Finite Group Databasebeta

← #34022105

Subgroups — #34022105

Order 18688.

1002 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989593C211yesmaximal, characteristic
2C2×#17989593C211yesmaximal, characteristic
3C2×#17989590C211yesmaximal, characteristic
4C2×#17989551C211yesmaximal, characteristic
5#22626033C211yesmaximal, characteristic, complemented
6#22626033C211yesmaximal, characteristic, complemented
7#22626032C211yesmaximal, characteristic, complemented
8#22626031C211yesmaximal, characteristic, complemented
9#22626030C211yesmaximal, characteristic, complemented
10#22626030C211yesmaximal, characteristic, complemented
11#22626029C211yesmaximal, characteristic, complemented
12#22626028C211yesmaximal, characteristic, complemented
13#22625597C211yesmaximal, characteristic
14C2×#17989453C211yesmaximal, characteristic
15#22612410C211yesmaximal, characteristic
16C22×#11924037C2211yescharacteristic
17C22×#11924037C2211yescharacteristic
18C23×#10596833C2211yescharacteristic
19C23×#10596833C2211yescharacteristic
20#17989593—24no—
21#17989593—24no—
22#17989590—24no—
23C2×#14556658C2211yescharacteristic
24C2×#14556658C2211yescharacteristic
25C2×#14556658C2211yescharacteristic
26C2×#14556658C2211yescharacteristic
27C2×#14556658C2211yescharacteristic
28C2×#14556658C2211yescharacteristic
29C2×C4⋊#10596833C2211yescharacteristic
30C2×C4⋊#10596833C2211yescharacteristic
31C2×C4⋊#10596833C2211yescharacteristic
32C2×C4⋊#10596833C2211yescharacteristic
33C2×C4⋊#10596833C2211yescharacteristic
34C2×C4⋊#10596833C2211yescharacteristic
35#17989551—24no—
36C2×#14556646C2211yescharacteristic
37C2×#14556646C2211yescharacteristic
38C2×C4×#10596833C2211yescharacteristic
39C2×C4×#10596833C2211yescharacteristic
40#17989453C2211yescharacteristic
41#17989453C2211yescharacteristic
42#17989453C2211yescharacteristic
43#17989453C2211yescharacteristic
44#17989453C2211yescharacteristic
45#17989453C2211yescharacteristic
46#17989453C2211yescharacteristic
47#17989453C2211yescharacteristic
48C2×#14556547C2211yescharacteristic
49C2×#14556547C2211yescharacteristic
50C22×C4×Dic73C2211yescharacteristic
51#17988484C411yescharacteristic
52#17988484C411yescharacteristic
53#17988482C411yescharacteristic
54#17988482C411yescharacteristic
55#17988462C411yescharacteristic
56#17988460C411yescharacteristic
57#17988459C411yescharacteristic
58C4⋊C4×Dic73C2211yescharacteristic
59C4⋊C4×Dic73C2211yescharacteristic
60#17988438C2211yescharacteristic
61#17988438C2211yescharacteristic
62#385×C73C411yescharacteristic
63C2×#11924037—22no—
64C2×#11924037—22no—
65C2×#11924037—22no—
66C2×#11924037—22no—
67C2×#11924037—22no—
68C2×#11924037D414yes—
69C22×#10596833—22no—
70C22×#10596833—22no—
71C22×#10596833—22no—
72C22×#10596833—22no—
73C22×#10596833—22no—
74C22×#10596833Q814yes—
75C22×#10596833C2311yescharacteristic
76C22×#10596833C2311yescharacteristic
77C22×#10596833C2311yescharacteristic
78C22×#10596833C2311yescharacteristic
79C22×#10596833C2311yescharacteristic
80C22×#10596833C2311yescharacteristic
81C22×#10596833C2311yescharacteristic
