Finite Group Databasebeta

← #33087205

Subgroups — #33087205

Order 18688.

1986 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989098C211yesmaximal, characteristic
2C2×#17988581C212yesmaximal, complemented
3C22×C22⋊#10596831C211yesmaximal, characteristic, complemented
4#22610253C212yesmaximal, complemented
5#22610248C212yesmaximal, complemented
6#22610243C212yesmaximal, complemented
7#22610238C212yesmaximal, complemented
8#22609647C212yesmaximal, complemented
9C22×#117×C73C211yesmaximal, characteristic, complemented
10C22×#11924028C2211yescharacteristic
11C23×#10596831C2212yes—
12#17989098—24no—
13#17989098C418yescomplemented
14C2×#14556574—21no—
15C2×#14556570C2211yescharacteristic
16C2×#14556570C2211yescharacteristic
17C2×#14556565C2211yescharacteristic
18C2×#14556565C2211yescharacteristic
19C2×#14556560—21no—
20C2×C4×C22⋊D73—21no—
21C2×C4×C22⋊D73—21no—
22C2×#14556550C2211yescharacteristic
23#17988581—28no—
24C2×#14556514C2214yescomplemented
25C2×#14556511C2214yescomplemented
26C2×C22⋊#10596831—24no—
27C2×C22⋊#10596831C2212yes—
28C2×C22⋊#10596831C2212yes—
29#17988349C2214yescomplemented
30#17988349C2214yescomplemented
31C24×C4×C73C2211yescharacteristic
32C2×#117×C73—21no—
33C2×#117×C73—21no—
34C2×#117×C73—28no—
35C2×#117×C73C2212yes—
36C2×#117×C73C2212yes—
37C22×C4⋊C4×C73C2212yes—
38C22×C22⋊C4×C73—21no—
39C22×C42×C73—21no—
40C2×#11924028—22no—
41C2×#11924028—24no—
42C2×#11924028—24no—
43C2×#11924028C2×C414yescomplemented
44C22×Dic146C2311yescharacteristic
45C22×D292C2311yescharacteristic
46C22×C22⋊D73—21no—
47C22×C22⋊D73—21no—
48C22×C22⋊D73—21no—
49C22×C4×D73—21no—
50C22×#10596831—24no—
51C22×#10596831—24no—
52C22×#10596831—24no—
53C22×#10596831C2314yes—
54#14556574—28no—
55#14556570—22no—
56#14556570—22no—
57#14556570C2×C414yes—
58#14556570C2×C414yescomplemented
59#14556565—22no—
60#14556565—22no—
61#14556565C2×C414yes—
62#14556565C2×C414yescomplemented
63#14556560—28no—
64C4×C22⋊D73—28no—
65C4×C22⋊D73—28no—
66#14556550—22no—
67#14556550C2×C414yescomplemented
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×#11924007—21no—
71C2×#11924007—21no—
72C2×C22⋊Dic73—21no—
73C2×C22⋊Dic73—21no—
74C2×C22⋊Dic73—21no—
75C2×#11924005—21no—
76C2×#11924005—21no—
77C2×C4×Dic73—21no—
78#14556514—28no—
79#14556511—28no—
80C22⋊#10596831—24no—
81C22⋊#10596831—24no—
82C25×C73D411yescharacteristic
83C23×C4×C73—24no—
84C23×C4×C73—24no—
85C23×C4×C73—22no—
86C23×C4×C73C2×C414yes—
87C23×C4×C73D411yescharacteristic
88C23×C4×C73D411yescharacteristic
89C23×C4×C73D411yescharacteristic
90C23×C4×C73D414yes—
91C23×C4×C73C2311yescharacteristic
92C23×C4×C73C2311yescharacteristic
93C23×C4×C73C2311yescharacteristic
94#117×C73—28no—
95#117×C73—28no—
96#117×C73—24no—
97#117×C73—28no—
98#117×C73D418yescomplemented
99C2×C4⋊C4×C73—22no—
100C2×C4⋊C4×C73—28no—
101C2×C4⋊C4×C73C2314yes—
102C2×C22⋊C4×C73—28no—
103C2×C22⋊C4×C73—22no—
104C2×C22⋊C4×C73—22no—
105C2×C42×C73—28no—
106C2×C42×C73—22no—
107C2×C42×C73—22no—
108C23×D73—21no—
109#11924028—216no—
110#11924028—48no—
111C2×Dic146—22no—
112C2×Dic146—22no—
113C2×Dic146—21no—
114C2×Dic146C22×C412yes—
115C2×D292—22no—
116C2×D292—22no—
117C2×D292—21no—
118C2×D292C22×C412yes—
119C2×C22⋊D73—24no—
120C2×C22⋊D73—24no—
121C2×C22⋊D73—24no—
122C2×C22⋊D73—44no—
123C2×C22⋊D73—42no—
124C2×C22⋊D73—42no—
125C2×C22⋊D73—44no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C2×C4×D73—24no—
130C2×C4×D73—42no—
131C2×C4×D73—42no—
132C2×#10596831—24no—
133C2×#10596831—24no—
