Finite Group Databasebeta

← #33007963

Subgroups — #33007963

Order 18688.

2006 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556549C211yesmaximal, characteristic
2C2×#17988456C211yesmaximal, characteristic, complemented
3C2×#17988444C211yesmaximal, characteristic
4C2×C4×#11924005C211yesmaximal, characteristic
5#22608175C214yesmaximal
6#22608162C214yesmaximal, complemented
7#22608161C211yesmaximal, characteristic, complemented
8#22608160C211yesmaximal, characteristic, complemented
9C2×#382×C73C211yesmaximal, characteristic, complemented
10C23×C4×D73C2211yescharacteristic
11C2×#14556549—24no—
12C2×#14556549—24no—
13C2×#14556549—24no—
14C22×C4⋊Dic73C2211yescharacteristic
15C22×C4⋊Dic73C2211yescharacteristic
16C22×#11924007C2214yes—
17C22×#11924005C2212yes—
18C22×#11924005C2212yes—
19C22×#11924005C2212yes—
20C22×C4×Dic73C2212yes—
21C22×C4×Dic73C2211yescharacteristic
22#17988456—24no—
23#17988444—24no—
24C4×#11924005C418yescomplemented
25C2×#14556498C2214yes—
26C2×#14556498C2212yes—
27C2×#14556498C2214yes—
28C2×#14556498C2212yes—
29C22×C4⋊C4×C73C2211yescharacteristic
30C22×C4⋊C4×C73C2211yescharacteristic
31C22×C42×C73C2211yescharacteristic
32#382×C73—24no—
33C2×#95×C73C2214yes—
34C24×D73C2311yescharacteristic
35C23×Dic73C2312yes—
36C23×Dic73C2312yes—
37C23×Dic73C2311yescharacteristic
38C23×Dic73C2312yes—
39C22×C4×D73—22no—
40C22×C4×D73—22no—
41C22×C4×D73—22no—
42C22×C4×D73—22no—
43C22×C4×D73C2×C414yes—
44C22×C4×D73D414yes—
45C22×C4×D73Q814yes—
46#14556549—232no—
47C2×C4⋊Dic73—22no—
48C2×C4⋊Dic73—22no—
49C2×C4⋊Dic73—22no—
50C2×C4⋊Dic73—22no—
51C2×C4⋊Dic73—22no—
52C2×C4⋊Dic73D414yes—
53C2×#11924007—28no—
54C2×#11924007—28no—
55C2×#11924007—44no—
56C2×#11924005—24no—
57C2×#11924005—24no—
58C2×#11924005—24no—
59C2×#11924005—24no—
60C2×#11924005—24no—
61C2×#11924005—24no—
62C2×#11924005—44no—
63C2×#11924005C2×C418yes—
64C2×C4×Dic73—24no—
65C2×C4×Dic73—24no—
66C2×C4×Dic73—22no—
67C2×C4×Dic73—24no—
68C2×C4×Dic73—22no—
69C2×C4×Dic73C2×C414yes—
70#14556498—24no—
71#14556498—28no—
72#14556498—44no—
73#14556498C2×C418yes—
74C23×C4×C73C2312yes—
75C23×C4×C73C2312yes—
76C23×C4×C73C2312yes—
77C23×C4×C73C2311yescharacteristic
78C2×C4⋊C4×C73—22no—
79C2×C4⋊C4×C73—22no—
80C2×C4⋊C4×C73—22no—
81C2×C4⋊C4×C73—22no—
82C2×C4⋊C4×C73—22no—
83C2×C4⋊C4×C73D414yes—
84C2×C42×C73—22no—
85C2×C42×C73—22no—
86C2×C42×C73C2×C414yes—
87#95×C73—28no—
88C23×D73—21no—
89C23×D73—21no—
90C23×D73—22no—
91C23×D73—22no—
92C23×D73—21no—
93C23×D73—21no—
94C23×D73—22no—
95C23×D73—22no—
96C23×D73C22×C412yes—
97C23×D73C2×D412yes—
98C23×D73C2×Q812yes—
99C22×Dic73—21no—
100C22×Dic73—21no—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—21no—
