Finite Group Databasebeta

← #33103232

Subgroups — #33103232

Order 18688.

982 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988588C211yesmaximal, characteristic
2#22612810C211yesmaximal, 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#22610018C211yesmaximal, 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
19C22×C4⋊Dic73C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22#17988588C418yescomplemented
23C2×#14556515C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556501C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29#17988490—21no—
30#17988490—21no—
31#17988490C2211yescharacteristic
32#17988490C2211yescharacteristic
33#17988483C2211yescharacteristic
34#17988483C2211yescharacteristic
35#17988472—21no—
36#17988472—21no—
37#17988447—21no—
38#17988447C2211yescharacteristic
39#17988445—21no—
40C4⋊C4×Dic73C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42C22⋊C4×Dic73—21no—
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—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—22no—
74C2×C4⋊Dic73C2×C414yes—
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C22⋊Dic73—21no—
85C2×C22⋊Dic73—21no—
86C2×C22⋊Dic73D411yescharacteristic
87C2×C22⋊Dic73D411yescharacteristic
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73C2311yescharacteristic
93C2×C4×Dic73C2311yescharacteristic
94#14556524—41no—
95#14556522—41no—
96#14556515C2×C414yes—
97#14556515C2×C414yes—
98#14556512C2×C414yes—
99#14556512C2×C414yes—
100#14556501—41no—
101C8×Dic73—41no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498D411yescharacteristic
108#14556498D411yescharacteristic
109C2×#41×C73—22no—
110C2×#41×C73—22no—
111C2×#41×C73—22no—
112C2×#41×C73—41no—
113C2×#41×C73—41no—
114C2×#41×C73C2×C414yes—
115C22×Q8×C73C2311yescharacteristic
116C22×D4×C73—21no—
117C22×D4×C73C2311yescharacteristic
118C23×C4×C73—21no—
119C23×C4×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121C22×C8×C73C2311yescharacteristic
122C2×C4⋊C4×C73C2311yescharacteristic
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C22⋊C4×C73—21no—
125#103×C73—22no—
126#103×C73—22no—
127#102×C73—41no—
128#102×C73—41no—
129C22⋊C8×C73—22no—
130#95×C73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—24no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—81no—
137C22×Dic73—81no—
138C22×Dic73C2×D411yescharacteristic
139C22×Dic73C2×D411yescharacteristic
140C22×Dic73C2×D411yescharacteristic
141C22×Dic73C2×D411yescharacteristic
142#11924012—44no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831C22×C412yes—
148C2×#10596831C22×C412yes—
149C4⋊Dic73—24no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—42no—
157C4⋊Dic73—42no—
158C4⋊Dic73C22×C412yes—
159C4⋊Dic73C22×C412yes—
160#11924007—81no—
161#11924007—81no—
162C22⋊Dic73—22no—
163C22⋊Dic73—41no—
164C22⋊Dic73—42no—
165C22⋊Dic73—42no—
166C22⋊Dic73—81no—
167C22⋊Dic73—81no—
168C22⋊Dic73—81no—
169C4×Dic73—21no—
170C4×Dic73—21no—
171C4×Dic73—21no—
172C4×Dic73—21no—
173C4×Dic73—42no—
174C4×Dic73—42no—
175C4×Dic73—81no—
176C4×Dic73—81no—
177C24×C73—21no—
178C24×C73C2×D411yescharacteristic
179#41×C73—28no—
180#41×C73—44no—
181#41×C73—44no—
182#41×C73—44no—
183C2×Q8×C73—21no—
184C2×Q8×C73—21no—
185C2×Q8×C73—21no—
