Finite Group Databasebeta

← #33096489

Subgroups — #33096489

Order 18688.

774 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988648C211yesmaximal, characteristic
2#22612429C211yesmaximal, characteristic
3#22610652C211yesmaximal, characteristic, complemented
4#22610652C211yesmaximal, characteristic, complemented
5#22610647C211yesmaximal, characteristic, complemented
6#22610647C211yesmaximal, characteristic, complemented
7#22610114C211yesmaximal, characteristic
8#22610114C211yesmaximal, characteristic
9#22610109C211yesmaximal, characteristic
10#22610109C211yesmaximal, characteristic
11#22610084C211yesmaximal, characteristic, complemented
12#22610082C211yesmaximal, characteristic, complemented
13#22610050C211yesmaximal, characteristic, complemented
14#22610047C211yesmaximal, characteristic, complemented
15C2×#424×C73C211yesmaximal, characteristic
16#17988648—24no—
17C2×C4⋊#10596831C2211yescharacteristic
18C2×C4⋊#10596831C2211yescharacteristic
19C2×#14556520C2211yescharacteristic
20C2×#14556520C2211yescharacteristic
21C2×#14556520C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556520C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×#14556520C2211yescharacteristic
26C2×#14556520C2211yescharacteristic
27C2×C22⋊#10596831C2211yescharacteristic
28C2×C22⋊#10596831C2211yescharacteristic
29C2×#14556502C2211yescharacteristic
30C2×C4×#10596831C2211yescharacteristic
31#17988480C411yescharacteristic
32#17988480C411yescharacteristic
33#17988480C411yescharacteristic
34#17988480C411yescharacteristic
35#17988473C411yescharacteristic
36#17988473C411yescharacteristic
37#17988473C411yescharacteristic
38#17988473C411yescharacteristic
39C4×C4⋊Dic73C2211yescharacteristic
40C4×C4⋊Dic73C2211yescharacteristic
41C4×C22⋊Dic73C2211yescharacteristic
42C4×C22⋊Dic73C2211yescharacteristic
43#17988433C2211yescharacteristic
44C42×Dic73C2211yescharacteristic
45#17988341C2211yescharacteristic
46#17988341C2211yescharacteristic
47#17988341C2211yescharacteristic
48#17988341C2211yescharacteristic
49#17988341C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C22×M16×C73C2211yescharacteristic
54C23×C8×C73C2211yescharacteristic
55C2×#117×C73C2211yescharacteristic
56#424×C73—24no—
57C2×C4⋊C8×C73C2211yescharacteristic
58C2×C4⋊C8×C73C2211yescharacteristic
59C2×C4⋊C8×C73C2211yescharacteristic
60C2×C4⋊C8×C73C2211yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C2×C4⋊Dic73C2×C411yescharacteristic
66C2×C4⋊Dic73C2×C411yescharacteristic
67C2×C4⋊Dic73C2×C411yescharacteristic
68C2×C4⋊Dic73C2×C411yescharacteristic
69C2×C22⋊Dic73C2×C411yescharacteristic
70C2×C22⋊Dic73C2×C411yescharacteristic
71C2×C22⋊Dic73C2×C411yescharacteristic
72C2×C22⋊Dic73C2×C411yescharacteristic
73C2×C4×Dic73C2×C411yescharacteristic
74C2×C4×Dic73C2×C411yescharacteristic
75C2×C4×Dic73C2×C411yescharacteristic
76C2×C4×Dic73C2×C411yescharacteristic
77C2×C4×Dic73C2311yescharacteristic
78C2×C4×Dic73C2311yescharacteristic
79C2×C4×Dic73C2311yescharacteristic
80C2×C4×Dic73C2311yescharacteristic
81C4⋊#10596831—22no—
82C4⋊#10596831—22no—
83#14556520—41no—
84#14556520—41no—
85#14556520—41no—
86#14556520—41no—
87#14556520—41no—
88#14556520—41no—
89#14556520—41no—
90#14556520—41no—
91C22⋊#10596831—22no—
92C22⋊#10596831—22no—
93#14556502—22no—
94C4×#10596831—22no—
95#14556498C2×C411yescharacteristic
96#14556498C2×C411yescharacteristic
97#14556498C2×C411yescharacteristic
98#14556498C2×C411yescharacteristic
99#14556498C2×C411yescharacteristic
