Finite Group Databasebeta

← #33103292

Subgroups — #33103292

Order 18688.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988588C211yesmaximal, characteristic
2#22612802C211yesmaximal, characteristic, complemented
3#22612772C211yesmaximal, characteristic
4#22610625C211yesmaximal, characteristic, complemented
5#22610611C211yesmaximal, characteristic, complemented
6#22610561C211yesmaximal, characteristic, complemented
7#22610554C211yesmaximal, characteristic, complemented
8#22610197C211yesmaximal, characteristic, complemented
9#22610183C211yesmaximal, characteristic, complemented
10#22610022C211yesmaximal, characteristic, complemented
11#22609962C211yesmaximal, characteristic, complemented
12#22609763C211yesmaximal, characteristic, complemented
13#103×Dic73C211yesmaximal, characteristic, complemented
14#22609717C211yesmaximal, characteristic, complemented
15C73×#1788C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×Q8×Dic73C2211yescharacteristic
19C2×#14556547—21no—
20C22×C4⋊Dic73C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23#17988588C418yescomplemented
24C2×#14556515C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×#14556501C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30#17988490—21no—
31#17988490—21no—
32#17988490C2211yescharacteristic
33#17988490C2211yescharacteristic
34#17988483C2211yescharacteristic
35#17988483C2211yescharacteristic
36#17988470—21no—
37#17988470—21no—
38#17988448—21no—
39#17988447—21no—
40#17988447C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42C4⋊C4×Dic73C2211yescharacteristic
43C4×C22⋊Dic73—21no—
44C4×C22⋊Dic73C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C22×#41×C73C2211yescharacteristic
54C2×#103×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×#102×C73C2211yescharacteristic
57C2×#102×C73C2211yescharacteristic
58C2×C22⋊C8×C73C2211yescharacteristic
59#380×C73C2211yescharacteristic
60C23×Dic73—21no—
61C2×#11924012—41no—
62C2×#11924012—41no—
63C2×#11924012C2×C414yes—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66Q8×Dic73—22no—
67Q8×Dic73—22no—
68Q8×Dic73—41no—
69Q8×Dic73—41no—
70#14556547—44no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73—22no—
75C2×C4⋊Dic73C2×C414yes—
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C22⋊Dic73—21no—
85C2×C22⋊Dic73—21no—
86C2×C22⋊Dic73—21no—
87C2×C22⋊Dic73D411yescharacteristic
88C2×C22⋊Dic73D411yescharacteristic
89C2×C4×Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73C2311yescharacteristic
94C2×C4×Dic73C2311yescharacteristic
95#14556524—41no—
96#14556522—41no—
97#14556515C2×C414yes—
98#14556515C2×C414yes—
99#14556512C2×C414yes—
100#14556512C2×C414yes—
101#14556501—41no—
102C8×Dic73—41no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498D411yescharacteristic
109#14556498D411yescharacteristic
110C2×#41×C73—22no—
111C2×#41×C73—22no—
112C2×#41×C73—22no—
113C2×#41×C73—41no—
114C2×#41×C73—41no—
115C2×#41×C73C2×C414yes—
116C22×Q8×C73C2311yescharacteristic
117C22×D4×C73—21no—
118C22×D4×C73C2311yescharacteristic
119C23×C4×C73—21no—
120C23×C4×C73C2311yescharacteristic
121C22×C8×C73C2311yescharacteristic
122C22×C8×C73C2311yescharacteristic
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C4⋊C4×C73C2311yescharacteristic
125C2×C22⋊C4×C73—21no—
126#103×C73—22no—
127#103×C73—22no—
128#102×C73—41no—
129#102×C73—41no—
130C22⋊C8×C73—22no—
131#95×C73—21no—
132C22×Dic73—24no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—42no—
138C22×Dic73—42no—
139C22×Dic73C2×D411yescharacteristic
140C22×Dic73C2×D411yescharacteristic
141C22×Dic73C2×D411yescharacteristic
142C22×Dic73C2×D411yescharacteristic
