Finite Group Databasebeta

← #32977873

Subgroups — #32977873

Order 18688.

706 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22610876C211yesmaximal, characteristic, complemented
2#22609761C211yesmaximal, characteristic, complemented
3#22608546C211yesmaximal, characteristic, complemented
4#22608519C211yesmaximal, characteristic
5#22608456C211yesmaximal, characteristic
6#22608378C211yesmaximal, characteristic, complemented
7C73×#3523C211yesmaximal, characteristic
8#17988644C2211yescharacteristic
9C2×#14556515—21no—
10C4×#11924012C2211yescharacteristic
11C2×#14556502C2211yescharacteristic
12C4×C4⋊Dic73C2211yescharacteristic
13#17988349—21no—
14#17988346—22no—
15#17988329—21no—
16#17988329—22no—
17#17988327—21no—
18#17988324—21no—
19#17988321—21no—
20Q8⋊#10596831—21no—
21Q8⋊#10596831—21no—
22#557×C73—21no—
23#557×C73C411yescharacteristic, complemented
24#557×C73C411yescharacteristic, complemented
25#556×C73—21no—
26#556×C73C411yescharacteristic, complemented
27#552×C73—22no—
28#552×C73—21no—
29D4×Q8×C73—21no—
30D4×Q8×C73C411yescharacteristic, complemented
31#541×C73—22no—
32#541×C73—21no—
33#533×C73—21no—
34#533×C73C2211yescharacteristic
35#529×C73—21no—
36#529×C73C2211yescharacteristic
37#519×C73—21no—
38C2×C4×Q8×C73C2211yescharacteristic
39C2×#11924012—21no—
40C22×#10596831—21no—
41C2×C4⋊Dic73—21no—
42C2×C4×Dic73—21no—
43C4⋊#10596831—21no—
44#14556515—24no—
45C22⋊#10596831—21no—
46#14556502—21no—
47#14556502—21no—
48#14556502—21no—
49#14556502—21no—
50C4×#10596831—21no—
51#14556498—21no—
52C22×Q8×C73—21no—
53C22×Q8×C73D411yescharacteristic
54C4⋊Q8×C73—22no—
55C4⋊Q8×C73—21no—
56C4⋊Q8×C73—21no—
57C4⋊Q8×C73—21no—
58C4⋊Q8×C73C2×C411yescharacteristic
59C4⋊Q8×C73C2×C411yescharacteristic
60#126×C73—22no—
61#126×C73—22no—
62#126×C73—22no—
63#126×C73—22no—
64#126×C73—21no—
65#125×C73—22no—
66#125×C73—22no—
67#125×C73—22no—
68#125×C73—21no—
69#125×C73—22no—
70#125×C73—22no—
71#125×C73—21no—
72#125×C73D411yescharacteristic
73#125×C73D412yes—
74#124×C73—21no—
75#124×C73D411yescharacteristic
76#123×C73—22no—
77#123×C73—21no—
78C22⋊Q8×C73—22no—
79C22⋊Q8×C73—22no—
80C22⋊Q8×C73—21no—
81C22⋊Q8×C73—21no—
82C22⋊Q8×C73—21no—
83C22⋊Q8×C73—21no—
84C22⋊Q8×C73—22no—
85C22⋊Q8×C73D411yescharacteristic
86#121×C73D411yescharacteristic
87C4×Q8×C73—22no—
88C4×Q8×C73—22no—
89C4×Q8×C73—21no—
90C4×Q8×C73—21no—
91C4×Q8×C73—21no—
92C4×Q8×C73—21no—
93C4×Q8×C73—21no—
94C4×Q8×C73—21no—
95C4×Q8×C73—21no—
96C4×Q8×C73—21no—
97C4×Q8×C73—21no—
98C4×Q8×C73C2×C411yescharacteristic
99C4×Q8×C73C2×C411yescharacteristic
100C4×Q8×C73C2×C411yescharacteristic
101C4×Q8×C73D411yescharacteristic
102C4×Q8×C73D411yescharacteristic
103C4×D4×C73—22no—
104C4×D4×C73—21no—
105C4×D4×C73—21no—
106C4×D4×C73C2×C411yescharacteristic
107#117×C73—22no—
108#117×C73—22no—
109#117×C73—22no—
110#117×C73—21no—
111#117×C73—21no—
112#117×C73D412yes—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73D411yescharacteristic
115C2×C42×C73—21no—
