Finite Group Databasebeta

← #33008603

Subgroups — #33008603

Order 18688.

1630 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×#17988481C212yesmaximal
2C2×C4×C22⋊Dic73C211yesmaximal, characteristic
3#22608186C212yesmaximal, complemented
4#22608174C212yesmaximal
5#22608162C212yesmaximal, complemented
6#22608159C211yesmaximal, characteristic, complemented
7#22608159C211yesmaximal, characteristic, complemented
8#22608156C211yesmaximal, characteristic, complemented
9#22608154C211yesmaximal, characteristic
10#95×Dic73C211yesmaximal, characteristic, complemented
11C73×C22⋊#95C211yesmaximal, characteristic
12C22×#11924007C2212yes—
13C22×#11924007C2212yes—
14C22×C22⋊Dic73C2212yes—
15C22×C22⋊Dic73C2211yescharacteristic
16C22×C22⋊Dic73C2211yescharacteristic
17C22×C4×Dic73C2212yes—
18C22×C4×Dic73C2211yescharacteristic
19C22×C4×Dic73C2211yescharacteristic
20#17988481—28no—
21C4×C22⋊Dic73C418yescomplemented
22C2×#14556498C2211yescharacteristic
23C2×#14556498C2212yes—
24C2×#14556498C2211yescharacteristic
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2211yescharacteristic
31C2×#14556498C2211yescharacteristic
32C24×C4×C73C2211yescharacteristic
33C22×C22⋊C4×C73C2212yes—
34C2×#95×C73C2211yescharacteristic
35C2×#95×C73C2211yescharacteristic
36C2×#95×C73C2212yes—
37C23×Dic73C2312yes—
38C23×Dic73C2311yescharacteristic
39C23×Dic73C2312yes—
40C23×Dic73C2311yescharacteristic
41C23×Dic73C2311yescharacteristic
42C23×Dic73C2311yescharacteristic
43C2×#11924007—24no—
44C2×#11924007—24no—
45C2×#11924007—24no—
46C2×#11924007—24no—
47C2×#11924007—42no—
48C2×#11924007—42no—
49C2×C22⋊Dic73—22no—
50C2×C22⋊Dic73—22no—
51C2×C22⋊Dic73—24no—
52C2×C22⋊Dic73—24no—
53C2×C22⋊Dic73—42no—
54C2×C22⋊Dic73C2×C414yes—
55C2×C22⋊Dic73C2×C414yes—
56C2×C22⋊Dic73D414yes—
57C2×C22⋊Dic73Q814yes—
58C2×C4×Dic73—24no—
59C2×C4×Dic73—24no—
60C2×C4×Dic73—42no—
61C2×C4×Dic73—42no—
62C2×C4×Dic73—42no—
63C2×C4×Dic73C2×C414yes—
64C2×C4×Dic73C2×C414yes—
65#14556498—24no—
66#14556498—24no—
67#14556498—24no—
68#14556498—22no—
69#14556498—22no—
70#14556498—24no—
71#14556498—42no—
72#14556498—42no—
73#14556498C2×C414yes—
74#14556498C2×C414yes—
75C25×C73C2311yescharacteristic
76C23×C4×C73—22no—
77C23×C4×C73—22no—
78C23×C4×C73—42no—
79C23×C4×C73—42no—
80C23×C4×C73C2×C414yes—
81C23×C4×C73C2312yes—
82C23×C4×C73C2311yescharacteristic
83C23×C4×C73C2312yes—
84C23×C4×C73C2311yescharacteristic
85C2×C22⋊C4×C73—24no—
86C2×C22⋊C4×C73—24no—
87C2×C22⋊C4×C73—42no—
88#95×C73—22no—
89#95×C73—22no—
90#95×C73—42no—
91C22×Dic73—22no—
92C22×Dic73—22no—
93C22×Dic73—22no—
94C22×Dic73—22no—
95C22×Dic73—22no—
96C22×Dic73—22no—
97C22×Dic73—22no—
98C22×Dic73—22no—
99C22×Dic73—22no—
100C22×Dic73—22no—
101C22×Dic73—21no—
102C22×Dic73—21no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73C22×C412yes—
115C22×Dic73C22×C412yes—
116C22×Dic73C22×C412yes—
117C22×Dic73C22×C412yes—
118C22×Dic73C2×D412yes—
119C22×Dic73C2×D412yes—
120C22×Dic73C2×Q812yes—
121C22×Dic73C2×Q812yes—
122C22×Dic73#4114yes—
123C22×Dic73#4112yes—
124C22×Dic73#4112yes—
125#11924007—48no—
126#11924007—48no—
127C22⋊Dic73—28no—
128C22⋊Dic73—48no—
129C22⋊Dic73C4⋊C4116yescomplemented
130C4×Dic73—44no—
131C4×Dic73—44no—
132C4×Dic73—48no—
133C24×C73—22no—
134C24×C73—21no—