186C2×Q8×C73C22×C412yes—
187C2×Q8×C73#4112yes—
188C2×Q8×C73#4112yes—
189C2×D4×C73—24no—
190C2×D4×C73—21no—
191C2×D4×C73—21no—
192C2×D4×C73—21no—
193C2×D4×C73—21no—
194C2×D4×C73—21no—
195C2×D4×C73—42no—
196C2×D4×C73—42no—
197C2×D4×C73C22×C412yes—
198C22×C4×C73—21no—
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73—24no—
204C22×C4×C73—21no—
205C22×C4×C73—21no—
206C22×C4×C73—42no—
207C22×C4×C73—42no—
208C22×C4×C73C22×C412yes—
209C22×C4×C73C2×D411yescharacteristic
210C22×C4×C73#4112yes—
211C22×C4×C73C2411yescharacteristic
212C2×C8×C73—21no—
213C2×C8×C73—21no—
214C2×C8×C73—21no—
215C2×C8×C73—21no—
216C2×C8×C73—21no—
217C2×C8×C73—21no—
218C4⋊C4×C73—21no—
219C4⋊C4×C73—21no—
220C4⋊C4×C73—21no—
221C4⋊C4×C73—21no—
222C22⋊C4×C73—42no—
223C23×C73—21no—
224C23×C73—21no—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73—22no—
230C23×C73—41no—
231C23×C73—41no—
232C23×C73—41no—
233C23×C73—41no—
234C23×C73—41no—
235C23×C73—41no—
236C23×C73C4×D412yes—
237C23×C73C22×D411yescharacteristic
238Q8×C73—22no—
239Q8×C73—22no—
240Q8×C73—22no—
241Q8×C73#11714yes—
242D4×C73—22no—
243D4×C73—22no—
244D4×C73—48no—
245D4×C73—41no—
246D4×C73—41no—
247C2×C4×C73—22no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—22no—
251C2×C4×C73—22no—
252C2×C4×C73—22no—
253C2×C4×C73—22no—
254C2×C4×C73—22no—
255C2×C4×C73—22no—
256C2×C4×C73—48no—
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×C73C4×D412yes—
264C2×C4×C73C23×C411yescharacteristic
265C2×C4×C73C2×#4111yescharacteristic
266C2×C4×C73C2×#4111yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×C4×C73H(5,𝔽2)11yescharacteristic
269C2×C4×C73H(5,𝔽2)11yescharacteristic
270C2×Dic73—21no—
271C2×Dic73—21no—
272C2×Dic73—22no—
273C2×Dic73—22no—
274C2×Dic73—21no—
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—41no—
284C2×Dic73—41no—
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—84no—
294C2×Dic73#13712yes—
295C8×C73—41no—
296C8×C73—41no—
297#10596831—22no—
298#10596831—22no—
299C22×C73—22no—
300C22×C73—22no—
301C22×C73—22no—
302C22×C73—41no—
303C22×C73—41no—
304C22×C73—41no—
305C22×C73—41no—
306C22×C73—42no—
307C22×C73—42no—
308C22×C73—42no—
309C22×C73—81no—
310C22×C73—81no—
311C22×C73—81no—
312C22×C73—81no—
313C22×C73C2×C4×D411yescharacteristic
314C22×C73#54611yescharacteristic
315C22×C73#54611yescharacteristic
316C22×C73C2×#13711yescharacteristic
317C22×C73#57511yescharacteristic
318C22×C73#57611yescharacteristic
319C22×C73#57711yescharacteristic
320C4×C73—22no—
321C4×C73—24no—
322C4×C73—81no—
323C4×C73—81no—
324C4×C73C2×#11711yescharacteristic
325C4×C73#51811yescharacteristic
326C4×C73#55111yescharacteristic
327C4×C73#55111yescharacteristic
328Dic73—22no—
329Dic73—41no—
330Dic73—42no—
331Dic73—42no—
332Dic73—41no—
333Dic73—41no—
334Dic73—81no—
335Dic73—81no—
336#22579—731nomaximal, Sylow
337C2×C73—42no—
338C2×C73—82no—
339C2×C73#234211yescharacteristic
340C2×C73C4×#13711yescharacteristic
341C2×C73#294611yescharacteristic
342C2×C73#330911yescharacteristic
343C2×C73#331611yescharacteristic
344C2×C73#332611yescharacteristic
345C2×C73#333311yescharacteristic
346C2×#417—731no—
347#2345—731no—
348#2341—731no—
349#1936—731no—
350#1934—731no—
351#1926—731no—
352#1925—731no—
353#1865—731no—
354#1863—731no—