116C2×C42×C73C2311yescharacteristic
117C22×Dic73—41no—
118#11924012—22no—
119#11924012—41no—
120C2×#10596831—22no—
121C2×#10596831—41no—
122C2×#10596831—41no—
123C2×#10596831—41no—
124C2×#10596831—41no—
125C4⋊Dic73—22no—
126C4⋊Dic73—41no—
127C4×Dic73—42no—
128C2×Q8×C73—22no—
129C2×Q8×C73—21no—
130C2×Q8×C73—21no—
131C2×Q8×C73—21no—
132C2×Q8×C73—22no—
133C2×Q8×C73—41no—
134C2×Q8×C73—42no—
135C2×Q8×C73—42no—
136C2×Q8×C73C22⋊C411yescharacteristic
137C2×Q8×C73C22⋊C411yescharacteristic
138C2×Q8×C73C22⋊C411yescharacteristic
139C2×D4×C73—21no—
140C2×D4×C73C22⋊C411yescharacteristic
141C22×C4×C73—22no—
142C22×C4×C73—21no—
143C22×C4×C73—21no—
144C22×C4×C73—21no—
145C22×C4×C73C2×D411yescharacteristic
146C22×C4×C73C2×D411yescharacteristic
147C22×C4×C73C2×D411yescharacteristic
148C4⋊C4×C73—22no—
149C4⋊C4×C73—22no—
150C4⋊C4×C73—22no—
151C4⋊C4×C73—22no—
152C4⋊C4×C73—22no—
153C4⋊C4×C73—22no—
154C4⋊C4×C73—22no—
155C4⋊C4×C73—22no—
156C4⋊C4×C73—21no—
157C4⋊C4×C73—21no—
158C4⋊C4×C73—21no—
159C4⋊C4×C73—21no—
160C4⋊C4×C73—21no—
161C4⋊C4×C73—21no—
162C4⋊C4×C73—21no—
163C4⋊C4×C73—21no—
164C4⋊C4×C73—22no—
165C4⋊C4×C73—21no—
166C4⋊C4×C73—21no—
167C4⋊C4×C73—21no—
168C4⋊C4×C73—22no—
169C4⋊C4×C73C22⋊C411yescharacteristic
170C4⋊C4×C73C22⋊C411yescharacteristic
171C4⋊C4×C73C22⋊C412yes—
172C4⋊C4×C73C22⋊C412yes—
173C4⋊C4×C73C22⋊C411yescharacteristic
174C4⋊C4×C73C22⋊C411yescharacteristic
175C22⋊C4×C73—22no—
176C22⋊C4×C73—22no—
177C22⋊C4×C73—22no—
178C22⋊C4×C73—21no—
179C22⋊C4×C73—22no—
180C22⋊C4×C73—21no—
181C22⋊C4×C73—21no—
182C22⋊C4×C73—21no—
183C22⋊C4×C73—21no—
184C42×C73—22no—
185C42×C73—22no—
186C42×C73—22no—
187C42×C73—21no—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—21no—
191C42×C73—21no—
192C42×C73—22no—
193C42×C73C22×C411yescharacteristic
194C42×C73C2×D411yescharacteristic
195C42×C73C2×D411yescharacteristic
196C42×C73C2×D411yescharacteristic
197C23×C73—21no—
198C23×C73#12011yescharacteristic
199Q8×C73—21no—
200Q8×C73—21no—
201Q8×C73—21no—
202Q8×C73—42no—
203Q8×C73—41no—
204Q8×C73—42no—
205Q8×C73—44no—
206Q8×C73—41no—
207Q8×C73—41no—
208Q8×C73#10412yes—
209D4×C73—41no—
210D4×C73—41no—
211D4×C73—41no—
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—22no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—22no—
225C2×C4×C73—41no—
226C2×C4×C73—42no—
227C2×C4×C73—44no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73C2×C22⋊C411yescharacteristic
233C2×C4×C73C2×C22⋊C411yescharacteristic
234C2×C4×C73C2×C22⋊C411yescharacteristic
235C2×C4×C73#12011yescharacteristic
236C2×C4×C73#12011yescharacteristic
237C2×C4×C73#12011yescharacteristic
238C2×Dic73—42no—
239C2×Dic73—81no—
240C2×Dic73—81no—
241#10596831—42no—
242#10596831—42no—
243#10596831—81no—
244C22×C73—21no—
245C22×C73—21no—
246C22×C73—41no—
247C22×C73—41no—
248C22×C73—41no—
249C22×C73#37911yescharacteristic
250C22×C73#45411yescharacteristic
251C22×C73#45611yescharacteristic
252C4×C73—21no—
253C4×C73—21no—
254C4×C73—21no—
255C4×C73—21no—
256C4×C73—21no—
