Finite Group Databasebeta

← #33104149

Subgroups — #33104149

Order 18688.

970 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988658C211yesmaximal, characteristic
2C2×#17988626C211yesmaximal, characteristic, complemented
3#22612881C211yesmaximal, characteristic
4#22612861C211yesmaximal, characteristic
5#22610696C211yesmaximal, characteristic, complemented
6#22610686C211yesmaximal, characteristic, complemented
7#22610130C211yesmaximal, characteristic, complemented
8#22610130C211yesmaximal, characteristic, complemented
9#22610127C211yesmaximal, characteristic, complemented
10#22610117C211yesmaximal, characteristic, complemented
11#22610096C211yesmaximal, characteristic, complemented
12#22609961C211yesmaximal, characteristic, complemented
13#22609960C211yesmaximal, characteristic, complemented
14#22609831C211yesmaximal, characteristic, complemented
15C2×#427×C73C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C2×C4×C22⋊D73C2211yescharacteristic
18C2×#14556549C2211yescharacteristic
19C2×C4⋊C4×D73C2211yescharacteristic
20#17988658C418yescomplemented
21#17988626—24no—
22C2×#14556525C2211yescharacteristic
23C2×#14556525C2211yescharacteristic
24C2×#14556524C2211yescharacteristic
25C2×#14556524C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×#14556522C2211yescharacteristic
28C2×#14556520C2211yescharacteristic
29C2×C22⋊#10596831C2211yescharacteristic
30C2×#14556503C2211yescharacteristic
31C2×#14556501C2211yescharacteristic
32#17988487—21no—
33#17988481—21no—
34#17988478—21no—
35#17988475—21no—
36#17988471—21no—
37#17988463—21no—
38#17988456C2211yescharacteristic
39#17988452C2211yescharacteristic
40#17988447C2211yescharacteristic
41#17988447C2211yescharacteristic
42#17988444C2211yescharacteristic
43#17988444C2211yescharacteristic
44C4⋊C4×Dic73C2211yescharacteristic
45C4⋊C4×Dic73C2211yescharacteristic
46C4×#11924007—21no—
47C4×#11924005—21no—
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988349C2211yescharacteristic
54C23×C8×C73C2211yescharacteristic
55C2×#117×C73C2211yescharacteristic
56C2×#117×C73C2211yescharacteristic
57#427×C73—24no—
58C2×#107×C73C2211yescharacteristic
59C2×#107×C73C2211yescharacteristic
60C2×#106×C73C2211yescharacteristic
61C2×#106×C73C2211yescharacteristic
62C22×C22⋊D73D411yescharacteristic
63C22×C4×D73—21no—
64C22×C4×D73C2311yescharacteristic
65C2×#11924011—22no—
66C2×#11924011—22no—
67C2×#11924011C2×C414yes—
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C4×C22⋊D73C2×C414yes—
71C4×C22⋊D73Q814yes—
72#14556549—22no—
73#14556549—41no—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007D411yescharacteristic
82C4⋊C4×D73—22no—
83C4⋊C4×D73—41no—
84C2×C22⋊Dic73—21no—
85C2×C22⋊Dic73—21no—
86C2×C22⋊Dic73D411yescharacteristic
87C2×#11924005—21no—
88C2×#11924005—21no—
89C2×#11924005—21no—
90C2×#11924005D411yescharacteristic
91C2×C4×Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73C2311yescharacteristic
95#14556525—22no—
96#14556525—22no—
97#14556524—22no—
98#14556524—41no—
99#14556522—22no—
100#14556522—41no—
101#14556520C2×C414yescomplemented
102C22⋊#10596831C2×C414yescomplemented
103#14556503C2×C414yes—
104#14556501C2×C414yescomplemented
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111#14556498—21no—
112#14556498—21no—
113C23×C4×C73C2311yescharacteristic
114C22×C8×C73—22no—
115C22×C8×C73—22no—
116C22×C8×C73C2×C414yescomplemented
117C22×C8×C73C2311yescharacteristic
118C22×C8×C73C2311yescharacteristic
119#117×C73—22no—
120#117×C73—22no—
121#117×C73—41no—
122#117×C73—41no—
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C4⋊C4×C73C2311yescharacteristic
