Finite Group Databasebeta

← #33094013

Subgroups — #33094013

Order 18688.

1022 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988932C211yesmaximal, characteristic
2C4×#14556590C211yesmaximal, characteristic, complemented
3C4×#14556587C211yesmaximal, characteristic
4#22610890C211yesmaximal, characteristic, complemented
5#22610735C211yesmaximal, characteristic
6#22610630C211yesmaximal, characteristic, complemented
7#22610629C211yesmaximal, characteristic, complemented
8#22610626C211yesmaximal, characteristic, complemented
9#22610625C211yesmaximal, characteristic, complemented
10#22610057C211yesmaximal, characteristic, complemented
11#22609759C211yesmaximal, characteristic, complemented
12#22609757C211yesmaximal, characteristic, complemented
13#22609756C211yesmaximal, characteristic, complemented
14#97×Dic73C211yesmaximal, characteristic, complemented
15C73×#1762C211yesmaximal, characteristic, complemented
16C2×#14556590C2211yescharacteristic
17C2×#14556587C2211yescharacteristic
18C2×C4×Dic146—21no—
19C2×C4×D292C2211yescharacteristic
20C2×C4×C22⋊D73—21no—
21C2×C4×C22⋊D73—21no—
22#17988932C418yescomplemented
23C2×#14556513C2211yescharacteristic
24C2×#14556513C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556511C2211yescharacteristic
28C2×#14556511C2211yescharacteristic
29C2×#14556502C2211yescharacteristic
30C2×#14556501C2211yescharacteristic
31C2×#14556501C2211yescharacteristic
32C2×C8×Dic73C2211yescharacteristic
33C2×C8×Dic73C2211yescharacteristic
34#17988491—21no—
35#17988490C2211yescharacteristic
36#17988488C2211yescharacteristic
37#17988487—21no—
38#17988487—21no—
39#17988486C2211yescharacteristic
40#17988484—21no—
41#17988483C2211yescharacteristic
42#17988456C2211yescharacteristic
43#17988456C2211yescharacteristic
44C4×#11924007—21no—
45C4×C4⋊Dic73C2211yescharacteristic
46C42×Dic73C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56C2×#97×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×Dic146—21no—
61C22×D292C2311yescharacteristic
62C22×C22⋊D73—21no—
63C22×C22⋊D73—21no—
64C22×C4×D73—21no—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556590—22no—
68#14556590C2×C414yescomplemented
69#14556587—22no—
70#14556587C2×C414yes—
71C4×Dic146—44no—
72C4×D292—22no—
73C4×D292—22no—
74C4×C22⋊D73—44no—
75C4×C22⋊D73—44no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×C22⋊Dic73—21no—
87C2×C22⋊Dic73—21no—
88C2×#11924005—21no—
89C2×#11924005—21no—
90C2×#11924005—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73D411yescharacteristic
93C2×C4×Dic73D411yescharacteristic
94C2×C4×Dic73D411yescharacteristic
95C2×C4×Dic73D411yescharacteristic
96C2×C4×Dic73C2311yescharacteristic
97C2×C4×Dic73C2311yescharacteristic
98#14556513—22no—
99#14556513C2×C414yescomplemented
100#14556512—22no—
101#14556512C2×C414yescomplemented
102#14556511—22no—
103#14556511C2×C414yescomplemented
104#14556502—22no—
105#14556501—22no—
106#14556501—22no—
107C8×Dic73—22no—
108C8×Dic73C2×C414yescomplemented
109#14556498—21no—
110#14556498—21no—
111#14556498—21no—
112C22×D4×C73C2311yescharacteristic
113C23×C4×C73—21no—
114C22×C8×C73C2311yescharacteristic
115C22×C8×C73C2311yescharacteristic
116C4×D4×C73—22no—
117C4×D4×C73—22no—
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C4⋊C4×C73C2311yescharacteristic
121C2×C22⋊C4×C73—21no—
122C2×C42×C73—21no—
123C2×C42×C73C2311yescharacteristic
124#102×C73—22no—
125#102×C73C2×C414yescomplemented
126#97×C73—22no—
127#95×C73—21no—
128C23×D73—21no—
129C2×Dic146—24no—
130C2×Dic146—42no—
131C2×Dic146—42no—
132C2×D292—21no—
133C2×D292—21no—
134C2×D292—21no—
135C2×D292—21no—
136C2×D292—21no—
137C2×D292C22×C412yes—
138C2×C22⋊D73—24no—
139C2×C22⋊D73—24no—
