Finite Group Databasebeta

← #33105424

Subgroups — #33105424

Order 18688.

866 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988685C211yesmaximal, characteristic
2#22612882C211yesmaximal, characteristic
3#22612852C211yesmaximal, characteristic, complemented
4#22610629C211yesmaximal, characteristic
5#22610615C211yesmaximal, characteristic
6#22610526C211yesmaximal, characteristic, complemented
7#22610505C211yesmaximal, characteristic, complemented
8#22610284C211yesmaximal, characteristic
9#22610284C211yesmaximal, characteristic
10#22610066C211yesmaximal, characteristic, complemented
11#22609942C211yesmaximal, characteristic
12#22609808C211yesmaximal, characteristic, complemented
13#22609763C211yesmaximal, characteristic, complemented
14#103×Dic73C211yesmaximal, characteristic, complemented
15C73×#1803C211yesmaximal, characteristic, complemented
16C22×#11924015C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×Q8×Dic73C2211yescharacteristic
19C2×C4×D292C2211yescharacteristic
20C2×#14556549—21no—
21#17988685C418yescomplemented
22C2×#14556524C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556513C2211yescharacteristic
25C2×#14556513C2211yescharacteristic
26C2×#14556511C2211yescharacteristic
27C2×#14556502C2211yescharacteristic
28C2×#14556501C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30#17988488C2211yescharacteristic
31#17988488C2211yescharacteristic
32#17988486—21no—
33#17988486—21no—
34#17988486C2211yescharacteristic
35#17988486C2211yescharacteristic
36#17988477—21no—
37#17988477—21no—
38#17988453—21no—
39#17988453C2211yescharacteristic
40#17988444—21no—
41C4⋊C4×Dic73C2211yescharacteristic
42C4⋊C4×Dic73C2211yescharacteristic
43C4×C4⋊Dic73—21no—
44C4×C4⋊Dic73C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×C4×Q8×C73C2211yescharacteristic
54C2×C4⋊C8×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×#103×C73C2211yescharacteristic
57#384×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×D292C2311yescharacteristic
61C22×C4×D73—21no—
62C2×#11924015—22no—
63C2×#11924015—22no—
64C2×#11924015C2×C414yes—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67Q8×Dic73—22no—
68Q8×Dic73—22no—
69Q8×Dic73—41no—
70Q8×Dic73—41no—
71C4×D292—22no—
72C4×D292C2×C414yes—
73#14556549—44no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73—21no—
78C2×C4⋊Dic73D411yescharacteristic
79C2×C4⋊Dic73D411yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×#11924005—21no—
84C2×#11924005—21no—
85C2×#11924005—21no—
86C2×#11924005—21no—
87C2×#11924005—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73C2311yescharacteristic
93C2×C4×Dic73C2311yescharacteristic
94#14556524C2×C414yes—
95#14556515C2×C414yes—
96#14556513—41no—
97#14556513—41no—
98#14556511C2×C414yes—
99#14556502C2×C414yes—
100#14556501—41no—
101C8×Dic73—41no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498D411yescharacteristic
109#14556498D411yescharacteristic
110C22×Q8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C4×Q8×C73—22no—
114C4×Q8×C73C2×C414yes—
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73—21no—
117C2×C4⋊C4×C73—21no—
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C4⋊C4×C73C2311yescharacteristic
121C2×C42×C73—21no—
122C2×C42×C73C2311yescharacteristic
123C4⋊C8×C73—22no—
124#103×C73—22no—
125#103×C73—22no—
126C23×D73—21no—
127C2×D292—21no—
128C2×D292—21no—
129C2×D292—21no—
130C2×D292—21no—
131C2×D292—21no—
132C2×D292C22×C412yes—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C22×Dic73C2×D411yescharacteristic
140C2×C4×D73—24no—
141C2×C4×D73—42no—
142C2×C4×D73—42no—
143#11924015—28no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831C22×C412yes—
149C2×#10596831C22×C412yes—
150C4⋊Dic73—21no—
151C4⋊Dic73—22no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—41no—
157C4⋊Dic73—42no—
158C4⋊Dic73—42no—
159C4⋊Dic73—42no—
160C4⋊Dic73—42no—
161C4⋊Dic73C22×C412yes—
162#11924005—24no—
163#11924005—42no—
164#11924005—42no—
165#11924005—81no—
166#11924005—81no—
167C4×Dic73—21no—
168C4×Dic73—21no—
169C4×Dic73—21no—
170C4×Dic73—21no—
171C4×Dic73—42no—
172C4×Dic73—42no—
173C4×Dic73—42no—
174C4×Dic73—42no—
175C2×Q8×C73—21no—