355#1823—731no—
356#1809—731no—
357#1788D7311yescharacteristic
358#1763—731no—
359C4×#103—731no—
360#1753—731no—
361C73#2257911yescharacteristic, complemented, Sylow
362C22×#41C2×D7311yescharacteristic
363C22×M16—731no—
364C2×C4×Q8—731no—
365C2×C4×Q8—731no—
366C22×C4⋊C4—731no—
367C2×#107—731no—
368C2×#106—731no—
369#417—738no—
370C2×#103—731no—
371C2×#103—731no—
372C2×#103C2×D7311yescharacteristic
373C2×#103C2×D7311yescharacteristic
374C2×#102—731no—
375C2×#102—731no—
376C2×#102C2×D7311yescharacteristic
377C2×#102C2×D7311yescharacteristic
378C2×C22⋊C8C2×D7311yescharacteristic
379C2×#97—731no—
380C2×C4×C8—731no—
381#390—731no—
382#390—731no—
383#390—1461no—
384#390—1461no—
385#389—731no—
386#389—731no—
387#387—1461no—
388#387—1461no—
389#380—731no—
390#380—1461no—
391#380—1461no—
392#380C2×D7311yescharacteristic
393C4×C4⋊C4—731no—
394C4×C4⋊C4—731no—
395C4×C22⋊C4—731no—
396C4×C22⋊C4—1461no—
397C4×C22⋊C4—1461no—
398#340—731no—
399#340—731no—
400#340—731no—
401#340—731no—
402#340—731no—
403#340—731no—
404#336—731no—
405#336—731no—
406C2×#41—22no—
407C2×#41—22no—
408C2×#41—22no—
409C2×#41—41no—
410C2×#41—41no—
411C2×#41C4×D7314yes—
412C22×Q8C22×D7311yescharacteristic
413C22×D4—21no—
414C22×D4C22×D7311yescharacteristic
415C23×C4—21no—
416C23×C4—1461no—
417C23×C4C22×D7311yescharacteristic
418C2×M16—2921no—
419C2×M16—2921no—
420C2×M16—734no—
421C22×C8—731no—
422C22×C8—731no—
423C22×C8C22×D7311yescharacteristic
424C22×C8C22×D7311yescharacteristic
425C4×Q8—2921no—
426C4×Q8—2921no—
427C4×Q8—1462no—
428C4×Q8—1462no—
429C2×C4⋊C4—734no—
430C2×C4⋊C4—731no—
431C2×C4⋊C4—731no—
432C2×C4⋊C4—731no—
433C2×C4⋊C4—731no—
434C2×C4⋊C4—1461no—
435C2×C4⋊C4—1461no—
436C2×C4⋊C4—1461no—
437C2×C4⋊C4—1461no—
438C2×C4⋊C4—1461no—
439C2×C4⋊C4—1462no—
440C2×C4⋊C4C22×D7311yescharacteristic
441C2×C4⋊C4C22×D7311yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—731no—
444C2×C22⋊C4—731no—
445C2×C22⋊C4—1461no—
446C2×C22⋊C4—1461no—
447C2×C22⋊C4—1461no—
448C2×C22⋊C4—1461no—
449C2×C22⋊C4—1461no—
450C2×C42—731no—
451C2×C42—731no—
452C2×C42—1461no—
453C2×C42—1461no—
454C2×C42—1461no—
455C2×C42—1461no—
456#107—2921no—
457#106—2921no—
458#103—22no—
459#103—22no—
460#103—734no—
461#103—734no—
462#102—41no—
463#102—41no—
464#102—734no—
465#102—734no—
466C22⋊C8—22no—
467#97—2921no—
468C4×C8—2921no—
469#95—21no—
470#95—731no—
471#95—731no—
472#95—1461no—
473#95—1461no—
474#95—1461no—
475#95—1461no—
476#95—1461no—
477C24—21no—
478C24D4×D7311yescharacteristic
479#41—28no—
480#41—44no—
481#41—44no—
482#41—44no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8C2×C4×D7312yes—
487C2×Q8#1192402712yes—
488C2×Q8#1192402712yes—
489C2×D4—24no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—42no—
496C2×D4—42no—
497C2×D4C2×C4×D7312yes—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—24no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—5841no—
509C22×C4—5841no—
510C22×C4—731no—
511C22×C4—731no—
512C22×C4—731no—
513C22×C4—731no—
514C22×C4—1461no—
515C22×C4—1461no—
516C22×C4—1461no—
517C22×C4—1461no—
518C22×C4—1464no—
519C22×C4C2×C4×D7312yes—
520C22×C4D4×D7311yescharacteristic
521C22×C4#1192402712yes—
522C22×C4C23×D7311yescharacteristic
523M16—2924no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—732no—
