Finite Group Databasebeta

← #33012641

Subgroups — #33012641

Order 18688.

1414 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556581C211yesmaximal, characteristic
2C2×#17988454C211yesmaximal, characteristic
3C2×C4⋊C4×Dic73C211yesmaximal, characteristic
4#22608197C212yesmaximal
5#22608197C212yesmaximal
6#22608185C211yesmaximal, characteristic
7#22608184C211yesmaximal, characteristic
8#22608181C211yesmaximal, characteristic
9#22608165C212yesmaximal
10#22608162C212yesmaximal
11C73×#1441C211yesmaximal, characteristic
12C2×#14556581—24no—
13C2×#14556581—24no—
14C2×#14556581C418yescomplemented
15C22×C4⋊Dic73C2211yescharacteristic
16C22×C4⋊Dic73C2211yescharacteristic
17C22×#11924007C2211yescharacteristic
18C22×#11924007C2211yescharacteristic
19C22×#11924007C2211yescharacteristic
20C22×#11924007C2211yescharacteristic
21C22×C4×Dic73C2212yes—
22C22×C4×Dic73C2211yescharacteristic
23#17988454—24no—
24C4⋊C4×Dic73—24no—
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2211yescharacteristic
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C2×#14556498C2212yes—
35C22×C4⋊C4×C73C2211yescharacteristic
36C22×C4⋊C4×C73C2211yescharacteristic
37C2×#95×C73C2212yes—
38C2×#95×C73C2211yescharacteristic
39C2×#95×C73C2212yes—
40C23×Dic73C2312yes—
41C23×Dic73C2311yescharacteristic
42C23×Dic73C2311yescharacteristic
43C23×Dic73C2312yes—
44C23×Dic73C2311yescharacteristic
45C23×Dic73C2311yescharacteristic
46#14556581—232no—
47C2×C4⋊Dic73—22no—
48C2×C4⋊Dic73—22no—
49C2×C4⋊Dic73—24no—
50C2×C4⋊Dic73—22no—
51C2×C4⋊Dic73C2×C414yes—
52C2×#11924007—22no—
53C2×#11924007—22no—
54C2×#11924007—22no—
55C2×#11924007—22no—
56C2×#11924007—22no—
57C2×#11924007—22no—
58C2×#11924007—22no—
59C2×#11924007—22no—
60C2×#11924007C2×C414yes—
61C2×#11924007C2×C414yes—
62C2×#11924007C2×C414yes—
63C2×#11924007C2×C414yes—
64C2×C4×Dic73—24no—
65C2×C4×Dic73—42no—
66C2×C4×Dic73—42no—
67C2×C4×Dic73C2×C414yes—
68C2×C4×Dic73D414yes—
69C2×C4×Dic73D414yes—
70#14556498—24no—
71#14556498—24no—
72#14556498—24no—
73#14556498—24no—
74#14556498—24no—
75#14556498—22no—
76#14556498—24no—
77#14556498—42no—
78#14556498—42no—
79#14556498—42no—
80C23×C4×C73C2312yes—
81C23×C4×C73C2311yescharacteristic
82C23×C4×C73C2311yescharacteristic
83C23×C4×C73C2312yes—
84C23×C4×C73C2311yescharacteristic
85C2×C4⋊C4×C73—22no—
86C2×C4⋊C4×C73—22no—
87C2×C4⋊C4×C73—22no—
88C2×C4⋊C4×C73—24no—
89C2×C4⋊C4×C73C2×C414yes—
90#95×C73—22no—
91#95×C73—24no—
92#95×C73—42no—
93C22×Dic73—21no—
94C22×Dic73—21no—
95C22×Dic73—22no—
96C22×Dic73—21no—
97C22×Dic73—22no—
98C22×Dic73—21no—
99C22×Dic73—22no—
100C22×Dic73—21no—
101C22×Dic73—21no—
102C22×Dic73—21no—
103C22×Dic73—21no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—21no—
108C22×Dic73—21no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—21no—
112C22×Dic73—21no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73C22×C412yes—
123C22×Dic73C22×C412yes—
124C22×Dic73C22×C412yes—
125C22×Dic73C22×C412yes—
126C22×Dic73C2×D412yes—
127C22×Dic73C2×D412yes—
128C22×Dic73C2×D412yes—
129C22×Dic73C2×D412yes—
