Finite Group Databasebeta

← #33092547

Subgroups — #33092547

Order 18688.

1014 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988826C211yesmaximal, characteristic
2C2×#17988691C211yesmaximal, characteristic, complemented
3C22×#14556501C211yesmaximal, characteristic
4C22⋊Q8×Dic73C211yesmaximal, characteristic
5#22612459C211yesmaximal, characteristic
6#22610799C211yesmaximal, characteristic, complemented
7#22610668C211yesmaximal, characteristic, complemented
8#22610622C211yesmaximal, characteristic, complemented
9#22610618C211yesmaximal, characteristic, complemented
10#22610354C211yesmaximal, characteristic
11#22609954C211yesmaximal, characteristic
12#22609827C211yesmaximal, characteristic, complemented
13#22609763C211yesmaximal, characteristic, complemented
14#22609760C211yesmaximal, characteristic, complemented
15C73×#1892C211yesmaximal, characteristic
16C22×#11924018C2211yescharacteristic
17C23×#10596831C2211yescharacteristic
18C2×#14556570C2211yescharacteristic
19C2×Q8×Dic73C2211yescharacteristic
20C2×#14556556C2211yescharacteristic
21C22×C4×Dic73C2211yescharacteristic
22#17988826C418yescomplemented
23#17988691—24no—
24C2×#14556524C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×#14556516C2211yescharacteristic
28C2×#14556516C2211yescharacteristic
29C2×#14556515C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×#14556501—24no—
32C2×#14556501C2211yescharacteristic
33C2×#14556501C2211yescharacteristic
34C2×#14556501C2211yescharacteristic
35C2×#14556501C2211yescharacteristic
36#17988494C2211yescharacteristic
37#17988492C2211yescharacteristic
38#17988486C2211yescharacteristic
39#17988483C2211yescharacteristic
40#17988473—21no—
41#17988469—21no—
42#17988462—21no—
43#17988459—21no—
44#17988447C2211yescharacteristic
45C4⋊C4×Dic73—21no—
46C4⋊C4×Dic73C2211yescharacteristic
47C4⋊C4×Dic73C2211yescharacteristic
48#17988438—21no—
49C22⋊C4×Dic73—21no—
50C22⋊C4×Dic73—21no—
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988349C2211yescharacteristic
55C23×C8×C73C2211yescharacteristic
56C2×C22⋊Q8×C73C2211yescharacteristic
57C2×#103×C73C2211yescharacteristic
58C2×#103×C73C2211yescharacteristic
59#380×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#340×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C23×Dic73D411yescharacteristic
64C23×Dic73D411yescharacteristic
65C2×#11924018—22no—
66C2×#11924018—22no—
67C2×#11924018C2×C414yescomplemented
68C22×#10596831—22no—
69C22×#10596831—22no—
70C22×#10596831C2×C414yes—
71C22×#10596831C2311yescharacteristic
72C22×#10596831C2311yescharacteristic
73#14556570—22no—
74#14556570C2×C414yes—
75Q8×Dic73—22no—
76Q8×Dic73—22no—
77#14556556—22no—
78#14556556—22no—
79C2×C4⋊Dic73—21no—
80C2×C4⋊Dic73—21no—
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×C4⋊Dic73C2311yescharacteristic
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×#11924007—21no—
87C2×C22⋊Dic73—21no—
88C2×C22⋊Dic73—21no—
89C2×C4×Dic73—22no—
90C2×C4×Dic73—22no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73—21no—
95C2×C4×Dic73D411yescharacteristic
96C2×C4×Dic73D411yescharacteristic
97C2×C4×Dic73C2311yescharacteristic
98C2×C4×Dic73C2311yescharacteristic
99#14556524C2×C414yescomplemented
100#14556522—22no—
101#14556522—22no—
102#14556516—22no—
103#14556516—22no—
104#14556515C2×C414yescomplemented
105#14556514C2×C414yescomplemented
106#14556501—22no—
107#14556501—22no—
108#14556501—22no—
109#14556501D414yes—
110#14556498—21no—
111#14556498—21no—
112#14556498—21no—
113#14556498—21no—
114#14556498—21no—
115#14556498—21no—
116#14556498—21no—
117#14556498—21no—
118#14556498—21no—
119C22×Q8×C73C2311yescharacteristic
