Finite Group Databasebeta

← #33136279

Subgroups — #33136279

Order 18688.

860 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988879C211yesmaximal, characteristic, complemented
2C2×#17988659C211yesmaximal, characteristic
3#22614840C211yesmaximal, characteristic
4#22611362C211yesmaximal, characteristic, complemented
5#22611357C211yesmaximal, characteristic, complemented
6#22611134C211yesmaximal, characteristic, complemented
7#22611127C211yesmaximal, characteristic, complemented
8#22610905C211yesmaximal, characteristic, complemented
9#22610699C211yesmaximal, characteristic, complemented
10#22610362C211yesmaximal, characteristic, complemented
11#22610352C211yesmaximal, characteristic, complemented
12#22609962C211yesmaximal, characteristic, complemented
13#22609958C211yesmaximal, characteristic, complemented
14#22609764C211yesmaximal, characteristic, complemented
15C73×#2547C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C23×#10596831C2211yescharacteristic
18C2×#14556570C2211yescharacteristic
19C2×#14556570C2211yescharacteristic
20C2×C4×Dic146C2211yescharacteristic
21#17988879—24no—
22#17988659—24no—
23C2×#14556524C2211yescharacteristic
24C2×#14556524C2211yescharacteristic
25C2×#14556524C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×#14556514C2211yescharacteristic
32C2×C22⋊#10596831C2211yescharacteristic
33C2×C22⋊#10596831C2211yescharacteristic
34C2×#14556502C2211yescharacteristic
35#17988518—21no—
36#17988501—21no—
37#17988475—21no—
38#17988475—21no—
39#17988453C2211yescharacteristic
40#17988447C2211yescharacteristic
41#17988447C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50C2×C4⋊Q8×C73C2211yescharacteristic
51C2×#124×C73—21no—
52C2×C4×D4×C73C2211yescharacteristic
53#397×C73—21no—
54#397×C73—21no—
55#391×C73—21no—
56#380×C73C2211yescharacteristic
57#380×C73C2211yescharacteristic
58#380×C73C2211yescharacteristic
59#380×C73C2211yescharacteristic
60C4×C4⋊C4×C73C2211yescharacteristic
61C22×Dic146C2311yescharacteristic
62C2×#11924012—22no—
63C2×#11924012—22no—
64C2×#11924012—41no—
65C22×#10596831—22no—
66C22×#10596831—22no—
67C22×#10596831—41no—
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72#14556570—22no—
73#14556570—22no—
74#14556570—41no—
75#14556570—41no—
76C4×Dic146—22no—
77C4×Dic146—22no—
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×C22⋊Dic73—21no—
85C2×C22⋊Dic73—21no—
86C2×C22⋊Dic73—21no—
87C2×C22⋊Dic73—21no—
88C2×C4×Dic73—21no—
89#14556524—22no—
90#14556524—22no—
91#14556524—41no—
92#14556522—41no—
93C4⋊#10596831D414yes—
94#14556514—22no—
95#14556514—22no—
96#14556514—22no—
97#14556514—22no—
98C22⋊#10596831—22no—
99C22⋊#10596831—22no—
100#14556502—22no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105C22×Q8×C73—21no—
106C22×D4×C73D411yescharacteristic
107C23×C4×C73C2311yescharacteristic
108C23×C4×C73C2311yescharacteristic
109C4⋊Q8×C73—22no—
110C4⋊Q8×C73—22no—
111#124×C73—44no—
112C4×D4×C73—22no—
113C4×D4×C73—22no—
114C2×C4⋊C4×C73—21no—
115C2×C4⋊C4×C73D411yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C22⋊C4×C73—21no—
121C2×C22⋊C4×C73—21no—
122C2×C22⋊C4×C73—21no—
123C2×C22⋊C4×C73—21no—
124C2×C22⋊C4×C73D411yescharacteristic
125C2×C22⋊C4×C73D411yescharacteristic
126C2×C42×C73—21no—
