Finite Group Databasebeta

← #33128521

Subgroups — #33128521

Order 18688.

858 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988862C211yesmaximal, characteristic, complemented
2#22619563C212yesmaximal
3C2×#17988828C211yesmaximal, characteristic, complemented
4C22×#14556527C211yesmaximal, characteristic, complemented
5#22611297C212yesmaximal, complemented
6#22611229C212yesmaximal, complemented
7#22610846C211yesmaximal, characteristic, complemented
8#22610716C211yesmaximal, characteristic, complemented
9#22610397C212yesmaximal, complemented
10#22609995C211yesmaximal, characteristic, complemented
11C73×#2929C211yesmaximal, characteristic
12C2×#14556621C2212yes—
13C22×Dic292C2211yescharacteristic
14C22×#11924012C2212yes—
15C2×#14556570C2212yes—
16#17988862—24no—
17#17988843—22no—
18#17988831—22no—
19#17988828—21no—
20#17988828—21no—
21#17988828—21no—
22#17988828—21no—
23#17988826—22no—
24#17988814—22no—
25C2×#14556527—24no—
26C2×#14556527C2212yes—
27C2×#14556527C2211yescharacteristic
28C2×#14556527C2211yescharacteristic
29#17988630C2212yes—
30C2×#14556525C2211yescharacteristic
31#17988595C2212yes—
32#17988588C2212yes—
33C2×#14556516C2212yes—
34C2×#14556509C2212yes—
35C2×#14556509C2212yes—
36#17988355C2212yes—
37#17988355C2212yes—
38C2×#131×C73C2211yescharacteristic
39C23×C8×C73C2211yescharacteristic
40#525×C73C2211yescharacteristic
41#435×C73—21no—
42#435×C73—21no—
43C8×D4×C73—21no—
44C8×D4×C73—21no—
45#424×C73C2211yescharacteristic
46#408×C73C2211yescharacteristic
47#407×C73C2211yescharacteristic
48#405×C73C2211yescharacteristic
49C22×Dic146C2312yes—
50#14556621—24no—
51#14556621—42no—
52C2×Dic292—21no—
53C2×Dic292—21no—
54C2×Dic292—22no—
55C2×Dic292—22no—
56C2×#11924018—22no—
57C2×#11924016—22no—
58C2×#11924012—24no—
59C2×#11924012—22no—
60C2×#11924012—42no—
61C2×#11924012C2312yes—
62C2×#11924012C2312yes—
63C22×#10596831—22no—
64#14556570—22no—
65#14556570—22no—
66#14556570—22no—
67#14556570—22no—
68C2×C4⋊Dic73C2312yes—
69C2×#11924007—22no—
70C2×C22⋊Dic73—22no—
71#14556527—24no—
72#14556527—22no—
73#14556527D414yescomplemented
74#14556525—21no—
75#14556525—21no—
76#14556524—22no—
77#14556522—22no—
78#14556516—22no—
79#14556516—22no—
80#14556515—22no—
81#14556514—22no—
82#14556514—22no—
83#14556512—22no—
84#14556509—24no—
85#14556509—42no—
86C2×#41×C73C2311yescharacteristic
87C23×C4×C73C2311yescharacteristic
88#131×C73—22no—
89#131×C73—22no—
90C2×M16×C73—21no—
91C2×M16×C73C2311yescharacteristic
92C2×M16×C73C2311yescharacteristic
93C22×C8×C73—22no—
94C22×C8×C73—22no—
95C22×C8×C73—21no—
96C22×C8×C73D411yescharacteristic
97C22×C8×C73D411yescharacteristic
98C22×C8×C73D411yescharacteristic
99C22×C8×C73D411yescharacteristic
100C22×C8×C73C2311yescharacteristic
101C22×C8×C73C2311yescharacteristic
102#123×C73—21no—
103C22⋊Q8×C73D411yescharacteristic
104C22⋊Q8×C73D411yescharacteristic
105#121×C73D411yescharacteristic
106#121×C73D411yescharacteristic
107#120×C73—21no—
108C4×D4×C73—21no—
109C4×D4×C73—21no—
110#117×C73C2311yescharacteristic
111C4⋊C8×C73—21no—
112C4⋊C8×C73—21no—
113C22⋊C8×C73—21no—
114C22⋊C8×C73—21no—
115C22⋊C8×C73—21no—
116C22⋊C8×C73—21no—
117#97×C73—21no—
118C4×C8×C73—21no—
119C2×Dic146—22no—
120C2×Dic146—22no—
121C2×Dic146—22no—