100#14556498C2×C411yescharacteristic
101#14556498C2×C411yescharacteristic
102#14556498C2×C411yescharacteristic
103#14556498C2×C411yescharacteristic
104#14556498C2×C411yescharacteristic
105#14556498C2×C411yescharacteristic
106#14556498C2×C411yescharacteristic
107C23×C4×C73C2311yescharacteristic
108C2×M16×C73—22no—
109C2×M16×C73—41no—
110C2×M16×C73—41no—
111C22×C8×C73—22no—
112C22×C8×C73—41no—
113C22×C8×C73—41no—
114C22×C8×C73C2311yescharacteristic
115C22×C8×C73C2311yescharacteristic
116C22×C8×C73C2311yescharacteristic
117C22×C8×C73C2311yescharacteristic
118#117×C73D414yes—
119#117×C73Q814yes—
120C2×C4⋊C4×C73C2×C411yescharacteristic
121C2×C4⋊C4×C73C2×C411yescharacteristic
122C2×C22⋊C4×C73C2×C411yescharacteristic
123C2×C22⋊C4×C73C2×C411yescharacteristic
124C2×C42×C73C2311yescharacteristic
125C2×C42×C73C2311yescharacteristic
126C4⋊C8×C73—22no—
127C4⋊C8×C73—22no—
128C4⋊C8×C73—22no—
129C4⋊C8×C73—22no—
130C22×Dic73C22×C411yescharacteristic
131C22×Dic73C22×C411yescharacteristic
132C22×Dic73C22×C411yescharacteristic
133C22×Dic73C22×C411yescharacteristic
134C22×Dic73C22×C411yescharacteristic
135C22×Dic73C22×C411yescharacteristic
136C22×Dic73C22×C411yescharacteristic
137C22×Dic73C22×C411yescharacteristic
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C4⋊Dic73—22no—
151C4⋊Dic73—22no—
152C4⋊Dic73—41no—
153C4⋊Dic73—41no—
154C22⋊Dic73—22no—
155C22⋊Dic73—22no—
156C22⋊Dic73—41no—
157C22⋊Dic73—41no—
158C4×Dic73—21no—
159C4×Dic73—21no—
160C4×Dic73—21no—
161C4×Dic73—21no—
162C4×Dic73—21no—
163C4×Dic73—21no—
164C4×Dic73—21no—
165C4×Dic73—21no—
166C4×Dic73—22no—
167C4×Dic73—22no—
168C4×Dic73—41no—
169C4×Dic73—41no—
170C24×C73C22×C411yescharacteristic
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73C22×C411yescharacteristic
176C22×C4×C73C22×C411yescharacteristic
177C22×C4×C73C22×C411yescharacteristic
178C22×C4×C73C22×C411yescharacteristic
179C22×C4×C73C22×C411yescharacteristic
180C22×C4×C73C2×D412yes—
181C22×C4×C73C2×Q812yes—
182C22×C4×C73C2411yescharacteristic
183M16×C73—44no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
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—44no—
197C4⋊C4×C73C4⋊C414yes—
198C4⋊C4×C73C4⋊C414yes—
199C22⋊C4×C73C4⋊C414yes—
200C22⋊C4×C73C4⋊C414yes—
201C42×C73C2×D412yes—
202C42×C73C2×D412yes—
203C42×C73C2×Q812yes—
204C42×C73C2×Q812yes—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73C2×C4⋊C412yes—
212C23×C73C23×C411yescharacteristic
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×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—22no—
227C2×C4×C73C2×C4⋊C412yes—
228C2×C4×C73C2×C4⋊C412yes—
229C2×C4×C73C2×C4⋊C412yes—
230C2×C4×C73C2×C4⋊C412yes—
231C2×C4×C73C2×C4⋊C412yes—
232C2×C4×C73C22×D411yescharacteristic
233C2×C4×C73C22×Q811yescharacteristic
234C2×C4×C73H(5,𝔽2)11yescharacteristic
235C2×C4×C73H(5,𝔽2)11yescharacteristic
236C2×C4×C732−1+411yescharacteristic
237C2×C4×C732−1+411yescharacteristic
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
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#13112yes—
261C2×Dic73#13112yes—
262C8×C73—41no—
263C8×C73—41no—
264C8×C73—41no—
265C8×C73—41no—
266#10596831—41no—
267#10596831—41no—
268#10596831—41no—
269#10596831—41no—
270C22×C73—22no—
271C22×C73—22no—
272C22×C73—22no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73C22×C4⋊C411yescharacteristic
278C22×C73#52011yescharacteristic
279C22×C73#52011yescharacteristic
280C22×C73C2×#13111yescharacteristic
