C4×C73—42no—
323C4×C73—42no—
324C4×C73—42no—
325C4×C73—42no—
326C4×C73#54812yes—
327C4×C73C22×#4111yescharacteristic
328C4×C73C2×2−1+411yescharacteristic
329Dic73—24no—
330Dic73—22no—
331Dic73—22no—
332Dic73—22no—
333Dic73—42no—
334Dic73—42no—
335Dic73—42no—
336Dic73C2×#13714yes—
337C2×#3374—731nomaximal, Sylow
338C2×C73#337414yescomplemented
339C2×C73C2×#54811yescharacteristic
340C2×C73C22×#13711yescharacteristic
341C2×C73C2×#57811yescharacteristic
342C2×#557—731no—
343C2×#557—731no—
344#3374—7316no—
345C2×#491—731no—
346C2×#491—731no—
347C2×#487—731no—
348C2×#479—731no—
349C2×#479D7311yescharacteristic
350C2×#475—731no—
351C2×#462—731no—
352C2×#462—731no—
353C2×#462—731no—
354C2×#446—731no—
355C2×#441—731no—
356C2×#441—731no—
357C2×C4×Q16—731no—
358C73C2×#337411yescharacteristic, complemented, Sylow
359C22×Q16—731no—
360C22×Q16—731no—
361C22×Q16—731no—
362#557—738no—
363#557—738no—
364#557—1464no—
365#557—1464no—
366C2×C4⋊Q8—731no—
367C2×C4⋊Q8—731no—
368C2×C4⋊Q8—731no—
369C2×C4⋊Q8—731no—
370C2×C4⋊Q8—1461no—
371C2×#125—731no—
372C2×#125—1461no—
373C2×#125—1461no—
374C2×#125—1461no—
375C2×#125—1461no—
376C2×#125—1461no—
377C2×#125C2×D7311yescharacteristic
378C2×C4×Q8—731no—
379C2×C4×Q8—731no—
380C2×C4×Q8—731no—
381C2×C4×Q8—731no—
382C2×C4×Q8—731no—
383C2×C4×Q8—731no—
384C2×C4×Q8—1461no—
385C2×C4×Q8—1461no—
386C2×C4×Q8C2×D7311yescharacteristic
387#491—738no—
388#491—738no—
389#487—738no—
390#479—738no—
391#479C2×D7318yes—
392#475—738no—
393#462—738no—
394#462—738no—
395#462—738no—
396#446—738no—
397#441—738no—
398#441—738no—
399C4×Q16—738no—
400C2×#107C2×D7311yescharacteristic
401C2×#106—731no—
402C2×#106C2×D7311yescharacteristic
403C2×C4⋊C8—731no—
404C2×C4⋊C8—731no—
405C2×C4⋊C8C2×D7311yescharacteristic
406C2×#103—731no—
407C2×#103—731no—
408C2×#103—731no—
409C2×#103—731no—
410C2×#103—731no—
411C2×#103—731no—
412C2×#103—731no—
413C2×#103—731no—
414C2×#103C2×D7311yescharacteristic
415C2×#103C2×D7311yescharacteristic
416C2×#97—731no—
417C2×#97—731no—
418C2×C4×C8—731no—
419C22×Q8—731no—
420C22×Q8—731no—
421C22×Q8—731no—
422C22×Q8—731no—
423C22×Q8—1461no—
424C22×Q8C22×D7311yescharacteristic
425C2×Q16—734no—
426C2×Q16—734no—
427C2×Q16—734no—
428C2×Q16—1464no—
429C2×Q16—1464no—
430C2×Q16—1464no—
431C22×C8—731no—
432C22×C8—731no—
433C22×C8C22×D7311yescharacteristic
434C22×C8C22×D7311yescharacteristic
435C4⋊Q8—734no—
436C4⋊Q8—734no—
437C4⋊Q8—734no—
438C4⋊Q8—734no—
439C4⋊Q8—1462no—
440C4⋊Q8—1462no—
