Finite Group Databasebeta

← #33012187

Subgroups — #33012187

Order 18688.

1470 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988494C211yesmaximal, characteristic
2C2×C4⋊C4×Dic73C211yesmaximal, characteristic
3#22608202C211yesmaximal, characteristic
4#22608196C211yesmaximal, characteristic
5#22608184C211yesmaximal, characteristic
6#22608165C212yesmaximal
7#22608165C212yesmaximal
8#22608163C212yesmaximal
9#22608160C212yesmaximal
10#95×Dic73C211yesmaximal, characteristic
11C2×#395×C73C211yesmaximal, characteristic
12C22×C4⋊Dic73C2212yes—
13C22×C4⋊Dic73C2211yescharacteristic
14C22×C4⋊Dic73C2211yescharacteristic
15C22×C4×Dic73C2211yescharacteristic
16C22×C4×Dic73C2212yes—
17C22×C4×Dic73C2211yescharacteristic
18C22×C4×Dic73C2211yescharacteristic
19#17988494—24no—
20C4⋊C4×Dic73—24no—
21C2×#14556498C2211yescharacteristic
22C2×#14556498C2211yescharacteristic
23C2×#14556498C2212yes—
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2211yescharacteristic
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C23×Q8×C73C2211yescharacteristic
33C22×C4⋊C4×C73C2211yescharacteristic
34C22×C4⋊C4×C73C2212yes—
35#395×C73C418yescomplemented
36C2×#95×C73C2211yescharacteristic
37C2×#95×C73C2212yes—
38C23×Dic73C2312yes—
39C23×Dic73C2312yes—
40C23×Dic73C2311yescharacteristic
41C23×Dic73C2311yescharacteristic
42C23×Dic73C2311yescharacteristic
43C23×Dic73C2311yescharacteristic
44C2×C4⋊Dic73—22no—
45C2×C4⋊Dic73—22no—
46C2×C4⋊Dic73—24no—
47C2×C4⋊Dic73—22no—
48C2×C4⋊Dic73—24no—
49C2×C4⋊Dic73—42no—
50C2×C4⋊Dic73—42no—
51C2×C4⋊Dic73Q814yescomplemented
52C2×C4×Dic73—22no—
53C2×C4×Dic73—22no—
54C2×C4×Dic73—22no—
55C2×C4×Dic73—24no—
56C2×C4×Dic73—22no—
57C2×C4×Dic73—24no—
58C2×C4×Dic73—42no—
59C2×C4×Dic73—42no—
60C2×C4×Dic73—42no—
61C2×C4×Dic73D414yes—
62#14556498—22no—
63#14556498—24no—
64#14556498—24no—
65#14556498—22no—
66#14556498—22no—
67#14556498—24no—
68#14556498—24no—
69#14556498—42no—
70#14556498—42no—
71#14556498—42no—
72#14556498Q818yescomplemented
73C22×Q8×C73—22no—
74C22×Q8×C73—22no—
75C22×Q8×C73—42no—
76C22×Q8×C73—42no—
77C22×Q8×C73C2×C414yes—
78C23×C4×C73C2312yes—
79C23×C4×C73C2311yescharacteristic
80C23×C4×C73C2311yescharacteristic
81C23×C4×C73C2311yescharacteristic
82C23×C4×C73C2312yes—
83C2×C4⋊C4×C73—22no—
84C2×C4⋊C4×C73—22no—
85C2×C4⋊C4×C73—42no—
86C2×C4⋊C4×C73—42no—
87C2×C4⋊C4×C73C2×C414yes—
88C2×C4⋊C4×C73C2×C418yes—
89#95×C73C2×C418yes—
90#95×C73C2×C414yes—
91C22×Dic73—21no—
92C22×Dic73—21no—
93C22×Dic73—22no—
94C22×Dic73—22no—
95C22×Dic73—22no—
96C22×Dic73—21no—
97C22×Dic73—21no—
98C22×Dic73—22no—
99C22×Dic73—22no—
100C22×Dic73—22no—
101C22×Dic73—22no—
102C22×Dic73—21no—
103C22×Dic73—22no—
104C22×Dic73—21no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73C2×D412yes—
118C22×Dic73C2×D412yes—
119C22×Dic73C2×Q814yes—
120C22×Dic73C2×Q812yes—
121C22×Dic73C2×Q812yes—
122C22×Dic73C2×Q814yes—
123C22×Dic73#4112yes—
124C22×Dic73#4112yes—
125C4⋊Dic73—28no—
126C4⋊Dic73—48no—
127C4⋊Dic73—44no—
128C4×Dic73—28no—
129C4×Dic73—44no—
130C4×Dic73—48no—
131C4×Dic73—44no—
