Finite Group Databasebeta

← #33130472

Subgroups — #33130472

Order 18688.

776 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614248C211yesmaximal, characteristic
2#22612860C211yesmaximal, characteristic
3#22611500C211yesmaximal, characteristic, complemented
4#22611414C211yesmaximal, characteristic, complemented
5#22611364C211yesmaximal, characteristic, complemented
6#22611354C211yesmaximal, characteristic, complemented
7#22610510C211yesmaximal, characteristic, complemented
8#22610503C211yesmaximal, characteristic, complemented
9#22610132C211yesmaximal, characteristic, complemented
10#22610121C211yesmaximal, characteristic, complemented
11#22610027C211yesmaximal, characteristic, complemented
12#22610022C211yesmaximal, characteristic, complemented
13#22609956C211yesmaximal, characteristic, complemented
14C2×#17988350C211yesmaximal, characteristic, complemented
15C73×#1869C211yesmaximal, characteristic
16C2×#14556550C2211yescharacteristic
17C22×C4⋊Dic73C2211yescharacteristic
18C22×C4⋊Dic73C2211yescharacteristic
19C2×#14556525C2211yescharacteristic
20C2×#14556525C2211yescharacteristic
21C2×#14556523C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23#17988526—21no—
24#17988523—21no—
25#17988485C2211yescharacteristic
26#17988485C2211yescharacteristic
27#17988480—21no—
28#17988473—21no—
29#17988467—21no—
30#17988467—21no—
31#17988459—21no—
32#17988453C2211yescharacteristic
33#17988447C2211yescharacteristic
34#17988447C2211yescharacteristic
35#17988447C2211yescharacteristic
36#17988447C2211yescharacteristic
37#17988447C2211yescharacteristic
38C4×C4⋊Dic73C2211yescharacteristic
39C22⋊C4×Dic73—21no—
40#17988350C2211yescharacteristic
41#17988350C2211yescharacteristic
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52C23×C8×C73C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54C2×C22⋊C8×C73C2211yescharacteristic
55C2×C22⋊C8×C73C2211yescharacteristic
56C2×C22⋊C8×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C23×Dic73—21no—
60C23×Dic73—21no—
61#14556550—22no—
62#14556550—41no—
63C2×C4⋊Dic73—22no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73—22no—
67C2×C4⋊Dic73—22no—
68C2×C4⋊Dic73—41no—
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73—21no—
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C22⋊Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87#14556525—22no—
88#14556525—41no—
89#14556523—22no—
90#14556523—41no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101C22×D4×C73D411yescharacteristic
102C23×C4×C73C2311yescharacteristic
103C23×C4×C73C2311yescharacteristic
104C22×C8×C73—22no—
105C22×C8×C73—22no—
106C22×C8×C73—41no—
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C4×D4×C73—22no—
112C4×D4×C73—22no—
113C2×C4⋊C4×C73D411yescharacteristic
114C2×C22⋊C4×C73D411yescharacteristic
115C2×C22⋊C4×C73D411yescharacteristic
116C2×C42×C73C2311yescharacteristic
117C22⋊C8×C73—22no—
118C22⋊C8×C73—22no—
119C22⋊C8×C73—41no—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—42no—
129C22×Dic73—42no—
130C22×Dic73—42no—
131C22×Dic73—81no—
132C22×Dic73—81no—
133C22×Dic73—81no—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—42no—
150C4⋊Dic73—42no—
151C4⋊Dic73#4112yes—
152#11924007—81no—
153#11924007—81no—
154C22⋊Dic73—42no—
155C22⋊Dic73—42no—
156C22⋊Dic73—42no—
157C22⋊Dic73—81no—
158C22⋊Dic73—81no—
159C22⋊Dic73—81no—
160C4×Dic73—42no—
161C4×Dic73—81no—
162C24×C73C2×D411yescharacteristic
163C24×C73C2×D411yescharacteristic
164C2×D4×C73—22no—
165C2×D4×C73—22no—
166C2×D4×C73—22no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73#4112yes—
182C22×C4×C73#4112yes—
183C22×C4×C73C2411yescharacteristic
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—44no—
196C2×C8×C73#4112yes—
