Finite Group Databasebeta

← #32980752

Subgroups — #32980752

Order 18688.

616 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988787C211yesmaximal, characteristic
2#22610858C211yesmaximal, characteristic
3#22610736C211yesmaximal, characteristic
4C2×Q8⋊#10596831C211yesmaximal, characteristic, complemented
5#22607814C211yesmaximal, characteristic, complemented
6#22607772C211yesmaximal, characteristic, complemented
7C73×#2355C211yesmaximal, characteristic, complemented
8C22×#11924015—21no—
9C2×#14556588C2211yescharacteristic
10#17988787—22no—
11#17988787C414yescomplemented
12C2×C4⋊#10596831C2211yescharacteristic
13C2×C4⋊#10596831C2211yescharacteristic
14C2×#14556511—21no—
15C2×#14556511—22no—
16C2×#14556511—21no—
17C2×#14556511—21no—
18C2×C4×#10596831C2211yescharacteristic
19Q8⋊#10596831—22no—
20C2×C4×Q8×C73C2211yescharacteristic
21#391×C73—21no—
22#387×C73—22no—
23#384×C73C2211yescharacteristic
24#382×C73—21no—
25C4×C4⋊C4×C73—21no—
26C4×C4⋊C4×C73—21no—
27C4×C4⋊C4×C73C2211yescharacteristic
28C22×D292—21no—
29C22×D292—21no—
30C22×D292—21no—
31C2×#11924015—24no—
32C2×#11924015—42no—
33C2×#11924015—42no—
34C22×#10596831—21no—
35C22×#10596831—21no—
36C22×#10596831—21no—
37#14556588—21no—
38#14556588—21no—
39#14556588—21no—
40#14556588C2×C412yes—
41C4⋊#10596831—21no—
42C4⋊#10596831—21no—
43C4⋊#10596831—21no—
44C4⋊#10596831C2×C412yes—
45#14556511—24no—
46#14556511—42no—
47#14556511—44no—
48#14556511—42no—
49C4×#10596831—21no—
50C4×#10596831—21no—
51C22×Q8×C73D411yescharacteristic
52C4×Q8×C73—21no—
53C4×Q8×C73—21no—
54C4×Q8×C73C2×C412yes—
55C4×Q8×C73D412yes—
56C2×C4⋊C4×C73—21no—
57C2×C4⋊C4×C73—21no—
58C2×C4⋊C4×C73—21no—
59C2×C4⋊C4×C73D411yescharacteristic
60C2×C4⋊C4×C73D411yescharacteristic
61C2×C4⋊C4×C73D411yescharacteristic
62C2×C4⋊C4×C73D412yes—
63C2×C42×C73—21no—
64C2×C42×C73—21no—
65C2×C42×C73—22no—
66C2×C42×C73C2311yescharacteristic
67#95×C73—22no—
68#95×C73—22no—
69#95×C73—22no—
70#95×C73—21no—
71C23×D73—41no—
72C2×D292—22no—
73C2×D292—22no—
74C2×D292—22no—
75C2×D292—42no—
76C2×D292—42no—
77C2×D292—41no—
78C2×D292—41no—
79C2×D292—41no—
80C2×D292—41no—
81C2×D292—41no—
82C2×D292—41no—
83C2×D292—41no—
84C2×D292—41no—
85C2×D292—41no—
86C2×D292—41no—
87C2×D292—41no—
88#11924015—48no—
89C2×#10596831—22no—
90C2×#10596831—22no—
91C2×#10596831—41no—
92C2×#10596831—41no—
93C2×#10596831—41no—
94C2×#10596831—41no—
95C2×#10596831—41no—
96C2×#10596831—41no—
97C2×#10596831—41no—
98C2×Q8×C73—21no—
99C2×Q8×C73—21no—
100C2×Q8×C73—21no—
101C2×Q8×C73—21no—
102C2×Q8×C73—21no—
103C2×Q8×C73C22⋊C412yes—
104C22×C4×C73—21no—
105C22×C4×C73—21no—
106C22×C4×C73—21no—
107C22×C4×C73—21no—
108C22×C4×C73—22no—
109C22×C4×C73C2×D411yescharacteristic
110C22×C4×C73C2×D411yescharacteristic
111C22×C4×C73C2×D411yescharacteristic
112C4⋊C4×C73—22no—
113C4⋊C4×C73—22no—
114C4⋊C4×C73—21no—
115C4⋊C4×C73—21no—
116C4⋊C4×C73—21no—
117C4⋊C4×C73—21no—
118C4⋊C4×C73—42no—
119C4⋊C4×C73—41no—
120C4⋊C4×C73—41no—
121C4⋊C4×C73—41no—
122C4⋊C4×C73—41no—
123C4⋊C4×C73C22⋊C412yes—
124C4⋊C4×C73SD1612yescomplemented
125C42×C73—22no—
126C42×C73—22no—
127C42×C73—41no—
128C42×C73—41no—
129C42×C73—82no—
130C42×C73C22×C411yescharacteristic
131C42×C73C2×D411yescharacteristic
132C42×C73#4111yescharacteristic
133C42×C73#4111yescharacteristic
134C23×C73#12011yescharacteristic
135C22×D73—42no—
136C22×D73—81no—
137C22×D73—81no—
138C22×D73—81no—
139C22×D73—81no—
140C22×D73—81no—
141C22×D73—81no—
142Q8×C73—22no—
143Q8×C73—22no—
144Q8×C73—41no—
145Q8×C73—41no—
146C2×C4×C73—21no—
147C2×C4×C73—21no—
148C2×C4×C73—22no—
