Finite Group Databasebeta

← #34026939

Subgroups — #34026939

Order 18688.

1116 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989617C212yesmaximal
2C2×#17989592C211yesmaximal, characteristic, complemented
3C2×#17989588C211yesmaximal, characteristic, complemented
4#22626033C212yesmaximal, complemented
5#22625877C212yesmaximal, complemented
6#22625876C212yesmaximal, complemented
7#22625864C211yesmaximal, characteristic, complemented
8#22625788C211yesmaximal, characteristic, complemented
9#22625788C211yesmaximal, characteristic, complemented
10#22625716C211yesmaximal, characteristic, complemented
11C4×#14556553C211yesmaximal, characteristic
12C22×#11924038C2212yes—
13C22×#11924036C2212yes—
14#17989617—28no—
15#17989592—24no—
16C2×#14556658C2211yescharacteristic
17C2×#14556658C2211yescharacteristic
18C2×#14556658C2212yes—
19C2×#14556658C2212yes—
20C2×#14556658C2211yescharacteristic
21C2×#14556658C2211yescharacteristic
22#17989588—24no—
23C2×C4⋊#10596833C2212yes—
24C2×C4⋊#10596833C2212yes—
25C2×#14556648C2212yes—
26C2×#14556648C2212yes—
27C2×#14556646C2212yes—
28#17989453C2212yes—
29#17989453C2212yes—
30#17989453C2212yes—
31C2×C4×D292C411yescharacteristic
32C2×#14556553C411yescharacteristic
33C2×#14556553C2211yescharacteristic
34C2×C42×D73C2211yescharacteristic
35C2×C42×D73C2211yescharacteristic
36C4⋊C4×Dic73C411yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4×#11924005C411yescharacteristic
39C4×#11924005C411yescharacteristic
40#17988434C411yescharacteristic
41#17988434C411yescharacteristic
42#17988434C2211yescharacteristic
43#17988434C2211yescharacteristic
44C42×Dic73C2211yescharacteristic
45C4×C4⋊C4×C73C411yescharacteristic
46C2×#11924038—24no—
47C2×#11924038—24no—
48C2×#11924038—42no—
49C2×#11924036—24no—
50C2×#11924036—24no—
51C2×#11924036—42no—
52C22×#10596833C2312yes—
53C22×#10596833C2312yes—
54C22×#10596833C2312yes—
55C22×#10596833C2312yes—
56#14556658—24no—
57#14556658—22no—
58#14556658—22no—
59#14556658—24no—
60#14556658—22no—
61#14556658D414yescomplemented
62C4⋊#10596833—24no—
63C4⋊#10596833—42no—
64#14556648—24no—
65#14556648—24no—
66#14556646—24no—
67C22×D292C2×C411yescharacteristic
68C22×C4×D73C2×C411yescharacteristic
69C22×C4×D73C2×C411yescharacteristic
70C22×C4×D73C2×C411yescharacteristic
71C22×C4×D73C2×C411yescharacteristic
72C22×C4×D73C2311yescharacteristic
73C22×C4×D73C2311yescharacteristic
74C4×D292—24no—
75#14556553—24no—
76#14556553D414yes—
77#14556553Q814yes—
78C2×C4⋊Dic73C2×C411yescharacteristic
79C2×#11924005C2×C411yescharacteristic
80C2×#11924005C2×C411yescharacteristic
81C2×#11924005C2×C411yescharacteristic
82C2×#11924005C2×C411yescharacteristic
83C2×#11924005C2×C411yescharacteristic
84C2×#11924005C2×C411yescharacteristic
85C2×C4×Dic73C2×C411yescharacteristic
86C2×C4×Dic73C2×C411yescharacteristic