197C4⋊C4×C73—22no—
198C22⋊C4×C73—22no—
199C22⋊C4×C73—22no—
200C42×C73—21no—
201C42×C73—21no—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73#12312yes—
214C23×C73#13512yes—
215C23×C73#13712yes—
216C23×C73C22×D411yescharacteristic
217D4×C73—44no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—22no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—21no—
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—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73#12312yes—
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C2×C4×C73H(5,𝔽2)11yescharacteristic
243C2×C4×C732−1+411yescharacteristic
244C2×Dic73—22no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—84no—
269C2×Dic73—84no—
270C8×C73—22no—
271C8×C73—41no—
272C8×C73—41no—
273C8×C73—41no—
274C22×C73—22no—
275C22×C73—22no—
276C22×C73—22no—
277C22×C73—22no—
278C22×C73—22no—
279C22×C73—22no—
280C22×C73—41no—
281C22×C73—41no—
282C22×C73—41no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73#48514yes—
288C22×C73C2×#12311yescharacteristic
289C22×C73#54611yescharacteristic
290C22×C73#54711yescharacteristic
291C22×C73C2×#13511yescharacteristic
292C22×C73C2×#13711yescharacteristic
293C22×C73#57611yescharacteristic
294C22×C73#57711yescharacteristic
295C4×C73—22no—
296C4×C73—41no—
297C4×C73—41no—
298C4×C73—41no—
299C4×C73#52911yescharacteristic
300C4×C73#53911yescharacteristic
301C4×C73#55111yescharacteristic
302C4×C73#55211yescharacteristic
303Dic73—42no—
304Dic73—81no—
305Dic73—81no—
306Dic73—81no—
307Dic73—81no—
308Dic73—81no—
309Dic73—81no—
310Dic73—81no—
311#24943—731nomaximal, Sylow
312C2×C73—24no—
313C2×C73—81no—
314C2×C73#241911yescharacteristic
315C2×C73C2×#48511yescharacteristic
316C2×C73#309611yescharacteristic
317C2×C73#329711yescharacteristic
318C2×C73#331611yescharacteristic
319C2×C73#333711yescharacteristic
320C2×C73#334611yescharacteristic
321#2487—731no—
322#2352—731no—
323#2101—731no—
324#2085—731no—
325#2077—731no—
326#2075—731no—
327#1919—731no—
328#1918—731no—
329#1869D7311yescharacteristic
330#1854—731no—
331#1852—731no—
332#1825—731no—
333#1824—731no—
334#1807—731no—
335C2×#340—731no—
336C73#2494311yescharacteristic, complemented, Sylow
337C23×C8C2×D7311yescharacteristic
338C2×C4×D4C2×D7311yescharacteristic
339C2×#117—731no—
340C22×C4⋊C4—731no—
341C22×C4⋊C4—731no—
342C2×#107—731no—
343C2×#107—731no—
344C2×#106—731no—
345C2×#106—731no—
346C2×C22⋊C8C2×D7311yescharacteristic
347C2×C22⋊C8C2×D7311yescharacteristic
348C2×C22⋊C8C2×D7311yescharacteristic
349#400—1461no—
350#399—1461no—
351#389—731no—
352#389—731no—
353#388—1461no—
354#387—1461no—
355#386—1461no—
356#386—1461no—
357#385—1461no—
358#382—731no—
359#380—731no—
360#380—731no—
361#380—731no—
362#380—731no—
363#380—731no—
364C4×C4⋊C4—731no—
365C4×C22⋊C4—1461no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#340—731no—
374#340—731no—
375#340—731no—
376#340—731no—
377#340—731no—
378#336C2×D7311yescharacteristic
379#336C2×D7311yescharacteristic
380C22×D4D29211yescharacteristic
381C23×C4—1461no—
382C23×C4—1461no—
383C23×C4C22×D7311yescharacteristic
384C23×C4C22×D7311yescharacteristic
385C22×C8—22no—
386C22×C8—22no—
387C22×C8—41no—
388C22×C8C22×D7311yescharacteristic
389C22×C8C22×D7311yescharacteristic
390C22×C8C22×D7311yescharacteristic
391C22×C8C22×D7311yescharacteristic
392C4×D4—22no—
393C4×D4—22no—
394#117—2921no—
395#117—1462no—
396C2×C4⋊C4—2921no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—731no—
403C2×C4⋊C4—731no—
404C2×C4⋊C4—731no—
405C2×C4⋊C4—1461no—
406C2×C4⋊C4—1461no—
407C2×C4⋊C4—1462no—
408C2×C4⋊C4—1462no—
409C2×C4⋊C4—1462no—
410C2×C4⋊C4—1461no—
411C2×C4⋊C4—1461no—
412C2×C4⋊C4D29211yescharacteristic
413C2×C22⋊C4—1461no—
414C2×C22⋊C4—1461no—