135C24×C73—21no—
136C24×C73—21no—
137C24×C73—21no—
138C24×C73—22no—
139C24×C73—22no—
140C24×C73—22no—
141C24×C73C22×C412yes—
142C24×C73C2×D412yes—
143C24×C73C2×Q812yes—
144C24×C73C2411yescharacteristic
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73—21no—
150C22×C4×C73—21no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—28no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—44no—
166C22×C4×C73—44no—
167C22×C4×C73—48no—
168C22×C4×C73—48no—
169C22×C4×C73C22×C412yes—
170C22×C4×C73C22×C412yes—
171C22×C4×C73#4112yes—
172C22×C4×C73#4112yes—
173C22×C4×C73#4114yes—
174C22⋊C4×C73—48no—
175C23×C73—22no—
176C23×C73—22no—
177C23×C73—22no—
178C23×C73—22no—
179C23×C73—22no—
180C23×C73—22no—
181C23×C73—22no—
182C23×C73—22no—
183C23×C73—22no—
184C23×C73—22no—
185C23×C73—24no—
186C23×C73—24no—
187C23×C73—24no—
188C23×C73—24no—
189C23×C73—42no—
190C23×C73—42no—
191C23×C73—42no—
192C23×C73—42no—
193C23×C73—42no—
194C23×C73—42no—
195C23×C73—42no—
196C23×C73—42no—
197C23×C73C2×C4⋊C414yes—
198C23×C73C23×C411yescharacteristic
199C23×C73C22×D411yescharacteristic
200C23×C73C22×Q811yescharacteristic
201C23×C73C2×#4111yescharacteristic
202C23×C73C2×#4111yescharacteristic
203C23×C73C2×#4112yes—
204C23×C73C2×#4111yescharacteristic
205C23×C73C2×#4111yescharacteristic
206C23×C73C2×#4112yes—
207C23×C73H(5,𝔽2)12yes—
208C23×C73H(5,𝔽2)12yes—
209C2×C4×C73—22no—
210C2×C4×C73—22no—
211C2×C4×C73—24no—
212C2×C4×C73—24no—
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—22no—
216C2×C4×C73—24no—
217C2×C4×C73—24no—
218C2×C4×C73—24no—
219C2×C4×C73—416no—
220C2×C4×C73—42no—
221C2×C4×C73—42no—
222C2×C4×C73—42no—
223C2×C4×C73—42no—
224C2×C4×C73—42no—
225C2×C4×C73—42no—
226C2×C4×C73—42no—
227C2×C4×C73—42no—
228C2×C4×C73—42no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73—42no—
232C2×C4×C73—42no—
233C2×C4×C73#11714yes—
234C2×Dic73—22no—
235C2×Dic73—22no—
236C2×Dic73—22no—
237C2×Dic73—22no—
238C2×Dic73—22no—
239C2×Dic73—22no—
240C2×Dic73—24no—
241C2×Dic73—24no—
242C2×Dic73—24no—
243C2×Dic73—24no—
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—22no—
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73—24no—
250C2×Dic73—24no—
251C2×Dic73—24no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—42no—
255C2×Dic73—42no—
256C2×Dic73—42no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73C2×C4⋊C414yes—
266C2×Dic73C2×C4⋊C414yes—
267C2×Dic73#11714yes—
268C22×C73—24no—
269C22×C73—24no—
270C22×C73—24no—
271C22×C73—42no—
272C22×C73—44no—
273C22×C73—44no—
274C22×C73—42no—
275C22×C73—42no—
276C22×C73—42no—
277C22×C73—44no—
278C22×C73—44no—
279C22×C73C22×C4⋊C411yescharacteristic
280C22×C73C2×#11711yescharacteristic
281C22×C73C2×#11711yescharacteristic
282C22×C73C4×#4111yescharacteristic
283C22×C73C4×#4111yescharacteristic
284C22×C73#51811yescharacteristic
285C22×C73#51811yescharacteristic
286C22×C73#52912yes—
287C22×C73#52912yes—
288C22×C73#53211yescharacteristic
289C22×C73#53211yescharacteristic
290C22×C73#53311yescharacteristic
291C22×C73#53311yescharacteristic
292C22×C73#53812yes—
293C22×C73#53812yes—
294C22×C73#54612yes—
295C22×C73#54612yes—
296C22×C73#55112yes—
297C22×C73#55312yes—
298C22×C73#55312yes—
299C22×C73#55412yes—
300C22×C73#55412yes—
301C22×C73#55912yes—