441C4⋊Q8—1462no—
442C4⋊Q8—1468no—
443C4⋊Q8—1462no—
444#125—22no—
445#125—734no—
446#125—1468no—
447#125—1462no—
448#125—1468no—
449#125—1468no—
450#125—1468no—
451#125—1468no—
452#125C22×D7314yes—
453C4×Q8—22no—
454C4×Q8—734no—
455C4×Q8—734no—
456C4×Q8—734no—
457C4×Q8—734no—
458C4×Q8—734no—
459C4×Q8—734no—
460C4×Q8—734no—
461C4×Q8—1468no—
462C4×Q8—1462no—
463C4×Q8—1462no—
464C4×Q8—1462no—
465C4×Q8—1462no—
466C4×Q8—1468no—
467C4×Q8—1462no—
468C4×Q8C22×D7314yes—
469C2×C4⋊C4—21no—
470C2×C4⋊C4—21no—
471C2×C4⋊C4—21no—
472C2×C4⋊C4—731no—
473C2×C4⋊C4—731no—
474C2×C4⋊C4—731no—
475C2×C4⋊C4—731no—
476C2×C4⋊C4—1461no—
477C2×C4⋊C4—1461no—
478C2×C4⋊C4—1461no—
479C2×C4⋊C4—1461no—
480C2×C4⋊C4—1461no—
481C2×C4⋊C4—1461no—
482C2×C4⋊C4—1461no—
483C2×C4⋊C4—1461no—
484C2×C4⋊C4—1461no—
485C2×C4⋊C4—1461no—
486C2×C4⋊C4—1461no—
487C2×C4⋊C4—1461no—
488C2×C4⋊C4—1461no—
489C2×C4⋊C4—1461no—
490C2×C4⋊C4C22×D7311yescharacteristic
491C2×C4⋊C4C22×D7311yescharacteristic
492C2×C4⋊C4C22×D7311yescharacteristic
493C2×C42—21no—
494C2×C42—731no—
495C2×C42—731no—
496C2×C42—1461no—
497C2×C42—1461no—
498C2×C42—1461no—
499C2×C42—1461no—
500C2×C42—1461no—
501C2×C42C22×D7311yescharacteristic
502#107C22×D7314yes—
503#106—734no—
504#106C22×D7314yes—
505C4⋊C8—734no—
506C4⋊C8—734no—
507C4⋊C8C22×D7314yes—
508#103—734no—
509#103—734no—
510#103—734no—
511#103—734no—
512#103—734no—
513#103—734no—
514#103—734no—
515#103—734no—
516#103C22×D7314yes—
517#103C22×D7314yes—
518#97—734no—
519#97—734no—
520C4×C8—734no—
521C2×Q8—21no—
522C2×Q8—22no—
523C2×Q8—22no—
524C2×Q8—2924no—
525C2×Q8—734no—
526C2×Q8—734no—
527C2×Q8—732no—
528C2×Q8—732no—
529C2×Q8—732no—
530C2×Q8—1464no—
531C2×Q8—1464no—
532C2×Q8—1462no—
533C2×Q8—1462no—
534C2×Q8—1462no—
535C2×Q8—1462no—
536C2×Q8—1461no—
537C2×Q8—1461no—
538C2×Q8—1461no—
539C2×Q8—1462no—
540C2×Q8C23×D7312yes—
541C22×C4—21no—
542C22×C4—21no—
543C22×C4—21no—
544C22×C4—21no—
545C22×C4—731no—
546C22×C4—731no—
547C22×C4—731no—
548C22×C4—731no—
549C22×C4—1461no—
550C22×C4—1461no—
551C22×C4—1461no—
552C22×C4—1461no—
553C22×C4D4×D7311yescharacteristic
554C22×C4D4×D7311yescharacteristic
555C22×C4C23×D7311yescharacteristic
556Q16—1468no—
557Q16—1468no—
558Q16—1468no—
559C2×C8—22no—
560C2×C8—22no—
561C2×C8—732no—
562C2×C8—732no—
563C2×C8—1462no—
564C2×C8—1462no—
565C2×C8C23×D7312yes—
566C2×C8C23×D7312yes—
567C4⋊C4—24no—
568C4⋊C4—24no—
569C4⋊C4—24no—
570C4⋊C4—21no—
571C4⋊C4—21no—
572C4⋊C4—21no—
573C4⋊C4—734no—
574C4⋊C4—734no—
