Finite Group Databasebeta

← #34026377

Subgroups — #34026377

Order 18688.

1292 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989639C212yesmaximal, complemented
2#22626105C212yesmaximal, complemented
3#22626104C212yesmaximal, complemented
4#22626102C212yesmaximal, complemented
5#22625788C211yesmaximal, characteristic
6#22625788C211yesmaximal, characteristic
7#22625786C211yesmaximal, characteristic
8#22625658C211yesmaximal, characteristic
9#22625658C211yesmaximal, characteristic
10C4×#14556648C211yesmaximal, characteristic
11C2×C4×Q8×D73C211yesmaximal, characteristic
12#17989639—28no—
13C2×#14556658C2212yes—
14C2×#14556658C2212yes—
15C2×C4⋊#10596833C2212yes—
16C2×#14556648C2211yescharacteristic
17C2×#14556648C2211yescharacteristic
18C2×#14556648C2211yescharacteristic
19C2×#14556648C2211yescharacteristic
20C2×#14556646C2212yes—
21C2×#14556646C2212yes—
22C2×C4×#10596833C2212yes—
23#17989453C2212yes—
24#17989453C2212yes—
25#17989453C2212yes—
26#17989453C2212yes—
27#17989453C2212yes—
28#17989453C2212yes—
29C22×Q8×D73C411yescharacteristic
30C2×Q8×Dic73C2211yescharacteristic
31C2×C4×Dic146C411yescharacteristic
32C2×C4×Dic146C411yescharacteristic
33C2×C4×Dic146C411yescharacteristic
34C2×#14556556C2211yescharacteristic
35C2×#14556556C2211yescharacteristic
36C2×#14556556C2211yescharacteristic
37C4×Q8×D73—28no—
38C2×C4⋊C4×D73C411yescharacteristic
39C2×C4⋊C4×D73C411yescharacteristic
40C2×C4⋊C4×D73C411yescharacteristic
41C2×C42×D73C2211yescharacteristic
42C2×C42×D73C2211yescharacteristic
43C2×C42×D73C2211yescharacteristic
44C2×C4×Q8×C73C411yescharacteristic
45C22×#10596833C2312yes—
46C22×#10596833C2312yes—
47C22×#10596833C2312yes—
48C22×#10596833C2312yes—
49#14556658—24no—
50#14556658—24no—
51C4⋊#10596833—24no—
52#14556648—22no—
53#14556648—22no—
54#14556648—22no—
55#14556648Q814yescomplemented
56#14556646—24no—
57#14556646—24no—
58C4×#10596833—24no—
59C22×Dic146C2×C411yescharacteristic
60C22×Dic146C2×C411yescharacteristic
61C22×Dic146C2×C411yescharacteristic
62C2×Q8×D73—24no—
63C2×Q8×D73—24no—
64C2×Q8×D73—24no—
65C22×C4×D73C2×C411yescharacteristic
66C22×C4×D73C2×C411yescharacteristic
67C22×C4×D73C2×C411yescharacteristic
68C22×C4×D73C2×C411yescharacteristic
69C22×C4×D73C2×C411yescharacteristic
70C22×C4×D73C2×C411yescharacteristic
71C22×C4×D73C2311yescharacteristic
72Q8×Dic73—24no—
73C4×Dic146—24no—
74C4×Dic146—24no—
75C4×Dic146—24no—
76#14556556—24no—
77#14556556—24no—
78#14556556D418yes—
79C2×C4⋊Dic73C2×C411yescharacteristic
80C2×C4⋊Dic73C2×C411yescharacteristic
81C2×C4⋊Dic73C2×C411yescharacteristic
82C2×#11924007C2×C411yescharacteristic
83C2×#11924007C2×C411yescharacteristic
84C2×#11924007C2×C411yescharacteristic