122C2×Dic146—22no—
123C2×Dic146—22no—
124C2×Dic146—22no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127Dic292—42no—
128Dic292—42no—
129#11924018—44no—
130#11924016—44no—
131#11924012—22no—
132#11924012—22no—
133#11924012—22no—
134#11924012—44no—
135#11924012C2×D414yes—
136C2×#10596831—22no—
137C2×#10596831—22no—
138C2×#10596831—44no—
139C4⋊Dic73—22no—
140C4⋊Dic73—22no—
141#11924007—42no—
142#11924007—42no—
143C22⋊Dic73—42no—
144C22⋊Dic73—42no—
145C24×C73#4111yescharacteristic
146#41×C73—21no—
147#41×C73—21no—
148#41×C73—21no—
149#41×C73—21no—
150C2×Q8×C73C2×D411yescharacteristic
151C2×D4×C73—21no—
152C2×D4×C73—21no—
153C2×D4×C73—21no—
154C2×D4×C73C2×D411yescharacteristic
155C22×C4×C73—21no—
156C22×C4×C73—21no—
157C22×C4×C73—21no—
158C22×C4×C73—21no—
159C22×C4×C73—21no—
160C22×C4×C73C2×D411yescharacteristic
161C22×C4×C73C2×D411yescharacteristic
162C22×C4×C73C2×D411yescharacteristic
163C22×C4×C73C2×D411yescharacteristic
164C22×C4×C73#4111yescharacteristic
165C22×C4×C73C2411yescharacteristic
166M16×C73—21no—
167M16×C73—21no—
168M16×C73—21no—
169M16×C73—21no—
170M16×C73—42no—
171C2×C8×C73—21no—
172C2×C8×C73—21no—
173C2×C8×C73—22no—
174C2×C8×C73—22no—
175C2×C8×C73—22no—
176C2×C8×C73—22no—
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—42no—
180C2×C8×C73C2×D412yes—
181C2×C8×C73C2×D411yescharacteristic
182C2×C8×C73C2×D411yescharacteristic
183C2×C8×C73C2×D411yescharacteristic
184C2×C8×C73C2×D411yescharacteristic
185C2×C8×C73#4112yes—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73C2×D411yescharacteristic
189C4⋊C4×C73C2×D411yescharacteristic
190C22⋊C4×C73—21no—
191C22⋊C4×C73—21no—
192C22⋊C4×C73—21no—
193C22⋊C4×C73—21no—
194C22⋊C4×C73—21no—
195C42×C73—21no—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73—22no—
199C23×C73—21no—
200C23×C73—41no—
201C23×C73#12111yescharacteristic
202C23×C73#12111yescharacteristic
203C23×C73C2×#4111yescharacteristic
204Q8×C73—21no—
205Q8×C73—21no—
206D4×C73—21no—
207D4×C73—21no—
208D4×C73—42no—
209D4×C73—41no—
210D4×C73—41no—
211D4×C73—42no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73#12111yescharacteristic
228C2×C4×C73#12111yescharacteristic
229C2×C4×C73C22×D411yescharacteristic
230C2×C4×C73C22×D411yescharacteristic
231C2×C4×C73C22×D411yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73H(5,𝔽2)11yescharacteristic
234C2×C4×C732−1+411yescharacteristic
235C2×Dic73—42no—
236C2×Dic73—42no—
237C2×Dic73—42no—
238C2×Dic73—42no—
239C2×Dic73—82no—
240Dic146—42no—
241Dic146—42no—
242Dic146—42no—
243Dic146—42no—
244C8×C73—21no—
245C8×C73—21no—
246C8×C73—41no—
247C8×C73—41no—
248C8×C73#12112yes—
249C8×C73#12112yes—
250#10596831—24no—
251#10596831—42no—
252C22×C73—21no—
253C22×C73—21no—
254C22×C73—21no—
255C22×C73—21no—
256C22×C73—21no—
257C22×C73—21no—
258C22×C73—21no—
259C22×C73—42no—
260C22×C73—42no—
261C22×C73—41no—
262C22×C73—41no—
263C22×C73—41no—
264C22×C73#47312yescomplemented
265C22×C73C2×#12111yescharacteristic
266C22×C73#54811yescharacteristic
267C22×C73#55311yescharacteristic
268C4×C73—21no—
269C4×C73—21no—
270C4×C73—41no—
271C4×C73—41no—
272C4×C73—41no—
273C4×C73C2×#12111yescharacteristic
274C4×C73#53711yescharacteristic
275C4×C73D4211yescharacteristic
