Finite Group Databasebeta

← #33008809

Subgroups — #33008809

Order 18688.

1918 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×#17988462C211yesmaximal, characteristic
2C2×#17988448C211yesmaximal, characteristic
3C2×#17988448C211yesmaximal, characteristic
4C2×C22⋊C4×Dic73C211yesmaximal, characteristic
5C22×#14556498C211yesmaximal, characteristic
6#22608191C212yesmaximal, complemented
7#22608187C212yesmaximal, complemented
8#22608165C212yesmaximal, complemented
9#22608155C212yesmaximal, complemented
10#22608154C211yesmaximal, characteristic
11C73×C22⋊#95C211yesmaximal, characteristic
12C24×Dic73C2211yescharacteristic
13C24×Dic73C2211yescharacteristic
14C22×C4⋊Dic73C2211yescharacteristic
15C22×C4⋊Dic73C2211yescharacteristic
16C22×C22⋊Dic73C2211yescharacteristic
17C22×C22⋊Dic73C2211yescharacteristic
18C22×C4×Dic73C2212yes—
19#17988462—24no—
20#17988448—24no—
21#17988448C418yescomplemented
22C22⋊C4×Dic73—24no—
23C2×#14556498—24no—
24C2×#14556498C2212yes—
25C2×#14556498C2211yescharacteristic
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2211yescharacteristic
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C2×#14556498C2212yes—
35C24×C4×C73C2211yescharacteristic
36C22×C22⋊C4×C73C2211yescharacteristic
37C22×C22⋊C4×C73C2211yescharacteristic
38C2×#95×C73C2212yes—
39C2×#95×C73C2212yes—
40C23×Dic73—24no—
41C23×Dic73—22no—
42C23×Dic73—22no—
43C23×Dic73—22no—
44C23×Dic73—22no—
45C23×Dic73—22no—
46C23×Dic73—22no—
47C23×Dic73—22no—
48C23×Dic73—22no—
49C23×Dic73—42no—
50C23×Dic73C2×C414yes—
51C23×Dic73D414yes—
52C23×Dic73C2311yescharacteristic
53C23×Dic73C2311yescharacteristic
54C23×Dic73C2312yes—
55C23×Dic73C2312yes—
56C23×Dic73C2311yescharacteristic
57C23×Dic73C2311yescharacteristic
58C2×C4⋊Dic73—22no—
59C2×C4⋊Dic73—22no—
60C2×C4⋊Dic73—24no—
61C2×C4⋊Dic73—42no—
62C2×C22⋊Dic73—22no—
63C2×C22⋊Dic73—22no—
64C2×C22⋊Dic73—22no—
65C2×C22⋊Dic73—22no—
66C2×C22⋊Dic73—22no—
67C2×C22⋊Dic73C2×C414yes—
68C2×C4×Dic73—24no—
69C2×C4×Dic73—24no—
70C2×C4×Dic73—42no—
71#14556498—24no—
72#14556498—24no—
73#14556498—22no—
74#14556498—22no—
75#14556498—24no—
76#14556498—24no—
77#14556498—24no—
78#14556498—24no—
79#14556498—42no—
80#14556498C2×C418yescomplemented
81#14556498C2×C418yescomplemented
82C25×C73C2311yescharacteristic
83C23×C4×C73—22no—
84C23×C4×C73—24no—
85C23×C4×C73—22no—
86C23×C4×C73—22no—
87C23×C4×C73—42no—
88C23×C4×C73C2311yescharacteristic
89C23×C4×C73C2311yescharacteristic
90C23×C4×C73C2312yes—
91C23×C4×C73C2312yes—
92C2×C22⋊C4×C73—22no—
93C2×C22⋊C4×C73—22no—
94C2×C22⋊C4×C73—22no—
95C2×C22⋊C4×C73—22no—
96C2×C22⋊C4×C73C2×C414yes—
97C2×C22⋊C4×C73D414yes—
98#95×C73—24no—
99#95×C73—42no—
100C22×Dic73—21no—
101C22×Dic73—21no—
102C22×Dic73—22no—
103C22×Dic73—28no—
104C22×Dic73—28no—
105C22×Dic73—28no—