130C22×Dic73#4114yes—
131C4⋊Dic73—28no—
132C4⋊Dic73—28no—
133#11924007—28no—
134#11924007—28no—
135#11924007—28no—
136#11924007—28no—
137C4×Dic73—48no—
138C4×Dic73C22⋊C4116yes—
139C24×C73C2411yescharacteristic
140C22×C4×C73—22no—
141C22×C4×C73—22no—
142C22×C4×C73—22no—
143C22×C4×C73—21no—
144C22×C4×C73—21no—
145C22×C4×C73—21no—
146C22×C4×C73—21no—
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73—22no—
150C22×C4×C73—21no—
151C22×C4×C73—22no—
152C22×C4×C73—21no—
153C22×C4×C73—22no—
154C22×C4×C73—21no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—21no—
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×C73C22×C412yes—
166C22×C4×C73C22×C412yes—
167C22×C4×C73C22×C412yes—
168C22×C4×C73C2×D412yes—
169C22×C4×C73C2×D412yes—
170C22×C4×C73#4114yes—
171C4⋊C4×C73—28no—
172C4⋊C4×C73—28no—
173C23×C73C23×C411yescharacteristic
174C23×C73C22×D411yescharacteristic
175C23×C73C22×D411yescharacteristic
176C23×C73C2×#4112yes—
177C23×C73C2×#4112yes—
178C23×C73H(5,𝔽2)11yescharacteristic
179C23×C73H(5,𝔽2)11yescharacteristic
180C23×C732−1+412yes—
181C23×C732−1+411yescharacteristic
182C23×C732−1+412yes—
183C23×C732−1+411yescharacteristic
184C2×C4×C73—24no—
185C2×C4×C73—22no—
186C2×C4×C73—22no—
187C2×C4×C73—24no—
188C2×C4×C73—22no—
189C2×C4×C73—22no—
190C2×C4×C73—24no—
191C2×C4×C73—22no—
192C2×C4×C73—22no—
193C2×C4×C73—24no—
194C2×C4×C73—24no—
195C2×C4×C73—24no—
196C2×C4×C73—41no—
197C2×C4×C73—41no—
198C2×C4×C73—42no—
199C2×C4×C73—41no—
200C2×C4×C73—42no—
201C2×C4×C73—42no—
202C2×C4×C73—41no—
203C2×C4×C73—41no—
204C2×C4×C73—41no—
205C2×C4×C73—42no—
206C2×C4×C73—42no—
207C2×C4×C73—42no—
208C2×C4×C73—42no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—42no—
213C2×C4×C73—42no—
214C2×C4×C73C2×C22⋊C414yes—
215C2×Dic73—24no—
216C2×Dic73—22no—
217C2×Dic73—22no—
218C2×Dic73—22no—
219C2×Dic73—22no—
220C2×Dic73—24no—
221C2×Dic73—22no—
222C2×Dic73—22no—
223C2×Dic73—22no—
224C2×Dic73—22no—
225C2×Dic73—22no—
226C2×Dic73—22no—
227C2×Dic73—24no—
228C2×Dic73—22no—
229C2×Dic73—22no—
230C2×Dic73—24no—
231C2×Dic73—24no—
232C2×Dic73—24no—
233C2×Dic73—41no—
234C2×Dic73—41no—
235C2×Dic73—42no—
236C2×Dic73—42no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—42no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—42no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—42no—
249C2×Dic73—42no—
250C2×Dic73—42no—
251C2×Dic73C2×C22⋊C414yes—
252C2×Dic73C2×C22⋊C414yes—
253C22×C73C22×C22⋊C411yescharacteristic
254C22×C73C4×#4112yes—
255C22×C73#51912yes—
256C22×C73#52011yescharacteristic
257C22×C73#52011yescharacteristic
258C22×C73#52511yescharacteristic
259C22×C73#52511yescharacteristic
260C22×C73#53212yes—
261C22×C73#53511yescharacteristic
262C22×C73#53611yescharacteristic
263C22×C73#53612yes—
264C22×C73#53611yescharacteristic
265C22×C73#53611yescharacteristic
266C22×C73#53711yescharacteristic
267C22×C73#53711yescharacteristic
268C22×C73#53912yes—
269C22×C73#53912yes—
270C22×C73#53912yes—
271C22×C73#53912yes—
272C22×C73#54112yes—
273C22×C73#54112yes—
274C22×C73#54112yes—
275C22×C73#54112yes—
