Finite Group Databasebeta

← #33096554

Subgroups — #33096554

Order 18688.

830 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988643C211yesmaximal, characteristic
2#22612336C211yesmaximal, characteristic
3#22612335C211yesmaximal, characteristic
4#22610692C211yesmaximal, characteristic, complemented
5#22610690C211yesmaximal, characteristic, complemented
6#22610689C211yesmaximal, characteristic, complemented
7#22610133C212yesmaximal
8#22610129C212yesmaximal
9#22610096C212yesmaximal, complemented
10#22610063C212yesmaximal, complemented
11C73×#1948C211yesmaximal, characteristic, complemented
12C2×#14556549C2212yes—
13C2×#14556548—21no—
14C2×C42×D73C2211yescharacteristic
15#17988643C418yescomplemented
16C2×#14556525C2212yes—
17C2×#14556524C2212yes—
18C2×#14556522C2212yes—
19C2×C4⋊#10596831C2211yescharacteristic
20C2×#14556503C2212yes—
21C2×#14556501C2212yes—
22#17988457C2211yescharacteristic
23#17988456C2211yescharacteristic
24#17988456C2211yescharacteristic
25#17988452C2212yes—
26#17988451—21no—
27#17988446—21no—
28#17988444—21no—
29#17988444—21no—
30#17988444C2211yescharacteristic
31#17988444C2211yescharacteristic
32C4⋊C4×Dic73—21no—
33C4⋊C4×Dic73—21no—
34C4×#11924007—21no—
35C4×C4⋊Dic73C2211yescharacteristic
36#17988350C2212yes—
37#17988349C2212yes—
38#17988349C2212yes—
39C2×#106×C73C2212yes—
40C2×C4⋊C8×C73C2211yescharacteristic
41#384×C73C2211yescharacteristic
42C4×C4⋊C4×C73C2211yescharacteristic
43#340×C73C2212yes—
44C22×C4×D73D411yescharacteristic
45C22×C4×D73D411yescharacteristic
46C22×C4×D73C2311yescharacteristic
47C22×#10596831C2312yes—
48#14556549—42no—
49#14556549—42no—
50#14556548—44no—
51C2×C4⋊Dic73—21no—
52C2×C4⋊Dic73—21no—
53C2×C4⋊Dic73C2311yescharacteristic
54C2×C4⋊Dic73C2312yes—
55C2×C4⋊Dic73C2311yescharacteristic
56C2×#11924007—21no—
57C2×#11924007—21no—
58C2×#11924007—21no—
59C2×#11924007—21no—
60C2×#11924005—22no—
61C2×#11924005—21no—
62C2×#11924005—21no—
63C2×C4×Dic73—22no—
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#14556522—42no—
73C4⋊#10596831C2×C414yes—
74#14556503C2×C418yes—
75#14556501C2×C418yes—
76#14556498—21no—
77#14556498—21no—
78#14556498—22no—
79#14556498—22no—
80#14556498—21no—
81C22×C8×C73C2312yes—
82C2×C4⋊C4×C73—21no—
83C2×C4⋊C4×C73—21no—
84C2×C4⋊C4×C73C2311yescharacteristic
85C2×C4⋊C4×C73C2311yescharacteristic
86C2×C4⋊C4×C73C2312yes—
87C2×C42×C73—21no—
88C2×C42×C73C2311yescharacteristic
89#106×C73—42no—
90C4⋊C8×C73C2×C414yes—
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—21no—
102C2×C4×D73—21no—
103C2×C4×D73—21no—
104C2×C4×D73—21no—
105C2×C4×D73—41no—
106C2×C4×D73—41no—
107C2×C4×D73C4⋊C414yes—
108C2×C4×D73C4⋊C414yes—
109C2×C4×D73C22×C412yes—
110C2×C4×D73C2×Q812yes—
111C2×#10596831—22no—
112C2×#10596831—22no—
113C2×#10596831C22×C414yes—
114C4⋊Dic73—21no—