302C4×C73—24no—
303C4×C73—42no—
304C4×C73—44no—
305C4×C73—82no—
306Dic73—24no—
307Dic73—24no—
308Dic73—24no—
309Dic73—42no—
310Dic73—44no—
311Dic73—82no—
312#13948—731nomaximal, Sylow
313C2×C73—44no—
314C2×C73#228511yescharacteristic
315C2×C73#229811yescharacteristic
316C2×C73#231211yescharacteristic
317C2×C73#234911yescharacteristic
318C2×C73#234911yescharacteristic
319C2×C73#235311yescharacteristic
320C2×C73#235711yescharacteristic
321C2×C73#255712yes—
322C2×C73#255912yes—
323C2×C73#258712yes—
324C2×C73#258912yes—
325C2×#388—732no—
326C2×C4×C22⋊C4—731no—
327#1442—732no—
328#1439—732no—
329#1436—732no—
330C4⋊#95—731no—
331C4⋊#95—731no—
332#1434—731no—
333C22⋊#95—731no—
334C22⋊#95D7311yescharacteristic
335C4×#95—731no—
336C73#1394811yescharacteristic, complemented, Sylow
337C24×C4C2×D7311yescharacteristic
338C22×C4⋊C4—732no—
339C22×C4⋊C4—732no—
340C22×C22⋊C4—732no—
341C22×C22⋊C4—731no—
342C22×C22⋊C4—731no—
343C22×C22⋊C4C2×D7312yes—
344C22×C42—731no—
345C22×C42—731no—
346C22×C42—732no—
347#388—1468no—
348C4×C22⋊C4—738no—
349C2×#95—732no—
350C2×#95—732no—
351C2×#95—731no—
352C2×#95—731no—
353C2×#95—731no—
354C2×#95—732no—
355C2×#95—731no—
356C2×#95—732no—
357C2×#95—732no—
358C2×#95—732no—
359C2×#95C2×D7311yescharacteristic
360C2×#95C2×D7311yescharacteristic
361C2×#95C2×D7312yes—
362C25C22×D7311yescharacteristic
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—42no—
366C23×C4—42no—
367C23×C4—731no—
368C23×C4—731no—
369C23×C4—731no—
370C23×C4—732no—
371C23×C4—731no—
372C23×C4—732no—
373C23×C4C4×D7314yes—
374C23×C4C22×D7312yes—
375C23×C4C22×D7311yescharacteristic
376C23×C4C22×D7312yes—
377C23×C4C22×D7311yescharacteristic
378C2×C4⋊C4—2922no—
379C2×C4⋊C4—2922no—
380C2×C4⋊C4—1464no—
381C2×C4⋊C4—1464no—
382C2×C4⋊C4—1464no—
383C2×C4⋊C4—1464no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—2922no—
388C2×C22⋊C4—734no—
389C2×C22⋊C4—734no—
390C2×C22⋊C4—734no—
391C2×C22⋊C4—734no—
392C2×C22⋊C4—1464no—
393C2×C22⋊C4—1464no—
394C2×C22⋊C4—1462no—
395C2×C22⋊C4—1462no—
396C2×C42—2922no—
397C2×C42—2922no—
398C2×C42—2922no—
399C2×C42—734no—
400C2×C42—734no—
401C2×C42—1464no—
402C2×C42—1464no—
403#95—22no—
404#95—22no—
405#95—42no—
406#95—2922no—
407#95—2922no—
408#95—734no—
409#95—734no—
410#95—1464no—
411#95—1464no—
412#95—1464no—
413#95—1462no—
414#95—1462no—
415#95—1464no—
416C24—21no—
417C24—21no—
418C24—22no—
419C24—21no—
420C24—21no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24C2×C4×D7312yes—
425C24D4×D7312yes—
426C24Q8×D7312yes—
427C24C23×D7311yescharacteristic
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—28no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—732no—
453C22×C4—732no—
454C22×C4—732no—
455C22×C4—732no—
456C22×C4—732no—
457C22×C4—732no—
458C22×C4—732no—
459C22×C4—732no—
460C22×C4—732no—
461C22×C4—732no—
462C22×C4—734no—
463C22×C4—1462no—
464C22×C4—1462no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4—1462no—
470C22×C4—1462no—
471C22×C4—1461no—
472C22×C4—1461no—
473C22×C4—1462no—
474C22×C4—1462no—
475C22×C4—1462no—
476C22×C4—1462no—
477C22×C4—1462no—
478C22×C4—1462no—
479C22×C4—1462no—
480C22×C4—1462no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4—1462no—
486C22×C4C2×C4×D7312yes—
487C22×C4C2×C4×D7312yes—
