Finite Group Databasebeta

← #33007874

Subgroups — #33007874

Order 18688.

2438 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556547C211yesmaximal, characteristic
2C23×C4⋊Dic73C211yesmaximal, characteristic
3C2×#17988448C211yesmaximal, characteristic
4C2×#17988448C211yesmaximal, characteristic
5C2×#17988448C211yesmaximal, characteristic
6#22608183C214yesmaximal, complemented
7#22608165C214yesmaximal, complemented
8#22608154C211yesmaximal, characteristic
9C73×C22⋊#95C211yesmaximal, characteristic
10C24×Dic73C2211yescharacteristic
11C24×Dic73C2211yescharacteristic
12C2×#14556547—24no—
13C2×#14556547—24no—
14C2×#14556547—24no—
15C22×C4⋊Dic73—24no—
16C22×C4⋊Dic73—24no—
17C22×C4⋊Dic73C418yescomplemented
18C22×C4⋊Dic73C2212yes—
19C22×C4⋊Dic73C2212yes—
20C22×#11924007C2212yes—
21C22×C22⋊Dic73C2211yescharacteristic
22C22×C22⋊Dic73C2211yescharacteristic
23C22×C4×Dic73C2212yes—
24#17988448—24no—
25#17988448—24no—
26#17988448—24no—
27C2×#14556498C2214yes—
28C2×#14556498C2212yes—
29C2×#14556498C2214yes—
30C2×#14556498C2214yes—
31C2×#14556498C2212yes—
32C24×C4×C73C2211yescharacteristic
33C22×C22⋊C4×C73C2211yescharacteristic
34C22×C22⋊C4×C73C2211yescharacteristic
35C2×#95×C73C2214yes—
36C23×Dic73—22no—
37C23×Dic73—22no—
38C23×Dic73—22no—
39C23×Dic73—22no—
40C23×Dic73—22no—
41C23×Dic73—22no—
42C23×Dic73—22no—
43C23×Dic73—22no—
44C23×Dic73—22no—
45C23×Dic73—22no—
46C23×Dic73C2×C414yes—
47C23×Dic73C2×C414yes—
48C23×Dic73D414yes—
49C23×Dic73D414yes—
50C23×Dic73C2312yes—
51C23×Dic73C2312yes—
52C23×Dic73C2312yes—
53C23×Dic73C2311yescharacteristic
54C23×Dic73C2311yescharacteristic
55#14556547—232no—
56C2×C4⋊Dic73—232no—
57C2×C4⋊Dic73—24no—
58C2×C4⋊Dic73—24no—
59C2×C4⋊Dic73—24no—
60C2×C4⋊Dic73—24no—
61C2×C4⋊Dic73C2×C418yes—
62C2×C4⋊Dic73C2×C418yes—
63C2×#11924007—24no—
64C2×#11924007—24no—
65C2×#11924007—24no—
66C2×C22⋊Dic73—22no—
67C2×C22⋊Dic73—22no—
68C2×C22⋊Dic73—22no—
69C2×C22⋊Dic73—22no—
70C2×C22⋊Dic73—22no—
71C2×C22⋊Dic73D414yes—
72C2×C4×Dic73—24no—
73C2×C4×Dic73—24no—
74C2×C4×Dic73—24no—
75#14556498—28no—
76#14556498—24no—
77#14556498—24no—
78#14556498—28no—
79#14556498—28no—
80C25×C73C2311yescharacteristic
81C23×C4×C73—22no—
82C23×C4×C73—22no—
83C23×C4×C73—28no—
84C23×C4×C73C2×C414yes—
85C23×C4×C73C2312yes—
86C23×C4×C73C2312yes—
87C23×C4×C73C2312yes—
88C2×C22⋊C4×C73—22no—
89C2×C22⋊C4×C73—22no—
90C2×C22⋊C4×C73—22no—
91C2×C22⋊C4×C73—22no—
92C2×C22⋊C4×C73—22no—
93C2×C22⋊C4×C73D414yes—
94#95×C73—28no—
95C22×Dic73—28no—
96C22×Dic73—28no—
97C22×Dic73—28no—
98C22×Dic73—28no—
99C22×Dic73—28no—
100C22×Dic73—28no—
101C22×Dic73—21no—
102C22×Dic73—21no—
103C22×Dic73—21no—
104C22×Dic73—21no—
105C22×Dic73—21no—
106C22×Dic73—21no—
107C22×Dic73—21no—
108C22×Dic73—21no—
109C22×Dic73—28no—
110C22×Dic73—28no—
111C22×Dic73—28no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—24no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—24no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—24no—
125C22×Dic73—22no—
126C22×Dic73—24no—
127C22×Dic73—44no—
128C22×Dic73—44no—
129C22×Dic73—44no—
130C22×Dic73—44no—
131C22×Dic73C22⋊C4116yes—
132C22×Dic73C22×C414yes—