104C22×Dic73—21no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—24no—
110C22×Dic73—24no—
111C22×Dic73—22no—
112C22×Dic73—24no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—24no—
116C22×Dic73C22×C412yes—
117C22×Dic73C22×C414yes—
118C22×Dic73C2×D414yes—
119C22×Dic73C2×D412yes—
120C22×Dic73C2×Q812yes—
121C22×Dic73#4114yes—
122C22×Dic73#4114yes—
123C2×C4×D73—28no—
124C2×C4×D73—28no—
125C2×C4×D73—28no—
126C2×C4×D73—28no—
127C2×C4×D73—28no—
128C2×C4×D73—28no—
129C2×C4×D73C4⋊C4116yescomplemented
130C4⋊Dic73—28no—
131C4⋊Dic73—44no—
132#11924007—416no—
133#11924005—216no—
134#11924005—216no—
135#11924005—48no—
136C4×Dic73—216no—
137C4×Dic73—28no—
138C24×C73C2411yescharacteristic
139C22×C4×C73—22no—
140C22×C4×C73—22no—
141C22×C4×C73—22no—
142C22×C4×C73—22no—
143C22×C4×C73—24no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73—24no—
149C22×C4×C73—22no—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—24no—
153C22×C4×C73—24no—
154C22×C4×C73C22×C414yes—
155C22×C4×C73C22×C412yes—
156C22×C4×C73C2×D412yes—
157C22×C4×C73C2×D412yes—
158C22×C4×C73C2×D414yes—
159C22×C4×C73C2×D412yes—
160C22×C4×C73C2×Q812yes—
161C22×C4×C73#4112yes—
162C22×C4×C73#4112yes—
163C4⋊C4×C73—28no—
164C4⋊C4×C73—44no—
165C42×C73—28no—
166C23×C73C23×C411yescharacteristic
167C23×C73C22×D411yescharacteristic
168C23×C73C22×D411yescharacteristic
169C23×C73C22×D411yescharacteristic
170C23×C73C22×Q811yescharacteristic
171C23×C73C2×#4112yes—
172C23×C73C2×#4112yes—
173C23×C73C2×#4111yescharacteristic
174C23×C73C2×#4111yescharacteristic
175C23×C73H(5,𝔽2)12yes—
176C23×C732−1+412yes—
177C22×D73—24no—
178C22×D73—22no—
179C22×D73—22no—
180C22×D73—24no—
181C22×D73—22no—
182C22×D73—22no—
183C22×D73—22no—
184C22×D73—22no—
185C22×D73—22no—
186C22×D73—22no—
187C22×D73—24no—
188C22×D73—22no—
189C22×D73—22no—
190C22×D73—24no—
191C22×D73—42no—
192C22×D73—42no—
193C22×D73—42no—
194C22×D73—42no—
195C22×D73—42no—
196C22×D73—42no—
197C22×D73—42no—
198C22×D73—42no—
199C22×D73C2×C4⋊C414yes—
200C2×C4×C73—24no—
201C2×C4×C73—24no—
202C2×C4×C73—24no—
203C2×C4×C73—24no—
204C2×C4×C73—28no—
205C2×C4×C73—42no—
206C2×C4×C73—42no—
207C2×C4×C73—42no—
208C2×C4×C73—44no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—42no—
213C2×C4×C73—42no—
214C2×C4×C73—42no—
215C2×C4×C73—44no—
216C2×C4×C73—44no—
217C2×C4×C73C2×C4⋊C414yes—
218C2×C4×C73C4×D414yes—
219C2×C4×C73C4×D414yes—
220C2×C4×C73#12014yes—
221C2×C4×C73C22⋊Q814yes—
222C2×C4×C73C22⋊Q814yes—
223C2×C4×C73#12314yes—
224C2×Dic73—24no—
225C2×Dic73—22no—
226C2×Dic73—22no—
227C2×Dic73—24no—
228C2×Dic73—22no—
229C2×Dic73—22no—
230C2×Dic73—22no—
231C2×Dic73—22no—