257C4×C73—22no—
258C4×C73—41no—
259C4×C73—41no—
260C4×C73—41no—
261C4×C73—41no—
262C4×C73—42no—
263C4×C73—41no—
264C4×C73—42no—
265C4×C73—41no—
266C4×C73C2×#10411yescharacteristic
267C4×C73#42111yescharacteristic
268Dic73—82no—
269#8417—731nomaximal, Sylow
270C2×C73—21no—
271C2×C73—21no—
272C2×C73—81no—
273C2×C73#178911yescharacteristic
274C2×C73#179311yescharacteristic
275C2×C73#179611yescharacteristic
276#3523D7311yescharacteristic
277#1986—731no—
278#1763—731no—
279#1531—731no—
280#1524—731no—
281#1509—731no—
282#1491—731no—
283C73#841711yescharacteristic, complemented, Sylow
284#557—21no—
285#557Dic7311yescharacteristic, complemented
286#557Dic7311yescharacteristic, complemented
287#556—21no—
288#556Dic7311yescharacteristic, complemented
289#552—22no—
290#552—21no—
291D4×Q8—21no—
292D4×Q8Dic7311yescharacteristic, complemented
293#541—22no—
294#541—21no—
295#533—21no—
296#533C2×D7311yescharacteristic
297#529—21no—
298#529C2×D7311yescharacteristic
299#519—21no—
300C2×C4×Q8C2×D7311yescharacteristic
301#432—731no—
302C2×#103—1461no—
303C4×M16—731no—
304C2×#97—731no—
305C4×C4⋊C4—731no—
306#340—1461no—
307#339—1462no—
308#330—1461no—
309#330—1462no—
310#329—1461no—
311#328—1461no—
312#327—1461no—
313Q8⋊C8—1461no—
314Q8⋊C8—1461no—
315C22×Q8—21no—
316C22×Q8C22⋊D7311yescharacteristic
317C2×M16—1461no—
318C22×C8—1461no—
319C4⋊Q8—22no—
320C4⋊Q8—21no—
321C4⋊Q8—21no—
322C4⋊Q8—21no—
323C4⋊Q8C2×Dic7311yescharacteristic
324C4⋊Q8C2×Dic7311yescharacteristic
325#126—22no—
326#126—22no—
327#126—22no—
328#126—22no—
329#126—21no—
330#125—22no—
331#125—22no—
332#125—22no—
333#125—21no—
334#125—22no—
335#125—22no—
336#125—21no—
337#125C22⋊D7311yescharacteristic
338#125C22⋊D7312yes—
339#124—21no—
340#124C22⋊D7311yescharacteristic
341#123—22no—
342#123—21no—
343C22⋊Q8—22no—
344C22⋊Q8—22no—
345C22⋊Q8—21no—
346C22⋊Q8—21no—
347C22⋊Q8—21no—
348C22⋊Q8—21no—
349C22⋊Q8—22no—
350C22⋊Q8C22⋊D7311yescharacteristic
351#121C22⋊D7311yescharacteristic
352C4×Q8—22no—
353C4×Q8—22no—
354C4×Q8—21no—
355C4×Q8—21no—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×Q8—21no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8—21no—
363C4×Q8C2×Dic7311yescharacteristic
364C4×Q8C2×Dic7311yescharacteristic
365C4×Q8C2×Dic7311yescharacteristic
366C4×Q8C22⋊D7311yescharacteristic
367C4×Q8C22⋊D7311yescharacteristic
368C4×D4—22no—
369C4×D4—21no—
370C4×D4—21no—
371C4×D4C2×Dic7311yescharacteristic
372#117—22no—
373#117—22no—
374#117—22no—
375#117—21no—
376#117—21no—
377#117C22⋊D7312yes—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4C22⋊D7311yescharacteristic
381C2×C42—21no—
382C2×C42—1461no—
383C2×C42C22×D7311yescharacteristic
384C4⋊C8—1461no—
385#103—1464no—
386C22⋊C8—1461no—
387#97—1461no—
388#97—1461no—
389#97—1461no—
390#97—1461no—
391C4×C8—1461no—
392#95—1461no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—41no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8C22⋊Dic7311yescharacteristic
402C2×Q8C22⋊Dic7311yescharacteristic
