Finite Group Databasebeta

← #33022627

Subgroups — #33022627

Order 18688.

1814 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988523C211yesmaximal, characteristic
2C2×#17988482C212yesmaximal, complemented
3C2×#17988456C211yesmaximal, characteristic, complemented
4C2×#17988445C212yesmaximal
5C2×C22⋊C4×Dic73C211yesmaximal, characteristic
6#22608199C211yesmaximal, characteristic, complemented
7#22608198C211yesmaximal, characteristic, complemented
8#22608194C212yesmaximal, complemented
9#22608174C212yesmaximal
10#22608159C211yesmaximal, characteristic, complemented
11C2×#399×C73C211yesmaximal, characteristic
12C24×Dic73C2211yescharacteristic
13C22×C4⋊Dic73C2211yescharacteristic
14C22×#11924007C2212yes—
15C22×#11924007C2212yes—
16C22×#11924007C2212yes—
17C22×#11924007C2212yes—
18C22×C22⋊Dic73C2212yes—
19C22×C22⋊Dic73C2211yescharacteristic
20C22×C4×Dic73C2211yescharacteristic
21C22×C4×Dic73C2211yescharacteristic
22#17988523—24no—
23#17988482—28no—
24#17988456—24no—
25#17988445—28no—
26C22⋊C4×Dic73—24no—
27C2×#14556498C2212yes—
28C2×#14556498C2211yescharacteristic
29C2×#14556498C2211yescharacteristic
30C2×#14556498C2211yescharacteristic
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C2×#14556498C2212yes—
35C22×C4⋊C4×C73C2211yescharacteristic
36C22×C4⋊C4×C73C2212yes—
37C22×C22⋊C4×C73C2212yes—
38C22×C22⋊C4×C73C2211yescharacteristic
39#399×C73—24no—
40C2×#95×C73C2211yescharacteristic
41C23×Dic73—22no—
42C23×Dic73—24no—
43C23×Dic73—24no—
44C23×Dic73—22no—
45C23×Dic73Q814yes—
46C23×Dic73C2312yes—
47C23×Dic73C2312yes—
48C23×Dic73C2311yescharacteristic
49C23×Dic73C2311yescharacteristic
50C23×Dic73C2311yescharacteristic
51C23×Dic73C2311yescharacteristic
52C2×C4⋊Dic73—22no—
53C2×C4⋊Dic73—22no—
54C2×C4⋊Dic73—22no—
55C2×#11924007—24no—
56C2×#11924007—24no—
57C2×#11924007—24no—
58C2×#11924007—24no—
59C2×#11924007—24no—
60C2×#11924007—24no—
61C2×#11924007—24no—
62C2×#11924007—24no—
63C2×#11924007—42no—
64C2×#11924007—42no—
65C2×#11924007—42no—
66C2×#11924007D418yes—
67C2×C22⋊Dic73—22no—
68C2×C22⋊Dic73—22no—
69C2×C22⋊Dic73—24no—
70C2×C22⋊Dic73—24no—
71C2×C22⋊Dic73—42no—
72C2×C22⋊Dic73Q814yes—
73C2×C4×Dic73—24no—
74C2×C4×Dic73—22no—
75C2×C4×Dic73—24no—
76C2×C4×Dic73D414yes—
77#14556498—22no—
78#14556498—24no—
79#14556498—24no—
80#14556498—24no—
81#14556498—22no—
82#14556498—24no—
83#14556498—24no—
84#14556498—22no—
85C25×C73C2311yescharacteristic
86C23×C4×C73C2312yes—
87C23×C4×C73C2312yes—
88C23×C4×C73C2311yescharacteristic
89C23×C4×C73C2311yescharacteristic
90C2×C4⋊C4×C73—24no—
91C2×C4⋊C4×C73—22no—
92C2×C4⋊C4×C73—22no—
93C2×C4⋊C4×C73—24no—
94C2×C4⋊C4×C73—22no—
95C2×C4⋊C4×C73—42no—
96C2×C22⋊C4×C73—22no—
97C2×C22⋊C4×C73—22no—
98C2×C22⋊C4×C73—24no—
99C2×C22⋊C4×C73—24no—
100C2×C22⋊C4×C73—42no—
101C2×C22⋊C4×C73D414yes—
