Finite Group Databasebeta

← #33104087

Subgroups — #33104087

Order 18688.

1010 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988650C211yesmaximal, characteristic
2C2×#17988623C211yesmaximal, characteristic, complemented
3C2×#106×D73C211yesmaximal, characteristic, complemented
4#22612861C211yesmaximal, characteristic
5#22612853C211yesmaximal, characteristic
6#22610700C211yesmaximal, characteristic, complemented
7#22610698C211yesmaximal, characteristic, complemented
8#22610132C211yesmaximal, characteristic, complemented
9#22610120C211yesmaximal, characteristic, complemented
10#22610117C211yesmaximal, characteristic, complemented
11#22610094C211yesmaximal, characteristic, complemented
12#22609955C211yesmaximal, characteristic, complemented
13#22609954C211yesmaximal, characteristic, complemented
14#22609819C211yesmaximal, characteristic, complemented
15C73×#1854C211yesmaximal, characteristic, complemented
16C23×#10596831C2211yescharacteristic
17C22×C8×D73C2211yescharacteristic
18C2×C4×C22⋊D73C2211yescharacteristic
19C2×#14556550C2211yescharacteristic
20C2×C4⋊C4×D73C2211yescharacteristic
21C2×C4⋊C4×D73C2211yescharacteristic
22#17988650C418yescomplemented
23#17988623—24no—
24C2×#14556525C2211yescharacteristic
25C2×#14556524C2211yescharacteristic
26C2×#14556524C2211yescharacteristic
27C2×#14556523C2211yescharacteristic
28C2×#14556522C2211yescharacteristic
29C2×#14556522C2211yescharacteristic
30C2×#14556522C2211yescharacteristic
31C2×#14556522C2211yescharacteristic
32#106×D73—24no—
33C2×#14556520C2211yescharacteristic
34C2×#14556503C2211yescharacteristic
35C2×C8×Dic73C2211yescharacteristic
36#17988481—21no—
37#17988475—21no—
38#17988471—21no—
39#17988469—21no—
40#17988463—21no—
41#17988463—21no—
42#17988452C2211yescharacteristic
43#17988452C2211yescharacteristic
44#17988447C2211yescharacteristic
45#17988444C2211yescharacteristic
46#17988444C2211yescharacteristic
47C4⋊C4×Dic73C2211yescharacteristic
48C4⋊C4×Dic73C2211yescharacteristic
49C4×#11924005—21no—
50#17988434—21no—
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988349C2211yescharacteristic
55C2×#117×C73C2211yescharacteristic
56C2×#107×C73C2211yescharacteristic
57C2×#106×C73C2211yescharacteristic
58C2×C22⋊C8×C73C2211yescharacteristic
59#380×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#340×C73C2211yescharacteristic
62C22×C22⋊D73D411yescharacteristic
63C22×C4×D73—21no—
64C22×C4×D73—21no—
65C22×C4×D73C2311yescharacteristic
66C22×#10596831—22no—
67C22×#10596831—22no—
68C22×#10596831C2×C414yescomplemented
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C2×C8×D73—22no—
72C2×C8×D73—22no—
73C2×C8×D73C2×C414yes—
74C4×C22⋊D73C2×C414yes—
75C4×C22⋊D73Q814yes—
76#14556550—22no—
77#14556550—41no—
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007D411yescharacteristic
85C4⋊C4×D73—22no—
86C4⋊C4×D73—22no—
87C4⋊C4×D73—41no—
88C4⋊C4×D73—41no—
89C2×C22⋊Dic73—21no—
90C2×C22⋊Dic73—21no—
91C2×C22⋊Dic73D411yescharacteristic
92C2×#11924005—21no—
93C2×#11924005—21no—
94C2×#11924005D411yescharacteristic
95C2×C4×Dic73—21no—
96C2×C4×Dic73—21no—
97C2×C4×Dic73—21no—
98C2×C4×Dic73C2311yescharacteristic
99#14556525—41no—
100#14556524—22no—
101#14556524—22no—
102#14556523—22no—
103#14556522—22no—
104#14556522—22no—
105#14556522—22no—
106#14556522D414yescomplemented
107#14556520C2×C414yescomplemented
108#14556503C2×C414yes—
109C8×Dic73C2×C414yescomplemented
110#14556498—21no—
111#14556498—21no—
112#14556498—21no—
113#14556498—21no—
114#14556498—21no—
115#14556498—21no—
116#14556498—21no—
117#14556498—21no—
118#14556498—21no—
119C23×C4×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121C22×C8×C73C2311yescharacteristic
122#117×C73—22no—
