Finite Group Databasebeta

← #33086535

Subgroups — #33086535

Order 18688.

886 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988539C211yesmaximal, characteristic
2#22612336C211yesmaximal, characteristic
3#22612324C211yesmaximal, characteristic
4#22610099C211yesmaximal, characteristic, complemented
5#22610066C211yesmaximal, characteristic, complemented
6#22610064C211yesmaximal, characteristic, complemented
7#22609965C212yesmaximal
8#22609961C212yesmaximal
9#107×Dic73C212yesmaximal, complemented
10#106×Dic73C212yesmaximal, complemented
11C73×#1847C211yesmaximal, characteristic, complemented
12C22×#11924011C2212yes—
13C2×#14556549—21no—
14C2×C42×D73C2211yescharacteristic
15C2×#14556525C2212yes—
16C2×#14556523C2212yes—
17#17988539C418yescomplemented
18C2×#14556502C2211yescharacteristic
19C2×C8×Dic73C2212yes—
20#17988457C2211yescharacteristic
21#17988454—21no—
22#17988453C2211yescharacteristic
23#17988452—21no—
24#17988449—21no—
25#17988444—21no—
26#17988444C2211yescharacteristic
27#17988444C2212yes—
28#17988444C2211yescharacteristic
29C4⋊C4×Dic73—21no—
30C4⋊C4×Dic73C2211yescharacteristic
31C4⋊C4×Dic73C2211yescharacteristic
32C4⋊C4×Dic73C2212yes—
33#17988434—21no—
34#17988432—21no—
35#17988349C2212yes—
36#17988349C2212yes—
37#17988349C2212yes—
38#17988349C2212yes—
39C2×#107×C73C2212yes—
40C2×#106×C73C2212yes—
41C2×#97×C73C2211yescharacteristic
42#384×C73C2211yescharacteristic
43#382×C73C2211yescharacteristic
44C22×C4×D73D411yescharacteristic
45C22×C4×D73D411yescharacteristic
46C22×C4×D73C2311yescharacteristic
47C2×#11924011—24no—
48C2×#11924011—24no—
49C2×#11924011C2×C418yes—
50C22×#10596831C2312yes—
51#14556549—44no—
52C2×C4⋊Dic73—21no—
53C2×C4⋊Dic73—21no—
54C2×C4⋊Dic73C2311yescharacteristic
55C2×C4⋊Dic73C2312yes—
56C2×C4⋊Dic73C2311yescharacteristic
57C2×#11924005—21no—
58C2×#11924005—21no—
59C2×#11924005—22no—
60C2×C4×Dic73—21no—
61C2×C4×Dic73—21no—
62C2×C4×Dic73—22no—
63C2×C4×Dic73D411yescharacteristic
64C2×C4×Dic73D411yescharacteristic
65C2×C4×Dic73C2311yescharacteristic
66C42×D73C2×C414yes—
67C42×D73Q814yes—
68#14556525—24no—
69#14556523—24no—
70#14556502C2×C414yes—
71C8×Dic73C2×C418yes—
72#14556498—21no—
73#14556498—21no—
74#14556498—22no—
75#14556498—22no—
76#14556498—21no—
77#14556498—21no—
78#14556498—21no—
79#14556498—21no—
80C22×C8×C73C2312yes—
81C2×C4⋊C4×C73—21no—
82C2×C4⋊C4×C73—21no—
83C2×C4⋊C4×C73C2311yescharacteristic
84C2×C4⋊C4×C73C2311yescharacteristic
85C2×C4⋊C4×C73C2312yes—
86C2×C42×C73C2311yescharacteristic
87#107×C73—24no—
88#106×C73—24no—
89#97×C73C2×C414yes—
90#95×C73—21no—
91#95×C73—21no—
92C23×D73C2×D411yescharacteristic
93C22×Dic73—21no—
94C22×Dic73—21no—
95C22×Dic73—22no—