276C4×C73—24no—
277C4×C73—42no—
278C4×C73—44no—
279C4×C73—42no—
280C4×C73—82no—
281Dic73—24no—
282Dic73—24no—
283Dic73—42no—
284Dic73—44no—
285Dic73—42no—
286Dic73—82no—
287#14289—731nomaximal, Sylow
288C2×C73#229711yescharacteristic
289C2×C73#231011yescharacteristic
290C2×C73#231111yescharacteristic
291C2×C73#233612yes—
292C2×C73#233612yes—
293C2×C73#263311yescharacteristic
294C2×C73#263311yescharacteristic
295C2×C73#263411yescharacteristic
296C2×C73#263411yescharacteristic
297C2×C73#266412yes—
298C2×C73#266412yes—
299C22×#125—731no—
300C2×#383—731no—
301C2×C4×C4⋊C4—731no—
302#1445—732no—
303#1445—732no—
304#1441—731no—
305#1441—731no—
306#1441—731no—
307#1441D7311yescharacteristic
308#1436—732no—
309#1436—732no—
310C73#1428911yescharacteristic, complemented, Sylow
311C2×#125—738no—
312C2×#125—1464no—
313C2×#125—1464no—
314C22×C4⋊C4—731no—
315C22×C4⋊C4—731no—
316C22×C4⋊C4—731no—
317C22×C4⋊C4—731no—
318C22×C4⋊C4—731no—
319C22×C4⋊C4—731no—
320C22×C4⋊C4C2×D7311yescharacteristic
321C22×C4⋊C4C2×D7311yescharacteristic
322C22×C42—732no—
323C22×C42—731no—
324#383—1464no—
325C4×C4⋊C4—1464no—
326C2×#95—732no—
327C2×#95—732no—
328C2×#95—732no—
329C2×#95—732no—
330C2×#95—732no—
331C2×#95—732no—
332C2×#95—732no—
333C2×#95—731no—
334C2×#95—732no—
335C2×#95—732no—
336C2×#95C2×D7312yes—
337C2×#95C2×D7311yescharacteristic
338C2×#95C2×D7312yes—
339C23×C4—731no—
340C23×C4—731no—
341C23×C4—731no—
342C23×C4—731no—
343C23×C4—732no—
344C23×C4—732no—
345C23×C4C22×D7312yes—
346C23×C4C22×D7311yescharacteristic
347C23×C4C22×D7311yescharacteristic
348C23×C4C22×D7312yes—
349C23×C4C22×D7311yescharacteristic
350#125—14632no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—734no—
356C2×C4⋊C4—734no—
357C2×C4⋊C4—734no—
358C2×C4⋊C4—734no—
359C2×C4⋊C4—734no—
360C2×C4⋊C4—1462no—
361C2×C4⋊C4—1462no—
362C2×C4⋊C4—1462no—
363C2×C4⋊C4—1462no—
364C2×C4⋊C4—1462no—
365C2×C4⋊C4—1462no—
366C2×C4⋊C4—1462no—
367C2×C4⋊C4—1462no—
368C2×C4⋊C4—1464no—
369C2×C4⋊C4—1462no—
370C2×C4⋊C4—1462no—
371C2×C4⋊C4—1462no—
372C2×C4⋊C4C4×D7314yes—
373C2×C42—2922no—
374C2×C42—2922no—
375C2×C42—734no—
376C2×C42—734no—
377C2×C42—734no—
378C2×C42—1464no—
379#95—22no—
380#95—24no—
381#95—42no—
382#95—2922no—
383#95—2922no—
384#95—2922no—
385#95—1464no—
386#95—1464no—
387#95—1464no—
388#95—1464no—
389#95—1464no—
390#95—1462no—
391#95—1464no—
392C24C23×D7311yescharacteristic
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—732no—
419C22×C4—732no—
420C22×C4—732no—
421C22×C4—732no—
422C22×C4—732no—
423C22×C4—732no—
424C22×C4—732no—
425C22×C4—732no—
426C22×C4—734no—
427C22×C4—1462no—
428C22×C4—1462no—
429C22×C4—1462no—
430C22×C4—1462no—
431C22×C4—1462no—
432C22×C4—1462no—
433C22×C4—1462no—
434C22×C4—1462no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4—1461no—
438C22×C4—1461no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4—1461no—
443C22×C4—1462no—
444C22×C4—1461no—
445C22×C4—1462no—
446C22×C4—1461no—
447C22×C4—1461no—
448C22×C4—1462no—
449C22×C4—1462no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1462no—