123#117×C73—41no—
124C2×C4⋊C4×C73C2311yescharacteristic
125C2×C4⋊C4×C73C2311yescharacteristic
126C2×C4⋊C4×C73C2311yescharacteristic
127C2×C4⋊C4×C73C2311yescharacteristic
128C2×C22⋊C4×C73—21no—
129C2×C22⋊C4×C73—21no—
130C2×C42×C73—21no—
131#107×C73—41no—
132#106×C73—22no—
133C22⋊C8×C73C2×C414yescomplemented
134#95×C73—21no—
135C23×D73C2×D411yescharacteristic
136C2×C22⋊D73—22no—
137C2×C22⋊D73—22no—
138C2×C22⋊D73C4⋊C414yes—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73C2×D411yescharacteristic
144C22×Dic73C2×D411yescharacteristic
145C22×Dic73C2×D411yescharacteristic
146C2×C4×D73—21no—
147C2×C4×D73—21no—
148C2×C4×D73—21no—
149C2×C4×D73—21no—
150C2×C4×D73—42no—
151C2×C4×D73—42no—
152C2×C4×D73—42no—
153C2×C4×D73—42no—
154C2×C4×D73—81no—
155C2×C4×D73—81no—
156C2×C4×D73C22×C412yes—
157C2×C4×D73C2×Q812yes—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—28no—
161C2×#10596831C22×C412yes—
162C2×#10596831C22×C412yes—
163C2×#10596831C2×D412yes—
164C2×#10596831#4112yes—
165C8×D73—28no—
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73C2×D412yes—
174#11924007—42no—
175#11924007—42no—
176#11924007C4⋊C414yes—
177C22⋊Dic73—42no—
178C22⋊Dic73—81no—
179C22⋊Dic73C4⋊C414yes—
180#11924005—42no—
181#11924005—42no—
182#11924005C4⋊C414yes—
183C4×Dic73—42no—
184C4×Dic73—81no—
185C4×Dic73—81no—
186C4×Dic73C22×C412yes—
187C4×Dic73C2×Q812yes—
188C24×C73C2×D411yescharacteristic
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73—21no—
192C22×C4×C73—21no—
193C22×C4×C73—21no—
194C22×C4×C73—21no—
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73C22×C412yes—
198C22×C4×C73C2×D411yescharacteristic
199C22×C4×C73C2×Q812yes—
200C22×C4×C73C2411yescharacteristic
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73C22×C412yes—
206C2×C8×C73C22×C412yes—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—21no—
211C4⋊C4×C73—21no—
212C4⋊C4×C73—21no—
213C4⋊C4×C73—21no—
214C4⋊C4×C73C2×D412yes—
215C22⋊C4×C73—42no—
216C22⋊C4×C73—81no—
217C42×C73—42no—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—21no—
222C23×C73—21no—
223C23×C73—21no—
224C23×C73C2×C4⋊C412yes—
225C23×C73C22×D411yescharacteristic
226C22×D73—21no—
227C22×D73—21no—
228C22×D73—21no—
229C22×D73—21no—
230C22×D73C2×C4⋊C412yes—
231C22×D73#13512yes—
232C22×D73#13512yes—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—22no—
236C2×C4×C73—22no—
237C2×C4×C73—22no—
238C2×C4×C73—22no—
239C2×C4×C73—22no—
240C2×C4×C73—41no—
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×C73C2×C4⋊C412yes—
252C2×C4×C73C23×C411yescharacteristic
253C2×C4×C73C22×D411yescharacteristic
254C2×C4×C73C22×Q811yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73H(5,𝔽2)11yescharacteristic
257C2×C4×C732−1+411yescharacteristic
258C22⋊D73—28no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—22no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
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×Dic73C2×C4⋊C412yes—
278C2×Dic73C2×C4⋊C412yes—
279C2×Dic73C2×C4⋊C412yes—
280C4×D73—22no—
281C4×D73—22no—
282C4×D73—22no—
283C4×D73—22no—
284C4×D73—84no—
285C4×D73—84no—
286C8×C73—22no—
287C8×C73—22no—
288#10596831—22no—
289#10596831C4×D414yes—
290C22×C73—22no—
291C22×C73—22no—
292C22×C73—22no—
293C22×C73—41no—
294C22×C73—41no—
295C22×C73—41no—
296C22×C73—41no—
297C22×C73C22×C4⋊C411yescharacteristic
298C22×C73D4211yescharacteristic
299C22×C73#54711yescharacteristic
300C22×C73C2×#13511yescharacteristic
301C22×C73C2×#13511yescharacteristic
302C22×C73#57711yescharacteristic
303C22×C73#57711yescharacteristic
304C2×D73—22no—
305C2×D73—22no—
306C2×D73—22no—
307C2×D73—22no—
308C2×D73—22no—
309C2×D73—22no—
310C2×D73#42714yescomplemented
311C4×C73—22no—
312C4×C73—42no—
