Finite Group Databasebeta

← #33096549

Subgroups — #33096549

Order 18688.

902 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988642C211yesmaximal, characteristic
2#384×D73C211yesmaximal, characteristic
3#22612324C211yesmaximal, characteristic
4#22610693C212yesmaximal, complemented
5#22610688C212yesmaximal, complemented
6#22610125C212yesmaximal
7#22610120C212yesmaximal
8#22610099C211yesmaximal, characteristic, complemented
9#22610097C211yesmaximal, characteristic, complemented
10#22610064C211yesmaximal, characteristic, complemented
11C73×#1836C211yesmaximal, characteristic, complemented
12C2×C4⋊C4×D73—21no—
13C2×C4⋊C4×D73—21no—
14C2×C4⋊C4×D73C2212yes—
15C2×C42×D73C2211yescharacteristic
16#17988642C418yescomplemented
17C2×#14556525C2212yes—
18C2×#14556524C2212yes—
19C2×#14556523C2212yes—
20C2×#14556522C2212yes—
21C2×#14556520C2212yes—
22C2×#14556503C2212yes—
23C2×#14556502C2211yescharacteristic
24#17988457C2211yescharacteristic
25#17988456—21no—
26#17988456—21no—
27#17988456C2212yes—
28#17988454—21no—
29#17988453C2211yescharacteristic
30#17988449—21no—
31#17988444C2211yescharacteristic
32#17988444C2211yescharacteristic
33C4⋊C4×Dic73C2211yescharacteristic
34C4⋊C4×Dic73C2211yescharacteristic
35#17988434—21no—
36#17988432—21no—
37#17988349C2212yes—
38#17988349C2212yes—
39C2×#97×C73C2211yescharacteristic
40#384×C73C2211yescharacteristic
41#382×C73C2211yescharacteristic
42#340×C73C2212yes—
43#340×C73C2212yes—
44C22×C4×D73—22no—
45C22×C4×D73D411yescharacteristic
46C22×C4×D73D411yescharacteristic
47C22×C4×D73C2311yescharacteristic
48C22×#10596831C2312yes—
49C2×C4⋊Dic73—21no—
50C2×C4⋊Dic73—21no—
51C2×C4⋊Dic73C2311yescharacteristic
52C2×C4⋊Dic73C2312yes—
53C2×C4⋊Dic73C2311yescharacteristic
54C2×#11924007—22no—
55C2×#11924007—22no—
56C2×#11924007—22no—
57C4⋊C4×D73—42no—
58C4⋊C4×D73—42no—
59C4⋊C4×D73—44no—
60C4⋊C4×D73—44no—
61C2×#11924005—21no—
62C2×#11924005—21no—
63C2×C4×Dic73—21no—
64C2×C4×Dic73—21no—
65C2×C4×Dic73D411yescharacteristic
66C2×C4×Dic73D411yescharacteristic
67C2×C4×Dic73C2311yescharacteristic
68C42×D73C2×C414yes—
69C42×D73Q814yes—
70#14556525—42no—
71#14556524—42no—
72#14556523—42no—
73#14556522—42no—
74#14556520C2×C418yescomplemented
75#14556503C2×C418yes—
76#14556502C2×C414yescomplemented
77#14556498—21no—
78#14556498—21no—
79#14556498—21no—
80#14556498—21no—
81#14556498—21no—
82#14556498—21no—
83C22×C8×C73C2312yes—
84C2×C4⋊C4×C73—21no—
85C2×C4⋊C4×C73—21no—
86C2×C4⋊C4×C73C2311yescharacteristic
87C2×C4⋊C4×C73C2311yescharacteristic
88C2×C4⋊C4×C73C2312yes—
89C2×C42×C73C2311yescharacteristic
90#97×C73C2×C414yescomplemented
91#95×C73—21no—
92#95×C73—21no—
93C23×D73C2×D411yescharacteristic
94C22×Dic73—21no—
95C22×Dic73—21no—
96C22×Dic73—22no—
97C22×Dic73C2×D411yescharacteristic