149C2×C4×C73—22no—
150C2×C4×C73—22no—
151C2×C4×C73—41no—
152C2×C4×C73—41no—
153C2×C4×C73—41no—
154C2×C4×C73—41no—
155C2×C4×C73—41no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73—41no—
159C2×C4×C73—42no—
160C2×C4×C73—42no—
161C2×C4×C73—42no—
162C2×C4×C73—41no—
163C2×C4×C73C2×C22⋊C411yescharacteristic
164C2×C4×C73C4×D411yescharacteristic
165C2×C4×C73C4×D411yescharacteristic
166C2×C4×C73#12111yescharacteristic
167C2×C4×C73#12111yescharacteristic
168C2×C4×C73#12311yescharacteristic
169C2×C4×C73C2×SD1611yescharacteristic
170C2×C4×C73#13511yescharacteristic
171C2×C4×C73#13511yescharacteristic
172C2×C4×C73#13511yescharacteristic
173C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C73#13711yescharacteristic
177D292—44no—
178D292—42no—
179D292—42no—
180D292—42no—
181D292—42no—
182D292—82no—
183D292—81no—
184D292—81no—
185D292—81no—
186D292—81no—
187#10596831—42no—
188#10596831—42no—
189#10596831—81no—
190C22×C73#38611yescharacteristic
191C22×C73#44811yescharacteristic
192C22×C73#44911yescharacteristic
193C22×C73#44911yescharacteristic
194C22×C73#44911yescharacteristic
195C22×C73#45311yescharacteristic
196C22×C73#45411yescharacteristic
197C2×D73—82no—
198C2×D73—82no—
199C2×D73—82no—
200C2×D73—161no—
201C2×D73—161no—
202C2×D73—161no—
203C2×D73—161no—
204C4×C73—21no—
205C4×C73—21no—
206C4×C73—42no—
207C4×C73—42no—
208C4×C73—42no—
209C4×C73—82no—
210C4×C73—81no—
211C4×C73#41611yescharacteristic
212C4×C73#41811yescharacteristic
213C4×C73C4×SD1611yescharacteristic
214C4×C73#44011yescharacteristic
215C4×C73#46011yescharacteristic
216C4×C73#46411yescharacteristic
217C4×C73#48211yescharacteristic
218C4×C73#48211yescharacteristic
219#8978—731nomaximal, Sylow
220C2×C73#162011yescharacteristic
221C2×C73#162211yescharacteristic
222C2×C73#162611yescharacteristic
223C2×C73#162811yescharacteristic
224C2×C73#187511yescharacteristic
225C2×C73#188011yescharacteristic
226C2×C73#188311yescharacteristic
227D73—162no—
228C2×#460—731no—
229#2355D7311yescharacteristic, complemented
230#1982—731no—
231#1957—731no—
232C2×Q8⋊C8—731no—
233#1296—731no—
234#1279—731no—
235C73#897811yescharacteristic, complemented, Sylow
236C22×SD16—1461no—
237C2×#127—731no—
238C2×C4×Q8C2×D7311yescharacteristic
239#460—734no—
240#460—1462no—
241C2×C4⋊C8—731no—
242C2×C4⋊C8—731no—
243C2×#102—1462no—
244C2×#102—1461no—
245C2×#102—1461no—
246C2×#102—1461no—
247C2×C4×C8—731no—
248#391—21no—
249#387—22no—
250#384C2×D7311yescharacteristic
251#382—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—21no—
254C4×C4⋊C4C2×D7311yescharacteristic
255Q8⋊C8—1462no—
256C22×Q8C22⋊D7311yescharacteristic
257C22×D4—1461no—
258C22×D4—1461no—
259C22×D4—1461no—
260C2×SD16—2922no—
261C2×SD16—2922no—
262C2×SD16—1464no—
263C22×C8—1461no—
264C22×C8—1461no—
265C22×C8—1461no—
266#127—732no—
267#127—1461no—
268#127—1461no—
269#127—1461no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8C4×D7312yes—
273C4×Q8D29212yes—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4C22⋊D7311yescharacteristic
278C2×C4⋊C4C22⋊D7311yescharacteristic
279C2×C4⋊C4C22⋊D7311yescharacteristic
280C2×C4⋊C4C22⋊D7312yes—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42C22×D7311yescharacteristic
285C4⋊C8—732no—
286C4⋊C8—1461no—
287C4⋊C8—1461no—
288C4⋊C8—1461no—
289#102—2922no—
290#102—2924no—
291#102—2922no—
292#102—1464no—
293C4×C8—1461no—
294C4×C8—1461no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—21no—
299C24—2921no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8#1192400512yes—
306C2×D4—2922no—
307C2×D4—2922no—
308C2×D4—2921no—