415C2×C22⋊C4—1461no—
416C2×C22⋊C4—1461no—
417C2×C22⋊C4—1461no—
418C2×C22⋊C4—1461no—
419C2×C22⋊C4D29211yescharacteristic
420C2×C22⋊C4D29211yescharacteristic
421C2×C42—1461no—
422C2×C42—1461no—
423C2×C42C22×D7311yescharacteristic
424#107—2921no—
425#107—1462no—
426#106—2921no—
427#106—1462no—
428C22⋊C8—22no—
429C22⋊C8—22no—
430C22⋊C8—41no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440#95—1461no—
441C24C2×D29211yescharacteristic
442C24C2×D29211yescharacteristic
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—2922no—
457C22×C4—2922no—
458C22×C4—2922no—
459C22×C4—5841no—
460C22×C4—5841no—
461C22×C4—5841no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4C2×D29211yescharacteristic
471C22×C4C2×D29211yescharacteristic
472C22×C4C2×D29211yescharacteristic
473C22×C4C2×D29211yescharacteristic
474C22×C4#1192402712yes—
475C22×C4#1192402712yes—
476C22×C4C23×D7311yescharacteristic
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—44no—
489C2×C8#1192402712yes—
490C4⋊C4—22no—
491C4⋊C4—2922no—
492C4⋊C4—2922no—
493C4⋊C4—5841no—
494C4⋊C4—5841no—
495C4⋊C4—732no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4—1461no—
506C4⋊C4—1461no—
507C4⋊C4—1461no—
508C4⋊C4—1461no—
509C4⋊C4—1461no—
510C4⋊C4—1461no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—2922no—
514C22⋊C4—2922no—
515C22⋊C4—2922no—
516C22⋊C4—5841no—
517C22⋊C4—5841no—
518C22⋊C4—5841no—
519C42—21no—
520C42—21no—
521C42—2922no—
522C42—5841no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23#1455657512yes—
535C23#1455661312yes—
536C23#1455662412yes—
537C23C22×D29211yescharacteristic
538D4—44no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—5844no—
582C2×C4—5844no—
583C2×C4—1462no—
584C2×C4#1455657512yes—
585C2×C4#41×D7311yescharacteristic
586C2×C4C2×#1192402711yescharacteristic
587C2×C4C2×#1192402711yescharacteristic
588C2×C4C2×#1192402711yescharacteristic
589C2×C4#1455663611yescharacteristic
590C2×C4#1455663811yescharacteristic
591C8—22no—
592C8—41no—
593C8—41no—
594C8—41no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22#1798892914yes—
609C22C2×#1455657511yescharacteristic
610C22#1798923211yescharacteristic
611C22#1798924111yescharacteristic
612C22C2×#1455661311yescharacteristic
613C22C2×#1455662411yescharacteristic
614C22#1798939211yescharacteristic
615C22#1798939911yescharacteristic
616C4—22no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—2922no—
621C4—5841no—
622C4—5841no—
623C4—5841no—
624C4—5841no—
625C4—5841no—
626C4—5841no—
627C4—5841no—
628C4#1798911811yescharacteristic
629C4#1798918211yescharacteristic
630C4#1798926811yescharacteristic
631C4#1798927411yescharacteristic
632C2—24no—
633C2—81no—
634C2#2261340611yescharacteristic
635C2C2×#1798892911yescharacteristic
636C2#2261989011yescharacteristic
637C2#2262178311yescharacteristic
638C2#2262204411yescharacteristic
639C2#2262231211yescharacteristic
640C2#2262241811yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1869C2
socleC23×C73C22×D4
prime radicalC22×D4×C73D4
hypercenter#1869D73
hypocenterC73#24943
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
3. #22611500C2semidirect
4. #22611414C2semidirect
5. #22611364C2semidirect
6. #22611354C2semidirect
7. #22610510C2semidirect
8. #22610503C2semidirect
9. #22610132C2semidirect
10. #22610121C2semidirect
11. #22610027C2semidirect
12. #22610022C2semidirect
13. #22609956C2semidirect
14. C2×#17988350C2semidirect
336. C73#24943semidirect

Maximal subgroups

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

subgroupindex
#226142482
#226128602
#226115002
#226114142
#226113642
#226113542
#226105102
#226105032
#226101322
#226101212
#226100272
#226100222
#226099562
C2×#179883502
C73×#18692
#2494373