140C2×C22⋊D73—42no—
141C2×C22⋊D73—42no—
142C2×C22⋊D73—42no—
143C2×C22⋊D73—42no—
144C22×Dic73—21no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147C22×Dic73C2×D411yescharacteristic
148C22×Dic73C2×D411yescharacteristic
149C22×Dic73C2×D411yescharacteristic
150C22×Dic73C2×D411yescharacteristic
151C2×C4×D73—42no—
152C2×C4×D73—42no—
153C2×C4×D73—42no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831C22×C412yes—
159C2×#10596831#4112yes—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—42no—
166C4⋊Dic73C22×C412yes—
167#11924007—24no—
168#11924007—42no—
169#11924007—42no—
170#11924007—42no—
171#11924007—42no—
172#11924007—42no—
173C22⋊Dic73—42no—
174C22⋊Dic73—42no—
175#11924005—42no—
176#11924005—42no—
177#11924005—42no—
178C4×Dic73—21no—
179C4×Dic73—21no—
180C4×Dic73—22no—
181C4×Dic73—22no—
182C4×Dic73—22no—
183C4×Dic73—22no—
184C4×Dic73—42no—
185C4×Dic73C22×C412yes—
186C4×Dic73#4112yes—
187C24×C73—21no—
188C2×D4×C73—21no—
189C2×D4×C73—21no—
190C2×D4×C73—21no—
191C2×D4×C73—21no—
192C2×D4×C73—21no—
193C2×D4×C73C22×C412yes—
194C22×C4×C73—21no—
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73—42no—
198C22×C4×C73—42no—
199C22×C4×C73—42no—
200C22×C4×C73C2×D411yescharacteristic
201C22×C4×C73C2×D411yescharacteristic
202C22×C4×C73C2411yescharacteristic
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73—21no—
206C2×C8×C73—21no—
207C2×C8×C73C22×C412yes—
208C2×C8×C73#4112yes—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—21no—
211C4⋊C4×C73—21no—
212C4⋊C4×C73—21no—
213C4⋊C4×C73—21no—
214C4⋊C4×C73C22×C412yes—
215C22⋊C4×C73—42no—
216C42×C73—42no—
217C42×C73#4112yes—
218C42×C73#4112yes—
219C23×C73—22no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73—41no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73—41no—
226C23×C73C22×D411yescharacteristic
227C22×D73—22no—
228C22×D73—41no—
229C22×D73—41no—
230C22×D73—41no—
231C22×D73—41no—
232C22×D73—41no—
233C22×D73—41no—
234D4×C73—22no—
235D4×C73—22no—
236D4×C73—41no—
237D4×C73—41no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—22no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—84no—
252C2×C4×C73#12412yes—
253C2×C4×C73#12412yes—
254C2×C4×C73C23×C411yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73C2×#4111yescharacteristic
259C2×C4×C73C2×#4111yescharacteristic
260C22⋊D73—48no—
261C22⋊D73—48no—
262C2×Dic73—22no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—21no—
266C2×Dic73—21no—
267C2×Dic73—21no—
268C2×Dic73—21no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73C4×D412yes—
278C2×Dic73C4×D412yes—
279C2×Dic73AGL(1,ℤ/8)12yes—
280C2×Dic73AGL(1,ℤ/8)12yes—
281C2×Dic73AGL(1,ℤ/8)12yes—
282C2×Dic73#13712yes—
283D292—22no—
284D292—22no—
285D292—41no—
286D292—41no—
287C4×D73—84no—
288Dic146—48no—
289C8×C73—22no—
290C8×C73—22no—
291#10596831—22no—
292#10596831—22no—
293C22×C73—42no—
294C22×C73—42no—
295C22×C73—42no—
296C22×C73—81no—
297C22×C73—81no—
298C22×C73—81no—
299C22×C73—81no—
300C22×C73C2×C4×D411yescharacteristic
301C22×C73C2×#12411yescharacteristic
302C22×C73#53211yescharacteristic
303C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
304C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C73C2×#13711yescharacteristic
307C2×D73—42no—
308C2×D73—42no—
309C2×D73—42no—
310C2×D73—81no—
311C2×D73—81no—
312C2×D73—81no—
313C2×D73—81no—
314C4×C73—22no—
315C4×C73—22no—
316C4×C73—42no—
317C4×C73—81no—
318C4×C73—81no—
319C4×C73C4×#4111yescharacteristic
320C4×C73C4×#4111yescharacteristic
321C4×C73#52911yescharacteristic
322C4×C73#52911yescharacteristic
323Dic73—41no—
324Dic73—41no—
325Dic73—42no—
326Dic73—81no—
327Dic73—81no—
328Dic73#44014yes—
329Dic73#44214yes—
330#21646—731nomaximal, Sylow