102#95×C73—22no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—21no—
107C22×Dic73—22no—
108C22×Dic73—21no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—21no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—21no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—28no—
120C22×Dic73—216no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73—22no—
128C22×Dic73—44no—
129C22×Dic73—48no—
130C22×Dic73—44no—
131C22×Dic73C2×D414yes—
132C22×Dic73C2×D412yes—
133C22×Dic73C2×D412yes—
134C22×Dic73C2×D414yes—
135C22×Dic73C2×Q812yes—
136C22×Dic73C2×Q812yes—
137C22×Dic73C2×Q812yes—
138C22×Dic73C2×Q812yes—
139C22×Dic73#4112yes—
140C22×Dic73#4114yes—
141C22×Dic73#4112yes—
142C4⋊Dic73—44no—
143#11924007—216no—
144#11924007—48no—
145#11924007—48no—
146#11924007—48no—
147C22⋊Dic73—28no—
148C22⋊Dic73—48no—
149C4×Dic73—28no—
150C4×Dic73—44no—
151C24×C73—21no—
152C24×C73—22no—
153C24×C73—21no—
154C24×C73—21no—
155C24×C73—21no—
156C24×C73—22no—
157C24×C73—22no—
158C24×C73—22no—
159C24×C73C2×D412yes—
160C24×C73C2×Q812yes—
161C24×C73C2×Q812yes—
162C24×C73C2411yescharacteristic
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—21no—
167C22×C4×C73—22no—
168C22×C4×C73—21no—
169C22×C4×C73—22no—
170C22×C4×C73—22no—
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×C73C2×D414yes—
183C22×C4×C73C2×D412yes—
184C22×C4×C73C2×D412yes—
185C22×C4×C73C2×D412yes—
186C22×C4×C73#4112yes—
187C4⋊C4×C73—48no—
188C4⋊C4×C73—44no—
189C22⋊C4×C73—28no—
190C22⋊C4×C73—48no—
191C23×C73—22no—
192C23×C73—22no—
193C23×C73—22no—
194C23×C73—24no—
195C23×C73—22no—
196C23×C73—22no—
197C23×C73—22no—
198C23×C73—24no—
199C23×C73—24no—
200C23×C73—24no—
201C23×C73—24no—
202C23×C73—24no—
203C23×C73—42no—
204C23×C73—41no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73—41no—
208C23×C73—42no—
209C23×C73—42no—
210C23×C73—42no—
211C23×C73—42no—
212C23×C73—42no—
213C23×C73C4⋊Q814yes—
214C23×C73C22×D411yescharacteristic
215C23×C73C22×D412yes—
216C23×C73C22×D411yescharacteristic
217C23×C73C22×Q811yescharacteristic
218C23×C73C22×Q811yescharacteristic
219C23×C73C2×#4111yescharacteristic
220C23×C73C2×#4112yes—
221C23×C73C2×#4111yescharacteristic
222C23×C73C2×#4111yescharacteristic
223C23×C73H(5,𝔽2)12yes—
224C23×C732−1+412yes—
225C2×C4×C73—22no—
226C2×C4×C73—22no—
227C2×C4×C73—24no—
228C2×C4×C73—24no—
229C2×C4×C73—22no—
230C2×C4×C73—24no—
231C2×C4×C73—24no—
232C2×C4×C73—24no—
233C2×C4×C73—24no—
234C2×C4×C73—41no—
235C2×C4×C73—42no—
236C2×C4×C73—41no—
237C2×C4×C73—42no—
238C2×C4×C73—42no—
239C2×C4×C73—42no—
240C2×C4×C73—42no—
241C2×C4×C73—42no—
242C2×C4×C73—42no—
243C2×C4×C73—42no—
244C2×C4×C73—42no—
245C2×C4×C73—42no—
246C2×C4×C73—42no—
247C2×C4×C73—42no—
248C2×C4×C73#12114yes—
249C2×Dic73—22no—
250C2×Dic73—22no—
251C2×Dic73—24no—
252C2×Dic73—24no—