106C22×Dic73—28no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—21no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—21no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73—28no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—22no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—22no—
132C22×Dic73—22no—
133C22×Dic73—22no—
134C22×Dic73—44no—
135C22×Dic73—48no—
136C22×Dic73—48no—
137C22×Dic73—44no—
138C22×Dic73—44no—
139C22×Dic73—44no—
140C22×Dic73—44no—
141C22×Dic73C22×C414yes—
142C22×Dic73C22×C414yes—
143C22×Dic73C2×D412yes—
144C22×Dic73C2×D412yes—
145C22×Dic73#4114yes—
146C4⋊Dic73—44no—
147C4⋊Dic73—44no—
148C22⋊Dic73—28no—
149C22⋊Dic73—44no—
150C4×Dic73—48no—
151C24×C73—22no—
152C24×C73—22no—
153C24×C73—22no—
154C24×C73—22no—
155C24×C73C22×C412yes—
156C24×C73C2×D412yes—
157C24×C73C2×D412yes—
158C24×C73#4112yes—
159C24×C73#4112yes—
160C24×C73#4112yes—
161C24×C73#4112yes—
162C24×C73C2411yescharacteristic
163C22×C4×C73—28no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—22no—
172C22×C4×C73—22no—
173C22×C4×C73—22no—
174C22×C4×C73—22no—
175C22×C4×C73—22no—
176C22×C4×C73—22no—
177C22×C4×C73—22no—
178C22×C4×C73—22no—
179C22×C4×C73—22no—
180C22×C4×C73—22no—
181C22×C4×C73—22no—
182C22×C4×C73—22no—
183C22×C4×C73—22no—
184C22×C4×C73—48no—
185C22×C4×C73—48no—
186C22×C4×C73—44no—
187C22×C4×C73—44no—
188C22×C4×C73C22×C412yes—
189C22×C4×C73C22×C412yes—
190C22×C4×C73C2×D414yes—
191C22×C4×C73#4114yes—
192C22⋊C4×C73—28no—
193C22⋊C4×C73—28no—
194C23×C73—24no—
195C23×C73—24no—
196C23×C73—22no—
197C23×C73—22no—
198C23×C73—24no—
199C23×C73—22no—
200C23×C73—22no—
201C23×C73—24no—
202C23×C73—24no—
203C23×C73—24no—
204C23×C73—24no—
205C23×C73—24no—
206C23×C73—24no—
207C23×C73—24no—
208C23×C73—24no—
209C23×C73—24no—
210C23×C73#11714yes—
211C23×C73C4×D414yes—
212C23×C73C4×D414yes—
213C23×C73#12114yes—
214C23×C73#12314yes—
215C23×C73#12414yes—
216C23×C73#12614yes—
217C23×C73C23×C411yescharacteristic
218C23×C73C22×D411yescharacteristic
219C23×C73C22×D411yescharacteristic
220C23×C73C2×#4112yes—
221C23×C73C2×#4111yescharacteristic
222C23×C73C2×#4111yescharacteristic
223C23×C73C2×#4112yes—
224C23×C73C2×#4111yescharacteristic
225C23×C73C2×#4111yescharacteristic
226C23×C73H(5,𝔽2)12yes—
227C23×C732−1+412yes—
228C2×C4×C73—22no—
229C2×C4×C73—22no—
230C2×C4×C73—24no—
231C2×C4×C73—24no—
232C2×C4×C73—24no—
233C2×C4×C73—24no—
234C2×C4×C73—24no—
235C2×C4×C73—24no—
236C2×C4×C73—24no—
237C2×C4×C73—24no—
238C2×C4×C73—416no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—42no—
242C2×C4×C73—42no—
243C2×C4×C73—42no—
244C2×C4×C73—42no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—42no—
248C2×C4×C73—42no—
249C2×C4×C73—42no—
250C2×C4×C73—42no—
251C2×C4×C73—42no—
252C2×C4×C73—42no—
253C2×Dic73—22no—
254C2×Dic73—22no—
255C2×Dic73—24no—
256C2×Dic73—24no—
