Finite Group Databasebeta

← #34023893

Subgroups — #34023893

Order 18688.

1434 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989606C211yesmaximal, characteristic, complemented
2C2×#17989588C211yesmaximal, characteristic, complemented
3#22626102C212yesmaximal, complemented
4#22626054C212yesmaximal, complemented
5#22625907C212yesmaximal, complemented
6#22625905C212yesmaximal, complemented
7#22625786C212yesmaximal, complemented
8#22625785C212yesmaximal, complemented
9C2×#17989055C211yesmaximal, characteristic
10C22×#11924036C2212yes—
11#17989606—24no—
12C2×#14556658C2212yes—
13#17989588—24no—
14C2×C4⋊#10596833C2212yes—
15C2×C4⋊#10596833C2212yes—
16C2×#14556648C2212yes—
17C2×#14556648C2214yes—
18C2×C22⋊#10596833C2214yes—
19C2×C4×#10596833C2212yes—
20#17989453C2214yes—
21#17989453C2214yes—
22C22×#11924028C411yescharacteristic
23#17989055—24no—
24#17989055—24no—
25C2×#14556556C411yescharacteristic
26C2×#14556556C2211yescharacteristic
27C2×#14556553C411yescharacteristic
28C2×#14556553C2211yescharacteristic
29C2×#14556551C411yescharacteristic
30C2×#14556551C411yescharacteristic
31C2×#14556551C2211yescharacteristic
32C2×#14556551C2211yescharacteristic
33C2×#14556548C411yescharacteristic
34C2×#14556548C411yescharacteristic
35C22×C4×Dic73C2211yescharacteristic
36C2×C42×D73C2211yescharacteristic
37C2×C42×D73C2211yescharacteristic
38C2×#117×C73C411yescharacteristic
39C2×#11924036—24no—
40C2×#11924036—44no—
41C22×#10596833C2314yes—
42C22×#10596833C2314yes—
43#14556658—24no—
44C4⋊#10596833—24no—
45C4⋊#10596833—24no—
46#14556648—24no—
47#14556648—44no—
48C22⋊#10596833—44no—
49C4×#10596833—24no—
50C2×#11924028—24no—
51C2×#11924028—24no—
52C2×#11924028C818yescomplemented
53C22×Dic146C2×C411yescharacteristic
54C22×D292C2×C411yescharacteristic
55C22×C22⋊D73C2×C411yescharacteristic
56C22×C22⋊D73C2×C411yescharacteristic
57C23×Dic73C2×C411yescharacteristic
58C23×Dic73C2311yescharacteristic
59C22×C4×D73C2×C411yescharacteristic
60C22×C4×D73C2×C411yescharacteristic
61C22×C4×D73C2×C411yescharacteristic
62C22×C4×D73C2×C411yescharacteristic
63C22×C4×D73C2311yescharacteristic
64C22×C4×D73C2311yescharacteristic
65#14556556—24no—
66#14556556—22no—
67#14556556—22no—
68#14556553—24no—
69#14556553—22no—
70#14556553—22no—
71#14556551—24no—
72#14556551—22no—
73#14556551—22no—
74#14556551—24no—
75#14556551—22no—
76#14556551—22no—
77#14556548—24no—
78#14556548—24no—
79C2×#11924007C2×C411yescharacteristic
80C2×#11924007C2×C411yescharacteristic
81C2×#11924007C2×C411yescharacteristic
82C2×#11924007C2×C411yescharacteristic
83C2×#11924005C2×C411yescharacteristic
84C2×#11924005C2×C411yescharacteristic
85C2×#11924005C2×C411yescharacteristic
86C2×#11924005C2×C411yescharacteristic
87C2×C4×Dic73—22no—
88C2×C4×Dic73—22no—
89C2×C4×Dic73C2×C411yescharacteristic
90C2×C4×Dic73C2×C411yescharacteristic