143#11924012—44no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831C22×C412yes—
149C2×#10596831C22×C412yes—
150C4⋊Dic73—24no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—42no—
158C4⋊Dic73—42no—
159C4⋊Dic73C22×C412yes—
160C4⋊Dic73C22×C412yes—
161#11924007—42no—
162#11924007—42no—
163C22⋊Dic73—22no—
164C22⋊Dic73—41no—
165C22⋊Dic73—42no—
166C22⋊Dic73—42no—
167C22⋊Dic73—81no—
168C22⋊Dic73—81no—
169C22⋊Dic73—81no—
170C4×Dic73—21no—
171C4×Dic73—21no—
172C4×Dic73—21no—
173C4×Dic73—21no—
174C4×Dic73—42no—
175C4×Dic73—42no—
176C4×Dic73—42no—
177C4×Dic73—42no—
178C24×C73—21no—
179C24×C73C2×D411yescharacteristic
180#41×C73—28no—
181#41×C73—44no—
182#41×C73—44no—
183#41×C73—44no—
184C2×Q8×C73—21no—
185C2×Q8×C73—21no—
186C2×Q8×C73—21no—
187C2×Q8×C73C22×C412yes—
188C2×Q8×C73#4112yes—
189C2×Q8×C73#4112yes—
190C2×D4×C73—24no—
191C2×D4×C73—21no—
192C2×D4×C73—21no—
193C2×D4×C73—21no—
194C2×D4×C73—21no—
195C2×D4×C73—21no—
196C2×D4×C73—42no—
197C2×D4×C73—42no—
198C2×D4×C73C22×C412yes—
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73—21no—
204C22×C4×C73—24no—
205C22×C4×C73—21no—
206C22×C4×C73—21no—
207C22×C4×C73—42no—
208C22×C4×C73—42no—
209C22×C4×C73C22×C412yes—
210C22×C4×C73C2×D411yescharacteristic
211C22×C4×C73#4112yes—
212C22×C4×C73C2411yescharacteristic
213C2×C8×C73—21no—
214C2×C8×C73—21no—
215C2×C8×C73—21no—
216C2×C8×C73—21no—
217C2×C8×C73—21no—
218C2×C8×C73—21no—
219C4⋊C4×C73—21no—
220C4⋊C4×C73—21no—
221C4⋊C4×C73—21no—
222C4⋊C4×C73—21no—
223C22⋊C4×C73—42no—
224C23×C73—21no—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—22no—
231C23×C73—41no—
232C23×C73—41no—
233C23×C73—41no—
234C23×C73—41no—
235C23×C73—41no—
236C23×C73—41no—
237C23×C73C4×D412yes—
238C23×C73C22×D411yescharacteristic
239Q8×C73—22no—
240Q8×C73—22no—
241Q8×C73—22no—
242Q8×C73#11714yescomplemented
243D4×C73—22no—
244D4×C73—22no—
245D4×C73—48no—
246D4×C73—41no—
247D4×C73—41no—
248C2×C4×C73—22no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
251C2×C4×C73—22no—
252C2×C4×C73—22no—
253C2×C4×C73—22no—
254C2×C4×C73—22no—
255C2×C4×C73—22no—
256C2×C4×C73—22no—
257C2×C4×C73—48no—
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×C73C4×D412yes—
265C2×C4×C73C23×C411yescharacteristic
266C2×C4×C73C2×#4111yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×C4×C73C2×#4111yescharacteristic
269C2×C4×C73H(5,𝔽2)11yescharacteristic
270C2×C4×C73H(5,𝔽2)11yescharacteristic
271C2×Dic73—21no—
272C2×Dic73—21no—
273C2×Dic73—22no—
274C2×Dic73—22no—
275C2×Dic73—21no—
276C2×Dic73—21no—
277C2×Dic73—21no—
278C2×Dic73—21no—
279C2×Dic73—21no—
280C2×Dic73—21no—
281C2×Dic73—21no—
282C2×Dic73—21no—
283C2×Dic73—21no—
284C2×Dic73—48no—
285C2×Dic73—41no—
286C2×Dic73—41no—
287C2×Dic73—41no—
288C2×Dic73—41no—
289C2×Dic73—41no—
290C2×Dic73—41no—
291C2×Dic73—41no—
292C2×Dic73—41no—
293C2×Dic73—41no—
294C2×Dic73—41no—
295C2×Dic73AGL(1,ℤ/8)12yes—
296C8×C73—41no—
297C8×C73—41no—
298#10596831—22no—
299#10596831—22no—
300C22×C73—22no—
301C22×C73—22no—
302C22×C73—22no—
303C22×C73—41no—
304C22×C73—41no—
305C22×C73—41no—
306C22×C73—41no—
307C22×C73—42no—
308C22×C73—42no—
309C22×C73—42no—
310C22×C73—81no—
311C22×C73—81no—
312C22×C73—81no—
313C22×C73—81no—
314C22×C73C2×C4×D411yescharacteristic
315C22×C73#54611yescharacteristic
316C22×C73#54611yescharacteristic
317C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
318C22×C73#57511yescharacteristic
319C22×C73#57711yescharacteristic
320C22×C73#57811yescharacteristic
321C4×C73—22no—
322C4×C73—24no—
323C4×C73—81no—
324C4×C73—81no—
325C4×C73C2×#11711yescharacteristic
326C4×C73#51811yescharacteristic
327C4×C73#55111yescharacteristic
328C4×C73#55111yescharacteristic