403C2×Q8C22⋊Dic7311yescharacteristic
404C2×D4—21no—
405C2×D4C22⋊Dic7311yescharacteristic
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—2921no—
411C22×C4C2×C22⋊D7311yescharacteristic
412C22×C4C2×C22⋊D7311yescharacteristic
413C22×C4C2×C22⋊D7311yescharacteristic
414M16—2921no—
415M16—1462no—
416C2×C8—2921no—
417C2×C8—2921no—
418C2×C8—2921no—
419C2×C8—2921no—
420C2×C8—1462no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—2921no—
443C4⋊C4—1462no—
444C4⋊C4C22⋊Dic7311yescharacteristic
445C4⋊C4C22⋊Dic7311yescharacteristic
446C4⋊C4C22⋊Dic7312yes—
447C4⋊C4C22⋊Dic7312yes—
448C4⋊C4C22⋊Dic7311yescharacteristic
449C4⋊C4C22⋊Dic7311yescharacteristic
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—21no—
454C22⋊C4—22no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—22no—
468C42—2922no—
469C42C22×Dic7311yescharacteristic
470C42C2×C22⋊D7311yescharacteristic
471C42C2×C22⋊D7311yescharacteristic
472C42C2×C22⋊D7311yescharacteristic
473C23—21no—
474C23#1455656011yescharacteristic
475Q8—21no—
476Q8—21no—
477Q8—21no—
478Q8—42no—
479Q8—41no—
480Q8—44no—
481Q8—42no—
482Q8—41no—
483Q8—41no—
484Q8#1455651812yes—
485D4—41no—
486D4—41no—
487D4—41no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—2922no—
509C2×C4—5841no—
510C2×C4—5841no—
511C2×C4C2×C22⋊Dic7311yescharacteristic
512C2×C4C2×C22⋊Dic7311yescharacteristic
513C2×C4C2×C22⋊Dic7311yescharacteristic
514C2×C4#1455656011yescharacteristic
515C2×C4#1455656011yescharacteristic
516C2×C4#1455656011yescharacteristic
517C8—2922no—
518C8—2922no—
519C8—5841no—
520C22—21no—
521C22—21no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22#1798844311yescharacteristic
526C22#1798876011yescharacteristic
527C22#1798877011yescharacteristic
528C4—21no—
529C4—22no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—41no—
535C4—42no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—42no—
541C4—41no—
542C4—5842no—
543C4C2×#1455651811yescharacteristic
544C4#1798860411yescharacteristic
545C2—21no—
546C2—21no—
547C2—81no—
548C2#2260987511yescharacteristic
549C2#2260989011yescharacteristic
550C2#2260990611yescharacteristic
subgroupG/H
centerC22#17988443
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#3523C2
socleC22×C73#379
prime radical#121×C73D4
cosocleC2×C42×C73C23
hypercenter#3523D73
hypocenterC73#8417
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3523

Complements

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

NH ≅ G/Ntype
1. #22610876C2semidirect
2. #22609761C2semidirect
3. #22608546C2semidirect
6. #22608378C2semidirect
23. #557×C73C4semidirect
24. #557×C73C4semidirect
26. #556×C73C4semidirect
30. D4×Q8×C73C4semidirect
283. C73#8417semidirect
285. #557Dic73semidirect
286. #557Dic73semidirect
288. #556Dic73semidirect
292. D4×Q8Dic73semidirect

Maximal subgroups

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

subgroupindex
#226108762
#226097612
#226085462
#226085192
#226084562
#226083782
C73×#35232
#841773