134C2×#10596831—24no—
135C2×#10596831—48no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73C22×C412yes—
139C4⋊Dic73C22×C412yes—
140#11924007—24no—
141#11924007—24no—
142C22⋊Dic73—24no—
143C22⋊Dic73—24no—
144C22⋊Dic73—24no—
145#11924005—24no—
146#11924005—24no—
147C4×Dic73—24no—
148C24×C73—22no—
149C24×C73—22no—
150C24×C73—24no—
151C24×C73—42no—
152C24×C73C22⋊C414yes—
153C24×C73C2×D411yescharacteristic
154C24×C73C2×D411yescharacteristic
155C24×C73C2×D411yescharacteristic
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—216no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—216no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73—28no—
169C22×C4×C73—24no—
170C22×C4×C73—44no—
171C22×C4×C73—44no—
172C22×C4×C73C22⋊C414yes—
173C22×C4×C73C22⋊C414yes—
174C22×C4×C73C22⋊C414yes—
175C22×C4×C73C22×C412yes—
176C22×C4×C73C22×C412yes—
177C22×C4×C73C22×C412yes—
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D414yes—
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D412yes—
183C22×C4×C73C2×D412yes—
184C22×C4×C73C2×D412yes—
185C22×C4×C73#4114yes—
186C22×C4×C73C2411yescharacteristic
187C4⋊C4×C73—28no—
188C4⋊C4×C73—24no—
189C4⋊C4×C73C2×D418yes—
190C22⋊C4×C73—28no—
191C22⋊C4×C73—28no—
192C42×C73—28no—
193C42×C73—28no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73—22no—
197C23×C73—22no—
198C23×C73—22no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73—22no—
202C23×C73—22no—
203C23×C73—28no—
204C23×C73—28no—
205C23×C73—44no—
206C23×C73—48no—
207C23×C73—48no—
208C23×C73C2×C22⋊C412yes—
209C23×C73C2×C22⋊C412yes—
210C23×C73C2×C22⋊C412yes—
211C23×C73#12012yes—
212C23×C73#12312yes—
213C23×C73#13514yes—
214C23×C73C22×D411yescharacteristic
215C22×D73—22no—
216C22×D73—41no—
217C22×D73—41no—
218C22×D73—42no—
219C22×D73—42no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—28no—
225C2×C4×C73—24no—
226C2×C4×C73—24no—
227C2×C4×C73—24no—
228C2×C4×C73—24no—
229C2×C4×C73—28no—
230C2×C4×C73—28no—
231C2×C4×C73—28no—
232C2×C4×C73—416no—
233C2×C4×C73—44no—
234C2×C4×C73—44no—
235C2×C4×C73C2×C22⋊C412yes—
236C2×C4×C73C2×C22⋊C412yes—
237C2×C4×C73C2×C22⋊C412yes—
238C2×C4×C73C4×D418yes—
239C2×C4×C73#12012yes—
240C2×C4×C73#12118yes—
241C2×C4×C73#12312yes—
242C2×C4×C73C23×C411yescharacteristic
243C2×C4×C73C22×D412yes—
244C2×C4×C73C22×D411yescharacteristic
245C2×C4×C73C2×#4112yes—
246C22⋊D73—48no—
247C22⋊D73—48no—
248C22⋊D73—48no—
249C2×Dic73—22no—
250C2×Dic73—22no—
251C2×Dic73—22no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—42no—
255C2×Dic73—42no—
256D292—24no—
257D292—42no—
258C4×D73—48no—
259Dic146—24no—
260Dic146—42no—
261#10596831—44no—
262C22×C73—24no—
263C22×C73—24no—
264C22×C73—22no—
265C22×C73—22no—
266C22×C73—22no—
267C22×C73—24no—
268C22×C73—24no—
269C22×C73—22no—
270C22×C73—22no—
271C22×C73—22no—
272C22×C73—22no—
273C22×C73—416no—
274C22×C73—42no—
275C22×C73—42no—
276C22×C73#38614yes—
277C22×C73#41614yescomplemented
278C22×C73C22×C22⋊C411yescharacteristic
279C22×C73C2×#12011yescharacteristic
280C22×C73C2×#12311yescharacteristic
281C22×C73C2×#13512yes—
282C22×C73#57512yes—
283C2×D73—42no—
284C2×D73—42no—
285C2×D73—42no—
286C2×D73—82no—
287C2×D73—82no—
288C4×C73—22no—
289C4×C73—22no—
290C4×C73—48no—
291C4×C73#38614yes—
292C4×C73C2×C4×D412yes—
293C4×C73C2×#12112yes—
294Dic73—42no—
295Dic73—42no—
296Dic73—42no—
297#20908—731nomaximal, Sylow
298C2×C73—24no—
299C2×C73—24no—
300C2×C73—42no—
301C2×C73C2×#38611yescharacteristic
302C2×C73C2×#41611yescharacteristic