120C23×C4×C73C2311yescharacteristic
121C22×C8×C73—22no—
122C22×C8×C73—22no—
123C22×C8×C73—22no—
124C22×C8×C73C2311yescharacteristic
125C22×C8×C73C2311yescharacteristic
126C22⋊Q8×C73—22no—
127C22⋊Q8×C73C2×C414yes—
128C2×C4⋊C4×C73—21no—
129C2×C4⋊C4×C73C2311yescharacteristic
130C2×C4⋊C4×C73C2311yescharacteristic
131C2×C4⋊C4×C73C2311yescharacteristic
132C2×C22⋊C4×C73—21no—
133C2×C22⋊C4×C73—21no—
134#103×C73—22no—
135#103×C73—22no—
136#95×C73—21no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146C22×C412yes—
143C22×Dic73—22no—
144C22×Dic73—22no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147C22×Dic73—22no—
148C22×Dic73—22no—
149C22×Dic73—21no—
150C22×Dic73—21no—
151C22×Dic73—41no—
152C22×Dic73—41no—
153C22×Dic73C2×D411yescharacteristic
154C22×Dic73C2×D411yescharacteristic
155C22×Dic73C2×D411yescharacteristic
156C22×Dic73C2×D411yescharacteristic
157#11924018—28no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—28no—
161C2×#10596831C22×C412yes—
162C2×#10596831C22×C412yes—
163C2×#10596831C2×D412yes—
164C2×#10596831#4112yes—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73—42no—
171C4⋊Dic73—42no—
172C4⋊Dic73C22×C412yes—
173#11924007—24no—
174#11924007—42no—
175#11924007—42no—
176C22⋊Dic73—24no—
177C22⋊Dic73—42no—
178C4×Dic73—21no—
179C4×Dic73—21no—
180C4×Dic73—22no—
181C4×Dic73—22no—
182C4×Dic73—42no—
183C4×Dic73—42no—
184C4×Dic73—42no—
185C4×Dic73—42no—
186C4×Dic73C2×D412yes—
187C4×Dic73#4112yes—
188C24×C73C2×D411yescharacteristic
189C2×Q8×C73—21no—
190C2×Q8×C73—21no—
191C2×Q8×C73—21no—
192C2×Q8×C73—21no—
193C2×Q8×C73—21no—
194C2×Q8×C73C22×C412yes—
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73—21no—
198C22×C4×C73—21no—
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73C22×C412yes—
204C22×C4×C73C2×D411yescharacteristic
205C22×C4×C73#4112yes—
206C22×C4×C73C2411yescharacteristic
207C2×C8×C73—28no—
208C2×C8×C73—21no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73—21no—
212C2×C8×C73C2×D412yes—
213C2×C8×C73#4112yes—
214C4⋊C4×C73—24no—
215C4⋊C4×C73—21no—
216C4⋊C4×C73—21no—
217C4⋊C4×C73—21no—
218C4⋊C4×C73—21no—
219C4⋊C4×C73—21no—
220C4⋊C4×C73C22×C412yes—
221C22⋊C4×C73—24no—
222C22⋊C4×C73—42no—
223C23×C73—21no—
224C23×C73—21no—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73C4×D412yes—
228C23×C73#13712yes—
229C23×C73#13712yes—
230C23×C73C22×D411yescharacteristic
231Q8×C73—22no—
232Q8×C73—22no—
233Q8×C73—41no—
234Q8×C73—41no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—22no—
238C2×C4×C73—22no—
239C2×C4×C73—22no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73C4×D412yes—
254C2×C4×C73C23×C411yescharacteristic
255C2×C4×C73C22×D411yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73C2×#4111yescharacteristic
259C2×C4×C73C2×#4111yescharacteristic
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—22no—
265C2×Dic73—22no—
266C2×Dic73—21no—
267C2×Dic73—21no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—21no—
271C2×Dic73—21no—
272C2×Dic73—44no—
273C2×Dic73—44no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—41no—
284C2×Dic73#12112yes—
285C2×Dic73#12112yes—
286Dic146—22no—
287Dic146—22no—
288Dic146—41no—
289Dic146—41no—
290C8×C73—22no—
291C8×C73—22no—
292#10596831—22no—
293#10596831C4×D414yes—
294C22×C73—22no—
295C22×C73—22no—
296C22×C73—22no—
297C22×C73—22no—
298C22×C73—22no—
299C22×C73—22no—
300C22×C73#44114yescomplemented
301C22×C73C2×C4×D411yescharacteristic
302C22×C73C2×#12111yescharacteristic
303C22×C73#53211yescharacteristic
304C22×C73C2×#13711yescharacteristic
305C22×C73C2×#13711yescharacteristic
306C22×C73#57511yescharacteristic