82C22×#10596833C2311yescharacteristic
83#14556658—22no—
84#14556658—22no—
85#14556658—22no—
86#14556658—22no—
87#14556658—22no—
88#14556658—22no—
89C4⋊#10596833—22no—
90C4⋊#10596833—22no—
91C4⋊#10596833—22no—
92C4⋊#10596833—22no—
93C4⋊#10596833—22no—
94C4⋊#10596833—22no—
95#14556646—22no—
96#14556646—22no—
97C4×#10596833—22no—
98C4×#10596833—22no—
99C23×Dic73C2×C411yescharacteristic
100C23×Dic73C2311yescharacteristic
101#14556547—22no—
102#14556547—22no—
103#14556547D414yes—
104#14556547Q814yes—
105C2×C4⋊Dic73C2×C411yescharacteristic
106C2×C4⋊Dic73C2×C411yescharacteristic
107C2×#11924007C2×C411yescharacteristic
108C2×#11924007C2×C411yescharacteristic
109C2×#11924007C2×C411yescharacteristic
110C2×#11924007C2×C411yescharacteristic
111C2×#11924007C2×C411yescharacteristic
112C2×#11924007C2×C411yescharacteristic
113C2×#11924007C2×C411yescharacteristic
114C2×#11924007C2×C411yescharacteristic
115C2×C22⋊Dic73C2×C411yescharacteristic
116C2×C22⋊Dic73C2×C411yescharacteristic
117C2×C4×Dic73—22no—
118C2×C4×Dic73—22no—
119C2×C4×Dic73C2×C411yescharacteristic
120C2×C4×Dic73C2×C411yescharacteristic
121C2×C4×Dic73C2311yescharacteristic
122C2×C4×Dic73C2311yescharacteristic
123C2×C4×Dic73C2311yescharacteristic
124C2×C4×Dic73C2311yescharacteristic
125C2×C4×Dic73C2311yescharacteristic
126C2×C4×Dic73C2311yescharacteristic
127#14556498C2×C411yescharacteristic
128#14556498C2×C411yescharacteristic
129#14556498C2×C411yescharacteristic
130#14556498C2×C411yescharacteristic
131#14556498C2×C411yescharacteristic
132#14556498C2×C411yescharacteristic
133C23×C4×C73C2×C411yescharacteristic
134C2×C4⋊C4×C73C2×C411yescharacteristic
135C2×C4⋊C4×C73C2×C411yescharacteristic
136C2×C22⋊C4×C73C2×C411yescharacteristic
137C2×C22⋊C4×C73C2×C411yescharacteristic
138#95×C73C2×C411yescharacteristic
139#95×C73C2×C411yescharacteristic
140#11924037—28no—
141#11924037—44no—
142C2×#10596833—28no—
143C2×#10596833—21no—
144C2×#10596833—21no—
145C2×#10596833—21no—
146C2×#10596833—21no—
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—21no—
154C2×#10596833—21no—
155C2×#10596833—21no—
156C2×#10596833—21no—
157C2×#10596833—21no—
158C2×#10596833—21no—
159C2×#10596833—21no—
160C2×#10596833—21no—
161C2×#10596833—21no—
162C2×#10596833—21no—
163C2×#10596833—44no—
164C2×#10596833C2×D412yes—
165C2×#10596833C2×D412yes—
166C2×#10596833C2×Q812yes—
167C2×#10596833C2×Q812yes—
168C22×Dic73—22no—
169C22×Dic73—22no—
170C22×Dic73—41no—
171C22×Dic73C22×C411yescharacteristic
172C22×Dic73C22×C411yescharacteristic
173C22×Dic73C22×C411yescharacteristic
174C22×Dic73C22×C411yescharacteristic
175C22×Dic73C22×C411yescharacteristic
176C22×Dic73C22×C411yescharacteristic
177C22×Dic73C22×C411yescharacteristic
178C22×Dic73C2×D412yes—
179C22×Dic73C2×D412yes—
180C22×Dic73C2×Q812yes—
181C22×Dic73C2×Q812yes—
182C22×Dic73#4112yes—
183C22×Dic73#4112yes—
184C22×Dic73C2411yescharacteristic
185C4⋊Dic73—22no—
186C4⋊Dic73—41no—
187#11924007—22no—
188#11924007—22no—
189#11924007—22no—
190#11924007—22no—
191#11924007—41no—
192#11924007—41no—
193#11924007C4⋊C414yes—