309C2×D4—2921no—
310C2×D4—2921no—
311C2×D4—2921no—
312C2×D4—2921no—
313C2×D4—2921no—
314C2×D4—2921no—
315C2×D4—2921no—
316C2×D4—2921no—
317C2×D4—2921no—
318C2×D4—2921no—
319C2×D4—1462no—
320C2×D4—1462no—
321C2×D4—1462no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4C2×C22⋊D7311yescharacteristic
328C22×C4C2×C22⋊D7311yescharacteristic
329C22×C4C2×C22⋊D7311yescharacteristic
330SD16—2928no—
331C2×C8—2921no—
332C2×C8—2921no—
333C2×C8—2921no—
334C2×C8—2921no—
335C2×C8—2921no—
336C2×C8—2921no—
337C2×C8—2921no—
338C2×C8—1462no—
339C2×C8—1462no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4#1192400512yes—
352C4⋊C4#1192401512yescomplemented
353C42—22no—
354C42—22no—
355C42—41no—
356C42—41no—
357C42—82no—
358C42C2×C4×D7311yescharacteristic
359C42C2×D29211yescharacteristic
360C42#1192402811yescharacteristic
361C42#1192402811yescharacteristic
362C23—2922no—
363C23—5841no—
364C23—5841no—
365C23—5841no—
366C23—5841no—
367C23—5841no—
368C23—5841no—
369C23#1455656011yescharacteristic
370Q8—22no—
371Q8—22no—
372Q8—41no—
373Q8—41no—
374D4—2924no—
375D4—2922no—
376D4—2922no—
377D4—2922no—
378D4—2922no—
379D4—5842no—
380D4—5841no—
381D4—5841no—
382D4—5841no—
383D4—5841no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4C2×#1192400511yescharacteristic
402C2×C4C4×C22⋊D7311yescharacteristic
403C2×C4C4×C22⋊D7311yescharacteristic
404C2×C4#1455656511yescharacteristic
405C2×C4#1455656511yescharacteristic
406C2×C4#1455657411yescharacteristic
407C2×C4C2×#1192401511yescharacteristic
408C2×C4#1455661111yescharacteristic
409C2×C4#1455661111yescharacteristic
410C2×C4#1455661111yescharacteristic
411C2×C4#1455661611yescharacteristic
412C2×C4#1455661611yescharacteristic
413C2×C4#1455661611yescharacteristic
414C2×C4#1455662211yescharacteristic
415C8—2922no—
416C8—2922no—
417C8—5841no—
418C22—5842no—
419C22—5842no—
420C22—5842no—
421C22—11681no—
422C22—11681no—
423C22—11681no—
424C22—11681no—
425C22#1798846611yescharacteristic
426C22#1798872611yescharacteristic
427C22#1798873211yescharacteristic
428C22#1798873211yescharacteristic
429C22#1798873211yescharacteristic
430C22#1798875511yescharacteristic
431C22#1798876011yescharacteristic
432C4—21no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—82no—
438C4—81no—
439C4#1798858111yescharacteristic
440C4#1798859111yescharacteristic
441C4C4×#1192401511yescharacteristic
442C4#1798868511yescharacteristic
443C4#1798878711yescharacteristic
444C4#1798880911yescharacteristic
445C4#1798891111yescharacteristic
446C4#1798891111yescharacteristic
447C2—11682no—
448C2#2260899011yescharacteristic
449C2#2260900311yescharacteristic
450C2#2260902711yescharacteristic
451C2#2260904111yescharacteristic
452C2#2261025311yescharacteristic
453C2#2261028611yescharacteristic
454C2#2261030711yescharacteristic
subgroupG/H
centerC23#14556560
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#2355C2
socleC23×C73#120
prime radicalC2×#14556588C22
cosocleC2×C42×C73C23
hypercenter#2355D73
hypocenterC73#8978
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2355

Complements

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

NH ≅ G/Ntype
4. C2×Q8⋊#10596831C2semidirect
5. #22607814C2semidirect
6. #22607772C2semidirect
7. C73×#2355C2semidirect
11. #17988787C4semidirect
124. C4⋊C4×C73SD16semidirect
229. #2355D73semidirect
235. C73#8978semidirect
352. C4⋊C4#11924015semidirect

Maximal subgroups

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

subgroupindex
C2×#179887872
#226108582
#226107362
C2×Q8⋊#105968312
#226078142
#226077722
C73×#23552
#897873