132C24×C73C2411yescharacteristic
133C2×Q8×C73—28no—
134C2×Q8×C73—48no—
135C2×Q8×C73—44no—
136C2×Q8×C73—44no—
137C2×Q8×C73—48no—
138C22×C4×C73—21no—
139C22×C4×C73—21no—
140C22×C4×C73—21no—
141C22×C4×C73—22no—
142C22×C4×C73—21no—
143C22×C4×C73—22no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73—21no—
149C22×C4×C73—22no—
150C22×C4×C73—21no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73C22×C414yes—
159C22×C4×C73C22×C412yes—
160C22×C4×C73C22×C412yes—
161C22×C4×C73C22×C412yes—
162C22×C4×C73C22×C414yes—
163C22×C4×C73C2×D412yes—
164C22×C4×C73C2×Q814yes—
165C22×C4×C73C2×Q812yes—
166C22×C4×C73#4112yes—
167C22×C4×C73#4112yes—
168C22×C4×C73#4114yes—
169C4⋊C4×C73—28no—
170C4⋊C4×C73—48no—
171C23×C73C23×C411yescharacteristic
172C23×C73C22×D411yescharacteristic
173C23×C73C22×Q811yescharacteristic
174C23×C73C22×Q812yes—
175C23×C73C2×#4111yescharacteristic
176C23×C73C2×#4111yescharacteristic
177C23×C73C2×#4111yescharacteristic
178C23×C73C2×#4111yescharacteristic
179C23×C73C2×#4112yes—
180C23×C73H(5,𝔽2)12yes—
181C23×C73H(5,𝔽2)12yes—
182Q8×C73—416no—
183C2×C4×C73—22no—
184C2×C4×C73—24no—
185C2×C4×C73—22no—
186C2×C4×C73—24no—
187C2×C4×C73—24no—
188C2×C4×C73—22no—
189C2×C4×C73—22no—
190C2×C4×C73—22no—
191C2×C4×C73—22no—
192C2×C4×C73—22no—
193C2×C4×C73—22no—
194C2×C4×C73—24no—
195C2×C4×C73—24no—
196C2×C4×C73—24no—
197C2×C4×C73—24no—
198C2×C4×C73—24no—
199C2×C4×C73—22no—
200C2×C4×C73—24no—
201C2×C4×C73—24no—
202C2×C4×C73—24no—
203C2×C4×C73—24no—
204C2×C4×C73—42no—
205C2×C4×C73—41no—
206C2×C4×C73—41no—
207C2×C4×C73—42no—
208C2×C4×C73—42no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73C4×D414yes—
212C2×C4×C73C4×Q818yes—
213C2×C4×C73C4×Q814yes—
214C2×Dic73—22no—
215C2×Dic73—22no—
216C2×Dic73—22no—
217C2×Dic73—22no—
218C2×Dic73—24no—
219C2×Dic73—24no—
220C2×Dic73—24no—
221C2×Dic73—24no—
222C2×Dic73—22no—
223C2×Dic73—22no—
224C2×Dic73—24no—
225C2×Dic73—24no—
226C2×Dic73—24no—
227C2×Dic73—24no—
228C2×Dic73—24no—
229C2×Dic73—42no—
230C2×Dic73—41no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73—42no—
235C2×Dic73—42no—
236C2×Dic73—42no—
237C2×Dic73—42no—
238C2×Dic73—42no—
239C2×Dic73—42no—
240C2×Dic73—42no—
241C2×Dic73—42no—
242C2×Dic73—42no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73C22⋊Q814yes—
247C2×Dic73C4⋊Q814yes—
248C22×C73C2×C4×D411yescharacteristic
249C22×C73C2×C4×Q812yes—
250C22×C73C2×C4×Q811yescharacteristic
251C22×C73C4×#4111yescharacteristic
252C22×C73#51812yes—
253C22×C73C2×C22⋊Q811yescharacteristic
254C22×C73#52511yescharacteristic
255C22×C73#52911yescharacteristic
256C22×C73C2×C4⋊Q811yescharacteristic
257C22×C73#53211yescharacteristic
258C22×C73#53311yescharacteristic
259C22×C73#53312yes—
260C22×C73#53411yescharacteristic
261C22×C73#53511yescharacteristic
262C22×C73#53812yes—
263C22×C73#55112yes—
264C22×C73#55112yes—
265C22×C73#55412yes—
266C22×C73#55412yes—
267C22×C73#55412yes—
268C22×C73#55412yes—
269C22×C73#55512yes—
270C22×C73Q8212yes—
271C4×C73—24no—
272C4×C73—28no—
273C4×C73—24no—
274C4×C73—42no—
275C4×C73—44no—
276Dic73—24no—
