Finite Group Databasebeta

← #34020740

Subgroups — #34020740

Order 18688.

954 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989619C211yesmaximal, characteristic
2C2×#17989616C211yesmaximal, characteristic
3#22625730C211yesmaximal, characteristic, complemented
4#22625730C211yesmaximal, characteristic, complemented
5#22625717C211yesmaximal, characteristic, complemented
6#22625673C211yesmaximal, characteristic, complemented
7#22625672C211yesmaximal, characteristic, complemented
8#22625669C211yesmaximal, characteristic, complemented
9#22625668C211yesmaximal, characteristic, complemented
10#22625661C211yesmaximal, characteristic, complemented
11#22625660C211yesmaximal, characteristic, complemented
12#22625648C211yesmaximal, characteristic, complemented
13#22625598C211yesmaximal, characteristic, complemented
14#22625597C211yesmaximal, characteristic, complemented
15#22612391C211yesmaximal, characteristic
16C22×#11924038C2211yescharacteristic
17C22×#11924037C2211yescharacteristic
18C22×#11924036C2211yescharacteristic
19C23×#10596833C2211yescharacteristic
20#17989619C418yescomplemented
21#17989616C418yescomplemented
22C2×C4⋊#10596833C2211yescharacteristic
23C2×C4⋊#10596833C2211yescharacteristic
24C2×#14556648C2211yescharacteristic
25C2×#14556648C2211yescharacteristic
26C2×C22⋊#10596833C2211yescharacteristic
27C2×C22⋊#10596833C2211yescharacteristic
28C2×#14556646C2211yescharacteristic
29C2×#14556646C2211yescharacteristic
30C2×#14556646C2211yescharacteristic
31C2×#14556646C2211yescharacteristic
32C2×#14556646C2211yescharacteristic
33C2×#14556646C2211yescharacteristic
34C2×C4×#10596833C2211yescharacteristic
35C2×C4×#10596833C2211yescharacteristic
36#17989453C2211yescharacteristic
37#17989453C2211yescharacteristic
38#17989453C2211yescharacteristic
39#17989453C2211yescharacteristic
40#17989453C2211yescharacteristic
41#17989453C2211yescharacteristic
42#17989453C2211yescharacteristic
43#17989453C2211yescharacteristic
44#17989453C2211yescharacteristic
45#17989453C2211yescharacteristic
46C2×#14556551C2211yescharacteristic
47C2×#14556549C2211yescharacteristic
48C2×#14556547C2211yescharacteristic
49#17988487C411yescharacteristic
50#17988487C411yescharacteristic
51#17988484C411yescharacteristic
52#17988482C411yescharacteristic
53#17988480C411yescharacteristic
54#17988478C411yescharacteristic
55#17988476C411yescharacteristic
56#17988438C2211yescharacteristic
57C22⋊C4×Dic73C2211yescharacteristic
58#17988434C2211yescharacteristic
59C42×Dic73C2211yescharacteristic
60#388×C73C411yescharacteristic
61C2×#11924038—41no—
62C2×#11924038—41no—
63C2×#11924038C2×C414yes—
64C2×#11924037—41no—
65C2×#11924037—41no—
66C2×#11924037C2×C414yes—
67C2×#11924036—41no—
68C2×#11924036—41no—
69C2×#11924036C2×C414yes—
70C22×#10596833—41no—
71C22×#10596833—41no—
72C22×#10596833C2×C414yes—
73C22×#10596833C2311yescharacteristic
74C22×#10596833C2311yescharacteristic
75C22×#10596833C2311yescharacteristic
76C22×#10596833C2311yescharacteristic
77C22×#10596833C2311yescharacteristic
78C22×#10596833C2311yescharacteristic
79C22×#10596833C2311yescharacteristic
80C22×#10596833C2311yescharacteristic
81C4⋊#10596833C2×C414yes—
82C4⋊#10596833C2×C414yes—
83#14556648C2×C414yes—
84#14556648C2×C414yes—