194#11924007C4⋊C414yes—
195C22⋊Dic73—22no—
196C22⋊Dic73C4⋊C414yes—
197C4×Dic73—22no—
198C4×Dic73—22no—
199C4×Dic73—21no—
200C4×Dic73—21no—
201C4×Dic73—21no—
202C4×Dic73—21no—
203C4×Dic73—21no—
204C4×Dic73—21no—
205C4×Dic73—21no—
206C4×Dic73—21no—
207C4×Dic73C2×D412yes—
208C4×Dic73C2×D412yes—
209C4×Dic73C2×Q812yes—
210C4×Dic73C2×Q812yes—
211C24×C73C22×C411yescharacteristic
212C22×C4×C73—22no—
213C22×C4×C73—22no—
214C22×C4×C73—41no—
215C22×C4×C73C22×C411yescharacteristic
216C22×C4×C73C22×C411yescharacteristic
217C22×C4×C73C22×C411yescharacteristic
218C22×C4×C73C22×C411yescharacteristic
219C22×C4×C73C22×C411yescharacteristic
220C22×C4×C73C22×C411yescharacteristic
221C4⋊C4×C73—22no—
222C4⋊C4×C73—41no—
223C22⋊C4×C73—22no—
224C22⋊C4×C73C4⋊C414yes—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73C2×C4⋊C412yes—
230C23×C73C4×D412yes—
231C23×C73C4×Q812yes—
232C23×C73C23×C411yescharacteristic
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—44no—
249C2×C4×C73C2×C4⋊C412yes—
250C2×C4×C73C2×C4⋊C412yes—
251C2×C4×C73#13112yes—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—21no—
266C2×Dic73—21no—
267C2×Dic73—44no—
268C2×Dic73C2×C4⋊C412yes—
269C2×Dic73C2×C4⋊C412yes—
270C2×Dic73C2×C4⋊C412yes—
271C2×Dic73C4×D412yes—
272C2×Dic73C4×Q812yes—
273C2×Dic73#12114yes—
274C2×Dic73C22⋊Q814yes—
275C2×Dic73#12514yes—
276C2×Dic73C4⋊Q814yes—
277C2×Dic73#13112yes—
278C2×Dic73C22×D411yescharacteristic
279C2×Dic73C22×D411yescharacteristic
280C2×Dic73C22×Q811yescharacteristic
281C2×Dic73C22×Q811yescharacteristic
282C2×Dic73C2×#4111yescharacteristic
283C2×Dic73C2×#4111yescharacteristic
284#10596833—22no—
285#10596833—22no—
286#10596833—22no—
287#10596833—22no—
288#10596833—41no—
289#10596833—41no—
290#10596833—41no—
291#10596833—41no—
292C22×C73—22no—
293C22×C73—22no—
294C22×C73—22no—
295C22×C73—22no—
296C22×C73—22no—
297C22×C73—22no—
298C22×C73#38914yes—
299C22×C73C22×C4⋊C411yescharacteristic
300C22×C73C2×C4×D411yescharacteristic
301C22×C73C2×C4×Q811yescharacteristic
302C22×C73C2×#13111yescharacteristic
303C22×C73C2×#13111yescharacteristic
304C22×C73#56811yescharacteristic
305C22×C73#56811yescharacteristic
306C4×C73—22no—
307C4×C73—22no—
308C4×C73—22no—
309C4×C73—41no—
310C4×C73—41no—
311C4×C73—41no—
312Dic73—22no—
313Dic73—22no—
314Dic73—22no—
315Dic73—41no—
316Dic73—41no—
317Dic73—41no—
318Dic73#38914yes—
319Dic73C2×#12111yescharacteristic
320Dic73C2×C22⋊Q811yescharacteristic
321Dic73C2×#12511yescharacteristic
322Dic73C2×C4⋊Q811yescharacteristic
323#21148—731nomaximal, Sylow
324C2×C73—24no—
325C2×C73C2×#38911yescharacteristic
326C2×C73#290611yescharacteristic
327C2×C73#290611yescharacteristic
328C2×C73#293311yescharacteristic
329C2×C73#293311yescharacteristic
330C2×C73#296011yescharacteristic
331C2×C73#296111yescharacteristic
332C2×#424—731no—
333C2×#424—731no—
334C2×#423—731no—
335C2×#405—731no—
336#2296—731no—
337#1940—731no—
338#1940—731no—
339#1940—731no—
340#1940—731no—
341#1939—731no—
342#1939—731no—
343#1939—731no—
344#1939—731no—
345#1729—731no—
346C2×#336—731no—