331C2×C73—82no—
332C2×C73C4×#12411yescharacteristic
333C2×C73C2×#44011yescharacteristic
334C2×C73C2×#44211yescharacteristic
335C2×C73#313811yescharacteristic
336C2×C73#314111yescharacteristic
337C2×C73#318211yescharacteristic
338C2×C73#318411yescharacteristic
339D73—82no—
340C2×#486—731no—
341C4×C4⋊Q8—731no—
342C4×#127—731no—
343#1990—731no—
344#1957—731no—
345#1936—731no—
346#1936—731no—
347#1936—731no—
348#1936—731no—
349#1834—731no—
350#1762—731no—
351#1762—731no—
352#1762—731no—
353#1762D7311yescharacteristic, complemented
354C4×#97—731no—
355C73#2164611yescharacteristic, complemented, Sylow
356C2×C4⋊Q8—731no—
357C2×#127—731no—
358C2×C4×Q8—1461no—
359C2×C4×D4—731no—
360C2×C4×D4—1461no—
361C2×C4×D4—1461no—
362C2×C4×D4C2×D7311yescharacteristic
363#486—738no—
364C2×#102—731no—
365C2×#102—731no—
366C2×#102—731no—
367C2×#102—731no—
368C2×#102—731no—
369C2×#102—731no—
370C2×#102C2×D7311yescharacteristic
371C2×#102C2×D7311yescharacteristic
372C2×#97—731no—
373C2×#97—731no—
374C2×#97—731no—
375C2×#97C2×D7311yescharacteristic
376C2×C4×C8—731no—
377C2×C4×C8—731no—
378#391—1461no—
379#390—731no—
380#390—731no—
381#390—1461no—
382#390—1461no—
383#389—731no—
384#389—731no—
385#389—1461no—
386#384—731no—
387#384—731no—
388C4×C4⋊C4—731no—
389C4×C4⋊C4—1461no—
390C4×C4⋊C4C2×D7311yescharacteristic
391C43—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#340—731no—
396#340—731no—
397#340—731no—
398#340C2×D7311yescharacteristic
399#340C2×D7311yescharacteristic
400C22×Q8—1461no—
401C22×D4—731no—
402C22×D4—1461no—
403C22×D4—1461no—
404C22×D4C22×D7311yescharacteristic
405C23×C4—21no—
406C23×C4—1461no—
407C22×C8—731no—
408C22×C8—731no—
409C22×C8C22×D7311yescharacteristic
410C22×C8C22×D7311yescharacteristic
411C4⋊Q8—734no—
412C4⋊Q8—1462no—
413#127—734no—
414#127—1462no—
415C4×Q8—2924no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—2924no—
419C4×D4—2924no—
420C4×D4—1462no—
421C4×D4—1462no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—1461no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4—1461no—
431C2×C4⋊C4—1461no—
432C2×C4⋊C4C22×D7311yescharacteristic
433C2×C4⋊C4C22×D7311yescharacteristic
434C2×C4⋊C4C22×D7311yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1461no—
437C2×C22⋊C4—1461no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4—1461no—
440C2×C22⋊C4—1461no—
441C2×C42—21no—
442C2×C42—731no—
443C2×C42—731no—
444C2×C42—731no—
445C2×C42—731no—
446C2×C42—731no—
447C2×C42—731no—
448C2×C42—1461no—
449C2×C42C22×D7311yescharacteristic
450#102—22no—
451#102—734no—
452#102—734no—
453#102—734no—
454#102—1462no—
455#102—1462no—
456#102—1462no—
457#102C4×D7314yescomplemented
458#97—22no—
459#97—1462no—
460#97—1462no—
461#97—1462no—
462C4×C8—734no—
463C4×C8—1462no—
464#95—21no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468C24—21no—
469C24—1461no—
470C2×Q8—2922no—
471C2×Q8—2922no—
472C2×Q8—1464no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—2922no—
479C2×D4—2922no—
480C2×D4—2922no—
481C2×D4—2922no—
482C2×D4—732no—
483C2×D4—1464no—
484C2×D4—1464no—
485C2×D4—1461no—
486C2×D4—1461no—
487C2×D4—1461no—
488C2×D4—1461no—
489C2×D4—1461no—
490C2×D4C2×C4×D7312yes—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—2922no—
498C22×C4—2922no—
499C22×C4—2922no—
500C22×C4—731no—
501C22×C4—731no—
502C22×C4—731no—
503C22×C4—731no—
504C22×C4—1461no—
505C22×C4—1461no—
506C22×C4—1461no—
507C22×C4D4×D7311yescharacteristic
508C22×C4D4×D7311yescharacteristic
509C22×C4C23×D7311yescharacteristic
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—732no—
515C2×C8—732no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8C2×C4×D7312yes—
