Finite Group Databasebeta

← #33135391

Subgroups — #33135391

Order 18688.

822 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988946C211yesmaximal, characteristic
2C2×#17988882C211yesmaximal, characteristic, complemented
3#22612989C211yesmaximal, characteristic, complemented
4#22611326C211yesmaximal, characteristic, complemented
5#22611319C211yesmaximal, characteristic
6#22611174C211yesmaximal, characteristic, complemented
7#22611167C211yesmaximal, characteristic
8#22610864C211yesmaximal, characteristic
9#22610864C211yesmaximal, characteristic
10#22610855C211yesmaximal, characteristic, complemented
11#22610655C211yesmaximal, characteristic, complemented
12#22610569C211yesmaximal, characteristic, complemented
13#22610548C211yesmaximal, characteristic
14#22609761C211yesmaximal, characteristic, complemented
15C73×#2545C211yesmaximal, characteristic
16C2×#14556591C2211yescharacteristic
17C2×#14556556—21no—
18#17988946—24no—
19#17988882—24no—
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×#14556515C2211yescharacteristic
30C2×#14556515C2211yescharacteristic
31C2×#14556515C2211yescharacteristic
32C2×#14556515C2211yescharacteristic
33C2×#14556512C2211yescharacteristic
34C2×#14556512C2211yescharacteristic
35C2×#14556512C2211yescharacteristic
36C2×#14556512C2211yescharacteristic
37C2×#14556502C2211yescharacteristic
38#17988492—21no—
39#17988485C2211yescharacteristic
40#17988485C2211yescharacteristic
41#17988472—21no—
42#17988472—21no—
43C4×C4⋊Dic73C2211yescharacteristic
44C4×C4⋊Dic73C2211yescharacteristic
45C4×C4⋊Dic73C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50C2×C4⋊Q8×C73C2211yescharacteristic
51C2×#124×C73C2211yescharacteristic
52C2×#124×C73C2211yescharacteristic
53C2×C4×Q8×C73C2211yescharacteristic
54#396×C73—21no—
55#396×C73—21no—
56#391×C73C2211yescharacteristic
57#390×C73C2211yescharacteristic
58#388×C73—21no—
59#388×C73—21no—
60C4×C4⋊C4×C73C2211yescharacteristic
61C22×Dic146—21no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66#14556591—22no—
67#14556591D414yes—
68#14556556—44no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×C4×Dic73—21no—
77C2×C4×Dic73—21no—
78C2×C4×Dic73—21no—
79#14556524—22no—
80#14556524—22no—
81C4⋊#10596831—22no—
82C4⋊#10596831—41no—
83C4⋊#10596831—41no—
84#14556515—22no—
85#14556515—22no—
86#14556515—22no—
87#14556515—22no—
88#14556515—22no—
89#14556515—22no—
90#14556515—22no—
91#14556515—41no—
92#14556512—22no—
93#14556512—22no—
94#14556512—22no—
95#14556512—41no—
96#14556502—22no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102C22×Q8×C73C2311yescharacteristic
103C22×Q8×C73C2311yescharacteristic
104C22×D4×C73C2311yescharacteristic
105C4⋊Q8×C73—22no—
106C4⋊Q8×C73—41no—
107#124×C73—22no—
108#124×C73—22no—
109#124×C73—22no—
110#124×C73—41no—
111C4×Q8×C73—22no—
112C4×Q8×C73—22no—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73—21no—
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73D411yescharacteristic
117C2×C4⋊C4×C73D411yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C22⋊C4×C73—21no—
122C2×C22⋊C4×C73—21no—
123C2×C22⋊C4×C73—21no—
124C2×C22⋊C4×C73—21no—
125C2×C42×C73—21no—
126C2×C42×C73C2311yescharacteristic
127C2×C42×C73C2311yescharacteristic
128C2×C42×C73C2311yescharacteristic
129#95×C73—21no—
130#95×C73—21no—
131#95×C73D411yescharacteristic
132#95×C73D411yescharacteristic
133C2×Dic146—24no—
134C2×Dic146—42no—
135C2×Dic146—42no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—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—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C4⋊Dic73—24no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73C2×D412yes—
159C4⋊Dic73C2×D412yes—
160C4⋊Dic73#4112yes—
161#11924007—42no—
162#11924007—42no—
163C4×Dic73—42no—
164C4×Dic73—42no—
165C4×Dic73—42no—
166C24×C73—21no—
167C2×Q8×C73—21no—
168C2×Q8×C73—21no—
169C2×Q8×C73—21no—
170C2×Q8×C73—21no—
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73—21no—
176C2×Q8×C73—21no—
177C2×Q8×C73—21no—
178C2×Q8×C73#4112yes—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73—21no—
182C2×D4×C73—21no—
183C2×D4×C73—21no—
184C2×D4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2411yescharacteristic
