Finite Group Databasebeta

← #32978696

Subgroups — #32978696

Order 18688.

548 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
1#22612392C211yesmaximal, characteristic
2#22610047C211yesmaximal, characteristic, complemented
3C4×#14556520C211yesmaximal, characteristic, complemented
4C2×#17988316C211yesmaximal, characteristic
5C2×#17988314C211yesmaximal, characteristic
6#22607772C211yesmaximal, characteristic, complemented
7C73×#1279C211yesmaximal, characteristic, complemented
8C2×C4×D292C2211yescharacteristic
9C2×#14556548—21no—
10C2×C4⋊#10596831C2211yescharacteristic
11C2×#14556520—21no—
12C2×C4×#10596831C2211yescharacteristic
13#17988488C411yescharacteristic
14#17988488C411yescharacteristic
15#17988483C411yescharacteristic
16#17988483C411yescharacteristic
17#17988476—21no—
18#17988476—21no—
19C4×#11924005—21no—
20C42×Dic73C2211yescharacteristic
21#17988341—21no—
22#17988341—21no—
23#17988341—21no—
24#17988316—22no—
25#17988314—22no—
26C2×C4⋊C8×C73C2211yescharacteristic
27C2×C4×C8×C73C2211yescharacteristic
28C4×C4⋊C4×C73C2211yescharacteristic
29C22×D292C2×C411yescharacteristic
30C22×C4×D73—21no—
31C22×#10596831—21no—
32C22×#10596831—21no—
33C4×D292—21no—
34C4×D292—21no—
35C4×D292D412yes—
36C4×D292Q812yes—
37#14556548—42no—
38#14556548—42no—
39C2×C4⋊Dic73C2×C411yescharacteristic
40C2×#11924007—21no—
41C2×#11924007—21no—
42C2×#11924005—21no—
43C2×#11924005—21no—
44C2×#11924005—21no—
45C2×#11924005—21no—
46C2×#11924005—21no—
47C2×C4×Dic73C2×C411yescharacteristic
48C2×C4×Dic73C2×C411yescharacteristic
49C2×C4×Dic73D411yescharacteristic
50C2×C4×Dic73D411yescharacteristic
51C2×C4×Dic73D411yescharacteristic
52C2×C4×Dic73D411yescharacteristic
53C4⋊#10596831—21no—
54C4⋊#10596831—21no—
55#14556520—24no—
56C4×#10596831—21no—
57C4×#10596831—21no—
58#14556498—21no—
59#14556498—21no—
60C22×C8×C73—21no—
61C22×C8×C73—21no—
62C2×C4⋊C4×C73C2×C411yescharacteristic
63C2×C4⋊C4×C73C2×C411yescharacteristic
64C2×C42×C73—21no—
65C2×C42×C73C2311yescharacteristic
66C4⋊C8×C73—21no—
67C4⋊C8×C73—21no—
68C4×C8×C73—21no—
69C4×C8×C73—21no—
70#95×C73—21no—
71C23×D73—21no—
72C2×D292—21no—
73C2×D292—21no—
74C2×D292—21no—
75C2×D292—21no—
76C2×D292—21no—
77C2×D292C4⋊C412yes—
78C22×Dic73—21no—
79C22×Dic73C22⋊C411yescharacteristic
80C22×Dic73C22⋊C411yescharacteristic
81C22×Dic73C22⋊C411yescharacteristic
82C22×Dic73C22⋊C411yescharacteristic
83C2×C4×D73—22no—
84C2×C4×D73—22no—
85C2×C4×D73—41no—
86C2×C4×D73—41no—
87C2×C4×D73—41no—
88C2×C4×D73—41no—
89C2×#10596831—22no—
90C2×#10596831—41no—
91C2×#10596831—41no—
92C2×#10596831—41no—
93C2×#10596831—41no—
94C2×#10596831—41no—
95C4⋊Dic73—21no—
96C4⋊Dic73C4⋊C412yes—
97#11924007—42no—
98#11924007—42no—
99#11924005—22no—
100#11924005—22no—
101#11924005—42no—
102#11924005—42no—
103#11924005—42no—
