Finite Group Databasebeta

← #33092548

Subgroups — #33092548

Order 18688.

1014 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988827C211yesmaximal, characteristic
2C2×#17988692C211yesmaximal, characteristic, complemented
3C2×#97×D73C211yesmaximal, characteristic
4C4×#14556571C211yesmaximal, characteristic
5C4×#14556567C211yesmaximal, characteristic
6#22610799C211yesmaximal, characteristic, complemented
7#22610668C211yesmaximal, characteristic, complemented
8#22610623C211yesmaximal, characteristic, complemented
9#22610619C211yesmaximal, characteristic, complemented
10#22610353C211yesmaximal, characteristic
11#22610351C211yesmaximal, characteristic
12#22609955C211yesmaximal, characteristic
13#22609829C211yesmaximal, characteristic, complemented
14#22609762C211yesmaximal, characteristic, complemented
15C73×#1763C211yesmaximal, characteristic, complemented
16C22×#11924018C2211yescharacteristic
17C22×C8×D73C2211yescharacteristic
18C22×C8×D73C2211yescharacteristic
19C2×#14556571C2211yescharacteristic
20C2×#14556567C2211yescharacteristic
21C2×C4×Dic146C2211yescharacteristic
22C2×C42×D73C2211yescharacteristic
23#17988827C418yescomplemented
24#17988692—24no—
25C2×#14556524C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×#14556522C2211yescharacteristic
28C2×#14556516C2211yescharacteristic
29C2×#14556516C2211yescharacteristic
30C2×#14556515C2211yescharacteristic
31C2×#14556514C2211yescharacteristic
32C2×#14556502C2211yescharacteristic
33C2×#14556501C2211yescharacteristic
34C2×#14556501C2211yescharacteristic
35#97×D73—24no—
36#17988494C2211yescharacteristic
37#17988492C2211yescharacteristic
38#17988486C2211yescharacteristic
39#17988483C2211yescharacteristic
40#17988472—21no—
41#17988468—21no—
42#17988461—21no—
43#17988458—21no—
44#17988444C2211yescharacteristic
45#17988444C2211yescharacteristic
46C4×#11924007—21no—
47C4×C4⋊Dic73—21no—
48C4×C4⋊Dic73C2211yescharacteristic
49C4×#11924005—21no—
50C4×#11924005—21no—
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988349C2211yescharacteristic
55C2×C4×Q8×C73C2211yescharacteristic
56C2×#103×C73C2211yescharacteristic
57C2×#103×C73C2211yescharacteristic
58C2×#97×C73C2211yescharacteristic
59C4×C4⋊C4×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#340×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C22×C4×D73D411yescharacteristic
64C22×C4×D73D411yescharacteristic
65C22×C4×D73C2311yescharacteristic
66C2×#11924018—22no—
67C2×#11924018—22no—
68C2×#11924018C2×C414yescomplemented
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C2×C8×D73—22no—
72C2×C8×D73—22no—
73C2×C8×D73—22no—
74C2×C8×D73—22no—
75C2×C8×D73—22no—
76C2×C8×D73C2×C414yes—
77#14556571—22no—
78#14556571C2×C414yes—
79#14556567—22no—
80#14556567C2×C414yes—
81C4×Dic146—22no—
82C4×Dic146—22no—
83C2×C4⋊Dic73—21no—
84C2×C4⋊Dic73—21no—
85C2×C4⋊Dic73C2311yescharacteristic
86C2×C4⋊Dic73C2311yescharacteristic
87C2×C4⋊Dic73C2311yescharacteristic