96C22×Dic73C2×D411yescharacteristic
97C22×Dic73C2×D411yescharacteristic
98C22×Dic73C2×D411yescharacteristic
99C2×C4×D73—22no—
100C2×C4×D73—22no—
101C2×C4×D73—22no—
102C2×C4×D73—22no—
103C2×C4×D73—41no—
104C2×C4×D73—41no—
105C2×C4×D73C4⋊C414yes—
106C2×C4×D73C4⋊C414yes—
107C2×C4×D73C22×C412yes—
108C2×C4×D73C2×Q812yes—
109#11924011—216no—
110C2×#10596831—22no—
111C2×#10596831—22no—
112C2×#10596831C22×C414yes—
113C4⋊Dic73—22no—
114C4⋊Dic73—22no—
115C4⋊Dic73—22no—
116C4⋊Dic73—22no—
117C4⋊Dic73—42no—
118C4⋊Dic73—42no—
119#11924005—42no—
120#11924005—42no—
121#11924005—44no—
122C4×Dic73—42no—
123C4×Dic73—44no—
124C4×Dic73—42no—
125C4×Dic73C4⋊C414yes—
126C4×Dic73C4⋊C414yes—
127C4×Dic73C22×C412yes—
128C4×Dic73C2×Q812yes—
129C22×C4×C73—21no—
130C22×C4×C73—21no—
131C22×C4×C73—22no—
132C22×C4×C73C2×D411yescharacteristic
133C22×C4×C73C2×D411yescharacteristic
134C22×C4×C73C2411yescharacteristic
135C2×C8×C73C22×C414yes—
136C2×C8×C73#4114yes—
137C2×C8×C73#4114yes—
138C4⋊C4×C73—22no—
139C4⋊C4×C73—22no—
140C4⋊C4×C73—22no—
141C4⋊C4×C73—22no—
142C4⋊C4×C73—42no—
143C4⋊C4×C73—42no—
144C42×C73C22×C412yes—
145C42×C73C2×Q812yes—
146C23×C73C22×D411yescharacteristic
147C22×D73—21no—
148C22×D73—21no—
149C22×D73—21no—
150C22×D73—21no—
151C22×D73—21no—
152C22×D73—21no—
153C22×D73C2×C4⋊C412yes—
154C2×C4×C73—21no—
155C2×C4×C73—21no—
156C2×C4×C73—21no—
157C2×C4×C73—21no—
158C2×C4×C73—42no—
159C2×C4×C73—41no—
160C2×C4×C73—41no—
161C2×C4×C73—42no—
162C2×C4×C73—41no—
163C2×C4×C73—41no—
164C2×C4×C73—42no—
165C2×C4×C73—41no—
166C2×C4×C73—41no—
167C2×C4×C73C2×C4⋊C412yes—
168C2×C4×C73C2×C4⋊C412yes—
169C2×C4×C73C23×C411yescharacteristic
170C2×C4×C73C22×Q811yescharacteristic
171C2×C4×C73C2×#4112yes—
172C2×C4×C73C2×#4112yes—
173C2×Dic73—21no—
174C2×Dic73—21no—
175C2×Dic73—21no—
176C2×Dic73—21no—
177C2×Dic73—42no—
178C2×Dic73—41no—
179C2×Dic73—41no—
180C2×Dic73—42no—
181C2×Dic73—41no—
182C2×Dic73—41no—
183C2×Dic73—41no—
184C2×Dic73—41no—
185C2×Dic73—42no—
186C2×Dic73C2×C4⋊C412yes—
187C2×Dic73C2×C4⋊C412yes—
188C2×Dic73C2×C4⋊C412yes—
189C2×Dic73AGL(1,ℤ/8)12yes—
190C2×Dic73#13712yes—
191C4×D73—24no—
192C4×D73—24no—
193C4×D73—28no—
194C4×D73—44no—
195C8×C73#11718yescomplemented
196#10596831—24no—
197C22×C73C22×C4⋊C411yescharacteristic
198C22×C73#53211yescharacteristic
199C22×C73#53211yescharacteristic
200C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
201C22×C73C2×#13711yescharacteristic
202C22×C73#57511yescharacteristic
203C22×C73#57511yescharacteristic
204C2×D73—22no—
205C2×D73—22no—
206C2×D73—22no—
207C2×D73—41no—
208C2×D73—41no—
209C2×D73—41no—
210C2×D73—41no—
211C4×C73—22no—
212C4×C73—22no—
213C4×C73—82no—
214C4×C73—81no—
215C4×C73—81no—
216C4×C73C2×#11712yes—
