Finite Group Databasebeta

← #33158585

Subgroups — #33158585

Order 18688.

726 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22613775C211yesmaximal, characteristic
2#22613026C211yesmaximal, characteristic, complemented
3#22611303C211yesmaximal, characteristic
4#22611268C211yesmaximal, characteristic, complemented
5#22611262C211yesmaximal, characteristic, complemented
6#22611075C211yesmaximal, characteristic
7#22610881C211yesmaximal, characteristic, complemented
8#22610666C211yesmaximal, characteristic, complemented
9#22610561C211yesmaximal, characteristic
10#22610554C211yesmaximal, characteristic
11#22610518C211yesmaximal, characteristic, complemented
12#22610517C211yesmaximal, characteristic, complemented
13#22610493C211yesmaximal, characteristic, complemented
14#22609756C211yesmaximal, characteristic
15C73×#1864C211yesmaximal, characteristic
16C2×#14556590C2211yescharacteristic
17C2×#14556582C2211yescharacteristic
18C2×#14556570—21no—
19C2×#14556524C2211yescharacteristic
20C2×#14556523C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556502C2211yescharacteristic
26#17988510—21no—
27#17988499—21no—
28#17988490C2211yescharacteristic
29#17988490C2211yescharacteristic
30#17988486C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988483C2211yescharacteristic
33#17988482—21no—
34#17988472—21no—
35#17988469—21no—
36#17988460—21no—
37C4⋊C4×Dic73—21no—
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4×#11924007C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988341C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×C4×D4×C73C2211yescharacteristic
53C2×#106×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56#389×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×Dic146—21no—
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62#14556590—22no—
63#14556590—41no—
64#14556582—22no—
65#14556582—22no—
66#14556570—44no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007D411yescharacteristic
78C2×#11924007D411yescharacteristic
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73—21no—
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73C2311yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88#14556524—22no—
89#14556523—22no—
90#14556522—41no—
91#14556520—41no—
92#14556512—22no—
93#14556512Q814yescomplemented
94#14556502—41no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498D411yescharacteristic
102#14556498D411yescharacteristic
103C22×D4×C73C2311yescharacteristic
104C23×C4×C73—21no—
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C4×D4×C73—22no—
108C4×D4×C73—41no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C22⋊C4×C73—21no—
114C2×C42×C73C2311yescharacteristic
115#106×C73—22no—
116#102×C73—22no—
117#102×C73—41no—
118#95×C73—21no—
119C2×Dic146—42no—
120C2×Dic146—42no—
121C2×Dic146—81no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73C2×D411yescharacteristic
127C22×Dic73C2×D411yescharacteristic
128C22×Dic73C2×D411yescharacteristic
129C22×Dic73C2×D411yescharacteristic
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831C2×Q812yes—
136C4⋊Dic73—24no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—42no—
144C4⋊Dic73C2×Q812yes—
145C4⋊Dic73#4112yes—
146#11924007—24no—
147#11924007—22no—
148#11924007—41no—
149#11924007—42no—
150#11924007—42no—
151#11924007—81no—
152C22⋊Dic73—42no—
153C22⋊Dic73—42no—
154C22⋊Dic73—42no—
155C22⋊Dic73—81no—
156C4×Dic73—21no—
157C4×Dic73—21no—
158C4×Dic73—21no—
159C4×Dic73—21no—
160C4×Dic73—42no—
161C4×Dic73—81no—
162C4×Dic73—81no—
163C24×C73—21no—
164C2×D4×C73—21no—
165C2×D4×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73C2×Q812yes—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—42no—
174C22×C4×C73—42no—