98C22×Dic73C2×D411yescharacteristic
99C22×Dic73C2×D411yescharacteristic
100C2×C4×D73—22no—
101C2×C4×D73—22no—
102C2×C4×D73—22no—
103C2×C4×D73—22no—
104C2×C4×D73—22no—
105C2×C4×D73—22no—
106C2×C4×D73—44no—
107C2×C4×D73—44no—
108C2×C4×D73—44no—
109C2×C4×D73C4⋊C414yes—
110C2×C4×D73C4⋊C414yes—
111C2×C4×D73C22×C412yes—
112C2×C4×D73C2×Q812yes—
113C2×#10596831—22no—
114C2×#10596831—22no—
115C2×#10596831C22×C414yes—
116C4⋊Dic73—22no—
117C4⋊Dic73—22no—
118C4⋊Dic73—22no—
119C4⋊Dic73—22no—
120C4⋊Dic73—42no—
121C4⋊Dic73—42no—
122#11924007—44no—
123#11924007—44no—
124#11924007—44no—
125#11924005—42no—
126#11924005—42no—
127C4×Dic73—42no—
128C4×Dic73—42no—
129C4×Dic73C4⋊C414yes—
130C4×Dic73C4⋊C414yes—
131C4×Dic73C22×C412yes—
132C4×Dic73C2×Q812yes—
133C22×C4×C73—21no—
134C22×C4×C73—21no—
135C22×C4×C73—22no—
136C22×C4×C73C2×D411yescharacteristic
137C22×C4×C73C2×D411yescharacteristic
138C22×C4×C73C2411yescharacteristic
139C2×C8×C73—22no—
140C2×C8×C73—22no—
141C2×C8×C73C22×C414yes—
142C4⋊C4×C73—22no—
143C4⋊C4×C73—22no—
144C4⋊C4×C73—22no—
145C4⋊C4×C73—22no—
146C4⋊C4×C73—42no—
147C4⋊C4×C73—42no—
148C42×C73C22×C412yes—
149C42×C73C2×Q812yes—
150C23×C73C22×D411yescharacteristic
151C22×D73—21no—
152C22×D73—21no—
153C22×D73—21no—
154C22×D73—21no—
155C22×D73C2×C4⋊C412yes—
156C22×D73AGL(1,ℤ/8)12yes—
157C22×D73#13712yes—
158C2×C4×C73—21no—
159C2×C4×C73—21no—
160C2×C4×C73—21no—
161C2×C4×C73—21no—
162C2×C4×C73—42no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73—42no—
166C2×C4×C73—42no—
167C2×C4×C73—41no—
168C2×C4×C73—41no—
169C2×C4×C73—41no—
170C2×C4×C73—41no—
171C2×C4×C73C2×C4⋊C412yes—
172C2×C4×C73C2×C4⋊C412yes—
173C2×C4×C73C23×C411yescharacteristic
174C2×C4×C73C22×Q811yescharacteristic
175C2×C4×C73H(5,𝔽2)12yes—
176C2×C4×C732−1+412yes—
177C2×Dic73—21no—
178C2×Dic73—21no—
179C2×Dic73—21no—
180C2×Dic73—21no—
181C2×Dic73—42no—
182C2×Dic73—41no—
183C2×Dic73—41no—
184C2×Dic73—41no—
185C2×Dic73—41no—
186C2×Dic73—42no—
187C2×Dic73—42no—
188C2×Dic73—41no—
189C2×Dic73—41no—
190C2×Dic73C2×C4⋊C412yes—
191C2×Dic73C2×C4⋊C412yes—
192C2×Dic73C2×C4⋊C412yes—
193C2×Dic73AGL(1,ℤ/8)12yes—
194C2×Dic73#13712yes—
195C4×D73—28no—
196C4×D73—24no—
197C4×D73—24no—
198C4×D73—28no—
199C4×D73—88no—
200C8×C73—24no—
201#10596831—24no—
202C22×C73C22×C4⋊C411yescharacteristic
203C22×C73#53511yescharacteristic
204C22×C73#53611yescharacteristic
205C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
206C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
207C22×C73C2×#13711yescharacteristic
208C22×C73C2×#13711yescharacteristic
209C2×D73—22no—
210C2×D73—22no—
211C2×D73—22no—
212C2×D73—22no—
213C2×D73—22no—
214C2×D73—22no—
215C2×D73#42814yes—
216C4×C73—22no—
217C4×C73—22no—
218C4×C73—82no—
219C4×C73—81no—
220C4×C73—81no—
221C4×C73#52012yes—
222C4×C73#54412yes—