87C2×C4×Dic73C2×C411yescharacteristic
88C2×C4×Dic73C2×C411yescharacteristic
89C2×C4×Dic73C2×C411yescharacteristic
90C2×C4×Dic73C2311yescharacteristic
91C2×C4×Dic73C2311yescharacteristic
92C2×C4×Dic73C2311yescharacteristic
93C2×C4×Dic73C2311yescharacteristic
94C2×C4×Dic73C2311yescharacteristic
95C42×D73—22no—
96C42×D73—22no—
97C42×D73—22no—
98C42×D73—22no—
99#14556498C2×C411yescharacteristic
100#14556498C2×C411yescharacteristic
101#14556498C2×C411yescharacteristic
102#14556498C2×C411yescharacteristic
103C2×C4⋊C4×C73C2×C411yescharacteristic
104C2×C4⋊C4×C73C2×C411yescharacteristic
105C2×C42×C73C2×C411yescharacteristic
106C2×C42×C73C2×C411yescharacteristic
107C2×C42×C73C2×C411yescharacteristic
108#95×C73C2×C411yescharacteristic
109#95×C73C2×C411yescharacteristic
110#11924038—48no—
111#11924036—48no—
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×#10596833—22no—
123C2×#10596833C2×D414yes—
124C23×D73C22×C411yescharacteristic
125C23×D73C22×C411yescharacteristic
126C2×D292—22no—
127C2×D292—22no—
128C2×D292C4⋊C414yes—
129C22×Dic73C22×C411yescharacteristic
130C22×Dic73C22×C411yescharacteristic
131C22×Dic73C22×C411yescharacteristic
132C22×Dic73C22×C411yescharacteristic
133C22×Dic73C22×C411yescharacteristic
134C22×Dic73C2411yescharacteristic
135C2×C4×D73—22no—
136C2×C4×D73—22no—
137C2×C4×D73—21no—
138C2×C4×D73—21no—
139C2×C4×D73—21no—
140C2×C4×D73—21no—
141C2×C4×D73—21no—
142C2×C4×D73—21no—
143C2×C4×D73—21no—
144C2×C4×D73—21no—
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—41no—
153C2×C4×D73—41no—
154C2×C4×D73M1614yescomplemented
155C2×C4×D73C2×D412yes—
156C2×C4×D73C2×D412yes—
157C2×C4×D73C2×Q812yes—
158C2×C4×D73C2×Q812yes—
159C4⋊Dic73—22no—
160#11924005—22no—
161#11924005—22no—
162#11924005—22no—
163#11924005C4⋊C414yes—
164#11924005C4⋊C414yes—
165#11924005M1614yescomplemented
166C4×Dic73—22no—
167C4×Dic73—22no—
168C4×Dic73—22no—
169C4×Dic73—22no—
170C4×Dic73—21no—
171C4×Dic73—21no—
172C4×Dic73—21no—
173C4×Dic73—21no—
174C4×Dic73—21no—
175C4×Dic73—21no—
176C4×Dic73—41no—
177C4×Dic73C2×D412yes—
178C4×Dic73C2×D412yes—
179C4×Dic73C2×Q812yes—
180C4×Dic73#4112yes—
181C22×C4×C73C22×C411yescharacteristic
182C22×C4×C73C22×C411yescharacteristic
183C22×C4×C73C22×C411yescharacteristic
184C22×C4×C73C22×C411yescharacteristic
185C22×C4×C73C22×C411yescharacteristic
186C22×C4×C73C22×C411yescharacteristic
187C22×C4×C73C22×C411yescharacteristic
188C4⋊C4×C73—22no—
189C4⋊C4×C73C4⋊C414yes—
190C42×C73—22no—
191C42×C73—22no—
192C42×C73—22no—
193C23×C73C23×C411yescharacteristic
194C22×D73—21no—
195C22×D73—21no—
196C22×D73—21no—
197C22×D73—21no—
198C22×D73—21no—
199C22×D73—21no—
200C22×D73—21no—
201C22×D73—21no—
202C22×D73—21no—
203C22×D73—21no—
204C22×D73C2×C4⋊C412yes—
205C22×D73C2×C4⋊C412yes—
206C22×D73C2×M1612yes—
207C22×D73C2×M1612yes—