253C2×Dic73—22no—
254C2×Dic73—22no—
255C2×Dic73—24no—
256C2×Dic73—24no—
257C2×Dic73—24no—
258C2×Dic73—24no—
259C2×Dic73—24no—
260C2×Dic73—42no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—42no—
265C2×Dic73—41no—
266C2×Dic73—42no—
267C2×Dic73—42no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73—416no—
271C2×Dic73—42no—
272C2×Dic73—42no—
273C2×Dic73—42no—
274C2×Dic73—42no—
275C2×Dic73—42no—
276C2×Dic73—42no—
277C2×Dic73—42no—
278C2×Dic73C22⋊Q814yes—
279C2×Dic73C22⋊Q814yes—
280C2×Dic73C22⋊Q818yes—
281C2×Dic73#12314yes—
282C2×Dic73#12318yes—
283C2×Dic73#12714yes—
284C22×C73—24no—
285C22×C73—28no—
286C22×C73—42no—
287C22×C73—42no—
288C22×C73—44no—
289C22×C73—44no—
290C22×C73—44no—
291C22×C73—42no—
292C22×C73—42no—
293C22×C73—42no—
294C22×C73—42no—
295C22×C73C2×#12111yescharacteristic
296C22×C73C2×C22⋊Q811yescharacteristic
297C22×C73C2×C22⋊Q811yescharacteristic
298C22×C73C2×C22⋊Q812yes—
299C22×C73C2×#12312yes—
300C22×C73C2×#12311yescharacteristic
301C22×C73C2×#12711yescharacteristic
302C22×C73C2×C4⋊Q811yescharacteristic
303C22×C73#53211yescharacteristic
304C22×C73#53311yescharacteristic
305C22×C73#53311yescharacteristic
306C22×C73#53511yescharacteristic
307C22×C73#53611yescharacteristic
308C22×C73#54212yes—
309C22×C73#54412yes—
310C22×C73D4212yes—
311C22×C73#54712yes—
312C22×C73#54712yes—
313C22×C73D4×Q812yes—
314C22×C73#55012yes—
315C22×C73#55212yes—
316C22×C73#55312yes—
317C22×C73#55412yes—
318C4×C73—24no—
319C4×C73—44no—
320C4×C73—42no—
321C4×C73—82no—
322Dic73—24no—
323Dic73—42no—
324Dic73—44no—
325Dic73—42no—
326Dic73—82no—
327Dic73#39918yescomplemented
328#15135—731nomaximal, Sylow
329C2×C73—44no—
330C2×C73C2×#39911yescharacteristic
331C2×C73#242212yes—
332C2×C73#242312yes—
333C2×C73#244012yes—
334C2×C73#244112yes—
335C2×C73#248611yescharacteristic
336C2×C73#248711yescharacteristic
337C2×C73#254111yescharacteristic
338C2×C73#254211yescharacteristic
339C2×C73#262811yescharacteristic
340C2×C73#285711yescharacteristic
341C2×#399—731no—
342C2×#399D7311yescharacteristic
343C2×#389—732no—
344C2×#384—731no—
345C2×#380—732no—
346C2×C4×C22⋊C4—731no—
347#1445—731no—
348#1445—731no—
349#1444—732no—
350#1439—732no—
351C4⋊#95—731no—
352C73#1513511yescharacteristic, complemented, Sylow
353C24×C4—731no—
354C22×C4⋊C4—732no—
355C22×C4⋊C4—732no—
356C22×C4⋊C4—732no—
357C22×C4⋊C4—731no—
358C22×C4⋊C4—732no—
359C22×C4⋊C4C2×D7311yescharacteristic
360C22×C4⋊C4C2×D7312yes—
361C22×C22⋊C4—732no—
362C22×C22⋊C4—731no—
363C22×C22⋊C4C2×D7312yes—
364C22×C22⋊C4C2×D7311yescharacteristic
365C22×C42—731no—
366C22×C42—731no—
367#399—24no—
368#399—1464no—
369#389—1468no—
370#384—1464no—
371#380—1468no—
372C4×C22⋊C4—1464no—
373C2×#95—731no—
374C2×#95—732no—
375C2×#95—732no—
376C2×#95—732no—
377C2×#95—731no—
378C2×#95—732no—
379C2×#95—732no—