223Dic73—22no—
224Dic73—22no—
225Dic73—82no—
226Dic73—81no—
227Dic73—81no—
228Dic73#42814yes—
229#21987—731nomaximal, Sylow
230C2×C73#231711yescharacteristic
231C2×C73C2×#42811yescharacteristic
232C2×C73C2×#42811yescharacteristic
233C2×C73#321711yescharacteristic
234C2×C73#321811yescharacteristic
235C2×C73#323811yescharacteristic
236C2×C73#323911yescharacteristic
237D73—24no—
238C2×#432—731no—
239C2×#384—731no—
240#2283—731no—
241#1948—732no—
242#1947—732no—
243#1853—732no—
244#1852—732no—
245#1847—731no—
246#1847—731no—
247#1836—731no—
248#1836D7311yescharacteristic, complemented
249C73#2198711yescharacteristic, complemented, Sylow
250C22×C4⋊C4—732no—
251C22×C4⋊C4—1461no—
252C22×C4⋊C4—1461no—
253C22×C42—731no—
254#432—738no—
255C2×#107—732no—
256C2×#107—732no—
257C2×#106—732no—
258C2×#106—732no—
259C2×C4⋊C8—732no—
260C2×C22⋊C8—732no—
261C2×#97—731no—
262C2×#97C2×D7311yescharacteristic
263#384—732no—
264#384—731no—
265#384—1461no—
266#384—1461no—
267#384C2×D7311yescharacteristic
268#383—1461no—
269#382—731no—
270#382C2×D7311yescharacteristic
271#381—1461no—
272#380—731no—
273#380—731no—
274C4×C4⋊C4—731no—
275C4×C4⋊C4—731no—
276C4×C22⋊C4—1461no—
277#376—1461no—
278#340—732no—
279#340—732no—
280#340C2×D7312yes—
281#340C2×D7312yes—
282C23×C4—731no—
283C23×C4—731no—
284C23×C4—731no—
285C23×C4—1462no—
286C22×C8—732no—
287C22×C8C22×D7312yes—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—2922no—
291C2×C4⋊C4—2922no—
292C2×C4⋊C4—2924no—
293C2×C4⋊C4—2924no—
294C2×C4⋊C4—732no—
295C2×C4⋊C4—731no—
296C2×C4⋊C4—731no—
297C2×C4⋊C4—1462no—
298C2×C4⋊C4—1462no—
299C2×C4⋊C4—1462no—
300C2×C4⋊C4—1461no—
301C2×C4⋊C4—1461no—
302C2×C4⋊C4C22×D7312yes—
303C2×C4⋊C4C22×D7311yescharacteristic
304C2×C4⋊C4C22×D7311yescharacteristic
305C2×C22⋊C4—1461no—
306C2×C22⋊C4—1461no—
307C2×C42—731no—
308C2×C42—731no—
309C2×C42—731no—
310C2×C42—734no—
311C2×C42—734no—
312C2×C42—1461no—
313C2×C42—1461no—
314C2×C42C22×D7311yescharacteristic
315#107—2922no—
316#107—2922no—
317#106—2922no—
318#106—2922no—
319C4⋊C8—738no—
320C22⋊C8—738no—
321#97—734no—
322#97C4×D7314yescomplemented
323#95—21no—
324#95—21no—
325#95—1461no—
326#95—1461no—
327#95—1461no—
328#95—1461no—
329#95—1461no—
330#95—1461no—
331C24—731no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—2924no—
336C22×C4—2924no—
337C22×C4—2924no—
338C22×C4—731no—
339C22×C4—731no—
340C22×C4—731no—
341C22×C4—734no—
342C22×C4—734no—
343C22×C4—732no—
344C22×C4—732no—
345C22×C4—1462no—
346C22×C4—1462no—
347C22×C4—1462no—
348C22×C4—1462no—
349C22×C4—1461no—
350C22×C4—1461no—
351C22×C4—1462no—
352C22×C4—1462no—
353C22×C4—1462no—
354C22×C4D4×D7311yescharacteristic
355C22×C4D4×D7311yescharacteristic
356C22×C4C23×D7311yescharacteristic
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—734no—