85C2×#11924007C2×C411yescharacteristic
86C2×#11924007C2×C411yescharacteristic
87C2×#11924007C2×C411yescharacteristic
88C4⋊C4×D73—24no—
89C4⋊C4×D73—24no—
90C4⋊C4×D73—24no—
91C2×C4×Dic73C2×C411yescharacteristic
92C2×C4×Dic73C2×C411yescharacteristic
93C2×C4×Dic73C2×C411yescharacteristic
94C2×C4×Dic73C2311yescharacteristic
95C2×C4×Dic73C2311yescharacteristic
96C2×C4×Dic73C2311yescharacteristic
97C2×C4×Dic73C2311yescharacteristic
98C2×C4×Dic73C2311yescharacteristic
99C2×C4×Dic73C2311yescharacteristic
100C42×D73—24no—
101C42×D73—24no—
102C42×D73—24no—
103C22×Q8×C73C2×C411yescharacteristic
104C4×Q8×C73—24no—
105C2×C4⋊C4×C73C2×C411yescharacteristic
106C2×C4⋊C4×C73C2×C411yescharacteristic
107C2×C4⋊C4×C73C2×C411yescharacteristic
108C2×C42×C73C2×C411yescharacteristic
109C2×C42×C73C2×C411yescharacteristic
110C2×C42×C73C2×C411yescharacteristic
111C2×#10596833—22no—
112C2×#10596833—22no—
113C2×#10596833—22no—
114C2×#10596833—22no—
115C2×#10596833—22no—
116C2×#10596833—22no—
117C2×#10596833—22no—
118C2×#10596833—22no—
119C2×#10596833—22no—
120C2×#10596833—22no—
121C2×#10596833—22no—
122C2×#10596833C2×Q814yescomplemented
123C23×D73C22×C411yescharacteristic
124C2×Dic146—22no—
125C2×Dic146—22no—
126C2×Dic146—22no—
127C2×Dic146—22no—
128C2×Dic146—22no—
129C2×Dic146—22no—
130C2×Dic146—22no—
131C2×Dic146C22⋊C414yes—
132C2×Dic146M1614yes—
133Q8×D73—232no—
134C22×Dic73C22×C411yescharacteristic
135C22×Dic73C22×C411yescharacteristic
136C22×Dic73C22×C411yescharacteristic
137C22×Dic73C22×C411yescharacteristic
138C22×Dic73C22×C411yescharacteristic
139C22×Dic73C22×C411yescharacteristic
140C22×Dic73C2411yescharacteristic
141C2×C4×D73—22no—
142C2×C4×D73—22no—
143C2×C4×D73—22no—
144C2×C4×D73—22no—
145C2×C4×D73—22no—
146C2×C4×D73—22no—
147C2×C4×D73—22no—
148C2×C4×D73—22no—
149C2×C4×D73—22no—
150C2×C4×D73—22no—
151C2×C4×D73—22no—
152C2×C4×D73—22no—
153C2×C4×D73—22no—
154C2×C4×D73—22no—
155C2×C4×D73—22no—
156C2×C4×D73—22no—
157C2×C4×D73—41no—
158C2×C4×D73—41no—
159C2×C4×D73—41no—
160C2×C4×D73M1614yes—
161C2×C4×D73C2×Q812yes—
162C2×C4×D73#4112yes—
163C4⋊Dic73—22no—
164C4⋊Dic73—22no—
165C4⋊Dic73—22no—
166#11924007—22no—
167#11924007—22no—
168#11924007—22no—
169#11924007—22no—
170#11924007C22⋊C414yes—
171#11924007C22⋊C414yes—
172C4×Dic73—22no—
173C4×Dic73—22no—
174C4×Dic73—22no—
175C4×Dic73—22no—
176C4×Dic73—22no—
177C4×Dic73—22no—
178C4×Dic73C2×D414yes—
179C4×Dic73C2×D414yes—
180C4×Dic73C2×D414yes—
181C2×Q8×C73—22no—
182C2×Q8×C73—22no—
183C2×Q8×C73—22no—
184C22×C4×C73C22×C411yescharacteristic
185C22×C4×C73C22×C411yescharacteristic
186C22×C4×C73C22×C411yescharacteristic
187C22×C4×C73C22×C411yescharacteristic
188C22×C4×C73C22×C411yescharacteristic