91C2×C4×Dic73C2×C411yescharacteristic
92C2×C4×Dic73C2×C411yescharacteristic
93C2×C4×Dic73C2311yescharacteristic
94C2×C4×Dic73C2311yescharacteristic
95C2×C4×Dic73C2311yescharacteristic
96C2×C4×Dic73C2311yescharacteristic
97C42×D73—22no—
98C42×D73—22no—
99C42×D73D414yes—
100C42×D73D414yes—
101C23×C4×C73C2×C411yescharacteristic
102#117×C73—24no—
103C2×C4⋊C4×C73C2×C411yescharacteristic
104C2×C4⋊C4×C73C2×C411yescharacteristic
105C2×C22⋊C4×C73C2×C411yescharacteristic
106C2×C22⋊C4×C73C2×C411yescharacteristic
107C2×C42×C73C2×C411yescharacteristic
108C2×C42×C73C2×C411yescharacteristic
109#11924036—48no—
110C2×#10596833—24no—
111C2×#10596833—24no—
112C2×#10596833—24no—
113C2×#10596833—24no—
114C2×#10596833—24no—
115C2×#10596833—24no—
116C23×D73C22×C411yescharacteristic
117C23×D73C22×C411yescharacteristic
118#11924028—232no—
119C2×Dic146—22no—
120C2×Dic146—22no—
121C2×Dic146C2×C814yescomplemented
122C2×D292—22no—
123C2×D292—22no—
124C2×D292C2×C814yes—
125C2×C22⋊D73—22no—
126C2×C22⋊D73—22no—
127C2×C22⋊D73—22no—
128C2×C22⋊D73—22no—
129C2×C22⋊D73C2×C814yes—
130C2×C22⋊D73C2×C814yescomplemented
131C22×Dic73—22no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—22no—
135C22×Dic73—22no—
136C22×Dic73—42no—
137C22×Dic73C22×C411yescharacteristic
138C22×Dic73C22×C411yescharacteristic
139C22×Dic73C22×C411yescharacteristic
140C22×Dic73C22×C411yescharacteristic
141C22×Dic73C22×C411yescharacteristic
142C22×Dic73C2411yescharacteristic
143C2×C4×D73—22no—
144C2×C4×D73—22no—
145C2×C4×D73—22no—
146C2×C4×D73—22no—
147C2×C4×D73—22no—
148C2×C4×D73—21no—
149C2×C4×D73—21no—
150C2×C4×D73—22no—
151C2×C4×D73—22no—
152C2×C4×D73—42no—
153C2×C4×D73—42no—
154C2×C4×D73C22⋊C414yes—
155C2×C4×D73C22⋊C414yes—
156C2×C4×D73C2×C814yes—
157C2×C4×D73C2×C814yescomplemented
158C2×C4×D73M1614yes—
159C2×C4×D73C2×D412yes—
160C2×C4×D73C2×D412yes—
161#11924007—22no—
162#11924007—22no—
163#11924007—22no—
164#11924007—22no—
165#11924005—22no—
166#11924005—22no—
167#11924005—22no—
168#11924005—22no—
169C4×Dic73—22no—
170C4×Dic73—22no—
171C4×Dic73—22no—
172C4×Dic73—21no—
173C4×Dic73—21no—
174C4×Dic73—21no—
175C4×Dic73—21no—
176C4×Dic73C22⋊C414yes—
177C4×Dic73C2×D412yes—
178C4×Dic73C2×D412yes—
179C4×Dic73C2×D412yes—
180C4×Dic73C2×D412yes—
181C24×C73C22×C411yescharacteristic
182C22×C4×C73—22no—
183C22×C4×C73—22no—
184C22×C4×C73C2×C814yescomplemented
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—
193C22⋊C4×C73—22no—
194C22⋊C4×C73—22no—
195C42×C73—22no—
196C42×C73C22⋊C414yes—
197C23×C73—21no—
198C23×C73—22no—
199C23×C73—21no—
200C23×C73—22no—
201C23×C73C22×C812yes—
202C23×C73C23×C411yescharacteristic
203C22×D73—22no—
204C22×D73—22no—
205C22×D73—21no—
206C22×D73—22no—
207C22×D73—21no—
208C22×D73—22no—
209C22×D73C2×C22⋊C412yes—
210C22×D73C22×C812yes—
211C22×D73C22×C812yes—
212C22×D73C2×M1612yes—
213C2×C4×C73—21no—
214C2×C4×C73—22no—
215C2×C4×C73—21no—