360C2×C8—1462no—
361C2×C8—1462no—
362C2×C8C2×C4×D7314yes—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—2922no—
370C4⋊C4—2922no—
371C4⋊C4—2924no—
372C4⋊C4—2924no—
373C4⋊C4—2924no—
374C4⋊C4—1462no—
375C4⋊C4—1462no—
376C4⋊C4—1462no—
377C4⋊C4—1462no—
378C22⋊C4—2922no—
379C22⋊C4—2922no—
380C42—2922no—
381C42—2922no—
382C42—734no—
383C42—734no—
384C42—732no—
385C42—732no—
386C42C2×C4×D7312yes—
387C42Q8×D7312yes—
388C23—732no—
389C23—732no—
390C23—732no—
391C23—1461no—
392C23—1461no—
393C23—1461no—
394C23—1461no—
395C23C2×D4×D7311yescharacteristic
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—2921no—
410C2×C4—2921no—
411C2×C4—2922no—
412C2×C4—2921no—
413C2×C4—2921no—
414C2×C4—2921no—
415C2×C4—2921no—
416C2×C4—2922no—
417C2×C4—2922no—
418C2×C4—5848no—
419C2×C4—732no—
420C2×C4—732no—
421C2×C4—732no—
422C2×C4—732no—
423C2×C4—732no—
424C2×C4—1468no—
425C2×C4—1468no—
426C2×C4—1461no—
427C2×C4—1461no—
428C2×C4—1461no—
429C2×C4—1461no—
430C2×C4—1464no—
431C2×C4—1464no—
432C2×C4C4⋊C4×D7312yes—
433C2×C4C4⋊C4×D7312yes—
434C2×C4C22×C4×D7311yescharacteristic
435C2×C4C2×Q8×D7311yescharacteristic
436C2×C4#1455663712yes—
437C2×C4#1455663812yes—
438C8—24no—
439C8—1464no—
440C22—734no—
441C22—1462no—
442C22—1462no—
443C22—1462no—
444C22—1462no—
445C22—1462no—
446C22—1462no—
447C22C2×C4⋊C4×D7311yescharacteristic
448C22#1798915511yescharacteristic
449C22#1798915711yescharacteristic
450C22AGL(1,ℤ/8)×D7311yescharacteristic
451C22AGL(1,ℤ/8)×D7311yescharacteristic
452C22#137×D7311yescharacteristic
453C22#137×D7311yescharacteristic
454C4—22no—
455C4—22no—
456C4—82no—
457C4—81no—
458C4—81no—
459C4—5842no—
460C4—5841no—
461C4—5841no—
462C4—734no—
463C4—1462no—
464C4—1462no—
465C4#1798906712yes—
466C4#1798921712yes—
467C2—1464no—
468C2#2261255611yescharacteristic
469C2#428×D7311yescharacteristic
470C2#428×D7311yescharacteristic
471C2#2262095011yescharacteristic
472C2#2262095911yescharacteristic
473C2#2262114811yescharacteristic
474C2#2262115711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1836C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1836D73
hypocenterC73#21987
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

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

NH ≅ G/Ntype
4. #22610693C2semidirect
5. #22610688C2semidirect
8. #22610099C2semidirect
9. #22610097C2semidirect
10. #22610064C2semidirect
11. C73×#1836C2semidirect
16. #17988642C4semidirect
74. #14556520C2×C4semidirect
76. #14556502C2×C4semidirect
90. #97×C73C2×C4semidirect
248. #1836D73semidirect
249. C73#21987semidirect
322. #97C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886422
#384×D732
#226123242
#226106932
#226106882
#226101252
#226101202
#226100992
#226100972
#226100642
C73×#18362
#2198773