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×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73C2×C4⋊C412yes—
224C2×C4×C73C2×C4⋊C412yes—
225C2×C4×C73C2×C4⋊C412yes—
226C2×C4×C73C4×D412yes—
227C2×C4×C73C2×M1612yes—
228C2×C4×C73C2×M1612yes—
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×C4⋊C412yes—
240C2×Dic73C4×D412yes—
241C2×Dic73C2×M1612yes—
242C2×Dic73C2×M1612yes—
243C2×Dic73#13112yes—
244C2×Dic73C22×D411yescharacteristic
245C2×Dic73C22×D411yescharacteristic
246C2×Dic73C22×Q811yescharacteristic
247C2×Dic73C2×#4111yescharacteristic
248C2×Dic73H(5,𝔽2)11yescharacteristic
249C2×Dic732−1+411yescharacteristic
250D292—28no—
251C4×D73—22no—
252C4×D73—22no—
253C4×D73—22no—
254C4×D73—22no—
255C4×D73—22no—
256C4×D73—22no—
257C4×D73—22no—
258C4×D73—22no—
259C4×D73—28no—
260C4×D73—44no—
261C4×D73—44no—
262C4×D73—44no—
263#10596833—24no—
264#10596833—42no—
265#10596833—42no—
266#10596833—42no—
267C22×C73C22×C4⋊C411yescharacteristic
268C22×C73C2×C4×D411yescharacteristic
269C22×C73#52011yescharacteristic
270C22×C73C22×M1611yescharacteristic
271C22×C73C22×M1611yescharacteristic
272C22×C73C2×#13111yescharacteristic
273C22×C73#56811yescharacteristic
274C2×D73—22no—
275C2×D73—22no—
276C2×D73—22no—
277C2×D73—22no—
278C2×D73—22no—
279C2×D73—22no—
280C2×D73—22no—
281C2×D73—22no—
282C2×D73—22no—
283C2×D73—41no—
284C2×D73—41no—
285C2×D73—41no—
286C2×D73—41no—
287C2×D73#42314yescomplemented
288C4×C73—22no—
289C4×C73—22no—
290C4×C73—22no—
291C4×C73—22no—
292C4×C73—22no—
293C4×C73—22no—
294C4×C73—41no—
295Dic73—22no—
296Dic73—22no—
297Dic73—41no—
298Dic73—41no—
299Dic73#43514yescomplemented
300Dic73D4211yescharacteristic
301Dic73#54711yescharacteristic
302Dic73D4×Q811yescharacteristic
303Dic73#55411yescharacteristic
304#22648—731nomaximal, Sylow
305C2×C73D4×C4⋊C411yescharacteristic
306C2×C73C2×#42311yescharacteristic
307C2×C73#290611yescharacteristic
308C2×C73C2×#43511yescharacteristic
309C2×C73D4×M1611yescharacteristic
310C2×C73#298911yescharacteristic
311C2×C73#299011yescharacteristic
312D73—24no—
313D73—42no—
314C2×#435—732no—
315C2×#423—731no—
316C22×C4⋊C8—731no—
317C42×D4—731no—
318#1940—732no—
319#1851—732no—
320#1850—732no—
321#1842—731no—
322#1797—731no—
323#1797—731no—
324#1767—731no—
325C73#2264811yescharacteristic, complemented, Sylow
326C22×M16—732no—
327C23×C8—732no—
328C2×C4×D4—731no—
329C2×C4×D4—731no—
330C2×C4×D4—731no—
331C22×C42—731no—
332C22×C42—731no—
333#435—1468no—
334#423—1464no—
335C2×C4⋊C8—732no—
336C2×C4⋊C8—731no—
337C2×C4⋊C8—731no—
338C2×C4⋊C8—731no—
339C2×C4⋊C8—731no—
340C2×C4⋊C8—732no—
341C2×C4⋊C8—732no—
342C2×C4⋊C8—732no—
343C2×C4⋊C8—1464no—
344C2×C22⋊C8—732no—
345C2×C22⋊C8—732no—
346C2×#97—732no—
347C4×C4⋊C4—731no—
348C4×C4⋊C4—731no—
349C4×C4⋊C4#6440811yescharacteristic
350C4×C22⋊C4—731no—