276C4×C73#54711yescharacteristic
277Dic73—82no—
278Dic73—82no—
279#24804—731nomaximal, Sylow
280C2×C73—21no—
281C2×C73—21no—
282C2×C73—22no—
283C2×C73—41no—
284C2×C73—81no—
285C2×C73#241611yescharacteristic
286C2×C73C2×#47311yescharacteristic
287C2×C73#306711yescharacteristic
288C2×#473—731no—
289#3056—732no—
290C2×#467—731no—
291#2929D7311yescharacteristic
292C22×#108—731no—
293#2064—732no—
294#2052—732no—
295#1979—731no—
296#1952—731no—
297#1901—732no—
298#1816—731no—
299C73#2480411yescharacteristic, complemented, Sylow
300C2×#137—732no—
301C22×Q16—731no—
302C2×#131C2×D7311yescharacteristic
303C22×M16—732no—
304C23×C8C2×D7311yescharacteristic
305#525C2×D7311yescharacteristic
306C2×C22⋊Q8—732no—
307#473—1464no—
308#470—1462no—
309#468—1462no—
310#467—1461no—
311#467—1461no—
312#467—1461no—
313#467—1461no—
314#467—1462no—
315#465—1462no—
316#435—21no—
317#435—21no—
318C8×D4—21no—
319C8×D4—21no—
320C2×#108—731no—
321C2×#108—732no—
322C2×#108—731no—
323C2×#108—1464no—
324#428—732no—
325C2×#107—731no—
326#424C2×D7311yescharacteristic
327#419—732no—
328#417—732no—
329C2×#103—732no—
330C2×#101—732no—
331C2×#101—732no—
332#408C2×D7311yescharacteristic
333#407C2×D7311yescharacteristic
334#405C2×D7311yescharacteristic
335#343—732no—
336#343—732no—
337C2×#41C22×D7311yescharacteristic
338C22×Q8—732no—
339C23×C4C22×D7311yescharacteristic
340#137—2922no—
341#137—1464no—
342C2×Q16—1462no—
343C2×Q16—1462no—
344C2×Q16—1461no—
345C2×Q16—1461no—
346C2×Q16—1462no—
347C2×SD16—1462no—
348#131—22no—
349#131—22no—
350C2×M16—21no—
351C2×M16—2922no—
352C2×M16—732no—
353C2×M16—732no—
354C2×M16—1464no—
355C2×M16—1462no—
356C2×M16C22×D7311yescharacteristic
357C2×M16C22×D7311yescharacteristic
358C22×C8—22no—
359C22×C8—22no—
360C22×C8—21no—
361C22×C8—1462no—
362C22×C8C22⋊D7311yescharacteristic
363C22×C8C22⋊D7311yescharacteristic
364C22×C8C22⋊D7311yescharacteristic
365C22×C8C22⋊D7311yescharacteristic
366C22×C8C22×D7311yescharacteristic
367C22×C8C22×D7311yescharacteristic
368#123—21no—
369C22⋊Q8—1462no—
370C22⋊Q8—1462no—
371C22⋊Q8—1462no—
372C22⋊Q8—1462no—
373C22⋊Q8D29211yescharacteristic
374C22⋊Q8D29211yescharacteristic
375#121D29211yescharacteristic
376#121D29211yescharacteristic
377#120—21no—
378C4×D4—21no—
379C4×D4—21no—
380#117C22×D7311yescharacteristic
381C2×C4⋊C4—732no—
382C2×C4⋊C4—1462no—
383C2×C22⋊C4—1462no—
384#108—734no—
385#108—1464no—
386#108—1462no—
387#107—1461no—
388#107—1461no—
389#107—1462no—
390#106—1462no—
391C4⋊C8—21no—
392C4⋊C8—21no—
393#103—1462no—
394#103—1462no—
395#103—1462no—
396#103—1462no—
397#103—1462no—
398#102—1462no—
399#101—2922no—
400#101—1464no—
401C22⋊C8—21no—
402C22⋊C8—21no—
403C22⋊C8—21no—
404C22⋊C8—21no—
405#97—21no—
406C4×C8—21no—
407C24#1192402811yescharacteristic
408#41—21no—
409#41—21no—
410#41—21no—
411#41—21no—
412C2×Q8—1462no—
413C2×Q8—1462no—
414C2×Q8—1462no—
415C2×Q8—1462no—
416C2×Q8—1462no—
417C2×Q8—1462no—
418C2×Q8C2×D29211yescharacteristic
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×D29211yescharacteristic
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—1462no—
429C22×C4—1462no—
430C22×C4C2×C22⋊D7311yescharacteristic
431C22×C4C2×C22⋊D7311yescharacteristic
432C22×C4C2×D29211yescharacteristic
433C22×C4C2×D29211yescharacteristic