88C2×#11924007—21no—
89C2×#11924007—21no—
90C2×#11924007—21no—
91C2×#11924005—21no—
92C2×#11924005—21no—
93C2×#11924005—21no—
94C2×#11924005—21no—
95C2×C4×Dic73—21no—
96C2×C4×Dic73—21no—
97C2×C4×Dic73D411yescharacteristic
98C2×C4×Dic73D411yescharacteristic
99C2×C4×Dic73C2311yescharacteristic
100C42×D73—22no—
101C42×D73—22no—
102#14556524C2×C414yescomplemented
103#14556522—22no—
104#14556522—22no—
105#14556516—22no—
106#14556516C2×C414yescomplemented
107#14556515—22no—
108#14556514—22no—
109#14556502D414yes—
110#14556501—22no—
111#14556501—22no—
112#14556498—21no—
113#14556498—21no—
114#14556498—21no—
115#14556498—21no—
116#14556498—21no—
117#14556498—21no—
118#14556498—21no—
119#14556498—21no—
120#14556498—21no—
121C22×Q8×C73C2311yescharacteristic
122C22×C8×C73C2311yescharacteristic
123C22×C8×C73C2311yescharacteristic
124C4×Q8×C73—22no—
125C4×Q8×C73—22no—
126C2×C4⋊C4×C73—21no—
127C2×C4⋊C4×C73C2311yescharacteristic
128C2×C4⋊C4×C73C2311yescharacteristic
129C2×C4⋊C4×C73C2311yescharacteristic
130C2×C42×C73—21no—
131C2×C42×C73—21no—
132C2×C42×C73C2311yescharacteristic
133#103×C73—22no—
134#103×C73C2×C414yescomplemented
135#97×C73—22no—
136#95×C73—21no—
137C23×D73C2×D411yescharacteristic
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146—21no—
143C2×Dic146C22×C412yes—
144C22×Dic73—21no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147C22×Dic73—21no—
148C22×Dic73C2×D411yescharacteristic
149C22×Dic73C2×D411yescharacteristic
150C22×Dic73C2×D411yescharacteristic
151C2×C4×D73—21no—
152C2×C4×D73—21no—
153C2×C4×D73—21no—
154C2×C4×D73—21no—
155C2×C4×D73—22no—
156C2×C4×D73—22no—
157C2×C4×D73—22no—
158C2×C4×D73—22no—
159C2×C4×D73—41no—
160C2×C4×D73—41no—
161C2×C4×D73C22×C412yes—
162C2×C4×D73#4112yes—
163#11924018—28no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831C22×C412yes—
167C2×#10596831C2×D412yes—
168C2×#10596831C2×D412yes—
169C2×#10596831#4112yes—
170C8×D73—28no—
171C8×D73—28no—
172C4⋊Dic73—21no—
173C4⋊Dic73—24no—
174C4⋊Dic73—21no—
175C4⋊Dic73—21no—
176C4⋊Dic73—21no—
177C4⋊Dic73—21no—
178C4⋊Dic73—42no—
179C4⋊Dic73C22×C412yes—
180#11924007—24no—
181#11924007—42no—
182#11924007—42no—
183#11924005—24no—
184#11924005—24no—
185#11924005—42no—
186#11924005—42no—
187C4×Dic73—22no—
188C4×Dic73—22no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—42no—
192C4×Dic73—42no—
193C2×Q8×C73—21no—
194C2×Q8×C73—21no—
195C2×Q8×C73—21no—
196C2×Q8×C73—21no—
197C2×Q8×C73—21no—
198C2×Q8×C73C22×C412yes—
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73C2×D411yescharacteristic
204C22×C4×C73C2×D411yescharacteristic
205C22×C4×C73C2411yescharacteristic
206C2×C8×C73—21no—
207C2×C8×C73—21no—
208C2×C8×C73—21no—
209C2×C8×C73—21no—
210C2×C8×C73C22×C412yes—
211C2×C8×C73#4112yes—
212C4⋊C4×C73—21no—
213C4⋊C4×C73—21no—
214C4⋊C4×C73—21no—
215C4⋊C4×C73—21no—
216C4⋊C4×C73—21no—
217C4⋊C4×C73—42no—
218C4⋊C4×C73C22×C412yes—
219C42×C73—42no—
220C42×C73—42no—