217C4×C73#53312yes—
218Dic73—22no—
219Dic73—22no—
220Dic73—82no—
221Dic73—81no—
222Dic73—81no—
223Dic73#42814yes—
224#20819—731nomaximal, Sylow
225C2×C73#228511yescharacteristic
226C2×C73C2×#42811yescharacteristic
227C2×C73#291111yescharacteristic
228C2×C73#317111yescharacteristic
229C2×C73#317111yescharacteristic
230C2×C73#317211yescharacteristic
231C2×C73#317211yescharacteristic
232D73—42no—
233C2×C4×M16—731no—
234#2285—731no—
235#2283—731no—
236#1847—731no—
237#1847D7311yescharacteristic, complemented
238#1836—731no—
239#1836—731no—
240#1809—732no—
241#1809—732no—
242C4×#107—732no—
243C4×#106—732no—
244C73#2081911yescharacteristic, complemented, Sylow
245C22×M16—732no—
246C2×#117—1461no—
247C22×C42—731no—
248C2×#107—732no—
249C2×#107C2×D7312yes—
250C2×#106—732no—
251C2×#106C2×D7312yes—
252C4×M16—738no—
253C2×#97—731no—
254C2×#97C2×D7311yescharacteristic
255C2×C4×C8—732no—
256#384—731no—
257#384C2×D7311yescharacteristic
258#383—1461no—
259#382—731no—
260#382—1461no—
261#382C2×D7311yescharacteristic
262#381—1461no—
263#380—731no—
264#380—731no—
265#380—732no—
266#380—1461no—
267C4×C4⋊C4—731no—
268C4×C4⋊C4—731no—
269C4×C4⋊C4—732no—
270C4×C4⋊C4—1461no—
271C4×C22⋊C4—1461no—
272#376—1461no—
273#340—732no—
274#340—732no—
275#340—732no—
276#340—732no—
277C23×C4—731no—
278C23×C4—731no—
279C23×C4—731no—
280C2×M16—738no—
281C2×M16—1464no—
282C2×M16—1464no—
283C22×C8—732no—
284C22×C8C22×D7312yes—
285#117—2924no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—731no—
289C2×C4⋊C4—731no—
290C2×C4⋊C4—732no—
291C2×C4⋊C4—1461no—
292C2×C4⋊C4—1461no—
293C2×C4⋊C4C22×D7311yescharacteristic
294C2×C4⋊C4C22×D7311yescharacteristic
295C2×C4⋊C4C22×D7312yes—
296C2×C22⋊C4—1462no—
297C2×C22⋊C4—1461no—
298C2×C22⋊C4—1461no—
299C2×C42—731no—
300C2×C42—731no—
301C2×C42—731no—
302C2×C42—734no—
303C2×C42—734no—
304C2×C42—1462no—
305C2×C42—1461no—
306C2×C42—1461no—
307C2×C42C22×D7311yescharacteristic
308#107—24no—
309#107—1464no—
310#106—24no—
311#106—1464no—
312#97—734no—
313#97C4×D7314yes—
314C4×C8—738no—
315#95—21no—
316#95—21no—
317#95—1461no—
318#95—1461no—
319#95—1462no—
320#95—1461no—
321#95—1461no—
322#95—1462no—
323#95—1461no—
324#95—1461no—
325C24—731no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—2921no—
330C22×C4—2921no—
331C22×C4—734no—
332C22×C4—731no—
333C22×C4—731no—
334C22×C4—731no—
335C22×C4—734no—
336C22×C4—732no—
337C22×C4—732no—
338C22×C4—1462no—
339C22×C4—1462no—
340C22×C4—1461no—
341C22×C4—1461no—
342C22×C4—1462no—
343C22×C4—1462no—
344C22×C4—1462no—
345C22×C4D4×D7311yescharacteristic
346C22×C4D4×D7311yescharacteristic
347C22×C4C23×D7311yescharacteristic
348M16—14616no—
349C2×C8—734no—
350C2×C8—1462no—
351C2×C8—1462no—
