Finite Group Databasebeta

← #33085779

Subgroups — #33085779

Order 18688.

1774 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D73C211yesmaximal, characteristic
2C2×#17988549C211yesmaximal, characteristic
3#22612313C211yesmaximal, characteristic
4#22610020C214yesmaximal, complemented
5#22610018C211yesmaximal, characteristic, complemented
6#22610017C211yesmaximal, characteristic, complemented
7#22610015C211yesmaximal, characteristic, complemented
8#22609649C214yesmaximal, complemented
9C73×#1823C211yesmaximal, characteristic, complemented
10C23×C4×D73—21no—
11C23×C4×D73C2211yescharacteristic
12C22×#11924011C2212yes—
13C2×#14556549C2212yes—
14C22×C4⋊Dic73C2211yescharacteristic
15C22×#11924007—21no—
16C2×C4⋊C4×D73—21no—
17C2×C4⋊C4×D73—21no—
18C2×C4⋊C4×D73—24no—
19C2×C4⋊C4×D73C418yescomplemented
20C2×C4⋊C4×D73C2212yes—
21C2×C4⋊C4×D73C2212yes—
22#17988549—24no—
23C2×C22⋊#10596831C2211yescharacteristic
24C2×#14556503C2212yes—
25#17988449—21no—
26#17988447C2211yescharacteristic
27#17988444C2212yes—
28#17988442—21no—
29C22⋊C4×Dic73—21no—
30#17988434—21no—
31#17988350C2214yescomplemented
32#17988349C2214yescomplemented
33#17988349C2214yescomplemented
34C22×C4⋊C4×C73C2211yescharacteristic
35C2×C22⋊C8×C73C2211yescharacteristic
36#380×C73C2211yescharacteristic
37#340×C73C2214yescomplemented
38C24×D73D411yescharacteristic
39C23×Dic73—21no—
40C23×Dic73D411yescharacteristic
41C22×C4×D73—28no—
42C22×C4×D73—28no—
43C22×C4×D73—22no—
44C22×C4×D73—22no—
45C22×C4×D73—44no—
46C22×C4×D73—44no—
47C22×C4×D73C2×C414yes—
48C22×C4×D73D411yescharacteristic
49C22×C4×D73D411yescharacteristic
50C22×C4×D73D414yes—
51C22×C4×D73C2311yescharacteristic
52C22×C4×D73C2311yescharacteristic
53C2×#11924011—24no—
54C2×#11924011—28no—
55C22×#10596831C2312yes—
56#14556549—28no—
57C2×C4⋊Dic73—21no—
58C2×C4⋊Dic73—22no—
59C2×C4⋊Dic73C2×C414yes—
60C2×C4⋊Dic73C2312yes—
61C2×C4⋊Dic73C2312yes—
62C2×#11924007—28no—
63C2×#11924007—22no—
64C2×#11924007—22no—
65C4⋊C4×D73—28no—
66C4⋊C4×D73—28no—
67C4⋊C4×D73—24no—
68C4⋊C4×D73—24no—
69C4⋊C4×D73C2×C418yes—
70C4⋊C4×D73C2×C418yes—
71C2×C22⋊Dic73—21no—
72C2×#11924005—22no—
73C2×#11924005—22no—
74C2×C4×Dic73—22no—
75C22⋊#10596831—22no—
76#14556503—24no—
77#14556498—22no—
78#14556498—21no—
79C23×C4×C73—21no—
80C23×C4×C73C2311yescharacteristic
81C22×C8×C73C2312yes—
82C2×C4⋊C4×C73—21no—
83C2×C4⋊C4×C73—22no—
84C2×C4⋊C4×C73C2×C414yes—
85C2×C4⋊C4×C73C2312yes—
86C2×C4⋊C4×C73C2312yes—
87C2×C22⋊C4×C73—21no—
88C22⋊C8×C73—22no—
89#95×C73—21no—
90C23×D73—22no—
91C23×D73—22no—
92C23×D73—24no—
93C23×D73—24no—
94C23×D73C22⋊C414yes—
95C23×D73C2×D411yescharacteristic
96C23×D73C2×D411yescharacteristic
97C22×Dic73—24no—
98C22×Dic73—24no—
99C22×Dic73—22no—
100C22×Dic73—22no—
101C22×Dic73—42no—
102C22×Dic73—42no—
103C22×Dic73C22⋊C414yes—
104C22×Dic73C2×D411yescharacteristic