351C4×C22⋊C4—731no—
352C4×C22⋊C4—731no—
353C4×C22⋊C4—731no—
354C4×C22⋊C4—731no—
355C4×C22⋊C4—731no—
356C43—731no—
357#336—732no—
358#336—732no—
359#336—732no—
360C22×D4—731no—
361C23×C4—731no—
362C23×C4—731no—
363C23×C4—731no—
364C23×C4—731no—
365C23×C4—731no—
366C23×C4—731no—
367C2×M16—2922no—
368C2×M16—1464no—
369C2×M16—1464no—
370C22×C8—2922no—
371C22×C8—732no—
372C22×C8—732no—
373C22×C8—732no—
374C22×C8—732no—
375C22×C8—1464no—
376C22×C8—1464no—
377C4×D4—734no—
378C4×D4—734no—
379C4×D4—1464no—
380C4×D4—1464no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4C2×#6440811yescharacteristic
383C2×C4⋊C4C2×#6440811yescharacteristic
384C2×C22⋊C4—731no—
385C2×C22⋊C4—731no—
386C2×C22⋊C4—731no—
387C2×C22⋊C4—731no—
388C2×C22⋊C4—731no—
389C2×C22⋊C4—731no—
390C2×C42—731no—
391C2×C42—731no—
392C2×C42—731no—
393C2×C42—731no—
394C2×C42—731no—
395C2×C42—731no—
396C2×C42—731no—
397C2×C42—731no—
398C2×C42—731no—
399C2×C42—731no—
400C2×C42—1462no—
401C2×C42—1462no—
402C2×C42—1462no—
403C2×C42—1462no—
404C2×C42C2×#6440811yescharacteristic
405C2×C42C2×#6440811yescharacteristic
406C2×C42C2×#6440811yescharacteristic
407C4⋊C8—2922no—
408C4⋊C8—734no—
409C4⋊C8—1462no—
410C4⋊C8—1462no—
411C4⋊C8—1464no—
412C4⋊C8—1462no—
413C4⋊C8—1464no—
414C4⋊C8—1464no—
415C22⋊C8—1464no—
416C22⋊C8—1464no—
417#97—1464no—
418#95—731no—
419#95—731no—
420#95—731no—
421#95—731no—
422#95C2×#6440811yescharacteristic
423#95C2×#6440811yescharacteristic
424C24—731no—
425C24—731no—
426C2×D4—734no—
427C2×D4—1462no—
428C2×D4—1462no—
429C22×C4—2921no—
430C22×C4—2921no—
431C22×C4—731no—
432C22×C4—731no—
433C22×C4—731no—
434C22×C4—731no—
435C22×C4—734no—
436C22×C4—732no—
437C22×C4—732no—
438C22×C4—732no—
439C22×C4—732no—
440C22×C4—731no—
441C22×C4—731no—
442C22×C4—1462no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4—1461no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1462no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1462no—
457C22×C4—1462no—
458C22×C4—1462no—
459C22×C4C22×#6440811yescharacteristic
460C22×C4C22×#6440811yescharacteristic
461C22×C4C22×#6440811yescharacteristic
462C22×C4C22×#6440811yescharacteristic
463C22×C4C22×#6440811yescharacteristic
464C22×C4C22×#6440811yescharacteristic
465C22×C4C22×#6440811yescharacteristic
466M16—2928no—
467C2×C8—2928no—
468C2×C8—734no—
469C2×C8—1462no—
470C2×C8—1462no—
471C2×C8—1462no—
472C2×C8—1462no—
473C2×C8—1462no—
474C2×C8—1462no—
475C2×C8—1462no—
476C2×C8—1462no—
477C2×C8—1462no—
478C2×C8—1462no—
479C2×C8—1462no—
480C4⋊C4—22no—
481C4⋊C4—1462no—
482C4⋊C4C4⋊#6440814yes—
483C22⋊C4—734no—
484C22⋊C4—734no—
485C22⋊C4—734no—
486C22⋊C4—1462no—
487C22⋊C4—1462no—
488C22⋊C4—1462no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—2921no—
493C42—732no—
494C42—732no—
495C42—732no—
496C42—732no—
497C42—1461no—