303C2×C73#289611yescharacteristic
304C2×C73#300212yes—
305C2×C73#309112yes—
306D73—82no—
307C2×#525—731no—
308C22×#117D7311yescharacteristic, complemented
309C2×#416—732no—
310C22×C22⋊C8—731no—
311#1875—732no—
312#1874—732no—
313#1873—732no—
314#1872—732no—
315#1735—732no—
316C73#2090811yescharacteristic, complemented, Sylow
317C22×#41—731no—
318C24×C4C2×D7311yescharacteristic
319C23×C8—732no—
320#525—738no—
321#525—1464no—
322C2×#123—1461no—
323C2×C22⋊Q8—731no—
324C2×C22⋊Q8—731no—
325C2×#121—731no—
326C2×#121—731no—
327C2×#120—1461no—
328C2×C4×D4—1461no—
329C2×C4×D4—1461no—
330C2×#117—21no—
331C2×#117—21no—
332C2×#117—28no—
333C2×#117—731no—
334C2×#117C2×D7312yes—
335C2×#117C2×D7312yes—
336C22×C4⋊C4C2×D7312yes—
337C22×C22⋊C4—21no—
338C22×C42—21no—
339#416—1468no—
340C2×#103—734no—
341C2×#102—734no—
342C2×C22⋊C8—732no—
343C2×C22⋊C8—732no—
344C2×C22⋊C8—1464no—
345#340—734no—
346#340—734no—
347C25C22⋊D7311yescharacteristic
348C2×#41—734no—
349C2×#41—1464no—
350C2×#41—1464no—
351C2×#41—1462no—
352C22×Q8—731no—
353C22×D4—731no—
354C22×D4—1461no—
355C22×D4—1461no—
356C22×D4—1461no—
357C23×C4—24no—
358C23×C4—24no—
359C23×C4—22no—
360C23×C4—1461no—
361C23×C4C4×D7314yes—
362C23×C4D29214yes—
363C23×C4C22⋊D7311yescharacteristic
364C23×C4C22⋊D7311yescharacteristic
365C23×C4C22⋊D7311yescharacteristic
366C23×C4C22×D7311yescharacteristic
367C23×C4C22×D7311yescharacteristic
368C23×C4C22×D7311yescharacteristic
369C22×C8—734no—
370C22×C8—1464no—
371C22×C8—1464no—
372C22×C8—1464no—
373#123—1468no—
374C22⋊Q8—734no—
375C22⋊Q8—734no—
376C22⋊Q8—1462no—
377C22⋊Q8—1462no—
378#121—734no—
379#121—734no—
380#121—1462no—
381#121—1462no—
382#120—1468no—
383C4×D4—1468no—
384C4×D4—1468no—
385#117—28no—
386#117—28no—
387#117—24no—
388#117—28no—
389#117—734no—
390#117—1462no—
391#117C22⋊D7318yescomplemented
392C2×C4⋊C4—22no—
393C2×C4⋊C4—28no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4—1461no—
398C2×C4⋊C4C22×D7314yes—
399C2×C22⋊C4—28no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—1461no—
403C2×C22⋊C4—1461no—
404C2×C22⋊C4—1461no—
405C2×C22⋊C4—1461no—
406C2×C22⋊C4—1461no—
407C2×C42—28no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42—1461no—
411#103—1468no—
412#102—1468no—
413C22⋊C8—1464no—
414C22⋊C8—1464no—
415C24—22no—
416C24—22no—
417C24—24no—
418C24—42no—
419C24—1461no—
420C24#1192400514yes—
421C24C2×C22⋊D7311yescharacteristic
422C24C2×C22⋊D7311yescharacteristic
423C24C2×C22⋊D7311yescharacteristic
424#41—2928no—
425#41—14616no—
426C2×Q8—732no—
427C2×Q8—1462no—
428C2×Q8—1462no—
429C2×Q8—1461no—
430C2×D4—2924no—
431C2×D4—2922no—
432C2×D4—2922no—
433C2×D4—2924no—
434C2×D4—732no—
435C2×D4—1464no—
436C2×D4—1464no—
437C2×D4—1464no—
438C2×D4—1462no—
439C2×D4—1461no—
440C2×D4—1462no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—216no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—216no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—2922no—
458C22×C4—2922no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1464no—
463C22×C4#1192400514yes—
464C22×C4#1192400514yes—
465C22×C4#1192400514yes—
466C22×C4C2×C4×D7312yes—
467C22×C4C2×C4×D7312yes—
468C22×C4C2×C4×D7312yes—
469C22×C4C2×C22⋊D7311yescharacteristic
470C22×C4C2×C22⋊D7311yescharacteristic
471C22×C4C2×C22⋊D7314yes—
472C22×C4C2×C22⋊D7311yescharacteristic
473C22×C4C2×D29212yes—
474C22×C4C2×D29212yes—
475C22×C4C2×D29212yes—
476C22×C4#1192402814yes—
477C22×C4C23×D7311yescharacteristic
478C2×C8—2928no—
479C2×C8—1464no—