221C42×C73C2×D412yes—
222C42×C73#4112yes—
223C23×C73C22×D411yescharacteristic
224C22×D73—21no—
225C22×D73—21no—
226C22×D73—21no—
227C22×D73—21no—
228C22×D73C4×D412yes—
229C22×D73#13712yes—
230C22×D73#13712yes—
231Q8×C73—22no—
232Q8×C73—22no—
233Q8×C73—41no—
234Q8×C73—41no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—22no—
240C2×C4×C73—22no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
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#12112yes—
252C2×C4×C73#12112yes—
253C2×C4×C73C23×C411yescharacteristic
254C2×C4×C73C22×D411yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73C2×#4111yescharacteristic
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—22no—
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×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73C4×D412yes—
280C4×D73—22no—
281C4×D73—22no—
282C4×D73—22no—
283C4×D73—22no—
284C4×D73—44no—
285C4×D73—44no—
286Dic146—22no—
287Dic146—22no—
288Dic146—41no—
289Dic146—41no—
290C8×C73—22no—
291C8×C73—22no—
292#10596831—22no—
293#10596831C4×D414yes—
294C22×C73C2×C4×D411yescharacteristic
295C22×C73C2×#12111yescharacteristic
296C22×C73#53211yescharacteristic
297C22×C73C2×#13711yescharacteristic
298C22×C73C2×#13711yescharacteristic
299C22×C73#57511yescharacteristic
300C22×C73#57511yescharacteristic
301C2×D73—22no—
302C2×D73—22no—
303C2×D73—22no—
304C2×D73—22no—
305C2×D73—22no—
306C2×D73—22no—
307C2×D73#44114yescomplemented
308C4×C73—22no—
309C4×C73—22no—
310C4×C73—42no—
311C4×C73—42no—
312C4×C73—81no—
313C4×C73—81no—
314C4×C73C2×C4×D411yescharacteristic
315C4×C73C4×#4111yescharacteristic
316C4×C73#52511yescharacteristic
317C4×C73#52911yescharacteristic
318Dic73—22no—
319Dic73—42no—
320Dic73—41no—
321Dic73—41no—
322Dic73—42no—
323Dic73—81no—
324Dic73—81no—
325#21527—731nomaximal, Sylow
326C2×C73C4×#12111yescharacteristic
327C2×C73C2×#44111yescharacteristic
328C2×C73#294311yescharacteristic
329C2×C73#305611yescharacteristic
330C2×C73#305611yescharacteristic
331C2×C73#318111yescharacteristic
332C2×C73#318111yescharacteristic
333D73—24no—
334C2×#467—731no—
335C2×#441—731no—
336C22×#97—731no—
337C4×C22⋊Q8—731no—
338C4×C22⋊Q8—731no—
339#1971—731no—
340#1943—731no—
341#1935—731no—
342#1935—731no—
343#1892—731no—
344#1892—731no—
345#1807—731no—
346#1776—731no—
347#1763—731no—
348#1763D7311yescharacteristic, complemented
349C73#2152711yescharacteristic, complemented, Sylow
350C22×Q16—731no—
351C23×C8—731no—
352C23×C8—731no—
353C2×C22⋊Q8—731no—
354C2×C22⋊Q8—731no—
355C2×C4×Q8—731no—
356C2×C4×Q8C2×D7311yescharacteristic
357C22×C42—731no—
358#467—738no—
359#441—1464no—
360C2×#107—731no—
361C2×#106—731no—
362C2×#106—731no—
363C2×#103—731no—
364C2×#103—731no—
365C2×#103—731no—
366C2×#103—731no—
367C2×#103C2×D7311yescharacteristic
368C2×#103C2×D7311yescharacteristic
369C2×#97—731no—
370C2×#97—731no—
371C2×#97—731no—
372C2×#97—1464no—
373C2×#97C2×D7311yescharacteristic
374#391—731no—
