Finite Group Databasebeta

← #33093545

Subgroups — #33093545

Order 18688.

1000 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988866C211yesmaximal, characteristic
2#22612502C211yesmaximal, characteristic, complemented
3C4×#14556553C211yesmaximal, characteristic
4#22610881C211yesmaximal, characteristic, complemented
5#22610688C211yesmaximal, characteristic, complemented
6#22610502C211yesmaximal, characteristic, complemented
7#22610181C211yesmaximal, characteristic
8#22610177C211yesmaximal, characteristic
9#22609930C211yesmaximal, characteristic
10#107×Dic73C211yesmaximal, characteristic, complemented
11#22609823C211yesmaximal, characteristic, complemented
12#22609779C211yesmaximal, characteristic, complemented
13#22609745C211yesmaximal, characteristic
14#22609745C211yesmaximal, characteristic
15C73×#1918C211yesmaximal, characteristic, complemented
16C2×#14556590C2211yescharacteristic
17C2×#14556556—21no—
18C2×C4×D292C2211yescharacteristic
19C2×#14556553C2211yescharacteristic
20C2×#14556553C2211yescharacteristic
21C2×C42×D73—21no—
22#17988866C418yescomplemented
23C2×#14556525C2211yescharacteristic
24C2×#14556524C2211yescharacteristic
25C2×#14556524C2211yescharacteristic
26C2×#14556520C2211yescharacteristic
27C2×#14556520C2211yescharacteristic
28C2×#14556513C2211yescharacteristic
29C2×#14556513C2211yescharacteristic
30C2×#14556511C2211yescharacteristic
31C2×#14556511C2211yescharacteristic
32C2×C8×Dic73C2211yescharacteristic
33C2×C8×Dic73C2211yescharacteristic
34#17988491—21no—
35#17988485C2211yescharacteristic
36#17988482—21no—
37#17988456C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42#17988438—21no—
43C4×#11924005—21no—
44#17988434—21no—
45#17988434—21no—
46C42×Dic73C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988342C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54#389×C73C2211yescharacteristic
55C4×C4⋊C4×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×Dic146—21no—
61C22×D292C2311yescharacteristic
62C22×C4×D73—21no—
63C22×C4×D73—21no—
64C22×C4×D73—21no—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556590—22no—
68#14556590C2×C414yescomplemented
69#14556556—44no—
70C4×D292—22no—
71C4×D292—22no—
72#14556553—22no—
73#14556553—22no—
74#14556553C2×C414yes—
75#14556553Q814yescomplemented
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924005—21no—
84C2×#11924005—21no—
85C2×#11924005—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73D411yescharacteristic
91C2×C4×Dic73D411yescharacteristic
92C2×C4×Dic73D411yescharacteristic
93C2×C4×Dic73D411yescharacteristic
94C2×C4×Dic73C2311yescharacteristic
95C2×C4×Dic73C2311yescharacteristic
96C42×D73—44no—
97#14556525—22no—
98#14556524—22no—
99#14556524C2×C414yescomplemented
100#14556520—22no—
101#14556520C2×C414yescomplemented
102#14556513—22no—
103#14556513C2×C414yes—
104#14556511—22no—
105#14556511C2×C414yes—
106C8×Dic73—22no—
107C8×Dic73—22no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111#14556498—21no—
112#14556498—21no—
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C42×C73—21no—
121C2×C42×C73C2311yescharacteristic
122#107×C73C2×C414yescomplemented
123#95×C73—21no—