104#11924005—42no—
105C4×Dic73—21no—
106C4×Dic73—21no—
107C4×Dic73—21no—
108C4×Dic73—21no—
109C4×Dic73—21no—
110C4×Dic73—21no—
111C4×Dic73—21no—
112C4×Dic73—21no—
113C4×Dic73—22no—
114C4×Dic73—41no—
115C22×C4×C73—21no—
116C22×C4×C73—21no—
117C22×C4×C73C22×C411yescharacteristic
118C22×C4×C73C2×D411yescharacteristic
119C22×C4×C73C2×D411yescharacteristic
120C2×C8×C73—22no—
121C2×C8×C73—41no—
122C2×C8×C73—41no—
123C2×C8×C73—41no—
124C2×C8×C73—41no—
125C2×C8×C73—41no—
126C4⋊C4×C73—21no—
127C4⋊C4×C73—21no—
128C4⋊C4×C73—21no—
129C4⋊C4×C73—21no—
130C42×C73—41no—
131C42×C73—41no—
132C42×C73C2×D411yescharacteristic
133C42×C73C2×Q811yescharacteristic
134C42×C73#4111yescharacteristic
135C42×C73#4111yescharacteristic
136C23×C73C2×C22⋊C411yescharacteristic
137C22×D73—22no—
138C22×D73—41no—
139C22×D73—41no—
140C22×D73—41no—
141C22×D73—41no—
142C22×D73—41no—
143C22×D73—41no—
144C2×C4×C73—21no—
145C2×C4×C73—21no—
146C2×C4×C73—41no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—41no—
152C2×C4×C73C2×C4⋊C411yescharacteristic
153C2×C4×C73#11711yescharacteristic
154C2×C4×C73#12111yescharacteristic
155C2×C4×C73#12111yescharacteristic
156C2×C4×C73C22⋊Q811yescharacteristic
157C2×C4×C73C22⋊Q811yescharacteristic
158C2×C4×C73#13111yescharacteristic
159C2×C4×C73#13111yescharacteristic
160C2×C4×C73#13111yescharacteristic
161C2×C4×C73#13111yescharacteristic
162C2×C4×C73#13511yescharacteristic
163C2×C4×C73#13511yescharacteristic
164C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
166C2×Dic73—21no—
167C2×Dic73—21no—
168C2×Dic73—22no—
169C2×Dic73—21no—
170C2×Dic73—21no—
171C2×Dic73—21no—
172C2×Dic73—21no—
173C2×Dic73—21no—
174C2×Dic73—21no—
175C2×Dic73—21no—
176C2×Dic73—21no—
177C2×Dic73—21no—
178C2×Dic73—41no—
179C2×Dic73—41no—
180C2×Dic73#10412yes—
181D292—22no—
182D292—22no—
183D292—41no—
184D292—41no—
185C4×D73—42no—
186C4×D73—42no—
187C4×D73—42no—
188C4×D73—42no—
189C8×C73—42no—
190C8×C73—81no—
191#10596831—42no—
192#10596831—81no—
193C22×C73#38011yescharacteristic
194C22×C73#40811yescharacteristic
195C22×C73#40811yescharacteristic
196C22×C73#41611yescharacteristic
197C22×C73#41811yescharacteristic
198C22×C73C2×#10411yescharacteristic
199C22×C73#42111yescharacteristic
200C2×D73—42no—
201C2×D73—42no—
202C2×D73—42no—
203C2×D73—81no—
204C2×D73—81no—
205C2×D73—81no—
206C2×D73—81no—
207C4×C73—21no—
208C4×C73—21no—
209C4×C73—81no—
210C4×C73—81no—
211C4×C73#42411yescharacteristic
212C4×C73#42411yescharacteristic
213C4×C73#43311yescharacteristic
214C4×C73#43311yescharacteristic
215C4×C73#46311yescharacteristic
216C4×C73#46311yescharacteristic
217C4×C73#47811yescharacteristic
218C4×C73#47811yescharacteristic
219Dic73—22no—
220Dic73—42no—
221Dic73—41no—
222Dic73—41no—
223Dic73—41no—
224#8569—731nomaximal, Sylow
225C2×C73#148211yescharacteristic