216C2×C4×C73—22no—
217C2×C4×C73—28no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—22no—
221C2×C4×C73—22no—
222C2×C4×C73C2×C22⋊C412yes—
223C2×C4×C73C2×C22⋊C412yes—
224C2×C4×C73C2×C22⋊C412yes—
225C2×C4×C73C22×C812yes—
226C2×C4×C73C22×C812yes—
227C2×C4×C73C2×M1612yes—
228C22⋊D73—28no—
229C22⋊D73—28no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—22no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—22no—
236C2×Dic73—22no—
237C2×Dic73—44no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—42no—
241C2×Dic73C2×C22⋊C412yes—
242C2×Dic73C2×C22⋊C412yes—
243C2×Dic73C22×C812yes—
244C2×Dic73C22×C812yes—
245C2×Dic73C2×M1612yes—
246C2×Dic73C22×D411yescharacteristic
247C2×Dic73C22×D411yescharacteristic
248C2×Dic73H(5,𝔽2)12yes—
249C2×Dic732−1+412yes—
250D292—28no—
251C4×D73—28no—
252C4×D73—22no—
253C4×D73—22no—
254C4×D73—24no—
255C4×D73—44no—
256C4×D73—44no—
257C4×D73—41no—
258C4×D73—41no—
259C4×D73—42no—
260C4×D73C22⋊C8116yescomplemented
261Dic146—28no—
262#10596833—44no—
263#10596833—44no—
264C22×C73—22no—
265C22×C73—24no—
266C22×C73—41no—
267C22×C73—41no—
268C22×C73—42no—
269C22×C73C22×C22⋊C411yescharacteristic
270C22×C73#52012yes—
271C22×C73C23×C811yescharacteristic
272C22×C73C22×M1611yescharacteristic
273C22×C73#56812yes—
274C2×D73—22no—
275C2×D73—22no—
276C2×D73—24no—
277C2×D73—24no—
278C2×D73—22no—
279C2×D73—24no—
280C2×D73—41no—
281C2×D73—41no—
282C2×D73—42no—
283C2×D73C2×C22⋊C814yes—
284C4×C73—22no—
285C4×C73—22no—
286C4×C73—22no—
287C4×C73—41no—
288C4×C73—41no—
289C4×C73C2×C22⋊C814yes—
290Dic73—22no—
291Dic73—22no—
292Dic73—41no—
293Dic73—41no—
294Dic73C2×C22⋊C814yes—
295Dic73#53511yescharacteristic
296Dic73#53611yescharacteristic
297Dic73#53712yes—
298#21847—731nomaximal, Sylow
299C2×C73—42no—
300C2×C73#231111yescharacteristic
301C2×C73C22×C22⋊C811yescharacteristic
302C2×C73#287911yescharacteristic
303C2×C73#298812yes—
304C2×C73#298912yes—
305D73—24no—
306D73—42no—
307C2×C4×#41—731no—
308C2×#431—731no—
309C22×C4⋊C8—731no—
310#1984—732no—
311#1954—732no—
312#1869—732no—
313#1869—732no—
314#1797—732no—
315#1797—732no—
316C73#2184711yescharacteristic, complemented, Sylow
317C22×#41—731no—
318C23×C8—732no—
319C4×#41—1464no—
320C4×#41—1464no—
321C2×C4×Q8—731no—
322C2×C4×Q8—731no—
323C2×C4×D4—731no—
324C2×C4×D4—731no—
325C2×C4×D4—731no—
326C2×C4×D4—731no—
327C2×C4×D4—731no—
328C2×C4×D4—731no—
329C2×#117—731no—
330C2×#117—731no—
331C2×#117#6440811yescharacteristic
332C22×C42—731no—
333C22×C42—731no—
334C22×C42—731no—
335#431—1464no—
336C2×C4⋊C8—732no—
337C2×C4⋊C8—732no—
338C2×C4⋊C8—732no—
339C2×C4⋊C8—1464no—
340C2×C22⋊C8—734no—
341C2×C22⋊C8—734no—
342C2×C22⋊C8—732no—
343C2×C4×C8—732no—
344#336—734no—
345#336—734no—
346C2×#41—738no—
347C2×#41—1464no—
348C2×#41—1464no—
349C22×Q8—731no—
350C22×D4—731no—
351C22×D4—731no—
352C22×D4—731no—
353C23×C4—731no—