115C4⋊Dic73—22no—
116C4⋊Dic73—21no—
117C4⋊Dic73—22no—
118C4⋊Dic73—21no—
119C4⋊Dic73—21no—
120C4⋊Dic73—42no—
121C4⋊Dic73—42no—
122#11924007—42no—
123#11924007—42no—
124#11924007—42no—
125#11924007—42no—
126#11924005—42no—
127#11924005—42no—
128#11924005—44no—
129C4×Dic73—44no—
130C4×Dic73—42no—
131C4×Dic73C4⋊C414yes—
132C4×Dic73C4⋊C414yes—
133C4×Dic73C22×C412yes—
134C4×Dic73C2×Q812yes—
135C22×C4×C73—21no—
136C22×C4×C73—21no—
137C22×C4×C73—22no—
138C22×C4×C73C2×D411yescharacteristic
139C22×C4×C73C2×D411yescharacteristic
140C22×C4×C73C2411yescharacteristic
141C2×C8×C73—22no—
142C2×C8×C73—22no—
143C2×C8×C73C22×C414yes—
144C4⋊C4×C73—21no—
145C4⋊C4×C73—22no—
146C4⋊C4×C73—22no—
147C4⋊C4×C73—21no—
148C4⋊C4×C73—21no—
149C4⋊C4×C73—21no—
150C4⋊C4×C73—42no—
151C4⋊C4×C73—42no—
152C42×C73—42no—
153C42×C73C22×C412yes—
154C42×C73C2×Q812yes—
155C23×C73C22×D411yescharacteristic
156C22×D73—21no—
157C22×D73—21no—
158C22×D73—21no—
159C22×D73—21no—
160C22×D73—21no—
161C22×D73—21no—
162C22×D73C2×C4⋊C412yes—
163C2×C4×C73—21no—
164C2×C4×C73—21no—
165C2×C4×C73—41no—
166C2×C4×C73—42no—
167C2×C4×C73—41no—
168C2×C4×C73—41no—
169C2×C4×C73—41no—
170C2×C4×C73—42no—
171C2×C4×C73—42no—
172C2×C4×C73—41no—
173C2×C4×C73—41no—
174C2×C4×C73C2×C4⋊C412yes—
175C2×C4×C73C2×C4⋊C412yes—
176C2×C4×C73AGL(1,ℤ/8)12yes—
177C2×C4×C73#13712yes—
178C2×C4×C73C23×C411yescharacteristic
179C2×C4×C73C22×Q811yescharacteristic
180C2×C4×C73H(5,𝔽2)11yescharacteristic
181C2×C4×C73H(5,𝔽2)11yescharacteristic
182C2×C4×C732−1+411yescharacteristic
183C2×C4×C732−1+411yescharacteristic
184C2×Dic73—21no—
185C2×Dic73—21no—
186C2×Dic73—21no—
187C2×Dic73—21no—
188C2×Dic73—21no—
189C2×Dic73—21no—
190C2×Dic73—42no—
191C2×Dic73—41no—
192C2×Dic73—41no—
193C2×Dic73—41no—
194C2×Dic73—41no—
195C2×Dic73—42no—
196C2×Dic73—41no—
197C2×Dic73—41no—
198C2×Dic73—42no—
199C2×Dic73C2×C4⋊C412yes—
200C2×Dic73C2×C4⋊C412yes—
201C2×Dic73C2×C4⋊C412yes—
202C4×D73—22no—
203C4×D73—22no—
204C4×D73—22no—
205C4×D73—22no—
206C4×D73—28no—
207C4×D73—44no—
208C8×C73—24no—
209#10596831—24no—
210C22×C73C22×C4⋊C411yescharacteristic
211C22×C73#53511yescharacteristic
212C22×C73#53611yescharacteristic
213C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
214C22×C73C2×#13711yescharacteristic
215C22×C73#57511yescharacteristic
216C22×C73#57511yescharacteristic
217C2×D73—22no—
218C2×D73—22no—
219C2×D73—22no—
220C2×D73—41no—
221C2×D73—41no—
222C2×D73—41no—
223C2×D73—41no—
224C4×C73—22no—
225C4×C73—81no—
226C4×C73—82no—
227C4×C73—81no—
228C4×C73#42814yes—
229C4×C73#52011yescharacteristic
230C4×C73#52011yescharacteristic
231C4×C73#54411yescharacteristic
232C4×C73#54411yescharacteristic
233Dic73—22no—
234Dic73—22no—
235Dic73—22no—
236Dic73—82no—
237Dic73—81no—
238Dic73—81no—
239#21988—731nomaximal, Sylow