380C2×#95—731no—
381C2×#95C2×D7311yescharacteristic
382C25C22×D7311yescharacteristic
383C23×C4—731no—
384C23×C4—731no—
385C23×C4—734no—
386C23×C4—732no—
387C23×C4—731no—
388C23×C4—731no—
389C23×C4—732no—
390C23×C4—1462no—
391C23×C4—1464no—
392C23×C4—1464no—
393C23×C4—1462no—
394C23×C4C22×D7312yes—
395C23×C4C22×D7312yes—
396C23×C4C22×D7311yescharacteristic
397C23×C4C22×D7311yescharacteristic
398C2×C4⋊C4—22no—
399C2×C4⋊C4—24no—
400C2×C4⋊C4—24no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—42no—
404C2×C4⋊C4—2922no—
405C2×C4⋊C4—2922no—
406C2×C4⋊C4—2922no—
407C2×C4⋊C4—738no—
408C2×C4⋊C4—1464no—
409C2×C4⋊C4—1464no—
410C2×C4⋊C4—1464no—
411C2×C4⋊C4—1464no—
412C2×C4⋊C4—1464no—
413C2×C4⋊C4—1462no—
414C2×C4⋊C4—1464no—
415C2×C4⋊C4—1462no—
416C2×C4⋊C4—1462no—
417C2×C4⋊C4—1464no—
418C2×C4⋊C4—1464no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—22no—
421C2×C22⋊C4—24no—
422C2×C22⋊C4—24no—
423C2×C22⋊C4—42no—
424C2×C22⋊C4—2922no—
425C2×C22⋊C4—734no—
426C2×C22⋊C4—1464no—
427C2×C22⋊C4—1464no—
428C2×C22⋊C4—1462no—
429C2×C22⋊C4—1462no—
430C2×C22⋊C4C22⋊D7314yes—
431C2×C42—734no—
432C2×C42—1462no—
433C2×C42—1464no—
434C2×C42—1464no—
435#95—22no—
436#95—1464no—
437#95—1462no—
438#95—1464no—
439#95—1464no—
440#95—1462no—
441#95—1464no—
442#95—1464no—
443#95—1462no—
444C24—22no—
445C24—21no—
446C24—21no—
447C24—21no—
448C24—22no—
449C24—21no—
450C24—22no—
451C24—22no—
452C24C2×C22⋊D7312yes—
453C24Q8×D7312yes—
454C24Q8×D7312yes—
455C24C23×D7311yescharacteristic
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—2924no—
476C22×C4—2924no—
477C22×C4—2928no—
478C22×C4—732no—
479C22×C4—734no—
480C22×C4—732no—
481C22×C4—734no—
482C22×C4—732no—
483C22×C4—732no—
484C22×C4—732no—
485C22×C4—732no—
486C22×C4—734no—
487C22×C4—732no—
488C22×C4—732no—
489C22×C4—1462no—
490C22×C4—1462no—
491C22×C4—1462no—
492C22×C4—1462no—
493C22×C4—1462no—
494C22×C4—1462no—
495C22×C4—1462no—
496C22×C4—1468no—
497C22×C4—14616no—
498C22×C4—1462no—
499C22×C4—1461no—
500C22×C4—1462no—
501C22×C4—1461no—
502C22×C4—1462no—
503C22×C4—1462no—
504C22×C4—1461no—
505C22×C4—1462no—
506C22×C4—1462no—
507C22×C4—1461no—
508C22×C4—1462no—
509C22×C4—1462no—
510C22×C4—1462no—
511C22×C4—1462no—
512C22×C4—1462no—
513C22×C4—1462no—
514C22×C4D4×D7314yes—
515C22×C4D4×D7312yes—
516C22×C4C2×C22⋊D7312yes—
517C22×C4C2×C22⋊D7312yes—
518C22×C4#1192402712yes—
519C4⋊C4—48no—
520C4⋊C4—44no—
521C4⋊C4—2928no—
522C4⋊C4—2928no—
523C4⋊C4—2928no—
524C4⋊C4—2924no—
525C4⋊C4—14616no—
526C22⋊C4—28no—
527C22⋊C4—48no—
528C22⋊C4—2928no—
529C22⋊C4—1468no—
530C42—2924no—
531C42—1468no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—41no—
554C23#1455658914yes—
555C23C2×D4×D7311yescharacteristic
556C23C2×D4×D7312yes—
557C23C22×C22⋊D7311yescharacteristic
558C23C2×Q8×D7311yescharacteristic