277Dic73—24no—
278Dic73—42no—
279Dic73—44no—
280Dic73—42no—
281Dic73—82no—
282#14241—731nomaximal, Sylow
283C2×C73C4×C22⋊Q811yescharacteristic
284C2×C73C4×C4⋊Q811yescharacteristic
285C2×C73#232111yescharacteristic
286C2×C73#236812yes—
287C2×C73#236912yes—
288C2×C73#256411yescharacteristic
289C2×C73#256711yescharacteristic
290C2×C73#257412yes—
291C2×C73#258311yescharacteristic
292C2×C73#258612yes—
293C2×C73#261011yescharacteristic
294C2×#395D7311yescharacteristic
295C2×#391—731no—
296C2×C4×C4⋊C4—731no—
297#1446—731no—
298#1444—731no—
299#1441—731no—
300#1436—732no—
301#1436—732no—
302#1436—732no—
303C4⋊#95—732no—
304C4×#95—731no—
305C73#1424111yescharacteristic, complemented, Sylow
306C23×Q8C2×D7311yescharacteristic
307C22×C4⋊C4—731no—
308C22×C4⋊C4—732no—
309C22×C4⋊C4—731no—
310C22×C4⋊C4C2×D7311yescharacteristic
311C22×C4⋊C4C2×D7312yes—
312C22×C42—731no—
313C22×C42—731no—
314C22×C42—732no—
315C22×C42—731no—
316#395Dic7318yescomplemented
317#391—1464no—
318C4×C4⋊C4—1464no—
319C2×#95—732no—
320C2×#95—732no—
321C2×#95—732no—
322C2×#95—732no—
323C2×#95—731no—
324C2×#95—731no—
325C2×#95—732no—
326C2×#95—732no—
327C2×#95—732no—
328C2×#95—732no—
329C2×#95—731no—
330C2×#95C2×D7311yescharacteristic
331C2×#95C2×D7312yes—
332C22×Q8—22no—
333C22×Q8—22no—
334C22×Q8—42no—
335C22×Q8—42no—
336C22×Q8C2×Dic7314yes—
337C23×C4—731no—
338C23×C4—731no—
339C23×C4—731no—
340C23×C4—731no—
341C23×C4—732no—
342C23×C4—732no—
343C23×C4C22×D7311yescharacteristic
344C23×C4C22×D7312yes—
345C23×C4C22×D7311yescharacteristic
346C23×C4C22×D7311yescharacteristic
347C23×C4C22×D7312yes—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—2922no—
353C2×C4⋊C4—2922no—
354C2×C4⋊C4—734no—
355C2×C4⋊C4—1464no—
356C2×C4⋊C4—1462no—
357C2×C4⋊C4—1464no—
358C2×C4⋊C4—1462no—
359C2×C4⋊C4—1462no—
360C2×C4⋊C4C2×Dic7314yes—
361C2×C4⋊C4C2×Dic7318yes—
362C2×C42—2922no—
363C2×C42—2922no—
364C2×C42—2922no—
365C2×C42—734no—
366C2×C42—1462no—
367C2×C42—1464no—
368C2×C42—1464no—
369C2×C42—1462no—
370C2×C42—1462no—
371C2×C42—1462no—
372#95—2922no—
373#95—2922no—
374#95—2922no—
375#95—738no—
376#95—1462no—
377#95—1464no—
378#95—1464no—
379#95—1462no—
380#95—1462no—
381#95—1464no—
382#95—1464no—
383#95C2×Dic7318yes—
384#95C2×Dic7314yes—
385C24C23×D7311yescharacteristic
386C2×Q8—28no—
387C2×Q8—48no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—48no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—732no—
412C22×C4—732no—
413C22×C4—734no—
414C22×C4—732no—
415C22×C4—732no—
416C22×C4—734no—
417C22×C4—732no—
418C22×C4—732no—
419C22×C4—1462no—
420C22×C4—1462no—
421C22×C4—1462no—
422C22×C4—1462no—
423C22×C4—1462no—
424C22×C4—1462no—
425C22×C4—1462no—
426C22×C4—1461no—
427C22×C4—1461no—
428C22×C4—1462no—
429C22×C4—1461no—
430C22×C4—1462no—
431C22×C4—1462no—
432C22×C4—1462no—
433C22×C4—1462no—
434C22×C4—1461no—
435C22×C4—1462no—
436C22×C4—1462no—
437C22×C4—1462no—
438C22×C4—1462no—
439C22×C4—1462no—
440C22×C4—1461no—
441C22×C4—1462no—
442C22×C4—1461no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4C22×Dic7314yes—