176C2×Q8×C73—21no—
177C2×Q8×C73—21no—
178C2×Q8×C73C22×C412yes—
179C2×Q8×C73#4112yes—
180C2×Q8×C73#4112yes—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2411yescharacteristic
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C4⋊C4×C73—24no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—42no—
201C4⋊C4×C73—81no—
202C4⋊C4×C73C22×C412yes—
203C42×C73—24no—
204C42×C73C22×C412yes—
205C42×C73#4112yes—
206C23×C73C22×D411yescharacteristic
207C22×D73—22no—
208C22×D73—41no—
209C22×D73—41no—
210C22×D73—41no—
211C22×D73—41no—
212C22×D73—41no—
213C22×D73—41no—
214Q8×C73—22no—
215Q8×C73—22no—
216Q8×C73—22no—
217Q8×C73#11714yescomplemented
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—22no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73C4×D412yes—
235C2×C4×C73C4×D412yes—
236C2×C4×C73C23×C411yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73H(5,𝔽2)11yescharacteristic
241C2×C4×C732−1+411yescharacteristic
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—22no—
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—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73AGL(1,ℤ/8)12yes—
263D292—22no—
264D292—22no—
265D292—41no—
266D292—41no—
267C4×D73—48no—
268C8×C73—41no—
269C8×C73—41no—
270#10596831—22no—
271#10596831—22no—
272C22×C73C2×C4×D411yescharacteristic
273C22×C73#54611yescharacteristic
274C22×C73#54811yescharacteristic
275C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C73#57511yescharacteristic
277C22×C73#57711yescharacteristic
278C22×C73#57811yescharacteristic
279C2×D73—42no—
280C2×D73—42no—
281C2×D73—42no—
282C2×D73—81no—
283C2×D73—81no—
284C2×D73—81no—
285C2×D73—81no—
286C4×C73—24no—
287C4×C73—22no—
288C4×C73—22no—
289C4×C73—42no—
290C4×C73—81no—
291C4×C73—81no—
292C4×C73C2×#11711yescharacteristic
293C4×C73#52011yescharacteristic
294C4×C73#55211yescharacteristic
295C4×C73#55311yescharacteristic
296Dic73—22no—
297Dic73—42no—
298Dic73—41no—
299Dic73—41no—
300Dic73—41no—
301Dic73—81no—
302Dic73—81no—
303#22781—731nomaximal, Sylow
304C2×C73#235211yescharacteristic
305C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
306C2×C73#294711yescharacteristic
307C2×C73#330811yescharacteristic
308C2×C73#332111yescharacteristic
309C2×C73#336811yescharacteristic
310C2×C73#337711yescharacteristic
311D73—82no—
312C2×#440—731no—
313#2354—731no—
314#2351—731no—
315#1936—731no—
316#1934—731no—
317#1921—731no—
318#1918—731no—
319#1880—731no—
320#1880—731no—
321#1836—731no—
322#1804—731no—
323#1803D7311yescharacteristic, complemented
324#1771—731no—
325#1763—731no—
326C4×#103—731no—
327C73#2278111yescharacteristic, complemented, Sylow
328C22×SD16—731no—
329C2×C4×Q8—731no—
330C2×C4×Q8—731no—
331C2×C4×Q8C2×D7311yescharacteristic
332C2×C4×D4—731no—
333C2×#117—1461no—
334#440—738no—
335C2×#107—731no—
336C2×C4⋊C8C2×D7311yescharacteristic
337C2×#103—731no—
338C2×#103C2×D7311yescharacteristic
339C2×#103C2×D7311yescharacteristic
340C2×#102—731no—
341C2×#102—731no—
342C2×#102—731no—
343C2×#97—731no—
344C2×#97—731no—
345C2×C4×C8—731no—
346#390—731no—
347#390—731no—
348#389—731no—
349#389—731no—
350#389—1461no—
351#389—1461no—
352#388—1461no—
353#388—1461no—
354#384C2×D7311yescharacteristic
355#382—731no—
356#382—1461no—
357#380—1461no—
358C4×C4⋊C4—731no—
359C4×C4⋊C4—731no—
360C4×C4⋊C4—731no—
361C4×C4⋊C4—1461no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340C2×D7311yescharacteristic
369#340C2×D7311yescharacteristic
370#336—731no—
371#336—731no—
372C22×Q8C22×D7311yescharacteristic
373C22×D4—731no—
374C23×C4—1461no—
375C2×SD16—734no—
376C2×SD16—1462no—
377C2×SD16—1462no—
378C22×C8—731no—
379C22×C8—731no—
380C22×C8C22×D7311yescharacteristic
381C22×C8C22×D7311yescharacteristic
382C4×Q8—22no—
383C4×Q8—2921no—
384C4×Q8—2921no—
385C4×Q8—1462no—
386C4×Q8—1462no—
387C4×Q8C4×D7314yes—
388C4×D4—734no—
389C4×D4—1462no—
390#117—2924no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—1461no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4—1461no—
403C2×C4⋊C4C22×D7311yescharacteristic
404C2×C4⋊C4C22×D7311yescharacteristic
405C2×C4⋊C4C22×D7311yescharacteristic
406C2×C22⋊C4—1461no—
407C2×C22⋊C4—1461no—