125C2×C4⋊C4×C73C2311yescharacteristic
126C2×C4⋊C4×C73C2311yescharacteristic
127C2×C22⋊C4×C73—21no—
128C2×C22⋊C4×C73—21no—
129C2×C42×C73—21no—
130C2×C42×C73—21no—
131#107×C73—22no—
132#107×C73—22no—
133#106×C73—22no—
134#106×C73D414yescomplemented
135C23×D73C2×D411yescharacteristic
136C2×C22⋊D73—41no—
137C2×C22⋊D73—41no—
138C2×C22⋊D73C4⋊C414yes—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73C2×D411yescharacteristic
144C22×Dic73C2×D411yescharacteristic
145C22×Dic73C2×D411yescharacteristic
146C2×C4×D73—21no—
147C2×C4×D73—21no—
148C2×C4×D73—21no—
149C2×C4×D73—21no—
150C2×C4×D73—42no—
151C2×C4×D73—42no—
152C2×C4×D73—81no—
153C2×C4×D73C22×C412yes—
154C2×C4×D73C2×Q812yes—
155#11924011—28no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831C22×C412yes—
161C2×#10596831C22×C412yes—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170#11924007—42no—
171#11924007—42no—
172#11924007—81no—
173#11924007C4⋊C414yes—
174C22⋊Dic73—81no—
175C22⋊Dic73—81no—
176C22⋊Dic73C4⋊C414yes—
177#11924005—42no—
178#11924005—42no—
179#11924005—81no—
180#11924005C4⋊C414yes—
181C4×Dic73—42no—
182C4×Dic73—42no—
183C4×Dic73—81no—
184C4×Dic73C22×C412yes—
185C4×Dic73C2×Q812yes—
186C24×C73C2×D411yescharacteristic
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73—21no—
192C22×C4×C73—21no—
193C22×C4×C73—21no—
194C22×C4×C73—21no—
195C22×C4×C73C22×C412yes—
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2×Q812yes—
198C22×C4×C73C2411yescharacteristic
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73—28no—
202C2×C8×C73C22×C412yes—
203C2×C8×C73C22×C412yes—
204C2×C8×C73C2×D412yes—
205C2×C8×C73#4112yes—
206C4⋊C4×C73—21no—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—21no—
211C4⋊C4×C73—21no—
212C4⋊C4×C73C2×D412yes—
213C4⋊C4×C73C2×D412yes—
214C22⋊C4×C73—42no—
215C22⋊C4×C73—42no—
216C42×C73—42no—
217C42×C73—42no—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—21no—
222C23×C73—21no—
223C23×C73—21no—
224C23×C73C2×C4⋊C412yes—
225C23×C73C22×D411yescharacteristic
226C22×D73—21no—
227C22×D73—21no—
228C22×D73—21no—
229C22×D73—21no—
230C22×D73—21no—
231C22×D73—21no—
232C22×D73C2×C4⋊C412yes—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—22no—
236C2×C4×C73—22no—
237C2×C4×C73—22no—
238C2×C4×C73—22no—
239C2×C4×C73—22no—
240C2×C4×C73—22no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73C2×C4⋊C412yes—
252C2×C4×C73C23×C411yescharacteristic
253C2×C4×C73C22×D411yescharacteristic
254C2×C4×C73C22×Q811yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73H(5,𝔽2)11yescharacteristic
257C2×C4×C732−1+411yescharacteristic
258C22⋊D73—44no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73C2×C4⋊C412yes—
278C2×Dic73C2×C4⋊C412yes—
279C2×Dic73C2×C4⋊C412yes—
280C4×D73—22no—
281C4×D73—22no—
282C4×D73—22no—
283C4×D73—22no—
284C4×D73—84no—
285C8×C73—22no—
286C8×C73C4×D414yes—
287#10596831—22no—
288#10596831—22no—
289C22×C73—22no—
290C22×C73—22no—
291C22×C73—22no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73—41no—
295C22×C73—41no—
296C22×C73C22×C4⋊C411yescharacteristic
297C22×C73D4211yescharacteristic
298C22×C73#54711yescharacteristic
299C22×C73#57511yescharacteristic
300C22×C73#57511yescharacteristic
301C22×C73#57611yescharacteristic
302C22×C73#57811yescharacteristic
303C2×D73—22no—
304C2×D73—22no—
305C2×D73—22no—
306C2×D73—41no—
307C2×D73—41no—
308C2×D73—41no—
309C2×D73—41no—
310C4×C73—22no—
311C4×C73—42no—
312C4×C73—42no—
313C4×C73—81no—
314C4×C73—81no—
315C4×C73C2×C4×D411yescharacteristic
316C4×C73#52011yescharacteristic
317C4×C73D4×Q811yescharacteristic