434C22×C4#1192402811yescharacteristic
435C22×C4C23×D7311yescharacteristic
436Q16—2924no—
437Q16—2922no—
438Q16—2922no—
439SD16—2924no—
440M16—21no—
441M16—21no—
442M16—21no—
443M16—21no—
444M16—42no—
445M16—2924no—
446M16—734no—
447M16—1462no—
448M16—1462no—
449M16—1462no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—42no—
459C2×C8—2924no—
460C2×C8—1462no—
461C2×C8—1462no—
462C2×C8D4×D7312yes—
463C2×C8C2×C22⋊D7311yescharacteristic
464C2×C8C2×C22⋊D7311yescharacteristic
465C2×C8C2×C22⋊D7311yescharacteristic
466C2×C8C2×C22⋊D7311yescharacteristic
467C2×C8#1192402712yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—2922no—
471C4⋊C4—2922no—
472C4⋊C4—1462no—
473C4⋊C4—1462no—
474C4⋊C4C2×D29211yescharacteristic
475C4⋊C4C2×D29211yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—2922no—
482C22⋊C4—2922no—
483C42—21no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—21no—
488C23—41no—
489C23#1455656511yescharacteristic
490C23#1455656511yescharacteristic
491C23C2×#1192402811yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—2922no—
495Q8—2922no—
496Q8—2922no—
497Q8—2922no—
498D4—21no—
499D4—21no—
500D4—42no—
501D4—41no—
502D4—41no—
503D4—42no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—2922no—
520C2×C4—2922no—
521C2×C4—2922no—
522C2×C4—2922no—
523C2×C4—5842no—
524C2×C4#1455656511yescharacteristic
525C2×C4#1455656511yescharacteristic
526C2×C4C2×D4×D7311yescharacteristic
527C2×C4C22×C22⋊D7311yescharacteristic
528C2×C4C22×D29211yescharacteristic
529C2×C4C2×#1192402711yescharacteristic
530C2×C4#1455663711yescharacteristic
531C2×C4#1455663811yescharacteristic
532C8—21no—
533C8—21no—
534C8—41no—
535C8—41no—
536C8—2922no—
537C8—1464no—
538C8#1455656312yes—
539C8#1455656312yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—42no—
548C22—42no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22#1798886212yescomplemented
553C22C2×#1455656511yescharacteristic
554C22#1798924611yescharacteristic
555C22#1798928311yescharacteristic
556C4—21no—
557C4—21no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—5842no—
562C4—5842no—
563C4C2×#1455656311yescharacteristic
564C4#1798916511yescharacteristic
565C4D4×D29211yescharacteristic
566C4#1798924111yescharacteristic
567C2—21no—
568C2—21no—
569C2—22no—
570C2—41no—
571C2—81no—
572C2#2261337111yescharacteristic
573C2C2×#1798886211yescharacteristic
574C2#2261965111yescharacteristic
subgroupG/H
centerC22C2×#14556565
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2929C2
socleC22×C73C2×#121
prime radical#525×C73C22
hypercenter#2929D73
hypocenterC73#24804
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. C2×#17988862C2semidirect
3. C2×#17988828C2semidirect
4. C22×#14556527C2semidirect
5. #22611297C2semidirect
6. #22611229C2semidirect
7. #22610846C2semidirect
8. #22610716C2semidirect
9. #22610397C2semidirect
10. #22609995C2semidirect
73. #14556527D4semidirect
264. C22×C73#473semidirect
299. C73#24804semidirect
552. C22#17988862semidirect

Maximal subgroups

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

subgroupindex
C2×#179888622
#226195632
C2×#179888282
C22×#145565272
#226112972
#226112292
#226108462
#226107162
#226103972
#226099952
C73×#29292
#2480473