226C2×C73#148411yescharacteristic
227C2×C73#149411yescharacteristic
228C2×C73#149611yescharacteristic
229C2×C73#179811yescharacteristic
230C2×C73#180411yescharacteristic
231C2×C73#180611yescharacteristic
232D73—82no—
233#2293—731no—
234#1831—731no—
235C4×C4⋊C8—731no—
236C2×D4⋊C8—731no—
237C2×D4⋊C8—731no—
238#1279—731no—
239#1279D7311yescharacteristic, complemented
240C73#856911yescharacteristic, complemented, Sylow
241C2×C4×D4—731no—
242C2×#117—1461no—
243C2×C4⋊C8—731no—
244C2×C4⋊C8—1461no—
245C2×C4⋊C8C2×D7311yescharacteristic
246C2×C4×C8—731no—
247C2×C4×C8C2×D7311yescharacteristic
248#390—731no—
249#390—731no—
250#389—731no—
251#389—731no—
252#388—1461no—
253#388—1461no—
254C4×C4⋊C4C2×D7311yescharacteristic
255C4×C22⋊C4—1461no—
256C43—731no—
257#336—1461no—
258#336—1461no—
259#336—1461no—
260D4⋊C8—1462no—
261D4⋊C8—1462no—
262C22×D4—731no—
263C23×C4—1461no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—1461no—
267C22×C8—1461no—
268C4×D4—732no—
269C4×D4—732no—
270C4×D4—1461no—
271C4×D4—1461no—
272#117—2922no—
273#117—2922no—
274C2×C4⋊C4—731no—
275C2×C4⋊C4—1461no—
276C2×C4⋊C4—1461no—
277C2×C4⋊C4C4×D7311yescharacteristic
278C2×C4⋊C4C4×D7311yescharacteristic
279C2×C22⋊C4—1461no—
280C2×C22⋊C4—1461no—
281C2×C22⋊C4—1461no—
282C2×C22⋊C4—1461no—
283C2×C22⋊C4—1461no—
284C2×C42—21no—
285C2×C42—731no—
286C2×C42—731no—
287C2×C42—731no—
288C2×C42—731no—
289C2×C42—731no—
290C2×C42—731no—
291C2×C42C22×D7311yescharacteristic
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—1464no—
295C4⋊C8—1461no—
296C4⋊C8—1461no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—1461no—
300C4×C8—1461no—
301#95—21no—
302#95—1461no—
303#95—1461no—
304C24—1461no—
305C2×D4—732no—
306C2×D4—1461no—
307C2×D4—1461no—
308C2×D4—1461no—
309C2×D4—1461no—
310C2×D4—1461no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—2921no—
314C22×C4—2921no—
315C22×C4—2921no—
316C22×C4—2921no—
317C22×C4—731no—
318C22×C4—731no—
319C22×C4—731no—
320C22×C4—731no—
321C22×C4—1461no—
322C22×C4—1462no—
323C22×C4—1462no—
324C22×C4C2×C4×D7311yescharacteristic
325C22×C4D4×D7311yescharacteristic
326C22×C4D4×D7311yescharacteristic
327C2×C8—22no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—2921no—
334C2×C8—2921no—
335C2×C8—2921no—
336C2×C8—2921no—
337C2×C8—2921no—
338C2×C8—1462no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—2922no—
344C4⋊C4—2922no—
345C4⋊C4—732no—
346C4⋊C4—1461no—
347C22⋊C4—2922no—
348C22⋊C4—2922no—
349C22⋊C4—2922no—
350C22⋊C4—2922no—
351C22⋊C4—1462no—
352C22⋊C4—1462no—
353C42—41no—
354C42—41no—
355C42—2921no—
356C42—1462no—
357C42—1461no—
358C42—1461no—
359C42—1461no—
360C42—1461no—
361C42—1461no—
362C42—1461no—
363C42—1461no—
364C42—1461no—
365C42D4×D7311yescharacteristic
366C42Q8×D7311yescharacteristic
367C42#1192402811yescharacteristic