354C23×C4—731no—
355C23×C4—731no—
356C23×C4—731no—
357C23×C4—731no—
358C23×C4—731no—
359C23×C4—731no—
360C23×C4—731no—
361C23×C4C2×#6440811yescharacteristic
362C22×C8—2924no—
363C22×C8—734no—
364C22×C8—734no—
365C22×C8—1464no—
366C4×Q8—1464no—
367C4×Q8—1462no—
368C4×Q8—1462no—
369C4×D4—1464no—
370C4×D4—1462no—
371C4×D4—1462no—
372C4×D4—1464no—
373C4×D4—1462no—
374C4×D4—1462no—
375C4×D4—1464no—
376C4×D4—1462no—
377C4×D4—1462no—
378#117—24no—
379#117—1464no—
380#117—1464no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—731no—
383C2×C4⋊C4—731no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4C2×#6440811yescharacteristic
386C2×C4⋊C4C2×#6440811yescharacteristic
387C2×C22⋊C4—731no—
388C2×C22⋊C4—731no—
389C2×C22⋊C4—731no—
390C2×C22⋊C4—731no—
391C2×C22⋊C4C2×#6440811yescharacteristic
392C2×C22⋊C4C2×#6440811yescharacteristic
393C2×C42—731no—
394C2×C42—731no—
395C2×C42—731no—
396C2×C42—734no—
397C2×C42—734no—
398C2×C42—731no—
399C2×C42—731no—
400C2×C42—731no—
401C2×C42—731no—
402C2×C42—731no—
403C2×C42—1462no—
404C2×C42—1462no—
405C2×C42—1462no—
406C2×C42—1462no—
407C2×C42C2×#6440811yescharacteristic
408C2×C42C2×#6440811yescharacteristic
409C4⋊C8—1464no—
410C4⋊C8—1464no—
411C4⋊C8—1464no—
412C22⋊C8—2924no—
413C22⋊C8—2924no—
414C22⋊C8—1464no—
415C4×C8—1464no—
416C24—731no—
417C24—731no—
418C24C22×#6440811yescharacteristic
419#41—14632no—
420C2×Q8—734no—
421C2×Q8—1462no—
422C2×Q8—1462no—
423C2×D4—734no—
424C2×D4—734no—
425C2×D4—734no—
426C2×D4—1462no—
427C2×D4—1462no—
428C2×D4—1462no—
429C2×D4—1462no—
430C2×D4—1462no—
431C2×D4—1462no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—2922no—
435C22×C4—2922no—
436C22×C4—2922no—
437C22×C4—734no—
438C22×C4—731no—
439C22×C4—731no—
440C22×C4—734no—
441C22×C4—734no—
442C22×C4—734no—
443C22×C4—731no—
444C22×C4—731no—
445C22×C4—734no—
446C22×C4—732no—
447C22×C4—732no—
448C22×C4—731no—
449C22×C4—731no—
450C22×C4—1462no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1462no—
454C22×C4—1462no—
455C22×C4—1462no—
456C22×C4—1462no—
457C22×C4—1462no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1462no—
461C22×C4—1462no—
462C22×C4—1462no—
463C22×C4—1462no—
464C22×C4#1192403614yescomplemented
465C22×C4C22×#6440811yescharacteristic
466C22×C4C22×#6440811yescharacteristic
467C22×C4C22×#6440811yescharacteristic
468C22×C4C22×#6440811yescharacteristic
469C22×C4C22×#6440811yescharacteristic
470C22×C4C22×#6440811yescharacteristic
471C2×C8—2928no—
472C2×C8—1464no—
473C2×C8—1464no—
474C2×C8—1464no—
475C2×C8—1464no—
476C2×C8—1464no—
477C2×C8—1464no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—1462no—
481C4⋊C4—1462no—
482C4⋊C4—1462no—
483C4⋊C4—1462no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—1462no—
487C22⋊C4—1462no—
488C22⋊C4—1462no—
489C22⋊C4—1462no—
490C42—22no—
491C42—734no—
492C42—732no—
493C42—732no—
494C42—732no—
495C42—732no—