124#95×C73—21no—
125C23×D73—21no—
126C2×Dic146—24no—
127C2×Dic146—42no—
128C2×Dic146—42no—
129C2×D292—21no—
130C2×D292—21no—
131C2×D292—21no—
132C2×D292C22×C412yes—
133C2×D292C2×Q812yes—
134C2×D292#4112yes—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73C2×D411yescharacteristic
139C22×Dic73C2×D411yescharacteristic
140C22×Dic73C2×D411yescharacteristic
141C22×Dic73C2×D411yescharacteristic
142C2×C4×D73—24no—
143C2×C4×D73—24no—
144C2×C4×D73—24no—
145C2×C4×D73—42no—
146C2×C4×D73—42no—
147C2×C4×D73—42no—
148C2×C4×D73—42no—
149C2×C4×D73—42no—
150C2×C4×D73—42no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831C22×C412yes—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73C22×C412yes—
162C4⋊Dic73#4112yes—
163#11924007—24no—
164#11924007—42no—
165#11924007—42no—
166#11924007—42no—
167#11924005—24no—
168#11924005—24no—
169#11924005—24no—
170C4×Dic73—22no—
171C4×Dic73—22no—
172C4×Dic73—41no—
173C4×Dic73—42no—
174C4×Dic73—42no—
175C4×Dic73—42no—
176C4×Dic73—42no—
177C4×Dic73D814yes—
178C4×Dic73C22×C412yes—
179C4×Dic73C2×Q812yes—
180C4×Dic73#4112yes—
181C4×Dic73#4112yes—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2411yescharacteristic
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73C22×C412yes—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—42no—
201C4⋊C4×C73C22×C412yes—
202C4⋊C4×C73C22×C412yes—
203C4⋊C4×C73C2×Q812yes—
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—42no—
207C23×C73C22×D411yescharacteristic
208C22×D73—22no—
209C22×D73—22no—
210C22×D73—22no—
211C22×D73—41no—
212C22×D73—41no—
213C22×D73—41no—
214C22×D73—41no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73C2×D812yes—
233C2×C4×C73C23×C411yescharacteristic
234C2×C4×C73C22×Q811yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—22no—
244C2×Dic73—22no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73C4×D412yes—
253C2×Dic73C4×D412yes—
254C2×Dic73C22⋊Q812yes—
255C2×Dic73C22⋊Q812yes—
256C2×Dic73C2×D812yes—
257C2×Dic73C2×D812yes—
258C2×Dic73#13512yes—
259C2×Dic73#13712yes—
260D292—22no—
261D292—22no—
262D292—22no—
263D292C4×Q814yescomplemented
264C4×D73—48no—
265C4×D73—48no—
266C4×D73—48no—
267Dic146—48no—
268C8×C73—22no—
269C8×C73—41no—
270#10596831—22no—
271#10596831—41no—
272C22×C73C2×C4×D411yescharacteristic
273C22×C73C2×C22⋊Q811yescharacteristic
274C22×C73#52511yescharacteristic
275C22×C73C22×D811yescharacteristic
276C22×C73C2×#13511yescharacteristic
277C22×C73C2×#13711yescharacteristic
278C22×C73#57511yescharacteristic
279C2×D73—24no—
280C2×D73—42no—
281C2×D73—42no—
282C2×D73—42no—
283C2×D73—42no—
284C2×D73—42no—
285C2×D73—42no—
286C4×C73—22no—
287C4×C73—41no—
288C4×C73—42no—
289C4×C73—42no—
290C4×C73—42no—
291C4×C73—81no—
292C4×C73C2×C4×Q811yescharacteristic
293C4×C73C4×#4111yescharacteristic
294C4×C73#52911yescharacteristic
295C4×C73#53311yescharacteristic
296Dic73—22no—
297Dic73—22no—
298Dic73—42no—
299Dic73—42no—
300Dic73—81no—
301Dic73C4×D814yes—
302Dic73#47414yescomplemented
303C4×#474—731nomaximal, Sylow
304C2×C73C4×C22⋊Q811yescharacteristic
305C2×C73C2×C4×D811yescharacteristic
306C2×C73C4×#13711yescharacteristic
307C2×C73C2×#47411yescharacteristic
308C2×C73#307611yescharacteristic
