Finite Group Databasebeta

← #33112523

Subgroups — #33112523

Order 18688.

702 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22613214C211yesmaximal, characteristic
2#22610872C211yesmaximal, characteristic, complemented
3#22610544C211yesmaximal, characteristic, complemented
4#22610544C211yesmaximal, characteristic, complemented
5#22610542C211yesmaximal, characteristic, complemented
6#22610542C211yesmaximal, characteristic, complemented
7#22610227C211yesmaximal, characteristic
8#22610222C211yesmaximal, characteristic
9#22610115C211yesmaximal, characteristic, complemented
10#22610112C211yesmaximal, characteristic, complemented
11#22610110C211yesmaximal, characteristic, complemented
12#22610107C211yesmaximal, characteristic, complemented
13#22610049C211yesmaximal, characteristic, complemented
14#22610049C211yesmaximal, characteristic, complemented
15C73×#1985C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C22×C8×D73C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19C2×C4⋊#10596831C2211yescharacteristic
20C2×#14556520C2211yescharacteristic
21C2×#14556520C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556520C2211yescharacteristic
24C2×#14556503C2211yescharacteristic
25C2×#14556503C2211yescharacteristic
26C2×#14556503C2211yescharacteristic
27C2×#14556503C2211yescharacteristic
28C2×#14556503C2211yescharacteristic
29C2×#14556503C2211yescharacteristic
30C2×#14556502C2211yescharacteristic
31#17988526C411yescharacteristic
32#17988526C411yescharacteristic
33#17988513C411yescharacteristic
34#17988513C411yescharacteristic
35#17988504C411yescharacteristic
36#17988504C411yescharacteristic
37#17988498C411yescharacteristic
38#17988498C411yescharacteristic
39C4×#11924005C2211yescharacteristic
40C4×#11924005C2211yescharacteristic
41C4×#11924005C2211yescharacteristic
42C4×#11924005C2211yescharacteristic
43#17988433C2211yescharacteristic
44#17988342C2211yescharacteristic
45#17988342C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988341C2211yescharacteristic
49#17988341C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×C4×Q8×C73C2211yescharacteristic
53C2×C4⋊C8×C73C2211yescharacteristic
54C2×#97×C73C2211yescharacteristic
55#336×C73C2211yescharacteristic
56#336×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×D292C2×C411yescharacteristic
60C22×C4×D73C2311yescharacteristic
61C22×C4×D73C2311yescharacteristic
62C2×#11924011—41no—
63C2×#11924011—41no—
64C2×#11924011—41no—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C2×C8×D73—41no—
70C2×C8×D73—41no—
71C2×C8×D73—41no—
72C4×D292—41no—
73C4×D292—41no—
74C2×C4⋊Dic73C2×C411yescharacteristic
75C2×#11924005C2×C411yescharacteristic
76C2×#11924005C2×C411yescharacteristic
77C2×#11924005C2×C411yescharacteristic
78C2×#11924005C2×C411yescharacteristic
79C2×#11924005C2×C411yescharacteristic
80C2×#11924005C2×C411yescharacteristic
81C2×#11924005C2×C411yescharacteristic
82C2×#11924005C2×C411yescharacteristic
83C2×#11924005C2×C411yescharacteristic
84C2×#11924005C2×C411yescharacteristic
85C2×C4×Dic73C2311yescharacteristic
86C2×C4×Dic73C2311yescharacteristic
87C4⋊#10596831—41no—
88#14556520—41no—
89#14556520—41no—
90#14556520—41no—
91#14556520—41no—
92#14556503—41no—
93#14556503—41no—
94#14556503—41no—
95#14556503—41no—
96#14556503—41no—
97#14556503—41no—
98#14556502—41no—
99#14556498C2×C411yescharacteristic
100#14556498C2×C411yescharacteristic
101#14556498C2×C411yescharacteristic
102#14556498C2×C411yescharacteristic
103#14556498C2×C411yescharacteristic