480C2×C8—1464no—
481C2×C8—1464no—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—732no—
485C4⋊C4—732no—
486C4⋊C4—1464no—
487C4⋊C4—1464no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4C2×C22⋊D7318yes—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—1464no—
494C22⋊C4—1464no—
495C22⋊C4—1464no—
496C22⋊C4—1464no—
497C22⋊C4—1464no—
498C42—28no—
499C42—28no—
500C42—1464no—
501C23—22no—
502C23—22no—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—22no—
507C23—28no—
508C23—21no—
509C23—21no—
510C23—28no—
511C23—22no—
512C23—44no—
513C23—48no—
514C23—48no—
515C23—2922no—
516C23—2921no—
517C23—2921no—
518C23—2922no—
519C23—1462no—
520C23C2×#1192400512yes—
521C23C2×#1192400512yes—
522C23C2×#1192400512yes—
523C23#1455656012yes—
524C23#1455657412yes—
525C23#1455661114yes—
526C23C22×C22⋊D7311yescharacteristic
527Q8—2922no—
528Q8—1464no—
529D4—2928no—
530D4—2928no—
531D4—2928no—
532D4—2922no—
533D4—1464no—
534C2×C4—28no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—416no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—2922no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2922no—
553C2×C4—2928no—
554C2×C4—1462no—
555C2×C4—1462no—
556C2×C4—1462no—
557C2×C4C2×#1192400512yes—
558C2×C4C2×#1192400512yes—
559C2×C4C2×#1192400512yes—
560C2×C4C4×C22⋊D7318yes—
561C2×C4#1455656012yes—
562C2×C4#1455656518yes—
563C2×C4#1455657412yes—
564C2×C4C22×C4×D7311yescharacteristic
565C2×C4C22×C22⋊D7312yes—
566C2×C4C22×D29211yescharacteristic
567C2×C4C2×#1192402812yes—
568C8—2924no—
569C22—24no—
570C22—24no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—24no—
576C22—24no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—416no—
581C22—42no—
582C22—42no—
583C22—2922no—
584C22—2922no—
585C22—2922no—
586C22—5842no—
587C22—5842no—
588C22#1798846614yes—
589C22#1798858114yescomplemented
590C22C22×#1192400511yescharacteristic
591C22C2×#1455656011yescharacteristic
592C22C2×#1455657411yescharacteristic
593C22C2×#1455661112yes—
594C22#1798938512yes—
595C4—22no—
596C4—22no—
597C4—48no—
598C4—2922no—
599C4—2922no—
600C4—2922no—
601C4#1798846614yes—
602C4C2×C4×C22⋊D7312yes—
603C4C2×#1455656512yes—
604C2—24no—
605C2—24no—
606C2—42no—
607C2—5842no—
608C2C2×#1798846611yescharacteristic
609C2C2×#1798858111yescharacteristic
610C2#2261844811yescharacteristic
611C2#2261914912yes—
612C2#2261984412yes—
subgroupG/H
centerC22×C4C2×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC22×#117×C73C2
socleC23×C73C22×D4
prime radicalC2×#17989098C2
cosocleC22×C4×C73C24
hypercenterC22×#117D73
hypocenterC73#20908
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C2×#17988581C2semidirect
3. C22×C22⋊#10596831C2semidirect
4. #22610253C2semidirect
5. #22610248C2semidirect
6. #22610243C2semidirect
7. #22610238C2semidirect
8. #22609647C2semidirect
9. C22×#117×C73C2semidirect
13. #17989098C4semidirect
24. C2×#14556514C22semidirect
25. C2×#14556511C22semidirect
29. #17988349C22semidirect
30. #17988349C22semidirect
43. C2×#11924028C2×C4semidirect
58. #14556570C2×C4semidirect
62. #14556565C2×C4semidirect
67. #14556550C2×C4semidirect
98. #117×C73D4semidirect
277. C22×C73#416semidirect
308. C22×#117D73semidirect
316. C73#20908semidirect
391. #117C22⋊D73semidirect
589. C22#17988581semidirect

Maximal subgroups

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

subgroupindex
C2×#179890982
C2×#179885812
C22×C22⋊#105968312
#226102532
#226102482
#226102432
#226102382
#226096472
C22×#117×C732
#2090873