496C42—1461no—
497C42—1461no—
498C42—1462no—
499C42—1461no—
500C42—1461no—
501C42—1462no—
502C42—1462no—
503C42C22⋊#6440814yes—
504C23—21no—
505C23—22no—
506C23—21no—
507C23—22no—
508C23—732no—
509C23—732no—
510C23—732no—
511C23—732no—
512C23—1461no—
513C23—1462no—
514C23—1461no—
515C23—1462no—
516C23—1462no—
517C23—1462no—
518C23C2×#1192403612yes—
519C23C23×#6440811yescharacteristic
520Q8—1468no—
521D4—1468no—
522D4—1468no—
523D4—1468no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—2924no—
534C2×C4—2921no—
535C2×C4—2922no—
536C2×C4—2921no—
537C2×C4—2924no—
538C2×C4—2921no—
539C2×C4—2922no—
540C2×C4—2921no—
541C2×C4—2924no—
542C2×C4—731no—
543C2×C4—731no—
544C2×C4—732no—
545C2×C4—732no—
546C2×C4—732no—
547C2×C4—732no—
548C2×C4—732no—
549C2×C4—7316no—
550C2×C4—732no—
551C2×C4—732no—
552C2×C4—1462no—
553C2×C4—1464no—
554C2×C4—1462no—
555C2×C4—1461no—
556C2×C4—1461no—
557C2×C4—1462no—
558C2×C4—1462no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1462no—
562C2×C4—1468no—
563C2×C4C2×C22⋊#6440812yes—
564C2×C4C2×C22⋊#6440812yes—
565C2×C4C2×C22⋊#6440812yes—
566C2×C4C2×#1192403612yes—
567C2×C4C2×#1192403612yes—
568C2×C4D73⋊M1612yes—
569C8—2924no—
570C8—2924no—
571C22—24no—
572C22—22no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—2922no—
577C22—2921no—
578C22—2921no—
579C22—734no—
580C22—1464no—
581C22—1462no—
582C22—1464no—
583C22—1464no—
584C22—1462no—
585C22—1462no—
586C22C22×C22⋊#6440811yescharacteristic
587C22#1798965612yes—
588C22C22×#1192403611yescharacteristic
589C22C2×D73⋊M1611yescharacteristic
590C22#1798966612yes—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—2921no—
597C4—2921no—
598C4—731no—
599C4—732no—
600C4—731no—
601C4—734no—
602C4—1462no—
603C4—1462no—
604C4#1798955514yes—
605C2—42no—
606C2—2922no—
607C2—1464no—
608C2#2262640611yescharacteristic
609C2C2×#1798955511yescharacteristic
610C2#2262651611yescharacteristic
611C2#2262684812yes—
612C2#2262685312yes—
subgroupG/H
center, FrattiniC23C23×#64408
commutatorC22×C73C23×C8
FittingC2×#117×C73C4
socleC23×C73C23×C4
prime radicalC22×#11924028C4
cosocleC22×Dic73C24
hypercenterC2×#117#64408
hypocenterC73#21847
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#17989606C2semidirect
2. C2×#17989588C2semidirect
3. #22626102C2semidirect
4. #22626054C2semidirect
5. #22625907C2semidirect
6. #22625905C2semidirect
7. #22625786C2semidirect
8. #22625785C2semidirect
52. C2×#11924028C8semidirect
121. C2×Dic146C2×C8semidirect
130. C2×C22⋊D73C2×C8semidirect
157. C2×C4×D73C2×C8semidirect
184. C22×C4×C73C2×C8semidirect
260. C4×D73C22⋊C8semidirect
316. C73#21847semidirect
464. C22×C4#11924036semidirect

Maximal subgroups

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

subgroupindex
C2×#179896062
C2×#179895882
#226261022
#226260542
#226259072
#226259052
#226257862
#226257852
C2×#179890552
#2184773