521C2×C8#1192402812yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—2922no—
528C4⋊C4—2922no—
529C4⋊C4—2922no—
530C4⋊C4—2922no—
531C4⋊C4—2922no—
532C4⋊C4—2922no—
533C4⋊C4—732no—
534C4⋊C4—1464no—
535C4⋊C4—1461no—
536C4⋊C4—1461no—
537C4⋊C4—1461no—
538C4⋊C4—1461no—
539C4⋊C4—1461no—
540C4⋊C4C2×C4×D7312yes—
541C22⋊C4—42no—
542C22⋊C4—2922no—
543C22⋊C4—2922no—
544C22⋊C4—2922no—
545C22⋊C4—2922no—
546C22⋊C4—2922no—
547C42—42no—
548C42—2922no—
549C42—732no—
550C42—732no—
551C42—1462no—
552C42—1462no—
553C42—1462no—
554C42—1462no—
555C42—1461no—
556C42—1461no—
557C42#1192402712yes—
558C42#1192402812yes—
559C23—22no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—2921no—
567C23—2921no—
568C23—2921no—
569C23—2921no—
570C23—2921no—
571C23—2921no—
572C23—1462no—
573C23C2×D4×D7311yescharacteristic
574Q8—2928no—
575D4—22no—
576D4—22no—
577D4—41no—
578D4—41no—
579D4—2928no—
580D4—2928no—
581D4—2921no—
582D4—2921no—
583D4—1462no—
584D4—1462no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—84no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—2921no—
603C2×C4—2921no—
604C2×C4—2921no—
605C2×C4—2921no—
606C2×C4—2921no—
607C2×C4—5844no—
608C2×C4—732no—
609C2×C4—732no—
610C2×C4—732no—
611C2×C4—732no—
612C2×C4—732no—
613C2×C4—732no—
614C2×C4—1461no—
615C2×C4—1461no—
616C2×C4—1461no—
617C2×C4—1462no—
618C2×C4—1461no—
619C2×C4—1461no—
620C2×C4—1461no—
621C2×C4#1455657712yes—
622C2×C4#1455657712yes—
623C2×C4C22×C4×D7311yescharacteristic
624C2×C4#41×D7311yescharacteristic
625C2×C4#41×D7311yescharacteristic
626C2×C4C2×#1192402711yescharacteristic
627C2×C4C2×#1192402811yescharacteristic
628C2×C4C2×#1192402811yescharacteristic
629C8—22no—
630C8—22no—
631C8—1462no—
632C8—1462no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—2922no—
641C22—2922no—
642C22—2922no—
643C22—5841no—
644C22—5841no—
645C22—5841no—
646C22—5841no—
647C22C4×D4×D7311yescharacteristic
648C22C2×#1455657711yescharacteristic
649C22#1798913411yescharacteristic
650C22AGL(1,ℤ/8)×D7311yescharacteristic
651C22AGL(1,ℤ/8)×D7311yescharacteristic
652C22AGL(1,ℤ/8)×D7311yescharacteristic
653C22#137×D7311yescharacteristic
654C4—22no—
655C4—22no—
656C4—42no—
657C4—81no—
658C4—81no—
659C4—2922no—
660C4—2921no—
661C4—2921no—
662C4—5841no—
663C4—5841no—
664C4—734no—
665C4—734no—
666C4C4×#1192402711yescharacteristic
667C4C4×#1192402811yescharacteristic
668C4#1798911511yescharacteristic
669C4#1798911711yescharacteristic
670C2—82no—
671C2—5842no—
672C2C4×#1455657711yescharacteristic
673C2#440×D7311yescharacteristic
674C2#442×D7311yescharacteristic
675C2#2262030311yescharacteristic
676C2#2262032211yescharacteristic
677C2#2262064611yescharacteristic
678C2#2262066611yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1762C2
socleC23×C73C22×D4
prime radicalC2×#14556587C22
cosocleC22×C4×C73C24
hypercenter#1762D73
hypocenterC73#21646
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C4×#14556590C2semidirect
4. #22610890C2semidirect
6. #22610630C2semidirect
7. #22610629C2semidirect
8. #22610626C2semidirect
9. #22610625C2semidirect
10. #22610057C2semidirect
11. #22609759C2semidirect
12. #22609757C2semidirect
13. #22609756C2semidirect
14. #97×Dic73C2semidirect
15. C73×#1762C2semidirect
22. #17988932C4semidirect
68. #14556590C2×C4semidirect
99. #14556513C2×C4semidirect
101. #14556512C2×C4semidirect
103. #14556511C2×C4semidirect
108. C8×Dic73C2×C4semidirect
125. #102×C73C2×C4semidirect
353. #1762D73semidirect
355. C73#21646semidirect
457. #102C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179889322
C4×#145565902
C4×#145565872
#226108902
#226107352
#226106302
#226106292
#226106262
#226106252
#226100572
#226097592
#226097572
#226097562
#97×Dic732
C73×#17622
#2164673