195C4⋊C4×C73—22no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—41no—
199C4⋊C4×C73—42no—
200C4⋊C4×C73—42no—
201C4⋊C4×C73—42no—
202C22⋊C4×C73—42no—
203C22⋊C4×C73—42no—
204C22⋊C4×C73—42no—
205C22⋊C4×C73—42no—
206C22⋊C4×C73—42no—
207C22⋊C4×C73—42no—
208C42×C73—21no—
209C42×C73—21no—
210C42×C73—21no—
211C42×C73—21no—
212C42×C73—42no—
213C42×C73C2×D412yes—
214C42×C73#4112yes—
215C23×C73—41no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—41no—
219C23×C73—41no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73C22×D411yescharacteristic
223Q8×C73—22no—
224Q8×C73—22no—
225Q8×C73—41no—
226Q8×C73—41no—
227Q8×C73—41no—
228Q8×C73—41no—
229Q8×C73—41no—
230Q8×C73—41no—
231D4×C73—41no—
232D4×C73—41no—
233D4×C73—41no—
234D4×C73—41no—
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×C73—21no—
244C2×C4×C73—22no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
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#12112yes—
260C2×C4×C73#12112yes—
261C2×C4×C73AGL(1,ℤ/8)12yes—
262C2×C4×C73C22×D411yescharacteristic
263C2×C4×C73C2×#4111yescharacteristic
264C2×C4×C73C2×#4111yescharacteristic
265C2×C4×C73C2×#4111yescharacteristic
266C2×C4×C73H(5,𝔽2)11yescharacteristic
267C2×C4×C732−1+411yescharacteristic
268C2×Dic73—22no—
269C2×Dic73—22no—
270C2×Dic73—22no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280Dic146—48no—
281#10596831—41no—
282#10596831—41no—
283#10596831—41no—
284#10596831—41no—
285C22×C73—81no—
286C22×C73—81no—
287C22×C73—81no—
288C22×C73—81no—
289C22×C73—81no—
290C22×C73—81no—
291C22×C73—81no—
292C22×C73C2×#12111yescharacteristic
293C22×C73#54611yescharacteristic
294C22×C73#54811yescharacteristic
295C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C73#57511yescharacteristic
297C22×C73#57811yescharacteristic
298C22×C73#57811yescharacteristic
299C4×C73—22no—
300C4×C73—22no—
301C4×C73—22no—
302C4×C73—42no—
303C4×C73—41no—
304C4×C73—41no—
305C4×C73—41no—
306C4×C73—81no—
307C4×C73—81no—
308C4×C73#52911yescharacteristic
309C4×C73#54211yescharacteristic
310C4×C73#54811yescharacteristic
311C4×C73#55011yescharacteristic
312Dic73—42no—
313Dic73—42no—
314Dic73—42no—
315Dic73—81no—
316#25286—731nomaximal, Sylow
317C2×C73—161no—
318C2×C73#242611yescharacteristic
319C2×C73#303611yescharacteristic
320C2×C73#304411yescharacteristic
321C2×C73#330811yescharacteristic
322C2×C73#331311yescharacteristic
323C2×C73#336711yescharacteristic
324C2×C73#337711yescharacteristic
325C2×#489—731no—
326C2×#477—731no—
327#2545D7311yescharacteristic
328#2365—731no—
329#2068—731no—
330#2067—731no—
331#2043—731no—
332#2042—731no—
333#1983—731no—
334#1983—731no—
335#1981—731no—
336#1940—731no—
337#1928—731no—
338#1925—731no—
339#1763—731no—
340C73#2528611yescharacteristic, complemented, Sylow
341C2×C4⋊Q8—731no—
342C2×C4⋊Q8C2×D7311yescharacteristic
343C2×#124C2×D7311yescharacteristic
344C2×#124C2×D7311yescharacteristic
345C2×C4×Q8—1461no—
346C2×C4×Q8C2×D7311yescharacteristic
347#489—1464no—
348#477—1464no—
349C2×#107—731no—
350C2×#107—731no—
351C2×C4⋊C8—731no—
352C2×C4⋊C8—731no—
353C2×C4⋊C8—731no—
354C2×#103—731no—
355C2×#103—731no—
356C2×#103—731no—
357C2×#103—731no—
358C2×#103—731no—
359C2×#103—731no—
360C2×#103—731no—
361C2×#103—731no—
362C2×#102—731no—
363C2×#102—731no—
364C2×#102—731no—
365C2×#102—731no—
366C2×#97—731no—
367#396—21no—
368#396—21no—
369#391—1461no—
370#391C2×D7311yescharacteristic
371#390C2×D7311yescharacteristic
372#389—731no—
373#389—731no—
374#388—21no—
375#388—21no—
376#387—1461no—
377#387—1461no—
378C4×C4⋊C4—731no—
379C4×C4⋊C4—731no—
380C4×C4⋊C4—731no—
381C4×C4⋊C4C2×D7311yescharacteristic
382#340—731no—
383#340—731no—
384#336—731no—
385#336—731no—
386C22×Q8—1461no—
387C22×Q8C22×D7311yescharacteristic
388C22×Q8C22×D7311yescharacteristic
389C22×D4C22×D7311yescharacteristic
390C22×C8—731no—
391C22×C8—731no—
392C22×C8—731no—
393C22×C8—731no—
394C4⋊Q8—22no—
395C4⋊Q8—41no—
396C4⋊Q8—734no—
397C4⋊Q8—1462no—
398#124—22no—
399#124—22no—
400#124—22no—
401#124—41no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×Q8—2924no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—731no—
409C2×C4⋊C4—731no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4—1461no—
413C2×C4⋊C4—1461no—
414C2×C4⋊C4—1461no—