368C42#1192402811yescharacteristic
369C23—2921no—
370C23—2921no—
371C23—2921no—
372C23—2921no—
373C23—2921no—
374C23—2921no—
375C23—1462no—
376C23C22⋊C4×D7311yescharacteristic
377D4—2921no—
378D4—2921no—
379D4—1462no—
380D4—1462no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—2921no—
390C2×C4—2921no—
391C2×C4—2922no—
392C2×C4—2922no—
393C2×C4—2922no—
394C2×C4—2922no—
395C2×C4—732no—
396C2×C4—1462no—
397C2×C4—1461no—
398C2×C4—1461no—
399C2×C4—1461no—
400C2×C4—1461no—
401C2×C4—1461no—
402C2×C4—1461no—
403C2×C4—1461no—
404C2×C4—1461no—
405C2×C4—1461no—
406C2×C4—1461no—
407C2×C4—1461no—
408C2×C4C4⋊C4×D7311yescharacteristic
409C2×C4#1455654811yescharacteristic
410C2×C4#1455656411yescharacteristic
411C2×C4#1455656411yescharacteristic
412C2×C4#1455656911yescharacteristic
413C2×C4#1455656911yescharacteristic
414C2×C4#1455659711yescharacteristic
415C2×C4#1455659711yescharacteristic
416C2×C4#1455659911yescharacteristic
417C2×C4#1455659911yescharacteristic
418C2×C4#1455661211yescharacteristic
419C2×C4#1455661411yescharacteristic
420C2×C4#1455661511yescharacteristic
421C2×C4#1455661811yescharacteristic
422C8—42no—
423C8—81no—
424C8—2922no—
425C8—5841no—
426C22—2922no—
427C22—2922no—
428C22—2922no—
429C22—5841no—
430C22—5841no—
431C22—5841no—
432C22—5841no—
433C22#1798844611yescharacteristic
434C22#1798855211yescharacteristic
435C22#1798855211yescharacteristic
436C22#1798858211yescharacteristic
437C22#1798859011yescharacteristic
438C22#104×D7311yescharacteristic
439C22#1798860211yescharacteristic
440C4—21no—
441C4—21no—
442C4—81no—
443C4—81no—
444C4—2921no—
445C4—2922no—
446C4—2921no—
447C4—2921no—
448C4—1462no—
449C4#1798861211yescharacteristic
450C4#1798861211yescharacteristic
451C4#1798864611yescharacteristic
452C4#1798864911yescharacteristic
453C4#1798880111yescharacteristic
454C4#1798880511yescharacteristic
455C4#1798888611yescharacteristic
456C4#1798889011yescharacteristic
457C2—5842no—
458C2#2260833811yescharacteristic
459C2#2260834611yescharacteristic
460C2#2260839011yescharacteristic
461C2#2260839811yescharacteristic
462C2#2260991411yescharacteristic
463C2#2260994011yescharacteristic
464C2#2260995011yescharacteristic
subgroupG/H
centerC22×C4D4×D73
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1279C2
socleC23×C73C2×C22⋊C4
prime radicalC22×D292C2×C4
cosocleC2×C42×C73C23
hypercenter#1279D73
hypocenterC73#8569
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
2. #22610047C2semidirect
3. C4×#14556520C2semidirect
6. #22607772C2semidirect
7. C73×#1279C2semidirect
239. #1279D73semidirect
240. C73#8569semidirect

Maximal subgroups

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

subgroupindex
#226123922
#226100472
C4×#145565202
C2×#179883162
C2×#179883142
#226077722
C73×#12792
#856973