313C4×C73—81no—
314C4×C73—81no—
315C4×C73—81no—
316C4×C73C2×C4×D411yescharacteristic
317C4×C73#52011yescharacteristic
318C4×C73D4×Q811yescharacteristic
319C4×C73#55411yescharacteristic
320Dic73—22no—
321Dic73—22no—
322Dic73—22no—
323Dic73—42no—
324Dic73—81no—
325Dic73—81no—
326Dic73—81no—
327#22654—731nomaximal, Sylow
328C2×C73—42no—
329C2×C73D4×C4⋊C411yescharacteristic
330C2×C73C2×#42711yescharacteristic
331C2×C73#291211yescharacteristic
332C2×C73#328211yescharacteristic
333C2×C73#328211yescharacteristic
334C2×C73#333711yescharacteristic
335C2×C73#333711yescharacteristic
336D73—24no—
337C2×C8×D4—731no—
338C2×#427—731no—
339C22×#106—731no—
340#2352—731no—
341#2352—731no—
342#1949—731no—
343#1949—731no—
344#1854—731no—
345#1854D7311yescharacteristic, complemented
346#1852—731no—
347#1852—731no—
348#1846—731no—
349#1807—731no—
350#1807—731no—
351C4×#106—731no—
352C73#2265411yescharacteristic, complemented, Sylow
353C23×C8—731no—
354C23×C8—731no—
355C2×C4×D4—731no—
356C2×#117—731no—
357C2×#117C2×D7311yescharacteristic
358C22×C4⋊C4—731no—
359C22×C4⋊C4—731no—
360C8×D4—738no—
361#427—1464no—
362C2×#107—731no—
363C2×#107—731no—
364C2×#107—731no—
365C2×#107C2×D7311yescharacteristic
366C2×#106—731no—
367C2×#106—731no—
368C2×#106—731no—
369C2×#106—731no—
370C2×#106—731no—
371C2×#106—1464no—
372C2×#106C2×D7311yescharacteristic
373C2×C4⋊C8—731no—
374C2×C22⋊C8—731no—
375C2×C22⋊C8C2×D7311yescharacteristic
376C2×C4×C8—731no—
377#388—1461no—
378#388—1461no—
379#387—1461no—
380#387—1461no—
381#386—1461no—
382#386—1461no—
383#382—731no—
384#382—731no—
385#380—731no—
386#380—731no—
387#380—731no—
388#380C2×D7311yescharacteristic
389C4×C4⋊C4—731no—
390C4×C4⋊C4—731no—
391C4×C22⋊C4—1461no—
392C4×C22⋊C4—1461no—
393#340—731no—
394#340—731no—
395#340—731no—
396#340—731no—
397#340C2×D7311yescharacteristic
398#340C2×D7311yescharacteristic
399C22×D4—731no—
400C23×C4—731no—
401C23×C4—1461no—
402C23×C4—1461no—
403C23×C4C22×D7311yescharacteristic
404C22×C8—731no—
405C22×C8—734no—
406C22×C8—731no—
407C22×C8—734no—
408C22×C8—1462no—
409C22×C8—1462no—
410C22×C8—1462no—
411C22×C8—1462no—
412C22×C8C22×D7311yescharacteristic
413C22×C8C22×D7311yescharacteristic
414C4×D4—734no—
415C4×D4—734no—
416#117—22no—
417#117—41no—
418#117—2921no—
419#117—1462no—
420C2×C4⋊C4—2921no—
421C2×C4⋊C4—2921no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—731no—
427C2×C4⋊C4—1462no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4—1462no—
431C2×C4⋊C4C22×D7311yescharacteristic
432C2×C4⋊C4C22×D7311yescharacteristic
433C2×C4⋊C4C22×D7311yescharacteristic
434C2×C4⋊C4C22×D7311yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—731no—
438C2×C22⋊C4—731no—
439C2×C22⋊C4—1461no—
440C2×C22⋊C4—1461no—
441C2×C22⋊C4—1461no—
442C2×C22⋊C4—1461no—
443C2×C42—21no—
444C2×C42—731no—
445C2×C42—1461no—
446C2×C42—1461no—
447C2×C42—1461no—
448#107—41no—
449#107—2921no—
450#107—1462no—
451#107—1462no—
452#106—22no—
453#106—734no—
454#106—1462no—
455#106—1462no—
456#106—1462no—
457#106—1462no—
458C4⋊C8—734no—
459C22⋊C8—734no—
460C22⋊C8C4×D7314yescomplemented
461C4×C8—734no—
462#95—21no—
463#95—1461no—
464#95—1461no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470#95—1461no—
471#95—1461no—
472C24—731no—
473C24D4×D7311yescharacteristic
474C2×D4—734no—
475C2×D4—1462no—
476C2×D4—1462no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—2922no—
486C22×C4—2922no—
487C22×C4—2922no—
488C22×C4—2922no—
489C22×C4—5841no—
490C22×C4—5841no—
491C22×C4—731no—
492C22×C4—731no—
493C22×C4—731no—
494C22×C4—732no—
495C22×C4—732no—
496C22×C4—1461no—
497C22×C4—1461no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4—1461no—
503C22×C4—1461no—
504C22×C4C2×C4×D7312yes—