329Dic73—22no—
330Dic73—41no—
331Dic73—42no—
332Dic73—42no—
333Dic73—41no—
334Dic73—41no—
335Dic73—81no—
336Dic73—81no—
337#22583—731nomaximal, Sylow
338C2×C73—42no—
339C2×C73—82no—
340C2×C73#234211yescharacteristic
341C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
342C2×C73#294711yescharacteristic
343C2×C73#330811yescharacteristic
344C2×C73#331711yescharacteristic
345C2×C73#332711yescharacteristic
346C2×C73#333211yescharacteristic
347C2×#417—731no—
348#2344—731no—
349#2341—731no—
350#1936—731no—
351#1934—731no—
352#1926—731no—
353#1925—731no—
354#1865—731no—
355#1863—731no—
356#1824—731no—
357#1809—731no—
358#1788D7311yescharacteristic
359#1763—731no—
360C4×#103—731no—
361#1753—731no—
362C73#2258311yescharacteristic, complemented, Sylow
363C22×#41C2×D7311yescharacteristic
364C22×M16—731no—
365C2×C4×Q8—731no—
366C2×C4×Q8—731no—
367C2×#117—1461no—
368C22×C4⋊C4—731no—
369C2×#107—731no—
370C2×#106—731no—
371#417—738no—
372C2×#103—731no—
373C2×#103—731no—
374C2×#103C2×D7311yescharacteristic
375C2×#103C2×D7311yescharacteristic
376C2×#102—731no—
377C2×#102—731no—
378C2×#102C2×D7311yescharacteristic
379C2×#102C2×D7311yescharacteristic
380C2×C22⋊C8C2×D7311yescharacteristic
381C2×#97—731no—
382C2×C4×C8—731no—
383#390—731no—
384#390—731no—
385#390—1461no—
386#390—1461no—
387#389—731no—
388#389—731no—
389#387—1461no—
390#387—1461no—
391#381—1461no—
392#380—731no—
393#380—1461no—
394#380C2×D7311yescharacteristic
395C4×C4⋊C4—731no—
396C4×C4⋊C4—731no—
397C4×C22⋊C4—731no—
398C4×C22⋊C4—1461no—
399#340—731no—
400#340—731no—
401#340—731no—
402#340—731no—
403#340—731no—
404#340—731no—
405#336—731no—
406#336—731no—
407C2×#41—22no—
408C2×#41—22no—
409C2×#41—22no—
410C2×#41—41no—
411C2×#41—41no—
412C2×#41C4×D7314yes—
413C22×Q8C22×D7311yescharacteristic
414C22×D4—21no—
415C22×D4C22×D7311yescharacteristic
416C23×C4—21no—
417C23×C4—1461no—
418C23×C4C22×D7311yescharacteristic
419C2×M16—2921no—
420C2×M16—2921no—
421C2×M16—734no—
422C22×C8—731no—
423C22×C8—731no—
424C22×C8C22×D7311yescharacteristic
425C22×C8C22×D7311yescharacteristic
426C4×Q8—2921no—
427C4×Q8—2921no—
428C4×Q8—1462no—
429C4×Q8—1462no—
430#117—2924no—
431C2×C4⋊C4—734no—
432C2×C4⋊C4—731no—
433C2×C4⋊C4—731no—
434C2×C4⋊C4—731no—
435C2×C4⋊C4—731no—
436C2×C4⋊C4—1461no—
437C2×C4⋊C4—1461no—
438C2×C4⋊C4—1461no—
439C2×C4⋊C4—1461no—
440C2×C4⋊C4—1461no—
441C2×C4⋊C4—1462no—
442C2×C4⋊C4C22×D7311yescharacteristic
443C2×C4⋊C4C22×D7311yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—731no—
446C2×C22⋊C4—731no—
447C2×C22⋊C4—1461no—
448C2×C22⋊C4—1461no—
449C2×C22⋊C4—1461no—
450C2×C22⋊C4—1461no—
451C2×C22⋊C4—1461no—
452C2×C42—731no—
453C2×C42—731no—
454C2×C42—1461no—
455C2×C42—1461no—
456C2×C42—1461no—
457C2×C42—1461no—
458#107—2921no—
459#106—2921no—
460#103—22no—
461#103—22no—
462#103—734no—
463#103—734no—
464#102—41no—
465#102—41no—
466#102—734no—
467#102—734no—
468C22⋊C8—22no—
469#97—2921no—
470C4×C8—2921no—
471#95—21no—
472#95—731no—
473#95—731no—
474#95—1461no—
475#95—1461no—
476#95—1461no—
477#95—1461no—
478#95—1461no—
479C24—21no—
480C24D4×D7311yescharacteristic
481#41—28no—
482#41—44no—
483#41—44no—
484#41—44no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8C2×C4×D7312yes—
489C2×Q8#1192402712yes—
490C2×Q8#1192402712yes—
491C2×D4—24no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—42no—
498C2×D4—42no—
499C2×D4C2×C4×D7312yes—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—24no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—2922no—
511C22×C4—2922no—
512C22×C4—731no—
513C22×C4—731no—
514C22×C4—731no—
515C22×C4—731no—
516C22×C4—1461no—
517C22×C4—1461no—