104#14556498C2×C411yescharacteristic
105#14556498C2×C411yescharacteristic
106#14556498C2×C411yescharacteristic
107#14556498C2×C411yescharacteristic
108#14556498C2×C411yescharacteristic
109#14556498C2×C411yescharacteristic
110#14556498C2×C411yescharacteristic
111C22×Q8×C73C2×C411yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C22×C8×C73C2311yescharacteristic
116C4×Q8×C73—41no—
117C4×Q8×C73—41no—
118C2×C4⋊C4×C73C2×C411yescharacteristic
119C2×C4⋊C4×C73C2×C411yescharacteristic
120C2×C4⋊C4×C73C2×C411yescharacteristic
121C2×C42×C73C2311yescharacteristic
122C2×C42×C73C2311yescharacteristic
123C2×C42×C73C2311yescharacteristic
124C4⋊C8×C73—41no—
125#97×C73—41no—
126C23×D73C22×C411yescharacteristic
127C23×D73C22×C411yescharacteristic
128C2×D292—22no—
129C2×D292—41no—
130C2×D292—41no—
131C22×Dic73C22×C411yescharacteristic
132C22×Dic73C22×C411yescharacteristic
133C22×Dic73C22×C411yescharacteristic
134C22×Dic73C22×C411yescharacteristic
135C22×Dic73C22×C411yescharacteristic
136C22×Dic73C22×C411yescharacteristic
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—21no—
146C2×C4×D73—21no—
147C2×C4×D73—21no—
148C2×C4×D73—21no—
149#11924011—44no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C8×D73—44no—
163C4⋊Dic73—41no—
164#11924005—22no—
165#11924005—22no—
166#11924005—41no—
167#11924005—41no—
168#11924005—41no—
169#11924005—41no—
170#11924005—41no—
171#11924005—41no—
172#11924005—41no—
173#11924005—41no—
174C4×Dic73—21no—
175C4×Dic73—21no—
176C4×Dic73—21no—
177C4×Dic73—21no—
178C2×Q8×C73—41no—
179C2×Q8×C73—41no—
180C2×Q8×C73—41no—
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×C73C2411yescharacteristic
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C4⋊C4×C73—41no—
201C4⋊C4×C73—41no—
202C4⋊C4×C73—41no—
203C42×C73—21no—
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—21no—
207C42×C73—21no—
208C42×C73—21no—
209C23×C73C23×C411yescharacteristic
210C22×D73—21no—
211C22×D73—21no—
212C22×D73—21no—
213C22×D73—21no—
214C22×D73—21no—
215C22×D73—21no—
216C22×D73—21no—
217C22×D73—21no—
218C22×D73—21no—
219C22×D73—21no—
220C22×D73—21no—
221C22×D73—21no—
222C22×D73—21no—
223C22×D73#13112yes—
224Q8×C73—44no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
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×C73H(5,𝔽2)11yescharacteristic
244C2×C4×C73H(5,𝔽2)11yescharacteristic
245C2×C4×C73H(5,𝔽2)11yescharacteristic
246C2×C4×C73H(5,𝔽2)11yescharacteristic
247C2×C4×C732−1+411yescharacteristic
248C2×C4×C732−1+411yescharacteristic
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
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—
267D292—44no—
268C4×D73—41no—
269C4×D73—41no—
270C4×D73—41no—
271C4×D73—41no—
272C4×D73—41no—
273C4×D73—41no—
274C4×D73—41no—
275C4×D73—41no—
276C8×C73—41no—
277C8×C73—41no—
278C8×C73—41no—
279C8×C73—41no—
280#10596831—41no—
281#10596831—41no—
282#10596831—41no—
283#10596831—41no—
284C22×C73#51811yescharacteristic
285C22×C73#52011yescharacteristic
286C22×C73#52011yescharacteristic
287C22×C73C2×#13111yescharacteristic
288C22×C73#56811yescharacteristic
289C22×C73#56811yescharacteristic
290C22×C73#56811yescharacteristic
291C2×D73—22no—
292C2×D73—22no—
293C2×D73—22no—
294C2×D73—41no—
295C2×D73—41no—
296C2×D73—41no—
297C2×D73—41no—
298C2×D73—41no—
299C2×D73—41no—
300C2×D73—41no—
301C2×D73—41no—
302C2×D73—41no—
303C2×D73—41no—
304C2×D73—41no—
305C4×C73—41no—
306C4×C73—41no—
307C4×C73—41no—
308C4×C73—41no—
309C4×C73—41no—
310C4×C73—41no—
311C4×C73H(3,𝔽4)11yescharacteristic