375#391—731no—
376#389—731no—
377#389—731no—
378#387—1461no—
379#387—1461no—
380#385—1461no—
381#385—1461no—
382#380—731no—
383#380—731no—
384C4×C4⋊C4—731no—
385C4×C4⋊C4—1461no—
386C4×C4⋊C4—1461no—
387C4×C4⋊C4C2×D7311yescharacteristic
388C4×C22⋊C4—1461no—
389C4×C22⋊C4—1461no—
390#340—731no—
391#340—731no—
392#340—731no—
393#340—731no—
394#340C2×D7311yescharacteristic
395#340C2×D7311yescharacteristic
396C22×Q8—731no—
397C22×Q8C22×D7311yescharacteristic
398C23×C4—731no—
399C23×C4—731no—
400C23×C4—731no—
401C2×Q16—734no—
402C2×Q16—1462no—
403C2×Q16—1462no—
404C22×C8—731no—
405C22×C8—731no—
406C22×C8—734no—
407C22×C8—1462no—
408C22×C8—1462no—
409C22×C8—1462no—
410C22×C8—1462no—
411C22×C8—1462no—
412C22×C8C22×D7311yescharacteristic
413C22×C8C22×D7311yescharacteristic
414C22⋊Q8—734no—
415C22⋊Q8—734no—
416C22⋊Q8—1462no—
417C22⋊Q8—1462no—
418C4×Q8—22no—
419C4×Q8—22no—
420C4×Q8—1462no—
421C4×Q8—1462no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4—1461no—
431C2×C4⋊C4C22×D7311yescharacteristic
432C2×C4⋊C4C22×D7311yescharacteristic
433C2×C4⋊C4C22×D7311yescharacteristic
434C2×C22⋊C4—1461no—
435C2×C22⋊C4—1461no—
436C2×C22⋊C4—1461no—
437C2×C22⋊C4—1461no—
438C2×C42—21no—
439C2×C42—21no—
440C2×C42—731no—
441C2×C42—731no—
442C2×C42—731no—
443C2×C42—1461no—
444C2×C42—1461no—
445C2×C42—1462no—
446C2×C42—1462no—
447C2×C42C22×D7311yescharacteristic
448#107—734no—
449#106—1462no—
450#106—1462no—
451#103—22no—
452#103—734no—
453#103—1462no—
454#103—1462no—
455#103—1462no—
456#103C4×D7314yescomplemented
457#97—22no—
458#97—734no—
459#97—1462no—
460#97—1462no—
461#95—21no—
462#95—1461no—
463#95—1461no—
464#95—1461no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470#95—1461no—
471C24—731no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—732no—
478C2×Q8—1461no—
479C2×Q8—1461no—
480C2×Q8—1461no—
481C2×Q8—1461no—
482C2×Q8—1461no—
483C2×Q8C2×C4×D7312yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—2921no—
489C22×C4—2921no—
490C22×C4—731no—
491C22×C4—732no—
492C22×C4—732no—
493C22×C4—731no—
494C22×C4—731no—
495C22×C4—1462no—
496C22×C4—1462no—
497C22×C4—1461no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1462no—
502C22×C4—1462no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4—1461no—
506C22×C4—1461no—
507C22×C4D4×D7311yescharacteristic
508C22×C4D4×D7311yescharacteristic
509C22×C4C23×D7311yescharacteristic
510Q16—1468no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—732no—
516C2×C8—732no—
517C2×C8—732no—
518C2×C8—732no—
519C2×C8—1468no—
520C2×C8—1468no—
521C2×C8—1461no—
522C2×C8—1461no—
523C2×C8C2×C4×D7312yes—
524C2×C8#1192402812yes—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—42no—
531C4⋊C4—2922no—
532C4⋊C4—2922no—
533C4⋊C4—2922no—
534C4⋊C4—732no—
535C4⋊C4—1464no—
536C4⋊C4—1464no—
537C4⋊C4—1461no—
538C4⋊C4—1461no—
539C4⋊C4—1461no—
540C4⋊C4—1461no—
541C4⋊C4—1461no—
542C4⋊C4C2×C4×D7312yes—
543C22⋊C4—2922no—
544C22⋊C4—2922no—