307C22×C73#57511yescharacteristic
308C4×C73—22no—
309C4×C73—42no—
310C4×C73—42no—
311C4×C73—81no—
312C4×C73—81no—
313C4×C73C2×C4×D411yescharacteristic
314C4×C73C4×#4111yescharacteristic
315C4×C73#52511yescharacteristic
316C4×C73#52911yescharacteristic
317Dic73—22no—
318Dic73—22no—
319Dic73—42no—
320Dic73—41no—
321Dic73—41no—
322Dic73—42no—
323Dic73—81no—
324Dic73—81no—
325#21527—731nomaximal, Sylow
326C2×C73—24no—
327C2×C73C4×#12111yescharacteristic
328C2×C73C2×#44111yescharacteristic
329C2×C73#294311yescharacteristic
330C2×C73#305611yescharacteristic
331C2×C73#305611yescharacteristic
332C2×C73#318111yescharacteristic
333C2×C73#318111yescharacteristic
334C2×#467—731no—
335C2×#441—731no—
336C22×#97—731no—
337C4×C22⋊Q8—731no—
338C4×C22⋊Q8—731no—
339#1971—731no—
340#1943—731no—
341#1935—731no—
342#1935—731no—
343#1892—731no—
344#1892D7311yescharacteristic
345#1807—731no—
346#1776—731no—
347#1763—731no—
348#1763—731no—
349C73#2152711yescharacteristic, complemented, Sylow
350C22×Q16—731no—
351C23×C8—731no—
352C23×C8C2×D7311yescharacteristic
353C2×C22⋊Q8—731no—
354C2×C22⋊Q8C2×D7311yescharacteristic
355C2×C4×Q8—731no—
356C2×C4×Q8—731no—
357C22×C42—731no—
358#467—738no—
359#441—1464no—
360C2×#107—731no—
361C2×#106—731no—
362C2×#106—731no—
363C2×#103—731no—
364C2×#103—731no—
365C2×#103—731no—
366C2×#103—731no—
367C2×#103C2×D7311yescharacteristic
368C2×#103C2×D7311yescharacteristic
369C2×#97—731no—
370C2×#97—731no—
371C2×#97—731no—
372C2×#97—731no—
373C2×#97—1464no—
374#391—731no—
375#391—731no—
376#389—731no—
377#389—731no—
378#387—1461no—
379#387—1461no—
380#385—1461no—
381#385—1461no—
382#380—731no—
383#380C2×D7311yescharacteristic
384C4×C4⋊C4—731no—
385C4×C4⋊C4—731no—
386C4×C4⋊C4—1461no—
387C4×C4⋊C4—1461no—
388C4×C22⋊C4—1461no—
389C4×C22⋊C4—1461no—
390#340—731no—
391#340—731no—
392#340—731no—
393#340—731no—
394#340C2×D7311yescharacteristic
395#340C2×D7311yescharacteristic
396C22×Q8—731no—
397C22×Q8C22×D7311yescharacteristic
398C23×C4—731no—
399C23×C4—731no—
400C23×C4C22×D7311yescharacteristic
401C2×Q16—734no—
402C2×Q16—1462no—
403C2×Q16—1462no—
404C22×C8—22no—
405C22×C8—22no—
406C22×C8—22no—
407C22×C8—731no—
408C22×C8—734no—
409C22×C8—731no—
410C22×C8—1462no—
411C22×C8—1462no—
412C22×C8C22×D7311yescharacteristic
413C22×C8C22×D7311yescharacteristic
414C22⋊Q8—22no—
415C22⋊Q8—734no—
416C22⋊Q8—1462no—
417C22⋊Q8C4×D7314yes—
418C4×Q8—1462no—
419C4×Q8—1462no—
420C4×Q8—1462no—
421C4×Q8—1462no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4—1461no—
431C2×C4⋊C4C22×D7311yescharacteristic
432C2×C4⋊C4C22×D7311yescharacteristic
433C2×C4⋊C4C22×D7311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1461no—
437C2×C22⋊C4—1461no—
438C2×C42—731no—
439C2×C42—731no—
440C2×C42—731no—
441C2×C42—731no—
442C2×C42—1461no—
443C2×C42—1461no—
444C2×C42—1461no—
445C2×C42—1461no—
446C2×C42—1462no—
447C2×C42—1462no—
448#107—734no—
449#106—1462no—
450#106—1462no—
451#103—22no—
452#103—22no—
453#103—734no—
454#103—734no—
455#103—1462no—
456#103—1462no—
457#97—734no—
458#97—1462no—
459#97—1462no—
460#97—1462no—
461#95—21no—
462#95—1461no—
463#95—1461no—
464#95—1461no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470#95—1461no—
471C24D4×D7311yescharacteristic
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—732no—
478C2×Q8—1461no—
479C2×Q8—1461no—
480C2×Q8—1461no—
481C2×Q8—1461no—
482C2×Q8—1461no—
483C2×Q8C2×C4×D7312yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—2921no—
493C22×C4—2921no—
494C22×C4—731no—
495C22×C4—731no—
496C22×C4—731no—
497C22×C4—731no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1462no—
501C22×C4—1462no—
502C22×C4—1462no—