127C2×C42×C73C2311yescharacteristic
128#95×C73—21no—
129#95×C73—21no—
130#95×C73—21no—
131#95×C73—21no—
132#95×C73—21no—
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146—21no—
136C2×Dic146—21no—
137C2×Dic146—21no—
138C2×Dic146#4112yes—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143#11924012—44no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—44no—
154C2×#10596831C2×D412yes—
155C2×#10596831C2×D412yes—
156C2×#10596831#4112yes—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163#11924007—42no—
164#11924007—42no—
165#11924007—42no—
166C22⋊Dic73—42no—
167C22⋊Dic73—42no—
168C22⋊Dic73—42no—
169C22⋊Dic73—42no—
170C4×Dic73—42no—
171C24×C73C2×D411yescharacteristic
172C24×C73C2×D411yescharacteristic
173C2×Q8×C73—42no—
174C2×Q8×C73—42no—
175C2×Q8×C73—42no—
176C2×D4×C73—22no—
177C2×D4×C73—22no—
178C2×D4×C73—41no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
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×C73C2×D411yescharacteristic
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2×D411yescharacteristic
199C22×C4×C73C2411yescharacteristic
200C4⋊C4×C73—22no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73—21no—
205C4⋊C4×C73—21no—
206C4⋊C4×C73—21no—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—42no—
210C22⋊C4×C73—22no—
211C22⋊C4×C73—22no—
212C22⋊C4×C73—42no—
213C22⋊C4×C73—42no—
214C22⋊C4×C73—42no—
215C22⋊C4×C73—42no—
216C42×C73—42no—
217C42×C73C2×D412yes—
218C42×C73#4112yes—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—21no—
222C23×C73—21no—
223C23×C73—21no—
224C23×C73—21no—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—21no—
231C23×C73—21no—
232C23×C73AGL(1,ℤ/8)12yes—
233C23×C73C22×D411yescharacteristic
234Q8×C73—84no—
235D4×C73—44no—
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×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—41no—
262C2×C4×C73—41no—
263C2×C4×C73—41no—
264C2×C4×C73—41no—
265C2×C4×C73—41no—
266C2×C4×C73#12112yes—
267C2×C4×C73#12112yes—
268C2×C4×C73C22×D411yescharacteristic
269C2×C4×C73C2×#4111yescharacteristic
270C2×C4×C73C2×#4111yescharacteristic
271C2×C4×C73C2×#4111yescharacteristic
272C2×C4×C73H(5,𝔽2)11yescharacteristic
273C2×C4×C732−1+411yescharacteristic
274C2×Dic73—22no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—41no—
284C2×Dic73—41no—
285C2×Dic73—41no—
286Dic146—22no—
287Dic146—22no—
288Dic146—41no—
289Dic146—41no—
290#10596831—22no—
291#10596831—22no—
292#10596831—22no—
293#10596831—41no—
294C22×C73—22no—
295C22×C73—22no—
296C22×C73—22no—
297C22×C73—41no—
298C22×C73—41no—
299C22×C73—41no—
300C22×C73—41no—
301C22×C73—41no—
302C22×C73—41no—
303C22×C73—41no—
304C22×C73—41no—
305C22×C73—41no—
306C22×C73—41no—
307C22×C73—41no—
308C22×C73C2×#12111yescharacteristic
309C22×C73#54611yescharacteristic
310C22×C73#54711yescharacteristic
311C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C73#57511yescharacteristic
313C22×C73#57811yescharacteristic
314C22×C73#57811yescharacteristic
315C4×C73—22no—
316C4×C73—22no—
317C4×C73—41no—
318C4×C73—41no—
319C4×C73—81no—
320C4×C73—81no—
321C4×C73—81no—
322C4×C73—81no—
323C4×C73#52911yescharacteristic
324C4×C73#54211yescharacteristic
325C4×C73#54611yescharacteristic
326C4×C73#54811yescharacteristic
327Dic73—42no—
328Dic73—81no—