309C2×C73#310011yescharacteristic
310C2×C73#310711yescharacteristic
311D73—44no—
312C2×#474—731no—
313C4×C4⋊Q8—731no—
314C42×D4—731no—
315#1988—731no—
316#1947—731no—
317#1918—731no—
318#1918D7311yescharacteristic, complemented
319#1862—731no—
320#1862—731no—
321#1802—731no—
322C4×#107—731no—
323C4×#107—731no—
324C4×C4⋊C8—731no—
325C4×#102—731no—
326C4×#102—731no—
327C73C4×#47411yescharacteristic, complemented, Sylow
328C2×C4⋊Q8—731no—
329C2×C4×Q8—1461no—
330C2×C4×D4—731no—
331C2×C4×D4—731no—
332C2×C4×D4—731no—
333C22×C42—1461no—
334#474—738no—
335C2×#107—731no—
336C2×#107—731no—
337C2×#107—731no—
338C2×#107C2×D7311yescharacteristic
339C2×C4⋊C8—731no—
340C2×C4⋊C8—731no—
341C2×#102—731no—
342C2×#102—731no—
343C2×#102—731no—
344C2×#102—731no—
345C2×C4×C8—731no—
346C2×C4×C8—731no—
347#391—1461no—
348#389—731no—
349#389—1461no—
350#389C2×D7311yescharacteristic
351#384—731no—
352C4×C4⋊C4—731no—
353C4×C4⋊C4—731no—
354C4×C4⋊C4—731no—
355C4×C4⋊C4—731no—
356C4×C4⋊C4—1461no—
357C4×C4⋊C4C2×D7311yescharacteristic
358C4×C22⋊C4—1461no—
359C4×C22⋊C4—1461no—
360C4×C22⋊C4—1461no—
361C43—731no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340C2×D7311yescharacteristic
368#340C2×D7311yescharacteristic
369#340C2×D7311yescharacteristic
370#336—731no—
371#336C2×D7311yescharacteristic
372C22×Q8—1461no—
373C22×D4—731no—
374C23×C4—1461no—
375C23×C4—1461no—
376C23×C4—1461no—
377C22×C8—731no—
378C22×C8—731no—
379C22×C8C22×D7311yescharacteristic
380C22×C8C22×D7311yescharacteristic
381C4⋊Q8—734no—
382C4⋊Q8—1462no—
383C4×Q8—2924no—
384C4×D4—734no—
385C4×D4—734no—
386C4×D4—1462no—
387C4×D4—1462no—
388C4×D4—1462no—
389C4×D4—1462no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4—1461no—
398C2×C4⋊C4C22×D7311yescharacteristic
399C2×C4⋊C4C22×D7311yescharacteristic
400C2×C4⋊C4C22×D7311yescharacteristic
401C2×C4⋊C4C22×D7311yescharacteristic
402C2×C22⋊C4—1461no—
403C2×C22⋊C4—1461no—
404C2×C22⋊C4—1461no—
405C2×C42—21no—
406C2×C42—2924no—
407C2×C42—731no—
408C2×C42—731no—
409C2×C42—731no—
410C2×C42—731no—
411C2×C42—731no—
412C2×C42—731no—
413C2×C42—1461no—
414C2×C42—1461no—
415C2×C42—1461no—
416C2×C42—1461no—
417C2×C42C22×D7311yescharacteristic
418#107—734no—
419#107—1462no—
420#107—1462no—
421#107C4×D7314yescomplemented
422C4⋊C8—734no—
423C4⋊C8—1462no—
424#102—734no—
425#102—734no—
426#102—1462no—
427#102—1462no—
428C4×C8—1462no—
429C4×C8—1462no—
430#95—21no—
431#95—21no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437C24—1461no—
438C2×Q8—2922no—
439C2×Q8—2922no—
440C2×Q8—1464no—
441C2×D4—732no—
442C2×D4—732no—
443C2×D4—732no—
444C2×D4—1461no—
445C2×D4—1461no—
446C2×D4—1461no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—2922no—
452C22×C4—2922no—
453C22×C4—2922no—
454C22×C4—2922no—
455C22×C4—2922no—
456C22×C4—2922no—
457C22×C4—731no—
458C22×C4—731no—
459C22×C4—731no—
460C22×C4—731no—
461C22×C4—1464no—
462C22×C4—1464no—
463C22×C4—1464no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4D4×D7311yescharacteristic
468C22×C4D4×D7311yescharacteristic
469C22×C4C23×D7311yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—732no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8C2×C4×D7312yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—2922no—