85C22⋊#10596833C2×C414yes—
86C22⋊#10596833C2×C414yes—
87#14556646—41no—
88#14556646—41no—
89#14556646—41no—
90#14556646—41no—
91#14556646C2×C414yes—
92#14556646C2×C414yes—
93C4×#10596833—41no—
94C4×#10596833—41no—
95C22×C22⋊D73C2×C411yescharacteristic
96C23×Dic73C2311yescharacteristic
97C22×C4×D73C2311yescharacteristic
98#14556551C2×C414yes—
99#14556551C2×C414yes—
100#14556549—41no—
101#14556549—41no—
102#14556547—41no—
103#14556547—41no—
104C2×C4⋊Dic73C2×C411yescharacteristic
105C2×#11924007C2×C411yescharacteristic
106C2×#11924007C2×C411yescharacteristic
107C2×#11924007C2×C411yescharacteristic
108C2×#11924007C2×C411yescharacteristic
109C2×#11924007C2×C411yescharacteristic
110C2×C22⋊Dic73C2×C411yescharacteristic
111C2×C22⋊Dic73C2×C411yescharacteristic
112C2×#11924005C2×C411yescharacteristic
113C2×#11924005C2×C411yescharacteristic
114C2×#11924005C2×C411yescharacteristic
115C2×#11924005C2×C411yescharacteristic
116C2×#11924005C2×C411yescharacteristic
117C2×C4×Dic73C2×C411yescharacteristic
118C2×C4×Dic73C2×C411yescharacteristic
119C2×C4×Dic73C2×C411yescharacteristic
120C2×C4×Dic73C2311yescharacteristic
121C2×C4×Dic73C2311yescharacteristic
122C2×C4×Dic73C2311yescharacteristic
123C2×C4×Dic73C2311yescharacteristic
124C2×C4×Dic73C2311yescharacteristic
125#14556498C2×C411yescharacteristic
126#14556498C2×C411yescharacteristic
127#14556498C2×C411yescharacteristic
128#14556498C2×C411yescharacteristic
129C2×C4⋊C4×C73C2×C411yescharacteristic
130C2×C22⋊C4×C73C2×C411yescharacteristic
131C2×C22⋊C4×C73C2×C411yescharacteristic
132C2×C22⋊C4×C73C2×C411yescharacteristic
133C2×C42×C73C2×C411yescharacteristic
134#95×C73C2×C411yescharacteristic
135#95×C73C2×C411yescharacteristic
136#11924038—44no—
137#11924037—44no—
138#11924036—44no—
139C2×#10596833—21no—
140C2×#10596833—21no—
141C2×#10596833—21no—
142C2×#10596833—21no—
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—44no—
156C2×#10596833C22×C412yes—
157C2×#10596833C22×C412yes—
158C2×#10596833C22×C412yes—
159C2×#10596833C22×C412yes—
160C2×#10596833C22×C412yes—
161C2×#10596833C22×C412yes—
162C2×#10596833C22×C412yes—
163C2×#10596833C22×C412yes—
164C23×D73C22×C411yescharacteristic
165C2×C22⋊D73—41no—
166C2×C22⋊D73—41no—
167C2×C22⋊D73C4214yes—
168C22×Dic73—21no—
169C22×Dic73—21no—
170C22×Dic73—21no—
171C22×Dic73—21no—
172C22×Dic73C22×C411yescharacteristic
173C22×Dic73C22×C411yescharacteristic
174C22×Dic73C22×C411yescharacteristic
175C22×Dic73C22×C411yescharacteristic
176C22×Dic73C22×C411yescharacteristic
177C22×Dic73C22×C411yescharacteristic
178C22×Dic73C22×C412yes—
179C22×Dic73C22×C412yes—
180C22×Dic73C2411yescharacteristic
181C2×C4×D73—21no—
182C2×C4×D73—21no—
183C2×C4×D73—21no—
184C2×C4×D73—21no—
185C2×C4×D73C22×C412yes—
186C2×C4×D73C22×C412yes—
187C4⋊Dic73—41no—
188#11924007—41no—
189#11924007—41no—
190#11924007—41no—
191#11924007—41no—
192#11924007C4214yes—
193C22⋊Dic73—41no—
194C22⋊Dic73—41no—
195#11924005—41no—
196#11924005—41no—
197#11924005—41no—
198#11924005—41no—
199#11924005C4214yes—
200C4×Dic73—21no—
201C4×Dic73—21no—
202C4×Dic73—21no—
203C4×Dic73—21no—
204C4×Dic73—21no—
205C4×Dic73—21no—
206C4×Dic73—21no—
207C4×Dic73—21no—
208C4×Dic73—41no—
209C4×Dic73—41no—
210C4×Dic73—41no—
211C4×Dic73C22×C412yes—
212C4×Dic73C22×C412yes—