318C4×C73#55411yescharacteristic
319Dic73—22no—
320Dic73—22no—
321Dic73—22no—
322Dic73—81no—
323Dic73—81no—
324Dic73—81no—
325Dic73—81no—
326#22660—731nomaximal, Sylow
327C2×C73—42no—
328C2×C73D4×C4⋊C411yescharacteristic
329C2×C73#291111yescharacteristic
330C2×C73#291311yescharacteristic
331C2×C73#326811yescharacteristic
332C2×C73#326911yescharacteristic
333C2×C73#334711yescharacteristic
334C2×C73#334911yescharacteristic
335D73—42no—
336C2×#435—731no—
337C2×#428—731no—
338C2×#427D7311yescharacteristic, complemented
339#2354—731no—
340#2352—731no—
341#1949—731no—
342#1947—731no—
343#1854—731no—
344#1854—731no—
345#1854—731no—
346#1852—731no—
347#1847—731no—
348#1809—731no—
349#1808—731no—
350#1776—731no—
351C73#2266011yescharacteristic, complemented, Sylow
352C22×M16—731no—
353C23×C8C2×D7311yescharacteristic
354C2×C4×D4—731no—
355C2×#117—731no—
356C2×#117C2×D7311yescharacteristic
357C2×#117C2×D7311yescharacteristic
358C22×C4⋊C4—731no—
359#435—738no—
360#428—1464no—
361#427—24no—
362C2×#107—731no—
363C2×#107—731no—
364C2×#107—731no—
365C2×#107—731no—
366C2×#107C2×D7311yescharacteristic
367C2×#107C2×D7311yescharacteristic
368C2×#106—731no—
369C2×#106—731no—
370C2×#106C2×D7311yescharacteristic
371C2×#106C2×D7311yescharacteristic
372C2×C4⋊C8—731no—
373C2×C22⋊C8—731no—
374C2×C22⋊C8—731no—
375C2×#97—731no—
376#390—1461no—
377#388—1461no—
378#388—1461no—
379#388—1461no—
380#387—1461no—
381#386—1461no—
382#384—731no—
383#382—731no—
384#380—731no—
385#380—731no—
386#380—731no—
387#380—731no—
388C4×C4⋊C4—731no—
389C4×C4⋊C4—731no—
390C4×C4⋊C4—1461no—
391C4×C22⋊C4—1461no—
392#340—731no—
393#340—731no—
394#340—731no—
395#340—731no—
396#340—731no—
397#340—731no—
398C22×D4—731no—
399C23×C4—731no—
400C23×C4—1461no—
401C23×C4C22×D7311yescharacteristic
402C2×M16—734no—
403C2×M16—1462no—
404C2×M16—1462no—
405C22×C8—22no—
406C22×C8—22no—
407C22×C8—731no—
408C22×C8—731no—
409C22×C8C4×D7314yescomplemented
410C22×C8C22×D7311yescharacteristic
411C22×C8C22×D7311yescharacteristic
412C4×D4—734no—
413C4×D4—734no—
414#117—22no—
415#117—22no—
416#117—41no—
417#117—41no—
418#117—2921no—
419#117—1462no—
420C2×C4⋊C4—2921no—
421C2×C4⋊C4—731no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1462no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4C22×D7311yescharacteristic
431C2×C4⋊C4C22×D7311yescharacteristic
432C2×C4⋊C4C22×D7311yescharacteristic
433C2×C4⋊C4C22×D7311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—731no—
437C2×C22⋊C4—731no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4—1461no—
440C2×C22⋊C4—1461no—
441C2×C22⋊C4—1461no—
442C2×C22⋊C4—1461no—
443C2×C42—21no—
444C2×C42—21no—
445C2×C42—731no—
446C2×C42—1461no—
447C2×C42—1461no—
448C2×C42—1461no—
449#107—22no—
450#107—22no—
451#107—2921no—
452#107—1462no—
453#107—1462no—
454#107—1462no—
455#106—22no—
456#106—2921no—
457#106—1462no—
458#106C22⋊D7314yescomplemented
459C4⋊C8—734no—
460C22⋊C8—734no—
461C22⋊C8—734no—
462#97—734no—
463#95—1461no—
464#95—1461no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470#95—1461no—
471C24—731no—
472C24D4×D7311yescharacteristic
473C2×D4—2921no—
474C2×D4—2921no—
475C2×D4—734no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—2922no—
485C22×C4—2922no—
486C22×C4—5841no—
487C22×C4—731no—
488C22×C4—731no—
489C22×C4—731no—
490C22×C4—732no—
491C22×C4—732no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4—1461no—
495C22×C4—1461no—
496C22×C4—1461no—
497C22×C4—1461no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4C2×C4×D7312yes—
501C22×C4D4×D7311yescharacteristic
502C22×C4Q8×D7312yes—
503C22×C4C23×D7311yescharacteristic