257C2×Dic73—24no—
258C2×Dic73—24no—
259C2×Dic73—24no—
260C2×Dic73—24no—
261C2×Dic73—24no—
262C2×Dic73—24no—
263C2×Dic73—22no—
264C2×Dic73—22no—
265C2×Dic73—22no—
266C2×Dic73—416no—
267C2×Dic73—42no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73—42no—
273C2×Dic73—42no—
274C2×Dic73—42no—
275C2×Dic73—42no—
276C2×Dic73—42no—
277C2×Dic73—416no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—41no—
284C2×Dic73—42no—
285C2×Dic73—42no—
286C2×Dic73—42no—
287C2×Dic73—42no—
288C2×Dic73#12314yes—
289C22×C73—24no—
290C22×C73—24no—
291C22×C73—28no—
292C22×C73—28no—
293C22×C73—24no—
294C22×C73—24no—
295C22×C73—28no—
296C22×C73—28no—
297C22×C73—24no—
298C22×C73—24no—
299C22×C73#38818yes—
300C22×C73C2×#11711yescharacteristic
301C22×C73C2×C4×D411yescharacteristic
302C22×C73C2×C4×D411yescharacteristic
303C22×C73C4×#4112yes—
304C22×C73#52012yes—
305C22×C73C2×#12111yescharacteristic
306C22×C73C2×#12311yescharacteristic
307C22×C73C2×#12311yescharacteristic
308C22×C73#52511yescharacteristic
309C22×C73C2×#12411yescharacteristic
310C22×C73C2×#12611yescharacteristic
311C22×C73#52912yes—
312C22×C73#53411yescharacteristic
313C22×C73#53611yescharacteristic
314C22×C73#53912yes—
315C22×C73#54612yes—
316C22×C73#54712yes—
317C22×C73#55112yes—
318C22×C73#55112yes—
319C22×C73#55112yes—
320C22×C73#55212yes—
321C22×C73#55212yes—
322C22×C73#55212yes—
323C4×C73—44no—
324C4×C73—42no—
325C4×C73—44no—
326C4×C73—42no—
327Dic73—24no—
328Dic73—44no—
329Dic73—42no—
330Dic73—44no—
331Dic73—81no—
332Dic73—81no—
333#13972—731nomaximal, Sylow
334C2×C73—28no—
335C2×C73C2×#38811yescharacteristic
336C2×C73C4×#12311yescharacteristic
337C2×C73#232211yescharacteristic
338C2×C73#234312yes—
339C2×C73#235912yes—
340C2×C73#241211yescharacteristic
341C2×C73#241311yescharacteristic
342C2×C73#241912yes—
343C2×C73#249312yes—
344C2×C73#250311yescharacteristic
345C2×C73#256111yescharacteristic
346C2×#385—731no—
347C2×#381—731no—
348C2×#381—731no—
349C2×C4×C22⋊C4—731no—
350C22×#95—731no—
351#1443—732no—
352#1442—732no—
353#1436—732no—
354#1434—732no—
355C22⋊#95—731no—
356C22⋊#95D7311yescharacteristic
357C73#1397211yescharacteristic, complemented, Sylow
358C24×C4—731no—
359C24×C4—731no—
360C24×C4C2×D7311yescharacteristic
361C22×C4⋊C4—731no—
362C22×C4⋊C4—731no—
363C22×C22⋊C4—731no—
364C22×C22⋊C4—731no—
365C22×C22⋊C4C2×D7311yescharacteristic
366C22×C22⋊C4C2×D7311yescharacteristic
367C22×C42—732no—
368#385—1464no—
369#381—738no—
370#381—1464no—
371C4×C22⋊C4—1464no—
372C2×#95—732no—
373C2×#95—732no—
374C2×#95—732no—
375C2×#95—732no—
376C2×#95—731no—
377C2×#95—732no—
378C2×#95—732no—
379C2×#95—732no—
380C2×#95—732no—
381C2×#95—732no—
382C2×#95—731no—
383C2×#95—1464no—
384C2×#95C2×D7312yes—
385C2×#95C2×D7312yes—
386C25C22×D7311yescharacteristic
387C23×C4—22no—
388C23×C4—24no—