189C22×C4×C73C22×C411yescharacteristic
190C22×C4×C73C22×C411yescharacteristic
191C4⋊C4×C73—22no—
192C4⋊C4×C73—22no—
193C4⋊C4×C73C22⋊C414yes—
194C42×C73—22no—
195C42×C73—22no—
196C42×C73—22no—
197C23×C73C23×C411yescharacteristic
198C22×D73—21no—
199C22×D73—21no—
200C22×D73—21no—
201C22×D73—21no—
202C22×D73C4×Q812yes—
203C22×D73C2×M1612yes—
204C22×D73#13112yes—
205Q8×C73—28no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73C2×C22⋊C412yes—
222C2×C4×C73C2×C22⋊C412yes—
223C2×C4×C73C2×C22⋊C412yes—
224C2×C4×C73C4×Q812yes—
225C2×C4×C73C2×M1612yes—
226C2×C4×C73C2×M1612yes—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73C2×C22⋊C412yes—
240C2×Dic73C2×C22⋊C412yes—
241C2×Dic73C2×C22⋊C412yes—
242C2×Dic73C2×M1612yes—
243C2×Dic73C2×M1612yes—
244C2×Dic73C2×M1612yes—
245C2×Dic73C22×D412yes—
246C2×Dic73C22×Q811yescharacteristic
247C2×Dic73C2×#4111yescharacteristic
248C2×Dic732−1+412yes—
249C4×D73—28no—
250C4×D73—28no—
251C4×D73—24no—
252C4×D73—24no—
253C4×D73—28no—
254C4×D73—44no—
255C4×D73—44no—
256C4×D73—44no—
257Dic146—28no—
258Dic146—28no—
259Dic146—28no—
260#10596833—24no—
261#10596833—42no—
262#10596833—42no—
263#10596833—42no—
264C22×C73C22×C22⋊C411yescharacteristic
265C22×C73C2×C4×Q811yescharacteristic
266C22×C73#51911yescharacteristic
267C22×C73C22×M1611yescharacteristic
268C22×C73C22×M1611yescharacteristic
269C22×C73C2×#13111yescharacteristic
270C22×C73#56811yescharacteristic
271C2×D73—22no—
272C2×D73—22no—
273C2×D73—22no—
274C2×D73—22no—
275C2×D73—22no—
276C2×D73—22no—
277C2×D73#44614yescomplemented
278C4×C73—22no—
279C4×C73—22no—
280C4×C73—22no—
281C4×C73—22no—
282C4×C73—22no—
283C4×C73—22no—
284C4×C73—41no—
285Dic73—22no—
286Dic73—22no—
287Dic73—22no—
288Dic73—41no—
289Dic73—41no—
290Dic73#40714yes—
291Dic73#54812yes—
292Dic73D4×Q812yes—
293#22510—731nomaximal, Sylow
294C2×C73Q8×C22⋊C411yescharacteristic
295C2×C73C2×#40711yescharacteristic
296C2×C73#288011yescharacteristic
297C2×C73C2×#44611yescharacteristic
298C2×C73Q8×M1611yescharacteristic
299C2×C73#298311yescharacteristic
300C2×C73#298411yescharacteristic
301D73—24no—
302C22×C4×Q8—731no—
303C2×#446—732no—
304#1985—732no—
305#1985—732no—
306#1984—732no—
307#1797—731no—
308#1797—731no—
309#1797—731no—
310#1749—731no—
311#1749—731no—
312C4×C22⋊C8—731no—
313C73#2251011yescharacteristic, complemented, Sylow
314C23×Q8—731no—
315C2×C4×Q8—731no—
316C2×C4×Q8—731no—
317C2×C4×Q8—731no—
318C2×C4×Q8—731no—
319C2×C4×Q8—731no—
320C2×C4×Q8—731no—
321C2×C4×Q8—731no—
322C2×C4×Q8—1468no—
323C2×C4×Q8#6440811yescharacteristic
324C22×C4⋊C4—731no—
325C22×C4⋊C4—731no—
326C22×C4⋊C4—731no—
327C22×C42—731no—
328C22×C42—731no—
329C22×C42—731no—
330#446—1468no—
331C2×C4⋊C8—732no—