504M16—1468no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—28no—
508C2×C8—732no—
509C2×C8—732no—
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8C2×C4×D7312yes—
515C2×C8C2×C4×D7312yes—
516C2×C8C2×C22⋊D7312yes—
517C2×C8#1192402812yes—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—2922no—
525C4⋊C4—2922no—
526C4⋊C4—5841no—
527C4⋊C4—734no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4—1461no—
536C4⋊C4C2×C22⋊D7312yes—
537C4⋊C4C2×C22⋊D7312yes—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—2922no—
541C22⋊C4—2922no—
542C22⋊C4—5841no—
543C22⋊C4—5841no—
544C22⋊C4—5841no—
545C22⋊C4—734no—
546C22⋊C4—734no—
547C42—42no—
548C42—42no—
549C42—2922no—
550C42—2922no—
551C42—5841no—
552C42—732no—
553C42—732no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—732no—
561C23—1461no—
562C23—1461no—
563C23—1461no—
564C23—1461no—
565C23—1461no—
566C23—1461no—
567C23C4⋊C4×D7312yes—
568C23C2×D4×D7311yescharacteristic
569D4—2924no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2921no—
600C2×C4—5844no—
601C2×C4—732no—
602C2×C4—732no—
603C2×C4—732no—
604C2×C4—1461no—
605C2×C4—1461no—
606C2×C4—1461no—
607C2×C4—1461no—
608C2×C4—1461no—
609C2×C4—1461no—
610C2×C4—1462no—
611C2×C4—1462no—
612C2×C4—1462no—
613C2×C4—1462no—
614C2×C4C4⋊C4×D7312yes—
615C2×C4C22×C4×D7311yescharacteristic
616C2×C4C22×C22⋊D7311yescharacteristic
617C2×C4C2×Q8×D7311yescharacteristic
618C2×C4C2×#1192402811yescharacteristic
619C2×C4#1455663611yescharacteristic
620C2×C4#1455663911yescharacteristic
621C8—22no—
622C8—1462no—
623C8—1462no—
624C8C4×C22⋊D7314yes—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—2921no—
633C22—2921no—
634C22—2921no—
635C22—2921no—
636C22—1462no—
637C22—1462no—
638C22—1462no—
639C22C2×C4⋊C4×D7311yescharacteristic
640C22D4×C22⋊D7311yescharacteristic
641C22#1798923811yescharacteristic
642C22#1798938411yescharacteristic
643C22#1798938411yescharacteristic
644C22#1798938911yescharacteristic
645C22#1798940111yescharacteristic
646C4—22no—
647C4—42no—
648C4—42no—
649C4—81no—
650C4—81no—
651C4—5841no—
652C4—5841no—
653C4—5841no—
654C4—5841no—
655C4—1462no—
656C4—1462no—
657C4—1462no—
658C4C2×C4×C22⋊D7311yescharacteristic
659C4#1798906811yescharacteristic
660C4Q8×C22⋊D7311yescharacteristic
661C4#1798929211yescharacteristic
662C2—42no—
663C2—2922no—
664C2C4⋊C4×C22⋊D7311yescharacteristic
665C2#2261851911yescharacteristic
666C2#2261852811yescharacteristic
667C2#2262147111yescharacteristic
668C2#2262148611yescharacteristic
669C2#2262242711yescharacteristic
670C2#2262245711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#427×C73C2
socleC23×C73C22×D4
prime radicalC22×C22⋊D73D4
cosocleC22×C4×C73C24
hypercenterC2×#427D73
hypocenterC73#22660
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. C2×#17988626C2semidirect
5. #22610696C2semidirect
6. #22610686C2semidirect
7. #22610130C2semidirect
8. #22610130C2semidirect
9. #22610127C2semidirect
10. #22610117C2semidirect
11. #22610096C2semidirect
12. #22609961C2semidirect
13. #22609960C2semidirect
14. #22609831C2semidirect
15. C2×#427×C73C2semidirect
20. #17988658C4semidirect
101. #14556520C2×C4semidirect
102. C22⋊#10596831C2×C4semidirect
104. #14556501C2×C4semidirect
116. C22×C8×C73C2×C4semidirect
134. #106×C73D4semidirect
338. C2×#427D73semidirect
351. C73#22660semidirect
409. C22×C8C4×D73semidirect
458. #106C22⋊D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886582
C2×#179886262
#226128812
#226128612
#226106962
#226106862
#226101302
#226101272
#226101172
#226100962
#226099612
#226099602
#226098312
C2×#427×C732
#2266073