389C23×C4—22no—
390C23×C4—22no—
391C23×C4—42no—
392C23×C4—2922no—
393C23×C4—734no—
394C23×C4—731no—
395C23×C4—734no—
396C23×C4—732no—
397C23×C4—731no—
398C23×C4—732no—
399C23×C4—731no—
400C23×C4—731no—
401C23×C4—1464no—
402C23×C4—1462no—
403C23×C4—1462no—
404C23×C4—1462no—
405C23×C4—1462no—
406C23×C4—1462no—
407C23×C4—1462no—
408C23×C4—1462no—
409C23×C4—1462no—
410C23×C4C22×D7311yescharacteristic
411C23×C4C22×D7311yescharacteristic
412C23×C4C22×D7312yes—
413C23×C4C22×D7312yes—
414C2×C4⋊C4—2922no—
415C2×C4⋊C4—1462no—
416C2×C4⋊C4—1462no—
417C2×C4⋊C4—1464no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—22no—
421C2×C22⋊C4—22no—
422C2×C22⋊C4—734no—
423C2×C22⋊C4—1462no—
424C2×C22⋊C4—1462no—
425C2×C22⋊C4—1462no—
426C2×C22⋊C4—1462no—
427C2×C22⋊C4—1462no—
428C2×C22⋊C4C4×D7314yes—
429C2×C22⋊C4D29214yes—
430C2×C42—2922no—
431C2×C42—1464no—
432C2×C42—1464no—
433#95—24no—
434#95—42no—
435#95—2922no—
436#95—738no—
437#95—738no—
438#95—1464no—
439#95—1464no—
440#95—1462no—
441#95—1462no—
442#95—1464no—
443#95—1464no—
444#95—1464no—
445#95—1464no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24C2×C4×D7312yes—
451C24D4×D7312yes—
452C24C2×D29212yes—
453C24#1192402712yes—
454C24#1192402712yes—
455C24#1192402812yes—
456C24#1192402912yes—
457C24C23×D7311yescharacteristic
458C22×C4—28no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—2924no—
484C22×C4—2924no—
485C22×C4—2924no—
486C22×C4—2924no—
487C22×C4—2928no—
488C22×C4—2928no—
489C22×C4—2924no—
490C22×C4—732no—
491C22×C4—734no—
492C22×C4—734no—
493C22×C4—734no—
494C22×C4—732no—
495C22×C4—1461no—
496C22×C4—1461no—
497C22×C4—1461no—
498C22×C4—1461no—
499C22×C4—1462no—
500C22×C4—1462no—
501C22×C4—1461no—
502C22×C4—1461no—
503C22×C4—1462no—
504C22×C4—1462no—
505C22×C4—1468no—
506C22×C4—1468no—
507C22×C4—1468no—
508C22×C4—1468no—
509C22×C4—1468no—
510C22×C4—1462no—
511C22×C4—1461no—
512C22×C4—1462no—
513C22×C4—1462no—
514C22×C4—1462no—
515C22×C4—1462no—
516C22×C4—1462no—
517C22×C4—1462no—
518C22×C4—1462no—
519C22×C4—1462no—
520C22×C4—1462no—
521C22×C4—1461no—
522C22×C4—1462no—
523C22×C4—1462no—
524C22×C4—1462no—
525C22×C4—1462no—
526C22×C4—1461no—
527C22×C4—1461no—
528C22×C4—1462no—
529C22×C4C2×C4×D7312yes—
530C22×C4C2×C4×D7312yes—
531C22×C4C2×D29214yes—
532C22×C4#1192402714yes—
533C4⋊C4—2924no—
534C4⋊C4—2924no—
535C22⋊C4—28no—
536C22⋊C4—28no—
537C22⋊C4—2924no—
538C22⋊C4—1468no—
539C42—2928no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23#1455654914yes—
557C23#1455655114yes—
558C23C4×D29214yes—
559C23#1455656614yes—
560C23#1455657514yes—
561C23#1455657714yes—
562C23#1455658614yes—
563C23C22×C4×D7311yescharacteristic
564C23C2×D4×D7311yescharacteristic
565C23C22×D29211yescharacteristic
566C23#41×D7312yes—
567C23C2×#1192402711yescharacteristic
568C23C2×#1192402712yes—