240C2×C73#231711yescharacteristic
241C2×C73C2×#42811yescharacteristic
242C2×C73#291111yescharacteristic
243C2×C73#321411yescharacteristic
244C2×C73#321611yescharacteristic
245C2×C73#323511yescharacteristic
246C2×C73#323711yescharacteristic
247D73—42no—
248C2×#432—731no—
249#2285—731no—
250#2285—731no—
251#1948—731no—
252#1948D7311yescharacteristic, complemented
253#1947—731no—
254#1947—731no—
255#1854—732no—
256#1854—732no—
257#1847—732no—
258#1836—732no—
259C73#2198811yescharacteristic, complemented, Sylow
260C2×#117—732no—
261C2×#117—1461no—
262C22×C42—731no—
263#432—738no—
264C2×#107—732no—
265C2×#107—732no—
266C2×#106—732no—
267C2×#106C2×D7312yes—
268C2×C4⋊C8—731no—
269C2×C4⋊C8C2×D7311yescharacteristic
270C2×C22⋊C8—732no—
271C2×#97—732no—
272#384—731no—
273#384—731no—
274#384—731no—
275#384C2×D7311yescharacteristic
276#382—732no—
277#382—1461no—
278#380—731no—
279#380—731no—
280#380—1461no—
281#380—1461no—
282#380—1461no—
283C4×C4⋊C4—731no—
284C4×C4⋊C4—1461no—
285C4×C4⋊C4—1461no—
286C4×C4⋊C4—1461no—
287C4×C4⋊C4C2×D7311yescharacteristic
288#340—732no—
289#340—732no—
290#340—732no—
291#340C2×D7312yes—
292C23×C4—731no—
293C23×C4—731no—
294C23×C4—731no—
295C22×C8—732no—
296C22×C8C22×D7312yes—
297#117—2924no—
298#117—2922no—
299#117—2922no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—732no—
303C2×C4⋊C4—731no—
304C2×C4⋊C4—731no—
305C2×C4⋊C4—1461no—
306C2×C4⋊C4—1461no—
307C2×C4⋊C4—1461no—
308C2×C4⋊C4—1461no—
309C2×C4⋊C4—1461no—
310C2×C4⋊C4—1461no—
311C2×C4⋊C4C22×D7312yes—
312C2×C4⋊C4C22×D7311yescharacteristic
313C2×C4⋊C4C22×D7311yescharacteristic
314C2×C22⋊C4—1462no—
315C2×C22⋊C4—1461no—
316C2×C22⋊C4—1461no—
317C2×C42—21no—
318C2×C42—731no—
319C2×C42—731no—
320C2×C42—731no—
321C2×C42—734no—
322C2×C42—734no—
323C2×C42—1462no—
324C2×C42—1461no—
325C2×C42C22×D7311yescharacteristic
326#107—2922no—
327#107—2922no—
328#106—42no—
329#106—2922no—
330C4⋊C8—734no—
331C4⋊C8C4×D7314yes—
332C22⋊C8—738no—
333#97—738no—
334#95—21no—
335#95—1461no—
336#95—1461no—
337#95—1462no—
338#95—1462no—
339#95—1461no—
340C24—731no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—2921no—
345C22×C4—2921no—
346C22×C4—734no—
347C22×C4—731no—
348C22×C4—731no—
349C22×C4—731no—
350C22×C4—734no—
351C22×C4—732no—
352C22×C4—732no—
353C22×C4—1461no—
354C22×C4—1462no—
355C22×C4—1461no—
356C22×C4—1462no—
357C22×C4—1462no—
358C22×C4—1461no—
359C22×C4—1461no—
360C22×C4—1461no—
361C22×C4—1461no—
362C22×C4D4×D7311yescharacteristic
363C22×C4D4×D7311yescharacteristic
364C22×C4C23×D7311yescharacteristic
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—734no—
368C2×C8—1462no—
369C2×C8—1462no—
370C2×C8C2×C4×D7314yes—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—2922no—
380C4⋊C4—2922no—
381C4⋊C4—2922no—
382C4⋊C4—2922no—
383C4⋊C4—2922no—
384C4⋊C4—2922no—