329Dic73—81no—
330Dic73—81no—
331#25352—731nomaximal, Sylow
332C2×C73—42no—
333C2×C73—81no—
334C2×C73#242711yescharacteristic
335C2×C73#305511yescharacteristic
336C2×C73#305911yescharacteristic
337C2×C73#330811yescharacteristic
338C2×C73#331311yescharacteristic
339C2×C73#334311yescharacteristic
340C2×C73#334911yescharacteristic
341C2×#477—731no—
342C2×#435—731no—
343#2547D7311yescharacteristic
344#2536—731no—
345#2076—731no—
346#2075—731no—
347#2036—731no—
348#2035—731no—
349#1993—731no—
350#1949—731no—
351#1894—731no—
352#1892—731no—
353#1809—731no—
354#1808—731no—
355#1763—731no—
356C73#2535211yescharacteristic, complemented, Sylow
357C22×M16—731no—
358C23×C8—731no—
359C2×C4⋊Q8C2×D7311yescharacteristic
360C2×#124—21no—
361C2×C22⋊Q8—731no—
362C2×C22⋊Q8—731no—
363C2×C4×Q8—731no—
364C2×C4×D4C2×D7311yescharacteristic
365#477—1464no—
366#435—1464no—
367C2×#107—731no—
368C2×#107—731no—
369C2×#107—731no—
370C2×#106—731no—
371C2×C4⋊C8—731no—
372C2×#103—731no—
373C2×#103—731no—
374C2×#103—731no—
375C2×#103—731no—
376C2×C22⋊C8—731no—
377C2×C22⋊C8—731no—
378C2×#97—731no—
379#398—1461no—
380#397—21no—
381#397—21no—
382#394—1461no—
383#391—21no—
384#388—1461no—
385#388—1461no—
386#382—731no—
387#380—731no—
388#380—731no—
389#380C2×D7311yescharacteristic
390#380C2×D7311yescharacteristic
391#380C2×D7311yescharacteristic
392#380C2×D7311yescharacteristic
393C4×C4⋊C4C2×D7311yescharacteristic
394#340—731no—
395#340—731no—
396#340—731no—
397#340—731no—
398#340—731no—
399#340—731no—
400#340—731no—
401#340—731no—
402C22×Q8—21no—
403C22×Q8—731no—
404C22×D4C22⋊D7311yescharacteristic
405C23×C4C22×D7311yescharacteristic
406C23×C4C22×D7311yescharacteristic
407C2×M16—2921no—
408C2×M16—1462no—
409C2×M16—1462no—
410C22×C8—2921no—
411C22×C8—731no—
412C22×C8—731no—
413C22×C8—731no—
414C22×C8—731no—
415C22×C8—1462no—
416C22×C8—1462no—
417C4⋊Q8—22no—
418C4⋊Q8—22no—
419#124—44no—
420C22⋊Q8—2921no—
421C22⋊Q8—2921no—
422C22⋊Q8—1462no—
423C22⋊Q8—1462no—
424C4×Q8—1462no—
425C4×Q8—1462no—
426C4×D4—22no—
427C4×D4—22no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—731no—
430C2×C4⋊C4—731no—
431C2×C4⋊C4—731no—
432C2×C4⋊C4—1461no—
433C2×C4⋊C4—1461no—
434C2×C4⋊C4—1461no—
435C2×C4⋊C4C22⋊D7311yescharacteristic
436C2×C4⋊C4C22×D7311yescharacteristic
437C2×C4⋊C4C22×D7311yescharacteristic
438C2×C4⋊C4C22×D7311yescharacteristic
439C2×C4⋊C4C22×D7311yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—1461no—
445C2×C22⋊C4—1461no—
446C2×C22⋊C4—1461no—
447C2×C22⋊C4—1461no—
448C2×C22⋊C4C22⋊D7311yescharacteristic
449C2×C22⋊C4C22⋊D7311yescharacteristic
450C2×C42—21no—
451C2×C42—1461no—
452C2×C42C22×D7311yescharacteristic
453#107—2921no—
454#107—1462no—
455#107—1462no—
456#106—2921no—
457C4⋊C8—734no—
458#103—1462no—
459#103—1462no—
460#103—1462no—
461#103—1462no—
462C22⋊C8—1462no—
463C22⋊C8—1462no—
464#97—1462no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—1461no—
471#95—1461no—
472#95—1461no—
473#95—1461no—
474C24C2×C22⋊D7311yescharacteristic
475C24C2×C22⋊D7311yescharacteristic
476C2×Q8—42no—
477C2×Q8—42no—
478C2×Q8—42no—
479C2×Q8—732no—
480C2×Q8—1461no—
481C2×Q8—1461no—
482C2×Q8—1461no—
483C2×Q8—1461no—
484C2×Q8—1461no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—41no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—1461no—