232C2×Dic73—24no—
233C2×Dic73—24no—
234C2×Dic73—24no—
235C2×Dic73—24no—
236C2×Dic73—24no—
237C2×Dic73—28no—
238C2×Dic73—42no—
239C2×Dic73—42no—
240C2×Dic73—44no—
241C2×Dic73—42no—
242C2×Dic73—42no—
243C2×Dic73—42no—
244C2×Dic73—44no—
245C2×Dic73—44no—
246C2×Dic73C2×C4⋊C414yes—
247C2×Dic73#11718yescomplemented
248C4×D73—232no—
249C22×C73C22×C4⋊C411yescharacteristic
250C22×C73C2×#11712yes—
251C22×C73C2×C4×D411yescharacteristic
252C22×C73C2×C4×D411yescharacteristic
253C22×C73#52012yes—
254C22×C73C2×#12011yescharacteristic
255C22×C73C2×C22⋊Q811yescharacteristic
256C22×C73C2×C22⋊Q811yescharacteristic
257C22×C73C2×#12311yescharacteristic
258C22×C73#53211yescharacteristic
259C22×C73#53211yescharacteristic
260C22×C73#53312yes—
261C22×C73#53511yescharacteristic
262C22×C73#53611yescharacteristic
263C22×C73#54412yes—
264C22×C73#54612yes—
265C22×C73#54712yes—
266C22×C73#54812yes—
267C22×C73#55012yes—
268C22×C73#55212yes—
269C22×C73#55312yes—
270C22×C73#55512yes—
271C22×C73#55612yes—
272C2×D73—24no—
273C2×D73—24no—
274C2×D73—24no—
275C2×D73—44no—
276C2×D73—44no—
277C2×D73—42no—
278C2×D73—42no—
279C2×D73—44no—
280C2×D73—44no—
281C2×D73—42no—
282C2×D73—42no—
283C4×C73—44no—
284C4×C73—44no—
285C4×C73—82no—
286C4×C73#38518yes—
287Dic73—28no—
288Dic73—24no—
289Dic73—44no—
290Dic73—82no—
291#13879—731nomaximal, Sylow
292C2×C73#228511yescharacteristic
293C2×C73C2×#38511yescharacteristic
294C2×C73#231711yescharacteristic
295C2×C73#235212yes—
296C2×C73#235412yes—
297C2×C73#243111yescharacteristic
298C2×C73#243211yescharacteristic
299C2×C73#243912yes—
300C2×C73#244112yes—
301C2×C73#249911yescharacteristic
302C2×C73#250011yescharacteristic
303D73—44no—
304C22×#117—731no—
305C2×#384—731no—
306C2×#382D7311yescharacteristic, complemented
307C2×#380—731no—
308C2×C4×C22⋊C4—731no—
309#1439—734no—
310#1436—734no—
311C4⋊#95—731no—
312C4⋊#95—731no—
313C73#1387911yescharacteristic, complemented, Sylow
314C24×C4—731no—
315C2×#117—1464no—
316C2×#117—1464no—
317C2×#117—1464no—
318C22×C4⋊C4—734no—
319C22×C4⋊C4—731no—
320C22×C4⋊C4—731no—
321C22×C4⋊C4C2×D7311yescharacteristic
322C22×C4⋊C4C2×D7311yescharacteristic
323C22×C22⋊C4—732no—
324C22×C22⋊C4—732no—
325C22×C22⋊C4—732no—
326C22×C42—731no—
327C22×C42—732no—
328C22×C42C2×D7311yescharacteristic
329#384—1464no—
330#382—24no—
331#380—1464no—
332C4×C22⋊C4—738no—
333C2×#95—734no—
334C2×#95—732no—
335C2×#95—734no—
336C2×#95—732no—
337C2×#95C2×D7314yes—
338C25—731no—
339C23×C4—732no—
340C23×C4—731no—
341C23×C4—734no—
342C23×C4—734no—
343C23×C4—734no—
344C23×C4—732no—
345C23×C4—732no—
346C23×C4—1462no—
347C23×C4—1462no—
348C23×C4—1462no—
349C23×C4—1462no—