175C22×C4×C73—81no—
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2411yescharacteristic
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—42no—
190C4⋊C4×C73#4112yes—
191C4⋊C4×C73#4112yes—
192C22⋊C4×C73—42no—
193C42×C73—21no—
194C42×C73—21no—
195C23×C73—22no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73C22×D411yescharacteristic
203D4×C73—22no—
204D4×C73—22no—
205D4×C73—41no—
206D4×C73—41no—
207C2×C4×C73—22no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—22no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—84no—
223C2×C4×C73C22×Q811yescharacteristic
224C2×C4×C73C2×#4111yescharacteristic
225C2×C4×C73C2×#4111yescharacteristic
226C2×C4×C73C2×#4111yescharacteristic
227C2×C4×C73H(5,𝔽2)11yescharacteristic
228C2×C4×C732−1+411yescharacteristic
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—22no—
234C2×Dic73—21no—
235C2×Dic73—22no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73AGL(1,ℤ/8)12yes—
253Dic146—84no—
254C8×C73—41no—
255C8×C73—41no—
256#10596831—22no—
257#10596831—41no—
258C22×C73—42no—
259C22×C73—42no—
260C22×C73—42no—
261C22×C73—81no—
262C22×C73—81no—
263C22×C73—81no—
264C22×C73—81no—
265C22×C73#52511yescharacteristic
266C22×C73#54611yescharacteristic
267C22×C73D4×Q811yescharacteristic
268C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C73#57511yescharacteristic
270C22×C73#57711yescharacteristic
271C22×C73#57711yescharacteristic
272C4×C73—42no—
273C4×C73—42no—
274C4×C73—41no—
275C4×C73—41no—
276C4×C73—81no—
277C4×C73#53311yescharacteristic
278C4×C73#54211yescharacteristic
279C4×C73#55411yescharacteristic
280C4×C73#55611yescharacteristic
281Dic73—22no—
282Dic73—41no—
283Dic73—41no—
284Dic73—42no—
285Dic73—42no—
286Dic73—41no—
287Dic73—81no—
288Dic73—81no—
289#27026—731nomaximal, Sylow
290C2×C73—82no—
291C2×C73#246511yescharacteristic
292C2×C73#311511yescharacteristic
293C2×C73#312511yescharacteristic
294C2×C73#331011yescharacteristic
295C2×C73#332611yescharacteristic
296C2×C73#338411yescharacteristic
297C2×C73#338811yescharacteristic
298#2452—731no—
299#2370—731no—
300#2065—731no—
301#2059—731no—
302#2058—731no—
303#2025—731no—
304#1988—731no—
305#1943—731no—
306#1926—731no—
307#1925—731no—
308#1920—731no—
309#1920—731no—
310#1917—731no—
311#1864D7311yescharacteristic
312#1762—731no—
313C73#2702611yescharacteristic, complemented, Sylow
314C2×C4⋊Q8—731no—
315C2×#125—731no—
316C2×C22⋊Q8—1461no—
317C2×C4×D4C2×D7311yescharacteristic
318C2×#107—731no—
319C2×#106—731no—
320C2×#106—731no—
321C2×#106C2×D7311yescharacteristic
322C2×C4⋊C8—731no—
323C2×#102—731no—
324C2×#102—731no—
325C2×#102C2×D7311yescharacteristic
326C2×#102C2×D7311yescharacteristic
327C2×#97—731no—
328#396—1461no—
329#393—1461no—
330#390—731no—
331#390—731no—
332#389—731no—
333#389—731no—
334#389—731no—
335#389—1461no—
336#389C2×D7311yescharacteristic
337#387—1461no—
338#387—1461no—
339#385—1461no—
340C4×C4⋊C4—731no—
341C4×C4⋊C4—731no—
342C4×C4⋊C4—731no—
343C4×C4⋊C4—731no—
344C4×C4⋊C4—1461no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#340C2×D7311yescharacteristic
353#336—731no—
354#336—731no—
355#336—731no—
356#336C2×D7311yescharacteristic
357C22×Q8—1461no—
358C22×D4C22×D7311yescharacteristic
359C23×C4—21no—
360C22×C8—731no—
361C22×C8—731no—
362C22×C8C22×D7311yescharacteristic
363C22×C8C22×D7311yescharacteristic
364C4⋊Q8—2921no—
365C4⋊Q8—1462no—
366#125—1462no—
367#125—1462no—
368C22⋊Q8—2924no—
369C4×D4—22no—
370C4×D4—41no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—1461no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4C22×D7311yescharacteristic
385C2×C4⋊C4C22×D7311yescharacteristic
386C2×C4⋊C4C22×D7311yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1461no—
389C2×C22⋊C4—1461no—