569C23C2×#1192402711yescharacteristic
570C23C2×#1192402811yescharacteristic
571C23C2×#1192402911yescharacteristic
572C23#1455663612yes—
573C23#1455663812yes—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—416no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—2922no—
600C2×C4—2922no—
601C2×C4—2922no—
602C2×C4—29216no—
603C2×C4—2922no—
604C2×C4—2922no—
605C2×C4—2922no—
606C2×C4—2922no—
607C2×C4—2922no—
608C2×C4—2922no—
609C2×C4—2922no—
610C2×C4—29216no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—2922no—
614C2×C4—2922no—
615C2×C4—2922no—
616C2×C4—2922no—
617C2×C4—2921no—
618C2×C4—2921no—
619C2×C4—2921no—
620C2×C4—2921no—
621C2×C4—734no—
622C2×C4—1462no—
623C2×C4—1464no—
624C2×C4—1464no—
625C2×C4—1464no—
626C2×C4—1464no—
627C2×C4—1464no—
628C2×C4—1464no—
629C2×C4—1462no—
630C2×C4—1462no—
631C2×C4—1462no—
632C2×C4—1464no—
633C2×C4—1464no—
634C2×C4—1462no—
635C22—24no—
636C22—28no—
637C22—28no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—24no—
645C22#1798847918yes—
646C22C2×#1455654911yescharacteristic
647C22C2×#1455655111yescharacteristic
648C22C2×C4×D29211yescharacteristic
649C22C4×#1192402712yes—
650C22#1798906612yes—
651C22C2×#1455656611yescharacteristic
652C22#123×D7311yescharacteristic
653C22C2×#1455657511yescharacteristic
654C22#1798909311yescharacteristic
655C22C2×#1455657711yescharacteristic
656C22C2×#1455658611yescharacteristic
657C22#1798911812yes—
658C22#1798914711yescharacteristic
659C22#1798915911yescharacteristic
660C22#1798918212yes—
661C22#1798923212yes—
662C22#1798924112yes—
663C22#1798926312yes—
664C22#1798926412yes—
665C22#1798926812yes—
666C22#1798927012yes—
667C22#1798927412yes—
668C22#1798927912yes—
669C4—44no—
670C4—42no—
671C4—44no—
672C4—42no—
673C4—2922no—
674C4—2924no—
675C4—2924no—
676C4—5841no—
677C4—5841no—
678C4—1464no—
679C2—28no—
680C2C2×#1798847911yescharacteristic
681C2#2261245611yescharacteristic
682C2#2261260511yescharacteristic
683C2#2261278912yes—
684C2#2261292512yes—
685C2#2261332511yescharacteristic
686C2#2261333611yescharacteristic
687C2#2261340612yes—
688C2#2261432212yes—
689C2#2261441411yescharacteristic
690C2#2261508611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×C22⋊#95C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC22⋊#95D73
hypocenterC73#13972
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
6. #22608191C2semidirect
7. #22608187C2semidirect
8. #22608165C2semidirect
9. #22608155C2semidirect
21. #17988448C4semidirect
80. #14556498C2×C4semidirect
81. #14556498C2×C4semidirect
357. C73#13972semidirect

Maximal subgroups

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

subgroupindex
C2×#179884622
C2×#179884482
C2×C22⋊C4×Dic732
C22×#145564982
#226081912
#226081872
#226081652
#226081552
#226081542
C73×C22⋊#952
#1397273