105C22×Dic73C2×D411yescharacteristic
106C2×C4×D73—24no—
107C2×C4×D73—24no—
108C2×C4×D73—24no—
109C2×C4×D73—216no—
110C2×C4×D73—216no—
111C2×C4×D73—28no—
112C2×C4×D73—28no—
113C2×C4×D73—44no—
114C2×C4×D73—44no—
115C2×C4×D73—44no—
116C2×C4×D73—44no—
117C2×C4×D73—432no—
118C2×C4×D73C22⋊C414yes—
119C2×C4×D73C22⋊C414yes—
120C2×C4×D73C22×C412yes—
121C2×C4×D73C22×C412yes—
122C2×C4×D73C2×D412yes—
123C2×C4×D73C2×D412yes—
124C2×C4×D73#4118yes—
125#11924011—48no—
126C2×#10596831—22no—
127C2×#10596831—24no—
128C4⋊Dic73—24no—
129C4⋊Dic73—22no—
130C4⋊Dic73—24no—
131C4⋊Dic73C22×C414yes—
132#11924007—28no—
133#11924007—28no—
134C22⋊Dic73—42no—
135#11924005—44no—
136#11924005—44no—
137C4×Dic73—44no—
138C24×C73C2×D411yescharacteristic
139C22×C4×C73—24no—
140C22×C4×C73—24no—
141C22×C4×C73—22no—
142C22×C4×C73—22no—
143C22×C4×C73—42no—
144C22×C4×C73—42no—
145C22×C4×C73C22×C412yes—
146C22×C4×C73C2×D411yescharacteristic
147C22×C4×C73C2×D412yes—
148C22×C4×C73C2411yescharacteristic
149C2×C8×C73—22no—
150C2×C8×C73—24no—
151C4⋊C4×C73—24no—
152C4⋊C4×C73—22no—
153C4⋊C4×C73—24no—
154C4⋊C4×C73C22×C414yes—
155C22⋊C4×C73—42no—
156C23×C73—22no—
157C23×C73—22no—
158C23×C73C2×C22⋊C412yes—
159C23×C73AGL(1,ℤ/8)12yes—
160C23×C73AGL(1,ℤ/8)12yes—
161C23×C73C22×D411yescharacteristic
162C22×D73—216no—
163C22×D73—216no—
164C22×D73—22no—
165C22×D73—21no—
166C22×D73—21no—
167C22×D73—21no—
168C22×D73—21no—
169C22×D73—22no—
170C22×D73—28no—
171C22×D73—28no—
172C22×D73—28no—
173C22×D73C2×C22⋊C412yes—
174C22×D73C2×C22⋊C412yes—
175C22×D73#12314yes—
176C22×D73#13714yes—
177C2×C4×C73—22no—
178C2×C4×C73—24no—
179C2×C4×C73—24no—
180C2×C4×C73—24no—
181C2×C4×C73—42no—
182C2×C4×C73—42no—
183C2×C4×C73—48no—
184C2×C4×C73—42no—
185C2×C4×C73—42no—
186C2×C4×C73—42no—
187C2×C4×C73C2×C22⋊C412yes—
188C2×C4×C73C23×C411yescharacteristic
189C2×C4×C73C22×D411yescharacteristic
190C2×C4×C73C2×#4114yes—
191C2×Dic73—22no—
192C2×Dic73—28no—
193C2×Dic73—24no—
194C2×Dic73—42no—
195C2×Dic73—42no—
196C2×Dic73—48no—
197C2×Dic73—42no—
198C2×Dic73—42no—
199C2×Dic73—42no—
200C2×Dic73C2×C22⋊C412yes—
201C2×Dic73C2×C22⋊C412yes—
202C2×Dic73#12314yes—
203C4×D73—28no—
204C4×D73—24no—
205C4×D73—24no—
206C4×D73—28no—
207C4×D73—416no—
208C4×D73—416no—
209C4×D73#11718yes—
210C4×D73#12318yes—
211C8×C73—42no—
212#10596831—42no—
213C22×C73—24no—
214C22×C73—24no—
215C22×C73—22no—
216C22×C73—22no—
217C22×C73#41814yes—
218C22×C73C22×C22⋊C411yescharacteristic
219C22×C73C2×#12312yes—
220C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
221C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
222C22×C73C2×#13712yes—
223C2×D73—232no—
224C2×D73—24no—
225C2×D73—24no—
226C2×D73—24no—
227C2×D73—24no—
228C2×D73—22no—
229C2×D73—22no—
230C2×D73—22no—
231C2×D73—22no—
232C2×D73#38114yes—
233C2×D73#41914yes—
234C4×C73—22no—
235C4×C73—44no—
236C4×C73—82no—
237C4×C73C2×#11712yes—