133C22×Dic73C22×C412yes—
134C22×Dic73C22×C412yes—
135C22×Dic73C2×D414yes—
136C22×Dic73C2×D412yes—
137C22×Dic73C2×D412yes—
138C22×Dic73C2×D412yes—
139C22×Dic73C2×D412yes—
140C4⋊Dic73—216no—
141C4⋊Dic73—216no—
142#11924007—216no—
143C22⋊Dic73—28no—
144C22⋊Dic73—44no—
145C4×Dic73—216no—
146C24×C73—24no—
147C24×C73—24no—
148C24×C73C22×C412yes—
149C24×C73C2×D412yes—
150C24×C73C2×D412yes—
151C24×C73C2×D412yes—
152C24×C73C2×D412yes—
153C24×C73#4112yes—
154C24×C73#4112yes—
155C24×C73C2411yescharacteristic
156C22×C4×C73—216no—
157C22×C4×C73—216no—
158C22×C4×C73—28no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—24no—
162C22×C4×C73—216no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—24no—
166C22×C4×C73—24no—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73—24no—
170C22×C4×C73—22no—
171C22×C4×C73—24no—
172C22×C4×C73C22×C414yes—
173C22×C4×C73C2×D414yes—
174C22×C4×C73#4118yes—
175C22⋊C4×C73—28no—
176C22⋊C4×C73—44no—
177C23×C73—24no—
178C23×C73—24no—
179C23×C73—24no—
180C23×C73—24no—
181C23×C73—24no—
182C23×C73—24no—
183C23×C73—28no—
184C23×C73—28no—
185C23×C73—28no—
186C23×C73—28no—
187C23×C73C2×C22⋊C414yes—
188C23×C73C4×D414yes—
189C23×C73C4×D414yes—
190C23×C73#12014yes—
191C23×C73#12114yes—
192C23×C73#12114yes—
193C23×C73#12314yes—
194C23×C73C23×C411yescharacteristic
195C23×C73C22×D411yescharacteristic
196C23×C73C22×D411yescharacteristic
197C23×C73C22×D411yescharacteristic
198C23×C73C22×D411yescharacteristic
199C23×C73C2×#4114yes—
200C23×C73C2×#4111yescharacteristic
201C23×C73C2×#4111yescharacteristic
202C23×C732−1+414yes—
203C2×C4×C73—232no—
204C2×C4×C73—24no—
205C2×C4×C73—24no—
206C2×C4×C73—28no—
207C2×C4×C73—28no—
208C2×C4×C73—24no—
209C2×C4×C73—28no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—44no—
213C2×C4×C73—42no—
214C2×C4×C73—42no—
215C2×C4×C73—42no—
216C2×C4×C73—44no—
217C2×C4×C73—44no—
218C2×C4×C73#11718yescomplemented
219C2×Dic73—22no—
220C2×Dic73—22no—
221C2×Dic73—22no—
222C2×Dic73—22no—
223C2×Dic73—22no—
224C2×Dic73—22no—
225C2×Dic73—22no—
226C2×Dic73—22no—
227C2×Dic73—22no—
228C2×Dic73—22no—
229C2×Dic73—22no—
230C2×Dic73—22no—
231C2×Dic73—232no—
232C2×Dic73—24no—
233C2×Dic73—24no—
234C2×Dic73—24no—
235C2×Dic73—24no—
236C2×Dic73—28no—
237C2×Dic73—42no—
238C2×Dic73—42no—
239C2×Dic73—42no—
240C2×Dic73—42no—
241C2×Dic73—42no—
242C2×Dic73—42no—
243C2×Dic73—44no—
244C2×Dic73—416no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—44no—
249C2×Dic73—44no—
250C2×Dic73C2×C22⋊C414yes—
251C2×Dic73C2×C22⋊C414yes—
252C22×C73—24no—
253C22×C73—28no—
254C22×C73—28no—
255C22×C73—28no—
256C22×C73—28no—
257C22×C73—28no—
258C22×C73—28no—
259C22×C73—24no—
260C22×C73#38618yes—
261C22×C73C22×C22⋊C411yescharacteristic
262C22×C73C2×#11712yes—
263C22×C73C2×C4×D411yescharacteristic
264C22×C73C2×C4×D411yescharacteristic
265C22×C73#51912yes—
266C22×C73C2×#12011yescharacteristic
267C22×C73C2×#12111yescharacteristic
268C22×C73C2×#12111yescharacteristic
269C22×C73C2×#12311yescharacteristic
270C22×C73#52512yes—
271C22×C73#52512yes—
272C22×C73#53612yes—
273C22×C73#53612yes—
274C22×C73#54714yes—
275C22×C73#54714yes—