385C4⋊C4—1461no—
386C4⋊C4—1462no—
387C4⋊C4—1461no—
388C4⋊C4—1462no—
389C4⋊C4—1461no—
390C4⋊C4—1461no—
391C22⋊C4—2924no—
392C22⋊C4—2922no—
393C22⋊C4—2922no—
394C42—42no—
395C42—2924no—
396C42—2922no—
397C42—734no—
398C42—734no—
399C42—732no—
400C42—732no—
401C42C2×C4×D7312yes—
402C42Q8×D7312yes—
403C23—732no—
404C23—1461no—
405C23—1461no—
406C23—1461no—
407C23—1461no—
408C23—1461no—
409C23—1461no—
410C23C2×D4×D7311yescharacteristic
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—2924no—
423C2×C4—2921no—
424C2×C4—2922no—
425C2×C4—2921no—
426C2×C4—2922no—
427C2×C4—2921no—
428C2×C4—2921no—
429C2×C4—2921no—
430C2×C4—2921no—
431C2×C4—2922no—
432C2×C4—732no—
433C2×C4—732no—
434C2×C4—732no—
435C2×C4—1461no—
436C2×C4—1461no—
437C2×C4—1461no—
438C2×C4—1461no—
439C2×C4—1461no—
440C2×C4—1461no—
441C2×C4—1468no—
442C2×C4—1462no—
443C2×C4—1462no—
444C2×C4—1462no—
445C2×C4—1462no—
446C2×C4C4⋊C4×D7312yes—
447C2×C4C4⋊C4×D7312yes—
448C2×C4#1455661812yes—
449C2×C4#1455662012yes—
450C2×C4C22×C4×D7311yescharacteristic
451C2×C4C2×Q8×D7311yescharacteristic
452C2×C4#1455663611yescharacteristic
453C2×C4#1455663711yescharacteristic
454C2×C4#1455663811yescharacteristic
455C2×C4#1455663911yescharacteristic
456C8—24no—
457C8—1464no—
458C22—2921no—
459C22—2921no—
460C22—2921no—
461C22—2921no—
462C22—1462no—
463C22—1462no—
464C22—1462no—
465C22C2×C4⋊C4×D7311yescharacteristic
466C22#1798915311yescharacteristic
467C22#1798916011yescharacteristic
468C22C2×#1455661811yescharacteristic
469C22C2×#1455662011yescharacteristic
470C22#1798938311yescharacteristic
471C22#1798938611yescharacteristic
472C4—22no—
473C4—82no—
474C4—81no—
475C4—81no—
476C4—5841no—
477C4—5842no—
478C4—5841no—
479C4—1462no—
480C4—1462no—
481C4—1462no—
482C4#1798862614yes—
483C4#1798906611yescharacteristic
484C4#1798906911yescharacteristic
485C4#1798921411yescharacteristic
486C4#1798921711yescharacteristic
487C2—2922no—
488C2#2261255311yescharacteristic
489C2C2×#1798862611yescharacteristic
490C2#2261851811yescharacteristic
491C2#2262092511yescharacteristic
492C2#2262094011yescharacteristic
493C2#2262112711yescharacteristic
494C2#2262114411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1948C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1948D73
hypocenterC73#21988
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
4. #22610692C2semidirect
5. #22610690C2semidirect
6. #22610689C2semidirect
9. #22610096C2semidirect
10. #22610063C2semidirect
11. C73×#1948C2semidirect
15. #17988643C4semidirect
252. #1948D73semidirect
259. C73#21988semidirect

Maximal subgroups

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

subgroupindex
C2×#179886432
#226123362
#226123352
#226106922
#226106902
#226106892
#226101332
#226101292
#226100962
#226100632
C73×#19482
#2198873