488C22×C4#1192402712yes—
489C22×C4#1192402712yes—
490C22×C4#1192402814yes—
491C4⋊C4—2928no—
492C4⋊C4—2928no—
493C22⋊C4—48no—
494C22⋊C4—2928no—
495C22⋊C4—7316no—
496C22⋊C4—1468no—
497C42—2924no—
498C42—2924no—
499C42—2928no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23C4⋊C4×D7314yes—
523C23C22×C4×D7311yescharacteristic
524C23C2×D4×D7311yescharacteristic
525C23C2×Q8×D7311yescharacteristic
526C23#41×D7311yescharacteristic
527C23#41×D7311yescharacteristic
528C23#41×D7312yes—
529C23C2×#1192402711yescharacteristic
530C23C2×#1192402711yescharacteristic
531C23C2×#1192402812yes—
532C23#1455663612yes—
533C23#1455663612yes—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—416no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—2922no—
559C2×C4—2922no—
560C2×C4—2922no—
561C2×C4—2922no—
562C2×C4—2922no—
563C2×C4—2922no—
564C2×C4—2922no—
565C2×C4—2922no—
566C2×C4—2922no—
567C2×C4—2922no—
568C2×C4—2922no—
569C2×C4—2922no—
570C2×C4—2922no—
571C2×C4—734no—
572C2×C4—734no—
573C2×C4—734no—
574C2×C4—1464no—
575C2×C4—1464no—
576C2×C4—1462no—
577C2×C4—1462no—
578C2×C4—1462no—
579C2×C4—1464no—
580C2×C4—1464no—
581C2×C4—1464no—
582C2×C4—1464no—
583C2×C4—1464no—
584C2×C4—1464no—
585C2×C4—1464no—
586C2×C4—1462no—
587C2×C4—1462no—
588C2×C4—1462no—
589C2×C4—1462no—
590C2×C4—1462no—
591C2×C4—1462no—
592C2×C4#1455654714yes—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—44no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—44no—
604C22C2×C4⋊C4×D7311yescharacteristic
605C22#117×D7311yescharacteristic
606C22C2×#1455654711yescharacteristic
607C22#1798905511yescharacteristic
608C22C4×#1192402811yescharacteristic
609C22#1798906011yescharacteristic
610C22#1798906011yescharacteristic
611C22#1798911712yes—
612C22#1798911912yes—
613C22#1798913211yescharacteristic
614C22#1798913211yescharacteristic
615C22#1798913911yescharacteristic
616C22#1798913911yescharacteristic
617C22#1798917212yes—
618C22#1798917212yes—
619C22#1798922612yes—
620C22#1798922712yes—
621C22#1798926512yes—
622C22#1798928212yes—
623C22#1798928612yes—
624C22#1798929112yes—
625C22#1798929212yes—
626C22#1798932012yes—
627C4—24no—
628C4—44no—
629C4—42no—
630C4—82no—
631C4—2922no—
632C4—2924no—
633C4—5842no—
634C4—1464no—
635C4—1464no—
636C4—1464no—
637C2—44no—
638C2#2261233311yescharacteristic
639C2#2261242711yescharacteristic
640C2#2261252711yescharacteristic
641C2#2261283511yescharacteristic
642C2#2261283811yescharacteristic
643C2#2261286811yescharacteristic
644C2#2261290511yescharacteristic
645C2#2261502912yes—
646C2#2261505712yes—
647C2#2261535112yes—
648C2#2261537912yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×C22⋊#95C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC22⋊#95D73
hypocenterC73#13948
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
3. #22608186C2semidirect
5. #22608162C2semidirect
6. #22608159C2semidirect
7. #22608159C2semidirect
8. #22608156C2semidirect
10. #95×Dic73C2semidirect
21. C4×C22⋊Dic73C4semidirect
129. C22⋊Dic73C4⋊C4semidirect
336. C73#13948semidirect

Maximal subgroups

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

subgroupindex
C2×#179884812
C2×C4×C22⋊Dic732
#226081862
#226081742
#226081622
#226081592
#226081562
#226081542
#95×Dic732
C73×C22⋊#952
#1394873