503C22×C4—1462no—
504C22×C4—1461no—
505C22×C4—1461no—
506C22×C4C2×C4×D7312yes—
507C22×C4D4×D7311yescharacteristic
508C22×C4#1192402712yes—
509C22×C4C23×D7311yescharacteristic
510Q16—1468no—
511C2×C8—28no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—732no—
517C2×C8—732no—
518C2×C8—732no—
519C2×C8—732no—
520C2×C8—1468no—
521C2×C8—1461no—
522C2×C8—1461no—
523C2×C8D4×D7312yes—
524C2×C8#1192402712yes—
525C4⋊C4—24no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—2922no—
532C4⋊C4—2922no—
533C4⋊C4—2922no—
534C4⋊C4—2922no—
535C4⋊C4—732no—
536C4⋊C4—1461no—
537C4⋊C4—1461no—
538C4⋊C4—1461no—
539C4⋊C4—1461no—
540C4⋊C4—1464no—
541C4⋊C4—1461no—
542C4⋊C4C2×C4×D7312yes—
543C22⋊C4—24no—
544C22⋊C4—42no—
545C22⋊C4—2922no—
546C22⋊C4—1464no—
547C42—2922no—
548C42—2922no—
549C42—2922no—
550C42—2922no—
551C42—732no—
552C42—732no—
553C42—1462no—
554C42—1462no—
555C42—1461no—
556C42—1461no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23#1455655112yes—
562C23#1455662012yes—
563C23#1455662312yes—
564C23C2×D4×D7311yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—2921no—
570Q8—2921no—
571Q8—1462no—
572Q8—1462no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2924no—
596C2×C4—2924no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—2921no—
603C2×C4—732no—
604C2×C4—732no—
605C2×C4—1461no—
606C2×C4—1461no—
607C2×C4—1461no—
608C2×C4—1461no—
609C2×C4—1462no—
610C2×C4—1462no—
611C2×C4—1461no—
612C2×C4—1461no—
613C2×C4—1461no—
614C2×C4—1461no—
615C2×C4—1461no—
616C2×C4—1461no—
617C2×C4#1455655112yes—
618C2×C4C22×C4×D7311yescharacteristic
619C2×C4C2×D4×D7311yescharacteristic
620C2×C4#41×D7311yescharacteristic
621C2×C4#41×D7311yescharacteristic
622C2×C4C2×#1192402711yescharacteristic
623C2×C4C2×#1192402711yescharacteristic
624C8—22no—
625C8—22no—
626C8—734no—
627C8—1462no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22#1798869114yescomplemented
635C22C2×#1455655111yescharacteristic
636C22#121×D7311yescharacteristic
637C22#1798913211yescharacteristic
638C22C2×#1455662011yescharacteristic
639C22C2×#1455662311yescharacteristic
640C22#1798938411yescharacteristic
641C22#1798938411yescharacteristic
642C4—22no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—81no—
647C4—2922no—
648C4—2921no—
649C4—2921no—
650C4—2922no—
651C4—5841no—
652C4—5841no—
653C4—1462no—
654C4—1462no—
655C4C4×D4×D7311yescharacteristic
656C4C4×#1192402711yescharacteristic
657C4#1798909311yescharacteristic
658C4#1798911411yescharacteristic
659C2—24no—
660C2#2261244111yescharacteristic
661C2C2×#1798869111yescharacteristic
662C2#2261872711yescharacteristic
663C2#2261955411yescharacteristic
664C2#2261955811yescharacteristic
665C2#2262063311yescharacteristic
666C2#2262063511yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1892C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#1892D73
hypocenterC73#21527
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
2. C2×#17988691C2semidirect
6. #22610799C2semidirect
7. #22610668C2semidirect
8. #22610622C2semidirect
9. #22610618C2semidirect
12. #22609827C2semidirect
13. #22609763C2semidirect
14. #22609760C2semidirect
22. #17988826C4semidirect
67. C2×#11924018C2×C4semidirect
99. #14556524C2×C4semidirect
104. #14556515C2×C4semidirect
105. #14556514C2×C4semidirect
300. C22×C73#441semidirect
349. C73#21527semidirect
634. C22#17988691semidirect

Maximal subgroups

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

subgroupindex
C2×#179888262
C2×#179886912
C22×#145565012
C22⋊Q8×Dic732
#226124592
#226107992
#226106682
#226106222
#226106182
#226103542
#226099542
#226098272
#226097632
#226097602
C73×#18922
#2152773