350C23×C4C22×D7312yes—
351C23×C4C22×D7312yes—
352C23×C4C22×D7312yes—
353C23×C4C22×D7311yescharacteristic
354#117—14632no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—2924no—
361C2×C4⋊C4—734no—
362C2×C4⋊C4—1468no—
363C2×C4⋊C4—1468no—
364C2×C4⋊C4—1462no—
365C2×C4⋊C4—1462no—
366C2×C4⋊C4—1462no—
367C2×C4⋊C4—1462no—
368C2×C4⋊C4—1462no—
369C2×C4⋊C4C22⋊D7314yes—
370C2×C22⋊C4—2924no—
371C2×C22⋊C4—738no—
372C2×C22⋊C4—1464no—
373C2×C22⋊C4—1464no—
374C2×C22⋊C4—1464no—
375C2×C22⋊C4—1464no—
376C2×C22⋊C4—1464no—
377C2×C22⋊C4—1464no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—734no—
381C2×C42—1464no—
382C2×C42—1464no—
383C2×C42—1462no—
384C2×C42—1464no—
385C2×C42—1462no—
386C2×C42C4×D7314yes—
387#95—28no—
388#95—2924no—
389#95—738no—
390#95—1468no—
391#95—1464no—
392C24—732no—
393C24—732no—
394C24—732no—
395C24—1461no—
396C24—1461no—
397C24—1462no—
398C24—1462no—
399C24—1461no—
400C24—1461no—
401C24—1462no—
402C24—1462no—
403C24C23×D7311yescharacteristic
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—732no—
420C22×C4—734no—
421C22×C4—732no—
422C22×C4—732no—
423C22×C4—734no—
424C22×C4—734no—
425C22×C4—7316no—
426C22×C4—734no—
427C22×C4—1462no—
428C22×C4—1464no—
429C22×C4—1462no—
430C22×C4—1462no—
431C22×C4—1464no—
432C22×C4—1462no—
433C22×C4—1462no—
434C22×C4—1464no—
435C22×C4—1464no—
436C22×C4—1461no—
437C22×C4—1462no—
438C22×C4—1461no—
439C22×C4—1461no—
440C22×C4—1462no—
441C22×C4—1461no—
442C22×C4—1462no—
443C22×C4—1462no—
444C22×C4—1468no—
445C22×C4—1468no—
446C22×C4—1468no—
447C22×C4—1468no—
448C22×C4—1468no—
449C22×C4—1468no—
450C22×C4C2×C4×D7314yes—
451C22×C4C2×C4×D7312yes—
452C22×C4D4×D7312yes—
453C22×C4D4×D7312yes—
454C22×C4C2×C22⋊D7314yes—
455C22×C4C2×C22⋊D7312yes—
456C22×C4Q8×D7312yes—
457C22×C4#1192402812yes—
458C22×C4#1192402912yes—
459C4⋊C4—28no—
460C4⋊C4—44no—
461C4⋊C4—29216no—
462C4⋊C4—2924no—
463C4⋊C4—1468no—
464C22⋊C4—2928no—
465C22⋊C4—14616no—
466C22⋊C4—14616no—
467C42—28no—
468C42—14616no—
469C42—1468no—
470C23—2922no—
471C23—2922no—
472C23—2922no—
473C23—2922no—
474C23—2922no—
475C23—2922no—
476C23—2922no—
477C23—2922no—
478C23—734no—
479C23—1462no—
480C23—1462no—
481C23—1464no—
482C23—1462no—
483C23—1462no—
484C23—1464no—
485C23—1464no—
486C23—1462no—
487C23—1462no—
488C23—1464no—
489C23—1462no—
490C23—1462no—
491C23—1462no—
492C23—1462no—
493C23C22×C4×D7311yescharacteristic
494C23C2×D4×D7311yescharacteristic
495C23C2×D4×D7311yescharacteristic
496C23C22×C22⋊D7311yescharacteristic
497C23C2×Q8×D7311yescharacteristic
498C23#41×D7312yes—
499C23#41×D7312yes—
500C23C2×#1192402811yescharacteristic
501C23C2×#1192402911yescharacteristic
502C23#1455663712yes—