408C2×C22⋊C4—1461no—
409C2×C22⋊C4—1461no—
410C2×C22⋊C4—1461no—
411C2×C42—21no—
412C2×C42—731no—
413C2×C42—731no—
414C2×C42—1461no—
415C2×C42—1461no—
416C2×C42—1461no—
417C2×C42—1461no—
418C2×C42C22×D7311yescharacteristic
419#107—734no—
420C4⋊C8—22no—
421#103—22no—
422#103—22no—
423#103—734no—
424#102—2921no—
425#102—2921no—
426#102—734no—
427#97—2921no—
428#97—734no—
429C4×C8—2921no—
430#95—731no—
431#95—731no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438C24—1461no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D7312yes—
443C2×Q8#1192402712yes—
444C2×Q8#1192402912yes—
445C2×D4—732no—
446C2×D4—1461no—
447C2×D4—1461no—
448C2×D4—1461no—
449C2×D4—1461no—
450C2×D4—1461no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—2922no—
456C22×C4—2922no—
457C22×C4—731no—
458C22×C4—731no—
459C22×C4—731no—
460C22×C4—731no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1464no—
465C22×C4D4×D7311yescharacteristic
466C22×C4D4×D7311yescharacteristic
467C22×C4C23×D7311yescharacteristic
468SD16—1468no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—732no—
476C2×C8—732no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—2922no—
490C4⋊C4—2922no—
491C4⋊C4—2921no—
492C4⋊C4—2922no—
493C4⋊C4—2922no—
494C4⋊C4—732no—
495C4⋊C4—1462no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4C2×C4×D7312yes—
502C22⋊C4—2922no—
503C22⋊C4—2922no—
504C22⋊C4—5841no—
505C22⋊C4—5841no—
506C22⋊C4—1464no—
507C42—24no—
508C42—2922no—
509C42—2922no—
510C42—2922no—
511C42—2922no—
512C42—1461no—
513C42—1461no—
514C42—1461no—
515C42—1461no—
516C42C2×C4×D7312yes—
517C42#1192402712yes—
518C23—2921no—
519C23—2921no—
520C23—2921no—
521C23—2921no—
522C23—2921no—
523C23—2921no—
524C23—1462no—
525C23C2×D4×D7311yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—22no—
529Q8#1455654914yescomplemented
530D4—2921no—
531D4—2921no—
532D4—1462no—
533D4—1462no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—2928no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—732no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1462no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4#1455655112yes—
573C2×C4#1455655112yes—
574C2×C4C22×C4×D7311yescharacteristic
575C2×C4C2×#1192402711yescharacteristic
576C2×C4C2×#1192402711yescharacteristic
577C2×C4C2×#1192402911yescharacteristic
578C2×C4#1455663711yescharacteristic
579C2×C4#1455663811yescharacteristic
580C8—41no—
581C8—41no—
582C8—1462no—
583C8—1462no—
584C22—2922no—
585C22—2922no—
586C22—2922no—
587C22—5841no—
588C22—5841no—
589C22—5841no—
590C22—5841no—
591C22C2×#1455655111yescharacteristic
592C22#1798922811yescharacteristic
593C22#1798924311yescharacteristic
594C22AGL(1,ℤ/8)×D7311yescharacteristic
595C22#1798938611yescharacteristic
596C22#1798939311yescharacteristic
597C22#1798940011yescharacteristic
598C4—22no—
599C4—24no—
600C4—22no—
601C4—42no—
602C4—81no—
603C4—81no—
604C4—2922no—
605C4—2921no—
606C4—2921no—
607C4—2921no—
608C4—5841no—
609C4—5841no—
610C4—1462no—
611C4C2×#1455654911yescharacteristic
612C4#1798906711yescharacteristic
613C4#1798927911yescharacteristic
614C4#1798928911yescharacteristic
615C2—5842no—
616C2#2261286211yescharacteristic
617C2#2261874411yescharacteristic
618C2#2261876811yescharacteristic
619C2#2262191911yescharacteristic
620C2#2262211411yescharacteristic
621C2#2262265511yescharacteristic
622C2#2262276611yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1803C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1803D73
hypocenterC73#22781
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
3. #22612852C2semidirect
6. #22610526C2semidirect
7. #22610505C2semidirect
10. #22610066C2semidirect
12. #22609808C2semidirect
13. #22609763C2semidirect
14. #103×Dic73C2semidirect
15. C73×#1803C2semidirect
21. #17988685C4semidirect
217. Q8×C73#117semidirect
323. #1803D73semidirect
327. C73#22781semidirect
529. Q8#14556549semidirect

Maximal subgroups

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

subgroupindex
C2×#179886852
#226128822
#226128522
#226106292
#226106152
#226105262
#226105052
#226102842
#226100662
#226099422
#226098082
#226097632
#103×Dic732
C73×#18032
#2278173