575C4⋊C4—734no—
576C4⋊C4—732no—
577C4⋊C4—1464no—
578C4⋊C4—1464no—
579C4⋊C4—1464no—
580C4⋊C4—1464no—
581C4⋊C4—1464no—
582C4⋊C4—1464no—
583C4⋊C4—1464no—
584C4⋊C4—1464no—
585C4⋊C4—1464no—
586C4⋊C4—1464no—
587C4⋊C4—1464no—
588C4⋊C4—1464no—
589C4⋊C4—1461no—
590C4⋊C4—1464no—
591C4⋊C4—1464no—
592C4⋊C4C23×D7312yes—
593C4⋊C4C23×D7312yes—
594C4⋊C4C23×D7312yes—
595C42—24no—
596C42—732no—
597C42—732no—
598C42—732no—
599C42—732no—
600C42—1464no—
601C42—1464no—
602C42—1464no—
603C42—1464no—
604C42—1464no—
605C42D4×D7312yes—
606C42C23×D7312yes—
607C23C2×D4×D7311yescharacteristic
608Q8—22no—
609Q8—22no—
610Q8—42no—
611Q8—2928no—
612Q8—2922no—
613Q8—2922no—
614Q8—734no—
615Q8—1468no—
616Q8—1462no—
617Q8—1462no—
618Q8—1462no—
619Q8—1462no—
620Q8—1462no—
621Q8—1464no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—2922no—
633C2×C4—2922no—
634C2×C4—2922no—
635C2×C4—734no—
636C2×C4—732no—
637C2×C4—732no—
638C2×C4—732no—
639C2×C4—732no—
640C2×C4—1462no—
641C2×C4—1462no—
642C2×C4—1462no—
643C2×C4—1462no—
644C2×C4—1462no—
645C2×C4—1462no—
646C2×C4—1462no—
647C2×C4—1464no—
648C2×C4C2×D4×D7312yes—
649C2×C4C2×D4×D7312yes—
650C2×C4C2×D4×D7311yescharacteristic
651C2×C4#41×D7312yes—
652C2×C4#1455663912yes—
653C2×C4C24×D7311yescharacteristic
654C8—22no—
655C8—22no—
656C8—1462no—
657C8—1462no—
658C22#1798924712yes—
659C22#137×D7312yes—
660C22#1798940112yes—
661C22C22×D4×D7311yescharacteristic
662C4—22no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—2922no—
669C4—2922no—
670C4—2922no—
671C4—734no—
672C4—1462no—
673C4—1464no—
674C4—1462no—
675C4—1462no—
676C4#1798924712yes—
677C4C2×#41×D7311yescharacteristic
678C4C2×#1455663911yescharacteristic
679C2#2262273714yesdirect product factor
680C2C2×#1798924711yescharacteristic
681C2C2×#137×D7311yescharacteristic
682C2C2×#1798940111yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC2×#479×C73C2
socle, prime radicalC23×C73C22×D4
hypercenterC2×#479D73
hypocenterC73C2×#3374
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#479

Complements

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

NH ≅ G/Ntype
3. #22622737C2direct (N × H)
338. C2×C73#3374semidirect
358. C73C2×#3374semidirect
679. C2#22622737direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179893152
C2×#179893142
#226227372
C2×#179889572
C2×#179889542
C2×#179889362
C2×#179888932
C2×#179888712
C2×#179887982
C2×#179887962
C2×#179887162
C2×#179886932
C2×#179886922
C2×#179886812
C2×#479×C732
C2×#337473