213C24×C73C22×C411yescharacteristic
214C22×C4×C73C22×C411yescharacteristic
215C22×C4×C73C22×C411yescharacteristic
216C22×C4×C73C22×C411yescharacteristic
217C22×C4×C73C22×C411yescharacteristic
218C22×C4×C73C22×C411yescharacteristic
219C22×C4×C73C22×C411yescharacteristic
220C4⋊C4×C73—41no—
221C22⋊C4×C73—41no—
222C22⋊C4×C73—41no—
223C22⋊C4×C73C4214yes—
224C42×C73—41no—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—21no—
231C23×C73C2×C4212yes—
232C23×C73C23×C411yescharacteristic
233C22×D73—21no—
234C22×D73—21no—
235C22×D73—21no—
236C22×D73—21no—
237C22×D73—21no—
238C22×D73—21no—
239C22×D73C2×C4212yes—
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—21no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
251C2×C4×C73—21no—
252C2×C4×C73—21no—
253C2×C4×C73—21no—
254C2×C4×C73—21no—
255C2×C4×C73—21no—
256C2×C4×C73C2×C4212yes—
257C2×C4×C73C2×C4212yes—
258C22⋊D73—44no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—22no—
264C2×Dic73—22no—
265C2×Dic73—22no—
266C2×Dic73—22no—
267C2×Dic73—21no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—21no—
271C2×Dic73—21no—
272C2×Dic73—21no—
273C2×Dic73—21no—
274C2×Dic73—21no—
275C2×Dic73—21no—
276C2×Dic73—21no—
277C2×Dic73—21no—
278C2×Dic73—21no—
279C2×Dic73C2×C4212yes—
280C2×Dic73C2×C4212yes—
281C2×Dic73C23×C411yescharacteristic
282C2×Dic73C23×C411yescharacteristic
283C2×Dic73H(5,𝔽2)11yescharacteristic
284C2×Dic73H(5,𝔽2)11yescharacteristic
285C2×Dic73H(5,𝔽2)11yescharacteristic
286C2×Dic732−1+411yescharacteristic
287C4×D73—22no—
288C4×D73—22no—
289C4×D73—22no—
290C4×D73—22no—
291#10596833—22no—
292#10596833—22no—
293#10596833—22no—
294#10596833—22no—
295#10596833—22no—
296#10596833—22no—
297#10596833—22no—
298#10596833—22no—
299C22×C73—22no—
300C22×C73—22no—
301C22×C73—22no—
302C22×C73—41no—
303C22×C73—41no—
304C22×C73—41no—
305C22×C73—41no—
306C22×C73C22×C4211yescharacteristic
307C22×C73#51811yescharacteristic
308C22×C73#52011yescharacteristic
309C22×C73#56811yescharacteristic
310C22×C73#56811yescharacteristic
311C22×C73#56811yescharacteristic
312C22×C73#56811yescharacteristic
313C2×D73—22no—
314C2×D73—22no—
315C2×D73—22no—
316C2×D73—41no—
317C2×D73—41no—
318C2×D73—41no—
319C2×D73—41no—
320C4×C73—22no—
321C4×C73—22no—
322C4×C73—41no—
323C4×C73—41no—
324C4×C73—41no—
325C4×C73—41no—
326Dic73—22no—
327Dic73—22no—
328Dic73—41no—
329Dic73—41no—
330Dic73—41no—
331Dic73—41no—
332Dic73#51811yescharacteristic
333Dic73#51811yescharacteristic
334Dic73#52011yescharacteristic
335Dic73#52011yescharacteristic
336#20563—731nomaximal, Sylow
337C2×C73—42no—
338C2×C73#227511yescharacteristic
339C2×C73#287311yescharacteristic
340C2×C73#287511yescharacteristic
341C2×C73#298011yescharacteristic
342C2×C73#298011yescharacteristic
343C2×C73#299511yescharacteristic
344C2×C73#299511yescharacteristic
345D73—42no—
346C2×#435—731no—
347C2×#435—731no—
348#2293—731no—
349#1771—731no—
350#1771—731no—
351#1767—731no—
352#1753—731no—
353#1753—731no—
354#1752—731no—
355#1752—731no—
356#1750—731no—
357#1750—731no—
358#1745—731no—
359#1729—731no—
360#1729—731no—
361C73#2056311yescharacteristic, complemented, Sylow
362C22×M16—731no—
363C22×M16—731no—
364C23×C8—731no—
365C23×C8—731no—
366C2×C4×D4—731no—
367C2×#117—731no—
368C2×#117—731no—