347C73#2114811yescharacteristic, complemented, Sylow
348C22×M16—731no—
349C22×M16—731no—
350C23×C8—731no—
351C23×C8—731no—
352C2×#117—731no—
353C2×#117—731no—
354C22×C42—731no—
355#424—1464no—
356#424—1464no—
357#423—1464no—
358C2×C4⋊C8—731no—
359C2×C4⋊C8—731no—
360C2×C4⋊C8—731no—
361C2×C4⋊C8—731no—
362C2×C4⋊C8—731no—
363C2×C4⋊C8—731no—
364C2×C4⋊C8—731no—
365C2×C4⋊C8—731no—
366C2×C4⋊C8—731no—
367C2×C4⋊C8—731no—
368C2×C4⋊C8—731no—
369C2×C4⋊C8—731no—
370#405—1464no—
371C2×#97—731no—
372C2×#97—731no—
373C2×C4×C8—731no—
374C2×C4×C8—731no—
375#389—731no—
376#389—731no—
377#389—731no—
378#389—731no—
379#385—731no—
380#385—731no—
381#385—731no—
382#385#6440811yescharacteristic
383C4×C4⋊C4—731no—
384C4×C4⋊C4—731no—
385C4×C4⋊C4—731no—
386C4×C4⋊C4—731no—
387#336—731no—
388#336—731no—
389#336—731no—
390#336—731no—
391#336—731no—
392#336—731no—
393#336—731no—
394#336—731no—
395C23×C4—731no—
396C23×C4—731no—
397C23×C4C2×#6440811yescharacteristic
398C2×M16—734no—
399C2×M16—1462no—
400C2×M16—1462no—
401C2×M16—1462no—
402C2×M16—1462no—
403C2×M16—1462no—
404C22×C8—734no—
405C22×C8—731no—
406C22×C8—731no—
407C22×C8—731no—
408C22×C8—731no—
409C22×C8—731no—
410C22×C8—731no—
411C22×C8—731no—
412C22×C8—731no—
413C22×C8—1462no—
414C22×C8—1462no—
415C22×C8—1462no—
416C22×C8—1462no—
417C22×C8—1462no—
418#117—734no—
419#117—734no—
420#117—1462no—
421#117—1462no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—731no—
427C2×C4⋊C4—731no—
428C2×C4⋊C4—731no—
429C2×C4⋊C4—731no—
430C2×C4⋊C4—731no—
431C2×C4⋊C4—731no—
432C2×C4⋊C4C2×#6440811yescharacteristic
433C2×C4⋊C4C2×#6440811yescharacteristic
434C2×C22⋊C4—731no—
435C2×C22⋊C4—731no—
436C2×C22⋊C4C2×#6440811yescharacteristic
437C2×C22⋊C4C2×#6440811yescharacteristic
438C2×C42—731no—
439C2×C42—731no—
440C2×C42—731no—
441C2×C42—731no—
442C2×C42—731no—
443C2×C42—731no—
444C2×C42—731no—
445C2×C42—731no—
446C2×C42—1462no—
447C2×C42—1462no—
448C4⋊C8—1462no—
449C4⋊C8—1462no—
450C4⋊C8—1462no—
451C4⋊C8—1462no—
452C4⋊C8—1462no—
453C4⋊C8—1462no—
454C4⋊C8—1462no—
455C4⋊C8—1462no—
456C4⋊C8—1462no—
457C4⋊C8—1462no—
458C4⋊C8—1462no—
459C4⋊C8—1462no—
460#97—1462no—
461#97—1462no—
462C4×C8—1462no—
463C4×C8—1462no—
464#95—731no—
465#95—731no—
466#95—731no—
467#95—731no—
468#95—731no—
469#95—731no—
470#95C2×#6440811yescharacteristic
471#95C2×#6440811yescharacteristic
472C24C22×#6440811yescharacteristic
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—41no—
476C22×C4—2921no—
477C22×C4—732no—
478C22×C4—732no—
479C22×C4—731no—
480C22×C4—731no—
481C22×C4—732no—
482C22×C4—732no—
483C22×C4—732no—
484C22×C4—732no—
485C22×C4—731no—
486C22×C4—731no—
487C22×C4—731no—
488C22×C4—731no—
489C22×C4—731no—
490C22×C4—731no—
491C22×C4—1462no—
492C22×C4—1462no—
493C22×C4C22×#6440811yescharacteristic
494C22×C4C22×#6440811yescharacteristic
495C22×C4C22×#6440811yescharacteristic
496C22×C4C22×#6440811yescharacteristic
497C22×C4C22×#6440811yescharacteristic
498C22×C4C22×#6440811yescharacteristic
499M16—2924no—
500M16—1468no—
501C2×C8—2924no—
502C2×C8—732no—
503C2×C8—732no—
504C2×C8—732no—
505C2×C8—732no—
506C2×C8—1461no—