281C22×C73C2×#13111yescharacteristic
282C22×C73#56811yescharacteristic
283C22×C73#56811yescharacteristic
284C4×C73—22no—
285C4×C73—22no—
286C4×C73—22no—
287C4×C73—22no—
288C4×C73—22no—
289C4×C73#53511yescharacteristic
290C4×C73#53611yescharacteristic
291C4×C73#54411yescharacteristic
292C4×C73#54411yescharacteristic
293Dic73—22no—
294Dic73—22no—
295Dic73—41no—
296Dic73—41no—
297Dic73—41no—
298Dic73—41no—
299Dic73—41no—
300Dic73—41no—
301#21980—731nomaximal, Sylow
302C2×C73—42no—
303C2×C73#231711yescharacteristic
304C2×C73#290611yescharacteristic
305C2×C73#290611yescharacteristic
306C2×C73#299211yescharacteristic
307C2×C73#299311yescharacteristic
308C2×C73#299411yescharacteristic
309C2×C73#299511yescharacteristic
310C2×#433—731no—
311C2×#424D7311yescharacteristic
312#2298—731no—
313#1940—731no—
314#1940—731no—
315#1939—731no—
316#1939—731no—
317#1851—731no—
318#1851—731no—
319#1850—731no—
320#1850—731no—
321#1843—731no—
322#1842—731no—
323#1832—731no—
324#1831—731no—
325C73#2198011yescharacteristic, complemented, Sylow
326C22×M16C2×D7311yescharacteristic
327C23×C8C2×D7311yescharacteristic
328C2×#117C2×D7311yescharacteristic
329#433—1464no—
330#424—24no—
331C2×C4⋊C8—731no—
332C2×C4⋊C8—731no—
333C2×C4⋊C8—731no—
334C2×C4⋊C8—731no—
335C2×C4⋊C8—731no—
336C2×C4⋊C8—731no—
337C2×C4⋊C8—731no—
338C2×C4⋊C8—731no—
339C2×C4⋊C8—731no—
340C2×C4⋊C8—731no—
341C2×C4⋊C8C2×D7311yescharacteristic
342C2×C4⋊C8C2×D7311yescharacteristic
343C2×C4⋊C8C2×D7311yescharacteristic
344C2×C4⋊C8C2×D7311yescharacteristic
345C2×C22⋊C8—731no—
346C2×C22⋊C8—731no—
347C2×#97—731no—
348C2×C4×C8—731no—
349#388—731no—
350#388—731no—
351#388—731no—
352#388—731no—
353#387—731no—
354#387—731no—
355#387—731no—
356#387—731no—
357C4×C4⋊C4—731no—
358C4×C4⋊C4—731no—
359C4×C22⋊C4—731no—
360C4×C22⋊C4—731no—
361#376—731no—
362C43—731no—
363#336—731no—
364#336—731no—
365#336—731no—
366#336—731no—
367#336—731no—
368#336—731no—
369#336—731no—
370#336—731no—
371C23×C4C22×D7311yescharacteristic
372C2×M16—22no—
373C2×M16—41no—
374C2×M16—41no—
375C22×C8—22no—
376C22×C8—41no—
377C22×C8—41no—
378C22×C8—731no—
379C22×C8—731no—
380C22×C8—731no—
381C22×C8—731no—
382C22×C8C22×D7311yescharacteristic
383C22×C8C22×D7311yescharacteristic
384C22×C8C22×D7311yescharacteristic
385C22×C8C22×D7311yescharacteristic
386#117Dic14614yes—
387#117C22⋊D7314yes—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4C4×D7311yescharacteristic
393C2×C4⋊C4C4×D7311yescharacteristic
394C2×C22⋊C4—731no—
395C2×C22⋊C4—731no—
396C2×C22⋊C4—731no—
397C2×C22⋊C4—731no—
398C2×C22⋊C4C4×D7311yescharacteristic
399C2×C22⋊C4C4×D7311yescharacteristic
400C2×C42—731no—
401C2×C42—731no—
402C2×C42—731no—
403C2×C42—731no—
404C2×C42—731no—
405C2×C42—731no—
406C2×C42—731no—
407C2×C42—731no—
408C2×C42C22×D7311yescharacteristic
409C2×C42C22×D7311yescharacteristic
410C4⋊C8—22no—
411C4⋊C8—22no—
412C4⋊C8—22no—
413C4⋊C8—22no—
414C4⋊C8—2921no—
415C4⋊C8—2921no—
416C4⋊C8—2921no—
417C4⋊C8—2921no—
418C4⋊C8—2921no—
419C4⋊C8—2921no—
420C4⋊C8—2921no—
421C4⋊C8—2921no—
422C4⋊C8—1462no—
423C4⋊C8—1462no—
424C22⋊C8—1462no—
425C22⋊C8—1462no—
426#97—1462no—
427C4×C8—1462no—
428#95—731no—
429#95—731no—
430#95—731no—
431#95—731no—
432#95—731no—
433#95—731no—
434#95—731no—
435#95—731no—
436#95—731no—
437#95—731no—
438#95—731no—
439#95—731no—
440C24C2×C4×D7311yescharacteristic
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—731no—
446C22×C4—731no—
447C22×C4—731no—
448C22×C4—731no—