332C2×C4⋊C8—732no—
333C2×C4⋊C8—732no—
334C2×C22⋊C8—731no—
335C2×C22⋊C8—731no—
336C2×C22⋊C8—731no—
337C2×C22⋊C8—731no—
338C2×#97—732no—
339C2×#97—732no—
340C2×C4×C8—732no—
341#336—732no—
342#336—732no—
343#336—732no—
344#336—732no—
345#336—732no—
346#336—732no—
347C22×Q8—731no—
348C22×Q8—731no—
349C22×Q8—731no—
350C22×Q8—1464no—
351C22×Q8—1464no—
352C22×Q8—1464no—
353C22×Q8C2×#6440811yescharacteristic
354C23×C4—731no—
355C23×C4—731no—
356C23×C4—731no—
357C23×C4—731no—
358C23×C4—731no—
359C23×C4—731no—
360C23×C4—731no—
361C22×C8—732no—
362C22×C8—732no—
363C22×C8—732no—
364C22×C8—732no—
365C4×Q8—24no—
366C4×Q8—738no—
367C4×Q8—1464no—
368C4×Q8—1464no—
369C4×Q8—1464no—
370C4×Q8—1464no—
371C4×Q8—1464no—
372C4×Q8—1464no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—731no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—731no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—1464no—
383C2×C4⋊C4—1464no—
384C2×C4⋊C4—1464no—
385C2×C4⋊C4C2×#6440811yescharacteristic
386C2×C4⋊C4C2×#6440811yescharacteristic
387C2×C4⋊C4C2×#6440811yescharacteristic
388C2×C42—731no—
389C2×C42—731no—
390C2×C42—731no—
391C2×C42—731no—
392C2×C42—731no—
393C2×C42—731no—
394C2×C42—731no—
395C2×C42—731no—
396C2×C42—731no—
397C2×C42—1464no—
398C2×C42—1464no—
399C2×C42—1464no—
400C2×C42C2×#6440811yescharacteristic
401C2×C42C2×#6440811yescharacteristic
402C2×C42C2×#6440811yescharacteristic
403C4⋊C8—1464no—
404C4⋊C8—1464no—
405C4⋊C8—1464no—
406C22⋊C8—734no—
407C22⋊C8—1462no—
408C22⋊C8—1462no—
409C22⋊C8—1462no—
410#97—1464no—
411#97—1464no—
412C4×C8—1464no—
413C24—731no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—734no—
418C2×Q8—734no—
419C2×Q8—1462no—
420C2×Q8—1462no—
421C2×Q8—1462no—
422C2×Q8—1462no—
423C2×Q8—1462no—
424C2×Q8—1462no—
425C2×Q8—14632no—
426C2×Q8—1462no—
427C22×C4—2921no—
428C22×C4—2921no—
429C22×C4—2921no—
430C22×C4—731no—
431C22×C4—731no—
432C22×C4—731no—
433C22×C4—731no—
434C22×C4—731no—
435C22×C4—732no—
436C22×C4—732no—
437C22×C4—734no—
438C22×C4—731no—
439C22×C4—731no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4—1462no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4—1462no—
448C22×C4—1462no—
449C22×C4—1462no—
450C22×C4—1462no—
451C22×C4—1462no—
452C22×C4—1462no—
453C22×C4—1462no—
454C22×C4—1462no—
455C22×C4—1462no—
456C22×C4C22×#6440811yescharacteristic
457C22×C4C22×#6440811yescharacteristic
458C22×C4C22×#6440811yescharacteristic
459C22×C4C22×#6440811yescharacteristic
460C22×C4C22×#6440811yescharacteristic
461C22×C4C22×#6440811yescharacteristic
462C22×C4C22×#6440811yescharacteristic
463C2×C8—734no—
464C2×C8—1462no—
465C2×C8—1462no—
466C2×C8—1462no—
467C2×C8—1462no—
468C2×C8—1462no—