518C22×C4—1464no—
519C22×C4—1461no—
520C22×C4—1461no—
521C22×C4C2×C4×D7312yes—
522C22×C4D4×D7311yescharacteristic
523C22×C4#1192402712yes—
524C22×C4C23×D7311yescharacteristic
525M16—2924no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—732no—
533C2×C8—732no—
534C2×C8—1461no—
535C2×C8—1461no—
536C2×C8—1461no—
537C2×C8—1461no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—2922no—
543C4⋊C4—2922no—
544C4⋊C4—2922no—
545C4⋊C4—2922no—
546C4⋊C4—732no—
547C4⋊C4—732no—
548C4⋊C4—1461no—
549C4⋊C4—1461no—
550C4⋊C4—1461no—
551C4⋊C4—1461no—
552C4⋊C4—1464no—
553C4⋊C4—1461no—
554C4⋊C4—1461no—
555C22⋊C4—42no—
556C22⋊C4—2922no—
557C22⋊C4—2922no—
558C22⋊C4—2921no—
559C22⋊C4—5841no—
560C22⋊C4—5841no—
561C22⋊C4—5841no—
562C22⋊C4—1462no—
563C42—2922no—
564C42—2922no—
565C42—2922no—
566C42—2922no—
567C42—1461no—
568C42—1461no—
569C42—1461no—
570C42—1461no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—22no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23#1455655112yes—
585C23C2×D4×D7311yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—22no—
589Q8#1455654714yescomplemented
590D4—22no—
591D4—22no—
592D4—48no—
593D4—41no—
594D4—41no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—48no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—2928no—
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—732no—
623C2×C4—1461no—
624C2×C4—1461no—
625C2×C4—1461no—
626C2×C4—1461no—
627C2×C4—1461no—
628C2×C4—1461no—
629C2×C4—1461no—
630C2×C4—1461no—
631C2×C4—1462no—
632C2×C4—1462no—
633C2×C4—1461no—
634C2×C4—1461no—
635C2×C4—1461no—
636C2×C4#1455655112yes—
637C2×C4C22×C4×D7311yescharacteristic
638C2×C4C2×#1192402711yescharacteristic
639C2×C4C2×#1192402711yescharacteristic
640C2×C4C2×#1192402711yescharacteristic
641C2×C4#1455663611yescharacteristic
642C2×C4#1455663611yescharacteristic
643C8—41no—
644C8—41no—
645C8—1462no—
646C8—1462no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22C2×#1455655111yescharacteristic
662C22#1798922411yescharacteristic
663C22#1798922411yescharacteristic
664C22AGL(1,ℤ/8)×D7311yescharacteristic
665C22#1798938611yescharacteristic
666C22#1798939411yescharacteristic
667C22#1798940111yescharacteristic
668C4—22no—
669C4—24no—
670C4—81no—
671C4—81no—
672C4—2921no—
673C4—2921no—
674C4—2922no—
675C4—2922no—
676C4—2921no—
677C4—5841no—
678C4—5841no—
679C4—1462no—
680C4C2×#1455654711yescharacteristic
681C4#1798906011yescharacteristic
682C4#1798926811yescharacteristic
683C4#1798926811yescharacteristic
684C2—42no—
685C2—82no—
686C2#2261278211yescharacteristic
687C2#2261874411yescharacteristic
688C2#2261877711yescharacteristic
689C2#2262191511yescharacteristic
690C2#2262205011yescharacteristic
691C2#2262219211yescharacteristic
692C2#2262225111yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1788C2
socleC23×C73C22×D4
prime radicalC22×#41×C73C22
cosocleC22×C4×C73C24
hypercenter#1788D73
hypocenterC73#22583
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
2. #22612802C2semidirect
4. #22610625C2semidirect
5. #22610611C2semidirect
6. #22610561C2semidirect
7. #22610554C2semidirect
8. #22610197C2semidirect
9. #22610183C2semidirect
10. #22610022C2semidirect
11. #22609962C2semidirect
12. #22609763C2semidirect
13. #103×Dic73C2semidirect
14. #22609717C2semidirect
23. #17988588C4semidirect
242. Q8×C73#117semidirect
362. C73#22583semidirect
589. Q8#14556547semidirect

Maximal subgroups

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

subgroupindex
C2×#179885882
#226128022
#226127722
#226106252
#226106112
#226105612
#226105542
#226101972
#226101832
#226100222
#226099622
#226097632
#103×Dic732
#226097172
C73×#17882
#2258373