449C22×C4—731no—
450C22×C4—731no—
451C22×C4—731no—
452C22×C4—731no—
453C22×C4C2×C4×D7311yescharacteristic
454C22×C4C2×C4×D7311yescharacteristic
455C22×C4C2×C4×D7311yescharacteristic
456C22×C4C2×C4×D7311yescharacteristic
457C22×C4C2×C4×D7311yescharacteristic
458C22×C4C2×C22⋊D7312yes—
459C22×C4C2×Dic14612yes—
460C22×C4C23×D7311yescharacteristic
461M16—44no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—44no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C4⋊C4—2921no—
488C4⋊C4—2921no—
489C4⋊C4—1462no—
490C4⋊C4—1462no—
491C4⋊C4#1192400714yes—
492C4⋊C4#1192400714yes—
493C22⋊C4—2921no—
494C22⋊C4—2921no—
495C22⋊C4—1462no—
496C22⋊C4—1462no—
497C22⋊C4#1192400714yes—
498C22⋊C4#1192400714yes—
499C42—2921no—
500C42—2921no—
501C42—1462no—
502C42—1462no—
503C42—1461no—
504C42—1461no—
505C42—1461no—
506C42—1461no—
507C42—1461no—
508C42—1461no—
509C42—1461no—
510C42—1461no—
511C42C2×C22⋊D7312yes—
512C42C2×C22⋊D7312yes—
513C42C2×Dic14612yes—
514C42C2×Dic14612yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23C2×#1192400712yes—
522C23C22×C4×D7311yescharacteristic
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—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—732no—
538C2×C4—732no—
539C2×C4—1461no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4—1461no—
543C2×C4—1461no—
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×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4C2×#1192400712yes—
562C2×C4C2×#1192400712yes—
563C2×C4C2×#1192400712yes—
564C2×C4C2×#1192400712yes—
565C2×C4C2×#1192400712yes—
566C2×C4C22×C22⋊D7311yescharacteristic
567C2×C4C22×Dic14611yescharacteristic
568C2×C4#1455663611yescharacteristic
569C2×C4#1455663611yescharacteristic
570C2×C4#1455663911yescharacteristic
571C2×C4#1455663911yescharacteristic
572C8—41no—
573C8—41no—
574C8—41no—
575C8—41no—
576C8—2921no—
577C8—2921no—
578C8—2921no—
579C8—2921no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C22×#1192400711yescharacteristic
588C22#1798906811yescharacteristic
589C22#1798906811yescharacteristic
590C22#131×D7311yescharacteristic
591C22#131×D7311yescharacteristic
592C22#1798935211yescharacteristic
593C22#1798935211yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—2921no—
600C4—2921no—
601C4—2921no—
602C4—2921no—
603C4—2921no—
604C4—2921no—
605C4—1462no—
606C4—1462no—
607C4#1798915411yescharacteristic
608C4#1798916111yescharacteristic
609C4#1798921811yescharacteristic
610C4#1798921811yescharacteristic
611C2—42no—
612C2#2261255511yescharacteristic
613C2#2261849711yescharacteristic
614C2#2261849711yescharacteristic
615C2#2261906811yescharacteristic
616C2#2261907811yescharacteristic
617C2#2261908911yescharacteristic
618C2#2261909911yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC2×#424×C73C2
prime radicalC24×C73C22×C4
cosocleC22×C4×C73C24
hypercenterC2×#424D73
hypocenterC73#21980
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C2×#424

Complements

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

NH ≅ G/Ntype
3. #22610652C2semidirect
4. #22610652C2semidirect
5. #22610647C2semidirect
6. #22610647C2semidirect
11. #22610084C2semidirect
12. #22610082C2semidirect
13. #22610050C2semidirect
14. #22610047C2semidirect
325. C73#21980semidirect

Maximal subgroups

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

subgroupindex
C2×#179886482
#226124292
#226106522
#226106472
#226101142
#226101092
#226100842
#226100822
#226100502
#226100472
C2×#424×C732
#2198073