559C23C2×Q8×D7311yescharacteristic
560C23#41×D7311yescharacteristic
561C23#41×D7311yescharacteristic
562C23#41×D7312yes—
563C23C2×#1192402711yescharacteristic
564C23#1455663612yes—
565C23#1455663812yes—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2922no—
592C2×C4—2921no—
593C2×C4—2922no—
594C2×C4—2922no—
595C2×C4—2922no—
596C2×C4—2922no—
597C2×C4—2922no—
598C2×C4—2921no—
599C2×C4—29216no—
600C2×C4—2922no—
601C2×C4—2922no—
602C2×C4—2922no—
603C2×C4—2922no—
604C2×C4—2922no—
605C2×C4—2922no—
606C2×C4—2922no—
607C2×C4—734no—
608C2×C4—738no—
609C2×C4—734no—
610C2×C4—734no—
611C2×C4—738no—
612C2×C4—734no—
613C2×C4—1464no—
614C2×C4—1464no—
615C2×C4—1464no—
616C2×C4—1462no—
617C2×C4—1464no—
618C2×C4—1462no—
619C2×C4—1464no—
620C2×C4—1462no—
621C2×C4—1464no—
622C2×C4—1464no—
623C2×C4—1462no—
624C2×C4#1455656314yes—
625C22—28no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—44no—
630C22—42no—
631C22—44no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22C2×#1455656311yescharacteristic
637C22C22⋊Q8×D7311yescharacteristic
638C22C22⋊Q8×D7312yes—
639C22C22⋊Q8×D7311yescharacteristic
640C22#123×D7312yes—
641C22#123×D7311yescharacteristic
642C22#127×D7311yescharacteristic
643C22C2×#1455658911yescharacteristic
644C22#1798913311yescharacteristic
645C22#1798913911yescharacteristic
646C22#1798913911yescharacteristic
647C22#1798915011yescharacteristic
648C22#1798915911yescharacteristic
649C22#1798920512yes—
650C22#1798921612yes—
651C22D4×C22⋊D7312yes—
652C22#1798923512yes—
653C22#1798923612yes—
654C22#1798925012yes—
655C22#1798925712yes—
656C22#1798927412yes—
657C22#1798928212yes—
658C22#1798929112yes—
659C4—24no—
660C4—42no—
661C4—44no—
662C4—82no—
663C4—2922no—
664C4—2922no—
665C4—2924no—
666C4—5842no—
667C4—738no—
668C4—1464no—
669C2—44no—
670C2#399×D7311yescharacteristic
671C2#2261344212yes—
672C2#2261344912yes—
673C2#2261363012yes—
674C2#2261363712yes—
675C2#2261423611yescharacteristic
676C2#2261424111yescharacteristic
677C2#2261487511yescharacteristic
678C2#2261487911yescharacteristic
679C2#2261575711yescharacteristic
680C2#2261827911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutator, socle, cosocleC24×C73C24
FittingC2×#399×C73C2
prime radicalC25×C73C23
hypercenterC2×#399D73
hypocenterC73#15135
LCSG ▷ C24×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
2. C2×#17988482C2semidirect
3. C2×#17988456C2semidirect
6. #22608199C2semidirect
7. #22608198C2semidirect
8. #22608194C2semidirect
10. #22608159C2semidirect
327. Dic73#399semidirect
352. C73#15135semidirect

Maximal subgroups

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

subgroupindex
C2×#179885232
C2×#179884822
C2×#179884562
C2×#179884452
C2×C22⋊C4×Dic732
#226081992
#226081982
#226081942
#226081742
#226081592
C2×#399×C732
#1513573