498C42—1461no—
499C42—1461no—
500C42—1461no—
501C42—1462no—
502C42—1461no—
503C42—1462no—
504C42—1462no—
505C42—1461no—
506C42—1462no—
507C23—732no—
508C23—732no—
509C23—732no—
510C23—732no—
511C23—1461no—
512C23—1461no—
513C23—1461no—
514C23—1461no—
515C23—1461no—
516C23—1461no—
517C23—1461no—
518C23—1461no—
519C23—1461no—
520C23—1461no—
521C23C23×#6440811yescharacteristic
522D4—1468no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—2924no—
539C2×C4—2924no—
540C2×C4—2924no—
541C2×C4—731no—
542C2×C4—731no—
543C2×C4—731no—
544C2×C4—731no—
545C2×C4—732no—
546C2×C4—731no—
547C2×C4—731no—
548C2×C4—732no—
549C2×C4—732no—
550C2×C4—732no—
551C2×C4—732no—
552C2×C4—1462no—
553C2×C4—1462no—
554C2×C4—1468no—
555C2×C4—1462no—
556C2×C4—1462no—
557C2×C4—1462no—
558C2×C4—1462no—
559C2×C4—1462no—
560C2×C4—1462no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4C2×C4⋊#6440812yes—
572C2×C4C2×C4⋊#6440812yes—
573C2×C4C2×C4⋊#6440812yes—
574C2×C4D4×#6440812yes—
575C2×C4D73⋊M1612yes—
576C2×C4D73⋊M1612yes—
577C8—2922no—
578C8—2922no—
579C8—2922no—
580C8—1464no—
581C22—2921no—
582C22—2921no—
583C22—2921no—
584C22—2921no—
585C22—734no—
586C22—1462no—
587C22—1462no—
588C22—1462no—
589C22—1462no—
590C22—1462no—
591C22—1462no—
592C22—1462no—
593C22—1462no—
594C22—1462no—
595C22C22×C4⋊#6440811yescharacteristic
596C22C2×D4×#6440811yescharacteristic
597C22#1798965611yescharacteristic
598C22C2×D73⋊M1611yescharacteristic
599C22C2×D73⋊M1611yescharacteristic
600C22D73⋊#13111yescharacteristic
601C22#1798966711yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—22no—
608C4—41no—
609C4—2921no—
610C4—2921no—
611C4—734no—
612C4—731no—
613C4—731no—
614C4—731no—
615C4—731no—
616C4—1462no—
617C4—1462no—
618C2—2922no—
619C2—1464no—
620C2D4×C4⋊#6440811yescharacteristic
621C2D73⋊#42311yescharacteristic
622C2#2262657911yescharacteristic
623C2D73⋊#43511yescharacteristic
624C2#2262667111yescharacteristic
625C2#2262685511yescharacteristic
626C2#2262686011yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
FittingC4×C4⋊C4×C73C4
prime radicalC22×D292C2×C4
cosocleC22×Dic73C24
hypercenterC4×C4⋊C4#64408
hypocenterC73#22648
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#17989592C2semidirect
3. C2×#17989588C2semidirect
4. #22626033C2semidirect
5. #22625877C2semidirect
6. #22625876C2semidirect
7. #22625864C2semidirect
8. #22625788C2semidirect
9. #22625788C2semidirect
10. #22625716C2semidirect
61. #14556658D4semidirect
154. C2×C4×D73M16semidirect
165. #11924005M16semidirect
287. C2×D73#423semidirect
299. Dic73#435semidirect
325. C73#22648semidirect

Maximal subgroups

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

subgroupindex
C2×#179896172
C2×#179895922
C2×#179895882
#226260332
#226258772
#226258762
#226258642
#226257882
#226257162
C4×#145565532
#2264873