312C4×C73#56211yescharacteristic
313C4×C73#56211yescharacteristic
314C4×C73#56311yescharacteristic
315Dic73—41no—
316Dic73—41no—
317Dic73—41no—
318Dic73—41no—
319Dic73—41no—
320Dic73—41no—
321#23354—731nomaximal, Sylow
322C2×C73#237911yescharacteristic
323C2×C73#297611yescharacteristic
324C2×C73#297811yescharacteristic
325C2×C73#299211yescharacteristic
326C2×C73#299211yescharacteristic
327C2×C73#299511yescharacteristic
328C2×C73#299511yescharacteristic
329D73—42no—
330D73—81no—
331#2398—731no—
332#1985—731no—
333#1985D7311yescharacteristic, complemented
334#1924—731no—
335#1924—731no—
336#1924—731no—
337#1924—731no—
338#1870—731no—
339#1869—731no—
340#1851—731no—
341#1851—731no—
342#1850—731no—
343#1850—731no—
344#1832—731no—
345#1832—731no—
346C73#2335411yescharacteristic, complemented, Sylow
347C22×M16—731no—
348C23×C8—731no—
349C2×C4×Q8C2×D7311yescharacteristic
350C2×C4×D4—731no—
351C2×C4⋊C8—731no—
352C2×C4⋊C8—731no—
353C2×C4⋊C8—731no—
354C2×C4⋊C8—731no—
355C2×C4⋊C8—731no—
356C2×C4⋊C8C2×D7311yescharacteristic
357C2×C22⋊C8—731no—
358C2×C22⋊C8—731no—
359C2×C22⋊C8—731no—
360C2×C22⋊C8—731no—
361C2×C22⋊C8—731no—
362C2×C22⋊C8—731no—
363C2×#97—731no—
364C2×#97C2×D7311yescharacteristic
365#400—731no—
366#400—731no—
367#397—731no—
368#397—731no—
369#394—731no—
370#394—731no—
371#393—731no—
372#393—731no—
373C4×C22⋊C4—731no—
374C4×C22⋊C4—731no—
375C4×C22⋊C4—731no—
376C4×C22⋊C4—731no—
377#376—731no—
378#336—731no—
379#336—731no—
380#336—731no—
381#336—731no—
382#336—731no—
383#336—731no—
384#336—731no—
385#336—731no—
386#336C2×D7311yescharacteristic
387#336C2×D7311yescharacteristic
388#336C2×D7311yescharacteristic
389#336C2×D7311yescharacteristic
390C22×Q8C4×D7311yescharacteristic
391C22×D4—731no—
392C23×C4—731no—
393C23×C4—731no—
394C2×M16—2921no—
395C2×M16—2921no—
396C2×M16—2921no—
397C22×C8—2921no—
398C22×C8—2921no—
399C22×C8—2921no—
400C22×C8—731no—
401C22×C8—731no—
402C22×C8—731no—
403C22×C8—731no—
404C22×C8C22×D7311yescharacteristic
405C22×C8C22×D7311yescharacteristic
406C22×C8C22×D7311yescharacteristic
407C22×C8C22×D7311yescharacteristic
408C4×Q8—41no—
409C4×Q8—41no—
410C4×D4—2921no—
411C4×D4—2921no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4C4×D7311yescharacteristic
414C2×C4⋊C4C4×D7311yescharacteristic
415C2×C4⋊C4C4×D7311yescharacteristic
416C2×C22⋊C4—731no—
417C2×C22⋊C4—731no—
418C2×C22⋊C4—731no—
419C2×C22⋊C4—731no—
420C2×C22⋊C4—731no—
421C2×C22⋊C4—731no—
422C2×C22⋊C4—731no—
423C2×C22⋊C4—731no—
424C2×C22⋊C4—731no—
425C2×C22⋊C4—731no—
426C2×C42—731no—
427C2×C42—731no—
428C2×C42C22×D7311yescharacteristic
429C2×C42C22×D7311yescharacteristic
430C2×C42C22×D7311yescharacteristic
431C4⋊C8—41no—
432C4⋊C8—2921no—
433C4⋊C8—2921no—
434C4⋊C8—2921no—
435C4⋊C8—2921no—
436C4⋊C8—2921no—
437C22⋊C8—2921no—
438C22⋊C8—2921no—
439C22⋊C8—2921no—
440C22⋊C8—2921no—
441C22⋊C8—2921no—
442C22⋊C8—2921no—
443#97—41no—
444#97—2921no—
445#95—731no—
446#95—731no—
447#95—731no—
448#95—731no—
449#95—731no—
450#95—731no—
451#95—731no—
452#95—731no—
453#95—731no—
454#95—731no—
455#95—731no—
456#95—731no—
457C24—731no—
458C24—731no—
459C2×Q8—41no—
460C2×Q8—41no—
461C2×Q8—41no—
462C2×D4—2921no—
463C2×D4—2921no—
464C2×D4—1462no—
465C22×C4—731no—
466C22×C4—731no—
467C22×C4—731no—
468C22×C4—731no—
469C22×C4—731no—
470C22×C4—731no—
471C22×C4—1461no—
472C22×C4—1461no—
473C22×C4—1461no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4C2×C4×D7311yescharacteristic
484C22×C4C2×C4×D7311yescharacteristic
485C22×C4C2×C4×D7311yescharacteristic