453C22×C4—1461no—
454C22×C4—1462no—
455C22×C4—1461no—
456C22×C4C2×C4×D7312yes—
457C22×C4C2×C4×D7312yes—
458C22×C4C2×C4×D7312yes—
459C22×C4D4×D7312yes—
460C22×C4D4×D7312yes—
461C22×C4#1192402714yes—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—1468no—
465C4⋊C4—1468no—
466C4⋊C4—1468no—
467C4⋊C4—1468no—
468C4⋊C4—1468no—
469C4⋊C4—1468no—
470C42—2928no—
471C42—7316no—
472C23C22×C4×D7311yescharacteristic
473C23C2×D4×D7311yescharacteristic
474C23C2×D4×D7311yescharacteristic
475C23#41×D7312yes—
476C23C2×#1192402712yes—
477C23#1455663611yescharacteristic
478C23#1455663711yescharacteristic
479C23#1455663812yes—
480C23#1455663811yescharacteristic
481C23#1455663912yes—
482C23#1455663911yescharacteristic
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—2922no—
514C2×C4—2922no—
515C2×C4—2921no—
516C2×C4—2921no—
517C2×C4—2922no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—2922no—
521C2×C4—2922no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2922no—
525C2×C4—2922no—
526C2×C4—2922no—
527C2×C4—2922no—
528C2×C4—2922no—
529C2×C4—2922no—
530C2×C4—2922no—
531C2×C4—734no—
532C2×C4—734no—
533C2×C4—1464no—
534C2×C4—1464no—
535C2×C4—1462no—
536C2×C4—1462no—
537C2×C4—1464no—
538C2×C4—1462no—
539C2×C4—1462no—
540C2×C4—1464no—
541C2×C4—1462no—
542C2×C4—1462no—
543C2×C4—1464no—
544C2×C4—1462no—
545C2×C4—1462no—
546C2×C4—1462no—
547C2×C4—1462no—
548C2×C4—1464no—
549C2×C4—1462no—
550C2×C4—1462no—
551C2×C4C22⋊C4×D7314yes—
552C22C2×C22⋊C4×D7311yescharacteristic
553C22C4×#1192402712yes—
554C22#1798906312yes—
555C22#1798906711yescharacteristic
556C22#1798906811yescharacteristic
557C22#1798909311yescharacteristic
558C22#1798909511yescharacteristic
559C22#1798913212yes—
560C22#1798915311yescharacteristic
561C22#1798915711yescharacteristic
562C22#1798916011yescharacteristic
563C22#1798916012yes—
564C22#1798916211yescharacteristic
565C22#1798916611yescharacteristic
566C22#1798916711yescharacteristic
567C22#1798918112yes—
568C22#1798918212yes—
569C22#1798918312yes—
570C22#1798918512yes—
571C22#1798919412yes—
572C22#1798919612yes—
573C22#1798919712yes—
574C22#1798919812yes—
575C4—24no—
576C4—42no—
577C4—42no—
578C4—44no—
579C4—82no—
580C4—2922no—
581C4—2922no—
582C4—2924no—
583C4—5842no—
584C4—1464no—
585C4—1464no—
586C2#2261241611yescharacteristic
587C2#2261251411yescharacteristic
588C2#2261252011yescharacteristic
589C2#2261272912yes—
590C2#2261273612yes—
591C2#2261579811yescharacteristic
592C2#2261580111yescharacteristic
593C2#2261580611yescharacteristic
594C2#2261580911yescharacteristic
595C2#2261608912yes—
596C2#2261609112yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1441C2
socle, prime radical, cosocleC24×C73C24
hypercenter#1441D73
hypocenterC73#14289
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
14. C2×#14556581C4semidirect
310. C73#14289semidirect

Maximal subgroups

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

subgroupindex
C22×#145565812
C2×#179884542
C2×C4⋊C4×Dic732
#226081972
#226081852
#226081842
#226081812
#226081652
#226081622
C73×#14412
#1428973