390C2×C22⋊C4—1461no—
391C2×C22⋊C4—1461no—
392C2×C42—731no—
393C2×C42—731no—
394C2×C42—1461no—
395C2×C42—1461no—
396C2×C42—1461no—
397C2×C42C22×D7311yescharacteristic
398#107—1462no—
399#106—22no—
400#106—2921no—
401#106—1462no—
402C4⋊C8—2921no—
403#102—22no—
404#102—41no—
405#102—734no—
406#102—1462no—
407#97—2921no—
408#95—21no—
409#95—731no—
410#95—731no—
411#95—1461no—
412#95—1461no—
413#95—1461no—
414#95—1461no—
415#95—1461no—
416#95—1461no—
417C24—21no—
418C2×Q8—2922no—
419C2×Q8—2922no—
420C2×Q8—5841no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4Q8×D7312yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—81no—
433C22×C4—731no—
434C22×C4—731no—
435C22×C4—731no—
436C22×C4—731no—
437C22×C4—1461no—
438C22×C4—1461no—
439C22×C4—1461no—
440C22×C4—1461no—
441C22×C4D4×D7311yescharacteristic
442C22×C4D4×D7311yescharacteristic
443C22×C4C23×D7311yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—732no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—2922no—
462C4⋊C4—2922no—
463C4⋊C4—2921no—
464C4⋊C4—2922no—
465C4⋊C4—5841no—
466C4⋊C4—732no—
467C4⋊C4—732no—
468C4⋊C4—1462no—
469C4⋊C4—1464no—
470C4⋊C4—1464no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4#1192402712yes—
478C4⋊C4#1192402812yes—
479C22⋊C4—42no—
480C22⋊C4—2922no—
481C22⋊C4—2922no—
482C22⋊C4—2922no—
483C22⋊C4—5841no—
484C42—21no—
485C42—21no—
486C42—2922no—
487C42—5841no—
488C42—5841no—
489C42—1461no—
490C42—1461no—
491C42—1461no—
492C42—1461no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23C2×D4×D7311yescharacteristic
501Q8—5844no—
502D4—22no—
503D4—22no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—84no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—732no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4—1461no—
539C2×C4—1462no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4—1462no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4C2×Q8×D7311yescharacteristic
547C2×C4#41×D7311yescharacteristic
548C2×C4C2×#1192402711yescharacteristic
549C2×C4C2×#1192402811yescharacteristic
550C2×C4#1455663611yescharacteristic
551C2×C4#1455663811yescharacteristic
552C8—41no—
553C8—41no—
554C8—2921no—
555C8—1462no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22#1798909611yescharacteristic
564C22#1798922411yescharacteristic
565C22#1798925011yescharacteristic
566C22AGL(1,ℤ/8)×D7311yescharacteristic
567C22#1798938311yescharacteristic
568C22#1798939311yescharacteristic
569C22#1798939411yescharacteristic
570C4—42no—
571C4—41no—
572C4—41no—
573C4—42no—
574C4—81no—
575C4—2922no—
576C4—2921no—
577C4—2922no—
578C4—2921no—
579C4—2921no—
580C4—5841no—
581C4—5841no—
582C4—1462no—
583C4#1798913911yescharacteristic
584C4#1798920511yescharacteristic
585C4#1798929211yescharacteristic
586C4#1798930711yescharacteristic
587C2—82no—
588C2#2261395611yescharacteristic
589C2#2262007811yescharacteristic
590C2#2262021411yescharacteristic
591C2#2262194511yescharacteristic
592C2#2262217711yescharacteristic
593C2#2262283311yescharacteristic
594C2#2262286911yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1864C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#1864D73
hypocenterC73#27026
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #22613026C2semidirect
4. #22611268C2semidirect
5. #22611262C2semidirect
7. #22610881C2semidirect
8. #22610666C2semidirect
11. #22610518C2semidirect
12. #22610517C2semidirect
13. #22610493C2semidirect
93. #14556512Q8semidirect
313. C73#27026semidirect

Maximal subgroups

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

subgroupindex
#226137752
#226130262
#226113032
#226112682
#226112622
#226110752
#226108812
#226106662
#226105612
#226105542
#226105182
#226105172
#226104932
#226097562
C73×#18642
#2702673