369#435—738no—
370#435—738no—
371C2×C4⋊C8—731no—
372C2×C4⋊C8—731no—
373C2×C22⋊C8—731no—
374C2×C22⋊C8—731no—
375C2×C22⋊C8—731no—
376C2×C22⋊C8—731no—
377C2×#97—731no—
378C2×#97—731no—
379C2×#97—731no—
380C2×#97—731no—
381C2×#97—731no—
382C2×#97—731no—
383C2×C4×C8—731no—
384C2×C4×C8—731no—
385#390—731no—
386#390—731no—
387#389—731no—
388#389—731no—
389#388—731no—
390#388—731no—
391#388—731no—
392#388#6440811yescharacteristic
393C4×C4⋊C4—731no—
394C4×C22⋊C4—731no—
395C4×C22⋊C4—731no—
396C43—731no—
397#336—731no—
398#336—731no—
399#336—731no—
400#336—731no—
401#336—731no—
402#336—731no—
403#336—731no—
404#336—731no—
405#336—731no—
406#336—731no—
407C22×D4—731no—
408C23×C4—731no—
409C23×C4—731no—
410C2×M16—2921no—
411C2×M16—2921no—
412C2×M16—2921no—
413C2×M16—2921no—
414C2×M16—734no—
415C2×M16—734no—
416C22×C8—2921no—
417C22×C8—2921no—
418C22×C8—2921no—
419C22×C8—2921no—
420C22×C8—734no—
421C22×C8—734no—
422C22×C8—731no—
423C22×C8—731no—
424C22×C8—731no—
425C22×C8—731no—
426C22×C8—731no—
427C22×C8—731no—
428C22×C8—731no—
429C22×C8—731no—
430C4×D4—734no—
431C4×D4—734no—
432#117—2921no—
433#117—2921no—
434#117—2921no—
435#117—2921no—
436C2×C4⋊C4—731no—
437C2×C4⋊C4—731no—
438C2×C4⋊C4—731no—
439C2×C4⋊C4—731no—
440C2×C4⋊C4—731no—
441C2×C4⋊C4—731no—
442C2×C4⋊C4C2×#6440811yescharacteristic
443C2×C22⋊C4—731no—
444C2×C22⋊C4—731no—
445C2×C22⋊C4—731no—
446C2×C22⋊C4—731no—
447C2×C22⋊C4—731no—
448C2×C22⋊C4—731no—
449C2×C22⋊C4—731no—
450C2×C22⋊C4C2×#6440811yescharacteristic
451C2×C22⋊C4C2×#6440811yescharacteristic
452C2×C22⋊C4C2×#6440811yescharacteristic
453C2×C42—731no—
454C2×C42—731no—
455C2×C42—731no—
456C2×C42—731no—
457C2×C42—731no—
458C2×C42—731no—
459C2×C42—731no—
460C2×C42—731no—
461C2×C42C2×#6440811yescharacteristic
462C4⋊C8—734no—
463C4⋊C8—734no—
464C22⋊C8—734no—
465C22⋊C8—734no—
466C22⋊C8—734no—
467C22⋊C8—734no—
468#97—2921no—
469#97—2921no—
470#97—2921no—
471#97—2921no—
472#97—734no—
473#97—734no—
474C4×C8—2921no—
475C4×C8—2921no—
476#95—731no—
477#95—731no—
478#95—731no—
479#95—731no—
480#95C2×#6440811yescharacteristic
481#95C2×#6440811yescharacteristic
482C24—731no—
483C24C22×#6440811yescharacteristic
484C2×D4—2921no—
485C2×D4—2921no—
486C2×D4—734no—
487C22×C4—731no—
488C22×C4—731no—
489C22×C4—731no—
490C22×C4—732no—
491C22×C4—732no—
492C22×C4—732no—
493C22×C4—732no—
494C22×C4—731no—
495C22×C4—731no—
496C22×C4—731no—
497C22×C4—731no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4—1461no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4—1461no—
506C22×C4C22×#6440811yescharacteristic
507C22×C4C22×#6440811yescharacteristic
508C22×C4C22×#6440811yescharacteristic
509C22×C4C22×#6440811yescharacteristic
510C22×C4C22×#6440811yescharacteristic
511C22×C4C22×#6440811yescharacteristic
512M16—2924no—
513M16—2924no—
514C2×C8—2924no—
515C2×C8—2924no—
516C2×C8—732no—
517C2×C8—732no—
518C2×C8—732no—
519C2×C8—732no—
520C2×C8—732no—
521C2×C8—732no—
522C2×C8—732no—
523C2×C8—732no—
524C2×C8—1461no—
525C2×C8—1461no—
526C2×C8—1461no—
527C2×C8—1461no—
528C2×C8—1461no—
529C2×C8—1461no—
530C2×C8—1461no—
531C2×C8—1461no—
532C2×C8—1461no—
533C2×C8—1461no—
534C2×C8—1461no—
535C2×C8—1461no—
536C2×C8—1461no—
537C2×C8—1461no—
538C2×C8—1461no—
539C2×C8—1461no—
540C4⋊C4—41no—
541C4⋊C4—2921no—