415C2×C4⋊C4C22⋊D7311yescharacteristic
416C2×C4⋊C4C22⋊D7311yescharacteristic
417C2×C4⋊C4C22×D7311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C42—21no—
425C2×C42—1461no—
426C2×C42—1461no—
427C2×C42—1461no—
428C2×C42C22×D7311yescharacteristic
429C2×C42C22×D7311yescharacteristic
430C2×C42C22×D7311yescharacteristic
431#107—1462no—
432#107—1462no—
433C4⋊C8—2921no—
434C4⋊C8—2921no—
435C4⋊C8—1462no—
436#103—2921no—
437#103—1462no—
438#103—1462no—
439#103—1462no—
440#103—1462no—
441#103—1462no—
442#103—1462no—
443#103—1462no—
444#102—2921no—
445#102—1462no—
446#102—1462no—
447#102—1462no—
448#97—1462no—
449#95—21no—
450#95—21no—
451#95—1461no—
452#95—1461no—
453#95—1461no—
454#95—1461no—
455#95—1461no—
456#95C22⋊D7311yescharacteristic
457#95C22⋊D7311yescharacteristic
458C24—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—2922no—
471C2×Q8—2922no—
472C2×Q8—1464no—
473C2×Q8#1192402712yes—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—1461no—
484C22×C4—1461no—
485C22×C4—1461no—
486C22×C4—1461no—
487C22×C4C2×C22⋊D7311yescharacteristic
488C22×C4C2×C22⋊D7311yescharacteristic
489C22×C4C2×C22⋊D7311yescharacteristic
490C22×C4C2×C22⋊D7311yescharacteristic
491C22×C4C2×C22⋊D7311yescharacteristic
492C22×C4C2×C22⋊D7311yescharacteristic
493C22×C4C23×D7311yescharacteristic
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8—1461no—
497C2×C8—1461no—
498C2×C8—1461no—
499C2×C8—1461no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8—1461no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—41no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—2922no—
514C4⋊C4—2922no—
515C4⋊C4—732no—
516C4⋊C4—732no—
517C4⋊C4—732no—
518C4⋊C4—1461no—
519C4⋊C4—1461no—
520C4⋊C4—1464no—
521C4⋊C4—1461no—
522C4⋊C4—1461no—
523C4⋊C4—1461no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—42no—
535C42—2922no—
536C42—2922no—
537C42—2922no—
538C42D4×D7312yes—
539C42#1192402712yes—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C22×C22⋊D7311yescharacteristic
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—41no—
556Q8—2928no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—41no—
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—22no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—1462no—
595C2×C4—1462no—
596C2×C4—1462no—
597C2×C4#1455656212yes—
598C2×C4#1455656212yes—
599C2×C4#1455661612yes—
600C2×C4C2×D4×D7311yescharacteristic
601C2×C4#41×D7311yescharacteristic
602C2×C4C2×#1192402711yescharacteristic
603C2×C4C2×#1192402711yescharacteristic
604C2×C4#1455663611yescharacteristic
605C2×C4#1455663811yescharacteristic
606C8—2921no—
607C8—2921no—
608C8—2921no—
609C8—2921no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×#1455656211yescharacteristic
618C22#1798923111yescharacteristic
619C22#1798924411yescharacteristic
620C22C2×#1455661611yescharacteristic
621C22#1798938511yescharacteristic
622C22#1798940211yescharacteristic
623C22#1798940211yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—81no—
632C4—81no—
633C4—2922no—
634C4—2922no—
635C4—2922no—
636C4—5841no—
637C4#1798911811yescharacteristic
638C4#1798920411yescharacteristic
639C4#1798924311yescharacteristic
640C4#1798925511yescharacteristic
641C2—161no—
642C2#2261349311yescharacteristic
643C2#2261939111yescharacteristic
644C2#2261945711yescharacteristic
645C2#2262192211yescharacteristic
646C2#2262199711yescharacteristic
647C2#2262264511yescharacteristic
648C2#2262276711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2545C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2545D73
hypocenterC73#25286
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2545

Complements

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

NH ≅ G/Ntype
2. C2×#17988882C2semidirect
3. #22612989C2semidirect
4. #22611326C2semidirect
6. #22611174C2semidirect
10. #22610855C2semidirect
11. #22610655C2semidirect
12. #22610569C2semidirect
14. #22609761C2semidirect
340. C73#25286semidirect

Maximal subgroups

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

subgroupindex
C2×#179889462
C2×#179888822
#226129892
#226113262
#226113192
#226111742
#226111672
#226108642
#226108552
#226106552
#226105692
#226105482
#226097612
C73×#25452
#2528673