505C22×C4—1461no—
506C22×C4—1461no—
507C22×C4—1461no—
508C22×C4C2×C22⋊D7311yescharacteristic
509C22×C4C2×C22⋊D7311yescharacteristic
510C22×C4C2×C22⋊D7311yescharacteristic
511C22×C4C2×C22⋊D7311yescharacteristic
512C22×C4C23×D7311yescharacteristic
513M16—2924no—
514C2×C8—2924no—
515C2×C8—732no—
516C2×C8—732no—
517C2×C8—732no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C2×C8—1461no—
523C2×C8—1461no—
524C2×C8—1461no—
525C2×C8—1461no—
526C2×C8—1461no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—42no—
537C4⋊C4—2922no—
538C4⋊C4—2922no—
539C4⋊C4—2922no—
540C4⋊C4—1461no—
541C4⋊C4—1461no—
542C4⋊C4—1461no—
543C4⋊C4—1461no—
544C4⋊C4—1461no—
545C4⋊C4—1461no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—2922no—
553C22⋊C4—2922no—
554C22⋊C4—2922no—
555C22⋊C4—2922no—
556C42—42no—
557C42—2922no—
558C42D4×D7312yes—
559C42#1192402712yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23#1455661712yes—
574C23C22×C22⋊D7311yescharacteristic
575Q8—84no—
576Q8—2921no—
577Q8—2921no—
578Q8—1462no—
579Q8—1462no—
580D4—44no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—2921no—
614C2×C4—2921no—
615C2×C4—2921no—
616C2×C4—2921no—
617C2×C4—2921no—
618C2×C4—2921no—
619C2×C4—2921no—
620C2×C4—2921no—
621C2×C4—2921no—
622C2×C4—1462no—
623C2×C4#1455656312yes—
624C2×C4#1455656312yes—
625C2×C4C2×D4×D7311yescharacteristic
626C2×C4#41×D7311yescharacteristic
627C2×C4#41×D7311yescharacteristic
628C2×C4C2×#1192402711yescharacteristic
629C2×C4#1455663711yescharacteristic
630C2×C4#1455663811yescharacteristic
631C8—2921no—
632C8—1462no—
633C8—1462no—
634C8—1462no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22C2×#1455656311yescharacteristic
650C22#1798922511yescharacteristic
651C22#1798923611yescharacteristic
652C22C2×#1455661711yescharacteristic
653C22#1798938511yescharacteristic
654C22#1798940211yescharacteristic
655C22#1798940211yescharacteristic
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—2922no—
665C4—5841no—
666C4—5841no—
667C4—5841no—
668C4#1798911611yescharacteristic
669C4#1798920611yescharacteristic
670C4#1798923011yescharacteristic
671C4#1798924511yescharacteristic
672C2—42no—
673C2—81no—
674C2#2261351111yescharacteristic
675C2#2261954611yescharacteristic
676C2#2261958311yescharacteristic
677C2#2262191611yescharacteristic
678C2#2262199111yescharacteristic
679C2#2262236511yescharacteristic
680C2#2262245511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2547C2
socleC23×C73C22×D4
prime radicalC22×D4×C73D4
hypercenter#2547D73
hypocenterC73#25352
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
1. C2×#17988879C2semidirect
4. #22611362C2semidirect
5. #22611357C2semidirect
6. #22611134C2semidirect
7. #22611127C2semidirect
8. #22610905C2semidirect
9. #22610699C2semidirect
10. #22610362C2semidirect
11. #22610352C2semidirect
12. #22609962C2semidirect
13. #22609958C2semidirect
14. #22609764C2semidirect
356. C73#25352semidirect

Maximal subgroups

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

subgroupindex
C2×#179888792
C2×#179886592
#226148402
#226113622
#226113572
#226111342
#226111272
#226109052
#226106992
#226103622
#226103522
#226099622
#226099582
#226097642
C73×#25472
#2535273