238C4×C73C2×#12312yes—
239Dic73—22no—
240Dic73—44no—
241Dic73—82no—
242Dic73#38114yes—
243#20700—731nomaximal, Sylow
244C2×C73—24no—
245C2×C73C2×#38111yescharacteristic
246C2×C73C2×#41811yescharacteristic
247C2×C73C2×#41911yescharacteristic
248C2×C73#309212yes—
249C2×C73#309212yes—
250D73—24no—
251D73—24no—
252C23×C4⋊C4—731no—
253C2×#407—731no—
254#2280—731no—
255#1824—734no—
256#1823—731no—
257#1823—731no—
258#1823—731no—
259#1823D7311yescharacteristic, complemented
260#1736—734no—
261C73#2070011yescharacteristic, complemented, Sylow
262C24×C4—731no—
263C24×C4—1461no—
264C22×M16—732no—
265C2×#117—732no—
266C22×C4⋊C4—731no—
267C22×C4⋊C4—738no—
268C22×C4⋊C4—732no—
269C22×C4⋊C4—732no—
270C22×C4⋊C4—1461no—
271C22×C4⋊C4—1461no—
272C22×C4⋊C4—1464no—
273C22×C4⋊C4—1461no—
274C22×C4⋊C4C2×D7311yescharacteristic
275#407—1464no—
276C2×C22⋊C8—731no—
277C2×C22⋊C8—732no—
278C2×C22⋊C8C2×D7311yescharacteristic
279#381—1461no—
280#380—731no—
281#380—732no—
282#380C2×D7311yescharacteristic
283#379—1461no—
284C4×C22⋊C4—1461no—
285C4×C22⋊C4—1461no—
286#340—734no—
287#340—734no—
288#340—734no—
289#340C2×D7314yescomplemented
290C25—731no—
291C23×C4—21no—
292C23×C4—2924no—
293C23×C4—2924no—
294C23×C4—731no—
295C23×C4—731no—
296C23×C4—734no—
297C23×C4—731no—
298C23×C4—731no—
299C23×C4—734no—
300C23×C4—731no—
301C23×C4—1468no—
302C23×C4—1462no—
303C23×C4—1462no—
304C23×C4—1468no—
305C23×C4—1461no—
306C23×C4C22×D7311yescharacteristic
307C2×M16—1464no—
308C2×M16—1468no—
309C22×C8—732no—
310C22×C8C22×D7312yes—
311#117—1468no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—734no—
315C2×C4⋊C4—732no—
316C2×C4⋊C4—738no—
317C2×C4⋊C4—738no—
318C2×C4⋊C4—732no—
319C2×C4⋊C4—1468no—
320C2×C4⋊C4—1468no—
321C2×C4⋊C4—1464no—
322C2×C4⋊C4—1464no—
323C2×C4⋊C4—1468no—
324C2×C4⋊C4—1462no—
325C2×C4⋊C4—1462no—
326C2×C4⋊C4—1461no—
327C2×C4⋊C4—1462no—
328C2×C4⋊C4C4×D7314yes—
329C2×C4⋊C4C22×D7312yes—
330C2×C4⋊C4C22×D7312yes—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—1462no—
333C2×C22⋊C4—1462no—
334C2×C22⋊C4—1461no—
335C2×C42—1462no—
336C22⋊C8—22no—
337C22⋊C8—1464no—
338C22⋊C8—1462no—
339#95—21no—
340#95—1462no—
341#95—1461no—
342C24—731no—
343C24—731no—
344C24—734no—
345C24—1464no—
346C24—1464no—
347C24—1462no—
348C24—1462no—
349C24D4×D7311yescharacteristic
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—29232no—
357C22×C4—2924no—
358C22×C4—2924no—
359C22×C4—2924no—
360C22×C4—2924no—
361C22×C4—2922no—
362C22×C4—2922no—
363C22×C4—731no—
364C22×C4—732no—
365C22×C4—732no—
366C22×C4—738no—
367C22×C4—734no—
368C22×C4—731no—
369C22×C4—734no—
370C22×C4—734no—
371C22×C4—732no—
372C22×C4—732no—
373C22×C4—1468no—
374C22×C4—1468no—
375C22×C4—1464no—
376C22×C4—14616no—
377C22×C4—14616no—
378C22×C4—1464no—
379C22×C4—1462no—
380C22×C4—1464no—
381C22×C4—1462no—
382C22×C4—1464no—
383C22×C4—1464no—
384C22×C4C2×C4×D7312yes—