489C4⋊C4—2922no—
490C4⋊C4—2922no—
491C4⋊C4—732no—
492C4⋊C4—732no—
493C4⋊C4—1464no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4C2×C4×D7312yes—
499C4⋊C4C2×C4×D7312yes—
500C4⋊C4Q8×D7312yes—
501C22⋊C4—1464no—
502C22⋊C4—1464no—
503C22⋊C4—1464no—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—2921no—
508C42—2922no—
509C42—2922no—
510C42—2922no—
511C42—2922no—
512C42—734no—
513C42—732no—
514C42—732no—
515C42—732no—
516C42—732no—
517C42—1462no—
518C42—1462no—
519C23—2921no—
520C23—2921no—
521C23—2921no—
522C23—2921no—
523C23—1462no—
524C23—1462no—
525C23—1462no—
526C23C2×D4×D7311yescharacteristic
527Q8—2928no—
528D4—734no—
529D4—1462no—
530D4—1462no—
531D4—1462no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—2928no—
550C2×C4—2928no—
551C2×C4—2928no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—732no—
560C2×C4—732no—
561C2×C4—732no—
562C2×C4—732no—
563C2×C4—732no—
564C2×C4—732no—
565C2×C4—732no—
566C2×C4—732no—
567C2×C4—1462no—
568C2×C4—1462no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4D8×D7312yes—
574C2×C4C22×C4×D7311yescharacteristic
575C2×C4C2×Q8×D7311yescharacteristic
576C2×C4#41×D7311yescharacteristic
577C2×C4#41×D7311yescharacteristic
578C2×C4#41×D7311yescharacteristic
579C2×C4#41×D7311yescharacteristic
580C8—22no—
581C8—41no—
582C8—2921no—
583C8—1462no—
584C22—2922no—
585C22—2922no—
586C22—2922no—
587C22—2922no—
588C22—2922no—
589C22—2922no—
590C22—1464no—
591C22C4×D4×D7311yescharacteristic
592C22C22⋊Q8×D7311yescharacteristic
593C22#1798909511yescharacteristic
594C22C2×D8×D7311yescharacteristic
595C22#135×D7311yescharacteristic
596C22#137×D7311yescharacteristic
597C22#1798938611yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—81no—
604C4—2922no—
605C4—2922no—
606C4—5841no—
607C4—734no—
608C4—734no—
609C4—1462no—
610C4—1462no—
611C4C4×Q8×D7311yescharacteristic
612C4#1798905511yescharacteristic
613C4#1798912011yescharacteristic
614C4#1798914311yescharacteristic
615C2—2924no—
616C2#2261246511yescharacteristic
617C2C4×D8×D7311yescharacteristic
618C2#2261874711yescharacteristic
619C2#474×D7311yescharacteristic
620C2#2261971811yescharacteristic
621C2#2261992811yescharacteristic
622C2#2261999711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1918C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1918D73
hypocenterC73C4×#474
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
2. #22612502C2semidirect
4. #22610881C2semidirect
5. #22610688C2semidirect
6. #22610502C2semidirect
10. #107×Dic73C2semidirect
11. #22609823C2semidirect
12. #22609779C2semidirect
15. C73×#1918C2semidirect
22. #17988866C4semidirect
68. #14556590C2×C4semidirect
75. #14556553Q8semidirect
99. #14556524C2×C4semidirect
101. #14556520C2×C4semidirect
122. #107×C73C2×C4semidirect
263. D292C4×Q8semidirect
302. Dic73#474semidirect
318. #1918D73semidirect
327. C73C4×#474semidirect
421. #107C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179888662
#226125022
C4×#145565532
#226108812
#226106882
#226105022
#226101812
#226101772
#226099302
#107×Dic732
#226098232
#226097792
#226097452
C73×#19182
C4×#47473