505C22×C4D4×D7311yescharacteristic
506C22×C4Q8×D7312yes—
507C22×C4C23×D7311yescharacteristic
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—732no—
513C2×C8—732no—
514C2×C8—732no—
515C2×C8—732no—
516C2×C8—1468no—
517C2×C8—1468no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8C2×C4×D7312yes—
521C2×C8C2×C4×D7312yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—2922no—
530C4⋊C4—2922no—
531C4⋊C4—734no—
532C4⋊C4—732no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4—1461no—
536C4⋊C4—1461no—
537C4⋊C4—1461no—
538C4⋊C4—1461no—
539C4⋊C4—1461no—
540C4⋊C4D4×D7312yes—
541C22⋊C4—42no—
542C22⋊C4—81no—
543C22⋊C4—2922no—
544C22⋊C4—2922no—
545C22⋊C4—2922no—
546C22⋊C4—5841no—
547C22⋊C4—734no—
548C22⋊C4—734no—
549C42—42no—
550C42—2922no—
551C42—5841no—
552C42—5841no—
553C42—732no—
554C42—732no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—732no—
562C23—732no—
563C23—732no—
564C23—1461no—
565C23—1461no—
566C23—1461no—
567C23—1461no—
568C23C4⋊C4×D7312yes—
569C23C2×D4×D7311yescharacteristic
570D4—1468no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
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—2921no—
590C2×C4—2921no—
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—5844no—
601C2×C4—5844no—
602C2×C4—732no—
603C2×C4—732no—
604C2×C4—732no—
605C2×C4—1462no—
606C2×C4—1461no—
607C2×C4—1461no—
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×C4C4⋊C4×D7312yes—
617C2×C4C22×C4×D7311yescharacteristic
618C2×C4C2×D4×D7311yescharacteristic
619C2×C4C2×Q8×D7311yescharacteristic
620C2×C4#41×D7311yescharacteristic
621C2×C4#1455663611yescharacteristic
622C2×C4#1455663811yescharacteristic
623C8—22no—
624C8—22no—
625C8—734no—
626C8—1462no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—734no—
635C22—1462no—
636C22—1462no—
637C22—1462no—
638C22—1462no—
639C22—1462no—
640C22—1462no—
641C22C2×C4⋊C4×D7311yescharacteristic
642C22#1798921911yescharacteristic
643C22#1798923511yescharacteristic
644C22#135×D7311yescharacteristic
645C22#135×D7311yescharacteristic
646C22#1798939611yescharacteristic
647C22#1798939711yescharacteristic
648C4—22no—
649C4—42no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—2922no—
654C4—5841no—
655C4—5841no—
656C4—5841no—
657C4—1462no—
658C4—1462no—
659C4—1462no—
660C4C4×D4×D7311yescharacteristic
661C4#1798906611yescharacteristic
662C4#1798925011yescharacteristic
663C4#1798929111yescharacteristic
664C2—42no—
665C2—1464no—
666C2#2261282911yescharacteristic
667C2#427×D7311yescharacteristic
668C2#2261852511yescharacteristic
669C2#2262163611yescharacteristic
670C2#2262164111yescharacteristic
671C2#2262230311yescharacteristic
672C2#2262231311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1854C2
socleC23×C73C22×D4
prime radicalC22×C22⋊D73D4
cosocleC22×C4×C73C24
hypercenter#1854D73
hypocenterC73#22654
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
2. C2×#17988623C2semidirect
3. C2×#106×D73C2semidirect
6. #22610700C2semidirect
7. #22610698C2semidirect
8. #22610132C2semidirect
9. #22610120C2semidirect
10. #22610117C2semidirect
11. #22610094C2semidirect
12. #22609955C2semidirect
13. #22609954C2semidirect
14. #22609819C2semidirect
15. C73×#1854C2semidirect
22. #17988650C4semidirect
68. C22×#10596831C2×C4semidirect
106. #14556522D4semidirect
107. #14556520C2×C4semidirect
109. C8×Dic73C2×C4semidirect
133. C22⋊C8×C73C2×C4semidirect
310. C2×D73#427semidirect
345. #1854D73semidirect
352. C73#22654semidirect
460. C22⋊C8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886502
C2×#179886232
C2×#106×D732
#226128612
#226128532
#226107002
#226106982
#226101322
#226101202
#226101172
#226100942
#226099552
#226099542
#226098192
C73×#18542
#2265473