385C22×C4D4×D7311yescharacteristic
386C22×C4D4×D7312yes—
387C22×C4C23×D7311yescharacteristic
388M16—2928no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—1462no—
392C2×C8—1464no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—734no—
397C4⋊C4—1468no—
398C4⋊C4—1468no—
399C4⋊C4—1464no—
400C4⋊C4—1464no—
401C4⋊C4—1462no—
402C4⋊C4C2×C4×D7314yes—
403C22⋊C4—42no—
404C22⋊C4—2924no—
405C22⋊C4—2924no—
406C22⋊C4—2922no—
407C42—2924no—
408C23—22no—
409C23—22no—
410C23—734no—
411C23—732no—
412C23—732no—
413C23—734no—
414C23—1468no—
415C23—1468no—
416C23—14616no—
417C23—14616no—
418C23—1462no—
419C23—1462no—
420C23—1468no—
421C23—1461no—
422C23—1461no—
423C23—1461no—
424C23—1461no—
425C23C22⋊C4×D7312yes—
426C23#1455661512yes—
427C23#1455661812yes—
428C23C2×D4×D7311yescharacteristic
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—48no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—29216no—
440C2×C4—29216no—
441C2×C4—2928no—
442C2×C4—2922no—
443C2×C4—2922no—
444C2×C4—2922no—
445C2×C4—2922no—
446C2×C4—2922no—
447C2×C4—732no—
448C2×C4—734no—
449C2×C4—738no—
450C2×C4—738no—
451C2×C4—732no—
452C2×C4—1468no—
453C2×C4—1468no—
454C2×C4—1464no—
455C2×C4—1468no—
456C2×C4—1462no—
457C2×C4—1464no—
458C2×C4—1464no—
459C2×C4C22⋊C4×D7312yes—
460C2×C4C22×C4×D7311yescharacteristic
461C2×C4C2×D4×D7311yescharacteristic
462C2×C4#41×D7314yes—
463C8—42no—
464C8—2922no—
465C22—22no—
466C22—22no—
467C22—24no—
468C22—24no—
469C22—734no—
470C22—734no—
471C22—14632no—
472C22—1462no—
473C22—1462no—
474C22—1462no—
475C22—1462no—
476C22—1464no—
477C22—1464no—
478C22—1464no—
479C22—1464no—
480C22#1798859014yes—
481C22C2×C22⋊C4×D7311yescharacteristic
482C22#123×D7312yes—
483C22C2×#1455661511yescharacteristic
484C22C2×#1455661811yescharacteristic
485C22#137×D7312yes—
486C4—22no—
487C4—44no—
488C4—82no—
489C4—2924no—
490C4—5842no—
491C4—734no—
492C4—1462no—
493C4#117×D7312yes—
494C4#123×D7312yes—
495C2—24no—
496C2—1464no—
497C2—1464no—
498C2#381×D7311yescharacteristic
499C2C2×#1798859011yescharacteristic
500C2#419×D7311yescharacteristic
501C2#2261985012yes—
502C2#2261985512yes—
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1823C2
socleC23×C73C22×D4
prime radicalC24×D73D4
cosocleC22×C4×C73C24
hypercenter#1823D73
hypocenterC73#20700
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
4. #22610020C2semidirect
5. #22610018C2semidirect
6. #22610017C2semidirect
7. #22610015C2semidirect
8. #22609649C2semidirect
9. C73×#1823C2semidirect
19. C2×C4⋊C4×D73C4semidirect
31. #17988350C22semidirect
32. #17988349C22semidirect
33. #17988349C22semidirect
37. #340×C73C22semidirect
259. #1823D73semidirect
261. C73#20700semidirect
289. #340C2×D73semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D732
C2×#179885492
#226123132
#226100202
#226100182
#226100172
#226100152
#226096492
C73×#18232
#2070073