352C2×C8C2×C4×D7314yes—
353C2×C8#1192402714yes—
354C2×C8#1192402914yes—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—2922no—
362C4⋊C4—2922no—
363C4⋊C4—1462no—
364C4⋊C4—1462no—
365C4⋊C4—1462no—
366C4⋊C4—1462no—
367C22⋊C4—2922no—
368C22⋊C4—2922no—
369C22⋊C4—2924no—
370C42—2922no—
371C42—2922no—
372C42—2924no—
373C42—734no—
374C42—734no—
375C42—732no—
376C42—732no—
377C42C2×C4×D7312yes—
378C42Q8×D7312yes—
379C23—732no—
380C23—1461no—
381C23—1461no—
382C23—1461no—
383C23—1461no—
384C23—1461no—
385C23—1461no—
386C23C2×D4×D7311yescharacteristic
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—2924no—
401C2×C4—2921no—
402C2×C4—2921no—
403C2×C4—2921no—
404C2×C4—2922no—
405C2×C4—2922no—
406C2×C4—2921no—
407C2×C4—2921no—
408C2×C4—2922no—
409C2×C4—2921no—
410C2×C4—732no—
411C2×C4—732no—
412C2×C4—732no—
413C2×C4—732no—
414C2×C4—732no—
415C2×C4—1468no—
416C2×C4—1461no—
417C2×C4—1461no—
418C2×C4—1461no—
419C2×C4—1461no—
420C2×C4—1464no—
421C2×C4—1464no—
422C2×C4C4⋊C4×D7312yes—
423C2×C4C4⋊C4×D7312yes—
424C2×C4C22×C4×D7311yescharacteristic
425C2×C4C2×Q8×D7311yescharacteristic
426C2×C4C2×#1192402712yes—
427C2×C4C2×#1192402912yes—
428C8—1464no—
429C8#1455654918yescomplemented
430C22—2921no—
431C22—2921no—
432C22—2921no—
433C22—2921no—
434C22—1462no—
435C22—1462no—
436C22—1462no—
437C22C2×C4⋊C4×D7311yescharacteristic
438C22#1798913511yescharacteristic
439C22#1798913811yescharacteristic
440C22AGL(1,ℤ/8)×D7311yescharacteristic
441C22#137×D7311yescharacteristic
442C22#1798938311yescharacteristic
443C22#1798938611yescharacteristic
444C4—22no—
445C4—22no—
446C4—81no—
447C4—82no—
448C4—81no—
449C4—5841no—
450C4—5842no—
451C4—5841no—
452C4—734no—
453C4—1462no—
454C4—1462no—
455C4C2×#1455654912yes—
456C4#1798914212yes—
457C2—2922no—
458C2#2261233611yescharacteristic
459C2#428×D7311yescharacteristic
460C2#2261851811yescharacteristic
461C2#2262052311yescharacteristic
462C2#2262052811yescharacteristic
463C2#2262053411yescharacteristic
464C2#2262053911yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1847C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1847D73
hypocenterC73#20819
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
4. #22610099C2semidirect
5. #22610066C2semidirect
6. #22610064C2semidirect
9. #107×Dic73C2semidirect
10. #106×Dic73C2semidirect
11. C73×#1847C2semidirect
17. #17988539C4semidirect
195. C8×C73#117semidirect
237. #1847D73semidirect
244. C73#20819semidirect
429. C8#14556549semidirect

Maximal subgroups

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

subgroupindex
C2×#179885392
#226123362
#226123242
#226100992
#226100662
#226100642
#226099652
#226099612
#107×Dic732
#106×Dic732
C73×#18472
#2081973