542C4⋊C4—2921no—
543C4⋊C4—2921no—
544C4⋊C4—2921no—
545C4⋊C4—2921no—
546C4⋊C4—734no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C22⋊C4—2921no—
550C22⋊C4—2921no—
551C22⋊C4—2921no—
552C22⋊C4—2921no—
553C22⋊C4—2921no—
554C22⋊C4—2921no—
555C22⋊C4—734no—
556C22⋊C4C4×#6440814yes—
557C42—41no—
558C42—2921no—
559C42—2921no—
560C42—2921no—
561C42—732no—
562C42—732no—
563C42—1461no—
564C42—1461no—
565C42—1461no—
566C42—1461no—
567C42—1461no—
568C42—1461no—
569C42—1461no—
570C42—1461no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—732no—
578C23—1461no—
579C23—1461no—
580C23—1461no—
581C23—1461no—
582C23—1461no—
583C23—1461no—
584C23C2×C4×#6440812yes—
585C23C23×#6440811yescharacteristic
586D4—2924no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—731no—
604C2×C4—731no—
605C2×C4—731no—
606C2×C4—731no—
607C2×C4—731no—
608C2×C4—731no—
609C2×C4—732no—
610C2×C4—732no—
611C2×C4—1462no—
612C2×C4—1462no—
613C2×C4—1462no—
614C2×C4—1462no—
615C2×C4—1461no—
616C2×C4—1461no—
617C2×C4—1461no—
618C2×C4—1461no—
619C2×C4—1461no—
620C2×C4—1461no—
621C2×C4—1461no—
622C2×C4—1461no—
623C2×C4—1461no—
624C2×C4—1461no—
625C2×C4—1461no—
626C2×C4—1461no—
627C2×C4—1462no—
628C2×C4—1462no—
629C2×C4—1462no—
630C2×C4—1462no—
631C2×C4—1461no—
632C2×C4—1461no—
633C2×C4—1461no—
634C2×C4—1461no—
635C2×C4C2×C4×#6440812yes—
636C2×C4C2×C4×#6440812yes—
637C8—1462no—
638C8—1462no—
639C8—1462no—
640C8—1462no—
641C8—1462no—
642C8—1462no—
643C8—1462no—
644C8—1462no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—2921no—
653C22—2921no—
654C22—2921no—
655C22—2921no—
656C22—1462no—
657C22—1462no—
658C22—1462no—
659C22C22×C4×#6440811yescharacteristic
660C22#1798965411yescharacteristic
661C22#1798965611yescharacteristic
662C22#1798966511yescharacteristic
663C22#1798966611yescharacteristic
664C22#1798966611yescharacteristic
665C22#1798966611yescharacteristic
666C4—22no—
667C4—22no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—2921no—
673C4—2921no—
674C4—2921no—
675C4—2921no—
676C4—731no—
677C4—731no—
678C4—731no—
679C4—731no—
680C4—1462no—
681C4—1462no—
682C2—42no—
683C2—2922no—
684C2#2262637011yescharacteristic
685C2#2262649611yescharacteristic
686C2#2262650211yescharacteristic
687C2#2262681511yescharacteristic
688C2#2262681711yescharacteristic
689C2#2262688511yescharacteristic
690C2#2262688611yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutatorC22×C73C22×C42
Fitting#388×C73C4
socleC23×C73C23×C4
prime radicalC22×C22⋊D73C2×C4
cosocleC22×Dic73C24
hypercenter#388#64408
hypocenterC73#20563
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
3. #22625730C2semidirect
4. #22625730C2semidirect
5. #22625717C2semidirect
6. #22625673C2semidirect
7. #22625672C2semidirect
8. #22625669C2semidirect
9. #22625668C2semidirect
10. #22625661C2semidirect
11. #22625660C2semidirect
12. #22625648C2semidirect
13. #22625598C2semidirect
14. #22625597C2semidirect
20. #17989619C4semidirect
21. #17989616C4semidirect
361. C73#20563semidirect

Maximal subgroups

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

subgroupindex
C2×#179896192
C2×#179896162
#226257302
#226257172
#226256732
#226256722
#226256692
#226256682
#226256612
#226256602
#226256482
#226255982
#226255972
#226123912
#2056373