486C22×C4C2×C4×D7311yescharacteristic
487C22×C4C2×C4×D7311yescharacteristic
488C22×C4C2×C4×D7311yescharacteristic
489C22×C4C23×D7311yescharacteristic
490M16—2924no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—2924no—
504C2×C8—1461no—
505C2×C8—1461no—
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—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—2921no—
520C22⋊C4—2921no—
521C22⋊C4—2921no—
522C22⋊C4—2921no—
523C22⋊C4—2921no—
524C22⋊C4—2921no—
525C22⋊C4—2921no—
526C22⋊C4—2921no—
527C22⋊C4—2921no—
528C22⋊C4—1462no—
529C22⋊C4—1462no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—1461no—
537C42—1461no—
538C42—1461no—
539C42—1461no—
540C23—732no—
541C23—1461no—
542C23—1461no—
543C23—1461no—
544C23—1461no—
545C23—1461no—
546C23—1461no—
547C23—1461no—
548C23—1461no—
549C23—1461no—
550C23—1461no—
551C23—1461no—
552C23—1461no—
553C23—1461no—
554C23C22×C4×D7311yescharacteristic
555Q8—44no—
556D4—2924no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
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—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—1461no—
584C2×C4—1461no—
585C2×C4—1461no—
586C2×C4—1461no—
587C2×C4—1461no—
588C2×C4—1461no—
589C2×C4—1461no—
590C2×C4—1461no—
591C2×C4—1461no—
592C2×C4—1461no—
593C2×C4—1461no—
594C2×C4—1461no—
595C2×C4—1461no—
596C2×C4—1461no—
597C2×C4—1461no—
598C2×C4—1461no—
599C2×C4—1461no—
600C2×C4—1461no—
601C2×C4#1455663611yescharacteristic
602C2×C4#1455663611yescharacteristic
603C2×C4#1455663711yescharacteristic
604C2×C4#1455663711yescharacteristic
605C2×C4#1455663911yescharacteristic
606C2×C4#1455663911yescharacteristic
607C8—41no—
608C8—41no—
609C8—41no—
610C8—41no—
611C8—2921no—
612C8—2921no—
613C8—2921no—
614C8—2921no—
615C22—2921no—
616C22—2921no—
617C22—2921no—
618C22—2921no—
619C22—2921no—
620C22—2921no—
621C22—2921no—
622C22—2921no—
623C22—2921no—
624C22—2921no—
625C22—2921no—
626C22—1462no—
627C22—1462no—
628C22—1462no—
629C22#1798906011yescharacteristic
630C22#1798906911yescharacteristic
631C22#1798906911yescharacteristic
632C22#131×D7311yescharacteristic
633C22#1798935211yescharacteristic
634C22#1798935311yescharacteristic
635C22#1798935311yescharacteristic
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—2921no—
643C4—2921no—
644C4—2921no—
645C4—2921no—
646C4—2921no—
647C4—2921no—
648C4#1798932911yescharacteristic
649C4#1798933311yescharacteristic
650C4#1798933311yescharacteristic
651C4#1798933611yescharacteristic
652C2—2922no—
653C2—5841no—
654C2#2261310311yescharacteristic
655C2#2261897411yescharacteristic
656C2#2261899111yescharacteristic
657C2#2261906911yescharacteristic
658C2#2261906911yescharacteristic
659C2#2261910011yescharacteristic
660C2#2261910011yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC73×#1985C2
prime radicalC22×D292C2×C4
cosocleC22×C4×C73C24
hypercenter#1985D73
hypocenterC73#23354
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #22610872C2semidirect
3. #22610544C2semidirect
4. #22610544C2semidirect
5. #22610542C2semidirect
6. #22610542C2semidirect
9. #22610115C2semidirect
10. #22610112C2semidirect
11. #22610110C2semidirect
12. #22610107C2semidirect
13. #22610049C2semidirect
14. #22610049C2semidirect
15. C73×#1985C2semidirect
333. #1985D73semidirect
346. C73#23354semidirect

Maximal subgroups

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

subgroupindex
#226132142
#226108722
#226105442
#226105422
#226102272
#226102222
#226101152
#226101122
#226101102
#226101072
#226100492
C73×#19852
#2335473