503C23#1455663812yes—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—2922no—
522C2×C4—2922no—
523C2×C4—2924no—
524C2×C4—2922no—
525C2×C4—2922no—
526C2×C4—2924no—
527C2×C4—2924no—
528C2×C4—2922no—
529C2×C4—734no—
530C2×C4—738no—
531C2×C4—1468no—
532C2×C4—1464no—
533C2×C4—1464no—
534C2×C4—1464no—
535C2×C4—1464no—
536C2×C4—1464no—
537C2×C4—1462no—
538C2×C4—1464no—
539C2×C4—1462no—
540C2×C4—1464no—
541C2×C4—1462no—
542C2×C4—1462no—
543C2×C4—1462no—
544C2×C4—1462no—
545C2×C4—14632no—
546C2×C4C4⋊C4×D7314yes—
547C2×C4C4×C22⋊D7314yes—
548C2×C4#1455655314yes—
549C2×C4#1455655914yes—
550C2×C4#1455656714yes—
551C2×C4#1455656914yes—
552C2×C4#1455657614yes—
553C22—2922no—
554C22—2922no—
555C22—2922no—
556C22—2924no—
557C22—2924no—
558C22—2922no—
559C22—2924no—
560C22—2924no—
561C22—1464no—
562C22—1464no—
563C22—1464no—
564C22C2×C4⋊C4×D7311yescharacteristic
565C22#117×D7312yes—
566C22C2×C4×C22⋊D7311yescharacteristic
567C22C2×#1455655311yescharacteristic
568C22#1798906712yes—
569C22C2×#1455655911yescharacteristic
570C22C2×#1455656711yescharacteristic
571C22C2×#1455656911yescharacteristic
572C22C2×#1455657611yescharacteristic
573C22#1798913611yescharacteristic
574C22#1798913611yescharacteristic
575C22#1798914312yes—
576C22#1798915511yescharacteristic
577C22#1798915711yescharacteristic
578C22#1798921712yes—
579C22#1798923012yes—
580C22#1798923612yes—
581C22#1798924512yes—
582C22#1798925612yes—
583C22#1798927512yes—
584C22#1798929012yes—
585C22#1798930012yes—
586C22#1798930512yes—
587C4—44no—
588C4—44no—
589C4—82no—
590C4—2924no—
591C4—5842no—
592C4—1464no—
593C4—1468no—
594C4#1798845818yes—
595C2—2924no—
596C2#2261233711yescharacteristic
597C2C2×#1798845811yescharacteristic
598C2#2261255611yescharacteristic
599C2#2261285712yes—
600C2#2261288112yes—
601C2#2261355311yescharacteristic
602C2#2261356211yescharacteristic
603C2#2261362512yes—
604C2#2261364012yes—
605C2#2261438711yescharacteristic
606C2#2261439611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×#382×C73C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenterC2×#382D73
hypocenterC73#13879
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. C2×#17988456C2semidirect
6. #22608162C2semidirect
7. #22608161C2semidirect
8. #22608160C2semidirect
9. C2×#382×C73C2semidirect
24. C4×#11924005C4semidirect
129. C2×C4×D73C4⋊C4semidirect
247. C2×Dic73#117semidirect
306. C2×#382D73semidirect
313. C73#13879semidirect

Maximal subgroups

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

subgroupindex
C22×#145565492
C2×#179884562
C2×#179884442
C2×C4×#119240052
#226081752
#226081622
#226081612
#226081602
C2×#382×C732
#1387973