545C22⋊C4—1464no—
546C22⋊C4—1464no—
547C42—42no—
548C42—42no—
549C42—2922no—
550C42—2922no—
551C42—1462no—
552C42—1462no—
553C42—1461no—
554C42—1461no—
555C42D4×D7312yes—
556C42#1192402812yes—
557C23—732no—
558C23—732no—
559C23—732no—
560C23—1461no—
561C23—1461no—
562C23—1461no—
563C23—1461no—
564C23C2×D4×D7311yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—2921no—
570Q8—2921no—
571Q8—1462no—
572Q8—1462no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—2924no—
590C2×C4—2924no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—732no—
602C2×C4—1461no—
603C2×C4—1461no—
604C2×C4—1461no—
605C2×C4—1461no—
606C2×C4—1462no—
607C2×C4—1462no—
608C2×C4—1461no—
609C2×C4—1461no—
610C2×C4—1461no—
611C2×C4—1461no—
612C2×C4—1462no—
613C2×C4—1462no—
614C2×C4—1462no—
615C2×C4—1462no—
616C2×C4#1455656412yes—
617C2×C4#1455656412yes—
618C2×C4C22×C4×D7311yescharacteristic
619C2×C4C2×D4×D7311yescharacteristic
620C2×C4#41×D7311yescharacteristic
621C2×C4#41×D7311yescharacteristic
622C2×C4C2×#1192402811yescharacteristic
623C2×C4C2×#1192402811yescharacteristic
624C8—22no—
625C8—22no—
626C8—734no—
627C8—1462no—
628C22—734no—
629C22—1462no—
630C22—1462no—
631C22—1462no—
632C22—1462no—
633C22—1462no—
634C22—1462no—
635C22C4×D4×D7311yescharacteristic
636C22C2×#1455656411yescharacteristic
637C22#1798913411yescharacteristic
638C22#137×D7311yescharacteristic
639C22#137×D7311yescharacteristic
640C22#1798938311yescharacteristic
641C22#1798938611yescharacteristic
642C4—22no—
643C4—22no—
644C4—42no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4—2922no—
649C4—2921no—
650C4—2921no—
651C4—2922no—
652C4—5841no—
653C4—5841no—
654C4—1462no—
655C4C4×D4×D7311yescharacteristic
656C4C4×#1192402811yescharacteristic
657C4#1798909611yescharacteristic
658C4#1798911511yescharacteristic
659C2—1464no—
660C2C4×#1455656411yescharacteristic
661C2#441×D7311yescharacteristic
662C2#2261872611yescharacteristic
663C2#2261955911yescharacteristic
664C2#2261956411yescharacteristic
665C2#2262063611yescharacteristic
666C2#2262063811yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1763C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1763D73
hypocenterC73#21527
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
2. C2×#17988692C2semidirect
6. #22610799C2semidirect
7. #22610668C2semidirect
8. #22610623C2semidirect
9. #22610619C2semidirect
13. #22609829C2semidirect
14. #22609762C2semidirect
15. C73×#1763C2semidirect
23. #17988827C4semidirect
68. C2×#11924018C2×C4semidirect
102. #14556524C2×C4semidirect
106. #14556516C2×C4semidirect
134. #103×C73C2×C4semidirect
307. C2×D73#441semidirect
348. #1763D73semidirect
349. C73#21527semidirect
456. #103C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179888272
C2×#179886922
C2×#97×D732
C4×#145565712
C4×#145565672
#226107992
#226106682
#226106232
#226106192
#226103532
#226103512
#226099552
#226098292
#226097622
C73×#17632
#2152773