446C22×C4C22×Dic7312yes—
447C22×C4C22×Dic7312yes—
448C22×C4C22×Dic7312yes—
449C22×C4C22×Dic7314yes—
450C22×C4D4×D7312yes—
451C22×C4Q8×D7314yes—
452C22×C4Q8×D7312yes—
453C22×C4#1192402712yes—
454C22×C4#1192402912yes—
455C22×C4#1192402914yes—
456C4⋊C4—28no—
457C4⋊C4—48no—
458C4⋊C4—2924no—
459C4⋊C4—2928no—
460C4⋊C4—1468no—
461C42—2928no—
462C42—2924no—
463C42—2924no—
464C42—1468no—
465C23C23×Dic7311yescharacteristic
466C23C2×D4×D7311yescharacteristic
467C23C2×Q8×D7312yes—
468C23C2×Q8×D7311yescharacteristic
469C23#41×D7311yescharacteristic
470C23#41×D7311yescharacteristic
471C23C2×#1192402711yescharacteristic
472C23C2×#1192402911yescharacteristic
473C23C2×#1192402912yes—
474C23#1455663612yes—
475C23#1455663612yes—
476Q8—416no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—2922no—
506C2×C4—2921no—
507C2×C4—2921no—
508C2×C4—2921no—
509C2×C4—2921no—
510C2×C4—2922no—
511C2×C4—2922no—
512C2×C4—2922no—
513C2×C4—2922no—
514C2×C4—2922no—
515C2×C4—2922no—
516C2×C4—2922no—
517C2×C4—2922no—
518C2×C4—2922no—
519C2×C4—2922no—
520C2×C4—2922no—
521C2×C4—2922no—
522C2×C4—734no—
523C2×C4—734no—
524C2×C4—1464no—
525C2×C4—1464no—
526C2×C4—1464no—
527C2×C4—1464no—
528C2×C4—1464no—
529C2×C4—1462no—
530C2×C4—1462no—
531C2×C4—1462no—
532C2×C4—1462no—
533C2×C4—1462no—
534C2×C4—1464no—
535C2×C4—1464no—
536C2×C4—1464no—
537C2×C4—1464no—
538C2×C4—1462no—
539C2×C4D4×Dic7314yes—
540C2×C4Q8×Dic7318yes—
541C2×C4Q8×Dic7314yes—
542C22C2×D4×Dic7311yescharacteristic
543C22C2×Q8×Dic7312yes—
544C22C2×Q8×Dic7311yescharacteristic
545C22#41×Dic7311yescharacteristic
546C22#1798906212yes—
547C22C22⋊Q8×D7311yescharacteristic
548C22#1798910011yescharacteristic
549C22#1798912311yescharacteristic
550C22C4⋊Q8×D7311yescharacteristic
551C22#1798913811yescharacteristic
552C22#1798913911yescharacteristic
553C22#1798914212yes—
554C22#1798914711yescharacteristic
555C22#1798915211yescharacteristic
556C22#1798917212yes—
557C22#1798926412yes—
558C22#1798926412yes—
559C22#1798929112yes—
560C22#1798929112yes—
561C22#1798929812yes—
562C22#1798929812yes—
563C22#1798930312yes—
564C22#1798931712yes—
565C4—28no—
566C4—24no—
567C4—24no—
568C4—44no—
569C4—42no—
570C4—2922no—
571C4—2922no—
572C4—2924no—
573C4—5842no—
574C4—1464no—
575C4—1464no—
576C2C22⋊Q8×Dic7311yescharacteristic
577C2C4⋊Q8×Dic7311yescharacteristic
578C2#2261259211yescharacteristic
579C2#2261301612yes—
580C2#2261302112yes—
581C2#2261511211yescharacteristic
582C2#2261513511yescharacteristic
583C2#2261520012yes—
584C2#2261530711yescharacteristic
585C2#2261533812yes—
586C2#2261561311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×#395×C73C2
socle, prime radical, cosocleC24×C73C24
hypercenterC2×#395D73
hypocenterC73#14241
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
35. #395×C73C4semidirect
51. C2×C4⋊Dic73Q8semidirect
72. #14556498Q8semidirect
305. C73#14241semidirect
316. #395Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179884942
C2×C4⋊C4×Dic732
#226082022
#226081962
#226081842
#226081652
#226081632
#226081602
#95×Dic732
C2×#395×C732
#1424173