507C2×C8—1461no—
508C2×C8—1461no—
509C2×C8—1461no—
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C2×C8—1461no—
523C2×C8—1461no—
524C2×C8—1468no—
525C2×C8—1461no—
526C2×C8—1461no—
527C4⋊C4—22no—
528C4⋊C4—41no—
529C4⋊C4—2921no—
530C4⋊C4—2921no—
531C4⋊C4—2921no—
532C4⋊C4—734no—
533C4⋊C4—734no—
534C4⋊C4—1462no—
535C4⋊C4—1462no—
536C4⋊C4—1462no—
537C4⋊C4—1462no—
538C4⋊C4—1462no—
539C22⋊C4—22no—
540C22⋊C4—734no—
541C22⋊C4—1462no—
542C22⋊C4C4⋊#6440814yes—
543C42—732no—
544C42—732no—
545C42—732no—
546C42—732no—
547C42—1461no—
548C42—1461no—
549C42—1461no—
550C42—1461no—
551C42—1461no—
552C42—1462no—
553C42—1461no—
554C42—1462no—
555C42—1461no—
556C42—1461no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23C2×C4⋊#6440812yes—
562C23D4×#6440812yes—
563C23Q8×#6440812yes—
564C23C23×#6440811yescharacteristic
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—44no—
581C2×C4—2924no—
582C2×C4—731no—
583C2×C4—731no—
584C2×C4—732no—
585C2×C4—731no—
586C2×C4—731no—
587C2×C4—732no—
588C2×C4—731no—
589C2×C4—731no—
590C2×C4—732no—
591C2×C4—734no—
592C2×C4—734no—
593C2×C4—734no—
594C2×C4—734no—
595C2×C4—732no—
596C2×C4—732no—
597C2×C4—732no—
598C2×C4—1461no—
599C2×C4—1461no—
600C2×C4—1461no—
601C2×C4—1461no—
602C2×C4—1461no—
603C2×C4—1461no—
604C2×C4—1461no—
605C2×C4—1461no—
606C2×C4—1461no—
607C2×C4—1461no—
608C2×C4—1461no—
609C2×C4—1461no—
610C2×C4—1461no—
611C2×C4—1461no—
612C2×C4—1461no—
613C2×C4C2×C4⋊#6440812yes—
614C2×C4C2×C4⋊#6440812yes—
615C2×C4#1455667812yes—
616C8—2921no—
617C8—2921no—
618C8—2921no—
619C8—2921no—
620C8—1462no—
621C8—1462no—
622C8—1462no—
623C8—1462no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22#1798953814yes—
631C22C22×C4⋊#6440811yescharacteristic
632C22C2×D4×#6440811yescharacteristic
633C22C2×Q8×#6440811yescharacteristic
634C22C2×#1455667811yescharacteristic
635C22D73⋊#13111yescharacteristic
636C22#1798966511yescharacteristic
637C22#1798966611yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—2921no—
645C4—2921no—
646C4—2921no—
647C4—731no—
648C4—731no—
649C4—731no—
650C4—731no—
651C4—734no—
652C4—1462no—
653C4—1462no—
654C4—1462no—
655C2—24no—
656C2C2×#1798953811yescharacteristic
657C2#2262657611yescharacteristic
658C2#2262657811yescharacteristic
659C2#2262666311yescharacteristic
660C2#2262666511yescharacteristic
661C2#2262671911yescharacteristic
662C2#2262672211yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
Fitting#385×C73C4
prime radicalC24×C73C22×C4
cosocleC22×Dic73C24
hypercenter#385#64408
hypocenterC73#21148
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
5. #22626033C2semidirect
6. #22626033C2semidirect
7. #22626032C2semidirect
8. #22626031C2semidirect
9. #22626030C2semidirect
10. #22626030C2semidirect
11. #22626029C2semidirect
12. #22626028C2semidirect
347. C73#21148semidirect

Maximal subgroups

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

subgroupindex
C2×#179895932
C2×#179895902
C2×#179895512
#226260332
#226260322
#226260312
#226260302
#226260292
#226260282
#226255972
C2×#179894532
#226124102
#2114873