276C22×C73#55214yes—
277C22×C73#55214yes—
278C4×C73—28no—
279C4×C73—44no—
280C4×C73—82no—
281Dic73—24no—
282Dic73—24no—
283Dic73—44no—
284Dic73—44no—
285Dic73—82no—
286#13870—731nomaximal, Sylow
287C2×C73—28no—
288C2×C73#228011yescharacteristic
289C2×C73C2×#38611yescharacteristic
290C2×C73#231011yescharacteristic
291C2×C73#234412yes—
292C2×C73#234412yes—
293C2×C73#240612yes—
294C2×C73#241312yes—
295C2×C73#241312yes—
296C2×C73#248412yes—
297C22×#117—731no—
298C23×C4⋊C4—731no—
299C2×#381—731no—
300C2×#381—731no—
301C2×#381—731no—
302#1441—734no—
303#1436—734no—
304C22⋊#95—731no—
305C22⋊#95D7311yescharacteristic
306C73#1387011yescharacteristic, complemented, Sylow
307C24×C4—731no—
308C24×C4—731no—
309C24×C4C2×D7311yescharacteristic
310C2×#117—1464no—
311C2×#117—1464no—
312C2×#117—1464no—
313C22×C4⋊C4—738no—
314C22×C4⋊C4—732no—
315C22×C4⋊C4—732no—
316C22×C4⋊C4—732no—
317C22×C4⋊C4—1464no—
318C22×C4⋊C4—1464no—
319C22×C22⋊C4—731no—
320C22×C22⋊C4—731no—
321C22×C22⋊C4C2×D7311yescharacteristic
322C22×C22⋊C4C2×D7311yescharacteristic
323C22×C42—732no—
324#381—1464no—
325#381—1464no—
326#381—1464no—
327C2×#95—734no—
328C2×#95—732no—
329C2×#95—734no—
330C2×#95—734no—
331C2×#95—732no—
332C2×#95C2×D7314yes—
333C25C22×D7311yescharacteristic
334C23×C4—22no—
335C23×C4—22no—
336C23×C4—28no—
337C23×C4—734no—
338C23×C4—734no—
339C23×C4—734no—
340C23×C4—731no—
341C23×C4—731no—
342C23×C4—734no—
343C23×C4—732no—
344C23×C4—732no—
345C23×C4—732no—
346C23×C4—1462no—
347C23×C4—1462no—
348C23×C4—1462no—
349C23×C4—1462no—
350C23×C4—1462no—
351C23×C4—1462no—
352C23×C4—1462no—
353C23×C4—1462no—
354C23×C4—1462no—
355C23×C4—1462no—
356C23×C4C4×D7314yes—
357C23×C4C22×D7312yes—
358C23×C4C22×D7312yes—
359C23×C4C22×D7312yes—
360#117—14632no—
361C2×C4⋊C4—738no—
362C2×C4⋊C4—738no—
363C2×C4⋊C4—14632no—
364C2×C4⋊C4—1464no—
365C2×C4⋊C4—1464no—
366C2×C4⋊C4—1464no—
367C2×C4⋊C4—1464no—
368C2×C4⋊C4—1464no—
369C2×C4⋊C4—1464no—
370C2×C4⋊C4—1464no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—734no—
377C2×C22⋊C4—1462no—
378C2×C22⋊C4—1462no—
379C2×C22⋊C4—1462no—
380C2×C22⋊C4—1462no—
381C2×C22⋊C4—1462no—
382C2×C22⋊C4D29214yes—
383C2×C42—1464no—
384C2×C42—1464no—
385C2×C42—1464no—
386#95—28no—
387#95—1464no—
388#95—1464no—
389#95—1468no—
390#95—1468no—
391#95—1468no—
392C24—24no—
393C24—24no—
394C24C2×C4×D7312yes—
395C24D4×D7312yes—
396C24D4×D7312yes—
397C24D4×D7312yes—
398C24C2×D29212yes—
399C24#1192402812yes—
400C24#1192402912yes—
401C24C23×D7311yescharacteristic
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—216no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—2924no—
419C22×C4—2924no—
420C22×C4—2924no—
421C22×C4—2924no—
422C22×C4—7316no—
423C22×C4—732no—
424C22×C4—732no—
425C22×C4—732no—
426C22×C4—732no—
427C22×C4—732no—
428C22×C4—732no—
429C22×C4—734no—
430C22×C4—734no—
431C22×C4—1462no—
432C22×C4—1462no—
433C22×C4—1464no—
434C22×C4—1462no—
435C22×C4—1464no—
436C22×C4—1468no—
437C22×C4—1468no—
438C22×C4—1468no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4—1462no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4—1464no—
446C22×C4—1462no—