531C2×C8—732no—
532C2×C8—1461no—
533C2×C8—1461no—
534C2×C8—1461no—
535C2×C8—1461no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—2922no—
541C4⋊C4—2922no—
542C4⋊C4—5841no—
543C4⋊C4—5841no—
544C4⋊C4—732no—
545C4⋊C4—732no—
546C4⋊C4—1461no—
547C4⋊C4—1461no—
548C4⋊C4—1461no—
549C4⋊C4—1461no—
550C4⋊C4—1464no—
551C4⋊C4—1461no—
552C4⋊C4—1461no—
553C22⋊C4—42no—
554C22⋊C4—2922no—
555C22⋊C4—2922no—
556C22⋊C4—2921no—
557C22⋊C4—5841no—
558C22⋊C4—5841no—
559C22⋊C4—5841no—
560C22⋊C4—1462no—
561C42—2922no—
562C42—2922no—
563C42—5841no—
564C42—5841no—
565C42—1461no—
566C42—1461no—
567C42—1461no—
568C42—1461no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23#1455655112yes—
583C23C2×D4×D7311yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—22no—
587Q8#1455654714yes—
588D4—22no—
589D4—22no—
590D4—48no—
591D4—41no—
592D4—41no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—48no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—2921no—
610C2×C4—2921no—
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—5844no—
620C2×C4—732no—
621C2×C4—1461no—
622C2×C4—1461no—
623C2×C4—1461no—
624C2×C4—1461no—
625C2×C4—1461no—
626C2×C4—1461no—
627C2×C4—1461no—
628C2×C4—1461no—
629C2×C4—1461no—
630C2×C4—1462no—
631C2×C4—1462no—
632C2×C4—1461no—
633C2×C4—1461no—
634C2×C4#1455655112yes—
635C2×C4C22×C4×D7311yescharacteristic
636C2×C4C2×#1192402711yescharacteristic
637C2×C4C2×#1192402711yescharacteristic
638C2×C4C2×#1192402711yescharacteristic
639C2×C4#1455663611yescharacteristic
640C2×C4#1455663611yescharacteristic
641C8—41no—
642C8—41no—
643C8—1462no—
644C8—1462no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22C2×#1455655111yescharacteristic
660C22#1798922411yescharacteristic
661C22#1798922411yescharacteristic
662C22#137×D7311yescharacteristic
663C22#1798938311yescharacteristic
664C22#1798938911yescharacteristic
665C22#1798939411yescharacteristic
666C4—22no—
667C4—24no—
668C4—81no—
669C4—81no—
670C4—2921no—
671C4—2922no—
672C4—2922no—
673C4—2921no—
674C4—2921no—
675C4—5841no—
676C4—5841no—
677C4—1462no—
678C4C2×#1455654711yescharacteristic
679C4#1798906011yescharacteristic
680C4#1798926811yescharacteristic
681C4#1798926811yescharacteristic
682C2—42no—
683C2—82no—
684C2#2261278211yescharacteristic
685C2#2261875511yescharacteristic
686C2#2261876611yescharacteristic
687C2#2262193011yescharacteristic
688C2#2262203511yescharacteristic
689C2#2262217711yescharacteristic
690C2#2262226211yescharacteristic
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#22579
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. #22612810C2semidirect
4. #22610625C2semidirect
5. #22610611C2semidirect
6. #22610561C2semidirect
7. #22610554C2semidirect
8. #22610197C2semidirect
9. #22610183C2semidirect
10. #22610018C2semidirect
11. #22609962C2semidirect
12. #22609763C2semidirect
13. #103×Dic73C2semidirect
14. #22609717C2semidirect
22. #17988588C4semidirect
361. C73#22579semidirect

Maximal subgroups

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

subgroupindex
C2×#179885882
#226128102
#226127722
#226106252
#226106112
#226105612
#226105542
#226101972
#226101832
#226100182
#226099622
#226097632
#103×Dic732
#226097172
C73×#17882
#2257973