469C2×C8—1462no—
470C2×C8—1462no—
471C2×C8—1462no—
472C2×C8—1462no—
473C2×C8—1462no—
474C2×C8—1462no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—734no—
478C4⋊C4—734no—
479C4⋊C4—1462no—
480C4⋊C4—1462no—
481C4⋊C4—1462no—
482C4⋊C4—1462no—
483C4⋊C4—1462no—
484C4⋊C4—1462no—
485C4⋊C4—1462no—
486C4⋊C4C22⋊#6440814yes—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—734no—
491C42—734no—
492C42—734no—
493C42—1462no—
494C42—1462no—
495C42—1462no—
496C42—1462no—
497C42—1462no—
498C42—1462no—
499C23—732no—
500C23—732no—
501C23—732no—
502C23—1461no—
503C23—1461no—
504C23—1461no—
505C23—1461no—
506C23C23×#6440811yescharacteristic
507Q8—28no—
508Q8—1468no—
509Q8—1468no—
510Q8—1468no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—2924no—
527C2×C4—2924no—
528C2×C4—2924no—
529C2×C4—732no—
530C2×C4—732no—
531C2×C4—731no—
532C2×C4—731no—
533C2×C4—732no—
534C2×C4—732no—
535C2×C4—732no—
536C2×C4—732no—
537C2×C4—732no—
538C2×C4—732no—
539C2×C4—1468no—
540C2×C4—1464no—
541C2×C4—1464no—
542C2×C4—1468no—
543C2×C4—1468no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4C2×C22⋊#6440812yes—
557C2×C4C2×C22⋊#6440812yes—
558C2×C4C2×C22⋊#6440812yes—
559C2×C4Q8×#6440812yes—
560C2×C4D73⋊M1612yes—
561C2×C4D73⋊M1612yes—
562C8—2922no—
563C8—2922no—
564C8—2922no—
565C8—1464no—
566C22—734no—
567C22—1462no—
568C22—1462no—
569C22—1462no—
570C22—1462no—
571C22—1462no—
572C22—1462no—
573C22C22×C22⋊#6440811yescharacteristic
574C22C2×Q8×#6440811yescharacteristic
575C22#1798965511yescharacteristic
576C22C2×D73⋊M1611yescharacteristic
577C22C2×D73⋊M1611yescharacteristic
578C22D73⋊#13111yescharacteristic
579C22#1798966611yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—41no—
587C4—2921no—
588C4—2921no—
589C4—732no—
590C4—732no—
591C4—734no—
592C4—1462no—
593C4—1462no—
594C4—1462no—
595C2—1464no—
596C2Q8×C22⋊#6440811yescharacteristic
597C2D73⋊#40711yescharacteristic
598C2#2262652111yescharacteristic
599C2D73⋊#44611yescharacteristic
600C2#2262673311yescharacteristic
601C2#2262682811yescharacteristic
602C2#2262683211yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
FittingC2×C4×Q8×C73C4
prime radicalC23×D73C22×C4
cosocleC22×Dic73C24
hypercenterC2×C4×Q8#64408
hypocenterC73#22510
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#17989639C2semidirect
2. #22626105C2semidirect
3. #22626104C2semidirect
4. #22626102C2semidirect
55. #14556648Q8semidirect
122. C2×#10596833C2×Q8semidirect
277. C2×D73#446semidirect
313. C73#22510semidirect

Maximal subgroups

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

subgroupindex
C2×#179896392
#226261052
#226261042
#226261022
#226257882
#226257862
#226256582
C4×#145566482
C2×C4×Q8×D732
#2251073