447C22×C4—1462no—
448C22×C4—1464no—
449C22×C4—1468no—
450C22×C4—1468no—
451C22×C4—1468no—
452C22×C4—1468no—
453C22×C4—1468no—
454C22×C4—1468no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4C2×C4×D7314yes—
464C22×C4C2×D29214yes—
465C22×C4#1192402718yes—
466C4⋊C4—14616no—
467C4⋊C4—14616no—
468C4⋊C4—14616no—
469C22⋊C4—28no—
470C22⋊C4—44no—
471C22⋊C4—2924no—
472C22⋊C4—1468no—
473C42—14616no—
474C23—24no—
475C23—24no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23C22⋊C4×D7314yes—
485C23#1455655314yes—
486C23C4×D29214yes—
487C23#1455656114yes—
488C23#1455656414yes—
489C23#1455656614yes—
490C23#1455657614yes—
491C23C22×C4×D7311yescharacteristic
492C23C2×D4×D7311yescharacteristic
493C23C2×D4×D7311yescharacteristic
494C23C2×D4×D7311yescharacteristic
495C23C22×D29211yescharacteristic
496C23C2×#1192402714yes—
497C23C2×#1192402811yescharacteristic
498C23C2×#1192402911yescharacteristic
499C23#1455663814yes—
500C2×C4—232no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—2922no—
516C2×C4—29216no—
517C2×C4—2924no—
518C2×C4—2922no—
519C2×C4—2922no—
520C2×C4—2922no—
521C2×C4—2922no—
522C2×C4—2922no—
523C2×C4—2922no—
524C2×C4—2924no—
525C2×C4—2924no—
526C2×C4—2922no—
527C2×C4—2922no—
528C2×C4—734no—
529C2×C4—734no—
530C2×C4—1468no—
531C2×C4—1464no—
532C2×C4—1464no—
533C2×C4—1464no—
534C2×C4—1464no—
535C2×C4—14632no—
536C2×C4—1462no—
537C2×C4—1462no—
538C2×C4—1462no—
539C2×C4—1462no—
540C2×C4—1462no—
541C2×C4—1462no—
542C2×C4—1462no—
543C2×C4—1462no—
544C2×C4—1462no—
545C2×C4—1462no—
546C2×C4—1462no—
547C2×C4—1462no—
548C2×C4#1455654718yescomplemented
549C22—28no—
550C22—24no—
551C22—28no—
552C22—28no—
553C22—24no—
554C22—28no—
555C22—28no—
556C22—28no—
557C22#1798846518yes—
558C22C2×C22⋊C4×D7311yescharacteristic
559C22C2×#1455654712yes—
560C22C2×#1455655311yescharacteristic
561C22C2×C4×D29211yescharacteristic
562C22#1798906412yes—
563C22C2×#1455656111yescharacteristic
564C22C2×#1455656411yescharacteristic
565C22C2×#1455656611yescharacteristic
566C22C2×#1455657611yescharacteristic
567C22#1798909312yes—
568C22#1798909312yes—
569C22#1798915912yes—
570C22#1798915912yes—
571C22#1798923714yes—
572C22#1798924114yes—
573C22#1798927714yes—
574C22#1798927914yes—
575C4—28no—
576C4—44no—
577C4—82no—
578C4—2924no—
579C4—2924no—
580C4—5842no—
581C4—1464no—
582C4—1464no—
583C2—28no—
584C2#2261230911yescharacteristic
585C2C2×#1798846511yescharacteristic
586C2#2261251611yescharacteristic
587C2#2261279812yes—
588C2#2261280212yes—
589C2#2261326512yes—
590C2#2261333612yes—
591C2#2261334112yes—
592C2#2261421712yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×C22⋊#95C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC22⋊#95D73
hypocenterC73#13870
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
6. #22608183C2semidirect
7. #22608165C2semidirect
17. C22×C4⋊Dic73C4semidirect
218. C2×C4×C73#117semidirect
306. C73#13870semidirect
548. C2×C4#14556547semidirect

Maximal subgroups

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

subgroupindex
C22×#145565472
C23×C4⋊Dic732
C2×#179884482
#226081832
#226081652
#226081542
C73×C22⋊#952
#1387073